Subgroups — #27163116

Order 12704.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263731	C₂	1	1	yes	maximal, characteristic, complemented
2	#20263721	C₂	1	1	yes	maximal, characteristic, complemented
3	#20263719	C₂	1	1	yes	maximal, characteristic, complemented
4	#20263718	C₂	1	1	yes	maximal, characteristic, complemented
5	#20263716	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15691787	C₂	1	1	yes	maximal, characteristic, complemented
7	#41×C₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	Dic₇₉₄	C₂²	1	1	yes	characteristic
9	D₁₅₈₈	C₂²	1	1	yes	characteristic
10	C₂²⋊D₃₉₇	—	2	1	no	—
11	C₄×D₃₉₇	—	2	1	no	—
12	#15691787	C₂²	1	1	yes	characteristic
13	#15691787	C₂²	1	1	yes	characteristic
14	Q₈×C₃₉₇	C₂²	1	1	yes	characteristic
15	D₄×C₃₉₇	—	2	1	no	—
16	D₄×C₃₉₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₃₉₇	—	2	1	no	—
18	C₂×C₄×C₃₉₇	C₂²	1	1	yes	characteristic
19	C₂²×C₃₉₇	—	2	1	no	—
20	C₂²×C₃₉₇	D₄	1	1	yes	characteristic
21	C₂×D₃₉₇	—	2	1	no	—
22	C₄×C₃₉₇	—	2	1	no	—
23	C₄×C₃₉₇	D₄	1	1	yes	characteristic
24	C₄×C₃₉₇	C₂³	1	1	yes	characteristic
25	Dic₃₉₇	—	2	1	no	—
26	C₂×C₃₉₇	—	2	1	no	—
27	C₂×C₃₉₇	—	4	1	no	—
28	C₂×C₃₉₇	C₂×D₄	1	1	yes	characteristic
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	D₃₉₇	1	1	yes	characteristic, complemented
34	Q₁₆	—	397	1	no	—
35	SD₁₆	—	397	1	no	—
36	SD₁₆	—	397	1	no	—
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	—	2	1	no	—
42	D₄	—	794	1	no	—
43	D₄	—	397	1	no	—
44	D₄	C₂×D₃₉₇	1	1	yes	characteristic
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	794	1	no	—
47	C₂×C₄	C₂×D₃₉₇	1	1	yes	characteristic
48	C₈	—	397	1	no	—
49	C₈	—	397	1	no	—
50	C₂²	—	2	1	no	—
51	C₂²	—	794	1	no	—
52	C₂²	C₂²⋊D₃₉₇	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	794	1	no	—
55	C₄	C₂²⋊D₃₉₇	1	1	yes	characteristic
56	C₄	C₂²×D₃₉₇	1	1	yes	characteristic
57	C₂	—	2	1	no	—
58	C₂	—	4	1	no	—
59	C₂	—	1588	1	no	—
60	C₂	C₂×C₂²⋊D₃₉₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊D₃₉₇
commutator, cosocle	C₄×C₃₉₇	C₂³
Frattini	C₄	C₂²×D₃₉₇
Fitting	#41×C₃₉₇	C₂
socle	C₂×C₃₉₇	C₂×D₄
hypercenter	#41	D₃₉₇
hypocenter	C₃₉₇	#135
LCS	G ▷ C₄×C₃₉₇ ▷ C₂×C₃₉₇ ▷ C₃₉₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. #20263731	C₂	semidirect
2. #20263721	C₂	semidirect
3. #20263719	C₂	semidirect
4. #20263718	C₂	semidirect
5. #20263716	C₂	semidirect
6. C₂×#15691787	C₂	semidirect
7. #41×C₃₉₇	C₂	semidirect
30. C₃₉₇	#135	semidirect
33. #41	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
#20263721	2
#20263719	2
#20263718	2
#20263716	2
C₂×#15691787	2
#41×C₃₉₇	2
#135	397