Subgroups — #38762898

Order 25408.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20263716	C₂	1	4	yes	maximal, complemented
2	#27163093	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#15691787	C₂	1	1	yes	maximal, characteristic, complemented
4	#127×C₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₅₈₈	—	2	2	no	—
6	C₂×D₁₅₈₈	C₂²	1	2	yes	—
7	#20263716	—	2	8	no	—
8	C₂×#15691787	C₂²	1	2	yes	—
9	C₂×D₄×C₃₉₇	—	2	2	no	—
10	C₂×D₄×C₃₉₇	C₂²	1	2	yes	—
11	C₄²×C₃₉₇	C₂²	1	1	yes	characteristic
12	C₂²×D₃₉₇	—	2	2	no	—
13	D₁₅₈₈	—	2	4	no	—
14	D₁₅₈₈	—	4	4	no	—
15	#15691787	D₄	1	4	yes	complemented
16	C₂³×C₃₉₇	—	2	2	no	—
17	D₄×C₃₉₇	—	2	4	no	—
18	D₄×C₃₉₇	—	4	4	no	—
19	C₂×C₄×C₃₉₇	D₄	1	2	yes	—
20	C₂×C₄×C₃₉₇	C₂³	1	1	yes	characteristic
21	C₂²×C₃₉₇	—	4	2	no	—
22	C₂²×C₃₉₇	—	4	4	no	—
23	C₂²×C₃₉₇	C₂×D₄	1	1	yes	characteristic
24	C₂×D₃₉₇	—	4	2	no	—
25	C₂×D₃₉₇	—	4	4	no	—
26	C₄×C₃₉₇	D₈	1	4	yes	complemented
27	C₄×C₃₉₇	C₂×D₄	1	2	yes	—
28	C₂×C₃₉₇	—	8	2	no	—
29	C₂×C₃₉₇	#127	1	1	yes	characteristic
30	C₂×C₃₉₇	C₂×D₈	1	2	yes	—
31	D₃₉₇	—	8	2	no	—
32	C₃₉₇	#493	1	1	yes	characteristic, complemented, Sylow
33	#493	—	397	1	no	maximal, Sylow
34	C₂×D₈	—	397	4	no	—
35	#127	—	397	1	no	—
36	#127	D₃₉₇	1	1	yes	characteristic, complemented
37	C₄×C₈	—	397	1	no	—
38	C₂×D₄	—	2	2	no	—
39	C₂×D₄	—	794	2	no	—
40	C₂×D₄	—	397	2	no	—
41	C₂×D₄	C₂×D₃₉₇	1	2	yes	—
42	D₈	—	794	8	no	—
43	C₂×C₈	—	397	2	no	—
44	C₄²	C₂×D₃₉₇	1	1	yes	characteristic
45	C₂³	—	2	2	no	—
46	C₂³	—	794	2	no	—
47	D₄	—	2	4	no	—
48	D₄	—	4	4	no	—
49	D₄	—	794	4	no	—
50	D₄	—	1588	4	no	—
51	C₂×C₄	C₂²⋊D₃₉₇	1	2	yes	—
52	C₂×C₄	C₂²×D₃₉₇	1	1	yes	characteristic
53	C₈	—	397	4	no	—
54	C₂²	—	4	2	no	—
55	C₂²	—	4	4	no	—
56	C₂²	—	1588	2	no	—
57	C₂²	—	1588	4	no	—
58	C₂²	C₂×C₂²⋊D₃₉₇	1	1	yes	characteristic
59	C₄	#20263716	1	4	yes	complemented
60	C₄	D₄×D₃₉₇	1	2	yes	—
61	C₂	—	8	2	no	—
62	C₂	—	3176	2	no	—
63	C₂	#27163092	1	1	yes	characteristic
64	C₂	C₂×#20263716	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₃₉₇
commutator, cosocle	C₂×C₄×C₃₉₇	C₂³
Frattini	C₂×C₄	C₂²×D₃₉₇
Fitting	#127×C₃₉₇	C₂
socle	C₂²×C₃₉₇	C₂×D₄
hypercenter	#127	D₃₉₇
hypocenter	C₃₉₇	#493
LCS	G ▷ C₂×C₄×C₃₉₇ ▷ C₂×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₂×#20263716	C₂	semidirect
2. #27163093	C₂	semidirect
3. C₄×#15691787	C₂	semidirect
4. #127×C₃₉₇	C₂	semidirect
15. #15691787	D₄	semidirect
26. C₄×C₃₉₇	D₈	semidirect
32. C₃₉₇	#493	semidirect
36. #127	D₃₉₇	semidirect
59. C₄	#20263716	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20263716	2
#27163093	2
C₄×#15691787	2
#127×C₃₉₇	2
#493	397