Subgroups — C₆₇×#34307 #30713558

Order 17152.

2136 conjugacy classes of subgroups.

The subgroup lattice has 970 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#556×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2373	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2373	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2372	C₂	1	2	yes	maximal
5	C₆₇×#2369	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2362	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2360	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2356	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2356	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2352	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#2347	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₇×#2346	C₂	1	1	yes	maximal, characteristic, complemented
13	C₆₇×#2337	C₂	1	2	yes	maximal
14	C₆₇×#2334	C₂	1	2	yes	maximal, complemented
15	C₆₇×#2331	C₂	1	1	yes	maximal, characteristic, complemented
16	C₆₇×#2323	C₂	1	1	yes	maximal, characteristic, complemented
17	C₄×#126×C₆₇	C₂	1	2	yes	maximal, complemented
18	C₄×C₂²⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
19	C₆₇×#2295	C₂	1	2	yes	maximal, complemented
20	C₆₇×#2290	C₂	1	1	yes	maximal, characteristic
21	C₆₇×#2276	C₂	1	1	yes	maximal, characteristic, complemented
22	C₆₇×#2274	C₂	1	1	yes	maximal, characteristic, complemented
23	C₄×#117×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
24	#556×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×#126×C₆₇	C₂²	1	2	yes	—
28	C₂×#124×C₆₇	C₂²	1	2	yes	—
29	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
35	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
40	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
41	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
42	#391×C₆₇	C₂²	1	2	yes	—
43	#391×C₆₇	C₂²	1	2	yes	—
44	#391×C₆₇	C₂²	1	1	yes	characteristic
45	#391×C₆₇	C₂²	1	1	yes	characteristic
46	#391×C₆₇	C₂²	1	2	yes	—
47	#391×C₆₇	C₂²	1	2	yes	—
48	#389×C₆₇	C₂²	1	2	yes	—
49	#389×C₆₇	C₂²	1	2	yes	—
50	#389×C₆₇	C₂²	1	1	yes	characteristic
51	#389×C₆₇	C₂²	1	1	yes	characteristic
52	#389×C₆₇	C₂²	1	1	yes	characteristic
53	#389×C₆₇	C₂²	1	1	yes	characteristic
54	#389×C₆₇	C₂²	1	2	yes	—
55	#388×C₆₇	C₂²	1	1	yes	characteristic
56	#388×C₆₇	C₂²	1	1	yes	characteristic
57	#388×C₆₇	C₂²	1	2	yes	—
58	#388×C₆₇	C₂²	1	2	yes	—
59	#388×C₆₇	C₂²	1	2	yes	—
60	#388×C₆₇	C₂²	1	2	yes	—
61	#388×C₆₇	C₂²	1	2	yes	—
62	#388×C₆₇	C₂²	1	2	yes	—
63	#388×C₆₇	C₂²	1	2	yes	—
64	#388×C₆₇	C₂²	1	2	yes	—
65	#388×C₆₇	C₂²	1	2	yes	—
66	#387×C₆₇	C₂²	1	1	yes	characteristic
67	#387×C₆₇	C₂²	1	1	yes	characteristic
68	#387×C₆₇	C₂²	1	2	yes	—
69	#387×C₆₇	C₂²	1	2	yes	—
70	#387×C₆₇	C₂²	1	2	yes	—
71	#387×C₆₇	C₂²	1	2	yes	—
72	#387×C₆₇	C₂²	1	2	yes	—
73	#387×C₆₇	C₂²	1	2	yes	—
74	#387×C₆₇	C₂²	1	2	yes	—
75	#387×C₆₇	C₂²	1	2	yes	—
76	#387×C₆₇	C₂²	1	2	yes	—
77	#386×C₆₇	C₂²	1	1	yes	characteristic
78	#386×C₆₇	C₂²	1	1	yes	characteristic
79	#386×C₆₇	C₂²	1	2	yes	—
80	#385×C₆₇	C₂²	1	1	yes	characteristic
81	#385×C₆₇	C₂²	1	1	yes	characteristic
82	#385×C₆₇	C₂²	1	2	yes	—
83	#385×C₆₇	C₂²	1	2	yes	—
84	#383×C₆₇	C₂²	1	2	yes	—
85	#383×C₆₇	C₂²	1	2	yes	—
86	#382×C₆₇	C₂²	1	1	yes	characteristic
87	#382×C₆₇	C₂²	1	2	yes	—
88	#380×C₆₇	C₂²	1	1	yes	characteristic
89	#380×C₆₇	C₂²	1	1	yes	characteristic
90	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
91	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
92	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
93	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
94	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
95	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
96	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
97	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
98	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
99	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
100	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
101	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
102	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
103	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
104	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
105	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
106	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
107	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
108	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
109	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
110	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
111	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
112	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
113	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
114	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
115	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
116	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
117	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
118	#376×C₆₇	C₂²	1	2	yes	—
119	#376×C₆₇	C₂²	1	1	yes	characteristic
120	#376×C₆₇	C₂²	1	2	yes	—
121	#376×C₆₇	C₂²	1	2	yes	—
122	#376×C₆₇	C₂²	1	2	yes	—
123	C₄³×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	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
128	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
129	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
130	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
131	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
132	C₄⋊Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
133	#126×C₆₇	C₂×C₄	1	16	yes	complemented
134	#126×C₆₇	C₂×C₄	1	16	yes	complemented
135	#124×C₆₇	C₂×C₄	1	16	yes	—
136	C₂²⋊Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
137	C₂²⋊Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
138	C₄×Q₈×C₆₇	—	2	4	no	—
139	C₄×Q₈×C₆₇	—	2	4	no	—
140	C₄×Q₈×C₆₇	C₂×C₄	1	16	yes	complemented
141	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
142	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
143	#117×C₆₇	—	2	4	no	—
144	#117×C₆₇	—	2	4	no	—
145	#117×C₆₇	C₂×C₄	1	8	yes	complemented
146	#117×C₆₇	C₂×C₄	1	8	yes	complemented
147	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
148	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
149	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
150	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
151	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
152	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
153	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
154	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
155	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
156	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
157	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
158	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
159	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
160	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
161	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
162	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
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	2	yes	—
169	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
170	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
171	C₂×C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
172	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
173	C₂×C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
174	C₂×C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
175	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
176	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
177	C₂×C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
178	C₂×C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
179	C₂×C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
180	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
181	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
182	C₂×C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
183	C₂×C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
184	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
185	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
186	C₂×C₄²×C₆₇	—	2	4	no	—
187	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
188	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
189	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
190	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
191	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
192	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
193	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
194	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
195	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
196	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
197	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
198	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
199	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
200	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
201	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
202	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
203	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
204	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
205	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
206	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
207	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
208	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
209	#95×C₆₇	C₂³	1	1	yes	characteristic
210	#95×C₆₇	C₂³	1	2	yes	—
211	#95×C₆₇	C₂³	1	1	yes	characteristic
212	#95×C₆₇	C₂³	1	1	yes	characteristic
213	#95×C₆₇	C₂³	1	2	yes	—
214	#95×C₆₇	C₂³	1	2	yes	—
215	#95×C₆₇	C₂³	1	2	yes	—
216	#95×C₆₇	C₂³	1	2	yes	—
217	#95×C₆₇	C₂³	1	2	yes	—
218	#95×C₆₇	C₂³	1	2	yes	—
219	#95×C₆₇	C₂³	1	1	yes	characteristic
220	#95×C₆₇	C₂³	1	1	yes	characteristic
221	#95×C₆₇	C₂³	1	1	yes	characteristic
222	#95×C₆₇	C₂³	1	2	yes	—
223	#95×C₆₇	C₂³	1	2	yes	—
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	2	yes	—
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	1	yes	characteristic
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	1	yes	characteristic
237	#95×C₆₇	C₂³	1	2	yes	—
238	#95×C₆₇	C₂³	1	2	yes	—
239	#95×C₆₇	C₂³	1	2	yes	—
240	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
241	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
242	C₂×Q₈×C₆₇	—	2	2	no	—
243	C₂×Q₈×C₆₇	—	2	2	no	—
244	C₂×Q₈×C₆₇	—	4	2	no	—
245	C₂×Q₈×C₆₇	—	4	2	no	—
246	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
247	C₂×Q₈×C₆₇	C₂²×C₄	1	8	yes	—
248	C₂×D₄×C₆₇	—	4	1	no	—
249	C₂×D₄×C₆₇	—	4	1	no	—
250	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
251	C₂²×C₄×C₆₇	—	2	2	no	—
252	C₂²×C₄×C₆₇	—	2	2	no	—
253	C₂²×C₄×C₆₇	—	2	2	no	—
254	C₂²×C₄×C₆₇	—	2	2	no	—
255	C₂²×C₄×C₆₇	—	2	2	no	—
256	C₂²×C₄×C₆₇	—	2	2	no	—
257	C₂²×C₄×C₆₇	—	2	2	no	—
258	C₂²×C₄×C₆₇	—	2	4	no	—
259	C₂²×C₄×C₆₇	—	4	1	no	—
260	C₂²×C₄×C₆₇	—	4	1	no	—
261	C₂²×C₄×C₆₇	—	4	1	no	—
262	C₂²×C₄×C₆₇	—	4	2	no	—
263	C₂²×C₄×C₆₇	—	4	2	no	—
264	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
265	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
266	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
267	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
268	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
269	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
270	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
271	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
272	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
273	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
274	C₂²×C₄×C₆₇	C₂⁴	1	2	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	2	yes	—
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	1	yes	characteristic
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	C₄⋊C₄×C₆₇	—	2	4	no	—
288	C₄⋊C₄×C₆₇	—	2	2	no	—
289	C₄⋊C₄×C₆₇	—	2	2	no	—
290	C₄⋊C₄×C₆₇	—	2	2	no	—
291	C₄⋊C₄×C₆₇	—	2	2	no	—
292	C₄⋊C₄×C₆₇	—	2	4	no	—
293	C₄⋊C₄×C₆₇	—	2	4	no	—
294	C₄⋊C₄×C₆₇	—	2	2	no	—
295	C₄⋊C₄×C₆₇	—	2	2	no	—
296	C₄⋊C₄×C₆₇	—	4	1	no	—
297	C₄⋊C₄×C₆₇	—	4	2	no	—
298	C₄⋊C₄×C₆₇	—	4	2	no	—
299	C₄⋊C₄×C₆₇	—	4	2	no	—
300	C₄⋊C₄×C₆₇	—	4	2	no	—
301	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
302	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
303	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
304	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
305	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
306	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
307	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
308	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
309	C₄⋊C₄×C₆₇	#41	1	4	yes	—
310	C₂²⋊C₄×C₆₇	—	2	2	no	—
311	C₂²⋊C₄×C₆₇	—	2	4	no	—
312	C₂²⋊C₄×C₆₇	—	2	2	no	—
313	C₂²⋊C₄×C₆₇	—	2	2	no	—
314	C₂²⋊C₄×C₆₇	—	4	2	no	—
315	C₂²⋊C₄×C₆₇	—	4	1	no	—
316	C₂²⋊C₄×C₆₇	—	4	1	no	—
317	C₂²⋊C₄×C₆₇	—	4	2	no	—
318	C₂²⋊C₄×C₆₇	—	4	1	no	—
319	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
320	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
321	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
322	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
323	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
324	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
325	C₄²×C₆₇	—	2	2	no	—
326	C₄²×C₆₇	—	2	2	no	—
327	C₄²×C₆₇	—	2	2	no	—
328	C₄²×C₆₇	—	2	4	no	—
329	C₄²×C₆₇	—	2	2	no	—
330	C₄²×C₆₇	—	2	4	no	—
331	C₄²×C₆₇	—	2	4	no	—
332	C₄²×C₆₇	—	2	2	no	—
333	C₄²×C₆₇	—	2	4	no	—
334	C₄²×C₆₇	—	4	2	no	—
335	C₄²×C₆₇	—	4	2	no	—
336	C₄²×C₆₇	—	4	2	no	—
337	C₄²×C₆₇	—	4	2	no	—
338	C₄²×C₆₇	—	4	2	no	—
339	C₄²×C₆₇	—	4	2	no	—
340	C₄²×C₆₇	—	4	2	no	—
341	C₄²×C₆₇	—	4	2	no	—
342	C₄²×C₆₇	C₂²×C₄	1	8	yes	—
343	C₄²×C₆₇	C₂²×C₄	1	8	yes	—
344	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
345	C₄²×C₆₇	C₂²×C₄	1	8	yes	—
346	C₄²×C₆₇	#41	1	4	yes	—
347	C₂³×C₆₇	—	2	1	no	—
348	C₂³×C₆₇	—	2	1	no	—
349	C₂³×C₆₇	—	2	1	no	—
350	C₂³×C₆₇	—	2	1	no	—
351	C₂³×C₆₇	—	2	1	no	—
352	C₂³×C₆₇	—	2	1	no	—
353	C₂³×C₆₇	—	2	1	no	—
354	C₂³×C₆₇	—	2	1	no	—
355	C₂³×C₆₇	—	2	1	no	—
356	C₂³×C₆₇	—	2	1	no	—
357	C₂³×C₆₇	—	2	1	no	—
358	C₂³×C₆₇	—	2	1	no	—
359	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
360	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
361	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
362	Q₈×C₆₇	—	2	8	no	—
363	Q₈×C₆₇	—	4	8	no	—
364	D₄×C₆₇	—	4	4	no	—
365	C₂×C₄×C₆₇	—	2	1	no	—
366	C₂×C₄×C₆₇	—	2	1	no	—
367	C₂×C₄×C₆₇	—	2	1	no	—
368	C₂×C₄×C₆₇	—	2	2	no	—
369	C₂×C₄×C₆₇	—	2	2	no	—
370	C₂×C₄×C₆₇	—	2	1	no	—
371	C₂×C₄×C₆₇	—	2	1	no	—
372	C₂×C₄×C₆₇	—	2	2	no	—
373	C₂×C₄×C₆₇	—	2	1	no	—
374	C₂×C₄×C₆₇	—	2	1	no	—
375	C₂×C₄×C₆₇	—	2	2	no	—
376	C₂×C₄×C₆₇	—	2	1	no	—
377	C₂×C₄×C₆₇	—	2	1	no	—
378	C₂×C₄×C₆₇	—	2	1	no	—
379	C₂×C₄×C₆₇	—	2	1	no	—
380	C₂×C₄×C₆₇	—	2	1	no	—
381	C₂×C₄×C₆₇	—	2	2	no	—
382	C₂×C₄×C₆₇	—	2	2	no	—
383	C₂×C₄×C₆₇	—	2	2	no	—
384	C₂×C₄×C₆₇	—	2	2	no	—
385	C₂×C₄×C₆₇	—	2	2	no	—
386	C₂×C₄×C₆₇	—	2	2	no	—
387	C₂×C₄×C₆₇	—	2	1	no	—
388	C₂×C₄×C₆₇	—	2	1	no	—
389	C₂×C₄×C₆₇	—	2	8	no	—
390	C₂×C₄×C₆₇	—	2	8	no	—
391	C₂×C₄×C₆₇	—	2	1	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	2	no	—
399	C₂×C₄×C₆₇	—	2	2	no	—
400	C₂×C₄×C₆₇	—	2	1	no	—
401	C₂×C₄×C₆₇	—	2	1	no	—
402	C₂×C₄×C₆₇	—	2	2	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	2	no	—
408	C₂×C₄×C₆₇	—	2	2	no	—
409	C₂×C₄×C₆₇	—	2	1	no	—
410	C₂×C₄×C₆₇	—	2	2	no	—
411	C₂×C₄×C₆₇	—	2	1	no	—
412	C₂×C₄×C₆₇	—	2	1	no	—
413	C₂×C₄×C₆₇	—	2	1	no	—
414	C₂×C₄×C₆₇	—	2	1	no	—
415	C₂×C₄×C₆₇	—	4	4	no	—
416	C₂×C₄×C₆₇	—	4	4	no	—
417	C₂×C₄×C₆₇	—	8	4	no	—
418	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
419	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
420	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
421	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
422	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
423	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
424	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
425	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
426	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
427	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
428	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
429	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
430	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
431	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
432	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
433	C₂²×C₆₇	—	2	2	no	—
434	C₂²×C₆₇	—	2	2	no	—
435	C₂²×C₆₇	—	2	2	no	—
436	C₂²×C₆₇	—	2	2	no	—
437	C₂²×C₆₇	—	2	2	no	—
438	C₂²×C₆₇	—	2	2	no	—
439	C₂²×C₆₇	—	4	1	no	—
440	C₂²×C₆₇	—	4	1	no	—
441	C₂²×C₆₇	—	4	1	no	—
442	C₂²×C₆₇	—	4	1	no	—
443	C₂²×C₆₇	—	4	1	no	—
444	C₂²×C₆₇	—	4	1	no	—
445	C₂²×C₆₇	—	4	1	no	—
446	C₂²×C₆₇	—	4	1	no	—
447	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
448	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
449	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
450	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
451	C₂²×C₆₇	#585	1	1	yes	characteristic
452	C₂²×C₆₇	#585	1	1	yes	characteristic
453	C₂²×C₆₇	#585	1	1	yes	characteristic
454	C₄×C₆₇	—	2	2	no	—
455	C₄×C₆₇	—	2	2	no	—
456	C₄×C₆₇	—	2	4	no	—
457	C₄×C₆₇	—	2	2	no	—
458	C₄×C₆₇	—	2	4	no	—
459	C₄×C₆₇	—	2	4	no	—
460	C₄×C₆₇	—	2	4	no	—
461	C₄×C₆₇	—	2	2	no	—
462	C₄×C₆₇	—	2	2	no	—
463	C₄×C₆₇	—	2	2	no	—
464	C₄×C₆₇	—	2	2	no	—
465	C₄×C₆₇	—	2	4	no	—
466	C₄×C₆₇	—	4	1	no	—
467	C₄×C₆₇	—	4	2	no	—
468	C₄×C₆₇	—	4	1	no	—
469	C₄×C₆₇	—	4	1	no	—
470	C₄×C₆₇	—	4	1	no	—
471	C₄×C₆₇	—	4	2	no	—
472	C₄×C₆₇	—	4	1	no	—
473	C₄×C₆₇	—	4	2	no	—
474	C₄×C₆₇	C₄×#41	1	4	yes	—
475	#34307	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
476	C₂×C₆₇	—	4	2	no	—
477	C₂×C₆₇	—	4	2	no	—
478	C₂×C₆₇	C₂×C₄×#41	1	1	yes	characteristic
479	C₂×C₆₇	#3428	1	1	yes	characteristic
480	C₂×C₆₇	C₄×2₋¹⁺⁴	1	1	yes	characteristic
481	C₂×C₆₇	#3493	1	1	yes	characteristic
482	C₂×C₆₇	#3495	1	1	yes	characteristic
483	C₂×C₆₇	#3518	1	1	yes	characteristic
484	C₂×C₆₇	#3521	1	1	yes	characteristic
485	C₂×#556	C₂×C₆₇	1	1	yes	characteristic
486	#2373	C₂×C₆₇	1	2	yes	complemented
487	#2373	C₂×C₆₇	1	2	yes	complemented
488	#2372	C₂×C₆₇	1	2	yes	—
489	#2369	C₂×C₆₇	1	1	yes	characteristic, complemented
490	#2362	C₂×C₆₇	1	1	yes	characteristic, complemented
491	#2360	C₂×C₆₇	1	1	yes	characteristic, complemented
492	#2356	C₂×C₆₇	1	1	yes	characteristic, complemented
493	#2356	C₂×C₆₇	1	2	yes	complemented
494	#2352	C₂×C₆₇	1	1	yes	characteristic
495	#2347	C₂×C₆₇	1	1	yes	characteristic, complemented
496	#2346	C₂×C₆₇	1	1	yes	characteristic, complemented
497	#2337	C₂×C₆₇	1	2	yes	—
498	#2334	C₂×C₆₇	1	2	yes	complemented
499	#2331	C₂×C₆₇	1	1	yes	characteristic, complemented
500	#2323	C₂×C₆₇	1	1	yes	characteristic, complemented
501	C₄×#126	C₂×C₆₇	1	2	yes	complemented
502	C₄×C₂²⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
503	#2295	C₂×C₆₇	1	2	yes	complemented
504	#2290	C₂×C₆₇	1	1	yes	characteristic
505	#2276	C₂×C₆₇	1	1	yes	characteristic, complemented
506	#2274	C₂×C₆₇	1	1	yes	characteristic, complemented
507	C₄×#117	C₂×C₆₇	1	1	yes	characteristic, complemented
508	C₆₇	#34307	1	1	yes	characteristic, direct product factor, Sylow
509	#556	C₄×C₆₇	1	16	yes	complemented
510	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
511	C₂×#126	C₂²×C₆₇	1	2	yes	—
512	C₂×#126	C₂²×C₆₇	1	2	yes	—
513	C₂×#124	C₂²×C₆₇	1	2	yes	—
514	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
515	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
516	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
517	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
518	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
519	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
520	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
521	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
522	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
523	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
524	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
525	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
526	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
527	#391	C₂²×C₆₇	1	1	yes	characteristic
528	#391	C₂²×C₆₇	1	2	yes	—
529	#391	C₂²×C₆₇	1	2	yes	—
530	#391	C₂²×C₆₇	1	2	yes	—
531	#391	C₂²×C₆₇	1	1	yes	characteristic
532	#391	C₂²×C₆₇	1	2	yes	—
533	#389	C₂²×C₆₇	1	2	yes	—
534	#389	C₂²×C₆₇	1	1	yes	characteristic
535	#389	C₂²×C₆₇	1	1	yes	characteristic
536	#389	C₂²×C₆₇	1	1	yes	characteristic
537	#389	C₂²×C₆₇	1	2	yes	—
538	#389	C₂²×C₆₇	1	1	yes	characteristic
539	#389	C₂²×C₆₇	1	2	yes	—
540	#388	C₂²×C₆₇	1	1	yes	characteristic
541	#388	C₂²×C₆₇	1	2	yes	—
542	#388	C₂²×C₆₇	1	2	yes	—
543	#388	C₂²×C₆₇	1	2	yes	—
544	#388	C₂²×C₆₇	1	2	yes	—
545	#388	C₂²×C₆₇	1	2	yes	—
546	#388	C₂²×C₆₇	1	2	yes	—
547	#388	C₂²×C₆₇	1	2	yes	—
548	#388	C₂²×C₆₇	1	2	yes	—
549	#388	C₂²×C₆₇	1	2	yes	—
550	#388	C₂²×C₆₇	1	1	yes	characteristic
551	#387	C₂²×C₆₇	1	2	yes	—
552	#387	C₂²×C₆₇	1	2	yes	—
553	#387	C₂²×C₆₇	1	2	yes	—
554	#387	C₂²×C₆₇	1	2	yes	—
555	#387	C₂²×C₆₇	1	2	yes	—
556	#387	C₂²×C₆₇	1	2	yes	—
557	#387	C₂²×C₆₇	1	2	yes	—
558	#387	C₂²×C₆₇	1	2	yes	—
559	#387	C₂²×C₆₇	1	1	yes	characteristic
560	#387	C₂²×C₆₇	1	2	yes	—
561	#387	C₂²×C₆₇	1	1	yes	characteristic
562	#386	C₂²×C₆₇	1	1	yes	characteristic
563	#386	C₂²×C₆₇	1	2	yes	—
564	#386	C₂²×C₆₇	1	1	yes	characteristic
565	#385	C₂²×C₆₇	1	1	yes	characteristic
566	#385	C₂²×C₆₇	1	2	yes	—
567	#385	C₂²×C₆₇	1	2	yes	—
568	#385	C₂²×C₆₇	1	1	yes	characteristic
569	#383	C₂²×C₆₇	1	2	yes	—
570	#383	C₂²×C₆₇	1	2	yes	—
571	#382	C₂²×C₆₇	1	1	yes	characteristic
572	#382	C₂²×C₆₇	1	2	yes	—
573	#380	C₂²×C₆₇	1	1	yes	characteristic
574	#380	C₂²×C₆₇	1	1	yes	characteristic
575	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
576	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
577	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
578	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
579	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
580	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
581	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
582	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
583	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
584	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
585	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
586	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
587	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
588	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
589	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
590	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
591	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
592	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
593	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
594	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
595	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
596	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
597	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
598	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
599	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
600	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
601	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
602	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
603	#376	C₂²×C₆₇	1	1	yes	characteristic
604	#376	C₂²×C₆₇	1	2	yes	—
605	#376	C₂²×C₆₇	1	2	yes	—
606	#376	C₂²×C₆₇	1	2	yes	—
607	#376	C₂²×C₆₇	1	2	yes	—
608	C₄³	C₂²×C₆₇	1	2	yes	—
609	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
610	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
611	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
612	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
613	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
614	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
615	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
616	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
617	C₄⋊Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
618	#126	C₂×C₄×C₆₇	1	16	yes	complemented
619	#126	C₂×C₄×C₆₇	1	16	yes	complemented
620	#124	C₂×C₄×C₆₇	1	16	yes	—
621	C₂²⋊Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
622	C₂²⋊Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
623	C₄×Q₈	—	2	4	no	—
624	C₄×Q₈	—	2	4	no	—
625	C₄×Q₈	C₂×C₄×C₆₇	1	16	yes	complemented
626	C₄×Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
627	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	—
628	#117	—	2	4	no	—
629	#117	—	2	4	no	—
630	#117	C₂×C₄×C₆₇	1	8	yes	complemented
631	#117	C₂×C₄×C₆₇	1	8	yes	complemented
632	C₂×C₄⋊C₄	—	2	4	no	—
633	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
634	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
635	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
636	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
637	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
638	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
639	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
640	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
641	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
642	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
643	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
644	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
645	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
646	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
647	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
648	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
649	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
650	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
651	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
652	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
653	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
654	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
655	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
656	C₂×C₂²⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
657	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
658	C₂×C₂²⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
659	C₂×C₂²⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
660	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
661	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
662	C₂×C₂²⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
663	C₂×C₂²⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
664	C₂×C₂²⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
665	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
666	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
667	C₂×C₂²⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
668	C₂×C₂²⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
669	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
670	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
671	C₂×C₄²	—	2	4	no	—
672	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
673	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
674	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
675	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
676	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
677	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
678	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
679	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
680	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
681	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
682	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
683	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
684	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
685	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
686	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
687	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
688	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
689	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
690	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
691	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
692	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
693	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
694	#95	C₂³×C₆₇	1	1	yes	characteristic
695	#95	C₂³×C₆₇	1	2	yes	—
696	#95	C₂³×C₆₇	1	1	yes	characteristic
697	#95	C₂³×C₆₇	1	1	yes	characteristic
698	#95	C₂³×C₆₇	1	2	yes	—
699	#95	C₂³×C₆₇	1	2	yes	—
700	#95	C₂³×C₆₇	1	2	yes	—
701	#95	C₂³×C₆₇	1	2	yes	—
702	#95	C₂³×C₆₇	1	2	yes	—
703	#95	C₂³×C₆₇	1	2	yes	—
704	#95	C₂³×C₆₇	1	1	yes	characteristic
705	#95	C₂³×C₆₇	1	1	yes	characteristic
706	#95	C₂³×C₆₇	1	1	yes	characteristic
707	#95	C₂³×C₆₇	1	2	yes	—
708	#95	C₂³×C₆₇	1	2	yes	—
709	#95	C₂³×C₆₇	1	2	yes	—
710	#95	C₂³×C₆₇	1	2	yes	—
711	#95	C₂³×C₆₇	1	2	yes	—
712	#95	C₂³×C₆₇	1	2	yes	—
713	#95	C₂³×C₆₇	1	1	yes	characteristic
714	#95	C₂³×C₆₇	1	2	yes	—
715	#95	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	1	yes	characteristic
719	#95	C₂³×C₆₇	1	2	yes	—
720	#95	C₂³×C₆₇	1	2	yes	—
721	#95	C₂³×C₆₇	1	1	yes	characteristic
722	#95	C₂³×C₆₇	1	2	yes	—
723	#95	C₂³×C₆₇	1	2	yes	—
724	#95	C₂³×C₆₇	1	2	yes	—
725	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
726	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
727	C₂×Q₈	—	2	2	no	—
728	C₂×Q₈	—	2	2	no	—
729	C₂×Q₈	—	4	2	no	—
730	C₂×Q₈	—	4	2	no	—
731	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
732	C₂×Q₈	C₂²×C₄×C₆₇	1	8	yes	—
733	C₂×D₄	—	4	1	no	—
734	C₂×D₄	—	4	1	no	—
735	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
736	C₂²×C₄	—	2	2	no	—
737	C₂²×C₄	—	2	2	no	—
738	C₂²×C₄	—	2	2	no	—
739	C₂²×C₄	—	2	2	no	—
740	C₂²×C₄	—	2	2	no	—
741	C₂²×C₄	—	2	2	no	—
742	C₂²×C₄	—	2	2	no	—
743	C₂²×C₄	—	2	4	no	—
744	C₂²×C₄	—	4	1	no	—
745	C₂²×C₄	—	4	1	no	—
746	C₂²×C₄	—	4	1	no	—
747	C₂²×C₄	—	4	2	no	—
748	C₂²×C₄	—	4	2	no	—
749	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
750	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
751	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
752	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
753	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
754	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
755	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
756	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
757	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
758	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
759	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
760	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
761	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
762	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
763	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
764	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
765	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
766	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
767	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
768	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
769	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
770	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
771	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
772	C₄⋊C₄	—	2	4	no	—
773	C₄⋊C₄	—	2	2	no	—
774	C₄⋊C₄	—	2	2	no	—
775	C₄⋊C₄	—	2	2	no	—
776	C₄⋊C₄	—	2	2	no	—
777	C₄⋊C₄	—	2	4	no	—
778	C₄⋊C₄	—	2	4	no	—
779	C₄⋊C₄	—	2	2	no	—
780	C₄⋊C₄	—	2	2	no	—
781	C₄⋊C₄	—	4	1	no	—
782	C₄⋊C₄	—	4	2	no	—
783	C₄⋊C₄	—	4	2	no	—
784	C₄⋊C₄	—	4	2	no	—
785	C₄⋊C₄	—	4	2	no	—
786	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
787	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
788	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
789	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
790	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
791	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
792	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
793	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
794	C₄⋊C₄	#41×C₆₇	1	4	yes	—
795	C₂²⋊C₄	—	2	2	no	—
796	C₂²⋊C₄	—	2	4	no	—
797	C₂²⋊C₄	—	2	2	no	—
798	C₂²⋊C₄	—	2	2	no	—
799	C₂²⋊C₄	—	4	2	no	—
800	C₂²⋊C₄	—	4	1	no	—
801	C₂²⋊C₄	—	4	1	no	—
802	C₂²⋊C₄	—	4	2	no	—
803	C₂²⋊C₄	—	4	1	no	—
804	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
805	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
806	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
807	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
808	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
809	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
810	C₄²	—	2	2	no	—
811	C₄²	—	2	2	no	—
812	C₄²	—	2	2	no	—
813	C₄²	—	2	4	no	—
814	C₄²	—	2	2	no	—
815	C₄²	—	2	4	no	—
816	C₄²	—	2	4	no	—
817	C₄²	—	2	2	no	—
818	C₄²	—	2	4	no	—
819	C₄²	—	4	2	no	—
820	C₄²	—	4	2	no	—
821	C₄²	—	4	2	no	—
822	C₄²	—	4	2	no	—
823	C₄²	—	4	2	no	—
824	C₄²	—	4	2	no	—
825	C₄²	—	4	2	no	—
826	C₄²	—	4	2	no	—
827	C₄²	C₂²×C₄×C₆₇	1	8	yes	—
828	C₄²	C₂²×C₄×C₆₇	1	8	yes	—
829	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
830	C₄²	C₂²×C₄×C₆₇	1	8	yes	—
831	C₄²	#41×C₆₇	1	4	yes	—
832	C₂³	—	2	1	no	—
833	C₂³	—	2	1	no	—
834	C₂³	—	2	1	no	—
835	C₂³	—	2	1	no	—
836	C₂³	—	2	1	no	—
837	C₂³	—	2	1	no	—
838	C₂³	—	2	1	no	—
839	C₂³	—	2	1	no	—
840	C₂³	—	2	1	no	—
841	C₂³	—	2	1	no	—
842	C₂³	—	2	1	no	—
843	C₂³	—	2	1	no	—
844	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
845	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
846	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
847	Q₈	—	2	8	no	—
848	Q₈	—	4	8	no	—
849	D₄	—	4	4	no	—
850	C₂×C₄	—	2	1	no	—
851	C₂×C₄	—	2	1	no	—
852	C₂×C₄	—	2	1	no	—
853	C₂×C₄	—	2	2	no	—
854	C₂×C₄	—	2	2	no	—
855	C₂×C₄	—	2	1	no	—
856	C₂×C₄	—	2	1	no	—
857	C₂×C₄	—	2	2	no	—
858	C₂×C₄	—	2	1	no	—
859	C₂×C₄	—	2	1	no	—
860	C₂×C₄	—	2	2	no	—
861	C₂×C₄	—	2	1	no	—
862	C₂×C₄	—	2	2	no	—
863	C₂×C₄	—	2	2	no	—
864	C₂×C₄	—	2	1	no	—
865	C₂×C₄	—	2	1	no	—
866	C₂×C₄	—	2	1	no	—
867	C₂×C₄	—	2	1	no	—
868	C₂×C₄	—	2	2	no	—
869	C₂×C₄	—	2	2	no	—
870	C₂×C₄	—	2	2	no	—
871	C₂×C₄	—	2	2	no	—
872	C₂×C₄	—	2	1	no	—
873	C₂×C₄	—	2	1	no	—
874	C₂×C₄	—	2	8	no	—
875	C₂×C₄	—	2	8	no	—
876	C₂×C₄	—	2	1	no	—
877	C₂×C₄	—	2	1	no	—
878	C₂×C₄	—	2	1	no	—
879	C₂×C₄	—	2	1	no	—
880	C₂×C₄	—	2	2	no	—
881	C₂×C₄	—	2	1	no	—
882	C₂×C₄	—	2	2	no	—
883	C₂×C₄	—	2	2	no	—
884	C₂×C₄	—	2	2	no	—
885	C₂×C₄	—	2	1	no	—
886	C₂×C₄	—	2	1	no	—
887	C₂×C₄	—	2	1	no	—
888	C₂×C₄	—	2	2	no	—
889	C₂×C₄	—	2	1	no	—
890	C₂×C₄	—	2	1	no	—
891	C₂×C₄	—	2	2	no	—
892	C₂×C₄	—	2	2	no	—
893	C₂×C₄	—	2	2	no	—
894	C₂×C₄	—	2	1	no	—
895	C₂×C₄	—	2	1	no	—
896	C₂×C₄	—	2	2	no	—
897	C₂×C₄	—	2	1	no	—
898	C₂×C₄	—	2	1	no	—
899	C₂×C₄	—	2	1	no	—
900	C₂×C₄	—	4	4	no	—
901	C₂×C₄	—	4	4	no	—
902	C₂×C₄	—	8	4	no	—
903	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
904	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
905	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
906	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
907	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
908	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
909	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
910	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
911	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
912	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
913	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
914	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
915	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
916	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
917	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
918	C₂²	—	2	2	no	—
919	C₂²	—	2	2	no	—
920	C₂²	—	2	2	no	—
921	C₂²	—	2	2	no	—
922	C₂²	—	2	2	no	—
923	C₂²	—	2	2	no	—
924	C₂²	—	4	1	no	—
925	C₂²	—	4	1	no	—
926	C₂²	—	4	1	no	—
927	C₂²	—	4	1	no	—
928	C₂²	—	4	1	no	—
929	C₂²	—	4	1	no	—
930	C₂²	—	4	1	no	—
931	C₂²	—	4	1	no	—
932	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
933	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
934	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
935	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
936	C₂²	#585×C₆₇	1	1	yes	characteristic
937	C₂²	#585×C₆₇	1	1	yes	characteristic
938	C₂²	#585×C₆₇	1	1	yes	characteristic
939	C₄	—	2	4	no	—
940	C₄	—	2	2	no	—
941	C₄	—	2	2	no	—
942	C₄	—	2	4	no	—
943	C₄	—	2	2	no	—
944	C₄	—	2	4	no	—
945	C₄	—	2	4	no	—
946	C₄	—	2	4	no	—
947	C₄	—	2	2	no	—
948	C₄	—	2	2	no	—
949	C₄	—	2	2	no	—
950	C₄	—	2	2	no	—
951	C₄	—	4	2	no	—
952	C₄	—	4	1	no	—
953	C₄	—	4	1	no	—
954	C₄	—	4	1	no	—
955	C₄	—	4	2	no	—
956	C₄	—	4	1	no	—
957	C₄	—	4	1	no	—
958	C₄	—	4	2	no	—
959	C₄	C₄×#41×C₆₇	1	4	yes	—
960	C₂	—	4	2	no	—
961	C₂	—	4	2	no	—
962	C₂	C₂×C₄×#41×C₆₇	1	1	yes	characteristic
963	C₂	C₆₇×#3428	1	1	yes	characteristic
964	C₂	C₄×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
965	C₂	C₆₇×#3493	1	1	yes	characteristic
966	C₂	C₆₇×#3495	1	1	yes	characteristic
967	C₂	C₆₇×#3518	1	1	yes	characteristic
968	C₂	C₆₇×#3521	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂²×D₄×C₆₇	C₂³
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×#2373	C₂	semidirect
3. C₆₇×#2373	C₂	semidirect
5. C₆₇×#2369	C₂	semidirect
6. C₆₇×#2362	C₂	semidirect
7. C₆₇×#2360	C₂	semidirect
8. C₆₇×#2356	C₂	semidirect
9. C₆₇×#2356	C₂	semidirect
11. C₆₇×#2347	C₂	semidirect
12. C₆₇×#2346	C₂	semidirect
14. C₆₇×#2334	C₂	semidirect
15. C₆₇×#2331	C₂	semidirect
16. C₆₇×#2323	C₂	semidirect
17. C₄×#126×C₆₇	C₂	semidirect
18. C₄×C₂²⋊Q₈×C₆₇	C₂	semidirect
19. C₆₇×#2295	C₂	semidirect
21. C₆₇×#2276	C₂	semidirect
22. C₆₇×#2274	C₂	semidirect
23. C₄×#117×C₆₇	C₂	semidirect
24. #556×C₆₇	C₄	semidirect
132. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
133. #126×C₆₇	C₂×C₄	semidirect
134. #126×C₆₇	C₂×C₄	semidirect
136. C₂²⋊Q₈×C₆₇	C₂×C₄	semidirect
137. C₂²⋊Q₈×C₆₇	C₂×C₄	semidirect
140. C₄×Q₈×C₆₇	C₂×C₄	semidirect
141. C₄×Q₈×C₆₇	C₂×C₄	semidirect
145. #117×C₆₇	C₂×C₄	semidirect
146. #117×C₆₇	C₂×C₄	semidirect
475. #34307	C₆₇	direct (N × H)
486. #2373	C₂×C₆₇	semidirect
487. #2373	C₂×C₆₇	semidirect
489. #2369	C₂×C₆₇	semidirect
490. #2362	C₂×C₆₇	semidirect
491. #2360	C₂×C₆₇	semidirect
492. #2356	C₂×C₆₇	semidirect
493. #2356	C₂×C₆₇	semidirect
495. #2347	C₂×C₆₇	semidirect
496. #2346	C₂×C₆₇	semidirect
498. #2334	C₂×C₆₇	semidirect
499. #2331	C₂×C₆₇	semidirect
500. #2323	C₂×C₆₇	semidirect
501. C₄×#126	C₂×C₆₇	semidirect
502. C₄×C₂²⋊Q₈	C₂×C₆₇	semidirect
503. #2295	C₂×C₆₇	semidirect
505. #2276	C₂×C₆₇	semidirect
506. #2274	C₂×C₆₇	semidirect
507. C₄×#117	C₂×C₆₇	semidirect
508. C₆₇	#34307	direct (N × H)
509. #556	C₄×C₆₇	semidirect
617. C₄⋊Q₈	C₂×C₄×C₆₇	semidirect
618. #126	C₂×C₄×C₆₇	semidirect
619. #126	C₂×C₄×C₆₇	semidirect
621. C₂²⋊Q₈	C₂×C₄×C₆₇	semidirect
622. C₂²⋊Q₈	C₂×C₄×C₆₇	semidirect
625. C₄×Q₈	C₂×C₄×C₆₇	semidirect
626. C₄×Q₈	C₂×C₄×C₆₇	semidirect
630. #117	C₂×C₄×C₆₇	semidirect
631. #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×#556×C₆₇	2
C₆₇×#2373	2
C₆₇×#2372	2
C₆₇×#2369	2
C₆₇×#2362	2
C₆₇×#2360	2
C₆₇×#2356	2
C₆₇×#2352	2
C₆₇×#2347	2
C₆₇×#2346	2
C₆₇×#2337	2
C₆₇×#2334	2
C₆₇×#2331	2
C₆₇×#2323	2
C₄×#126×C₆₇	2
C₄×C₂²⋊Q₈×C₆₇	2
C₆₇×#2295	2
C₆₇×#2290	2
C₆₇×#2276	2
C₆₇×#2274	2
C₄×#117×C₆₇	2
#34307	67

Subgroups — C67×#34307 #30713558

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34307 #30713558