Group #31913875 — subgroups

Subgroups — #31913875

Order 18176.

278 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 119 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	#22590105	C₂	1	2	yes	maximal
2	#22590100	C₂	1	2	yes	maximal
3	#22590099	C₂	1	1	yes	maximal, characteristic
4	#22588095	C₂	1	1	yes	maximal, characteristic
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	#17985819	—	2	4	no	—
10	C₄⋊(C₇₁⋊C₁₆)	—	2	2	no	—
11	Dic₇₁⋊C₁₆	—	2	4	no	—
12	C₇₁⋊#338	C₄	1	1	yes	characteristic
13	#17985756	C₄	1	1	yes	characteristic
14	#17985755	C₄	1	2	yes	—
15	C₂×M₃₂×C₇₁	C₂²	1	2	yes	—
16	#405×C₇₁	C₂²	1	1	yes	characteristic
17	#364×C₇₁	—	2	2	no	—
18	C₂×#11765849	C₂×C₄	1	2	yes	—
19	#14556087	C₂×C₄	1	2	yes	—
20	#14556068	—	2	2	no	—
21	C₂×C₇₁⋊C₁₆	—	2	2	no	—
22	C₂×C₇₁⋊C₁₆	—	2	1	no	—
23	C₂×C₇₁⋊C₁₆	—	2	1	no	—
24	#14556038	Q₈	1	4	yes	—
25	C₈×Dic₇₁	D₄	1	4	yes	—
26	C₂×M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
28	#117×C₇₁	C₂×C₄	1	1	yes	characteristic
29	M₃₂×C₇₁	—	2	2	no	—
30	M₃₂×C₇₁	—	2	2	no	—
31	C₂×C₁₆×C₇₁	—	2	2	no	—
32	#97×C₇₁	Q₈	1	2	yes	—
33	C₄×C₈×C₇₁	D₄	1	2	yes	—
34	#11765849	—	2	2	no	—
35	#11765849	—	2	2	no	—
36	C₂×C₇₁⋊C₈	C₄⋊C₄	1	4	yes	—
37	C₄⋊Dic₇₁	—	2	2	no	—
38	C₂²⋊Dic₇₁	—	2	2	no	—
39	C₄×Dic₇₁	C₄⋊C₄	1	4	yes	—
40	C₇₁⋊C₁₆	—	4	1	no	—
41	C₇₁⋊C₁₆	—	4	1	no	—
42	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
43	M₁₆×C₇₁	—	2	2	no	—
44	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
46	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
48	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
49	C₄⋊C₄×C₇₁	—	2	1	no	—
50	C₂²⋊C₄×C₇₁	—	2	1	no	—
51	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
52	C₁₆×C₇₁	—	4	2	no	—
53	C₂³×C₇₁	#131	1	1	yes	characteristic
54	C₂×C₄×C₇₁	—	2	2	no	—
55	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
56	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₇₁	#131	1	2	yes	—
58	C₂×C₄×C₇₁	#131	1	1	yes	characteristic
59	C₂×Dic₇₁	—	2	4	no	—
60	C₈×C₇₁	—	2	2	no	—
61	C₈×C₇₁	#127	1	1	yes	characteristic
62	C₈×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
63	C₈×C₇₁	C₄⋊Q₈	1	2	yes	—
64	C₇₁⋊C₈	—	2	4	no	—
65	C₂²×C₇₁	—	4	1	no	—
66	C₂²×C₇₁	#424	1	2	yes	—
67	C₂²×C₇₁	#424	1	1	yes	characteristic
68	C₄×C₇₁	—	4	2	no	—
69	C₄×C₇₁	#384	1	1	yes	characteristic
70	C₄×C₇₁	#424	1	2	yes	—
71	C₄×C₇₁	#424	1	1	yes	characteristic
72	Dic₇₁	—	4	4	no	—
73	#12902	—	71	1	no	maximal, Sylow
74	C₂×C₇₁	—	2	2	no	—
75	C₂×C₇₁	—	2	1	no	—
76	C₂×C₇₁	#1843	1	1	yes	characteristic
77	#2153	—	71	2	no	—
78	#2153	D₇₁	1	1	yes	characteristic
79	#2151	—	71	2	no	—
80	#2151	—	71	1	no	—
81	#1747	—	71	1	no	—
82	C₇₁	#12902	1	1	yes	characteristic, complemented, Sylow
83	C₂×M₃₂	—	71	1	no	—
84	C₂×M₃₂	C₂×D₇₁	1	2	yes	—
85	C₂²×C₁₆	—	71	1	no	—
86	#405	—	71	2	no	—
87	#405	C₂×D₇₁	1	1	yes	characteristic
88	#364	—	2	2	no	—
89	#364	—	142	4	no	—
90	C₄⋊C₁₆	—	142	4	no	—
91	C₄⋊C₁₆	—	142	2	no	—
92	#338	—	71	1	no	—
93	H(3,ℤ/4)	—	71	1	no	—
94	H(3,ℤ/4)	—	71	2	no	—
95	C₂×M₁₆	—	71	2	no	—
96	C₂×M₁₆	C₄×D₇₁	1	1	yes	characteristic
97	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
98	#117	—	71	2	no	—
99	#117	C₄×D₇₁	1	1	yes	characteristic
100	M₃₂	—	2	2	no	—
101	M₃₂	—	2	2	no	—
102	M₃₂	—	142	2	no	—
103	C₂×C₁₆	—	2	2	no	—
104	C₂×C₁₆	—	142	2	no	—
105	C₂×C₁₆	—	142	1	no	—
106	C₂×C₁₆	—	142	1	no	—
107	#97	—	71	4	no	—
108	#97	Dic₁₄₂	1	2	yes	—
109	C₄×C₈	—	71	4	no	—
110	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
111	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
112	M₁₆	—	2	2	no	—
113	M₁₆	—	142	2	no	—
114	M₁₆	—	142	2	no	—
115	C₂×C₈	—	71	4	no	—
116	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
117	C₂×C₈	D₄×D₇₁	1	2	yes	—
118	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
119	C₂×C₈	Q₈×D₇₁	1	2	yes	—
120	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	142	2	no	—
123	C₂²⋊C₄	—	2	1	no	—
124	C₂²⋊C₄	—	142	2	no	—
125	C₄²	—	71	4	no	—
126	C₄²	Dic₇₁⋊C₄	1	2	yes	—
127	C₁₆	—	4	2	no	—
128	C₁₆	—	284	1	no	—
129	C₁₆	—	284	1	no	—
130	C₂³	#14556136	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	142	4	no	—
133	C₂×C₄	C₄⋊C₄×D₇₁	1	2	yes	—
134	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
135	C₂×C₄	#14556134	1	2	yes	—
136	C₂×C₄	#14556136	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	142	4	no	—
139	C₈	#14556124	1	1	yes	characteristic
140	C₈	#14556126	1	1	yes	characteristic
141	C₈	#14556127	1	2	yes	—
142	C₂²	—	4	1	no	—
143	C₂²	#17986024	1	2	yes	—
144	C₂²	#17986027	1	1	yes	characteristic
145	C₄	—	4	2	no	—
146	C₄	—	284	4	no	—
147	C₄	#17985868	1	1	yes	characteristic
148	C₄	#17986024	1	2	yes	—
149	C₄	#17986025	1	1	yes	characteristic
150	C₂	—	2	1	no	—
151	C₂	—	2	2	no	—
152	C₂	#22588493	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590105

C₂

yes

maximal

#22590100

C₂

yes

maximal

#22590099

C₂

yes

maximal, characteristic

#22588095

C₂

yes

maximal, characteristic

C₇₁×#2153

C₂

yes

maximal, characteristic

C₂×#14556068

C₂²

yes

characteristic

C₂²×C₇₁⋊C₁₆

C₂²

yes

characteristic

#17985954

C₂²

yes

—

#17985819

—

C₄⋊(C₇₁⋊C₁₆)

—

Dic₇₁⋊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

Q₈

yes

—

C₈×Dic₇₁

D₄

yes

—

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⋊C₄

yes

—

C₄⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

C₄⋊C₄

yes

—

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

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

—

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

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

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

#131

yes

—

C₂×C₄×C₇₁

#131

yes

characteristic

C₂×Dic₇₁

—

C₈×C₇₁

—

C₈×C₇₁

#127

yes

characteristic

C₈×C₇₁

C₄⋊Q₈

yes

characteristic

C₈×C₇₁

C₄⋊Q₈

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

#424

yes

—

C₂²×C₇₁

#424

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

#384

yes

characteristic

C₄×C₇₁

#424

yes

—

C₄×C₇₁

#424

yes

characteristic

Dic₇₁

—

#12902

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

—

D₇₁

yes

—

—

—

C₇₁

#12902

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

C₂×D₇₁

yes

—

C₂²×C₁₆

—

#405

—

#405

C₂×D₇₁

yes

characteristic

#364

—

#364

—

142

—

C₄⋊C₁₆

—

142

—

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

100

M₃₂

—

101

M₃₂

—

102

M₃₂

—

142

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

142

—

105

C₂×C₁₆

—

142

—

106

C₂×C₁₆

—

142

—

107

#97

—

108

#97

Dic₁₄₂

yes

—

109

C₄×C₈

—

110

C₄×C₈

C₂²⋊D₇₁

yes

—

111

C₂²×C₄

C₂×C₄×D₇₁

yes

characteristic

112

M₁₆

—

113

M₁₆

—

142

—

114

M₁₆

—

142

—

115

C₂×C₈

—

116

C₂×C₈

Dic₇₁⋊C₄

yes

—

117

C₂×C₈

D₄×D₇₁

yes

—

118

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

119

C₂×C₈

Q₈×D₇₁

yes

—

120

C₂×C₈

C₂×Dic₁₄₂

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

142

—

123

C₂²⋊C₄

—

124

C₂²⋊C₄

—

142

—

125

C₄²

—

126

C₄²

Dic₇₁⋊C₄

yes

—

127

C₁₆

—

128

C₁₆

—

284

—

129

C₁₆

—

284

—

130

C₂³

#14556136

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

142

—

133

C₂×C₄

C₄⋊C₄×D₇₁

yes

—

134

C₂×C₄

C₂×Dic₇₁⋊C₄

yes

characteristic

135

C₂×C₄

#14556134

yes

—

136

C₂×C₄

#14556136

yes

characteristic

137

C₈

—

138

C₈

—

142

—

139

C₈

#14556124

yes

characteristic

140

C₈

#14556126

yes

characteristic

141

C₈

#14556127

yes

—

142

C₂²

—

143

C₂²

#17986024

yes

—

144

C₂²

#17986027

yes

characteristic

145

C₄

—

146

C₄

—

284

—

147

C₄

#17985868

yes

characteristic

148

C₄

#17986024

yes

—

149

C₄

#17986025

yes

characteristic

150

C₂

—

151

C₂

—

152

C₂

#22588493

yes

characteristic

	subgroup	G/H
center	C₈	#14556124
commutator	C₂²×C₄×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#2153	C₂
socle	C₂×C₇₁	#1843
prime radical	C₂³×C₇₁	#131
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#2153	D₇₁
hypocenter	C₇₁	#12902
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2153

subgroup

G/H

center

C₈

#14556124

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
82. C₇₁	#12902	semidirect

H ≅ G/N

type

82. C₇₁

#12902

semidirect

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

subgroup	index
#22590105	2
#22590100	2
#22590099	2
#22588095	2
C₇₁×#2153	2
#12902	71

subgroup

index

C₇₁×#2153

Subgroups — #31913875

Conjugacy classes of subgroups

Complements

Maximal subgroups