Subgroups — #27163058

Order 12704.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₅₈₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₃₉₇	C₂	1	2	yes	maximal, complemented
3	#20263708	C₂	1	2	yes	maximal, complemented
4	C₄×Dic₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊C₄×C₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₉₇	C₂²	1	2	yes	—
7	D₁₅₈₈	C₄	1	4	yes	complemented
8	C₂×Dic₃₉₇	C₂²	1	1	yes	characteristic
9	C₂×Dic₃₉₇	C₂²	1	1	yes	characteristic
10	C₄×D₃₉₇	—	2	4	no	—
11	C₂×C₄×C₃₉₇	C₂²	1	2	yes	—
12	C₂×C₄×C₃₉₇	C₂²	1	1	yes	characteristic
13	C₂²×C₃₉₇	C₂³	1	1	yes	characteristic
14	C₂×D₃₉₇	—	2	2	no	—
15	C₂×D₃₉₇	—	2	2	no	—
16	C₂×D₃₉₇	C₂×C₄	1	4	yes	complemented
17	C₄×C₃₉₇	—	2	2	no	—
18	C₄×C₃₉₇	C₂×C₄	1	2	yes	complemented
19	Dic₃₉₇	—	2	1	no	—
20	Dic₃₉₇	D₄	1	2	yes	complemented
21	C₂×C₃₉₇	C₂²×C₄	1	1	yes	characteristic
22	C₂×C₃₉₇	C₂×D₄	1	1	yes	characteristic
23	C₂×C₃₉₇	#41	1	1	yes	characteristic
24	D₃₉₇	—	2	4	no	—
25	C₃₉₇	C₄×D₄	1	1	yes	characteristic, complemented, Sylow
26	C₄×D₄	—	397	1	no	maximal, Sylow
27	C₂×D₄	—	397	1	no	—
28	C₂²×C₄	—	397	2	no	—
29	C₄⋊C₄	D₃₉₇	1	1	yes	characteristic, complemented
30	C₂²⋊C₄	—	397	2	no	—
31	C₄²	—	397	1	no	—
32	C₂³	—	397	2	no	—
33	D₄	—	397	4	no	—
34	C₂×C₄	—	794	4	no	—
35	C₂×C₄	—	397	1	no	—
36	C₂×C₄	—	397	1	no	—
37	C₂×C₄	C₂×D₃₉₇	1	2	yes	—
38	C₂×C₄	C₂×D₃₉₇	1	1	yes	characteristic
39	C₂²	—	794	2	no	—
40	C₂²	—	794	2	no	—
41	C₂²	—	397	4	no	—
42	C₂²	C₂²×D₃₉₇	1	1	yes	characteristic
43	C₄	—	2	2	no	—
44	C₄	—	794	1	no	—
45	C₄	—	397	2	no	—
46	C₄	C₄×D₃₉₇	1	2	yes	complemented
47	C₂	—	794	4	no	—
48	C₂	C₂×C₄×D₃₉₇	1	1	yes	characteristic
49	C₂	D₄×D₃₉₇	1	1	yes	characteristic
50	C₂	#20263732	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃₉₇
commutator	C₂×C₃₉₇	C₂²×C₄
Fitting	C₄⋊C₄×C₃₉₇	C₂
socle, cosocle	C₂²×C₃₉₇	C₂³
prime radical	C₂×D₁₅₈₈	C₂
hypercenter	C₄⋊C₄	D₃₉₇
hypocenter	C₃₉₇	C₄×D₄
LCS	G ▷ C₂×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₃₉₇	C₂	semidirect
3. #20263708	C₂	semidirect
4. C₄×Dic₃₉₇	C₂	semidirect
5. C₄⋊C₄×C₃₉₇	C₂	semidirect
7. D₁₅₈₈	C₄	semidirect
16. C₂×D₃₉₇	C₂×C₄	semidirect
18. C₄×C₃₉₇	C₂×C₄	semidirect
20. Dic₃₉₇	D₄	semidirect
25. C₃₉₇	C₄×D₄	semidirect
29. C₄⋊C₄	D₃₉₇	semidirect
46. C₄	C₄×D₃₉₇	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×Dic₃₉₇	2
C₄⋊C₄×C₃₉₇	2
C₄×D₄	397