Subgroups — C₆₇×#34419 #30713670

Order 17152.

2182 conjugacy classes of subgroups.

The subgroup lattice has 998 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#542×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2379	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2376	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2369	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2367	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2367	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2367	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2365	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2359	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2355	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2354	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₇×#2351	C₂	1	1	yes	maximal, characteristic, complemented
13	C₆₇×#2349	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₇×#2334	C₂	1	2	yes	maximal, complemented
15	C₆₇×#2334	C₂	1	2	yes	maximal, complemented
16	C₆₇×#2329	C₂	1	2	yes	maximal, complemented
17	C₆₇×#2324	C₂	1	1	yes	maximal, characteristic
18	C₆₇×#2323	C₂	1	1	yes	maximal, characteristic, complemented
19	C₆₇×#2323	C₂	1	2	yes	maximal, complemented
20	C₆₇×#2314	C₂	1	1	yes	maximal, characteristic, complemented
21	C₄×C₂²⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
22	C₄×#121×C₆₇	C₂	1	1	yes	maximal, characteristic
23	C₆₇×#2296	C₂	1	1	yes	maximal, characteristic, complemented
24	#542×C₆₇	C₄	1	16	yes	complemented
25	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#126×C₆₇	C₂²	1	2	yes	—
27	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#124×C₆₇	C₂²	1	2	yes	—
29	C₂×#123×C₆₇	C₂²	1	2	yes	—
30	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
32	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
36	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
39	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
42	C₂×#117×C₆₇	C₂²	1	2	yes	—
43	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
44	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
45	#391×C₆₇	C₂²	1	1	yes	characteristic
46	#391×C₆₇	C₂²	1	1	yes	characteristic
47	#391×C₆₇	C₂²	1	2	yes	—
48	#391×C₆₇	C₂²	1	2	yes	—
49	#391×C₆₇	C₂²	1	2	yes	—
50	#390×C₆₇	C₂²	1	1	yes	characteristic
51	#390×C₆₇	C₂²	1	1	yes	characteristic
52	#390×C₆₇	C₂²	1	2	yes	—
53	#389×C₆₇	C₂²	1	1	yes	characteristic
54	#389×C₆₇	C₂²	1	1	yes	characteristic
55	#389×C₆₇	C₂²	1	1	yes	characteristic
56	#389×C₆₇	C₂²	1	1	yes	characteristic
57	#389×C₆₇	C₂²	1	1	yes	characteristic
58	#389×C₆₇	C₂²	1	1	yes	characteristic
59	#389×C₆₇	C₂²	1	2	yes	—
60	#389×C₆₇	C₂²	1	1	yes	characteristic
61	#389×C₆₇	C₂²	1	2	yes	—
62	#389×C₆₇	C₂²	1	1	yes	characteristic
63	#389×C₆₇	C₂²	1	2	yes	—
64	#389×C₆₇	C₂²	1	2	yes	—
65	#388×C₆₇	C₂²	1	2	yes	—
66	#388×C₆₇	C₂²	1	2	yes	—
67	#388×C₆₇	C₂²	1	2	yes	—
68	#388×C₆₇	C₂²	1	2	yes	—
69	#388×C₆₇	C₂²	1	2	yes	—
70	#388×C₆₇	C₂²	1	2	yes	—
71	#388×C₆₇	C₂²	1	2	yes	—
72	#388×C₆₇	C₂²	1	2	yes	—
73	#388×C₆₇	C₂²	1	2	yes	—
74	#388×C₆₇	C₂²	1	2	yes	—
75	#388×C₆₇	C₂²	1	2	yes	—
76	#388×C₆₇	C₂²	1	2	yes	—
77	#387×C₆₇	C₂²	1	2	yes	—
78	#387×C₆₇	C₂²	1	2	yes	—
79	#387×C₆₇	C₂²	1	2	yes	—
80	#387×C₆₇	C₂²	1	2	yes	—
81	#387×C₆₇	C₂²	1	2	yes	—
82	#387×C₆₇	C₂²	1	2	yes	—
83	#387×C₆₇	C₂²	1	2	yes	—
84	#387×C₆₇	C₂²	1	2	yes	—
85	#386×C₆₇	C₂²	1	2	yes	—
86	#386×C₆₇	C₂²	1	2	yes	—
87	#385×C₆₇	C₂²	1	1	yes	characteristic
88	#385×C₆₇	C₂²	1	1	yes	characteristic
89	#385×C₆₇	C₂²	1	1	yes	characteristic
90	#385×C₆₇	C₂²	1	1	yes	characteristic
91	#385×C₆₇	C₂²	1	2	yes	—
92	#385×C₆₇	C₂²	1	2	yes	—
93	#385×C₆₇	C₂²	1	2	yes	—
94	#385×C₆₇	C₂²	1	2	yes	—
95	#385×C₆₇	C₂²	1	2	yes	—
96	#385×C₆₇	C₂²	1	2	yes	—
97	#384×C₆₇	C₂²	1	1	yes	characteristic
98	#382×C₆₇	C₂²	1	1	yes	characteristic
99	#382×C₆₇	C₂²	1	2	yes	—
100	#382×C₆₇	C₂²	1	2	yes	—
101	#380×C₆₇	C₂²	1	1	yes	characteristic
102	#380×C₆₇	C₂²	1	1	yes	characteristic
103	#380×C₆₇	C₂²	1	2	yes	—
104	#380×C₆₇	C₂²	1	2	yes	—
105	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
106	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
107	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
108	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
109	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
110	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
111	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
112	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
113	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
114	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
115	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
116	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
117	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
118	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
119	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
120	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
121	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
122	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
123	C₂×#95×C₆₇	C₂²	1	2	yes	—
124	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
125	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
126	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
127	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
128	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
129	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
130	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
131	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
132	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
133	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
134	C₄⋊Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
135	#126×C₆₇	C₂×C₄	1	16	yes	complemented
136	#124×C₆₇	C₂×C₄	1	8	yes	complemented
137	#124×C₆₇	C₂×C₄	1	16	yes	complemented
138	#123×C₆₇	C₂×C₄	1	16	yes	complemented
139	C₂²⋊Q₈×C₆₇	C₂×C₄	1	16	yes	complemented
140	C₂²⋊Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
141	#121×C₆₇	C₂×C₄	1	8	yes	—
142	C₄×Q₈×C₆₇	—	4	4	no	—
143	C₄×Q₈×C₆₇	—	4	2	no	—
144	C₄×Q₈×C₆₇	—	4	2	no	—
145	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
146	C₄×D₄×C₆₇	—	4	2	no	—
147	C₄×D₄×C₆₇	—	4	2	no	—
148	C₄×D₄×C₆₇	—	4	4	no	—
149	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	complemented
150	#117×C₆₇	—	4	2	no	—
151	#117×C₆₇	—	4	4	no	—
152	#117×C₆₇	C₂×C₄	1	8	yes	complemented
153	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
154	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
155	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
156	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
157	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
158	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
159	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
160	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
161	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
162	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
163	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
164	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
165	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
166	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
167	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
168	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
169	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
170	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
171	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
172	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
173	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
174	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
175	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
176	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
177	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
178	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
179	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
180	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
181	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
182	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
183	C₂×C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
184	C₂×C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
185	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
186	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
187	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
188	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
189	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
190	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
191	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
192	C₂×C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
193	C₂×C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
194	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
195	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
196	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
197	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
198	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
199	C₂×C₄²×C₆₇	—	4	2	no	—
200	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
201	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
202	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
203	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
204	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
205	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
206	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
207	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
208	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
209	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
210	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
211	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
212	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
213	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
214	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
215	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
216	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
217	#95×C₆₇	C₂³	1	1	yes	characteristic
218	#95×C₆₇	C₂³	1	2	yes	—
219	#95×C₆₇	C₂³	1	2	yes	—
220	#95×C₆₇	C₂³	1	2	yes	—
221	#95×C₆₇	C₂³	1	1	yes	characteristic
222	#95×C₆₇	C₂³	1	2	yes	—
223	#95×C₆₇	C₂³	1	1	yes	characteristic
224	#95×C₆₇	C₂³	1	2	yes	—
225	#95×C₆₇	C₂³	1	2	yes	—
226	#95×C₆₇	C₂³	1	2	yes	—
227	#95×C₆₇	C₂³	1	1	yes	characteristic
228	#95×C₆₇	C₂³	1	1	yes	characteristic
229	#95×C₆₇	C₂³	1	2	yes	—
230	#95×C₆₇	C₂³	1	2	yes	—
231	#95×C₆₇	C₂³	1	2	yes	—
232	#95×C₆₇	C₂³	1	2	yes	—
233	#95×C₆₇	C₂³	1	1	yes	characteristic
234	#95×C₆₇	C₂³	1	2	yes	—
235	#95×C₆₇	C₂³	1	2	yes	—
236	#95×C₆₇	C₂³	1	2	yes	—
237	#95×C₆₇	C₂³	1	2	yes	—
238	#95×C₆₇	C₂³	1	2	yes	—
239	#95×C₆₇	C₂³	1	1	yes	characteristic
240	#95×C₆₇	C₂³	1	2	yes	—
241	#95×C₆₇	C₂³	1	1	yes	characteristic
242	#95×C₆₇	C₂³	1	2	yes	—
243	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
244	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
245	C₂×Q₈×C₆₇	—	2	2	no	—
246	C₂×Q₈×C₆₇	—	2	2	no	—
247	C₂×Q₈×C₆₇	—	4	2	no	—
248	C₂×Q₈×C₆₇	—	4	2	no	—
249	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
250	C₂×Q₈×C₆₇	C₂²×C₄	1	8	yes	—
251	C₂×D₄×C₆₇	—	2	2	no	—
252	C₂×D₄×C₆₇	—	2	2	no	—
253	C₂×D₄×C₆₇	—	4	2	no	—
254	C₂×D₄×C₆₇	—	4	2	no	—
255	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
256	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
257	C₂²×C₄×C₆₇	—	2	2	no	—
258	C₂²×C₄×C₆₇	—	2	2	no	—
259	C₂²×C₄×C₆₇	—	2	2	no	—
260	C₂²×C₄×C₆₇	—	2	2	no	—
261	C₂²×C₄×C₆₇	—	2	4	no	—
262	C₂²×C₄×C₆₇	—	4	1	no	—
263	C₂²×C₄×C₆₇	—	4	1	no	—
264	C₂²×C₄×C₆₇	—	4	2	no	—
265	C₂²×C₄×C₆₇	—	4	2	no	—
266	C₂²×C₄×C₆₇	—	4	1	no	—
267	C₂²×C₄×C₆₇	—	4	1	no	—
268	C₂²×C₄×C₆₇	—	4	2	no	—
269	C₂²×C₄×C₆₇	—	4	2	no	—
270	C₂²×C₄×C₆₇	—	4	2	no	—
271	C₂²×C₄×C₆₇	—	4	2	no	—
272	C₂²×C₄×C₆₇	—	4	2	no	—
273	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
274	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
275	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
276	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
277	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
278	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
279	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
280	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
281	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
282	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
283	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
284	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
285	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
286	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
287	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
288	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
289	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
290	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
291	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
292	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
293	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
294	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
295	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
296	C₄⋊C₄×C₆₇	—	2	2	no	—
297	C₄⋊C₄×C₆₇	—	2	4	no	—
298	C₄⋊C₄×C₆₇	—	2	2	no	—
299	C₄⋊C₄×C₆₇	—	2	2	no	—
300	C₄⋊C₄×C₆₇	—	2	2	no	—
301	C₄⋊C₄×C₆₇	—	2	2	no	—
302	C₄⋊C₄×C₆₇	—	2	2	no	—
303	C₄⋊C₄×C₆₇	—	2	2	no	—
304	C₄⋊C₄×C₆₇	—	2	2	no	—
305	C₄⋊C₄×C₆₇	—	4	1	no	—
306	C₄⋊C₄×C₆₇	—	4	2	no	—
307	C₄⋊C₄×C₆₇	—	4	1	no	—
308	C₄⋊C₄×C₆₇	—	4	1	no	—
309	C₄⋊C₄×C₆₇	—	4	2	no	—
310	C₄⋊C₄×C₆₇	—	4	2	no	—
311	C₄⋊C₄×C₆₇	—	4	1	no	—
312	C₄⋊C₄×C₆₇	—	4	1	no	—
313	C₄⋊C₄×C₆₇	—	4	1	no	—
314	C₄⋊C₄×C₆₇	—	4	2	no	—
315	C₄⋊C₄×C₆₇	—	4	2	no	—
316	C₄⋊C₄×C₆₇	—	4	2	no	—
317	C₄⋊C₄×C₆₇	—	4	2	no	—
318	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
319	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
320	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
321	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
322	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
323	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
324	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
325	C₂²⋊C₄×C₆₇	—	2	4	no	—
326	C₂²⋊C₄×C₆₇	—	4	2	no	—
327	C₂²⋊C₄×C₆₇	—	4	2	no	—
328	C₂²⋊C₄×C₆₇	—	4	2	no	—
329	C₂²⋊C₄×C₆₇	—	4	1	no	—
330	C₂²⋊C₄×C₆₇	—	4	2	no	—
331	C₂²⋊C₄×C₆₇	—	4	1	no	—
332	C₂²⋊C₄×C₆₇	—	4	2	no	—
333	C₂²⋊C₄×C₆₇	—	4	2	no	—
334	C₂²⋊C₄×C₆₇	—	4	2	no	—
335	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
336	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
337	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
338	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
339	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
340	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
341	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
342	C₄²×C₆₇	—	2	2	no	—
343	C₄²×C₆₇	—	2	2	no	—
344	C₄²×C₆₇	—	2	2	no	—
345	C₄²×C₆₇	—	2	4	no	—
346	C₄²×C₆₇	—	2	2	no	—
347	C₄²×C₆₇	—	4	1	no	—
348	C₄²×C₆₇	—	4	1	no	—
349	C₄²×C₆₇	—	4	2	no	—
350	C₄²×C₆₇	—	4	2	no	—
351	C₄²×C₆₇	—	4	2	no	—
352	C₄²×C₆₇	—	4	2	no	—
353	C₄²×C₆₇	—	4	2	no	—
354	C₄²×C₆₇	—	4	2	no	—
355	C₄²×C₆₇	—	4	2	no	—
356	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
357	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
358	C₄²×C₆₇	C₂²×C₄	1	8	yes	—
359	C₂³×C₆₇	—	2	2	no	—
360	C₂³×C₆₇	—	2	1	no	—
361	C₂³×C₆₇	—	2	1	no	—
362	C₂³×C₆₇	—	2	2	no	—
363	C₂³×C₆₇	—	2	1	no	—
364	C₂³×C₆₇	—	2	1	no	—
365	C₂³×C₆₇	—	2	2	no	—
366	C₂³×C₆₇	—	2	2	no	—
367	C₂³×C₆₇	—	2	1	no	—
368	C₂³×C₆₇	—	2	2	no	—
369	C₂³×C₆₇	—	2	2	no	—
370	C₂³×C₆₇	—	2	1	no	—
371	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
372	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
373	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
374	Q₈×C₆₇	—	2	8	no	—
375	Q₈×C₆₇	—	4	8	no	—
376	D₄×C₆₇	—	2	8	no	—
377	D₄×C₆₇	—	4	8	no	—
378	C₂×C₄×C₆₇	—	2	1	no	—
379	C₂×C₄×C₆₇	—	2	2	no	—
380	C₂×C₄×C₆₇	—	2	1	no	—
381	C₂×C₄×C₆₇	—	2	2	no	—
382	C₂×C₄×C₆₇	—	2	1	no	—
383	C₂×C₄×C₆₇	—	2	1	no	—
384	C₂×C₄×C₆₇	—	2	2	no	—
385	C₂×C₄×C₆₇	—	2	1	no	—
386	C₂×C₄×C₆₇	—	2	1	no	—
387	C₂×C₄×C₆₇	—	2	1	no	—
388	C₂×C₄×C₆₇	—	2	1	no	—
389	C₂×C₄×C₆₇	—	2	1	no	—
390	C₂×C₄×C₆₇	—	2	1	no	—
391	C₂×C₄×C₆₇	—	2	2	no	—
392	C₂×C₄×C₆₇	—	2	1	no	—
393	C₂×C₄×C₆₇	—	2	1	no	—
394	C₂×C₄×C₆₇	—	2	1	no	—
395	C₂×C₄×C₆₇	—	2	2	no	—
396	C₂×C₄×C₆₇	—	2	2	no	—
397	C₂×C₄×C₆₇	—	2	1	no	—
398	C₂×C₄×C₆₇	—	2	8	no	—
399	C₂×C₄×C₆₇	—	2	1	no	—
400	C₂×C₄×C₆₇	—	2	2	no	—
401	C₂×C₄×C₆₇	—	2	1	no	—
402	C₂×C₄×C₆₇	—	2	1	no	—
403	C₂×C₄×C₆₇	—	2	1	no	—
404	C₂×C₄×C₆₇	—	2	1	no	—
405	C₂×C₄×C₆₇	—	2	1	no	—
406	C₂×C₄×C₆₇	—	2	2	no	—
407	C₂×C₄×C₆₇	—	2	1	no	—
408	C₂×C₄×C₆₇	—	2	1	no	—
409	C₂×C₄×C₆₇	—	2	2	no	—
410	C₂×C₄×C₆₇	—	2	2	no	—
411	C₂×C₄×C₆₇	—	2	2	no	—
412	C₂×C₄×C₆₇	—	2	1	no	—
413	C₂×C₄×C₆₇	—	2	1	no	—
414	C₂×C₄×C₆₇	—	2	2	no	—
415	C₂×C₄×C₆₇	—	2	2	no	—
416	C₂×C₄×C₆₇	—	2	2	no	—
417	C₂×C₄×C₆₇	—	2	2	no	—
418	C₂×C₄×C₆₇	—	2	2	no	—
419	C₂×C₄×C₆₇	—	2	1	no	—
420	C₂×C₄×C₆₇	—	2	2	no	—
421	C₂×C₄×C₆₇	—	2	1	no	—
422	C₂×C₄×C₆₇	—	2	1	no	—
423	C₂×C₄×C₆₇	—	2	1	no	—
424	C₂×C₄×C₆₇	—	2	1	no	—
425	C₂×C₄×C₆₇	—	2	1	no	—
426	C₂×C₄×C₆₇	—	2	2	no	—
427	C₂×C₄×C₆₇	—	2	1	no	—
428	C₂×C₄×C₆₇	—	2	1	no	—
429	C₂×C₄×C₆₇	—	4	8	no	—
430	C₂×C₄×C₆₇	—	4	4	no	—
431	C₂×C₄×C₆₇	—	4	4	no	—
432	C₂×C₄×C₆₇	—	8	4	no	—
433	C₂×C₄×C₆₇	—	8	4	no	—
434	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
435	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
436	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
437	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
438	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
439	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
440	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
441	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
442	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
443	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
444	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
445	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
446	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
447	C₂²×C₆₇	—	2	2	no	—
448	C₂²×C₆₇	—	2	4	no	—
449	C₂²×C₆₇	—	2	4	no	—
450	C₂²×C₆₇	—	2	2	no	—
451	C₂²×C₆₇	—	2	2	no	—
452	C₂²×C₆₇	—	2	4	no	—
453	C₂²×C₆₇	—	4	1	no	—
454	C₂²×C₆₇	—	4	2	no	—
455	C₂²×C₆₇	—	4	1	no	—
456	C₂²×C₆₇	—	4	2	no	—
457	C₂²×C₆₇	—	4	2	no	—
458	C₂²×C₆₇	—	4	1	no	—
459	C₂²×C₆₇	—	4	1	no	—
460	C₂²×C₆₇	—	4	2	no	—
461	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
462	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
463	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
464	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
465	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
466	C₂²×C₆₇	#585	1	1	yes	characteristic
467	C₂²×C₆₇	#585	1	1	yes	characteristic
468	C₄×C₆₇	—	2	2	no	—
469	C₄×C₆₇	—	2	4	no	—
470	C₄×C₆₇	—	2	2	no	—
471	C₄×C₆₇	—	2	4	no	—
472	C₄×C₆₇	—	2	4	no	—
473	C₄×C₆₇	—	2	2	no	—
474	C₄×C₆₇	—	2	2	no	—
475	C₄×C₆₇	—	2	2	no	—
476	C₄×C₆₇	—	2	2	no	—
477	C₄×C₆₇	—	2	2	no	—
478	C₄×C₆₇	—	4	1	no	—
479	C₄×C₆₇	—	4	1	no	—
480	C₄×C₆₇	—	4	2	no	—
481	C₄×C₆₇	—	4	1	no	—
482	C₄×C₆₇	—	4	1	no	—
483	C₄×C₆₇	—	4	1	no	—
484	C₄×C₆₇	—	4	2	no	—
485	C₄×C₆₇	—	4	1	no	—
486	C₄×C₆₇	—	4	2	no	—
487	C₄×C₆₇	—	4	2	no	—
488	C₄×C₆₇	#520	1	4	yes	—
489	#34419	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
490	C₂×C₆₇	—	4	2	no	—
491	C₂×C₆₇	—	4	4	no	—
492	C₂×C₆₇	C₂×#520	1	1	yes	characteristic
493	C₂×C₆₇	C₄×H(5,𝔽₂)	1	1	yes	characteristic
494	C₂×C₆₇	C₄×2₋¹⁺⁴	1	1	yes	characteristic
495	C₂×C₆₇	#3536	1	1	yes	characteristic
496	C₂×C₆₇	#3540	1	1	yes	characteristic
497	C₂×C₆₇	#3554	1	1	yes	characteristic
498	C₂×C₆₇	#3555	1	1	yes	characteristic
499	C₂×#542	C₂×C₆₇	1	1	yes	characteristic
500	#2379	C₂×C₆₇	1	2	yes	complemented
501	#2376	C₂×C₆₇	1	2	yes	complemented
502	#2369	C₂×C₆₇	1	1	yes	characteristic, complemented
503	#2367	C₂×C₆₇	1	1	yes	characteristic, complemented
504	#2367	C₂×C₆₇	1	1	yes	characteristic, complemented
505	#2367	C₂×C₆₇	1	2	yes	complemented
506	#2365	C₂×C₆₇	1	1	yes	characteristic, complemented
507	#2359	C₂×C₆₇	1	2	yes	complemented
508	#2355	C₂×C₆₇	1	1	yes	characteristic, complemented
509	#2354	C₂×C₆₇	1	1	yes	characteristic, complemented
510	#2351	C₂×C₆₇	1	1	yes	characteristic, complemented
511	#2349	C₂×C₆₇	1	1	yes	characteristic, complemented
512	#2334	C₂×C₆₇	1	2	yes	complemented
513	#2334	C₂×C₆₇	1	2	yes	complemented
514	#2329	C₂×C₆₇	1	2	yes	complemented
515	#2324	C₂×C₆₇	1	1	yes	characteristic
516	#2323	C₂×C₆₇	1	1	yes	characteristic, complemented
517	#2323	C₂×C₆₇	1	2	yes	complemented
518	#2314	C₂×C₆₇	1	1	yes	characteristic, complemented
519	C₄×C₂²⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
520	C₄×#121	C₂×C₆₇	1	1	yes	characteristic
521	#2296	C₂×C₆₇	1	1	yes	characteristic, complemented
522	C₆₇	#34419	1	1	yes	characteristic, direct product factor, Sylow
523	#542	C₄×C₆₇	1	16	yes	complemented
524	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
525	C₂×#126	C₂²×C₆₇	1	2	yes	—
526	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
527	C₂×#124	C₂²×C₆₇	1	2	yes	—
528	C₂×#123	C₂²×C₆₇	1	2	yes	—
529	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
530	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
531	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
532	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
533	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
534	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
535	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
536	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
537	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
538	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
539	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
540	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
541	C₂×#117	C₂²×C₆₇	1	2	yes	—
542	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
543	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
544	#391	C₂²×C₆₇	1	1	yes	characteristic
545	#391	C₂²×C₆₇	1	2	yes	—
546	#391	C₂²×C₆₇	1	2	yes	—
547	#391	C₂²×C₆₇	1	2	yes	—
548	#391	C₂²×C₆₇	1	1	yes	characteristic
549	#390	C₂²×C₆₇	1	1	yes	characteristic
550	#390	C₂²×C₆₇	1	2	yes	—
551	#390	C₂²×C₆₇	1	1	yes	characteristic
552	#389	C₂²×C₆₇	1	2	yes	—
553	#389	C₂²×C₆₇	1	1	yes	characteristic
554	#389	C₂²×C₆₇	1	1	yes	characteristic
555	#389	C₂²×C₆₇	1	2	yes	—
556	#389	C₂²×C₆₇	1	1	yes	characteristic
557	#389	C₂²×C₆₇	1	1	yes	characteristic
558	#389	C₂²×C₆₇	1	2	yes	—
559	#389	C₂²×C₆₇	1	1	yes	characteristic
560	#389	C₂²×C₆₇	1	1	yes	characteristic
561	#389	C₂²×C₆₇	1	2	yes	—
562	#389	C₂²×C₆₇	1	1	yes	characteristic
563	#389	C₂²×C₆₇	1	1	yes	characteristic
564	#388	C₂²×C₆₇	1	2	yes	—
565	#388	C₂²×C₆₇	1	2	yes	—
566	#388	C₂²×C₆₇	1	2	yes	—
567	#388	C₂²×C₆₇	1	2	yes	—
568	#388	C₂²×C₆₇	1	2	yes	—
569	#388	C₂²×C₆₇	1	2	yes	—
570	#388	C₂²×C₆₇	1	2	yes	—
571	#388	C₂²×C₆₇	1	2	yes	—
572	#388	C₂²×C₆₇	1	2	yes	—
573	#388	C₂²×C₆₇	1	2	yes	—
574	#388	C₂²×C₆₇	1	2	yes	—
575	#388	C₂²×C₆₇	1	2	yes	—
576	#387	C₂²×C₆₇	1	2	yes	—
577	#387	C₂²×C₆₇	1	2	yes	—
578	#387	C₂²×C₆₇	1	2	yes	—
579	#387	C₂²×C₆₇	1	2	yes	—
580	#387	C₂²×C₆₇	1	2	yes	—
581	#387	C₂²×C₆₇	1	2	yes	—
582	#387	C₂²×C₆₇	1	2	yes	—
583	#387	C₂²×C₆₇	1	2	yes	—
584	#386	C₂²×C₆₇	1	2	yes	—
585	#386	C₂²×C₆₇	1	2	yes	—
586	#385	C₂²×C₆₇	1	2	yes	—
587	#385	C₂²×C₆₇	1	2	yes	—
588	#385	C₂²×C₆₇	1	2	yes	—
589	#385	C₂²×C₆₇	1	2	yes	—
590	#385	C₂²×C₆₇	1	2	yes	—
591	#385	C₂²×C₆₇	1	1	yes	characteristic
592	#385	C₂²×C₆₇	1	1	yes	characteristic
593	#385	C₂²×C₆₇	1	1	yes	characteristic
594	#385	C₂²×C₆₇	1	2	yes	—
595	#385	C₂²×C₆₇	1	1	yes	characteristic
596	#384	C₂²×C₆₇	1	1	yes	characteristic
597	#382	C₂²×C₆₇	1	1	yes	characteristic
598	#382	C₂²×C₆₇	1	2	yes	—
599	#382	C₂²×C₆₇	1	2	yes	—
600	#380	C₂²×C₆₇	1	1	yes	characteristic
601	#380	C₂²×C₆₇	1	1	yes	characteristic
602	#380	C₂²×C₆₇	1	2	yes	—
603	#380	C₂²×C₆₇	1	2	yes	—
604	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
605	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
606	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
607	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
608	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
609	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
610	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
611	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
612	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
613	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
614	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
615	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
616	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
617	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
618	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
619	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
620	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
621	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
622	C₂×#95	C₂²×C₆₇	1	2	yes	—
623	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
624	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
625	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
626	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
627	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
628	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
629	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
630	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
631	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
632	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
633	C₄⋊Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
634	#126	C₂×C₄×C₆₇	1	16	yes	complemented
635	#124	C₂×C₄×C₆₇	1	8	yes	complemented
636	#124	C₂×C₄×C₆₇	1	16	yes	complemented
637	#123	C₂×C₄×C₆₇	1	16	yes	complemented
638	C₂²⋊Q₈	C₂×C₄×C₆₇	1	16	yes	complemented
639	C₂²⋊Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
640	#121	C₂×C₄×C₆₇	1	8	yes	—
641	C₄×Q₈	—	4	4	no	—
642	C₄×Q₈	—	4	2	no	—
643	C₄×Q₈	—	4	2	no	—
644	C₄×Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
645	C₄×D₄	—	4	2	no	—
646	C₄×D₄	—	4	2	no	—
647	C₄×D₄	—	4	4	no	—
648	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	complemented
649	#117	—	4	2	no	—
650	#117	—	4	4	no	—
651	#117	C₂×C₄×C₆₇	1	8	yes	complemented
652	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
653	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
654	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
655	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
656	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
657	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
658	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
659	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
660	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
661	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
662	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
663	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
664	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
665	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
666	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
667	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
668	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
669	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
670	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
671	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
672	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
673	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
674	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
675	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
676	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
677	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
678	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
679	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
680	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
681	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
682	C₂×C₂²⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
683	C₂×C₂²⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
684	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
685	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
686	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
687	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
688	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
689	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
690	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
691	C₂×C₂²⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
692	C₂×C₂²⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
693	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
694	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
695	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
696	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
697	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
698	C₂×C₄²	—	4	2	no	—
699	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
700	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
701	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
702	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
703	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
704	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
705	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
706	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
707	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
708	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
709	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
710	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
711	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
712	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
713	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
714	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
715	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
716	#95	C₂³×C₆₇	1	1	yes	characteristic
717	#95	C₂³×C₆₇	1	2	yes	—
718	#95	C₂³×C₆₇	1	2	yes	—
719	#95	C₂³×C₆₇	1	2	yes	—
720	#95	C₂³×C₆₇	1	1	yes	characteristic
721	#95	C₂³×C₆₇	1	2	yes	—
722	#95	C₂³×C₆₇	1	1	yes	characteristic
723	#95	C₂³×C₆₇	1	2	yes	—
724	#95	C₂³×C₆₇	1	2	yes	—
725	#95	C₂³×C₆₇	1	2	yes	—
726	#95	C₂³×C₆₇	1	1	yes	characteristic
727	#95	C₂³×C₆₇	1	1	yes	characteristic
728	#95	C₂³×C₆₇	1	2	yes	—
729	#95	C₂³×C₆₇	1	2	yes	—
730	#95	C₂³×C₆₇	1	2	yes	—
731	#95	C₂³×C₆₇	1	2	yes	—
732	#95	C₂³×C₆₇	1	1	yes	characteristic
733	#95	C₂³×C₆₇	1	2	yes	—
734	#95	C₂³×C₆₇	1	2	yes	—
735	#95	C₂³×C₆₇	1	2	yes	—
736	#95	C₂³×C₆₇	1	2	yes	—
737	#95	C₂³×C₆₇	1	2	yes	—
738	#95	C₂³×C₆₇	1	1	yes	characteristic
739	#95	C₂³×C₆₇	1	2	yes	—
740	#95	C₂³×C₆₇	1	1	yes	characteristic
741	#95	C₂³×C₆₇	1	2	yes	—
742	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
743	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
744	C₂×Q₈	—	2	2	no	—
745	C₂×Q₈	—	2	2	no	—
746	C₂×Q₈	—	4	2	no	—
747	C₂×Q₈	—	4	2	no	—
748	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
749	C₂×Q₈	C₂²×C₄×C₆₇	1	8	yes	—
750	C₂×D₄	—	2	2	no	—
751	C₂×D₄	—	2	2	no	—
752	C₂×D₄	—	4	2	no	—
753	C₂×D₄	—	4	2	no	—
754	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
755	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
756	C₂²×C₄	—	2	2	no	—
757	C₂²×C₄	—	2	2	no	—
758	C₂²×C₄	—	2	2	no	—
759	C₂²×C₄	—	2	2	no	—
760	C₂²×C₄	—	2	4	no	—
761	C₂²×C₄	—	4	1	no	—
762	C₂²×C₄	—	4	1	no	—
763	C₂²×C₄	—	4	2	no	—
764	C₂²×C₄	—	4	2	no	—
765	C₂²×C₄	—	4	1	no	—
766	C₂²×C₄	—	4	1	no	—
767	C₂²×C₄	—	4	2	no	—
768	C₂²×C₄	—	4	2	no	—
769	C₂²×C₄	—	4	2	no	—
770	C₂²×C₄	—	4	2	no	—
771	C₂²×C₄	—	4	2	no	—
772	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
773	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
774	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
775	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
776	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
777	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
778	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
779	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
780	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
781	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
782	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
783	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
784	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
785	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
786	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
787	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
788	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
789	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
790	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
791	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
792	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
793	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
794	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
795	C₄⋊C₄	—	2	2	no	—
796	C₄⋊C₄	—	2	4	no	—
797	C₄⋊C₄	—	2	2	no	—
798	C₄⋊C₄	—	2	2	no	—
799	C₄⋊C₄	—	2	2	no	—
800	C₄⋊C₄	—	2	2	no	—
801	C₄⋊C₄	—	2	2	no	—
802	C₄⋊C₄	—	2	2	no	—
803	C₄⋊C₄	—	2	2	no	—
804	C₄⋊C₄	—	4	1	no	—
805	C₄⋊C₄	—	4	1	no	—
806	C₄⋊C₄	—	4	2	no	—
807	C₄⋊C₄	—	4	2	no	—
808	C₄⋊C₄	—	4	2	no	—
809	C₄⋊C₄	—	4	2	no	—
810	C₄⋊C₄	—	4	1	no	—
811	C₄⋊C₄	—	4	2	no	—
812	C₄⋊C₄	—	4	1	no	—
813	C₄⋊C₄	—	4	1	no	—
814	C₄⋊C₄	—	4	2	no	—
815	C₄⋊C₄	—	4	2	no	—
816	C₄⋊C₄	—	4	1	no	—
817	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
818	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
819	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
820	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
821	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
822	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
823	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
824	C₂²⋊C₄	—	2	4	no	—
825	C₂²⋊C₄	—	4	1	no	—
826	C₂²⋊C₄	—	4	2	no	—
827	C₂²⋊C₄	—	4	1	no	—
828	C₂²⋊C₄	—	4	2	no	—
829	C₂²⋊C₄	—	4	2	no	—
830	C₂²⋊C₄	—	4	2	no	—
831	C₂²⋊C₄	—	4	2	no	—
832	C₂²⋊C₄	—	4	2	no	—
833	C₂²⋊C₄	—	4	2	no	—
834	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
835	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
836	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
837	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
838	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
839	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
840	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
841	C₄²	—	2	2	no	—
842	C₄²	—	2	2	no	—
843	C₄²	—	2	2	no	—
844	C₄²	—	2	4	no	—
845	C₄²	—	2	2	no	—
846	C₄²	—	4	1	no	—
847	C₄²	—	4	1	no	—
848	C₄²	—	4	2	no	—
849	C₄²	—	4	2	no	—
850	C₄²	—	4	2	no	—
851	C₄²	—	4	2	no	—
852	C₄²	—	4	2	no	—
853	C₄²	—	4	2	no	—
854	C₄²	—	4	2	no	—
855	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
856	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
857	C₄²	C₂²×C₄×C₆₇	1	8	yes	—
858	C₂³	—	2	2	no	—
859	C₂³	—	2	1	no	—
860	C₂³	—	2	1	no	—
861	C₂³	—	2	2	no	—
862	C₂³	—	2	1	no	—
863	C₂³	—	2	1	no	—
864	C₂³	—	2	2	no	—
865	C₂³	—	2	2	no	—
866	C₂³	—	2	1	no	—
867	C₂³	—	2	2	no	—
868	C₂³	—	2	2	no	—
869	C₂³	—	2	1	no	—
870	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
871	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
872	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
873	Q₈	—	2	8	no	—
874	Q₈	—	4	8	no	—
875	D₄	—	2	8	no	—
876	D₄	—	4	8	no	—
877	C₂×C₄	—	2	1	no	—
878	C₂×C₄	—	2	2	no	—
879	C₂×C₄	—	2	1	no	—
880	C₂×C₄	—	2	2	no	—
881	C₂×C₄	—	2	1	no	—
882	C₂×C₄	—	2	1	no	—
883	C₂×C₄	—	2	2	no	—
884	C₂×C₄	—	2	1	no	—
885	C₂×C₄	—	2	1	no	—
886	C₂×C₄	—	2	1	no	—
887	C₂×C₄	—	2	1	no	—
888	C₂×C₄	—	2	1	no	—
889	C₂×C₄	—	2	1	no	—
890	C₂×C₄	—	2	2	no	—
891	C₂×C₄	—	2	1	no	—
892	C₂×C₄	—	2	1	no	—
893	C₂×C₄	—	2	1	no	—
894	C₂×C₄	—	2	2	no	—
895	C₂×C₄	—	2	2	no	—
896	C₂×C₄	—	2	1	no	—
897	C₂×C₄	—	2	8	no	—
898	C₂×C₄	—	2	1	no	—
899	C₂×C₄	—	2	2	no	—
900	C₂×C₄	—	2	1	no	—
901	C₂×C₄	—	2	1	no	—
902	C₂×C₄	—	2	1	no	—
903	C₂×C₄	—	2	1	no	—
904	C₂×C₄	—	2	1	no	—
905	C₂×C₄	—	2	2	no	—
906	C₂×C₄	—	2	1	no	—
907	C₂×C₄	—	2	1	no	—
908	C₂×C₄	—	2	2	no	—
909	C₂×C₄	—	2	2	no	—
910	C₂×C₄	—	2	2	no	—
911	C₂×C₄	—	2	1	no	—
912	C₂×C₄	—	2	1	no	—
913	C₂×C₄	—	2	2	no	—
914	C₂×C₄	—	2	2	no	—
915	C₂×C₄	—	2	2	no	—
916	C₂×C₄	—	2	2	no	—
917	C₂×C₄	—	2	2	no	—
918	C₂×C₄	—	2	1	no	—
919	C₂×C₄	—	2	2	no	—
920	C₂×C₄	—	2	1	no	—
921	C₂×C₄	—	2	1	no	—
922	C₂×C₄	—	2	1	no	—
923	C₂×C₄	—	2	1	no	—
924	C₂×C₄	—	2	1	no	—
925	C₂×C₄	—	2	2	no	—
926	C₂×C₄	—	2	1	no	—
927	C₂×C₄	—	2	1	no	—
928	C₂×C₄	—	4	8	no	—
929	C₂×C₄	—	4	4	no	—
930	C₂×C₄	—	4	4	no	—
931	C₂×C₄	—	8	4	no	—
932	C₂×C₄	—	8	4	no	—
933	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
934	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
935	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
936	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
937	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
938	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
939	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
940	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
941	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
942	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
943	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
944	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
945	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
946	C₂²	—	2	2	no	—
947	C₂²	—	2	4	no	—
948	C₂²	—	2	4	no	—
949	C₂²	—	2	2	no	—
950	C₂²	—	2	2	no	—
951	C₂²	—	2	4	no	—
952	C₂²	—	4	1	no	—
953	C₂²	—	4	2	no	—
954	C₂²	—	4	1	no	—
955	C₂²	—	4	2	no	—
956	C₂²	—	4	2	no	—
957	C₂²	—	4	1	no	—
958	C₂²	—	4	1	no	—
959	C₂²	—	4	2	no	—
960	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
961	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
962	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
963	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
964	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
965	C₂²	#585×C₆₇	1	1	yes	characteristic
966	C₂²	#585×C₆₇	1	1	yes	characteristic
967	C₄	—	2	2	no	—
968	C₄	—	2	2	no	—
969	C₄	—	2	2	no	—
970	C₄	—	2	4	no	—
971	C₄	—	2	2	no	—
972	C₄	—	2	4	no	—
973	C₄	—	2	4	no	—
974	C₄	—	2	2	no	—
975	C₄	—	2	2	no	—
976	C₄	—	2	2	no	—
977	C₄	—	4	2	no	—
978	C₄	—	4	1	no	—
979	C₄	—	4	1	no	—
980	C₄	—	4	1	no	—
981	C₄	—	4	1	no	—
982	C₄	—	4	1	no	—
983	C₄	—	4	2	no	—
984	C₄	—	4	1	no	—
985	C₄	—	4	2	no	—
986	C₄	—	4	2	no	—
987	C₄	#520×C₆₇	1	4	yes	—
988	C₂	—	4	2	no	—
989	C₂	—	4	4	no	—
990	C₂	C₂×#520×C₆₇	1	1	yes	characteristic
991	C₂	C₄×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
992	C₂	C₄×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
993	C₂	C₆₇×#3536	1	1	yes	characteristic
994	C₂	C₆₇×#3540	1	1	yes	characteristic
995	C₂	C₆₇×#3554	1	1	yes	characteristic
996	C₂	C₆₇×#3555	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂×#121×C₆₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₆₇×#2379	C₂	semidirect
3. C₆₇×#2376	C₂	semidirect
4. C₆₇×#2369	C₂	semidirect
5. C₆₇×#2367	C₂	semidirect
6. C₆₇×#2367	C₂	semidirect
7. C₆₇×#2367	C₂	semidirect
8. C₆₇×#2365	C₂	semidirect
9. C₆₇×#2359	C₂	semidirect
10. C₆₇×#2355	C₂	semidirect
11. C₆₇×#2354	C₂	semidirect
12. C₆₇×#2351	C₂	semidirect
13. C₆₇×#2349	C₂	semidirect
14. C₆₇×#2334	C₂	semidirect
15. C₆₇×#2334	C₂	semidirect
16. C₆₇×#2329	C₂	semidirect
18. C₆₇×#2323	C₂	semidirect
19. C₆₇×#2323	C₂	semidirect
20. C₆₇×#2314	C₂	semidirect
21. C₄×C₂²⋊Q₈×C₆₇	C₂	semidirect
23. C₆₇×#2296	C₂	semidirect
24. #542×C₆₇	C₄	semidirect
134. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
135. #126×C₆₇	C₂×C₄	semidirect
136. #124×C₆₇	C₂×C₄	semidirect
137. #124×C₆₇	C₂×C₄	semidirect
138. #123×C₆₇	C₂×C₄	semidirect
139. C₂²⋊Q₈×C₆₇	C₂×C₄	semidirect
140. C₂²⋊Q₈×C₆₇	C₂×C₄	semidirect
145. C₄×Q₈×C₆₇	C₂×C₄	semidirect
149. C₄×D₄×C₆₇	C₂×C₄	semidirect
152. #117×C₆₇	C₂×C₄	semidirect
489. #34419	C₆₇	direct (N × H)
500. #2379	C₂×C₆₇	semidirect
501. #2376	C₂×C₆₇	semidirect
502. #2369	C₂×C₆₇	semidirect
503. #2367	C₂×C₆₇	semidirect
504. #2367	C₂×C₆₇	semidirect
505. #2367	C₂×C₆₇	semidirect
506. #2365	C₂×C₆₇	semidirect
507. #2359	C₂×C₆₇	semidirect
508. #2355	C₂×C₆₇	semidirect
509. #2354	C₂×C₆₇	semidirect
510. #2351	C₂×C₆₇	semidirect
511. #2349	C₂×C₆₇	semidirect
512. #2334	C₂×C₆₇	semidirect
513. #2334	C₂×C₆₇	semidirect
514. #2329	C₂×C₆₇	semidirect
516. #2323	C₂×C₆₇	semidirect
517. #2323	C₂×C₆₇	semidirect
518. #2314	C₂×C₆₇	semidirect
519. C₄×C₂²⋊Q₈	C₂×C₆₇	semidirect
521. #2296	C₂×C₆₇	semidirect
522. C₆₇	#34419	direct (N × H)
523. #542	C₄×C₆₇	semidirect
633. C₄⋊Q₈	C₂×C₄×C₆₇	semidirect
634. #126	C₂×C₄×C₆₇	semidirect
635. #124	C₂×C₄×C₆₇	semidirect
636. #124	C₂×C₄×C₆₇	semidirect
637. #123	C₂×C₄×C₆₇	semidirect
638. C₂²⋊Q₈	C₂×C₄×C₆₇	semidirect
639. C₂²⋊Q₈	C₂×C₄×C₆₇	semidirect
644. C₄×Q₈	C₂×C₄×C₆₇	semidirect
648. C₄×D₄	C₂×C₄×C₆₇	semidirect
651. #117	C₂×C₄×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#542×C₆₇	2
C₆₇×#2379	2
C₆₇×#2376	2
C₆₇×#2369	2
C₆₇×#2367	2
C₆₇×#2365	2
C₆₇×#2359	2
C₆₇×#2355	2
C₆₇×#2354	2
C₆₇×#2351	2
C₆₇×#2349	2
C₆₇×#2334	2
C₆₇×#2329	2
C₆₇×#2324	2
C₆₇×#2323	2
C₆₇×#2314	2
C₄×C₂²⋊Q₈×C₆₇	2
C₄×#121×C₆₇	2
C₆₇×#2296	2
#34419	67

Subgroups — C67×#34419 #30713670

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34419 #30713670