Subgroups — #13086372

Order 1440.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624114	C₂	1	2	yes	maximal
2	C₈×C₃⋊D₁₅	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃⋊D₁₅	C₂²	1	1	yes	characteristic
4	#67127	—	2	2	no	—
5	C₃²×C₅×C₈	C₄	1	1	yes	characteristic
6	C₃⋊(C₃⋊#168)	C₄	1	1	yes	characteristic
7	#63770	—	5	1	no	maximal, Hall
8	C₈×D₁₅	—	6	2	no	—
9	C₂×C₃⋊D₁₅	Q₈	1	1	yes	characteristic
10	C₃²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
11	C₃⋊Dic₁₅	D₄	1	1	yes	characteristic
12	#3995	—	9	1	no	maximal, Hall
13	#3788	—	5	2	no	—
14	C₈×C₃⋊D₃	—	5	1	no	—
15	C₄×D₁₅	—	6	2	no	—
16	C₃×C₅×C₈	—	2	2	no	—
17	C₃⋊#168	—	6	2	no	—
18	C₂×C₃²×C₅	C₄⋊C₄	1	1	yes	characteristic
19	C₃⋊D₁₅	—	2	1	no	—
20	#700	—	9	2	no	—
21	C₈×D₅	—	9	1	no	—
22	C₄×C₃⋊D₃	—	5	1	no	—
23	#622	—	10	2	no	—
24	C₃²×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
25	C₃⋊(C₃⋊C₈)	—	5	1	no	—
26	C₂×D₁₅	—	6	2	no	—
27	C₃×C₄×C₅	—	2	2	no	—
28	Dic₁₅	—	6	2	no	—
29	D₃×C₈	—	30	2	no	—
30	C₃²×C₅	#108	1	1	yes	characteristic, complemented, Hall
31	C₄×D₅	—	9	1	no	—
32	#170	—	18	2	no	—
33	C₅×C₈	#157	1	1	yes	characteristic
34	#168	—	9	1	no	—
35	C₂×C₃⋊D₃	—	5	1	no	—
36	C₃²×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
37	C₃⋊Dic₃	—	5	1	no	—
38	#108	—	45	1	no	Sylow
39	C₂×C₃×C₅	—	2	2	no	—
40	D₁₅	—	6	4	no	—
41	C₄×D₃	—	30	2	no	—
42	C₃×C₈	—	2	2	no	—
43	C₃⋊C₈	—	30	2	no	—
44	C₂×D₅	—	9	1	no	—
45	C₄×C₅	C₂×#157	1	1	yes	characteristic
46	Dic₅	—	9	1	no	—
47	C₂×C₃²	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
48	C₃⋊D₃	—	10	1	no	—
49	M₁₆	—	45	2	no	—
50	C₂×C₈	—	45	1	no	—
51	C₃×C₅	—	2	2	no	—
52	C₂×D₃	—	30	2	no	—
53	C₃×C₄	—	2	2	no	—
54	Dic₃	—	30	2	no	—
55	C₂×C₅	C₄⋊#157	1	1	yes	characteristic
56	D₅	—	18	1	no	—
57	C₃²	#3995	1	1	yes	characteristic, complemented, Sylow
58	C₂×C₄	—	45	1	no	—
59	C₈	—	45	1	no	—
60	C₈	—	90	2	no	—
61	C₈	#4380	1	1	yes	characteristic
62	C₂×C₃	—	2	2	no	—
63	D₃	—	30	4	no	—
64	C₅	#63770	1	1	yes	characteristic, complemented, Sylow
65	C₂²	—	45	1	no	—
66	C₄	—	45	1	no	—
67	C₄	C₂×#4380	1	1	yes	characteristic
68	C₃	—	2	2	no	—
69	C₂	—	90	1	no	—
70	C₂	C₄⋊#4380	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₄⋊#4380
commutator	C₃²×C₄×C₅	C₂×C₄
Frattini	C₄	C₂×#4380
Fitting	C₃²×C₅×C₈	C₄
socle	C₂×C₃²×C₅	C₄⋊C₄
prime radical	C₂×C₃⋊D₁₅	Q₈
cosocle	C₄×C₃⋊D₁₅	C₂²
hypercenter	C₈	#4380
hypocenter	C₃²×C₅	#108
LCS	G ▷ C₃²×C₄×C₅ ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
30. C₃²×C₅	#108	semidirect
57. C₃²	#3995	semidirect
64. C₅	#63770	semidirect

Maximal subgroups

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

subgroup	index
#10624114	2
C₈×C₃⋊D₁₅	2
#63770	5
#3995	9