Subgroups — #38762932

Order 25408.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27163096	C₂	1	1	yes	maximal, characteristic
2	#27163029	C₂	1	2	yes	maximal
3	#27163027	C₂	1	2	yes	maximal
4	#27163007	C₂	1	1	yes	maximal, characteristic
5	#125×C₃₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₉₄	—	2	1	no	—
7	C₂×#15691787	C₂²	1	2	yes	—
8	C₄⋊Dic₃₉₇	—	2	1	no	—
9	C₄⋊Dic₃₉₇	C₂²	1	2	yes	—
10	C₄⋊C₄×C₃₉₇	—	2	1	no	—
11	C₄⋊C₄×C₃₉₇	—	2	1	no	—
12	C₄⋊C₄×C₃₉₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₃₉₇	C₂²	1	1	yes	characteristic
14	C₄²×C₃₉₇	C₂²	1	1	yes	characteristic
15	Dic₇₉₄	—	4	2	no	—
16	C₂×Dic₃₉₇	—	2	2	no	—
17	#15691787	Q₈	1	4	yes	complemented
18	C₂×C₄×C₃₉₇	—	2	1	no	—
19	C₂×C₄×C₃₉₇	—	2	1	no	—
20	C₂×C₄×C₃₉₇	D₄	1	1	yes	characteristic
21	C₂×C₄×C₃₉₇	D₄	1	1	yes	characteristic
22	C₂×C₄×C₃₉₇	C₂³	1	1	yes	characteristic
23	C₂²×C₃₉₇	C₂×D₄	1	1	yes	characteristic
24	C₄×C₃₉₇	—	2	1	no	—
25	C₄×C₃₉₇	—	2	1	no	—
26	C₄×C₃₉₇	—	4	1	no	—
27	C₄×C₃₉₇	—	4	1	no	—
28	C₄×C₃₉₇	C₂×Q₈	1	2	yes	—
29	Dic₃₉₇	—	4	2	no	—
30	C₂×C₃₉₇	C₄⋊Q₈	1	1	yes	characteristic
31	C₂×C₃₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic
32	C₂×C₃₉₇	#137	1	1	yes	characteristic
33	C₃₉₇	#501	1	1	yes	characteristic, complemented, Sylow
34	#501	—	397	1	no	maximal, Sylow
35	C₄⋊Q₈	—	397	1	no	—
36	#125	D₃₉₇	1	1	yes	characteristic
37	#107	—	397	2	no	—
38	#106	—	397	2	no	—
39	#97	—	397	1	no	—
40	C₂×Q₈	—	794	1	no	—
41	C₂×C₈	—	397	2	no	—
42	C₄⋊C₄	—	2	1	no	—
43	C₄⋊C₄	—	2	1	no	—
44	C₄⋊C₄	—	794	1	no	—
45	C₄⋊C₄	—	397	2	no	—
46	C₄⋊C₄	C₂×D₃₉₇	1	1	yes	characteristic
47	C₄⋊C₄	C₂×D₃₉₇	1	1	yes	characteristic
48	C₄²	C₂×D₃₉₇	1	1	yes	characteristic
49	Q₈	—	1588	2	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	794	2	no	—
53	C₂×C₄	C₂²⋊D₃₉₇	1	1	yes	characteristic
54	C₂×C₄	C₂²⋊D₃₉₇	1	1	yes	characteristic
55	C₂×C₄	C₂²×D₃₉₇	1	1	yes	characteristic
56	C₈	—	397	4	no	—
57	C₂²	C₂×C₂²⋊D₃₉₇	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	1	no	—
60	C₄	—	4	1	no	—
61	C₄	—	4	1	no	—
62	C₄	—	1588	2	no	—
63	C₄	Q₈×D₃₉₇	1	2	yes	—
64	C₂	#27163094	1	1	yes	characteristic
65	C₂	#27163121	1	1	yes	characteristic
66	C₂	#27163126	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₃₉₇
commutator, cosocle	C₂×C₄×C₃₉₇	C₂³
Frattini	C₂×C₄	C₂²×D₃₉₇
Fitting	#125×C₃₉₇	C₂
socle, prime radical	C₂²×C₃₉₇	C₂×D₄
hypercenter	#125	D₃₉₇
hypocenter	C₃₉₇	#501
LCS	G ▷ C₂×C₄×C₃₉₇ ▷ C₂×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
17. #15691787	Q₈	semidirect
33. C₃₉₇	#501	semidirect

Maximal subgroups

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

subgroup	index
#27163096	2
#27163029	2
#27163027	2
#27163007	2
#125×C₃₉₇	2
#501	397