Subgroups — #88162

Order 416.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6127	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₅₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	D₁₀₄	C₂	1	2	yes	maximal, complemented
5	#6095	C₂	1	2	yes	maximal, complemented
6	C₂²×D₁₃	—	2	1	no	—
7	C₂×C₄×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²⋊D₁₃	—	2	1	no	—
9	D₅₂	—	2	1	no	—
10	D₅₂	C₂²	1	2	yes	—
11	D₅₂	C₂²	1	1	yes	characteristic, complemented
12	C₄×D₁₃	—	2	1	no	—
13	Dic₂₆	C₂²	1	1	yes	characteristic, complemented
14	C₈×C₁₃	C₂²	1	2	yes	complemented
15	C₂²×C₁₃	D₄	1	1	yes	characteristic
16	C₂×D₁₃	—	2	1	no	—
17	C₂×D₁₃	—	2	2	no	—
18	C₂×D₁₃	—	4	2	no	—
19	C₄×C₁₃	D₄	1	1	yes	characteristic
20	C₄×C₁₃	C₂³	1	1	yes	characteristic
21	Dic₁₃	—	2	1	no	—
22	AGL(1,ℤ/8)	—	13	1	no	maximal, Sylow
23	C₂×C₁₃	—	2	1	no	—
24	C₂×C₁₃	C₂×D₄	1	1	yes	characteristic
25	D₁₃	—	4	1	no	—
26	D₁₃	—	4	2	no	—
27	#41	—	13	1	no	—
28	C₂×D₄	—	13	1	no	—
29	SD₁₆	—	13	2	no	—
30	D₈	—	13	2	no	—
31	M₁₆	D₁₃	1	1	yes	characteristic, complemented
32	C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
33	C₂³	—	26	1	no	—
34	Q₈	—	13	1	no	—
35	D₄	—	13	1	no	—
36	D₄	—	13	2	no	—
37	D₄	—	26	1	no	—
38	D₄	—	26	1	no	—
39	C₂×C₄	—	26	1	no	—
40	C₂×C₄	C₂×D₁₃	1	1	yes	characteristic
41	C₈	C₂×D₁₃	1	2	yes	complemented
42	C₂²	—	26	1	no	—
43	C₂²	—	26	2	no	—
44	C₂²	—	52	2	no	—
45	C₂²	D₅₂	1	1	yes	characteristic
46	C₄	—	26	1	no	—
47	C₄	D₅₂	1	1	yes	characteristic
48	C₄	C₂²×D₁₃	1	1	yes	characteristic
49	C₂	—	2	1	no	—
50	C₂	—	52	2	no	—
51	C₂	—	52	1	no	—
52	C₂	C₂×D₅₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₅₂
commutator, cosocle	C₄×C₁₃	C₂³
Frattini	C₄	C₂²×D₁₃
Fitting	C₁₃×M₁₆	C₂
socle	C₂×C₁₃	C₂×D₄
hypercenter	M₁₆	D₁₃
hypocenter	C₁₃	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
1. #6127	C₂	semidirect
2. C₂×D₅₂	C₂	semidirect
3. C₁₃×M₁₆	C₂	semidirect
4. D₁₀₄	C₂	semidirect
5. #6095	C₂	semidirect
11. D₅₂	C₂²	semidirect
13. Dic₂₆	C₂²	semidirect
14. C₈×C₁₃	C₂²	semidirect
31. M₁₆	D₁₃	semidirect
32. C₁₃	AGL(1,ℤ/8)	semidirect
41. C₈	C₂×D₁₃	semidirect

Maximal subgroups

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

subgroup	index
#6127	2
C₂×D₅₂	2
C₁₃×M₁₆	2
D₁₀₄	2
#6095	2
AGL(1,ℤ/8)	13