Subgroups — #31738312

Order 17152.

1900 conjugacy classes of subgroups.

The subgroup lattice has 756 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22574172	C₂	1	2	yes	maximal, complemented
2	#22573222	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×D₅₃₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#22573003	C₂	1	2	yes	maximal, complemented
5	#22572640	C₂	1	1	yes	maximal, characteristic, complemented
6	#22572632	C₂	1	2	yes	maximal, complemented
7	#22572441	C₂	1	2	yes	maximal, complemented
8	#22572367	C₂	1	4	yes	maximal, complemented
9	#22571896	C₂	1	2	yes	maximal, complemented
10	D₄×D₅₃₆	C₂	1	1	yes	maximal, characteristic, complemented
11	#22570848	C₂	1	2	yes	maximal, complemented
12	#22570788	C₂	1	1	yes	maximal, characteristic, complemented
13	#22570395	C₂	1	4	yes	maximal, complemented
14	#22570361	C₂	1	2	yes	maximal, complemented
15	#22569062	C₂	1	2	yes	maximal, complemented
16	C₄×#14524065	C₂	1	1	yes	maximal, characteristic, complemented
17	C₆₇×#2966	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×#14524065	C₂²	1	1	yes	characteristic
19	C₂²×D₅₃₆	C₂²	1	2	yes	—
20	#16864131	C₂²	1	2	yes	complemented
21	#16864116	C₂²	1	2	yes	—
22	#16864111	C₂²	1	4	yes	complemented
23	#16864071	—	2	4	no	—
24	Q₈×D₂₆₈	C₂²	1	2	yes	complemented
25	#16864056	—	2	2	no	—
26	#16864043	—	2	2	no	—
27	#16864042	C₂²	1	4	yes	complemented
28	D₄×D₂₆₈	C₂²	1	2	yes	—
29	#16864017	C₂²	1	4	yes	complemented
30	#16863982	—	2	4	no	—
31	#16863946	C₂²	1	2	yes	complemented
32	#16863931	—	2	2	no	—
33	#124×D₆₇	—	2	2	no	—
34	#16863870	C₂²	1	4	yes	complemented
35	C₄×#11761628	C₂²	1	2	yes	complemented
36	#16863801	C₂²	1	1	yes	characteristic, complemented
37	#16863801	C₂²	1	2	yes	complemented
38	#16863780	C₂²	1	2	yes	complemented
39	#16863773	C₂²	1	1	yes	characteristic
40	#16863756	C₂²	1	4	yes	complemented
41	#16863743	C₂²	1	2	yes	complemented
42	#16863738	C₂²	1	4	yes	complemented
43	#16863721	C₂²	1	4	yes	complemented
44	#16863721	C₂²	1	4	yes	complemented
45	#16863687	C₂²	1	4	yes	complemented
46	#16863682	C₂²	1	2	yes	complemented
47	#16863677	C₂²	1	4	yes	complemented
48	#16863677	C₂²	1	2	yes	complemented
49	#16863642	C₂²	1	4	yes	complemented
50	#16863637	C₂²	1	2	yes	complemented
51	#16863632	C₂²	1	2	yes	—
52	#16863632	C₂²	1	2	yes	complemented
53	#16863632	C₂²	1	2	yes	complemented
54	#16863612	C₂²	1	4	yes	complemented
55	#16863612	C₂²	1	4	yes	complemented
56	#16863597	C₂²	1	2	yes	complemented
57	#16863592	C₂²	1	2	yes	—
58	#16863549	C₂²	1	4	yes	complemented
59	#16863543	C₂²	1	4	yes	complemented
60	#16863532	C₂²	1	4	yes	—
61	#16863509	C₂²	1	4	yes	complemented
62	#16863509	C₂²	1	4	yes	complemented
63	#16863498	C₂²	1	4	yes	complemented
64	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic, complemented
65	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic, complemented
66	C₄×#11761617	C₂²	1	2	yes	complemented
67	C₄×#11761617	C₂²	1	2	yes	complemented
68	C₄×D₅₃₆	C₂²	1	1	yes	characteristic
69	C₄×D₅₃₆	C₂²	1	2	yes	—
70	C₄×D₅₃₆	C₂²	1	1	yes	characteristic
71	C₄×D₅₃₆	C₂²	1	2	yes	complemented
72	C₄×D₅₃₆	C₂²	1	2	yes	complemented
73	C₄×D₅₃₆	C₂²	1	2	yes	complemented
74	#16863434	C₂²	1	1	yes	characteristic, complemented
75	#16863434	C₂²	1	2	yes	complemented
76	#16863402	C₂²	1	4	yes	complemented
77	#16863393	C₂²	1	2	yes	complemented
78	#529×C₆₇	C₂²	1	1	yes	characteristic
79	#446×C₆₇	C₂²	1	1	yes	characteristic, complemented
80	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
81	#436×C₆₇	C₂²	1	1	yes	characteristic
82	#435×C₆₇	C₂²	1	2	yes	complemented
83	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
84	C₈×D₄×C₆₇	C₂²	1	2	yes	complemented
85	#433×C₆₇	C₂²	1	2	yes	complemented
86	#432×C₆₇	C₂²	1	1	yes	characteristic, complemented
87	#405×C₆₇	C₂²	1	2	yes	complemented
88	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
89	C₂×#11761628	—	2	2	no	—
90	C₂×#11761628	C₂³	1	2	yes	—
91	C₂×Q₈×D₆₇	—	2	2	no	—
92	C₂²×D₂₆₈	C₂³	1	4	yes	—
93	C₂×D₄×D₆₇	—	2	2	no	—
94	#14524065	—	2	4	no	—
95	C₂×Dic₂₆₈	C₂³	1	1	yes	characteristic
96	C₂×#11761617	C₂³	1	2	yes	—
97	C₂×#11761617	C₂³	1	4	yes	—
98	C₂×D₅₃₆	—	2	8	no	—
99	C₂×D₅₃₆	C₂³	1	4	yes	—
100	C₂×D₅₃₆	C₂³	1	2	yes	—
101	C₂×D₅₃₆	C₂³	1	2	yes	—
102	C₂×D₅₃₆	C₂³	1	1	yes	characteristic
103	#14524043	C₂³	1	4	yes	—
104	#14524043	C₂³	1	2	yes	—
105	#14524040	C₂³	1	2	yes	—
106	#14524038	—	2	4	no	—
107	#14524036	—	2	4	no	—
108	#14524032	—	2	2	no	—
109	#14524031	—	2	2	no	—
110	#14524031	C₂³	1	4	yes	—
111	#14524030	—	2	2	no	—
112	#14524029	—	2	4	no	—
113	#14524029	—	2	4	no	—
114	#14524029	—	2	4	no	—
115	#14524028	—	2	4	no	—
116	#14524028	—	2	4	no	—
117	#14524025	—	2	4	no	—
118	#14524023	—	2	2	no	—
119	#14524023	—	2	4	no	—
120	#14524022	C₂³	1	4	yes	—
121	#14524021	—	2	4	no	—
122	#14524018	—	2	2	no	—
123	#14524017	—	2	2	no	—
124	#14524017	C₂³	1	4	yes	—
125	#14524017	C₂³	1	4	yes	—
126	#14524016	—	2	4	no	—
127	#14524014	—	2	4	no	—
128	#14524013	—	2	4	no	—
129	#14524011	—	2	4	no	—
130	C₄×Dic₁₃₄	C₂³	1	2	yes	—
131	C₄×Dic₁₃₄	C₂³	1	2	yes	—
132	#14524008	—	2	2	no	—
133	C₄×D₂₆₈	—	2	2	no	—
134	C₄×D₂₆₈	C₂³	1	4	yes	—
135	C₄×D₂₆₈	C₂³	1	2	yes	—
136	C₄×D₂₆₈	C₂³	1	4	yes	—
137	C₄×D₂₆₈	C₂³	1	4	yes	—
138	C₄×D₂₆₈	C₂³	1	2	yes	—
139	#14524005	—	2	2	no	—
140	#14524005	—	2	4	no	—
141	#14524005	—	2	4	no	—
142	C₄×C₂²⋊D₆₇	—	2	2	no	—
143	C₄×C₂²⋊D₆₇	—	2	2	no	—
144	#14524002	C₂³	1	2	yes	—
145	#14524002	C₂³	1	4	yes	—
146	#14524001	—	2	4	no	—
147	#14524000	—	2	2	no	—
148	C₂²⋊C₄×D₆₇	—	2	4	no	—
149	C₂²⋊C₄×D₆₇	—	2	4	no	—
150	C₂²⋊C₄×D₆₇	—	2	4	no	—
151	C₄²×D₆₇	—	2	2	no	—
152	#14523977	C₂³	1	1	yes	characteristic
153	#14523977	C₂³	1	1	yes	characteristic
154	#14523977	C₂³	1	4	yes	—
155	#14523977	C₂³	1	2	yes	—
156	#14523977	C₂³	1	2	yes	—
157	#14523975	C₂³	1	4	yes	—
158	#14523975	C₂³	1	2	yes	—
159	#14523968	C₂³	1	2	yes	—
160	#14523968	C₂³	1	2	yes	—
161	#14523968	C₂³	1	4	yes	—
162	#14523965	C₂³	1	4	yes	—
163	#14523965	C₂³	1	4	yes	—
164	#14523965	C₂³	1	2	yes	—
165	#14523965	C₂³	1	4	yes	—
166	#14523965	C₂³	1	4	yes	—
167	#14523965	C₂³	1	2	yes	—
168	#14523965	C₂³	1	4	yes	—
169	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
170	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
171	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
172	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
173	#126×C₆₇	D₄	1	2	yes	—
174	#125×C₆₇	D₄	1	1	yes	characteristic
175	#124×C₆₇	D₄	1	1	yes	characteristic
176	#123×C₆₇	D₄	1	2	yes	—
177	C₂²⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
178	#121×C₆₇	D₄	1	1	yes	characteristic
179	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
180	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
181	C₄×D₄×C₆₇	C₂³	1	2	yes	—
182	#117×C₆₇	C₂³	1	2	yes	—
183	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
184	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
185	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
186	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
187	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
188	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
189	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
190	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
191	#97×C₆₇	C₂³	1	2	yes	—
192	#97×C₆₇	C₂³	1	2	yes	—
193	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
194	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
195	C₄×C₈×C₆₇	C₂³	1	2	yes	—
196	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
197	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
198	C₂³×D₆₇	—	2	4	no	—
199	#11761628	—	2	4	no	—
200	#11761628	—	4	2	no	—
201	#11761628	—	8	4	no	—
202	C₂×Dic₁₃₄	—	2	2	no	—
203	C₂×Dic₁₃₄	—	2	4	no	—
204	C₂×Dic₁₃₄	C₂⁴	1	2	yes	—
205	Q₈×D₆₇	—	8	4	no	—
206	C₂×D₂₆₈	—	2	8	no	—
207	C₂×D₂₆₈	—	2	4	no	—
208	C₂×D₂₆₈	—	2	2	no	—
209	C₂×D₂₆₈	—	2	2	no	—
210	C₂×D₂₆₈	—	4	4	no	—
211	C₂×D₂₆₈	C₂⁴	1	4	yes	—
212	C₂×D₂₆₈	C₂⁴	1	2	yes	—
213	C₂×C₂²⋊D₆₇	—	2	2	no	—
214	C₂×C₂²⋊D₆₇	—	2	4	no	—
215	C₂×C₂²⋊D₆₇	—	2	4	no	—
216	D₄×D₆₇	—	8	4	no	—
217	C₂×C₄×D₆₇	—	2	2	no	—
218	C₂×C₄×D₆₇	—	2	2	no	—
219	C₂×C₄×D₆₇	—	2	2	no	—
220	C₂×C₄×D₆₇	—	2	4	no	—
221	C₂×C₄×D₆₇	—	2	4	no	—
222	C₂×C₄×D₆₇	—	2	4	no	—
223	Dic₂₆₈	—	2	2	no	—
224	#11761617	—	2	4	no	—
225	#11761617	—	4	4	no	—
226	D₅₃₆	—	2	8	no	—
227	D₅₃₆	—	2	4	no	—
228	D₅₃₆	—	4	2	no	—
229	D₅₃₆	#41	1	4	yes	complemented
230	C₄⋊Dic₆₇	—	2	4	no	—
231	C₄⋊Dic₆₇	—	2	2	no	—
232	C₄⋊Dic₆₇	C₂⁴	1	2	yes	—
233	C₄⋊Dic₆₇	C₂⁴	1	4	yes	—
234	C₄⋊Dic₆₇	C₂⁴	1	2	yes	—
235	Dic₆₇⋊C₄	—	2	2	no	—
236	Dic₆₇⋊C₄	—	2	2	no	—
237	Dic₆₇⋊C₄	—	2	4	no	—
238	C₂²⋊Dic₆₇	—	2	4	no	—
239	C₂²⋊Dic₆₇	—	2	4	no	—
240	C₂²⋊Dic₆₇	—	2	2	no	—
241	(C₂×D₆₇)⋊C₄	—	2	2	no	—
242	(C₂×D₆₇)⋊C₄	—	2	4	no	—
243	(C₂×D₆₇)⋊C₄	—	2	4	no	—
244	(C₂×D₆₇)⋊C₄	—	2	4	no	—
245	(C₂×D₆₇)⋊C₄	—	2	4	no	—
246	(C₂×D₆₇)⋊C₄	—	2	4	no	—
247	(C₂×D₆₇)⋊C₄	—	2	2	no	—
248	(C₂×D₆₇)⋊C₄	—	2	4	no	—
249	(C₂×D₆₇)⋊C₄	—	2	4	no	—
250	C₄×Dic₆₇	—	2	2	no	—
251	C₄×Dic₆₇	—	2	2	no	—
252	C₄×Dic₆₇	—	2	2	no	—
253	C₄×Dic₆₇	—	2	4	no	—
254	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
255	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
256	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
257	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
258	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
259	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
260	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
261	M₁₆×C₆₇	—	4	2	no	—
262	C₂×C₈×C₆₇	—	2	4	no	—
263	C₂×C₈×C₆₇	—	2	2	no	—
264	C₂×C₈×C₆₇	—	4	1	no	—
265	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
266	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
267	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
268	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
269	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
270	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
271	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
272	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
273	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
274	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
275	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
276	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
277	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
278	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
279	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
280	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
281	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
282	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
283	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
284	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
285	C₄²×C₆₇	C₂⁴	1	2	yes	—
286	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
287	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
288	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
289	C₂²×D₆₇	—	2	2	no	—
290	C₂²×D₆₇	—	2	4	no	—
291	C₂²×D₆₇	—	4	8	no	—
292	C₂²×D₆₇	—	8	4	no	—
293	C₂²×D₆₇	—	8	4	no	—
294	Q₈×C₆₇	—	2	2	no	—
295	D₄×C₆₇	—	2	2	no	—
296	D₄×C₆₇	—	4	2	no	—
297	C₂×C₄×C₆₇	—	2	2	no	—
298	C₂×C₄×C₆₇	—	2	2	no	—
299	C₂×C₄×C₆₇	—	2	1	no	—
300	C₂×C₄×C₆₇	—	2	1	no	—
301	C₂×C₄×C₆₇	—	2	2	no	—
302	C₂×C₄×C₆₇	—	4	1	no	—
303	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
304	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
305	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
306	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
307	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
308	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
309	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
310	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
311	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
312	C₂²⋊D₆₇	—	4	4	no	—
313	C₂²⋊D₆₇	—	8	4	no	—
314	C₂²⋊D₆₇	—	8	4	no	—
315	C₂×Dic₆₇	—	2	2	no	—
316	C₂×Dic₆₇	—	2	2	no	—
317	C₂×Dic₆₇	—	2	2	no	—
318	C₂×Dic₆₇	—	2	4	no	—
319	D₂₆₈	—	2	4	no	—
320	D₂₆₈	—	2	2	no	—
321	D₂₆₈	—	2	4	no	—
322	D₂₆₈	—	4	8	no	—
323	D₂₆₈	—	4	4	no	—
324	D₂₆₈	—	8	2	no	—
325	D₂₆₈	C₂×#41	1	4	yes	—
326	C₄×D₆₇	—	4	4	no	—
327	C₄×D₆₇	—	4	4	no	—
328	C₄×D₆₇	—	8	2	no	—
329	C₄×D₆₇	—	8	4	no	—
330	C₄×D₆₇	—	8	4	no	—
331	C₄×D₆₇	—	8	4	no	—
332	Dic₁₃₄	—	2	2	no	—
333	Dic₁₃₄	—	2	2	no	—
334	Dic₁₃₄	—	4	4	no	—
335	Dic₁₃₄	—	8	4	no	—
336	C₈×C₆₇	—	2	2	no	—
337	C₈×C₆₇	—	2	1	no	—
338	C₈×C₆₇	—	2	1	no	—
339	C₈×C₆₇	—	2	1	no	—
340	C₈×C₆₇	—	2	2	no	—
341	C₈×C₆₇	C₂×#41	1	2	yes	—
342	C₂²×C₆₇	—	2	1	no	—
343	C₂²×C₆₇	—	2	2	no	—
344	C₂²×C₆₇	—	2	1	no	—
345	C₂²×C₆₇	—	2	2	no	—
346	C₂²×C₆₇	—	2	1	no	—
347	C₂²×C₆₇	—	2	2	no	—
348	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
349	C₂×D₆₇	—	2	4	no	—
350	C₂×D₆₇	—	4	2	no	—
351	C₂×D₆₇	—	4	2	no	—
352	C₂×D₆₇	—	4	4	no	—
353	C₂×D₆₇	—	4	4	no	—
354	C₂×D₆₇	—	4	4	no	—
355	C₂×D₆₇	—	16	8	no	—
356	C₄×C₆₇	—	2	1	no	—
357	C₄×C₆₇	—	2	2	no	—
358	C₄×C₆₇	—	2	1	no	—
359	C₄×C₆₇	—	2	1	no	—
360	C₄×C₆₇	—	2	2	no	—
361	C₄×C₆₇	—	2	2	no	—
362	C₄×C₆₇	—	2	1	no	—
363	C₄×C₆₇	#576	1	2	yes	—
364	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
365	C₄×C₆₇	#585	1	1	yes	characteristic
366	Dic₆₇	—	4	2	no	—
367	Dic₆₇	—	4	2	no	—
368	Dic₆₇	—	4	4	no	—
369	Dic₆₇	—	4	2	no	—
370	#61475	—	67	1	no	maximal, Sylow
371	C₂×C₆₇	—	4	2	no	—
372	C₂×C₆₇	—	4	1	no	—
373	C₂×C₆₇	D₄×#41	1	1	yes	characteristic
374	C₂×C₆₇	C₂×#576	1	1	yes	characteristic
375	C₂×C₆₇	#3584	1	1	yes	characteristic
376	D₆₇	—	4	4	no	—
377	D₆₇	—	8	4	no	—
378	#3514	—	67	2	no	—
379	#3396	—	67	1	no	—
380	Q₈×D₈	—	67	1	no	—
381	#3373	—	67	2	no	—
382	#3342	—	67	1	no	—
383	#3341	—	67	2	no	—
384	#3324	—	67	2	no	—
385	#3319	—	67	4	no	—
386	#3280	—	67	2	no	—
387	D₄×D₈	—	67	1	no	—
388	#3175	—	67	2	no	—
389	#3169	—	67	1	no	—
390	#3118	—	67	4	no	—
391	#3115	—	67	2	no	—
392	#2966	D₆₇	1	1	yes	characteristic, complemented
393	#2948	—	67	2	no	—
394	C₄×#135	—	67	1	no	—
395	C₆₇	#61475	1	1	yes	characteristic, complemented, Sylow
396	C₂×#135	—	67	1	no	—
397	C₂²×D₈	—	67	2	no	—
398	#559	—	67	2	no	—
399	#556	—	67	2	no	—
400	#555	—	67	4	no	—
401	#551	—	134	4	no	—
402	D₄×Q₈	—	67	2	no	—
403	#548	—	134	2	no	—
404	#546	—	67	4	no	—
405	#546	—	134	2	no	—
406	D₄²	—	67	2	no	—
407	#542	—	67	4	no	—
408	#538	—	134	4	no	—
409	#532	—	67	2	no	—
410	#529	—	134	2	no	—
411	#529	C₂×D₆₇	1	1	yes	characteristic
412	C₂×#124	—	134	2	no	—
413	#518	—	67	4	no	—
414	C₄×#41	—	67	2	no	—
415	#500	—	67	2	no	—
416	#500	—	67	1	no	—
417	#495	—	67	2	no	—
418	#493	—	67	1	no	—
419	#489	—	67	4	no	—
420	#486	—	67	2	no	—
421	#485	—	67	4	no	—
422	#482	—	67	4	no	—
423	#482	—	67	4	no	—
424	#476	—	67	4	no	—
425	#475	—	67	2	no	—
426	#474	—	67	2	no	—
427	#474	—	67	4	no	—
428	#468	—	67	4	no	—
429	#467	—	67	2	no	—
430	#466	—	67	2	no	—
431	#466	—	67	2	no	—
432	#466	—	67	2	no	—
433	#463	—	67	4	no	—
434	#463	—	67	4	no	—
435	#460	—	67	2	no	—
436	#459	—	67	2	no	—
437	#450	—	67	4	no	—
438	#449	—	67	4	no	—
439	#447	—	67	4	no	—
440	#446	C₂×D₆₇	1	1	yes	characteristic, complemented
441	C₈×Q₈	C₂×D₆₇	1	1	yes	characteristic, complemented
442	#442	—	67	4	no	—
443	#442	—	67	4	no	—
444	#440	—	67	4	no	—
445	C₄×Q₁₆	—	67	1	no	—
446	C₄×Q₁₆	—	67	1	no	—
447	C₄×SD₁₆	—	67	2	no	—
448	C₄×SD₁₆	—	67	2	no	—
449	C₄×D₈	—	67	2	no	—
450	C₄×D₈	—	67	2	no	—
451	C₄×D₈	—	67	1	no	—
452	C₄×D₈	—	67	2	no	—
453	C₄×D₈	—	67	2	no	—
454	C₄×D₈	—	67	1	no	—
455	#436	C₂×D₆₇	1	1	yes	characteristic
456	#435	C₂×D₆₇	1	2	yes	complemented
457	C₈×D₄	C₂×D₆₇	1	1	yes	characteristic
458	C₈×D₄	C₂×D₆₇	1	2	yes	complemented
459	#433	C₂×D₆₇	1	2	yes	complemented
460	#432	C₂×D₆₇	1	1	yes	characteristic, complemented
461	#427	—	67	1	no	—
462	#427	—	67	2	no	—
463	#418	—	67	4	no	—
464	#416	—	67	2	no	—
465	#405	C₂×D₆₇	1	2	yes	complemented
466	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic, complemented
467	C₂×#41	—	67	2	no	—
468	C₂×#41	—	134	2	no	—
469	C₂²×Q₈	—	134	2	no	—
470	C₂²×D₄	—	67	4	no	—
471	C₂²×D₄	—	134	2	no	—
472	#135	—	134	4	no	—
473	C₂×Q₁₆	—	67	1	no	—
474	C₂×SD₁₆	—	67	4	no	—
475	C₂×SD₁₆	—	67	2	no	—
476	C₂×D₈	—	67	2	no	—
477	C₂×D₈	—	67	1	no	—
478	C₂×D₈	—	67	2	no	—
479	C₂×D₈	—	67	4	no	—
480	C₂×D₈	—	134	8	no	—
481	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
482	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
483	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
484	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
485	C₄⋊Q₈	—	67	2	no	—
486	C₄⋊Q₈	—	67	4	no	—
487	#127	—	67	2	no	—
488	#126	—	134	4	no	—
489	#126	—	134	4	no	—
490	#126	D₂₆₈	1	2	yes	—
491	#125	D₂₆₈	1	1	yes	characteristic
492	#124	—	67	4	no	—
493	#124	—	134	2	no	—
494	#124	—	134	2	no	—
495	#124	—	134	4	no	—
496	#124	—	134	4	no	—
497	#124	—	134	4	no	—
498	#124	—	134	2	no	—
499	#124	D₂₆₈	1	1	yes	characteristic
500	#123	—	134	4	no	—
501	#123	—	134	4	no	—
502	#123	—	134	4	no	—
503	#123	D₂₆₈	1	2	yes	—
504	C₂²⋊Q₈	—	67	4	no	—
505	C₂²⋊Q₈	—	134	2	no	—
506	C₂²⋊Q₈	—	134	4	no	—
507	C₂²⋊Q₈	—	134	4	no	—
508	C₂²⋊Q₈	D₂₆₈	1	1	yes	characteristic
509	#121	—	67	4	no	—
510	#121	—	67	4	no	—
511	#121	—	134	4	no	—
512	#121	—	134	2	no	—
513	#121	—	134	2	no	—
514	#121	—	134	4	no	—
515	#121	D₂₆₈	1	1	yes	characteristic
516	#120	—	134	4	no	—
517	#120	—	134	4	no	—
518	C₄×Q₈	—	67	2	no	—
519	C₄×Q₈	—	67	2	no	—
520	C₄×Q₈	—	134	2	no	—
521	C₄×Q₈	C₂²×D₆₇	1	1	yes	characteristic
522	C₄×D₄	—	67	2	no	—
523	C₄×D₄	—	67	2	no	—
524	C₄×D₄	—	67	4	no	—
525	C₄×D₄	—	67	4	no	—
526	C₄×D₄	—	67	4	no	—
527	C₄×D₄	—	134	2	no	—
528	C₄×D₄	—	134	2	no	—
529	C₄×D₄	—	134	4	no	—
530	C₄×D₄	—	134	4	no	—
531	C₄×D₄	—	134	2	no	—
532	C₄×D₄	—	134	2	no	—
533	C₄×D₄	C₂²×D₆₇	1	1	yes	characteristic
534	C₄×D₄	C₂²×D₆₇	1	2	yes	—
535	#117	—	67	2	no	—
536	#117	—	67	4	no	—
537	#117	—	134	4	no	—
538	#117	—	134	2	no	—
539	#117	C₂²×D₆₇	1	2	yes	—
540	C₂×C₂²⋊C₄	—	134	4	no	—
541	C₂×C₂²⋊C₄	—	134	4	no	—
542	C₂×C₂²⋊C₄	—	134	4	no	—
543	C₂×C₄²	—	134	2	no	—
544	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
545	#107	—	67	1	no	—
546	#107	—	67	2	no	—
547	#107	—	67	1	no	—
548	#107	—	67	2	no	—
549	#107	—	67	4	no	—
550	#106	—	67	2	no	—
551	#106	—	67	4	no	—
552	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
553	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
554	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
555	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
556	#103	—	67	2	no	—
557	#103	—	67	2	no	—
558	#103	—	67	4	no	—
559	#102	—	67	2	no	—
560	#102	—	67	2	no	—
561	#102	—	67	4	no	—
562	#102	—	67	4	no	—
563	#102	—	67	4	no	—
564	#102	—	67	4	no	—
565	#102	—	67	4	no	—
566	C₂²⋊C₈	C₂²×D₆₇	1	2	yes	—
567	C₂²⋊C₈	C₂²×D₆₇	1	2	yes	—
568	C₂²⋊C₈	C₂²×D₆₇	1	2	yes	—
569	#97	C₂²×D₆₇	1	2	yes	—
570	#97	C₂²×D₆₇	1	2	yes	—
571	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
572	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
573	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
574	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
575	C₄×C₈	C₂²×D₆₇	1	2	yes	—
576	C₂⁴	—	134	4	no	—
577	#41	—	268	2	no	—
578	#41	—	536	4	no	—
579	#41	—	134	4	no	—
580	C₂×Q₈	—	536	4	no	—
581	C₂×Q₈	—	67	2	no	—
582	C₂×Q₈	—	134	2	no	—
583	C₂×Q₈	—	134	4	no	—
584	C₂×Q₈	C₂×D₂₆₈	1	1	yes	characteristic
585	C₂×D₄	—	268	4	no	—
586	C₂×D₄	—	536	4	no	—
587	C₂×D₄	—	67	2	no	—
588	C₂×D₄	—	67	4	no	—
589	C₂×D₄	—	134	8	no	—
590	C₂×D₄	—	134	2	no	—
591	C₂×D₄	—	134	4	no	—
592	C₂×D₄	—	134	4	no	—
593	C₂×D₄	—	134	4	no	—
594	C₂×D₄	—	134	2	no	—
595	C₂×D₄	—	134	2	no	—
596	C₂×D₄	C₂×D₂₆₈	1	1	yes	characteristic
597	C₂×D₄	C₂×D₂₆₈	1	2	yes	—
598	C₂²×C₄	—	134	2	no	—
599	C₂²×C₄	—	134	2	no	—
600	C₂²×C₄	—	134	2	no	—
601	C₂²×C₄	—	134	4	no	—
602	C₂²×C₄	—	134	4	no	—
603	C₂²×C₄	—	134	4	no	—
604	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
605	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
606	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
607	C₂²×C₄	C₂³×D₆₇	1	2	yes	—
608	Q₁₆	—	134	2	no	—
609	SD₁₆	—	268	4	no	—
610	SD₁₆	—	134	4	no	—
611	D₈	—	268	2	no	—
612	D₈	—	67	4	no	—
613	D₈	—	134	8	no	—
614	D₈	—	134	4	no	—
615	M₁₆	—	4	2	no	—
616	C₂×C₈	—	2	4	no	—
617	C₂×C₈	—	2	2	no	—
618	C₂×C₈	—	4	1	no	—
619	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
620	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
621	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
622	C₂×C₈	C₂³×D₆₇	1	2	yes	—
623	C₂×C₈	C₂³×D₆₇	1	2	yes	—
624	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
625	C₄⋊C₄	—	67	2	no	—
626	C₄⋊C₄	—	67	4	no	—
627	C₄⋊C₄	—	67	2	no	—
628	C₄⋊C₄	—	134	2	no	—
629	C₄⋊C₄	—	134	2	no	—
630	C₄⋊C₄	—	134	4	no	—
631	C₄⋊C₄	—	134	2	no	—
632	C₄⋊C₄	—	134	4	no	—
633	C₄⋊C₄	C₂×D₂₆₈	1	2	yes	—
634	C₄⋊C₄	C₂×D₂₆₈	1	2	yes	—
635	C₄⋊C₄	C₂×D₂₆₈	1	2	yes	—
636	C₄⋊C₄	C₂×D₂₆₈	1	1	yes	characteristic
637	C₄⋊C₄	C₂×D₂₆₈	1	1	yes	characteristic
638	C₄⋊C₄	C₂×D₂₆₈	1	2	yes	—
639	C₂²⋊C₄	—	134	4	no	—
640	C₂²⋊C₄	—	134	4	no	—
641	C₂²⋊C₄	—	134	2	no	—
642	C₂²⋊C₄	—	134	2	no	—
643	C₂²⋊C₄	—	134	4	no	—
644	C₂²⋊C₄	—	134	4	no	—
645	C₂²⋊C₄	—	134	4	no	—
646	C₂²⋊C₄	—	134	4	no	—
647	C₂²⋊C₄	—	134	4	no	—
648	C₂²⋊C₄	—	134	2	no	—
649	C₂²⋊C₄	—	134	4	no	—
650	C₂²⋊C₄	—	134	4	no	—
651	C₂²⋊C₄	C₂×D₂₆₈	1	2	yes	—
652	C₂²⋊C₄	C₂×D₂₆₈	1	2	yes	—
653	C₂²⋊C₄	C₂×D₂₆₈	1	2	yes	—
654	C₂²⋊C₄	C₂×D₂₆₈	1	2	yes	—
655	C₂²⋊C₄	C₂×D₂₆₈	1	2	yes	—
656	C₄²	—	134	2	no	—
657	C₄²	—	134	2	no	—
658	C₄²	—	134	2	no	—
659	C₄²	—	134	4	no	—
660	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
661	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
662	C₄²	C₂³×D₆₇	1	1	yes	characteristic
663	C₄²	C₂³×D₆₇	1	1	yes	characteristic
664	C₄²	C₂³×D₆₇	1	2	yes	—
665	C₂³	—	268	8	no	—
666	C₂³	—	536	4	no	—
667	C₂³	—	536	4	no	—
668	C₂³	—	134	4	no	—
669	C₂³	—	134	2	no	—
670	C₂³	C₂²×D₂₆₈	1	2	yes	—
671	C₂³	C₂²×D₂₆₈	1	1	yes	characteristic
672	Q₈	—	2	2	no	—
673	Q₈	—	268	4	no	—
674	Q₈	—	536	4	no	—
675	Q₈	—	134	2	no	—
676	Q₈	—	134	2	no	—
677	D₄	—	2	2	no	—
678	D₄	—	4	2	no	—
679	D₄	—	268	8	no	—
680	D₄	—	268	4	no	—
681	D₄	—	268	4	no	—
682	D₄	—	536	2	no	—
683	D₄	—	536	4	no	—
684	D₄	—	536	4	no	—
685	D₄	—	67	4	no	—
686	D₄	—	134	4	no	—
687	D₄	—	134	4	no	—
688	D₄	—	134	2	no	—
689	C₂×C₄	—	2	2	no	—
690	C₂×C₄	—	2	2	no	—
691	C₂×C₄	—	2	1	no	—
692	C₂×C₄	—	2	1	no	—
693	C₂×C₄	—	2	2	no	—
694	C₂×C₄	—	4	1	no	—
695	C₂×C₄	—	268	4	no	—
696	C₂×C₄	—	268	4	no	—
697	C₂×C₄	—	536	4	no	—
698	C₂×C₄	—	536	4	no	—
699	C₂×C₄	—	536	2	no	—
700	C₂×C₄	—	536	4	no	—
701	C₂×C₄	—	134	2	no	—
702	C₂×C₄	—	134	2	no	—
703	C₂×C₄	—	134	2	no	—
704	C₂×C₄	—	134	4	no	—
705	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
706	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
707	C₂×C₄	C₂²×D₂₆₈	1	2	yes	—
708	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
709	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
710	C₂×C₄	C₂²×D₂₆₈	1	2	yes	—
711	C₂×C₄	C₂²×D₂₆₈	1	2	yes	—
712	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
713	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
714	C₈	—	2	2	no	—
715	C₈	—	2	1	no	—
716	C₈	—	2	1	no	—
717	C₈	—	2	1	no	—
718	C₈	—	2	2	no	—
719	C₈	#41×D₆₇	1	2	yes	—
720	C₂²	—	2	1	no	—
721	C₂²	—	2	2	no	—
722	C₂²	—	2	1	no	—
723	C₂²	—	2	2	no	—
724	C₂²	—	2	1	no	—
725	C₂²	—	2	2	no	—
726	C₂²	—	268	2	no	—
727	C₂²	—	268	2	no	—
728	C₂²	—	268	4	no	—
729	C₂²	—	268	4	no	—
730	C₂²	—	268	4	no	—
731	C₂²	—	1072	8	no	—
732	C₂²	—	134	4	no	—
733	C₂²	C₂³×D₂₆₈	1	1	yes	characteristic
734	C₄	—	2	1	no	—
735	C₄	—	2	2	no	—
736	C₄	—	2	1	no	—
737	C₄	—	2	2	no	—
738	C₄	—	2	1	no	—
739	C₄	—	2	1	no	—
740	C₄	—	2	2	no	—
741	C₄	—	268	2	no	—
742	C₄	—	268	4	no	—
743	C₄	—	268	2	no	—
744	C₄	—	268	2	no	—
745	C₄	#16864201	1	2	yes	—
746	C₄	C₂×#41×D₆₇	1	1	yes	characteristic
747	C₄	#16864232	1	1	yes	characteristic
748	C₂	—	4	2	no	—
749	C₂	—	4	1	no	—
750	C₂	—	268	4	no	—
751	C₂	—	536	4	no	—
752	C₂	#41×D₂₆₈	1	1	yes	characteristic
753	C₂	C₂×#16864201	1	1	yes	characteristic
754	C₂	#22574983	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×D₂₆₈
commutator, cosocle	C₂×C₄×C₆₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₆₇×#2966	C₂
socle	C₂²×C₆₇	C₂³×D₄
hypercenter	#2966	D₆₇
hypocenter	C₆₇	#61475
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ #529 ◁ #2966

Complements

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

N	H ≅ G/N	type
1. #22574172	C₂	semidirect
2. #22573222	C₂	semidirect
3. Q₈×D₅₃₆	C₂	semidirect
4. #22573003	C₂	semidirect
5. #22572640	C₂	semidirect
6. #22572632	C₂	semidirect
7. #22572441	C₂	semidirect
8. #22572367	C₂	semidirect
9. #22571896	C₂	semidirect
10. D₄×D₅₃₆	C₂	semidirect
11. #22570848	C₂	semidirect
12. #22570788	C₂	semidirect
13. #22570395	C₂	semidirect
14. #22570361	C₂	semidirect
15. #22569062	C₂	semidirect
16. C₄×#14524065	C₂	semidirect
17. C₆₇×#2966	C₂	semidirect
20. #16864131	C₂²	semidirect
22. #16864111	C₂²	semidirect
24. Q₈×D₂₆₈	C₂²	semidirect
27. #16864042	C₂²	semidirect
29. #16864017	C₂²	semidirect
31. #16863946	C₂²	semidirect
34. #16863870	C₂²	semidirect
35. C₄×#11761628	C₂²	semidirect
36. #16863801	C₂²	semidirect
37. #16863801	C₂²	semidirect
38. #16863780	C₂²	semidirect
40. #16863756	C₂²	semidirect
41. #16863743	C₂²	semidirect
42. #16863738	C₂²	semidirect
43. #16863721	C₂²	semidirect
44. #16863721	C₂²	semidirect
45. #16863687	C₂²	semidirect
46. #16863682	C₂²	semidirect
47. #16863677	C₂²	semidirect
48. #16863677	C₂²	semidirect
49. #16863642	C₂²	semidirect
50. #16863637	C₂²	semidirect
52. #16863632	C₂²	semidirect
53. #16863632	C₂²	semidirect
54. #16863612	C₂²	semidirect
55. #16863612	C₂²	semidirect
56. #16863597	C₂²	semidirect
58. #16863549	C₂²	semidirect
59. #16863543	C₂²	semidirect
61. #16863509	C₂²	semidirect
62. #16863509	C₂²	semidirect
63. #16863498	C₂²	semidirect
64. C₄×Dic₂₆₈	C₂²	semidirect
65. C₄×Dic₂₆₈	C₂²	semidirect
66. C₄×#11761617	C₂²	semidirect
67. C₄×#11761617	C₂²	semidirect
71. C₄×D₅₃₆	C₂²	semidirect
72. C₄×D₅₃₆	C₂²	semidirect
73. C₄×D₅₃₆	C₂²	semidirect
74. #16863434	C₂²	semidirect
75. #16863434	C₂²	semidirect
76. #16863402	C₂²	semidirect
77. #16863393	C₂²	semidirect
79. #446×C₆₇	C₂²	semidirect
80. C₈×Q₈×C₆₇	C₂²	semidirect
82. #435×C₆₇	C₂²	semidirect
84. C₈×D₄×C₆₇	C₂²	semidirect
85. #433×C₆₇	C₂²	semidirect
86. #432×C₆₇	C₂²	semidirect
87. #405×C₆₇	C₂²	semidirect
88. C₂×C₄×C₈×C₆₇	C₂²	semidirect
229. D₅₃₆	#41	semidirect
392. #2966	D₆₇	semidirect
395. C₆₇	#61475	semidirect
440. #446	C₂×D₆₇	semidirect
441. C₈×Q₈	C₂×D₆₇	semidirect
456. #435	C₂×D₆₇	semidirect
458. C₈×D₄	C₂×D₆₇	semidirect
459. #433	C₂×D₆₇	semidirect
460. #432	C₂×D₆₇	semidirect
465. #405	C₂×D₆₇	semidirect
466. C₂×C₄×C₈	C₂×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22574172	2
#22573222	2
Q₈×D₅₃₆	2
#22573003	2
#22572640	2
#22572632	2
#22572441	2
#22572367	2
#22571896	2
D₄×D₅₃₆	2
#22570848	2
#22570788	2
#22570395	2
#22570361	2
#22569062	2
C₄×#14524065	2
C₆₇×#2966	2
#61475	67