Group #34000278 — subgroups

Subgroups — #34000278

Order 18688.

274 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 88 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	#22626132	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556660	C₂	1	1	yes	maximal, characteristic
3	#334×D₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556664	—	2	1	no	—
5	C₂×#14556660	C₄	1	2	yes	—
6	C₄×C₄⋊#64408	—	2	1	no	—
7	#17989476	C₄	1	2	yes	complemented
8	#17989475	—	2	1	no	—
9	#17989461	—	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, complemented
13	#17988337	—	2	1	no	—
14	#334×C₇₃	C₄	1	1	yes	characteristic, complemented
15	C₂×C₄⋊#64408	—	2	2	no	—
16	C₂×C₄×#64408	—	4	1	no	—
17	#14556664	—	2	4	no	—
18	C₂×#11924031	—	2	2	no	—
19	#14556660	—	2	2	no	—
20	#14556660	C₈	1	4	yes	complemented
21	(C₂×#64408)⋊C₄	—	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	C₂²×#64408	—	4	2	no	—
34	C₄⋊#64408	—	2	4	no	—
35	C₄⋊#64408	—	4	2	no	—
36	C₄×#64408	—	8	2	no	—
37	#11924031	—	2	4	no	—
38	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
40	C₂×C₄×D₇₃	SD₁₆	1	1	yes	characteristic
41	C₂×#10596831	—	4	1	no	—
42	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
44	C₈×D₇₃	—	2	2	no	—
45	C₈×D₇₃	—	8	2	no	—
46	C₈×D₇₃	C₂×C₈	1	2	yes	—
47	C₈×D₇₃	M₁₆	1	2	yes	—
48	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
49	C₄×Dic₇₃	SD₁₆	1	1	yes	characteristic
50	C₄×Dic₇₃	Q₁₆	1	1	yes	characteristic
51	C₂×C₈×C₇₃	—	4	1	no	—
52	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
53	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
54	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×D₇₃	#102	1	1	yes	characteristic
56	C₂×#64408	—	4	4	no	—
57	C₂×#64408	—	8	2	no	—
58	C₂×#64408	—	8	2	no	—
59	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
60	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
61	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
62	C₂×Dic₇₃	#103	1	1	yes	characteristic
63	C₂×Dic₇₃	#106	1	1	yes	characteristic
64	C₂×Dic₇₃	#107	1	1	yes	characteristic
65	C₄×D₇₃	—	2	2	no	—
66	C₄×D₇₃	—	2	1	no	—
67	C₄×D₇₃	—	2	1	no	—
68	C₄×D₇₃	C₂²⋊C₈	1	2	yes	—
69	C₄×D₇₃	#104	1	2	yes	—
70	C₈×C₇₃	—	2	1	no	—
71	C₈×C₇₃	—	8	1	no	—
72	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
73	C₈×C₇₃	#97	1	1	yes	characteristic
74	#10596831	—	2	1	no	—
75	#10596831	—	8	1	no	—
76	#10596831	C₄⋊C₈	1	2	yes	—
77	C₂²×C₇₃	#340	1	1	yes	characteristic
78	C₂×D₇₃	D₄⋊C₈	1	2	yes	—
79	C₂×D₇₃	#360	1	2	yes	—
80	C₂×D₇₃	#361	1	1	yes	characteristic
81	C₂×D₇₃	#362	1	1	yes	characteristic
82	#64408	—	8	4	no	—
83	C₄×C₇₃	—	2	1	no	—
84	C₄×C₇₃	—	2	1	no	—
85	C₄×C₇₃	#336	1	1	yes	characteristic
86	C₄×C₇₃	#339	1	1	yes	characteristic
87	Dic₇₃	—	2	1	no	—
88	Dic₇₃	—	2	1	no	—
89	Dic₇₃	Q₈⋊C₈	1	2	yes	—
90	#7178	—	73	1	no	maximal, Sylow
91	C₂×C₇₃	#1279	1	1	yes	characteristic
92	C₂×C₇₃	#1386	1	1	yes	characteristic
93	C₂×C₇₃	#1387	1	1	yes	characteristic
94	D₇₃	#1333	1	2	yes	—
95	D₇₃	#1334	1	2	yes	—
96	C₂×#346	—	73	1	no	—
97	C₄×#107	—	73	1	no	—
98	C₂×#334	—	73	1	no	—
99	C₇₃	#7178	1	1	yes	characteristic, complemented, Sylow
100	C₂²×C₁₆	—	146	1	no	—
101	C₂×#107	—	73	2	no	—
102	C₂×C₄⋊C₈	—	146	1	no	—
103	C₂×C₄×C₈	—	73	1	no	—
104	C₄×C₄⋊C₄	—	146	1	no	—
105	#346	—	73	2	no	—
106	#346	—	146	1	no	—
107	#340	—	146	1	no	—
108	#334	—	73	1	no	—
109	#334	—	146	1	no	—
110	#334	#64408	1	1	yes	characteristic, complemented
111	C₂²×C₈	—	292	1	no	—
112	C₂²×C₈	—	73	1	no	—
113	C₂²×C₈	—	73	1	no	—
114	C₂×C₄⋊C₄	—	146	2	no	—
115	C₂×C₄²	—	292	1	no	—
116	C₂×C₄²	—	73	1	no	—
117	C₂×C₁₆	—	146	2	no	—
118	C₂×C₁₆	—	146	4	no	—
119	#107	—	73	4	no	—
120	#107	—	146	2	no	—
121	C₄⋊C₈	—	2	1	no	—
122	C₄⋊C₈	—	292	1	no	—
123	C₄⋊C₈	—	146	1	no	—
124	C₄×C₈	—	73	1	no	—
125	C₄×C₈	—	73	1	no	—
126	C₄×C₈	—	73	1	no	—
127	C₄×C₈	C₂×#64408	1	1	yes	characteristic
128	#95	—	292	1	no	—
129	C₂²×C₄	—	292	2	no	—
130	C₂²×C₄	—	73	1	no	—
131	C₂²×C₄	—	73	1	no	—
132	C₂²×C₄	—	73	1	no	—
133	C₂×C₈	—	4	1	no	—
134	C₂×C₈	—	292	1	no	—
135	C₂×C₈	—	584	2	no	—
136	C₂×C₈	—	73	1	no	—
137	C₂×C₈	—	73	2	no	—
138	C₂×C₈	—	73	2	no	—
139	C₂×C₈	—	73	1	no	—
140	C₂×C₈	—	146	2	no	—
141	C₂×C₈	C₄×#64408	1	1	yes	characteristic
142	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
143	C₄⋊C₄	—	292	2	no	—
144	C₄⋊C₄	—	146	4	no	—
145	C₄²	—	584	2	no	—
146	C₄²	—	73	1	no	—
147	C₄²	—	73	1	no	—
148	C₄²	—	73	1	no	—
149	C₄²	C₂²⋊#64408	1	1	yes	characteristic
150	C₁₆	—	146	4	no	—
151	C₂³	—	73	1	no	—
152	C₂×C₄	—	292	4	no	—
153	C₂×C₄	—	584	2	no	—
154	C₂×C₄	—	584	2	no	—
155	C₂×C₄	—	73	1	no	—
156	C₂×C₄	—	73	2	no	—
157	C₂×C₄	—	73	2	no	—
158	C₂×C₄	—	73	1	no	—
159	C₂×C₄	—	73	1	no	—
160	C₂×C₄	—	146	1	no	—
161	C₂×C₄	—	146	2	no	—
162	C₂×C₄	—	146	1	no	—
163	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
164	C₂×C₄	#14556653	1	1	yes	characteristic
165	C₂×C₄	#14556654	1	1	yes	characteristic
166	C₈	—	2	1	no	—
167	C₈	—	8	1	no	—
168	C₈	—	584	1	no	—
169	C₈	—	73	2	no	—
170	C₈	—	146	1	no	—
171	C₈	C₈×#64408	1	1	yes	characteristic
172	C₈	#14556645	1	1	yes	characteristic
173	C₂²	—	73	2	no	—
174	C₂²	—	73	1	no	—
175	C₂²	—	73	1	no	—
176	C₂²	—	73	2	no	—
177	C₂²	#17989460	1	1	yes	characteristic
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	—	584	4	no	—
181	C₄	—	73	2	no	—
182	C₄	—	146	1	no	—
183	C₄	—	146	1	no	—
184	C₄	#17989452	1	1	yes	characteristic
185	C₄	#17989459	1	1	yes	characteristic
186	C₂	—	73	2	no	—
187	C₂	—	73	2	no	—
188	C₂	#22624765	1	1	yes	characteristic
189	C₂	#22625004	1	1	yes	characteristic
190	C₂	#22625007	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626132

C₂

yes

maximal, characteristic

C₄×#14556660

C₂

yes

maximal, characteristic

#334×D₇₃

C₂

yes

maximal, characteristic

C₂×#14556664

—

C₂×#14556660

C₄

yes

—

C₄×C₄⋊#64408

—

#17989476

C₄

yes

complemented

#17989475

—

#17989461

—

C₄⋊C₈×D₇₃

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988338

C₄

yes

characteristic, complemented

#17988337

—

#334×C₇₃

C₄

yes

characteristic, complemented

C₂×C₄⋊#64408

—

C₂×C₄×#64408

—

—

—

—

C₈

yes

complemented

(C₂×#64408)⋊C₄

—

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

characteristic

C₂²×#64408

—

—

—

—

—

C₂×C₄×D₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₇₃

D₈

yes

characteristic

C₂×C₄×D₇₃

SD₁₆

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

SD₁₆

yes

characteristic

C₄×Dic₇₃

Q₁₆

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

—

C₄×D₇₃

C₂²⋊C₈

yes

—

C₄×D₇₃

#104

yes

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

C₄×C₈

yes

C₈×C₇₃

yes

—

—

C₄⋊C₈

yes

—

C₂²×C₇₃

#340

yes

characteristic

C₂×D₇₃

D₄⋊C₈

yes

—

C₂×D₇₃

#360

yes

—

C₂×D₇₃

yes

C₂×D₇₃

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

Q₈⋊C₈

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

yes

—

D₇₃

#1334

yes

—

C₂×#346

—

C₄×#107

—

C₂×#334

—

C₇₃

#7178

yes

characteristic, complemented, Sylow

100

C₂²×C₁₆

—

146

—

101

C₂×#107

—

102

C₂×C₄⋊C₈

—

146

—

103

C₂×C₄×C₈

—

104

C₄×C₄⋊C₄

—

146

—

105

#346

—

106

#346

—

146

—

107

#340

—

146

—

108

#334

—

109

#334

—

146

—

110

#334

#64408

yes

characteristic, complemented

111

C₂²×C₈

—

292

—

112

C₂²×C₈

—

113

C₂²×C₈

—

114

C₂×C₄⋊C₄

—

146

—

115

C₂×C₄²

—

292

—

116

C₂×C₄²

—

117

C₂×C₁₆

—

146

—

118

C₂×C₁₆

—

146

—

119

#107

—

120

#107

—

146

—

121

C₄⋊C₈

—

122

C₄⋊C₈

—

292

—

123

C₄⋊C₈

—

146

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₄×C₈

—

127

C₄×C₈

C₂×#64408

yes

characteristic

128

#95

—

292

—

129

C₂²×C₄

—

292

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂×C₈

—

134

C₂×C₈

—

292

—

135

C₂×C₈

—

584

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

146

—

141

C₂×C₈

C₄×#64408

yes

characteristic

142

C₂×C₈

C₄⋊#64408

yes

characteristic

143

C₄⋊C₄

—

292

—

144

C₄⋊C₄

—

146

—

145

C₄²

—

584

—

146

C₄²

—

147

C₄²

—

148

C₄²

—

149

C₄²

C₂²⋊#64408

yes

characteristic

150

C₁₆

—

146

—

151

C₂³

—

152

C₂×C₄

—

292

—

153

C₂×C₄

—

584

—

154

C₂×C₄

—

584

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

146

—

161

C₂×C₄

—

146

—

162

C₂×C₄

—

146

—

163

C₂×C₄

(C₂×#64408)⋊C₄

yes

characteristic

164

C₂×C₄

#14556653

yes

characteristic

165

C₂×C₄

#14556654

yes

characteristic

166

C₈

—

167

C₈

—

168

C₈

—

584

—

169

C₈

—

170

C₈

—

146

—

171

C₈

C₈×#64408

yes

characteristic

172

C₈

#14556645

yes

characteristic

173

C₂²

—

174

C₂²

—

175

C₂²

—

176

C₂²

—

177

C₂²

#17989460

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

—

584

—

181

C₄

—

182

C₄

—

146

—

183

C₄

—

146

—

184

C₄

#17989452

yes

characteristic

185

C₄

#17989459

yes

characteristic

186

C₂

—

187

C₂

—

188

C₂

#22624765

yes

characteristic

189

C₂

#22625004

yes

characteristic

190

C₂

#22625007

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	#102
cosocle	C₄×C₈×D₇₃	C₂²
hypercenter	#334	#64408
hypocenter	C₇₃	#7178
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
7. #17989476	C₄	semidirect
12. #17988338	C₄	semidirect
14. #334×C₇₃	C₄	semidirect
20. #14556660	C₈	semidirect
99. C₇₃	#7178	semidirect
110. #334	#64408	semidirect

H ≅ G/N

type

7. #17989476

C₄

semidirect

12. #17988338

C₄

semidirect

14. #334×C₇₃

C₄

semidirect

20. #14556660

C₈

semidirect

99. C₇₃

#7178

semidirect

110. #334

#64408

semidirect

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

subgroup	index
#22626132	2
C₄×#14556660	2
#334×D₇₃	2
#7178	73

subgroup

index

#22626132

C₄×#14556660

#334×D₇₃

#7178

Subgroups — #34000278

Conjugacy classes of subgroups

Complements

Maximal subgroups