Group #11998697 — subgroups

Subgroups — #11998697

Order 1280.

478 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 119 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×#3147	C₂	1	1	yes	maximal, characteristic
2	#10602607	C₂	1	4	yes	maximal, complemented
3	#10600556	C₂	1	1	yes	maximal, characteristic, complemented
4	#10600519	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#135	—	2	2	no	—
6	C₅×#532	C₂²	1	2	yes	—
7	C₅×#495	—	2	2	no	—
8	C₅×#494	C₄	1	1	yes	characteristic, complemented
9	C₅×#493	C₄	1	1	yes	characteristic, complemented
10	C₅×#492	C₄	1	2	yes	complemented
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	#65225	C₂²	1	2	yes	—
13	#65214	C₂²	1	2	yes	—
14	#64771	—	2	2	no	—
15	#64757	—	2	8	no	—
16	#64674	—	2	2	no	—
17	#9760	—	5	1	no	maximal, Sylow
18	C₂×C₅×#41	—	2	2	no	—
19	C₅×#135	—	4	8	no	—
20	C₂×C₅×Q₁₆	—	2	2	no	—
21	C₂×C₅×SD₁₆	—	2	4	no	—
22	C₂×C₅×D₈	—	2	2	no	—
23	C₂²×C₅×C₈	D₄	1	2	yes	—
24	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
25	C₅×#127	C₂×C₄	1	2	yes	—
26	C₅×#124	—	2	2	no	—
27	C₅×#121	—	2	4	no	—
28	C₄×C₅×D₄	—	2	4	no	—
29	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊Dic₅	—	2	2	no	—
31	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
33	C₄×C₅×C₈	D₄	1	2	yes	—
34	#3963	—	2	4	no	—
35	C₂²⋊#168	D₄	1	4	yes	—
36	C₄⋊#168	—	2	2	no	—
37	C₄×#168	—	2	2	no	—
38	#3147	D₅	1	1	yes	characteristic
39	#1835	—	5	1	no	—
40	#1517	—	5	4	no	—
41	#1452	—	5	1	no	—
42	C₅×#41	—	4	8	no	—
43	C₂×C₅×Q₈	—	2	2	no	—
44	C₂×C₅×D₄	—	2	2	no	—
45	C₂×C₅×D₄	—	2	4	no	—
46	C₂×C₅×D₄	—	4	2	no	—
47	C₂²×C₄×C₅	—	4	2	no	—
48	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₅	—	4	2	no	—
51	C₅×Q₁₆	—	4	4	no	—
52	C₅×SD₁₆	—	4	8	no	—
53	C₅×D₈	—	4	4	no	—
54	C₂×C₅×C₈	—	2	4	no	—
55	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
56	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
57	C₅×C₄⋊C₄	—	2	4	no	—
58	C₅×C₂²⋊C₄	—	4	4	no	—
59	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₅	#41	1	2	yes	—
62	C₄⋊Dic₅	—	4	4	no	—
63	C₂×#168	—	2	4	no	—
64	C₂×#135	—	2	2	no	—
65	#532	C₂×D₅	1	2	yes	—
66	#495	—	2	2	no	—
67	#494	Dic₅	1	1	yes	characteristic, complemented
68	#493	Dic₅	1	1	yes	characteristic, complemented
69	#492	Dic₅	1	2	yes	complemented
70	#431	—	5	2	no	—
71	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
72	#382	—	5	2	no	—
73	#340	—	10	2	no	—
74	#327	—	10	8	no	—
75	#322	—	10	2	no	—
76	C₂³×C₅	—	4	2	no	—
77	C₂³×C₅	#127	1	1	yes	characteristic
78	C₅×Q₈	—	4	4	no	—
79	C₅×D₄	—	4	4	no	—
80	C₅×D₄	—	8	4	no	—
81	C₅×D₄	—	8	4	no	—
82	C₂×C₄×C₅	—	4	2	no	—
83	C₂×C₄×C₅	—	4	2	no	—
84	C₂×C₄×C₅	—	8	4	no	—
85	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
87	C₂×C₄×C₅	#121	1	2	yes	—
88	C₂×C₄×C₅	#124	1	1	yes	characteristic
89	C₂×C₄×C₅	AGL(1,ℤ/8)	1	4	yes	—
90	C₂×Dic₅	—	8	4	no	—
91	C₂×Dic₅	—	8	2	no	—
92	C₅×C₈	—	2	2	no	—
93	C₅×C₈	—	2	2	no	—
94	#168	—	4	4	no	—
95	C₂×#41	—	2	2	no	—
96	#135	—	4	8	no	—
97	C₂×Q₁₆	—	2	2	no	—
98	C₂×SD₁₆	—	2	4	no	—
99	C₂×D₈	—	2	2	no	—
100	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
101	C₄⋊Q₈	C₂×Dic₅	1	2	yes	—
102	#127	C₂×Dic₅	1	2	yes	—
103	#124	—	2	2	no	—
104	#121	—	2	4	no	—
105	C₄×D₄	—	2	4	no	—
106	C₂×C₄⋊C₄	—	10	2	no	—
107	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
108	C₄⋊C₈	—	10	2	no	—
109	C₂²⋊C₈	—	5	4	no	—
110	C₄×C₈	—	10	2	no	—
111	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
112	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
113	C₄×C₈	C₂²⋊D₅	1	2	yes	—
114	#95	—	10	4	no	—
115	C₂²×C₅	—	2	2	no	—
116	C₂²×C₅	—	8	2	no	—
117	C₂²×C₅	—	8	4	no	—
118	C₂²×C₅	#390	1	1	yes	characteristic
119	C₂²×C₅	#496	1	2	yes	—
120	C₄×C₅	—	2	2	no	—
121	C₄×C₅	—	2	2	no	—
122	C₄×C₅	—	8	2	no	—
123	C₄×C₅	#418	1	2	yes	—
124	C₄×C₅	#490	1	2	yes	—
125	Dic₅	—	16	2	no	—
126	#41	—	4	8	no	—
127	C₂×Q₈	—	2	2	no	—
128	C₂×D₄	—	2	2	no	—
129	C₂×D₄	—	2	4	no	—
130	C₂×D₄	—	4	2	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	20	2	no	—
133	C₂²×C₄	D₄×D₅	1	2	yes	—
134	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
135	Q₁₆	—	4	4	no	—
136	SD₁₆	—	4	8	no	—
137	D₈	—	4	4	no	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	10	4	no	—
140	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
141	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	20	4	no	—
144	C₂²⋊C₄	—	4	4	no	—
145	C₄²	#711	1	2	yes	—
146	C₄²	C₂²×Dic₅	1	1	yes	characteristic
147	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
148	C₂×C₅	—	2	2	no	—
149	C₂×C₅	—	16	2	no	—
150	C₂×C₅	#1961	1	1	yes	characteristic
151	C₂×C₅	#1976	1	2	yes	—
152	C₂³	—	4	2	no	—
153	C₂³	#4086	1	1	yes	characteristic
154	Q₈	—	4	4	no	—
155	D₄	—	4	4	no	—
156	D₄	—	8	4	no	—
157	D₄	—	8	4	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	8	4	no	—
161	C₂×C₄	—	40	4	no	—
162	C₂×C₄	—	40	2	no	—
163	C₂×C₄	#4078	1	4	yes	—
164	C₂×C₄	D₄×Dic₅	1	2	yes	—
165	C₂×C₄	#4082	1	1	yes	characteristic
166	C₂×C₄	#4084	1	2	yes	—
167	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
168	C₈	—	2	2	no	—
169	C₈	—	2	2	no	—
170	C₈	—	20	4	no	—
171	C₅	#9760	1	1	yes	characteristic, complemented, Sylow
172	C₂²	—	2	2	no	—
173	C₂²	—	8	2	no	—
174	C₂²	—	8	4	no	—
175	C₂²	#65363	1	2	yes	—
176	C₂²	#65510	1	1	yes	characteristic
177	C₄	—	2	2	no	—
178	C₄	—	2	2	no	—
179	C₄	—	8	2	no	—
180	C₄	—	80	2	no	—
181	C₄	#65360	1	2	yes	—
182	C₄	#65504	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	—	16	2	no	—
185	C₂	#10602558	1	1	yes	characteristic
186	C₂	#10602879	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#3147

C₂

yes

maximal, characteristic

#10602607

C₂

yes

maximal, complemented

#10600556

C₂

yes

maximal, characteristic, complemented

#10600519

C₂

yes

maximal, characteristic, complemented

C₂×C₅×#135

—

C₅×#532

C₂²

yes

—

C₅×#495

—

C₅×#494

C₄

yes

characteristic, complemented

C₅×#493

C₄

yes

characteristic, complemented

C₅×#492

C₄

yes

complemented

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

#65225

C₂²

yes

—

#65214

C₂²

yes

—

—

—

—

—

C₂×C₅×#41

—

C₅×#135

—

C₂×C₅×Q₁₆

—

C₂×C₅×SD₁₆

—

C₂×C₅×D₈

—

C₂²×C₅×C₈

D₄

yes

—

C₅×C₄⋊Q₈

C₂×C₄

yes

—

C₅×#127

C₂×C₄

yes

—

C₅×#124

—

C₅×#121

—

C₄×C₅×D₄

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂×C₄⋊Dic₅

—

C₄×C₅×C₈

C₂×C₄

yes

characteristic

C₄×C₅×C₈

C₂×C₄

yes

characteristic

C₄×C₅×C₈

D₄

yes

—

#3963

—

C₂²⋊#168

D₄

yes

—

C₄⋊#168

—

C₄×#168

—

#3147

D₅

yes

—

—

—

C₅×#41

—

C₂×C₅×Q₈

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×D₄

yes

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×Dic₅

—

C₅×Q₁₆

—

C₅×SD₁₆

—

C₅×D₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂²⋊C₄

yes

—

C₂×C₅×C₈

C₂²⋊C₄

yes

—

C₅×C₄⋊C₄

—

C₅×C₂²⋊C₄

—

C₄²×C₅

C₂²×C₄

yes

characteristic

C₄²×C₅

C₂×D₄

yes

characteristic

C₄²×C₅

#41

yes

—

C₄⋊Dic₅

—

C₂×#168

—

C₂×#135

—

#532

C₂×D₅

yes

—

#495

—

#494

Dic₅

yes

characteristic, complemented

#493

Dic₅

yes

characteristic, complemented

#492

Dic₅

yes

complemented

#431

—

C₂×C₄×C₈

C₂×D₅

yes

—

—

—

—

C₂³×C₅

—

C₂³×C₅

#127

yes

characteristic

C₅×Q₈

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅

C₄×D₄

yes

—

C₂×C₄×C₅

#121

yes

—

C₂×C₄×C₅

#124

yes

characteristic

C₂×C₄×C₅

AGL(1,ℤ/8)

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₅×C₈

—

C₅×C₈

—

#168

—

C₂×#41

—

#135

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

100

C₂²×C₈

C₂²⋊D₅

yes

—

101

C₄⋊Q₈

C₂×Dic₅

yes

—

102

#127

C₂×Dic₅

yes

—

103

#124

—

104

#121

—

105

C₄×D₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄²

C₂²×D₅

yes

characteristic

108

C₄⋊C₈

—

109

C₂²⋊C₈

—

110

C₄×C₈

—

111

C₄×C₈

C₂×Dic₅

yes

characteristic

112

C₄×C₈

C₂×Dic₅

yes

characteristic

113

C₄×C₈

C₂²⋊D₅

yes

—

114

#95

—

115

C₂²×C₅

—

116

C₂²×C₅

—

117

C₂²×C₅

—

118

C₂²×C₅

#390

yes

characteristic

119

C₂²×C₅

#496

yes

—

120

C₄×C₅

—

121

C₄×C₅

—

122

C₄×C₅

—

123

C₄×C₅

#418

yes

—

124

C₄×C₅

#490

yes

—

125

Dic₅

—

126

#41

—

127

C₂×Q₈

—

128

C₂×D₄

—

129

C₂×D₄

—

130

C₂×D₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

D₄×D₅

yes

—

134

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

135

Q₁₆

—

136

SD₁₆

—

137

D₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

C₂²⋊Dic₅

yes

—

141

C₂×C₈

C₂²⋊Dic₅

yes

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₂²⋊C₄

—

145

C₄²

#711

yes

—

146

C₄²

C₂²×Dic₅

yes

characteristic

147

C₄²

C₂×C₂²⋊D₅

yes

characteristic

148

C₂×C₅

—

149

C₂×C₅

—

150

C₂×C₅

#1961

yes

characteristic

151

C₂×C₅

#1976

yes

—

152

C₂³

—

153

C₂³

#4086

yes

characteristic

154

Q₈

—

155

D₄

—

156

D₄

—

157

D₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

#4078

yes

—

164

C₂×C₄

D₄×Dic₅

yes

—

165

C₂×C₄

#4082

yes

characteristic

166

C₂×C₄

#4084

yes

—

167

C₂×C₄

C₂×C₂²⋊Dic₅

yes

characteristic

168

C₈

—

169

C₈

—

170

C₈

—

171

C₅

#9760

yes

characteristic, complemented, Sylow

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

#65363

yes

—

176

C₂²

#65510

yes

characteristic

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

#65360

yes

—

182

C₄

#65504

yes

—

183

C₂

—

184

C₂

—

185

C₂

#10602558

yes

characteristic

186

C₂

#10602879

yes

—

	subgroup	G/H
center	C₂²	#65510
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting, prime radical	C₅×#3147	C₂
socle	C₂²×C₅	#390
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#3147	D₅
hypocenter	C₅	#9760
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3147

subgroup

G/H

center

C₂²

#65510

commutator

C₄²×C₅

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅

Fitting, prime radical

C₂

C₂²×C₅

C₂×C₄²×C₅

C₂³

D₅

C₅

G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3147

N	H ≅ G/N	type
2. #10602607	C₂	semidirect
3. #10600556	C₂	semidirect
4. #10600519	C₂	semidirect
8. C₅×#494	C₄	semidirect
9. C₅×#493	C₄	semidirect
10. C₅×#492	C₄	semidirect
67. #494	Dic₅	semidirect
68. #493	Dic₅	semidirect
69. #492	Dic₅	semidirect
171. C₅	#9760	semidirect

H ≅ G/N

type

2. #10602607

C₂

semidirect

3. #10600556

C₂

semidirect

4. #10600519

C₂

semidirect

8. C₅×#494

C₄

semidirect

9. C₅×#493

C₄

semidirect

10. C₅×#492

C₄

semidirect

67. #494

Dic₅

semidirect

68. #493

Dic₅

semidirect

69. #492

Dic₅

semidirect

171. C₅

#9760

semidirect

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

subgroup	index
C₅×#3147	2
#10602607	2
#10600556	2
#10600519	2
#9760	5

subgroup

index

Subgroups — #11998697

Conjugacy classes of subgroups

Complements

Maximal subgroups