Group #10614754 — subgroups

Subgroups — #10614754

Order 640.

444 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 237 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	C₂×C₅×#125	C₂	1	1	yes	maximal, characteristic
2	#65274	C₂	1	2	yes	maximal
3	#65274	C₂	1	4	yes	maximal
4	#65270	C₂	1	4	yes	maximal
5	C₄⋊C₄×Dic₅	C₂	1	2	yes	maximal
6	#65224	C₂	1	1	yes	maximal, characteristic
7	#65221	C₂	1	1	yes	maximal, characteristic
8	C₅×#125	C₄	1	8	yes	complemented
9	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₅×C₄⋊C₄	C₂²	1	4	yes	—
11	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₅	C₂²	1	4	yes	—
14	C₂×C₄⋊Dic₅	C₂²	1	4	yes	—
15	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
16	C₂×C₄×Dic₅	C₂²	1	4	yes	—
17	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
18	#3963	C₂²	1	2	yes	—
19	#3963	C₂²	1	2	yes	—
20	#3963	C₂²	1	4	yes	—
21	#3963	C₂²	1	4	yes	—
22	#2330	—	5	1	no	maximal, Sylow
23	C₂²×C₄×C₅	C₂³	1	2	yes	—
24	C₂²×C₄×C₅	C₂³	1	4	yes	—
25	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂²×Dic₅	C₂³	1	1	yes	characteristic
27	C₂²×Dic₅	C₂³	1	1	yes	characteristic
28	C₂²×Dic₅	C₂³	1	4	yes	—
29	C₂²×Dic₅	C₂³	1	2	yes	—
30	C₅×C₄⋊C₄	C₂×C₄	1	16	yes	—
31	C₅×C₄⋊C₄	C₂×C₄	1	8	yes	—
32	C₄²×C₅	C₂×C₄	1	4	yes	—
33	C₄⋊Dic₅	—	2	4	no	—
34	C₄⋊Dic₅	—	4	4	no	—
35	C₄⋊Dic₅	—	4	4	no	—
36	C₄×Dic₅	—	2	2	no	—
37	C₄×Dic₅	—	4	4	no	—
38	C₄×Dic₅	Q₈	1	4	yes	—
39	C₂×#125	D₅	1	1	yes	characteristic
40	#389	—	5	4	no	—
41	#389	—	5	2	no	—
42	#387	—	5	4	no	—
43	#384	—	5	1	no	—
44	C₄×C₄⋊C₄	—	5	2	no	—
45	#376	—	5	1	no	—
46	C₂³×C₅	C₂⁴	1	1	yes	characteristic
47	C₂×C₄×C₅	—	2	4	no	—
48	C₂×C₄×C₅	—	2	2	no	—
49	C₂×C₄×C₅	—	2	2	no	—
50	C₂×C₄×C₅	—	2	4	no	—
51	C₂×C₄×C₅	C₂²×C₄	1	4	yes	—
52	C₂×C₄×C₅	C₂²×C₄	1	8	yes	—
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×Dic₅	—	2	2	no	—
57	C₂×Dic₅	—	2	2	no	—
58	C₂×Dic₅	—	2	1	no	—
59	C₂×Dic₅	—	2	1	no	—
60	C₂×Dic₅	—	2	4	no	—
61	C₂×Dic₅	—	2	4	no	—
62	C₂×Dic₅	—	2	2	no	—
63	C₂×Dic₅	—	2	4	no	—
64	C₂×Dic₅	—	2	2	no	—
65	C₂×Dic₅	C₂×Q₈	1	4	yes	—
66	#125	Dic₅	1	8	yes	complemented
67	C₂×C₄⋊C₄	—	5	2	no	—
68	C₂×C₄⋊C₄	—	5	4	no	—
69	C₂×C₄⋊C₄	—	5	4	no	—
70	C₂×C₄⋊C₄	C₂×D₅	1	2	yes	—
71	C₂×C₄⋊C₄	C₂×D₅	1	4	yes	—
72	C₂×C₄²	—	5	1	no	—
73	C₂×C₄²	—	5	1	no	—
74	C₂×C₄²	—	5	4	no	—
75	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
76	#95	—	5	2	no	—
77	#95	—	5	2	no	—
78	#95	—	5	4	no	—
79	#95	—	5	4	no	—
80	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
81	C₂²×C₅	C₂²×Q₈	1	1	yes	characteristic
82	C₂²×C₅	C₂×#41	1	1	yes	characteristic
83	C₂²×C₅	H(5,𝔽₂)	1	2	yes	—
84	C₂²×C₅	2₋¹⁺⁴	1	2	yes	—
85	C₄×C₅	—	2	8	no	—
86	C₄×C₅	—	2	4	no	—
87	C₄×C₅	C₄×Q₈	1	4	yes	—
88	Dic₅	—	2	4	no	—
89	Dic₅	—	4	1	no	—
90	Dic₅	—	4	4	no	—
91	Dic₅	—	4	1	no	—
92	C₂²×C₄	—	5	2	no	—
93	C₂²×C₄	—	5	1	no	—
94	C₂²×C₄	—	5	1	no	—
95	C₂²×C₄	—	5	4	no	—
96	C₂²×C₄	C₂²×D₅	1	4	yes	—
97	C₂²×C₄	C₂²×D₅	1	2	yes	—
98	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
99	C₄⋊C₄	—	10	4	no	—
100	C₄⋊C₄	—	20	4	no	—
101	C₄⋊C₄	—	20	4	no	—
102	C₄⋊C₄	C₂×Dic₅	1	16	yes	—
103	C₄⋊C₄	C₂×Dic₅	1	8	yes	—
104	C₄²	—	5	4	no	—
105	C₄²	—	10	2	no	—
106	C₄²	—	20	4	no	—
107	C₄²	C₂×Dic₅	1	4	yes	—
108	C₂×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
109	C₂×C₅	#520	1	2	yes	—
110	C₂×C₅	#540	1	1	yes	characteristic
111	C₂×C₅	#541	1	1	yes	characteristic
112	C₂×C₅	#544	1	2	yes	—
113	C₂³	C₂³×D₅	1	1	yes	characteristic
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	5	4	no	—
119	C₂×C₄	—	10	2	no	—
120	C₂×C₄	—	10	2	no	—
121	C₂×C₄	—	10	1	no	—
122	C₂×C₄	—	10	1	no	—
123	C₂×C₄	—	10	4	no	—
124	C₂×C₄	—	10	4	no	—
125	C₂×C₄	—	10	2	no	—
126	C₂×C₄	—	10	4	no	—
127	C₂×C₄	—	10	2	no	—
128	C₂×C₄	Q₈×D₅	1	2	yes	—
129	C₂×C₄	#713	1	2	yes	—
130	C₂×C₄	C₂²×Dic₅	1	4	yes	—
131	C₂×C₄	C₂²×Dic₅	1	8	yes	—
132	C₂×C₄	C₂²×Dic₅	1	2	yes	—
133	C₅	#2330	1	1	yes	characteristic, complemented, Sylow
134	C₂²	C₂×Q₈×D₅	1	1	yes	characteristic
135	C₂²	C₂×#713	1	1	yes	characteristic
136	C₂²	#4149	1	2	yes	—
137	C₂²	#4150	1	2	yes	—
138	C₂²	C₂³×Dic₅	1	1	yes	characteristic
139	C₄	—	2	8	no	—
140	C₄	—	2	4	no	—
141	C₄	—	10	4	no	—
142	C₄	—	20	1	no	—
143	C₄	—	20	1	no	—
144	C₄	—	20	4	no	—
145	C₄	Q₈×Dic₅	1	4	yes	—
146	C₂	#66023	1	2	yes	—
147	C₂	#66031	1	1	yes	characteristic
148	C₂	#66032	1	1	yes	characteristic
149	C₂	C₂×Q₈×Dic₅	1	1	yes	characteristic
150	C₂	#66161	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅×#125

C₂

yes

maximal, characteristic

#65274

C₂

yes

maximal

#65274

C₂

yes

maximal

#65270

C₂

yes

maximal

C₄⋊C₄×Dic₅

C₂

yes

maximal

#65224

C₂

yes

maximal, characteristic

#65221

C₂

yes

maximal, characteristic

C₅×#125

C₄

yes

complemented

C₂×C₅×C₄⋊C₄

C₂²

yes

—

C₂×C₅×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₅

C₂²

yes

characteristic

C₂×C₄⋊Dic₅

C₂²

yes

—

C₂×C₄⋊Dic₅

C₂²

yes

—

C₂×C₄⋊Dic₅

C₂²

yes

—

C₂×C₄×Dic₅

C₂²

yes

characteristic

C₂×C₄×Dic₅

C₂²

yes

—

C₂×C₄×Dic₅

C₂²

yes

characteristic

#3963

C₂²

yes

—

#3963

C₂²

yes

—

#3963

C₂²

yes

—

#3963

C₂²

yes

—

#2330

—

maximal, Sylow

C₂²×C₄×C₅

C₂³

yes

—

C₂²×C₄×C₅

C₂³

yes

—

C₂²×C₄×C₅

C₂³

yes

characteristic

C₂²×Dic₅

C₂³

yes

characteristic

C₂²×Dic₅

C₂³

yes

characteristic

C₂²×Dic₅

C₂³

yes

—

C₂²×Dic₅

C₂³

yes

—

C₅×C₄⋊C₄

C₂×C₄

yes

—

C₅×C₄⋊C₄

C₂×C₄

yes

—

C₄²×C₅

C₂×C₄

yes

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

Q₈

yes

—

C₂×#125

D₅

yes

—

—

—

—

C₄×C₄⋊C₄

—

#376

—

C₂³×C₅

C₂⁴

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×C₄

yes

—

C₂×C₄×C₅

C₂²×C₄

yes

—

C₂×C₄×C₅

C₂²×C₄

yes

—

C₂×C₄×C₅

C₂×Q₈

yes

—

C₂×C₄×C₅

#41

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

C₂×Q₈

yes

—

#125

Dic₅

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅

yes

—

C₂×C₄⋊C₄

C₂×D₅

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅

yes

—

—

—

—

C₂²×C₅

C₂³×C₄

yes

characteristic

C₂²×C₅

C₂²×Q₈

yes

characteristic

C₂²×C₅

C₂×#41

yes

characteristic

C₂²×C₅

H(5,𝔽₂)

yes

—

C₂²×C₅

2₋¹⁺⁴

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₄×Q₈

yes

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅

yes

—

C₂²×C₄

C₂²×D₅

yes

—

C₂²×C₄

C₂²×D₅

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

C₂×Dic₅

yes

—

103

C₄⋊C₄

C₂×Dic₅

yes

—

104

C₄²

—

105

C₄²

—

106

C₄²

—

107

C₄²

C₂×Dic₅

yes

—

108

C₂×C₅

C₂×C₄×Q₈

yes

characteristic

109

C₂×C₅

#520

yes

—

110

C₂×C₅

#540

yes

characteristic

111

C₂×C₅

#541

yes

characteristic

112

C₂×C₅

#544

yes

—

113

C₂³

C₂³×D₅

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

Q₈×D₅

yes

—

129

C₂×C₄

#713

yes

—

130

C₂×C₄

C₂²×Dic₅

yes

—

131

C₂×C₄

C₂²×Dic₅

yes

—

132

C₂×C₄

C₂²×Dic₅

yes

—

133

C₅

#2330

yes

characteristic, complemented, Sylow

134

C₂²

C₂×Q₈×D₅

yes

characteristic

135

C₂²

C₂×#713

yes

characteristic

136

C₂²

#4149

yes

—

137

C₂²

#4150

yes

—

138

C₂²

C₂³×Dic₅

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

Q₈×Dic₅

yes

—

146

C₂

#66023

yes

—

147

C₂

#66031

yes

characteristic

148

C₂

#66032

yes

characteristic

149

C₂

C₂×Q₈×Dic₅

yes

characteristic

150

C₂

#66161

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅
commutator	C₂²×C₅	C₂³×C₄
Fitting	C₂×C₅×#125	C₂
socle, prime radical, cosocle	C₂³×C₅	C₂⁴
hypercenter	C₂×#125	D₅
hypocenter	C₅	#2330
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅

commutator

C₂²×C₅

C₂³×C₄

Fitting

C₂×C₅×#125

C₂

socle, prime radical, cosocle

C₂³×C₅

C₂⁴

D₅

C₅

G ▷ C₂²×C₅ ▷ C₅

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
8. C₅×#125	C₄	semidirect
66. #125	Dic₅	semidirect
133. C₅	#2330	semidirect

H ≅ G/N

type

8. C₅×#125

C₄

semidirect

66. #125

Dic₅

semidirect

133. C₅

#2330

semidirect

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

subgroup	index
C₂×C₅×#125	2
#65274	2
#65270	2
C₄⋊C₄×Dic₅	2
#65224	2
#65221	2
#2330	5

subgroup

index

C₂×C₅×#125

C₄⋊C₄×Dic₅

Subgroups — #10614754

Conjugacy classes of subgroups

Complements

Maximal subgroups