Group #31884373 — subgroups

Subgroups — #31884373

Order 18176.

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

normal subgroups only

This group has 246 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	C₄×#14556064	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985750	C₂	1	1	yes	maximal, characteristic
3	C₄×M₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556064	C₄	1	2	yes	—
5	C₂×C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
6	C₄×#11765849	—	2	1	no	—
7	#17985764	—	2	1	no	—
8	#17985750	—	2	1	no	—
9	#17985750	C₄	1	2	yes	—
10	C₂×M₃₂×C₇₁	—	2	1	no	—
11	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
12	#346×C₇₁	—	2	1	no	—
13	C₄×C₁₆×C₇₁	C₄	1	2	yes	—
14	C₂×#11765849	—	2	2	no	—
15	C₂²×C₇₁⋊C₈	—	4	1	no	—
16	#14556064	—	2	2	no	—
17	#14556064	C₈	1	4	yes	complemented
18	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
19	C₄⋊(C₇₁⋊C₈)	—	4	1	no	—
20	#14556039	—	4	1	no	—
21	C₄×C₇₁⋊C₈	—	4	1	no	—
22	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
25	M₃₂×C₇₁	—	2	4	no	—
26	C₂×C₁₆×C₇₁	—	2	2	no	—
27	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
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	#11765849	—	2	4	no	—
32	#11765849	—	4	2	no	—
33	C₂×C₇₁⋊C₈	—	4	2	no	—
34	C₂×C₇₁⋊C₈	—	4	2	no	—
35	C₂×C₇₁⋊C₈	—	8	2	no	—
36	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
38	C₂²×C₄×C₇₁	Q₁₆	1	1	yes	characteristic
39	C₂×C₈×C₇₁	—	2	2	no	—
40	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
45	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
46	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₇₁	D₈	1	1	yes	characteristic
49	C₄²×C₇₁	SD₁₆	1	1	yes	characteristic
50	C₁₆×C₇₁	—	2	4	no	—
51	C₂³×C₇₁	#103	1	1	yes	characteristic
52	C₂×C₄×C₇₁	—	2	2	no	—
53	C₂×C₄×C₇₁	—	4	1	no	—
54	C₂×C₄×C₇₁	#95	1	1	yes	characteristic
55	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
56	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
57	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
58	C₂×C₄×C₇₁	#103	1	1	yes	characteristic
59	C₂×C₄×C₇₁	#104	1	2	yes	—
60	C₂×C₄×C₇₁	#106	1	1	yes	characteristic
61	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
62	C₈×C₇₁	—	2	1	no	—
63	C₈×C₇₁	—	2	1	no	—
64	C₈×C₇₁	C₄×C₈	1	1	yes	characteristic
65	C₈×C₇₁	#97	1	1	yes	characteristic
66	C₈×C₇₁	C₄⋊C₈	1	2	yes	—
67	C₇₁⋊C₈	—	4	4	no	—
68	C₇₁⋊C₈	—	8	2	no	—
69	C₂²×C₇₁	—	2	1	no	—
70	C₂²×C₇₁	—	2	1	no	—
71	C₂²×C₇₁	Q₈⋊C₈	1	2	yes	—
72	C₂²×C₇₁	#340	1	1	yes	characteristic
73	C₄×C₇₁	—	2	1	no	—
74	C₄×C₇₁	—	2	1	no	—
75	C₄×C₇₁	—	2	1	no	—
76	C₄×C₇₁	D₄⋊C₈	1	2	yes	—
77	C₄×C₇₁	#336	1	1	yes	characteristic
78	C₄×C₇₁	#339	1	1	yes	characteristic
79	C₄×C₇₁	#365	1	2	yes	—
80	#7173	—	71	1	no	maximal, Sylow
81	C₂×C₇₁	—	2	2	no	—
82	C₂×C₇₁	#1279	1	1	yes	characteristic
83	C₂×C₇₁	#1383	1	1	yes	characteristic
84	C₂×C₇₁	#1385	1	1	yes	characteristic
85	C₄×M₃₂	D₇₁	1	1	yes	characteristic
86	C₄×#108	—	71	1	no	—
87	C₂×#334	—	71	1	no	—
88	C₇₁	#7173	1	1	yes	characteristic, complemented, Sylow
89	C₂×M₃₂	—	2	1	no	—
90	C₂×#108	—	71	2	no	—
91	C₂×C₄⋊C₈	—	142	1	no	—
92	C₄×M₁₆	—	142	1	no	—
93	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
94	#346	—	2	1	no	—
95	C₄×C₁₆	Dic₇₁	1	2	yes	—
96	#341	—	142	1	no	—
97	#334	—	71	2	no	—
98	#334	—	142	1	no	—
99	C₂×M₁₆	—	142	2	no	—
100	C₂²×C₈	—	284	1	no	—
101	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
102	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
103	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
104	M₃₂	—	2	4	no	—
105	C₂×C₁₆	—	2	2	no	—
106	#108	—	71	4	no	—
107	#108	—	142	2	no	—
108	C₄⋊C₈	—	284	1	no	—
109	C₄⋊C₈	—	142	2	no	—
110	#97	—	284	1	no	—
111	C₄×C₈	—	284	1	no	—
112	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
113	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
114	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
115	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
116	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
117	C₂²×C₄	#11765854	1	1	yes	characteristic
118	C₂²×C₄	Dic₂₈₄	1	1	yes	characteristic
119	M₁₆	—	284	2	no	—
120	M₁₆	—	142	4	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	284	2	no	—
123	C₂×C₈	—	284	2	no	—
124	C₂×C₈	—	568	2	no	—
125	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
126	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
127	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
128	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
129	C₂×C₈	C₈×D₇₁	1	2	yes	—
130	C₂×C₈	#11765848	1	2	yes	—
131	C₄²	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
132	C₄²	C₄⋊Dic₇₁	1	1	yes	characteristic
133	C₄²	D₅₆₈	1	1	yes	characteristic
134	C₄²	#11765854	1	1	yes	characteristic
135	C₁₆	—	2	4	no	—
136	C₂³	#14556053	1	1	yes	characteristic
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	C₇₁⋊#95	1	1	yes	characteristic
140	C₂×C₄	(C₂×D₇₁)⋊C₈	1	2	yes	—
141	C₂×C₄	#14556050	1	1	yes	characteristic
142	C₂×C₄	#14556050	1	1	yes	characteristic
143	C₂×C₄	#14556053	1	1	yes	characteristic
144	C₂×C₄	#14556056	1	2	yes	—
145	C₂×C₄	#14556060	1	1	yes	characteristic
146	C₂×C₄	#14556062	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	—	2	1	no	—
149	C₈	—	284	4	no	—
150	C₈	—	568	2	no	—
151	C₈	C₈×Dic₇₁	1	1	yes	characteristic
152	C₈	#14556038	1	1	yes	characteristic
153	C₈	Dic₇₁⋊C₈	1	2	yes	—
154	C₂²	—	2	1	no	—
155	C₂²	—	2	1	no	—
156	C₂²	#17985731	1	2	yes	—
157	C₂²	#17985762	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	#17985728	1	2	yes	—
162	C₄	#17985753	1	1	yes	characteristic
163	C₄	#17985759	1	1	yes	characteristic
164	C₄	#17985822	1	2	yes	—
165	C₂	—	2	2	no	—
166	C₂	#22586180	1	1	yes	characteristic
167	C₂	#22586439	1	1	yes	characteristic
168	C₂	#22586444	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14556064

C₂

yes

maximal, characteristic

C₂×#17985750

C₂

yes

maximal, characteristic

C₄×M₃₂×C₇₁

C₂

yes

maximal, characteristic

C₂×#14556064

C₄

yes

—

C₂×C₄⋊(C₇₁⋊C₈)

—

—

—

—

C₄

yes

—

C₂×M₃₂×C₇₁

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#346×C₇₁

—

C₄×C₁₆×C₇₁

C₄

yes

—

C₂×#11765849

—

C₂²×C₇₁⋊C₈

—

#14556064

—

#14556064

C₈

yes

complemented

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

#14556039

—

C₄×C₇₁⋊C₈

—

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂×C₄²×C₇₁

D₄

yes

characteristic

M₃₂×C₇₁

—

C₂×C₁₆×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

Q₈

yes

characteristic

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂²×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₁

SD₁₆

yes

characteristic

C₂²×C₄×C₇₁

Q₁₆

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄²

yes

characteristic

C₂×C₈×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×C₈

yes

—

C₂×C₈×C₇₁

M₁₆

yes

—

C₄²×C₇₁

C₂²⋊C₄

yes

characteristic

C₄²×C₇₁

C₄⋊C₄

yes

characteristic

C₄²×C₇₁

D₈

yes

characteristic

C₄²×C₇₁

SD₁₆

yes

characteristic

C₁₆×C₇₁

—

C₂³×C₇₁

#103

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#95

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₈

yes

—

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

—

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

C₄×C₈

yes

characteristic

C₈×C₇₁

#97

yes

characteristic

C₈×C₇₁

C₄⋊C₈

yes

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

Q₈⋊C₈

yes

—

C₂²×C₇₁

#340

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

D₄⋊C₈

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

#7173

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₄×M₃₂

D₇₁

yes

characteristic

C₄×#108

—

C₂×#334

—

C₇₁

#7173

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×#108

—

C₂×C₄⋊C₈

—

142

—

C₄×M₁₆

—

142

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

#346

—

C₄×C₁₆

Dic₇₁

yes

—

#341

—

142

—

#334

—

#334

—

142

—

C₂×M₁₆

—

142

—

100

C₂²×C₈

—

284

—

101

C₂²×C₈

C₄×D₇₁

yes

characteristic

102

C₂²×C₈

C₂²⋊D₇₁

yes

characteristic

103

C₂×C₄²

D₂₈₄

yes

characteristic

104

M₃₂

—

105

C₂×C₁₆

—

106

#108

—

107

#108

—

142

—

108

C₄⋊C₈

—

284

—

109

C₄⋊C₈

—

142

—

110

#97

—

284

—

111

C₄×C₈

—

284

—

112

C₄×C₈

Dic₁₄₂

yes

characteristic

113

C₄×C₈

C₄×D₇₁

yes

characteristic

114

C₄×C₈

C₂×Dic₇₁

yes

characteristic

115

C₄×C₈

C₂²⋊D₇₁

yes

characteristic

116

C₂²×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

117

C₂²×C₄

#11765854

yes

characteristic

118

C₂²×C₄

Dic₂₈₄

yes

characteristic

119

M₁₆

—

284

—

120

M₁₆

—

142

—

121

C₂×C₈

—

122

C₂×C₈

—

284

—

123

C₂×C₈

—

284

—

124

C₂×C₈

—

568

—

125

C₂×C₈

C₄×Dic₇₁

yes

characteristic

126

C₂×C₈

C₂²⋊Dic₇₁

yes

characteristic

127

C₂×C₈

Dic₇₁⋊C₄

yes

characteristic

128

C₂×C₈

Dic₇₁⋊C₄

yes

characteristic

129

C₂×C₈

C₈×D₇₁

yes

—

130

C₂×C₈

#11765848

yes

—

131

C₄²

(C₂×D₇₁)⋊C₄

yes

characteristic

132

C₄²

C₄⋊Dic₇₁

yes

characteristic

133

C₄²

D₅₆₈

yes

characteristic

134

C₄²

#11765854

yes

characteristic

135

C₁₆

—

136

C₂³

#14556053

yes

characteristic

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

C₇₁⋊#95

yes

characteristic

140

C₂×C₄

(C₂×D₇₁)⋊C₈

yes

—

141

C₂×C₄

#14556050

yes

characteristic

142

C₂×C₄

#14556050

yes

characteristic

143

C₂×C₄

#14556053

yes

characteristic

144

C₂×C₄

#14556056

yes

—

145

C₂×C₄

#14556060

yes

characteristic

146

C₂×C₄

#14556062

yes

characteristic

147

C₈

—

148

C₈

—

149

C₈

—

284

—

150

C₈

—

568

—

151

C₈

C₈×Dic₇₁

yes

characteristic

152

C₈

#14556038

yes

characteristic

153

C₈

Dic₇₁⋊C₈

yes

—

154

C₂²

—

155

C₂²

—

156

C₂²

#17985731

yes

—

157

C₂²

#17985762

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

#17985728

yes

—

162

C₄

#17985753

yes

characteristic

163

C₄

#17985759

yes

characteristic

164

C₄

#17985822

yes

—

165

C₂

—

166

C₂

#22586180

yes

characteristic

167

C₂

#22586439

yes

characteristic

168

C₂

#22586444

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556060
commutator	C₈×C₇₁	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₁
Fitting	C₄×M₃₂×C₇₁	C₂
socle	C₂²×C₇₁	#340
prime radical	C₂³×C₇₁	#103
cosocle	C₂×C₄×C₈×C₇₁	C₂²
hypercenter	C₄×M₃₂	D₇₁
hypocenter	C₇₁	#7173
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

subgroup

G/H

center

C₂×C₄

#14556060

commutator

C₈×C₇₁

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₇₁

Fitting

C₄×M₃₂×C₇₁

C₂

C₂²×C₇₁

C₂³×C₇₁

C₂×C₄×C₈×C₇₁

C₂²

hypercenter

C₄×M₃₂

D₇₁

hypocenter

C₇₁

#7173

LCS

G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

N	H ≅ G/N	type
17. #14556064	C₈	semidirect
88. C₇₁	#7173	semidirect

H ≅ G/N

type

17. #14556064

C₈

semidirect

88. C₇₁

#7173

semidirect

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

subgroup	index
C₄×#14556064	2
C₂×#17985750	2
C₄×M₃₂×C₇₁	2
#7173	71

subgroup

index

C₄×#14556064

C₂×#17985750

C₄×M₃₂×C₇₁

#7173

Subgroups — #31884373

Conjugacy classes of subgroups

Complements

Maximal subgroups