Subgroups — C₆₇×#34425 #30713676

Order 17152.

2202 conjugacy classes of subgroups.

The subgroup lattice has 894 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#539×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2383	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2378	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2373	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2360	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2359	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2359	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2353	C₂	1	2	yes	maximal, complemented
9	D₄×C₄⋊C₄×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2336	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2335	C₂	1	2	yes	maximal, complemented
12	C₆₇×#2330	C₂	1	2	yes	maximal, complemented
13	C₆₇×#2329	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₇×#2325	C₂	1	1	yes	maximal, characteristic
15	C₆₇×#2322	C₂	1	2	yes	maximal, complemented
16	C₆₇×#2316	C₂	1	1	yes	maximal, characteristic, complemented
17	C₄×C₂²⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
18	C₄×#121×C₆₇	C₂	1	1	yes	maximal, characteristic
19	C₂×#387×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
20	#539×C₆₇	C₄	1	16	yes	complemented
21	C₂×#126×C₆₇	C₂²	1	2	yes	—
22	C₂×#125×C₆₇	C₂²	1	2	yes	—
23	C₂×#123×C₆₇	C₂²	1	2	yes	—
24	C₂×#123×C₆₇	C₂²	1	2	yes	—
25	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
27	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
31	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
32	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
33	C₂×#117×C₆₇	C₂²	1	2	yes	—
34	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
35	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
36	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
37	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
38	#391×C₆₇	C₂²	1	1	yes	characteristic
39	#391×C₆₇	C₂²	1	2	yes	—
40	#390×C₆₇	C₂²	1	1	yes	characteristic
41	#389×C₆₇	C₂²	1	1	yes	characteristic
42	#389×C₆₇	C₂²	1	1	yes	characteristic
43	#389×C₆₇	C₂²	1	2	yes	—
44	#389×C₆₇	C₂²	1	2	yes	—
45	#389×C₆₇	C₂²	1	2	yes	—
46	#389×C₆₇	C₂²	1	2	yes	—
47	#389×C₆₇	C₂²	1	2	yes	—
48	#389×C₆₇	C₂²	1	2	yes	—
49	#389×C₆₇	C₂²	1	2	yes	—
50	#389×C₆₇	C₂²	1	2	yes	—
51	#389×C₆₇	C₂²	1	2	yes	—
52	#388×C₆₇	C₂²	1	2	yes	—
53	#388×C₆₇	C₂²	1	2	yes	—
54	#388×C₆₇	C₂²	1	2	yes	—
55	#388×C₆₇	C₂²	1	2	yes	—
56	#388×C₆₇	C₂²	1	2	yes	—
57	#388×C₆₇	C₂²	1	2	yes	—
58	#388×C₆₇	C₂²	1	2	yes	—
59	#388×C₆₇	C₂²	1	2	yes	—
60	#387×C₆₇	C₂²	1	2	yes	—
61	#387×C₆₇	C₂²	1	2	yes	—
62	#387×C₆₇	C₂²	1	2	yes	—
63	#387×C₆₇	C₂²	1	2	yes	—
64	#387×C₆₇	C₂²	1	2	yes	—
65	#387×C₆₇	C₂²	1	2	yes	—
66	#387×C₆₇	C₂²	1	2	yes	—
67	#387×C₆₇	C₂²	1	2	yes	—
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	#386×C₆₇	C₂²	1	2	yes	—
75	#386×C₆₇	C₂²	1	2	yes	—
76	#386×C₆₇	C₂²	1	2	yes	—
77	#385×C₆₇	C₂²	1	2	yes	—
78	#385×C₆₇	C₂²	1	2	yes	—
79	#385×C₆₇	C₂²	1	2	yes	—
80	#385×C₆₇	C₂²	1	2	yes	—
81	#385×C₆₇	C₂²	1	2	yes	—
82	#385×C₆₇	C₂²	1	2	yes	—
83	#385×C₆₇	C₂²	1	2	yes	—
84	#384×C₆₇	C₂²	1	2	yes	—
85	#383×C₆₇	C₂²	1	2	yes	—
86	#381×C₆₇	C₂²	1	2	yes	—
87	#380×C₆₇	C₂²	1	1	yes	characteristic
88	#380×C₆₇	C₂²	1	1	yes	characteristic
89	#380×C₆₇	C₂²	1	2	yes	—
90	#380×C₆₇	C₂²	1	2	yes	—
91	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
92	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
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	2	yes	—
96	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
97	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
98	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
99	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
100	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
101	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
102	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
103	#376×C₆₇	C₂²	1	2	yes	—
104	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
105	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
106	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
107	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
108	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
109	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
110	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
111	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
112	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
113	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
114	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
115	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
116	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
117	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
118	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
119	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
120	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
121	#126×C₆₇	C₂×C₄	1	16	yes	complemented
122	#125×C₆₇	C₂×C₄	1	16	yes	complemented
123	#123×C₆₇	C₂×C₄	1	16	yes	complemented
124	#123×C₆₇	C₂×C₄	1	16	yes	complemented
125	C₂²⋊Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
126	C₂²⋊Q₈×C₆₇	C₂×C₄	1	16	yes	complemented
127	#121×C₆₇	C₂×C₄	1	8	yes	—
128	C₄×Q₈×C₆₇	—	4	2	no	—
129	C₄×D₄×C₆₇	—	4	2	no	—
130	C₄×D₄×C₆₇	—	4	4	no	—
131	C₄×D₄×C₆₇	—	4	4	no	—
132	C₄×D₄×C₆₇	C₂×C₄	1	16	yes	complemented
133	#117×C₆₇	—	4	4	no	—
134	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
135	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
136	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	8	yes	complemented
137	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
138	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
139	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
140	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
141	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
142	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
143	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
144	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
145	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
146	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
147	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
148	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
149	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
150	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
151	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
152	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
153	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
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×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
162	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
163	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
164	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
165	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
166	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
167	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
168	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
169	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
170	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
171	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
172	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
173	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
174	C₂×C₄²×C₆₇	—	4	2	no	—
175	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
176	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
177	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
178	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
179	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
180	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
181	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
182	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
183	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
184	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
185	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
186	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
187	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
188	#95×C₆₇	C₂³	1	2	yes	—
189	#95×C₆₇	C₂³	1	2	yes	—
190	#95×C₆₇	C₂³	1	2	yes	—
191	#95×C₆₇	C₂³	1	2	yes	—
192	#95×C₆₇	C₂³	1	2	yes	—
193	#95×C₆₇	C₂³	1	2	yes	—
194	#95×C₆₇	C₂³	1	2	yes	—
195	#95×C₆₇	C₂³	1	2	yes	—
196	#95×C₆₇	C₂³	1	2	yes	—
197	#95×C₆₇	C₂³	1	2	yes	—
198	#95×C₆₇	C₂³	1	2	yes	—
199	#95×C₆₇	C₂³	1	2	yes	—
200	#95×C₆₇	C₂³	1	2	yes	—
201	#95×C₆₇	C₂³	1	2	yes	—
202	#95×C₆₇	C₂³	1	2	yes	—
203	#95×C₆₇	C₂³	1	2	yes	—
204	#95×C₆₇	C₂³	1	2	yes	—
205	#95×C₆₇	C₂³	1	2	yes	—
206	#95×C₆₇	C₂³	1	2	yes	—
207	#95×C₆₇	C₂³	1	2	yes	—
208	#95×C₆₇	C₂³	1	2	yes	—
209	#95×C₆₇	C₂³	1	2	yes	—
210	#95×C₆₇	C₂³	1	2	yes	—
211	#95×C₆₇	C₂³	1	2	yes	—
212	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
213	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
214	C₂×Q₈×C₆₇	—	4	1	no	—
215	C₂×Q₈×C₆₇	—	4	1	no	—
216	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
217	C₂×D₄×C₆₇	—	2	4	no	—
218	C₂×D₄×C₆₇	—	2	4	no	—
219	C₂×D₄×C₆₇	—	4	1	no	—
220	C₂×D₄×C₆₇	—	4	1	no	—
221	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
222	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
223	C₂²×C₄×C₆₇	—	2	2	no	—
224	C₂²×C₄×C₆₇	—	2	2	no	—
225	C₂²×C₄×C₆₇	—	2	2	no	—
226	C₂²×C₄×C₆₇	—	2	2	no	—
227	C₂²×C₄×C₆₇	—	2	2	no	—
228	C₂²×C₄×C₆₇	—	2	2	no	—
229	C₂²×C₄×C₆₇	—	2	2	no	—
230	C₂²×C₄×C₆₇	—	2	2	no	—
231	C₂²×C₄×C₆₇	—	2	4	no	—
232	C₂²×C₄×C₆₇	—	4	1	no	—
233	C₂²×C₄×C₆₇	—	4	1	no	—
234	C₂²×C₄×C₆₇	—	4	2	no	—
235	C₂²×C₄×C₆₇	—	4	1	no	—
236	C₂²×C₄×C₆₇	—	4	2	no	—
237	C₂²×C₄×C₆₇	—	4	1	no	—
238	C₂²×C₄×C₆₇	—	4	1	no	—
239	C₂²×C₄×C₆₇	—	4	1	no	—
240	C₂²×C₄×C₆₇	—	4	2	no	—
241	C₂²×C₄×C₆₇	—	4	1	no	—
242	C₂²×C₄×C₆₇	—	4	2	no	—
243	C₂²×C₄×C₆₇	—	4	2	no	—
244	C₂²×C₄×C₆₇	—	4	2	no	—
245	C₂²×C₄×C₆₇	—	4	1	no	—
246	C₂²×C₄×C₆₇	—	4	1	no	—
247	C₂²×C₄×C₆₇	—	4	2	no	—
248	C₂²×C₄×C₆₇	—	4	1	no	—
249	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
250	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
251	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
252	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
253	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
254	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
255	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
256	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
257	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
258	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
259	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
260	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
261	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
262	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
263	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
264	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
265	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
266	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
267	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
268	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
269	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
270	C₄⋊C₄×C₆₇	—	2	4	no	—
271	C₄⋊C₄×C₆₇	—	2	4	no	—
272	C₄⋊C₄×C₆₇	—	4	2	no	—
273	C₄⋊C₄×C₆₇	—	4	2	no	—
274	C₄⋊C₄×C₆₇	—	4	2	no	—
275	C₄⋊C₄×C₆₇	—	4	2	no	—
276	C₄⋊C₄×C₆₇	—	4	2	no	—
277	C₄⋊C₄×C₆₇	—	4	2	no	—
278	C₄⋊C₄×C₆₇	—	4	1	no	—
279	C₄⋊C₄×C₆₇	—	4	1	no	—
280	C₄⋊C₄×C₆₇	—	4	2	no	—
281	C₄⋊C₄×C₆₇	—	4	2	no	—
282	C₄⋊C₄×C₆₇	—	4	1	no	—
283	C₄⋊C₄×C₆₇	—	4	2	no	—
284	C₄⋊C₄×C₆₇	—	4	2	no	—
285	C₄⋊C₄×C₆₇	—	4	1	no	—
286	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
287	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
288	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
289	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
290	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
291	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
292	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
293	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
294	C₂²⋊C₄×C₆₇	—	2	4	no	—
295	C₂²⋊C₄×C₆₇	—	2	4	no	—
296	C₂²⋊C₄×C₆₇	—	2	4	no	—
297	C₂²⋊C₄×C₆₇	—	4	2	no	—
298	C₂²⋊C₄×C₆₇	—	4	2	no	—
299	C₂²⋊C₄×C₆₇	—	4	2	no	—
300	C₂²⋊C₄×C₆₇	—	4	2	no	—
301	C₂²⋊C₄×C₆₇	—	4	2	no	—
302	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
303	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
304	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
305	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
306	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
307	C₄²×C₆₇	—	2	2	no	—
308	C₄²×C₆₇	—	2	2	no	—
309	C₄²×C₆₇	—	2	4	no	—
310	C₄²×C₆₇	—	4	1	no	—
311	C₄²×C₆₇	—	4	1	no	—
312	C₄²×C₆₇	—	4	2	no	—
313	C₄²×C₆₇	—	4	2	no	—
314	C₄²×C₆₇	—	4	2	no	—
315	C₄²×C₆₇	—	4	2	no	—
316	C₄²×C₆₇	—	4	2	no	—
317	C₄²×C₆₇	—	4	2	no	—
318	C₄²×C₆₇	—	4	2	no	—
319	C₄²×C₆₇	C₂²×C₄	1	8	yes	—
320	C₂³×C₆₇	—	2	1	no	—
321	C₂³×C₆₇	—	2	1	no	—
322	C₂³×C₆₇	—	2	1	no	—
323	C₂³×C₆₇	—	2	1	no	—
324	C₂³×C₆₇	—	2	2	no	—
325	C₂³×C₆₇	—	2	2	no	—
326	C₂³×C₆₇	—	2	2	no	—
327	C₂³×C₆₇	—	2	2	no	—
328	C₂³×C₆₇	—	2	2	no	—
329	C₂³×C₆₇	—	2	2	no	—
330	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
331	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
332	C₂³×C₆₇	H(5,𝔽₂)	1	2	yes	—
333	C₂³×C₆₇	2₋¹⁺⁴	1	2	yes	—
334	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
335	Q₈×C₆₇	—	4	4	no	—
336	D₄×C₆₇	—	2	16	no	—
337	D₄×C₆₇	—	4	4	no	—
338	C₂×C₄×C₆₇	—	2	8	no	—
339	C₂×C₄×C₆₇	—	2	1	no	—
340	C₂×C₄×C₆₇	—	2	1	no	—
341	C₂×C₄×C₆₇	—	2	1	no	—
342	C₂×C₄×C₆₇	—	2	2	no	—
343	C₂×C₄×C₆₇	—	2	2	no	—
344	C₂×C₄×C₆₇	—	2	1	no	—
345	C₂×C₄×C₆₇	—	2	2	no	—
346	C₂×C₄×C₆₇	—	2	2	no	—
347	C₂×C₄×C₆₇	—	2	2	no	—
348	C₂×C₄×C₆₇	—	2	2	no	—
349	C₂×C₄×C₆₇	—	2	1	no	—
350	C₂×C₄×C₆₇	—	2	2	no	—
351	C₂×C₄×C₆₇	—	2	2	no	—
352	C₂×C₄×C₆₇	—	2	2	no	—
353	C₂×C₄×C₆₇	—	2	2	no	—
354	C₂×C₄×C₆₇	—	2	1	no	—
355	C₂×C₄×C₆₇	—	2	1	no	—
356	C₂×C₄×C₆₇	—	2	1	no	—
357	C₂×C₄×C₆₇	—	2	1	no	—
358	C₂×C₄×C₆₇	—	2	2	no	—
359	C₂×C₄×C₆₇	—	2	2	no	—
360	C₂×C₄×C₆₇	—	2	1	no	—
361	C₂×C₄×C₆₇	—	2	2	no	—
362	C₂×C₄×C₆₇	—	2	2	no	—
363	C₂×C₄×C₆₇	—	2	2	no	—
364	C₂×C₄×C₆₇	—	2	2	no	—
365	C₂×C₄×C₆₇	—	2	1	no	—
366	C₂×C₄×C₆₇	—	2	2	no	—
367	C₂×C₄×C₆₇	—	2	2	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	2	no	—
374	C₂×C₄×C₆₇	—	2	2	no	—
375	C₂×C₄×C₆₇	—	2	2	no	—
376	C₂×C₄×C₆₇	—	2	2	no	—
377	C₂×C₄×C₆₇	—	2	2	no	—
378	C₂×C₄×C₆₇	—	2	2	no	—
379	C₂×C₄×C₆₇	—	2	1	no	—
380	C₂×C₄×C₆₇	—	2	2	no	—
381	C₂×C₄×C₆₇	—	4	4	no	—
382	C₂×C₄×C₆₇	—	4	8	no	—
383	C₂×C₄×C₆₇	—	4	4	no	—
384	C₂×C₄×C₆₇	—	4	4	no	—
385	C₂×C₄×C₆₇	—	4	4	no	—
386	C₂×C₄×C₆₇	—	8	4	no	—
387	C₂×C₄×C₆₇	—	8	2	no	—
388	C₂×C₄×C₆₇	—	8	2	no	—
389	C₂×C₄×C₆₇	—	8	4	no	—
390	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
391	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
392	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
393	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
394	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
395	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
396	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
397	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
398	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
399	C₂²×C₆₇	—	2	2	no	—
400	C₂²×C₆₇	—	2	2	no	—
401	C₂²×C₆₇	—	2	2	no	—
402	C₂²×C₆₇	—	2	2	no	—
403	C₂²×C₆₇	—	2	2	no	—
404	C₂²×C₆₇	—	2	2	no	—
405	C₂²×C₆₇	—	2	4	no	—
406	C₂²×C₆₇	—	2	4	no	—
407	C₂²×C₆₇	—	2	4	no	—
408	C₂²×C₆₇	—	4	2	no	—
409	C₂²×C₆₇	—	4	2	no	—
410	C₂²×C₆₇	—	4	2	no	—
411	C₂²×C₆₇	—	4	2	no	—
412	C₂²×C₆₇	#520	1	4	yes	—
413	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
414	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
415	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
416	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
417	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
418	C₂²×C₆₇	#585	1	1	yes	characteristic
419	C₂²×C₆₇	#585	1	1	yes	characteristic
420	C₄×C₆₇	—	2	4	no	—
421	C₄×C₆₇	—	2	4	no	—
422	C₄×C₆₇	—	2	4	no	—
423	C₄×C₆₇	—	2	2	no	—
424	C₄×C₆₇	—	2	2	no	—
425	C₄×C₆₇	—	2	4	no	—
426	C₄×C₆₇	—	2	4	no	—
427	C₄×C₆₇	—	2	2	no	—
428	C₄×C₆₇	—	2	2	no	—
429	C₄×C₆₇	—	4	1	no	—
430	C₄×C₆₇	—	4	2	no	—
431	C₄×C₆₇	—	4	2	no	—
432	C₄×C₆₇	—	4	2	no	—
433	C₄×C₆₇	—	4	1	no	—
434	C₄×C₆₇	—	4	2	no	—
435	C₄×C₆₇	—	4	2	no	—
436	C₄×C₆₇	—	4	2	no	—
437	#34425	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
438	C₂×C₆₇	—	2	4	no	—
439	C₂×C₆₇	—	4	4	no	—
440	C₂×C₆₇	C₂×#520	1	1	yes	characteristic
441	C₂×C₆₇	C₄×H(5,𝔽₂)	1	1	yes	characteristic
442	C₂×C₆₇	C₄×2₋¹⁺⁴	1	1	yes	characteristic
443	C₂×C₆₇	#3534	1	1	yes	characteristic
444	C₂×C₆₇	#3538	1	1	yes	characteristic
445	C₂×C₆₇	#3559	1	1	yes	characteristic
446	C₂×C₆₇	#3564	1	1	yes	characteristic
447	C₂×#539	C₂×C₆₇	1	1	yes	characteristic
448	#2383	C₂×C₆₇	1	2	yes	complemented
449	#2378	C₂×C₆₇	1	2	yes	complemented
450	#2373	C₂×C₆₇	1	2	yes	complemented
451	#2360	C₂×C₆₇	1	2	yes	complemented
452	#2359	C₂×C₆₇	1	2	yes	complemented
453	#2359	C₂×C₆₇	1	2	yes	complemented
454	#2353	C₂×C₆₇	1	2	yes	complemented
455	D₄×C₄⋊C₄	C₂×C₆₇	1	2	yes	complemented
456	#2336	C₂×C₆₇	1	2	yes	complemented
457	#2335	C₂×C₆₇	1	2	yes	complemented
458	#2330	C₂×C₆₇	1	2	yes	complemented
459	#2329	C₂×C₆₇	1	1	yes	characteristic, complemented
460	#2325	C₂×C₆₇	1	1	yes	characteristic
461	#2322	C₂×C₆₇	1	2	yes	complemented
462	#2316	C₂×C₆₇	1	1	yes	characteristic, complemented
463	C₄×C₂²⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
464	C₄×#121	C₂×C₆₇	1	1	yes	characteristic
465	C₂×#387	C₂×C₆₇	1	1	yes	characteristic, complemented
466	C₆₇	#34425	1	1	yes	characteristic, direct product factor, Sylow
467	#539	C₄×C₆₇	1	16	yes	complemented
468	C₂×#126	C₂²×C₆₇	1	2	yes	—
469	C₂×#125	C₂²×C₆₇	1	2	yes	—
470	C₂×#123	C₂²×C₆₇	1	2	yes	—
471	C₂×#123	C₂²×C₆₇	1	2	yes	—
472	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
473	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
474	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
475	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
476	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
477	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
478	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
479	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
480	C₂×#117	C₂²×C₆₇	1	2	yes	—
481	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
482	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
483	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
484	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
485	#391	C₂²×C₆₇	1	2	yes	—
486	#391	C₂²×C₆₇	1	1	yes	characteristic
487	#390	C₂²×C₆₇	1	1	yes	characteristic
488	#389	C₂²×C₆₇	1	2	yes	—
489	#389	C₂²×C₆₇	1	2	yes	—
490	#389	C₂²×C₆₇	1	2	yes	—
491	#389	C₂²×C₆₇	1	2	yes	—
492	#389	C₂²×C₆₇	1	2	yes	—
493	#389	C₂²×C₆₇	1	1	yes	characteristic
494	#389	C₂²×C₆₇	1	2	yes	—
495	#389	C₂²×C₆₇	1	2	yes	—
496	#389	C₂²×C₆₇	1	2	yes	—
497	#389	C₂²×C₆₇	1	2	yes	—
498	#389	C₂²×C₆₇	1	1	yes	characteristic
499	#388	C₂²×C₆₇	1	2	yes	—
500	#388	C₂²×C₆₇	1	2	yes	—
501	#388	C₂²×C₆₇	1	2	yes	—
502	#388	C₂²×C₆₇	1	2	yes	—
503	#388	C₂²×C₆₇	1	2	yes	—
504	#388	C₂²×C₆₇	1	2	yes	—
505	#388	C₂²×C₆₇	1	2	yes	—
506	#388	C₂²×C₆₇	1	2	yes	—
507	#387	C₂²×C₆₇	1	2	yes	—
508	#387	C₂²×C₆₇	1	2	yes	—
509	#387	C₂²×C₆₇	1	2	yes	—
510	#387	C₂²×C₆₇	1	2	yes	—
511	#387	C₂²×C₆₇	1	2	yes	—
512	#387	C₂²×C₆₇	1	2	yes	—
513	#387	C₂²×C₆₇	1	2	yes	—
514	#387	C₂²×C₆₇	1	2	yes	—
515	#387	C₂²×C₆₇	1	2	yes	—
516	#387	C₂²×C₆₇	1	2	yes	—
517	#387	C₂²×C₆₇	1	2	yes	—
518	#387	C₂²×C₆₇	1	2	yes	—
519	#387	C₂²×C₆₇	1	2	yes	—
520	#387	C₂²×C₆₇	1	2	yes	—
521	#386	C₂²×C₆₇	1	2	yes	—
522	#386	C₂²×C₆₇	1	2	yes	—
523	#386	C₂²×C₆₇	1	2	yes	—
524	#385	C₂²×C₆₇	1	2	yes	—
525	#385	C₂²×C₆₇	1	2	yes	—
526	#385	C₂²×C₆₇	1	2	yes	—
527	#385	C₂²×C₆₇	1	2	yes	—
528	#385	C₂²×C₆₇	1	2	yes	—
529	#385	C₂²×C₆₇	1	2	yes	—
530	#385	C₂²×C₆₇	1	2	yes	—
531	#384	C₂²×C₆₇	1	2	yes	—
532	#383	C₂²×C₆₇	1	2	yes	—
533	#381	C₂²×C₆₇	1	2	yes	—
534	#380	C₂²×C₆₇	1	1	yes	characteristic
535	#380	C₂²×C₆₇	1	1	yes	characteristic
536	#380	C₂²×C₆₇	1	2	yes	—
537	#380	C₂²×C₆₇	1	2	yes	—
538	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
539	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
540	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
541	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
542	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
543	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
544	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
545	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
546	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
547	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
548	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
549	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
550	#376	C₂²×C₆₇	1	2	yes	—
551	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
552	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
553	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
554	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
555	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
556	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
557	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
558	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
559	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
560	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
561	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
562	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
563	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
564	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
565	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
566	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
567	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
568	#126	C₂×C₄×C₆₇	1	16	yes	complemented
569	#125	C₂×C₄×C₆₇	1	16	yes	complemented
570	#123	C₂×C₄×C₆₇	1	16	yes	complemented
571	#123	C₂×C₄×C₆₇	1	16	yes	complemented
572	C₂²⋊Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
573	C₂²⋊Q₈	C₂×C₄×C₆₇	1	16	yes	complemented
574	#121	C₂×C₄×C₆₇	1	8	yes	—
575	C₄×Q₈	—	4	2	no	—
576	C₄×D₄	—	4	2	no	—
577	C₄×D₄	—	4	4	no	—
578	C₄×D₄	—	4	4	no	—
579	C₄×D₄	C₂×C₄×C₆₇	1	16	yes	complemented
580	#117	—	4	4	no	—
581	C₂×C₄⋊C₄	—	4	2	no	—
582	C₂×C₄⋊C₄	—	4	4	no	—
583	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	8	yes	complemented
584	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
585	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
586	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
587	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
588	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
589	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
590	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
591	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
592	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
593	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
594	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
595	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
596	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
597	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
598	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
599	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
600	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
601	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
602	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
603	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
604	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
605	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
606	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
607	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
608	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
609	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
610	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
611	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
612	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
613	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
614	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
615	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
616	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
617	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
618	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
619	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
620	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
621	C₂×C₄²	—	4	2	no	—
622	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
623	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
624	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
625	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
626	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
627	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
628	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
629	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
630	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
631	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
632	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
633	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
634	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
635	#95	C₂³×C₆₇	1	2	yes	—
636	#95	C₂³×C₆₇	1	2	yes	—
637	#95	C₂³×C₆₇	1	2	yes	—
638	#95	C₂³×C₆₇	1	2	yes	—
639	#95	C₂³×C₆₇	1	2	yes	—
640	#95	C₂³×C₆₇	1	2	yes	—
641	#95	C₂³×C₆₇	1	2	yes	—
642	#95	C₂³×C₆₇	1	2	yes	—
643	#95	C₂³×C₆₇	1	2	yes	—
644	#95	C₂³×C₆₇	1	2	yes	—
645	#95	C₂³×C₆₇	1	2	yes	—
646	#95	C₂³×C₆₇	1	2	yes	—
647	#95	C₂³×C₆₇	1	2	yes	—
648	#95	C₂³×C₆₇	1	2	yes	—
649	#95	C₂³×C₆₇	1	2	yes	—
650	#95	C₂³×C₆₇	1	2	yes	—
651	#95	C₂³×C₆₇	1	2	yes	—
652	#95	C₂³×C₆₇	1	2	yes	—
653	#95	C₂³×C₆₇	1	2	yes	—
654	#95	C₂³×C₆₇	1	2	yes	—
655	#95	C₂³×C₆₇	1	2	yes	—
656	#95	C₂³×C₆₇	1	2	yes	—
657	#95	C₂³×C₆₇	1	2	yes	—
658	#95	C₂³×C₆₇	1	2	yes	—
659	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
660	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
661	C₂×Q₈	—	4	1	no	—
662	C₂×Q₈	—	4	1	no	—
663	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
664	C₂×D₄	—	2	4	no	—
665	C₂×D₄	—	2	4	no	—
666	C₂×D₄	—	4	1	no	—
667	C₂×D₄	—	4	1	no	—
668	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
669	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
670	C₂²×C₄	—	2	2	no	—
671	C₂²×C₄	—	2	2	no	—
672	C₂²×C₄	—	2	2	no	—
673	C₂²×C₄	—	2	2	no	—
674	C₂²×C₄	—	2	2	no	—
675	C₂²×C₄	—	2	2	no	—
676	C₂²×C₄	—	2	2	no	—
677	C₂²×C₄	—	2	2	no	—
678	C₂²×C₄	—	2	4	no	—
679	C₂²×C₄	—	4	1	no	—
680	C₂²×C₄	—	4	1	no	—
681	C₂²×C₄	—	4	2	no	—
682	C₂²×C₄	—	4	1	no	—
683	C₂²×C₄	—	4	2	no	—
684	C₂²×C₄	—	4	1	no	—
685	C₂²×C₄	—	4	1	no	—
686	C₂²×C₄	—	4	1	no	—
687	C₂²×C₄	—	4	2	no	—
688	C₂²×C₄	—	4	1	no	—
689	C₂²×C₄	—	4	2	no	—
690	C₂²×C₄	—	4	2	no	—
691	C₂²×C₄	—	4	2	no	—
692	C₂²×C₄	—	4	1	no	—
693	C₂²×C₄	—	4	1	no	—
694	C₂²×C₄	—	4	2	no	—
695	C₂²×C₄	—	4	1	no	—
696	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
697	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
698	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
699	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
700	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
701	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
702	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
703	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
704	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
705	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
706	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
707	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
708	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
709	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
710	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
711	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
712	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
713	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
714	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
715	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
716	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
717	C₄⋊C₄	—	2	4	no	—
718	C₄⋊C₄	—	2	4	no	—
719	C₄⋊C₄	—	4	2	no	—
720	C₄⋊C₄	—	4	2	no	—
721	C₄⋊C₄	—	4	2	no	—
722	C₄⋊C₄	—	4	2	no	—
723	C₄⋊C₄	—	4	2	no	—
724	C₄⋊C₄	—	4	1	no	—
725	C₄⋊C₄	—	4	2	no	—
726	C₄⋊C₄	—	4	2	no	—
727	C₄⋊C₄	—	4	1	no	—
728	C₄⋊C₄	—	4	2	no	—
729	C₄⋊C₄	—	4	2	no	—
730	C₄⋊C₄	—	4	2	no	—
731	C₄⋊C₄	—	4	1	no	—
732	C₄⋊C₄	—	4	1	no	—
733	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
734	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
735	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
736	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
737	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
738	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
739	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
740	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
741	C₂²⋊C₄	—	2	4	no	—
742	C₂²⋊C₄	—	2	4	no	—
743	C₂²⋊C₄	—	2	4	no	—
744	C₂²⋊C₄	—	4	2	no	—
745	C₂²⋊C₄	—	4	2	no	—
746	C₂²⋊C₄	—	4	2	no	—
747	C₂²⋊C₄	—	4	2	no	—
748	C₂²⋊C₄	—	4	2	no	—
749	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
750	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
751	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
752	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
753	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
754	C₄²	—	2	2	no	—
755	C₄²	—	2	2	no	—
756	C₄²	—	2	4	no	—
757	C₄²	—	4	1	no	—
758	C₄²	—	4	1	no	—
759	C₄²	—	4	2	no	—
760	C₄²	—	4	2	no	—
761	C₄²	—	4	2	no	—
762	C₄²	—	4	2	no	—
763	C₄²	—	4	2	no	—
764	C₄²	—	4	2	no	—
765	C₄²	—	4	2	no	—
766	C₄²	C₂²×C₄×C₆₇	1	8	yes	—
767	C₂³	—	2	1	no	—
768	C₂³	—	2	1	no	—
769	C₂³	—	2	1	no	—
770	C₂³	—	2	1	no	—
771	C₂³	—	2	2	no	—
772	C₂³	—	2	2	no	—
773	C₂³	—	2	2	no	—
774	C₂³	—	2	2	no	—
775	C₂³	—	2	2	no	—
776	C₂³	—	2	2	no	—
777	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
778	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
779	C₂³	H(5,𝔽₂)×C₆₇	1	2	yes	—
780	C₂³	2₋¹⁺⁴×C₆₇	1	2	yes	—
781	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
782	Q₈	—	4	4	no	—
783	D₄	—	2	16	no	—
784	D₄	—	4	4	no	—
785	C₂×C₄	—	2	8	no	—
786	C₂×C₄	—	2	1	no	—
787	C₂×C₄	—	2	1	no	—
788	C₂×C₄	—	2	1	no	—
789	C₂×C₄	—	2	2	no	—
790	C₂×C₄	—	2	2	no	—
791	C₂×C₄	—	2	1	no	—
792	C₂×C₄	—	2	2	no	—
793	C₂×C₄	—	2	2	no	—
794	C₂×C₄	—	2	2	no	—
795	C₂×C₄	—	2	1	no	—
796	C₂×C₄	—	2	2	no	—
797	C₂×C₄	—	2	2	no	—
798	C₂×C₄	—	2	2	no	—
799	C₂×C₄	—	2	2	no	—
800	C₂×C₄	—	2	2	no	—
801	C₂×C₄	—	2	1	no	—
802	C₂×C₄	—	2	1	no	—
803	C₂×C₄	—	2	1	no	—
804	C₂×C₄	—	2	1	no	—
805	C₂×C₄	—	2	2	no	—
806	C₂×C₄	—	2	2	no	—
807	C₂×C₄	—	2	1	no	—
808	C₂×C₄	—	2	2	no	—
809	C₂×C₄	—	2	2	no	—
810	C₂×C₄	—	2	2	no	—
811	C₂×C₄	—	2	2	no	—
812	C₂×C₄	—	2	1	no	—
813	C₂×C₄	—	2	2	no	—
814	C₂×C₄	—	2	2	no	—
815	C₂×C₄	—	2	2	no	—
816	C₂×C₄	—	2	2	no	—
817	C₂×C₄	—	2	1	no	—
818	C₂×C₄	—	2	1	no	—
819	C₂×C₄	—	2	2	no	—
820	C₂×C₄	—	2	2	no	—
821	C₂×C₄	—	2	2	no	—
822	C₂×C₄	—	2	2	no	—
823	C₂×C₄	—	2	2	no	—
824	C₂×C₄	—	2	2	no	—
825	C₂×C₄	—	2	2	no	—
826	C₂×C₄	—	2	1	no	—
827	C₂×C₄	—	2	2	no	—
828	C₂×C₄	—	4	4	no	—
829	C₂×C₄	—	4	8	no	—
830	C₂×C₄	—	4	4	no	—
831	C₂×C₄	—	4	4	no	—
832	C₂×C₄	—	4	4	no	—
833	C₂×C₄	—	8	4	no	—
834	C₂×C₄	—	8	2	no	—
835	C₂×C₄	—	8	2	no	—
836	C₂×C₄	—	8	4	no	—
837	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
838	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
839	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
840	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
841	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
842	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
843	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
844	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
845	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
846	C₂²	—	2	2	no	—
847	C₂²	—	2	2	no	—
848	C₂²	—	2	2	no	—
849	C₂²	—	2	2	no	—
850	C₂²	—	2	2	no	—
851	C₂²	—	2	2	no	—
852	C₂²	—	2	4	no	—
853	C₂²	—	2	4	no	—
854	C₂²	—	2	4	no	—
855	C₂²	—	4	2	no	—
856	C₂²	—	4	2	no	—
857	C₂²	—	4	2	no	—
858	C₂²	—	4	2	no	—
859	C₂²	#520×C₆₇	1	4	yes	—
860	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
861	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
862	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
863	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
864	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
865	C₂²	#585×C₆₇	1	1	yes	characteristic
866	C₂²	#585×C₆₇	1	1	yes	characteristic
867	C₄	—	2	2	no	—
868	C₄	—	2	2	no	—
869	C₄	—	2	4	no	—
870	C₄	—	2	4	no	—
871	C₄	—	2	4	no	—
872	C₄	—	2	2	no	—
873	C₄	—	2	2	no	—
874	C₄	—	2	4	no	—
875	C₄	—	2	4	no	—
876	C₄	—	4	1	no	—
877	C₄	—	4	2	no	—
878	C₄	—	4	2	no	—
879	C₄	—	4	2	no	—
880	C₄	—	4	1	no	—
881	C₄	—	4	2	no	—
882	C₄	—	4	2	no	—
883	C₄	—	4	2	no	—
884	C₂	—	2	4	no	—
885	C₂	—	4	4	no	—
886	C₂	C₂×#520×C₆₇	1	1	yes	characteristic
887	C₂	C₄×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
888	C₂	C₄×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
889	C₂	C₆₇×#3534	1	1	yes	characteristic
890	C₂	C₆₇×#3538	1	1	yes	characteristic
891	C₂	C₆₇×#3559	1	1	yes	characteristic
892	C₂	C₆₇×#3564	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×#2383	C₂	semidirect
3. C₆₇×#2378	C₂	semidirect
4. C₆₇×#2373	C₂	semidirect
5. C₆₇×#2360	C₂	semidirect
6. C₆₇×#2359	C₂	semidirect
7. C₆₇×#2359	C₂	semidirect
8. C₆₇×#2353	C₂	semidirect
9. D₄×C₄⋊C₄×C₆₇	C₂	semidirect
10. C₆₇×#2336	C₂	semidirect
11. C₆₇×#2335	C₂	semidirect
12. C₆₇×#2330	C₂	semidirect
13. C₆₇×#2329	C₂	semidirect
15. C₆₇×#2322	C₂	semidirect
16. C₆₇×#2316	C₂	semidirect
17. C₄×C₂²⋊Q₈×C₆₇	C₂	semidirect
19. C₂×#387×C₆₇	C₂	semidirect
20. #539×C₆₇	C₄	semidirect
121. #126×C₆₇	C₂×C₄	semidirect
122. #125×C₆₇	C₂×C₄	semidirect
123. #123×C₆₇	C₂×C₄	semidirect
124. #123×C₆₇	C₂×C₄	semidirect
125. C₂²⋊Q₈×C₆₇	C₂×C₄	semidirect
126. C₂²⋊Q₈×C₆₇	C₂×C₄	semidirect
132. C₄×D₄×C₆₇	C₂×C₄	semidirect
136. C₂×C₄⋊C₄×C₆₇	C₂×C₄	semidirect
437. #34425	C₆₇	direct (N × H)
448. #2383	C₂×C₆₇	semidirect
449. #2378	C₂×C₆₇	semidirect
450. #2373	C₂×C₆₇	semidirect
451. #2360	C₂×C₆₇	semidirect
452. #2359	C₂×C₆₇	semidirect
453. #2359	C₂×C₆₇	semidirect
454. #2353	C₂×C₆₇	semidirect
455. D₄×C₄⋊C₄	C₂×C₆₇	semidirect
456. #2336	C₂×C₆₇	semidirect
457. #2335	C₂×C₆₇	semidirect
458. #2330	C₂×C₆₇	semidirect
459. #2329	C₂×C₆₇	semidirect
461. #2322	C₂×C₆₇	semidirect
462. #2316	C₂×C₆₇	semidirect
463. C₄×C₂²⋊Q₈	C₂×C₆₇	semidirect
465. C₂×#387	C₂×C₆₇	semidirect
466. C₆₇	#34425	direct (N × H)
467. #539	C₄×C₆₇	semidirect
568. #126	C₂×C₄×C₆₇	semidirect
569. #125	C₂×C₄×C₆₇	semidirect
570. #123	C₂×C₄×C₆₇	semidirect
571. #123	C₂×C₄×C₆₇	semidirect
572. C₂²⋊Q₈	C₂×C₄×C₆₇	semidirect
573. C₂²⋊Q₈	C₂×C₄×C₆₇	semidirect
579. C₄×D₄	C₂×C₄×C₆₇	semidirect
583. C₂×C₄⋊C₄	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×#539×C₆₇	2
C₆₇×#2383	2
C₆₇×#2378	2
C₆₇×#2373	2
C₆₇×#2360	2
C₆₇×#2359	2
C₆₇×#2353	2
D₄×C₄⋊C₄×C₆₇	2
C₆₇×#2336	2
C₆₇×#2335	2
C₆₇×#2330	2
C₆₇×#2329	2
C₆₇×#2325	2
C₆₇×#2322	2
C₆₇×#2316	2
C₄×C₂²⋊Q₈×C₆₇	2
C₄×#121×C₆₇	2
C₂×#387×C₆₇	2
#34425	67

Subgroups — C67×#34425 #30713676

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34425 #30713676