Subgroups — #38762333

Order 25408.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27163118	C₂	1	1	yes	maximal, characteristic
2	#27163028	C₂	1	1	yes	maximal, characteristic, complemented
3	M₃₂×C₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#20263731	—	2	1	no	—
5	Dic₁₅₈₈	C₄	1	1	yes	characteristic, complemented
6	#20263720	—	2	1	no	—
7	D₃₁₇₆	C₄	1	1	yes	characteristic, complemented
8	C₄⋊Dic₃₉₇	—	2	1	no	—
9	C₂×C₈×C₃₉₇	C₂²	1	1	yes	characteristic
10	C₁₆×C₃₉₇	—	2	1	no	—
11	Dic₇₉₄	—	2	1	no	—
12	D₁₅₈₈	—	2	1	no	—
13	C₂²⋊D₃₉₇	—	4	1	no	—
14	C₂×Dic₃₉₇	—	4	1	no	—
15	C₄×D₃₉₇	—	4	1	no	—
16	C₂×C₄×C₃₉₇	D₄	1	1	yes	characteristic
17	C₈×C₃₉₇	C₂×C₄	1	1	yes	characteristic
18	C₈×C₃₉₇	D₄	1	1	yes	characteristic
19	C₂²×C₃₉₇	D₈	1	1	yes	characteristic
20	C₂×D₃₉₇	—	4	1	no	—
21	C₄×C₃₉₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₃₉₇	SD₁₆	1	1	yes	characteristic
23	Dic₃₉₇	—	4	1	no	—
24	Dic₃₉₇	—	8	1	no	—
25	C₂×C₃₉₇	—	2	1	no	—
26	C₂×C₃₉₇	#102	1	1	yes	characteristic
27	D₃₉₇	—	8	1	no	—
28	C₃₉₇	#360	1	1	yes	characteristic, complemented, Sylow
29	#360	—	397	1	no	maximal, Sylow
30	#135	—	397	1	no	—
31	M₃₂	D₃₉₇	1	1	yes	characteristic, complemented
32	#106	—	397	1	no	—
33	#41	—	794	1	no	—
34	Q₁₆	—	397	1	no	—
35	SD₁₆	—	794	1	no	—
36	D₈	—	397	1	no	—
37	C₂×C₈	C₂×D₃₉₇	1	1	yes	characteristic
38	C₄⋊C₄	—	794	1	no	—
39	C₁₆	—	2	1	no	—
40	Q₈	—	794	1	no	—
41	D₄	—	794	1	no	—
42	D₄	—	1588	1	no	—
43	C₂×C₄	—	1588	1	no	—
44	C₂×C₄	—	1588	1	no	—
45	C₂×C₄	D₁₅₈₈	1	1	yes	characteristic
46	C₈	C₄×D₃₉₇	1	1	yes	characteristic
47	C₈	C₂²⋊D₃₉₇	1	1	yes	characteristic
48	C₂²	—	1588	1	no	—
49	C₂²	D₃₁₇₆	1	1	yes	characteristic
50	C₄	—	1588	1	no	—
51	C₄	—	3176	1	no	—
52	C₄	#20263708	1	1	yes	characteristic
53	C₄	#20263720	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	3176	1	no	—
56	C₂	#27163018	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27163018
commutator	C₈×C₃₉₇	C₂×C₄
Frattini	C₂×C₈	C₂×D₃₉₇
Fitting	M₃₂×C₃₉₇	C₂
socle	C₂×C₃₉₇	#102
prime radical	#27163118	C₂
cosocle	C₂×C₈×C₃₉₇	C₂²
hypercenter	M₃₂	D₃₉₇
hypocenter	C₃₉₇	#360
LCS	G ▷ C₈×C₃₉₇ ▷ C₄×C₃₉₇ ▷ C₂×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. #27163028	C₂	semidirect
3. M₃₂×C₃₉₇	C₂	semidirect
5. Dic₁₅₈₈	C₄	semidirect
7. D₃₁₇₆	C₄	semidirect
28. C₃₉₇	#360	semidirect
31. M₃₂	D₃₉₇	semidirect

Maximal subgroups

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

subgroup	index
#27163118	2
#27163028	2
M₃₂×C₃₉₇	2
#360	397