Subgroups — #27163175

Order 12704.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#13097544	C₂	1	2	yes	maximal, complemented
2	C₂²⋊#13097544	C₂	1	2	yes	maximal
3	C₄×#13097544	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×D₃₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13097544	C₂²	1	4	yes	—
6	C₂²×D₃₉₇	C₂²	1	1	yes	characteristic
7	C₂×Dic₃₉₇	C₄	1	1	yes	characteristic, complemented
8	C₄×D₃₉₇	C₄	1	1	yes	characteristic, complemented
9	C₄×D₃₉₇	C₄	1	1	yes	characteristic, complemented
10	C₄×D₃₉₇	C₂²	1	1	yes	characteristic
11	C₄×D₃₉₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₃₉₇	C₄	1	1	yes	characteristic, complemented
13	C₂²×C₃₉₇	C₂×C₄	1	1	yes	characteristic
14	C₂×D₃₉₇	—	2	2	no	—
15	C₂×D₃₉₇	C₂×C₄	1	1	yes	characteristic
16	C₂×D₃₉₇	C₂³	1	1	yes	characteristic
17	#13097544	—	2	4	no	—
18	C₄×C₃₉₇	C₂×C₄	1	1	yes	characteristic
19	C₄×C₃₉₇	C₂×C₄	1	1	yes	characteristic
20	Dic₃₉₇	C₂×C₄	1	1	yes	characteristic
21	Dic₃₉₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃₉₇	—	2	1	no	—
23	C₂×C₃₉₇	C₂²×C₄	1	1	yes	characteristic
24	D₃₉₇	—	2	1	no	—
25	D₃₉₇	#41	1	2	yes	—
26	C₃₉₇	#117	1	1	yes	characteristic, complemented, Sylow
27	#117	—	397	1	no	maximal, Sylow
28	C₂²×C₄	—	397	1	no	—
29	C₄⋊C₄	—	397	2	no	—
30	C₂²⋊C₄	—	397	2	no	—
31	C₄²	—	397	2	no	—
32	C₂³	—	397	1	no	—
33	C₂×C₄	—	397	4	no	—
34	C₂×C₄	—	397	1	no	—
35	C₂×C₄	—	397	1	no	—
36	C₂×C₄	—	397	1	no	—
37	C₂×C₄	—	397	1	no	—
38	C₂×C₄	—	397	1	no	—
39	C₂×C₄	#13097544	1	1	yes	characteristic, complemented
40	C₂²	—	794	2	no	—
41	C₂²	—	397	1	no	—
42	C₂²	—	397	1	no	—
43	C₂²	C₂×#13097544	1	1	yes	characteristic
44	C₄	—	794	4	no	—
45	C₄	—	397	1	no	—
46	C₄	—	397	1	no	—
47	C₄	C₂×#13097544	1	1	yes	characteristic
48	C₄	C₂×#13097544	1	1	yes	characteristic
49	C₂	—	2	1	no	—
50	C₂	—	794	1	no	—
51	C₂	—	397	2	no	—
52	C₂	C₂²×#13097544	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#13097544
commutator, socle	C₂×C₃₉₇	C₂²×C₄
Frattini	C₂	C₂²×#13097544
Fitting	C₂×C₄×C₃₉₇	C₄
prime radical	C₂²×D₃₉₇	C₂²
cosocle	C₂×D₃₉₇	C₂³
hypercenter	C₂×C₄	#13097544
hypocenter	C₃₉₇	#117
LCS	G ▷ C₂×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
1. C₄⋊#13097544	C₂	semidirect
3. C₄×#13097544	C₂	semidirect
7. C₂×Dic₃₉₇	C₄	semidirect
8. C₄×D₃₉₇	C₄	semidirect
9. C₄×D₃₉₇	C₄	semidirect
12. C₂×C₄×C₃₉₇	C₄	semidirect
26. C₃₉₇	#117	semidirect
39. C₂×C₄	#13097544	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#13097544	2
C₂²⋊#13097544	2
C₄×#13097544	2
C₂×C₄×D₃₉₇	2
#117	397