Subgroups — #6772

Order 240.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Dic₃₀	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×D₅	C₂	1	1	yes	maximal, characteristic, complemented
4	#1201	C₂	1	2	yes	maximal, complemented
5	D₃×Dic₅	C₂	1	2	yes	maximal, complemented
6	#711	—	3	1	no	maximal, Hall
7	C₂×C₅×D₃	C₂²	1	2	yes	—
8	C₂×C₃×D₅	C₂²	1	1	yes	characteristic
9	C₃×C₄×C₅	C₂²	1	1	yes	characteristic
10	Dic₁₅	C₂²	1	2	yes	—
11	C₃×Dic₅	C₂²	1	1	yes	characteristic
12	#235	—	5	1	no	maximal, Hall
13	C₅×D₄	—	3	1	no	—
14	C₂²⋊D₅	—	3	2	no	—
15	C₂×Dic₅	—	3	2	no	—
16	C₄×D₅	D₃	1	1	yes	characteristic, complemented
17	Dic₁₀	—	3	1	no	—
18	C₂×C₃×C₅	C₂³	1	1	yes	characteristic
19	C₃×D₅	—	2	1	no	—
20	C₅×D₃	—	2	2	no	—
21	C₂×C₃×C₄	—	5	1	no	—
22	#67	—	5	2	no	—
23	D₁₂	D₅	1	1	yes	characteristic, complemented
24	C₄×D₃	—	5	2	no	—
25	Dic₆	—	5	1	no	—
26	C₂²×C₅	—	3	2	no	—
27	C₂×D₅	C₂×D₃	1	1	yes	characteristic
28	C₄×C₅	C₂×D₃	1	1	yes	characteristic
29	Dic₅	—	3	2	no	—
30	Dic₅	C₂×D₃	1	1	yes	characteristic
31	#41	—	15	1	no	Sylow
32	C₃×C₅	#41	1	1	yes	characteristic, complemented, Hall
33	C₂²×C₃	—	5	1	no	—
34	C₂×D₃	C₂×D₅	1	2	yes	—
35	C₃×C₄	—	5	1	no	—
36	C₃×C₄	C₂×D₅	1	1	yes	characteristic
37	Dic₃	—	5	2	no	—
38	C₂×C₅	—	6	2	no	—
39	C₂×C₅	C₂²×D₃	1	1	yes	characteristic
40	D₅	—	2	1	no	—
41	Q₈	—	15	1	no	—
42	D₄	—	3	1	no	—
43	D₄	—	15	2	no	—
44	C₂×C₄	—	5	1	no	—
45	C₂×C₄	—	15	2	no	—
46	C₂×C₃	—	10	1	no	—
47	C₂×C₃	C₂²×D₅	1	1	yes	characteristic
48	D₃	—	2	2	no	—
49	C₅	#235	1	1	yes	characteristic, complemented, Sylow
50	C₂²	—	3	2	no	—
51	C₂²	—	5	1	no	—
52	C₄	—	5	1	no	—
53	C₄	—	15	2	no	—
54	C₄	D₃×D₅	1	1	yes	characteristic
55	C₃	#711	1	1	yes	characteristic, complemented, Sylow
56	C₂	—	6	2	no	—
57	C₂	—	10	1	no	—
58	C₂	C₂×D₃×D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₃×D₅
commutator, socle, cosocle	C₂×C₃×C₅	C₂³
Fitting	C₃×C₄×C₅	C₂²
hypercenter	C₄	D₃×D₅
hypocenter	C₃×C₅	#41
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. Dic₃₀	C₂	semidirect
2. C₅×D₁₂	C₂	semidirect
3. C₃×C₄×D₅	C₂	semidirect
4. #1201	C₂	semidirect
5. D₃×Dic₅	C₂	semidirect
16. C₄×D₅	D₃	semidirect
23. D₁₂	D₅	semidirect
32. C₃×C₅	#41	semidirect
49. C₅	#235	semidirect
55. C₃	#711	semidirect

Maximal subgroups

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

subgroup	index
Dic₃₀	2
C₅×D₁₂	2
C₃×C₄×D₅	2
#1201	2
D₃×Dic₅	2
#711	3
#235	5