Group #10614970 — subgroups

Subgroups — #10614970

Order 640.

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

normal subgroups only

This group has 540 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	C₂×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#4090	C₂	1	2	yes	maximal
3	C₂×C₄×Dic₁₀	C₂	1	1	yes	maximal, characteristic
4	#65519	C₂	1	2	yes	maximal
5	#65269	C₂	1	4	yes	maximal
6	#65262	C₂	1	4	yes	maximal
7	C₄×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₅×Q₈	C₂²	1	2	yes	—
9	C₂²×Dic₁₀	C₂²	1	1	yes	characteristic
10	C₅×C₄⋊Q₈	—	2	4	no	—
11	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
14	#4090	—	2	8	no	—
15	C₂×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₅	C₂²	1	2	yes	—
17	C₂×#683	C₂²	1	2	yes	—
18	C₂×#683	C₂²	1	8	yes	—
19	C₂×C₄×Dic₅	C₂²	1	2	yes	—
20	C₂×C₄×Dic₅	C₂²	1	2	yes	—
21	C₄×Dic₁₀	—	2	4	no	—
22	#3963	C₂²	1	2	yes	—
23	#3963	C₂²	1	8	yes	—
24	#2546	—	5	1	no	maximal, Sylow
25	C₂×C₅×Q₈	—	2	4	no	—
26	C₂×C₅×Q₈	—	2	4	no	—
27	C₂×C₅×Q₈	—	2	4	no	—
28	C₂²×C₄×C₅	C₂³	1	4	yes	—
29	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂²×Dic₅	C₂³	1	2	yes	—
33	C₂²×Dic₅	C₂³	1	2	yes	—
34	C₂²×Dic₅	C₂³	1	4	yes	—
35	C₂×Dic₁₀	—	2	2	no	—
36	C₂×Dic₁₀	—	2	2	no	—
37	C₂×Dic₁₀	D₄	1	4	yes	—
38	C₅×C₄⋊C₄	—	2	4	no	—
39	C₅×C₄⋊C₄	—	2	4	no	—
40	C₄²×C₅	D₄	1	4	yes	—
41	C₄⋊Dic₅	—	2	2	no	—
42	C₄⋊Dic₅	Q₈	1	8	yes	complemented
43	#683	—	2	4	no	—
44	#683	—	2	16	no	—
45	C₄×Dic₅	—	2	4	no	—
46	C₄×Dic₅	—	2	4	no	—
47	C₂×C₄⋊Q₈	—	5	2	no	—
48	C₂×C₄⋊Q₈	D₅	1	1	yes	characteristic
49	C₂×C₄×Q₈	—	5	1	no	—
50	#398	—	5	4	no	—
51	#391	—	5	2	no	—
52	#389	—	5	4	no	—
53	C₄×C₄⋊C₄	—	5	1	no	—
54	C₂³×C₅	C₂⁴	1	1	yes	characteristic
55	C₅×Q₈	—	4	8	no	—
56	C₂×C₄×C₅	—	2	2	no	—
57	C₂×C₄×C₅	—	2	4	no	—
58	C₂×C₄×C₅	—	2	4	no	—
59	C₂×C₄×C₅	—	2	4	no	—
60	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₅	C₂×Q₈	1	4	yes	—
65	C₂×C₄×C₅	#41	1	2	yes	—
66	C₂×Dic₅	—	2	4	no	—
67	C₂×Dic₅	—	2	2	no	—
68	C₂×Dic₅	—	2	2	no	—
69	C₂×Dic₅	—	2	4	no	—
70	C₂×Dic₅	—	2	2	no	—
71	C₂×Dic₅	—	2	4	no	—
72	C₂×Dic₅	C₂×D₄	1	4	yes	—
73	C₂×Dic₅	C₂×Q₈	1	8	yes	—
74	Dic₁₀	—	2	8	no	—
75	C₂²×Q₈	—	5	1	no	—
76	C₂²×Q₈	C₂×D₅	1	2	yes	—
77	C₄⋊Q₈	—	2	4	no	—
78	C₄⋊Q₈	—	10	8	no	—
79	C₄×Q₈	—	10	4	no	—
80	C₂×C₄⋊C₄	—	5	2	no	—
81	C₂×C₄⋊C₄	—	5	1	no	—
82	C₂×C₄⋊C₄	—	5	2	no	—
83	C₂×C₄⋊C₄	—	5	8	no	—
84	C₂×C₄⋊C₄	C₂×D₅	1	2	yes	—
85	C₂×C₄⋊C₄	C₂×D₅	1	2	yes	—
86	C₂×C₄²	—	5	2	no	—
87	C₂×C₄²	—	5	2	no	—
88	C₂×C₄²	C₂×D₅	1	1	yes	characteristic
89	#95	—	5	2	no	—
90	#95	—	5	8	no	—
91	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
92	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
93	C₂²×C₅	C₂²×Q₈	1	2	yes	—
94	C₂²×C₅	C₂×#41	1	1	yes	characteristic
95	C₂²×C₅	2₋¹⁺⁴	1	2	yes	—
96	C₄×C₅	—	2	2	no	—
97	C₄×C₅	—	4	4	no	—
98	C₄×C₅	#121	1	4	yes	—
99	C₄×C₅	C₄⋊Q₈	1	4	yes	complemented
100	Dic₅	—	2	4	no	—
101	Dic₅	—	2	8	no	—
102	Dic₅	—	4	2	no	—
103	C₂×Q₈	—	2	4	no	—
104	C₂×Q₈	—	2	4	no	—
105	C₂×Q₈	—	2	4	no	—
106	C₂×Q₈	—	5	4	no	—
107	C₂×Q₈	—	10	2	no	—
108	C₂×Q₈	—	10	2	no	—
109	C₂²×C₄	—	5	4	no	—
110	C₂²×C₄	—	5	2	no	—
111	C₂²×C₄	—	5	2	no	—
112	C₂²×C₄	C₂²×D₅	1	4	yes	—
113	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
114	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
115	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	—	2	4	no	—
118	C₄⋊C₄	—	5	8	no	—
119	C₄⋊C₄	—	10	16	no	—
120	C₄⋊C₄	—	10	4	no	—
121	C₄⋊C₄	—	10	2	no	—
122	C₄²	—	10	4	no	—
123	C₄²	—	10	4	no	—
124	C₄²	C₂²⋊D₅	1	4	yes	—
125	C₂×C₅	C₂×#121	1	1	yes	characteristic
126	C₂×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
127	C₂×C₅	#536	1	1	yes	characteristic
128	C₂×C₅	D₄×Q₈	1	2	yes	—
129	C₂×C₅	#557	1	2	yes	—
130	C₂³	C₂³×D₅	1	1	yes	characteristic
131	Q₈	—	4	8	no	—
132	Q₈	—	10	8	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	5	4	no	—
138	C₂×C₄	—	5	8	no	—
139	C₂×C₄	—	10	4	no	—
140	C₂×C₄	—	10	2	no	—
141	C₂×C₄	—	10	4	no	—
142	C₂×C₄	—	10	2	no	—
143	C₂×C₄	—	10	4	no	—
144	C₂×C₄	—	10	2	no	—
145	C₂×C₄	D₄×D₅	1	2	yes	—
146	C₂×C₄	#711	1	2	yes	—
147	C₂×C₄	Q₈×D₅	1	4	yes	—
148	C₂×C₄	C₂×C₂²⋊D₅	1	2	yes	—
149	C₂×C₄	C₂×C₂²⋊D₅	1	2	yes	—
150	C₂×C₄	C₂×C₂²⋊D₅	1	2	yes	—
151	C₅	#2546	1	1	yes	characteristic, complemented, Sylow
152	C₂²	C₂×D₄×D₅	1	1	yes	characteristic
153	C₂²	C₂×#711	1	1	yes	characteristic
154	C₂²	C₂×Q₈×D₅	1	2	yes	—
155	C₂²	#4147	1	2	yes	—
156	C₂²	C₂²×C₂²⋊D₅	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	4	4	no	—
159	C₄	—	10	8	no	—
160	C₄	—	10	4	no	—
161	C₄	—	20	2	no	—
162	C₄	#4084	1	4	yes	—
163	C₄	#4090	1	4	yes	complemented
164	C₂	#66055	1	2	yes	—
165	C₂	#66061	1	2	yes	—
166	C₂	C₂×#4084	1	1	yes	characteristic
167	C₂	C₂×#4090	1	1	yes	characteristic
168	C₂	#66150	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂×#4090

C₂

yes

maximal

C₂×C₄×Dic₁₀

C₂

yes

maximal, characteristic

#65519

C₂

yes

maximal

#65269

C₂

yes

maximal

#65262

C₂

yes

maximal

C₄×C₄⋊Dic₅

C₂

yes

maximal, characteristic

C₂²×C₅×Q₈

C₂²

yes

—

C₂²×Dic₁₀

C₂²

yes

characteristic

C₅×C₄⋊Q₈

—

C₂×C₅×C₄⋊C₄

C₂²

yes

—

C₂×C₅×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₅

C₂²

yes

characteristic

#4090

—

C₂×C₄⋊Dic₅

C₂²

yes

characteristic

C₂×C₄⋊Dic₅

C₂²

yes

—

C₂×#683

C₂²

yes

—

C₂×#683

C₂²

yes

—

C₂×C₄×Dic₅

C₂²

yes

—

C₂×C₄×Dic₅

C₂²

yes

—

C₄×Dic₁₀

—

#3963

C₂²

yes

—

#3963

C₂²

yes

—

#2546

—

maximal, Sylow

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂²×C₄×C₅

C₂³

yes

—

C₂²×C₄×C₅

C₂³

yes

characteristic

C₂²×C₄×C₅

C₂³

yes

characteristic

C₂²×C₄×C₅

C₂³

yes

characteristic

C₂²×Dic₅

C₂³

yes

—

C₂²×Dic₅

C₂³

yes

—

C₂²×Dic₅

C₂³

yes

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

D₄

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

D₄

yes

—

C₄⋊Dic₅

—

C₄⋊Dic₅

Q₈

yes

complemented

#683

—

#683

—

C₄×Dic₅

—

C₄×Dic₅

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

D₅

yes

characteristic

C₂×C₄×Q₈

—

#398

—

#391

—

#389

—

C₄×C₄⋊C₄

—

C₂³×C₅

C₂⁴

yes

characteristic

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×D₄

yes

—

C₂×C₄×C₅

C₂×D₄

yes

—

C₂×C₄×C₅

C₂×D₄

yes

—

C₂×C₄×C₅

C₂×D₄

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×D₄

yes

—

C₂×Dic₅

C₂×Q₈

yes

—

Dic₁₀

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₅

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

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×D₅

yes

characteristic

#95

—

#95

—

C₂²×C₅

C₂²×D₄

yes

characteristic

C₂²×C₅

C₂²×D₄

yes

characteristic

C₂²×C₅

C₂²×Q₈

yes

—

C₂²×C₅

C₂×#41

yes

characteristic

C₂²×C₅

2₋¹⁺⁴

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#121

yes

—

C₄×C₅

C₄⋊Q₈

yes

complemented

100

Dic₅

—

101

Dic₅

—

102

Dic₅

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

C₂²×D₅

yes

—

113

C₂²×C₄

C₂²×D₅

yes

characteristic

114

C₂²×C₄

C₂²×D₅

yes

characteristic

115

C₂²×C₄

C₂²×D₅

yes

characteristic

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄²

—

123

C₄²

—

124

C₄²

C₂²⋊D₅

yes

—

125

C₂×C₅

C₂×#121

yes

characteristic

126

C₂×C₅

C₂×C₄⋊Q₈

yes

characteristic

127

C₂×C₅

#536

yes

characteristic

128

C₂×C₅

D₄×Q₈

yes

—

129

C₂×C₅

#557

yes

—

130

C₂³

C₂³×D₅

yes

characteristic

131

Q₈

—

132

Q₈

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

D₄×D₅

yes

—

146

C₂×C₄

#711

yes

—

147

C₂×C₄

Q₈×D₅

yes

—

148

C₂×C₄

C₂×C₂²⋊D₅

yes

—

149

C₂×C₄

C₂×C₂²⋊D₅

yes

—

150

C₂×C₄

C₂×C₂²⋊D₅

yes

—

151

C₅

#2546

yes

characteristic, complemented, Sylow

152

C₂²

C₂×D₄×D₅

yes

characteristic

153

C₂²

C₂×#711

yes

characteristic

154

C₂²

C₂×Q₈×D₅

yes

—

155

C₂²

#4147

yes

—

156

C₂²

C₂²×C₂²⋊D₅

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

#4084

yes

—

163

C₄

#4090

yes

complemented

164

C₂

#66055

yes

—

165

C₂

#66061

yes

—

166

C₂

C₂×#4084

yes

characteristic

167

C₂

C₂×#4090

yes

characteristic

168

C₂

#66150

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅
commutator, socle, prime radical, cosocle	C₂³×C₅	C₂⁴
Fitting	C₂×C₅×C₄⋊Q₈	C₂
hypercenter	C₂×C₄⋊Q₈	D₅
hypocenter	C₅	#2546
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅

commutator, socle, prime radical, cosocle

C₂³×C₅

C₂⁴

Fitting

C₂×C₅×C₄⋊Q₈

C₂

hypercenter

C₂×C₄⋊Q₈

D₅

C₅

G ▷ C₂³×C₅ ▷ C₅

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
42. C₄⋊Dic₅	Q₈	semidirect
99. C₄×C₅	C₄⋊Q₈	semidirect
151. C₅	#2546	semidirect
163. C₄	#4090	semidirect

H ≅ G/N

type

42. C₄⋊Dic₅

Q₈

semidirect

99. C₄×C₅

C₄⋊Q₈

semidirect

151. C₅

#2546

semidirect

163. C₄

#4090

semidirect

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

subgroup	index
C₂×C₅×C₄⋊Q₈	2
C₂×#4090	2
C₂×C₄×Dic₁₀	2
#65519	2
#65269	2
#65262	2
C₄×C₄⋊Dic₅	2
#2546	5

subgroup

index

C₂×C₅×C₄⋊Q₈

C₂×C₄×Dic₁₀

C₄×C₄⋊Dic₅

Subgroups — #10614970

Conjugacy classes of subgroups

Complements

Maximal subgroups