Subgroups — #13086370

Order 1440.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#4380	C₂	1	2	yes	maximal
2	C₈×C₃⋊D₁₅	C₂	1	1	yes	maximal, characteristic
3	C₂×#4380	—	2	2	no	—
4	C₄×C₃⋊D₁₅	C₂²	1	1	yes	characteristic
5	C₃²×C₅×C₈	C₄	1	1	yes	characteristic, complemented
6	C₃⋊(C₃⋊#168)	C₄	1	1	yes	characteristic, complemented
7	#63768	—	5	1	no	maximal, Hall
8	C₈×D₁₅	—	6	2	no	—
9	C₂×C₃⋊D₁₅	D₄	1	1	yes	characteristic
10	#4380	—	4	2	no	—
11	C₃²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
12	C₃⋊Dic₁₅	Q₈	1	1	yes	characteristic
13	#3993	—	9	1	no	maximal, Hall
14	C₄⋊#157	—	5	2	no	—
15	C₈×C₃⋊D₃	—	5	1	no	—
16	C₄×D₁₅	—	6	2	no	—
17	C₃×C₅×C₈	—	2	2	no	—
18	C₃⋊#168	—	6	2	no	—
19	C₂×C₃²×C₅	C₄⋊C₄	1	1	yes	characteristic
20	C₃⋊D₁₅	SD₁₆	1	2	yes	—
21	C₄⋊AGL(1,𝔽₅)	—	9	2	no	—
22	C₈×D₅	—	9	1	no	—
23	C₂×#157	—	10	2	no	—
24	C₄×C₃⋊D₃	—	5	1	no	—
25	C₃²×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
26	C₃⋊(C₃⋊C₈)	—	5	1	no	—
27	C₂×D₁₅	—	6	2	no	—
28	C₃×C₄×C₅	—	2	2	no	—
29	Dic₁₅	—	6	2	no	—
30	D₃×C₈	—	30	2	no	—
31	C₃²×C₅	#106	1	1	yes	characteristic, complemented, Hall
32	C₂×AGL(1,𝔽₅)	—	18	2	no	—
33	C₄×D₅	—	9	1	no	—
34	C₅×C₈	#157	1	1	yes	characteristic, complemented
35	#168	—	9	1	no	—
36	C₂×C₃⋊D₃	—	5	1	no	—
37	#157	—	20	2	no	—
38	C₃²×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
39	C₃⋊Dic₃	—	5	1	no	—
40	#106	—	45	1	no	Sylow
41	C₂×C₃×C₅	—	2	2	no	—
42	D₁₅	—	6	4	no	—
43	C₄×D₃	—	30	2	no	—
44	C₃×C₈	—	2	2	no	—
45	C₃⋊C₈	—	30	2	no	—
46	C₂×D₅	—	9	1	no	—
47	AGL(1,𝔽₅)	—	36	2	no	—
48	C₄×C₅	C₂×#157	1	1	yes	characteristic
49	Dic₅	—	9	1	no	—
50	C₂×C₃²	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
51	C₃⋊D₃	—	5	2	no	—
52	C₂×C₈	—	45	1	no	—
53	C₄⋊C₄	—	45	2	no	—
54	C₃×C₅	—	2	2	no	—
55	C₂×D₃	—	30	2	no	—
56	C₃×C₄	—	2	2	no	—
57	Dic₃	—	30	2	no	—
58	C₂×C₅	C₄⋊#157	1	1	yes	characteristic
59	D₅	—	9	2	no	—
60	C₃²	#3993	1	1	yes	characteristic, complemented, Sylow
61	C₂×C₄	—	45	1	no	—
62	C₂×C₄	—	90	2	no	—
63	C₈	—	45	1	no	—
64	C₈	#4380	1	1	yes	characteristic, complemented
65	C₂×C₃	—	2	2	no	—
66	D₃	—	30	4	no	—
67	C₅	#63768	1	1	yes	characteristic, complemented, Sylow
68	C₂²	—	45	1	no	—
69	C₄	—	45	1	no	—
70	C₄	—	180	2	no	—
71	C₄	C₂×#4380	1	1	yes	characteristic
72	C₃	—	2	2	no	—
73	C₂	—	45	2	no	—
74	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	D₄
cosocle	C₄×C₃⋊D₁₅	C₂²
hypercenter	C₈	#4380
hypocenter	C₃²×C₅	#106
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
5. C₃²×C₅×C₈	C₄	semidirect
6. C₃⋊(C₃⋊#168)	C₄	semidirect
25. C₃²×C₈	AGL(1,𝔽₅)	semidirect
31. C₃²×C₅	#106	semidirect
34. C₅×C₈	#157	semidirect
60. C₃²	#3993	semidirect
64. C₈	#4380	semidirect
67. C₅	#63768	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#4380	2
C₈×C₃⋊D₁₅	2
#63768	5
#3993	9