Group #31909083 — subgroups

Subgroups — #31909083

Order 18176.

226 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 107 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	C₂×#17985775	C₂	1	1	yes	maximal, characteristic
2	#22587103	C₂	1	1	yes	maximal, characteristic
3	#22586405	C₂	1	4	yes	maximal, complemented
4	C₇₁×#1566	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₁⋊C₁₆	C₄	1	2	yes	—
6	#17986056	—	2	1	no	—
7	#17985775	C₄	1	1	yes	characteristic
8	#17985775	C₄	1	1	yes	characteristic
9	#17985775	C₂²	1	2	yes	—
10	#17985752	—	2	1	no	—
11	#17985750	C₂²	1	2	yes	—
12	#423×C₇₁	—	2	1	no	—
13	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	#334×C₇₁	—	2	1	no	—
15	#334×C₇₁	C₂²	1	2	yes	—
16	C₂×C₇₁⋊C₁₆	—	2	4	no	—
17	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
18	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
19	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
20	C₂²⋊(C₇₁⋊C₈)	—	4	1	no	—
21	#14556039	—	4	1	no	—
22	C₂×M₁₆×C₇₁	—	4	1	no	—
23	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₇₁	Q₈	1	1	yes	characteristic
25	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
26	C₄⋊C₈×C₇₁	—	2	2	no	—
27	C₄⋊C₈×C₇₁	—	4	1	no	—
28	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
30	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
31	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
32	C₂×C₇₁⋊C₈	—	4	2	no	—
33	C₇₁⋊C₁₆	—	2	2	no	—
34	C₇₁⋊C₁₆	M₁₆	1	4	yes	complemented
35	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
37	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
38	M₁₆×C₇₁	—	8	2	no	—
39	C₂×C₈×C₇₁	—	2	2	no	—
40	C₂×C₈×C₇₁	—	2	2	no	—
41	C₂×C₈×C₇₁	—	4	2	no	—
42	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
46	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₇₁	SD₁₆	1	1	yes	characteristic
48	C₄²×C₇₁	SD₁₆	1	1	yes	characteristic
49	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
50	C₂³×C₇₁	#106	1	1	yes	characteristic
51	C₂×C₄×C₇₁	—	2	2	no	—
52	C₂×C₄×C₇₁	—	2	1	no	—
53	C₂×C₄×C₇₁	—	2	1	no	—
54	C₂×C₄×C₇₁	—	4	1	no	—
55	C₂×C₄×C₇₁	#106	1	1	yes	characteristic
56	C₂×C₄×C₇₁	#106	1	1	yes	characteristic
57	C₂×C₄×C₇₁	#106	1	1	yes	characteristic
58	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
60	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
61	C₈×C₇₁	—	2	1	no	—
62	C₈×C₇₁	—	2	1	no	—
63	C₈×C₇₁	—	8	2	no	—
64	C₈×C₇₁	#108	1	2	yes	—
65	C₈×C₇₁	C₂×M₁₆	1	2	yes	—
66	C₇₁⋊C₈	—	8	2	no	—
67	C₂²×C₇₁	—	2	1	no	—
68	C₂²×C₇₁	—	2	1	no	—
69	C₂²×C₇₁	#365	1	2	yes	—
70	C₂²×C₇₁	C₂×#106	1	1	yes	characteristic
71	C₄×C₇₁	—	2	1	no	—
72	C₄×C₇₁	—	2	1	no	—
73	C₄×C₇₁	—	2	1	no	—
74	C₄×C₇₁	—	2	1	no	—
75	C₄×C₇₁	—	2	1	no	—
76	C₄×C₇₁	#423	1	1	yes	characteristic
77	C₄×C₇₁	C₂×#108	1	1	yes	characteristic
78	#12019	—	71	1	no	maximal, Sylow
79	C₂×C₇₁	—	2	2	no	—
80	C₂×C₇₁	#1566	1	1	yes	characteristic
81	C₂×C₇₁	C₂×#365	1	1	yes	characteristic
82	C₂×C₇₁	#2155	1	1	yes	characteristic
83	C₂×#346	—	71	1	no	—
84	#1568	—	71	1	no	—
85	#1566	D₇₁	1	1	yes	characteristic
86	#1368	—	71	4	no	—
87	C₇₁	#12019	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₁₆	—	71	2	no	—
89	#432	—	142	1	no	—
90	#423	—	2	1	no	—
91	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
92	#346	—	71	1	no	—
93	#346	—	71	1	no	—
94	#346	—	71	2	no	—
95	#335	—	142	1	no	—
96	#334	—	2	1	no	—
97	#334	—	71	2	no	—
98	#334	C₂×D₇₁	1	2	yes	—
99	C₂×M₁₆	—	4	1	no	—
100	C₂²×C₈	Dic₁₄₂	1	1	yes	characteristic
101	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
102	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
103	C₂×C₁₆	—	71	2	no	—
104	C₂×C₁₆	—	71	2	no	—
105	C₂×C₁₆	—	142	4	no	—
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	—	4	1	no	—
108	C₄⋊C₈	—	142	2	no	—
109	C₂²⋊C₈	—	284	1	no	—
110	#97	—	284	1	no	—
111	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
112	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
113	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
114	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
115	C₂²×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
116	C₂²×C₄	#11765852	1	1	yes	characteristic
117	C₂²×C₄	#11765853	1	1	yes	characteristic
118	M₁₆	—	8	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	—	284	2	no	—
123	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
124	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
125	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
126	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
127	C₄²	Dic₇₁⋊C₄	1	1	yes	characteristic
128	C₄²	#11765852	1	1	yes	characteristic
129	C₄²	#11765853	1	1	yes	characteristic
130	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
131	C₁₆	—	71	4	no	—
132	C₁₆	—	142	2	no	—
133	C₂³	#14556059	1	1	yes	characteristic
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	#14556059	1	1	yes	characteristic
139	C₂×C₄	#14556059	1	1	yes	characteristic
140	C₂×C₄	#14556059	1	1	yes	characteristic
141	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
142	C₂×C₄	C₂×#11765852	1	1	yes	characteristic
143	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	8	2	no	—
147	C₈	—	568	2	no	—
148	C₈	#14556063	1	2	yes	—
149	C₈	M₁₆×D₇₁	1	2	yes	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#17985821	1	2	yes	—
153	C₂²	C₂×#14556059	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	#17986022	1	1	yes	characteristic
160	C₄	C₂×#14556063	1	1	yes	characteristic
161	C₂	—	2	2	no	—
162	C₂	#22587095	1	1	yes	characteristic
163	C₂	C₂×#17985821	1	1	yes	characteristic
164	C₂	#22590114	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17985775

C₂

yes

maximal, characteristic

#22587103

C₂

yes

maximal, characteristic

#22586405

C₂

yes

maximal, complemented

C₇₁×#1566

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₁₆

C₄

yes

—

#17986056

—

#17985775

C₄

yes

characteristic

#17985775

C₄

yes

characteristic

#17985775

C₂²

yes

—

#17985752

—

#17985750

C₂²

yes

—

#423×C₇₁

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#334×C₇₁

—

#334×C₇₁

C₂²

yes

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

—

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

—

C₄⋊(C₇₁⋊C₈)

—

C₂²⋊(C₇₁⋊C₈)

—

#14556039

—

C₂×M₁₆×C₇₁

—

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₁

Q₈

yes

characteristic

C₂×C₄²×C₇₁

D₄

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

Q₈

yes

characteristic

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₇₁⋊C₈

—

C₇₁⋊C₁₆

—

C₇₁⋊C₁₆

M₁₆

yes

complemented

C₂²×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₁

SD₁₆

yes

characteristic

C₂²×C₄×C₇₁

SD₁₆

yes

characteristic

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

C₄²×C₇₁

C₄⋊C₄

yes

characteristic

C₄²×C₇₁

SD₁₆

yes

characteristic

C₄²×C₇₁

SD₁₆

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₂³×C₇₁

#106

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

#108

yes

—

C₈×C₇₁

C₂×M₁₆

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#365

yes

—

C₂²×C₇₁

C₂×#106

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

—

D₇₁

yes

characteristic

#1368

—

C₇₁

#12019

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

#432

—

142

—

#423

—

C₂×C₄×C₈

C₂×D₇₁

yes

—

—

—

—

142

—

#334

—

#334

—

#334

C₂×D₇₁

yes

—

C₂×M₁₆

—

100

C₂²×C₈

Dic₁₄₂

yes

characteristic

101

C₂²×C₈

C₄×D₇₁

yes

characteristic

102

C₂×C₄²

C₂²⋊D₇₁

yes

characteristic

103

C₂×C₁₆

—

104

C₂×C₁₆

—

105

C₂×C₁₆

—

142

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

142

—

109

C₂²⋊C₈

—

284

—

110

#97

—

284

—

111

C₄×C₈

Dic₁₄₂

yes

characteristic

112

C₄×C₈

C₄×D₇₁

yes

characteristic

113

C₄×C₈

C₂²⋊D₇₁

yes

characteristic

114

C₄×C₈

C₂²×D₇₁

yes

characteristic

115

C₂²×C₄

Dic₇₁⋊C₄

yes

characteristic

116

C₂²×C₄

#11765852

yes

characteristic

117

C₂²×C₄

#11765853

yes

characteristic

118

M₁₆

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

284

—

123

C₂×C₈

Dic₇₁⋊C₄

yes

characteristic

124

C₂×C₈

Dic₇₁⋊C₄

yes

characteristic

125

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

126

C₂×C₈

C₂×Dic₁₄₂

yes

characteristic

127

C₄²

Dic₇₁⋊C₄

yes

characteristic

128

C₄²

#11765852

yes

characteristic

129

C₄²

#11765853

yes

characteristic

130

C₄²

C₂×C₂²⋊D₇₁

yes

characteristic

131

C₁₆

—

132

C₁₆

—

142

—

133

C₂³

#14556059

yes

characteristic

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

#14556059

yes

characteristic

139

C₂×C₄

#14556059

yes

characteristic

140

C₂×C₄

#14556059

yes

characteristic

141

C₂×C₄

C₂×Dic₇₁⋊C₄

yes

characteristic

142

C₂×C₄

C₂×#11765852

yes

characteristic

143

C₂×C₄

C₂×#11765853

yes

characteristic

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

568

—

148

C₈

#14556063

yes

—

149

C₈

M₁₆×D₇₁

yes

—

150

C₂²

—

151

C₂²

—

152

C₂²

#17985821

yes

—

153

C₂²

C₂×#14556059

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

#17986022

yes

characteristic

160

C₄

C₂×#14556063

yes

characteristic

161

C₂

—

162

C₂

#22587095

yes

characteristic

163

C₂

C₂×#17985821

yes

characteristic

164

C₂

#22590114

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#14556059
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1566	C₂
socle	C₂²×C₇₁	C₂×#106
prime radical	C₂³×C₇₁	#106
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	#1566	D₇₁
hypocenter	C₇₁	#12019
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1566

subgroup

G/H

center

C₂²

C₂×#14556059

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#1566

C₂

C₂²×C₇₁

C₂³×C₇₁

C₄×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1566

N	H ≅ G/N	type
3. #22586405	C₂	semidirect
34. C₇₁⋊C₁₆	M₁₆	semidirect
87. C₇₁	#12019	semidirect

H ≅ G/N

type

3. #22586405

C₂

semidirect

34. C₇₁⋊C₁₆

M₁₆

semidirect

87. C₇₁

#12019

semidirect

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

subgroup	index
C₂×#17985775	2
#22587103	2
#22586405	2
C₇₁×#1566	2
#12019	71

subgroup

index

C₇₁×#1566

Subgroups — #31909083

Conjugacy classes of subgroups

Complements

Maximal subgroups