Subgroups — #22574506

Order 8576.

1032 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16864147	C₂	1	2	yes	maximal, complemented
2	#16864142	C₂	1	2	yes	maximal, complemented
3	#16864117	C₂	1	1	yes	maximal, characteristic, complemented
4	#16864101	C₂	1	2	yes	maximal, complemented
5	#16864101	C₂	1	1	yes	maximal, characteristic, complemented
6	#16864094	C₂	1	2	yes	maximal, complemented
7	#16864094	C₂	1	1	yes	maximal, characteristic, complemented
8	#16864080	C₂	1	1	yes	maximal, characteristic, complemented
9	Q₈×C₂²⋊D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#16864057	C₂	1	1	yes	maximal, characteristic, complemented
11	#16864056	C₂	1	1	yes	maximal, characteristic, complemented
12	#16864048	C₂	1	1	yes	maximal, characteristic, complemented
13	#16864019	C₂	1	1	yes	maximal, characteristic, complemented
14	#16864016	C₂	1	2	yes	maximal, complemented
15	#16864006	C₂	1	1	yes	maximal, characteristic, complemented
16	#16863992	C₂	1	2	yes	maximal, complemented
17	#16863984	C₂	1	2	yes	maximal, complemented
18	#16863984	C₂	1	2	yes	maximal, complemented
19	#16863976	C₂	1	1	yes	maximal, characteristic, complemented
20	#16863969	C₂	1	1	yes	maximal, characteristic, complemented
21	#16863907	C₂	1	1	yes	maximal, characteristic, complemented
22	#16863907	C₂	1	1	yes	maximal, characteristic, complemented
23	C₂×C₂²⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
24	C₂×#11761629	C₂²	1	1	yes	characteristic, complemented
25	C₂×#11761628	C₂²	1	1	yes	characteristic
26	C₂×#11761628	C₂²	1	1	yes	characteristic
27	C₂×Q₈×D₆₇	C₂²	1	1	yes	characteristic, complemented
28	#14524042	C₂²	1	1	yes	characteristic, complemented
29	#14524042	C₂²	1	2	yes	complemented
30	#14524041	C₂²	1	1	yes	characteristic, complemented
31	#14524041	C₂²	1	2	yes	complemented
32	#14524038	C₂²	1	2	yes	complemented
33	#14524038	C₂²	1	2	yes	complemented
34	#14524038	C₂²	1	2	yes	complemented
35	#14524036	C₂²	1	2	yes	—
36	#14524036	C₂²	1	2	yes	—
37	#14524036	C₂²	1	2	yes	—
38	#14524036	C₂²	1	2	yes	—
39	#14524034	C₂²	1	1	yes	characteristic, complemented
40	#14524034	C₂²	1	2	yes	complemented
41	#14524033	C₂²	1	2	yes	complemented
42	#14524033	C₂²	1	2	yes	complemented
43	#14524033	C₂²	1	2	yes	complemented
44	#14524032	C₂²	1	2	yes	complemented
45	#14524032	C₂²	1	1	yes	characteristic, complemented
46	#14524029	C₂²	1	2	yes	—
47	#14524029	C₂²	1	2	yes	—
48	#14524029	C₂²	1	2	yes	—
49	#14524029	C₂²	1	2	yes	—
50	#14524028	C₂²	1	2	yes	complemented
51	#14524028	C₂²	1	2	yes	complemented
52	#14524028	C₂²	1	2	yes	complemented
53	#14524026	C₂²	1	1	yes	characteristic
54	#14524026	C₂²	1	2	yes	—
55	#14524026	C₂²	1	1	yes	characteristic
56	#14524025	C₂²	1	2	yes	—
57	#14524025	C₂²	1	2	yes	—
58	#14524025	C₂²	1	2	yes	—
59	#14524025	C₂²	1	2	yes	—
60	#14524023	C₂²	1	2	yes	complemented
61	#14524023	C₂²	1	1	yes	characteristic, complemented
62	#14524022	C₂²	1	1	yes	characteristic
63	#14524022	C₂²	1	1	yes	characteristic
64	#14524022	C₂²	1	2	yes	—
65	#14524022	C₂²	1	1	yes	characteristic
66	#14524022	C₂²	1	2	yes	—
67	#14524022	C₂²	1	1	yes	characteristic
68	#14524021	C₂²	1	2	yes	complemented
69	#14524021	C₂²	1	2	yes	complemented
70	#14524021	C₂²	1	2	yes	complemented
71	#14524019	C₂²	1	1	yes	characteristic, complemented
72	#14524019	C₂²	1	2	yes	complemented
73	#14524019	C₂²	1	1	yes	characteristic, complemented
74	#14524019	C₂²	1	2	yes	complemented
75	#14524018	C₂²	1	1	yes	characteristic, complemented
76	#14524018	C₂²	1	2	yes	complemented
77	#14524017	C₂²	1	1	yes	characteristic
78	#14524017	C₂²	1	1	yes	characteristic
79	#14524017	C₂²	1	2	yes	—
80	#14524017	C₂²	1	1	yes	characteristic
81	#14524017	C₂²	1	2	yes	—
82	#14524017	C₂²	1	1	yes	characteristic
83	#14524016	C₂²	1	2	yes	—
84	#14524016	C₂²	1	2	yes	—
85	#14524016	C₂²	1	2	yes	—
86	#14524016	C₂²	1	2	yes	—
87	#14524012	C₂²	1	1	yes	characteristic
88	#14524012	C₂²	1	2	yes	—
89	#14524012	C₂²	1	1	yes	characteristic
90	Q₈×Dic₆₇	C₂²	1	1	yes	characteristic, complemented
91	Q₈×Dic₆₇	C₂²	1	1	yes	characteristic, complemented
92	#14524008	C₂²	1	1	yes	characteristic, complemented
93	#14524008	C₂²	1	2	yes	complemented
94	#14524005	C₂²	1	1	yes	characteristic, complemented
95	#14524005	C₂²	1	2	yes	complemented
96	C₄×C₂²⋊D₆₇	C₂²	1	2	yes	—
97	C₄×C₂²⋊D₆₇	C₂²	1	1	yes	characteristic
98	C₄×C₂²⋊D₆₇	C₂²	1	1	yes	characteristic
99	C₄×C₂²⋊D₆₇	C₂²	1	1	yes	characteristic
100	C₄×C₂²⋊D₆₇	C₂²	1	2	yes	—
101	C₄×C₂²⋊D₆₇	C₂²	1	1	yes	characteristic
102	C₄×C₂²⋊D₆₇	C₂²	1	2	yes	—
103	C₄×C₂²⋊D₆₇	C₂²	1	2	yes	—
104	C₄×C₂²⋊D₆₇	C₂²	1	2	yes	—
105	C₄×C₂²⋊D₆₇	C₂²	1	2	yes	—
106	#14524002	C₂²	1	1	yes	characteristic
107	#14524002	C₂²	1	1	yes	characteristic
108	#14524001	C₂²	1	2	yes	complemented
109	#14524001	C₂²	1	1	yes	characteristic, complemented
110	C₄⋊C₄×D₆₇	C₂²	1	2	yes	complemented
111	C₄⋊C₄×D₆₇	C₂²	1	1	yes	characteristic, complemented
112	C₂²×Q₈×C₆₇	C₂²	1	1	yes	characteristic
113	C₂³×C₄×C₆₇	C₂²	1	1	yes	characteristic
114	C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
115	C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
116	C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
117	C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
118	C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
119	C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
120	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
121	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
122	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
123	#11761629	—	4	2	no	—
124	#11761628	—	4	2	no	—
125	#11761628	—	4	2	no	—
126	C₂×Dic₁₃₄	C₂³	1	1	yes	characteristic
127	C₂×Dic₁₃₄	C₂³	1	2	yes	—
128	C₂×Dic₁₃₄	C₂³	1	1	yes	characteristic
129	C₂×Dic₁₃₄	C₂³	1	2	yes	—
130	Q₈×D₆₇	—	4	2	no	—
131	C₂×D₂₆₈	C₂³	1	1	yes	characteristic
132	C₂×D₂₆₈	C₂³	1	2	yes	—
133	C₂×D₂₆₈	C₂³	1	2	yes	—
134	C₂×D₂₆₈	C₂³	1	1	yes	characteristic
135	C₂×C₂²⋊D₆₇	C₂³	1	1	yes	characteristic
136	C₂×C₂²⋊D₆₇	C₂³	1	2	yes	—
137	C₂×C₂²⋊D₆₇	C₂³	1	1	yes	characteristic
138	C₂×C₂²⋊D₆₇	C₂³	1	1	yes	characteristic
139	C₂×C₂²⋊D₆₇	C₂³	1	2	yes	—
140	C₂×C₂²⋊D₆₇	C₂³	1	1	yes	characteristic
141	C₂×C₂²⋊D₆₇	C₂³	1	1	yes	characteristic
142	C₂×C₂²⋊D₆₇	C₂³	1	2	yes	—
143	C₂×C₂²⋊D₆₇	C₂³	1	1	yes	characteristic
144	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
145	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
146	C₂×C₄×D₆₇	C₂³	1	2	yes	—
147	C₂×C₄×D₆₇	C₂³	1	2	yes	—
148	C₂×C₄×D₆₇	C₂³	1	2	yes	—
149	C₂×C₄×D₆₇	C₂³	1	2	yes	—
150	C₂×C₄×D₆₇	C₂³	1	2	yes	—
151	C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
152	C₄⋊Dic₆₇	C₂³	1	2	yes	—
153	C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
154	C₄⋊Dic₆₇	C₂³	1	2	yes	—
155	C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
156	C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
157	C₄⋊Dic₆₇	C₂³	1	2	yes	—
158	C₄⋊Dic₆₇	C₂³	1	2	yes	—
159	Dic₆₇⋊C₄	C₂³	1	1	yes	characteristic
160	Dic₆₇⋊C₄	C₂³	1	2	yes	—
161	Dic₆₇⋊C₄	C₂³	1	2	yes	—
162	Dic₆₇⋊C₄	C₂³	1	1	yes	characteristic
163	Dic₆₇⋊C₄	C₂³	1	1	yes	characteristic
164	Dic₆₇⋊C₄	C₂³	1	2	yes	—
165	Dic₆₇⋊C₄	C₂³	1	2	yes	—
166	Dic₆₇⋊C₄	C₂³	1	2	yes	—
167	Dic₆₇⋊C₄	C₂³	1	1	yes	characteristic
168	Dic₆₇⋊C₄	C₂³	1	2	yes	—
169	Dic₆₇⋊C₄	C₂³	1	2	yes	—
170	Dic₆₇⋊C₄	C₂³	1	2	yes	—
171	Dic₆₇⋊C₄	C₂³	1	2	yes	—
172	Dic₆₇⋊C₄	C₂³	1	2	yes	—
173	C₂²⋊Dic₆₇	C₂³	1	1	yes	characteristic
174	C₂²⋊Dic₆₇	C₂³	1	2	yes	—
175	C₂²⋊Dic₆₇	C₂³	1	1	yes	characteristic
176	C₂²⋊Dic₆₇	C₂³	1	1	yes	characteristic
177	C₂²⋊Dic₆₇	C₂³	1	2	yes	—
178	C₂²⋊Dic₆₇	C₂³	1	1	yes	characteristic
179	C₂²⋊Dic₆₇	C₂³	1	1	yes	characteristic
180	C₂²⋊Dic₆₇	C₂³	1	2	yes	—
181	C₂²⋊Dic₆₇	C₂³	1	1	yes	characteristic
182	(C₂×D₆₇)⋊C₄	C₂³	1	1	yes	characteristic
183	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
184	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
185	(C₂×D₆₇)⋊C₄	C₂³	1	1	yes	characteristic
186	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
187	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
188	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
189	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
190	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
191	(C₂×D₆₇)⋊C₄	C₂³	1	1	yes	characteristic
192	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
193	(C₂×D₆₇)⋊C₄	C₂³	1	1	yes	characteristic
194	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
195	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
196	C₄×Dic₆₇	C₂³	1	2	yes	—
197	C₄×Dic₆₇	C₂³	1	2	yes	—
198	C₄×Dic₆₇	C₂³	1	2	yes	—
199	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
200	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
201	C₄×Dic₆₇	C₂³	1	2	yes	—
202	C₄×Dic₆₇	C₂³	1	2	yes	—
203	C₂⁴×C₆₇	C₂³	1	1	yes	characteristic
204	C₂×Q₈×C₆₇	—	4	2	no	—
205	C₂×Q₈×C₆₇	C₂³	1	1	yes	characteristic
206	C₂×Q₈×C₆₇	C₂³	1	1	yes	characteristic
207	C₂×Q₈×C₆₇	C₂³	1	2	yes	—
208	C₂²×C₄×C₆₇	—	4	2	no	—
209	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
210	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
211	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
212	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
213	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
214	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
215	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
216	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
217	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
218	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
219	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
220	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
221	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
222	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
223	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
224	C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
225	C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
226	C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
227	C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
228	C₂³×C₆₇	—	2	2	no	—
229	C₂³×C₆₇	—	4	1	no	—
230	C₂³×C₆₇	—	4	1	no	—
231	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
232	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
233	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
234	C₂²×D₆₇	C₂⁴	1	1	yes	characteristic
235	C₂²×D₆₇	C₂⁴	1	1	yes	characteristic
236	C₂²×D₆₇	C₂⁴	1	2	yes	—
237	Q₈×C₆₇	—	2	2	no	—
238	Q₈×C₆₇	—	2	2	no	—
239	Q₈×C₆₇	—	4	2	no	—
240	C₂×C₄×C₆₇	—	2	2	no	—
241	C₂×C₄×C₆₇	—	2	2	no	—
242	C₂×C₄×C₆₇	—	2	2	no	—
243	C₂×C₄×C₆₇	—	4	1	no	—
244	C₂×C₄×C₆₇	—	4	1	no	—
245	C₂×C₄×C₆₇	—	4	1	no	—
246	C₂×C₄×C₆₇	—	4	2	no	—
247	C₂×C₄×C₆₇	—	4	2	no	—
248	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
249	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
250	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
251	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
252	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
253	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
254	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
255	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
256	C₂²⋊D₆₇	—	2	2	no	—
257	C₂²⋊D₆₇	—	2	4	no	—
258	C₂²⋊D₆₇	—	2	2	no	—
259	C₂²⋊D₆₇	—	4	1	no	—
260	C₂²⋊D₆₇	—	4	2	no	—
261	C₂²⋊D₆₇	—	4	1	no	—
262	C₂²⋊D₆₇	—	4	2	no	—
263	C₂²⋊D₆₇	—	4	1	no	—
264	C₂²⋊D₆₇	—	4	1	no	—
265	C₂×Dic₆₇	C₂⁴	1	2	yes	—
266	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
267	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
268	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
269	C₂×Dic₆₇	C₂⁴	1	2	yes	—
270	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
271	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
272	C₂×Dic₆₇	C₂⁴	1	2	yes	—
273	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
274	D₂₆₈	—	2	2	no	—
275	D₂₆₈	—	4	2	no	—
276	D₂₆₈	—	4	1	no	—
277	D₂₆₈	—	4	2	no	—
278	C₄×D₆₇	—	2	2	no	—
279	C₄×D₆₇	—	2	2	no	—
280	C₄×D₆₇	—	4	2	no	—
281	C₄×D₆₇	—	4	2	no	—
282	C₄×D₆₇	—	4	2	no	—
283	C₄×D₆₇	—	4	2	no	—
284	C₄×D₆₇	—	4	2	no	—
285	Dic₁₃₄	—	2	2	no	—
286	Dic₁₃₄	—	4	2	no	—
287	Dic₁₃₄	—	4	2	no	—
288	Dic₁₃₄	—	4	1	no	—
289	C₂²×C₆₇	—	2	1	no	—
290	C₂²×C₆₇	—	2	1	no	—
291	C₂²×C₆₇	—	2	1	no	—
292	C₂²×C₆₇	—	2	1	no	—
293	C₂²×C₆₇	—	2	1	no	—
294	C₂²×C₆₇	—	2	1	no	—
295	C₂²×C₆₇	—	2	1	no	—
296	C₂²×C₆₇	—	2	1	no	—
297	C₂²×C₆₇	—	4	4	no	—
298	C₂²×C₆₇	2₋¹⁺⁴	1	2	yes	—
299	C₂²×C₆₇	C₂⁵	1	1	yes	characteristic
300	C₂×D₆₇	—	2	1	no	—
301	C₂×D₆₇	—	2	1	no	—
302	C₂×D₆₇	—	2	2	no	—
303	C₂×D₆₇	—	2	1	no	—
304	C₂×D₆₇	—	2	1	no	—
305	C₂×D₆₇	—	2	2	no	—
306	C₂×D₆₇	—	2	1	no	—
307	C₂×D₆₇	—	2	2	no	—
308	C₂×D₆₇	—	2	1	no	—
309	C₄×C₆₇	—	2	2	no	—
310	C₄×C₆₇	—	2	1	no	—
311	C₄×C₆₇	—	2	1	no	—
312	C₄×C₆₇	—	2	2	no	—
313	C₄×C₆₇	—	2	1	no	—
314	C₄×C₆₇	—	2	2	no	—
315	C₄×C₆₇	—	2	2	no	—
316	C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
317	Dic₆₇	—	2	2	no	—
318	Dic₆₇	—	2	1	no	—
319	Dic₆₇	—	2	1	no	—
320	Dic₆₇	—	2	2	no	—
321	Dic₆₇	—	2	1	no	—
322	Dic₆₇	—	2	1	no	—
323	Dic₆₇	—	2	1	no	—
324	Dic₆₇	—	2	2	no	—
325	Dic₆₇	—	2	1	no	—
326	C₂×C₆₇	—	2	2	no	—
327	C₂×C₆₇	—	4	1	no	—
328	C₂×C₆₇	—	4	1	no	—
329	C₂×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
330	C₂×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
331	C₂×C₆₇	#585	1	1	yes	characteristic
332	D₆₇	—	4	1	no	—
333	D₆₇	—	4	1	no	—
334	D₆₇	—	4	2	no	—
335	#3541	—	67	1	no	maximal, Sylow
336	C₆₇	#3541	1	1	yes	characteristic, complemented, Sylow
337	#563	—	67	2	no	—
338	#562	—	67	2	no	—
339	#556	—	67	1	no	—
340	#554	—	67	1	no	—
341	#554	—	67	2	no	—
342	#553	—	67	1	no	—
343	#553	—	67	2	no	—
344	#552	—	67	1	no	—
345	D₄×Q₈	—	67	1	no	—
346	#548	—	67	1	no	—
347	#548	—	67	1	no	—
348	#547	—	67	1	no	—
349	#542	—	67	2	no	—
350	#542	—	67	1	no	—
351	#541	—	67	1	no	—
352	#539	—	67	2	no	—
353	#538	—	67	2	no	—
354	#538	—	67	2	no	—
355	#537	—	67	1	no	—
356	#536	—	67	1	no	—
357	#525	—	67	1	no	—
358	#525	—	67	1	no	—
359	C₂×C₂²⋊Q₈	D₆₇	1	1	yes	characteristic, complemented
360	C₂×#41	—	67	1	no	—
361	C₂×#41	—	67	1	no	—
362	C₂×#41	—	67	1	no	—
363	C₂²×Q₈	—	67	1	no	—
364	C₂²×Q₈	C₂×D₆₇	1	1	yes	characteristic
365	C₂³×C₄	C₂×D₆₇	1	1	yes	characteristic
366	C₄⋊Q₈	—	67	1	no	—
367	C₄⋊Q₈	—	67	2	no	—
368	C₄⋊Q₈	—	67	2	no	—
369	C₄⋊Q₈	—	67	1	no	—
370	#126	—	67	2	no	—
371	#126	—	67	2	no	—
372	#126	—	67	2	no	—
373	#126	—	67	2	no	—
374	#126	—	67	2	no	—
375	#126	—	67	2	no	—
376	#126	—	67	2	no	—
377	#125	—	67	2	no	—
378	#125	—	67	2	no	—
379	#125	—	67	1	no	—
380	#125	—	67	2	no	—
381	#125	—	67	2	no	—
382	#124	—	67	1	no	—
383	#124	—	67	2	no	—
384	#124	—	67	2	no	—
385	#124	—	67	2	no	—
386	#124	—	67	2	no	—
387	#124	—	67	2	no	—
388	#123	—	67	1	no	—
389	#123	—	67	1	no	—
390	#123	—	67	2	no	—
391	#123	—	67	2	no	—
392	#123	—	67	2	no	—
393	#123	—	67	2	no	—
394	#123	—	67	2	no	—
395	#123	—	67	2	no	—
396	#123	—	67	2	no	—
397	#123	—	67	2	no	—
398	C₂²⋊Q₈	—	67	1	no	—
399	C₂²⋊Q₈	—	67	1	no	—
400	C₂²⋊Q₈	—	67	2	no	—
401	C₂²⋊Q₈	—	67	2	no	—
402	C₂²⋊Q₈	—	67	1	no	—
403	C₂²⋊Q₈	—	67	1	no	—
404	C₂²⋊Q₈	—	67	1	no	—
405	C₂²⋊Q₈	—	67	1	no	—
406	C₂²⋊Q₈	—	67	2	no	—
407	C₂²⋊Q₈	—	67	2	no	—
408	C₂²⋊Q₈	—	67	1	no	—
409	C₂²⋊Q₈	—	67	2	no	—
410	C₂²⋊Q₈	—	67	2	no	—
411	C₂²⋊Q₈	—	67	2	no	—
412	C₂²⋊Q₈	—	67	2	no	—
413	C₂²⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
414	C₂²⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
415	C₂²⋊Q₈	C₂×D₆₇	1	2	yes	—
416	C₂²⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
417	C₂²⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
418	C₂²⋊Q₈	C₂×D₆₇	1	2	yes	—
419	#121	—	67	1	no	—
420	#121	—	67	1	no	—
421	#121	—	67	1	no	—
422	#121	—	67	1	no	—
423	#121	—	67	2	no	—
424	#121	—	67	2	no	—
425	#121	—	67	1	no	—
426	#121	—	67	2	no	—
427	#121	—	67	2	no	—
428	#121	—	67	2	no	—
429	#121	—	67	2	no	—
430	#121	—	67	2	no	—
431	#120	—	67	1	no	—
432	#120	—	67	2	no	—
433	#120	—	67	1	no	—
434	C₄×Q₈	—	67	1	no	—
435	C₄×Q₈	—	67	1	no	—
436	C₄×Q₈	—	67	1	no	—
437	C₄×Q₈	—	67	2	no	—
438	C₄×D₄	—	67	2	no	—
439	C₄×D₄	—	67	1	no	—
440	C₄×D₄	—	67	1	no	—
441	C₄×D₄	—	67	1	no	—
442	C₄×D₄	—	67	1	no	—
443	C₄×D₄	—	67	2	no	—
444	C₄×D₄	—	67	2	no	—
445	C₄×D₄	—	67	2	no	—
446	C₄×D₄	—	67	2	no	—
447	C₄×D₄	—	67	2	no	—
448	C₄×D₄	—	67	2	no	—
449	C₄×D₄	—	67	1	no	—
450	#117	—	67	1	no	—
451	#117	—	67	1	no	—
452	#117	—	67	1	no	—
453	#117	—	67	2	no	—
454	C₂×C₄⋊C₄	—	67	1	no	—
455	C₂×C₄⋊C₄	—	67	2	no	—
456	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
457	C₂×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
458	C₂×C₂²⋊C₄	C₂×D₆₇	1	2	yes	—
459	C₂⁴	C₂²×D₆₇	1	1	yes	characteristic
460	#41	—	268	2	no	—
461	#41	—	268	2	no	—
462	#41	—	268	2	no	—
463	C₂×Q₈	—	4	2	no	—
464	C₂×Q₈	—	268	2	no	—
465	C₂×Q₈	—	67	2	no	—
466	C₂×Q₈	—	67	1	no	—
467	C₂×Q₈	—	67	2	no	—
468	C₂×Q₈	—	67	1	no	—
469	C₂×Q₈	C₂²×D₆₇	1	1	yes	characteristic
470	C₂×Q₈	C₂²×D₆₇	1	1	yes	characteristic
471	C₂×Q₈	C₂²×D₆₇	1	2	yes	—
472	C₂×D₄	—	67	1	no	—
473	C₂×D₄	—	67	2	no	—
474	C₂×D₄	—	67	1	no	—
475	C₂×D₄	—	67	1	no	—
476	C₂×D₄	—	67	2	no	—
477	C₂×D₄	—	67	1	no	—
478	C₂×D₄	—	67	1	no	—
479	C₂×D₄	—	67	1	no	—
480	C₂×D₄	—	67	2	no	—
481	C₂×D₄	—	67	2	no	—
482	C₂×D₄	—	67	1	no	—
483	C₂×D₄	—	67	2	no	—
484	C₂×D₄	—	67	1	no	—
485	C₂²×C₄	—	4	2	no	—
486	C₂²×C₄	—	67	2	no	—
487	C₂²×C₄	—	67	2	no	—
488	C₂²×C₄	—	67	2	no	—
489	C₂²×C₄	—	67	2	no	—
490	C₂²×C₄	—	67	2	no	—
491	C₂²×C₄	—	67	1	no	—
492	C₂²×C₄	—	67	1	no	—
493	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
494	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
495	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
496	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
497	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
498	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
499	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
500	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
501	C₄⋊C₄	—	67	1	no	—
502	C₄⋊C₄	—	67	1	no	—
503	C₄⋊C₄	—	67	2	no	—
504	C₄⋊C₄	—	67	2	no	—
505	C₄⋊C₄	—	67	2	no	—
506	C₄⋊C₄	—	67	1	no	—
507	C₄⋊C₄	—	67	1	no	—
508	C₄⋊C₄	—	67	1	no	—
509	C₄⋊C₄	—	67	2	no	—
510	C₄⋊C₄	—	67	1	no	—
511	C₄⋊C₄	—	67	1	no	—
512	C₄⋊C₄	—	67	2	no	—
513	C₄⋊C₄	—	67	2	no	—
514	C₄⋊C₄	—	67	2	no	—
515	C₄⋊C₄	—	67	1	no	—
516	C₄⋊C₄	—	67	2	no	—
517	C₄⋊C₄	—	67	2	no	—
518	C₄⋊C₄	—	67	2	no	—
519	C₄⋊C₄	—	67	2	no	—
520	C₄⋊C₄	—	67	2	no	—
521	C₄⋊C₄	—	67	2	no	—
522	C₄⋊C₄	—	67	2	no	—
523	C₄⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
524	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
525	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
526	C₄⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
527	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
528	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
529	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
530	C₂²⋊C₄	—	67	1	no	—
531	C₂²⋊C₄	—	67	2	no	—
532	C₂²⋊C₄	—	67	1	no	—
533	C₂²⋊C₄	—	67	1	no	—
534	C₂²⋊C₄	—	67	2	no	—
535	C₂²⋊C₄	—	67	1	no	—
536	C₂²⋊C₄	—	67	1	no	—
537	C₂²⋊C₄	—	67	2	no	—
538	C₂²⋊C₄	—	67	1	no	—
539	C₂²⋊C₄	—	67	1	no	—
540	C₂²⋊C₄	—	67	2	no	—
541	C₂²⋊C₄	—	67	2	no	—
542	C₂²⋊C₄	—	67	1	no	—
543	C₂²⋊C₄	—	67	2	no	—
544	C₂²⋊C₄	—	67	2	no	—
545	C₂²⋊C₄	—	67	2	no	—
546	C₂²⋊C₄	—	67	1	no	—
547	C₂²⋊C₄	—	67	2	no	—
548	C₂²⋊C₄	—	67	1	no	—
549	C₂²⋊C₄	—	67	2	no	—
550	C₂²⋊C₄	—	67	2	no	—
551	C₂²⋊C₄	—	67	2	no	—
552	C₂²⋊C₄	—	67	2	no	—
553	C₂²⋊C₄	C₂²×D₆₇	1	2	yes	—
554	C₂²⋊C₄	C₂²×D₆₇	1	2	yes	—
555	C₂²⋊C₄	C₂²×D₆₇	1	2	yes	—
556	C₂²⋊C₄	C₂²×D₆₇	1	2	yes	—
557	C₄²	—	67	2	no	—
558	C₄²	—	67	1	no	—
559	C₄²	—	67	1	no	—
560	C₄²	—	67	2	no	—
561	C₄²	—	67	2	no	—
562	C₄²	—	67	2	no	—
563	C₄²	—	67	2	no	—
564	C₂³	—	2	2	no	—
565	C₂³	—	4	1	no	—
566	C₂³	—	4	1	no	—
567	C₂³	—	67	1	no	—
568	C₂³	—	67	2	no	—
569	C₂³	—	67	1	no	—
570	C₂³	C₂³×D₆₇	1	1	yes	characteristic
571	C₂³	C₂³×D₆₇	1	1	yes	characteristic
572	C₂³	C₂³×D₆₇	1	1	yes	characteristic
573	Q₈	—	2	2	no	—
574	Q₈	—	2	2	no	—
575	Q₈	—	4	2	no	—
576	Q₈	—	268	2	no	—
577	Q₈	—	268	1	no	—
578	Q₈	—	268	2	no	—
579	Q₈	—	134	2	no	—
580	D₄	—	268	1	no	—
581	D₄	—	268	2	no	—
582	D₄	—	268	2	no	—
583	D₄	—	268	1	no	—
584	D₄	—	268	2	no	—
585	D₄	—	268	1	no	—
586	D₄	—	268	1	no	—
587	D₄	—	268	2	no	—
588	D₄	—	268	1	no	—
589	D₄	—	134	2	no	—
590	D₄	—	134	2	no	—
591	D₄	—	134	4	no	—
592	D₄	—	134	2	no	—
593	C₂×C₄	—	2	2	no	—
594	C₂×C₄	—	2	2	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	4	1	no	—
597	C₂×C₄	—	4	1	no	—
598	C₂×C₄	—	4	1	no	—
599	C₂×C₄	—	4	2	no	—
600	C₂×C₄	—	4	2	no	—
601	C₂×C₄	—	268	2	no	—
602	C₂×C₄	—	268	2	no	—
603	C₂×C₄	—	268	2	no	—
604	C₂×C₄	—	268	2	no	—
605	C₂×C₄	—	268	2	no	—
606	C₂×C₄	—	67	2	no	—
607	C₂×C₄	—	67	1	no	—
608	C₂×C₄	—	67	1	no	—
609	C₂×C₄	—	67	1	no	—
610	C₂×C₄	—	67	2	no	—
611	C₂×C₄	—	67	1	no	—
612	C₂×C₄	—	67	1	no	—
613	C₂×C₄	—	67	2	no	—
614	C₂×C₄	—	67	1	no	—
615	C₂×C₄	—	134	2	no	—
616	C₂×C₄	—	134	2	no	—
617	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
618	C₂×C₄	C₂³×D₆₇	1	2	yes	—
619	C₂×C₄	C₂³×D₆₇	1	2	yes	—
620	C₂×C₄	C₂³×D₆₇	1	2	yes	—
621	C₂×C₄	C₂³×D₆₇	1	2	yes	—
622	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
623	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
624	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
625	C₂²	—	2	1	no	—
626	C₂²	—	2	1	no	—
627	C₂²	—	2	1	no	—
628	C₂²	—	2	1	no	—
629	C₂²	—	2	1	no	—
630	C₂²	—	2	1	no	—
631	C₂²	—	2	1	no	—
632	C₂²	—	2	1	no	—
633	C₂²	—	4	4	no	—
634	C₂²	—	134	1	no	—
635	C₂²	—	134	2	no	—
636	C₂²	—	134	1	no	—
637	C₂²	—	134	1	no	—
638	C₂²	—	134	2	no	—
639	C₂²	—	134	1	no	—
640	C₂²	—	134	1	no	—
641	C₂²	—	134	2	no	—
642	C₂²	—	134	1	no	—
643	C₂²	#14524091	1	2	yes	—
644	C₂²	C₂⁴×D₆₇	1	1	yes	characteristic
645	C₄	—	2	1	no	—
646	C₄	—	2	1	no	—
647	C₄	—	2	1	no	—
648	C₄	—	2	2	no	—
649	C₄	—	2	2	no	—
650	C₄	—	2	2	no	—
651	C₄	—	2	2	no	—
652	C₄	—	134	1	no	—
653	C₄	—	134	2	no	—
654	C₄	—	134	1	no	—
655	C₄	—	134	2	no	—
656	C₄	—	134	1	no	—
657	C₄	—	134	1	no	—
658	C₄	—	134	1	no	—
659	C₄	—	134	2	no	—
660	C₄	—	134	1	no	—
661	C₄	#14524088	1	2	yes	—
662	C₂	—	2	2	no	—
663	C₂	—	4	1	no	—
664	C₂	—	4	1	no	—
665	C₂	—	268	1	no	—
666	C₂	—	268	2	no	—
667	C₂	—	268	1	no	—
668	C₂	C₂×#14524088	1	1	yes	characteristic
669	C₂	C₂×#14524091	1	1	yes	characteristic
670	C₂	#16864232	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₆₇
commutator, socle	C₂²×C₆₇	C₂⁵
Fitting	C₂×C₂²⋊Q₈×C₆₇	C₂
hypercenter	C₂×C₂²⋊Q₈	D₆₇
hypocenter	C₆₇	#3541
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #16864147	C₂	semidirect
2. #16864142	C₂	semidirect
3. #16864117	C₂	semidirect
4. #16864101	C₂	semidirect
5. #16864101	C₂	semidirect
6. #16864094	C₂	semidirect
7. #16864094	C₂	semidirect
8. #16864080	C₂	semidirect
9. Q₈×C₂²⋊D₆₇	C₂	semidirect
10. #16864057	C₂	semidirect
11. #16864056	C₂	semidirect
12. #16864048	C₂	semidirect
13. #16864019	C₂	semidirect
14. #16864016	C₂	semidirect
15. #16864006	C₂	semidirect
16. #16863992	C₂	semidirect
17. #16863984	C₂	semidirect
18. #16863984	C₂	semidirect
19. #16863976	C₂	semidirect
20. #16863969	C₂	semidirect
21. #16863907	C₂	semidirect
22. #16863907	C₂	semidirect
23. C₂×C₂²⋊Q₈×C₆₇	C₂	semidirect
24. C₂×#11761629	C₂²	semidirect
27. C₂×Q₈×D₆₇	C₂²	semidirect
28. #14524042	C₂²	semidirect
29. #14524042	C₂²	semidirect
30. #14524041	C₂²	semidirect
31. #14524041	C₂²	semidirect
32. #14524038	C₂²	semidirect
33. #14524038	C₂²	semidirect
34. #14524038	C₂²	semidirect
39. #14524034	C₂²	semidirect
40. #14524034	C₂²	semidirect
41. #14524033	C₂²	semidirect
42. #14524033	C₂²	semidirect
43. #14524033	C₂²	semidirect
44. #14524032	C₂²	semidirect
45. #14524032	C₂²	semidirect
50. #14524028	C₂²	semidirect
51. #14524028	C₂²	semidirect
52. #14524028	C₂²	semidirect
60. #14524023	C₂²	semidirect
61. #14524023	C₂²	semidirect
68. #14524021	C₂²	semidirect
69. #14524021	C₂²	semidirect
70. #14524021	C₂²	semidirect
71. #14524019	C₂²	semidirect
72. #14524019	C₂²	semidirect
73. #14524019	C₂²	semidirect
74. #14524019	C₂²	semidirect
75. #14524018	C₂²	semidirect
76. #14524018	C₂²	semidirect
90. Q₈×Dic₆₇	C₂²	semidirect
91. Q₈×Dic₆₇	C₂²	semidirect
92. #14524008	C₂²	semidirect
93. #14524008	C₂²	semidirect
94. #14524005	C₂²	semidirect
95. #14524005	C₂²	semidirect
108. #14524001	C₂²	semidirect
109. #14524001	C₂²	semidirect
110. C₄⋊C₄×D₆₇	C₂²	semidirect
111. C₄⋊C₄×D₆₇	C₂²	semidirect
336. C₆₇	#3541	semidirect
359. C₂×C₂²⋊Q₈	D₆₇	semidirect