Subgroups — #27161481

Order 12608.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263318	C₂	1	2	yes	maximal
2	#20263260	C₂	1	2	yes	maximal
3	#20263258	C₂	1	2	yes	maximal
4	C₄×C₈×C₁₉₇	C₂	1	1	yes	maximal, characteristic
5	C₄⋊Dic₁₉₇	C₂²	1	4	yes	—
6	#15691337	—	2	4	no	—
7	C₂×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
8	C₂×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
9	C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
10	C₂×Dic₁₉₇	—	2	4	no	—
11	C₂×C₄×C₁₉₇	D₄	1	2	yes	—
12	C₂×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
13	C₈×C₁₉₇	Q₈	1	2	yes	—
14	C₈×C₁₉₇	Q₈	1	2	yes	—
15	C₂²×C₁₉₇	C₂×D₄	1	1	yes	characteristic
16	C₄×C₁₉₇	—	2	2	no	—
17	C₄×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
18	C₄×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
19	Dic₁₉₇	—	4	4	no	—
20	C₂×C₁₉₇	C₄⋊Q₈	1	1	yes	characteristic
21	C₂×C₁₉₇	#135	1	2	yes	—
22	C₁₉₇	#499	1	1	yes	characteristic, complemented, Sylow
23	#499	—	197	1	no	maximal, Sylow
24	#125	—	197	2	no	—
25	#107	—	197	2	no	—
26	#106	—	197	2	no	—
27	C₄×C₈	D₁₉₇	1	1	yes	characteristic
28	C₂×C₈	C₂×D₁₉₇	1	1	yes	characteristic
29	C₂×C₈	C₂×D₁₉₇	1	1	yes	characteristic
30	C₄⋊C₄	—	394	4	no	—
31	C₄⋊C₄	—	197	4	no	—
32	C₄²	C₂×D₁₉₇	1	1	yes	characteristic
33	C₂×C₄	—	394	4	no	—
34	C₂×C₄	D₇₈₈	1	2	yes	—
35	C₂×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
36	C₈	Dic₃₉₄	1	2	yes	—
37	C₈	Dic₃₉₄	1	2	yes	—
38	C₂²	C₂×D₇₈₈	1	1	yes	characteristic
39	C₄	—	2	2	no	—
40	C₄	—	788	4	no	—
41	C₄	C₂×Dic₃₉₄	1	1	yes	characteristic
42	C₄	C₂×Dic₃₉₄	1	1	yes	characteristic
43	C₂	#20263326	1	1	yes	characteristic
44	C₂	#20263348	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₇₈₈
commutator, cosocle	C₂×C₄×C₁₉₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₇
Fitting	C₄×C₈×C₁₉₇	C₂
socle, prime radical	C₂²×C₁₉₇	C₂×D₄
hypercenter	C₄×C₈	D₁₉₇
hypocenter	C₁₉₇	#499
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
22. C₁₉₇	#499	semidirect

Maximal subgroups

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

subgroup	index
#20263318	2
#20263260	2
#20263258	2
C₄×C₈×C₁₉₇	2
#499	197