Group #34000263 — subgroups

Subgroups — #34000263

Order 18688.

246 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 82 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	#22626138	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556661	C₂	1	1	yes	maximal, characteristic
3	#334×D₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556666	—	2	1	no	—
5	C₂×#14556661	C₄	1	2	yes	—
6	C₄×#11924038	—	2	1	no	—
7	#17989475	—	2	1	no	—
8	#17989473	C₄	1	2	yes	—
9	#17989463	—	2	1	no	—
10	C₄⋊C₈×D₇₃	—	2	1	no	—
11	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
12	#17988338	C₄	1	1	yes	characteristic
13	#17988337	—	2	1	no	—
14	#334×C₇₃	C₄	1	1	yes	characteristic
15	C₂×#11924038	—	2	2	no	—
16	#14556666	—	2	4	no	—
17	C₂×#11924031	—	2	2	no	—
18	#14556661	—	2	2	no	—
19	#14556661	C₈	1	4	yes	complemented
20	#14556646	—	4	1	no	—
21	C₄×#10596833	—	4	1	no	—
22	C₂×C₈×D₇₃	—	4	1	no	—
23	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
25	C₄²×D₇₃	D₄	1	1	yes	characteristic
26	C₄⋊#10596831	—	2	1	no	—
27	#14556520	—	4	1	no	—
28	C₄×#10596831	D₄	1	1	yes	characteristic
29	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
30	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
31	C₄⋊C₈×C₇₃	—	2	1	no	—
32	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
33	#11924038	—	2	4	no	—
34	#11924038	—	4	2	no	—
35	C₂×#10596833	—	4	2	no	—
36	#11924031	—	2	4	no	—
37	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×D₇₃	SD₁₆	1	1	yes	characteristic
39	C₂×C₄×D₇₃	Q₁₆	1	1	yes	characteristic
40	C₂×#10596831	—	4	1	no	—
41	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
43	C₈×D₇₃	—	2	2	no	—
44	C₈×D₇₃	—	8	2	no	—
45	C₈×D₇₃	C₂×C₈	1	2	yes	—
46	C₈×D₇₃	M₁₆	1	2	yes	—
47	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
48	C₄×Dic₇₃	D₈	1	1	yes	characteristic
49	C₄×Dic₇₃	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₇₃	—	4	1	no	—
51	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
52	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
53	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×D₇₃	#103	1	1	yes	characteristic
55	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
57	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
58	C₂×Dic₇₃	#102	1	1	yes	characteristic
59	C₂×Dic₇₃	#106	1	1	yes	characteristic
60	C₂×Dic₇₃	#107	1	1	yes	characteristic
61	C₄×D₇₃	—	2	2	no	—
62	C₄×D₇₃	—	4	1	no	—
63	C₄×D₇₃	C₂²⋊C₈	1	2	yes	—
64	C₄×D₇₃	#104	1	2	yes	—
65	#10596833	—	4	4	no	—
66	C₈×C₇₃	—	2	1	no	—
67	C₈×C₇₃	—	8	1	no	—
68	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
69	C₈×C₇₃	#97	1	1	yes	characteristic
70	#10596831	—	2	1	no	—
71	#10596831	—	8	1	no	—
72	#10596831	C₄⋊C₈	1	2	yes	—
73	C₂²×C₇₃	#340	1	1	yes	characteristic
74	C₂×D₇₃	—	2	1	no	—
75	C₂×D₇₃	—	2	1	no	—
76	C₂×D₇₃	Q₈⋊C₈	1	2	yes	—
77	C₄×C₇₃	—	2	1	no	—
78	C₄×C₇₃	—	2	1	no	—
79	C₄×C₇₃	#336	1	1	yes	characteristic
80	C₄×C₇₃	#339	1	1	yes	characteristic
81	Dic₇₃	—	2	1	no	—
82	Dic₇₃	D₄⋊C₈	1	2	yes	—
83	Dic₇₃	#365	1	2	yes	—
84	#7173	—	73	1	no	maximal, Sylow
85	C₂×C₇₃	#1279	1	1	yes	characteristic
86	C₂×C₇₃	#1383	1	1	yes	characteristic
87	C₂×C₇₃	#1385	1	1	yes	characteristic
88	D₇₃	—	2	2	no	—
89	C₄×M₃₂	—	73	1	no	—
90	C₄×#108	—	73	1	no	—
91	C₂×#334	—	73	1	no	—
92	C₇₃	#7173	1	1	yes	characteristic, complemented, Sylow
93	C₂×M₃₂	—	146	1	no	—
94	C₂×#108	—	73	2	no	—
95	C₂×C₄⋊C₈	—	146	1	no	—
96	C₄×M₁₆	—	146	1	no	—
97	C₂×C₄×C₈	—	73	1	no	—
98	#346	—	146	1	no	—
99	C₄×C₁₆	—	73	2	no	—
100	#341	—	146	1	no	—
101	#334	—	73	1	no	—
102	#334	—	146	1	no	—
103	#334	#64408	1	1	yes	characteristic
104	C₂×M₁₆	—	146	2	no	—
105	C₂²×C₈	—	292	1	no	—
106	C₂²×C₈	—	73	1	no	—
107	C₂²×C₈	—	73	1	no	—
108	C₂×C₄²	—	73	1	no	—
109	M₃₂	—	146	4	no	—
110	C₂×C₁₆	—	146	2	no	—
111	#108	—	73	4	no	—
112	#108	—	146	2	no	—
113	C₄⋊C₈	—	2	1	no	—
114	C₄⋊C₈	—	292	1	no	—
115	C₄⋊C₈	—	146	1	no	—
116	#97	—	292	1	no	—
117	C₄×C₈	—	292	1	no	—
118	C₄×C₈	—	73	1	no	—
119	C₄×C₈	—	73	1	no	—
120	C₄×C₈	—	73	1	no	—
121	C₄×C₈	C₂×#64408	1	1	yes	characteristic
122	C₂²×C₄	—	73	1	no	—
123	C₂²×C₄	—	73	1	no	—
124	C₂²×C₄	—	73	1	no	—
125	M₁₆	—	292	2	no	—
126	M₁₆	—	146	4	no	—
127	C₂×C₈	—	4	1	no	—
128	C₂×C₈	—	292	2	no	—
129	C₂×C₈	—	292	1	no	—
130	C₂×C₈	—	584	2	no	—
131	C₂×C₈	—	73	1	no	—
132	C₂×C₈	—	73	2	no	—
133	C₂×C₈	—	73	2	no	—
134	C₂×C₈	—	73	1	no	—
135	C₂×C₈	—	146	2	no	—
136	C₂×C₈	C₄×#64408	1	1	yes	characteristic
137	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
138	C₄²	—	73	1	no	—
139	C₄²	—	73	1	no	—
140	C₄²	—	73	1	no	—
141	C₄²	C₂²⋊#64408	1	1	yes	characteristic
142	C₁₆	—	146	4	no	—
143	C₂³	—	73	1	no	—
144	C₂×C₄	—	292	1	no	—
145	C₂×C₄	—	73	1	no	—
146	C₂×C₄	—	73	1	no	—
147	C₂×C₄	—	73	2	no	—
148	C₂×C₄	—	73	2	no	—
149	C₂×C₄	—	73	1	no	—
150	C₂×C₄	—	146	2	no	—
151	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
152	C₂×C₄	#14556653	1	1	yes	characteristic
153	C₂×C₄	#14556654	1	1	yes	characteristic
154	C₈	—	2	1	no	—
155	C₈	—	8	1	no	—
156	C₈	—	292	4	no	—
157	C₈	—	584	1	no	—
158	C₈	—	73	2	no	—
159	C₈	—	146	1	no	—
160	C₈	C₈×#64408	1	1	yes	characteristic
161	C₈	#14556645	1	1	yes	characteristic
162	C₂²	—	73	2	no	—
163	C₂²	—	146	1	no	—
164	C₂²	—	146	1	no	—
165	C₂²	#17989460	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	73	2	no	—
169	C₄	—	73	2	no	—
170	C₄	—	146	1	no	—
171	C₄	#17989452	1	1	yes	characteristic
172	C₄	#17989459	1	1	yes	characteristic
173	C₂	—	146	2	no	—
174	C₂	#22624765	1	1	yes	characteristic
175	C₂	#22624995	1	1	yes	characteristic
176	C₂	#22625000	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626138

C₂

yes

maximal, characteristic

C₄×#14556661

C₂

yes

maximal, characteristic

#334×D₇₃

C₂

yes

maximal, characteristic

C₂×#14556666

—

C₂×#14556661

C₄

yes

—

C₄×#11924038

—

#17989475

—

#17989473

C₄

yes

—

#17989463

—

C₄⋊C₈×D₇₃

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988338

C₄

yes

characteristic

#17988337

—

#334×C₇₃

C₄

yes

—

—

—

—

C₈

yes

complemented

#14556646

—

C₄×#10596833

—

C₂×C₈×D₇₃

—

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₂×C₈×D₇₃

D₄

yes

characteristic

C₄²×D₇₃

D₄

yes

—

—

D₄

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

Q₈

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

—

—

—

C₂×C₄×D₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₇₃

SD₁₆

yes

characteristic

C₂×C₄×D₇₃

Q₁₆

yes

characteristic

C₂×#10596831

—

C₂×#10596831

C₂²⋊C₄

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₈×D₇₃

C₂×C₈

yes

—

C₈×D₇₃

M₁₆

yes

—

C₄×Dic₇₃

C₄⋊C₄

yes

characteristic

C₄×Dic₇₃

D₈

yes

characteristic

C₄×Dic₇₃

SD₁₆

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₄²×C₇₃

C₂²⋊C₄

yes

C₂²×D₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₂²⋊C₈

yes

—

C₄×D₇₃

#104

yes

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

C₄×C₈

yes

C₈×C₇₃

yes

—

—

C₄⋊C₈

yes

—

C₂²×C₇₃

#340

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

Q₈⋊C₈

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

D₄⋊C₈

yes

—

Dic₇₃

#365

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

C₄×M₃₂

—

C₄×#108

—

C₂×#334

—

C₇₃

#7173

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

146

—

C₂×#108

—

C₂×C₄⋊C₈

—

146

—

C₄×M₁₆

—

146

—

C₂×C₄×C₈

—

#346

—

146

—

C₄×C₁₆

—

100

#341

—

146

—

101

#334

—

102

#334

—

146

—

103

#334

#64408

yes

characteristic

104

C₂×M₁₆

—

146

—

105

C₂²×C₈

—

292

—

106

C₂²×C₈

—

107

C₂²×C₈

—

108

C₂×C₄²

—

109

M₃₂

—

146

—

110

C₂×C₁₆

—

146

—

111

#108

—

112

#108

—

146

—

113

C₄⋊C₈

—

114

C₄⋊C₈

—

292

—

115

C₄⋊C₈

—

146

—

116

#97

—

292

—

117

C₄×C₈

—

292

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

C₂×#64408

yes

characteristic

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

M₁₆

—

292

—

126

M₁₆

—

146

—

127

C₂×C₈

—

128

C₂×C₈

—

292

—

129

C₂×C₈

—

292

—

130

C₂×C₈

—

584

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

146

—

136

C₂×C₈

C₄×#64408

yes

characteristic

137

C₂×C₈

C₄⋊#64408

yes

characteristic

138

C₄²

—

139

C₄²

—

140

C₄²

—

141

C₄²

C₂²⋊#64408

yes

characteristic

142

C₁₆

—

146

—

143

C₂³

—

144

C₂×C₄

—

292

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

146

—

151

C₂×C₄

(C₂×#64408)⋊C₄

yes

characteristic

152

C₂×C₄

#14556653

yes

characteristic

153

C₂×C₄

#14556654

yes

characteristic

154

C₈

—

155

C₈

—

156

C₈

—

292

—

157

C₈

—

584

—

158

C₈

—

159

C₈

—

146

—

160

C₈

C₈×#64408

yes

characteristic

161

C₈

#14556645

yes

characteristic

162

C₂²

—

163

C₂²

—

146

—

164

C₂²

—

146

—

165

C₂²

#17989460

yes

characteristic

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

146

—

171

C₄

#17989452

yes

characteristic

172

C₄

#17989459

yes

characteristic

173

C₂

—

146

—

174

C₂

#22624765

yes

characteristic

175

C₂

#22624995

yes

characteristic

176

C₂

#22625000

yes

characteristic

	subgroup	G/H
center	C₂²	#17989460
commutator	C₈×C₇₃	C₄×C₈
Frattini	C₄×C₈	C₂×#64408
Fitting	#334×C₇₃	C₄
socle	C₂²×C₇₃	#340
prime radical	C₂²×D₇₃	#103
cosocle	C₄×C₈×D₇₃	C₂²
hypercenter	#334	#64408
hypocenter	C₇₃	#7173
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

subgroup

G/H

center

C₂²

#17989460

commutator

C₈×C₇₃

C₄×C₈

Frattini

C₄×C₈

C₂×#64408

Fitting

#334×C₇₃

C₄

C₂²×C₇₃

C₂²×D₇₃

C₄×C₈×D₇₃

C₂²

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

N	H ≅ G/N	type
19. #14556661	C₈	semidirect
92. C₇₃	#7173	semidirect

H ≅ G/N

type

19. #14556661

C₈

semidirect

92. C₇₃

#7173

semidirect

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

subgroup	index
#22626138	2
C₄×#14556661	2
#334×D₇₃	2
#7173	73

subgroup

index

#22626138

C₄×#14556661

#334×D₇₃

#7173

Subgroups — #34000263

Conjugacy classes of subgroups

Complements

Maximal subgroups