Subgroups — C₂×C₅×#126 #66192

Order 320.

246 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#126	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₅×C₄⋊C₄	C₂	1	3	yes	maximal, complemented
3	C₂×C₅×C₂²⋊C₄	C₂	1	3	yes	maximal
4	C₂×C₄²×C₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×C₅	C₂²	1	1	yes	characteristic
6	C₂²×C₄×C₅	C₂²	1	3	yes	—
7	C₂²×C₄×C₅	C₂²	1	3	yes	—
8	C₅×C₄⋊C₄	C₂²	1	12	yes	complemented
9	C₅×C₂²⋊C₄	C₂²	1	12	yes	—
10	C₄²×C₅	C₂²	1	4	yes	complemented
11	C₂×#126	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
12	C₂³×C₅	—	2	6	no	—
13	C₂³×C₅	C₂³	1	2	yes	—
14	C₂³×C₅	C₂³	1	1	yes	characteristic
15	C₂×C₄×C₅	—	2	6	no	—
16	C₂×C₄×C₅	—	2	6	no	—
17	C₂×C₄×C₅	C₂³	1	6	yes	—
18	C₂×C₄×C₅	C₂³	1	6	yes	—
19	#126	C₂×C₅	1	8	yes	direct product factor
20	C₂×C₄⋊C₄	C₂×C₅	1	3	yes	complemented
21	C₂×C₂²⋊C₄	C₂×C₅	1	3	yes	—
22	C₂×C₄²	C₂×C₅	1	1	yes	characteristic, complemented
23	C₂²×C₅	—	2	6	no	—
24	C₂²×C₅	—	4	4	no	—
25	C₂²×C₅	#41	1	6	yes	—
26	C₂²×C₅	C₂⁴	1	1	yes	characteristic
27	C₄×C₅	—	2	6	no	—
28	C₄×C₅	—	2	6	no	—
29	C₂⁴	C₂²×C₅	1	1	yes	characteristic
30	C₂²×C₄	C₂²×C₅	1	3	yes	—
31	C₂²×C₄	C₂²×C₅	1	3	yes	—
32	C₄⋊C₄	C₂²×C₅	1	12	yes	complemented
33	C₂²⋊C₄	C₂²×C₅	1	12	yes	—
34	C₄²	C₂²×C₅	1	4	yes	complemented
35	C₂×C₅	—	4	2	no	—
36	C₂×C₅	#126	1	4	yes	direct product factor
37	C₂×C₅	C₂×#41	1	3	yes	—
38	C₂³	—	2	6	no	—
39	C₂³	C₂³×C₅	1	2	yes	—
40	C₂³	C₂³×C₅	1	1	yes	characteristic
41	C₂×C₄	—	2	6	no	—
42	C₂×C₄	—	2	6	no	—
43	C₂×C₄	C₂³×C₅	1	6	yes	—
44	C₂×C₄	C₂³×C₅	1	6	yes	—
45	C₅	C₂×#126	1	1	yes	characteristic, direct product factor, Sylow
46	C₂²	—	2	6	no	—
47	C₂²	—	4	4	no	—
48	C₂²	C₅×#41	1	6	yes	—
49	C₂²	C₂⁴×C₅	1	1	yes	characteristic
50	C₄	—	2	6	no	—
51	C₄	—	2	6	no	—
52	C₂	—	4	2	no	—
53	C₂	C₅×#126	1	4	yes	direct product factor
54	C₂	C₂×C₅×#41	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₅×#126	C₂	direct (N × H)
2. C₂×C₅×C₄⋊C₄	C₂	semidirect
4. C₂×C₄²×C₅	C₂	semidirect
8. C₅×C₄⋊C₄	C₂²	semidirect
10. C₄²×C₅	C₂²	semidirect
11. C₂×#126	C₅	direct (N × H)
19. #126	C₂×C₅	direct (N × H)
20. C₂×C₄⋊C₄	C₂×C₅	semidirect
22. C₂×C₄²	C₂×C₅	semidirect
32. C₄⋊C₄	C₂²×C₅	semidirect
34. C₄²	C₂²×C₅	semidirect
36. C₂×C₅	#126	direct (N × H)
45. C₅	C₂×#126	direct (N × H)
53. C₂	C₅×#126	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#126	2
C₂×C₅×C₄⋊C₄	2
C₂×C₅×C₂²⋊C₄	2
C₂×C₄²×C₅	2
C₂×#126	5

Subgroups — C2×C5×#126 #66192

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#126 #66192