Subgroups — #12014012

Order 1280.

398 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10599853	C₂	1	4	yes	maximal, complemented
2	#10598855	C₂	1	2	yes	maximal, complemented
3	C₅×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₈₀	—	2	8	no	—
5	#64984	—	2	4	no	—
6	#64984	C₂²	1	4	yes	—
7	C₄×C₅×C₁₆	C₂²	1	2	yes	—
8	C₅×C₈²	C₂²	1	1	yes	characteristic
9	#12515	—	5	1	no	maximal, Sylow
10	C₂×D₄₀	—	2	8	no	—
11	C₂×D₄₀	—	4	8	no	—
12	#4020	—	2	4	no	—
13	C₂×C₅×C₁₆	D₄	1	4	yes	—
14	C₄×C₅×C₈	D₄	1	2	yes	—
15	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
16	D₈₀	—	4	16	no	—
17	#2246	—	5	4	no	—
18	#1712	—	5	2	no	—
19	C₈×C₁₆	D₅	1	1	yes	characteristic, complemented
20	C₂×D₂₀	—	4	4	no	—
21	C₂×D₂₀	—	4	8	no	—
22	C₂×C₅×C₈	D₈	1	4	yes	—
23	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
24	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
25	D₄₀	—	4	16	no	—
26	D₄₀	—	8	16	no	—
27	C₅×C₁₆	D₈	1	8	yes	complemented
28	C₂×D₁₆	—	10	8	no	—
29	#493	—	5	4	no	—
30	#493	—	10	4	no	—
31	C₄×C₁₆	C₂×D₅	1	2	yes	—
32	C₈²	C₂×D₅	1	1	yes	characteristic
33	C₂²×D₅	—	8	4	no	—
34	C₂×C₄×C₅	#127	1	1	yes	characteristic
35	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
36	D₂₀	—	8	8	no	—
37	D₂₀	—	8	16	no	—
38	C₅×C₈	D₁₆	1	8	yes	complemented
39	C₅×C₈	C₂×D₈	1	4	yes	—
40	C₂×D₈	—	10	8	no	—
41	C₂×D₈	—	20	8	no	—
42	#127	—	10	4	no	—
43	D₁₆	—	20	16	no	—
44	C₂×C₁₆	D₂₀	1	4	yes	—
45	C₄×C₈	D₂₀	1	2	yes	—
46	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
47	C₂²×C₅	#493	1	1	yes	characteristic
48	C₂×D₅	—	16	4	no	—
49	C₂×D₅	—	16	8	no	—
50	C₄×C₅	#493	1	2	yes	—
51	C₄×C₅	C₂×D₁₆	1	4	yes	—
52	C₂×D₄	—	20	4	no	—
53	C₂×D₄	—	20	8	no	—
54	D₈	—	20	16	no	—
55	D₈	—	40	16	no	—
56	C₂×C₈	D₄₀	1	4	yes	—
57	C₂×C₈	C₂×D₂₀	1	2	yes	—
58	C₄²	C₂×D₂₀	1	1	yes	characteristic
59	C₁₆	D₄₀	1	8	yes	complemented
60	C₂×C₅	#1712	1	1	yes	characteristic
61	C₂×C₅	#2246	1	2	yes	—
62	D₅	—	32	4	no	—
63	C₂³	—	40	4	no	—
64	D₄	—	40	16	no	—
65	D₄	—	40	8	no	—
66	C₂×C₄	#4020	1	1	yes	characteristic
67	C₂×C₄	C₂×D₄₀	1	2	yes	—
68	C₈	D₈₀	1	8	yes	complemented
69	C₈	C₂×D₄₀	1	4	yes	—
70	C₅	#12515	1	1	yes	characteristic, complemented, Sylow
71	C₂²	—	80	8	no	—
72	C₂²	—	80	4	no	—
73	C₂²	#64984	1	1	yes	characteristic
74	C₄	#64984	1	2	yes	—
75	C₄	C₂×D₈₀	1	4	yes	—
76	C₂	—	160	4	no	—
77	C₂	#10598855	1	1	yes	characteristic
78	C₂	#10599853	1	2	yes	—

	subgroup	G/H
center	C₂²	#64984
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×C₈×C₁₆	C₂
socle	C₂²×C₅	#493
hypercenter	C₈×C₁₆	D₅
hypocenter	C₅	#12515
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
1. #10599853	C₂	semidirect
2. #10598855	C₂	semidirect
3. C₅×C₈×C₁₆	C₂	semidirect
19. C₈×C₁₆	D₅	semidirect
27. C₅×C₁₆	D₈	semidirect
38. C₅×C₈	D₁₆	semidirect
59. C₁₆	D₄₀	semidirect
68. C₈	D₈₀	semidirect
70. C₅	#12515	semidirect

Maximal subgroups

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

subgroup	index
#10599853	2
#10598855	2
C₅×C₈×C₁₆	2
#12515	5