Group #14568584 — subgroups

Subgroups — #14568584

Order 2432.

676 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 205 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×#10597315	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#10597343	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×D₇₆	C₂	1	1	yes	maximal, characteristic
4	#11926481	C₂	1	4	yes	maximal, complemented
5	#11926446	C₂	1	2	yes	maximal
6	#11926414	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊C₄×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₇₆	C₂²	1	1	yes	characteristic
9	C₂²×C₄×D₁₉	C₂²	1	2	yes	—
10	C₂²×Dic₃₈	C₂²	1	2	yes	—
11	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
14	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	8	yes	—
15	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₁₉	C₂²	1	8	yes	—
17	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
20	#10597343	—	2	16	no	—
21	C₄×D₇₆	—	2	4	no	—
22	#10597315	—	2	4	no	—
23	C₁₉×#95	C₂²	1	2	yes	—
24	C₂³×D₁₉	C₂³	1	2	yes	—
25	C₂²×C₄×C₁₉	C₂³	1	4	yes	—
26	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
29	C₂²×Dic₁₉	C₂³	1	4	yes	—
30	C₂²×Dic₁₉	C₂³	1	2	yes	—
31	C₂×D₇₆	—	2	2	no	—
32	C₂×D₇₆	—	2	2	no	—
33	C₂×D₇₆	Q₈	1	4	yes	complemented
34	C₂×C₄×D₁₉	—	2	4	no	—
35	C₂×C₄×D₁₉	—	2	8	no	—
36	C₂×Dic₃₈	—	2	4	no	—
37	C₂×Dic₃₈	—	2	8	no	—
38	C₄⋊C₄×C₁₉	D₄	1	8	yes	complemented
39	C₄²×C₁₉	—	2	4	no	—
40	C₄²×C₁₉	D₄	1	4	yes	—
41	(C₂×D₁₉)⋊C₄	—	2	4	no	—
42	(C₂×D₁₉)⋊C₄	—	2	16	no	—
43	C₄⋊Dic₁₉	—	2	4	no	—
44	C₄⋊Dic₁₉	—	2	4	no	—
45	C₄⋊Dic₁₉	—	2	2	no	—
46	C₄⋊Dic₁₉	—	2	16	no	—
47	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
48	C₂²×D₁₉	—	2	4	no	—
49	C₂²×D₁₉	—	2	2	no	—
50	C₂²×D₁₉	—	2	2	no	—
51	C₂²×D₁₉	—	2	4	no	—
52	C₂²×D₁₉	C₂×Q₈	1	4	yes	—
53	C₂×C₄×C₁₉	—	2	4	no	—
54	C₂×C₄×C₁₉	—	2	2	no	—
55	C₂×C₄×C₁₉	—	2	4	no	—
56	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₁₉	C₂×D₄	1	8	yes	—
58	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₉	C₂×D₄	1	4	yes	—
61	C₂×C₄×C₁₉	C₂×Q₈	1	2	yes	—
62	C₂×C₄×C₁₉	#41	1	2	yes	—
63	C₂×Dic₁₉	—	2	4	no	—
64	C₂×Dic₁₉	—	2	4	no	—
65	C₂×Dic₁₉	—	2	8	no	—
66	C₂×Dic₁₉	—	2	2	no	—
67	D₇₆	—	2	8	no	—
68	C₄×D₁₉	—	4	8	no	—
69	Dic₃₈	—	4	8	no	—
70	#2542	—	19	1	no	maximal, Sylow
71	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
72	C₂²×C₁₉	C₂²×D₄	1	2	yes	—
73	C₂²×C₁₉	C₂²×Q₈	1	1	yes	characteristic
74	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
75	C₂²×C₁₉	2₋¹⁺⁴	1	2	yes	—
76	C₂×D₁₉	—	2	4	no	—
77	C₂×D₁₉	—	2	8	no	—
78	C₂×D₁₉	—	4	2	no	—
79	C₂×D₁₉	—	4	2	no	—
80	C₂×D₁₉	—	4	4	no	—
81	C₄×C₁₉	—	2	2	no	—
82	C₄×C₁₉	—	2	8	no	—
83	C₄×C₁₉	C₂²⋊Q₈	1	4	yes	complemented
84	C₄×C₁₉	#127	1	4	yes	—
85	Dic₁₉	—	4	4	no	—
86	Dic₁₉	—	4	2	no	—
87	C₂×C₄⋊Q₈	—	19	1	no	—
88	C₂×C₂²⋊Q₈	—	19	4	no	—
89	C₂×C₄×D₄	—	19	1	no	—
90	#399	—	19	4	no	—
91	#390	—	19	2	no	—
92	#384	—	19	2	no	—
93	C₄×C₄⋊C₄	D₁₉	1	1	yes	characteristic, complemented
94	C₂×C₁₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₁₉	C₂×#127	1	1	yes	characteristic
96	C₂×C₁₉	#536	1	1	yes	characteristic
97	C₂×C₁₉	#547	1	2	yes	—
98	C₂×C₁₉	D₄×Q₈	1	2	yes	—
99	D₁₉	—	4	4	no	—
100	C₂²×Q₈	—	19	2	no	—
101	C₂²×D₄	—	19	1	no	—
102	C₂³×C₄	—	19	2	no	—
103	C₄⋊Q₈	—	38	4	no	—
104	C₂²⋊Q₈	—	38	16	no	—
105	C₄×D₄	—	38	4	no	—
106	C₂×C₄⋊C₄	—	19	1	no	—
107	C₂×C₄⋊C₄	—	19	8	no	—
108	C₂×C₄⋊C₄	—	19	2	no	—
109	C₂×C₄⋊C₄	—	19	2	no	—
110	C₂×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
111	C₂×C₂²⋊C₄	—	19	2	no	—
112	C₂×C₂²⋊C₄	—	19	8	no	—
113	C₂×C₄²	C₂×D₁₉	1	2	yes	—
114	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
115	#95	C₂×D₁₉	1	2	yes	—
116	C₁₉	#2542	1	1	yes	characteristic, complemented, Sylow
117	C₂⁴	—	19	2	no	—
118	C₂×Q₈	—	38	8	no	—
119	C₂×Q₈	—	38	4	no	—
120	C₂×D₄	—	19	4	no	—
121	C₂×D₄	—	38	2	no	—
122	C₂×D₄	—	38	2	no	—
123	C₂²×C₄	—	19	4	no	—
124	C₂²×C₄	—	19	2	no	—
125	C₂²×C₄	—	38	4	no	—
126	C₂²×C₄	—	38	8	no	—
127	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
128	C₂²×C₄	C₂²×D₁₉	1	4	yes	—
129	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
130	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
131	C₄⋊C₄	—	38	2	no	—
132	C₄⋊C₄	—	38	16	no	—
133	C₄⋊C₄	—	38	4	no	—
134	C₄⋊C₄	—	38	4	no	—
135	C₄⋊C₄	D₇₆	1	8	yes	complemented
136	C₂²⋊C₄	—	38	4	no	—
137	C₂²⋊C₄	—	38	16	no	—
138	C₄²	—	2	4	no	—
139	C₄²	D₇₆	1	4	yes	—
140	C₂³	—	19	4	no	—
141	C₂³	—	38	4	no	—
142	C₂³	—	38	2	no	—
143	C₂³	—	38	2	no	—
144	C₂³	—	38	4	no	—
145	C₂³	C₂³×D₁₉	1	1	yes	characteristic
146	Q₈	—	76	8	no	—
147	D₄	—	38	8	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	38	2	no	—
152	C₂×C₄	—	38	8	no	—
153	C₂×C₄	—	38	4	no	—
154	C₂×C₄	—	38	4	no	—
155	C₂×C₄	—	76	8	no	—
156	C₂×C₄	C₂×D₇₆	1	2	yes	—
157	C₂×C₄	C₂×D₇₆	1	8	yes	—
158	C₂×C₄	C₂×D₇₆	1	2	yes	—
159	C₂×C₄	C₂×D₇₆	1	4	yes	—
160	C₂×C₄	C₂×D₇₆	1	2	yes	—
161	C₂×C₄	#64543	1	2	yes	—
162	C₂×C₄	Q₈×D₁₉	1	2	yes	—
163	C₂²	—	38	4	no	—
164	C₂²	—	38	8	no	—
165	C₂²	—	76	2	no	—
166	C₂²	—	76	2	no	—
167	C₂²	—	76	4	no	—
168	C₂²	C₂²×D₇₆	1	1	yes	characteristic
169	C₂²	C₂²×D₇₆	1	2	yes	—
170	C₂²	C₂×#64543	1	1	yes	characteristic
171	C₂²	C₂×Q₈×D₁₉	1	1	yes	characteristic
172	C₂²	#10597436	1	2	yes	—
173	C₄	—	2	2	no	—
174	C₄	—	2	8	no	—
175	C₄	—	76	2	no	—
176	C₄	—	76	4	no	—
177	C₄	#10597320	1	4	yes	—
178	C₄	#10597343	1	4	yes	complemented
179	C₂	—	76	4	no	—
180	C₂	C₂×#10597343	1	1	yes	characteristic
181	C₂	C₂×#10597320	1	1	yes	characteristic
182	C₂	#11927120	1	1	yes	characteristic
183	C₂	#11927198	1	2	yes	—
184	C₂	Q₈×D₇₆	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10597315

C₂

yes

maximal, characteristic, complemented

C₂×#10597343

C₂

yes

maximal, complemented

C₂×C₄×D₇₆

C₂

yes

maximal, characteristic

#11926481

C₂

yes

maximal, complemented

#11926446

C₂

yes

maximal

#11926414

C₂

yes

maximal, complemented

C₄×C₄⋊C₄×C₁₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₇₆

C₂²

yes

characteristic

C₂²×C₄×D₁₉

C₂²

yes

—

C₂²×Dic₃₈

C₂²

yes

—

C₂×C₄⋊C₄×C₁₉

C₂²

yes

—

C₂×C₄²×C₁₉

C₂²

yes

—

C₂×C₄²×C₁₉

C₂²

yes

characteristic

C₂×(C₂×D₁₉)⋊C₄

C₂²

yes

—

C₂×(C₂×D₁₉)⋊C₄

C₂²

yes

—

C₂×C₄⋊Dic₁₉

C₂²

yes

—

C₂×C₄⋊Dic₁₉

C₂²

yes

—

C₂×C₄⋊Dic₁₉

C₂²

yes

—

C₂×C₄⋊Dic₁₉

C₂²

yes

characteristic

#10597343

—

C₄×D₇₆

—

#10597315

—

C₁₉×#95

C₂²

yes

—

C₂³×D₁₉

C₂³

yes

—

C₂²×C₄×C₁₉

C₂³

yes

—

C₂²×C₄×C₁₉

C₂³

yes

characteristic

C₂²×C₄×C₁₉

C₂³

yes

characteristic

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

—

C₂×Dic₃₈

—

C₂×Dic₃₈

—

C₄⋊C₄×C₁₉

D₄

yes

complemented

C₄²×C₁₉

—

C₄²×C₁₉

D₄

yes

—

(C₂×D₁₉)⋊C₄

—

(C₂×D₁₉)⋊C₄

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₄⋊Dic₁₉

—

C₂³×C₁₉

C₂⁴

yes

characteristic

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

—

C₂²×D₁₉

C₂×Q₈

yes

—

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×D₄

yes

—

C₂×C₄×C₁₉

C₂×D₄

yes

—

C₂×C₄×C₁₉

C₂×Q₈

yes

—

C₂×C₄×C₁₉

#41

yes

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

C₂×Dic₁₉

—

D₇₆

—

C₄×D₁₉

—

Dic₃₈

—

#2542

—

maximal, Sylow

C₂²×C₁₉

C₂²×D₄

yes

characteristic

C₂²×C₁₉

C₂²×D₄

yes

—

C₂²×C₁₉

C₂²×Q₈

yes

characteristic

C₂²×C₁₉

C₂×#41

yes

characteristic

C₂²×C₁₉

2₋¹⁺⁴

yes

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₂×D₁₉

—

C₄×C₁₉

—

C₄×C₁₉

—

C₄×C₁₉

C₂²⋊Q₈

yes

complemented

C₄×C₁₉

#127

yes

—

Dic₁₉

—

Dic₁₉

—

C₂×C₄⋊Q₈

—

C₂×C₂²⋊Q₈

—

C₂×C₄×D₄

—

#399

—

#390

—

#384

—

C₄×C₄⋊C₄

D₁₉

yes

characteristic, complemented

C₂×C₁₉

C₂×C₂²⋊Q₈

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

C₂×C₁₉

yes

—

C₂×C₁₉

D₄×Q₈

yes

—

D₁₉

—

100

C₂²×Q₈

—

101

C₂²×D₄

—

102

C₂³×C₄

—

103

C₄⋊Q₈

—

104

C₂²⋊Q₈

—

105

C₄×D₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

C₂×D₁₉

yes

—

111

C₂×C₂²⋊C₄

—

112

C₂×C₂²⋊C₄

—

113

C₂×C₄²

C₂×D₁₉

yes

—

114

C₂×C₄²

C₂×D₁₉

yes

characteristic

115

#95

C₂×D₁₉

yes

—

116

C₁₉

#2542

yes

characteristic, complemented, Sylow

117

C₂⁴

—

118

C₂×Q₈

—

119

C₂×Q₈

—

120

C₂×D₄

—

121

C₂×D₄

—

122

C₂×D₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

C₂²×D₁₉

yes

characteristic

128

C₂²×C₄

C₂²×D₁₉

yes

—

129

C₂²×C₄

C₂²×D₁₉

yes

characteristic

130

C₂²×C₄

C₂²×D₁₉

yes

characteristic

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

D₇₆

yes

complemented

136

C₂²⋊C₄

—

137

C₂²⋊C₄

—

138

C₄²

—

139

C₄²

D₇₆

yes

—

140

C₂³

—

141

C₂³

—

142

C₂³

—

143

C₂³

—

144

C₂³

—

145

C₂³

C₂³×D₁₉

yes

characteristic

146

Q₈

—

147

D₄

—

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

C₂×D₇₆

yes

—

157

C₂×C₄

C₂×D₇₆

yes

—

158

C₂×C₄

C₂×D₇₆

yes

—

159

C₂×C₄

C₂×D₇₆

yes

—

160

C₂×C₄

C₂×D₇₆

yes

—

161

C₂×C₄

#64543

yes

—

162

C₂×C₄

Q₈×D₁₉

yes

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

C₂²×D₇₆

yes

characteristic

169

C₂²

C₂²×D₇₆

yes

—

170

C₂²

C₂×#64543

yes

characteristic

171

C₂²

C₂×Q₈×D₁₉

yes

characteristic

172

C₂²

#10597436

yes

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

#10597320

yes

—

178

C₄

#10597343

yes

complemented

179

C₂

—

180

C₂

C₂×#10597343

yes

characteristic

181

C₂

C₂×#10597320

yes

characteristic

182

C₂

#11927120

yes

characteristic

183

C₂

#11927198

yes

—

184

C₂

Q₈×D₇₆

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉
commutator, socle, cosocle	C₂³×C₁₉	C₂⁴
Fitting	C₄×C₄⋊C₄×C₁₉	C₂
prime radical	C₂²×D₇₆	C₂²
hypercenter	C₄×C₄⋊C₄	D₁₉
hypocenter	C₁₉	#2542
LCS	G ▷ C₂³×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₉

commutator, socle, cosocle

C₂³×C₁₉

C₂⁴

Fitting

C₄×C₄⋊C₄×C₁₉

C₂

prime radical

C₂²×D₇₆

C₂²

hypercenter

C₄×C₄⋊C₄

D₁₉

C₁₉

G ▷ C₂³×C₁₉ ▷ C₁₉

C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
1. C₂×#10597315	C₂	semidirect
2. C₂×#10597343	C₂	semidirect
4. #11926481	C₂	semidirect
6. #11926414	C₂	semidirect
7. C₄×C₄⋊C₄×C₁₉	C₂	semidirect
33. C₂×D₇₆	Q₈	semidirect
38. C₄⋊C₄×C₁₉	D₄	semidirect
83. C₄×C₁₉	C₂²⋊Q₈	semidirect
93. C₄×C₄⋊C₄	D₁₉	semidirect
116. C₁₉	#2542	semidirect
135. C₄⋊C₄	D₇₆	semidirect
178. C₄	#10597343	semidirect

H ≅ G/N

type

1. C₂×#10597315

C₂

semidirect

2. C₂×#10597343

C₂

semidirect

4. #11926481

C₂

semidirect

6. #11926414

C₂

semidirect

7. C₄×C₄⋊C₄×C₁₉

C₂

semidirect

33. C₂×D₇₆

Q₈

semidirect

38. C₄⋊C₄×C₁₉

D₄

semidirect

83. C₄×C₁₉

C₂²⋊Q₈

semidirect

93. C₄×C₄⋊C₄

D₁₉

semidirect

116. C₁₉

#2542

semidirect

135. C₄⋊C₄

D₇₆

semidirect

178. C₄

#10597343

semidirect

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

subgroup	index
C₂×#10597315	2
C₂×#10597343	2
C₂×C₄×D₇₆	2
#11926481	2
#11926446	2
#11926414	2
C₄×C₄⋊C₄×C₁₉	2
#2542	19

subgroup

index

C₂×C₄×D₇₆

C₄×C₄⋊C₄×C₁₉

Subgroups — #14568584

Conjugacy classes of subgroups

Complements

Maximal subgroups