Subgroups — #27161436

Order 12608.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263314	C₂	1	1	yes	maximal, characteristic
2	#20263249	C₂	1	2	yes	maximal, complemented
3	#20263246	C₂	1	2	yes	maximal
4	C₄×#13096802	C₂	1	1	yes	maximal, characteristic, complemented
5	#125×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₃₉₄	C₂²	1	1	yes	characteristic
7	C₂×D₇₈₈	C₂²	1	1	yes	characteristic
8	C₂×#13096802	C₂²	1	2	yes	—
9	#15691335	—	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	C₂²×D₁₉₇	—	2	1	no	—
14	Dic₃₉₄	—	2	2	no	—
15	D₇₈₈	—	2	2	no	—
16	C₂×Dic₁₉₇	—	2	1	no	—
17	#13096802	—	2	2	no	—
18	C₂×C₄×C₁₉₇	—	2	2	no	—
19	C₂×C₄×C₁₉₇	D₄	1	2	yes	—
20	C₂×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
21	C₂²×C₁₉₇	C₂×D₄	1	1	yes	characteristic
22	C₂×D₁₉₇	—	4	1	no	—
23	C₂×D₁₉₇	—	4	2	no	—
24	C₄×C₁₉₇	—	2	2	no	—
25	C₄×C₁₉₇	—	4	2	no	—
26	C₄×C₁₉₇	#41	1	2	yes	—
27	Dic₁₉₇	—	4	1	no	—
28	C₂×C₁₉₇	#124	1	1	yes	characteristic
29	C₂×C₁₉₇	#135	1	2	yes	—
30	D₁₉₇	—	8	1	no	—
31	C₁₉₇	#488	1	1	yes	characteristic, complemented, Sylow
32	#488	—	197	1	no	maximal, Sylow
33	#125	D₁₉₇	1	1	yes	characteristic, complemented
34	#124	—	197	1	no	—
35	#103	—	197	2	no	—
36	#102	—	197	2	no	—
37	C₄×C₈	—	197	1	no	—
38	C₂×Q₈	—	197	1	no	—
39	C₂×D₄	—	197	1	no	—
40	C₂×C₈	—	197	2	no	—
41	C₄⋊C₄	—	2	2	no	—
42	C₄⋊C₄	C₂×D₁₉₇	1	2	yes	—
43	C₂²⋊C₄	—	394	2	no	—
44	C₄²	C₂×D₁₉₇	1	1	yes	characteristic
45	C₂³	—	394	1	no	—
46	Q₈	—	394	2	no	—
47	D₄	—	394	2	no	—
48	C₂×C₄	—	2	2	no	—
49	C₂×C₄	—	394	1	no	—
50	C₂×C₄	C₂²⋊D₁₉₇	1	2	yes	—
51	C₂×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
52	C₈	—	394	2	no	—
53	C₂²	—	788	1	no	—
54	C₂²	—	788	2	no	—
55	C₂²	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	4	2	no	—
58	C₄	—	788	1	no	—
59	C₄	#15691359	1	2	yes	—
60	C₂	—	1576	1	no	—
61	C₂	#20263315	1	1	yes	characteristic
62	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	C₂²×C₁₉₇	C₂×D₄
prime radical	C₂×D₇₈₈	C₂²
hypercenter	#125	D₁₉₇
hypocenter	C₁₉₇	#488
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
2. #20263249	C₂	semidirect
4. C₄×#13096802	C₂	semidirect
5. #125×C₁₉₇	C₂	semidirect
31. C₁₉₇	#488	semidirect
33. #125	D₁₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20263314	2
#20263249	2
#20263246	2
C₄×#13096802	2
#125×C₁₉₇	2
#488	197