Subgroups — C₅×#392 #65555

Order 320.

322 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×D₄	C₂	1	3	yes	maximal, complemented
2	C₂×C₅×C₂²⋊C₄	C₂	1	3	yes	maximal, complemented
3	C₅×#95	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×C₅	C₂²	1	3	yes	—
5	C₂×C₅×D₄	—	2	12	no	—
6	C₂×C₅×D₄	—	2	6	no	—
7	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₅	C₂²	1	3	yes	—
9	C₅×C₂²⋊C₄	—	2	6	no	—
10	#392	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂³×C₅	—	2	3	no	—
12	C₂³×C₅	—	2	6	no	—
13	C₂³×C₅	—	2	6	no	—
14	C₂³×C₅	—	2	3	no	—
15	C₂³×C₅	D₄	1	6	yes	complemented
16	C₂³×C₅	C₂³	1	1	yes	characteristic
17	C₅×D₄	—	2	24	no	—
18	C₂×C₄×C₅	—	2	6	no	—
19	C₂×C₄×C₅	—	2	3	no	—
20	C₂×C₄×C₅	D₄	1	6	yes	complemented
21	C₂²×D₄	C₂×C₅	1	3	yes	complemented
22	C₂×C₂²⋊C₄	C₂×C₅	1	3	yes	complemented
23	#95	C₂×C₅	1	1	yes	characteristic, complemented
24	C₂²×C₅	—	2	12	no	—
25	C₂²×C₅	—	2	6	no	—
26	C₂²×C₅	—	4	3	no	—
27	C₂²×C₅	—	4	6	no	—
28	C₂²×C₅	—	4	3	no	—
29	C₂²×C₅	C₂×D₄	1	3	yes	—
30	C₂²×C₅	C₂×D₄	1	3	yes	—
31	C₂²×C₅	#41	1	1	yes	characteristic
32	C₄×C₅	—	2	6	no	—
33	C₄×C₅	—	4	1	no	—
34	C₂⁴	C₂²×C₅	1	3	yes	—
35	C₂×D₄	—	2	12	no	—
36	C₂×D₄	—	2	6	no	—
37	C₂²×C₄	C₂²×C₅	1	1	yes	characteristic
38	C₂²×C₄	C₂²×C₅	1	3	yes	—
39	C₂²⋊C₄	—	2	6	no	—
40	C₂×C₅	—	4	6	no	—
41	C₂×C₅	#120	1	3	yes	—
42	C₂×C₅	#121	1	3	yes	—
43	C₂×C₅	#127	1	1	yes	characteristic
44	C₂³	—	2	3	no	—
45	C₂³	—	2	6	no	—
46	C₂³	—	2	6	no	—
47	C₂³	—	2	3	no	—
48	C₂³	C₅×D₄	1	6	yes	complemented
49	C₂³	C₂³×C₅	1	1	yes	characteristic
50	D₄	—	2	24	no	—
51	C₂×C₄	—	2	6	no	—
52	C₂×C₄	—	2	3	no	—
53	C₂×C₄	C₅×D₄	1	6	yes	complemented
54	C₅	#392	1	1	yes	characteristic, direct product factor, Sylow
55	C₂²	—	2	12	no	—
56	C₂²	—	2	6	no	—
57	C₂²	—	4	3	no	—
58	C₂²	—	4	6	no	—
59	C₂²	—	4	3	no	—
60	C₂²	C₂×C₅×D₄	1	3	yes	—
61	C₂²	C₂×C₅×D₄	1	3	yes	—
62	C₂²	C₅×#41	1	1	yes	characteristic
63	C₄	—	2	6	no	—
64	C₄	—	4	1	no	—
65	C₂	—	4	6	no	—
66	C₂	C₅×#120	1	3	yes	—
67	C₂	C₅×#121	1	3	yes	—
68	C₂	C₅×#127	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂³
commutator, Frattini, cosocle	C₂³	C₂³×C₅
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×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. C₂×C₅×C₂²⋊C₄	C₂	semidirect
3. C₅×#95	C₂	semidirect
10. #392	C₅	direct (N × H)
15. C₂³×C₅	D₄	semidirect
20. C₂×C₄×C₅	D₄	semidirect
21. C₂²×D₄	C₂×C₅	semidirect
22. C₂×C₂²⋊C₄	C₂×C₅	semidirect
23. #95	C₂×C₅	semidirect
48. C₂³	C₅×D₄	semidirect
53. C₂×C₄	C₅×D₄	semidirect
54. C₅	#392	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×C₅×D₄	2
C₂×C₅×C₂²⋊C₄	2
C₅×#95	2
#392	5

Subgroups — C5×#392 #65555

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#392 #65555