Subgroups — #38763437

Order 25408.

56 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	#27163167	C₂	1	1	yes	maximal, characteristic, complemented
3	#27163119	C₂	1	1	yes	maximal, characteristic
4	#20263741	—	2	1	no	—
5	#20263734	—	2	1	no	—
6	#20263732	—	2	1	no	—
7	#20263718	—	2	1	no	—
8	D₃₁₇₆	C₄	1	1	yes	characteristic
9	C₈×D₃₉₇	C₂²	1	1	yes	characteristic
10	Q₁₆×C₃₉₇	C₄	1	1	yes	characteristic
11	#15691794	—	4	1	no	—
12	D₁₅₈₈	—	2	1	no	—
13	D₁₅₈₈	—	4	1	no	—
14	C₄×D₃₉₇	—	4	1	no	—
15	C₄×D₃₉₇	D₄	1	1	yes	characteristic
16	#15691787	D₄	1	1	yes	characteristic
17	Q₈×C₃₉₇	—	2	1	no	—
18	C₈×C₃₉₇	C₂×C₄	1	1	yes	characteristic
19	C₂×D₃₉₇	—	4	1	no	—
20	C₂×D₃₉₇	SD₁₆	1	1	yes	characteristic
21	C₄×C₃₉₇	—	4	1	no	—
22	C₄×C₃₉₇	C₂²⋊C₄	1	1	yes	characteristic
23	Dic₃₉₇	D₈	1	1	yes	characteristic
24	C₂×C₃₉₇	#102	1	1	yes	characteristic
25	D₃₉₇	—	2	1	no	—
26	D₃₉₇	—	8	1	no	—
27	C₃₉₇	#359	1	1	yes	characteristic, complemented, Sylow
28	#359	—	397	1	no	maximal, Sylow
29	#135	—	397	1	no	—
30	C₂×C₁₆	—	397	1	no	—
31	#108	—	397	1	no	—
32	#41	—	794	1	no	—
33	Q₁₆	#13097544	1	1	yes	characteristic
34	SD₁₆	—	794	1	no	—
35	D₈	—	397	1	no	—
36	M₁₆	—	794	1	no	—
37	C₂×C₈	—	397	1	no	—
38	C₁₆	—	794	1	no	—
39	Q₈	—	2	1	no	—
40	D₄	—	794	1	no	—
41	D₄	—	1588	1	no	—
42	C₂×C₄	—	1588	1	no	—
43	C₂×C₄	—	397	1	no	—
44	C₈	—	1588	1	no	—
45	C₈	—	397	1	no	—
46	C₈	C₂×#13097544	1	1	yes	characteristic
47	C₂²	—	1588	1	no	—
48	C₂²	—	397	1	no	—
49	C₄	—	4	1	no	—
50	C₄	—	397	1	no	—
51	C₄	C₂²⋊#13097544	1	1	yes	characteristic
52	C₂	—	794	1	no	—
53	C₂	—	3176	1	no	—
54	C₂	#27163158	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27163158
commutator	C₈×C₃₉₇	C₂×C₄
Frattini	C₈	C₂×#13097544
Fitting	Q₁₆×C₃₉₇	C₄
socle	C₂×C₃₉₇	#102
prime radical	#27163119	C₂
cosocle	C₈×D₃₉₇	C₂²
hypercenter	Q₁₆	#13097544
hypocenter	C₃₉₇	#359
LCS	G ▷ C₈×C₃₉₇ ▷ C₄×C₃₉₇ ▷ C₂×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
1. D₃₉₇⋊C₁₆	C₂	semidirect
2. #27163167	C₂	semidirect
27. C₃₉₇	#359	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
#27163167	2
#27163119	2
#359	397