Subgroups — #27133067

Order 12224.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259321	C₂	1	2	yes	maximal
2	#20259256	C₂	1	1	yes	maximal, characteristic
3	#20259255	C₂	1	2	yes	maximal
4	C₈×Dic₁₉₁	C₂	1	1	yes	maximal, characteristic
5	#107×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₈₂	—	2	2	no	—
7	C₂×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
9	Dic₁₉₁⋊C₄	—	2	2	no	—
10	C₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₁₉₁	C₂²	1	2	yes	—
13	Dic₃₈₂	—	4	4	no	—
14	C₂×Dic₁₉₁	—	2	2	no	—
15	C₂×Dic₁₉₁	D₄	1	1	yes	characteristic
16	C₂×Dic₁₉₁	D₄	1	1	yes	characteristic
17	C₁₉₁⋊C₈	Q₈	1	2	yes	complemented
18	C₂×C₄×C₁₉₁	—	2	2	no	—
19	C₂×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
20	C₈×C₁₉₁	Q₈	1	2	yes	complemented
21	C₂²×C₁₉₁	C₂×D₄	1	1	yes	characteristic
22	C₄×C₁₉₁	—	4	2	no	—
23	C₄×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
24	C₄×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
25	Dic₁₉₁	—	4	2	no	—
26	Dic₁₉₁	D₈	1	2	yes	—
27	Dic₁₉₁	Q₁₆	1	2	yes	—
28	C₂×C₁₉₁	C₄⋊Q₈	1	1	yes	characteristic
29	C₂×C₁₉₁	C₂×D₈	1	1	yes	characteristic
30	C₂×C₁₉₁	C₂×Q₁₆	1	1	yes	characteristic
31	C₁₉₁	#500	1	1	yes	characteristic, complemented, Sylow
32	#500	—	191	1	no	maximal, Sylow
33	C₄⋊Q₈	—	191	2	no	—
34	#107	—	191	1	no	—
35	#107	—	191	2	no	—
36	#107	D₁₉₁	1	1	yes	characteristic
37	C₄×C₈	—	191	1	no	—
38	C₂×Q₈	—	382	2	no	—
39	C₂×C₈	—	191	1	no	—
40	C₂×C₈	C₂×D₁₉₁	1	1	yes	characteristic
41	C₄⋊C₄	—	382	2	no	—
42	C₄⋊C₄	—	191	2	no	—
43	C₄⋊C₄	C₂×D₁₉₁	1	2	yes	—
44	C₄²	—	191	1	no	—
45	Q₈	—	764	4	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	382	2	no	—
48	C₂×C₄	—	191	1	no	—
49	C₂×C₄	—	191	1	no	—
50	C₂×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
51	C₈	—	191	2	no	—
52	C₈	Dic₃₈₂	1	2	yes	complemented
53	C₂²	D₄×D₁₉₁	1	1	yes	characteristic
54	C₄	—	4	2	no	—
55	C₄	—	191	2	no	—
56	C₄	—	191	2	no	—
57	C₄	—	764	2	no	—
58	C₄	Q₈×D₁₉₁	1	1	yes	characteristic
59	C₄	C₂×Dic₃₈₂	1	1	yes	characteristic
60	C₂	#20259321	1	1	yes	characteristic
61	C₂	D₈×D₁₉₁	1	1	yes	characteristic
62	C₂	Q₁₆×D₁₉₁	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, prime radical	C₂²×C₁₉₁	C₂×D₄
hypercenter	#107	D₁₉₁
hypocenter	C₁₉₁	#500
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
17. C₁₉₁⋊C₈	Q₈	semidirect
20. C₈×C₁₉₁	Q₈	semidirect
31. C₁₉₁	#500	semidirect
52. C₈	Dic₃₈₂	semidirect

Maximal subgroups

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

subgroup	index
#20259321	2
#20259256	2
#20259255	2
C₈×Dic₁₉₁	2
#107×C₁₉₁	2
#500	191