Subgroups — #13088366

Order 1440.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅×#622	C₂	1	1	yes	maximal, characteristic
2	#10623979	C₂	1	2	yes	maximal
3	C₂×C₃×ΓL(1,𝔽₁₆)	—	2	2	no	—
4	D₅×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
5	#67126	C₄	1	1	yes	characteristic, complemented
6	C₅×#622	C₄	1	1	yes	characteristic, complemented
7	#63743	—	5	1	no	maximal, Hall
8	Dic₃⋊AGL(1,𝔽₅)	—	6	2	no	—
9	C₂×C₃²×D₅	D₄	1	1	yes	characteristic
10	C₃×ΓL(1,𝔽₁₆)	—	4	2	no	—
11	C₃⋊Dic₁₅	Q₈	1	1	yes	characteristic
12	C₅×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
13	#3994	—	9	1	no	maximal, Hall
14	C₂×#622	—	5	1	no	—
15	#3724	—	5	2	no	—
16	C₂×ΓL(1,𝔽₁₆)	—	2	2	no	—
17	C₂×C₃×AGL(1,𝔽₅)	—	6	2	no	—
18	D₅×Dic₃	—	6	2	no	—
19	C₂×C₃²×C₅	C₄⋊C₄	1	1	yes	characteristic
20	C₃²×D₅	D₈	1	1	yes	characteristic
21	C₃²×D₅	Q₁₆	1	1	yes	characteristic
22	C₄⋊AGL(1,𝔽₅)	—	9	2	no	—
23	C₈×D₅	—	9	1	no	—
24	C₂×C₃⋊Dic₃	—	5	1	no	—
25	C₂×C₃×Dic₃	—	10	2	no	—
26	#622	—	5	1	no	—
27	#622	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
28	C₂×C₃×D₅	—	2	2	no	—
29	ΓL(1,𝔽₁₆)	—	4	2	no	—
30	C₃×AGL(1,𝔽₅)	—	12	2	no	—
31	Dic₁₅	—	6	2	no	—
32	C₅×Dic₃	—	6	2	no	—
33	Dic₃⋊C₄	—	30	2	no	—
34	C₃²×C₅	#107	1	1	yes	characteristic, complemented, Hall
35	C₂×AGL(1,𝔽₅)	—	6	2	no	—
36	C₄×D₅	—	9	1	no	—
37	C₅×C₈	—	9	1	no	—
38	#168	—	9	1	no	—
39	C₂²×C₃²	—	5	1	no	—
40	C₃⋊Dic₃	—	5	1	no	—
41	C₃⋊Dic₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
42	C₃×Dic₃	—	20	2	no	—
43	#107	—	45	1	no	Sylow
44	C₂×C₃×C₅	—	2	2	no	—
45	C₃×D₅	—	2	2	no	—
46	C₃×D₅	—	2	2	no	—
47	C₂×C₃×C₄	—	30	2	no	—
48	C₂×Dic₃	—	10	2	no	—
49	C₂×Dic₃	—	30	2	no	—
50	C₂×D₅	GO⁺(4,𝔽₂)	1	1	yes	characteristic
51	AGL(1,𝔽₅)	—	12	2	no	—
52	C₄×C₅	—	9	1	no	—
53	Dic₅	—	9	1	no	—
54	C₂×C₃²	—	5	1	no	—
55	C₂×C₃²	—	5	1	no	—
56	C₂×C₃²	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
57	C₂×C₈	—	45	1	no	—
58	C₄⋊C₄	—	45	2	no	—
59	C₃×C₅	—	2	2	no	—
60	C₂²×C₃	—	10	2	no	—
61	C₃×C₄	—	60	2	no	—
62	Dic₃	—	6	2	no	—
63	Dic₃	—	20	2	no	—
64	Dic₃	—	30	2	no	—
65	C₂×C₅	#3773	1	1	yes	characteristic
66	D₅	#3774	1	1	yes	characteristic
67	D₅	#3776	1	1	yes	characteristic
68	C₃²	#3994	1	1	yes	characteristic, complemented, Sylow
69	C₂×C₄	—	30	2	no	—
70	C₂×C₄	—	45	1	no	—
71	C₈	—	9	1	no	—
72	C₈	—	45	1	no	—
73	C₂×C₃	—	2	2	no	—
74	C₂×C₃	—	10	2	no	—
75	C₂×C₃	—	10	2	no	—
76	C₅	#63743	1	1	yes	characteristic, complemented, Sylow
77	C₂²	—	5	1	no	—
78	C₄	—	9	1	no	—
79	C₄	—	45	1	no	—
80	C₄	—	60	2	no	—
81	C₃	—	2	2	no	—
82	C₂	—	5	1	no	—
83	C₂	—	5	1	no	—
84	C₂	#10624306	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	#10624306
commutator	C₅×C₃⋊Dic₃	C₂×C₄
Fitting, socle	C₂×C₃²×C₅	C₄⋊C₄
prime radical	C₂×C₃²×D₅	D₄
cosocle	D₅×C₃⋊Dic₃	C₂²
hypocenter	C₃²×C₅	#107
LCS	G ▷ C₅×C₃⋊Dic₃ ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
5. #67126	C₄	semidirect
6. C₅×#622	C₄	semidirect
27. #622	AGL(1,𝔽₅)	semidirect
34. C₃²×C₅	#107	semidirect
68. C₃²	#3994	semidirect
76. C₅	#63743	semidirect

Maximal subgroups

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

subgroup	index
D₅×#622	2
#10623979	2
#63743	5
#3994	9