Subgroups — #38763434

Order 25408.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃₉₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#27163165	C₂	1	1	yes	maximal, characteristic, complemented
3	D₈×D₃₉₇	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#13097544	—	2	1	no	—
5	#20263734	—	2	1	no	—
6	D₄×D₃₉₇	—	2	1	no	—
7	D₃₁₇₆	C₄	1	1	yes	characteristic, complemented
8	#20263716	—	2	1	no	—
9	C₈×D₃₉₇	C₂²	1	1	yes	characteristic
10	D₈×C₃₉₇	C₄	1	1	yes	characteristic, complemented
11	C₂×#13097544	—	4	1	no	—
12	C₂²×D₃₉₇	—	4	1	no	—
13	D₁₅₈₈	—	2	1	no	—
14	C₂²⋊D₃₉₇	—	4	1	no	—
15	C₄×D₃₉₇	D₄	1	1	yes	characteristic
16	#15691787	D₄	1	1	yes	characteristic
17	D₄×C₃₉₇	—	2	1	no	—
18	C₈×C₃₉₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₃₉₇	—	4	1	no	—
20	C₂×D₃₉₇	—	4	1	no	—
21	C₂×D₃₉₇	—	8	1	no	—
22	C₂×D₃₉₇	—	8	1	no	—
23	C₂×D₃₉₇	D₈	1	1	yes	characteristic
24	#13097544	—	8	1	no	—
25	C₄×C₃₉₇	C₂²⋊C₄	1	1	yes	characteristic
26	Dic₃₉₇	SD₁₆	1	1	yes	characteristic
27	C₂×C₃₉₇	—	8	1	no	—
28	C₂×C₃₉₇	#102	1	1	yes	characteristic
29	D₃₉₇	—	8	1	no	—
30	D₃₉₇	D₁₆	1	1	yes	characteristic
31	D₃₉₇	SD₃₂	1	1	yes	characteristic
32	C₃₉₇	#357	1	1	yes	characteristic, complemented, Sylow
33	#357	—	397	1	no	maximal, Sylow
34	C₂×D₈	—	397	1	no	—
35	C₂×C₁₆	—	397	1	no	—
36	#107	—	397	1	no	—
37	C₂×D₄	—	794	1	no	—
38	D₈	—	794	1	no	—
39	D₈	—	397	1	no	—
40	D₈	#13097544	1	1	yes	characteristic, complemented
41	C₂×C₈	—	397	1	no	—
42	C₄⋊C₄	—	794	1	no	—
43	C₁₆	—	794	1	no	—
44	C₂³	—	1588	1	no	—
45	D₄	—	2	1	no	—
46	D₄	—	794	1	no	—
47	D₄	—	1588	1	no	—
48	C₂×C₄	—	1588	1	no	—
49	C₂×C₄	—	397	1	no	—
50	C₈	—	397	1	no	—
51	C₈	C₂×#13097544	1	1	yes	characteristic
52	C₂²	—	4	1	no	—
53	C₂²	—	1588	1	no	—
54	C₂²	—	3176	1	no	—
55	C₂²	—	3176	1	no	—
56	C₂²	—	397	1	no	—
57	C₄	—	3176	1	no	—
58	C₄	—	397	1	no	—
59	C₄	C₂²⋊#13097544	1	1	yes	characteristic
60	C₂	—	8	1	no	—
61	C₂	—	3176	1	no	—
62	C₂	—	397	1	no	—
63	C₂	—	397	1	no	—
64	C₂	#27163158	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27163158
commutator	C₈×C₃₉₇	C₂×C₄
Frattini	C₈	C₂×#13097544
Fitting	D₈×C₃₉₇	C₄
socle	C₂×C₃₉₇	#102
prime radical	D₈×D₃₉₇	C₂
cosocle	C₈×D₃₉₇	C₂²
hypercenter	D₈	#13097544
hypocenter	C₃₉₇	#357
LCS	G ▷ C₈×C₃₉₇ ▷ C₄×C₃₉₇ ▷ C₂×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. D₃₉₇⋊C₁₆	C₂	semidirect
2. #27163165	C₂	semidirect
7. D₃₁₇₆	C₄	semidirect
10. D₈×C₃₉₇	C₄	semidirect
32. C₃₉₇	#357	semidirect
40. D₈	#13097544	semidirect

Maximal subgroups

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

subgroup	index
D₃₉₇⋊C₁₆	2
#27163165	2
D₈×D₃₉₇	2
#357	397