Subgroups — C₂×C₅×#117 #66178

Order 320.

330 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×C₅	C₂	1	1	yes	maximal, characteristic
2	C₅×#117	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₅×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
4	C₂×C₅×C₂²⋊C₄	C₂	1	2	yes	maximal
5	C₂×C₄²×C₅	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₅	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₅	C₄	1	8	yes	complemented
8	C₂²×C₄×C₅	C₂²	1	4	yes	—
9	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₅	C₂²	1	4	yes	—
11	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
12	C₅×C₄⋊C₄	C₂²	1	8	yes	complemented
13	C₅×C₂²⋊C₄	C₂²	1	8	yes	—
14	C₄²×C₅	C₂²	1	8	yes	complemented
15	C₂×#117	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂³×C₅	—	2	4	no	—
17	C₂³×C₅	C₂×C₄	1	4	yes	—
18	C₂³×C₅	C₂³	1	2	yes	—
19	C₂³×C₅	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₅	—	2	8	no	—
21	C₂×C₄×C₅	C₂×C₄	1	4	yes	—
22	C₂×C₄×C₅	C₂×C₄	1	4	yes	—
23	C₂×C₄×C₅	C₂×C₄	1	16	yes	complemented
24	C₂×C₄×C₅	C₂³	1	8	yes	—
25	C₂×C₄×C₅	C₂³	1	2	yes	—
26	C₂×C₄×C₅	C₂³	1	2	yes	—
27	C₂³×C₄	C₂×C₅	1	1	yes	characteristic
28	#117	C₂×C₅	1	8	yes	direct product factor
29	C₂×C₄⋊C₄	C₂×C₅	1	2	yes	complemented
30	C₂×C₂²⋊C₄	C₂×C₅	1	2	yes	—
31	C₂×C₄²	C₂×C₅	1	2	yes	complemented
32	C₂²×C₅	—	2	8	no	—
33	C₂²×C₅	—	2	4	no	—
34	C₂²×C₅	C₂²×C₄	1	4	yes	—
35	C₂²×C₅	C₂²×C₄	1	2	yes	—
36	C₂²×C₅	#41	1	4	yes	—
37	C₂²×C₅	C₂⁴	1	1	yes	characteristic
38	C₄×C₅	—	2	8	no	—
39	C₄×C₅	C₂²×C₄	1	4	yes	—
40	C₄×C₅	C₂²×C₄	1	4	yes	—
41	C₂⁴	C₂²×C₅	1	1	yes	characteristic
42	C₂²×C₄	C₄×C₅	1	8	yes	complemented
43	C₂²×C₄	C₂²×C₅	1	4	yes	—
44	C₂²×C₄	C₂²×C₅	1	1	yes	characteristic
45	C₂²×C₄	C₂²×C₅	1	1	yes	characteristic
46	C₂²×C₄	C₂²×C₅	1	4	yes	—
47	C₄⋊C₄	C₂²×C₅	1	8	yes	complemented
48	C₂²⋊C₄	C₂²×C₅	1	8	yes	—
49	C₄²	C₂²×C₅	1	8	yes	complemented
50	C₂×C₅	—	2	4	no	—
51	C₂×C₅	#117	1	4	yes	direct product factor
52	C₂×C₅	C₂³×C₄	1	1	yes	characteristic
53	C₂×C₅	C₂×#41	1	2	yes	—
54	C₂³	—	2	4	no	—
55	C₂³	C₂×C₄×C₅	1	4	yes	—
56	C₂³	C₂³×C₅	1	2	yes	—
57	C₂³	C₂³×C₅	1	1	yes	characteristic
58	C₂×C₄	—	2	8	no	—
59	C₂×C₄	C₂×C₄×C₅	1	4	yes	—
60	C₂×C₄	C₂×C₄×C₅	1	4	yes	—
61	C₂×C₄	C₂×C₄×C₅	1	16	yes	complemented
62	C₂×C₄	C₂³×C₅	1	8	yes	—
63	C₂×C₄	C₂³×C₅	1	2	yes	—
64	C₂×C₄	C₂³×C₅	1	2	yes	—
65	C₅	C₂×#117	1	1	yes	characteristic, direct product factor, Sylow
66	C₂²	—	2	8	no	—
67	C₂²	—	2	4	no	—
68	C₂²	C₂²×C₄×C₅	1	4	yes	—
69	C₂²	C₂²×C₄×C₅	1	2	yes	—
70	C₂²	C₅×#41	1	4	yes	—
71	C₂²	C₂⁴×C₅	1	1	yes	characteristic
72	C₄	—	2	8	no	—
73	C₄	C₂²×C₄×C₅	1	4	yes	—
74	C₄	C₂²×C₄×C₅	1	4	yes	—
75	C₂	—	2	4	no	—
76	C₂	C₅×#117	1	4	yes	direct product factor
77	C₂	C₂³×C₄×C₅	1	1	yes	characteristic
78	C₂	C₂×C₅×#41	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₅×#117	C₂	direct (N × H)
3. C₂×C₅×C₄⋊C₄	C₂	semidirect
5. C₂×C₄²×C₅	C₂	semidirect
7. C₂²×C₄×C₅	C₄	semidirect
12. C₅×C₄⋊C₄	C₂²	semidirect
14. C₄²×C₅	C₂²	semidirect
15. C₂×#117	C₅	direct (N × H)
23. C₂×C₄×C₅	C₂×C₄	semidirect
28. #117	C₂×C₅	direct (N × H)
29. C₂×C₄⋊C₄	C₂×C₅	semidirect
31. C₂×C₄²	C₂×C₅	semidirect
42. C₂²×C₄	C₄×C₅	semidirect
47. C₄⋊C₄	C₂²×C₅	semidirect
49. C₄²	C₂²×C₅	semidirect
51. C₂×C₅	#117	direct (N × H)
61. C₂×C₄	C₂×C₄×C₅	semidirect
65. C₅	C₂×#117	direct (N × H)
76. C₂	C₅×#117	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄×C₅	2
C₅×#117	2
C₂×C₅×C₄⋊C₄	2
C₂×C₅×C₂²⋊C₄	2
C₂×C₄²×C₅	2
C₂×#117	5

Subgroups — C2×C5×#117 #66178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#117 #66178