Group #41013404 — subgroups

Subgroups — #41013404

Order 27008.

308 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 125 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×C₄×Dic₄₂₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#20269902	C₂	1	2	yes	maximal
3	#27213016	C₂	1	1	yes	maximal, characteristic
4	#27212909	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₄₂₂	C₂²	1	1	yes	characteristic
7	C₄×Dic₄₂₂	—	2	2	no	—
8	C₄×Dic₄₂₂	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₂₁₁	—	2	1	no	—
10	C₂×C₄⋊Dic₂₁₁	—	2	1	no	—
11	C₂×C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₂₁₁	C₂²	1	1	yes	characteristic
13	C₂×Dic₂₁₁⋊C₄	—	2	1	no	—
14	C₂×C₄×Dic₂₁₁	—	2	1	no	—
15	#20269902	—	2	4	no	—
16	C₂²×C₈×C₂₁₁	C₂²	1	2	yes	—
17	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
18	C₄⋊C₈×C₂₁₁	—	2	2	no	—
19	C₂×Dic₄₂₂	—	2	1	no	—
20	C₂×Dic₄₂₂	—	2	1	no	—
21	C₂×Dic₄₂₂	—	2	1	no	—
22	C₂×Dic₄₂₂	C₂×C₄	1	2	yes	—
23	C₂×Dic₄₂₂	D₄	1	2	yes	—
24	C₂×Dic₄₂₂	Q₈	1	2	yes	—
25	C₂²×Dic₂₁₁	—	2	1	no	—
26	C₂²×Dic₂₁₁	—	2	1	no	—
27	C₂²×Dic₂₁₁	—	2	2	no	—
28	C₄⋊Dic₂₁₁	—	2	1	no	—
29	C₄⋊Dic₂₁₁	—	2	2	no	—
30	C₄⋊Dic₂₁₁	—	2	2	no	—
31	C₄⋊Dic₂₁₁	—	4	2	no	—
32	C₄⋊Dic₂₁₁	—	4	2	no	—
33	C₄⋊Dic₂₁₁	C₂×C₄	1	2	yes	—
34	Dic₂₁₁⋊C₄	—	2	4	no	—
35	C₄×Dic₂₁₁	—	2	4	no	—
36	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
38	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
39	C₂×C₈×C₂₁₁	—	2	2	no	—
40	C₂×C₈×C₂₁₁	—	2	2	no	—
41	C₂×C₈×C₂₁₁	—	2	2	no	—
42	C₄²×C₂₁₁	C₂×C₄	1	2	yes	—
43	C₄²×C₂₁₁	D₄	1	2	yes	—
44	Dic₄₂₂	—	2	2	no	—
45	Dic₄₂₂	—	2	2	no	—
46	Dic₄₂₂	—	2	2	no	—
47	Dic₄₂₂	C₄⋊C₄	1	4	yes	complemented
48	C₂×Dic₂₁₁	—	2	2	no	—
49	C₂×Dic₂₁₁	—	2	2	no	—
50	C₂×Dic₂₁₁	—	2	2	no	—
51	C₂×Dic₂₁₁	—	2	2	no	—
52	C₂×Dic₂₁₁	—	4	1	no	—
53	C₂×Dic₂₁₁	—	4	1	no	—
54	C₂×Dic₂₁₁	—	4	2	no	—
55	C₂×Dic₂₁₁	—	4	2	no	—
56	C₂×Dic₂₁₁	—	4	2	no	—
57	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₂₁₁	—	2	1	no	—
59	C₂×C₄×C₂₁₁	—	2	1	no	—
60	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
61	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
62	C₂×C₄×C₂₁₁	SD₁₆	1	2	yes	—
63	C₂×C₄×C₂₁₁	Q₁₆	1	2	yes	—
64	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
69	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
70	C₈×C₂₁₁	—	4	2	no	—
71	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₂₁₁	#121	1	1	yes	characteristic
73	C₂²×C₂₁₁	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂²×C₂₁₁	C₂×SD₁₆	1	1	yes	characteristic
75	C₂²×C₂₁₁	C₂×Q₁₆	1	1	yes	characteristic
76	C₂²×C₂₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₂²×C₂₁₁	#137	1	1	yes	characteristic
78	C₄×C₂₁₁	—	2	2	no	—
79	C₄×C₂₁₁	#103	1	4	yes	complemented
80	C₄×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₄×C₂₁₁	#117	1	1	yes	characteristic
82	C₄×C₂₁₁	#121	1	1	yes	characteristic
83	C₄×C₂₁₁	C₂²⋊Q₈	1	1	yes	characteristic
84	Dic₂₁₁	—	2	4	no	—
85	Dic₂₁₁	—	4	2	no	—
86	Dic₂₁₁	—	8	2	no	—
87	C₂×C₂₁₁	#380	1	1	yes	characteristic
88	C₂×C₂₁₁	C₂×#103	1	1	yes	characteristic
89	C₂×C₂₁₁	#417	1	1	yes	characteristic
90	C₂×C₂₁₁	#460	1	1	yes	characteristic
91	C₂×C₂₁₁	#462	1	1	yes	characteristic
92	C₂×C₂₁₁	#475	1	1	yes	characteristic
93	C₂×C₂₁₁	#477	1	1	yes	characteristic
94	C₂₁₁	#1803	1	1	yes	characteristic, complemented, Sylow
95	#1803	—	211	1	no	maximal, Sylow
96	C₂×C₄×Q₈	—	211	1	no	—
97	C₂×C₄⋊C₈	D₂₁₁	1	1	yes	characteristic
98	C₂×#103	—	211	2	no	—
99	#384	—	211	1	no	—
100	#340	—	211	2	no	—
101	C₂²×Q₈	—	211	1	no	—
102	C₂²×C₈	C₂×D₂₁₁	1	2	yes	—
103	C₄×Q₈	—	422	2	no	—
104	C₄×Q₈	—	211	4	no	—
105	C₂×C₄⋊C₄	—	422	1	no	—
106	C₂×C₄⋊C₄	—	422	1	no	—
107	C₂×C₄⋊C₄	—	422	1	no	—
108	C₂×C₄⋊C₄	—	211	2	no	—
109	C₂×C₄⋊C₄	—	211	1	no	—
110	C₂×C₄²	—	422	1	no	—
111	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
112	C₄⋊C₈	—	2	2	no	—
113	#103	—	422	4	no	—
114	C₂×Q₈	—	422	1	no	—
115	C₂×Q₈	—	422	1	no	—
116	C₂×Q₈	—	422	1	no	—
117	C₂×Q₈	—	211	2	no	—
118	C₂×Q₈	—	211	2	no	—
119	C₂×Q₈	—	211	2	no	—
120	C₂²×C₄	—	422	2	no	—
121	C₂²×C₄	—	422	1	no	—
122	C₂²×C₄	—	422	1	no	—
123	C₂²×C₄	D₈₄₄	1	1	yes	characteristic
124	C₂²×C₄	D₈₄₄	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₄⋊C₄	—	844	2	no	—
130	C₄⋊C₄	—	844	2	no	—
131	C₄⋊C₄	—	422	2	no	—
132	C₄⋊C₄	—	422	2	no	—
133	C₄⋊C₄	—	422	4	no	—
134	C₄⋊C₄	—	422	1	no	—
135	C₄⋊C₄	—	211	2	no	—
136	C₄²	—	422	4	no	—
137	C₄²	C₄×D₂₁₁	1	2	yes	—
138	C₄²	C₂²⋊D₂₁₁	1	2	yes	—
139	C₂³	C₂×D₈₄₄	1	1	yes	characteristic
140	Q₈	—	422	2	no	—
141	Q₈	—	422	2	no	—
142	Q₈	—	422	2	no	—
143	Q₈	—	211	4	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	844	1	no	—
147	C₂×C₄	—	844	1	no	—
148	C₂×C₄	—	844	2	no	—
149	C₂×C₄	—	844	2	no	—
150	C₂×C₄	—	844	2	no	—
151	C₂×C₄	—	422	2	no	—
152	C₂×C₄	—	422	2	no	—
153	C₂×C₄	—	422	2	no	—
154	C₂×C₄	—	422	2	no	—
155	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	2	yes	—
156	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	2	yes	—
157	C₂×C₄	#16809951	1	2	yes	—
158	C₂×C₄	Dic₈₄₄	1	2	yes	—
159	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
160	C₂×C₄	D₄×D₂₁₁	1	1	yes	characteristic
161	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
162	C₂×C₄	Q₈×D₂₁₁	1	1	yes	characteristic
163	C₂×C₄	#16809961	1	1	yes	characteristic
164	C₂×C₄	#16809963	1	1	yes	characteristic
165	C₈	—	4	2	no	—
166	C₂²	C₂×(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
167	C₂²	#20269952	1	1	yes	characteristic
168	C₂²	#20269957	1	1	yes	characteristic
169	C₂²	C₂×#16809951	1	1	yes	characteristic
170	C₂²	C₂×Dic₈₄₄	1	1	yes	characteristic
171	C₂²	#20270005	1	1	yes	characteristic
172	C₂²	#20270010	1	1	yes	characteristic
173	C₄	—	2	2	no	—
174	C₄	—	844	2	no	—
175	C₄	—	1688	2	no	—
176	C₄	—	422	4	no	—
177	C₄	#20269902	1	4	yes	complemented
178	C₄	C₄⋊C₄×D₂₁₁	1	1	yes	characteristic
179	C₄	#20269935	1	1	yes	characteristic
180	C₄	#20269948	1	1	yes	characteristic
181	C₄	#20269953	1	1	yes	characteristic
182	C₂	#27213003	1	1	yes	characteristic
183	C₂	C₂×#20269902	1	1	yes	characteristic
184	C₂	#27213148	1	1	yes	characteristic
185	C₂	#27213347	1	1	yes	characteristic
186	C₂	#27213357	1	1	yes	characteristic
187	C₂	#27213432	1	1	yes	characteristic
188	C₂	#27213442	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₄₂₂

C₂

yes

maximal, characteristic

C₂×#20269902

C₂

yes

maximal

#27213016

C₂

yes

maximal, characteristic

#27212909

C₂

yes

maximal

C₂×C₄⋊C₈×C₂₁₁

C₂

yes

maximal, characteristic

C₂²×Dic₄₂₂

C₂²

yes

characteristic

C₄×Dic₄₂₂

—

C₄×Dic₄₂₂

C₄

yes

complemented

C₂×C₄⋊Dic₂₁₁

—

C₂×C₄⋊Dic₂₁₁

—

C₂×C₄⋊Dic₂₁₁

C₂²

yes

—

C₂×C₄⋊Dic₂₁₁

C₂²

yes

characteristic

C₂×Dic₂₁₁⋊C₄

—

C₂×C₄×Dic₂₁₁

—

#20269902

—

C₂²×C₈×C₂₁₁

C₂²

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₁₁

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

C₂×C₄

yes

—

C₂×Dic₄₂₂

D₄

yes

—

C₂×Dic₄₂₂

Q₈

yes

—

C₂²×Dic₂₁₁

—

C₂²×Dic₂₁₁

—

C₂²×Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

C₂×C₄

yes

—

Dic₂₁₁⋊C₄

—

C₄×Dic₂₁₁

—

C₂²×C₄×C₂₁₁

D₄

yes

characteristic

C₂²×C₄×C₂₁₁

D₄

yes

characteristic

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₄²×C₂₁₁

C₂×C₄

yes

—

C₄²×C₂₁₁

D₄

yes

—

Dic₄₂₂

—

Dic₄₂₂

—

Dic₄₂₂

—

Dic₄₂₂

C₄⋊C₄

yes

complemented

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂³×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₁₁

SD₁₆

yes

—

C₂×C₄×C₂₁₁

Q₁₆

yes

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×Q₈

yes

C₂×C₄×C₂₁₁

yes

C₂×C₄×C₂₁₁

yes

C₈×C₂₁₁

—

C₂²×C₂₁₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₁₁

#121

yes

characteristic

C₂²×C₂₁₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₁₁

C₂×SD₁₆

yes

characteristic

C₂²×C₂₁₁

C₂×Q₁₆

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

C₄×C₂₁₁

—

C₄×C₂₁₁

#103

yes

complemented

C₄×C₂₁₁

C₂×C₄⋊C₄

yes

C₄×C₂₁₁

yes

C₄×C₂₁₁

yes

C₄×C₂₁₁

C₂²⋊Q₈

yes

characteristic

Dic₂₁₁

—

Dic₂₁₁

—

Dic₂₁₁

—

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂₁₁

yes

characteristic, complemented, Sylow

#1803

—

211

maximal, Sylow

C₂×C₄×Q₈

—

211

—

C₂×C₄⋊C₈

D₂₁₁

yes

characteristic

C₂×#103

—

211

—

#384

—

211

—

100

#340

—

211

—

101

C₂²×Q₈

—

211

—

102

C₂²×C₈

C₂×D₂₁₁

yes

—

103

C₄×Q₈

—

422

—

104

C₄×Q₈

—

211

—

105

C₂×C₄⋊C₄

—

422

—

106

C₂×C₄⋊C₄

—

422

—

107

C₂×C₄⋊C₄

—

422

—

108

C₂×C₄⋊C₄

—

211

—

109

C₂×C₄⋊C₄

—

211

—

110

C₂×C₄²

—

422

—

111

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

112

C₄⋊C₈

—

113

#103

—

422

—

114

C₂×Q₈

—

422

—

115

C₂×Q₈

—

422

—

116

C₂×Q₈

—

422

—

117

C₂×Q₈

—

211

—

118

C₂×Q₈

—

211

—

119

C₂×Q₈

—

211

—

120

C₂²×C₄

—

422

—

121

C₂²×C₄

—

422

—

122

C₂²×C₄

—

422

—

123

C₂²×C₄

D₈₄₄

yes

characteristic

124

C₂²×C₄

D₈₄₄

yes

characteristic

125

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₄⋊C₄

—

844

—

130

C₄⋊C₄

—

844

—

131

C₄⋊C₄

—

422

—

132

C₄⋊C₄

—

422

—

133

C₄⋊C₄

—

422

—

134

C₄⋊C₄

—

422

—

135

C₄⋊C₄

—

211

—

136

C₄²

—

422

—

137

C₄²

C₄×D₂₁₁

yes

—

138

C₄²

C₂²⋊D₂₁₁

yes

—

139

C₂³

C₂×D₈₄₄

yes

characteristic

140

Q₈

—

422

—

141

Q₈

—

422

—

142

Q₈

—

422

—

143

Q₈

—

211

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

844

—

147

C₂×C₄

—

844

—

148

C₂×C₄

—

844

—

149

C₂×C₄

—

844

—

150

C₂×C₄

—

844

—

151

C₂×C₄

—

422

—

152

C₂×C₄

—

422

—

153

C₂×C₄

—

422

—

154

C₂×C₄

—

422

—

155

C₂×C₄

(C₂×D₂₁₁)⋊C₄

yes

—

156

C₂×C₄

(C₂×D₂₁₁)⋊C₄

yes

—

157

C₂×C₄

#16809951

yes

—

158

C₂×C₄

Dic₈₄₄

yes

—

159

C₂×C₄

C₂×C₄×D₂₁₁

yes

characteristic

160

C₂×C₄

D₄×D₂₁₁

yes

characteristic

161

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

characteristic

162

C₂×C₄

Q₈×D₂₁₁

yes

characteristic

163

C₂×C₄

#16809961

yes

characteristic

164

C₂×C₄

#16809963

yes

characteristic

165

C₈

—

166

C₂²

C₂×(C₂×D₂₁₁)⋊C₄

yes

characteristic

167

C₂²

#20269952

yes

characteristic

168

C₂²

#20269957

yes

characteristic

169

C₂²

C₂×#16809951

yes

characteristic

170

C₂²

C₂×Dic₈₄₄

yes

characteristic

171

C₂²

#20270005

yes

characteristic

172

C₂²

#20270010

yes

characteristic

173

C₄

—

174

C₄

—

844

—

175

C₄

—

1688

—

176

C₄

—

422

—

177

C₄

#20269902

yes

complemented

178

C₄

C₄⋊C₄×D₂₁₁

yes

characteristic

179

C₄

#20269935

yes

characteristic

180

C₄

#20269948

yes

characteristic

181

C₄

#20269953

yes

characteristic

182

C₂

#27213003

yes

characteristic

183

C₂

C₂×#20269902

yes

characteristic

184

C₂

#27213148

yes

characteristic

185

C₂

#27213347

yes

characteristic

186

C₂

#27213357

yes

characteristic

187

C₂

#27213432

yes

characteristic

188

C₂

#27213442

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₈₄₄
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	C₂×C₄⋊C₈×C₂₁₁	C₂
socle, prime radical	C₂³×C₂₁₁	C₂×D₄
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	C₂×C₄⋊C₈	D₂₁₁
hypocenter	C₂₁₁	#1803
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₈₄₄

commutator

C₂×C₄×C₂₁₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

C₂×C₄⋊C₈×C₂₁₁

C₂

socle, prime radical

C₂³×C₂₁₁

C₂×D₄

cosocle

C₂²×C₄×C₂₁₁

C₂³

hypercenter

C₂×C₄⋊C₈

D₂₁₁

hypocenter

C₂₁₁

#1803

LCS

G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
8. C₄×Dic₄₂₂	C₄	semidirect
47. Dic₄₂₂	C₄⋊C₄	semidirect
79. C₄×C₂₁₁	#103	semidirect
94. C₂₁₁	#1803	semidirect
177. C₄	#20269902	semidirect

H ≅ G/N

type

8. C₄×Dic₄₂₂

C₄

semidirect

47. Dic₄₂₂

C₄⋊C₄

semidirect

79. C₄×C₂₁₁

#103

semidirect

94. C₂₁₁

#1803

semidirect

177. C₄

#20269902

semidirect

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

subgroup	index
C₂×C₄×Dic₄₂₂	2
C₂×#20269902	2
#27213016	2
#27212909	2
C₂×C₄⋊C₈×C₂₁₁	2
#1803	211

subgroup

index

C₂×C₄×Dic₄₂₂

C₂×C₄⋊C₈×C₂₁₁

211

Subgroups — #41013404

Conjugacy classes of subgroups

Complements

Maximal subgroups