Subgroups — C₂²×#685 #66121

Order 320.

674 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄×C₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#685	C₂	1	12	yes	maximal, direct product factor
5	C₂⁴×C₅	C₂²	1	1	yes	characteristic
6	C₂³×D₅	—	2	4	no	—
7	C₂³×D₅	—	2	4	no	—
8	C₂³×D₅	C₄	1	8	yes	complemented
9	C₂³×D₅	C₂²	1	6	yes	—
10	C₂²×C₄×C₅	—	2	4	no	—
11	C₂²×C₄×C₅	C₂²	1	6	yes	complemented
12	C₂²×Dic₅	—	2	4	no	—
13	C₂²×Dic₅	C₂²	1	6	yes	complemented
14	#685	C₂²	1	16	yes	direct product factor
15	C₂²×C₂²⋊C₄	—	5	1	no	maximal, Sylow
16	C₂³×C₅	C₂×C₄	1	4	yes	—
17	C₂³×C₅	D₄	1	4	yes	—
18	C₂³×C₅	D₄	1	4	yes	—
19	C₂³×C₅	C₂³	1	3	yes	—
20	C₂²×D₅	—	2	32	no	—
21	C₂²×D₅	—	2	12	no	—
22	C₂²×D₅	—	2	12	no	—
23	C₂²×D₅	C₂×C₄	1	24	yes	complemented
24	C₂²×D₅	C₂³	1	4	yes	—
25	C₂×C₄×C₅	—	2	12	no	—
26	C₂×C₄×C₅	C₂³	1	4	yes	complemented
27	C₂×Dic₅	—	2	12	no	—
28	C₂×Dic₅	C₂³	1	4	yes	complemented
29	C₂⁵	—	5	1	no	—
30	C₂³×C₄	—	5	1	no	—
31	C₂³×C₄	D₅	1	1	yes	characteristic, complemented
32	C₂×C₂²⋊C₄	—	5	12	no	—
33	C₂²×C₅	C₂²⋊C₄	1	16	yes	complemented
34	C₂²×C₅	C₂²×C₄	1	6	yes	—
35	C₂²×C₅	C₂×D₄	1	6	yes	—
36	C₂²×C₅	C₂×D₄	1	6	yes	—
37	C₂²×C₅	C₂⁴	1	1	yes	characteristic
38	C₂×D₅	—	2	48	no	—
39	C₂×D₅	—	2	4	no	—
40	C₂×D₅	—	2	4	no	—
41	C₂×D₅	C₂²×C₄	1	8	yes	complemented
42	C₄×C₅	—	2	4	no	—
43	Dic₅	—	2	4	no	—
44	C₂⁴	—	5	8	no	—
45	C₂⁴	—	5	6	no	—
46	C₂⁴	—	10	4	no	—
47	C₂⁴	—	10	4	no	—
48	C₂⁴	C₂×D₅	1	1	yes	characteristic
49	C₂²×C₄	—	2	4	no	—
50	C₂²×C₄	—	5	6	no	—
51	C₂²×C₄	—	10	4	no	—
52	C₂²×C₄	C₂×D₅	1	6	yes	complemented
53	C₂²⋊C₄	—	5	16	no	—
54	C₂×C₅	C₂×C₂²⋊C₄	1	12	yes	complemented
55	C₂×C₅	C₂³×C₄	1	1	yes	characteristic
56	C₂×C₅	C₂²×D₄	1	1	yes	characteristic
57	C₂×C₅	C₂²×D₄	1	1	yes	characteristic
58	D₅	—	2	8	no	—
59	C₂³	—	5	24	no	—
60	C₂³	—	5	4	no	—
61	C₂³	—	10	32	no	—
62	C₂³	—	10	12	no	—
63	C₂³	—	10	12	no	—
64	C₂³	C₄×D₅	1	4	yes	—
65	C₂³	D₂₀	1	4	yes	—
66	C₂³	C₂²⋊D₅	1	4	yes	—
67	C₂³	C₂²×D₅	1	3	yes	—
68	C₂×C₄	—	2	12	no	—
69	C₂×C₄	—	5	4	no	—
70	C₂×C₄	—	10	12	no	—
71	C₂×C₄	C₂²×D₅	1	4	yes	complemented
72	C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
73	C₂²	—	5	8	no	—
74	C₂²	—	10	4	no	—
75	C₂²	—	10	4	no	—
76	C₂²	—	10	48	no	—
77	C₂²	#685	1	16	yes	direct product factor
78	C₂²	C₂×C₄×D₅	1	6	yes	—
79	C₂²	C₂×D₂₀	1	6	yes	—
80	C₂²	C₂×C₂²⋊D₅	1	6	yes	—
81	C₂²	C₂³×D₅	1	1	yes	characteristic
82	C₄	—	2	4	no	—
83	C₄	—	10	4	no	—
84	C₂	—	10	8	no	—
85	C₂	C₂×#685	1	12	yes	direct product factor
86	C₂	C₂²×C₄×D₅	1	1	yes	characteristic
87	C₂	C₂²×D₂₀	1	1	yes	characteristic
88	C₂	C₂²×C₂²⋊D₅	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂³×C₄×C₅	C₂	semidirect
3. C₂³×Dic₅	C₂	semidirect
4. C₂×#685	C₂	direct (N × H)
8. C₂³×D₅	C₄	semidirect
11. C₂²×C₄×C₅	C₂²	semidirect
13. C₂²×Dic₅	C₂²	semidirect
14. #685	C₂²	direct (N × H)
23. C₂²×D₅	C₂×C₄	semidirect
26. C₂×C₄×C₅	C₂³	semidirect
28. C₂×Dic₅	C₂³	semidirect
31. C₂³×C₄	D₅	semidirect
33. C₂²×C₅	C₂²⋊C₄	semidirect
41. C₂×D₅	C₂²×C₄	semidirect
52. C₂²×C₄	C₂×D₅	semidirect
54. C₂×C₅	C₂×C₂²⋊C₄	semidirect
71. C₂×C₄	C₂²×D₅	semidirect
72. C₅	C₂²×C₂²⋊C₄	semidirect
77. C₂²	#685	direct (N × H)
85. C₂	C₂×#685	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₅	2
C₂³×C₄×C₅	2
C₂³×Dic₅	2
C₂×#685	2
C₂²×C₂²⋊C₄	5

Subgroups — C22×#685 #66121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#685 #66121