Subgroups — #27163142

Order 12704.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263732	C₂	1	2	yes	maximal, complemented
2	#20263731	C₂	1	3	yes	maximal, complemented
3	C₂×D₁₅₈₈	C₂	1	3	yes	maximal, complemented
4	D₄×D₃₉₇	C₂	1	6	yes	maximal, complemented
5	#41×C₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₉₇	C₂²	1	6	yes	complemented
7	Dic₇₉₄	C₂²	1	1	yes	characteristic, complemented
8	D₁₅₈₈	C₂²	1	6	yes	complemented
9	D₁₅₈₈	C₂²	1	3	yes	complemented
10	C₂²⋊D₃₉₇	C₂²	1	6	yes	complemented
11	C₄×D₃₉₇	C₂²	1	6	yes	complemented
12	Q₈×C₃₉₇	C₂²	1	1	yes	characteristic, complemented
13	D₄×C₃₉₇	C₂²	1	3	yes	complemented
14	C₂×C₄×C₃₉₇	C₂²	1	3	yes	complemented
15	C₂²×C₃₉₇	C₂³	1	3	yes	—
16	C₂×D₃₉₇	—	4	6	no	—
17	C₂×D₃₉₇	C₂³	1	6	yes	—
18	C₄×C₃₉₇	C₂³	1	1	yes	characteristic
19	C₄×C₃₉₇	C₂³	1	3	yes	—
20	Dic₃₉₇	C₂³	1	2	yes	—
21	C₂×C₃₉₇	—	2	3	no	—
22	C₂×C₃₉₇	C₂⁴	1	1	yes	characteristic
23	D₃₉₇	—	2	6	no	—
24	C₃₉₇	H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
25	H(5,𝔽₂)	—	397	1	no	maximal, Sylow
26	#41	—	397	2	no	—
27	#41	—	397	3	no	—
28	#41	D₃₉₇	1	1	yes	characteristic, complemented
29	C₂×D₄	—	397	6	no	—
30	C₂×D₄	—	397	3	no	—
31	C₂³	—	397	6	no	—
32	Q₈	—	397	1	no	—
33	Q₈	C₂×D₃₉₇	1	1	yes	characteristic, complemented
34	D₄	—	397	6	no	—
35	D₄	—	397	6	no	—
36	D₄	—	397	3	no	—
37	D₄	C₂×D₃₉₇	1	3	yes	complemented
38	C₂×C₄	—	397	6	no	—
39	C₂×C₄	C₂×D₃₉₇	1	3	yes	complemented
40	C₂²	—	1588	6	no	—
41	C₂²	—	397	6	no	—
42	C₂²	C₂²×D₃₉₇	1	3	yes	—
43	C₄	—	397	2	no	—
44	C₄	C₂²×D₃₉₇	1	3	yes	—
45	C₄	C₂²×D₃₉₇	1	1	yes	characteristic
46	C₂	—	2	3	no	—
47	C₂	—	794	6	no	—
48	C₂	C₂³×D₃₉₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×D₃₉₇
commutator, socle, cosocle	C₂×C₃₉₇	C₂⁴
Fitting	#41×C₃₉₇	C₂
hypercenter	#41	D₃₉₇
hypocenter	C₃₉₇	H(5,𝔽₂)
LCS	G ▷ C₂×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂ ◁ #41

Complements

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

N	H ≅ G/N	type
1. #20263732	C₂	semidirect
2. #20263731	C₂	semidirect
3. C₂×D₁₅₈₈	C₂	semidirect
4. D₄×D₃₉₇	C₂	semidirect
5. #41×C₃₉₇	C₂	semidirect
6. C₂²×D₃₉₇	C₂²	semidirect
7. Dic₇₉₄	C₂²	semidirect
8. D₁₅₈₈	C₂²	semidirect
9. D₁₅₈₈	C₂²	semidirect
10. C₂²⋊D₃₉₇	C₂²	semidirect
11. C₄×D₃₉₇	C₂²	semidirect
12. Q₈×C₃₉₇	C₂²	semidirect
13. D₄×C₃₉₇	C₂²	semidirect
14. C₂×C₄×C₃₉₇	C₂²	semidirect
24. C₃₉₇	H(5,𝔽₂)	semidirect
28. #41	D₃₉₇	semidirect
33. Q₈	C₂×D₃₉₇	semidirect
37. D₄	C₂×D₃₉₇	semidirect
39. 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
#20263732	2
#20263731	2
C₂×D₁₅₈₈	2
D₄×D₃₉₇	2
#41×C₃₉₇	2
H(5,𝔽₂)	397