Subgroups — #11752450

Order 960.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#4094	C₂	1	1	yes	maximal, characteristic
2	#91227	C₂	1	3	yes	maximal, complemented
3	C₂×C₃⋊#674	C₂	1	3	yes	maximal, complemented
4	#65791	—	3	1	no	maximal, Hall
5	C₃×C₅×#41	C₄	1	1	yes	characteristic
6	C₃×#681	C₄	1	3	yes	—
7	C₂×C₃×#168	C₂²	1	3	yes	—
8	C₃⋊#674	C₂²	1	1	yes	characteristic
9	C₃⋊#674	C₂²	1	3	yes	—
10	#5140	—	5	1	no	maximal, Hall
11	#4094	D₃	1	1	yes	characteristic
12	#3998	—	3	3	no	—
13	C₂×#674	—	3	3	no	—
14	C₃×C₅×Q₈	C₈	1	1	yes	characteristic
15	C₃×C₅×D₄	C₈	1	3	yes	—
16	C₂×C₃×C₄×C₅	C₂×C₄	1	3	yes	—
17	C₃×#168	C₂×C₄	1	3	yes	—
18	C₃×#168	C₂³	1	1	yes	characteristic
19	C₃×#131	—	5	1	no	—
20	#812	—	5	3	no	—
21	C₂×C₃⋊C₁₆	—	5	3	no	—
22	C₅×#41	Dic₃	1	1	yes	characteristic
23	#681	Dic₃	1	3	yes	—
24	C₂×#168	C₂×D₃	1	3	yes	—
25	#674	—	3	1	no	—
26	#674	—	3	3	no	—
27	#504	—	15	1	no	Sylow
28	C₂²×C₃×C₅	C₂×C₈	1	3	yes	—
29	C₃×C₄×C₅	C₂×C₈	1	3	yes	—
30	C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
31	C₃×#41	AGL(1,𝔽₅)	1	1	yes	characteristic
32	C₃×M₁₆	—	5	3	no	—
33	C₂×C₃×C₈	—	5	3	no	—
34	C₃⋊C₁₆	—	5	1	no	—
35	C₃⋊C₁₆	—	5	3	no	—
36	C₅×Q₈	C₃⋊C₈	1	1	yes	characteristic
37	C₅×D₄	C₃⋊C₈	1	3	yes	—
38	C₂×C₄×C₅	C₂×Dic₃	1	3	yes	—
39	#168	C₂×Dic₃	1	3	yes	—
40	#168	C₂²×D₃	1	1	yes	characteristic
41	#131	—	5	1	no	—
42	M₃₂	—	15	3	no	—
43	C₂×C₁₆	—	15	3	no	—
44	C₂×C₃×C₅	—	2	3	no	—
45	C₂×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
46	C₃×Q₈	#170	1	1	yes	characteristic
47	C₃×D₄	#170	1	3	yes	—
48	C₂×C₃×C₄	C₂×AGL(1,𝔽₅)	1	3	yes	—
49	C₃×C₈	—	5	3	no	—
50	C₃×C₈	—	5	1	no	—
51	C₂²×C₅	C₂×C₃⋊C₈	1	3	yes	—
52	C₄×C₅	C₂×C₃⋊C₈	1	3	yes	—
53	C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
54	#41	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
55	M₁₆	—	5	3	no	—
56	C₂×C₈	—	5	3	no	—
57	C₁₆	—	15	1	no	—
58	C₁₆	—	15	3	no	—
59	C₃×C₅	#504	1	1	yes	characteristic, complemented, Hall
60	C₂²×C₃	C₂×#170	1	3	yes	—
61	C₃×C₄	C₂×#170	1	3	yes	—
62	C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
63	C₂×C₅	—	2	3	no	—
64	C₂×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
65	Q₈	C₃⋊#170	1	1	yes	characteristic
66	D₄	C₃⋊#170	1	3	yes	—
67	C₂×C₄	C₂×ΓL(1,𝔽₁₆)	1	3	yes	—
68	C₈	—	5	1	no	—
69	C₈	—	5	3	no	—
70	C₂×C₃	—	2	3	no	—
71	C₂×C₃	C₂²×#170	1	1	yes	characteristic
72	C₅	#5140	1	1	yes	characteristic, complemented, Sylow
73	C₂²	C₂×C₃⋊#170	1	3	yes	—
74	C₄	C₂×C₃⋊#170	1	3	yes	—
75	C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
76	C₃	#65791	1	1	yes	characteristic, complemented, Sylow
77	C₂	—	2	3	no	—
78	C₂	C₂²×C₃⋊#170	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×ΓL(1,𝔽₁₆)
commutator, socle	C₂×C₃×C₅	C₂²×C₈
Fitting, prime radical	C₃×C₅×#41	C₄
cosocle	C₃×#168	C₂³
hypercenter	#41	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#504
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
2. #91227	C₂	semidirect
3. C₂×C₃⋊#674	C₂	semidirect
59. C₃×C₅	#504	semidirect
72. C₅	#5140	semidirect
76. C₃	#65791	semidirect

Maximal subgroups

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

subgroup	index
C₃×#4094	2
#91227	2
C₂×C₃⋊#674	2
#65791	3
#5140	5