Subgroups — C₂²×#1204 #92045

Order 480.

312 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₁₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₂×#1204	C₂	1	12	yes	maximal, direct product factor
5	C₂²×Dic₁₀	—	3	1	no	maximal, Hall
6	C₂³×C₃×C₅	C₂²	1	1	yes	characteristic
7	C₂×Dic₁₅	C₂²	1	6	yes	—
8	C₂×C₅×Dic₃	C₂²	1	6	yes	—
9	C₂×C₃×Dic₅	C₂²	1	6	yes	—
10	#1204	C₂²	1	16	yes	direct product factor
11	C₂²×Dic₆	—	5	1	no	maximal, Hall
12	C₂²×C₄×C₅	—	3	1	no	—
13	C₂²×Dic₅	—	3	1	no	—
14	C₂²×Dic₅	D₃	1	1	yes	characteristic
15	C₂×Dic₁₀	—	3	12	no	—
16	C₂²×C₃×C₅	Q₈	1	4	yes	complemented
17	C₂²×C₃×C₅	C₂³	1	3	yes	—
18	Dic₁₅	C₂³	1	4	yes	—
19	C₃×Dic₅	C₂³	1	4	yes	—
20	C₅×Dic₃	C₂³	1	4	yes	—
21	C₂²×C₃×C₄	—	5	1	no	—
22	C₂²×Dic₃	—	5	1	no	—
23	C₂²×Dic₃	D₅	1	1	yes	characteristic
24	C₂×Dic₆	—	5	12	no	—
25	C₂³×C₅	C₂×D₃	1	1	yes	characteristic
26	C₂×C₄×C₅	—	3	6	no	—
27	C₂×Dic₅	—	3	6	no	—
28	C₂×Dic₅	C₂×D₃	1	6	yes	—
29	Dic₁₀	—	3	16	no	—
30	C₂²×Q₈	—	15	1	no	Sylow
31	C₂×C₃×C₅	C₂×Q₈	1	6	yes	complemented
32	C₂×C₃×C₅	C₂⁴	1	1	yes	characteristic
33	C₂³×C₃	C₂×D₅	1	1	yes	characteristic
34	C₂×C₃×C₄	—	5	6	no	—
35	C₂×Dic₃	—	5	6	no	—
36	C₂×Dic₃	C₂×D₅	1	6	yes	—
37	Dic₆	—	5	16	no	—
38	C₂²×C₅	Dic₆	1	4	yes	complemented
39	C₂²×C₅	C₂²×D₃	1	3	yes	—
40	C₄×C₅	—	3	4	no	—
41	Dic₅	—	3	4	no	—
42	Dic₅	C₂²×D₃	1	4	yes	—
43	C₂×Q₈	—	15	12	no	—
44	C₂²×C₄	—	3	1	no	—
45	C₂²×C₄	—	5	1	no	—
46	C₂²×C₄	—	15	1	no	—
47	C₃×C₅	C₂²×Q₈	1	1	yes	characteristic, complemented, Hall
48	C₂²×C₃	Dic₁₀	1	4	yes	complemented
49	C₂²×C₃	C₂²×D₅	1	3	yes	—
50	C₃×C₄	—	5	4	no	—
51	Dic₃	—	5	4	no	—
52	Dic₃	C₂²×D₅	1	4	yes	—
53	C₂×C₅	C₂×Dic₆	1	6	yes	complemented
54	C₂×C₅	C₂³×D₃	1	1	yes	characteristic
55	C₂³	D₃×D₅	1	1	yes	characteristic
56	Q₈	—	15	16	no	—
57	C₂×C₄	—	3	6	no	—
58	C₂×C₄	—	5	6	no	—
59	C₂×C₄	—	15	6	no	—
60	C₂×C₃	C₂×Dic₁₀	1	6	yes	complemented
61	C₂×C₃	C₂³×D₅	1	1	yes	characteristic
62	C₅	C₂²×Dic₆	1	1	yes	characteristic, complemented, Sylow
63	C₂²	#1204	1	4	yes	direct product factor
64	C₂²	C₂×D₃×D₅	1	3	yes	—
65	C₄	—	3	4	no	—
66	C₄	—	5	4	no	—
67	C₄	—	15	4	no	—
68	C₃	C₂²×Dic₁₀	1	1	yes	characteristic, complemented, Sylow
69	C₂	C₂×#1204	1	6	yes	direct product factor
70	C₂	C₂²×D₃×D₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³	D₃×D₅
commutator, cosocle	C₂×C₃×C₅	C₂⁴
Frattini	C₂	C₂²×D₃×D₅
Fitting, socle, prime radical	C₂³×C₃×C₅	C₂²
hypocenter	C₃×C₅	C₂²×Q₈
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
4. C₂×#1204	C₂	direct (N × H)
10. #1204	C₂²	direct (N × H)
16. C₂²×C₃×C₅	Q₈	semidirect
31. C₂×C₃×C₅	C₂×Q₈	semidirect
38. C₂²×C₅	Dic₆	semidirect
47. C₃×C₅	C₂²×Q₈	semidirect
48. C₂²×C₃	Dic₁₀	semidirect
53. C₂×C₅	C₂×Dic₆	semidirect
60. C₂×C₃	C₂×Dic₁₀	semidirect
62. C₅	C₂²×Dic₆	semidirect
63. C₂²	#1204	direct (N × H)
68. C₃	C₂²×Dic₁₀	semidirect
69. C₂	C₂×#1204	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×Dic₁₅	2
C₂²×C₅×Dic₃	2
C₂²×C₃×Dic₅	2
C₂×#1204	2
C₂²×Dic₁₀	3
C₂²×Dic₆	5

Subgroups — C22×#1204 #92045

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#1204 #92045