Subgroups — C₅×D₄² #66209

Order 320.

428 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×D₄	C₂	1	4	yes	maximal, complemented
2	C₅×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#121	C₂	1	4	yes	maximal, complemented
4	C₅×#120	C₂	1	4	yes	maximal, complemented
5	C₄×C₅×D₄	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₅	C₂²	1	4	yes	complemented
7	C₂×C₅×D₄	—	2	16	no	—
8	C₂×C₅×D₄	C₂²	1	2	yes	—
9	C₂×C₅×D₄	C₂²	1	8	yes	complemented
10	C₂×C₅×D₄	C₂²	1	4	yes	complemented
11	C₂×C₅×D₄	C₂²	1	2	yes	complemented
12	C₂²×C₄×C₅	C₂²	1	4	yes	complemented
13	C₅×C₄⋊C₄	C₂²	1	2	yes	complemented
14	C₅×C₂²⋊C₄	C₂²	1	4	yes	complemented
15	C₅×C₂²⋊C₄	C₂²	1	4	yes	complemented
16	C₄²×C₅	C₂²	1	1	yes	characteristic, complemented
17	D₄²	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₅	—	2	16	no	—
19	C₂³×C₅	—	4	4	no	—
20	C₂³×C₅	C₂³	1	4	yes	—
21	C₂³×C₅	C₂³	1	4	yes	—
22	C₅×D₄	—	2	16	no	—
23	C₅×D₄	—	2	8	no	—
24	C₅×D₄	—	4	2	no	—
25	C₅×D₄	D₄	1	8	yes	direct product factor
26	C₂×C₄×C₅	—	2	8	no	—
27	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₅	C₂³	1	4	yes	—
29	C₂×C₄×C₅	C₂³	1	2	yes	—
30	C₂²×D₄	C₂×C₅	1	4	yes	complemented
31	#127	C₂×C₅	1	1	yes	characteristic, complemented
32	#121	C₂×C₅	1	4	yes	complemented
33	#120	C₂×C₅	1	4	yes	complemented
34	C₄×D₄	C₂×C₅	1	2	yes	complemented
35	C₂²×C₅	—	2	4	no	—
36	C₂²×C₅	—	2	4	no	—
37	C₂²×C₅	—	2	8	no	—
38	C₂²×C₅	—	2	4	no	—
39	C₂²×C₅	—	4	16	no	—
40	C₂²×C₅	C₂×D₄	1	8	yes	complemented
41	C₂²×C₅	C₂⁴	1	1	yes	characteristic
42	C₄×C₅	—	2	4	no	—
43	C₄×C₅	—	2	1	no	—
44	C₄×C₅	C₂×D₄	1	4	yes	complemented
45	C₂⁴	C₂²×C₅	1	4	yes	complemented
46	C₂×D₄	—	2	16	no	—
47	C₂×D₄	C₂²×C₅	1	2	yes	—
48	C₂×D₄	C₂²×C₅	1	8	yes	complemented
49	C₂×D₄	C₂²×C₅	1	4	yes	complemented
50	C₂×D₄	C₂²×C₅	1	2	yes	complemented
51	C₂²×C₄	C₂²×C₅	1	4	yes	complemented
52	C₄⋊C₄	C₂²×C₅	1	2	yes	complemented
53	C₂²⋊C₄	C₂²×C₅	1	4	yes	complemented
54	C₂²⋊C₄	C₂²×C₅	1	4	yes	complemented
55	C₄²	C₂²×C₅	1	1	yes	characteristic, complemented
56	C₂×C₅	—	2	8	no	—
57	C₂×C₅	—	4	4	no	—
58	C₂×C₅	C₂²×D₄	1	2	yes	—
59	C₂×C₅	H(5,𝔽₂)	1	1	yes	characteristic
60	C₂³	—	2	16	no	—
61	C₂³	—	4	4	no	—
62	C₂³	C₂³×C₅	1	4	yes	—
63	C₂³	C₂³×C₅	1	4	yes	—
64	D₄	—	2	16	no	—
65	D₄	—	2	8	no	—
66	D₄	—	4	2	no	—
67	D₄	C₅×D₄	1	8	yes	direct product factor
68	C₂×C₄	—	2	8	no	—
69	C₂×C₄	C₂³×C₅	1	1	yes	characteristic
70	C₂×C₄	C₂³×C₅	1	4	yes	—
71	C₂×C₄	C₂³×C₅	1	2	yes	—
72	C₅	D₄²	1	1	yes	characteristic, direct product factor, Sylow
73	C₂²	—	2	4	no	—
74	C₂²	—	2	4	no	—
75	C₂²	—	2	8	no	—
76	C₂²	—	2	4	no	—
77	C₂²	—	4	16	no	—
78	C₂²	C₂×C₅×D₄	1	8	yes	complemented
79	C₂²	C₂⁴×C₅	1	1	yes	characteristic
80	C₄	—	2	4	no	—
81	C₄	—	2	1	no	—
82	C₄	C₂×C₅×D₄	1	4	yes	complemented
83	C₂	—	2	8	no	—
84	C₂	—	4	4	no	—
85	C₂	C₂²×C₅×D₄	1	2	yes	—
86	C₂	C₅×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₅
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×D₄	C₂	semidirect
2. C₅×#127	C₂	semidirect
3. C₅×#121	C₂	semidirect
4. C₅×#120	C₂	semidirect
5. C₄×C₅×D₄	C₂	semidirect
6. C₂⁴×C₅	C₂²	semidirect
9. C₂×C₅×D₄	C₂²	semidirect
10. C₂×C₅×D₄	C₂²	semidirect
11. C₂×C₅×D₄	C₂²	semidirect
12. C₂²×C₄×C₅	C₂²	semidirect
13. C₅×C₄⋊C₄	C₂²	semidirect
14. C₅×C₂²⋊C₄	C₂²	semidirect
15. C₅×C₂²⋊C₄	C₂²	semidirect
16. C₄²×C₅	C₂²	semidirect
17. D₄²	C₅	direct (N × H)
25. C₅×D₄	D₄	direct (N × H)
30. C₂²×D₄	C₂×C₅	semidirect
31. #127	C₂×C₅	semidirect
32. #121	C₂×C₅	semidirect
33. #120	C₂×C₅	semidirect
34. C₄×D₄	C₂×C₅	semidirect
40. C₂²×C₅	C₂×D₄	semidirect
44. C₄×C₅	C₂×D₄	semidirect
45. C₂⁴	C₂²×C₅	semidirect
48. C₂×D₄	C₂²×C₅	semidirect
49. C₂×D₄	C₂²×C₅	semidirect
50. C₂×D₄	C₂²×C₅	semidirect
51. C₂²×C₄	C₂²×C₅	semidirect
52. C₄⋊C₄	C₂²×C₅	semidirect
53. C₂²⋊C₄	C₂²×C₅	semidirect
54. C₂²⋊C₄	C₂²×C₅	semidirect
55. C₄²	C₂²×C₅	semidirect
67. D₄	C₅×D₄	direct (N × H)
72. C₅	D₄²	direct (N × H)
78. C₂²	C₂×C₅×D₄	semidirect
82. C₄	C₂×C₅×D₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×D₄	2
C₅×#127	2
C₅×#121	2
C₅×#120	2
C₄×C₅×D₄	2
D₄²	5

Subgroups — C5×D42 #66209

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×D₄² #66209