Group #14568993 — subgroups

Subgroups — #14568993

Order 2432.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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×D₇₆	C₂	1	1	yes	maximal, characteristic
2	#11926479	C₂	1	2	yes	maximal, complemented
3	#11926455	C₂	1	2	yes	maximal, complemented
4	#11926446	C₂	1	2	yes	maximal
5	#11926442	C₂	1	2	yes	maximal, complemented
6	#11926415	C₂	1	4	yes	maximal, complemented
7	#11926410	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×Q₈×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₈×C₁₉	C₂²	1	1	yes	characteristic
10	C₂²×D₇₆	C₂²	1	1	yes	characteristic
11	C₂²×C₄×D₁₉	C₂²	1	2	yes	—
12	C₄×Q₈×C₁₉	—	2	4	no	—
13	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₁₉	C₂²	1	2	yes	—
16	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	4	yes	—
18	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
19	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	4	yes	—
20	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
21	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
22	C₂×Dic₁₉⋊C₄	C₂²	1	4	yes	—
23	C₄×D₇₆	—	2	4	no	—
24	C₁₉⋊#95	C₂²	1	2	yes	—
25	C₁₉⋊#95	C₂²	1	2	yes	—
26	C₁₉⋊#95	C₂²	1	4	yes	—
27	C₂³×D₁₉	C₂³	1	2	yes	—
28	C₂×Q₈×C₁₉	—	2	2	no	—
29	C₂×Q₈×C₁₉	—	2	4	no	—
30	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
31	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
34	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
35	C₂²×Dic₁₉	C₂³	1	2	yes	—
36	C₂²×Dic₁₉	C₂³	1	4	yes	—
37	C₂×D₇₆	—	2	2	no	—
38	C₂×D₇₆	—	2	2	no	—
39	C₂×D₇₆	Q₈	1	4	yes	complemented
40	C₂×C₄×D₁₉	—	2	4	no	—
41	C₂×C₄×D₁₉	—	4	4	no	—
42	C₄⋊C₄×C₁₉	—	2	4	no	—
43	C₄⋊C₄×C₁₉	—	2	2	no	—
44	C₄²×C₁₉	—	2	4	no	—
45	C₄²×C₁₉	D₄	1	4	yes	—
46	(C₂×D₁₉)⋊C₄	—	2	4	no	—
47	(C₂×D₁₉)⋊C₄	—	2	8	no	—
48	(C₂×D₁₉)⋊C₄	—	4	4	no	—
49	C₄⋊Dic₁₉	—	2	2	no	—
50	C₄⋊Dic₁₉	—	2	4	no	—
51	Dic₁₉⋊C₄	—	4	4	no	—
52	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
53	C₂²×D₁₉	—	2	2	no	—
54	C₂²×D₁₉	—	2	4	no	—
55	C₂²×D₁₉	—	2	2	no	—
56	C₂²×D₁₉	—	2	4	no	—
57	C₂²×D₁₉	C₂×Q₈	1	4	yes	—
58	Q₈×C₁₉	—	2	8	no	—
59	C₂×C₄×C₁₉	—	2	2	no	—
60	C₂×C₄×C₁₉	—	2	2	no	—
61	C₂×C₄×C₁₉	—	2	2	no	—
62	C₂×C₄×C₁₉	—	2	2	no	—
63	C₂×C₄×C₁₉	—	2	2	no	—
64	C₂×C₄×C₁₉	—	2	2	no	—
65	C₂×C₄×C₁₉	—	2	2	no	—
66	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₁₉	C₂×Q₈	1	2	yes	—
70	C₂×C₄×C₁₉	#41	1	4	yes	—
71	C₂×C₄×C₁₉	#41	1	2	yes	—
72	C₂×Dic₁₉	—	2	2	no	—
73	C₂×Dic₁₉	—	2	4	no	—
74	C₂×Dic₁₉	—	2	4	no	—
75	C₂×Dic₁₉	—	2	4	no	—
76	C₂×Dic₁₉	—	2	4	no	—
77	D₇₆	—	2	8	no	—
78	C₄×D₁₉	—	4	8	no	—
79	#2580	—	19	1	no	maximal, Sylow
80	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
81	C₂²×C₁₉	C₂²×Q₈	1	1	yes	characteristic
82	C₂²×C₁₉	C₂×#41	1	2	yes	—
83	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
84	C₂²×C₁₉	H(5,𝔽₂)	1	2	yes	—
85	C₂×D₁₉	—	2	4	no	—
86	C₂×D₁₉	—	2	8	no	—
87	C₂×D₁₉	—	4	2	no	—
88	C₂×D₁₉	—	4	2	no	—
89	C₂×D₁₉	—	4	4	no	—
90	C₄×C₁₉	—	2	2	no	—
91	C₄×C₁₉	—	2	4	no	—
92	C₄×C₁₉	—	2	2	no	—
93	C₄×C₁₉	—	4	2	no	—
94	C₄×C₁₉	C₂²⋊Q₈	1	4	yes	—
95	Dic₁₉	—	4	2	no	—
96	Dic₁₉	—	4	4	no	—
97	C₂×C₄×Q₈	D₁₉	1	1	yes	characteristic, complemented
98	C₂×C₄×D₄	—	19	1	no	—
99	#399	—	19	2	no	—
100	#393	—	19	2	no	—
101	#390	—	19	2	no	—
102	#389	—	19	2	no	—
103	#385	—	19	4	no	—
104	#382	—	19	1	no	—
105	C₂×C₁₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₁₉	#525	1	1	yes	characteristic
107	C₂×C₁₉	#535	1	1	yes	characteristic
108	C₂×C₁₉	#554	1	2	yes	—
109	C₂×C₁₉	#559	1	2	yes	—
110	D₁₉	—	4	4	no	—
111	C₂²×Q₈	C₂×D₁₉	1	1	yes	characteristic
112	C₂²×D₄	—	19	1	no	—
113	C₂³×C₄	—	19	2	no	—
114	C₄×Q₈	—	2	4	no	—
115	C₄×D₄	—	38	4	no	—
116	C₂×C₄⋊C₄	—	19	1	no	—
117	C₂×C₄⋊C₄	—	19	2	no	—
118	C₂×C₄⋊C₄	—	19	4	no	—
119	C₂×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
120	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
121	C₂×C₂²⋊C₄	—	19	2	no	—
122	C₂×C₂²⋊C₄	—	19	4	no	—
123	C₂×C₂²⋊C₄	—	19	4	no	—
124	C₂×C₄²	C₂×D₁₉	1	2	yes	—
125	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
126	#95	—	19	2	no	—
127	#95	—	19	2	no	—
128	#95	—	19	4	no	—
129	C₁₉	#2580	1	1	yes	characteristic, complemented, Sylow
130	C₂⁴	—	19	2	no	—
131	C₂×Q₈	—	2	4	no	—
132	C₂×Q₈	—	2	2	no	—
133	C₂×D₄	—	19	4	no	—
134	C₂×D₄	—	38	2	no	—
135	C₂×D₄	—	38	2	no	—
136	C₂²×C₄	—	19	4	no	—
137	C₂²×C₄	—	19	2	no	—
138	C₂²×C₄	—	38	4	no	—
139	C₂²×C₄	—	76	4	no	—
140	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
141	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
142	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
143	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
144	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	2	2	no	—
147	C₄⋊C₄	—	38	4	no	—
148	C₄⋊C₄	—	38	2	no	—
149	C₄⋊C₄	—	76	4	no	—
150	C₂²⋊C₄	—	38	8	no	—
151	C₂²⋊C₄	—	38	4	no	—
152	C₂²⋊C₄	—	76	4	no	—
153	C₄²	—	2	4	no	—
154	C₄²	C₂²⋊D₁₉	1	4	yes	—
155	C₂³	—	19	4	no	—
156	C₂³	—	38	4	no	—
157	C₂³	—	38	2	no	—
158	C₂³	—	38	4	no	—
159	C₂³	—	38	2	no	—
160	C₂³	C₂³×D₁₉	1	1	yes	characteristic
161	Q₈	—	2	8	no	—
162	D₄	—	38	8	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	38	4	no	—
171	C₂×C₄	—	38	4	no	—
172	C₂×C₄	—	38	4	no	—
173	C₂×C₄	—	38	4	no	—
174	C₂×C₄	—	38	2	no	—
175	C₂×C₄	—	76	8	no	—
176	C₂×C₄	#64541	1	4	yes	—
177	C₂×C₄	Q₈×D₁₉	1	2	yes	—
178	C₂×C₄	#64545	1	2	yes	—
179	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
180	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
181	C₂×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
182	C₂²	—	38	8	no	—
183	C₂²	—	38	4	no	—
184	C₂²	—	76	2	no	—
185	C₂²	—	76	2	no	—
186	C₂²	—	76	4	no	—
187	C₂²	C₂×#64541	1	2	yes	—
188	C₂²	C₂×Q₈×D₁₉	1	1	yes	characteristic
189	C₂²	C₂×#64545	1	1	yes	characteristic
190	C₂²	#10597435	1	2	yes	—
191	C₂²	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
192	C₄	—	2	4	no	—
193	C₄	—	2	2	no	—
194	C₄	—	2	2	no	—
195	C₄	—	4	2	no	—
196	C₄	—	76	2	no	—
197	C₄	—	76	4	no	—
198	C₄	#10597392	1	4	yes	—
199	C₂	—	76	4	no	—
200	C₂	C₂×#10597392	1	1	yes	characteristic
201	C₂	#11927055	1	1	yes	characteristic
202	C₂	#11927108	1	1	yes	characteristic
203	C₂	#11927254	1	2	yes	—
204	C₂	#11927279	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×D₇₆

C₂

yes

maximal, characteristic

#11926479

C₂

yes

maximal, complemented

#11926455

C₂

yes

maximal, complemented

#11926446

C₂

yes

maximal

#11926442

C₂

yes

maximal, complemented

#11926415

C₂

yes

maximal, complemented

#11926410

C₂

yes

maximal, characteristic, complemented

C₂×C₄×Q₈×C₁₉

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₁₉

C₂²

yes

characteristic

C₂²×D₇₆

C₂²

yes

characteristic

C₂²×C₄×D₁₉

C₂²

yes

—

C₄×Q₈×C₁₉

—

C₂×C₄⋊C₄×C₁₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₉

C₂²

yes

characteristic

C₂×C₄²×C₁₉

C₂²

yes

—

C₂×C₄²×C₁₉

C₂²

yes

characteristic

C₂×(C₂×D₁₉)⋊C₄

C₂²

yes

—

C₂×(C₂×D₁₉)⋊C₄

C₂²

yes

—

C₂×(C₂×D₁₉)⋊C₄

C₂²

yes

—

C₂×C₄⋊Dic₁₉

C₂²

yes

—

C₂×C₄⋊Dic₁₉

C₂²

yes

characteristic

C₂×Dic₁₉⋊C₄

C₂²

yes

—

C₄×D₇₆

—

C₁₉⋊#95

C₂²

yes

—

C₁₉⋊#95

C₂²

yes

—

C₁₉⋊#95

C₂²

yes

—

C₂³×D₁₉

C₂³

yes

—

C₂×Q₈×C₁₉

—

C₂×Q₈×C₁₉

—

C₂²×C₄×C₁₉

C₂³

yes

—

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂²×C₄×C₁₉

C₂³

yes

—

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂²×Dic₁₉

C₂³

yes

—

C₂²×Dic₁₉

C₂³

yes

—

C₂×D₇₆

—

C₂×D₇₆

—

C₂×D₇₆

Q₈

yes

complemented

C₂×C₄×D₁₉

—

C₂×C₄×D₁₉

—

C₄⋊C₄×C₁₉

—

C₄⋊C₄×C₁₉

—

C₄²×C₁₉

—

C₄²×C₁₉

D₄

yes

—

(C₂×D₁₉)⋊C₄

—

(C₂×D₁₉)⋊C₄

—

(C₂×D₁₉)⋊C₄

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

Dic₁₉⋊C₄

—

C₂³×C₁₉

C₂⁴

yes

characteristic

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

C₂×Q₈

yes

—

Q₈×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

—

C₂×C₄×C₁₉

C₂×D₄

yes

—

C₂×C₄×C₁₉

C₂×D₄

yes

—

C₂×C₄×C₁₉

C₂×D₄

yes

—

C₂×C₄×C₁₉

C₂×Q₈

yes

—

C₂×C₄×C₁₉

#41

yes

—

C₂×C₄×C₁₉

#41

yes

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

D₇₆

—

C₄×D₁₉

—

#2580

—

maximal, Sylow

C₂²×C₁₉

C₂²×D₄

yes

characteristic

C₂²×C₁₉

C₂²×Q₈

yes

characteristic

C₂²×C₁₉

C₂×#41

yes

—

C₂²×C₁₉

C₂×#41

yes

characteristic

C₂²×C₁₉

H(5,𝔽₂)

yes

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

C₂²⋊Q₈

yes

—

Dic₁₉

—

Dic₁₉

—

C₂×C₄×Q₈

D₁₉

yes

characteristic, complemented

C₂×C₄×D₄

—

#399

—

100

#393

—

101

#390

—

102

#389

—

103

#385

—

104

#382

—

105

C₂×C₁₉

C₂×C₂²⋊Q₈

yes

characteristic

106

C₂×C₁₉

#525

yes

characteristic

107

C₂×C₁₉

#535

yes

characteristic

108

C₂×C₁₉

#554

yes

—

109

C₂×C₁₉

#559

yes

—

110

D₁₉

—

111

C₂²×Q₈

C₂×D₁₉

yes

characteristic

112

C₂²×D₄

—

113

C₂³×C₄

—

114

C₄×Q₈

—

115

C₄×D₄

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

C₂×D₁₉

yes

—

120

C₂×C₄⋊C₄

C₂×D₁₉

yes

characteristic

121

C₂×C₂²⋊C₄

—

122

C₂×C₂²⋊C₄

—

123

C₂×C₂²⋊C₄

—

124

C₂×C₄²

C₂×D₁₉

yes

—

125

C₂×C₄²

C₂×D₁₉

yes

characteristic

126

#95

—

127

#95

—

128

#95

—

129

C₁₉

#2580

yes

characteristic, complemented, Sylow

130

C₂⁴

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×D₄

—

134

C₂×D₄

—

135

C₂×D₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

C₂²×D₁₉

yes

—

141

C₂²×C₄

C₂²×D₁₉

yes

—

142

C₂²×C₄

C₂²×D₁₉

yes

characteristic

143

C₂²×C₄

C₂²×D₁₉

yes

characteristic

144

C₂²×C₄

C₂²×D₁₉

yes

characteristic

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₂²⋊C₄

—

151

C₂²⋊C₄

—

152

C₂²⋊C₄

—

153

C₄²

—

154

C₄²

C₂²⋊D₁₉

yes

—

155

C₂³

—

156

C₂³

—

157

C₂³

—

158

C₂³

—

159

C₂³

—

160

C₂³

C₂³×D₁₉

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

#64541

yes

—

177

C₂×C₄

Q₈×D₁₉

yes

—

178

C₂×C₄

#64545

yes

—

179

C₂×C₄

C₂×C₂²⋊D₁₉

yes

—

180

C₂×C₄

C₂×C₂²⋊D₁₉

yes

—

181

C₂×C₄

C₂×C₂²⋊D₁₉

yes

—

182

C₂²

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

C₂×#64541

yes

—

188

C₂²

C₂×Q₈×D₁₉

yes

characteristic

189

C₂²

C₂×#64545

yes

characteristic

190

C₂²

#10597435

yes

—

191

C₂²

C₂²×C₂²⋊D₁₉

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

#10597392

yes

—

199

C₂

—

200

C₂

C₂×#10597392

yes

characteristic

201

C₂

#11927055

yes

characteristic

202

C₂

#11927108

yes

characteristic

203

C₂

#11927254

yes

—

204

C₂

#11927279

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉
commutator, socle, cosocle	C₂³×C₁₉	C₂⁴
Fitting	C₂×C₄×Q₈×C₁₉	C₂
prime radical	C₂²×D₇₆	C₂²
hypercenter	C₂×C₄×Q₈	D₁₉
hypocenter	C₁₉	#2580
LCS	G ▷ C₂³×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₉

commutator, socle, cosocle

C₂³×C₁₉

C₂⁴

Fitting

C₂×C₄×Q₈×C₁₉

C₂

prime radical

C₂²×D₇₆

C₂²

hypercenter

C₂×C₄×Q₈

D₁₉

C₁₉

G ▷ C₂³×C₁₉ ▷ C₁₉

C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
2. #11926479	C₂	semidirect
3. #11926455	C₂	semidirect
5. #11926442	C₂	semidirect
6. #11926415	C₂	semidirect
7. #11926410	C₂	semidirect
8. C₂×C₄×Q₈×C₁₉	C₂	semidirect
39. C₂×D₇₆	Q₈	semidirect
97. C₂×C₄×Q₈	D₁₉	semidirect
129. C₁₉	#2580	semidirect

H ≅ G/N

type

2. #11926479

C₂

semidirect

3. #11926455

C₂

semidirect

5. #11926442

C₂

semidirect

6. #11926415

C₂

semidirect

7. #11926410

C₂

semidirect

8. C₂×C₄×Q₈×C₁₉

C₂

semidirect

39. C₂×D₇₆

Q₈

semidirect

97. C₂×C₄×Q₈

D₁₉

semidirect

129. C₁₉

#2580

semidirect

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

subgroup	index
C₂×C₄×D₇₆	2
#11926479	2
#11926455	2
#11926446	2
#11926442	2
#11926415	2
#11926410	2
C₂×C₄×Q₈×C₁₉	2
#2580	19

subgroup

index

C₂×C₄×D₇₆

C₂×C₄×Q₈×C₁₉

Subgroups — #14568993

Conjugacy classes of subgroups

Complements

Maximal subgroups