Group #31888609 — subgroups

Subgroups — #31888609

Order 18176.

414 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 175 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	C₄²×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#22587086	C₂	1	2	yes	maximal
3	#22586624	C₂	1	2	yes	maximal
4	#22586173	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1276	C₂	1	1	yes	maximal, characteristic
6	#17986052	C₂²	1	2	yes	—
7	C₂×C₄×C₇₁⋊C₈	C₄	1	2	yes	—
8	C₂×C₄×C₇₁⋊C₈	C₄	1	2	yes	—
9	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
11	#17985751	C₄	1	4	yes	complemented
12	#17985749	C₄	1	4	yes	complemented
13	#17985721	—	2	2	no	—
14	C₈×C₇₁⋊C₈	—	2	2	no	—
15	#431×C₇₁	C₂²	1	2	yes	—
16	C₄³×C₇₁	C₂²	1	1	yes	characteristic
17	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
20	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	4	yes	—
21	C₂²⋊(C₇₁⋊C₈)	—	2	2	no	—
22	C₄×C₇₁⋊C₈	—	2	4	no	—
23	C₄×C₇₁⋊C₈	—	2	2	no	—
24	C₄×C₇₁⋊C₈	C₈	1	8	yes	complemented
25	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
26	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
27	C₄×C₇₁⋊C₈	D₄	1	1	yes	characteristic
28	C₄×C₇₁⋊C₈	D₄	1	1	yes	characteristic
29	C₄×C₇₁⋊C₈	Q₈	1	2	yes	—
30	C₂×C₄²×C₇₁	—	2	2	no	—
31	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₇₁	C₂×C₄	1	4	yes	—
34	C₂²⋊C₈×C₇₁	—	2	2	no	—
35	C₄×C₈×C₇₁	—	2	2	no	—
36	C₂×C₇₁⋊C₈	—	2	2	no	—
37	C₂×C₇₁⋊C₈	—	2	4	no	—
38	C₂×C₇₁⋊C₈	C₄²	1	2	yes	—
39	C₂×C₇₁⋊C₈	C₄²	1	1	yes	characteristic
40	C₂×C₇₁⋊C₈	C₄²	1	1	yes	characteristic
41	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
42	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
43	C₂×C₇₁⋊C₈	C₂×C₈	1	4	yes	—
44	C₂×C₇₁⋊C₈	C₂×C₈	1	4	yes	—
45	C₂×C₇₁⋊C₈	M₁₆	1	4	yes	—
46	C₂²×C₄×C₇₁	—	2	2	no	—
47	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₇₁	#41	1	2	yes	—
49	C₂×C₈×C₇₁	—	2	4	no	—
50	C₄²×C₇₁	—	2	8	no	—
51	C₄²×C₇₁	—	2	2	no	—
52	C₄²×C₇₁	C₂×C₈	1	4	yes	—
53	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
54	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
56	C₂³×C₇₁	#117	1	1	yes	characteristic
57	C₂×C₄×C₇₁	—	2	8	no	—
58	C₂×C₄×C₇₁	—	2	4	no	—
59	C₂×C₄×C₇₁	—	2	4	no	—
60	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
61	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
63	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
64	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
65	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
66	C₈×C₇₁	—	4	4	no	—
67	C₇₁⋊C₈	—	2	2	no	—
68	C₇₁⋊C₈	—	4	4	no	—
69	C₇₁⋊C₈	C₄×C₈	1	2	yes	—
70	C₇₁⋊C₈	C₄×C₈	1	2	yes	—
71	C₇₁⋊C₈	C₄⋊C₈	1	4	yes	—
72	C₇₁⋊C₈	C₄⋊C₈	1	4	yes	—
73	C₂²×C₇₁	—	2	2	no	—
74	C₂²×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₇₁	#431	1	2	yes	—
76	C₄×C₇₁	—	2	8	no	—
77	C₄×C₇₁	—	2	2	no	—
78	C₄×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
79	C₄×C₇₁	C₄×M₁₆	1	1	yes	characteristic
80	C₄×C₇₁	C₂×C₄⋊C₈	1	2	yes	—
81	C₄×C₇₁	#443	1	4	yes	—
82	#8266	—	71	1	no	maximal, Sylow
83	C₂×C₇₁	—	2	2	no	—
84	C₂×C₇₁	C₄×C₄⋊C₈	1	1	yes	characteristic
85	C₂×C₇₁	#1772	1	2	yes	—
86	C₄²×C₈	—	71	1	no	—
87	#1564	—	71	2	no	—
88	#1450	—	71	2	no	—
89	#1276	—	71	1	no	—
90	#1276	D₇₁	1	1	yes	characteristic
91	C₇₁	#8266	1	1	yes	characteristic, complemented, Sylow
92	#431	—	71	2	no	—
93	#431	C₂×D₇₁	1	2	yes	—
94	C₂×C₄×C₈	—	71	2	no	—
95	C₂×C₄×C₈	—	71	1	no	—
96	C₂×C₄×C₈	—	71	1	no	—
97	C₂×C₄×C₈	—	71	2	no	—
98	C₄³	C₂×D₇₁	1	1	yes	characteristic
99	#335	—	71	4	no	—
100	#334	—	71	4	no	—
101	#322	—	142	2	no	—
102	C₈²	—	142	2	no	—
103	C₂²×C₈	—	71	1	no	—
104	C₂²×C₈	—	71	2	no	—
105	C₂²×C₈	—	71	1	no	—
106	C₂×C₄²	—	2	2	no	—
107	C₂×C₄²	C₄×D₇₁	1	2	yes	—
108	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
109	C₄⋊C₈	—	71	4	no	—
110	C₄⋊C₈	C₄×D₇₁	1	4	yes	—
111	C₂²⋊C₈	—	2	2	no	—
112	C₂²⋊C₈	—	142	2	no	—
113	C₄×C₈	—	2	2	no	—
114	C₄×C₈	—	71	8	no	—
115	C₄×C₈	—	71	2	no	—
116	C₄×C₈	—	71	2	no	—
117	C₄×C₈	—	71	1	no	—
118	C₄×C₈	—	71	1	no	—
119	C₄×C₈	—	71	2	no	—
120	C₄×C₈	—	142	4	no	—
121	C₄×C₈	—	142	2	no	—
122	C₂²×C₄	—	2	2	no	—
123	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
124	C₂²×C₄	#11765864	1	2	yes	—
125	C₂×C₈	—	2	4	no	—
126	C₂×C₈	—	71	1	no	—
127	C₂×C₈	—	71	1	no	—
128	C₂×C₈	—	71	2	no	—
129	C₂×C₈	—	71	2	no	—
130	C₂×C₈	—	71	4	no	—
131	C₂×C₈	—	71	4	no	—
132	C₂×C₈	—	71	4	no	—
133	C₂×C₈	—	71	2	no	—
134	C₂×C₈	—	142	4	no	—
135	C₂×C₈	—	142	2	no	—
136	C₄²	—	2	8	no	—
137	C₄²	—	2	2	no	—
138	C₄²	C₈×D₇₁	1	4	yes	—
139	C₄²	C₂×C₄×D₇₁	1	2	yes	—
140	C₄²	D₄×D₇₁	1	1	yes	characteristic
141	C₄²	Q₈×D₇₁	1	1	yes	characteristic
142	C₂³	#14556084	1	1	yes	characteristic
143	C₂×C₄	—	2	8	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	C₄²×D₇₁	1	1	yes	characteristic
147	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
148	C₂×C₄	(C₂²⋊D₇₁)⋊C₄	1	2	yes	—
149	C₂×C₄	#14556093	1	2	yes	—
150	C₂×C₄	C₂×C₈×D₇₁	1	2	yes	—
151	C₂×C₄	M₁₆×D₇₁	1	2	yes	—
152	C₈	—	4	4	no	—
153	C₈	—	284	4	no	—
154	C₈	—	71	2	no	—
155	C₈	—	71	2	no	—
156	C₈	—	71	4	no	—
157	C₈	—	71	4	no	—
158	C₈	—	142	2	no	—
159	C₂²	—	2	2	no	—
160	C₂²	#17985850	1	1	yes	characteristic
161	C₂²	#17986049	1	2	yes	—
162	C₄	—	2	8	no	—
163	C₄	—	2	2	no	—
164	C₄	C₄×C₈×D₇₁	1	1	yes	characteristic
165	C₄	#17985951	1	1	yes	characteristic
166	C₄	C₄⋊C₈×D₇₁	1	2	yes	—
167	C₄	#17986113	1	4	yes	—
168	C₂	—	2	2	no	—
169	C₂	#22588188	1	1	yes	characteristic
170	C₂	#22588218	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄²×C₇₁⋊C₈

C₂

yes

maximal, characteristic

#22587086

C₂

yes

maximal

#22586624

C₂

yes

maximal

#22586173

C₂

yes

maximal, characteristic

C₇₁×#1276

C₂

yes

maximal, characteristic

#17986052

C₂²

yes

—

C₂×C₄×C₇₁⋊C₈

C₄

yes

—

C₂×C₄×C₇₁⋊C₈

C₄

yes

—

C₂×C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₇₁⋊C₈

C₂²

yes

characteristic

#17985751

C₄

yes

complemented

#17985749

C₄

yes

complemented

#17985721

—

C₈×C₇₁⋊C₈

—

#431×C₇₁

C₂²

yes

—

C₄³×C₇₁

C₂²

yes

characteristic

C₂²×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₇₁⋊C₈

C₂×C₄

yes

—

C₄⋊(C₇₁⋊C₈)

C₂×C₄

yes

—

C₂²⋊(C₇₁⋊C₈)

—

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

C₈

yes

complemented

C₄×C₇₁⋊C₈

C₂×C₄

yes

—

C₄×C₇₁⋊C₈

C₂×C₄

yes

—

C₄×C₇₁⋊C₈

D₄

yes

characteristic

C₄×C₇₁⋊C₈

D₄

yes

characteristic

C₄×C₇₁⋊C₈

Q₈

yes

—

C₂×C₄²×C₇₁

—

C₂×C₄²×C₇₁

C₂×C₄

yes

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄⋊C₈×C₇₁

C₂×C₄

yes

—

C₂²⋊C₈×C₇₁

—

C₄×C₈×C₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₄²

yes

—

C₂×C₇₁⋊C₈

C₄²

yes

characteristic

C₂×C₇₁⋊C₈

C₄²

yes

characteristic

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

—

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

—

C₂×C₇₁⋊C₈

C₂×C₈

yes

—

C₂×C₇₁⋊C₈

C₂×C₈

yes

—

C₂×C₇₁⋊C₈

M₁₆

yes

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₁

#41

yes

—

C₂×C₈×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

C₂×C₈

yes

—

C₄²×C₇₁

C₂²×C₄

yes

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×Q₈

yes

characteristic

C₂³×C₇₁

#117

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄×D₄

yes

—

C₂×C₄×C₇₁

C₄×Q₈

yes

—

C₂×C₄×C₇₁

C₂²×C₈

yes

—

C₂×C₄×C₇₁

C₂×M₁₆

yes

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

C₄×C₈

yes

—

C₇₁⋊C₈

C₄×C₈

yes

—

C₇₁⋊C₈

C₄⋊C₈

yes

—

C₇₁⋊C₈

C₄⋊C₈

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

C₄×C₄⋊C₄

yes

characteristic

C₂²×C₇₁

#431

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×C₄×C₈

yes

characteristic

C₄×C₇₁

C₄×M₁₆

yes

characteristic

C₄×C₇₁

C₂×C₄⋊C₈

yes

—

C₄×C₇₁

#443

yes

—

#8266

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

C₄×C₄⋊C₈

yes

characteristic

C₂×C₇₁

#1772

yes

—

C₄²×C₈

—

—

—

—

D₇₁

yes

characteristic

C₇₁

#8266

yes

characteristic, complemented, Sylow

#431

—

#431

C₂×D₇₁

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄³

C₂×D₇₁

yes

characteristic

#335

—

100

#334

—

101

#322

—

142

—

102

C₈²

—

142

—

103

C₂²×C₈

—

104

C₂²×C₈

—

105

C₂²×C₈

—

106

C₂×C₄²

—

107

C₂×C₄²

C₄×D₇₁

yes

—

108

C₂×C₄²

C₂²×D₇₁

yes

characteristic

109

C₄⋊C₈

—

110

C₄⋊C₈

C₄×D₇₁

yes

—

111

C₂²⋊C₈

—

112

C₂²⋊C₈

—

142

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₄×C₈

—

142

—

121

C₄×C₈

—

142

—

122

C₂²×C₄

—

123

C₂²×C₄

C₂×C₄×D₇₁

yes

characteristic

124

C₂²×C₄

#11765864

yes

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

142

—

135

C₂×C₈

—

142

—

136

C₄²

—

137

C₄²

—

138

C₄²

C₈×D₇₁

yes

—

139

C₄²

C₂×C₄×D₇₁

yes

—

140

C₄²

D₄×D₇₁

yes

characteristic

141

C₄²

Q₈×D₇₁

yes

characteristic

142

C₂³

#14556084

yes

characteristic

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₄²×D₇₁

yes

characteristic

147

C₂×C₄

C₄⋊C₄×D₇₁

yes

characteristic

148

C₂×C₄

(C₂²⋊D₇₁)⋊C₄

yes

—

149

C₂×C₄

#14556093

yes

—

150

C₂×C₄

C₂×C₈×D₇₁

yes

—

151

C₂×C₄

M₁₆×D₇₁

yes

—

152

C₈

—

153

C₈

—

284

—

154

C₈

—

155

C₈

—

156

C₈

—

157

C₈

—

158

C₈

—

142

—

159

C₂²

—

160

C₂²

#17985850

yes

characteristic

161

C₂²

#17986049

yes

—

162

C₄

—

163

C₄

—

164

C₄

C₄×C₈×D₇₁

yes

characteristic

165

C₄

#17985951

yes

characteristic

166

C₄

C₄⋊C₈×D₇₁

yes

—

167

C₄

#17986113

yes

—

168

C₂

—

169

C₂

#22588188

yes

characteristic

170

C₂

#22588218

yes

—

	subgroup	G/H
center	C₄²	D₄×D₇₁
commutator	C₄×C₇₁	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1276	C₂
socle	C₂²×C₇₁	C₄×C₄⋊C₄
prime radical	C₂³×C₇₁	#117
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1276	D₇₁
hypocenter	C₇₁	#8266
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1276

subgroup

G/H

center

C₄²

D₄×D₇₁

commutator

C₄×C₇₁

C₂×C₄×C₈

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#1276

C₂

socle

C₂²×C₇₁

C₄×C₄⋊C₄

prime radical

C₂³×C₇₁

#117

cosocle

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1276

N	H ≅ G/N	type
11. #17985751	C₄	semidirect
12. #17985749	C₄	semidirect
24. C₄×C₇₁⋊C₈	C₈	semidirect
91. C₇₁	#8266	semidirect

H ≅ G/N

type

11. #17985751

C₄

semidirect

12. #17985749

C₄

semidirect

24. C₄×C₇₁⋊C₈

C₈

semidirect

91. C₇₁

#8266

semidirect

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

subgroup	index
C₄²×C₇₁⋊C₈	2
#22587086	2
#22586624	2
#22586173	2
C₇₁×#1276	2
#8266	71

subgroup

index

C₄²×C₇₁⋊C₈

C₇₁×#1276

Subgroups — #31888609

Conjugacy classes of subgroups

Complements

Maximal subgroups