Subgroups — #65057

Order 320.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4082	C₂	1	1	yes	maximal, characteristic
2	#4036	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#102	C₂	1	1	yes	maximal, characteristic
4	#3964	C₂	1	1	yes	maximal, characteristic
5	#3948	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
7	#3942	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×D₄	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₀	C₂²	1	1	yes	characteristic
10	C₂×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂²⋊Dic₅	—	2	1	no	—
13	C₂²⋊Dic₅	—	2	1	no	—
14	C₄⋊Dic₅	—	2	1	no	—
15	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
16	#683	—	2	1	no	—
17	C₄×Dic₅	C₂²	1	1	yes	characteristic
18	C₂×#168	C₂²	1	1	yes	characteristic
19	#488	—	5	1	no	maximal, Sylow
20	C₂³×C₅	—	2	1	no	—
21	C₅×D₄	—	2	1	no	—
22	C₅×D₄	—	2	1	no	—
23	C₂×C₄×C₅	—	2	1	no	—
24	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂×Dic₅	—	2	1	no	—
26	C₂×Dic₅	—	2	1	no	—
27	C₂×Dic₅	D₄	1	1	yes	characteristic
28	C₂×Dic₅	D₄	1	1	yes	characteristic
29	Dic₁₀	—	2	1	no	—
30	Dic₁₀	—	2	1	no	—
31	C₅×C₈	—	2	1	no	—
32	#168	—	2	1	no	—
33	#125	—	5	1	no	—
34	#124	—	5	1	no	—
35	#103	—	5	1	no	—
36	#103	—	5	1	no	—
37	#102	—	5	1	no	—
38	#102	D₅	1	1	yes	characteristic
39	C₄×C₈	—	5	1	no	—
40	C₂²×C₅	—	4	1	no	—
41	C₂²×C₅	—	4	1	no	—
42	C₂²×C₅	—	4	1	no	—
43	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
44	C₄×C₅	—	4	1	no	—
45	C₄×C₅	#41	1	1	yes	characteristic
46	C₄×C₅	#41	1	1	yes	characteristic
47	Dic₅	—	2	1	no	—
48	Dic₅	—	2	1	no	—
49	Dic₅	—	4	1	no	—
50	Dic₅	—	4	1	no	—
51	C₂×Q₈	—	5	1	no	—
52	C₂×D₄	C₂×D₅	1	1	yes	characteristic
53	C₂×C₈	—	5	1	no	—
54	C₂×C₈	C₂×D₅	1	1	yes	characteristic
55	C₄⋊C₄	—	5	1	no	—
56	C₄⋊C₄	—	10	1	no	—
57	C₄⋊C₄	—	10	1	no	—
58	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
59	C₂²⋊C₄	—	10	1	no	—
60	C₂²⋊C₄	—	10	1	no	—
61	C₄²	—	5	1	no	—
62	C₂×C₅	—	8	1	no	—
63	C₂×C₅	#124	1	1	yes	characteristic
64	C₂×C₅	#135	1	1	yes	characteristic
65	C₂×C₅	#135	1	1	yes	characteristic
66	C₂³	—	2	1	no	—
67	Q₈	—	10	1	no	—
68	Q₈	—	10	1	no	—
69	D₄	—	2	1	no	—
70	D₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	5	1	no	—
73	C₂×C₄	—	5	1	no	—
74	C₂×C₄	—	10	1	no	—
75	C₂×C₄	—	10	1	no	—
76	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	10	1	no	—
79	C₅	#488	1	1	yes	characteristic, complemented, Sylow
80	C₂²	—	4	1	no	—
81	C₂²	—	4	1	no	—
82	C₂²	—	4	1	no	—
83	C₂²	D₄×D₅	1	1	yes	characteristic
84	C₄	—	4	1	no	—
85	C₄	—	10	1	no	—
86	C₄	—	10	1	no	—
87	C₄	—	20	1	no	—
88	C₄	—	20	1	no	—
89	C₄	#709	1	1	yes	characteristic
90	C₄	#711	1	1	yes	characteristic
91	C₂	—	8	1	no	—
92	C₂	#4031	1	1	yes	characteristic
93	C₂	#4058	1	1	yes	characteristic
94	C₂	#4062	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×#102	C₂
socle	C₂²×C₅	C₂×D₄
prime radical	C₂×C₅×D₄	C₂²
hypercenter	#102	D₅
hypocenter	C₅	#488
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

Complements

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

N	H ≅ G/N	type
2. #4036	C₂	semidirect
5. #3948	C₂	semidirect
6. C₈×Dic₅	C₂	semidirect
7. #3942	C₂	semidirect
79. C₅	#488	semidirect

Maximal subgroups

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

subgroup	index
#4082	2
#4036	2
C₅×#102	2
#3964	2
#3948	2
C₈×Dic₅	2
#3942	2
#488	5