Group #19135064 — subgroups

Subgroups — #19135064

Order 5504.

338 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 101 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	C₂²×#10621877	C₂	1	1	yes	maximal, characteristic
2	C₂×#13063244	C₂	1	2	yes	maximal, complemented
3	#14582863	C₂	1	1	yes	maximal, characteristic
4	#14582668	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#10621877	—	2	1	no	—
7	C₂×#10621877	—	2	4	no	—
8	C₂×#10621877	C₂²	1	2	yes	—
9	C₂×#10621877	C₂²	1	2	yes	—
10	C₂²×C₄₃⋊C₈	—	2	1	no	—
11	#13063244	—	2	4	no	—
12	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
13	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
14	C₂²×Q₈×C₄₃	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic
16	C₂²⋊Q₈×C₄₃	—	2	1	no	—
17	C₂²⋊Q₈×C₄₃	—	2	1	no	—
18	C₂²⋊Q₈×C₄₃	—	2	1	no	—
19	C₂²⋊Q₈×C₄₃	—	2	1	no	—
20	C₂×C₄⋊C₄×C₄₃	C₄	1	2	yes	—
21	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×C₄₃	C₄	1	2	yes	—
23	C₂⁴×C₄₃	C₂×C₄	1	1	yes	characteristic
24	C₂×Q₈×C₄₃	—	2	1	no	—
25	C₂×Q₈×C₄₃	—	2	1	no	—
26	C₂×Q₈×C₄₃	—	2	2	no	—
27	C₂×Q₈×C₄₃	—	2	2	no	—
28	C₂²×C₄×C₄₃	—	2	2	no	—
29	C₂²×C₄×C₄₃	—	2	2	no	—
30	C₂²×C₄×C₄₃	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₄₃	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₄₃	—	2	2	no	—
39	C₄⋊C₄×C₄₃	—	2	2	no	—
40	C₄⋊C₄×C₄₃	—	2	1	no	—
41	C₄⋊C₄×C₄₃	—	2	1	no	—
42	C₂²⋊C₄×C₄₃	—	2	2	no	—
43	C₂²⋊C₄×C₄₃	—	2	2	no	—
44	#10621877	—	2	4	no	—
45	#10621877	—	2	2	no	—
46	#10621877	—	2	2	no	—
47	#10621877	—	2	2	no	—
48	#10621877	D₄	1	4	yes	—
49	C₂×C₄₃⋊C₈	—	2	4	no	—
50	C₂×C₄₃⋊C₈	—	2	2	no	—
51	C₂×C₄₃⋊C₈	—	2	2	no	—
52	C₂³×C₄₃	—	2	2	no	—
53	C₂³×C₄₃	—	2	2	no	—
54	C₂³×C₄₃	—	4	1	no	—
55	C₂³×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂³×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
58	Q₈×C₄₃	—	4	1	no	—
59	Q₈×C₄₃	—	4	2	no	—
60	Q₈×C₄₃	—	4	1	no	—
61	C₂×C₄×C₄₃	—	2	2	no	—
62	C₂×C₄×C₄₃	—	2	1	no	—
63	C₂×C₄×C₄₃	—	2	1	no	—
64	C₂×C₄×C₄₃	—	2	2	no	—
65	C₂×C₄×C₄₃	—	2	2	no	—
66	C₂×C₄×C₄₃	—	2	1	no	—
67	C₂×C₄×C₄₃	—	2	1	no	—
68	C₂×C₄×C₄₃	—	2	2	no	—
69	C₂×C₄×C₄₃	—	2	2	no	—
70	C₂×C₄×C₄₃	—	2	4	no	—
71	C₂×C₄×C₄₃	—	4	2	no	—
72	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
75	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₄₃	#41	1	2	yes	—
78	C₄₃⋊C₈	—	2	4	no	—
79	C₄₃⋊C₈	—	4	2	no	—
80	C₂²×C₄₃	—	2	1	no	—
81	C₂²×C₄₃	—	2	1	no	—
82	C₂²×C₄₃	—	2	1	no	—
83	C₂²×C₄₃	—	2	1	no	—
84	C₂²×C₄₃	—	2	1	no	—
85	C₂²×C₄₃	—	2	1	no	—
86	C₂²×C₄₃	—	2	1	no	—
87	C₂²×C₄₃	—	4	4	no	—
88	C₂²×C₄₃	#101	1	2	yes	complemented
89	C₂²×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₄₃	C₄×D₄	1	2	yes	—
91	C₄×C₄₃	—	2	1	no	—
92	C₄×C₄₃	—	2	1	no	—
93	C₄×C₄₃	—	4	2	no	—
94	C₄×C₄₃	—	4	2	no	—
95	C₄×C₄₃	#120	1	1	yes	characteristic
96	C₄×C₄₃	#121	1	2	yes	—
97	C₄×C₄₃	#123	1	1	yes	characteristic
98	#1901	—	43	1	no	maximal, Sylow
99	C₂×C₄₃	—	2	2	no	—
100	C₂×C₄₃	—	2	2	no	—
101	C₂×C₄₃	—	4	1	no	—
102	C₂×C₄₃	#386	1	1	yes	characteristic
103	C₂×C₄₃	C₂×#101	1	1	yes	characteristic
104	C₂×C₄₃	#413	1	1	yes	characteristic
105	C₂²×M₁₆	—	43	1	no	—
106	C₂×C₂²⋊Q₈	D₄₃	1	1	yes	characteristic
107	C₂×#101	—	43	2	no	—
108	#407	—	43	1	no	—
109	#343	—	43	2	no	—
110	C₄₃	#1901	1	1	yes	characteristic, complemented, Sylow
111	C₂²×Q₈	C₂×D₄₃	1	1	yes	characteristic
112	C₂³×C₄	C₂×D₄₃	1	1	yes	characteristic
113	C₂×M₁₆	—	43	2	no	—
114	C₂×M₁₆	—	43	2	no	—
115	C₂×M₁₆	—	86	4	no	—
116	C₂×M₁₆	—	86	1	no	—
117	C₂²×C₈	—	86	1	no	—
118	C₂²⋊Q₈	—	2	1	no	—
119	C₂²⋊Q₈	—	2	1	no	—
120	C₂²⋊Q₈	—	2	1	no	—
121	C₂²⋊Q₈	—	2	1	no	—
122	C₂×C₄⋊C₄	Dic₄₃	1	2	yes	—
123	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
124	C₂×C₂²⋊C₄	Dic₄₃	1	2	yes	—
125	#101	—	86	4	no	—
126	C₂²⋊C₈	—	86	1	no	—
127	C₂²⋊C₈	—	86	1	no	—
128	C₂⁴	C₂×Dic₄₃	1	1	yes	characteristic
129	C₂×Q₈	—	2	2	no	—
130	C₂×Q₈	—	2	2	no	—
131	C₂×Q₈	—	2	1	no	—
132	C₂×Q₈	—	2	1	no	—
133	C₂²×C₄	—	2	2	no	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	C₂×Dic₄₃	1	2	yes	—
136	C₂²×C₄	C₂×Dic₄₃	1	1	yes	characteristic
137	C₂²×C₄	C₂×Dic₄₃	1	2	yes	—
138	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
139	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
140	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
141	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
142	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
143	M₁₆	—	43	4	no	—
144	M₁₆	—	86	4	no	—
145	M₁₆	—	86	2	no	—
146	M₁₆	—	86	2	no	—
147	M₁₆	—	86	2	no	—
148	C₂×C₈	—	86	4	no	—
149	C₂×C₈	—	86	2	no	—
150	C₂×C₈	—	86	2	no	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	—	2	1	no	—
154	C₄⋊C₄	—	2	1	no	—
155	C₂²⋊C₄	—	2	2	no	—
156	C₂²⋊C₄	—	2	2	no	—
157	C₂³	—	2	2	no	—
158	C₂³	—	2	2	no	—
159	C₂³	—	4	1	no	—
160	C₂³	C₂²⋊Dic₄₃	1	1	yes	characteristic
161	C₂³	C₂²⋊Dic₄₃	1	1	yes	characteristic
162	C₂³	C₂²×Dic₄₃	1	1	yes	characteristic
163	Q₈	—	4	1	no	—
164	Q₈	—	4	2	no	—
165	Q₈	—	4	1	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
178	C₂×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
179	C₂×C₄	D₄×D₄₃	1	2	yes	—
180	C₂×C₄	#10621900	1	2	yes	—
181	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
182	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
183	C₈	—	86	4	no	—
184	C₈	—	172	2	no	—
185	C₂²	—	2	1	no	—
186	C₂²	—	2	1	no	—
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	2	1	no	—
192	C₂²	—	4	4	no	—
193	C₂²	#13063244	1	2	yes	complemented
194	C₂²	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
195	C₂²	D₄×Dic₄₃	1	2	yes	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	4	2	no	—
199	C₄	—	4	2	no	—
200	C₄	#13063293	1	1	yes	characteristic
201	C₄	#13063297	1	2	yes	—
202	C₄	#13063306	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	—	2	2	no	—
205	C₂	—	4	1	no	—
206	C₂	#14582779	1	1	yes	characteristic
207	C₂	C₂×#13063244	1	1	yes	characteristic
208	C₂	#14582883	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#10621877

C₂

yes

maximal, characteristic

C₂×#13063244

C₂

yes

maximal, complemented

#14582863

C₂

yes

maximal, characteristic

#14582668

C₂

yes

maximal, complemented

C₂×C₂²⋊Q₈×C₄₃

C₂

yes

maximal, characteristic

C₂×#10621877

—

C₂×#10621877

—

C₂×#10621877

C₂²

yes

—

C₂×#10621877

C₂²

yes

—

C₂²×C₄₃⋊C₈

—

#13063244

—

C₂²⋊(C₄₃⋊C₈)

—

C₂²⋊(C₄₃⋊C₈)

—

C₂²×Q₈×C₄₃

C₂²

yes

characteristic

C₂³×C₄×C₄₃

C₂²

yes

characteristic

C₂²⋊Q₈×C₄₃

—

C₂²⋊Q₈×C₄₃

—

C₂²⋊Q₈×C₄₃

—

C₂²⋊Q₈×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₄

yes

—

C₂×C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₄₃

C₄

yes

—

C₂⁴×C₄₃

C₂×C₄

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×C₄

yes

—

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×C₄

yes

—

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

—

—

—

—

D₄

yes

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

C₂²⋊C₄

yes

characteristic

C₂³×C₄₃

C₂²⋊C₄

yes

characteristic

C₂³×C₄₃

C₂²×C₄

yes

characteristic

Q₈×C₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄×C₄₃

#41

yes

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#101

yes

complemented

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₃

C₄×D₄

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#120

yes

characteristic

C₄×C₄₃

#121

yes

—

C₄×C₄₃

yes

—

C₂×C₄₃

—

100

C₂×C₄₃

—

101

C₂×C₄₃

—

102

C₂×C₄₃

#386

yes

characteristic

103

C₂×C₄₃

C₂×#101

yes

characteristic

104

C₂×C₄₃

#413

yes

characteristic

105

C₂²×M₁₆

—

106

C₂×C₂²⋊Q₈

D₄₃

yes

characteristic

107

C₂×#101

—

108

#407

—

109

#343

—

110

C₄₃

#1901

yes

characteristic, complemented, Sylow

111

C₂²×Q₈

C₂×D₄₃

yes

characteristic

112

C₂³×C₄

C₂×D₄₃

yes

characteristic

113

C₂×M₁₆

—

114

C₂×M₁₆

—

115

C₂×M₁₆

—

116

C₂×M₁₆

—

117

C₂²×C₈

—

118

C₂²⋊Q₈

—

119

C₂²⋊Q₈

—

120

C₂²⋊Q₈

—

121

C₂²⋊Q₈

—

122

C₂×C₄⋊C₄

Dic₄₃

yes

—

123

C₂×C₄⋊C₄

C₂×D₄₃

yes

characteristic

124

C₂×C₂²⋊C₄

Dic₄₃

yes

—

125

#101

—

126

C₂²⋊C₈

—

127

C₂²⋊C₈

—

128

C₂⁴

C₂×Dic₄₃

yes

characteristic

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

C₂×Dic₄₃

yes

—

136

C₂²×C₄

C₂×Dic₄₃

yes

characteristic

137

C₂²×C₄

C₂×Dic₄₃

yes

—

138

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

139

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

140

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

141

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

142

C₂²×C₄

C₂²×D₄₃

yes

characteristic

143

M₁₆

—

144

M₁₆

—

145

M₁₆

—

146

M₁₆

—

147

M₁₆

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₂²⋊C₄

—

156

C₂²⋊C₄

—

157

C₂³

—

158

C₂³

—

159

C₂³

—

160

C₂³

C₂²⋊Dic₄₃

yes

characteristic

161

C₂³

C₂²⋊Dic₄₃

yes

characteristic

162

C₂³

C₂²×Dic₄₃

yes

characteristic

163

Q₈

—

164

Q₈

—

165

Q₈

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

C₂²⋊Dic₄₃

yes

characteristic

178

C₂×C₄

C₂²⋊Dic₄₃

yes

characteristic

179

C₂×C₄

D₄×D₄₃

yes

—

180

C₂×C₄

#10621900

yes

—

181

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

182

C₂×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

183

C₈

—

184

C₈

—

172

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

192

C₂²

—

193

C₂²

#13063244

yes

complemented

194

C₂²

C₂×C₂²⋊Dic₄₃

yes

characteristic

195

C₂²

D₄×Dic₄₃

yes

—

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

#13063293

yes

characteristic

201

C₄

#13063297

yes

—

202

C₄

#13063306

yes

characteristic

203

C₂

—

204

C₂

—

205

C₂

—

206

C₂

#14582779

yes

characteristic

207

C₂

C₂×#13063244

yes

characteristic

208

C₂

#14582883

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₄₃
commutator	C₂³×C₄₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₃
Fitting	C₂×C₂²⋊Q₈×C₄₃	C₂
socle	C₂²×C₄₃	C₂×C₂²⋊C₄
hypercenter	C₂×C₂²⋊Q₈	D₄₃
hypocenter	C₄₃	#1901
LCS	G ▷ C₂³×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×C₂²⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₄₃

commutator

C₂³×C₄₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₃

Fitting

C₂×C₂²⋊Q₈×C₄₃

C₂

socle

C₂²×C₄₃

C₂×C₂²⋊C₄

hypercenter

C₂×C₂²⋊Q₈

D₄₃

hypocenter

C₄₃

#1901

LCS

G ▷ C₂³×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×C₂²⋊Q₈

N	H ≅ G/N	type
2. C₂×#13063244	C₂	semidirect
4. #14582668	C₂	semidirect
88. C₂²×C₄₃	#101	semidirect
110. C₄₃	#1901	semidirect
193. C₂²	#13063244	semidirect

H ≅ G/N

type

2. C₂×#13063244

C₂

semidirect

4. #14582668

C₂

semidirect

88. C₂²×C₄₃

#101

semidirect

110. C₄₃

#1901

semidirect

193. C₂²

#13063244

semidirect

Subgroups — #19135064

Conjugacy classes of subgroups

Complements