Group #19137470 — subgroups

Subgroups — #19137470

Order 5504.

548 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 221 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×C₄⋊Dic₄₃	C₂	1	1	yes	maximal, characteristic
2	#14582797	C₂	1	2	yes	maximal, complemented
3	#14582782	C₂	1	4	yes	maximal, complemented
4	#14582759	C₂	1	1	yes	maximal, characteristic
5	#14582757	C₂	1	2	yes	maximal
6	C₄×C₄⋊Dic₄₃	C₂	1	2	yes	maximal, complemented
7	C₂²⋊C₄×Dic₄₃	C₂	1	2	yes	maximal
8	C₄×C₂²⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂³×Dic₄₃	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₄₃	C₄	1	8	yes	complemented
11	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
15	C₂×Dic₄₃⋊C₄	C₂²	1	4	yes	—
16	C₂×C₂²⋊Dic₄₃	C₂²	1	2	yes	—
17	C₂×C₄×Dic₄₃	C₂²	1	4	yes	—
18	C₄₃⋊#95	C₂²	1	4	yes	—
19	C₄₃⋊#95	C₂²	1	4	yes	—
20	C₄₃⋊#95	C₂²	1	2	yes	—
21	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₄₃	C₂²	1	2	yes	—
25	#95×C₄₃	C₂²	1	2	yes	—
26	C₂⁴×C₄₃	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₄₃	—	2	4	no	—
28	C₂²×C₄×C₄₃	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
30	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
32	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
33	C₂²×Dic₄₃	—	4	4	no	—
34	C₂²×Dic₄₃	C₂×C₄	1	8	yes	—
35	C₂²×Dic₄₃	C₂³	1	4	yes	—
36	C₂²×Dic₄₃	C₂³	1	2	yes	—
37	C₂²×Dic₄₃	C₂³	1	2	yes	—
38	C₂²⋊C₄×C₄₃	—	2	2	no	—
39	C₂²⋊C₄×C₄₃	Q₈	1	4	yes	—
40	C₄²×C₄₃	—	2	4	no	—
41	C₂²⋊Dic₄₃	—	2	4	no	—
42	C₄⋊Dic₄₃	—	2	4	no	—
43	C₄⋊Dic₄₃	C₂×C₄	1	8	yes	—
44	C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
45	C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
46	Dic₄₃⋊C₄	—	4	4	no	—
47	C₄×Dic₄₃	—	4	4	no	—
48	C₂³×C₄₃	—	2	2	no	—
49	C₂³×C₄₃	—	2	2	no	—
50	C₂³×C₄₃	C₂²×C₄	1	2	yes	—
51	C₂³×C₄₃	C₂×Q₈	1	2	yes	—
52	C₂³×C₄₃	#41	1	2	yes	—
53	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
54	C₂×C₄×C₄₃	—	2	2	no	—
55	C₂×C₄×C₄₃	—	2	2	no	—
56	C₂×C₄×C₄₃	—	2	2	no	—
57	C₂×C₄×C₄₃	—	2	8	no	—
58	C₂×C₄×C₄₃	—	2	2	no	—
59	C₂×C₄×C₄₃	—	2	2	no	—
60	C₂×C₄×C₄₃	—	2	2	no	—
61	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₄₃	C₂×Q₈	1	4	yes	—
64	C₂×C₄×C₄₃	#41	1	4	yes	—
65	C₂×Dic₄₃	—	2	4	no	—
66	C₂×Dic₄₃	—	2	4	no	—
67	C₂×Dic₄₃	—	2	4	no	—
68	C₂×Dic₄₃	—	2	4	no	—
69	C₂×Dic₄₃	—	2	4	no	—
70	C₂×Dic₄₃	—	4	8	no	—
71	C₂×Dic₄₃	C₂²×C₄	1	4	yes	—
72	C₂×Dic₄₃	C₂²×C₄	1	4	yes	—
73	C₂²×C₄₃	—	2	2	no	—
74	C₂²×C₄₃	—	2	2	no	—
75	C₂²×C₄₃	—	2	4	no	—
76	C₂²×C₄₃	—	2	4	no	—
77	C₂²×C₄₃	C₄×Q₈	1	4	yes	—
78	C₂²×C₄₃	C₂³×C₄	1	1	yes	characteristic
79	C₂²×C₄₃	C₂²×Q₈	1	1	yes	characteristic
80	C₂²×C₄₃	C₂×#41	1	2	yes	—
81	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
82	C₂²×C₄₃	H(5,𝔽₂)	1	2	yes	—
83	C₄×C₄₃	—	2	2	no	—
84	C₄×C₄₃	—	2	4	no	—
85	C₄×C₄₃	—	4	2	no	—
86	C₄×C₄₃	#117	1	4	yes	—
87	Dic₄₃	—	2	4	no	—
88	Dic₄₃	—	2	4	no	—
89	Dic₄₃	—	4	4	no	—
90	#2345	—	43	1	no	maximal, Sylow
91	C₂×C₄₃	—	2	4	no	—
92	C₂×C₄₃	C₂×#117	1	1	yes	characteristic
93	C₂×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
94	C₂×C₄₃	#518	1	1	yes	characteristic
95	C₂×C₄₃	#553	1	2	yes	—
96	C₂×C₄₃	#554	1	2	yes	—
97	C₂²×C₄⋊C₄	—	43	1	no	—
98	#389	—	43	2	no	—
99	#387	—	43	4	no	—
100	#380	—	43	1	no	—
101	#380	—	43	2	no	—
102	C₄×C₄⋊C₄	—	43	2	no	—
103	C₄×C₂²⋊C₄	—	43	2	no	—
104	C₄×C₂²⋊C₄	D₄₃	1	1	yes	characteristic
105	C₄₃	#2345	1	1	yes	characteristic, complemented, Sylow
106	C₂³×C₄	—	43	2	no	—
107	C₂³×C₄	C₂×D₄₃	1	1	yes	characteristic
108	C₂×C₄⋊C₄	—	43	1	no	—
109	C₂×C₄⋊C₄	—	43	2	no	—
110	C₂×C₄⋊C₄	—	43	1	no	—
111	C₂×C₄⋊C₄	—	43	8	no	—
112	C₂×C₄⋊C₄	—	43	2	no	—
113	C₂×C₄⋊C₄	—	43	4	no	—
114	C₂×C₂²⋊C₄	—	43	2	no	—
115	C₂×C₂²⋊C₄	C₂×D₄₃	1	1	yes	characteristic
116	C₂×C₂²⋊C₄	C₂×D₄₃	1	1	yes	characteristic
117	C₂×C₄²	—	43	4	no	—
118	C₂×C₄²	C₂×D₄₃	1	2	yes	—
119	#95	—	43	4	no	—
120	#95	—	43	4	no	—
121	#95	—	43	2	no	—
122	#95	C₂×D₄₃	1	2	yes	—
123	C₂⁴	C₂²×D₄₃	1	1	yes	characteristic
124	C₂²×C₄	—	2	4	no	—
125	C₂²×C₄	—	43	2	no	—
126	C₂²×C₄	—	43	2	no	—
127	C₂²×C₄	—	43	8	no	—
128	C₂²×C₄	—	43	4	no	—
129	C₂²×C₄	—	172	4	no	—
130	C₂²×C₄	C₄×D₄₃	1	4	yes	—
131	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
132	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
133	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
134	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
135	C₄⋊C₄	—	43	8	no	—
136	C₄⋊C₄	—	43	4	no	—
137	C₄⋊C₄	—	43	4	no	—
138	C₄⋊C₄	—	86	4	no	—
139	C₄⋊C₄	—	172	4	no	—
140	C₂²⋊C₄	—	2	2	no	—
141	C₂²⋊C₄	—	86	4	no	—
142	C₂²⋊C₄	Dic₈₆	1	4	yes	—
143	C₄²	—	2	4	no	—
144	C₄²	—	172	4	no	—
145	C₂³	—	2	2	no	—
146	C₂³	—	2	2	no	—
147	C₂³	C₂×Dic₈₆	1	2	yes	—
148	C₂³	C₂×C₄×D₄₃	1	2	yes	—
149	C₂³	#10621898	1	2	yes	—
150	C₂³	C₂³×D₄₃	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	8	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	43	4	no	—
159	C₂×C₄	—	43	4	no	—
160	C₂×C₄	—	86	4	no	—
161	C₂×C₄	—	86	4	no	—
162	C₂×C₄	—	86	4	no	—
163	C₂×C₄	—	86	4	no	—
164	C₂×C₄	—	86	4	no	—
165	C₂×C₄	—	172	8	no	—
166	C₂×C₄	C₂×Dic₈₆	1	4	yes	—
167	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
168	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
169	C₂×C₄	#10621900	1	4	yes	—
170	C₂²	—	2	4	no	—
171	C₂²	—	2	4	no	—
172	C₂²	—	2	2	no	—
173	C₂²	—	2	2	no	—
174	C₂²	C₄×Dic₈₆	1	4	yes	—
175	C₂²	C₂²×C₄×D₄₃	1	1	yes	characteristic
176	C₂²	C₂²×Dic₈₆	1	1	yes	characteristic
177	C₂²	C₂×#10621900	1	2	yes	—
178	C₂²	C₂×#10621898	1	1	yes	characteristic
179	C₂²	#13063371	1	2	yes	—
180	C₄	—	2	4	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	86	4	no	—
184	C₄	—	86	4	no	—
185	C₄	—	172	4	no	—
186	C₄	#13063281	1	4	yes	—
187	C₂	—	2	4	no	—
188	C₂	C₂×#13063281	1	1	yes	characteristic
189	C₂	C₂×C₄×Dic₈₆	1	1	yes	characteristic
190	C₂	#14583373	1	1	yes	characteristic
191	C₂	#14583595	1	2	yes	—
192	C₂	#14583606	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊Dic₄₃

C₂

yes

maximal, characteristic

#14582797

C₂

yes

maximal, complemented

#14582782

C₂

yes

maximal, complemented

#14582759

C₂

yes

maximal, characteristic

#14582757

C₂

yes

maximal

C₄×C₄⋊Dic₄₃

C₂

yes

maximal, complemented

C₂²⋊C₄×Dic₄₃

C₂

yes

maximal

C₄×C₂²⋊C₄×C₄₃

C₂

yes

maximal, characteristic

C₂³×Dic₄₃

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₄

yes

complemented

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×Dic₄₃⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₄₃

C₂²

yes

—

C₂×C₄×Dic₄₃

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₂³×C₄×C₄₃

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₄₃

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₄₃

C₂²

yes

characteristic

C₂×C₄²×C₄₃

C₂²

yes

—

#95×C₄₃

C₂²

yes

—

C₂⁴×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×C₄

yes

—

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

C₂×C₄

yes

—

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

—

C₂²⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

Q₈

yes

—

C₄²×C₄₃

—

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

C₄⋊Dic₄₃

C₂×C₄

yes

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

C₂²×C₄

yes

—

C₂³×C₄₃

C₂×Q₈

yes

—

C₂³×C₄₃

#41

yes

—

C₂³×C₄₃

C₂⁴

yes

characteristic

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

—

C₂×C₄×C₄₃

C₂²×C₄

yes

—

C₂×C₄×C₄₃

C₂×Q₈

yes

—

C₂×C₄×C₄₃

#41

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂²×C₄

yes

—

C₂×Dic₄₃

C₂²×C₄

yes

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₄×Q₈

yes

—

C₂²×C₄₃

C₂³×C₄

yes

characteristic

C₂²×C₄₃

C₂²×Q₈

yes

characteristic

C₂²×C₄₃

C₂×#41

yes

—

C₂²×C₄₃

C₂×#41

yes

characteristic

C₂²×C₄₃

H(5,𝔽₂)

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#117

yes

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

#2345

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

C₂×#117

yes

characteristic

C₂×C₄₃

C₂×C₄×Q₈

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄₃

#554

yes

—

C₂²×C₄⋊C₄

—

#389

—

#387

—

100

#380

—

101

#380

—

102

C₄×C₄⋊C₄

—

103

C₄×C₂²⋊C₄

—

104

C₄×C₂²⋊C₄

D₄₃

yes

characteristic

105

C₄₃

#2345

yes

characteristic, complemented, Sylow

106

C₂³×C₄

—

107

C₂³×C₄

C₂×D₄₃

yes

characteristic

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

114

C₂×C₂²⋊C₄

—

115

C₂×C₂²⋊C₄

C₂×D₄₃

yes

characteristic

116

C₂×C₂²⋊C₄

C₂×D₄₃

yes

characteristic

117

C₂×C₄²

—

118

C₂×C₄²

C₂×D₄₃

yes

—

119

#95

—

120

#95

—

121

#95

—

122

#95

C₂×D₄₃

yes

—

123

C₂⁴

C₂²×D₄₃

yes

characteristic

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

172

—

130

C₂²×C₄

C₄×D₄₃

yes

—

131

C₂²×C₄

C₂²×D₄₃

yes

—

132

C₂²×C₄

C₂²×D₄₃

yes

—

133

C₂²×C₄

C₂²×D₄₃

yes

characteristic

134

C₂²×C₄

C₂²×D₄₃

yes

characteristic

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

172

—

140

C₂²⋊C₄

—

141

C₂²⋊C₄

—

142

C₂²⋊C₄

Dic₈₆

yes

—

143

C₄²

—

144

C₄²

—

172

—

145

C₂³

—

146

C₂³

—

147

C₂³

C₂×Dic₈₆

yes

—

148

C₂³

C₂×C₄×D₄₃

yes

—

149

C₂³

#10621898

yes

—

150

C₂³

C₂³×D₄₃

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

172

—

166

C₂×C₄

C₂×Dic₈₆

yes

—

167

C₂×C₄

C₂×C₄×D₄₃

yes

—

168

C₂×C₄

C₂×C₄×D₄₃

yes

—

169

C₂×C₄

#10621900

yes

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

C₄×Dic₈₆

yes

—

175

C₂²

C₂²×C₄×D₄₃

yes

characteristic

176

C₂²

C₂²×Dic₈₆

yes

characteristic

177

C₂²

C₂×#10621900

yes

—

178

C₂²

C₂×#10621898

yes

characteristic

179

C₂²

#13063371

yes

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

172

—

186

C₄

#13063281

yes

—

187

C₂

—

188

C₂

C₂×#13063281

yes

characteristic

189

C₂

C₂×C₄×Dic₈₆

yes

characteristic

190

C₂

#14583373

yes

characteristic

191

C₂

#14583595

yes

—

192

C₂

#14583606

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂³×C₄
Fitting	C₄×C₂²⋊C₄×C₄₃	C₂
socle	C₂³×C₄₃	C₂⁴
hypercenter	C₄×C₂²⋊C₄	D₄₃
hypocenter	C₄₃	#2345
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₃

commutator

C₂²×C₄₃

C₂³×C₄

Fitting

C₄×C₂²⋊C₄×C₄₃

C₂

socle

C₂³×C₄₃

C₂⁴

hypercenter

C₄×C₂²⋊C₄

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₄₃

C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄

N	H ≅ G/N	type
2. #14582797	C₂	semidirect
3. #14582782	C₂	semidirect
6. C₄×C₄⋊Dic₄₃	C₂	semidirect
10. C₂×C₄⋊Dic₄₃	C₄	semidirect
105. C₄₃	#2345	semidirect

H ≅ G/N

type

2. #14582797

C₂

semidirect

3. #14582782

C₂

semidirect

6. C₄×C₄⋊Dic₄₃

C₂

semidirect

10. C₂×C₄⋊Dic₄₃

C₄

semidirect

105. C₄₃

#2345

semidirect

Subgroups — #19137470

Conjugacy classes of subgroups

Complements