Group #10601308 — subgroups

Subgroups — #10601308

Order 640.

244 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#65804	C₂	1	1	yes	maximal, characteristic
2	C₅×#423	C₂	1	1	yes	maximal, characteristic
3	#65221	C₂	1	1	yes	maximal, characteristic
4	#64775	C₂	1	4	yes	maximal
5	C₂×C₅×M₁₆	C₂²	1	2	yes	—
6	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
7	#4072	C₂²	1	2	yes	—
8	#4070	—	2	2	no	—
9	C₂×C₄×Dic₅	C₂²	1	2	yes	—
10	#4016	C₄	1	2	yes	complemented
11	#4015	C₄	1	2	yes	complemented
12	C₄×Dic₁₀	—	2	2	no	—
13	C₅×C₄⋊C₈	—	2	2	no	—
14	#3963	—	2	2	no	—
15	#1995	—	5	1	no	maximal, Sylow
16	C₂²×C₄×C₅	D₄	1	2	yes	—
17	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
18	C₂²×Dic₅	—	2	2	no	—
19	C₂×Dic₁₀	—	2	2	no	—
20	C₅×M₁₆	—	2	4	no	—
21	C₂×C₅×C₈	—	2	2	no	—
22	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₅	D₄	1	2	yes	—
25	C₂²⋊Dic₅	—	2	2	no	—
26	C₄⋊Dic₅	—	2	2	no	—
27	C₄⋊Dic₅	C₂×C₄	1	4	yes	—
28	#683	—	2	4	no	—
29	C₄×Dic₅	—	2	2	no	—
30	C₄×Dic₅	—	2	2	no	—
31	C₄×Dic₅	Q₈	1	4	yes	complemented
32	#533	—	5	1	no	—
33	#423	D₅	1	1	yes	characteristic
34	#376	—	5	1	no	—
35	#339	—	5	4	no	—
36	C₂³×C₅	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₅	—	4	2	no	—
38	C₂×C₄×C₅	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₅	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₅	#41	1	2	yes	—
44	C₂×Dic₅	—	2	4	no	—
45	C₂×Dic₅	—	2	4	no	—
46	C₂×Dic₅	—	4	4	no	—
47	Dic₁₀	—	4	4	no	—
48	C₅×C₈	—	4	2	no	—
49	C₂×M₁₆	C₂×D₅	1	2	yes	—
50	C₄⋊Q₈	—	5	2	no	—
51	#125	—	5	2	no	—
52	C₂²⋊Q₈	—	10	2	no	—
53	C₄×Q₈	—	10	2	no	—
54	#117	—	5	2	no	—
55	C₂×C₄²	—	5	2	no	—
56	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
57	C₄⋊C₈	—	2	2	no	—
58	#95	—	10	2	no	—
59	C₂²×C₅	—	2	2	no	—
60	C₂²×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₅	C₂²⋊Q₈	1	2	yes	—
62	C₄×C₅	—	2	2	no	—
63	C₄×C₅	—	2	2	no	—
64	C₄×C₅	C₄×Q₈	1	2	yes	—
65	C₄×C₅	#124	1	1	yes	characteristic
66	C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
67	Dic₅	—	4	4	no	—
68	Dic₅	—	4	4	no	—
69	C₂×Q₈	—	10	2	no	—
70	C₂²×C₄	—	10	2	no	—
71	C₂²×C₄	D₂₀	1	2	yes	—
72	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
73	M₁₆	—	2	4	no	—
74	C₂×C₈	—	2	2	no	—
75	C₄⋊C₄	—	5	4	no	—
76	C₄⋊C₄	—	10	4	no	—
77	C₄⋊C₄	—	10	2	no	—
78	C₂²⋊C₄	—	10	2	no	—
79	C₄²	—	5	4	no	—
80	C₄²	—	10	2	no	—
81	C₄²	—	10	2	no	—
82	C₄²	C₄×D₅	1	1	yes	characteristic
83	C₄²	C₄×D₅	1	1	yes	characteristic
84	C₄²	C₂²⋊D₅	1	2	yes	—
85	C₂×C₅	—	2	2	no	—
86	C₂×C₅	#391	1	1	yes	characteristic
87	C₂×C₅	#421	1	2	yes	—
88	C₂³	C₂×D₂₀	1	1	yes	characteristic
89	Q₈	—	20	4	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	—	10	4	no	—
92	C₂×C₄	—	10	4	no	—
93	C₂×C₄	—	20	4	no	—
94	C₂×C₄	#685	1	2	yes	—
95	C₂×C₄	#685	1	2	yes	—
96	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
97	C₂×C₄	#711	1	2	yes	—
98	C₂×C₄	Q₈×D₅	1	2	yes	—
99	C₂×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
100	C₈	—	4	2	no	—
101	C₅	#1995	1	1	yes	characteristic, complemented, Sylow
102	C₂²	—	2	2	no	—
103	C₂²	#4043	1	2	yes	—
104	C₂²	C₂×#685	1	1	yes	characteristic
105	C₄	—	2	2	no	—
106	C₄	—	2	2	no	—
107	C₄	—	20	4	no	—
108	C₄	—	20	4	no	—
109	C₄	#4033	1	2	yes	—
110	C₄	#4082	1	1	yes	characteristic
111	C₄	#4090	1	1	yes	characteristic
112	C₂	—	2	2	no	—
113	C₂	#65263	1	1	yes	characteristic
114	C₂	#65428	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#65804

C₂

yes

maximal, characteristic

C₅×#423

C₂

yes

maximal, characteristic

#65221

C₂

yes

maximal, characteristic

#64775

C₂

yes

maximal

C₂×C₅×M₁₆

C₂²

yes

—

C₂×C₄²×C₅

C₂²

yes

characteristic

#4072

C₂²

yes

—

#4070

—

C₂×C₄×Dic₅

C₂²

yes

—

#4016

C₄

yes

complemented

#4015

C₄

yes

complemented

C₄×Dic₁₀

—

C₅×C₄⋊C₈

—

#3963

—

#1995

—

maximal, Sylow

C₂²×C₄×C₅

D₄

yes

—

C₂²×C₄×C₅

C₂³

yes

characteristic

C₂²×Dic₅

—

C₂×Dic₁₀

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₄²×C₅

C₂×C₄

yes

characteristic

C₄²×C₅

C₂×C₄

yes

characteristic

C₄²×C₅

D₄

yes

—

C₂²⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

C₂×C₄

yes

—

#683

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

Q₈

yes

complemented

#533

—

#423

D₅

yes

characteristic

#376

—

#339

—

C₂³×C₅

C₂×D₄

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊C₄

yes

—

C₂×C₄×C₅

C₂²⋊C₄

yes

—

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₄×C₅

C₂×Q₈

yes

—

C₂×C₄×C₅

#41

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

C₅×C₈

—

C₂×M₁₆

C₂×D₅

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅

yes

characteristic

C₄⋊C₈

—

#95

—

C₂²×C₅

—

C₂²×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅

C₂²⋊Q₈

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₄×Q₈

yes

—

C₄×C₅

#124

yes

characteristic

C₄×C₅

C₄⋊Q₈

yes

characteristic

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

D₂₀

yes

—

C₂²×C₄

C₂²×D₅

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₄×D₅

yes

characteristic

C₄²

C₄×D₅

yes

characteristic

C₄²

C₂²⋊D₅

yes

—

C₂×C₅

—

C₂×C₅

#391

yes

characteristic

C₂×C₅

#421

yes

—

C₂³

C₂×D₂₀

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#685

yes

—

C₂×C₄

#685

yes

—

C₂×C₄

C₂×C₄×D₅

yes

characteristic

C₂×C₄

#711

yes

—

C₂×C₄

Q₈×D₅

yes

—

C₂×C₄

C₂×C₂²⋊D₅

yes

characteristic

100

C₈

—

101

C₅

#1995

yes

characteristic, complemented, Sylow

102

C₂²

—

103

C₂²

#4043

yes

—

104

C₂²

C₂×#685

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

#4033

yes

—

110

C₄

#4082

yes

characteristic

111

C₄

#4090

yes

characteristic

112

C₂

—

113

C₂

#65263

yes

characteristic

114

C₂

#65428

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅
commutator	C₂×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅
Fitting	C₅×#423	C₂
socle	C₂²×C₅	C₂×C₂²⋊C₄
prime radical	C₂³×C₅	C₂×D₄
cosocle	C₂²×C₄×C₅	C₂³
hypercenter	#423	D₅
hypocenter	C₅	#1995
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₅

commutator

C₂×C₄×C₅

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅

Fitting

C₅×#423

C₂

socle

C₂²×C₅

C₂×C₂²⋊C₄

prime radical

C₂³×C₅

C₂×D₄

cosocle

C₂²×C₄×C₅

C₂³

D₅

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
10. #4016	C₄	semidirect
11. #4015	C₄	semidirect
31. C₄×Dic₅	Q₈	semidirect
101. C₅	#1995	semidirect

H ≅ G/N

type

10. #4016

C₄

semidirect

11. #4015

C₄

semidirect

31. C₄×Dic₅

Q₈

semidirect

101. C₅

#1995

semidirect

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

subgroup	index
#65804	2
C₅×#423	2
#65221	2
#64775	2
#1995	5

subgroup

index

Subgroups — #10601308

Conjugacy classes of subgroups

Complements

Maximal subgroups