Group #19137805 — subgroups

Subgroups — C₂×#14582819 #19137805

Order 5504.

908 conjugacy classes of subgroups; only the 277 normal subgroups are shown.

normal subgroups only

This group has 908 conjugacy classes of subgroups — too many to draw. Only its 277 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	C₂³×Dic₈₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×Dic₄₃⋊C₄	C₂	1	2	yes	maximal
3	#14582819	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄₃⋊#95	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₃⋊#95	C₂	1	2	yes	maximal
6	C₂²×C₄⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₈₆	—	2	4	no	—
8	C₂²×Dic₈₆	—	2	8	no	—
9	C₂²×Dic₈₆	C₂²	1	4	yes	—
10	C₂³×Dic₄₃	C₂²	1	2	yes	—
11	C₂³×Dic₄₃	C₂²	1	2	yes	—
12	C₂×Dic₄₃⋊C₄	—	2	8	no	—
13	C₂×Dic₄₃⋊C₄	C₂²	1	8	yes	—
14	C₄₃⋊#95	C₂²	1	8	yes	—
15	C₄₃⋊#95	C₂²	1	4	yes	—
16	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic
18	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
20	C₂×C₄⋊C₄×C₄₃	C₂²	1	4	yes	—
21	C₂⁴×C₄₃	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₄₃	—	2	2	no	—
23	C₂²×C₄×C₄₃	—	2	4	no	—
24	C₂²×C₄×C₄₃	—	2	2	no	—
25	C₂²×C₄×C₄₃	—	2	4	no	—
26	C₂²×C₄×C₄₃	—	2	2	no	—
27	C₂²×C₄×C₄₃	D₄	1	4	yes	—
28	C₂²×C₄×C₄₃	Q₈	1	4	yes	complemented
29	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
30	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
31	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
32	C₂²×Dic₄₃	—	2	4	no	—
33	C₂²×Dic₄₃	—	2	4	no	—
34	C₂²×Dic₄₃	—	2	4	no	—
35	C₂²×Dic₄₃	—	2	4	no	—
36	C₂²×Dic₄₃	D₄	1	8	yes	—
37	C₂²×Dic₄₃	Q₈	1	8	yes	complemented
38	C₂²×Dic₄₃	C₂³	1	4	yes	—
39	C₂²×Dic₄₃	C₂³	1	4	yes	—
40	C₂×Dic₈₆	—	2	32	no	—
41	C₂×Dic₈₆	—	2	8	no	—
42	C₂×Dic₈₆	—	2	16	no	—
43	C₄⋊C₄×C₄₃	—	2	8	no	—
44	Dic₄₃⋊C₄	—	2	16	no	—
45	C₂³×C₄₃	C₂×D₄	1	4	yes	—
46	C₂³×C₄₃	C₂×D₄	1	2	yes	—
47	C₂³×C₄₃	C₂×Q₈	1	2	yes	—
48	C₂³×C₄₃	C₂×Q₈	1	4	yes	—
49	C₂³×C₄₃	#41	1	2	yes	—
50	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
51	C₂×C₄×C₄₃	—	2	8	no	—
52	C₂×C₄×C₄₃	—	2	8	no	—
53	C₂×C₄×C₄₃	—	2	2	no	—
54	C₂×C₄×C₄₃	—	2	4	no	—
55	C₂×C₄×C₄₃	—	2	2	no	—
56	C₂×C₄×C₄₃	—	2	4	no	—
57	C₂×C₄×C₄₃	—	2	2	no	—
58	C₂×C₄×C₄₃	—	4	4	no	—
59	C₂×C₄×C₄₃	C₂×D₄	1	4	yes	—
60	C₂×C₄×C₄₃	C₂×Q₈	1	4	yes	complemented
61	C₂×Dic₄₃	—	2	16	no	—
62	C₂×Dic₄₃	—	2	16	no	—
63	C₂×Dic₄₃	—	2	4	no	—
64	C₂×Dic₄₃	—	2	4	no	—
65	C₂×Dic₄₃	—	2	4	no	—
66	C₂×Dic₄₃	—	2	4	no	—
67	C₂×Dic₄₃	C₂×D₄	1	8	yes	—
68	C₂×Dic₄₃	C₂×Q₈	1	8	yes	complemented
69	Dic₈₆	—	2	32	no	—
70	C₂²×C₄₃	#120	1	4	yes	—
71	C₂²×C₄₃	C₂²⋊Q₈	1	8	yes	—
72	C₂²×C₄₃	C₂²⋊Q₈	1	4	yes	—
73	C₂²×C₄₃	C₄⋊Q₈	1	4	yes	—
74	C₂²×C₄₃	C₄⋊Q₈	1	8	yes	—
75	C₂²×C₄₃	C₂²×D₄	1	2	yes	—
76	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
77	C₂²×C₄₃	C₂²×Q₈	1	2	yes	—
78	C₂²×C₄₃	C₂²×Q₈	1	1	yes	characteristic
79	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
80	C₄×C₄₃	—	2	4	no	—
81	C₄×C₄₃	—	2	4	no	—
82	C₄×C₄₃	—	4	2	no	—
83	Dic₄₃	—	2	8	no	—
84	Dic₄₃	—	2	8	no	—
85	C₂×#395	—	43	1	no	maximal, Sylow
86	C₂×C₄₃	#395	1	8	yes	complemented
87	C₂×C₄₃	C₂×#120	1	1	yes	characteristic
88	C₂×C₄₃	C₂×C₂²⋊Q₈	1	2	yes	—
89	C₂×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
90	C₂×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
91	C₂×C₄₃	C₂×C₄⋊Q₈	1	2	yes	—
92	C₂³×Q₈	—	43	1	no	—
93	C₂²×C₄⋊C₄	—	43	2	no	—
94	C₂²×C₄⋊C₄	D₄₃	1	1	yes	characteristic
95	#395	—	43	8	no	—
96	C₂×#95	—	43	2	no	—
97	C₂×#95	—	43	1	no	—
98	C₄₃	C₂×#395	1	1	yes	characteristic, complemented, Sylow
99	C₂²×Q₈	—	43	4	no	—
100	C₂²×Q₈	—	86	8	no	—
101	C₂²×Q₈	—	86	4	no	—
102	C₂³×C₄	—	43	2	no	—
103	C₂³×C₄	—	43	2	no	—
104	C₂³×C₄	C₂×D₄₃	1	1	yes	characteristic
105	C₂³×C₄	C₂×D₄₃	1	1	yes	characteristic
106	C₂³×C₄	C₂×D₄₃	1	1	yes	characteristic
107	C₂×C₄⋊C₄	—	2	4	no	—
108	C₂×C₄⋊C₄	—	43	8	no	—
109	C₂×C₄⋊C₄	—	86	8	no	—
110	C₂×C₄⋊C₄	C₂×D₄₃	1	4	yes	—
111	#95	—	43	4	no	—
112	#95	—	43	8	no	—
113	C₂⁴	C₂²×D₄₃	1	1	yes	characteristic
114	C₂×Q₈	—	86	16	no	—
115	C₂×Q₈	—	86	32	no	—
116	C₂×Q₈	—	86	8	no	—
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	—	2	4	no	—
119	C₂²×C₄	—	2	2	no	—
120	C₂²×C₄	—	2	4	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	43	8	no	—
123	C₂²×C₄	—	43	8	no	—
124	C₂²×C₄	—	43	4	no	—
125	C₂²×C₄	—	43	4	no	—
126	C₂²×C₄	—	86	4	no	—
127	C₂²×C₄	—	86	4	no	—
128	C₂²×C₄	—	86	4	no	—
129	C₂²×C₄	—	86	4	no	—
130	C₂²×C₄	Dic₈₆	1	4	yes	complemented
131	C₂²×C₄	C₂²⋊D₄₃	1	4	yes	—
132	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
133	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
134	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
135	C₄⋊C₄	—	2	8	no	—
136	C₄⋊C₄	—	86	16	no	—
137	C₂³	C₂×Dic₈₆	1	2	yes	—
138	C₂³	#10621898	1	2	yes	—
139	C₂³	D₄×D₄₃	1	4	yes	—
140	C₂³	Q₈×D₄₃	1	4	yes	—
141	C₂³	C₂×C₂²⋊D₄₃	1	2	yes	—
142	C₂³	C₂³×D₄₃	1	1	yes	characteristic
143	Q₈	—	86	32	no	—
144	C₂×C₄	—	2	8	no	—
145	C₂×C₄	—	2	8	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	4	4	no	—
152	C₂×C₄	—	43	8	no	—
153	C₂×C₄	—	43	8	no	—
154	C₂×C₄	—	86	16	no	—
155	C₂×C₄	—	86	16	no	—
156	C₂×C₄	—	86	4	no	—
157	C₂×C₄	—	86	4	no	—
158	C₂×C₄	—	86	4	no	—
159	C₂×C₄	—	86	4	no	—
160	C₂×C₄	C₂×Dic₈₆	1	4	yes	complemented
161	C₂×C₄	C₂×C₂²⋊D₄₃	1	4	yes	—
162	C₂²	#13063293	1	4	yes	—
163	C₂²	#13063303	1	8	yes	—
164	C₂²	#13063304	1	4	yes	—
165	C₂²	#13063323	1	4	yes	—
166	C₂²	#13063324	1	8	yes	—
167	C₂²	C₂×D₄×D₄₃	1	2	yes	—
168	C₂²	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
169	C₂²	C₂×Q₈×D₄₃	1	2	yes	—
170	C₂²	C₂²×Dic₈₆	1	1	yes	characteristic
171	C₂²	C₂×#10621898	1	1	yes	characteristic
172	C₄	—	2	4	no	—
173	C₄	—	2	4	no	—
174	C₄	—	4	2	no	—
175	C₄	—	86	8	no	—
176	C₄	—	86	8	no	—
177	C₂	#14582819	1	8	yes	direct product factor
178	C₂	C₂×#13063293	1	1	yes	characteristic
179	C₂	C₂×#13063303	1	2	yes	—
180	C₂	C₂×#13063304	1	1	yes	characteristic
181	C₂	C₂×#13063323	1	1	yes	characteristic
182	C₂	C₂×#13063324	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×Dic₈₆

C₂

yes

maximal, characteristic

C₂²×Dic₄₃⋊C₄

C₂

yes

maximal

#14582819

C₂

yes

maximal, direct product factor

C₂×C₄₃⋊#95

C₂

yes

maximal, characteristic

C₂×C₄₃⋊#95

C₂

yes

maximal

C₂²×C₄⋊C₄×C₄₃

C₂

yes

maximal, characteristic

C₂²×Dic₈₆

—

C₂²×Dic₈₆

—

C₂²×Dic₈₆

C₂²

yes

—

C₂³×Dic₄₃

C₂²

yes

—

C₂³×Dic₄₃

C₂²

yes

—

C₂×Dic₄₃⋊C₄

—

C₂×Dic₄₃⋊C₄

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₄₃⋊#95

C₂²

yes

—

C₂³×C₄×C₄₃

C₂²

yes

characteristic

C₂³×C₄×C₄₃

C₂²

yes

characteristic

C₂³×C₄×C₄₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₂²

yes

—

C₂⁴×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

Q₈

yes

complemented

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂³

yes

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

D₄

yes

—

C₂²×Dic₄₃

Q₈

yes

complemented

C₂²×Dic₄₃

C₂³

yes

—

C₂²×Dic₄₃

C₂³

yes

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₄⋊C₄×C₄₃

—

Dic₄₃⋊C₄

—

C₂³×C₄₃

C₂×D₄

yes

—

C₂³×C₄₃

C₂×D₄

yes

—

C₂³×C₄₃

C₂×Q₈

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

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×Q₈

yes

complemented

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

C₂×D₄

yes

—

C₂×Dic₄₃

C₂×Q₈

yes

complemented

Dic₈₆

—

C₂²×C₄₃

#120

yes

—

C₂²×C₄₃

C₂²⋊Q₈

yes

—

C₂²×C₄₃

C₂²⋊Q₈

yes

—

C₂²×C₄₃

C₄⋊Q₈

yes

—

C₂²×C₄₃

C₄⋊Q₈

yes

—

C₂²×C₄₃

C₂²×D₄

yes

—

C₂²×C₄₃

C₂²×D₄

yes

characteristic

C₂²×C₄₃

C₂²×Q₈

yes

—

C₂²×C₄₃

C₂²×Q₈

yes

characteristic

C₂²×C₄₃

C₂×#41

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

Dic₄₃

—

Dic₄₃

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

C₂×C₂²⋊Q₈

yes

—

C₂×C₄₃

C₂×C₂²⋊Q₈

yes

characteristic

C₂×C₄₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₄₃

C₂×C₄⋊Q₈

yes

—

C₂³×Q₈

—

C₂²×C₄⋊C₄

—

C₂²×C₄⋊C₄

D₄₃

yes

characteristic

#395

—

C₂×#95

—

C₂×#95

—

C₄₃

C₂×#395

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

100

C₂²×Q₈

—

101

C₂²×Q₈

—

102

C₂³×C₄

—

103

C₂³×C₄

—

104

C₂³×C₄

C₂×D₄₃

yes

characteristic

105

C₂³×C₄

C₂×D₄₃

yes

characteristic

106

C₂³×C₄

C₂×D₄₃

yes

characteristic

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

C₂×D₄₃

yes

—

111

#95

—

112

#95

—

113

C₂⁴

C₂²×D₄₃

yes

characteristic

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

Dic₈₆

yes

complemented

131

C₂²×C₄

C₂²⋊D₄₃

yes

—

132

C₂²×C₄

C₂²×D₄₃

yes

—

133

C₂²×C₄

C₂²×D₄₃

yes

—

134

C₂²×C₄

C₂²×D₄₃

yes

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₂³

C₂×Dic₈₆

yes

—

138

C₂³

#10621898

yes

—

139

C₂³

D₄×D₄₃

yes

—

140

C₂³

Q₈×D₄₃

yes

—

141

C₂³

C₂×C₂²⋊D₄₃

yes

—

142

C₂³

C₂³×D₄₃

yes

characteristic

143

Q₈

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

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

C₂×Dic₈₆

yes

complemented

161

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

162

C₂²

#13063293

yes

—

163

C₂²

#13063303

yes

—

164

C₂²

#13063304

yes

—

165

C₂²

#13063323

yes

—

166

C₂²

#13063324

yes

—

167

C₂²

C₂×D₄×D₄₃

yes

—

168

C₂²

C₂²×C₂²⋊D₄₃

yes

characteristic

169

C₂²

C₂×Q₈×D₄₃

yes

—

170

C₂²

C₂²×Dic₈₆

yes

characteristic

171

C₂²

C₂×#10621898

yes

characteristic

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₂

#14582819

yes

direct product factor

178

C₂

C₂×#13063293

yes

characteristic

179

C₂

C₂×#13063303

yes

—

180

C₂

C₂×#13063304

yes

characteristic

181

C₂

C₂×#13063323

yes

characteristic

182

C₂

C₂×#13063324

yes

—

	subgroup	G/H
center	C₂⁴	C₂²×D₄₃
commutator	C₂³×C₄₃	C₂⁴
Frattini	C₂³	C₂³×D₄₃
Fitting	C₂²×C₄⋊C₄×C₄₃	C₂
socle	C₂⁴×C₄₃	C₂³
hypercenter	C₂²×C₄⋊C₄	D₄₃
hypocenter	C₄₃	C₂×#395
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center

C₂⁴

C₂²×D₄₃

commutator

C₂³×C₄₃

C₂⁴

Frattini

C₂³

C₂³×D₄₃

Fitting

C₂²×C₄⋊C₄×C₄₃

C₂

socle

C₂⁴×C₄₃

C₂³

hypercenter

C₂²×C₄⋊C₄

D₄₃

C₄₃

G ▷ C₂³×C₄₃ ▷ C₄₃

C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
3. #14582819	C₂	direct (N × H)
28. C₂²×C₄×C₄₃	Q₈	semidirect
37. C₂²×Dic₄₃	Q₈	semidirect
60. C₂×C₄×C₄₃	C₂×Q₈	semidirect
68. C₂×Dic₄₃	C₂×Q₈	semidirect
86. C₂×C₄₃	#395	semidirect
98. C₄₃	C₂×#395	semidirect
130. C₂²×C₄	Dic₈₆	semidirect
160. C₂×C₄	C₂×Dic₈₆	semidirect
177. C₂	#14582819	direct (N × H)

H ≅ G/N

type

3. #14582819

C₂

direct (N × H)

28. C₂²×C₄×C₄₃

Q₈

semidirect

37. C₂²×Dic₄₃

Q₈

semidirect

60. C₂×C₄×C₄₃

C₂×Q₈

semidirect

68. C₂×Dic₄₃

C₂×Q₈

semidirect

86. C₂×C₄₃

#395

semidirect

98. C₄₃

C₂×#395

semidirect

130. C₂²×C₄

Dic₈₆

semidirect

160. C₂×C₄

C₂×Dic₈₆

semidirect

177. C₂

#14582819

direct (N × H)

Subgroups — C₂×#14582819 #19137805

Conjugacy classes of subgroups

Complements

Subgroups — C2×#14582819 #19137805

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#14582819 #19137805