Group #10610775 — subgroups

Subgroups — #10610775

Order 640.

364 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 189 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	#65411	C₂	1	2	yes	maximal
4	#65408	C₂	1	2	yes	maximal
5	C₂×C₄×#168	C₂	1	1	yes	maximal, characteristic
6	#65120	C₂	1	4	yes	maximal, complemented
7	#65119	C₂	1	4	yes	maximal, complemented
8	C₂×C₅×M₁₆	C₂²	1	2	yes	—
9	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
10	#4072	C₂²	1	2	yes	—
11	#4070	C₄	1	4	yes	—
12	C₂²×#168	C₂²	1	1	yes	characteristic
13	C₂²×#168	C₂²	1	1	yes	characteristic
14	#4016	C₄	1	2	yes	—
15	#4015	C₄	1	2	yes	—
16	C₄×Dic₁₀	C₂²	1	4	yes	—
17	C₅×C₄⋊C₈	C₂²	1	4	yes	—
18	#3947	C₂²	1	4	yes	—
19	#3946	C₂²	1	8	yes	—
20	C₈×Dic₅	C₂²	1	4	yes	—
21	C₄×#168	C₂²	1	2	yes	—
22	C₄×#168	C₂²	1	2	yes	—
23	#2960	—	5	1	no	maximal, Sylow
24	C₂²×C₄×C₅	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂×Dic₁₀	C₂×C₄	1	4	yes	—
27	C₅×M₁₆	—	2	4	no	—
28	C₂×C₅×C₈	C₂³	1	4	yes	—
29	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₅	C₂³	1	2	yes	—
32	C₂²⋊Dic₅	C₂×C₄	1	4	yes	—
33	C₄⋊Dic₅	C₂×C₄	1	4	yes	—
34	C₄⋊Dic₅	C₂×C₄	1	4	yes	—
35	#683	C₂×C₄	1	8	yes	—
36	C₄×Dic₅	C₂³	1	4	yes	—
37	C₂×#168	—	2	2	no	—
38	C₂×#168	Q₈	1	4	yes	complemented
39	C₂×#168	C₂³	1	2	yes	—
40	C₂×#168	C₂³	1	1	yes	characteristic
41	C₂×#168	C₂³	1	1	yes	characteristic
42	#533	—	5	1	no	—
43	#446	—	5	4	no	—
44	C₈×Q₈	—	5	4	no	—
45	#424	—	5	2	no	—
46	#423	D₅	1	1	yes	characteristic
47	#405	—	5	2	no	—
48	C₂×C₄×C₈	—	5	1	no	—
49	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₅	—	2	4	no	—
51	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
52	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
54	C₂×C₄×C₅	C₂×Q₈	1	2	yes	—
55	C₂×C₄×C₅	#41	1	2	yes	—
56	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
57	C₂×Dic₅	C₂²×C₄	1	8	yes	—
58	Dic₁₀	—	2	8	no	—
59	C₅×C₈	—	2	4	no	—
60	#168	—	2	2	no	—
61	#168	C₂×Q₈	1	2	yes	—
62	#168	C₂×Q₈	1	2	yes	—
63	C₂×M₁₆	C₂×D₅	1	2	yes	—
64	C₂²×C₈	—	5	1	no	—
65	C₂²×C₈	—	5	1	no	—
66	C₄⋊Q₈	—	5	2	no	—
67	#125	—	5	2	no	—
68	C₂²⋊Q₈	—	5	4	no	—
69	C₄×Q₈	—	5	4	no	—
70	#117	—	5	2	no	—
71	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
72	C₄⋊C₈	—	5	8	no	—
73	C₄⋊C₈	C₂×D₅	1	4	yes	—
74	#97	—	5	4	no	—
75	C₄×C₈	—	5	4	no	—
76	C₄×C₈	—	5	2	no	—
77	C₄×C₈	—	5	2	no	—
78	C₂²×C₅	—	2	2	no	—
79	C₂²×C₅	C₄×Q₈	1	2	yes	—
80	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
81	C₄×C₅	—	2	2	no	—
82	C₄×C₅	C₄×Q₈	1	2	yes	—
83	C₄×C₅	#131	1	4	yes	—
84	C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
85	C₄×C₅	C₂×#41	1	1	yes	characteristic
86	Dic₅	—	2	8	no	—
87	C₂×Q₈	—	5	4	no	—
88	C₂²×C₄	C₄×D₅	1	2	yes	—
89	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
90	M₁₆	—	2	4	no	—
91	C₂×C₈	—	5	4	no	—
92	C₂×C₈	—	5	1	no	—
93	C₂×C₈	—	5	1	no	—
94	C₂×C₈	—	5	2	no	—
95	C₂×C₈	—	10	2	no	—
96	C₂×C₈	C₂²×D₅	1	4	yes	—
97	C₄⋊C₄	—	5	4	no	—
98	C₄⋊C₄	—	5	8	no	—
99	C₄⋊C₄	—	5	4	no	—
100	C₂²⋊C₄	—	5	4	no	—
101	C₄²	—	5	4	no	—
102	C₄²	C₄×D₅	1	1	yes	characteristic
103	C₄²	C₄×D₅	1	1	yes	characteristic
104	C₄²	C₂²×D₅	1	2	yes	—
105	C₂×C₅	—	2	2	no	—
106	C₂×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
107	C₂×C₅	C₂×#131	1	2	yes	—
108	C₂³	C₂×C₄×D₅	1	1	yes	characteristic
109	Q₈	—	10	8	no	—
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	—	5	8	no	—
112	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
113	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
114	C₂×C₄	C₂×C₄×D₅	1	2	yes	—
115	C₂×C₄	#711	1	2	yes	—
116	C₂×C₄	Q₈×D₅	1	2	yes	—
117	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
118	C₈	—	2	4	no	—
119	C₈	—	5	2	no	—
120	C₈	—	5	2	no	—
121	C₈	—	10	2	no	—
122	C₅	#2960	1	1	yes	characteristic, complemented, Sylow
123	C₂²	—	2	2	no	—
124	C₂²	#4033	1	2	yes	—
125	C₂²	C₂²×C₄×D₅	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	10	8	no	—
128	C₄	#4033	1	2	yes	—
129	C₄	#4053	1	4	yes	—
130	C₄	C₂×#711	1	1	yes	characteristic
131	C₄	C₂×Q₈×D₅	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×#4033	1	1	yes	characteristic
134	C₂	C₂×#4053	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#65804

C₂

yes

maximal, characteristic

C₅×#423

C₂

yes

maximal, characteristic

#65411

C₂

yes

maximal

#65408

C₂

yes

maximal

C₂×C₄×#168

C₂

yes

maximal, characteristic

#65120

C₂

yes

maximal, complemented

#65119

C₂

yes

maximal, complemented

C₂×C₅×M₁₆

C₂²

yes

—

C₂×C₄²×C₅

C₂²

yes

characteristic

#4072

C₂²

yes

—

#4070

C₄

yes

—

C₂²×#168

C₂²

yes

characteristic

C₂²×#168

C₂²

yes

characteristic

#4016

C₄

yes

—

#4015

C₄

yes

—

C₄×Dic₁₀

C₂²

yes

—

C₅×C₄⋊C₈

C₂²

yes

—

#3947

C₂²

yes

—

#3946

C₂²

yes

—

C₈×Dic₅

C₂²

yes

—

C₄×#168

C₂²

yes

—

C₄×#168

C₂²

yes

—

#2960

—

maximal, Sylow

C₂²×C₄×C₅

C₂×C₄

yes

—

C₂²×C₄×C₅

C₂³

yes

characteristic

C₂×Dic₁₀

C₂×C₄

yes

—

C₅×M₁₆

—

C₂×C₅×C₈

C₂³

yes

—

C₄²×C₅

C₂×C₄

yes

characteristic

C₄²×C₅

C₂×C₄

yes

characteristic

C₄²×C₅

C₂³

yes

—

C₂²⋊Dic₅

C₂×C₄

yes

—

C₄⋊Dic₅

C₂×C₄

yes

—

C₄⋊Dic₅

C₂×C₄

yes

—

#683

C₂×C₄

yes

—

C₄×Dic₅

C₂³

yes

—

C₂×#168

—

C₂×#168

Q₈

yes

complemented

C₂×#168

C₂³

yes

—

C₂×#168

C₂³

yes

characteristic

C₂×#168

C₂³

yes

characteristic

#533

—

#446

—

C₈×Q₈

—

#424

—

#423

D₅

yes

characteristic

#405

—

C₂×C₄×C₈

—

C₂³×C₅

C₂²×C₄

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×C₄

yes

—

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×C₄×C₅

C₂²×C₄

yes

—

C₂×C₄×C₅

C₂×Q₈

yes

—

C₂×C₄×C₅

#41

yes

—

C₂×C₄×C₅

C₂⁴

yes

characteristic

C₂×Dic₅

C₂²×C₄

yes

—

Dic₁₀

—

C₅×C₈

—

#168

—

#168

C₂×Q₈

yes

—

#168

C₂×Q₈

yes

—

C₂×M₁₆

C₂×D₅

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₅

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅

yes

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₅

—

C₂²×C₅

C₄×Q₈

yes

—

C₂²×C₅

C₂³×C₄

yes

characteristic

C₄×C₅

—

C₄×C₅

C₄×Q₈

yes

—

C₄×C₅

#131

yes

—

C₄×C₅

C₂²×Q₈

yes

characteristic

C₄×C₅

C₂×#41

yes

characteristic

Dic₅

—

C₂×Q₈

—

C₂²×C₄

C₄×D₅

yes

—

C₂²×C₄

C₂²×D₅

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×D₅

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₂²⋊C₄

—

101

C₄²

—

102

C₄²

C₄×D₅

yes

characteristic

103

C₄²

C₄×D₅

yes

characteristic

104

C₄²

C₂²×D₅

yes

—

105

C₂×C₅

—

106

C₂×C₅

C₂×C₄×Q₈

yes

characteristic

107

C₂×C₅

C₂×#131

yes

—

108

C₂³

C₂×C₄×D₅

yes

characteristic

109

Q₈

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂×C₄×D₅

yes

—

113

C₂×C₄

C₂×C₄×D₅

yes

characteristic

114

C₂×C₄

C₂×C₄×D₅

yes

—

115

C₂×C₄

#711

yes

—

116

C₂×C₄

Q₈×D₅

yes

—

117

C₂×C₄

C₂³×D₅

yes

characteristic

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

—

122

C₅

#2960

yes

characteristic, complemented, Sylow

123

C₂²

—

124

C₂²

#4033

yes

—

125

C₂²

C₂²×C₄×D₅

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

#4033

yes

—

129

C₄

#4053

yes

—

130

C₄

C₂×#711

yes

characteristic

131

C₄

C₂×Q₈×D₅

yes

characteristic

132

C₂

—

133

C₂

C₂×#4033

yes

characteristic

134

C₂

C₂×#4053

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×#423	C₂
prime radical	C₂³×C₅	C₂²×C₄
cosocle	C₂×C₄×C₅	C₂⁴
hypercenter	#423	D₅
hypocenter	C₅	#2960
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₅

commutator, socle

C₂²×C₅

C₂³×C₄

Fitting

C₅×#423

C₂

prime radical

C₂³×C₅

C₂²×C₄

cosocle

C₂×C₄×C₅

C₂⁴

D₅

C₅

G ▷ C₂²×C₅ ▷ C₅

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
6. #65120	C₂	semidirect
7. #65119	C₂	semidirect
38. C₂×#168	Q₈	semidirect
122. C₅	#2960	semidirect

H ≅ G/N

type

6. #65120

C₂

semidirect

7. #65119

C₂

semidirect

38. C₂×#168

Q₈

semidirect

122. C₅

#2960

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
#65411	2
#65408	2
C₂×C₄×#168	2
#65120	2
#65119	2
#2960	5

subgroup

index

C₂×C₄×#168

Subgroups — #10610775

Conjugacy classes of subgroups

Complements

Maximal subgroups