Subgroups — C₅×#235 #6807

Order 240.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₄×C₅	C₂	semidirect
2. C₅×#67	C₂	semidirect
3. C₅×D₁₂	C₂	semidirect
4. C₄×C₅×D₃	C₂	semidirect
5. C₅×Dic₆	C₂	semidirect
12. #235	C₅	direct (N × H)
17. C₂×C₄×C₅	D₃	semidirect
21. C₂×C₃×C₄	C₂×C₅	semidirect
22. #67	C₂×C₅	semidirect
23. D₁₂	C₂×C₅	semidirect
24. C₄×D₃	C₂×C₅	semidirect
25. Dic₆	C₂×C₅	semidirect
32. C₃×C₅	#41	semidirect
45. C₂×C₄	C₅×D₃	semidirect
49. C₅	#235	direct (N × H)
55. C₃	C₅×#41	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₅	2
C₅×#67	2
C₅×D₁₂	2
C₄×C₅×D₃	2
C₅×Dic₆	2
C₅×#41	3
#235	5

Subgroups — C5×#235 #6807

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#235 #6807