Subgroups — #27163123

Order 12704.

68 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	D₄×D₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#20263718	C₂	1	1	yes	maximal, characteristic, complemented
4	D₃₁₇₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#20263716	C₂	1	1	yes	maximal, characteristic, complemented
6	#20263714	C₂	1	1	yes	maximal, characteristic, complemented
7	SD₁₆×C₃₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₃₉₇	—	2	1	no	—
9	D₁₅₈₈	—	2	1	no	—
10	D₁₅₈₈	C₂²	1	1	yes	characteristic
11	D₁₅₈₈	C₂²	1	1	yes	characteristic, complemented
12	C₂²⋊D₃₉₇	—	2	1	no	—
13	C₄×D₃₉₇	—	2	1	no	—
14	C₄×D₃₉₇	C₂²	1	1	yes	characteristic
15	#15691787	C₂²	1	1	yes	characteristic, complemented
16	Q₈×C₃₉₇	C₂²	1	1	yes	characteristic, complemented
17	D₄×C₃₉₇	C₂²	1	1	yes	characteristic
18	C₈×C₃₉₇	C₂²	1	1	yes	characteristic, complemented
19	C₂²×C₃₉₇	—	2	1	no	—
20	C₂×D₃₉₇	—	2	1	no	—
21	C₂×D₃₉₇	—	2	1	no	—
22	C₂×D₃₉₇	—	4	2	no	—
23	C₂×D₃₉₇	D₄	1	1	yes	characteristic
24	C₄×C₃₉₇	—	2	1	no	—
25	C₄×C₃₉₇	C₂³	1	1	yes	characteristic
26	Dic₃₉₇	D₄	1	1	yes	characteristic
27	C₂×C₃₉₇	—	4	1	no	—
28	C₂×C₃₉₇	C₂×D₄	1	1	yes	characteristic
29	D₃₉₇	—	2	1	no	—
30	D₃₉₇	—	4	1	no	—
31	D₃₉₇	—	4	1	no	—
32	C₃₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
33	AGL(1,ℤ/8)	—	397	1	no	maximal, Sylow
34	#41	—	397	1	no	—
35	C₂×D₄	—	397	1	no	—
36	SD₁₆	—	397	1	no	—
37	SD₁₆	D₃₉₇	1	1	yes	characteristic, complemented
38	D₈	—	397	1	no	—
39	D₈	—	397	1	no	—
40	M₁₆	—	397	1	no	—
41	C₂³	—	794	1	no	—
42	Q₈	C₂×D₃₉₇	1	1	yes	characteristic, complemented
43	D₄	—	794	1	no	—
44	D₄	—	794	1	no	—
45	D₄	—	397	1	no	—
46	D₄	—	397	1	no	—
47	D₄	C₂×D₃₉₇	1	1	yes	characteristic
48	C₂×C₄	—	794	1	no	—
49	C₂×C₄	—	397	1	no	—
50	C₈	—	397	1	no	—
51	C₈	C₂×D₃₉₇	1	1	yes	characteristic, complemented
52	C₂²	—	2	1	no	—
53	C₂²	—	794	1	no	—
54	C₂²	—	794	1	no	—
55	C₂²	—	1588	2	no	—
56	C₂²	—	397	1	no	—
57	C₄	—	2	1	no	—
58	C₄	—	397	1	no	—
59	C₄	C₂²×D₃₉₇	1	1	yes	characteristic
60	C₂	—	4	1	no	—
61	C₂	—	794	1	no	—
62	C₂	—	1588	1	no	—
63	C₂	—	1588	1	no	—
64	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	AGL(1,ℤ/8)
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. D₄×D₃₉₇	C₂	semidirect
3. #20263718	C₂	semidirect
4. D₃₁₇₆	C₂	semidirect
5. #20263716	C₂	semidirect
6. #20263714	C₂	semidirect
7. SD₁₆×C₃₉₇	C₂	semidirect
11. D₁₅₈₈	C₂²	semidirect
15. #15691787	C₂²	semidirect
16. Q₈×C₃₉₇	C₂²	semidirect
18. C₈×C₃₉₇	C₂²	semidirect
32. C₃₉₇	AGL(1,ℤ/8)	semidirect
37. SD₁₆	D₃₉₇	semidirect
42. Q₈	C₂×D₃₉₇	semidirect
51. C₈	C₂×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
D₄×D₃₉₇	2
#20263718	2
D₃₁₇₆	2
#20263716	2
#20263714	2
SD₁₆×C₃₉₇	2
AGL(1,ℤ/8)	397