Group #11998736 — subgroups

Subgroups — #11998736

Order 1280.

414 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 159 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×#498	C₂	1	1	yes	maximal, characteristic
2	#10602568	C₂	1	2	yes	maximal
3	#10602567	C₂	1	2	yes	maximal
4	#10600585	C₂	1	1	yes	maximal, characteristic
5	#10600517	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₅×#498	—	2	1	no	—
8	C₅×#498	—	2	1	no	—
9	C₅×#498	C₄	1	4	yes	—
10	C₂×C₅×#106	—	2	2	no	—
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂×#3951	—	2	2	no	—
13	#65214	C₂²	1	2	yes	—
14	#65213	C₂²	1	1	yes	characteristic
15	#65213	C₂²	1	1	yes	characteristic
16	#64767	—	2	4	no	—
17	#64766	—	2	4	no	—
18	#64675	—	2	1	no	—
19	C₈×#168	—	2	1	no	—
20	#9767	—	5	1	no	maximal, Sylow
21	C₂²×C₅×Q₈	—	2	2	no	—
22	C₂²×C₅×C₈	D₄	1	2	yes	—
23	C₅×C₄⋊Q₈	—	2	2	no	—
24	C₅×C₄⋊Q₈	—	2	2	no	—
25	C₅×C₄⋊Q₈	—	2	2	no	—
26	C₅×C₄⋊Q₈	C₂×C₄	1	4	yes	—
27	C₂×C₅×C₄⋊C₄	—	2	2	no	—
28	C₂×C₅×C₄⋊C₄	—	2	2	no	—
29	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
30	C₂×#681	—	2	1	no	—
31	C₂×#681	—	2	1	no	—
32	C₅×#106	—	2	4	no	—
33	C₅×#106	—	2	4	no	—
34	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
35	C₄×C₅×C₈	D₄	1	1	yes	characteristic
36	C₄×C₅×C₈	D₄	1	1	yes	characteristic
37	C₂²⋊#168	—	2	2	no	—
38	#3951	—	4	4	no	—
39	C₄⋊#168	—	2	2	no	—
40	C₄⋊#168	—	2	2	no	—
41	C₄⋊#168	D₄	1	4	yes	—
42	C₄×#168	—	2	2	no	—
43	C₂×#498	D₅	1	1	yes	characteristic
44	#1848	—	5	1	no	—
45	#1549	—	5	2	no	—
46	#1548	—	5	2	no	—
47	#1450	—	5	1	no	—
48	C₂×C₅×Q₈	—	2	4	no	—
49	C₂×C₅×Q₈	—	4	2	no	—
50	C₂×C₅×Q₈	—	8	4	no	—
51	C₂²×C₄×C₅	—	4	2	no	—
52	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅	#41	1	2	yes	—
54	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
55	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
56	C₂×C₅×C₈	SD₁₆	1	4	yes	—
57	C₂×C₅×C₈	Q₁₆	1	4	yes	—
58	C₅×C₄⋊C₄	—	2	4	no	—
59	C₅×C₄⋊C₄	—	2	4	no	—
60	C₅×C₄⋊C₄	—	4	2	no	—
61	C₅×C₄⋊C₄	—	4	2	no	—
62	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₅	C₂×D₄	1	2	yes	—
64	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
65	#681	—	4	2	no	—
66	#681	—	4	2	no	—
67	C₂×#168	—	2	4	no	—
68	C₂×#168	—	4	1	no	—
69	C₂×#168	—	4	1	no	—
70	C₂×C₄⋊Q₈	C₂×D₅	1	2	yes	—
71	#498	—	2	1	no	—
72	#498	—	2	1	no	—
73	#498	Dic₅	1	4	yes	—
74	#431	—	5	2	no	—
75	C₂×#108	—	10	2	no	—
76	C₂×#106	—	2	2	no	—
77	#423	—	5	1	no	—
78	#423	—	5	1	no	—
79	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
80	#333	—	10	4	no	—
81	#332	—	10	4	no	—
82	#322	—	10	1	no	—
83	C₈²	—	10	1	no	—
84	C₂³×C₅	#124	1	1	yes	characteristic
85	C₅×Q₈	—	8	4	no	—
86	C₅×Q₈	—	8	4	no	—
87	C₂×C₄×C₅	—	2	4	no	—
88	C₂×C₄×C₅	—	4	4	no	—
89	C₂×C₄×C₅	—	8	4	no	—
90	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
92	C₂×C₄×C₅	#121	1	2	yes	—
93	C₂×C₄×C₅	#127	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
95	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
96	C₅×C₈	#103	1	4	yes	—
97	C₅×C₈	#103	1	4	yes	—
98	#168	—	4	4	no	—
99	#168	—	8	1	no	—
100	#168	—	8	1	no	—
101	C₂²×Q₈	—	2	2	no	—
102	C₂×M₁₆	—	10	1	no	—
103	C₂×M₁₆	—	10	1	no	—
104	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
105	C₄⋊Q₈	—	2	2	no	—
106	C₄⋊Q₈	—	2	2	no	—
107	C₄⋊Q₈	—	2	2	no	—
108	C₄⋊Q₈	C₂×Dic₅	1	4	yes	—
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄⋊C₄	—	2	2	no	—
111	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
112	#108	—	20	4	no	—
113	#106	—	2	4	no	—
114	#106	—	2	4	no	—
115	C₄⋊C₈	—	5	4	no	—
116	C₄⋊C₈	—	10	2	no	—
117	C₄⋊C₈	—	10	2	no	—
118	C₂²⋊C₈	—	10	2	no	—
119	C₄×C₈	—	10	2	no	—
120	C₄×C₈	C₂×Dic₅	1	2	yes	—
121	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
122	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
123	C₂²×C₅	—	2	2	no	—
124	C₂²×C₅	#390	1	1	yes	characteristic
125	C₂²×C₅	#487	1	2	yes	—
126	C₄×C₅	—	2	2	no	—
127	C₄×C₅	—	8	4	no	—
128	C₄×C₅	C₂×#103	1	2	yes	—
129	C₄×C₅	#471	1	4	yes	—
130	C₄×C₅	#492	1	1	yes	characteristic
131	C₄×C₅	#494	1	1	yes	characteristic
132	C₂×Q₈	—	2	4	no	—
133	C₂×Q₈	—	4	2	no	—
134	C₂×Q₈	—	8	4	no	—
135	C₂²×C₄	—	4	2	no	—
136	C₂²×C₄	#711	1	2	yes	—
137	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
138	M₁₆	—	20	2	no	—
139	M₁₆	—	20	2	no	—
140	C₂×C₈	—	10	4	no	—
141	C₂×C₈	—	20	1	no	—
142	C₂×C₈	—	20	1	no	—
143	C₂×C₈	#688	1	4	yes	—
144	C₂×C₈	#689	1	4	yes	—
145	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
146	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄²	D₄×D₅	1	2	yes	—
152	C₄²	C₂²×Dic₅	1	1	yes	characteristic
153	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
154	C₂×C₅	—	2	2	no	—
155	C₂×C₅	#1956	1	1	yes	characteristic
156	C₂×C₅	#1977	1	2	yes	—
157	C₂³	#4082	1	1	yes	characteristic
158	Q₈	—	8	4	no	—
159	Q₈	—	8	4	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	8	4	no	—
163	C₂×C₄	D₄×Dic₅	1	2	yes	—
164	C₂×C₄	#4084	1	2	yes	—
165	C₂×C₄	#4086	1	1	yes	characteristic
166	C₂×C₄	C₂×#688	1	2	yes	—
167	C₂×C₄	C₂×#689	1	2	yes	—
168	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
169	C₈	—	20	4	no	—
170	C₈	—	40	1	no	—
171	C₈	—	40	1	no	—
172	C₈	#3967	1	4	yes	—
173	C₈	#3967	1	4	yes	—
174	C₅	#9767	1	1	yes	characteristic, complemented, Sylow
175	C₂²	—	2	2	no	—
176	C₂²	#65370	1	2	yes	—
177	C₂²	#65510	1	1	yes	characteristic
178	C₄	—	2	2	no	—
179	C₄	—	8	4	no	—
180	C₄	#65374	1	1	yes	characteristic
181	C₄	#65381	1	1	yes	characteristic
182	C₄	#65466	1	4	yes	—
183	C₄	C₂×#3967	1	2	yes	—
184	C₂	—	2	2	no	—
185	C₂	#10602578	1	1	yes	characteristic
186	C₂	#10602845	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅×#498

C₂

yes

maximal, characteristic

#10602568

C₂

yes

maximal

#10602567

C₂

yes

maximal

#10600585

C₂

yes

maximal, characteristic

#10600517

C₂

yes

maximal, characteristic

C₂×C₅×C₄⋊Q₈

C₂²

yes

—

C₅×#498

—

C₅×#498

—

C₅×#498

C₄

yes

—

C₂×C₅×#106

—

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

C₂×#3951

—

#65214

C₂²

yes

—

#65213

C₂²

yes

characteristic

#65213

C₂²

yes

—

—

—

—

—

C₂²×C₅×Q₈

—

C₂²×C₅×C₈

D₄

yes

—

C₅×C₄⋊Q₈

—

C₅×C₄⋊Q₈

—

C₅×C₄⋊Q₈

—

C₅×C₄⋊Q₈

C₂×C₄

yes

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

C₂×C₄²×C₅

C₂³

yes

—

—

—

—

C₄×C₅×C₈

C₂×C₄

yes

—

C₄×C₅×C₈

D₄

yes

characteristic

C₄×C₅×C₈

D₄

yes

characteristic

C₂²⋊#168

—

—

—

—

D₄

yes

—

C₄×#168

—

C₂×#498

D₅

yes

—

—

—

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

#41

yes

—

C₂×C₅×C₈

C₂²⋊C₄

yes

—

C₂×C₅×C₈

C₂²⋊C₄

yes

—

C₂×C₅×C₈

SD₁₆

yes

—

C₂×C₅×C₈

Q₁₆

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂²×C₄

yes

characteristic

C₄²×C₅

C₂×D₄

yes

—

C₄²×C₅

C₂×D₄

yes

—

—

—

—

—

C₂×C₄⋊Q₈

C₂×D₅

yes

—

#498

—

#498

—

#498

Dic₅

yes

—

—

—

—

—

—

C₂×C₄×C₈

C₂×D₅

yes

—

—

—

C₈²

—

C₂³×C₅

#124

yes

characteristic

C₅×Q₈

—

C₅×Q₈

—

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

#127

yes

characteristic

C₂×C₄×C₅

C₂×SD₁₆

yes

—

C₂×C₄×C₅

C₂×Q₁₆

yes

—

C₅×C₈

#103

yes

—

C₅×C₈

#103

yes

—

#168

—

#168

—

100

#168

—

101

C₂²×Q₈

—

102

C₂×M₁₆

—

103

C₂×M₁₆

—

104

C₂²×C₈

C₂²⋊D₅

yes

—

105

C₄⋊Q₈

—

106

C₄⋊Q₈

—

107

C₄⋊Q₈

—

108

C₄⋊Q₈

C₂×Dic₅

yes

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄²

C₂²×D₅

yes

characteristic

112

#108

—

113

#106

—

114

#106

—

115

C₄⋊C₈

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

C₂²⋊C₈

—

119

C₄×C₈

—

120

C₄×C₈

C₂×Dic₅

yes

—

121

C₄×C₈

C₂²⋊D₅

yes

characteristic

122

C₄×C₈

C₂²⋊D₅

yes

characteristic

123

C₂²×C₅

—

124

C₂²×C₅

#390

yes

characteristic

125

C₂²×C₅

#487

yes

—

126

C₄×C₅

—

127

C₄×C₅

—

128

C₄×C₅

C₂×#103

yes

—

129

C₄×C₅

#471

yes

—

130

C₄×C₅

#492

yes

characteristic

131

C₄×C₅

#494

yes

characteristic

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂²×C₄

—

136

C₂²×C₄

#711

yes

—

137

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

138

M₁₆

—

139

M₁₆

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

#688

yes

—

144

C₂×C₈

#689

yes

—

145

C₂×C₈

C₂²⋊Dic₅

yes

—

146

C₂×C₈

C₂²⋊Dic₅

yes

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄²

D₄×D₅

yes

—

152

C₄²

C₂²×Dic₅

yes

characteristic

153

C₄²

C₂×C₂²⋊D₅

yes

characteristic

154

C₂×C₅

—

155

C₂×C₅

#1956

yes

characteristic

156

C₂×C₅

#1977

yes

—

157

C₂³

#4082

yes

characteristic

158

Q₈

—

159

Q₈

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

D₄×Dic₅

yes

—

164

C₂×C₄

#4084

yes

—

165

C₂×C₄

#4086

yes

characteristic

166

C₂×C₄

C₂×#688

yes

—

167

C₂×C₄

C₂×#689

yes

—

168

C₂×C₄

C₂×C₂²⋊Dic₅

yes

characteristic

169

C₈

—

170

C₈

—

171

C₈

—

172

C₈

#3967

yes

—

173

C₈

#3967

yes

—

174

C₅

#9767

yes

characteristic, complemented, Sylow

175

C₂²

—

176

C₂²

#65370

yes

—

177

C₂²

#65510

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

#65374

yes

characteristic

181

C₄

#65381

yes

characteristic

182

C₄

#65466

yes

—

183

C₄

C₂×#3967

yes

—

184

C₂

—

185

C₂

#10602578

yes

characteristic

186

C₂

#10602845

yes

—

	subgroup	G/H
center	C₂²	#65510
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₂×C₅×#498	C₂
socle	C₂²×C₅	#390
prime radical	C₂³×C₅	#124
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×#498	D₅
hypocenter	C₅	#9767
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#498

subgroup

G/H

center

C₂²

#65510

commutator

C₄²×C₅

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅

Fitting

C₂×C₅×#498

C₂

C₂²×C₅

C₂³×C₅

C₂×C₄²×C₅

C₂³

D₅

C₅

G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#498

N	H ≅ G/N	type
174. C₅	#9767	semidirect

H ≅ G/N

type

174. C₅

#9767

semidirect

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

subgroup	index
C₂×C₅×#498	2
#10602568	2
#10602567	2
#10600585	2
#10600517	2
#9767	5

subgroup

index

C₂×C₅×#498

Subgroups — #11998736

Conjugacy classes of subgroups

Complements

Maximal subgroups