Group #31912832 — subgroups

Subgroups — #31912832

Order 18176.

308 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 131 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	#22590114	C₂	1	2	yes	maximal
2	M₃₂×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	#22588222	C₂	1	2	yes	maximal
4	#22586442	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1385	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556068	C₂²	1	1	yes	characteristic
7	#17986037	C₂²	1	2	yes	—
8	#17985957	—	2	2	no	—
9	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
10	#17985823	C₄	1	8	yes	complemented
11	#17985774	C₄	1	2	yes	complemented
12	C₁₆×Dic₇₁	C₄	1	2	yes	complemented
13	#17985760	—	2	2	no	—
14	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×#108×C₇₁	C₂²	1	2	yes	—
16	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
17	#14556087	—	2	2	no	—
18	C₂×C₄×Dic₇₁	D₄	1	1	yes	characteristic
19	#14556068	C₂×C₄	1	4	yes	—
20	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
21	#14556062	—	2	2	no	—
22	#14556060	C₂×C₄	1	4	yes	—
23	#14556038	—	4	2	no	—
24	C₈×Dic₇₁	—	4	2	no	—
25	C₈×Dic₇₁	C₂×C₄	1	1	yes	characteristic
26	C₈×Dic₇₁	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₇₁	Q₈	1	1	yes	characteristic
28	C₈×Dic₇₁	Q₈	1	1	yes	characteristic
29	C₂×M₁₆×C₇₁	—	2	2	no	—
30	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
31	M₃₂×C₇₁	—	2	2	no	—
32	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
33	#108×C₇₁	—	2	2	no	—
34	#108×C₇₁	C₂×C₄	1	4	yes	—
35	C₂²×Dic₇₁	—	2	1	no	—
36	C₂×C₇₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₇₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₇₁⋊C₈	SD₁₆	1	2	yes	—
39	C₂×C₇₁⋊C₈	SD₁₆	1	2	yes	—
40	C₄⋊Dic₇₁	—	2	4	no	—
41	C₂²⋊Dic₇₁	—	4	2	no	—
42	C₄×Dic₇₁	—	2	1	no	—
43	C₄×Dic₇₁	—	4	2	no	—
44	C₄×Dic₇₁	C₄⋊C₄	1	1	yes	characteristic
45	C₄×Dic₇₁	C₄⋊C₄	1	1	yes	characteristic
46	C₇₁⋊C₁₆	C₄²	1	4	yes	—
47	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
48	M₁₆×C₇₁	—	2	4	no	—
49	M₁₆×C₇₁	—	4	2	no	—
50	C₂×C₈×C₇₁	—	4	2	no	—
51	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
53	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₇₁	#41	1	2	yes	—
55	C₁₆×C₇₁	—	2	2	no	—
56	C₂³×C₇₁	#135	1	1	yes	characteristic
57	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
59	C₂×C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
60	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
61	C₂×Dic₇₁	—	2	1	no	—
62	C₂×Dic₇₁	—	2	1	no	—
63	C₂×Dic₇₁	—	4	2	no	—
64	C₂×Dic₇₁	—	4	4	no	—
65	C₈×C₇₁	—	4	4	no	—
66	C₈×C₇₁	C₂×C₄²	1	1	yes	characteristic
67	C₈×C₇₁	#117	1	1	yes	characteristic
68	C₈×C₇₁	C₄×Q₈	1	2	yes	—
69	C₇₁⋊C₈	#106	1	1	yes	characteristic
70	C₇₁⋊C₈	#106	1	2	yes	—
71	C₇₁⋊C₈	#106	1	1	yes	characteristic
72	C₂²×C₇₁	—	4	1	no	—
73	C₂²×C₇₁	#427	1	1	yes	characteristic
74	C₂²×C₇₁	C₄×SD₁₆	1	2	yes	—
75	C₄×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
76	C₄×C₇₁	C₂×#106	1	1	yes	characteristic
77	C₄×C₇₁	C₄×SD₁₆	1	2	yes	—
78	Dic₇₁	—	2	2	no	—
79	Dic₇₁	—	4	1	no	—
80	Dic₇₁	—	8	4	no	—
81	#12668	—	71	1	no	maximal, Sylow
82	C₂×C₇₁	—	2	1	no	—
83	C₂×C₇₁	—	2	2	no	—
84	C₂×C₇₁	C₄×#106	1	1	yes	characteristic
85	#2155	—	71	2	no	—
86	C₄×M₃₂	—	71	1	no	—
87	#1772	—	71	2	no	—
88	#1385	D₇₁	1	1	yes	characteristic
89	#1384	—	71	1	no	—
90	C₇₁	#12668	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₃₂	—	71	1	no	—
92	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
93	C₂×#108	C₂×D₇₁	1	2	yes	—
94	#427	—	71	2	no	—
95	#405	—	142	2	no	—
96	C₂×C₄×C₈	—	71	1	no	—
97	#365	—	71	8	no	—
98	#346	—	71	2	no	—
99	C₄×C₁₆	—	71	2	no	—
100	#339	—	142	2	no	—
101	C₂×M₁₆	—	2	2	no	—
102	C₂²×C₈	—	71	1	no	—
103	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
104	#117	—	142	2	no	—
105	C₂×C₄²	—	71	1	no	—
106	M₃₂	—	2	2	no	—
107	M₃₂	—	71	4	no	—
108	C₂×C₁₆	—	71	2	no	—
109	C₂×C₁₆	C₄×D₇₁	1	2	yes	—
110	#108	—	2	2	no	—
111	#108	C₄×D₇₁	1	4	yes	—
112	#107	—	142	2	no	—
113	#106	—	71	4	no	—
114	#97	—	284	2	no	—
115	C₄×C₈	—	284	2	no	—
116	C₄×C₈	—	71	1	no	—
117	C₄×C₈	—	71	1	no	—
118	C₄×C₈	—	71	1	no	—
119	C₄×C₈	—	71	1	no	—
120	C₂²×C₄	—	142	1	no	—
121	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
122	M₁₆	—	2	4	no	—
123	M₁₆	—	4	2	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	—	71	1	no	—
126	C₂×C₈	—	71	1	no	—
127	C₂×C₈	—	71	2	no	—
128	C₂×C₈	—	71	2	no	—
129	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
130	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
131	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
132	C₂×C₈	#11765864	1	2	yes	—
133	C₄⋊C₄	—	142	4	no	—
134	C₂²⋊C₄	—	284	2	no	—
135	C₄²	—	284	2	no	—
136	C₄²	—	71	1	no	—
137	C₄²	—	71	1	no	—
138	C₄²	—	142	1	no	—
139	C₁₆	—	2	2	no	—
140	C₁₆	—	71	4	no	—
141	C₂³	#14556151	1	1	yes	characteristic
142	C₂×C₄	—	284	2	no	—
143	C₂×C₄	—	284	4	no	—
144	C₂×C₄	—	142	1	no	—
145	C₂×C₄	—	142	1	no	—
146	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
147	C₂×C₄	(C₂²⋊D₇₁)⋊C₄	1	2	yes	—
148	C₂×C₄	SD₁₆×D₇₁	1	2	yes	—
149	C₂×C₄	#14556147	1	1	yes	characteristic
150	C₈	—	4	4	no	—
151	C₈	—	71	1	no	—
152	C₈	—	71	2	no	—
153	C₈	—	71	1	no	—
154	C₈	C₄²×D₇₁	1	1	yes	characteristic
155	C₈	#14556084	1	1	yes	characteristic
156	C₈	#14556093	1	2	yes	—
157	C₂²	—	4	1	no	—
158	C₂²	#17986036	1	1	yes	characteristic
159	C₂²	#17986084	1	2	yes	—
160	C₄	—	284	1	no	—
161	C₄	—	568	4	no	—
162	C₄	—	142	2	no	—
163	C₄	#17985850	1	1	yes	characteristic
164	C₄	#106×D₇₁	1	1	yes	characteristic
165	C₄	#17986090	1	2	yes	—
166	C₂	—	2	1	no	—
167	C₂	—	2	2	no	—
168	C₂	#22588228	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590114

C₂

yes

maximal

M₃₂×Dic₇₁

C₂

yes

maximal, characteristic

#22588222

C₂

yes

maximal

#22586442

C₂

yes

maximal, characteristic

C₇₁×#1385

C₂

yes

maximal, characteristic

C₂×#14556068

C₂²

yes

characteristic

#17986037

C₂²

yes

—

#17985957

—

C₂×C₈×Dic₇₁

C₂²

yes

characteristic

#17985823

C₄

yes

complemented

#17985774

C₄

yes

complemented

C₁₆×Dic₇₁

C₄

yes

complemented

#17985760

—

C₂×M₃₂×C₇₁

C₂²

yes

characteristic

C₂×#108×C₇₁

C₂²

yes

—

C₂²×C₇₁⋊C₈

D₄

yes

characteristic

#14556087

—

C₂×C₄×Dic₇₁

D₄

yes

characteristic

#14556068

C₂×C₄

yes

—

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

—

#14556062

—

#14556060

C₂×C₄

yes

—

#14556038

—

C₈×Dic₇₁

—

C₈×Dic₇₁

C₂×C₄

yes

characteristic

C₈×Dic₇₁

C₂×C₄

yes

characteristic

C₈×Dic₇₁

Q₈

yes

characteristic

C₈×Dic₇₁

Q₈

yes

characteristic

C₂×M₁₆×C₇₁

—

C₂²×C₈×C₇₁

C₂³

yes

characteristic

M₃₂×C₇₁

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

#108×C₇₁

—

#108×C₇₁

C₂×C₄

yes

—

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₇₁⋊C₈

SD₁₆

yes

—

C₂×C₇₁⋊C₈

SD₁₆

yes

—

C₄⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

C₄⋊C₄

yes

characteristic

C₄×Dic₇₁

C₄⋊C₄

yes

characteristic

C₇₁⋊C₁₆

C₄²

yes

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂²×C₄

yes

—

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₁

#41

yes

—

C₁₆×C₇₁

—

C₂³×C₇₁

#135

yes

characteristic

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄×D₄

yes

—

C₂×C₄×C₇₁

C₂×SD₁₆

yes

—

C₂×C₄×C₇₁

#135

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₈×C₇₁

C₂×C₄²

yes

characteristic

C₈×C₇₁

#117

yes

characteristic

C₈×C₇₁

C₄×Q₈

yes

—

C₇₁⋊C₈

#106

yes

characteristic

C₇₁⋊C₈

#106

yes

—

C₇₁⋊C₈

#106

yes

characteristic

C₂²×C₇₁

—

C₂²×C₇₁

#427

yes

characteristic

C₂²×C₇₁

C₄×SD₁₆

yes

—

C₄×C₇₁

C₄×C₄⋊C₄

yes

characteristic

C₄×C₇₁

C₂×#106

yes

characteristic

C₄×C₇₁

C₄×SD₁₆

yes

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

#12668

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

C₄×#106

yes

characteristic

#2155

—

C₄×M₃₂

—

#1772

—

#1385

D₇₁

yes

characteristic

#1384

—

C₇₁

#12668

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

C₂×D₇₁

yes

characteristic

C₂×#108

C₂×D₇₁

yes

—

#427

—

#405

—

142

—

C₂×C₄×C₈

—

#365

—

#346

—

C₄×C₁₆

—

100

#339

—

142

—

101

C₂×M₁₆

—

102

C₂²×C₈

—

103

C₂²×C₈

C₂²×D₇₁

yes

characteristic

104

#117

—

142

—

105

C₂×C₄²

—

106

M₃₂

—

107

M₃₂

—

108

C₂×C₁₆

—

109

C₂×C₁₆

C₄×D₇₁

yes

—

110

#108

—

111

#108

C₄×D₇₁

yes

—

112

#107

—

142

—

113

#106

—

114

#97

—

284

—

115

C₄×C₈

—

284

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₂²×C₄

—

142

—

121

C₂²×C₄

D₄×D₇₁

yes

characteristic

122

M₁₆

—

123

M₁₆

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

130

C₂×C₈

C₂×C₄×D₇₁

yes

—

131

C₂×C₈

Q₈×D₇₁

yes

characteristic

132

C₂×C₈

#11765864

yes

—

133

C₄⋊C₄

—

142

—

134

C₂²⋊C₄

—

284

—

135

C₄²

—

284

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

142

—

139

C₁₆

—

140

C₁₆

—

141

C₂³

#14556151

yes

characteristic

142

C₂×C₄

—

284

—

143

C₂×C₄

—

284

—

144

C₂×C₄

—

142

—

145

C₂×C₄

—

142

—

146

C₂×C₄

C₄⋊C₄×D₇₁

yes

characteristic

147

C₂×C₄

(C₂²⋊D₇₁)⋊C₄

yes

—

148

C₂×C₄

SD₁₆×D₇₁

yes

—

149

C₂×C₄

#14556147

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

C₄²×D₇₁

yes

characteristic

155

C₈

#14556084

yes

characteristic

156

C₈

#14556093

yes

—

157

C₂²

—

158

C₂²

#17986036

yes

characteristic

159

C₂²

#17986084

yes

—

160

C₄

—

284

—

161

C₄

—

568

—

162

C₄

—

142

—

163

C₄

#17985850

yes

characteristic

164

C₄

#106×D₇₁

yes

characteristic

165

C₄

#17986090

yes

—

166

C₂

—

167

C₂

—

168

C₂

#22588228

yes

characteristic

	subgroup	G/H
center	C₄	#106×D₇₁
commutator	C₈×C₇₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1385	C₂
socle	C₂×C₇₁	C₄×#106
prime radical	C₂³×C₇₁	#135
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1385	D₇₁
hypocenter	C₇₁	#12668
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

subgroup

G/H

center

C₄

#106×D₇₁

commutator

C₈×C₇₁

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₇₁×#1385

C₂

C₂×C₇₁

C₂³×C₇₁

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

N	H ≅ G/N	type
10. #17985823	C₄	semidirect
11. #17985774	C₄	semidirect
12. C₁₆×Dic₇₁	C₄	semidirect
90. C₇₁	#12668	semidirect

H ≅ G/N

type

10. #17985823

C₄

semidirect

11. #17985774

C₄

semidirect

12. C₁₆×Dic₇₁

C₄

semidirect

90. C₇₁

#12668

semidirect

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

subgroup	index
#22590114	2
M₃₂×Dic₇₁	2
#22588222	2
#22586442	2
C₇₁×#1385	2
#12668	71

subgroup

index

M₃₂×Dic₇₁

C₇₁×#1385

Subgroups — #31912832

Conjugacy classes of subgroups

Complements

Maximal subgroups