Subgroups — C₃×(C₂×AGL(1,𝔽₅))⋊C₄ #91210

Order 480.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×AGL(1,𝔽₅)	C₂	1	2	yes	maximal
2	C₂×C₃×C₄×D₅	C₂	1	1	yes	maximal, characteristic
3	(C₂×AGL(1,𝔽₅))⋊C₄	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂²×C₃×D₅	C₂²	1	1	yes	characteristic
5	C₂×C₃×AGL(1,𝔽₅)	—	2	2	no	—
6	C₂×C₃×AGL(1,𝔽₅)	—	2	2	no	—
7	C₂×C₃×AGL(1,𝔽₅)	C₄	1	4	yes	complemented
8	C₂×C₃×C₄×C₅	C₄	1	1	yes	characteristic, complemented
9	C₂×C₃×Dic₅	C₄	1	1	yes	characteristic, complemented
10	C₃×C₄×D₅	—	2	2	no	—
11	C₃×#95	—	5	1	no	maximal, Hall
12	C₂²×AGL(1,𝔽₅)	C₂×C₃	1	2	yes	—
13	C₂×C₄×D₅	C₂×C₃	1	1	yes	characteristic
14	C₂²×C₃×C₅	C₂×C₄	1	1	yes	characteristic
15	C₂×C₃×D₅	C₂×C₄	1	2	yes	—
16	C₂×C₃×D₅	D₄	1	1	yes	characteristic
17	C₂×C₃×D₅	D₄	1	2	yes	—
18	C₂×C₃×D₅	Q₈	1	1	yes	characteristic
19	C₃×AGL(1,𝔽₅)	—	2	4	no	—
20	C₃×C₄×C₅	—	2	1	no	—
21	C₃×Dic₅	—	2	1	no	—
22	C₂²×C₃×C₄	—	5	2	no	—
23	C₂²×C₃×C₄	—	5	1	no	—
24	C₂²×D₅	C₂²×C₃	1	1	yes	characteristic
25	C₂×AGL(1,𝔽₅)	—	2	2	no	—
26	C₂×AGL(1,𝔽₅)	—	2	2	no	—
27	C₂×AGL(1,𝔽₅)	C₃×C₄	1	4	yes	complemented
28	C₂×C₄×C₅	C₃×C₄	1	1	yes	characteristic, complemented
29	C₂×Dic₅	C₃×C₄	1	1	yes	characteristic, complemented
30	C₄×D₅	—	2	2	no	—
31	#95	—	5	1	no	Sylow
32	C₂×C₃×C₅	C₄²	1	1	yes	characteristic
33	C₂×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₃×C₅	C₄⋊C₄	1	1	yes	characteristic
35	C₃×D₅	C₂²⋊C₄	1	2	yes	—
36	C₃×D₅	C₄⋊C₄	1	2	yes	—
37	C₂³×C₃	—	5	1	no	—
38	C₂×C₃×C₄	—	5	4	no	—
39	C₂×C₃×C₄	—	5	1	no	—
40	C₂×C₃×C₄	—	10	2	no	—
41	C₂×C₃×C₄	—	10	2	no	—
42	C₂×C₃×C₄	—	10	2	no	—
43	C₂×C₃×C₄	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
44	C₂²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
45	C₂×D₅	C₂×C₃×C₄	1	2	yes	—
46	C₂×D₅	C₃×D₄	1	1	yes	characteristic
47	C₂×D₅	C₃×D₄	1	2	yes	—
48	C₂×D₅	C₃×Q₈	1	1	yes	characteristic
49	AGL(1,𝔽₅)	—	2	4	no	—
50	C₄×C₅	—	2	1	no	—
51	Dic₅	—	2	1	no	—
52	C₂²×C₄	—	5	2	no	—
53	C₂²×C₄	—	5	1	no	—
54	C₃×C₅	#95	1	1	yes	characteristic, complemented, Hall
55	C₂²×C₃	—	5	2	no	—
56	C₂²×C₃	—	5	1	no	—
57	C₂²×C₃	—	5	1	no	—
58	C₂²×C₃	—	5	2	no	—
59	C₂²×C₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
60	C₃×C₄	—	2	1	no	—
61	C₃×C₄	—	10	4	no	—
62	C₃×C₄	—	10	1	no	—
63	C₂×C₅	C₃×C₄²	1	1	yes	characteristic
64	C₂×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
66	D₅	C₃×C₂²⋊C₄	1	2	yes	—
67	D₅	C₃×C₄⋊C₄	1	2	yes	—
68	C₂³	—	5	1	no	—
69	C₂×C₄	—	5	4	no	—
70	C₂×C₄	—	5	1	no	—
71	C₂×C₄	—	10	2	no	—
72	C₂×C₄	—	10	2	no	—
73	C₂×C₄	—	10	2	no	—
74	C₂×C₄	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
75	C₂×C₃	—	5	2	no	—
76	C₂×C₃	—	5	2	no	—
77	C₂×C₃	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
78	C₂×C₃	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
79	C₂×C₃	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
80	C₅	C₃×#95	1	1	yes	characteristic, complemented, Sylow
81	C₂²	—	5	2	no	—
82	C₂²	—	5	1	no	—
83	C₂²	—	5	1	no	—
84	C₂²	—	5	2	no	—
85	C₂²	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	10	4	no	—
88	C₄	—	10	1	no	—
89	C₃	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
90	C₂	—	5	2	no	—
91	C₂	—	5	2	no	—
92	C₂	C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
93	C₂	C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
94	C₂	C₃×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₃×C₄²
Frattini	C₂²	C₂×C₃×AGL(1,𝔽₅)
Fitting	C₂×C₃×C₄×C₅	C₄
socle	C₂²×C₃×C₅	C₂×C₄
prime radical	C₂²×C₃×D₅	C₂²
cosocle	C₂²×D₅	C₂²×C₃
hypercenter	C₂×C₃×C₄	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#95
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄

Complements

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

N	H ≅ G/N	type
3. (C₂×AGL(1,𝔽₅))⋊C₄	C₃	direct (N × H)
7. C₂×C₃×AGL(1,𝔽₅)	C₄	semidirect
8. C₂×C₃×C₄×C₅	C₄	semidirect
9. C₂×C₃×Dic₅	C₄	semidirect
27. C₂×AGL(1,𝔽₅)	C₃×C₄	semidirect
28. C₂×C₄×C₅	C₃×C₄	semidirect
29. C₂×Dic₅	C₃×C₄	semidirect
43. C₂×C₃×C₄	AGL(1,𝔽₅)	semidirect
54. C₃×C₅	#95	semidirect
74. C₂×C₄	C₃×AGL(1,𝔽₅)	semidirect
80. C₅	C₃×#95	semidirect
89. C₃	(C₂×AGL(1,𝔽₅))⋊C₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×AGL(1,𝔽₅)	2
C₂×C₃×C₄×D₅	2
(C₂×AGL(1,𝔽₅))⋊C₄	3
C₃×#95	5

Subgroups — C3×(C2×AGL(1,𝔽5))⋊C4 #91210

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×(C₂×AGL(1,𝔽₅))⋊C₄ #91210