Subgroups — #13086366

Order 1440.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×C₃⋊D₁₅	C₂	1	1	yes	maximal, characteristic
2	#10623725	C₂	1	2	yes	maximal, complemented
3	C₄×C₃⋊D₁₅	C₄	1	1	yes	characteristic
4	C₃²×C₅×C₈	C₄	1	1	yes	characteristic
5	C₃⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
6	#63764	—	5	1	no	maximal, Hall
7	C₈×D₁₅	—	6	2	no	—
8	C₂×C₃⋊D₁₅	C₈	1	1	yes	characteristic
9	C₃²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
10	C₃⋊Dic₁₅	C₈	1	1	yes	characteristic
11	D₅⋊C₁₆	—	9	1	no	maximal, Hall
12	C₈×C₃⋊D₃	—	5	1	no	—
13	#3708	—	5	2	no	—
14	C₄×D₁₅	—	6	2	no	—
15	C₃×C₅×C₈	—	2	2	no	—
16	C₃⋊#168	—	6	2	no	—
17	C₂×C₃²×C₅	C₂×C₈	1	1	yes	characteristic
18	C₃⋊D₁₅	C₁₆	1	2	yes	complemented
19	C₈×D₅	—	9	1	no	—
20	#674	—	9	2	no	—
21	C₄×C₃⋊D₃	—	5	1	no	—
22	C₃²×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
23	C₃⋊(C₃⋊C₈)	—	5	1	no	—
24	C₂×D₁₅	—	6	2	no	—
25	C₃×C₄×C₅	—	2	2	no	—
26	Dic₁₅	—	6	2	no	—
27	D₃×C₈	—	30	2	no	—
28	C₃²×C₅	C₂×C₁₆	1	1	yes	characteristic, complemented, Hall
29	C₄×D₅	—	9	1	no	—
30	C₅×C₈	#157	1	1	yes	characteristic
31	#168	—	9	1	no	—
32	C₂×C₃⋊D₃	—	5	1	no	—
33	C₃²×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
34	C₃⋊Dic₃	—	5	1	no	—
35	C₂×C₁₆	—	45	1	no	Sylow
36	C₂×C₃×C₅	—	2	2	no	—
37	D₁₅	—	6	4	no	—
38	C₄×D₃	—	30	2	no	—
39	C₃×C₈	—	2	2	no	—
40	C₃⋊C₈	—	30	2	no	—
41	C₂×D₅	—	9	1	no	—
42	C₄×C₅	C₂×#157	1	1	yes	characteristic
43	Dic₅	—	9	1	no	—
44	C₂×C₃²	D₅⋊C₈	1	1	yes	characteristic
45	C₃⋊D₃	—	5	2	no	—
46	C₂×C₈	—	45	1	no	—
47	C₁₆	—	45	2	no	—
48	C₃×C₅	—	2	2	no	—
49	C₂×D₃	—	30	2	no	—
50	C₃×C₄	—	2	2	no	—
51	Dic₃	—	30	2	no	—
52	C₂×C₅	#3787	1	1	yes	characteristic
53	D₅	—	9	2	no	—
54	C₃²	D₅⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
55	C₂×C₄	—	45	1	no	—
56	C₈	—	45	1	no	—
57	C₈	#4380	1	1	yes	characteristic
58	C₂×C₃	—	2	2	no	—
59	D₃	—	30	4	no	—
60	C₅	#63764	1	1	yes	characteristic, complemented, Sylow
61	C₂²	—	45	1	no	—
62	C₄	—	45	1	no	—
63	C₄	C₂×#4380	1	1	yes	characteristic
64	C₃	—	2	2	no	—
65	C₂	—	45	2	no	—
66	C₂	#10624113	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₈	#4380
commutator, hypocenter	C₃²×C₅	C₂×C₁₆
Frattini	C₄	C₂×#4380
Fitting	C₃²×C₅×C₈	C₄
socle	C₂×C₃²×C₅	C₂×C₈
prime radical	C₂×C₃⋊D₁₅	C₈
cosocle	C₃⋊(C₃⋊#168)	C₂²
LCS	G ▷ C₃²×C₅
UCS	C₁ ◁ C₈

Complements

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

N	H ≅ G/N	type
2. #10623725	C₂	semidirect
18. C₃⋊D₁₅	C₁₆	semidirect
28. C₃²×C₅	C₂×C₁₆	semidirect
54. C₃²	D₅⋊C₁₆	semidirect
60. C₅	#63764	semidirect

Maximal subgroups

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

subgroup	index
C₈×C₃⋊D₁₅	2
#10623725	2
#63764	5
D₅⋊C₁₆	9