Subgroups — #37583047

Order 24448.

340 conjugacy classes of subgroups.

The subgroup lattice has 310 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
2	#27133290	C₂	1	1	yes	maximal, characteristic
3	#27133072	C₂	1	1	yes	maximal, characteristic, complemented
4	#27133071	C₂	1	1	yes	maximal, characteristic, complemented
5	#27133070	C₂	1	1	yes	maximal, characteristic, complemented
6	#27133064	C₂	1	1	yes	maximal, characteristic, complemented
7	#27133062	C₂	1	1	yes	maximal, characteristic, complemented
8	#27133060	C₂	1	1	yes	maximal, characteristic, complemented
9	#27133059	C₂	1	1	yes	maximal, characteristic, complemented
10	#27132704	C₂	1	1	yes	maximal, characteristic
11	#27132702	C₂	1	1	yes	maximal, characteristic
12	#27132700	C₂	1	1	yes	maximal, characteristic
13	#106×D₁₉₁	C₂	1	1	yes	maximal, characteristic
14	#27132620	C₂	1	1	yes	maximal, characteristic
15	#501×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#15690270	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₁₉₁	C₂²	1	1	yes	characteristic
18	#20259322	C₂²	1	1	yes	characteristic
19	#20259321	—	2	1	no	—
20	#20259321	C₂²	1	1	yes	characteristic
21	#20259321	C₂²	1	1	yes	characteristic
22	#20259320	—	2	1	no	—
23	#20259314	C₂²	1	1	yes	characteristic
24	#20259313	C₂²	1	1	yes	characteristic
25	#20259313	C₂²	1	1	yes	characteristic
26	#20259312	—	2	1	no	—
27	#20259312	—	2	1	no	—
28	#20259302	—	2	1	no	—
29	#20259300	—	2	1	no	—
30	#20259300	—	2	1	no	—
31	#20259298	—	2	1	no	—
32	#20259280	C₂²	1	1	yes	characteristic
33	#20259280	C₂²	1	1	yes	characteristic
34	#20259279	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₁₉₁	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₁₉₁	C₂²	1	1	yes	characteristic
37	#20259256	C₂²	1	1	yes	characteristic
38	#20259256	C₂²	1	1	yes	characteristic
39	#20259255	C₂²	1	1	yes	characteristic
40	#20259255	C₂²	1	1	yes	characteristic
41	#20259254	C₂²	1	1	yes	characteristic
42	#20259254	C₂²	1	1	yes	characteristic
43	#20259253	C₂²	1	1	yes	characteristic
44	#20259253	C₂²	1	1	yes	characteristic
45	#20259253	C₂²	1	1	yes	characteristic
46	#20259253	C₂²	1	1	yes	characteristic
47	#20259253	C₂²	1	1	yes	characteristic
48	#20259253	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	#107×C₁₉₁	C₂²	1	1	yes	characteristic
55	#107×C₁₉₁	C₂²	1	1	yes	characteristic
56	#106×C₁₉₁	C₂²	1	1	yes	characteristic
57	#106×C₁₉₁	C₂²	1	1	yes	characteristic
58	#97×C₁₉₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₃₈₂	—	2	1	no	—
60	C₂×Dic₃₈₂	—	2	1	no	—
61	C₂×Dic₃₈₂	—	2	1	no	—
62	C₂×C₄×D₁₉₁	—	2	1	no	—
63	C₂×C₄×D₁₉₁	—	2	1	no	—
64	C₂×C₄×D₁₉₁	C₂³	1	1	yes	characteristic
65	#15690270	Q₈	1	4	yes	complemented
66	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
67	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
68	C₈×D₁₉₁	Q₈	1	4	yes	complemented
69	C₄⋊Dic₁₉₁	—	2	1	no	—
70	C₄⋊Dic₁₉₁	—	2	1	no	—
71	C₄⋊Dic₁₉₁	—	2	1	no	—
72	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
76	Dic₁₉₁⋊C₄	—	2	1	no	—
77	Dic₁₉₁⋊C₄	—	2	1	no	—
78	Dic₁₉₁⋊C₄	—	2	1	no	—
79	Dic₁₉₁⋊C₄	—	2	1	no	—
80	Dic₁₉₁⋊C₄	—	2	1	no	—
81	Dic₁₉₁⋊C₄	—	2	1	no	—
82	Dic₁₉₁⋊C₄	—	2	1	no	—
83	Dic₁₉₁⋊C₄	—	2	1	no	—
84	Dic₁₉₁⋊C₄	D₄	1	1	yes	characteristic
85	Dic₁₉₁⋊C₄	D₄	1	1	yes	characteristic
86	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
87	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
88	(C₂×D₁₉₁)⋊C₄	D₄	1	1	yes	characteristic
89	(C₂×D₁₉₁)⋊C₄	D₄	1	1	yes	characteristic
90	C₄×Dic₁₉₁	—	2	1	no	—
91	C₄×Dic₁₉₁	—	2	1	no	—
92	C₄×Dic₁₉₁	C₂³	1	1	yes	characteristic
93	C₂×Q₈×C₁₉₁	—	2	1	no	—
94	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
95	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
96	C₄⋊C₄×C₁₉₁	—	2	1	no	—
97	C₄⋊C₄×C₁₉₁	—	2	1	no	—
98	C₄⋊C₄×C₁₉₁	—	2	1	no	—
99	C₄⋊C₄×C₁₉₁	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₁₉₁	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₉₁	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₉₁	C₂³	1	1	yes	characteristic
103	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
104	C₂²×D₁₉₁	C₂×D₄	1	1	yes	characteristic
105	Dic₃₈₂	—	8	1	no	—
106	Dic₃₈₂	—	8	1	no	—
107	Dic₃₈₂	—	8	1	no	—
108	C₂×Dic₁₉₁	—	2	1	no	—
109	C₂×Dic₁₉₁	—	2	1	no	—
110	C₂×Dic₁₉₁	—	2	1	no	—
111	C₂×Dic₁₉₁	—	2	1	no	—
112	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
113	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
114	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
115	C₄×D₁₉₁	—	8	1	no	—
116	C₄×D₁₉₁	—	8	1	no	—
117	C₄×D₁₉₁	C₂×Q₈	1	2	yes	—
118	C₄×D₁₉₁	C₂×Q₈	1	2	yes	—
119	C₁₉₁⋊C₈	C₂×Q₈	1	2	yes	—
120	C₁₉₁⋊C₈	C₂×Q₈	1	2	yes	—
121	Q₈×C₁₉₁	—	8	1	no	—
122	C₂×C₄×C₁₉₁	—	2	1	no	—
123	C₂×C₄×C₁₉₁	—	2	1	no	—
124	C₂×C₄×C₁₉₁	—	2	1	no	—
125	C₂×C₄×C₁₉₁	—	2	1	no	—
126	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
129	C₈×C₁₉₁	C₂×Q₈	1	2	yes	—
130	C₈×C₁₉₁	C₂×Q₈	1	2	yes	—
131	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
132	C₂×D₁₉₁	—	2	1	no	—
133	C₂×D₁₉₁	—	2	1	no	—
134	C₂×D₁₉₁	C₄⋊Q₈	1	2	yes	—
135	C₄×C₁₉₁	—	2	1	no	—
136	C₄×C₁₉₁	—	2	1	no	—
137	C₄×C₁₉₁	—	4	1	no	—
138	C₄×C₁₉₁	—	4	1	no	—
139	C₄×C₁₉₁	—	4	1	no	—
140	C₄×C₁₉₁	—	4	1	no	—
141	C₄×C₁₉₁	C₂²×Q₈	1	1	yes	characteristic
142	C₄×C₁₉₁	C₂²×Q₈	1	1	yes	characteristic
143	Dic₁₉₁	—	2	1	no	—
144	Dic₁₉₁	—	2	1	no	—
145	Dic₁₉₁	—	4	1	no	—
146	Dic₁₉₁	—	4	1	no	—
147	Dic₁₉₁	—	4	1	no	—
148	Dic₁₉₁	—	4	1	no	—
149	Dic₁₉₁	C₄⋊Q₈	1	2	yes	—
150	C₂×C₁₉₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
151	C₂×C₁₉₁	#577	1	1	yes	characteristic
152	C₂×C₁₉₁	#577	1	1	yes	characteristic
153	D₁₉₁	—	2	2	no	—
154	C₁₉₁	#3163	1	1	yes	characteristic, complemented, Sylow
155	#3163	—	191	1	no	maximal, Sylow
156	#544	—	191	1	no	—
157	#544	—	191	1	no	—
158	#501	—	191	1	no	—
159	#501	—	191	1	no	—
160	#501	—	191	1	no	—
161	#501	D₁₉₁	1	1	yes	characteristic, complemented
162	#499	—	191	1	no	—
163	#499	—	191	1	no	—
164	#498	—	191	1	no	—
165	#498	—	191	1	no	—
166	#428	—	191	1	no	—
167	#428	—	191	1	no	—
168	#427	—	191	1	no	—
169	C₂×#106	—	191	1	no	—
170	#405	—	191	1	no	—
171	C₂×M₁₆	—	191	1	no	—
172	C₂²×C₈	—	191	1	no	—
173	C₄⋊Q₈	—	382	1	no	—
174	C₄⋊Q₈	—	382	1	no	—
175	C₄⋊Q₈	—	191	1	no	—
176	C₄⋊Q₈	—	191	1	no	—
177	C₄⋊Q₈	—	191	1	no	—
178	C₄⋊Q₈	C₂×D₁₉₁	1	1	yes	characteristic
179	#125	—	382	1	no	—
180	#125	—	382	1	no	—
181	#125	—	191	1	no	—
182	#125	—	191	1	no	—
183	#125	—	191	1	no	—
184	#125	C₂×D₁₉₁	1	1	yes	characteristic
185	C₂²⋊Q₈	—	382	1	no	—
186	C₂²⋊Q₈	—	382	1	no	—
187	C₂²⋊Q₈	—	382	1	no	—
188	C₂²⋊Q₈	—	382	1	no	—
189	#117	—	191	1	no	—
190	#117	—	191	1	no	—
191	#117	—	191	1	no	—
192	C₂×C₄⋊C₄	—	191	1	no	—
193	C₂×C₄⋊C₄	—	191	1	no	—
194	#107	—	191	1	no	—
195	#107	—	191	1	no	—
196	#107	—	191	1	no	—
197	#107	—	191	1	no	—
198	#107	C₂×D₁₉₁	1	1	yes	characteristic
199	#107	C₂×D₁₉₁	1	1	yes	characteristic
200	#106	—	191	1	no	—
201	#106	—	191	1	no	—
202	#106	—	191	1	no	—
203	#106	—	191	1	no	—
204	#106	—	191	1	no	—
205	#106	—	191	1	no	—
206	#106	—	191	1	no	—
207	#106	—	191	1	no	—
208	#106	C₂×D₁₉₁	1	1	yes	characteristic
209	#106	C₂×D₁₉₁	1	1	yes	characteristic
210	#97	—	191	1	no	—
211	#97	C₂×D₁₉₁	1	1	yes	characteristic
212	C₄×C₈	—	191	1	no	—
213	C₄×C₈	—	191	1	no	—
214	C₂×Q₈	—	2	1	no	—
215	C₂×Q₈	—	382	1	no	—
216	C₂×Q₈	—	382	1	no	—
217	C₂×Q₈	—	382	1	no	—
218	C₂²×C₄	—	382	1	no	—
219	C₂²×C₄	—	382	1	no	—
220	C₂²×C₄	—	191	1	no	—
221	M₁₆	—	191	4	no	—
222	C₂×C₈	—	191	1	no	—
223	C₂×C₈	—	191	1	no	—
224	C₂×C₈	—	191	4	no	—
225	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
226	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
227	C₄⋊C₄	—	2	1	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	382	1	no	—
231	C₄⋊C₄	—	382	1	no	—
232	C₄⋊C₄	—	382	1	no	—
233	C₄⋊C₄	—	382	1	no	—
234	C₄⋊C₄	—	382	1	no	—
235	C₄⋊C₄	—	382	1	no	—
236	C₄⋊C₄	—	382	1	no	—
237	C₄⋊C₄	—	382	1	no	—
238	C₄⋊C₄	—	382	1	no	—
239	C₄⋊C₄	—	382	1	no	—
240	C₄⋊C₄	—	382	1	no	—
241	C₄⋊C₄	—	191	1	no	—
242	C₄⋊C₄	—	191	1	no	—
243	C₄⋊C₄	—	191	1	no	—
244	C₄⋊C₄	—	191	1	no	—
245	C₄⋊C₄	—	191	1	no	—
246	C₄⋊C₄	—	191	1	no	—
247	C₄⋊C₄	C₂²×D₁₉₁	1	1	yes	characteristic
248	C₄⋊C₄	C₂²×D₁₉₁	1	1	yes	characteristic
249	C₄⋊C₄	C₂²×D₁₉₁	1	1	yes	characteristic
250	C₄⋊C₄	C₂²×D₁₉₁	1	1	yes	characteristic
251	C₂²⋊C₄	—	382	1	no	—
252	C₂²⋊C₄	—	382	1	no	—
253	C₂²⋊C₄	—	191	1	no	—
254	C₂²⋊C₄	—	191	1	no	—
255	C₄²	—	382	1	no	—
256	C₄²	—	382	1	no	—
257	C₄²	—	191	1	no	—
258	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
259	C₂³	—	191	1	no	—
260	Q₈	—	8	1	no	—
261	Q₈	—	1528	1	no	—
262	Q₈	—	1528	1	no	—
263	Q₈	—	1528	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	382	1	no	—
269	C₂×C₄	—	382	1	no	—
270	C₂×C₄	—	382	1	no	—
271	C₂×C₄	—	382	1	no	—
272	C₂×C₄	—	191	1	no	—
273	C₂×C₄	—	191	1	no	—
274	C₂×C₄	—	191	1	no	—
275	C₂×C₄	—	191	2	no	—
276	C₂×C₄	—	191	2	no	—
277	C₂×C₄	—	1528	1	no	—
278	C₂×C₄	—	1528	1	no	—
279	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
280	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
281	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
282	C₈	—	191	2	no	—
283	C₈	—	191	2	no	—
284	C₈	Q₈×D₁₉₁	1	2	yes	—
285	C₈	Q₈×D₁₉₁	1	2	yes	—
286	C₂²	—	382	1	no	—
287	C₂²	—	382	1	no	—
288	C₂²	—	191	2	no	—
289	C₂²	C₂×D₄×D₁₉₁	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	4	1	no	—
293	C₄	—	4	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	—	382	1	no	—
297	C₄	—	382	1	no	—
298	C₄	—	191	2	no	—
299	C₄	—	764	1	no	—
300	C₄	—	764	1	no	—
301	C₄	—	764	1	no	—
302	C₄	—	764	1	no	—
303	C₄	C₂×Q₈×D₁₉₁	1	1	yes	characteristic
304	C₄	C₂×Q₈×D₁₉₁	1	1	yes	characteristic
305	C₂	—	382	2	no	—
306	C₂	C₄⋊Q₈×D₁₉₁	1	1	yes	characteristic
307	C₂	#27133472	1	1	yes	characteristic
308	C₂	#27133473	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	#501×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂²×D₄
prime radical	C₂²×D₁₉₁	C₂×D₄
hypercenter	#501	D₁₉₁
hypocenter	C₁₉₁	#3163
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #501

Complements

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

N	H ≅ G/N	type
3. #27133072	C₂	semidirect
4. #27133071	C₂	semidirect
5. #27133070	C₂	semidirect
6. #27133064	C₂	semidirect
7. #27133062	C₂	semidirect
8. #27133060	C₂	semidirect
9. #27133059	C₂	semidirect
15. #501×C₁₉₁	C₂	semidirect
65. #15690270	Q₈	semidirect
68. C₈×D₁₉₁	Q₈	semidirect
154. C₁₉₁	#3163	semidirect
161. #501	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
#27133290	2
#27133072	2
#27133071	2
#27133070	2
#27133064	2
#27133062	2
#27133060	2
#27133059	2
#27132704	2
#27132702	2
#27132700	2
#106×D₁₉₁	2
#27132620	2
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