Group #31884667 — subgroups

Subgroups — #31884667

Order 18176.

298 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 75 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	#22589281	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556047	C₂	1	1	yes	maximal, characteristic
3	C₄×#101×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₇₁⋊C₈)	C₄	1	1	yes	characteristic
5	C₂×#14556047	—	2	1	no	—
6	C₄×#11765849	—	2	1	no	—
7	C₂×#14556039	C₄	1	1	yes	characteristic
8	#17985768	C₄	1	2	yes	—
9	#17985754	—	2	1	no	—
10	#17985754	—	2	1	no	—
11	C₂×C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
12	C₂×#101×C₇₁	—	2	1	no	—
13	C₄×M₁₆×C₇₁	—	2	1	no	—
14	#343×C₇₁	C₄	1	2	yes	—
15	C₂×#11765849	—	2	2	no	—
16	C₂²×C₇₁⋊C₈	—	2	1	no	—
17	C₂²×C₇₁⋊C₈	—	2	1	no	—
18	C₄⋊(C₇₁⋊C₈)	—	4	1	no	—
19	#14556047	—	2	4	no	—
20	#14556039	—	4	1	no	—
21	#14556039	—	4	1	no	—
22	C₄×C₇₁⋊C₈	—	4	1	no	—
23	C₂²×Q₈×C₇₁	D₄	1	1	yes	characteristic
24	C₂×M₁₆×C₇₁	—	2	2	no	—
25	C₄×Q₈×C₇₁	—	4	1	no	—
26	C₄×Q₈×C₇₁	—	4	1	no	—
27	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₇₁	Q₈	1	1	yes	characteristic
30	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
33	#101×C₇₁	—	2	4	no	—
34	#97×C₇₁	—	4	1	no	—
35	C₄×C₈×C₇₁	—	4	1	no	—
36	#11765849	—	2	4	no	—
37	#11765849	—	4	2	no	—
38	C₂×C₇₁⋊C₈	—	2	2	no	—
39	C₂×C₇₁⋊C₈	—	4	2	no	—
40	C₂×C₇₁⋊C₈	—	4	1	no	—
41	C₂×C₇₁⋊C₈	—	4	1	no	—
42	C₂×C₇₁⋊C₈	—	4	2	no	—
43	C₂×C₇₁⋊C₈	—	4	1	no	—
44	C₂×Q₈×C₇₁	—	2	2	no	—
45	C₂×Q₈×C₇₁	—	2	2	no	—
46	C₂×Q₈×C₇₁	—	4	1	no	—
47	C₂²×C₄×C₇₁	C₄²	1	1	yes	characteristic
48	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
54	M₁₆×C₇₁	—	2	4	no	—
55	M₁₆×C₇₁	—	4	2	no	—
56	C₂×C₈×C₇₁	—	4	2	no	—
57	C₄⋊C₄×C₇₁	—	4	1	no	—
58	C₄⋊C₄×C₇₁	—	4	1	no	—
59	C₄⋊C₄×C₇₁	—	4	1	no	—
60	C₄²×C₇₁	—	2	1	no	—
61	C₄²×C₇₁	—	2	1	no	—
62	C₄²×C₇₁	—	2	1	no	—
63	C₄²×C₇₁	—	2	1	no	—
64	C₄²×C₇₁	—	2	1	no	—
65	C₄²×C₇₁	—	2	1	no	—
66	C₂³×C₇₁	#95	1	1	yes	characteristic
67	Q₈×C₇₁	—	4	4	no	—
68	C₂×C₄×C₇₁	—	2	1	no	—
69	C₂×C₄×C₇₁	—	2	1	no	—
70	C₂×C₄×C₇₁	—	2	1	no	—
71	C₂×C₄×C₇₁	—	2	1	no	—
72	C₂×C₄×C₇₁	—	2	1	no	—
73	C₂×C₄×C₇₁	—	2	1	no	—
74	C₂×C₄×C₇₁	—	2	1	no	—
75	C₂×C₄×C₇₁	—	2	1	no	—
76	C₂×C₄×C₇₁	—	2	1	no	—
77	C₂×C₄×C₇₁	—	2	1	no	—
78	C₂×C₄×C₇₁	—	2	1	no	—
79	C₂×C₄×C₇₁	—	2	1	no	—
80	C₂×C₄×C₇₁	—	4	1	no	—
81	C₂×C₄×C₇₁	—	4	1	no	—
82	C₂×C₄×C₇₁	—	4	1	no	—
83	C₂×C₄×C₇₁	#99	1	1	yes	characteristic
84	C₂×C₄×C₇₁	#99	1	1	yes	characteristic
85	C₂×C₄×C₇₁	#104	1	2	yes	—
86	C₂×C₄×C₇₁	#104	1	2	yes	—
87	C₈×C₇₁	—	4	4	no	—
88	C₇₁⋊C₈	—	4	2	no	—
89	C₇₁⋊C₈	—	4	4	no	—
90	C₇₁⋊C₈	—	8	1	no	—
91	C₂²×C₇₁	—	2	1	no	—
92	C₂²×C₇₁	—	2	1	no	—
93	C₂²×C₇₁	#339	1	1	yes	characteristic
94	C₂²×C₇₁	#339	1	1	yes	characteristic
95	C₂²×C₇₁	#342	1	1	yes	characteristic
96	C₄×C₇₁	—	2	2	no	—
97	C₄×C₇₁	—	2	2	no	—
98	C₄×C₇₁	—	2	1	no	—
99	C₄×C₇₁	—	4	1	no	—
100	C₄×C₇₁	—	4	1	no	—
101	C₄×C₇₁	—	4	1	no	—
102	C₄×C₇₁	—	4	1	no	—
103	C₄×C₇₁	#355	1	1	yes	characteristic
104	C₄×C₇₁	#356	1	1	yes	characteristic
105	#7284	—	71	1	no	maximal, Sylow
106	C₂×C₇₁	—	2	1	no	—
107	C₂×C₇₁	—	2	1	no	—
108	C₂×C₇₁	#1297	1	1	yes	characteristic
109	C₂×C₇₁	#1392	1	1	yes	characteristic
110	C₂×C₇₁	#1394	1	1	yes	characteristic
111	#1985	—	71	1	no	—
112	C₄×#101	—	71	1	no	—
113	C₄×#101	D₇₁	1	1	yes	characteristic
114	C₇₁	#7284	1	1	yes	characteristic, complemented, Sylow
115	C₂×C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
116	C₂×C₄⋊C₈	—	71	1	no	—
117	C₂×#101	—	2	1	no	—
118	C₂×#101	—	142	1	no	—
119	C₄×M₁₆	—	2	1	no	—
120	C₄×M₁₆	—	142	1	no	—
121	C₂×#97	—	71	1	no	—
122	#343	—	71	2	no	—
123	#343	Dic₇₁	1	2	yes	—
124	#336	—	142	1	no	—
125	#336	—	142	1	no	—
126	C₂²×Q₈	C₂²⋊D₇₁	1	1	yes	characteristic
127	C₂×M₁₆	—	2	2	no	—
128	C₂×M₁₆	—	142	2	no	—
129	C₂²×C₈	—	142	1	no	—
130	C₂²×C₈	—	142	1	no	—
131	C₄×Q₈	—	4	1	no	—
132	C₄×Q₈	—	4	1	no	—
133	C₂×C₄⋊C₄	Dic₁₄₂	1	1	yes	characteristic
134	C₂×C₄⋊C₄	C₄×D₇₁	1	1	yes	characteristic
135	C₂×C₄⋊C₄	C₂×Dic₇₁	1	1	yes	characteristic
136	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
137	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
138	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
139	C₄⋊C₈	—	284	1	no	—
140	#101	—	2	4	no	—
141	#101	—	142	4	no	—
142	#97	—	4	1	no	—
143	#97	—	284	1	no	—
144	#97	—	284	1	no	—
145	C₄×C₈	—	4	1	no	—
146	C₄×C₈	—	284	1	no	—
147	C₂×Q₈	—	2	2	no	—
148	C₂×Q₈	—	2	2	no	—
149	C₂×Q₈	—	4	1	no	—
150	C₂²×C₄	C₄×Dic₇₁	1	1	yes	characteristic
151	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
152	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
153	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
154	C₂²×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
155	C₂²×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
156	C₂²×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
157	M₁₆	—	2	4	no	—
158	M₁₆	—	4	2	no	—
159	M₁₆	—	284	2	no	—
160	M₁₆	—	142	4	no	—
161	C₂×C₈	—	4	2	no	—
162	C₂×C₈	—	284	2	no	—
163	C₂×C₈	—	284	1	no	—
164	C₂×C₈	—	284	1	no	—
165	C₂×C₈	—	284	2	no	—
166	C₂×C₈	—	284	1	no	—
167	C₂×C₈	—	142	2	no	—
168	C₄⋊C₄	—	4	1	no	—
169	C₄⋊C₄	—	4	1	no	—
170	C₄⋊C₄	—	4	1	no	—
171	C₄²	—	2	1	no	—
172	C₄²	—	2	1	no	—
173	C₄²	—	2	1	no	—
174	C₄²	—	2	1	no	—
175	C₄²	—	2	1	no	—
176	C₄²	—	2	1	no	—
177	C₂³	C₇₁⋊#95	1	1	yes	characteristic
178	Q₈	—	4	4	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	#14556042	1	1	yes	characteristic
195	C₂×C₄	#14556042	1	1	yes	characteristic
196	C₂×C₄	#14556055	1	2	yes	—
197	C₂×C₄	#14556056	1	2	yes	—
198	C₈	—	4	4	no	—
199	C₈	—	284	2	no	—
200	C₈	—	284	4	no	—
201	C₈	—	568	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	2	1	no	—
204	C₂²	#17985758	1	1	yes	characteristic
205	C₂²	#17985759	1	1	yes	characteristic
206	C₂²	#17985765	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	2	2	no	—
209	C₄	—	2	2	no	—
210	C₄	—	4	1	no	—
211	C₄	—	4	1	no	—
212	C₄	—	4	1	no	—
213	C₄	—	4	1	no	—
214	C₄	#17985795	1	1	yes	characteristic
215	C₄	#17985797	1	1	yes	characteristic
216	C₂	—	2	1	no	—
217	C₂	—	2	1	no	—
218	C₂	#22586226	1	1	yes	characteristic
219	C₂	#22586462	1	1	yes	characteristic
220	C₂	#22586466	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22589281

C₂

yes

maximal, characteristic

C₄×#14556047

C₂

yes

maximal, characteristic

C₄×#101×C₇₁

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₇₁⋊C₈)

C₄

yes

—

—

C₄

yes

characteristic

#17985768

C₄

yes

—

#17985754

—

#17985754

—

C₂×C₄×Q₈×C₇₁

C₂²

yes

characteristic

C₂×#101×C₇₁

—

C₄×M₁₆×C₇₁

—

#343×C₇₁

C₄

yes

—

C₂×#11765849

—

C₂²×C₇₁⋊C₈

—

C₂²×C₇₁⋊C₈

—

C₄⋊(C₇₁⋊C₈)

—

#14556047

—

#14556039

—

#14556039

—

C₄×C₇₁⋊C₈

—

C₂²×Q₈×C₇₁

D₄

yes

characteristic

C₂×M₁₆×C₇₁

—

C₄×Q₈×C₇₁

—

C₄×Q₈×C₇₁

—

C₂×C₄⋊C₄×C₇₁

C₂×C₄

yes

characteristic

C₂×C₄⋊C₄×C₇₁

C₂×C₄

yes

characteristic

C₂×C₄⋊C₄×C₇₁

Q₈

yes

characteristic

C₂×C₄²×C₇₁

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₁

D₄

yes

characteristic

C₂×C₄²×C₇₁

D₄

yes

characteristic

#101×C₇₁

—

#97×C₇₁

—

C₄×C₈×C₇₁

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂²×C₄×C₇₁

C₄²

yes

characteristic

C₂²×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₁

C₄⋊C₄

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₂³×C₇₁

#95

yes

characteristic

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

—

C₂×C₄×C₇₁

#104

yes

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

100

C₄×C₇₁

—

101

C₄×C₇₁

—

102

C₄×C₇₁

—

103

C₄×C₇₁

#355

yes

characteristic

104

C₄×C₇₁

yes

105

—

106

C₂×C₇₁

—

107

C₂×C₇₁

—

108

C₂×C₇₁

#1297

yes

characteristic

109

C₂×C₇₁

#1392

yes

characteristic

110

C₂×C₇₁

#1394

yes

characteristic

111

#1985

—

112

C₄×#101

—

113

C₄×#101

D₇₁

yes

characteristic

114

C₇₁

#7284

yes

characteristic, complemented, Sylow

115

C₂×C₄×Q₈

C₂×D₇₁

yes

characteristic

116

C₂×C₄⋊C₈

—

117

C₂×#101

—

118

C₂×#101

—

142

—

119

C₄×M₁₆

—

120

C₄×M₁₆

—

142

—

121

C₂×#97

—

122

#343

—

123

#343

Dic₇₁

yes

—

124

#336

—

142

—

125

#336

—

142

—

126

C₂²×Q₈

C₂²⋊D₇₁

yes

characteristic

127

C₂×M₁₆

—

128

C₂×M₁₆

—

142

—

129

C₂²×C₈

—

142

—

130

C₂²×C₈

—

142

—

131

C₄×Q₈

—

132

C₄×Q₈

—

133

C₂×C₄⋊C₄

Dic₁₄₂

yes

characteristic

134

C₂×C₄⋊C₄

C₄×D₇₁

yes

characteristic

135

C₂×C₄⋊C₄

C₂×Dic₇₁

yes

characteristic

136

C₂×C₄²

C₄×D₇₁

yes

characteristic

137

C₂×C₄²

D₂₈₄

yes

characteristic

138

C₂×C₄²

C₂²⋊D₇₁

yes

characteristic

139

C₄⋊C₈

—

284

—

140

#101

—

141

#101

—

142

—

142

#97

—

143

#97

—

284

—

144

#97

—

284

—

145

C₄×C₈

—

146

C₄×C₈

—

284

—

147

C₂×Q₈

—

148

C₂×Q₈

—

149

C₂×Q₈

—

150

C₂²×C₄

C₄×Dic₇₁

yes

characteristic

151

C₂²×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

152

C₂²×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

153

C₂²×C₄

C₂²⋊Dic₇₁

yes

characteristic

154

C₂²×C₄

Dic₇₁⋊C₄

yes

characteristic

155

C₂²×C₄

Dic₇₁⋊C₄

yes

characteristic

156

C₂²×C₄

C₄⋊Dic₇₁

yes

characteristic

157

M₁₆

—

158

M₁₆

—

159

M₁₆

—

284

—

160

M₁₆

—

142

—

161

C₂×C₈

—

162

C₂×C₈

—

284

—

163

C₂×C₈

—

284

—

164

C₂×C₈

—

284

—

165

C₂×C₈

—

284

—

166

C₂×C₈

—

284

—

167

C₂×C₈

—

142

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄²

—

172

C₄²

—

173

C₄²

—

174

C₄²

—

175

C₄²

—

176

C₄²

—

177

C₂³

C₇₁⋊#95

yes

characteristic

178

Q₈

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

#14556042

yes

characteristic

195

C₂×C₄

#14556042

yes

characteristic

196

C₂×C₄

#14556055

yes

—

197

C₂×C₄

#14556056

yes

—

198

C₈

—

199

C₈

—

284

—

200

C₈

—

284

—

201

C₈

—

568

—

202

C₂²

—

203

C₂²

—

204

C₂²

#17985758

yes

characteristic

205

C₂²

#17985759

yes

characteristic

206

C₂²

#17985765

yes

characteristic

207

C₄

—

208

C₄

—

209

C₄

—

210

C₄

—

211

C₄

—

212

C₄

—

213

C₄

—

214

C₄

#17985795

yes

characteristic

215

C₄

#17985797

yes

characteristic

216

C₂

—

217

C₂

—

218

C₂

#22586226

yes

characteristic

219

C₂

#22586462

yes

characteristic

220

C₂

#22586466

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556042
commutator	C₂²×C₄×C₇₁	C₄²
Frattini	C₂×C₄×Q₈	C₂×D₇₁
Fitting	C₄×#101×C₇₁	C₂
socle	C₂²×C₇₁	#342
prime radical	C₂³×C₇₁	#95
cosocle	C₂×C₄×Q₈×C₇₁	C₂²
hypercenter	C₄×#101	D₇₁
hypocenter	C₇₁	#7284
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂³×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ C₄×#101

subgroup

G/H

center

C₂×C₄

#14556042

commutator

C₂²×C₄×C₇₁

C₄²

Frattini

C₂×C₄×Q₈

C₂×D₇₁

Fitting

C₄×#101×C₇₁

C₂

C₂²×C₇₁

C₂³×C₇₁

C₂×C₄×Q₈×C₇₁

C₂²

D₇₁

C₇₁

G ▷ C₂²×C₄×C₇₁ ▷ C₂³×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ C₄×#101

N	H ≅ G/N	type
114. C₇₁	#7284	semidirect

H ≅ G/N

type

114. C₇₁

#7284

semidirect

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

subgroup	index
#22589281	2
C₄×#14556047	2
C₄×#101×C₇₁	2
#7284	71

subgroup

index

#22589281

C₄×#14556047

C₄×#101×C₇₁

#7284

Subgroups — #31884667

Conjugacy classes of subgroups

Complements

Maximal subgroups