Subgroups — #13085513

Order 1440.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624107	C₂	1	1	yes	maximal, characteristic
2	C₅×#3788	C₂	1	1	yes	maximal, characteristic
3	C₃⋊D₃×#168	C₂	1	1	yes	maximal, characteristic
4	C₄×C₅×C₃⋊D₃	C₂²	1	1	yes	characteristic
5	#67126	—	2	1	no	—
6	C₅×#622	—	2	1	no	—
7	C₃⋊(C₃⋊#168)	C₄	1	1	yes	characteristic
8	C₃²×#168	C₄	1	1	yes	characteristic
9	#63770	—	5	1	no	maximal, Hall
10	D₃×#168	—	6	2	no	—
11	C₂×C₅×C₃⋊D₃	Q₈	1	1	yes	characteristic
12	C₃²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
13	C₅×C₃⋊Dic₃	D₄	1	1	yes	characteristic
14	#3954	—	9	1	no	maximal, Hall
15	#3788	—	5	1	no	—
16	#3788	D₅	1	1	yes	characteristic
17	C₈×C₃⋊D₃	—	5	1	no	—
18	C₄×C₅×D₃	—	6	2	no	—
19	C₃⋊#168	—	6	2	no	—
20	C₃×#168	—	2	2	no	—
21	C₂×C₃²×C₅	C₄⋊C₄	1	1	yes	characteristic
22	C₅×C₃⋊D₃	—	2	1	no	—
23	C₅×M₁₆	—	9	1	no	—
24	#681	—	9	1	no	—
25	C₂×#168	—	9	1	no	—
26	C₄×C₃⋊D₃	C₂×D₅	1	1	yes	characteristic
27	#622	—	2	1	no	—
28	#622	—	10	1	no	—
29	C₃²×C₈	—	5	1	no	—
30	C₃⋊(C₃⋊C₈)	—	5	1	no	—
31	C₂×C₅×D₃	—	6	2	no	—
32	C₃×C₄×C₅	—	2	2	no	—
33	C₅×Dic₃	—	6	2	no	—
34	D₃×C₈	—	30	2	no	—
35	C₃²×C₅	#108	1	1	yes	characteristic, complemented, Hall
36	C₂×C₄×C₅	—	9	1	no	—
37	C₅×C₈	—	18	1	no	—
38	#168	—	9	1	no	—
39	#168	—	18	1	no	—
40	#168	#157	1	1	yes	characteristic
41	C₂×C₃⋊D₃	Dic₁₀	1	1	yes	characteristic
42	C₃²×C₄	C₄×D₅	1	1	yes	characteristic
43	C₃⋊Dic₃	C₂²⋊D₅	1	1	yes	characteristic
44	#108	—	45	1	no	Sylow
45	C₂×C₃×C₅	—	2	2	no	—
46	C₅×D₃	—	6	4	no	—
47	C₄×D₃	—	6	2	no	—
48	C₃×C₈	—	10	2	no	—
49	C₃⋊C₈	—	30	2	no	—
50	C₂²×C₅	—	9	1	no	—
51	C₄×C₅	—	9	1	no	—
52	C₄×C₅	C₂×#157	1	1	yes	characteristic
53	C₂×C₃²	#683	1	1	yes	characteristic
54	C₃⋊D₃	—	2	1	no	—
55	M₁₆	—	9	1	no	—
56	M₁₆	—	45	1	no	—
57	C₂×C₈	—	45	1	no	—
58	C₃×C₅	—	2	2	no	—
59	C₂×D₃	—	6	2	no	—
60	C₃×C₄	—	2	2	no	—
61	Dic₃	—	6	2	no	—
62	C₂×C₅	—	18	1	no	—
63	C₂×C₅	C₄⋊#157	1	1	yes	characteristic
64	C₃²	#3954	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₄	—	9	1	no	—
66	C₈	—	5	1	no	—
67	C₈	—	18	1	no	—
68	C₈	—	45	1	no	—
69	C₈	—	90	1	no	—
70	C₂×C₃	—	2	2	no	—
71	D₃	—	6	4	no	—
72	C₅	#63770	1	1	yes	characteristic, complemented, Sylow
73	C₂²	—	9	1	no	—
74	C₄	—	9	1	no	—
75	C₄	D₅×#157	1	1	yes	characteristic
76	C₃	—	2	2	no	—
77	C₂	—	18	1	no	—
78	C₂	#10623986	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10623986
commutator, Fitting	C₃²×C₄×C₅	C₂×C₄
Frattini, hypercenter	C₄	D₅×#157
socle	C₂×C₃²×C₅	C₄⋊C₄
prime radical	C₂×C₅×C₃⋊D₃	Q₈
cosocle	C₄×C₅×C₃⋊D₃	C₂²
hypocenter	C₃²×C₅	#108
LCS	G ▷ C₃²×C₄×C₅ ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
35. C₃²×C₅	#108	semidirect
64. C₃²	#3954	semidirect
72. C₅	#63770	semidirect

Maximal subgroups

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

subgroup	index
#10624107	2
C₅×#3788	2
C₃⋊D₃×#168	2
#63770	5
#3954	9