Subgroups — #37581227

Order 24448.

500 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20259328	C₂	1	2	yes	maximal, complemented
2	#27133210	C₂	1	1	yes	maximal, characteristic
3	#27132741	C₂	1	1	yes	maximal, characteristic, complemented
4	#27132740	C₂	1	1	yes	maximal, characteristic, complemented
5	#27132738	C₂	1	2	yes	maximal, complemented
6	C₈×D₇₆₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₂²⋊D₁₉₁	C₂	1	2	yes	maximal, complemented
8	#27132685	C₂	1	1	yes	maximal, characteristic, complemented
9	#27132628	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15690287	C₂²	1	2	yes	—
13	#20259328	—	2	8	no	—
14	C₂×#15690271	C₂²	1	2	yes	—
15	C₂×#15690270	C₂²	1	2	yes	—
16	C₂×#15690270	C₂²	1	2	yes	—
17	C₂×C₈×D₁₉₁	C₂²	1	2	yes	—
18	C₂×C₈×D₁₉₁	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₉₁	C₂²	1	1	yes	characteristic
20	#20259321	C₄	1	1	yes	characteristic
21	#20259318	C₄	1	1	yes	characteristic
22	#20259308	C₄	1	2	yes	—
23	#20259297	C₄	1	1	yes	characteristic
24	#20259295	C₄	1	2	yes	—
25	#20259293	C₄	1	1	yes	characteristic
26	C₄×D₇₆₄	C₂²	1	1	yes	characteristic
27	C₄×C₂²⋊D₁₉₁	C₂²	1	2	yes	—
28	C₄²×D₁₉₁	C₂²	1	1	yes	characteristic
29	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	1	yes	characteristic
30	Dic₁₉₁⋊C₈	C₂²	1	2	yes	—
31	C₂²⋊(C₁₉₁⋊C₈)	C₂²	1	2	yes	—
32	(C₂×D₁₉₁)⋊C₈	C₂²	1	2	yes	—
33	(C₂×D₁₉₁)⋊C₈	C₂²	1	2	yes	—
34	C₄×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
35	C₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
36	C₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
37	C₂²×C₈×C₁₉₁	C₂²	1	2	yes	—
38	C₄×D₄×C₁₉₁	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₁₉₁	C₂²	1	1	yes	characteristic
40	C₂²⋊C₈×C₁₉₁	C₂²	1	2	yes	—
41	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
42	#15690287	—	2	4	no	—
43	#15690287	—	2	4	no	—
44	C₂×Dic₃₈₂	C₂×C₄	1	2	yes	—
45	C₂×D₇₆₄	C₂×C₄	1	2	yes	—
46	C₂×D₇₆₄	C₂×C₄	1	1	yes	characteristic
47	C₂×C₂²⋊D₁₉₁	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊D₁₉₁	C₂×C₄	1	2	yes	—
49	C₂×C₂²⋊D₁₉₁	C₂×C₄	1	2	yes	—
50	C₂×C₄×D₁₉₁	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₁₉₁	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₁₉₁	C₂³	1	2	yes	—
53	C₂×C₄×D₁₉₁	C₂³	1	1	yes	characteristic
54	#15690271	—	2	4	no	—
55	#15690270	—	2	4	no	—
56	#15690270	—	2	4	no	—
57	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
58	C₂×C₁₉₁⋊C₈	C₂³	1	2	yes	—
59	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
60	C₈×D₁₉₁	—	2	2	no	—
61	C₈×D₁₉₁	—	2	4	no	—
62	C₈×D₁₉₁	D₄	1	4	yes	complemented
63	C₄⋊Dic₁₉₁	C₂×C₄	1	1	yes	characteristic
64	Dic₁₉₁⋊C₄	C₂×C₄	1	2	yes	—
65	C₂²⋊Dic₁₉₁	C₂×C₄	1	2	yes	—
66	(C₂×D₁₉₁)⋊C₄	C₂×C₄	1	2	yes	—
67	(C₂×D₁₉₁)⋊C₄	C₂×C₄	1	2	yes	—
68	C₄×Dic₁₉₁	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₁₉₁	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₁₉₁	C₂³	1	1	yes	characteristic
71	C₂×D₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₉₁	C₂³	1	2	yes	—
73	C₂×C₈×C₁₉₁	—	2	4	no	—
74	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₁₉₁	C₂³	1	2	yes	—
76	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
78	C₂²⋊C₄×C₁₉₁	C₂×C₄	1	2	yes	—
79	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
80	C₂²×D₁₉₁	C₂²×C₄	1	2	yes	—
81	C₂²×D₁₉₁	C₂²×C₄	1	1	yes	characteristic
82	Dic₃₈₂	—	2	4	no	—
83	D₇₆₄	—	2	4	no	—
84	D₇₆₄	—	2	2	no	—
85	C₂²⋊D₁₉₁	—	2	4	no	—
86	C₂²⋊D₁₉₁	—	2	4	no	—
87	C₂²⋊D₁₉₁	—	2	4	no	—
88	C₂×Dic₁₉₁	C₂²×C₄	1	1	yes	characteristic
89	C₂×Dic₁₉₁	C₂²×C₄	1	2	yes	—
90	C₂×Dic₁₉₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×Dic₁₉₁	C₂²×C₄	1	1	yes	characteristic
92	C₄×D₁₉₁	—	2	2	no	—
93	C₄×D₁₉₁	—	2	2	no	—
94	C₄×D₁₉₁	—	2	2	no	—
95	C₄×D₁₉₁	—	2	2	no	—
96	C₄×D₁₉₁	C₂×D₄	1	2	yes	—
97	C₄×D₁₉₁	#41	1	2	yes	—
98	C₁₉₁⋊C₈	—	2	2	no	—
99	C₁₉₁⋊C₈	—	2	1	no	—
100	C₁₉₁⋊C₈	C₂×D₄	1	2	yes	—
101	C₂³×C₁₉₁	C₂²×C₄	1	2	yes	—
102	D₄×C₁₉₁	—	2	2	no	—
103	C₂×C₄×C₁₉₁	—	2	2	no	—
104	C₂×C₄×C₁₉₁	—	2	2	no	—
105	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	yes	—
107	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
109	C₈×C₁₉₁	—	2	2	no	—
110	C₈×C₁₉₁	—	2	1	no	—
111	C₈×C₁₉₁	C₂×D₄	1	2	yes	—
112	C₂²×C₁₉₁	—	2	2	no	—
113	C₂²×C₁₉₁	—	2	2	no	—
114	C₂²×C₁₉₁	—	2	2	no	—
115	C₂²×C₁₉₁	C₂³×C₄	1	1	yes	characteristic
116	C₂×D₁₉₁	—	2	1	no	—
117	C₂×D₁₉₁	—	2	1	no	—
118	C₂×D₁₉₁	—	2	2	no	—
119	C₂×D₁₉₁	—	2	2	no	—
120	C₂×D₁₉₁	—	2	2	no	—
121	C₂×D₁₉₁	C₄×D₄	1	2	yes	—
122	C₄×C₁₉₁	—	2	2	no	—
123	C₄×C₁₉₁	—	2	1	no	—
124	C₄×C₁₉₁	#131	1	2	yes	complemented
125	C₄×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
126	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
127	Dic₁₉₁	—	2	2	no	—
128	Dic₁₉₁	—	2	1	no	—
129	Dic₁₉₁	C₄×D₄	1	2	yes	—
130	Dic₁₉₁	#131	1	2	yes	complemented
131	C₂×C₁₉₁	—	4	2	no	—
132	C₂×C₁₉₁	C₂×C₄×D₄	1	1	yes	characteristic
133	C₂×C₁₉₁	C₂×#131	1	1	yes	characteristic
134	C₂×C₁₉₁	C₂×#131	1	1	yes	characteristic
135	D₁₉₁	—	2	2	no	—
136	D₁₉₁	—	4	2	no	—
137	C₁₉₁	#2931	1	1	yes	characteristic, complemented, Sylow
138	#2931	—	191	1	no	maximal, Sylow
139	C₂×#131	—	191	2	no	—
140	#532	—	191	1	no	—
141	#436	—	191	1	no	—
142	#436	—	191	1	no	—
143	#436	—	191	2	no	—
144	C₈×D₄	—	191	2	no	—
145	C₈×D₄	—	191	1	no	—
146	C₈×D₄	D₁₉₁	1	1	yes	characteristic, complemented
147	#423	—	191	1	no	—
148	#408	—	191	2	no	—
149	C₂×C₄×C₈	—	191	1	no	—
150	C₂×#41	—	191	2	no	—
151	#131	—	382	8	no	—
152	C₂×M₁₆	—	191	2	no	—
153	C₂×M₁₆	—	191	2	no	—
154	C₂×M₁₆	—	191	2	no	—
155	C₂²×C₈	—	191	1	no	—
156	C₂²×C₈	—	191	2	no	—
157	C₂²×C₈	—	191	1	no	—
158	C₂²×C₈	C₂×D₁₉₁	1	2	yes	—
159	C₄⋊Q₈	—	191	1	no	—
160	#127	—	191	1	no	—
161	#124	—	191	2	no	—
162	#121	—	191	1	no	—
163	#121	—	191	1	no	—
164	#121	—	191	2	no	—
165	C₄×D₄	—	191	2	no	—
166	C₄×D₄	—	191	1	no	—
167	C₄×D₄	C₂×D₁₉₁	1	1	yes	characteristic
168	C₂×C₄²	—	191	1	no	—
169	C₄⋊C₈	—	191	2	no	—
170	C₄⋊C₈	—	191	1	no	—
171	C₄⋊C₈	C₂×D₁₉₁	1	1	yes	characteristic
172	C₂²⋊C₈	—	191	2	no	—
173	C₂²⋊C₈	—	191	2	no	—
174	C₂²⋊C₈	—	191	2	no	—
175	C₂²⋊C₈	C₂×D₁₉₁	1	2	yes	—
176	C₄×C₈	—	191	1	no	—
177	C₄×C₈	—	191	1	no	—
178	C₄×C₈	—	191	1	no	—
179	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
180	#41	—	382	4	no	—
181	#41	—	382	4	no	—
182	C₂×Q₈	—	191	2	no	—
183	C₂×D₄	—	191	2	no	—
184	C₂×D₄	—	191	2	no	—
185	C₂×D₄	—	191	2	no	—
186	C₂×D₄	—	191	1	no	—
187	C₂×D₄	—	191	2	no	—
188	C₂×D₄	C₄×D₁₉₁	1	1	yes	characteristic
189	C₂²×C₄	—	191	1	no	—
190	C₂²×C₄	—	191	1	no	—
191	C₂²×C₄	—	191	2	no	—
192	C₂²×C₄	—	191	1	no	—
193	C₂²×C₄	C₂²×D₁₉₁	1	2	yes	—
194	M₁₆	—	382	4	no	—
195	M₁₆	—	382	4	no	—
196	M₁₆	—	382	4	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	382	2	no	—
199	C₂×C₈	—	382	4	no	—
200	C₂×C₈	—	191	1	no	—
201	C₂×C₈	—	191	2	no	—
202	C₂×C₈	—	191	1	no	—
203	C₂×C₈	—	191	4	no	—
204	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
205	C₂×C₈	C₂²×D₁₉₁	1	2	yes	—
206	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
207	C₄⋊C₄	—	191	2	no	—
208	C₄⋊C₄	—	191	1	no	—
209	C₄⋊C₄	C₄×D₁₉₁	1	1	yes	characteristic
210	C₂²⋊C₄	—	191	2	no	—
211	C₂²⋊C₄	—	191	2	no	—
212	C₂²⋊C₄	—	191	2	no	—
213	C₂²⋊C₄	C₄×D₁₉₁	1	2	yes	—
214	C₄²	—	191	1	no	—
215	C₄²	—	191	1	no	—
216	C₄²	—	191	1	no	—
217	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
218	C₂³	—	191	2	no	—
219	C₂³	—	191	1	no	—
220	C₂³	C₂×C₄×D₁₉₁	1	2	yes	—
221	Q₈	—	382	4	no	—
222	D₄	—	2	2	no	—
223	D₄	—	382	4	no	—
224	D₄	—	382	4	no	—
225	D₄	—	382	4	no	—
226	D₄	—	382	4	no	—
227	D₄	—	382	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	382	2	no	—
231	C₂×C₄	—	382	2	no	—
232	C₂×C₄	—	382	2	no	—
233	C₂×C₄	—	382	2	no	—
234	C₂×C₄	—	191	1	no	—
235	C₂×C₄	—	191	2	no	—
236	C₂×C₄	—	191	1	no	—
237	C₂×C₄	—	191	1	no	—
238	C₂×C₄	—	191	2	no	—
239	C₂×C₄	—	191	2	no	—
240	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
241	C₂×C₄	C₂×C₄×D₁₉₁	1	2	yes	—
242	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
243	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	2	1	no	—
246	C₈	—	382	2	no	—
247	C₈	—	382	1	no	—
248	C₈	—	191	2	no	—
249	C₈	D₄×D₁₉₁	1	2	yes	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	382	1	no	—
254	C₂²	—	382	1	no	—
255	C₂²	—	382	2	no	—
256	C₂²	—	382	2	no	—
257	C₂²	—	382	2	no	—
258	C₂²	—	191	2	no	—
259	C₂²	C₂²×C₄×D₁₉₁	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	1	no	—
262	C₄	—	382	2	no	—
263	C₄	—	382	1	no	—
264	C₄	—	191	2	no	—
265	C₄	—	191	2	no	—
266	C₄	#20259328	1	2	yes	complemented
267	C₄	C₂×D₄×D₁₉₁	1	1	yes	characteristic
268	C₄	#41×D₁₉₁	1	1	yes	characteristic
269	C₂	—	4	2	no	—
270	C₂	—	382	2	no	—
271	C₂	—	764	2	no	—
272	C₂	C₄×D₄×D₁₉₁	1	1	yes	characteristic
273	C₂	#131×D₁₉₁	1	1	yes	characteristic
274	C₂	C₂×#20259328	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₁₉₁
commutator, socle	C₂²×C₁₉₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	C₈×D₄×C₁₉₁	C₂
prime radical	#27133210	C₂
cosocle	C₂×C₄×C₁₉₁	C₂⁴
hypercenter	C₈×D₄	D₁₉₁
hypocenter	C₁₉₁	#2931
LCS	G ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂×C₈ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#20259328	C₂	semidirect
3. #27132741	C₂	semidirect
4. #27132740	C₂	semidirect
5. #27132738	C₂	semidirect
6. C₈×D₇₆₄	C₂	semidirect
7. C₈×C₂²⋊D₁₉₁	C₂	semidirect
8. #27132685	C₂	semidirect
9. #27132628	C₂	semidirect
10. C₄×C₈×D₁₉₁	C₂	semidirect
11. C₈×D₄×C₁₉₁	C₂	semidirect
62. C₈×D₁₉₁	D₄	semidirect
124. C₄×C₁₉₁	#131	semidirect
130. Dic₁₉₁	#131	semidirect
137. C₁₉₁	#2931	semidirect
146. C₈×D₄	D₁₉₁	semidirect
266. C₄	#20259328	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20259328	2
#27133210	2
#27132741	2
#27132740	2
#27132738	2
C₈×D₇₆₄	2
C₈×C₂²⋊D₁₉₁	2
#27132685	2
#27132628	2
C₄×C₈×D₁₉₁	2
C₈×D₄×C₁₉₁	2
#2931	191