Group #31917588 — subgroups

Subgroups — #31917588

Order 18176.

234 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 89 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	#22589093	C₂	1	1	yes	maximal, characteristic
2	#22588515	C₂	1	1	yes	maximal, characteristic
3	#22586455	C₂	1	2	yes	maximal
4	#22586452	C₂	1	2	yes	maximal
5	C₇₁×#1375	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556064	—	2	1	no	—
7	C₂×#14556064	C₂²	1	1	yes	characteristic
8	C₂×#14556064	C₂²	1	2	yes	—
9	#17986037	C₂²	1	1	yes	characteristic
10	#17986025	—	2	1	no	—
11	#17986023	—	2	1	no	—
12	#17985759	—	2	1	no	—
13	C₂×M₃₂×C₇₁	C₂²	1	2	yes	—
14	C₂×#97×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×#11765849	—	2	1	no	—
16	C₂×#11765849	—	2	2	no	—
17	#14556087	—	2	1	no	—
18	#14556064	—	2	2	no	—
19	#14556064	—	2	4	no	—
20	#14556064	—	4	2	no	—
21	#14556062	—	2	1	no	—
22	#14556060	Q₈	1	2	yes	—
23	C₄⋊(C₇₁⋊C₈)	—	4	2	no	—
24	Dic₇₁⋊C₈	—	4	2	no	—
25	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
28	M₃₂×C₇₁	—	2	2	no	—
29	M₃₂×C₇₁	—	2	2	no	—
30	C₂×C₁₆×C₇₁	—	2	2	no	—
31	#97×C₇₁	—	2	2	no	—
32	#11765849	—	4	1	no	—
33	#11765849	—	4	2	no	—
34	#11765849	—	4	2	no	—
35	#11765849	—	4	1	no	—
36	C₂×C₇₁⋊C₈	—	4	1	no	—
37	C₂×C₇₁⋊C₈	—	4	2	no	—
38	C₄⋊Dic₇₁	—	2	2	no	—
39	C₂²⋊Dic₇₁	—	4	1	no	—
40	C₄×Dic₇₁	—	4	1	no	—
41	C₂²×C₄×C₇₁	—	2	1	no	—
42	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₇₁	—	2	1	no	—
44	C₂×C₈×C₇₁	—	2	1	no	—
45	C₂×C₈×C₇₁	D₈	1	2	yes	—
46	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₇₁	#41	1	2	yes	—
48	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
49	C₂×C₈×C₇₁	#41	1	2	yes	—
50	C₄²×C₇₁	—	2	1	no	—
51	C₄²×C₇₁	SD₁₆	1	2	yes	—
52	C₁₆×C₇₁	—	4	2	no	—
53	C₂³×C₇₁	#137	1	1	yes	characteristic
54	C₂×C₄×C₇₁	—	2	2	no	—
55	C₂×C₄×C₇₁	—	8	1	no	—
56	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
57	C₂×C₄×C₇₁	#123	1	1	yes	characteristic
58	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
59	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
61	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
62	C₂×Dic₇₁	—	4	2	no	—
63	C₈×C₇₁	—	2	2	no	—
64	C₈×C₇₁	#125	1	2	yes	—
65	C₈×C₇₁	#126	1	2	yes	—
66	C₇₁⋊C₈	—	8	1	no	—
67	C₇₁⋊C₈	—	8	2	no	—
68	C₂²×C₇₁	—	4	1	no	—
69	C₂²×C₇₁	#474	1	1	yes	characteristic
70	C₂²×C₇₁	#483	1	1	yes	characteristic
71	C₂²×C₇₁	#491	1	1	yes	characteristic
72	C₄×C₇₁	—	4	2	no	—
73	C₄×C₇₁	#400	1	1	yes	characteristic
74	C₄×C₇₁	#475	1	1	yes	characteristic
75	C₄×C₇₁	#480	1	1	yes	characteristic
76	C₄×C₇₁	#486	1	1	yes	characteristic
77	Dic₇₁	—	8	2	no	—
78	#13540	—	71	1	no	maximal, Sylow
79	C₂×C₇₁	—	2	1	no	—
80	C₂×C₇₁	—	2	1	no	—
81	C₂×C₇₁	—	2	1	no	—
82	C₂×C₇₁	#2099	1	1	yes	characteristic
83	#1946	—	71	1	no	—
84	#1849	—	71	1	no	—
85	#1389	—	71	2	no	—
86	#1388	—	71	2	no	—
87	#1375	D₇₁	1	1	yes	characteristic
88	C₇₁	#13540	1	1	yes	characteristic, complemented, Sylow
89	C₂×M₃₂	C₂×D₇₁	1	2	yes	—
90	C₂×#108	—	71	2	no	—
91	C₂×#108	—	71	1	no	—
92	C₂×#108	—	142	1	no	—
93	#427	—	71	1	no	—
94	#424	—	142	1	no	—
95	#423	—	142	1	no	—
96	C₂×#97	C₂×D₇₁	1	1	yes	characteristic
97	#339	—	142	1	no	—
98	C₂×M₁₆	—	142	1	no	—
99	C₂×M₁₆	—	142	2	no	—
100	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
101	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
102	#117	—	142	1	no	—
103	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
104	M₃₂	—	2	2	no	—
105	M₃₂	—	2	2	no	—
106	C₂×C₁₆	—	2	2	no	—
107	#108	—	284	2	no	—
108	#108	—	142	2	no	—
109	#108	—	142	4	no	—
110	#107	—	142	1	no	—
111	#106	—	71	2	no	—
112	C₄⋊C₈	—	284	2	no	—
113	C₄⋊C₈	—	284	2	no	—
114	#97	—	2	2	no	—
115	C₂²×C₄	—	2	1	no	—
116	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
117	M₁₆	—	284	1	no	—
118	M₁₆	—	284	1	no	—
119	M₁₆	—	284	2	no	—
120	M₁₆	—	284	2	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	284	1	no	—
124	C₂×C₈	—	284	2	no	—
125	C₂×C₈	D₅₆₈	1	2	yes	—
126	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
127	C₂×C₈	#11765864	1	2	yes	—
128	C₂×C₈	#11765864	1	1	yes	characteristic
129	C₂×C₈	#11765865	1	2	yes	—
130	C₄⋊C₄	—	142	2	no	—
131	C₂²⋊C₄	—	284	1	no	—
132	C₄²	—	2	1	no	—
133	C₄²	—	284	1	no	—
134	C₄²	#11765854	1	2	yes	—
135	C₁₆	—	4	2	no	—
136	C₂³	#14556161	1	1	yes	characteristic
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	8	1	no	—
139	C₂×C₄	—	284	2	no	—
140	C₂×C₄	#14556108	1	1	yes	characteristic
141	C₂×C₄	#14556112	1	1	yes	characteristic
142	C₂×C₄	#14556116	1	1	yes	characteristic
143	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
144	C₂×C₄	C₂×#11765854	1	1	yes	characteristic
145	C₂×C₄	#14556161	1	1	yes	characteristic
146	C₈	—	2	2	no	—
147	C₈	—	568	1	no	—
148	C₈	—	568	2	no	—
149	C₈	#14556118	1	2	yes	—
150	C₈	#14556121	1	2	yes	—
151	C₂²	—	4	1	no	—
152	C₂²	#17986280	1	1	yes	characteristic
153	C₂²	#17986330	1	1	yes	characteristic
154	C₂²	#17986368	1	1	yes	characteristic
155	C₄	—	4	2	no	—
156	C₄	—	568	2	no	—
157	C₄	#17985939	1	1	yes	characteristic
158	C₄	#17986285	1	1	yes	characteristic
159	C₄	#17986313	1	1	yes	characteristic
160	C₄	#17986346	1	1	yes	characteristic
161	C₂	—	2	1	no	—
162	C₂	—	2	1	no	—
163	C₂	—	2	1	no	—
164	C₂	#22589899	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22589093

C₂

yes

maximal, characteristic

#22588515

C₂

yes

maximal, characteristic

#22586455

C₂

yes

maximal

#22586452

C₂

yes

maximal

C₇₁×#1375

C₂

yes

maximal, characteristic

C₂×#14556064

—

C₂×#14556064

C₂²

yes

characteristic

C₂×#14556064

C₂²

yes

—

#17986037

C₂²

yes

—

—

—

C₂×M₃₂×C₇₁

C₂²

yes

—

C₂×#97×C₇₁

C₂²

yes

—

—

—

—

—

—

—

Q₈

yes

—

C₄⋊(C₇₁⋊C₈)

—

Dic₇₁⋊C₈

—

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄²×C₇₁

D₄

yes

characteristic

M₃₂×C₇₁

—

M₃₂×C₇₁

—

C₂×C₁₆×C₇₁

—

#97×C₇₁

—

—

—

—

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

D₈

yes

—

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₁

#41

yes

—

C₂×C₈×C₇₁

#41

yes

characteristic

C₂×C₈×C₇₁

#41

yes

—

C₄²×C₇₁

—

C₄²×C₇₁

SD₁₆

yes

—

C₁₆×C₇₁

—

C₂³×C₇₁

#137

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊Q₈

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

C₂×D₈

yes

characteristic

C₂×C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₁

#137

yes

characteristic

C₂×Dic₇₁

—

C₈×C₇₁

—

C₈×C₇₁

#125

yes

—

C₈×C₇₁

#126

yes

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

#13540

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

—

—

—

—

D₇₁

yes

characteristic

C₇₁

#13540

yes

characteristic, complemented, Sylow

C₂×M₃₂

C₂×D₇₁

yes

—

C₂×#108

—

C₂×#108

—

C₂×#108

—

142

—

#427

—

#424

—

142

—

#423

—

142

—

C₂×#97

C₂×D₇₁

yes

characteristic

#339

—

142

—

C₂×M₁₆

—

142

—

C₂×M₁₆

—

142

—

100

C₂²×C₈

D₂₈₄

yes

characteristic

101

C₂²×C₈

C₂²×D₇₁

yes

characteristic

102

#117

—

142

—

103

C₂×C₄²

D₂₈₄

yes

characteristic

104

M₃₂

—

105

M₃₂

—

106

C₂×C₁₆

—

107

#108

—

284

—

108

#108

—

142

—

109

#108

—

142

—

110

#107

—

142

—

111

#106

—

112

C₄⋊C₈

—

284

—

113

C₄⋊C₈

—

284

—

114

#97

—

115

C₂²×C₄

—

116

C₂²×C₄

C₂×D₂₈₄

yes

characteristic

117

M₁₆

—

284

—

118

M₁₆

—

284

—

119

M₁₆

—

284

—

120

M₁₆

—

284

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

284

—

124

C₂×C₈

—

284

—

125

C₂×C₈

D₅₆₈

yes

—

126

C₂×C₈

Q₈×D₇₁

yes

characteristic

127

C₂×C₈

#11765864

yes

—

128

C₂×C₈

#11765864

yes

characteristic

129

C₂×C₈

#11765865

yes

—

130

C₄⋊C₄

—

142

—

131

C₂²⋊C₄

—

284

—

132

C₄²

—

133

C₄²

—

284

—

134

C₄²

#11765854

yes

—

135

C₁₆

—

136

C₂³

#14556161

yes

characteristic

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

284

—

140

C₂×C₄

#14556108

yes

characteristic

141

C₂×C₄

#14556112

yes

characteristic

142

C₂×C₄

#14556116

yes

characteristic

143

C₂×C₄

C₂×D₅₆₈

yes

characteristic

144

C₂×C₄

C₂×#11765854

yes

characteristic

145

C₂×C₄

#14556161

yes

characteristic

146

C₈

—

147

C₈

—

568

—

148

C₈

—

568

—

149

C₈

#14556118

yes

—

150

C₈

#14556121

yes

—

151

C₂²

—

152

C₂²

#17986280

yes

characteristic

153

C₂²

#17986330

yes

characteristic

154

C₂²

#17986368

yes

characteristic

155

C₄

—

156

C₄

—

568

—

157

C₄

#17985939

yes

characteristic

158

C₄

#17986285

yes

characteristic

159

C₄

#17986313

yes

characteristic

160

C₄

#17986346

yes

characteristic

161

C₂

—

162

C₂

—

163

C₂

—

164

C₂

#22589899

yes

characteristic

	subgroup	G/H
center	C₄	#17986346
commutator, cosocle	C₂²×C₈×C₇₁	C₂³
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1375	C₂
socle	C₂×C₇₁	#2099
prime radical	C₂³×C₇₁	#137
hypercenter	#1375	D₇₁
hypocenter	C₇₁	#13540
LCS	G ▷ C₂²×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1375

subgroup

G/H

center

C₄

#17986346

commutator, cosocle

C₂²×C₈×C₇₁

C₂³

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₇₁×#1375

C₂

C₂×C₇₁

C₂³×C₇₁

D₇₁

C₇₁

G ▷ C₂²×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1375

N	H ≅ G/N	type
88. C₇₁	#13540	semidirect

H ≅ G/N

type

88. C₇₁

#13540

semidirect

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

subgroup	index
#22589093	2
#22588515	2
#22586455	2
#22586452	2
C₇₁×#1375	2
#13540	71

subgroup

index

C₇₁×#1375

Subgroups — #31917588

Conjugacy classes of subgroups

Complements

Maximal subgroups