Subgroups — #27163118

Order 12704.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263731	C₂	1	2	yes	maximal, complemented
2	Dic₁₅₈₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#20263720	C₂	1	2	yes	maximal, complemented
4	D₃₁₇₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₈×C₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₇₉₄	C₂²	1	2	yes	—
7	D₁₅₈₈	C₂²	1	2	yes	—
8	C₂²⋊D₃₉₇	—	2	2	no	—
9	C₄×D₃₉₇	—	2	2	no	—
10	C₂×C₄×C₃₉₇	C₂²	1	1	yes	characteristic
11	C₈×C₃₉₇	C₂²	1	1	yes	characteristic
12	C₈×C₃₉₇	C₂²	1	1	yes	characteristic
13	C₂²×C₃₉₇	D₄	1	1	yes	characteristic
14	C₂×D₃₉₇	—	2	2	no	—
15	C₄×C₃₉₇	D₄	1	1	yes	characteristic
16	C₄×C₃₉₇	C₂³	1	1	yes	characteristic
17	Dic₃₉₇	—	2	2	no	—
18	C₂×C₃₉₇	—	2	1	no	—
19	C₂×C₃₉₇	C₂×D₄	1	1	yes	characteristic
20	D₃₉₇	—	4	2	no	—
21	C₃₉₇	#135	1	1	yes	characteristic, complemented, Sylow
22	#135	—	397	1	no	maximal, Sylow
23	#41	—	397	2	no	—
24	Q₁₆	—	397	1	no	—
25	SD₁₆	—	397	2	no	—
26	D₈	—	397	1	no	—
27	C₂×C₈	D₃₉₇	1	1	yes	characteristic, complemented
28	Q₈	—	397	2	no	—
29	D₄	—	794	2	no	—
30	D₄	—	397	2	no	—
31	C₂×C₄	—	794	2	no	—
32	C₂×C₄	C₂×D₃₉₇	1	1	yes	characteristic
33	C₈	C₂×D₃₉₇	1	1	yes	characteristic
34	C₈	C₂×D₃₉₇	1	1	yes	characteristic
35	C₂²	—	794	2	no	—
36	C₂²	D₁₅₈₈	1	1	yes	characteristic
37	C₄	—	794	2	no	—
38	C₄	D₁₅₈₈	1	1	yes	characteristic
39	C₄	C₂²×D₃₉₇	1	1	yes	characteristic
40	C₂	—	2	1	no	—
41	C₂	—	1588	2	no	—
42	C₂	C₂×D₁₅₈₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₁₅₈₈
commutator, cosocle	C₄×C₃₉₇	C₂³
Frattini	C₄	C₂²×D₃₉₇
Fitting	C₂×C₈×C₃₉₇	C₂
socle	C₂×C₃₉₇	C₂×D₄
hypercenter	C₂×C₈	D₃₉₇
hypocenter	C₃₉₇	#135
LCS	G ▷ C₄×C₃₉₇ ▷ C₂×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
1. #20263731	C₂	semidirect
2. Dic₁₅₈₈	C₂	semidirect
3. #20263720	C₂	semidirect
4. D₃₁₇₆	C₂	semidirect
5. C₂×C₈×C₃₉₇	C₂	semidirect
21. C₃₉₇	#135	semidirect
27. C₂×C₈	D₃₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20263731	2
Dic₁₅₈₈	2
#20263720	2
D₃₁₇₆	2
C₂×C₈×C₃₉₇	2
#135	397