Subgroups — (C₂×D₃)⋊AGL(1,𝔽₅) #6735

Order 240.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×ΓL(1,𝔽₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊AGL(1,𝔽₅)	—	3	1	no	maximal, Hall
5	C₂×D₁₅	C₄	1	1	yes	characteristic, complemented
6	C₂×C₅×D₃	C₄	1	1	yes	characteristic, complemented
7	C₂×C₃×D₅	C₂²	1	1	yes	characteristic
8	D₃×D₅	—	2	1	no	—
9	D₃×D₅	—	2	1	no	—
10	ΓL(1,𝔽₁₆)	—	2	1	no	—
11	C₃×AGL(1,𝔽₅)	—	2	1	no	—
12	(C₂×D₃)⋊C₄	—	5	1	no	maximal, Hall
13	C₂²×D₅	—	3	1	no	—
14	C₂×AGL(1,𝔽₅)	—	3	1	no	—
15	C₂×AGL(1,𝔽₅)	D₃	1	1	yes	characteristic, complemented
16	C₂×C₃×C₅	C₂×C₄	1	1	yes	characteristic
17	D₁₅	—	2	1	no	—
18	C₃×D₅	D₄	1	1	yes	characteristic
19	C₃×D₅	D₄	1	1	yes	characteristic
20	C₅×D₃	—	2	1	no	—
21	C₂²×D₃	—	5	1	no	—
22	C₂×C₃×C₄	—	5	1	no	—
23	C₂×Dic₃	—	5	1	no	—
24	C₂²×C₅	—	3	1	no	—
25	C₂×D₅	—	3	1	no	—
26	C₂×D₅	—	6	1	no	—
27	C₂×D₅	—	6	1	no	—
28	C₂×D₅	C₂×D₃	1	1	yes	characteristic
29	AGL(1,𝔽₅)	—	2	1	no	—
30	AGL(1,𝔽₅)	—	6	1	no	—
31	C₂²⋊C₄	—	15	1	no	Sylow
32	C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic, complemented, Hall
33	C₂²×C₃	—	5	1	no	—
34	C₂×D₃	—	5	1	no	—
35	C₂×D₃	—	10	1	no	—
36	C₂×D₃	—	10	1	no	—
37	C₂×D₃	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
38	C₃×C₄	—	10	1	no	—
39	Dic₃	—	10	1	no	—
40	C₂×C₅	—	6	1	no	—
41	C₂×C₅	C₄×D₃	1	1	yes	characteristic
42	D₅	—	6	1	no	—
43	D₅	D₁₂	1	1	yes	characteristic
44	D₅	#67	1	1	yes	characteristic
45	C₂³	—	15	1	no	—
46	C₂×C₄	—	5	1	no	—
47	C₂×C₄	—	15	1	no	—
48	C₂×C₃	—	5	1	no	—
49	C₂×C₃	—	5	1	no	—
50	C₂×C₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
51	D₃	—	2	1	no	—
52	D₃	—	10	1	no	—
53	C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic, complemented, Sylow
54	C₂²	—	3	1	no	—
55	C₂²	—	5	1	no	—
56	C₂²	—	15	1	no	—
57	C₂²	—	30	1	no	—
58	C₂²	—	30	1	no	—
59	C₄	—	10	1	no	—
60	C₄	—	30	1	no	—
61	C₃	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
62	C₂	—	5	1	no	—
63	C₂	—	5	1	no	—
64	C₂	—	6	1	no	—
65	C₂	—	30	1	no	—
66	C₂	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×AGL(1,𝔽₅)
commutator, Fitting, socle	C₂×C₃×C₅	C₂×C₄
prime radical	C₂×D₃×D₅	C₂
cosocle	C₂×C₃×D₅	C₂²
hypocenter	C₃×C₅	C₂²⋊C₄
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
2. C₂×ΓL(1,𝔽₁₆)	C₂	semidirect
3. C₂×C₃×AGL(1,𝔽₅)	C₂	semidirect
5. C₂×D₁₅	C₄	semidirect
6. C₂×C₅×D₃	C₄	semidirect
15. C₂×AGL(1,𝔽₅)	D₃	semidirect
32. C₃×C₅	C₂²⋊C₄	semidirect
37. C₂×D₃	AGL(1,𝔽₅)	semidirect
53. C₅	(C₂×D₃)⋊C₄	semidirect
61. C₃	C₂²⋊AGL(1,𝔽₅)	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃×D₅	2
C₂×ΓL(1,𝔽₁₆)	2
C₂×C₃×AGL(1,𝔽₅)	2
C₂²⋊AGL(1,𝔽₅)	3
(C₂×D₃)⋊C₄	5

Subgroups — (C2×D3)⋊AGL(1,𝔽5) #6735

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — (C₂×D₃)⋊AGL(1,𝔽₅) #6735