Subgroups — C₃²×#683 #10623842

Order 720.

156 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃²×Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₃×#683	C₃	1	4	yes	maximal, direct product factor
5	C₂²×C₃²×C₅	C₂²	1	1	yes	characteristic
6	C₃²×C₄×C₅	—	2	1	no	—
7	C₃²×Dic₅	—	2	1	no	—
8	C₃²×Dic₅	C₄	1	2	yes	complemented
9	C₃²×C₄⋊C₄	—	5	1	no	maximal, Hall
10	C₂×C₃×C₄×C₅	C₂×C₃	1	4	yes	—
11	C₂×C₃×Dic₅	C₂×C₃	1	4	yes	—
12	C₂×C₃×Dic₅	C₂×C₃	1	4	yes	—
13	C₂×C₃²×C₅	C₂×C₄	1	1	yes	characteristic
14	C₂×C₃²×C₅	D₄	1	1	yes	characteristic
15	C₂×C₃²×C₅	Q₈	1	1	yes	characteristic
16	#683	C₃²	1	1	yes	characteristic, direct product factor, Hall
17	C₂×C₃²×C₄	—	5	1	no	—
18	C₂×C₃²×C₄	—	5	1	no	—
19	C₂×C₃²×C₄	D₅	1	1	yes	characteristic
20	C₂²×C₃×C₅	C₂²×C₃	1	4	yes	—
21	C₃×C₄×C₅	—	2	4	no	—
22	C₃×Dic₅	—	2	4	no	—
23	C₃×Dic₅	C₃×C₄	1	8	yes	complemented
24	C₃×C₄⋊C₄	—	5	4	no	—
25	C₃²×C₅	C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
26	C₂×C₄×C₅	C₂×C₃²	1	1	yes	characteristic
27	C₂×Dic₅	C₂×C₃²	1	1	yes	characteristic
28	C₂×Dic₅	C₂×C₃²	1	1	yes	characteristic
29	C₂²×C₃²	C₂×D₅	1	1	yes	characteristic
30	C₃²×C₄	—	2	1	no	—
31	C₃²×C₄	—	5	2	no	—
32	C₃²×C₄	—	10	1	no	—
33	C₂×C₃×C₅	C₂×C₃×C₄	1	4	yes	—
34	C₂×C₃×C₅	C₃×D₄	1	4	yes	—
35	C₂×C₃×C₅	C₃×Q₈	1	4	yes	—
36	C₂×C₃×C₄	—	5	4	no	—
37	C₂×C₃×C₄	—	5	4	no	—
38	C₂×C₃×C₄	C₃×D₅	1	4	yes	—
39	C₂²×C₅	C₂²×C₃²	1	1	yes	characteristic
40	C₄×C₅	—	2	1	no	—
41	Dic₅	—	2	1	no	—
42	Dic₅	C₃²×C₄	1	2	yes	complemented
43	C₂×C₃²	Dic₁₀	1	1	yes	characteristic
44	C₂×C₃²	C₄×D₅	1	1	yes	characteristic
45	C₂×C₃²	C₂²⋊D₅	1	1	yes	characteristic
46	C₄⋊C₄	—	5	1	no	Sylow
47	C₃×C₅	C₃×C₄⋊C₄	1	4	yes	complemented
48	C₂²×C₃	C₂×C₃×D₅	1	4	yes	—
49	C₃×C₄	—	2	4	no	—
50	C₃×C₄	—	5	8	no	—
51	C₃×C₄	—	10	4	no	—
52	C₂×C₅	C₂×C₃²×C₄	1	1	yes	characteristic
53	C₂×C₅	C₃²×D₄	1	1	yes	characteristic
54	C₂×C₅	C₃²×Q₈	1	1	yes	characteristic
55	C₃²	#683	1	1	yes	characteristic, direct product factor, Sylow
56	C₂×C₄	—	5	1	no	—
57	C₂×C₄	—	5	1	no	—
58	C₂×C₄	C₃²×D₅	1	1	yes	characteristic
59	C₂×C₃	C₃×Dic₁₀	1	4	yes	—
60	C₂×C₃	C₃×C₄×D₅	1	4	yes	—
61	C₂×C₃	C₃×C₂²⋊D₅	1	4	yes	—
62	C₅	C₃²×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
63	C₂²	C₂×C₃²×D₅	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	5	2	no	—
66	C₄	—	10	1	no	—
67	C₃	C₃×#683	1	4	yes	direct product factor
68	C₂	C₃²×Dic₁₀	1	1	yes	characteristic
69	C₂	C₃²×C₄×D₅	1	1	yes	characteristic
70	C₂	C₃²×C₂²⋊D₅	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₃×#683	C₃	direct (N × H)
8. C₃²×Dic₅	C₄	semidirect
16. #683	C₃²	direct (N × H)
23. C₃×Dic₅	C₃×C₄	semidirect
25. C₃²×C₅	C₄⋊C₄	semidirect
42. Dic₅	C₃²×C₄	semidirect
47. C₃×C₅	C₃×C₄⋊C₄	semidirect
55. C₃²	#683	direct (N × H)
62. C₅	C₃²×C₄⋊C₄	semidirect
67. C₃	C₃×#683	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₄×C₅	2
C₂×C₃²×Dic₅	2
C₃×#683	3
C₃²×C₄⋊C₄	5

Subgroups — C32×#683 #10623842

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#683 #10623842