Subgroups — #27163071

Order 12704.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₅₈₈	C₂	1	2	yes	maximal, complemented
2	C₂×D₁₅₈₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#20263708	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₄×C₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₉₇	C₂²	1	2	yes	—
7	C₂²×D₃₉₇	C₂²	1	1	yes	characteristic
8	D₁₅₈₈	—	2	4	no	—
9	D₁₅₈₈	—	2	2	no	—
10	C₂×Dic₃₉₇	C₂²	1	1	yes	characteristic
11	C₄×D₃₉₇	—	2	2	no	—
12	C₂×C₄×C₃₉₇	C₂²	1	2	yes	—
13	C₂×C₄×C₃₉₇	C₂²	1	1	yes	characteristic
14	C₂²×C₃₉₇	C₂³	1	1	yes	characteristic
15	C₂×D₃₉₇	—	2	1	no	—
16	C₂×D₃₉₇	—	2	1	no	—
17	C₂×D₃₉₇	—	2	2	no	—
18	C₂×D₃₉₇	—	2	2	no	—
19	C₂×D₃₉₇	—	2	2	no	—
20	C₂×D₃₉₇	D₄	1	2	yes	complemented
21	C₄×C₃₉₇	—	2	2	no	—
22	C₄×C₃₉₇	D₄	1	2	yes	complemented
23	Dic₃₉₇	—	2	1	no	—
24	C₂×C₃₉₇	C₂×D₄	1	1	yes	characteristic
25	C₂×C₃₉₇	C₂×D₄	1	1	yes	characteristic
26	C₂×C₃₉₇	#41	1	1	yes	characteristic
27	D₃₉₇	—	2	2	no	—
28	D₃₉₇	—	4	2	no	—
29	C₃₉₇	#121	1	1	yes	characteristic, complemented, Sylow
30	#121	—	397	1	no	maximal, Sylow
31	C₂×D₄	—	397	2	no	—
32	C₂×D₄	—	397	1	no	—
33	C₂²×C₄	—	397	1	no	—
34	C₄⋊C₄	D₃₉₇	1	1	yes	characteristic, complemented
35	C₂²⋊C₄	—	397	2	no	—
36	C₂³	—	397	2	no	—
37	C₂³	—	397	1	no	—
38	D₄	—	794	4	no	—
39	D₄	—	794	2	no	—
40	C₂×C₄	—	794	2	no	—
41	C₂×C₄	—	397	1	no	—
42	C₂×C₄	C₂×D₃₉₇	1	2	yes	—
43	C₂×C₄	C₂×D₃₉₇	1	1	yes	characteristic
44	C₂²	—	794	1	no	—
45	C₂²	—	794	1	no	—
46	C₂²	—	794	2	no	—
47	C₂²	—	794	2	no	—
48	C₂²	—	794	2	no	—
49	C₂²	—	397	2	no	—
50	C₂²	C₂²×D₃₉₇	1	1	yes	characteristic
51	C₄	—	2	2	no	—
52	C₄	—	794	1	no	—
53	C₄	D₁₅₈₈	1	2	yes	complemented
54	C₂	—	794	2	no	—
55	C₂	—	1588	2	no	—
56	C₂	D₄×D₃₉₇	1	1	yes	characteristic
57	C₂	C₂×D₁₅₈₈	1	1	yes	characteristic
58	C₂	#20263732	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃₉₇
commutator, socle, cosocle	C₂²×C₃₉₇	C₂³
Fitting	C₄⋊C₄×C₃₉₇	C₂
hypercenter	C₄⋊C₄	D₃₉₇
hypocenter	C₃₉₇	#121
LCS	G ▷ C₂²×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×D₁₅₈₈	C₂	semidirect
2. C₂×D₁₅₈₈	C₂	semidirect
3. C₂×C₄×D₃₉₇	C₂	semidirect
4. #20263708	C₂	semidirect
5. C₄⋊C₄×C₃₉₇	C₂	semidirect
20. C₂×D₃₉₇	D₄	semidirect
22. C₄×C₃₉₇	D₄	semidirect
29. C₃₉₇	#121	semidirect
34. C₄⋊C₄	D₃₉₇	semidirect
53. C₄	D₁₅₈₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₅₈₈	2
C₂×C₄×D₃₉₇	2
#20263708	2
C₄⋊C₄×C₃₉₇	2
#121	397