Group #31911723 — subgroups

Subgroups — #31911723

Order 18176.

244 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 67 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	#22590248	C₂	1	1	yes	maximal, characteristic, complemented
2	#22587697	C₂	1	2	yes	maximal, complemented
3	#22586414	C₂	1	1	yes	maximal, characteristic, complemented
4	#22586386	C₂	1	2	yes	maximal, complemented
5	C₇₁×#2246	C₂	1	1	yes	maximal, characteristic
6	#17986196	—	2	2	no	—
7	C₈×Dic₁₄₂	C₂²	1	1	yes	characteristic
8	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	1	yes	characteristic
9	Dic₇₁⋊C₁₆	—	2	1	no	—
10	#17985774	—	2	1	no	—
11	#17985752	C₂²	1	2	yes	—
12	#17985743	—	2	2	no	—
13	C₂×D₁₆×C₇₁	—	2	1	no	—
14	C₂×D₁₆×C₇₁	—	2	1	no	—
15	#493×C₇₁	C₂²	1	2	yes	—
16	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×#11765853	—	4	2	no	—
18	C₄×Dic₁₄₂	D₄	1	1	yes	characteristic
19	C₂×C₇₁⋊C₁₆	—	2	1	no	—
20	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
21	C₄⋊(C₇₁⋊C₈)	D₄	1	1	yes	characteristic
22	Dic₇₁⋊C₈	—	2	1	no	—
23	#14556049	—	4	2	no	—
24	C₈×Dic₇₁	—	2	1	no	—
25	C₂×D₈×C₇₁	—	2	2	no	—
26	C₂×D₈×C₇₁	—	2	2	no	—
27	#127×C₇₁	—	2	2	no	—
28	D₁₆×C₇₁	—	4	2	no	—
29	D₁₆×C₇₁	—	4	2	no	—
30	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
32	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₄₂	D₈	1	1	yes	characteristic
34	#11765853	—	8	4	no	—
35	C₂×C₇₁⋊C₈	—	2	1	no	—
36	C₂×C₇₁⋊C₈	—	4	2	no	—
37	C₄⋊Dic₇₁	D₈	1	1	yes	characteristic
38	Dic₇₁⋊C₄	—	2	1	no	—
39	C₄×Dic₇₁	—	2	1	no	—
40	C₇₁⋊C₁₆	—	4	1	no	—
41	C₂×D₄×C₇₁	—	4	2	no	—
42	C₂×D₄×C₇₁	—	4	2	no	—
43	C₂×D₄×C₇₁	—	4	2	no	—
44	D₈×C₇₁	—	4	2	no	—
45	D₈×C₇₁	—	4	2	no	—
46	D₈×C₇₁	—	8	2	no	—
47	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
49	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
50	C₁₆×C₇₁	—	2	1	no	—
51	C₁₆×C₇₁	SD₁₆	1	2	yes	complemented
52	C₂³×C₇₁	—	8	2	no	—
53	D₄×C₇₁	—	8	2	no	—
54	D₄×C₇₁	—	8	2	no	—
55	D₄×C₇₁	—	8	2	no	—
56	D₄×C₇₁	—	8	2	no	—
57	D₄×C₇₁	—	16	2	no	—
58	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
59	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
61	C₂×Dic₇₁	—	2	1	no	—
62	C₂×Dic₇₁	—	2	1	no	—
63	Dic₁₄₂	—	2	2	no	—
64	C₈×C₇₁	—	2	1	no	—
65	C₈×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
66	C₈×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
67	C₇₁⋊C₈	—	4	1	no	—
68	C₇₁⋊C₈	—	8	2	no	—
69	C₂²×C₇₁	—	16	2	no	—
70	C₂²×C₇₁	—	16	2	no	—
71	C₂²×C₇₁	—	16	2	no	—
72	C₂²×C₇₁	#466	1	1	yes	characteristic
73	C₄×C₇₁	—	2	1	no	—
74	C₄×C₇₁	#460	1	1	yes	characteristic
75	C₄×C₇₁	#472	1	1	yes	characteristic
76	C₄×C₇₁	#509	1	2	yes	—
77	Dic₇₁	—	4	1	no	—
78	Dic₇₁	—	4	1	no	—
79	#12442	—	71	1	no	maximal, Sylow
80	C₂×C₇₁	—	32	2	no	—
81	C₂×C₇₁	#1668	1	1	yes	characteristic
82	C₂×C₇₁	#2218	1	1	yes	characteristic
83	C₂×C₇₁	#2221	1	1	yes	characteristic
84	#2246	D₇₁	1	1	yes	characteristic
85	#2183	—	71	1	no	—
86	#1668	—	71	2	no	—
87	#1372	—	71	1	no	—
88	#1361	—	71	2	no	—
89	C₇₁	#12442	1	1	yes	characteristic, complemented, Sylow
90	C₂×D₁₆	—	2	1	no	—
91	C₂×D₁₆	—	2	1	no	—
92	#493	C₂×D₇₁	1	2	yes	—
93	#460	—	142	2	no	—
94	C₈×Q₈	—	71	1	no	—
95	C₄⋊C₁₆	—	71	1	no	—
96	C₄⋊C₁₆	—	142	1	no	—
97	#346	—	142	1	no	—
98	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
99	#335	—	71	2	no	—
100	#331	—	142	2	no	—
101	C₂×SD₁₆	—	284	2	no	—
102	C₂×D₈	—	2	2	no	—
103	C₂×D₈	—	2	2	no	—
104	#127	—	2	2	no	—
105	C₄×Q₈	—	71	1	no	—
106	D₁₆	—	4	2	no	—
107	D₁₆	—	4	2	no	—
108	C₂×C₁₆	—	142	1	no	—
109	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
110	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
111	C₄⋊C₈	—	71	1	no	—
112	C₄⋊C₈	—	142	1	no	—
113	C₄⋊C₈	—	142	2	no	—
114	#102	—	284	2	no	—
115	C₄×C₈	—	142	1	no	—
116	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
117	C₂×Q₈	—	71	1	no	—
118	C₂×D₄	—	4	2	no	—
119	C₂×D₄	—	4	2	no	—
120	C₂×D₄	—	4	2	no	—
121	SD₁₆	—	568	4	no	—
122	D₈	—	4	2	no	—
123	D₈	—	4	2	no	—
124	D₈	—	8	2	no	—
125	C₂×C₈	—	284	2	no	—
126	C₂×C₈	—	142	1	no	—
127	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
128	C₂×C₈	#11765864	1	1	yes	characteristic
129	C₄⋊C₄	—	71	1	no	—
130	C₄⋊C₄	—	142	1	no	—
131	C₄²	—	142	1	no	—
132	C₄²	D₄×D₇₁	1	1	yes	characteristic
133	C₁₆	—	2	1	no	—
134	C₁₆	—	284	1	no	—
135	C₁₆	#11765853	1	2	yes	complemented
136	C₂³	—	8	2	no	—
137	Q₈	—	142	2	no	—
138	D₄	—	8	2	no	—
139	D₄	—	8	2	no	—
140	D₄	—	8	2	no	—
141	D₄	—	8	2	no	—
142	D₄	—	16	2	no	—
143	C₂×C₄	—	142	1	no	—
144	C₂×C₄	—	142	1	no	—
145	C₂×C₄	#14556099	1	1	yes	characteristic
146	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
147	C₂×C₄	#14556147	1	1	yes	characteristic
148	C₈	—	2	1	no	—
149	C₈	—	284	1	no	—
150	C₈	—	568	2	no	—
151	C₈	C₂×#11765853	1	1	yes	characteristic
152	C₈	#14556154	1	1	yes	characteristic
153	C₂²	—	16	2	no	—
154	C₂²	—	16	2	no	—
155	C₂²	—	16	2	no	—
156	C₂²	#17986232	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	284	1	no	—
159	C₄	—	284	1	no	—
160	C₄	#17986196	1	1	yes	characteristic
161	C₄	#17986266	1	1	yes	characteristic
162	C₄	#17986433	1	2	yes	—
163	C₂	—	32	2	no	—
164	C₂	#22587697	1	1	yes	characteristic
165	C₂	#22590440	1	1	yes	characteristic
166	C₂	#22590457	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590248

C₂

yes

maximal, characteristic, complemented

#22587697

C₂

yes

maximal, complemented

#22586414

C₂

yes

maximal, characteristic, complemented

#22586386

C₂

yes

maximal, complemented

C₇₁×#2246

C₂

yes

maximal, characteristic

#17986196

—

C₈×Dic₁₄₂

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₁₆)

C₂²

yes

characteristic

Dic₇₁⋊C₁₆

—

#17985774

—

#17985752

C₂²

yes

—

#17985743

—

C₂×D₁₆×C₇₁

—

C₂×D₁₆×C₇₁

—

#493×C₇₁

C₂²

yes

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×#11765853

—

C₄×Dic₁₄₂

D₄

yes

characteristic

C₂×C₇₁⋊C₁₆

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

D₄

yes

characteristic

Dic₇₁⋊C₈

—

#14556049

—

C₈×Dic₇₁

—

C₂×D₈×C₇₁

—

C₂×D₈×C₇₁

—

#127×C₇₁

—

D₁₆×C₇₁

—

D₁₆×C₇₁

—

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×Dic₁₄₂

D₈

yes

characteristic

#11765853

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

D₈

yes

characteristic

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₇₁⋊C₁₆

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

D₈×C₇₁

—

D₈×C₇₁

—

D₈×C₇₁

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

#41

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

SD₁₆

yes

complemented

C₂³×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

#121

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

#135

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

C₂×SD₁₆

yes

characteristic

C₈×C₇₁

AGL(1,ℤ/8)

yes

characteristic

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#466

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

Dic₇₁

—

Dic₇₁

—

#12442

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

yes

—

—

—

—

C₇₁

#12442

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

C₂×D₁₆

—

#493

C₂×D₇₁

yes

—

#460

—

142

—

C₈×Q₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

142

—

#346

—

142

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

#335

—

100

#331

—

142

—

101

C₂×SD₁₆

—

284

—

102

C₂×D₈

—

103

C₂×D₈

—

104

#127

—

105

C₄×Q₈

—

106

D₁₆

—

107

D₁₆

—

108

C₂×C₁₆

—

142

—

109

C₂×C₁₆

C₂²⋊D₇₁

yes

characteristic

110

C₂×C₁₆

C₂²⋊D₇₁

yes

characteristic

111

C₄⋊C₈

—

112

C₄⋊C₈

—

142

—

113

C₄⋊C₈

—

142

—

114

#102

—

284

—

115

C₄×C₈

—

142

—

116

C₄×C₈

C₂²×D₇₁

yes

characteristic

117

C₂×Q₈

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

—

121

SD₁₆

—

568

—

122

D₈

—

123

D₈

—

124

D₈

—

125

C₂×C₈

—

284

—

126

C₂×C₈

—

142

—

127

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

128

C₂×C₈

#11765864

yes

characteristic

129

C₄⋊C₄

—

130

C₄⋊C₄

—

142

—

131

C₄²

—

142

—

132

C₄²

D₄×D₇₁

yes

characteristic

133

C₁₆

—

134

C₁₆

—

284

—

135

C₁₆

#11765853

yes

complemented

136

C₂³

—

137

Q₈

—

142

—

138

D₄

—

139

D₄

—

140

D₄

—

141

D₄

—

142

D₄

—

143

C₂×C₄

—

142

—

144

C₂×C₄

—

142

—

145

C₂×C₄

#14556099

yes

characteristic

146

C₂×C₄

D₈×D₇₁

yes

characteristic

147

C₂×C₄

#14556147

yes

characteristic

148

C₈

—

149

C₈

—

284

—

150

C₈

—

568

—

151

C₈

C₂×#11765853

yes

characteristic

152

C₈

#14556154

yes

characteristic

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

#17986232

yes

characteristic

157

C₄

—

158

C₄

—

284

—

159

C₄

—

284

—

160

C₄

#17986196

yes

characteristic

161

C₄

#17986266

yes

characteristic

162

C₄

#17986433

yes

—

163

C₂

—

164

C₂

#22587697

yes

characteristic

165

C₂

#22590440

yes

characteristic

166

C₂

#22590457

yes

characteristic

	subgroup	G/H
center	C₂²	#17986232
commutator, cosocle	C₄×C₈×C₇₁	C₂³
Frattini	C₄×C₈	C₂²×D₇₁
Fitting, prime radical	C₇₁×#2246	C₂
socle	C₂²×C₇₁	#466
hypercenter	#2246	D₇₁
hypocenter	C₇₁	#12442
LCS	G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

subgroup

G/H

center

C₂²

#17986232

commutator, cosocle

C₄×C₈×C₇₁

C₂³

Frattini

C₄×C₈

C₂²×D₇₁

Fitting, prime radical

C₇₁×#2246

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

N	H ≅ G/N	type
1. #22590248	C₂	semidirect
2. #22587697	C₂	semidirect
3. #22586414	C₂	semidirect
4. #22586386	C₂	semidirect
51. C₁₆×C₇₁	SD₁₆	semidirect
89. C₇₁	#12442	semidirect
135. C₁₆	#11765853	semidirect

H ≅ G/N

type

1. #22590248

C₂

semidirect

2. #22587697

C₂

semidirect

3. #22586414

C₂

semidirect

4. #22586386

C₂

semidirect

51. C₁₆×C₇₁

SD₁₆

semidirect

89. C₇₁

#12442

semidirect

135. C₁₆

#11765853

semidirect

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

subgroup	index
#22590248	2
#22587697	2
#22586414	2
#22586386	2
C₇₁×#2246	2
#12442	71

subgroup

index

C₇₁×#2246

Subgroups — #31911723

Conjugacy classes of subgroups

Complements

Maximal subgroups