Group #31913821 — subgroups

Subgroups — #31913821

Order 18176.

234 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 103 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	#22590099	C₂	1	1	yes	maximal, characteristic
2	#22588095	C₂	1	1	yes	maximal, characteristic
3	#22586430	C₂	1	2	yes	maximal
4	#22586425	C₂	1	2	yes	maximal
5	C₇₁×#2153	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556068	C₂²	1	1	yes	characteristic
7	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
8	#17985954	C₂²	1	2	yes	—
9	C₄⋊(C₇₁⋊C₁₆)	—	2	2	no	—
10	C₇₁⋊#338	C₄	1	1	yes	characteristic
11	#17985756	C₄	1	1	yes	characteristic
12	#17985755	C₄	1	2	yes	—
13	C₂×M₃₂×C₇₁	C₂²	1	2	yes	—
14	#405×C₇₁	C₂²	1	1	yes	characteristic
15	#364×C₇₁	—	2	2	no	—
16	C₂×#11765849	C₂×C₄	1	2	yes	—
17	#14556087	C₂×C₄	1	2	yes	—
18	#14556068	—	2	2	no	—
19	C₂×C₇₁⋊C₁₆	—	2	1	no	—
20	C₂×C₇₁⋊C₁₆	—	2	1	no	—
21	C₂×C₇₁⋊C₁₆	—	2	2	no	—
22	#14556038	—	2	2	no	—
23	C₈×Dic₇₁	—	2	2	no	—
24	C₂×M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
26	#117×C₇₁	C₂×C₄	1	1	yes	characteristic
27	M₃₂×C₇₁	—	2	2	no	—
28	M₃₂×C₇₁	—	2	2	no	—
29	C₂×C₁₆×C₇₁	—	2	2	no	—
30	#97×C₇₁	Q₈	1	2	yes	—
31	C₄×C₈×C₇₁	D₄	1	2	yes	—
32	#11765849	—	2	2	no	—
33	#11765849	—	2	2	no	—
34	C₂×C₇₁⋊C₈	—	2	2	no	—
35	C₄⋊Dic₇₁	—	2	2	no	—
36	C₂²⋊Dic₇₁	—	2	2	no	—
37	C₄×Dic₇₁	—	2	2	no	—
38	C₇₁⋊C₁₆	—	4	1	no	—
39	C₇₁⋊C₁₆	—	4	1	no	—
40	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
41	M₁₆×C₇₁	—	2	2	no	—
42	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₇₁	#41	1	2	yes	—
46	C₂×C₈×C₇₁	#41	1	2	yes	—
47	C₄⋊C₄×C₇₁	—	2	1	no	—
48	C₂²⋊C₄×C₇₁	—	2	1	no	—
49	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
50	C₁₆×C₇₁	—	4	2	no	—
51	C₂³×C₇₁	#131	1	1	yes	characteristic
52	C₂×C₄×C₇₁	—	2	2	no	—
53	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₇₁	#117	1	2	yes	—
55	C₂×C₄×C₇₁	#131	1	2	yes	—
56	C₂×C₄×C₇₁	#131	1	1	yes	characteristic
57	C₂×Dic₇₁	—	4	2	no	—
58	C₈×C₇₁	—	2	2	no	—
59	C₈×C₇₁	#124	1	1	yes	characteristic
60	C₈×C₇₁	#124	1	1	yes	characteristic
61	C₈×C₇₁	#125	1	2	yes	—
62	C₇₁⋊C₈	—	4	2	no	—
63	C₂²×C₇₁	—	4	1	no	—
64	C₂²×C₇₁	#424	1	1	yes	characteristic
65	C₂²×C₇₁	#433	1	2	yes	—
66	C₄×C₇₁	—	4	2	no	—
67	C₄×C₇₁	#382	1	1	yes	characteristic
68	C₄×C₇₁	#424	1	1	yes	characteristic
69	C₄×C₇₁	#433	1	2	yes	—
70	Dic₇₁	—	8	2	no	—
71	#12885	—	71	1	no	maximal, Sylow
72	C₂×C₇₁	—	2	2	no	—
73	C₂×C₇₁	—	2	1	no	—
74	C₂×C₇₁	#1832	1	1	yes	characteristic
75	#2153	D₇₁	1	1	yes	characteristic
76	#2151	—	71	1	no	—
77	#1747	—	71	1	no	—
78	#1379	—	71	2	no	—
79	#1378	—	71	2	no	—
80	C₇₁	#12885	1	1	yes	characteristic, complemented, Sylow
81	C₂×M₃₂	—	71	1	no	—
82	C₂×M₃₂	C₂×D₇₁	1	2	yes	—
83	C₂²×C₁₆	—	71	1	no	—
84	#405	—	71	2	no	—
85	#405	C₂×D₇₁	1	1	yes	characteristic
86	#364	—	2	2	no	—
87	C₄⋊C₁₆	—	142	2	no	—
88	#338	—	71	1	no	—
89	H(3,ℤ/4)	—	71	1	no	—
90	H(3,ℤ/4)	—	71	2	no	—
91	C₂×M₁₆	—	71	2	no	—
92	C₂×M₁₆	C₄×D₇₁	1	1	yes	characteristic
93	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
94	#117	—	71	2	no	—
95	#117	C₄×D₇₁	1	1	yes	characteristic
96	M₃₂	—	2	2	no	—
97	M₃₂	—	2	2	no	—
98	M₃₂	—	142	2	no	—
99	C₂×C₁₆	—	2	2	no	—
100	C₂×C₁₆	—	142	1	no	—
101	C₂×C₁₆	—	142	1	no	—
102	C₂×C₁₆	—	142	2	no	—
103	#97	—	142	2	no	—
104	#97	Dic₁₄₂	1	2	yes	—
105	C₄×C₈	—	142	2	no	—
106	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
107	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
108	M₁₆	—	2	2	no	—
109	M₁₆	—	142	2	no	—
110	M₁₆	—	142	2	no	—
111	C₂×C₈	—	142	2	no	—
112	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
113	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
114	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
115	C₂×C₈	#11765864	1	2	yes	—
116	C₂×C₈	#11765866	1	2	yes	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	142	2	no	—
119	C₂²⋊C₄	—	2	1	no	—
120	C₂²⋊C₄	—	142	2	no	—
121	C₄²	—	142	2	no	—
122	C₄²	Dic₇₁⋊C₄	1	2	yes	—
123	C₁₆	—	4	2	no	—
124	C₁₆	—	284	1	no	—
125	C₁₆	—	284	1	no	—
126	C₂³	#14556136	1	1	yes	characteristic
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	284	2	no	—
129	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
130	C₂×C₄	#14556086	1	2	yes	—
131	C₂×C₄	#14556134	1	2	yes	—
132	C₂×C₄	#14556136	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	284	2	no	—
135	C₈	#14556115	1	1	yes	characteristic
136	C₈	#14556117	1	1	yes	characteristic
137	C₈	#14556119	1	2	yes	—
138	C₂²	—	4	1	no	—
139	C₂²	#17986027	1	1	yes	characteristic
140	C₂²	#17986059	1	2	yes	—
141	C₄	—	4	2	no	—
142	C₄	—	568	2	no	—
143	C₄	#17985864	1	1	yes	characteristic
144	C₄	#17986025	1	1	yes	characteristic
145	C₄	#17986059	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	2	2	no	—
148	C₂	#22588459	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590099

C₂

yes

maximal, characteristic

#22588095

C₂

yes

maximal, characteristic

#22586430

C₂

yes

maximal

#22586425

C₂

yes

maximal

C₇₁×#2153

C₂

yes

maximal, characteristic

C₂×#14556068

C₂²

yes

characteristic

C₂²×C₇₁⋊C₁₆

C₂²

yes

characteristic

#17985954

C₂²

yes

—

C₄⋊(C₇₁⋊C₁₆)

—

C₇₁⋊#338

C₄

yes

characteristic

#17985756

C₄

yes

characteristic

#17985755

C₄

yes

—

C₂×M₃₂×C₇₁

C₂²

yes

—

#405×C₇₁

C₂²

yes

characteristic

#364×C₇₁

—

C₂×#11765849

C₂×C₄

yes

—

#14556087

C₂×C₄

yes

—

#14556068

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

—

#14556038

—

C₈×Dic₇₁

—

C₂×M₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

#117×C₇₁

C₂×C₄

yes

characteristic

M₃₂×C₇₁

—

M₃₂×C₇₁

—

C₂×C₁₆×C₇₁

—

#97×C₇₁

Q₈

yes

—

C₄×C₈×C₇₁

D₄

yes

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₇₁⋊C₁₆

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

M₁₆×C₇₁

—

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₁

#41

yes

—

C₂×C₈×C₇₁

#41

yes

—

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

C₄⋊C₄

yes

—

C₁₆×C₇₁

—

C₂³×C₇₁

#131

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

#117

yes

—

C₂×C₄×C₇₁

#131

yes

—

C₂×C₄×C₇₁

#131

yes

characteristic

C₂×Dic₇₁

—

C₈×C₇₁

—

C₈×C₇₁

yes

C₈×C₇₁

yes

C₈×C₇₁

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

#424

yes

characteristic

C₂²×C₇₁

#433

yes

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

Dic₇₁

—

#12885

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#1832

yes

characteristic

#2153

D₇₁

yes

—

—

—

—

C₇₁

#12885

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

C₂×D₇₁

yes

—

C₂²×C₁₆

—

#405

—

#405

C₂×D₇₁

yes

characteristic

#364

—

C₄⋊C₁₆

—

142

—

#338

—

H(3,ℤ/4)

—

H(3,ℤ/4)

—

C₂×M₁₆

—

C₂×M₁₆

C₄×D₇₁

yes

characteristic

C₂²×C₈

C₂²×D₇₁

yes

characteristic

#117

—

#117

C₄×D₇₁

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

142

—

C₂×C₁₆

—

100

C₂×C₁₆

—

142

—

101

C₂×C₁₆

—

142

—

102

C₂×C₁₆

—

142

—

103

#97

—

142

—

104

#97

Dic₁₄₂

yes

—

105

C₄×C₈

—

142

—

106

C₄×C₈

C₂²⋊D₇₁

yes

—

107

C₂²×C₄

C₂×C₄×D₇₁

yes

characteristic

108

M₁₆

—

109

M₁₆

—

142

—

110

M₁₆

—

142

—

111

C₂×C₈

—

142

—

112

C₂×C₈

Dic₇₁⋊C₄

yes

—

113

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

114

C₂×C₈

C₂×Dic₁₄₂

yes

characteristic

115

C₂×C₈

#11765864

yes

—

116

C₂×C₈

#11765866

yes

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

142

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

142

—

121

C₄²

—

142

—

122

C₄²

Dic₇₁⋊C₄

yes

—

123

C₁₆

—

124

C₁₆

—

284

—

125

C₁₆

—

284

—

126

C₂³

#14556136

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

284

—

129

C₂×C₄

C₂×Dic₇₁⋊C₄

yes

characteristic

130

C₂×C₄

#14556086

yes

—

131

C₂×C₄

#14556134

yes

—

132

C₂×C₄

#14556136

yes

characteristic

133

C₈

—

134

C₈

—

284

—

135

C₈

#14556115

yes

characteristic

136

C₈

#14556117

yes

characteristic

137

C₈

#14556119

yes

—

138

C₂²

—

139

C₂²

#17986027

yes

characteristic

140

C₂²

#17986059

yes

—

141

C₄

—

142

C₄

—

568

—

143

C₄

#17985864

yes

characteristic

144

C₄

#17986025

yes

characteristic

145

C₄

#17986059

yes

—

146

C₂

—

147

C₂

—

148

C₂

#22588459

yes

characteristic

	subgroup	G/H
center	C₈	#14556115
commutator	C₂²×C₄×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#2153	C₂
socle	C₂×C₇₁	#1832
prime radical	C₂³×C₇₁	#131
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#2153	D₇₁
hypocenter	C₇₁	#12885
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2153

subgroup

G/H

center

C₈

#14556115

commutator

C₂²×C₄×C₇₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₇₁×#2153

C₂

C₂×C₇₁

C₂³×C₇₁

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2153

N	H ≅ G/N	type
80. C₇₁	#12885	semidirect

H ≅ G/N

type

80. C₇₁

#12885

semidirect

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

subgroup	index
#22590099	2
#22588095	2
#22586430	2
#22586425	2
C₇₁×#2153	2
#12885	71

subgroup

index

C₇₁×#2153

Subgroups — #31913821

Conjugacy classes of subgroups

Complements

Maximal subgroups