Subgroups — #13085607

Order 1440.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#157	C₂	1	1	yes	maximal, characteristic
2	#10623818	C₂	1	2	yes	maximal
3	C₂×C₅×#157	C₄	1	1	yes	characteristic
4	C₂×C₅×#157	C₄	1	1	yes	characteristic
5	C₄×C₅×C₃⋊D₃	C₂²	1	1	yes	characteristic
6	C₃×C₃⋊#168	—	2	2	no	—
7	#63732	—	5	1	no	maximal, Hall
8	D₃×#168	—	6	2	no	—
9	C₂×C₅×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
10	C₅×#157	—	2	1	no	—
11	C₅×#157	C₈	1	2	yes	complemented
12	C₃²×C₄×C₅	D₄	1	1	yes	characteristic
13	C₅×C₃⋊Dic₃	Q₈	1	1	yes	characteristic
14	C₄⋊#168	—	9	1	no	maximal, Hall
15	C₄×#157	D₅	1	1	yes	characteristic
16	#3710	—	5	2	no	—
17	C₄×C₅×D₃	—	6	2	no	—
18	C₃⋊#168	—	2	2	no	—
19	C₃×#168	—	6	2	no	—
20	C₂×C₃²×C₅	C₄⋊C₄	1	1	yes	characteristic
21	C₅×C₃⋊D₃	C₂×C₈	1	1	yes	characteristic
22	C₅×C₃⋊D₃	M₁₆	1	1	yes	characteristic
23	C₄²×C₅	—	9	1	no	—
24	C₂×#168	—	9	2	no	—
25	C₂×#157	Dic₅	1	1	yes	characteristic
26	C₂×#157	Dic₅	1	1	yes	characteristic
27	C₄×C₃⋊D₃	C₂×D₅	1	1	yes	characteristic
28	C₃×C₃⋊C₈	—	10	2	no	—
29	C₂×C₅×D₃	—	6	2	no	—
30	C₃×C₄×C₅	—	2	2	no	—
31	C₅×Dic₃	—	6	2	no	—
32	D₃×C₈	—	30	2	no	—
33	C₃²×C₅	C₄⋊C₈	1	1	yes	characteristic, complemented, Hall
34	C₂×C₄×C₅	—	9	1	no	—
35	C₂×C₄×C₅	—	9	1	no	—
36	C₂×C₄×C₅	—	9	1	no	—
37	#168	—	6	2	no	—
38	C₂×C₃⋊D₃	C₂×Dic₅	1	1	yes	characteristic
39	#157	—	2	1	no	—
40	#157	#168	1	2	yes	complemented
41	C₃²×C₄	D₂₀	1	1	yes	characteristic
42	C₃⋊Dic₃	Dic₁₀	1	1	yes	characteristic
43	C₄⋊C₈	—	45	1	no	Sylow
44	C₂×C₃×C₅	—	2	2	no	—
45	C₅×D₃	—	6	2	no	—
46	C₅×D₃	—	6	2	no	—
47	C₄×D₃	—	6	2	no	—
48	C₃×C₈	—	30	2	no	—
49	C₃⋊C₈	—	10	2	no	—
50	C₂²×C₅	—	9	1	no	—
51	C₄×C₅	—	9	2	no	—
52	C₄×C₅	—	9	1	no	—
53	C₄×C₅	—	18	1	no	—
54	C₄×C₅	GO⁺(4,𝔽₂)	1	1	yes	characteristic
55	C₂×C₃²	C₄⋊Dic₅	1	1	yes	characteristic
56	C₃⋊D₃	C₂×#168	1	1	yes	characteristic
57	C₃⋊D₃	#681	1	1	yes	characteristic
58	C₂×C₈	—	45	2	no	—
59	C₄²	—	9	1	no	—
60	C₃×C₅	—	2	2	no	—
61	C₂×D₃	—	6	2	no	—
62	C₃×C₄	—	2	2	no	—
63	Dic₃	—	6	2	no	—
64	C₂×C₅	—	9	1	no	—
65	C₂×C₅	—	9	1	no	—
66	C₂×C₅	#3773	1	1	yes	characteristic
67	C₃²	C₄⋊#168	1	1	yes	characteristic, complemented, Sylow
68	C₂×C₄	—	9	1	no	—
69	C₂×C₄	—	9	1	no	—
70	C₂×C₄	—	9	1	no	—
71	C₈	—	30	2	no	—
72	C₂×C₃	—	2	2	no	—
73	D₃	—	6	2	no	—
74	D₃	—	6	2	no	—
75	C₅	#63732	1	1	yes	characteristic, complemented, Sylow
76	C₂²	—	9	1	no	—
77	C₄	—	9	2	no	—
78	C₄	—	9	1	no	—
79	C₄	—	18	1	no	—
80	C₄	#157⋊D₅	1	1	yes	characteristic
81	C₃	—	2	2	no	—
82	C₂	—	9	1	no	—
83	C₂	—	9	1	no	—
84	C₂	#10624011	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₄	#157⋊D₅
commutator	C₅×C₃⋊D₃	C₂×C₈
Fitting	C₃²×C₄×C₅	D₄
socle	C₂×C₃²×C₅	C₄⋊C₄
prime radical	C₂×C₅×C₃⋊D₃	C₂×C₄
cosocle	C₄×C₅×C₃⋊D₃	C₂²
hypocenter	C₃²×C₅	C₄⋊C₈
LCS	G ▷ C₅×C₃⋊D₃ ▷ C₃²×C₅
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
11. C₅×#157	C₈	semidirect
33. C₃²×C₅	C₄⋊C₈	semidirect
40. #157	#168	semidirect
67. C₃²	C₄⋊#168	semidirect
75. C₅	#63732	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×#157	2
#10623818	2
#63732	5
C₄⋊#168	9