Group #37576164 — subgroups

Subgroups — #37576164

Order 24448.

800 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 213 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×Dic₃₈₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#20259322	C₂	1	2	yes	maximal, complemented
3	C₂×#20259301	C₂	1	1	yes	maximal, characteristic
4	#27132584	C₂	1	4	yes	maximal, complemented
5	#27132569	C₂	1	2	yes	maximal, complemented
6	#27132538	C₂	1	4	yes	maximal
7	C₄×C₂²⋊C₄×C₁₉₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₃₈₂	—	2	8	no	—
9	C₂²×Dic₃₈₂	—	2	4	no	—
10	C₂²×Dic₃₈₂	C₂²	1	1	yes	characteristic
11	C₂²×Dic₃₈₂	C₂²	1	2	yes	—
12	C₂²×Dic₃₈₂	C₂²	1	2	yes	—
13	C₂³×Dic₁₉₁	C₂²	1	2	yes	—
14	#20259322	—	2	8	no	—
15	#20259301	—	2	4	no	—
16	C₂×C₄⋊Dic₁₉₁	C₂²	1	8	yes	—
17	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
18	C₂×Dic₁₉₁⋊C₄	C₂²	1	2	yes	—
19	C₂×C₂²⋊Dic₁₉₁	C₂²	1	2	yes	—
20	C₁₉₁⋊#95	C₂²	1	8	yes	—
21	C₂³×C₄×C₁₉₁	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×C₁₉₁	C₂²	1	2	yes	—
23	C₂×C₄²×C₁₉₁	C₂²	1	2	yes	—
24	#95×C₁₉₁	C₂²	1	2	yes	—
25	C₂×Dic₃₈₂	—	2	32	no	—
26	C₂×Dic₃₈₂	—	2	8	no	—
27	C₂×Dic₃₈₂	—	2	4	no	—
28	C₂×Dic₃₈₂	—	2	4	no	—
29	C₂×Dic₃₈₂	—	2	4	no	—
30	C₂×Dic₃₈₂	—	2	2	no	—
31	C₂×Dic₃₈₂	—	2	4	no	—
32	C₂×Dic₃₈₂	D₄	1	8	yes	—
33	C₂²×Dic₁₉₁	—	2	8	no	—
34	C₂²×Dic₁₉₁	—	2	4	no	—
35	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
36	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
37	C₂²×Dic₁₉₁	C₂³	1	4	yes	—
38	C₄⋊Dic₁₉₁	—	2	16	no	—
39	C₄⋊Dic₁₉₁	—	2	2	no	—
40	Dic₁₉₁⋊C₄	—	2	4	no	—
41	C₂²⋊Dic₁₉₁	—	2	4	no	—
42	C₂⁴×C₁₉₁	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₁₉₁	—	2	4	no	—
44	C₂²×C₄×C₁₉₁	D₄	1	4	yes	—
45	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₉₁	C₂³	1	4	yes	—
47	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
48	C₂²⋊C₄×C₁₉₁	Q₈	1	8	yes	complemented
49	C₄²×C₁₉₁	—	2	4	no	—
50	Dic₃₈₂	—	2	16	no	—
51	Dic₃₈₂	—	2	16	no	—
52	Dic₃₈₂	—	4	4	no	—
53	C₂×Dic₁₉₁	—	2	16	no	—
54	C₂×Dic₁₉₁	—	2	4	no	—
55	C₂×Dic₁₉₁	—	2	4	no	—
56	C₂×Dic₁₉₁	—	2	2	no	—
57	C₂×Dic₁₉₁	—	2	2	no	—
58	C₂×Dic₁₉₁	—	2	4	no	—
59	C₂×Dic₁₉₁	—	2	4	no	—
60	C₂×Dic₁₉₁	C₂×D₄	1	8	yes	—
61	C₂³×C₁₉₁	—	2	2	no	—
62	C₂³×C₁₉₁	—	2	2	no	—
63	C₂³×C₁₉₁	C₂×D₄	1	2	yes	—
64	C₂³×C₁₉₁	C₂×Q₈	1	4	yes	—
65	C₂³×C₁₉₁	C₂⁴	1	1	yes	characteristic
66	C₂×C₄×C₁₉₁	—	2	4	no	—
67	C₂×C₄×C₁₉₁	—	2	2	no	—
68	C₂×C₄×C₁₉₁	—	2	4	no	—
69	C₂×C₄×C₁₉₁	—	2	8	no	—
70	C₂×C₄×C₁₉₁	C₂×D₄	1	4	yes	—
71	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
72	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
73	C₂×C₄×C₁₉₁	C₂×Q₈	1	8	yes	complemented
74	C₂²×C₁₉₁	—	2	2	no	—
75	C₂²×C₁₉₁	—	2	2	no	—
76	C₂²×C₁₉₁	—	2	8	no	—
77	C₂²×C₁₉₁	C₄⋊Q₈	1	4	yes	complemented
78	C₂²×C₁₉₁	C₂²×D₄	1	2	yes	—
79	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
80	C₂²×C₁₉₁	C₂²×Q₈	1	2	yes	—
81	C₂²×C₁₉₁	2₋¹⁺⁴	1	2	yes	—
82	C₄×C₁₉₁	—	2	2	no	—
83	C₄×C₁₉₁	—	2	8	no	—
84	C₄×C₁₉₁	#120	1	4	yes	—
85	Dic₁₉₁	—	2	8	no	—
86	Dic₁₉₁	—	4	2	no	—
87	Dic₁₉₁	—	4	2	no	—
88	C₂×C₁₉₁	—	2	4	no	—
89	C₂×C₁₉₁	C₂×#120	1	1	yes	characteristic
90	C₂×C₁₉₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
91	C₂×C₁₉₁	#536	1	1	yes	characteristic
92	C₂×C₁₉₁	D₄×Q₈	1	4	yes	—
93	C₁₉₁	#2434	1	1	yes	characteristic, complemented, Sylow
94	#2434	—	191	1	no	maximal, Sylow
95	C₂³×Q₈	—	191	1	no	—
96	C₂×C₄⋊Q₈	—	191	2	no	—
97	C₂×C₂²⋊Q₈	—	191	1	no	—
98	#395	—	191	4	no	—
99	#391	—	191	2	no	—
100	#385	—	191	4	no	—
101	C₄×C₂²⋊C₄	D₁₉₁	1	1	yes	characteristic
102	C₂²×Q₈	—	382	8	no	—
103	C₂²×Q₈	—	382	4	no	—
104	C₂²×Q₈	—	191	1	no	—
105	C₂²×Q₈	—	191	2	no	—
106	C₂²×Q₈	—	191	2	no	—
107	C₂³×C₄	—	191	2	no	—
108	C₂³×C₄	C₂×D₁₉₁	1	1	yes	characteristic
109	C₄⋊Q₈	—	382	8	no	—
110	C₂²⋊Q₈	—	382	4	no	—
111	C₂×C₄⋊C₄	—	191	1	no	—
112	C₂×C₄⋊C₄	—	191	2	no	—
113	C₂×C₄⋊C₄	—	191	8	no	—
114	C₂×C₂²⋊C₄	—	191	2	no	—
115	C₂×C₂²⋊C₄	C₂×D₁₉₁	1	2	yes	—
116	C₂×C₄²	C₂×D₁₉₁	1	2	yes	—
117	#95	—	191	8	no	—
118	#95	C₂×D₁₉₁	1	2	yes	—
119	C₂⁴	C₂²×D₁₉₁	1	1	yes	characteristic
120	C₂×Q₈	—	382	32	no	—
121	C₂×Q₈	—	382	8	no	—
122	C₂×Q₈	—	382	4	no	—
123	C₂×Q₈	—	382	4	no	—
124	C₂×Q₈	—	382	4	no	—
125	C₂×Q₈	—	382	4	no	—
126	C₂×Q₈	—	382	2	no	—
127	C₂×Q₈	—	191	8	no	—
128	C₂²×C₄	—	2	4	no	—
129	C₂²×C₄	—	382	4	no	—
130	C₂²×C₄	—	382	8	no	—
131	C₂²×C₄	—	191	2	no	—
132	C₂²×C₄	—	191	2	no	—
133	C₂²×C₄	—	191	4	no	—
134	C₂²×C₄	D₇₆₄	1	4	yes	—
135	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
136	C₂²×C₄	C₂²×D₁₉₁	1	4	yes	—
137	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
138	C₄⋊C₄	—	382	16	no	—
139	C₄⋊C₄	—	382	4	no	—
140	C₄⋊C₄	—	382	2	no	—
141	C₂²⋊C₄	—	382	4	no	—
142	C₂²⋊C₄	Dic₃₈₂	1	8	yes	complemented
143	C₄²	—	2	4	no	—
144	C₂³	—	2	2	no	—
145	C₂³	—	2	2	no	—
146	C₂³	C₂×D₇₆₄	1	2	yes	—
147	C₂³	C₂×Dic₃₈₂	1	4	yes	—
148	C₂³	C₂³×D₁₉₁	1	1	yes	characteristic
149	Q₈	—	382	16	no	—
150	Q₈	—	382	16	no	—
151	Q₈	—	764	4	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	2	8	no	—
156	C₂×C₄	—	382	16	no	—
157	C₂×C₄	—	382	4	no	—
158	C₂×C₄	—	382	4	no	—
159	C₂×C₄	—	382	4	no	—
160	C₂×C₄	—	382	2	no	—
161	C₂×C₄	—	382	2	no	—
162	C₂×C₄	—	382	4	no	—
163	C₂×C₄	—	191	8	no	—
164	C₂×C₄	D₄×D₁₉₁	1	4	yes	—
165	C₂×C₄	C₂×D₇₆₄	1	2	yes	—
166	C₂×C₄	C₂×D₇₆₄	1	2	yes	—
167	C₂×C₄	C₂×Dic₃₈₂	1	8	yes	complemented
168	C₂²	—	2	2	no	—
169	C₂²	—	2	8	no	—
170	C₂²	—	2	2	no	—
171	C₂²	#20259322	1	4	yes	complemented
172	C₂²	C₂×D₄×D₁₉₁	1	2	yes	—
173	C₂²	C₂²×D₇₆₄	1	1	yes	characteristic
174	C₂²	C₂²×Dic₃₈₂	1	2	yes	—
175	C₂²	#20259369	1	2	yes	—
176	C₄	—	2	8	no	—
177	C₄	—	2	2	no	—
178	C₄	—	382	8	no	—
179	C₄	—	764	2	no	—
180	C₄	—	764	2	no	—
181	C₄	#20259292	1	4	yes	—
182	C₂	—	2	4	no	—
183	C₂	C₂×#20259292	1	1	yes	characteristic
184	C₂	C₂×#20259322	1	1	yes	characteristic
185	C₂	#27133239	1	1	yes	characteristic
186	C₂	D₄×Dic₃₈₂	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×Dic₃₈₂

C₂

yes

maximal, characteristic

C₂×#20259322

C₂

yes

maximal, complemented

C₂×#20259301

C₂

yes

maximal, characteristic

#27132584

C₂

yes

maximal, complemented

#27132569

C₂

yes

maximal, complemented

#27132538

C₂

yes

maximal

C₄×C₂²⋊C₄×C₁₉₁

C₂

yes

maximal, characteristic

C₂²×Dic₃₈₂

—

C₂²×Dic₃₈₂

—

C₂²×Dic₃₈₂

C₂²

yes

characteristic

C₂²×Dic₃₈₂

C₂²

yes

—

C₂²×Dic₃₈₂

C₂²

yes

—

C₂³×Dic₁₉₁

C₂²

yes

—

#20259322

—

#20259301

—

C₂×C₄⋊Dic₁₉₁

C₂²

yes

—

C₂×C₄⋊Dic₁₉₁

C₂²

yes

characteristic

C₂×Dic₁₉₁⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₁₉₁

C₂²

yes

—

C₁₉₁⋊#95

C₂²

yes

—

C₂³×C₄×C₁₉₁

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₁₉₁

C₂²

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

—

#95×C₁₉₁

C₂²

yes

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

D₄

yes

—

C₂²×Dic₁₉₁

—

C₂²×Dic₁₉₁

—

C₂²×Dic₁₉₁

C₂³

yes

—

C₂²×Dic₁₉₁

C₂³

yes

—

C₂²×Dic₁₉₁

C₂³

yes

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

Dic₁₉₁⋊C₄

—

C₂²⋊Dic₁₉₁

—

C₂⁴×C₁₉₁

C₂³

yes

characteristic

C₂²×C₄×C₁₉₁

—

C₂²×C₄×C₁₉₁

D₄

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₂²×C₄×C₁₉₁

C₂³

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₂²⋊C₄×C₁₉₁

Q₈

yes

complemented

C₄²×C₁₉₁

—

Dic₃₈₂

—

Dic₃₈₂

—

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

C₂×D₄

yes

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

C₂×D₄

yes

—

C₂³×C₁₉₁

C₂×Q₈

yes

—

C₂³×C₁₉₁

C₂⁴

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×Q₈

yes

complemented

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₄⋊Q₈

yes

complemented

C₂²×C₁₉₁

C₂²×D₄

yes

—

C₂²×C₁₉₁

C₂²×D₄

yes

characteristic

C₂²×C₁₉₁

C₂²×Q₈

yes

—

C₂²×C₁₉₁

2₋¹⁺⁴

yes

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#120

yes

—

Dic₁₉₁

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

C₂×#120

yes

characteristic

C₂×C₁₉₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₉₁

#536

yes

characteristic

C₂×C₁₉₁

D₄×Q₈

yes

—

C₁₉₁

#2434

yes

characteristic, complemented, Sylow

#2434

—

191

maximal, Sylow

C₂³×Q₈

—

191

—

C₂×C₄⋊Q₈

—

191

—

C₂×C₂²⋊Q₈

—

191

—

#395

—

191

—

#391

—

191

—

100

#385

—

191

—

101

C₄×C₂²⋊C₄

D₁₉₁

yes

characteristic

102

C₂²×Q₈

—

382

—

103

C₂²×Q₈

—

382

—

104

C₂²×Q₈

—

191

—

105

C₂²×Q₈

—

191

—

106

C₂²×Q₈

—

191

—

107

C₂³×C₄

—

191

—

108

C₂³×C₄

C₂×D₁₉₁

yes

characteristic

109

C₄⋊Q₈

—

382

—

110

C₂²⋊Q₈

—

382

—

111

C₂×C₄⋊C₄

—

191

—

112

C₂×C₄⋊C₄

—

191

—

113

C₂×C₄⋊C₄

—

191

—

114

C₂×C₂²⋊C₄

—

191

—

115

C₂×C₂²⋊C₄

C₂×D₁₉₁

yes

—

116

C₂×C₄²

C₂×D₁₉₁

yes

—

117

#95

—

191

—

118

#95

C₂×D₁₉₁

yes

—

119

C₂⁴

C₂²×D₁₉₁

yes

characteristic

120

C₂×Q₈

—

382

—

121

C₂×Q₈

—

382

—

122

C₂×Q₈

—

382

—

123

C₂×Q₈

—

382

—

124

C₂×Q₈

—

382

—

125

C₂×Q₈

—

382

—

126

C₂×Q₈

—

382

—

127

C₂×Q₈

—

191

—

128

C₂²×C₄

—

129

C₂²×C₄

—

382

—

130

C₂²×C₄

—

382

—

131

C₂²×C₄

—

191

—

132

C₂²×C₄

—

191

—

133

C₂²×C₄

—

191

—

134

C₂²×C₄

D₇₆₄

yes

—

135

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

136

C₂²×C₄

C₂²×D₁₉₁

yes

—

137

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

138

C₄⋊C₄

—

382

—

139

C₄⋊C₄

—

382

—

140

C₄⋊C₄

—

382

—

141

C₂²⋊C₄

—

382

—

142

C₂²⋊C₄

Dic₃₈₂

yes

complemented

143

C₄²

—

144

C₂³

—

145

C₂³

—

146

C₂³

C₂×D₇₆₄

yes

—

147

C₂³

C₂×Dic₃₈₂

yes

—

148

C₂³

C₂³×D₁₉₁

yes

characteristic

149

Q₈

—

382

—

150

Q₈

—

382

—

151

Q₈

—

764

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

382

—

157

C₂×C₄

—

382

—

158

C₂×C₄

—

382

—

159

C₂×C₄

—

382

—

160

C₂×C₄

—

382

—

161

C₂×C₄

—

382

—

162

C₂×C₄

—

382

—

163

C₂×C₄

—

191

—

164

C₂×C₄

D₄×D₁₉₁

yes

—

165

C₂×C₄

C₂×D₇₆₄

yes

—

166

C₂×C₄

C₂×D₇₆₄

yes

—

167

C₂×C₄

C₂×Dic₃₈₂

yes

complemented

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

#20259322

yes

complemented

172

C₂²

C₂×D₄×D₁₉₁

yes

—

173

C₂²

C₂²×D₇₆₄

yes

characteristic

174

C₂²

C₂²×Dic₃₈₂

yes

—

175

C₂²

#20259369

yes

—

176

C₄

—

177

C₄

—

178

C₄

—

382

—

179

C₄

—

764

—

180

C₄

—

764

—

181

C₄

#20259292

yes

—

182

C₂

—

183

C₂

C₂×#20259292

yes

characteristic

184

C₂

C₂×#20259322

yes

characteristic

185

C₂

#27133239

yes

characteristic

186

C₂

D₄×Dic₃₈₂

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉₁
commutator, socle, cosocle	C₂³×C₁₉₁	C₂⁴
Fitting	C₄×C₂²⋊C₄×C₁₉₁	C₂
prime radical	C₂⁴×C₁₉₁	C₂³
hypercenter	C₄×C₂²⋊C₄	D₁₉₁
hypocenter	C₁₉₁	#2434
LCS	G ▷ C₂³×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₉₁

commutator, socle, cosocle

C₂³×C₁₉₁

C₂⁴

Fitting

C₄×C₂²⋊C₄×C₁₉₁

C₂

prime radical

C₂⁴×C₁₉₁

C₂³

hypercenter

C₄×C₂²⋊C₄

D₁₉₁

C₁₉₁

G ▷ C₂³×C₁₉₁ ▷ C₁₉₁

C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄

N	H ≅ G/N	type
2. C₂×#20259322	C₂	semidirect
4. #27132584	C₂	semidirect
5. #27132569	C₂	semidirect
48. C₂²⋊C₄×C₁₉₁	Q₈	semidirect
73. C₂×C₄×C₁₉₁	C₂×Q₈	semidirect
77. C₂²×C₁₉₁	C₄⋊Q₈	semidirect
93. C₁₉₁	#2434	semidirect
142. C₂²⋊C₄	Dic₃₈₂	semidirect
167. C₂×C₄	C₂×Dic₃₈₂	semidirect
171. C₂²	#20259322	semidirect

H ≅ G/N

type

2. C₂×#20259322

C₂

semidirect

4. #27132584

C₂

semidirect

5. #27132569

C₂

semidirect

48. C₂²⋊C₄×C₁₉₁

Q₈

semidirect

73. C₂×C₄×C₁₉₁

C₂×Q₈

semidirect

77. C₂²×C₁₉₁

C₄⋊Q₈

semidirect

93. C₁₉₁

#2434

semidirect

142. C₂²⋊C₄

Dic₃₈₂

semidirect

167. C₂×C₄

C₂×Dic₃₈₂

semidirect

171. C₂²

#20259322

semidirect

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

subgroup	index
C₂³×Dic₃₈₂	2
C₂×#20259322	2
C₂×#20259301	2
#27132584	2
#27132569	2
#27132538	2
C₄×C₂²⋊C₄×C₁₉₁	2
#2434	191

subgroup

index

C₂³×Dic₃₈₂

C₄×C₂²⋊C₄×C₁₉₁

191

Subgroups — #37576164

Conjugacy classes of subgroups

Complements

Maximal subgroups