Subgroups — #27163158

Order 12704.

50 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃₉₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄⋊#13097544	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×D₃₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13097544	—	2	1	no	—
5	#15691794	—	2	1	no	—
6	C₂²×D₃₉₇	—	2	1	no	—
7	D₁₅₈₈	C₄	1	1	yes	characteristic, complemented
8	C₂²⋊D₃₉₇	—	2	1	no	—
9	C₄×D₃₉₇	C₂²	1	1	yes	characteristic
10	D₄×C₃₉₇	C₄	1	1	yes	characteristic, complemented
11	C₂²×C₃₉₇	—	2	1	no	—
12	C₂×D₃₉₇	—	2	1	no	—
13	C₂×D₃₉₇	—	4	1	no	—
14	C₂×D₃₉₇	—	4	1	no	—
15	C₂×D₃₉₇	D₄	1	1	yes	characteristic
16	#13097544	—	4	1	no	—
17	C₄×C₃₉₇	C₂×C₄	1	1	yes	characteristic
18	Dic₃₉₇	D₄	1	1	yes	characteristic
19	C₂×C₃₉₇	—	4	1	no	—
20	C₂×C₃₉₇	C₂²⋊C₄	1	1	yes	characteristic
21	D₃₉₇	—	4	1	no	—
22	D₃₉₇	D₈	1	1	yes	characteristic
23	D₃₉₇	SD₁₆	1	1	yes	characteristic
24	C₃₉₇	#102	1	1	yes	characteristic, complemented, Sylow
25	#102	—	397	1	no	maximal, Sylow
26	C₂×D₄	—	397	1	no	—
27	C₂×C₈	—	397	1	no	—
28	C₄⋊C₄	—	397	1	no	—
29	C₂³	—	794	1	no	—
30	D₄	—	794	1	no	—
31	D₄	—	397	1	no	—
32	D₄	#13097544	1	1	yes	characteristic, complemented
33	C₂×C₄	—	794	1	no	—
34	C₂×C₄	—	397	1	no	—
35	C₈	—	794	1	no	—
36	C₂²	—	2	1	no	—
37	C₂²	—	794	1	no	—
38	C₂²	—	1588	1	no	—
39	C₂²	—	1588	1	no	—
40	C₂²	—	397	1	no	—
41	C₄	—	1588	1	no	—
42	C₄	—	397	1	no	—
43	C₄	C₂×#13097544	1	1	yes	characteristic
44	C₂	—	4	1	no	—
45	C₂	—	1588	1	no	—
46	C₂	—	397	1	no	—
47	C₂	—	397	1	no	—
48	C₂	C₂²⋊#13097544	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²⋊#13097544
commutator	C₄×C₃₉₇	C₂×C₄
Frattini	C₄	C₂×#13097544
Fitting	D₄×C₃₉₇	C₄
socle	C₂×C₃₉₇	C₂²⋊C₄
prime radical	D₄×D₃₉₇	C₂
cosocle	C₄×D₃₉₇	C₂²
hypercenter	D₄	#13097544
hypocenter	C₃₉₇	#102
LCS	G ▷ C₄×C₃₉₇ ▷ C₂×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₄

Complements

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

N	H ≅ G/N	type
1. D₃₉₇⋊C₈	C₂	semidirect
2. C₄⋊#13097544	C₂	semidirect
7. D₁₅₈₈	C₄	semidirect
10. D₄×C₃₉₇	C₄	semidirect
24. C₃₉₇	#102	semidirect
32. D₄	#13097544	semidirect

Maximal subgroups

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

subgroup	index
D₃₉₇⋊C₈	2
C₄⋊#13097544	2
D₄×D₃₉₇	2
#102	397