Group #34032911 — subgroups

Subgroups — #34032911

Order 18688.

216 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 67 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	#22627129	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556687	C₂	1	1	yes	maximal, characteristic
3	#22625138	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556687	—	2	1	no	—
5	#17989690	C₄	1	2	yes	—
6	C₂²⋊#11924040	—	2	1	no	—
7	#17989681	C₄	1	2	yes	—
8	C₄×#11924040	—	2	1	no	—
9	C₄×#11924040	—	2	1	no	—
10	#17989603	C₄	1	1	yes	characteristic
11	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
12	#17989429	—	2	1	no	—
13	#17989429	—	2	1	no	—
14	M₁₆×Dic₇₃	C₄	1	1	yes	characteristic
15	#14556687	—	2	4	no	—
16	C₂×#11924040	—	2	2	no	—
17	C₂×#11924040	—	2	2	no	—
18	C₂²×#10596833	D₄	1	1	yes	characteristic
19	C₂²×#10596833	Q₈	1	1	yes	characteristic
20	C₄⋊#10596833	C₈	1	2	yes	—
21	C₂²⋊#10596833	—	2	1	no	—
22	C₄×#10596833	—	2	1	no	—
23	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
24	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
25	C₄×#10596833	D₄	1	1	yes	characteristic
26	C₄×#10596833	D₄	1	1	yes	characteristic
27	C₂×#11924012	C₈	1	1	yes	characteristic
28	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
29	#14556501	—	2	1	no	—
30	C₈×Dic₇₃	—	2	1	no	—
31	C₂×M₁₆×C₇₃	C₈	1	1	yes	characteristic
32	#11924040	—	2	4	no	—
33	#11924040	—	4	2	no	—
34	C₂×#10596833	—	2	2	no	—
35	C₂×#10596833	—	2	2	no	—
36	C₂×#10596833	—	2	2	no	—
37	C₂×#10596833	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×#10596833	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×#10596833	C₄⋊C₄	1	1	yes	characteristic
40	C₂×#10596833	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×Dic₇₃	M₁₆	1	1	yes	characteristic
43	#11924012	—	4	1	no	—
44	C₂×#10596831	—	2	1	no	—
45	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
46	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
48	C₄×Dic₇₃	M₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
50	M₁₆×C₇₃	—	4	1	no	—
51	C₂×C₈×C₇₃	—	2	1	no	—
52	C₂³×C₇₃	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₇₃	—	2	1	no	—
54	C₂×C₄×C₇₃	—	2	1	no	—
55	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
56	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
57	C₂×Dic₇₃	—	2	1	no	—
58	C₂×Dic₇₃	—	2	1	no	—
59	C₂×Dic₇₃	—	2	2	no	—
60	C₂×Dic₇₃	#95	1	1	yes	characteristic
61	C₂×Dic₇₃	#97	1	1	yes	characteristic
62	C₂×Dic₇₃	C₂²⋊C₈	1	1	yes	characteristic
63	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
64	#10596833	—	2	1	no	—
65	#10596833	—	2	1	no	—
66	#10596833	—	4	2	no	—
67	#10596833	#104	1	2	yes	—
68	#10596833	#108	1	2	yes	—
69	C₈×C₇₃	—	4	1	no	—
70	#10596831	—	4	1	no	—
71	C₂²×C₇₃	—	2	1	no	—
72	C₂²×C₇₃	—	2	1	no	—
73	C₂²×C₇₃	—	2	1	no	—
74	C₂²×C₇₃	#336	1	1	yes	characteristic
75	C₄×C₇₃	—	2	1	no	—
76	C₄×C₇₃	#349	1	1	yes	characteristic
77	C₄×C₇₃	#349	1	1	yes	characteristic
78	Dic₇₃	—	2	1	no	—
79	Dic₇₃	—	2	1	no	—
80	Dic₇₃	#339	1	1	yes	characteristic
81	Dic₇₃	#341	1	1	yes	characteristic
82	Dic₇₃	#350	1	2	yes	—
83	#7166	—	73	1	no	maximal, Sylow
84	C₂×C₇₃	—	4	1	no	—
85	C₂×C₇₃	#1278	1	1	yes	characteristic
86	C₂×C₇₃	#1377	1	1	yes	characteristic
87	C₂×C₇₃	#1379	1	1	yes	characteristic
88	#2162	—	73	1	no	—
89	C₄×M₃₂	—	73	1	no	—
90	#1451	—	73	1	no	—
91	C₇₃	#7166	1	1	yes	characteristic, complemented, Sylow
92	C₂×M₃₂	—	146	1	no	—
93	#431	—	73	1	no	—
94	C₄×M₁₆	—	73	1	no	—
95	C₂×C₄×C₈	—	73	1	no	—
96	C₄⋊C₁₆	—	73	2	no	—
97	C₂²⋊C₁₆	—	146	1	no	—
98	#346	—	73	2	no	—
99	C₄×C₁₆	—	146	1	no	—
100	C₄×C₁₆	—	146	1	no	—
101	#322	—	146	1	no	—
102	#322	—	146	1	no	—
103	C₂×M₁₆	—	73	1	no	—
104	C₂×M₁₆	#10596842	1	1	yes	characteristic
105	C₂²×C₈	—	73	1	no	—
106	C₂²×C₈	—	73	1	no	—
107	C₂×C₄²	—	73	1	no	—
108	M₃₂	—	146	4	no	—
109	C₂×C₁₆	—	146	2	no	—
110	C₂×C₁₆	—	146	2	no	—
111	C₄⋊C₈	—	73	2	no	—
112	C₂²⋊C₈	—	146	1	no	—
113	#97	—	146	1	no	—
114	C₄×C₈	—	73	1	no	—
115	C₄×C₈	—	73	1	no	—
116	C₄×C₈	—	73	1	no	—
117	C₄×C₈	—	73	1	no	—
118	C₄×C₈	—	146	1	no	—
119	C₄×C₈	—	146	1	no	—
120	C₂²×C₄	—	73	1	no	—
121	C₂²×C₄	—	73	1	no	—
122	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
123	M₁₆	—	4	1	no	—
124	M₁₆	—	292	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	73	1	no	—
127	C₂×C₈	—	73	1	no	—
128	C₂×C₈	—	73	1	no	—
129	C₂×C₈	—	73	1	no	—
130	C₂×C₈	—	146	2	no	—
131	C₂×C₈	—	146	1	no	—
132	C₂×C₈	—	146	2	no	—
133	C₂×C₈	—	146	2	no	—
134	C₄²	—	73	1	no	—
135	C₄²	—	73	1	no	—
136	C₄²	—	73	1	no	—
137	C₄²	—	73	1	no	—
138	C₁₆	—	292	2	no	—
139	C₁₆	—	146	4	no	—
140	C₂³	C₄⋊#10596842	1	1	yes	characteristic
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	73	1	no	—
144	C₂×C₄	—	73	1	no	—
145	C₂×C₄	—	73	1	no	—
146	C₂×C₄	—	73	1	no	—
147	C₂×C₄	—	146	1	no	—
148	C₂×C₄	—	146	1	no	—
149	C₂×C₄	—	146	2	no	—
150	C₂×C₄	C₄×#10596842	1	1	yes	characteristic
151	C₂×C₄	C₂²⋊#10596842	1	1	yes	characteristic
152	C₈	—	4	1	no	—
153	C₈	—	292	2	no	—
154	C₈	—	292	1	no	—
155	C₈	—	73	2	no	—
156	C₈	—	73	2	no	—
157	C₈	—	146	1	no	—
158	C₈	—	146	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	—	2	1	no	—
162	C₂²	(C₂×#10596842)⋊C₄	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	73	1	no	—
165	C₄	—	73	1	no	—
166	C₄	—	73	2	no	—
167	C₄	—	146	1	no	—
168	C₄	—	146	1	no	—
169	C₄	#17989685	1	1	yes	characteristic
170	C₄	#17989686	1	1	yes	characteristic
171	C₂	—	4	1	no	—
172	C₂	#22626938	1	1	yes	characteristic
173	C₂	#22627007	1	1	yes	characteristic
174	C₂	#22627012	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22627129

C₂

yes

maximal, characteristic

C₄×#14556687

C₂

yes

maximal, characteristic

#22625138

C₂

yes

maximal, characteristic

C₂×#14556687

—

#17989690

C₄

yes

—

C₂²⋊#11924040

—

#17989681

C₄

yes

—

C₄×#11924040

—

C₄×#11924040

—

#17989603

C₄

yes

characteristic

C₂×C₄×#10596833

C₂²

yes

characteristic

#17989429

—

#17989429

—

M₁₆×Dic₇₃

C₄

yes

—

—

—

D₄

yes

characteristic

C₂²×#10596833

Q₈

yes

characteristic

C₄⋊#10596833

C₈

yes

—

C₂²⋊#10596833

—

C₄×#10596833

—

C₄×#10596833

C₂×C₄

yes

characteristic

C₄×#10596833

C₂×C₄

yes

characteristic

C₄×#10596833

D₄

yes

characteristic

C₄×#10596833

D₄

yes

characteristic

C₂×#11924012

C₈

yes

characteristic

C₂×C₄×Dic₇₃

C₂×C₄

yes

characteristic

#14556501

—

C₈×Dic₇₃

—

C₂×M₁₆×C₇₃

C₈

yes

—

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#10596833

C₂²⋊C₄

yes

characteristic

C₂×#10596833

C₄⋊C₄

yes

characteristic

C₂×#10596833

C₄⋊C₄

yes

characteristic

C₂²×Dic₇₃

C₄⋊C₄

yes

characteristic

C₂²×Dic₇₃

M₁₆

yes

characteristic

#11924012

—

C₂×#10596831

—

C₄×Dic₇₃

C₄²

yes

characteristic

C₄×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₇₃

C₂×C₈

yes

characteristic

C₄×Dic₇₃

M₁₆

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂³×C₇₃

C₄⋊C₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

C₂²⋊C₈

yes

characteristic

C₂×Dic₇₃

C₄⋊C₈

yes

—

—

—

yes

—

#10596833

#108

yes

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#336

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

yes

Dic₇₃

yes

Dic₇₃

yes

—

#7166

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₄×M₃₂

—

#1451

—

C₇₃

#7166

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

146

—

#431

—

C₄×M₁₆

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

146

—

#346

—

C₄×C₁₆

—

146

—

100

C₄×C₁₆

—

146

—

101

#322

—

146

—

102

#322

—

146

—

103

C₂×M₁₆

—

104

C₂×M₁₆

#10596842

yes

characteristic

105

C₂²×C₈

—

106

C₂²×C₈

—

107

C₂×C₄²

—

108

M₃₂

—

146

—

109

C₂×C₁₆

—

146

—

110

C₂×C₁₆

—

146

—

111

C₄⋊C₈

—

112

C₂²⋊C₈

—

146

—

113

#97

—

146

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

—

146

—

119

C₄×C₈

—

146

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

C₂×#10596842

yes

characteristic

123

M₁₆

—

124

M₁₆

—

292

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

146

—

131

C₂×C₈

—

146

—

132

C₂×C₈

—

146

—

133

C₂×C₈

—

146

—

134

C₄²

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₁₆

—

292

—

139

C₁₆

—

146

—

140

C₂³

C₄⋊#10596842

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

146

—

148

C₂×C₄

—

146

—

149

C₂×C₄

—

146

—

150

C₂×C₄

C₄×#10596842

yes

characteristic

151

C₂×C₄

C₂²⋊#10596842

yes

characteristic

152

C₈

—

153

C₈

—

292

—

154

C₈

—

292

—

155

C₈

—

156

C₈

—

157

C₈

—

146

—

158

C₈

—

146

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

(C₂×#10596842)⋊C₄

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

—

146

—

168

C₄

—

146

—

169

C₄

#17989685

yes

characteristic

170

C₄

#17989686

yes

characteristic

171

C₂

—

172

C₂

#22626938

yes

characteristic

173

C₂

#22627007

yes

characteristic

174

C₂

#22627012

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊#10596842
commutator	C₂×C₄×C₇₃	C₄×C₈
Frattini	C₂²×C₄	C₂×#10596842
Fitting	C₂×M₁₆×C₇₃	C₈
socle	C₂²×C₇₃	#336
prime radical	C₂³×C₇₃	C₄⋊C₈
cosocle	C₂×C₄×#10596833	C₂²
hypercenter	C₂×M₁₆	#10596842
hypocenter	C₇₃	#7166
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

subgroup

G/H

center

C₂×C₄

C₂²⋊#10596842

commutator

C₂×C₄×C₇₃

C₄×C₈

Frattini

C₂²×C₄

C₂×#10596842

Fitting

C₂×M₁₆×C₇₃

C₈

socle

C₂²×C₇₃

#336

prime radical

C₂³×C₇₃

C₄⋊C₈

C₂²

C₂×M₁₆

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
91. C₇₃	#7166	semidirect

H ≅ G/N

type

91. C₇₃

#7166

semidirect

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

subgroup	index
#22627129	2
C₄×#14556687	2
#22625138	2
#7166	73

subgroup

index

Subgroups — #34032911

Conjugacy classes of subgroups

Complements

Maximal subgroups