Subgroups — C₅×#537 #66201

Order 320.

294 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#41	C₂	1	2	yes	maximal, complemented
2	C₅×C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
3	C₅×#121	C₂	1	4	yes	maximal, complemented
4	C₅×#121	C₂	1	4	yes	maximal, complemented
5	C₂×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#41	—	2	8	no	—
7	C₂×C₅×Q₈	C₂²	1	2	yes	—
8	C₂×C₅×D₄	C₂²	1	4	yes	—
9	C₂×C₅×D₄	C₂²	1	2	yes	—
10	C₂×C₅×D₄	C₂²	1	4	yes	—
11	C₂²×C₄×C₅	C₂²	1	4	yes	—
12	C₂²×C₄×C₅	C₂²	1	2	yes	—
13	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
14	C₅×C₄⋊C₄	C₂²	1	4	yes	—
15	C₅×C₄⋊C₄	C₂²	1	4	yes	—
16	C₅×C₂²⋊C₄	C₂²	1	8	yes	—
17	#537	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₅	C₂³	1	4	yes	—
19	C₂³×C₅	C₂³	1	1	yes	characteristic
20	C₅×Q₈	—	2	4	no	—
21	C₅×D₄	—	2	8	no	—
22	C₅×D₄	—	2	4	no	—
23	C₅×D₄	—	4	4	no	—
24	C₂×C₄×C₅	—	2	8	no	—
25	C₂×C₄×C₅	—	4	2	no	—
26	C₂×C₄×C₅	D₄	1	4	yes	complemented
27	C₂×C₄×C₅	C₂³	1	4	yes	—
28	C₂×C₄×C₅	C₂³	1	4	yes	—
29	C₂×C₄×C₅	C₂³	1	2	yes	—
30	C₂×#41	C₂×C₅	1	2	yes	complemented
31	C₂²⋊Q₈	C₂×C₅	1	4	yes	complemented
32	#121	C₂×C₅	1	4	yes	complemented
33	#121	C₂×C₅	1	4	yes	complemented
34	C₂×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
35	C₂²×C₅	—	2	1	no	—
36	C₂²×C₅	—	2	1	no	—
37	C₂²×C₅	—	2	4	no	—
38	C₂²×C₅	—	2	4	no	—
39	C₂²×C₅	—	2	4	no	—
40	C₂²×C₅	C₂×D₄	1	2	yes	—
41	C₂²×C₅	C₂⁴	1	1	yes	characteristic
42	C₄×C₅	—	2	4	no	—
43	C₄×C₅	—	2	4	no	—
44	C₄×C₅	C₂×D₄	1	4	yes	—
45	#41	—	2	8	no	—
46	C₂×Q₈	C₂²×C₅	1	2	yes	—
47	C₂×D₄	C₂²×C₅	1	4	yes	—
48	C₂×D₄	C₂²×C₅	1	2	yes	—
49	C₂×D₄	C₂²×C₅	1	4	yes	—
50	C₂²×C₄	C₂²×C₅	1	4	yes	—
51	C₂²×C₄	C₂²×C₅	1	2	yes	—
52	C₂²×C₄	C₂²×C₅	1	1	yes	characteristic
53	C₄⋊C₄	C₂²×C₅	1	4	yes	—
54	C₄⋊C₄	C₂²×C₅	1	4	yes	—
55	C₂²⋊C₄	C₂²×C₅	1	8	yes	—
56	C₂×C₅	—	2	2	no	—
57	C₂×C₅	—	4	4	no	—
58	C₂×C₅	C₂²×D₄	1	1	yes	characteristic
59	C₂×C₅	H(5,𝔽₂)	1	1	yes	characteristic
60	C₂×C₅	2₋¹⁺⁴	1	1	yes	characteristic
61	C₂³	C₂³×C₅	1	4	yes	—
62	C₂³	C₂³×C₅	1	1	yes	characteristic
63	Q₈	—	2	4	no	—
64	D₄	—	2	8	no	—
65	D₄	—	2	4	no	—
66	D₄	—	4	4	no	—
67	C₂×C₄	—	2	8	no	—
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	C₅×D₄	1	4	yes	complemented
70	C₂×C₄	C₂³×C₅	1	4	yes	—
71	C₂×C₄	C₂³×C₅	1	4	yes	—
72	C₂×C₄	C₂³×C₅	1	2	yes	—
73	C₅	#537	1	1	yes	characteristic, direct product factor, Sylow
74	C₂²	—	2	1	no	—
75	C₂²	—	2	1	no	—
76	C₂²	—	2	4	no	—
77	C₂²	—	2	4	no	—
78	C₂²	—	2	4	no	—
79	C₂²	C₂×C₅×D₄	1	2	yes	—
80	C₂²	C₂⁴×C₅	1	1	yes	characteristic
81	C₄	—	2	4	no	—
82	C₄	—	2	4	no	—
83	C₄	C₂×C₅×D₄	1	4	yes	—
84	C₂	—	2	2	no	—
85	C₂	—	4	4	no	—
86	C₂	C₂²×C₅×D₄	1	1	yes	characteristic
87	C₂	C₅×H(5,𝔽₂)	1	1	yes	characteristic
88	C₂	C₅×2₋¹⁺⁴	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. C₂×C₅×#41	C₂	semidirect
2. C₅×C₂²⋊Q₈	C₂	semidirect
3. C₅×#121	C₂	semidirect
4. C₅×#121	C₂	semidirect
5. C₂×C₅×C₄⋊C₄	C₂	semidirect
17. #537	C₅	direct (N × H)
26. C₂×C₄×C₅	D₄	semidirect
30. C₂×#41	C₂×C₅	semidirect
31. C₂²⋊Q₈	C₂×C₅	semidirect
32. #121	C₂×C₅	semidirect
33. #121	C₂×C₅	semidirect
34. C₂×C₄⋊C₄	C₂×C₅	semidirect
69. C₂×C₄	C₅×D₄	semidirect
73. C₅	#537	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#41	2
C₅×C₂²⋊Q₈	2
C₅×#121	2
C₂×C₅×C₄⋊C₄	2
#537	5

Subgroups — C5×#537 #66201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#537 #66201