Group #12004110 — subgroups

Subgroups — #12004110

Order 1280.

294 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 87 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	C₅×#1960	C₂	1	1	yes	maximal, characteristic
2	#10602566	C₂	1	2	yes	maximal
3	#10600634	C₂	1	1	yes	maximal, characteristic
4	#10598152	C₂	1	1	yes	maximal, characteristic
5	#10598122	C₂	1	2	yes	maximal
6	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₅×#103	—	2	2	no	—
8	C₂×C₅×#97	C₂²	1	1	yes	characteristic
9	C₅×#382	C₂²	1	1	yes	characteristic
10	#65214	C₂²	1	2	yes	—
11	#65213	C₂²	1	2	yes	—
12	#64772	—	2	2	no	—
13	#64767	—	2	4	no	—
14	#10635	—	5	1	no	maximal, Sylow
15	C₂²×C₅×Q₈	—	2	1	no	—
16	C₂²×C₅×C₈	D₄	1	2	yes	—
17	C₅×C₄⋊Q₈	—	2	2	no	—
18	C₅×C₄⋊Q₈	—	2	1	no	—
19	C₅×C₄⋊Q₈	D₄	1	2	yes	—
20	C₂×C₅×C₄⋊C₄	—	2	2	no	—
21	C₂×C₅×C₄⋊C₄	—	2	1	no	—
22	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
23	C₂×#681	—	2	2	no	—
24	C₅×#103	—	4	4	no	—
25	C₅×#97	—	2	2	no	—
26	C₅×#95	—	2	2	no	—
27	C₂²⋊#168	—	2	2	no	—
28	C₄⋊#168	—	2	2	no	—
29	C₄⋊#168	—	2	2	no	—
30	C₄⋊#168	—	2	2	no	—
31	C₄×#168	—	2	2	no	—
32	#1960	D₅	1	1	yes	characteristic
33	#1838	—	5	1	no	—
34	#1546	—	5	2	no	—
35	#1286	—	5	2	no	—
36	#1281	—	5	1	no	—
37	C₂×C₅×Q₈	—	2	2	no	—
38	C₂×C₅×Q₈	—	4	1	no	—
39	C₂×C₅×Q₈	—	4	2	no	—
40	C₂×C₅×Q₈	—	4	2	no	—
41	C₂²×C₄×C₅	—	4	1	no	—
42	C₂²×C₄×C₅	—	4	1	no	—
43	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₅	#41	1	2	yes	—
45	C₂×C₅×C₈	—	2	2	no	—
46	C₂×C₅×C₈	—	2	2	no	—
47	C₂×C₅×C₈	—	2	2	no	—
48	C₅×C₄⋊C₄	—	2	4	no	—
49	C₅×C₄⋊C₄	—	4	2	no	—
50	C₅×C₄⋊C₄	—	4	2	no	—
51	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₅	#41	1	1	yes	characteristic
53	C₄²×C₅	#41	1	2	yes	—
54	#681	—	4	2	no	—
55	#681	—	4	2	no	—
56	C₂×#168	—	4	2	no	—
57	C₂×#168	—	4	2	no	—
58	C₂×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
59	#431	—	5	2	no	—
60	#423	—	5	2	no	—
61	C₂×#103	—	2	2	no	—
62	C₂×#97	C₂×D₅	1	1	yes	characteristic
63	#382	C₂×D₅	1	1	yes	characteristic
64	#341	—	10	2	no	—
65	#332	—	10	4	no	—
66	C₂³×C₅	#124	1	1	yes	characteristic
67	C₅×Q₈	—	4	4	no	—
68	C₅×Q₈	—	8	2	no	—
69	C₂×C₄×C₅	—	4	2	no	—
70	C₂×C₄×C₅	—	4	2	no	—
71	C₂×C₄×C₅	—	8	1	no	—
72	C₂×C₄×C₅	—	8	1	no	—
73	C₂×C₄×C₅	—	8	2	no	—
74	C₂×C₄×C₅	—	8	1	no	—
75	C₂×C₄×C₅	#121	1	1	yes	characteristic
76	C₂×C₄×C₅	#123	1	2	yes	—
77	C₂×C₄×C₅	#123	1	1	yes	characteristic
78	C₂×C₄×C₅	#126	1	2	yes	—
79	C₂×C₄×C₅	#135	1	2	yes	—
80	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
81	C₅×C₈	—	4	2	no	—
82	#168	—	8	2	no	—
83	#168	—	8	2	no	—
84	C₂²×Q₈	—	2	1	no	—
85	C₂×M₁₆	—	10	2	no	—
86	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
87	C₄⋊Q₈	—	2	1	no	—
88	C₄⋊Q₈	—	2	2	no	—
89	C₄⋊Q₈	D₂₀	1	2	yes	—
90	C₂×C₄⋊C₄	—	2	2	no	—
91	C₂×C₄⋊C₄	—	2	1	no	—
92	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
93	C₄⋊C₈	—	10	2	no	—
94	C₄⋊C₈	—	10	2	no	—
95	C₄⋊C₈	—	10	2	no	—
96	#103	—	4	4	no	—
97	C₂²⋊C₈	—	10	2	no	—
98	#97	—	2	2	no	—
99	C₄×C₈	—	10	2	no	—
100	#95	—	2	2	no	—
101	C₂²×C₅	—	2	2	no	—
102	C₂²×C₅	#397	1	1	yes	characteristic
103	C₂²×C₅	#488	1	1	yes	characteristic
104	C₂²×C₅	#490	1	1	yes	characteristic
105	C₄×C₅	—	2	2	no	—
106	C₄×C₅	—	2	2	no	—
107	C₄×C₅	—	8	2	no	—
108	C₄×C₅	—	16	1	no	—
109	C₄×C₅	#463	1	2	yes	—
110	C₄×C₅	#482	1	2	yes	—
111	C₂×Q₈	—	2	2	no	—
112	C₂×Q₈	—	4	1	no	—
113	C₂×Q₈	—	4	2	no	—
114	C₂×Q₈	—	4	2	no	—
115	C₂²×C₄	—	4	1	no	—
116	C₂²×C₄	—	4	1	no	—
117	C₂²×C₄	#711	1	2	yes	—
118	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
119	M₁₆	—	20	2	no	—
120	M₁₆	—	20	2	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	20	2	no	—
125	C₂×C₈	—	20	2	no	—
126	C₄⋊C₄	—	2	4	no	—
127	C₄⋊C₄	—	4	2	no	—
128	C₄⋊C₄	—	4	2	no	—
129	C₄²	C₂×D₂₀	1	1	yes	characteristic
130	C₄²	#709	1	1	yes	characteristic
131	C₄²	#711	1	2	yes	—
132	C₂×C₅	—	2	1	no	—
133	C₂×C₅	—	2	1	no	—
134	C₂×C₅	#2073	1	1	yes	characteristic
135	C₂×C₅	#2082	1	2	yes	—
136	C₂³	#4082	1	1	yes	characteristic
137	Q₈	—	4	4	no	—
138	Q₈	—	8	2	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	8	1	no	—
142	C₂×C₄	—	8	1	no	—
143	C₂×C₄	—	8	2	no	—
144	C₂×C₄	—	8	1	no	—
145	C₂×C₄	#4025	1	2	yes	—
146	C₂×C₄	#4032	1	2	yes	—
147	C₂×C₄	#4076	1	1	yes	characteristic
148	C₂×C₄	#4078	1	2	yes	—
149	C₂×C₄	#4081	1	1	yes	characteristic
150	C₂×C₄	#4096	1	2	yes	—
151	C₈	—	4	2	no	—
152	C₈	—	40	2	no	—
153	C₈	—	40	2	no	—
154	C₅	#10635	1	1	yes	characteristic, complemented, Sylow
155	C₂²	—	2	2	no	—
156	C₂²	#65242	1	1	yes	characteristic
157	C₂²	#65345	1	1	yes	characteristic
158	C₂²	#65360	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	8	2	no	—
162	C₄	—	16	1	no	—
163	C₄	#65305	1	2	yes	—
164	C₄	#65324	1	2	yes	—
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	#10600434	1	1	yes	characteristic
168	C₂	#10601134	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#1960

C₂

yes

maximal, characteristic

#10602566

C₂

yes

maximal

#10600634

C₂

yes

maximal, characteristic

#10598152

C₂

yes

maximal, characteristic

#10598122

C₂

yes

maximal

C₂×C₅×C₄⋊Q₈

C₂²

yes

characteristic

C₂×C₅×#103

—

C₂×C₅×#97

C₂²

yes

characteristic

C₅×#382

C₂²

yes

characteristic

#65214

C₂²

yes

—

#65213

C₂²

yes

—

—

—

—

C₂²×C₅×Q₈

—

C₂²×C₅×C₈

D₄

yes

—

C₅×C₄⋊Q₈

—

C₅×C₄⋊Q₈

—

C₅×C₄⋊Q₈

D₄

yes

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂×#681

—

C₅×#103

—

C₅×#97

—

C₅×#95

—

C₂²⋊#168

—

—

—

—

—

D₅

yes

—

—

—

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

#41

yes

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂×D₄

yes

C₄²×C₅

yes

C₄²×C₅

yes

—

—

—

—

—

C₂×C₄⋊Q₈

C₂×D₅

yes

—

—

—

C₂×#97

C₂×D₅

yes

characteristic

#382

C₂×D₅

yes

characteristic

#341

—

#332

—

C₂³×C₅

#124

yes

characteristic

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#121

yes

characteristic

C₂×C₄×C₅

#123

yes

—

C₂×C₄×C₅

#123

yes

characteristic

C₂×C₄×C₅

#126

yes

—

C₂×C₄×C₅

#135

yes

—

C₂×C₄×C₅

AGL(1,ℤ/8)

yes

—

C₅×C₈

—

#168

—

#168

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊D₅

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₂₀

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₅

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

100

#95

—

101

C₂²×C₅

—

102

C₂²×C₅

#397

yes

characteristic

103

C₂²×C₅

#488

yes

characteristic

104

C₂²×C₅

#490

yes

characteristic

105

C₄×C₅

—

106

C₄×C₅

—

107

C₄×C₅

—

108

C₄×C₅

—

109

C₄×C₅

#463

yes

—

110

C₄×C₅

#482

yes

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

#711

yes

—

118

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

119

M₁₆

—

120

M₁₆

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄²

C₂×D₂₀

yes

characteristic

130

C₄²

#709

yes

characteristic

131

C₄²

#711

yes

—

132

C₂×C₅

—

133

C₂×C₅

—

134

C₂×C₅

#2073

yes

characteristic

135

C₂×C₅

#2082

yes

—

136

C₂³

#4082

yes

characteristic

137

Q₈

—

138

Q₈

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

#4025

yes

—

146

C₂×C₄

#4032

yes

—

147

C₂×C₄

#4076

yes

characteristic

148

C₂×C₄

#4078

yes

—

149

C₂×C₄

#4081

yes

characteristic

150

C₂×C₄

#4096

yes

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₅

#10635

yes

characteristic, complemented, Sylow

155

C₂²

—

156

C₂²

#65242

yes

characteristic

157

C₂²

#65345

yes

characteristic

158

C₂²

#65360

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#65305

yes

—

164

C₄

#65324

yes

—

165

C₂

—

166

C₂

—

167

C₂

#10600434

yes

characteristic

168

C₂

#10601134

yes

—

	subgroup	G/H
center	C₂²	#65242
commutator, cosocle	C₂×C₄²×C₅	C₂³
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1960	C₂
socle	C₂²×C₅	#397
prime radical	C₂³×C₅	#124
hypercenter	#1960	D₅
hypocenter	C₅	#10635
LCS	G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1960

subgroup

G/H

center

C₂²

#65242

commutator, cosocle

C₂×C₄²×C₅

C₂³

Frattini

C₂×C₄²

C₂²×D₅

C₂

C₂²×C₅

C₂³×C₅

D₅

C₅

G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1960

N	H ≅ G/N	type
154. C₅	#10635	semidirect

H ≅ G/N

type

154. C₅

#10635

semidirect

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

subgroup	index
C₅×#1960	2
#10602566	2
#10600634	2
#10598152	2
#10598122	2
#10635	5

subgroup

index

Subgroups — #12004110

Conjugacy classes of subgroups

Complements

Maximal subgroups