Subgroups — #27163141

Order 12704.

166 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	#20263730	C₂	1	4	yes	maximal, complemented
3	C₂×C₂²⋊D₃₉₇	C₂	1	4	yes	maximal, complemented
4	D₄×D₃₉₇	C₂	1	4	yes	maximal, complemented
5	C₂×D₄×C₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₉₇	C₂²	1	4	yes	complemented
7	Dic₇₉₄	C₂²	1	2	yes	complemented
8	D₁₅₈₈	C₂²	1	2	yes	complemented
9	C₂²⋊D₃₉₇	C₂²	1	8	yes	complemented
10	C₂²⋊D₃₉₇	C₂²	1	4	yes	complemented
11	C₂×Dic₃₉₇	C₂²	1	4	yes	complemented
12	C₄×D₃₉₇	C₂²	1	4	yes	complemented
13	C₂³×C₃₉₇	C₂²	1	2	yes	complemented
14	D₄×C₃₉₇	C₂²	1	4	yes	complemented
15	C₂×C₄×C₃₉₇	C₂²	1	1	yes	characteristic, complemented
16	C₂²×C₃₉₇	—	4	2	no	—
17	C₂²×C₃₉₇	C₂³	1	1	yes	characteristic
18	C₂²×C₃₉₇	C₂³	1	4	yes	—
19	C₂×D₃₉₇	—	4	4	no	—
20	C₂×D₃₉₇	C₂³	1	4	yes	—
21	C₄×C₃₉₇	C₂³	1	2	yes	—
22	Dic₃₉₇	C₂³	1	4	yes	—
23	C₂×C₃₉₇	—	2	4	no	—
24	C₂×C₃₉₇	—	2	1	no	—
25	C₂×C₃₉₇	C₂⁴	1	1	yes	characteristic
26	D₃₉₇	—	2	4	no	—
27	C₃₉₇	H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
28	H(5,𝔽₂)	—	397	1	no	maximal, Sylow
29	#41	—	397	4	no	—
30	#41	—	397	2	no	—
31	C₂×D₄	—	397	4	no	—
32	C₂×D₄	—	397	4	no	—
33	C₂×D₄	D₃₉₇	1	1	yes	characteristic, complemented
34	C₂³	—	397	4	no	—
35	C₂³	C₂×D₃₉₇	1	2	yes	complemented
36	Q₈	—	397	2	no	—
37	D₄	—	397	8	no	—
38	D₄	—	397	4	no	—
39	D₄	—	397	2	no	—
40	D₄	C₂×D₃₉₇	1	4	yes	complemented
41	C₂×C₄	—	397	4	no	—
42	C₂×C₄	—	397	4	no	—
43	C₂×C₄	C₂×D₃₉₇	1	1	yes	characteristic, complemented
44	C₂²	—	4	2	no	—
45	C₂²	—	1588	4	no	—
46	C₂²	—	397	4	no	—
47	C₂²	C₂²×D₃₉₇	1	4	yes	—
48	C₂²	C₂²×D₃₉₇	1	1	yes	characteristic
49	C₄	—	397	4	no	—
50	C₄	C₂²×D₃₉₇	1	2	yes	—
51	C₂	—	2	1	no	—
52	C₂	—	2	4	no	—
53	C₂	—	794	4	no	—
54	C₂	C₂³×D₃₉₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×D₃₉₇
commutator, socle, cosocle	C₂×C₃₉₇	C₂⁴
Fitting	C₂×D₄×C₃₉₇	C₂
hypercenter	C₂×D₄	D₃₉₇
hypocenter	C₃₉₇	H(5,𝔽₂)
LCS	G ▷ C₂×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂ ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. #20263731	C₂	semidirect
2. #20263730	C₂	semidirect
3. C₂×C₂²⋊D₃₉₇	C₂	semidirect
4. D₄×D₃₉₇	C₂	semidirect
5. C₂×D₄×C₃₉₇	C₂	semidirect
6. C₂²×D₃₉₇	C₂²	semidirect
7. Dic₇₉₄	C₂²	semidirect
8. D₁₅₈₈	C₂²	semidirect
9. C₂²⋊D₃₉₇	C₂²	semidirect
10. C₂²⋊D₃₉₇	C₂²	semidirect
11. C₂×Dic₃₉₇	C₂²	semidirect
12. C₄×D₃₉₇	C₂²	semidirect
13. C₂³×C₃₉₇	C₂²	semidirect
14. D₄×C₃₉₇	C₂²	semidirect
15. C₂×C₄×C₃₉₇	C₂²	semidirect
27. C₃₉₇	H(5,𝔽₂)	semidirect
33. C₂×D₄	D₃₉₇	semidirect
35. C₂³	C₂×D₃₉₇	semidirect
40. D₄	C₂×D₃₉₇	semidirect
43. C₂×C₄	C₂×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
#20263730	2
C₂×C₂²⋊D₃₉₇	2
D₄×D₃₉₇	2
C₂×D₄×C₃₉₇	2
H(5,𝔽₂)	397