Group #11996823 — subgroups

Subgroups — #11996823

Order 1280.

384 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 103 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×#3139	C₂	1	1	yes	maximal, characteristic
2	#10615064	C₂	1	1	yes	maximal, characteristic
3	#10602616	C₂	1	1	yes	maximal, characteristic
4	#10602569	C₂	1	2	yes	maximal
5	#10598161	C₂	1	2	yes	maximal
6	C₅×#533	C₂²	1	1	yes	characteristic
7	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₅×#491	—	2	2	no	—
9	C₅×#487	C₄	1	4	yes	—
10	C₅×#419	—	2	2	no	—
11	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
12	#65313	—	2	4	no	—
13	#65289	—	2	2	no	—
14	#65215	C₂²	1	2	yes	—
15	#65215	C₂²	1	2	yes	—
16	#64767	—	2	4	no	—
17	#64763	—	2	2	no	—
18	#9422	—	5	1	no	maximal, Sylow
19	C₂²×C₅×Q₈	—	2	1	no	—
20	C₂×C₅×M₁₆	—	2	1	no	—
21	C₅×C₄⋊Q₈	—	2	2	no	—
22	C₅×C₄⋊Q₈	—	2	1	no	—
23	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
24	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
25	C₅×#125	D₄	1	2	yes	—
26	C₅×C₂²⋊Q₈	—	2	2	no	—
27	C₄×C₅×Q₈	—	2	2	no	—
28	C₅×#117	D₄	1	2	yes	—
29	C₂×C₅×C₄⋊C₄	—	2	2	no	—
30	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
31	C₅×#103	—	2	8	no	—
32	C₅×#97	—	2	1	no	—
33	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₂²⋊#168	—	2	2	no	—
35	C₂²⋊#168	—	2	2	no	—
36	C₄⋊#168	—	2	4	no	—
37	C₄⋊#168	—	2	2	no	—
38	C₄⋊#168	D₄	1	4	yes	—
39	#3935	—	2	2	no	—
40	#3935	—	2	2	no	—
41	#3935	—	2	2	no	—
42	C₄×#168	—	2	2	no	—
43	#3139	D₅	1	1	yes	characteristic
44	#2987	—	5	1	no	—
45	#1556	—	5	2	no	—
46	#1537	—	5	1	no	—
47	#1291	—	5	2	no	—
48	C₂×C₅×Q₈	—	2	2	no	—
49	C₂×C₅×Q₈	—	2	2	no	—
50	C₂×C₅×Q₈	—	4	4	no	—
51	C₂×C₅×Q₈	—	4	1	no	—
52	C₂²×C₄×C₅	—	4	1	no	—
53	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	#41	1	2	yes	—
55	C₅×M₁₆	—	4	2	no	—
56	C₂×C₅×C₈	—	2	2	no	—
57	C₅×C₄⋊C₄	—	2	2	no	—
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	C₂²⋊C₄	1	4	yes	—
62	C₅×C₂²⋊C₄	—	2	2	no	—
63	C₄²×C₅	—	2	2	no	—
64	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₅	C₂×D₄	1	2	yes	—
66	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂×#168	—	2	4	no	—
68	C₂×#168	—	4	2	no	—
69	#533	C₂×D₅	1	1	yes	characteristic
70	C₂×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
71	#491	—	2	2	no	—
72	#487	Dic₅	1	4	yes	—
73	#446	—	10	4	no	—
74	#433	—	10	2	no	—
75	#432	—	5	2	no	—
76	#432	—	5	2	no	—
77	#419	—	2	2	no	—
78	C₄×M₁₆	C₂×D₅	1	1	yes	characteristic
79	#332	—	10	4	no	—
80	#330	—	10	2	no	—
81	C₂³×C₅	#123	1	1	yes	characteristic
82	C₅×Q₈	—	4	4	no	—
83	C₅×Q₈	—	8	2	no	—
84	C₅×Q₈	—	8	2	no	—
85	C₂×C₄×C₅	—	2	2	no	—
86	C₂×C₄×C₅	—	2	4	no	—
87	C₂×C₄×C₅	—	4	2	no	—
88	C₂×C₄×C₅	—	4	2	no	—
89	C₂×C₄×C₅	—	8	2	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	#120	1	1	yes	characteristic
93	C₂×C₄×C₅	#121	1	2	yes	—
94	C₅×C₈	—	4	2	no	—
95	#168	—	4	4	no	—
96	#168	—	8	2	no	—
97	C₂²×Q₈	—	2	1	no	—
98	C₂×M₁₆	—	2	1	no	—
99	C₄⋊Q₈	—	2	2	no	—
100	C₄⋊Q₈	—	2	1	no	—
101	C₄⋊Q₈	C₂×Dic₅	1	2	yes	—
102	C₄⋊Q₈	C₂×Dic₅	1	2	yes	—
103	#125	C₂²⋊D₅	1	2	yes	—
104	C₂²⋊Q₈	—	2	2	no	—
105	C₄×Q₈	—	2	2	no	—
106	#117	C₂²⋊D₅	1	2	yes	—
107	C₂×C₄⋊C₄	—	2	2	no	—
108	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
109	C₄⋊C₈	—	5	4	no	—
110	C₄⋊C₈	—	10	4	no	—
111	C₄⋊C₈	—	10	2	no	—
112	#103	—	2	8	no	—
113	C₂²⋊C₈	—	10	2	no	—
114	C₂²⋊C₈	—	10	2	no	—
115	#97	—	2	1	no	—
116	#97	—	10	2	no	—
117	#97	—	10	2	no	—
118	#97	—	10	2	no	—
119	C₄×C₈	—	10	2	no	—
120	C₄×C₈	C₂×Dic₅	1	2	yes	—
121	C₂²×C₅	—	2	1	no	—
122	C₂²×C₅	—	2	2	no	—
123	C₂²×C₅	#386	1	1	yes	characteristic
124	C₄×C₅	—	2	1	no	—
125	C₄×C₅	—	2	2	no	—
126	C₄×C₅	—	2	2	no	—
127	C₄×C₅	—	4	4	no	—
128	C₄×C₅	—	8	2	no	—
129	C₄×C₅	#454	1	2	yes	—
130	C₂×Q₈	—	2	2	no	—
131	C₂×Q₈	—	2	2	no	—
132	C₂×Q₈	—	4	4	no	—
133	C₂×Q₈	—	4	1	no	—
134	C₂²×C₄	—	4	1	no	—
135	C₂²×C₄	#711	1	2	yes	—
136	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
137	M₁₆	—	4	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	10	4	no	—
140	C₂×C₈	—	20	2	no	—
141	C₄⋊C₄	—	2	2	no	—
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	C₂²⋊Dic₅	1	4	yes	—
146	C₂²⋊C₄	—	2	2	no	—
147	C₄²	—	2	2	no	—
148	C₄²	D₄×D₅	1	2	yes	—
149	C₄²	C₂²×Dic₅	1	1	yes	characteristic
150	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
151	C₂×C₅	—	4	1	no	—
152	C₂×C₅	#1888	1	1	yes	characteristic
153	C₂×C₅	#1902	1	2	yes	—
154	C₂³	#4081	1	1	yes	characteristic
155	Q₈	—	4	4	no	—
156	Q₈	—	8	2	no	—
157	Q₈	—	8	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	D₄×Dic₅	1	2	yes	—
164	C₂×C₄	#4083	1	1	yes	characteristic
165	C₂×C₄	#4085	1	2	yes	—
166	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
167	C₈	—	4	2	no	—
168	C₈	—	20	4	no	—
169	C₈	—	40	2	no	—
170	C₅	#9422	1	1	yes	characteristic, complemented, Sylow
171	C₂²	—	2	1	no	—
172	C₂²	—	2	2	no	—
173	C₂²	#65509	1	1	yes	characteristic
174	C₄	—	2	1	no	—
175	C₄	—	2	2	no	—
176	C₄	—	2	2	no	—
177	C₄	—	4	4	no	—
178	C₄	—	8	2	no	—
179	C₄	#65351	1	2	yes	—
180	C₂	—	4	1	no	—
181	C₂	#10602517	1	1	yes	characteristic
182	C₂	#10602895	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#3139

C₂

yes

maximal, characteristic

#10615064

C₂

yes

maximal, characteristic

#10602616

C₂

yes

maximal, characteristic

#10602569

C₂

yes

maximal

#10598161

C₂

yes

maximal

C₅×#533

C₂²

yes

characteristic

C₂×C₅×C₄⋊Q₈

C₂²

yes

characteristic

C₅×#491

—

C₅×#487

C₄

yes

—

C₅×#419

—

C₄×C₅×M₁₆

C₂²

yes

—

—

C₂²

yes

—

#65215

C₂²

yes

—

—

—

—

C₂²×C₅×Q₈

—

C₂×C₅×M₁₆

—

C₅×C₄⋊Q₈

—

C₅×C₄⋊Q₈

—

C₅×C₄⋊Q₈

C₂×C₄

yes

—

C₅×C₄⋊Q₈

C₂×C₄

yes

—

C₅×#125

D₄

yes

—

C₅×C₂²⋊Q₈

—

C₄×C₅×Q₈

—

C₅×#117

D₄

yes

—

C₂×C₅×C₄⋊C₄

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₅×#103

—

C₅×#97

—

C₄×C₅×C₈

C₂×C₄

yes

—

C₂²⋊#168

—

C₂²⋊#168

—

C₄⋊#168

—

C₄⋊#168

—

C₄⋊#168

D₄

yes

—

—

—

—

—

D₅

yes

—

—

—

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

#41

yes

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

C₂²⋊C₄

yes

—

C₅×C₂²⋊C₄

—

C₄²×C₅

—

C₄²×C₅

C₂²×C₄

yes

characteristic

C₄²×C₅

C₂×D₄

yes

—

C₄²×C₅

C₂×D₄

yes

—

—

C₂×D₅

yes

characteristic

C₂×C₄⋊Q₈

C₂×D₅

yes

characteristic

#491

—

#487

Dic₅

yes

—

—

—

—

—

—

C₄×M₁₆

C₂×D₅

yes

characteristic

#332

—

#330

—

C₂³×C₅

#123

yes

characteristic

C₅×Q₈

—

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

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

#120

yes

characteristic

C₂×C₄×C₅

#121

yes

—

C₅×C₈

—

#168

—

#168

—

C₂²×Q₈

—

C₂×M₁₆

—

C₄⋊Q₈

—

100

C₄⋊Q₈

—

101

C₄⋊Q₈

C₂×Dic₅

yes

—

102

C₄⋊Q₈

C₂×Dic₅

yes

—

103

#125

C₂²⋊D₅

yes

—

104

C₂²⋊Q₈

—

105

C₄×Q₈

—

106

#117

C₂²⋊D₅

yes

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄²

C₂²×D₅

yes

characteristic

109

C₄⋊C₈

—

110

C₄⋊C₈

—

111

C₄⋊C₈

—

112

#103

—

113

C₂²⋊C₈

—

114

C₂²⋊C₈

—

115

#97

—

116

#97

—

117

#97

—

118

#97

—

119

C₄×C₈

—

120

C₄×C₈

C₂×Dic₅

yes

—

121

C₂²×C₅

—

122

C₂²×C₅

—

123

C₂²×C₅

#386

yes

characteristic

124

C₄×C₅

—

125

C₄×C₅

—

126

C₄×C₅

—

127

C₄×C₅

—

128

C₄×C₅

—

129

C₄×C₅

#454

yes

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂²×C₄

—

135

C₂²×C₄

#711

yes

—

136

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

137

M₁₆

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

C₂²⋊Dic₅

yes

—

146

C₂²⋊C₄

—

147

C₄²

—

148

C₄²

D₄×D₅

yes

—

149

C₄²

C₂²×Dic₅

yes

characteristic

150

C₄²

C₂×C₂²⋊D₅

yes

characteristic

151

C₂×C₅

—

152

C₂×C₅

#1888

yes

characteristic

153

C₂×C₅

#1902

yes

—

154

C₂³

#4081

yes

characteristic

155

Q₈

—

156

Q₈

—

157

Q₈

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

D₄×Dic₅

yes

—

164

C₂×C₄

#4083

yes

characteristic

165

C₂×C₄

#4085

yes

—

166

C₂×C₄

C₂×C₂²⋊Dic₅

yes

characteristic

167

C₈

—

168

C₈

—

169

C₈

—

170

C₅

#9422

yes

characteristic, complemented, Sylow

171

C₂²

—

172

C₂²

—

173

C₂²

#65509

yes

characteristic

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

#65351

yes

—

180

C₂

—

181

C₂

#10602517

yes

characteristic

182

C₂

#10602895

yes

—

	subgroup	G/H
center	C₂²	#65509
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#3139	C₂
socle	C₂²×C₅	#386
prime radical	C₂³×C₅	#123
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#3139	D₅
hypocenter	C₅	#9422
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3139

subgroup

G/H

center

C₂²

#65509

commutator

C₄²×C₅

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅

C₂

C₂²×C₅

C₂³×C₅

C₂×C₄²×C₅

C₂³

D₅

C₅

G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3139

N	H ≅ G/N	type
170. C₅	#9422	semidirect

H ≅ G/N

type

170. C₅

#9422

semidirect

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

subgroup	index
C₅×#3139	2
#10615064	2
#10602616	2
#10602569	2
#10598161	2
#9422	5

subgroup

index

Subgroups — #11996823

Conjugacy classes of subgroups

Complements

Maximal subgroups