Subgroups — #13086300

Order 1440.

156 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#168	C₂	1	1	yes	maximal, characteristic
2	C₃⋊(C₃⋊#674)	C₂	1	2	yes	maximal, complemented
3	#91227	—	3	4	no	maximal
4	C₂×C₃²×C₄×C₅	C₄	1	1	yes	characteristic
5	C₃²×#168	C₄	1	1	yes	characteristic
6	C₃²×#168	C₂²	1	1	yes	characteristic
7	#63639	—	5	1	no	maximal, Hall
8	C₂×C₃×#168	D₃	1	4	yes	—
9	C₃⋊#674	—	3	8	no	—
10	C₂²×C₃²×C₅	C₈	1	1	yes	characteristic
11	C₃²×C₄×C₅	C₈	1	1	yes	characteristic
12	C₃²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
13	#3998	—	9	1	no	Hall
14	C₂×C₃²×C₈	—	5	1	no	—
15	C₃⋊(C₃⋊C₁₆)	—	5	2	no	—
16	C₂×C₃×C₄×C₅	Dic₃	1	4	yes	—
17	C₃×#168	Dic₃	1	4	yes	—
18	C₃×#168	C₂×D₃	1	4	yes	—
19	#812	—	15	4	no	—
20	C₂×C₃²×C₅	—	2	1	no	—
21	C₂×C₃²×C₅	C₂×C₈	1	1	yes	characteristic
22	C₂×#168	C₃⋊D₃	1	1	yes	characteristic
23	#674	—	9	2	no	—
24	C₂×C₃²×C₄	AGL(1,𝔽₅)	1	1	yes	characteristic
25	C₃²×C₈	—	5	1	no	—
26	C₃²×C₈	—	5	1	no	—
27	C₂²×C₃×C₅	C₃⋊C₈	1	4	yes	—
28	C₃×C₄×C₅	C₃⋊C₈	1	4	yes	—
29	C₃×C₄×C₅	C₂×Dic₃	1	4	yes	—
30	C₂×C₃×C₈	—	5	4	no	—
31	C₃⋊C₁₆	—	15	8	no	—
32	C₃²×C₅	M₃₂	1	1	yes	characteristic, complemented, Hall
33	C₂×C₄×C₅	C₃⋊Dic₃	1	1	yes	characteristic
34	#168	C₃⋊Dic₃	1	1	yes	characteristic
35	#168	C₂×C₃⋊D₃	1	1	yes	characteristic
36	C₂²×C₃²	#170	1	1	yes	characteristic
37	C₃²×C₄	#170	1	1	yes	characteristic
38	C₃²×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
39	M₃₂	—	45	1	no	Sylow
40	C₂×C₃×C₅	—	2	4	no	—
41	C₂×C₃×C₅	C₂×C₃⋊C₈	1	4	yes	—
42	C₂×C₃×C₄	ΓL(1,𝔽₁₆)	1	4	yes	—
43	C₃×C₈	—	5	4	no	—
44	C₃×C₈	—	5	4	no	—
45	C₂²×C₅	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
46	C₄×C₅	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
47	C₄×C₅	C₂×C₃⋊Dic₃	1	1	yes	characteristic
48	C₂×C₃²	—	2	1	no	—
49	C₂×C₃²	C₂×#170	1	1	yes	characteristic
50	C₂×C₈	—	5	1	no	—
51	C₁₆	—	45	2	no	—
52	C₃×C₅	#812	1	4	yes	complemented
53	C₂²×C₃	C₃⋊#170	1	4	yes	—
54	C₃×C₄	C₃⋊#170	1	4	yes	—
55	C₃×C₄	C₂×ΓL(1,𝔽₁₆)	1	4	yes	—
56	C₂×C₅	—	2	1	no	—
57	C₂×C₅	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
58	C₃²	#3998	1	1	yes	characteristic, complemented, Sylow
59	C₂×C₄	C₃⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
60	C₈	—	5	1	no	—
61	C₈	—	5	1	no	—
62	C₂×C₃	—	2	4	no	—
63	C₂×C₃	C₂×C₃⋊#170	1	4	yes	—
64	C₅	#63639	1	1	yes	characteristic, complemented, Sylow
65	C₂²	C₃⋊(C₃⋊#170)	1	1	yes	characteristic
66	C₄	C₃⋊(C₃⋊#170)	1	1	yes	characteristic
67	C₄	C₂×C₃⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
68	C₃	#91227	1	4	yes	complemented
69	C₂	—	2	1	no	—
70	C₂	C₂×C₃⋊(C₃⋊#170)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂×C₃⋊ΓL(1,𝔽₁₆)
commutator, socle	C₂×C₃²×C₅	C₂×C₈
Fitting	C₂×C₃²×C₄×C₅	C₄
prime radical	C₂²×C₃²×C₅	C₈
cosocle	C₃²×#168	C₂²
hypercenter	C₂×C₄	C₃⋊ΓL(1,𝔽₁₆)
hypocenter	C₃²×C₅	M₃₂
LCS	G ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. C₃⋊(C₃⋊#674)	C₂	semidirect
32. C₃²×C₅	M₃₂	semidirect
52. C₃×C₅	#812	semidirect
58. C₃²	#3998	semidirect
64. C₅	#63639	semidirect
68. C₃	#91227	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×#168	2
C₃⋊(C₃⋊#674)	2
#91227	3
#63639	5