Subgroups — #37574037

Order 24448.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27132712	C₂	1	3	yes	maximal, complemented
2	#27132626	C₂	1	3	yes	maximal
3	#338×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₄×D₁₉₁	C₂²	1	1	yes	characteristic
5	C₂×#15690271	C₂²	1	3	yes	—
6	#20259258	—	2	6	no	—
7	(C₂×D₁₉₁)⋊C₈	—	2	6	no	—
8	C₂×M₁₆×C₁₉₁	C₂²	1	3	yes	—
9	C₂³×D₁₉₁	Q₈	1	1	yes	characteristic
10	C₂²×Dic₁₉₁	Q₈	1	1	yes	characteristic
11	C₂×C₄×D₁₉₁	—	2	3	no	—
12	C₂×C₄×D₁₉₁	—	2	3	no	—
13	#15690271	—	2	6	no	—
14	C₂×C₁₉₁⋊C₈	—	2	3	no	—
15	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
16	M₁₆×C₁₉₁	—	2	6	no	—
17	C₂×C₈×C₁₉₁	D₄	1	6	yes	—
18	C₂²×D₁₉₁	—	2	3	no	—
19	C₂²×D₁₉₁	—	4	2	no	—
20	C₂×Dic₁₉₁	—	2	3	no	—
21	C₄×D₁₉₁	—	4	3	no	—
22	C₄×D₁₉₁	—	4	1	no	—
23	C₁₉₁⋊C₈	—	4	3	no	—
24	C₂³×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
25	C₂×C₄×C₁₉₁	C₂×D₄	1	3	yes	—
26	C₂×C₄×C₁₉₁	#41	1	3	yes	—
27	C₈×C₁₉₁	—	2	6	no	—
28	C₂²×C₁₉₁	—	4	1	no	—
29	C₂²×C₁₉₁	C₂²⋊Q₈	1	3	yes	—
30	C₂×D₁₉₁	—	2	6	no	—
31	C₂×D₁₉₁	—	4	3	no	—
32	C₂×D₁₉₁	—	4	1	no	—
33	C₄×C₁₉₁	#123	1	3	yes	—
34	C₄×C₁₉₁	#127	1	1	yes	characteristic
35	Dic₁₉₁	—	4	1	no	—
36	C₂×C₁₉₁	—	2	3	no	—
37	C₂×C₁₉₁	#399	1	1	yes	characteristic
38	D₁₉₁	—	4	2	no	—
39	C₁₉₁	#2091	1	1	yes	characteristic, complemented, Sylow
40	#2091	—	191	1	no	maximal, Sylow
41	#430	—	191	3	no	—
42	#407	—	191	3	no	—
43	#338	D₁₉₁	1	1	yes	characteristic, complemented
44	C₂³×C₄	—	191	1	no	—
45	C₂×M₁₆	—	191	3	no	—
46	C₂×M₁₆	C₂×D₁₉₁	1	3	yes	—
47	#108	—	382	6	no	—
48	C₂²⋊C₈	—	382	6	no	—
49	C₂⁴	—	191	1	no	—
50	C₂²×C₄	—	382	3	no	—
51	C₂²×C₄	—	382	3	no	—
52	C₂²×C₄	—	191	1	no	—
53	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
54	M₁₆	—	2	6	no	—
55	M₁₆	—	382	6	no	—
56	C₂×C₈	—	382	3	no	—
57	C₂×C₈	D₇₆₄	1	6	yes	—
58	C₂³	—	382	3	no	—
59	C₂³	—	764	2	no	—
60	C₂³	Q₈×D₁₉₁	1	1	yes	characteristic
61	C₂×C₄	—	382	3	no	—
62	C₂×C₄	—	764	3	no	—
63	C₂×C₄	—	764	1	no	—
64	C₂×C₄	C₂×D₇₆₄	1	3	yes	—
65	C₂×C₄	#15690286	1	3	yes	—
66	C₈	—	2	6	no	—
67	C₈	—	764	3	no	—
68	C₂²	—	4	1	no	—
69	C₂²	—	382	6	no	—
70	C₂²	—	764	3	no	—
71	C₂²	—	764	1	no	—
72	C₂²	#20259302	1	3	yes	—
73	C₄	—	764	1	no	—
74	C₄	#20259306	1	3	yes	—
75	C₄	#20259319	1	1	yes	characteristic
76	C₂	—	2	3	no	—
77	C₂	—	764	2	no	—
78	C₂	#27132600	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#20259319
commutator, cosocle	C₂²×C₄×C₁₉₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉₁
Fitting	#338×C₁₉₁	C₂
socle	C₂×C₁₉₁	#399
prime radical	C₂³×D₁₉₁	Q₈
hypercenter	#338	D₁₉₁
hypocenter	C₁₉₁	#2091
LCS	G ▷ C₂²×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
1. #27132712	C₂	semidirect
3. #338×C₁₉₁	C₂	semidirect
39. C₁₉₁	#2091	semidirect
43. #338	D₁₉₁	semidirect

Maximal subgroups

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

subgroup	index
#27132712	2
#27132626	2
#338×C₁₉₁	2
#2091	191