Subgroups — #27132905

Order 12224.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₇₆₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₈×D₁₉₁	C₂	1	1	yes	maximal, characteristic
3	#20259302	C₂	1	2	yes	maximal
4	#20259245	C₂	1	2	yes	maximal, complemented
5	#107×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₃₈₂	C₂²	1	2	yes	—
7	C₂×C₄×D₁₉₁	C₂²	1	1	yes	characteristic
8	Dic₇₆₄	—	2	2	no	—
9	C₂×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
10	C₈×D₁₉₁	—	2	2	no	—
11	C₄⋊Dic₁₉₁	—	2	2	no	—
12	(C₂×D₁₉₁)⋊C₄	—	2	2	no	—
13	C₂×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₁₉₁	C₂²	1	2	yes	—
15	C₂²×D₁₉₁	D₄	1	1	yes	characteristic
16	Dic₃₈₂	—	2	2	no	—
17	Dic₃₈₂	—	2	2	no	—
18	C₂×Dic₁₉₁	—	2	2	no	—
19	C₂×Dic₁₉₁	D₄	1	1	yes	characteristic
20	C₄×D₁₉₁	—	2	1	no	—
21	C₄×D₁₉₁	—	2	1	no	—
22	C₁₉₁⋊C₈	—	2	1	no	—
23	C₂×C₄×C₁₉₁	—	2	2	no	—
24	C₂×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
25	C₈×C₁₉₁	D₄	1	2	yes	—
26	C₂²×C₁₉₁	C₂×D₄	1	1	yes	characteristic
27	C₂×D₁₉₁	—	2	1	no	—
28	C₂×D₁₉₁	—	2	1	no	—
29	C₂×D₁₉₁	Q₁₆	1	2	yes	complemented
30	C₄×C₁₉₁	—	4	2	no	—
31	C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
32	C₄×C₁₉₁	#41	1	1	yes	characteristic
33	Dic₁₉₁	—	2	1	no	—
34	Dic₁₉₁	—	4	2	no	—
35	C₂×C₁₉₁	#121	1	1	yes	characteristic
36	C₂×C₁₉₁	C₂×Q₁₆	1	1	yes	characteristic
37	C₂×C₁₉₁	#135	1	1	yes	characteristic
38	D₁₉₁	—	2	2	no	—
39	C₁₉₁	#467	1	1	yes	characteristic, complemented, Sylow
40	#467	—	191	1	no	maximal, Sylow
41	C₂×Q₁₆	—	191	1	no	—
42	C₂²×C₈	—	191	1	no	—
43	C₂²⋊Q₈	—	191	2	no	—
44	#107	D₁₉₁	1	1	yes	characteristic, complemented
45	#103	—	191	2	no	—
46	C₂×Q₈	—	191	2	no	—
47	C₂²×C₄	—	191	1	no	—
48	Q₁₆	—	382	2	no	—
49	C₂×C₈	—	382	2	no	—
50	C₂×C₈	—	191	1	no	—
51	C₂×C₈	C₂×D₁₉₁	1	1	yes	characteristic
52	C₄⋊C₄	—	382	2	no	—
53	C₄⋊C₄	C₂×D₁₉₁	1	2	yes	—
54	C₂²⋊C₄	—	382	2	no	—
55	C₂³	—	191	1	no	—
56	Q₈	—	382	2	no	—
57	Q₈	—	382	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	382	1	no	—
60	C₂×C₄	—	382	1	no	—
61	C₂×C₄	—	382	2	no	—
62	C₂×C₄	—	191	1	no	—
63	C₂×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
64	C₈	—	382	1	no	—
65	C₈	D₇₆₄	1	2	yes	—
66	C₂²	—	382	1	no	—
67	C₂²	—	382	1	no	—
68	C₂²	—	191	2	no	—
69	C₂²	D₄×D₁₉₁	1	1	yes	characteristic
70	C₄	—	4	2	no	—
71	C₄	—	382	1	no	—
72	C₄	—	764	2	no	—
73	C₄	C₂×D₇₆₄	1	1	yes	characteristic
74	C₄	#15690288	1	1	yes	characteristic
75	C₂	—	382	2	no	—
76	C₂	#20259297	1	1	yes	characteristic
77	C₂	Q₁₆×D₁₉₁	1	1	yes	characteristic
78	C₂	#20259341	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₁
Fitting	#107×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂×D₄
prime radical	C₂²×D₁₉₁	D₄
hypercenter	#107	D₁₉₁
hypocenter	C₁₉₁	#467
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₇₆₄	C₂	semidirect
4. #20259245	C₂	semidirect
5. #107×C₁₉₁	C₂	semidirect
29. C₂×D₁₉₁	Q₁₆	semidirect
39. C₁₉₁	#467	semidirect
44. #107	D₁₉₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₇₆₄	2
C₂×C₈×D₁₉₁	2
#20259302	2
#20259245	2
#107×C₁₉₁	2
#467	191