Subgroups — #15777631

Order 3328.

398 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13117086	C₂	1	4	yes	maximal, complemented
2	#13114451	C₂	1	2	yes	maximal, complemented
3	C₈×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₂₀₈	—	2	8	no	—
5	#11718356	—	2	4	no	—
6	#11718356	C₂²	1	4	yes	—
7	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
8	C₈²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×D₁₀₄	—	2	8	no	—
10	C₂×D₁₀₄	—	4	8	no	—
11	#88128	—	2	4	no	—
12	C₂×C₁₃×C₁₆	D₄	1	4	yes	—
13	C₄×C₈×C₁₃	D₄	1	2	yes	—
14	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
15	D₂₀₈	—	4	16	no	—
16	#12515	—	13	1	no	maximal, Sylow
17	C₂×D₅₂	—	4	4	no	—
18	C₂×D₅₂	—	4	8	no	—
19	C₂×C₈×C₁₃	D₈	1	4	yes	—
20	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
21	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
22	D₁₀₄	—	4	16	no	—
23	D₁₀₄	—	8	16	no	—
24	C₁₃×C₁₆	D₈	1	8	yes	complemented
25	#2246	—	13	4	no	—
26	#1712	—	13	2	no	—
27	C₈×C₁₆	D₁₃	1	1	yes	characteristic, complemented
28	C₂²×D₁₃	—	8	4	no	—
29	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
30	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
31	D₅₂	—	8	8	no	—
32	D₅₂	—	8	16	no	—
33	C₈×C₁₃	D₁₆	1	8	yes	complemented
34	C₈×C₁₃	C₂×D₈	1	4	yes	—
35	C₂×D₁₆	—	26	8	no	—
36	#493	—	13	4	no	—
37	#493	—	26	4	no	—
38	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
39	C₈²	C₂×D₁₃	1	1	yes	characteristic
40	C₂²×C₁₃	#493	1	1	yes	characteristic
41	C₂×D₁₃	—	16	4	no	—
42	C₂×D₁₃	—	16	8	no	—
43	C₄×C₁₃	#493	1	2	yes	—
44	C₄×C₁₃	C₂×D₁₆	1	4	yes	—
45	C₂×D₈	—	26	8	no	—
46	C₂×D₈	—	52	8	no	—
47	#127	—	26	4	no	—
48	D₁₆	—	52	16	no	—
49	C₂×C₁₆	D₅₂	1	4	yes	—
50	C₄×C₈	D₅₂	1	2	yes	—
51	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
52	C₂×C₁₃	#1712	1	1	yes	characteristic
53	C₂×C₁₃	#2246	1	2	yes	—
54	D₁₃	—	32	4	no	—
55	C₂×D₄	—	52	4	no	—
56	C₂×D₄	—	52	8	no	—
57	D₈	—	52	16	no	—
58	D₈	—	104	16	no	—
59	C₂×C₈	D₁₀₄	1	4	yes	—
60	C₂×C₈	C₂×D₅₂	1	2	yes	—
61	C₄²	C₂×D₅₂	1	1	yes	characteristic
62	C₁₆	D₁₀₄	1	8	yes	complemented
63	C₁₃	#12515	1	1	yes	characteristic, complemented, Sylow
64	C₂³	—	104	4	no	—
65	D₄	—	104	16	no	—
66	D₄	—	104	8	no	—
67	C₂×C₄	#88128	1	1	yes	characteristic
68	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
69	C₈	D₂₀₈	1	8	yes	complemented
70	C₈	C₂×D₁₀₄	1	4	yes	—
71	C₂²	—	208	4	no	—
72	C₂²	—	208	8	no	—
73	C₂²	#11718356	1	1	yes	characteristic
74	C₄	#11718356	1	2	yes	—
75	C₄	C₂×D₂₀₈	1	4	yes	—
76	C₂	—	416	4	no	—
77	C₂	#13114451	1	1	yes	characteristic
78	C₂	#13117086	1	2	yes	—

	subgroup	G/H
center	C₂²	#11718356
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₈×C₁₃×C₁₆	C₂
socle	C₂²×C₁₃	#493
hypercenter	C₈×C₁₆	D₁₃
hypocenter	C₁₃	#12515
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
1. #13117086	C₂	semidirect
2. #13114451	C₂	semidirect
3. C₈×C₁₃×C₁₆	C₂	semidirect
24. C₁₃×C₁₆	D₈	semidirect
27. C₈×C₁₆	D₁₃	semidirect
33. C₈×C₁₃	D₁₆	semidirect
62. C₁₆	D₁₀₄	semidirect
63. C₁₃	#12515	semidirect
69. C₈	D₂₀₈	semidirect

Maximal subgroups

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

subgroup	index
#13117086	2
#13114451	2
C₈×C₁₃×C₁₆	2
#12515	13