Subgroups — C₂²×C₅×C₂²⋊C₄ #66176

Order 320.

674 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄×C₅	C₂	1	2	yes	maximal, complemented
3	C₂×C₅×C₂²⋊C₄	C₂	1	12	yes	maximal, direct product factor
4	C₂⁴×C₅	—	2	8	no	—
5	C₂⁴×C₅	C₄	1	8	yes	complemented
6	C₂⁴×C₅	C₂²	1	6	yes	—
7	C₂⁴×C₅	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₅	—	2	8	no	—
9	C₂²×C₄×C₅	C₂²	1	12	yes	complemented
10	C₅×C₂²⋊C₄	C₂²	1	16	yes	direct product factor
11	C₂²×C₂²⋊C₄	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
12	C₂³×C₅	—	2	32	no	—
13	C₂³×C₅	—	2	24	no	—
14	C₂³×C₅	C₂×C₄	1	4	yes	—
15	C₂³×C₅	C₂×C₄	1	24	yes	complemented
16	C₂³×C₅	D₄	1	8	yes	—
17	C₂³×C₅	C₂³	1	4	yes	—
18	C₂³×C₅	C₂³	1	3	yes	—
19	C₂×C₄×C₅	—	2	24	no	—
20	C₂×C₄×C₅	C₂³	1	8	yes	complemented
21	C₂⁵	C₂×C₅	1	1	yes	characteristic
22	C₂³×C₄	C₂×C₅	1	2	yes	complemented
23	C₂×C₂²⋊C₄	C₂×C₅	1	12	yes	direct product factor
24	C₂²×C₅	—	2	48	no	—
25	C₂²×C₅	—	2	8	no	—
26	C₂²×C₅	C₂²⋊C₄	1	16	yes	direct product factor
27	C₂²×C₅	C₂²×C₄	1	6	yes	—
28	C₂²×C₅	C₂²×C₄	1	8	yes	complemented
29	C₂²×C₅	C₂×D₄	1	12	yes	—
30	C₂²×C₅	C₂⁴	1	1	yes	characteristic
31	C₄×C₅	—	2	8	no	—
32	C₂⁴	—	2	8	no	—
33	C₂⁴	C₄×C₅	1	8	yes	complemented
34	C₂⁴	C₂²×C₅	1	6	yes	—
35	C₂⁴	C₂²×C₅	1	1	yes	characteristic
36	C₂²×C₄	—	2	8	no	—
37	C₂²×C₄	C₂²×C₅	1	12	yes	complemented
38	C₂²⋊C₄	C₂²×C₅	1	16	yes	direct product factor
39	C₂×C₅	—	2	8	no	—
40	C₂×C₅	C₂×C₂²⋊C₄	1	12	yes	direct product factor
41	C₂×C₅	C₂³×C₄	1	1	yes	characteristic
42	C₂×C₅	C₂²×D₄	1	2	yes	—
43	C₂³	—	2	32	no	—
44	C₂³	—	2	24	no	—
45	C₂³	C₂×C₄×C₅	1	4	yes	—
46	C₂³	C₂×C₄×C₅	1	24	yes	complemented
47	C₂³	C₅×D₄	1	8	yes	—
48	C₂³	C₂³×C₅	1	4	yes	—
49	C₂³	C₂³×C₅	1	3	yes	—
50	C₂×C₄	—	2	24	no	—
51	C₂×C₄	C₂³×C₅	1	8	yes	complemented
52	C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
53	C₂²	—	2	48	no	—
54	C₂²	—	2	8	no	—
55	C₂²	C₅×C₂²⋊C₄	1	16	yes	direct product factor
56	C₂²	C₂²×C₄×C₅	1	6	yes	—
57	C₂²	C₂²×C₄×C₅	1	8	yes	complemented
58	C₂²	C₂×C₅×D₄	1	12	yes	—
59	C₂²	C₂⁴×C₅	1	1	yes	characteristic
60	C₄	—	2	8	no	—
61	C₂	—	2	8	no	—
62	C₂	C₂×C₅×C₂²⋊C₄	1	12	yes	direct product factor
63	C₂	C₂³×C₄×C₅	1	1	yes	characteristic
64	C₂	C₂²×C₅×D₄	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅	C₂²
commutator	C₂	C₂³×C₄×C₅
Frattini, cosocle	C₂²	C₂⁴×C₅
prime radical	C₂⁵×C₅	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₄×C₅	C₂	semidirect
3. C₂×C₅×C₂²⋊C₄	C₂	direct (N × H)
5. C₂⁴×C₅	C₄	semidirect
9. C₂²×C₄×C₅	C₂²	semidirect
10. C₅×C₂²⋊C₄	C₂²	direct (N × H)
11. C₂²×C₂²⋊C₄	C₅	direct (N × H)
15. C₂³×C₅	C₂×C₄	semidirect
20. C₂×C₄×C₅	C₂³	semidirect
22. C₂³×C₄	C₂×C₅	semidirect
23. C₂×C₂²⋊C₄	C₂×C₅	direct (N × H)
26. C₂²×C₅	C₂²⋊C₄	direct (N × H)
28. C₂²×C₅	C₂²×C₄	semidirect
33. C₂⁴	C₄×C₅	semidirect
37. C₂²×C₄	C₂²×C₅	semidirect
38. C₂²⋊C₄	C₂²×C₅	direct (N × H)
40. C₂×C₅	C₂×C₂²⋊C₄	direct (N × H)
46. C₂³	C₂×C₄×C₅	semidirect
51. C₂×C₄	C₂³×C₅	semidirect
52. C₅	C₂²×C₂²⋊C₄	direct (N × H)
55. C₂²	C₅×C₂²⋊C₄	direct (N × H)
57. C₂²	C₂²×C₄×C₅	semidirect
62. C₂	C₂×C₅×C₂²⋊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
C₂⁵×C₅	2
C₂³×C₄×C₅	2
C₂×C₅×C₂²⋊C₄	2
C₂²×C₂²⋊C₄	5

Subgroups — C22×C5×C22⋊C4 #66176

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×C₂²⋊C₄ #66176