Group #31891262 — subgroups

Subgroups — #31891262

Order 18176.

338 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 147 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	#22588511	C₂	1	2	yes	maximal
2	#22587096	C₂	1	1	yes	maximal, characteristic
3	#22587095	C₂	1	2	yes	maximal
4	C₂×#17985721	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1566	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556063	—	2	4	no	—
7	#17986023	C₂²	1	2	yes	—
8	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
10	#17985750	—	2	2	no	—
11	#17985749	—	2	4	no	—
12	#17985721	C₄	1	4	yes	—
13	#423×C₇₁	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	#334×C₇₁	—	2	2	no	—
16	C₂×#11765849	—	2	2	no	—
17	C₂²×C₇₁⋊C₈	D₄	1	2	yes	—
18	C₂²×C₇₁⋊C₈	Q₈	1	2	yes	—
19	#14556063	—	4	8	no	—
20	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
21	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
22	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
23	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
24	C₄×C₇₁⋊C₈	D₄	1	2	yes	—
25	C₄×C₇₁⋊C₈	Q₈	1	2	yes	—
26	C₂×M₁₆×C₇₁	—	2	2	no	—
27	C₂²×C₈×C₇₁	Q₈	1	2	yes	—
28	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₇₁	—	2	2	no	—
30	C₄⋊C₈×C₇₁	—	2	2	no	—
31	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₇₁	D₄	1	2	yes	—
33	#11765849	—	4	4	no	—
34	C₂×C₇₁⋊C₈	—	4	2	no	—
35	C₂×C₇₁⋊C₈	—	4	2	no	—
36	C₂×C₇₁⋊C₈	C₄⋊C₄	1	4	yes	—
37	C₂×C₇₁⋊C₈	C₄⋊C₄	1	4	yes	—
38	C₂×C₇₁⋊C₈	SD₁₆	1	8	yes	—
39	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
42	M₁₆×C₇₁	—	4	4	no	—
43	C₂×C₈×C₇₁	—	4	2	no	—
44	C₂×C₈×C₇₁	—	4	2	no	—
45	C₂×C₈×C₇₁	C₄⋊C₄	1	4	yes	—
46	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
50	C₂³×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
51	C₂×C₄×C₇₁	—	2	2	no	—
52	C₂×C₄×C₇₁	—	2	2	no	—
53	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
57	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₇₁	C₂×SD₁₆	1	4	yes	—
60	C₈×C₇₁	—	2	4	no	—
61	C₈×C₇₁	—	8	2	no	—
62	C₇₁⋊C₈	—	2	4	no	—
63	C₇₁⋊C₈	—	8	2	no	—
64	C₇₁⋊C₈	#106	1	8	yes	—
65	C₂²×C₇₁	—	2	1	no	—
66	C₂²×C₇₁	—	2	1	no	—
67	C₂²×C₇₁	#384	1	1	yes	characteristic
68	C₂²×C₇₁	#498	1	2	yes	—
69	C₄×C₇₁	—	2	1	no	—
70	C₄×C₇₁	—	2	1	no	—
71	C₄×C₇₁	C₂×#106	1	2	yes	—
72	C₄×C₇₁	#430	1	2	yes	—
73	C₄×C₇₁	#430	1	2	yes	—
74	C₄×C₇₁	#492	1	2	yes	—
75	#8825	—	71	1	no	maximal, Sylow
76	C₂×C₇₁	—	2	2	no	—
77	C₂×C₇₁	#1844	1	1	yes	characteristic
78	C₂×C₇₁	#1849	1	1	yes	characteristic
79	C₂×C₇₁	#1849	1	1	yes	characteristic
80	#1848	—	71	2	no	—
81	#1566	—	71	2	no	—
82	#1566	—	71	1	no	—
83	#1566	D₇₁	1	1	yes	characteristic
84	C₂×#322	—	71	1	no	—
85	C₇₁	#8825	1	1	yes	characteristic, complemented, Sylow
86	C₂×#108	—	142	4	no	—
87	#423	—	71	2	no	—
88	#423	C₂×D₇₁	1	2	yes	—
89	C₂×C₄×C₈	—	71	1	no	—
90	C₂×C₄×C₈	—	71	1	no	—
91	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
92	#334	—	2	2	no	—
93	#334	—	142	4	no	—
94	#334	—	142	2	no	—
95	#322	—	71	4	no	—
96	C₂×M₁₆	—	2	2	no	—
97	C₂×M₁₆	—	142	2	no	—
98	C₂²×C₈	—	71	2	no	—
99	C₂²×C₈	—	71	2	no	—
100	C₂²×C₈	Dic₁₄₂	1	2	yes	—
101	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
102	#108	—	284	8	no	—
103	C₄⋊C₈	—	2	2	no	—
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	142	2	no	—
106	C₄⋊C₈	—	142	2	no	—
107	C₄×C₈	—	71	2	no	—
108	C₄×C₈	—	71	2	no	—
109	C₄×C₈	—	71	2	no	—
110	C₄×C₈	—	71	2	no	—
111	C₄×C₈	C₄×D₇₁	1	2	yes	—
112	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
113	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
114	C₂²×C₄	Q₈×D₇₁	1	1	yes	characteristic
115	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
116	M₁₆	—	4	4	no	—
117	M₁₆	—	284	4	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	4	2	no	—
120	C₂×C₈	—	284	2	no	—
121	C₂×C₈	—	284	2	no	—
122	C₂×C₈	—	71	4	no	—
123	C₂×C₈	—	71	8	no	—
124	C₂×C₈	—	71	4	no	—
125	C₂×C₈	Dic₇₁⋊C₄	1	4	yes	—
126	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
127	C₄²	D₄×D₇₁	1	1	yes	characteristic
128	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
129	C₄²	Q₈×D₇₁	1	1	yes	characteristic
130	C₂³	#14556127	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
134	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
135	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
136	C₂×C₄	#14556124	1	1	yes	characteristic
137	C₂×C₄	#14556126	1	1	yes	characteristic
138	C₂×C₄	#14556127	1	1	yes	characteristic
139	C₂×C₄	SD₁₆×D₇₁	1	4	yes	—
140	C₈	—	2	4	no	—
141	C₈	—	8	2	no	—
142	C₈	—	568	2	no	—
143	C₈	—	71	8	no	—
144	C₈	—	142	4	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	#17985868	1	1	yes	characteristic
148	C₂²	#17986396	1	2	yes	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	#106×D₇₁	1	2	yes	—
152	C₄	#17986046	1	2	yes	—
153	C₄	#17986047	1	2	yes	—
154	C₄	#17986370	1	2	yes	—
155	C₂	—	2	2	no	—
156	C₂	#22588495	1	1	yes	characteristic
157	C₂	#22588513	1	1	yes	characteristic
158	C₂	#22588514	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22588511

C₂

yes

maximal

#22587096

C₂

yes

maximal, characteristic

#22587095

C₂

yes

maximal

C₂×#17985721

C₂

yes

maximal, characteristic

C₇₁×#1566

C₂

yes

maximal, characteristic

C₂×#14556063

—

#17986023

C₂²

yes

—

C₂×C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₇₁⋊C₈

C₂²

yes

—

—

C₄

yes

—

#423×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#334×C₇₁

—

C₂×#11765849

—

C₂²×C₇₁⋊C₈

D₄

yes

—

C₂²×C₇₁⋊C₈

Q₈

yes

—

#14556063

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

C₄×C₇₁⋊C₈

C₂×C₄

yes

—

C₄×C₇₁⋊C₈

C₂×C₄

yes

—

C₄×C₇₁⋊C₈

D₄

yes

—

C₄×C₇₁⋊C₈

Q₈

yes

—

C₂×M₁₆×C₇₁

—

C₂²×C₈×C₇₁

Q₈

yes

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

D₄

yes

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

—

C₂×C₇₁⋊C₈

C₄⋊C₄

yes

—

C₂×C₇₁⋊C₈

SD₁₆

yes

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂²×C₄×C₇₁

C₂×Q₈

yes

characteristic

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₄²×C₇₁

C₂²×C₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×Q₈

yes

characteristic

C₂³×C₇₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

#127

yes

characteristic

C₂×C₄×C₇₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₁

C₂×SD₁₆

yes

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

#106

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#384

yes

characteristic

C₂²×C₇₁

#498

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×#106

yes

—

C₄×C₇₁

#430

yes

—

C₄×C₇₁

#430

yes

—

C₄×C₇₁

#492

yes

—

#8825

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

—

—

D₇₁

yes

characteristic

C₂×#322

—

C₇₁

#8825

yes

characteristic, complemented, Sylow

C₂×#108

—

142

—

#423

—

#423

C₂×D₇₁

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

#334

—

#334

—

142

—

#334

—

142

—

#322

—

C₂×M₁₆

—

C₂×M₁₆

—

142

—

C₂²×C₈

—

C₂²×C₈

—

100

C₂²×C₈

Dic₁₄₂

yes

—

101

C₂×C₄²

C₂²×D₇₁

yes

characteristic

102

#108

—

284

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

142

—

106

C₄⋊C₈

—

142

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

C₄×D₇₁

yes

—

112

C₄×C₈

C₂²⋊D₇₁

yes

—

113

C₂²×C₄

D₄×D₇₁

yes

characteristic

114

C₂²×C₄

Q₈×D₇₁

yes

characteristic

115

C₂²×C₄

C₂×Dic₁₄₂

yes

characteristic

116

M₁₆

—

117

M₁₆

—

284

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

284

—

121

C₂×C₈

—

284

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

Dic₇₁⋊C₄

yes

—

126

C₄²

C₂×C₄×D₇₁

yes

characteristic

127

C₄²

D₄×D₇₁

yes

characteristic

128

C₄²

C₂×C₂²⋊D₇₁

yes

characteristic

129

C₄²

Q₈×D₇₁

yes

characteristic

130

C₂³

#14556127

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₄⋊C₄×D₇₁

yes

characteristic

134

C₂×C₄

C₄⋊C₄×D₇₁

yes

characteristic

135

C₂×C₄

C₂×Dic₇₁⋊C₄

yes

characteristic

136

C₂×C₄

#14556124

yes

characteristic

137

C₂×C₄

#14556126

yes

characteristic

138

C₂×C₄

#14556127

yes

characteristic

139

C₂×C₄

SD₁₆×D₇₁

yes

—

140

C₈

—

141

C₈

—

142

C₈

—

568

—

143

C₈

—

144

C₈

—

142

—

145

C₂²

—

146

C₂²

—

147

C₂²

#17985868

yes

characteristic

148

C₂²

#17986396

yes

—

149

C₄

—

150

C₄

—

151

C₄

#106×D₇₁

yes

—

152

C₄

#17986046

yes

—

153

C₄

#17986047

yes

—

154

C₄

#17986370

yes

—

155

C₂

—

156

C₂

#22588495

yes

characteristic

157

C₂

#22588513

yes

characteristic

158

C₂

#22588514

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556124
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1566	C₂
socle	C₂²×C₇₁	#384
prime radical	C₂³×C₇₁	C₄⋊Q₈
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1566	D₇₁
hypocenter	C₇₁	#8825
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

subgroup

G/H

center

C₂×C₄

#14556124

commutator

C₄²×C₇₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#1566

C₂

socle

C₂²×C₇₁

#384

prime radical

C₂³×C₇₁

C₄⋊Q₈

cosocle

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

N	H ≅ G/N	type
85. C₇₁	#8825	semidirect

H ≅ G/N

type

85. C₇₁

#8825

semidirect

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

subgroup	index
#22588511	2
#22587096	2
#22587095	2
C₂×#17985721	2
C₇₁×#1566	2
#8825	71

subgroup

index

C₇₁×#1566

Subgroups — #31891262

Conjugacy classes of subgroups

Complements

Maximal subgroups