Subgroups — #13117086

Order 1664.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₀₈	C₂	1	4	yes	maximal, complemented
2	#11718356	C₂	1	2	yes	maximal, complemented
3	C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₀₄	—	2	2	no	—
5	C₂×D₁₀₄	C₂²	1	4	yes	—
6	#88128	—	2	2	no	—
7	C₂×C₁₃×C₁₆	C₂²	1	2	yes	—
8	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	D₂₀₈	—	2	8	no	—
10	C₂×D₅₂	—	2	4	no	—
11	C₂×D₅₂	—	4	4	no	—
12	C₂×C₈×C₁₃	D₄	1	1	yes	characteristic
13	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
14	C₄²×C₁₃	D₄	1	1	yes	characteristic
15	D₁₀₄	—	2	8	no	—
16	D₁₀₄	—	4	4	no	—
17	C₁₃×C₁₆	D₄	1	4	yes	complemented
18	#2246	—	13	1	no	maximal, Sylow
19	C₂²×D₁₃	—	4	4	no	—
20	C₂×C₄×C₁₃	D₈	1	2	yes	—
21	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
22	D₅₂	—	4	4	no	—
23	D₅₂	—	4	4	no	—
24	D₅₂	—	8	8	no	—
25	C₈×C₁₃	D₈	1	2	yes	—
26	C₈×C₁₃	C₂×D₄	1	2	yes	—
27	C₂×D₁₆	—	13	4	no	—
28	#493	—	13	2	no	—
29	C₄×C₁₆	D₁₃	1	1	yes	characteristic, complemented
30	C₂²×C₁₃	C₂×D₈	1	1	yes	characteristic
31	C₂×D₁₃	—	8	4	no	—
32	C₂×D₁₃	—	8	8	no	—
33	C₄×C₁₃	D₁₆	1	4	yes	complemented
34	C₄×C₁₃	#127	1	1	yes	characteristic
35	C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
36	C₂×D₈	—	13	4	no	—
37	C₂×D₈	—	26	2	no	—
38	#127	—	26	2	no	—
39	D₁₆	—	26	8	no	—
40	C₂×C₁₆	C₂×D₁₃	1	2	yes	—
41	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
42	C₂×C₁₃	#493	1	1	yes	characteristic
43	C₂×C₁₃	C₂×D₁₆	1	2	yes	—
44	D₁₃	—	16	4	no	—
45	C₂×D₄	—	26	4	no	—
46	C₂×D₄	—	52	4	no	—
47	D₈	—	26	8	no	—
48	D₈	—	52	4	no	—
49	C₂×C₈	D₅₂	1	1	yes	characteristic
50	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
51	C₄²	D₅₂	1	1	yes	characteristic
52	C₁₆	D₅₂	1	4	yes	complemented
53	C₁₃	#2246	1	1	yes	characteristic, complemented, Sylow
54	C₂³	—	52	4	no	—
55	D₄	—	52	4	no	—
56	D₄	—	52	4	no	—
57	D₄	—	104	8	no	—
58	C₂×C₄	D₁₀₄	1	2	yes	—
59	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
60	C₈	D₁₀₄	1	2	yes	—
61	C₈	C₂×D₅₂	1	2	yes	—
62	C₂²	—	104	8	no	—
63	C₂²	—	104	4	no	—
64	C₂²	C₂×D₁₀₄	1	1	yes	characteristic
65	C₄	D₂₀₈	1	4	yes	complemented
66	C₄	#88128	1	1	yes	characteristic
67	C₄	C₂×D₁₀₄	1	1	yes	characteristic
68	C₂	—	208	4	no	—
69	C₂	#11718356	1	1	yes	characteristic
70	C₂	C₂×D₂₀₈	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂³
Frattini	C₂×C₈	C₂²×D₁₃
Fitting	C₄×C₁₃×C₁₆	C₂
socle	C₂²×C₁₃	C₂×D₈
hypercenter	C₄×C₁₆	D₁₃
hypocenter	C₁₃	#2246
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×D₂₀₈	C₂	semidirect
2. #11718356	C₂	semidirect
3. C₄×C₁₃×C₁₆	C₂	semidirect
17. C₁₃×C₁₆	D₄	semidirect
29. C₄×C₁₆	D₁₃	semidirect
33. C₄×C₁₃	D₁₆	semidirect
52. C₁₆	D₅₂	semidirect
53. C₁₃	#2246	semidirect
65. C₄	D₂₀₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₀₈	2
#11718356	2
C₄×C₁₃×C₁₆	2
#2246	13