Subgroups — #27163053

Order 12704.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₃₉₇	C₂	1	1	yes	maximal, characteristic
2	#20263710	C₂	1	2	yes	maximal, complemented
3	#20263708	C₂	1	2	yes	maximal
4	C₄×Dic₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄²×C₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₉₇	C₂²	1	1	yes	characteristic
7	C₂×Dic₃₉₇	C₂²	1	2	yes	—
8	C₂×Dic₃₉₇	C₂²	1	1	yes	characteristic
9	C₄×D₃₉₇	C₄	1	4	yes	complemented
10	C₂×C₄×C₃₉₇	C₂²	1	2	yes	—
11	C₂×C₄×C₃₉₇	C₂²	1	1	yes	characteristic
12	C₂²×C₃₉₇	C₂³	1	1	yes	characteristic
13	C₂×D₃₉₇	—	2	2	no	—
14	C₂×D₃₉₇	C₂×C₄	1	2	yes	—
15	C₄×C₃₉₇	—	2	2	no	—
16	C₄×C₃₉₇	C₂×C₄	1	2	yes	—
17	Dic₃₉₇	—	2	2	no	—
18	Dic₃₉₇	C₂×C₄	1	2	yes	—
19	C₂×C₃₉₇	C₂²×C₄	1	1	yes	characteristic
20	C₂×C₃₉₇	#41	1	2	yes	—
21	D₃₉₇	—	2	2	no	—
22	C₃₉₇	#117	1	1	yes	characteristic, complemented, Sylow
23	#117	—	397	1	no	maximal, Sylow
24	C₂²×C₄	—	397	1	no	—
25	C₄⋊C₄	—	397	2	no	—
26	C₂²⋊C₄	—	397	2	no	—
27	C₄²	—	397	1	no	—
28	C₄²	D₃₉₇	1	1	yes	characteristic, complemented
29	C₂³	—	397	1	no	—
30	C₂×C₄	—	397	2	no	—
31	C₂×C₄	—	397	1	no	—
32	C₂×C₄	—	397	4	no	—
33	C₂×C₄	C₂×D₃₉₇	1	2	yes	—
34	C₂×C₄	C₂×D₃₉₇	1	1	yes	characteristic
35	C₂²	—	794	2	no	—
36	C₂²	—	397	2	no	—
37	C₂²	C₂²×D₃₉₇	1	1	yes	characteristic
38	C₄	—	2	2	no	—
39	C₄	—	794	2	no	—
40	C₄	—	397	2	no	—
41	C₄	C₄×D₃₉₇	1	2	yes	—
42	C₂	—	794	2	no	—
43	C₂	C₂×C₄×D₃₉₇	1	1	yes	characteristic
44	C₂	#20263731	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₃₉₇
commutator	C₂×C₃₉₇	C₂²×C₄
Frattini	C₂²	C₂²×D₃₉₇
Fitting	C₄²×C₃₉₇	C₂
socle, cosocle	C₂²×C₃₉₇	C₂³
prime radical	C₂²×D₃₉₇	C₂²
hypercenter	C₄²	D₃₉₇
hypocenter	C₃₉₇	#117
LCS	G ▷ C₂×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. #20263710	C₂	semidirect
4. C₄×Dic₃₉₇	C₂	semidirect
5. C₄²×C₃₉₇	C₂	semidirect
9. C₄×D₃₉₇	C₄	semidirect
22. C₃₉₇	#117	semidirect
28. C₄²	D₃₉₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₃₉₇	2
#20263710	2
#20263708	2
C₄×Dic₃₉₇	2
C₄²×C₃₉₇	2
#117	397