Group #19139949 — subgroups

Subgroups — #19139949

Order 5504.

564 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 169 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×Dic₈₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#14582835	C₂	1	2	yes	maximal, complemented
3	#14582811	C₂	1	2	yes	maximal, complemented
4	#14582802	C₂	1	2	yes	maximal
5	#14582795	C₂	1	2	yes	maximal, complemented
6	#14582772	C₂	1	4	yes	maximal, complemented
7	#14582765	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₄×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
10	C₄×Dic₈₆	—	2	4	no	—
11	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
13	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
14	C₂×Dic₄₃⋊C₄	C₂²	1	4	yes	—
15	C₂×C₂²⋊Dic₄₃	C₂²	1	4	yes	—
16	C₂×C₂²⋊Dic₄₃	C₂²	1	4	yes	—
17	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
18	C₄₃⋊#95	C₂²	1	4	yes	—
19	C₄₃⋊#95	C₂²	1	2	yes	—
20	C₄₃⋊#95	C₂²	1	2	yes	—
21	C₂²×D₄×C₄₃	C₂²	1	1	yes	characteristic
22	C₂³×C₄×C₄₃	C₂²	1	2	yes	—
23	C₄×D₄×C₄₃	—	2	4	no	—
24	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
26	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
27	C₂⁴×C₄₃	C₂³	1	2	yes	—
28	C₂×D₄×C₄₃	—	2	2	no	—
29	C₂×D₄×C₄₃	—	2	2	no	—
30	C₂×D₄×C₄₃	Q₈	1	4	yes	complemented
31	C₂²×C₄×C₄₃	—	2	4	no	—
32	C₂²×C₄×C₄₃	—	4	4	no	—
33	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
34	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
37	C₂²×Dic₄₃	C₂³	1	4	yes	—
38	C₂²×Dic₄₃	C₂³	1	2	yes	—
39	C₂²×Dic₄₃	C₂³	1	2	yes	—
40	C₂×Dic₈₆	—	2	2	no	—
41	C₂×Dic₈₆	—	2	4	no	—
42	C₄⋊C₄×C₄₃	—	2	2	no	—
43	C₂²⋊C₄×C₄₃	—	2	4	no	—
44	C₄²×C₄₃	D₄	1	4	yes	—
45	C₂²⋊Dic₄₃	—	2	8	no	—
46	C₂²⋊Dic₄₃	—	4	4	no	—
47	C₄⋊Dic₄₃	—	2	2	no	—
48	C₄⋊Dic₄₃	—	2	4	no	—
49	Dic₄₃⋊C₄	—	2	4	no	—
50	Dic₄₃⋊C₄	—	4	4	no	—
51	C₄×Dic₄₃	—	2	4	no	—
52	C₂³×C₄₃	—	2	2	no	—
53	C₂³×C₄₃	—	2	4	no	—
54	C₂³×C₄₃	—	2	2	no	—
55	C₂³×C₄₃	—	2	4	no	—
56	C₂³×C₄₃	C₂×Q₈	1	4	yes	—
57	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
58	D₄×C₄₃	—	2	8	no	—
59	C₂×C₄×C₄₃	—	2	2	no	—
60	C₂×C₄×C₄₃	—	2	2	no	—
61	C₂×C₄×C₄₃	—	2	2	no	—
62	C₂×C₄×C₄₃	—	2	4	no	—
63	C₂×C₄×C₄₃	—	4	8	no	—
64	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
68	C₂×C₄×C₄₃	#41	1	2	yes	—
69	C₂×Dic₄₃	—	2	2	no	—
70	C₂×Dic₄₃	—	2	2	no	—
71	C₂×Dic₄₃	—	2	2	no	—
72	C₂×Dic₄₃	—	2	4	no	—
73	C₂×Dic₄₃	—	2	4	no	—
74	C₂×Dic₄₃	—	2	2	no	—
75	C₂×Dic₄₃	—	2	4	no	—
76	C₂×Dic₄₃	—	2	2	no	—
77	C₂×Dic₄₃	#41	1	4	yes	—
78	Dic₈₆	—	2	8	no	—
79	C₂²×C₄₃	—	2	4	no	—
80	C₂²×C₄₃	—	2	8	no	—
81	C₂²×C₄₃	—	4	2	no	—
82	C₂²×C₄₃	—	4	2	no	—
83	C₂²×C₄₃	—	4	4	no	—
84	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
85	C₂²×C₄₃	C₂²×Q₈	1	1	yes	characteristic
86	C₂²×C₄₃	C₂×#41	1	2	yes	—
87	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
88	C₂²×C₄₃	H(5,𝔽₂)	1	2	yes	—
89	C₄×C₄₃	—	2	2	no	—
90	C₄×C₄₃	—	2	2	no	—
91	C₄×C₄₃	—	4	2	no	—
92	C₄×C₄₃	C₂²⋊Q₈	1	4	yes	—
93	Dic₄₃	—	2	4	no	—
94	Dic₄₃	—	4	2	no	—
95	Dic₄₃	—	4	4	no	—
96	#2580	—	43	1	no	maximal, Sylow
97	C₂×C₄₃	—	4	4	no	—
98	C₂×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₄₃	#525	1	1	yes	characteristic
100	C₂×C₄₃	#535	1	1	yes	characteristic
101	C₂×C₄₃	#554	1	2	yes	—
102	C₂×C₄₃	#559	1	2	yes	—
103	C₂×C₄×Q₈	—	43	1	no	—
104	C₂×C₄×D₄	D₄₃	1	1	yes	characteristic
105	#399	—	43	2	no	—
106	#393	—	43	2	no	—
107	#390	—	43	2	no	—
108	#389	—	43	2	no	—
109	#385	—	43	4	no	—
110	#382	—	43	1	no	—
111	C₄₃	#2580	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₈	—	43	1	no	—
113	C₂²×D₄	C₂×D₄₃	1	1	yes	characteristic
114	C₂³×C₄	C₂×D₄₃	1	2	yes	—
115	C₄×Q₈	—	86	4	no	—
116	C₄×D₄	—	2	4	no	—
117	C₂×C₄⋊C₄	—	43	2	no	—
118	C₂×C₄⋊C₄	—	43	1	no	—
119	C₂×C₄⋊C₄	—	43	2	no	—
120	C₂×C₄⋊C₄	—	43	4	no	—
121	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
122	C₂×C₂²⋊C₄	—	43	4	no	—
123	C₂×C₂²⋊C₄	—	43	4	no	—
124	C₂×C₂²⋊C₄	C₂×D₄₃	1	2	yes	—
125	C₂×C₄²	—	43	2	no	—
126	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
127	#95	—	43	2	no	—
128	#95	—	43	2	no	—
129	#95	—	43	4	no	—
130	C₂⁴	C₂²×D₄₃	1	2	yes	—
131	C₂×Q₈	—	86	4	no	—
132	C₂×Q₈	—	86	2	no	—
133	C₂×D₄	—	2	2	no	—
134	C₂×D₄	—	2	2	no	—
135	C₂×D₄	Dic₈₆	1	4	yes	complemented
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	—	4	4	no	—
138	C₂²×C₄	—	43	2	no	—
139	C₂²×C₄	—	43	4	no	—
140	C₂²×C₄	—	43	2	no	—
141	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
142	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
143	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
144	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	—	86	4	no	—
147	C₄⋊C₄	—	86	4	no	—
148	C₄⋊C₄	—	86	2	no	—
149	C₄⋊C₄	—	172	4	no	—
150	C₂²⋊C₄	—	2	4	no	—
151	C₂²⋊C₄	—	86	8	no	—
152	C₂²⋊C₄	—	172	4	no	—
153	C₄²	—	86	4	no	—
154	C₄²	C₂²⋊D₄₃	1	4	yes	—
155	C₂³	—	2	2	no	—
156	C₂³	—	2	4	no	—
157	C₂³	—	2	2	no	—
158	C₂³	—	2	4	no	—
159	C₂³	C₂×Dic₈₆	1	4	yes	—
160	C₂³	C₂³×D₄₃	1	1	yes	characteristic
161	Q₈	—	86	8	no	—
162	D₄	—	2	8	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	4	8	no	—
168	C₂×C₄	—	43	4	no	—
169	C₂×C₄	—	86	2	no	—
170	C₂×C₄	—	86	2	no	—
171	C₂×C₄	—	86	2	no	—
172	C₂×C₄	—	86	4	no	—
173	C₂×C₄	—	86	4	no	—
174	C₂×C₄	—	86	2	no	—
175	C₂×C₄	—	86	4	no	—
176	C₂×C₄	—	86	2	no	—
177	C₂×C₄	C₂×Dic₈₆	1	2	yes	—
178	C₂×C₄	#10621898	1	2	yes	—
179	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
180	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
181	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
182	C₂²	—	2	8	no	—
183	C₂²	—	2	4	no	—
184	C₂²	—	4	4	no	—
185	C₂²	—	4	2	no	—
186	C₂²	—	4	2	no	—
187	C₂²	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
188	C₂²	C₂²×Dic₈₆	1	1	yes	characteristic
189	C₂²	#41×D₄₃	1	2	yes	—
190	C₂²	C₂×#10621898	1	1	yes	characteristic
191	C₂²	#13063370	1	2	yes	—
192	C₄	—	2	2	no	—
193	C₄	—	2	2	no	—
194	C₄	—	4	2	no	—
195	C₄	—	86	4	no	—
196	C₄	—	172	4	no	—
197	C₄	—	172	2	no	—
198	C₄	#13063304	1	4	yes	—
199	C₂	—	4	4	no	—
200	C₂	C₂×#13063304	1	1	yes	characteristic
201	C₂	#14583406	1	1	yes	characteristic
202	C₂	#14583466	1	1	yes	characteristic
203	C₂	#14583608	1	2	yes	—
204	C₂	#14583632	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₈₆

C₂

yes

maximal, characteristic, complemented

#14582835

C₂

yes

maximal, complemented

#14582811

C₂

yes

maximal, complemented

#14582802

C₂

yes

maximal

#14582795

C₂

yes

maximal, complemented

#14582772

C₂

yes

maximal, complemented

#14582765

C₂

yes

maximal, characteristic, complemented

C₂×C₄×D₄×C₄₃

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

C₂²

yes

characteristic

C₄×Dic₈₆

—

C₂×C₄⋊Dic₄₃

C₂²

yes

—

C₂×C₄⋊Dic₄₃

C₂²

yes

characteristic

C₂×Dic₄₃⋊C₄

C₂²

yes

—

C₂×Dic₄₃⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₄₃

C₂²

yes

—

C₂×C₂²⋊Dic₄₃

C₂²

yes

—

C₂×C₄×Dic₄₃

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₂²×D₄×C₄₃

C₂²

yes

characteristic

C₂³×C₄×C₄₃

C₂²

yes

—

C₄×D₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₄₃

C₂²

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂⁴×C₄₃

C₂³

yes

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

Q₈

yes

complemented

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

C₂³

yes

characteristic

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₄⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

D₄

yes

—

C₂²⋊Dic₄₃

—

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

C₂×Q₈

yes

—

C₂³×C₄₃

C₂⁴

yes

characteristic

D₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

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

—

C₂×Dic₄₃

—

C₂×Dic₄₃

#41

yes

—

Dic₈₆

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂²×D₄

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

C₂²⋊Q₈

yes

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

#2580

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₄₃

#525

yes

characteristic

100

C₂×C₄₃

#535

yes

characteristic

101

C₂×C₄₃

#554

yes

—

102

C₂×C₄₃

#559

yes

—

103

C₂×C₄×Q₈

—

104

C₂×C₄×D₄

D₄₃

yes

characteristic

105

#399

—

106

#393

—

107

#390

—

108

#389

—

109

#385

—

110

#382

—

111

C₄₃

#2580

yes

characteristic, complemented, Sylow

112

C₂²×Q₈

—

113

C₂²×D₄

C₂×D₄₃

yes

characteristic

114

C₂³×C₄

C₂×D₄₃

yes

—

115

C₄×Q₈

—

116

C₄×D₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

C₂×D₄₃

yes

characteristic

122

C₂×C₂²⋊C₄

—

123

C₂×C₂²⋊C₄

—

124

C₂×C₂²⋊C₄

C₂×D₄₃

yes

—

125

C₂×C₄²

—

126

C₂×C₄²

C₂×D₄₃

yes

characteristic

127

#95

—

128

#95

—

129

#95

—

130

C₂⁴

C₂²×D₄₃

yes

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×D₄

—

134

C₂×D₄

—

135

C₂×D₄

Dic₈₆

yes

complemented

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

C₂²×D₄₃

yes

—

142

C₂²×C₄

C₂²×D₄₃

yes

characteristic

143

C₂²×C₄

C₂²×D₄₃

yes

characteristic

144

C₂²×C₄

C₂²×D₄₃

yes

characteristic

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

172

—

150

C₂²⋊C₄

—

151

C₂²⋊C₄

—

152

C₂²⋊C₄

—

172

—

153

C₄²

—

154

C₄²

C₂²⋊D₄₃

yes

—

155

C₂³

—

156

C₂³

—

157

C₂³

—

158

C₂³

—

159

C₂³

C₂×Dic₈₆

yes

—

160

C₂³

C₂³×D₄₃

yes

characteristic

161

Q₈

—

162

D₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

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

C₂×Dic₈₆

yes

—

178

C₂×C₄

#10621898

yes

—

179

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

180

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

181

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

182

C₂²

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

C₂²×C₂²⋊D₄₃

yes

characteristic

188

C₂²

C₂²×Dic₈₆

yes

characteristic

189

C₂²

#41×D₄₃

yes

—

190

C₂²

C₂×#10621898

yes

characteristic

191

C₂²

#13063370

yes

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

172

—

197

C₄

—

172

—

198

C₄

#13063304

yes

—

199

C₂

—

200

C₂

C₂×#13063304

yes

characteristic

201

C₂

#14583406

yes

characteristic

202

C₂

#14583466

yes

characteristic

203

C₂

#14583608

yes

—

204

C₂

#14583632

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator, socle	C₂³×C₄₃	C₂⁴
Fitting	C₂×C₄×D₄×C₄₃	C₂
hypercenter	C₂×C₄×D₄	D₄₃
hypocenter	C₄₃	#2580
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₃

commutator, socle

C₂³×C₄₃

C₂⁴

Fitting

C₂×C₄×D₄×C₄₃

C₂

hypercenter

C₂×C₄×D₄

D₄₃

C₄₃

G ▷ C₂³×C₄₃ ▷ C₄₃

C₁ ◁ C₂³ ◁ C₂×C₄×D₄

N	H ≅ G/N	type
1. C₂×C₄×Dic₈₆	C₂	semidirect
2. #14582835	C₂	semidirect
3. #14582811	C₂	semidirect
5. #14582795	C₂	semidirect
6. #14582772	C₂	semidirect
7. #14582765	C₂	semidirect
30. C₂×D₄×C₄₃	Q₈	semidirect
111. C₄₃	#2580	semidirect
135. C₂×D₄	Dic₈₆	semidirect

H ≅ G/N

type

1. C₂×C₄×Dic₈₆

C₂

semidirect

2. #14582835

C₂

semidirect

3. #14582811

C₂

semidirect

5. #14582795

C₂

semidirect

6. #14582772

C₂

semidirect

7. #14582765

C₂

semidirect

30. C₂×D₄×C₄₃

Q₈

semidirect

111. C₄₃

#2580

semidirect

135. C₂×D₄

Dic₈₆

semidirect

Subgroups — #19139949

Conjugacy classes of subgroups

Complements