Group #31909124 — subgroups

Subgroups — #31909124

Order 18176.

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

normal subgroups only

This group has 218 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	#22590143	C₂	1	1	yes	maximal, characteristic
2	#22587096	C₂	1	1	yes	maximal, characteristic
3	#22586405	C₂	1	4	yes	maximal, complemented
4	C₂×#334×C₇₁	C₂	1	1	yes	maximal, characteristic
5	#17986023	—	2	1	no	—
6	C₄⋊(C₇₁⋊C₁₆)	C₄	1	2	yes	—
7	C₂²⋊(C₇₁⋊C₁₆)	C₄	1	2	yes	—
8	#17985775	C₂²	1	2	yes	—
9	#17985750	—	2	1	no	—
10	#17985750	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₇₁	—	2	1	no	—
12	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
13	#334×C₇₁	—	2	1	no	—
14	#334×C₇₁	C₂²	1	2	yes	—
15	C₂×#11765849	—	4	1	no	—
16	C₂×C₇₁⋊C₁₆	C₂×C₄	1	4	yes	—
17	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
18	C₄⋊(C₇₁⋊C₈)	—	4	1	no	—
19	C₂²×C₈×C₇₁	—	4	1	no	—
20	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₇₁	Q₈	1	1	yes	characteristic
22	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
23	C₄⋊C₈×C₇₁	—	2	2	no	—
24	C₄⋊C₈×C₇₁	—	4	1	no	—
25	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
27	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
28	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
29	#11765849	—	8	2	no	—
30	C₂×C₇₁⋊C₈	—	4	2	no	—
31	C₇₁⋊C₁₆	—	2	4	no	—
32	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
34	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
35	C₂×C₈×C₇₁	—	2	2	no	—
36	C₂×C₈×C₇₁	—	4	1	no	—
37	C₂×C₈×C₇₁	—	4	2	no	—
38	C₂×C₈×C₇₁	—	8	2	no	—
39	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
43	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₇₁	SD₁₆	1	1	yes	characteristic
45	C₄²×C₇₁	SD₁₆	1	1	yes	characteristic
46	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
47	C₂³×C₇₁	#106	1	1	yes	characteristic
48	C₂×C₄×C₇₁	—	2	2	no	—
49	C₂×C₄×C₇₁	—	2	2	no	—
50	C₂×C₄×C₇₁	—	2	1	no	—
51	C₂×C₄×C₇₁	—	2	1	no	—
52	C₂×C₄×C₇₁	#106	1	1	yes	characteristic
53	C₂×C₄×C₇₁	#106	1	1	yes	characteristic
54	C₂×C₄×C₇₁	#106	1	1	yes	characteristic
55	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
58	C₈×C₇₁	—	2	1	no	—
59	C₈×C₇₁	—	2	1	no	—
60	C₈×C₇₁	—	2	1	no	—
61	C₈×C₇₁	—	8	2	no	—
62	C₈×C₇₁	#131	1	2	yes	—
63	C₇₁⋊C₈	—	8	2	no	—
64	C₂²×C₇₁	—	2	1	no	—
65	C₂²×C₇₁	—	2	1	no	—
66	C₂²×C₇₁	#365	1	2	yes	—
67	C₂²×C₇₁	C₂×#106	1	1	yes	characteristic
68	C₄×C₇₁	—	2	1	no	—
69	C₄×C₇₁	—	2	1	no	—
70	C₄×C₇₁	—	2	1	no	—
71	C₄×C₇₁	—	2	1	no	—
72	C₄×C₇₁	#365	1	2	yes	—
73	C₄×C₇₁	#424	1	1	yes	characteristic
74	C₄×C₇₁	#430	1	1	yes	characteristic
75	#12030	—	71	1	no	maximal, Sylow
76	C₂×C₇₁	—	2	2	no	—
77	C₂×C₇₁	#1570	1	1	yes	characteristic
78	C₂×C₇₁	C₂×#365	1	1	yes	characteristic
79	C₂×C₇₁	C₂×#365	1	1	yes	characteristic
80	#2163	—	71	1	no	—
81	#1566	—	71	1	no	—
82	C₂×#334	D₇₁	1	1	yes	characteristic
83	#1368	—	71	4	no	—
84	C₇₁	#12030	1	1	yes	characteristic, complemented, Sylow
85	#423	—	142	1	no	—
86	C₂×C₄⋊C₈	—	2	1	no	—
87	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
88	C₄⋊C₁₆	—	71	2	no	—
89	C₂²⋊C₁₆	—	71	2	no	—
90	#346	—	71	2	no	—
91	#334	—	2	1	no	—
92	#334	—	71	2	no	—
93	#334	—	142	1	no	—
94	#334	C₂×D₇₁	1	2	yes	—
95	C₂×M₁₆	—	284	1	no	—
96	C₂²×C₈	—	4	1	no	—
97	C₂²×C₈	Dic₁₄₂	1	1	yes	characteristic
98	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
99	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
100	C₂×C₁₆	—	71	4	no	—
101	C₄⋊C₈	—	2	2	no	—
102	C₄⋊C₈	—	4	1	no	—
103	C₄⋊C₈	—	284	1	no	—
104	C₄⋊C₈	—	142	2	no	—
105	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
106	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
107	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
108	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
109	C₂²×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
110	C₂²×C₄	#11765852	1	1	yes	characteristic
111	C₂²×C₄	#11765853	1	1	yes	characteristic
112	M₁₆	—	568	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	4	1	no	—
115	C₂×C₈	—	4	2	no	—
116	C₂×C₈	—	8	2	no	—
117	C₂×C₈	—	284	2	no	—
118	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
119	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
120	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
121	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
122	C₄²	Dic₇₁⋊C₄	1	1	yes	characteristic
123	C₄²	#11765852	1	1	yes	characteristic
124	C₄²	#11765853	1	1	yes	characteristic
125	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
126	C₁₆	—	142	4	no	—
127	C₂³	#14556059	1	1	yes	characteristic
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	#14556059	1	1	yes	characteristic
133	C₂×C₄	#14556059	1	1	yes	characteristic
134	C₂×C₄	#14556059	1	1	yes	characteristic
135	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
136	C₂×C₄	C₂×#11765852	1	1	yes	characteristic
137	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
138	C₈	—	2	1	no	—
139	C₈	—	2	1	no	—
140	C₈	—	2	1	no	—
141	C₈	—	8	2	no	—
142	C₈	—	568	2	no	—
143	C₈	#14556134	1	2	yes	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	#17985823	1	2	yes	—
147	C₂²	C₂×#14556059	1	1	yes	characteristic
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	#17985821	1	2	yes	—
153	C₄	#17986025	1	1	yes	characteristic
154	C₄	#17986047	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	#22587110	1	1	yes	characteristic
157	C₂	C₂×#17985821	1	1	yes	characteristic
158	C₂	C₂×#17985823	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22590143

C₂

yes

maximal, characteristic

#22587096

C₂

yes

maximal, characteristic

#22586405

C₂

yes

maximal, complemented

C₂×#334×C₇₁

C₂

yes

maximal, characteristic

#17986023

—

C₄⋊(C₇₁⋊C₁₆)

C₄

yes

—

C₂²⋊(C₇₁⋊C₁₆)

C₄

yes

—

#17985775

C₂²

yes

—

#17985750

—

#17985750

C₂²

yes

—

C₂×C₄⋊C₈×C₇₁

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#334×C₇₁

—

#334×C₇₁

C₂²

yes

—

C₂×#11765849

—

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

C₂²×C₈×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

#11765849

—

C₂×C₇₁⋊C₈

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₁

SD₁₆

yes

characteristic

C₂²×C₄×C₇₁

SD₁₆

yes

characteristic

C₂×C₈×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

—

C₈×C₇₁

#131

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

#365

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

—

D₇₁

yes

characteristic

#1368

—

C₇₁

#12030

yes

characteristic, complemented, Sylow

#423

—

142

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

—

—

—

—

142

—

#334

C₂×D₇₁

yes

—

C₂×M₁₆

—

284

—

C₂²×C₈

—

C₂²×C₈

Dic₁₄₂

yes

characteristic

C₂²×C₈

C₄×D₇₁

yes

characteristic

C₂×C₄²

C₂²⋊D₇₁

yes

characteristic

100

C₂×C₁₆

—

101

C₄⋊C₈

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

284

—

104

C₄⋊C₈

—

142

—

105

C₄×C₈

Dic₁₄₂

yes

characteristic

106

C₄×C₈

C₄×D₇₁

yes

characteristic

107

C₄×C₈

C₂²⋊D₇₁

yes

characteristic

108

C₄×C₈

C₂²×D₇₁

yes

characteristic

109

C₂²×C₄

Dic₇₁⋊C₄

yes

characteristic

110

C₂²×C₄

#11765852

yes

characteristic

111

C₂²×C₄

#11765853

yes

characteristic

112

M₁₆

—

568

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

284

—

118

C₂×C₈

Dic₇₁⋊C₄

yes

characteristic

119

C₂×C₈

Dic₇₁⋊C₄

yes

characteristic

120

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

121

C₂×C₈

C₂×Dic₁₄₂

yes

characteristic

122

C₄²

Dic₇₁⋊C₄

yes

characteristic

123

C₄²

#11765852

yes

characteristic

124

C₄²

#11765853

yes

characteristic

125

C₄²

C₂×C₂²⋊D₇₁

yes

characteristic

126

C₁₆

—

142

—

127

C₂³

#14556059

yes

characteristic

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

#14556059

yes

characteristic

133

C₂×C₄

#14556059

yes

characteristic

134

C₂×C₄

#14556059

yes

characteristic

135

C₂×C₄

C₂×Dic₇₁⋊C₄

yes

characteristic

136

C₂×C₄

C₂×#11765852

yes

characteristic

137

C₂×C₄

C₂×#11765853

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

568

—

143

C₈

#14556134

yes

—

144

C₂²

—

145

C₂²

—

146

C₂²

#17985823

yes

—

147

C₂²

C₂×#14556059

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

#17985821

yes

—

153

C₄

#17986025

yes

characteristic

154

C₄

#17986047

yes

characteristic

155

C₂

—

156

C₂

#22587110

yes

characteristic

157

C₂

C₂×#17985821

yes

characteristic

158

C₂

C₂×#17985823

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#14556059
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₂×#334×C₇₁	C₂
socle	C₂²×C₇₁	C₂×#106
prime radical	C₂³×C₇₁	#106
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	C₂×#334	D₇₁
hypocenter	C₇₁	#12030
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#334

subgroup

G/H

center

C₂²

C₂×#14556059

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₂×#334×C₇₁

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 ◁ C₂×#334

N	H ≅ G/N	type
3. #22586405	C₂	semidirect
84. C₇₁	#12030	semidirect

H ≅ G/N

type

3. #22586405

C₂

semidirect

84. C₇₁

#12030

semidirect

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

subgroup	index
#22590143	2
#22587096	2
#22586405	2
C₂×#334×C₇₁	2
#12030	71

subgroup

index

C₂×#334×C₇₁

Subgroups — #31909124

Conjugacy classes of subgroups

Complements

Maximal subgroups