Subgroups — #27163129

Order 12704.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263731	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₇₉₄	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₁₅₈₈	C₂	1	2	yes	maximal, complemented
4	#20263720	C₂	1	2	yes	maximal, complemented
5	M₁₆×C₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₇₉₄	—	2	1	no	—
7	Dic₇₉₄	C₂²	1	2	yes	—
8	Dic₇₉₄	C₂²	1	1	yes	characteristic
9	D₁₅₈₈	C₂²	1	1	yes	characteristic
10	C₂²⋊D₃₉₇	—	2	1	no	—
11	C₂×Dic₃₉₇	—	2	1	no	—
12	C₄×D₃₉₇	—	2	1	no	—
13	C₂×C₄×C₃₉₇	C₂²	1	1	yes	characteristic
14	C₈×C₃₉₇	C₂²	1	2	yes	—
15	C₂²×C₃₉₇	D₄	1	1	yes	characteristic
16	C₂×D₃₉₇	—	2	1	no	—
17	C₄×C₃₉₇	D₄	1	1	yes	characteristic
18	C₄×C₃₉₇	C₂³	1	1	yes	characteristic
19	Dic₃₉₇	—	2	1	no	—
20	Dic₃₉₇	—	2	2	no	—
21	C₂×C₃₉₇	—	2	1	no	—
22	C₂×C₃₉₇	C₂×D₄	1	1	yes	characteristic
23	D₃₉₇	—	4	1	no	—
24	C₃₉₇	#137	1	1	yes	characteristic, complemented, Sylow
25	#137	—	397	1	no	maximal, Sylow
26	#41	—	397	1	no	—
27	C₂×Q₈	—	397	1	no	—
28	Q₁₆	—	397	2	no	—
29	SD₁₆	—	397	2	no	—
30	M₁₆	D₃₉₇	1	1	yes	characteristic, complemented
31	Q₈	—	794	1	no	—
32	Q₈	—	397	2	no	—
33	Q₈	—	397	1	no	—
34	D₄	—	794	1	no	—
35	D₄	—	397	1	no	—
36	C₂×C₄	—	794	1	no	—
37	C₂×C₄	—	794	1	no	—
38	C₂×C₄	C₂×D₃₉₇	1	1	yes	characteristic
39	C₈	C₂×D₃₉₇	1	2	yes	—
40	C₂²	—	794	1	no	—
41	C₂²	D₁₅₈₈	1	1	yes	characteristic
42	C₄	—	794	1	no	—
43	C₄	—	794	2	no	—
44	C₄	D₁₅₈₈	1	1	yes	characteristic
45	C₄	C₂²×D₃₉₇	1	1	yes	characteristic
46	C₂	—	2	1	no	—
47	C₂	—	1588	1	no	—
48	C₂	C₂×D₁₅₈₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₁₅₈₈
commutator, cosocle	C₄×C₃₉₇	C₂³
Frattini	C₄	C₂²×D₃₉₇
Fitting	M₁₆×C₃₉₇	C₂
socle	C₂×C₃₉₇	C₂×D₄
prime radical	#20263731	C₂
hypercenter	M₁₆	D₃₉₇
hypocenter	C₃₉₇	#137
LCS	G ▷ C₄×C₃₉₇ ▷ C₂×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₇₉₄	C₂	semidirect
3. Dic₁₅₈₈	C₂	semidirect
4. #20263720	C₂	semidirect
5. M₁₆×C₃₉₇	C₂	semidirect
24. C₃₉₇	#137	semidirect
30. M₁₆	D₃₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20263731	2
C₂×Dic₇₉₄	2
Dic₁₅₈₈	2
#20263720	2
M₁₆×C₃₉₇	2
#137	397