Subgroups — #27161478

Order 12608.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263318	C₂	1	1	yes	maximal, characteristic
2	#20263259	C₂	1	2	yes	maximal
3	#20263257	C₂	1	2	yes	maximal
4	C₄×#13096802	C₂	1	1	yes	maximal, characteristic
5	#125×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13096802	C₂²	1	1	yes	characteristic
7	C₂×#13096802	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₁₉₇	C₂²	1	2	yes	—
9	#15691337	—	2	2	no	—
10	C₄⋊C₄×C₁₉₇	—	2	2	no	—
11	C₄⋊C₄×C₁₉₇	C₂²	1	2	yes	—
12	C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
13	C₂×Dic₁₉₇	—	2	2	no	—
14	#13096802	Q₈	1	2	yes	—
15	#13096802	Q₈	1	2	yes	—
16	C₂×C₄×C₁₉₇	—	2	2	no	—
17	C₂×C₄×C₁₉₇	D₄	1	2	yes	—
18	C₂×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
19	C₂²×C₁₉₇	C₂×D₄	1	1	yes	characteristic
20	C₄×C₁₉₇	—	2	2	no	—
21	C₄×C₁₉₇	—	4	2	no	—
22	C₄×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
23	C₄×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
24	Dic₁₉₇	—	4	2	no	—
25	C₂×C₁₉₇	C₄⋊Q₈	1	1	yes	characteristic
26	C₂×C₁₉₇	#135	1	2	yes	—
27	C₁₉₇	#499	1	1	yes	characteristic, complemented, Sylow
28	#499	—	197	1	no	maximal, Sylow
29	#125	—	197	1	no	—
30	#125	D₁₉₇	1	1	yes	characteristic
31	#107	—	197	2	no	—
32	#106	—	197	2	no	—
33	C₄×C₈	—	197	1	no	—
34	C₂×C₈	—	197	1	no	—
35	C₂×C₈	—	197	1	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	—	394	2	no	—
38	C₄⋊C₄	—	197	2	no	—
39	C₄⋊C₄	C₂×D₁₉₇	1	2	yes	—
40	C₄²	C₂×D₁₉₇	1	1	yes	characteristic
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	394	2	no	—
43	C₂×C₄	C₂²⋊D₁₉₇	1	2	yes	—
44	C₂×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
45	C₈	—	197	2	no	—
46	C₈	—	197	2	no	—
47	C₂²	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
48	C₄	—	2	2	no	—
49	C₄	—	4	2	no	—
50	C₄	—	788	2	no	—
51	C₄	Q₈×D₁₉₇	1	1	yes	characteristic
52	C₄	Q₈×D₁₉₇	1	1	yes	characteristic
53	C₂	#20263324	1	1	yes	characteristic
54	C₂	#20263346	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₉₇
commutator, cosocle	C₂×C₄×C₁₉₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₇
Fitting	#125×C₁₉₇	C₂
socle, prime radical	C₂²×C₁₉₇	C₂×D₄
hypercenter	#125	D₁₉₇
hypocenter	C₁₉₇	#499
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
27. 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
#20263259	2
#20263257	2
C₄×#13096802	2
#125×C₁₉₇	2
#499	197