Subgroups — #38763370

Order 25408.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#15691794	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#15691794	C₂	1	1	yes	maximal, characteristic, complemented
3	#27163058	C₂	1	1	yes	maximal, characteristic
4	C₂×#15691794	—	2	1	no	—
5	C₂×#15691794	—	2	1	no	—
6	C₂×D₁₅₈₈	C₄	1	1	yes	characteristic
7	C₂×C₄×D₃₉₇	—	2	1	no	—
8	#20263708	—	2	1	no	—
9	C₄×Dic₃₉₇	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₃₉₇	C₄	1	1	yes	characteristic
11	#15691794	—	2	2	no	—
12	#15691794	—	4	1	no	—
13	C₂²×D₃₉₇	—	2	1	no	—
14	D₁₅₈₈	—	2	1	no	—
15	D₁₅₈₈	C₈	1	2	yes	complemented
16	C₂×Dic₃₉₇	D₄	1	1	yes	characteristic
17	C₂×Dic₃₉₇	D₄	1	1	yes	characteristic
18	C₄×D₃₉₇	—	4	1	no	—
19	C₄×D₃₉₇	—	4	1	no	—
20	C₂×C₄×C₃₉₇	—	2	1	no	—
21	C₂×C₄×C₃₉₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₃₉₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×D₃₉₇	—	2	2	no	—
24	C₂×D₃₉₇	—	4	1	no	—
25	C₂×D₃₉₇	—	4	1	no	—
26	C₄×C₃₉₇	—	4	1	no	—
27	C₄×C₃₉₇	C₂×C₈	1	1	yes	characteristic
28	C₄×C₃₉₇	M₁₆	1	1	yes	characteristic
29	Dic₃₉₇	—	2	1	no	—
30	Dic₃₉₇	D₈	1	1	yes	characteristic
31	Dic₃₉₇	SD₁₆	1	1	yes	characteristic
32	C₂×C₃₉₇	C₂²⋊C₈	1	1	yes	characteristic
33	C₂×C₃₉₇	#102	1	1	yes	characteristic
34	C₂×C₃₉₇	#104	1	1	yes	characteristic
35	D₃₉₇	—	4	2	no	—
36	C₃₉₇	D₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
37	D₄⋊C₈	—	397	1	no	maximal, Sylow
38	C₄×D₄	—	397	1	no	—
39	C₄⋊C₈	—	397	1	no	—
40	C₄×C₈	—	397	1	no	—
41	C₂×D₄	—	397	1	no	—
42	C₂²×C₄	—	794	1	no	—
43	C₂×C₈	—	794	1	no	—
44	C₂×C₈	—	794	1	no	—
45	C₄⋊C₄	#13097544	1	1	yes	characteristic
46	C₂²⋊C₄	—	794	1	no	—
47	C₄²	—	397	1	no	—
48	C₂³	—	794	1	no	—
49	D₄	—	794	1	no	—
50	D₄	—	397	2	no	—
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	1588	1	no	—
53	C₂×C₄	—	1588	1	no	—
54	C₂×C₄	—	397	1	no	—
55	C₂×C₄	—	397	1	no	—
56	C₂×C₄	C₂×#13097544	1	1	yes	characteristic
57	C₈	—	794	2	no	—
58	C₈	—	1588	1	no	—
59	C₂²	—	794	2	no	—
60	C₂²	—	1588	1	no	—
61	C₂²	—	1588	1	no	—
62	C₂²	C₂²⋊#13097544	1	1	yes	characteristic
63	C₄	—	4	1	no	—
64	C₄	—	794	1	no	—
65	C₄	—	397	1	no	—
66	C₄	—	397	1	no	—
67	C₄	D₃₉₇⋊C₈	1	1	yes	characteristic
68	C₄	#20263741	1	1	yes	characteristic
69	C₂	—	1588	2	no	—
70	C₂	#27163153	1	1	yes	characteristic
71	C₂	#27163158	1	1	yes	characteristic
72	C₂	#27163161	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#13097544
commutator	C₄×C₃₉₇	C₂×C₈
Frattini	C₂×C₄	C₂×#13097544
Fitting	C₄⋊C₄×C₃₉₇	C₄
socle	C₂²×C₃₉₇	C₂²⋊C₄
prime radical	C₂×D₁₅₈₈	C₄
cosocle	C₄×Dic₃₉₇	C₂²
hypercenter	C₄⋊C₄	#13097544
hypocenter	C₃₉₇	D₄⋊C₈
LCS	G ▷ C₄×C₃₉₇ ▷ C₂×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₄⋊#15691794	C₂	semidirect
2. C₄×#15691794	C₂	semidirect
15. D₁₅₈₈	C₈	semidirect
36. C₃₉₇	D₄⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#15691794	2
C₄×#15691794	2
#27163058	2
D₄⋊C₈	397