Subgroups — #37582886

Order 24448.

372 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27133293	C₂	1	1	yes	maximal, characteristic, complemented
2	#27133238	C₂	1	1	yes	maximal, characteristic
3	#27133030	C₂	1	1	yes	maximal, characteristic, complemented
4	#27133024	C₂	1	1	yes	maximal, characteristic, complemented
5	#27133023	C₂	1	1	yes	maximal, characteristic, complemented
6	#27133020	C₂	1	1	yes	maximal, characteristic, complemented
7	#27133019	C₂	1	1	yes	maximal, characteristic, complemented
8	#27133013	C₂	1	1	yes	maximal, characteristic, complemented
9	#27133012	C₂	1	1	yes	maximal, characteristic, complemented
10	#27132670	C₂	1	1	yes	maximal, characteristic, complemented
11	#27132663	C₂	1	1	yes	maximal, characteristic
12	#27132658	C₂	1	1	yes	maximal, characteristic
13	#103×D₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
14	#27132620	C₂	1	1	yes	maximal, characteristic, complemented
15	#491×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#15690288	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₁₉₁	C₂²	1	1	yes	characteristic
18	C₂×#15690270	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₉₁	C₂²	1	1	yes	characteristic
20	#20259322	C₂²	1	1	yes	characteristic
21	#20259321	—	2	1	no	—
22	#20259321	C₂²	1	1	yes	characteristic
23	#20259320	C₂²	1	1	yes	characteristic
24	#20259313	C₂²	1	1	yes	characteristic
25	#20259312	—	2	1	no	—
26	#20259311	C₂²	1	1	yes	characteristic
27	#20259310	C₂²	1	1	yes	characteristic
28	#20259307	—	2	1	no	—
29	#20259307	—	2	1	no	—
30	#20259302	—	2	1	no	—
31	#20259300	—	2	1	no	—
32	#20259300	—	2	1	no	—
33	#20259300	—	2	1	no	—
34	#20259280	C₂²	1	1	yes	characteristic
35	#20259279	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₁₉₁	C₂²	1	1	yes	characteristic
37	#20259247	C₂²	1	1	yes	characteristic
38	#20259247	C₂²	1	1	yes	characteristic
39	#20259246	C₂²	1	1	yes	characteristic
40	#20259246	C₂²	1	1	yes	characteristic
41	#20259246	C₂²	1	1	yes	characteristic
42	#20259246	C₂²	1	1	yes	characteristic
43	#20259245	C₂²	1	1	yes	characteristic
44	#20259245	C₂²	1	1	yes	characteristic
45	#20259244	C₂²	1	1	yes	characteristic
46	#20259244	C₂²	1	1	yes	characteristic
47	#20259242	C₂²	1	1	yes	characteristic
48	#20259242	C₂²	1	1	yes	characteristic
49	#20259232	C₂²	1	1	yes	characteristic
50	C₄×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
53	#125×C₁₉₁	C₂²	1	1	yes	characteristic
54	#103×C₁₉₁	C₂²	1	1	yes	characteristic
55	#103×C₁₉₁	C₂²	1	1	yes	characteristic
56	#103×C₁₉₁	C₂²	1	1	yes	characteristic
57	#103×C₁₉₁	C₂²	1	1	yes	characteristic
58	#97×C₁₉₁	C₂²	1	1	yes	characteristic
59	#15690288	—	2	2	no	—
60	#15690288	—	2	2	no	—
61	C₂×Dic₃₈₂	—	2	1	no	—
62	C₂×Dic₃₈₂	—	2	1	no	—
63	C₂×Dic₃₈₂	—	2	1	no	—
64	C₂×Dic₃₈₂	C₂³	1	1	yes	characteristic
65	Q₈×D₁₉₁	—	2	2	no	—
66	Q₈×D₁₉₁	—	2	2	no	—
67	C₂×D₇₆₄	—	2	1	no	—
68	C₂×D₇₆₄	C₂³	1	1	yes	characteristic
69	C₂×C₄×D₁₉₁	—	2	1	no	—
70	C₂×C₄×D₁₉₁	—	2	1	no	—
71	C₂×C₄×D₁₉₁	—	2	1	no	—
72	C₂×C₄×D₁₉₁	C₂³	1	1	yes	characteristic
73	#15690270	—	2	2	no	—
74	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
75	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
76	C₈×D₁₉₁	—	2	2	no	—
77	C₄⋊Dic₁₉₁	—	2	1	no	—
78	C₄⋊Dic₁₉₁	—	2	1	no	—
79	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
81	Dic₁₉₁⋊C₄	—	2	1	no	—
82	Dic₁₉₁⋊C₄	—	2	1	no	—
83	Dic₁₉₁⋊C₄	—	2	1	no	—
84	Dic₁₉₁⋊C₄	—	2	1	no	—
85	Dic₁₉₁⋊C₄	—	2	1	no	—
86	Dic₁₉₁⋊C₄	D₄	1	1	yes	characteristic
87	Dic₁₉₁⋊C₄	D₄	1	1	yes	characteristic
88	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
89	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
90	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
91	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
92	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
93	(C₂×D₁₉₁)⋊C₄	D₄	1	1	yes	characteristic
94	(C₂×D₁₉₁)⋊C₄	D₄	1	1	yes	characteristic
95	C₄×Dic₁₉₁	—	2	1	no	—
96	C₄×Dic₁₉₁	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₁₉₁	C₂³	1	1	yes	characteristic
98	C₂×Q₈×C₁₉₁	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₉₁	—	2	1	no	—
102	C₄⋊C₄×C₁₉₁	—	2	1	no	—
103	C₄⋊C₄×C₁₉₁	—	2	1	no	—
104	C₄⋊C₄×C₁₉₁	—	2	1	no	—
105	C₄⋊C₄×C₁₉₁	C₂³	1	1	yes	characteristic
106	C₄⋊C₄×C₁₉₁	C₂³	1	1	yes	characteristic
107	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
108	C₂²×D₁₉₁	—	2	1	no	—
109	C₂²×D₁₉₁	C₂×D₄	1	1	yes	characteristic
110	Dic₃₈₂	—	2	1	no	—
111	Dic₃₈₂	—	2	1	no	—
112	Dic₃₈₂	—	4	2	no	—
113	Dic₃₈₂	—	8	1	no	—
114	Dic₃₈₂	—	8	1	no	—
115	D₇₆₄	—	2	1	no	—
116	D₇₆₄	—	2	1	no	—
117	D₇₆₄	—	4	2	no	—
118	C₂×Dic₁₉₁	—	2	1	no	—
119	C₂×Dic₁₉₁	—	2	1	no	—
120	C₂×Dic₁₉₁	—	2	1	no	—
121	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
124	C₄×D₁₉₁	—	4	2	no	—
125	C₄×D₁₉₁	—	4	2	no	—
126	C₄×D₁₉₁	—	8	1	no	—
127	C₄×D₁₉₁	#41	1	2	yes	—
128	C₄×D₁₉₁	#41	1	2	yes	—
129	C₁₉₁⋊C₈	—	2	1	no	—
130	C₁₉₁⋊C₈	—	2	1	no	—
131	Q₈×C₁₉₁	—	2	1	no	—
132	Q₈×C₁₉₁	—	2	1	no	—
133	Q₈×C₁₉₁	—	2	1	no	—
134	Q₈×C₁₉₁	—	2	1	no	—
135	C₂×C₄×C₁₉₁	—	2	1	no	—
136	C₂×C₄×C₁₉₁	—	2	1	no	—
137	C₂×C₄×C₁₉₁	—	2	1	no	—
138	C₂×C₄×C₁₉₁	—	2	1	no	—
139	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
142	C₈×C₁₉₁	—	2	1	no	—
143	C₈×C₁₉₁	—	2	1	no	—
144	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
145	C₂×D₁₉₁	—	2	1	no	—
146	C₂×D₁₉₁	—	2	1	no	—
147	C₂×D₁₉₁	—	4	1	no	—
148	C₂×D₁₉₁	—	4	1	no	—
149	C₂×D₁₉₁	—	4	1	no	—
150	C₂×D₁₉₁	#124	1	2	yes	—
151	C₄×C₁₉₁	—	2	1	no	—
152	C₄×C₁₉₁	—	2	1	no	—
153	C₄×C₁₉₁	—	4	1	no	—
154	C₄×C₁₉₁	—	4	1	no	—
155	C₄×C₁₉₁	—	4	1	no	—
156	C₄×C₁₉₁	—	4	1	no	—
157	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
158	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
159	Dic₁₉₁	—	2	1	no	—
160	Dic₁₉₁	—	2	1	no	—
161	Dic₁₉₁	—	4	1	no	—
162	Dic₁₉₁	—	4	1	no	—
163	Dic₁₉₁	—	4	1	no	—
164	Dic₁₉₁	#124	1	2	yes	—
165	C₂×C₁₉₁	C₂×#124	1	1	yes	characteristic
166	C₂×C₁₉₁	#577	1	1	yes	characteristic
167	C₂×C₁₉₁	#578	1	1	yes	characteristic
168	D₁₉₁	—	2	2	no	—
169	D₁₉₁	—	8	1	no	—
170	C₁₉₁	#3137	1	1	yes	characteristic, complemented, Sylow
171	#3137	—	191	1	no	maximal, Sylow
172	#544	—	191	1	no	—
173	#536	—	191	1	no	—
174	#491	—	191	1	no	—
175	#491	D₁₉₁	1	1	yes	characteristic, complemented
176	#489	—	191	1	no	—
177	#489	—	191	1	no	—
178	#488	—	191	1	no	—
179	#488	—	191	1	no	—
180	#487	—	191	1	no	—
181	#487	—	191	1	no	—
182	#419	—	191	1	no	—
183	#417	—	191	1	no	—
184	#416	—	191	1	no	—
185	C₂×#103	—	191	1	no	—
186	#405	—	191	1	no	—
187	C₂×#41	—	191	1	no	—
188	C₂²×Q₈	—	191	1	no	—
189	C₂×M₁₆	—	191	1	no	—
190	C₂²×C₈	—	191	1	no	—
191	C₄⋊Q₈	—	382	1	no	—
192	C₄⋊Q₈	—	191	1	no	—
193	C₄⋊Q₈	—	191	1	no	—
194	C₄⋊Q₈	—	191	1	no	—
195	C₄⋊Q₈	C₂×D₁₉₁	1	1	yes	characteristic
196	#125	—	382	1	no	—
197	#125	—	191	1	no	—
198	#125	C₂×D₁₉₁	1	1	yes	characteristic
199	#124	—	191	1	no	—
200	#124	—	191	1	no	—
201	#123	—	382	1	no	—
202	#123	—	382	1	no	—
203	C₂²⋊Q₈	—	382	1	no	—
204	C₂²⋊Q₈	—	382	1	no	—
205	C₂²⋊Q₈	—	382	1	no	—
206	C₂²⋊Q₈	—	382	1	no	—
207	#117	—	191	1	no	—
208	#117	—	191	1	no	—
209	C₂×C₄⋊C₄	—	191	1	no	—
210	#103	—	191	1	no	—
211	#103	—	191	1	no	—
212	#103	—	191	1	no	—
213	#103	—	191	1	no	—
214	#103	—	191	1	no	—
215	#103	—	191	1	no	—
216	#103	—	191	1	no	—
217	#103	—	191	1	no	—
218	#103	C₂×D₁₉₁	1	1	yes	characteristic
219	#103	C₂×D₁₉₁	1	1	yes	characteristic
220	#103	C₂×D₁₉₁	1	1	yes	characteristic
221	#103	C₂×D₁₉₁	1	1	yes	characteristic
222	#102	—	191	1	no	—
223	#102	—	191	1	no	—
224	#102	—	191	1	no	—
225	#102	—	191	1	no	—
226	#97	—	191	1	no	—
227	#97	C₂×D₁₉₁	1	1	yes	characteristic
228	C₄×C₈	—	191	1	no	—
229	C₄×C₈	—	191	1	no	—
230	#41	—	382	2	no	—
231	#41	—	382	2	no	—
232	C₂×Q₈	—	382	1	no	—
233	C₂×Q₈	—	382	1	no	—
234	C₂×Q₈	—	382	1	no	—
235	C₂×Q₈	—	382	2	no	—
236	C₂×Q₈	—	382	2	no	—
237	C₂×Q₈	—	191	1	no	—
238	C₂×Q₈	C₂²×D₁₉₁	1	1	yes	characteristic
239	C₂×Q₈	C₂²×D₁₉₁	1	1	yes	characteristic
240	C₂×D₄	—	382	1	no	—
241	C₂×D₄	—	191	1	no	—
242	C₂²×C₄	—	382	1	no	—
243	C₂²×C₄	—	382	1	no	—
244	C₂²×C₄	—	382	1	no	—
245	C₂²×C₄	—	191	1	no	—
246	M₁₆	—	382	2	no	—
247	C₂×C₈	—	382	2	no	—
248	C₂×C₈	—	191	1	no	—
249	C₂×C₈	—	191	1	no	—
250	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
251	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	382	1	no	—
257	C₄⋊C₄	—	382	1	no	—
258	C₄⋊C₄	—	382	1	no	—
259	C₄⋊C₄	—	382	1	no	—
260	C₄⋊C₄	—	382	1	no	—
261	C₄⋊C₄	—	382	1	no	—
262	C₄⋊C₄	—	382	1	no	—
263	C₄⋊C₄	—	191	1	no	—
264	C₄⋊C₄	—	191	1	no	—
265	C₄⋊C₄	—	191	1	no	—
266	C₄⋊C₄	—	191	1	no	—
267	C₄⋊C₄	C₂²×D₁₉₁	1	1	yes	characteristic
268	C₄⋊C₄	C₂²×D₁₉₁	1	1	yes	characteristic
269	C₂²⋊C₄	—	382	1	no	—
270	C₂²⋊C₄	—	382	1	no	—
271	C₂²⋊C₄	—	382	1	no	—
272	C₂²⋊C₄	—	382	1	no	—
273	C₂²⋊C₄	—	382	1	no	—
274	C₂²⋊C₄	—	191	1	no	—
275	C₂²⋊C₄	—	191	1	no	—
276	C₄²	—	382	1	no	—
277	C₄²	—	191	1	no	—
278	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
279	C₂³	—	382	1	no	—
280	C₂³	—	191	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	2	1	no	—
284	Q₈	—	2	1	no	—
285	Q₈	—	382	1	no	—
286	Q₈	—	382	1	no	—
287	Q₈	—	1528	1	no	—
288	Q₈	—	1528	1	no	—
289	Q₈	—	764	2	no	—
290	D₄	—	382	1	no	—
291	D₄	—	382	1	no	—
292	D₄	—	764	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	382	1	no	—
298	C₂×C₄	—	382	1	no	—
299	C₂×C₄	—	382	1	no	—
300	C₂×C₄	—	191	1	no	—
301	C₂×C₄	—	191	1	no	—
302	C₂×C₄	—	191	1	no	—
303	C₂×C₄	—	191	2	no	—
304	C₂×C₄	—	191	2	no	—
305	C₂×C₄	—	1528	1	no	—
306	C₂×C₄	—	764	2	no	—
307	C₂×C₄	—	764	2	no	—
308	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
309	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	382	1	no	—
314	C₈	—	382	1	no	—
315	C₂²	—	382	1	no	—
316	C₂²	—	382	1	no	—
317	C₂²	—	191	2	no	—
318	C₂²	—	764	1	no	—
319	C₂²	—	764	1	no	—
320	C₂²	—	764	1	no	—
321	C₂²	C₂×D₄×D₁₉₁	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	382	1	no	—
329	C₄	—	382	1	no	—
330	C₄	—	191	2	no	—
331	C₄	—	764	1	no	—
332	C₄	—	764	1	no	—
333	C₄	—	764	1	no	—
334	C₄	#41×D₁₉₁	1	1	yes	characteristic
335	C₄	#41×D₁₉₁	1	1	yes	characteristic
336	C₂	—	382	2	no	—
337	C₂	—	1528	1	no	—
338	C₂	#124×D₁₉₁	1	1	yes	characteristic
339	C₂	#27133472	1	1	yes	characteristic
340	C₂	#27133479	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	#491×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂²×D₄
prime radical	C₂×#15690288	C₂²
hypercenter	#491	D₁₉₁
hypocenter	C₁₉₁	#3137
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #491

Complements

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

N	H ≅ G/N	type
1. #27133293	C₂	semidirect
3. #27133030	C₂	semidirect
4. #27133024	C₂	semidirect
5. #27133023	C₂	semidirect
6. #27133020	C₂	semidirect
7. #27133019	C₂	semidirect
8. #27133013	C₂	semidirect
9. #27133012	C₂	semidirect
10. #27132670	C₂	semidirect
13. #103×D₁₉₁	C₂	semidirect
14. #27132620	C₂	semidirect
15. #491×C₁₉₁	C₂	semidirect
170. C₁₉₁	#3137	semidirect
175. #491	D₁₉₁	semidirect

Maximal subgroups

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

subgroup	index
#27133293	2
#27133238	2
#27133030	2
#27133024	2
#27133023	2
#27133020	2
#27133019	2
#27133013	2
#27133012	2
#27132670	2
#27132663	2
#27132658	2
#103×D₁₉₁	2
#27132620	2
#491×C₁₉₁	2
#3137	191