Subgroups — #27163117

Order 12704.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263732	C₂	1	1	yes	maximal, characteristic, complemented
2	#20263730	C₂	1	1	yes	maximal, characteristic, complemented
3	#20263721	C₂	1	1	yes	maximal, characteristic, complemented
4	#20263720	C₂	1	1	yes	maximal, characteristic, complemented
5	#20263716	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×D₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
7	SD₁₆×C₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	Dic₇₉₄	C₂²	1	1	yes	characteristic
9	D₁₅₈₈	—	2	1	no	—
10	D₁₅₈₈	C₂²	1	1	yes	characteristic
11	C₂²⋊D₃₉₇	—	2	1	no	—
12	C₂×Dic₃₉₇	—	2	1	no	—
13	C₄×D₃₉₇	—	2	1	no	—
14	C₄×D₃₉₇	C₂²	1	1	yes	characteristic
15	#15691787	C₂²	1	1	yes	characteristic
16	Q₈×C₃₉₇	C₂²	1	1	yes	characteristic
17	D₄×C₃₉₇	C₂²	1	1	yes	characteristic
18	C₈×C₃₉₇	C₂²	1	1	yes	characteristic
19	C₂²×C₃₉₇	—	2	1	no	—
20	C₂×D₃₉₇	—	2	1	no	—
21	C₂×D₃₉₇	D₄	1	1	yes	characteristic
22	C₄×C₃₉₇	—	2	1	no	—
23	C₄×C₃₉₇	C₂³	1	1	yes	characteristic
24	Dic₃₉₇	—	2	1	no	—
25	Dic₃₉₇	D₄	1	1	yes	characteristic
26	C₂×C₃₉₇	—	4	1	no	—
27	C₂×C₃₉₇	C₂×D₄	1	1	yes	characteristic
28	D₃₉₇	—	2	1	no	—
29	D₃₉₇	—	4	1	no	—
30	C₃₉₇	#135	1	1	yes	characteristic, complemented, Sylow
31	#135	—	397	1	no	maximal, Sylow
32	#41	—	397	1	no	—
33	#41	—	397	1	no	—
34	Q₁₆	—	397	1	no	—
35	SD₁₆	—	397	1	no	—
36	SD₁₆	D₃₉₇	1	1	yes	characteristic, complemented
37	D₈	—	397	1	no	—
38	C₂×C₈	—	397	1	no	—
39	Q₈	—	397	1	no	—
40	Q₈	C₂×D₃₉₇	1	1	yes	characteristic
41	D₄	—	794	1	no	—
42	D₄	—	794	1	no	—
43	D₄	—	397	1	no	—
44	D₄	C₂×D₃₉₇	1	1	yes	characteristic
45	C₂×C₄	—	794	1	no	—
46	C₂×C₄	—	794	1	no	—
47	C₂×C₄	—	397	1	no	—
48	C₈	—	397	1	no	—
49	C₈	C₂×D₃₉₇	1	1	yes	characteristic
50	C₂²	—	2	1	no	—
51	C₂²	—	794	1	no	—
52	C₂²	—	397	1	no	—
53	C₄	—	2	1	no	—
54	C₄	—	794	1	no	—
55	C₄	—	397	1	no	—
56	C₄	C₂²×D₃₉₇	1	1	yes	characteristic
57	C₂	—	4	1	no	—
58	C₂	—	794	1	no	—
59	C₂	—	1588	1	no	—
60	C₂	D₄×D₃₉₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₃₉₇
commutator, cosocle	C₄×C₃₉₇	C₂³
Frattini	C₄	C₂²×D₃₉₇
Fitting	SD₁₆×C₃₉₇	C₂
socle	C₂×C₃₉₇	C₂×D₄
hypercenter	SD₁₆	D₃₉₇
hypocenter	C₃₉₇	#135
LCS	G ▷ C₄×C₃₉₇ ▷ C₂×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

Complements

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

N	H ≅ G/N	type
1. #20263732	C₂	semidirect
2. #20263730	C₂	semidirect
3. #20263721	C₂	semidirect
4. #20263720	C₂	semidirect
5. #20263716	C₂	semidirect
6. C₈×D₃₉₇	C₂	semidirect
7. SD₁₆×C₃₉₇	C₂	semidirect
30. C₃₉₇	#135	semidirect
36. SD₁₆	D₃₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20263732	2
#20263730	2
#20263721	2
#20263720	2
#20263716	2
C₈×D₃₉₇	2
SD₁₆×C₃₉₇	2
#135	397