Subgroups — #65908

Order 320.

294 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#713	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	#4039	C₂	1	3	yes	maximal, complemented
5	D₅×C₄⋊C₄	C₂	1	3	yes	maximal, complemented
6	C₄×D₂₀	C₂	1	3	yes	maximal, complemented
7	#4018	C₂	1	3	yes	maximal, complemented
8	C₂×C₅×Q₈	C₂²	1	1	yes	characteristic
9	#713	C₄	1	8	yes	complemented
10	C₂×D₂₀	C₂²	1	3	yes	—
11	C₂×C₄×D₅	C₂²	1	6	yes	—
12	C₂×C₄×D₅	C₂²	1	3	yes	—
13	C₅×C₄⋊C₄	C₂²	1	3	yes	—
14	C₄²×C₅	C₂²	1	3	yes	—
15	#685	C₂²	1	3	yes	—
16	#685	C₂²	1	3	yes	—
17	C₄⋊Dic₅	C₂²	1	3	yes	—
18	#683	C₂²	1	1	yes	characteristic
19	#683	C₂²	1	3	yes	—
20	C₄×Dic₅	C₂²	1	3	yes	—
21	#520	—	5	1	no	maximal, Sylow
22	C₂²×D₅	C₂³	1	3	yes	—
23	C₅×Q₈	C₂×C₄	1	4	yes	complemented
24	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂×C₄×C₅	C₂³	1	3	yes	—
26	C₂×C₄×C₅	C₂³	1	3	yes	—
27	C₂×Dic₅	C₂³	1	1	yes	characteristic
28	C₂×Dic₅	C₂³	1	3	yes	—
29	C₂×Dic₅	C₂³	1	1	yes	characteristic
30	D₂₀	C₂×C₄	1	12	yes	complemented
31	C₄×D₅	—	4	6	no	—
32	C₄×D₅	C₂×C₄	1	12	yes	complemented
33	C₂×#41	—	5	1	no	—
34	C₄×Q₈	—	5	1	no	—
35	C₄×Q₈	D₅	1	1	yes	characteristic, complemented
36	C₄×D₄	—	5	3	no	—
37	C₄×D₄	—	5	3	no	—
38	#117	—	5	3	no	—
39	C₂×C₄⋊C₄	—	5	3	no	—
40	C₂²×C₅	C₂⁴	1	1	yes	characteristic
41	C₂×D₅	—	2	3	no	—
42	C₂×D₅	—	2	3	no	—
43	C₂×D₅	C₂²×C₄	1	6	yes	—
44	C₄×C₅	—	2	3	no	—
45	C₄×C₅	—	2	1	no	—
46	C₄×C₅	C₂²×C₄	1	6	yes	—
47	Dic₅	—	2	3	no	—
48	Dic₅	—	2	1	no	—
49	Dic₅	C₂²×C₄	1	2	yes	—
50	#41	—	5	8	no	—
51	C₂×Q₈	C₂×D₅	1	1	yes	characteristic
52	C₂×D₄	—	5	3	no	—
53	C₂²×C₄	—	5	3	no	—
54	C₂²×C₄	—	5	6	no	—
55	C₄⋊C₄	—	5	3	no	—
56	C₄⋊C₄	—	5	3	no	—
57	C₄⋊C₄	—	5	1	no	—
58	C₄⋊C₄	C₂×D₅	1	3	yes	—
59	C₂²⋊C₄	—	5	3	no	—
60	C₂²⋊C₄	—	5	3	no	—
61	C₄²	—	5	3	no	—
62	C₄²	C₂×D₅	1	3	yes	—
63	C₂×C₅	C₂³×C₄	1	1	yes	characteristic
64	C₂×C₅	H(5,𝔽₂)	1	1	yes	characteristic
65	C₂×C₅	2₋¹⁺⁴	1	1	yes	characteristic
66	D₅	—	2	6	no	—
67	C₂³	—	5	3	no	—
68	Q₈	C₄×D₅	1	4	yes	complemented
69	D₄	—	5	12	no	—
70	C₂×C₄	—	5	1	no	—
71	C₂×C₄	—	5	12	no	—
72	C₂×C₄	—	5	1	no	—
73	C₂×C₄	—	5	3	no	—
74	C₂×C₄	—	20	6	no	—
75	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
76	C₂×C₄	C₂²×D₅	1	3	yes	—
77	C₂×C₄	C₂²×D₅	1	3	yes	—
78	C₅	#520	1	1	yes	characteristic, complemented, Sylow
79	C₂²	—	5	6	no	—
80	C₂²	—	10	3	no	—
81	C₂²	—	10	3	no	—
82	C₂²	C₂³×D₅	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	2	3	no	—
85	C₄	—	5	2	no	—
86	C₄	—	10	1	no	—
87	C₄	—	10	3	no	—
88	C₄	C₂×C₄×D₅	1	6	yes	—
89	C₂	—	10	6	no	—
90	C₂	C₂²×C₄×D₅	1	1	yes	characteristic
91	C₂	#4147	1	1	yes	characteristic
92	C₂	#4149	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator	C₂×C₅	C₂³×C₄
Fitting	C₄×C₅×Q₈	C₂
socle, cosocle	C₂²×C₅	C₂⁴
prime radical	C₂×#713	C₂
hypercenter	C₄×Q₈	D₅
hypocenter	C₅	#520
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₄×C₅×Q₈	C₂	semidirect
3. Q₈×Dic₅	C₂	semidirect
4. #4039	C₂	semidirect
5. D₅×C₄⋊C₄	C₂	semidirect
6. C₄×D₂₀	C₂	semidirect
7. #4018	C₂	semidirect
9. #713	C₄	semidirect
23. C₅×Q₈	C₂×C₄	semidirect
30. D₂₀	C₂×C₄	semidirect
32. C₄×D₅	C₂×C₄	semidirect
35. C₄×Q₈	D₅	semidirect
68. Q₈	C₄×D₅	semidirect
78. C₅	#520	semidirect

Maximal subgroups

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

subgroup	index
C₂×#713	2
C₄×C₅×Q₈	2
Q₈×Dic₅	2
#4039	2
D₅×C₄⋊C₄	2
C₄×D₂₀	2
#4018	2
#520	5