Subgroups — #88165

Order 416.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6129	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#6105	C₂	1	1	yes	maximal, characteristic, complemented
5	#6104	C₂	1	1	yes	maximal, characteristic, complemented
6	#6095	C₂	1	1	yes	maximal, characteristic, complemented
7	#6094	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₁₃	—	2	1	no	—
9	D₄×C₁₃	C₂²	1	1	yes	characteristic
10	D₄×C₁₃	C₂²	1	1	yes	characteristic, complemented
11	C₂²⋊D₁₃	—	2	1	no	—
12	C₂²⋊D₁₃	—	2	1	no	—
13	C₂×Dic₁₃	—	2	1	no	—
14	D₅₂	C₂²	1	1	yes	characteristic
15	C₄×D₁₃	C₂²	1	1	yes	characteristic
16	Dic₂₆	C₂²	1	1	yes	characteristic, complemented
17	C₈×C₁₃	C₂²	1	1	yes	characteristic, complemented
18	#1055	C₂²	1	1	yes	characteristic, complemented
19	C₂²×C₁₃	—	2	1	no	—
20	C₂²×C₁₃	—	2	1	no	—
21	C₂×D₁₃	—	2	1	no	—
22	C₂×D₁₃	—	4	2	no	—
23	C₂×D₁₃	D₄	1	1	yes	characteristic
24	C₄×C₁₃	C₂³	1	1	yes	characteristic
25	Dic₁₃	—	2	1	no	—
26	Dic₁₃	D₄	1	1	yes	characteristic
27	AGL(1,ℤ/8)	—	13	1	no	maximal, Sylow
28	C₂×C₁₃	—	4	1	no	—
29	C₂×C₁₃	—	4	1	no	—
30	C₂×C₁₃	C₂×D₄	1	1	yes	characteristic
31	D₁₃	—	2	1	no	—
32	D₁₃	—	4	1	no	—
33	#41	—	13	1	no	—
34	C₂×D₄	—	13	1	no	—
35	SD₁₆	—	13	1	no	—
36	SD₁₆	—	13	1	no	—
37	D₈	—	13	1	no	—
38	D₈	D₁₃	1	1	yes	characteristic, complemented
39	M₁₆	—	13	1	no	—
40	C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
41	C₂³	—	26	1	no	—
42	Q₈	—	13	1	no	—
43	D₄	—	13	1	no	—
44	D₄	—	26	1	no	—
45	D₄	—	26	1	no	—
46	D₄	C₂×D₁₃	1	1	yes	characteristic
47	D₄	C₂×D₁₃	1	1	yes	characteristic, complemented
48	C₂×C₄	—	13	1	no	—
49	C₂×C₄	—	26	1	no	—
50	C₈	—	13	1	no	—
51	C₈	C₂×D₁₃	1	1	yes	characteristic, complemented
52	C₂²	—	2	1	no	—
53	C₂²	—	2	1	no	—
54	C₂²	—	13	1	no	—
55	C₂²	—	26	1	no	—
56	C₂²	—	52	2	no	—
57	C₄	—	13	1	no	—
58	C₄	—	26	1	no	—
59	C₄	C₂²×D₁₃	1	1	yes	characteristic
60	C₂	—	4	1	no	—
61	C₂	—	4	1	no	—
62	C₂	—	26	1	no	—
63	C₂	—	52	1	no	—
64	C₂	D₄×D₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₁₃
commutator, cosocle	C₄×C₁₃	C₂³
Frattini	C₄	C₂²×D₁₃
Fitting	C₁₃×D₈	C₂
socle	C₂×C₁₃	C₂×D₄
hypercenter	D₈	D₁₃
hypocenter	C₁₃	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #6129	C₂	semidirect
2. D₄×D₁₃	C₂	semidirect
3. C₁₃×D₈	C₂	semidirect
4. #6105	C₂	semidirect
5. #6104	C₂	semidirect
6. #6095	C₂	semidirect
7. #6094	C₂	semidirect
10. D₄×C₁₃	C₂²	semidirect
16. Dic₂₆	C₂²	semidirect
17. C₈×C₁₃	C₂²	semidirect
18. #1055	C₂²	semidirect
38. D₈	D₁₃	semidirect
40. C₁₃	AGL(1,ℤ/8)	semidirect
47. D₄	C₂×D₁₃	semidirect
51. 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
#6129	2
D₄×D₁₃	2
C₁₃×D₈	2
#6105	2
#6104	2
#6095	2
#6094	2
AGL(1,ℤ/8)	13