Subgroups — #16813646

Order 3616.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14268646	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×D₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#14268636	C₂	1	1	yes	maximal, characteristic, complemented
4	#14268635	C₂	1	1	yes	maximal, characteristic, complemented
5	#14268632	C₂	1	1	yes	maximal, characteristic, complemented
6	#14268630	C₂	1	1	yes	maximal, characteristic, complemented
7	D₈×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₁₁₃	—	2	1	no	—
9	D₄×C₁₁₃	C₂²	1	1	yes	characteristic
10	D₄×C₁₁₃	C₂²	1	1	yes	characteristic, complemented
11	C₂²⋊D₁₁₃	—	2	1	no	—
12	C₂²⋊D₁₁₃	—	2	1	no	—
13	C₂×Dic₁₁₃	—	2	1	no	—
14	D₄₅₂	C₂²	1	1	yes	characteristic
15	C₄×D₁₁₃	C₂²	1	1	yes	characteristic
16	Dic₂₂₆	C₂²	1	1	yes	characteristic, complemented
17	C₈×C₁₁₃	C₂²	1	1	yes	characteristic, complemented
18	#11744751	C₂²	1	1	yes	characteristic, complemented
19	C₂²×C₁₁₃	—	2	1	no	—
20	C₂²×C₁₁₃	—	2	1	no	—
21	C₂×D₁₁₃	—	2	1	no	—
22	C₂×D₁₁₃	—	4	2	no	—
23	C₂×D₁₁₃	D₄	1	1	yes	characteristic
24	C₄×C₁₁₃	C₂³	1	1	yes	characteristic
25	Dic₁₁₃	—	2	1	no	—
26	Dic₁₁₃	D₄	1	1	yes	characteristic
27	C₂×C₁₁₃	—	4	1	no	—
28	C₂×C₁₁₃	—	4	1	no	—
29	C₂×C₁₁₃	C₂×D₄	1	1	yes	characteristic
30	D₁₁₃	—	2	1	no	—
31	D₁₁₃	—	4	1	no	—
32	C₁₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
33	AGL(1,ℤ/8)	—	113	1	no	maximal, Sylow
34	#41	—	113	1	no	—
35	C₂×D₄	—	113	1	no	—
36	SD₁₆	—	113	1	no	—
37	SD₁₆	—	113	1	no	—
38	D₈	—	113	1	no	—
39	D₈	D₁₁₃	1	1	yes	characteristic, complemented
40	M₁₆	—	113	1	no	—
41	C₂³	—	226	1	no	—
42	Q₈	—	113	1	no	—
43	D₄	—	113	1	no	—
44	D₄	—	226	1	no	—
45	D₄	—	226	1	no	—
46	D₄	C₂×D₁₁₃	1	1	yes	characteristic
47	D₄	C₂×D₁₁₃	1	1	yes	characteristic, complemented
48	C₂×C₄	—	113	1	no	—
49	C₂×C₄	—	226	1	no	—
50	C₈	—	113	1	no	—
51	C₈	C₂×D₁₁₃	1	1	yes	characteristic, complemented
52	C₂²	—	2	1	no	—
53	C₂²	—	2	1	no	—
54	C₂²	—	113	1	no	—
55	C₂²	—	452	2	no	—
56	C₂²	—	226	1	no	—
57	C₄	—	113	1	no	—
58	C₄	—	226	1	no	—
59	C₄	C₂²×D₁₁₃	1	1	yes	characteristic
60	C₂	—	4	1	no	—
61	C₂	—	4	1	no	—
62	C₂	—	452	1	no	—
63	C₂	—	226	1	no	—
64	C₂	D₄×D₁₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₁₁₃
commutator, cosocle	C₄×C₁₁₃	C₂³
Frattini	C₄	C₂²×D₁₁₃
Fitting	D₈×C₁₁₃	C₂
socle	C₂×C₁₁₃	C₂×D₄
hypercenter	D₈	D₁₁₃
hypocenter	C₁₁₃	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #14268646	C₂	semidirect
2. D₄×D₁₁₃	C₂	semidirect
3. #14268636	C₂	semidirect
4. #14268635	C₂	semidirect
5. #14268632	C₂	semidirect
6. #14268630	C₂	semidirect
7. D₈×C₁₁₃	C₂	semidirect
10. D₄×C₁₁₃	C₂²	semidirect
16. Dic₂₂₆	C₂²	semidirect
17. C₈×C₁₁₃	C₂²	semidirect
18. #11744751	C₂²	semidirect
32. C₁₁₃	AGL(1,ℤ/8)	semidirect
39. D₈	D₁₁₃	semidirect
47. D₄	C₂×D₁₁₃	semidirect
51. C₈	C₂×D₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
#14268646	2
D₄×D₁₁₃	2
#14268636	2
#14268635	2
#14268632	2
#14268630	2
D₈×C₁₁₃	2
AGL(1,ℤ/8)	113