Group #31891237 — subgroups

Subgroups — #31891237

Order 18176.

438 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 227 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	#22588500	C₂	1	1	yes	maximal, characteristic
2	#22588497	C₂	1	1	yes	maximal, characteristic
3	#22588497	C₂	1	2	yes	maximal
4	#22588468	C₂	1	2	yes	maximal
5	C₂×#322×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556062	—	2	2	no	—
7	C₂×#14556060	—	2	4	no	—
8	C₂×#14556060	—	2	2	no	—
9	#17985869	C₂²	1	1	yes	characteristic
10	#17985869	C₂²	1	2	yes	—
11	#17985869	C₂²	1	1	yes	characteristic
12	#17985762	—	2	4	no	—
13	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	#322×C₇₁	C₄	1	4	yes	complemented
16	C₂×C₄⋊Dic₇₁	—	2	2	no	—
17	C₂×C₄⋊Dic₇₁	—	2	2	no	—
18	C₂×C₄⋊Dic₇₁	—	2	2	no	—
19	C₂×C₄⋊Dic₇₁	—	2	2	no	—
20	C₂×C₄⋊Dic₇₁	—	2	2	no	—
21	C₂×C₄⋊Dic₇₁	—	2	1	no	—
22	C₂×C₄⋊Dic₇₁	—	2	1	no	—
23	#14556062	—	4	4	no	—
24	#14556060	—	4	8	no	—
25	#14556060	—	4	4	no	—
26	C₂²×C₈×C₇₁	D₄	1	4	yes	—
27	C₂²×C₈×C₇₁	D₄	1	2	yes	—
28	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
29	C₄×C₈×C₇₁	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₇₁	Q₈	1	4	yes	—
32	C₄×C₈×C₇₁	Q₈	1	2	yes	—
33	C₂²×Dic₇₁	—	4	1	no	—
34	C₂²×Dic₇₁	—	4	1	no	—
35	C₂²×Dic₇₁	—	4	2	no	—
36	C₄⋊Dic₇₁	—	4	4	no	—
37	C₄⋊Dic₇₁	—	4	2	no	—
38	C₄⋊Dic₇₁	—	4	2	no	—
39	C₄⋊Dic₇₁	—	4	4	no	—
40	C₄⋊Dic₇₁	—	4	4	no	—
41	C₄⋊Dic₇₁	—	4	4	no	—
42	C₄⋊Dic₇₁	—	4	4	no	—
43	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
45	C₂×C₈×C₇₁	—	2	4	no	—
46	C₂×C₈×C₇₁	—	2	4	no	—
47	C₂×C₈×C₇₁	C₄⋊C₄	1	8	yes	—
48	C₂×C₈×C₇₁	C₄⋊C₄	1	4	yes	—
49	C₂×C₈×C₇₁	SD₁₆	1	4	yes	—
50	C₂×C₈×C₇₁	SD₁₆	1	4	yes	—
51	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
53	C₄²×C₇₁	C₂×Q₈	1	2	yes	—
54	C₂³×C₇₁	#127	1	1	yes	characteristic
55	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₇₁	C₄⋊Q₈	1	2	yes	—
59	C₂×C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
60	C₂×C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
61	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	4	yes	—
62	C₂×C₄×C₇₁	#137	1	4	yes	—
63	C₂×Dic₇₁	—	8	2	no	—
64	C₂×Dic₇₁	—	8	1	no	—
65	C₂×Dic₇₁	—	8	1	no	—
66	C₂×Dic₇₁	—	8	2	no	—
67	C₂×Dic₇₁	—	8	2	no	—
68	C₂×Dic₇₁	—	8	2	no	—
69	C₂×Dic₇₁	—	8	2	no	—
70	C₈×C₇₁	—	2	8	no	—
71	C₈×C₇₁	#106	1	8	yes	complemented
72	C₂²×C₇₁	#384	1	1	yes	characteristic
73	C₂²×C₇₁	#492	1	1	yes	characteristic
74	C₂²×C₇₁	#492	1	1	yes	characteristic
75	C₂²×C₇₁	#496	1	2	yes	—
76	C₂²×C₇₁	#497	1	2	yes	—
77	C₄×C₇₁	C₂×#106	1	2	yes	—
78	C₄×C₇₁	#428	1	4	yes	—
79	C₄×C₇₁	#498	1	2	yes	—
80	C₄×C₇₁	#501	1	4	yes	—
81	Dic₇₁	—	16	2	no	—
82	Dic₇₁	—	16	1	no	—
83	Dic₇₁	—	16	1	no	—
84	#8819	—	71	1	no	maximal, Sylow
85	C₂×C₇₁	#1714	1	2	yes	—
86	C₂×C₇₁	#1715	1	1	yes	characteristic
87	C₂×C₇₁	#1716	1	1	yes	characteristic
88	C₂×C₇₁	#1844	1	1	yes	characteristic
89	C₂×C₇₁	#1847	1	2	yes	—
90	#1845	—	71	1	no	—
91	#1844	—	71	1	no	—
92	#1844	—	71	2	no	—
93	#1834	—	71	2	no	—
94	C₂×#322	D₇₁	1	1	yes	characteristic
95	C₇₁	#8819	1	1	yes	characteristic, complemented, Sylow
96	C₂×#107	—	142	2	no	—
97	C₂×#106	—	142	4	no	—
98	C₂×#106	—	142	2	no	—
99	C₂×C₄×C₈	C₂×D₇₁	1	2	yes	—
100	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
101	#384	—	71	1	no	—
102	#384	—	71	2	no	—
103	#384	—	71	1	no	—
104	#340	—	142	4	no	—
105	#322	Dic₇₁	1	4	yes	complemented
106	C₂²×C₈	D₂₈₄	1	4	yes	—
107	C₂²×C₈	D₂₈₄	1	2	yes	—
108	C₂×C₄⋊C₄	—	142	2	no	—
109	C₂×C₄⋊C₄	—	142	2	no	—
110	C₂×C₄⋊C₄	—	142	2	no	—
111	C₂×C₄⋊C₄	—	142	2	no	—
112	C₂×C₄⋊C₄	—	142	2	no	—
113	C₂×C₄⋊C₄	—	142	1	no	—
114	C₂×C₄⋊C₄	—	142	1	no	—
115	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
116	#107	—	284	4	no	—
117	#106	—	284	8	no	—
118	#106	—	284	4	no	—
119	C₄×C₈	Dic₁₄₂	1	4	yes	—
120	C₄×C₈	Dic₁₄₂	1	2	yes	—
121	C₄×C₈	C₂×Dic₇₁	1	4	yes	—
122	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
123	C₂²×C₄	—	284	1	no	—
124	C₂²×C₄	—	284	1	no	—
125	C₂²×C₄	—	284	2	no	—
126	C₂²×C₄	C₂×D₂₈₄	1	2	yes	—
127	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	2	4	no	—
130	C₂×C₈	C₄⋊Dic₇₁	1	8	yes	—
131	C₂×C₈	C₄⋊Dic₇₁	1	4	yes	—
132	C₂×C₈	#11765854	1	4	yes	—
133	C₂×C₈	#11765854	1	4	yes	—
134	C₄⋊C₄	—	284	4	no	—
135	C₄⋊C₄	—	284	4	no	—
136	C₄⋊C₄	—	284	2	no	—
137	C₄⋊C₄	—	284	2	no	—
138	C₄⋊C₄	—	284	4	no	—
139	C₄⋊C₄	—	284	4	no	—
140	C₄⋊C₄	—	284	4	no	—
141	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
142	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
143	C₄²	C₂×Dic₁₄₂	1	2	yes	—
144	C₂³	#14556125	1	1	yes	characteristic
145	C₂×C₄	—	568	2	no	—
146	C₂×C₄	—	568	1	no	—
147	C₂×C₄	—	568	1	no	—
148	C₂×C₄	—	568	2	no	—
149	C₂×C₄	—	568	2	no	—
150	C₂×C₄	—	568	2	no	—
151	C₂×C₄	—	568	2	no	—
152	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄⋊Dic₇₁	1	2	yes	—
154	C₂×C₄	#14556128	1	1	yes	characteristic
155	C₂×C₄	#14556128	1	2	yes	—
156	C₂×C₄	C₂×#11765854	1	2	yes	—
157	C₂×C₄	C₂×#11765854	1	2	yes	—
158	C₂×C₄	#14556156	1	4	yes	—
159	C₂×C₄	#14556161	1	4	yes	—
160	C₈	—	2	8	no	—
161	C₈	#14556060	1	8	yes	complemented
162	C₂²	#17985869	1	1	yes	characteristic
163	C₂²	#17986372	1	1	yes	characteristic
164	C₂²	#17986372	1	1	yes	characteristic
165	C₂²	#17986388	1	2	yes	—
166	C₂²	#17986393	1	2	yes	—
167	C₄	—	1136	2	no	—
168	C₄	—	1136	1	no	—
169	C₄	—	1136	1	no	—
170	C₄	C₂×#14556060	1	2	yes	—
171	C₄	#17986041	1	4	yes	—
172	C₄	#17986397	1	2	yes	—
173	C₄	#17986408	1	4	yes	—
174	C₂	#22587962	1	2	yes	—
175	C₂	#22587968	1	1	yes	characteristic
176	C₂	#22587973	1	1	yes	characteristic
177	C₂	#22588497	1	1	yes	characteristic
178	C₂	#22588508	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22588500

C₂

yes

maximal, characteristic

#22588497

C₂

yes

maximal, characteristic

#22588497

C₂

yes

maximal

#22588468

C₂

yes

maximal

C₂×#322×C₇₁

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

characteristic

#17985869

C₂²

yes

—

#17985869

C₂²

yes

characteristic

#17985762

—

C₂×C₄×C₈×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#322×C₇₁

C₄

yes

complemented

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

—

#14556062

—

#14556060

—

#14556060

—

C₂²×C₈×C₇₁

D₄

yes

—

C₂²×C₈×C₇₁

D₄

yes

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

Q₈

yes

—

C₄×C₈×C₇₁

Q₈

yes

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₂×C₈×C₇₁

SD₁₆

yes

—

C₂×C₈×C₇₁

SD₁₆

yes

—

C₄²×C₇₁

C₂²×C₄

yes

characteristic

C₄²×C₇₁

C₂×Q₈

yes

characteristic

C₄²×C₇₁

C₂×Q₈

yes

—

C₂³×C₇₁

#127

yes

characteristic

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₇₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₁

C₄⋊Q₈

yes

—

C₂×C₄×C₇₁

C₂×SD₁₆

yes

—

C₂×C₄×C₇₁

C₂×SD₁₆

yes

—

C₂×C₄×C₇₁

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇₁

#137

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₈×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

—

C₂²×C₇₁

#497

yes

—

C₄×C₇₁

C₂×#106

yes

—

C₄×C₇₁

#428

yes

—

C₄×C₇₁

#498

yes

—

C₄×C₇₁

#501

yes

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

#8819

—

maximal, Sylow

C₂×C₇₁

#1714

yes

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

—

—

—

—

D₇₁

yes

characteristic

C₇₁

#8819

yes

characteristic, complemented, Sylow

C₂×#107

—

142

—

C₂×#106

—

142

—

C₂×#106

—

142

—

C₂×C₄×C₈

C₂×D₇₁

yes

—

100

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

101

#384

—

102

#384

—

103

#384

—

104

#340

—

142

—

105

#322

Dic₇₁

yes

complemented

106

C₂²×C₈

D₂₈₄

yes

—

107

C₂²×C₈

D₂₈₄

yes

—

108

C₂×C₄⋊C₄

—

142

—

109

C₂×C₄⋊C₄

—

142

—

110

C₂×C₄⋊C₄

—

142

—

111

C₂×C₄⋊C₄

—

142

—

112

C₂×C₄⋊C₄

—

142

—

113

C₂×C₄⋊C₄

—

142

—

114

C₂×C₄⋊C₄

—

142

—

115

C₂×C₄²

C₂²×D₇₁

yes

characteristic

116

#107

—

284

—

117

#106

—

284

—

118

#106

—

284

—

119

C₄×C₈

Dic₁₄₂

yes

—

120

C₄×C₈

Dic₁₄₂

yes

—

121

C₄×C₈

C₂×Dic₇₁

yes

—

122

C₄×C₈

C₂×Dic₇₁

yes

—

123

C₂²×C₄

—

284

—

124

C₂²×C₄

—

284

—

125

C₂²×C₄

—

284

—

126

C₂²×C₄

C₂×D₂₈₄

yes

—

127

C₂²×C₄

C₂×D₂₈₄

yes

characteristic

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

C₄⋊Dic₇₁

yes

—

131

C₂×C₈

C₄⋊Dic₇₁

yes

—

132

C₂×C₈

#11765854

yes

—

133

C₂×C₈

#11765854

yes

—

134

C₄⋊C₄

—

284

—

135

C₄⋊C₄

—

284

—

136

C₄⋊C₄

—

284

—

137

C₄⋊C₄

—

284

—

138

C₄⋊C₄

—

284

—

139

C₄⋊C₄

—

284

—

140

C₄⋊C₄

—

284

—

141

C₄²

C₂²×Dic₇₁

yes

characteristic

142

C₄²

C₂×Dic₁₄₂

yes

characteristic

143

C₄²

C₂×Dic₁₄₂

yes

—

144

C₂³

#14556125

yes

characteristic

145

C₂×C₄

—

568

—

146

C₂×C₄

—

568

—

147

C₂×C₄

—

568

—

148

C₂×C₄

—

568

—

149

C₂×C₄

—

568

—

150

C₂×C₄

—

568

—

151

C₂×C₄

—

568

—

152

C₂×C₄

C₂×C₄⋊Dic₇₁

yes

characteristic

153

C₂×C₄

C₂×C₄⋊Dic₇₁

yes

—

154

C₂×C₄

#14556128

yes

characteristic

155

C₂×C₄

#14556128

yes

—

156

C₂×C₄

C₂×#11765854

yes

—

157

C₂×C₄

C₂×#11765854

yes

—

158

C₂×C₄

#14556156

yes

—

159

C₂×C₄

#14556161

yes

—

160

C₈

—

161

C₈

#14556060

yes

complemented

162

C₂²

#17985869

yes

characteristic

163

C₂²

#17986372

yes

characteristic

164

C₂²

#17986372

yes

characteristic

165

C₂²

#17986388

yes

—

166

C₂²

#17986393

yes

—

167

C₄

—

1136

—

168

C₄

—

1136

—

169

C₄

—

1136

—

170

C₄

C₂×#14556060

yes

—

171

C₄

#17986041

yes

—

172

C₄

#17986397

yes

—

173

C₄

#17986408

yes

—

174

C₂

#22587962

yes

—

175

C₂

#22587968

yes

characteristic

176

C₂

#22587973

yes

characteristic

177

C₂

#22588497

yes

characteristic

178

C₂

#22588508

yes

—

	subgroup	G/H
center	C₂³	#14556125
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₂×#322×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	#127
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₂×#322	D₇₁
hypocenter	C₇₁	#8819
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂³

#14556125

commutator

C₄²×C₇₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₂×#322×C₇₁

C₂

socle, prime radical

C₂³×C₇₁

#127

cosocle

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
15. #322×C₇₁	C₄	semidirect
71. C₈×C₇₁	#106	semidirect
95. C₇₁	#8819	semidirect
105. #322	Dic₇₁	semidirect
161. C₈	#14556060	semidirect

H ≅ G/N

type

15. #322×C₇₁

C₄

semidirect

71. C₈×C₇₁

#106

semidirect

95. C₇₁

#8819

semidirect

105. #322

Dic₇₁

semidirect

161. C₈

#14556060

semidirect

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

subgroup	index
#22588500	2
#22588497	2
#22588468	2
C₂×#322×C₇₁	2
#8819	71

subgroup

index

C₂×#322×C₇₁

Subgroups — #31891237

Conjugacy classes of subgroups

Complements

Maximal subgroups