Group #31911727 — subgroups

Subgroups — #31911727

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	#22590590	C₂	1	1	yes	maximal, characteristic
2	#22590248	C₂	1	1	yes	maximal, characteristic, complemented
3	#22587701	C₂	1	2	yes	maximal, complemented
4	#22586385	C₂	1	2	yes	maximal, complemented
5	C₇₁×#1372	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₁₃₆	—	2	1	no	—
7	C₂×D₁₁₃₆	—	2	1	no	—
8	#17986376	C₂²	1	2	yes	—
9	#17986200	—	2	2	no	—
10	C₈×Dic₁₄₂	C₂²	1	1	yes	characteristic
11	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	1	yes	characteristic
12	Dic₇₁⋊C₁₆	—	2	1	no	—
13	#17985774	—	2	1	no	—
14	#17985742	—	2	2	no	—
15	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
16	#335×C₇₁	C₂²	1	2	yes	—
17	C₂×#11765854	—	4	2	no	—
18	C₂×D₅₆₈	—	2	2	no	—
19	C₂×D₅₆₈	—	2	2	no	—
20	#14556125	—	2	2	no	—
21	C₄×Dic₁₄₂	D₄	1	1	yes	characteristic
22	D₁₁₃₆	—	4	2	no	—
23	D₁₁₃₆	—	4	2	no	—
24	C₂×C₇₁⋊C₁₆	—	2	1	no	—
25	C₄⋊(C₇₁⋊C₈)	D₄	1	1	yes	characteristic
26	Dic₇₁⋊C₈	—	2	1	no	—
27	#14556050	—	4	2	no	—
28	C₈×Dic₇₁	—	2	1	no	—
29	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
30	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
31	C₄⋊C₈×C₇₁	—	2	2	no	—
32	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₄₂	D₈	1	1	yes	characteristic
34	C₂×D₂₈₄	—	4	2	no	—
35	C₂×D₂₈₄	—	4	2	no	—
36	C₂×D₂₈₄	—	4	2	no	—
37	#11765854	—	8	4	no	—
38	D₅₆₈	—	4	2	no	—
39	D₅₆₈	—	4	2	no	—
40	D₅₆₈	—	8	2	no	—
41	C₂×C₇₁⋊C₈	—	2	1	no	—
42	C₄⋊Dic₇₁	D₈	1	1	yes	characteristic
43	Dic₇₁⋊C₄	—	2	1	no	—
44	C₄×Dic₇₁	—	2	1	no	—
45	C₇₁⋊C₁₆	—	4	1	no	—
46	C₂×C₈×C₇₁	—	4	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	C₂²×D₇₁	—	8	2	no	—
53	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
54	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
56	C₂×Dic₇₁	—	2	1	no	—
57	C₂×Dic₇₁	—	2	1	no	—
58	D₂₈₄	—	8	2	no	—
59	D₂₈₄	—	8	2	no	—
60	D₂₈₄	—	8	2	no	—
61	D₂₈₄	—	8	2	no	—
62	D₂₈₄	—	16	2	no	—
63	Dic₁₄₂	—	2	2	no	—
64	C₈×C₇₁	—	2	1	no	—
65	C₈×C₇₁	—	8	2	no	—
66	C₈×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
67	C₈×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₇₁⋊C₈	—	4	1	no	—
69	C₂²×C₇₁	#466	1	1	yes	characteristic
70	C₂×D₇₁	—	16	2	no	—
71	C₂×D₇₁	—	16	2	no	—
72	C₂×D₇₁	—	16	2	no	—
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	#1668	1	1	yes	characteristic
81	C₂×C₇₁	#2218	1	1	yes	characteristic
82	C₂×C₇₁	#2221	1	1	yes	characteristic
83	D₇₁	—	32	2	no	—
84	#2246	—	71	1	no	—
85	#2183	—	71	1	no	—
86	#1668	—	71	2	no	—
87	#1372	D₇₁	1	1	yes	characteristic, complemented
88	#1361	—	71	2	no	—
89	C₇₁	#12442	1	1	yes	characteristic, complemented, Sylow
90	C₂×D₁₆	—	142	1	no	—
91	C₂×D₁₆	—	142	1	no	—
92	#493	—	71	2	no	—
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	C₂×D₇₁	1	2	yes	—
100	#331	—	142	2	no	—
101	C₂×SD₁₆	—	284	2	no	—
102	C₂×D₈	—	142	2	no	—
103	C₂×D₈	—	142	2	no	—
104	#127	—	142	2	no	—
105	C₄×Q₈	—	71	1	no	—
106	D₁₆	—	284	2	no	—
107	D₁₆	—	284	2	no	—
108	C₂×C₁₆	—	142	1	no	—
109	C₂×C₁₆	D₂₈₄	1	1	yes	characteristic
110	C₂×C₁₆	D₂₈₄	1	1	yes	characteristic
111	C₄⋊C₈	—	2	2	no	—
112	C₄⋊C₈	—	71	1	no	—
113	C₄⋊C₈	—	142	1	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	—	284	2	no	—
119	C₂×D₄	—	284	2	no	—
120	C₂×D₄	—	284	2	no	—
121	SD₁₆	—	568	4	no	—
122	D₈	—	284	2	no	—
123	D₈	—	284	2	no	—
124	D₈	—	568	2	no	—
125	C₂×C₈	—	4	2	no	—
126	C₂×C₈	—	142	1	no	—
127	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
128	C₂×C₈	#11765866	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	#11765854	1	2	yes	complemented
136	C₂³	—	568	2	no	—
137	Q₈	—	142	2	no	—
138	D₄	—	568	2	no	—
139	D₄	—	568	2	no	—
140	D₄	—	568	2	no	—
141	D₄	—	568	2	no	—
142	D₄	—	1136	2	no	—
143	C₂×C₄	—	142	1	no	—
144	C₂×C₄	—	142	1	no	—
145	C₂×C₄	#14556103	1	1	yes	characteristic
146	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
147	C₂×C₄	#14556151	1	1	yes	characteristic
148	C₈	—	2	1	no	—
149	C₈	—	8	2	no	—
150	C₈	—	284	1	no	—
151	C₈	C₂×#11765854	1	1	yes	characteristic
152	C₈	#14556156	1	1	yes	characteristic
153	C₂²	—	1136	2	no	—
154	C₂²	—	1136	2	no	—
155	C₂²	—	1136	2	no	—
156	C₂²	#17986236	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	284	1	no	—
159	C₄	—	284	1	no	—
160	C₄	#17986200	1	1	yes	characteristic
161	C₄	#17986270	1	1	yes	characteristic
162	C₄	#17986436	1	2	yes	—
163	C₂	—	2272	2	no	—
164	C₂	#22587701	1	1	yes	characteristic
165	C₂	#22590444	1	1	yes	characteristic
166	C₂	#22590461	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590590

C₂

yes

maximal, characteristic

#22590248

C₂

yes

maximal, characteristic, complemented

#22587701

C₂

yes

maximal, complemented

#22586385

C₂

yes

maximal, complemented

C₇₁×#1372

C₂

yes

maximal, characteristic, complemented

C₂×D₁₁₃₆

—

C₂×D₁₁₃₆

—

#17986376

C₂²

yes

—

#17986200

—

C₈×Dic₁₄₂

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₁₆)

C₂²

yes

characteristic

Dic₇₁⋊C₁₆

—

#17985774

—

#17985742

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

#335×C₇₁

C₂²

yes

—

C₂×#11765854

—

C₂×D₅₆₈

—

C₂×D₅₆₈

—

#14556125

—

C₄×Dic₁₄₂

D₄

yes

characteristic

D₁₁₃₆

—

D₁₁₃₆

—

C₂×C₇₁⋊C₁₆

—

C₄⋊(C₇₁⋊C₈)

D₄

yes

characteristic

Dic₇₁⋊C₈

—

#14556050

—

C₈×Dic₇₁

—

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

D₄

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×Dic₁₄₂

D₈

yes

characteristic

C₂×D₂₈₄

—

C₂×D₂₈₄

—

C₂×D₂₈₄

—

#11765854

—

D₅₆₈

—

D₅₆₈

—

D₅₆₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

D₈

yes

characteristic

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₇₁⋊C₁₆

—

C₂×C₈×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

C₂²×D₇₁

—

C₂×C₄×C₇₁

#121

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

#135

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

C₂×SD₁₆

yes

characteristic

C₈×C₇₁

AGL(1,ℤ/8)

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

#466

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

Dic₇₁

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

—

—

—

—

D₇₁

yes

characteristic, complemented

#1361

—

C₇₁

#12442

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

142

—

C₂×D₁₆

—

142

—

#493

—

#460

—

142

—

C₈×Q₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

142

—

#346

—

142

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

#335

C₂×D₇₁

yes

—

100

#331

—

142

—

101

C₂×SD₁₆

—

284

—

102

C₂×D₈

—

142

—

103

C₂×D₈

—

142

—

104

#127

—

142

—

105

C₄×Q₈

—

106

D₁₆

—

284

—

107

D₁₆

—

284

—

108

C₂×C₁₆

—

142

—

109

C₂×C₁₆

D₂₈₄

yes

characteristic

110

C₂×C₁₆

D₂₈₄

yes

characteristic

111

C₄⋊C₈

—

112

C₄⋊C₈

—

113

C₄⋊C₈

—

142

—

114

#102

—

284

—

115

C₄×C₈

—

142

—

116

C₄×C₈

C₂²×D₇₁

yes

characteristic

117

C₂×Q₈

—

118

C₂×D₄

—

284

—

119

C₂×D₄

—

284

—

120

C₂×D₄

—

284

—

121

SD₁₆

—

568

—

122

D₈

—

284

—

123

D₈

—

284

—

124

D₈

—

568

—

125

C₂×C₈

—

126

C₂×C₈

—

142

—

127

C₂×C₈

C₂×D₂₈₄

yes

characteristic

128

C₂×C₈

#11765866

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

#11765854

yes

complemented

136

C₂³

—

568

—

137

Q₈

—

142

—

138

D₄

—

568

—

139

D₄

—

568

—

140

D₄

—

568

—

141

D₄

—

568

—

142

D₄

—

1136

—

143

C₂×C₄

—

142

—

144

C₂×C₄

—

142

—

145

C₂×C₄

#14556103

yes

characteristic

146

C₂×C₄

D₈×D₇₁

yes

characteristic

147

C₂×C₄

#14556151

yes

characteristic

148

C₈

—

149

C₈

—

150

C₈

—

284

—

151

C₈

C₂×#11765854

yes

characteristic

152

C₈

#14556156

yes

characteristic

153

C₂²

—

1136

—

154

C₂²

—

1136

—

155

C₂²

—

1136

—

156

C₂²

#17986236

yes

characteristic

157

C₄

—

158

C₄

—

284

—

159

C₄

—

284

—

160

C₄

#17986200

yes

characteristic

161

C₄

#17986270

yes

characteristic

162

C₄

#17986436

yes

—

163

C₂

—

2272

—

164

C₂

#22587701

yes

characteristic

165

C₂

#22590444

yes

characteristic

166

C₂

#22590461

yes

characteristic

	subgroup	G/H
center	C₂²	#17986236
commutator, cosocle	C₄×C₈×C₇₁	C₂³
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1372	C₂
socle	C₂²×C₇₁	#466
prime radical	#22590590	C₂
hypercenter	#1372	D₇₁
hypocenter	C₇₁	#12442
LCS	G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1372

subgroup

G/H

center

C₂²

#17986236

commutator, cosocle

C₄×C₈×C₇₁

C₂³

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#1372

C₂

C₂²×C₇₁

C₂

D₇₁

C₇₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1372

N	H ≅ G/N	type
2. #22590248	C₂	semidirect
3. #22587701	C₂	semidirect
4. #22586385	C₂	semidirect
5. C₇₁×#1372	C₂	semidirect
51. C₁₆×C₇₁	SD₁₆	semidirect
87. #1372	D₇₁	semidirect
89. C₇₁	#12442	semidirect
135. C₁₆	#11765854	semidirect

H ≅ G/N

type

2. #22590248

C₂

semidirect

3. #22587701

C₂

semidirect

4. #22586385

C₂

semidirect

5. C₇₁×#1372

C₂

semidirect

51. C₁₆×C₇₁

SD₁₆

semidirect

87. #1372

D₇₁

semidirect

89. C₇₁

#12442

semidirect

135. C₁₆

#11765854

semidirect

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

subgroup	index
#22590590	2
#22590248	2
#22587701	2
#22586385	2
C₇₁×#1372	2
#12442	71

subgroup

index

C₇₁×#1372

Subgroups — #31911727

Conjugacy classes of subgroups

Complements

Maximal subgroups