Subgroups — #13088217

Order 1440.

372 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅×#41	C₂	1	1	yes	maximal, characteristic
2	#10623904	C₂	1	3	yes	maximal, complemented
3	C₂×C₃⋊(C₃⋊#168)	C₂	1	3	yes	maximal, complemented
4	#91837	—	3	4	no	maximal
5	C₃²×C₅×Q₈	C₄	1	1	yes	characteristic
6	C₃²×C₅×D₄	C₄	1	3	yes	—
7	C₂×C₃²×C₄×C₅	C₂²	1	3	yes	—
8	C₃⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
9	C₃⋊(C₃⋊#168)	C₂²	1	3	yes	—
10	#64157	—	5	1	no	maximal, Hall
11	C₃×C₅×#41	D₃	1	4	yes	—
12	#6710	—	3	12	no	—
13	C₂×C₃⋊#168	—	3	12	no	—
14	C₂²×C₃²×C₅	C₂×C₄	1	3	yes	—
15	C₃²×C₄×C₅	C₂×C₄	1	3	yes	—
16	C₃²×C₄×C₅	C₂³	1	1	yes	characteristic
17	#4094	—	9	1	no	Hall
18	C₃²×#41	D₅	1	1	yes	characteristic
19	#3748	—	5	3	no	—
20	C₂×C₃⋊(C₃⋊C₈)	—	5	3	no	—
21	C₃×C₅×Q₈	Dic₃	1	4	yes	—
22	C₃×C₅×D₄	Dic₃	1	12	yes	—
23	C₂×C₃×C₄×C₅	C₂×D₃	1	12	yes	—
24	C₃⋊#168	—	3	4	no	—
25	C₃⋊#168	—	3	12	no	—
26	#948	—	15	4	no	—
27	C₂×C₃²×C₅	—	2	3	no	—
28	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
29	C₅×#41	C₃⋊D₃	1	1	yes	characteristic
30	#681	—	9	3	no	—
31	C₂×#168	—	9	3	no	—
32	C₃²×Q₈	Dic₅	1	1	yes	characteristic
33	C₃²×D₄	Dic₅	1	3	yes	—
34	C₂×C₃²×C₄	C₂×D₅	1	3	yes	—
35	C₃⋊(C₃⋊C₈)	—	5	1	no	—
36	C₃⋊(C₃⋊C₈)	—	5	3	no	—
37	C₂²×C₃×C₅	C₂×Dic₃	1	12	yes	—
38	C₃×C₄×C₅	C₂×Dic₃	1	12	yes	—
39	C₃×C₄×C₅	C₂²×D₃	1	4	yes	—
40	C₃×#41	D₁₅	1	4	yes	—
41	#208	—	15	12	no	—
42	C₂×C₃⋊C₈	—	15	12	no	—
43	C₃²×C₅	#131	1	1	yes	characteristic, complemented, Hall
44	C₅×Q₈	C₃⋊Dic₃	1	1	yes	characteristic
45	C₅×D₄	C₃⋊Dic₃	1	3	yes	—
46	C₂×C₄×C₅	C₂×C₃⋊D₃	1	3	yes	—
47	#168	—	9	1	no	—
48	#168	—	9	3	no	—
49	C₂²×C₃²	C₂×Dic₅	1	3	yes	—
50	C₃²×C₄	C₂×Dic₅	1	3	yes	—
51	C₃²×C₄	C₂²×D₅	1	1	yes	characteristic
52	#131	—	45	1	no	Sylow
53	C₂×C₃×C₅	—	2	12	no	—
54	C₂×C₃×C₅	C₂²×Dic₃	1	4	yes	—
55	C₃×Q₈	Dic₁₅	1	4	yes	—
56	C₃×D₄	Dic₁₅	1	12	yes	—
57	C₂×C₃×C₄	C₂×D₁₅	1	12	yes	—
58	C₃⋊C₈	—	15	4	no	—
59	C₃⋊C₈	—	15	12	no	—
60	C₂²×C₅	C₂×C₃⋊Dic₃	1	3	yes	—
61	C₄×C₅	C₂×C₃⋊Dic₃	1	3	yes	—
62	C₄×C₅	C₂²×C₃⋊D₃	1	1	yes	characteristic
63	C₂×C₃²	—	2	3	no	—
64	C₂×C₃²	C₂²×Dic₅	1	1	yes	characteristic
65	#41	C₃⋊D₁₅	1	1	yes	characteristic
66	M₁₆	—	45	3	no	—
67	C₂×C₈	—	45	3	no	—
68	C₃×C₅	#948	1	4	yes	complemented
69	C₂²×C₃	C₂×Dic₁₅	1	12	yes	—
70	C₃×C₄	C₂×Dic₁₅	1	12	yes	—
71	C₃×C₄	C₂²×D₁₅	1	4	yes	—
72	C₂×C₅	—	2	3	no	—
73	C₂×C₅	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
74	C₃²	#4094	1	1	yes	characteristic, complemented, Sylow
75	Q₈	C₃⋊Dic₁₅	1	1	yes	characteristic
76	D₄	C₃⋊Dic₁₅	1	3	yes	—
77	C₂×C₄	C₂×C₃⋊D₁₅	1	3	yes	—
78	C₈	—	45	1	no	—
79	C₈	—	45	3	no	—
80	C₂×C₃	—	2	12	no	—
81	C₂×C₃	C₂²×Dic₁₅	1	4	yes	—
82	C₅	#64157	1	1	yes	characteristic, complemented, Sylow
83	C₂²	C₂×C₃⋊Dic₁₅	1	3	yes	—
84	C₄	C₂×C₃⋊Dic₁₅	1	3	yes	—
85	C₄	C₂²×C₃⋊D₁₅	1	1	yes	characteristic
86	C₃	#91837	1	4	yes	complemented
87	C₂	—	2	3	no	—
88	C₂	C₂²×C₃⋊Dic₁₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×C₃⋊D₁₅
commutator, socle	C₂×C₃²×C₅	C₂²×C₄
Fitting, prime radical	C₃²×C₅×#41	C₂
cosocle	C₃²×C₄×C₅	C₂³
hypercenter	#41	C₃⋊D₁₅
hypocenter	C₃²×C₅	#131
LCS	G ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
2. #10623904	C₂	semidirect
3. C₂×C₃⋊(C₃⋊#168)	C₂	semidirect
43. C₃²×C₅	#131	semidirect
68. C₃×C₅	#948	semidirect
74. C₃²	#4094	semidirect
82. C₅	#64157	semidirect
86. C₃	#91837	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₅×#41	2
#10623904	2
C₂×C₃⋊(C₃⋊#168)	2
#91837	3
#64157	5