Subgroups — C₂²×#1200 #92041

Order 480.

472 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₁₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#1200	C₂	1	12	yes	maximal, direct product factor
5	C₂²×C₄×D₅	—	3	1	no	maximal, Hall
6	C₂³×C₃×C₅	C₂²	1	1	yes	characteristic
7	C₂²×D₁₅	C₄	1	8	yes	complemented
8	C₂²×D₁₅	C₂²	1	6	yes	—
9	C₂×C₅×Dic₃	C₂²	1	6	yes	complemented
10	C₂×C₃×Dic₅	C₂²	1	6	yes	complemented
11	#1200	C₂²	1	16	yes	direct product factor
12	C₂²×C₄×D₃	—	5	1	no	maximal, Hall
13	C₂³×D₅	—	3	1	no	—
14	C₂²×C₄×C₅	—	3	1	no	—
15	C₂²×Dic₅	D₃	1	1	yes	characteristic, complemented
16	C₂×C₄×D₅	—	3	12	no	—
17	C₂²×C₃×C₅	C₂×C₄	1	4	yes	complemented
18	C₂²×C₃×C₅	C₂³	1	3	yes	—
19	C₂×D₁₅	C₂×C₄	1	24	yes	complemented
20	C₂×D₁₅	C₂³	1	4	yes	—
21	C₃×Dic₅	C₂³	1	4	yes	complemented
22	C₅×Dic₃	C₂³	1	4	yes	complemented
23	C₂³×D₃	—	5	1	no	—
24	C₂²×C₃×C₄	—	5	1	no	—
25	C₂²×Dic₃	D₅	1	1	yes	characteristic, complemented
26	C₂×C₄×D₃	—	5	12	no	—
27	C₂³×C₅	C₂×D₃	1	1	yes	characteristic
28	C₂²×D₅	—	3	8	no	—
29	C₂²×D₅	—	3	6	no	—
30	C₂×C₄×C₅	—	3	6	no	—
31	C₂×Dic₅	C₂×D₃	1	6	yes	complemented
32	C₄×D₅	—	3	16	no	—
33	C₂³×C₄	—	15	1	no	Sylow
34	C₂×C₃×C₅	C₂²×C₄	1	6	yes	complemented
35	C₂×C₃×C₅	C₂⁴	1	1	yes	characteristic
36	D₁₅	C₂²×C₄	1	8	yes	complemented
37	C₂³×C₃	C₂×D₅	1	1	yes	characteristic
38	C₂²×D₃	—	5	8	no	—
39	C₂²×D₃	—	5	6	no	—
40	C₂×C₃×C₄	—	5	6	no	—
41	C₂×Dic₃	C₂×D₅	1	6	yes	complemented
42	C₄×D₃	—	5	16	no	—
43	C₂²×C₅	C₄×D₃	1	4	yes	complemented
44	C₂²×C₅	C₂²×D₃	1	3	yes	—
45	C₂×D₅	—	3	24	no	—
46	C₂×D₅	—	3	4	no	—
47	C₄×C₅	—	3	4	no	—
48	Dic₅	C₂²×D₃	1	4	yes	complemented
49	C₂⁴	—	15	1	no	—
50	C₂²×C₄	—	3	1	no	—
51	C₂²×C₄	—	5	1	no	—
52	C₂²×C₄	—	15	12	no	—
53	C₃×C₅	C₂³×C₄	1	1	yes	characteristic, complemented, Hall
54	C₂²×C₃	C₄×D₅	1	4	yes	complemented
55	C₂²×C₃	C₂²×D₅	1	3	yes	—
56	C₂×D₃	—	5	24	no	—
57	C₂×D₃	—	5	4	no	—
58	C₃×C₄	—	5	4	no	—
59	Dic₃	C₂²×D₅	1	4	yes	complemented
60	C₂×C₅	C₂×C₄×D₃	1	6	yes	complemented
61	C₂×C₅	C₂³×D₃	1	1	yes	characteristic
62	D₅	—	3	8	no	—
63	C₂³	—	15	8	no	—
64	C₂³	—	15	6	no	—
65	C₂³	D₃×D₅	1	1	yes	characteristic
66	C₂×C₄	—	3	6	no	—
67	C₂×C₄	—	5	6	no	—
68	C₂×C₄	—	15	16	no	—
69	C₂×C₃	C₂×C₄×D₅	1	6	yes	complemented
70	C₂×C₃	C₂³×D₅	1	1	yes	characteristic
71	D₃	—	5	8	no	—
72	C₅	C₂²×C₄×D₃	1	1	yes	characteristic, complemented, Sylow
73	C₂²	—	15	24	no	—
74	C₂²	—	15	4	no	—
75	C₂²	#1200	1	4	yes	direct product factor
76	C₂²	C₂×D₃×D₅	1	3	yes	—
77	C₄	—	3	4	no	—
78	C₄	—	5	4	no	—
79	C₃	C₂²×C₄×D₅	1	1	yes	characteristic, complemented, Sylow
80	C₂	—	15	8	no	—
81	C₂	C₂×#1200	1	6	yes	direct product factor
82	C₂	C₂²×D₃×D₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³	D₃×D₅
commutator, hypocenter	C₃×C₅	C₂³×C₄
Frattini	C₂	C₂²×D₃×D₅
Fitting, socle	C₂³×C₃×C₅	C₂²
prime radical	C₂³×D₁₅	C₂
cosocle	C₂×C₃×C₅	C₂⁴
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅×Dic₃	C₂	semidirect
3. C₂²×C₃×Dic₅	C₂	semidirect
4. C₂×#1200	C₂	direct (N × H)
7. C₂²×D₁₅	C₄	semidirect
9. C₂×C₅×Dic₃	C₂²	semidirect
10. C₂×C₃×Dic₅	C₂²	semidirect
11. #1200	C₂²	direct (N × H)
15. C₂²×Dic₅	D₃	semidirect
17. C₂²×C₃×C₅	C₂×C₄	semidirect
19. C₂×D₁₅	C₂×C₄	semidirect
21. C₃×Dic₅	C₂³	semidirect
22. C₅×Dic₃	C₂³	semidirect
25. C₂²×Dic₃	D₅	semidirect
31. C₂×Dic₅	C₂×D₃	semidirect
34. C₂×C₃×C₅	C₂²×C₄	semidirect
36. D₁₅	C₂²×C₄	semidirect
41. C₂×Dic₃	C₂×D₅	semidirect
43. C₂²×C₅	C₄×D₃	semidirect
48. Dic₅	C₂²×D₃	semidirect
53. C₃×C₅	C₂³×C₄	semidirect
54. C₂²×C₃	C₄×D₅	semidirect
59. Dic₃	C₂²×D₅	semidirect
60. C₂×C₅	C₂×C₄×D₃	semidirect
69. C₂×C₃	C₂×C₄×D₅	semidirect
72. C₅	C₂²×C₄×D₃	semidirect
75. C₂²	#1200	direct (N × H)
79. C₃	C₂²×C₄×D₅	semidirect
81. C₂	C₂×#1200	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₁₅	2
C₂²×C₅×Dic₃	2
C₂²×C₃×Dic₅	2
C₂×#1200	2
C₂²×C₄×D₅	3
C₂²×C₄×D₃	5

Subgroups — C22×#1200 #92041

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#1200 #92041