Subgroups — #15693045

Order 3296.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13110104	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×D₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#13110094	C₂	1	1	yes	maximal, characteristic, complemented
4	#13110093	C₂	1	1	yes	maximal, characteristic, complemented
5	#13110090	C₂	1	1	yes	maximal, characteristic, complemented
6	#13110088	C₂	1	1	yes	maximal, characteristic, complemented
7	D₈×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₁₀₃	—	2	1	no	—
9	D₄×C₁₀₃	C₂²	1	1	yes	characteristic
10	D₄×C₁₀₃	C₂²	1	1	yes	characteristic, complemented
11	C₂²⋊D₁₀₃	—	2	1	no	—
12	C₂²⋊D₁₀₃	—	2	1	no	—
13	C₂×Dic₁₀₃	—	2	1	no	—
14	D₄₁₂	C₂²	1	1	yes	characteristic
15	C₄×D₁₀₃	C₂²	1	1	yes	characteristic
16	Dic₂₀₆	C₂²	1	1	yes	characteristic, complemented
17	C₈×C₁₀₃	C₂²	1	1	yes	characteristic, complemented
18	C₁₀₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
19	C₂²×C₁₀₃	—	2	1	no	—
20	C₂²×C₁₀₃	—	2	1	no	—
21	C₂×D₁₀₃	—	2	1	no	—
22	C₂×D₁₀₃	—	4	2	no	—
23	C₂×D₁₀₃	D₄	1	1	yes	characteristic
24	C₄×C₁₀₃	C₂³	1	1	yes	characteristic
25	Dic₁₀₃	—	2	1	no	—
26	Dic₁₀₃	D₄	1	1	yes	characteristic
27	C₂×C₁₀₃	—	4	1	no	—
28	C₂×C₁₀₃	—	4	1	no	—
29	C₂×C₁₀₃	C₂×D₄	1	1	yes	characteristic
30	D₁₀₃	—	2	1	no	—
31	D₁₀₃	—	4	1	no	—
32	C₁₀₃	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
33	AGL(1,ℤ/8)	—	103	1	no	maximal, Sylow
34	#41	—	103	1	no	—
35	C₂×D₄	—	103	1	no	—
36	SD₁₆	—	103	1	no	—
37	SD₁₆	—	103	1	no	—
38	D₈	—	103	1	no	—
39	D₈	D₁₀₃	1	1	yes	characteristic, complemented
40	M₁₆	—	103	1	no	—
41	C₂³	—	206	1	no	—
42	Q₈	—	103	1	no	—
43	D₄	—	103	1	no	—
44	D₄	—	206	1	no	—
45	D₄	—	206	1	no	—
46	D₄	C₂×D₁₀₃	1	1	yes	characteristic
47	D₄	C₂×D₁₀₃	1	1	yes	characteristic, complemented
48	C₂×C₄	—	103	1	no	—
49	C₂×C₄	—	206	1	no	—
50	C₈	—	103	1	no	—
51	C₈	C₂×D₁₀₃	1	1	yes	characteristic, complemented
52	C₂²	—	2	1	no	—
53	C₂²	—	2	1	no	—
54	C₂²	—	103	1	no	—
55	C₂²	—	412	2	no	—
56	C₂²	—	206	1	no	—
57	C₄	—	103	1	no	—
58	C₄	—	206	1	no	—
59	C₄	C₂²×D₁₀₃	1	1	yes	characteristic
60	C₂	—	4	1	no	—
61	C₂	—	4	1	no	—
62	C₂	—	412	1	no	—
63	C₂	—	206	1	no	—
64	C₂	D₄×D₁₀₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₁₀₃
commutator, cosocle	C₄×C₁₀₃	C₂³
Frattini	C₄	C₂²×D₁₀₃
Fitting	D₈×C₁₀₃	C₂
socle	C₂×C₁₀₃	C₂×D₄
hypercenter	D₈	D₁₀₃
hypocenter	C₁₀₃	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #13110104	C₂	semidirect
2. D₄×D₁₀₃	C₂	semidirect
3. #13110094	C₂	semidirect
4. #13110093	C₂	semidirect
5. #13110090	C₂	semidirect
6. #13110088	C₂	semidirect
7. D₈×C₁₀₃	C₂	semidirect
10. D₄×C₁₀₃	C₂²	semidirect
16. Dic₂₀₆	C₂²	semidirect
17. C₈×C₁₀₃	C₂²	semidirect
18. C₁₀₃⋊C₈	C₂²	semidirect
32. C₁₀₃	AGL(1,ℤ/8)	semidirect
39. D₈	D₁₀₃	semidirect
47. D₄	C₂×D₁₀₃	semidirect
51. C₈	C₂×D₁₀₃	semidirect

Maximal subgroups

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

subgroup	index
#13110104	2
D₄×D₁₀₃	2
#13110094	2
#13110093	2
#13110090	2
#13110088	2
D₈×C₁₀₃	2
AGL(1,ℤ/8)	103