Subgroups — #10624222

Order 720.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
2	C₃×Dic₃₀	C₂	1	2	yes	maximal, complemented
3	#67158	C₂	1	2	yes	maximal, complemented
4	#67155	C₂	1	2	yes	maximal
5	D₃×Dic₁₀	—	3	2	no	maximal
6	C₂×C₅×C₃⋊D₃	C₂²	1	1	yes	characteristic
7	C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₅×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
9	C₃×Dic₁₅	C₂²	1	4	yes	—
10	#3797	—	5	1	no	maximal, Hall
11	Dic₃₀	D₃	1	2	yes	complemented
12	C₄×C₅×D₃	—	3	2	no	—
13	C₄×C₅×D₃	—	6	1	no	—
14	C₃×Dic₁₀	—	3	2	no	—
15	#1204	—	3	4	no	—
16	D₃×Dic₅	—	3	4	no	—
17	C₂×C₃²×C₅	C₂³	1	1	yes	characteristic
18	C₅×C₃⋊D₃	Q₈	1	2	yes	complemented
19	C₂×Dic₁₀	—	9	1	no	Hall
20	C₄×C₃⋊D₃	D₅	1	1	yes	characteristic
21	C₃×Dic₆	—	5	2	no	—
22	#627	—	5	2	no	—
23	#624	—	5	2	no	—
24	C₂×C₅×D₃	—	3	2	no	—
25	C₂×C₅×D₃	—	6	1	no	—
26	C₃×C₄×C₅	—	2	1	no	—
27	C₃×C₄×C₅	C₂×D₃	1	2	yes	—
28	Dic₁₅	C₂×D₃	1	4	yes	—
29	C₃×Dic₅	—	3	4	no	—
30	C₅×Dic₃	—	3	2	no	—
31	C₅×Dic₃	—	6	1	no	—
32	D₃×Q₈	—	15	2	no	—
33	C₃²×C₅	C₂×Q₈	1	1	yes	characteristic, complemented, Hall
34	C₂×C₄×C₅	—	9	1	no	—
35	C₂×Dic₅	—	9	2	no	—
36	Dic₁₀	—	3	2	no	—
37	Dic₁₀	—	9	2	no	—
38	C₂×C₃⋊D₃	C₂×D₅	1	1	yes	characteristic
39	C₃²×C₄	C₂×D₅	1	1	yes	characteristic
40	C₃⋊Dic₃	C₂×D₅	1	1	yes	characteristic
41	C₃×Dic₃	—	5	4	no	—
42	C₂×C₃×C₅	—	2	1	no	—
43	C₂×C₃×C₅	C₂²×D₃	1	2	yes	—
44	C₅×D₃	—	3	4	no	—
45	C₅×D₃	—	6	2	no	—
46	C₃×Q₈	—	15	2	no	—
47	C₄×D₃	—	3	2	no	—
48	C₄×D₃	—	6	1	no	—
49	C₄×D₃	—	15	4	no	—
50	Dic₆	—	5	2	no	—
51	Dic₆	—	15	4	no	—
52	C₂²×C₅	—	9	1	no	—
53	C₄×C₅	—	9	1	no	—
54	C₄×C₅	D₃²	1	1	yes	characteristic
55	Dic₅	—	3	4	no	—
56	C₂×C₃²	C₂²×D₅	1	1	yes	characteristic
57	C₃⋊D₃	Dic₁₀	1	2	yes	complemented
58	C₂×Q₈	—	45	1	no	Sylow
59	C₃×C₅	—	2	1	no	—
60	C₃×C₅	D₃×Q₈	1	2	yes	complemented
61	C₂×D₃	—	3	2	no	—
62	C₂×D₃	—	6	1	no	—
63	C₃×C₄	—	2	1	no	—
64	C₃×C₄	—	15	4	no	—
65	C₃×C₄	D₃×D₅	1	2	yes	—
66	Dic₃	—	3	2	no	—
67	Dic₃	—	5	4	no	—
68	Dic₃	—	6	1	no	—
69	C₂×C₅	—	9	2	no	—
70	C₂×C₅	C₂×D₃²	1	1	yes	characteristic
71	C₃²	C₂×Dic₁₀	1	1	yes	characteristic, complemented, Sylow
72	Q₈	—	15	2	no	—
73	Q₈	—	45	2	no	—
74	C₂×C₄	—	9	1	no	—
75	C₂×C₄	—	45	2	no	—
76	C₂×C₃	—	2	1	no	—
77	C₂×C₃	C₂×D₃×D₅	1	2	yes	—
78	D₃	—	3	4	no	—
79	D₃	—	6	2	no	—
80	C₅	#3797	1	1	yes	characteristic, complemented, Sylow
81	C₂²	—	9	1	no	—
82	C₄	—	9	1	no	—
83	C₄	—	15	4	no	—
84	C₄	(C₃⋊D₃)⋊D₅	1	1	yes	characteristic
85	C₃	—	2	1	no	—
86	C₃	D₃×Dic₁₀	1	2	yes	complemented
87	C₂	—	9	2	no	—
88	C₂	C₂×(C₃⋊D₃)⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×(C₃⋊D₃)⋊D₅
commutator, socle, cosocle	C₂×C₃²×C₅	C₂³
Fitting	C₃²×C₄×C₅	C₂²
prime radical	C₂×C₅×C₃⋊D₃	C₂²
hypercenter	C₄	(C₃⋊D₃)⋊D₅
hypocenter	C₃²×C₅	C₂×Q₈
LCS	G ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. C₃×Dic₃₀	C₂	semidirect
3. #67158	C₂	semidirect
11. Dic₃₀	D₃	semidirect
18. C₅×C₃⋊D₃	Q₈	semidirect
33. C₃²×C₅	C₂×Q₈	semidirect
57. C₃⋊D₃	Dic₁₀	semidirect
60. C₃×C₅	D₃×Q₈	semidirect
71. C₃²	C₂×Dic₁₀	semidirect
80. C₅	#3797	semidirect
86. C₃	D₃×Dic₁₀	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×C₃⋊D₃	2
C₃×Dic₃₀	2
#67158	2
#67155	2
D₃×Dic₁₀	3
#3797	5