Subgroups — #27133033

Order 12224.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259322	C₂	1	1	yes	maximal, characteristic
2	#20259245	C₂	1	4	yes	maximal
3	#20259233	C₂	1	1	yes	maximal, characteristic
4	#125×C₁₉₁	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₃₈₂	C₂²	1	2	yes	—
6	C₂×C₁₉₁⋊C₈	C₂²	1	2	yes	—
7	C₄⋊Dic₁₉₁	—	2	2	no	—
8	C₄⋊C₄×C₁₉₁	—	2	2	no	—
9	C₄⋊C₄×C₁₉₁	C₂²	1	2	yes	—
10	C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
11	Dic₃₈₂	—	2	4	no	—
12	C₂×Dic₁₉₁	—	2	2	no	—
13	C₁₉₁⋊C₈	—	2	2	no	—
14	C₂×C₄×C₁₉₁	—	2	2	no	—
15	C₂×C₄×C₁₉₁	D₄	1	2	yes	—
16	C₂×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
17	C₂²×C₁₉₁	C₂×D₄	1	1	yes	characteristic
18	C₄×C₁₉₁	—	2	2	no	—
19	C₄×C₁₉₁	—	4	2	no	—
20	C₄×C₁₉₁	#41	1	2	yes	—
21	Dic₁₉₁	—	4	2	no	—
22	C₂×C₁₉₁	#124	1	1	yes	characteristic
23	C₂×C₁₉₁	#137	1	2	yes	—
24	C₁₉₁	#491	1	1	yes	characteristic, complemented, Sylow
25	#491	—	191	1	no	maximal, Sylow
26	C₄⋊Q₈	—	191	1	no	—
27	#125	D₁₉₁	1	1	yes	characteristic
28	#103	—	191	4	no	—
29	#97	—	191	1	no	—
30	C₂×Q₈	—	191	2	no	—
31	C₂×C₈	—	191	2	no	—
32	C₄⋊C₄	—	2	2	no	—
33	C₄⋊C₄	—	382	2	no	—
34	C₄⋊C₄	C₂×D₁₉₁	1	2	yes	—
35	C₄²	C₂×D₁₉₁	1	1	yes	characteristic
36	Q₈	—	382	4	no	—
37	C₂×C₄	—	2	2	no	—
38	C₂×C₄	—	382	2	no	—
39	C₂×C₄	C₂²⋊D₁₉₁	1	2	yes	—
40	C₂×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
41	C₈	—	382	2	no	—
42	C₂²	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
43	C₄	—	2	2	no	—
44	C₄	—	4	2	no	—
45	C₄	—	764	2	no	—
46	C₄	#15690288	1	2	yes	—
47	C₂	#20259311	1	1	yes	characteristic
48	C₂	#20259352	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₁
Fitting	#125×C₁₉₁	C₂
socle, prime radical	C₂²×C₁₉₁	C₂×D₄
hypercenter	#125	D₁₉₁
hypocenter	C₁₉₁	#491
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
24. C₁₉₁	#491	semidirect

Maximal subgroups

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

subgroup	index
#20259322	2
#20259245	2
#20259233	2
#125×C₁₉₁	2
#491	191