Group #34013873 — subgroups

Subgroups — #34013873

Order 18688.

290 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 124 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	C₂×#17989509	C₂	1	1	yes	maximal, characteristic
2	#22626266	C₂	1	2	yes	maximal
3	C₂×#17989476	C₂	1	1	yes	maximal, characteristic
4	#22624985	C₂	1	2	yes	maximal
5	#22609812	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556665	C₂²	1	2	yes	—
7	C₂²×#11924031	C₂²	1	1	yes	characteristic
8	C₂²×#11924031	C₂²	1	1	yes	characteristic
9	#17989509	—	2	2	no	—
10	C₄⋊#11924031	—	2	4	no	—
11	#17989476	—	2	2	no	—
12	C₂×#14556527	C₄	1	1	yes	characteristic
13	#17988543	C₂²	1	2	yes	—
14	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
15	#17988346	C₄	1	2	yes	—
16	#427×C₇₃	C₄	1	1	yes	characteristic
17	#14556665	—	2	2	no	—
18	#14556665	—	2	2	no	—
19	C₂×#11924031	—	2	1	no	—
20	C₂×#11924031	—	2	1	no	—
21	C₂×#11924031	—	2	2	no	—
22	C₂×#11924031	—	2	1	no	—
23	C₂×#11924031	—	2	1	no	—
24	C₂×#11924031	D₄	1	2	yes	—
25	C₂×#11924031	Q₈	1	2	yes	—
26	C₂×#11924012	C₂×C₄	1	2	yes	—
27	C₂²×#10596831	C₂³	1	1	yes	characteristic
28	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
29	#14556527	—	2	2	no	—
30	#14556502	—	2	2	no	—
31	C₄×#10596831	—	2	2	no	—
32	C₈×Dic₇₃	D₄	1	2	yes	—
33	C₈×Dic₇₃	Q₈	1	2	yes	—
34	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
35	#117×C₇₃	C₂×C₄	1	2	yes	—
36	#107×C₇₃	C₈	1	2	yes	—
37	#106×C₇₃	C₈	1	2	yes	—
38	#11924031	—	2	2	no	—
39	#11924031	—	2	2	no	—
40	#11924031	—	4	2	no	—
41	C₂²×Dic₇₃	—	2	1	no	—
42	#11924012	—	2	2	no	—
43	#11924012	—	2	2	no	—
44	C₂×#10596831	—	2	2	no	—
45	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
46	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
47	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
48	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
49	C₂×#10596831	#41	1	1	yes	characteristic
50	C₂×#10596831	#41	1	1	yes	characteristic
51	C₄×Dic₇₃	—	2	1	no	—
52	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
55	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
56	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
57	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
58	C₄⋊C₄×C₇₃	C₂×C₈	1	4	yes	—
59	C₂²⋊C₄×C₇₃	—	2	2	no	—
60	C₄²×C₇₃	—	2	2	no	—
61	C₂³×C₇₃	#131	1	1	yes	characteristic
62	C₂×C₄×C₇₃	—	2	4	no	—
63	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
65	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
66	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
68	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
69	C₂×Dic₇₃	—	2	2	no	—
70	C₂×Dic₇₃	—	4	2	no	—
71	C₈×C₇₃	C₄⋊C₈	1	2	yes	—
72	C₈×C₇₃	C₄⋊C₈	1	2	yes	—
73	#10596831	—	4	2	no	—
74	#10596831	#121	1	1	yes	characteristic
75	#10596831	C₂²⋊Q₈	1	1	yes	characteristic
76	#10596831	#125	1	1	yes	characteristic
77	#10596831	C₄⋊Q₈	1	1	yes	characteristic
78	C₂²×C₇₃	—	2	2	no	—
79	C₂²×C₇₃	#424	1	1	yes	characteristic
80	C₂²×C₇₃	C₈×D₄	1	1	yes	characteristic
81	C₂²×C₇₃	#446	1	1	yes	characteristic
82	C₄×C₇₃	—	4	4	no	—
83	C₄×C₇₃	#389	1	1	yes	characteristic
84	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
85	C₄×C₇₃	#436	1	1	yes	characteristic
86	C₄×C₇₃	C₈×Q₈	1	1	yes	characteristic
87	Dic₇₃	—	4	2	no	—
88	#13162	—	73	1	no	maximal, Sylow
89	C₂×C₇₃	—	2	1	no	—
90	C₂×C₇₃	—	2	1	no	—
91	C₂×C₇₃	#1939	1	1	yes	characteristic
92	C₂×C₇₃	#2171	1	2	yes	—
93	C₂×#364	—	73	1	no	—
94	#2151	—	73	2	no	—
95	C₂×#346	—	73	1	no	—
96	#1772	—	73	1	no	—
97	#1379	—	73	2	no	—
98	C₇₃	#13162	1	1	yes	characteristic, complemented, Sylow
99	C₂×M₃₂	—	73	2	no	—
100	C₂²×C₁₆	—	73	1	no	—
101	C₂²×C₁₆	—	73	1	no	—
102	C₂×#108	—	73	1	no	—
103	#427	#64408	1	1	yes	characteristic
104	#405	—	73	2	no	—
105	C₂×C₄×C₈	—	73	1	no	—
106	#364	—	146	2	no	—
107	C₄⋊C₁₆	—	146	4	no	—
108	#346	—	146	2	no	—
109	#339	—	73	2	no	—
110	C₂×M₁₆	—	73	2	no	—
111	C₂²×C₈	—	73	1	no	—
112	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
113	#117	C₂×#64408	1	2	yes	—
114	C₂×C₄²	—	73	1	no	—
115	M₃₂	—	146	2	no	—
116	M₃₂	—	146	2	no	—
117	C₂×C₁₆	—	73	2	no	—
118	C₂×C₁₆	—	73	2	no	—
119	C₂×C₁₆	—	146	2	no	—
120	C₂×C₁₆	—	146	1	no	—
121	C₂×C₁₆	—	146	1	no	—
122	C₂×C₁₆	—	146	1	no	—
123	C₂×C₁₆	—	146	1	no	—
124	#108	—	146	2	no	—
125	#107	#10596833	1	2	yes	—
126	#106	#10596833	1	2	yes	—
127	#97	—	146	2	no	—
128	C₄×C₈	—	73	2	no	—
129	C₄×C₈	—	73	2	no	—
130	C₄×C₈	—	146	2	no	—
131	C₂²×C₄	—	146	1	no	—
132	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
133	M₁₆	—	146	2	no	—
134	M₁₆	—	146	2	no	—
135	C₂×C₈	—	73	1	no	—
136	C₂×C₈	—	73	1	no	—
137	C₂×C₈	—	73	1	no	—
138	C₂×C₈	—	73	1	no	—
139	C₂×C₈	—	73	1	no	—
140	C₂×C₈	—	73	1	no	—
141	C₂×C₈	—	146	2	no	—
142	C₂×C₈	C₄⋊#64408	1	2	yes	—
143	C₂×C₈	C₂×#10596833	1	1	yes	characteristic
144	C₂×C₈	C₂×#10596833	1	1	yes	characteristic
145	C₂×C₈	#11924038	1	2	yes	—
146	C₄⋊C₄	C₂×#10596833	1	4	yes	—
147	C₂²⋊C₄	—	2	2	no	—
148	C₄²	—	2	2	no	—
149	C₄²	—	73	2	no	—
150	C₄²	—	146	1	no	—
151	C₁₆	—	292	2	no	—
152	C₁₆	—	146	2	no	—
153	C₁₆	—	146	2	no	—
154	C₂³	#14556678	1	1	yes	characteristic
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	292	2	no	—
157	C₂×C₄	—	146	2	no	—
158	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
159	C₂×C₄	D₄×#64408	1	1	yes	characteristic
160	C₂×C₄	Q₈×#64408	1	1	yes	characteristic
161	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
162	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
163	C₂×C₄	#14556679	1	1	yes	characteristic
164	C₈	—	292	2	no	—
165	C₈	—	73	1	no	—
166	C₈	—	73	1	no	—
167	C₈	—	73	1	no	—
168	C₈	—	73	1	no	—
169	C₈	C₄⋊#10596833	1	2	yes	—
170	C₈	C₄⋊#10596833	1	2	yes	—
171	C₂²	—	2	2	no	—
172	C₂²	#17989594	1	1	yes	characteristic
173	C₂²	D₄×#10596833	1	1	yes	characteristic
174	C₂²	#17989637	1	1	yes	characteristic
175	C₄	—	4	4	no	—
176	C₄	—	292	2	no	—
177	C₄	#17989538	1	1	yes	characteristic
178	C₄	C₂×C₄⋊#10596833	1	1	yes	characteristic
179	C₄	#17989621	1	1	yes	characteristic
180	C₄	Q₈×#10596833	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	1	no	—
183	C₂	#22626028	1	1	yes	characteristic
184	C₂	#22626321	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17989509

C₂

yes

maximal, characteristic

#22626266

C₂

yes

maximal

C₂×#17989476

C₂

yes

maximal, characteristic

#22624985

C₂

yes

maximal

#22609812

C₂

yes

maximal, characteristic

C₂×#14556665

C₂²

yes

—

C₂²×#11924031

C₂²

yes

characteristic

C₂²×#11924031

C₂²

yes

—

—

—

C₄

yes

characteristic

#17988543

C₂²

yes

—

C₂×C₈×Dic₇₃

C₂²

yes

characteristic

#17988346

C₄

yes

—

#427×C₇₃

C₄

yes

—

—

—

—

—

—

—

D₄

yes

—

C₂×#11924031

Q₈

yes

—

C₂×#11924012

C₂×C₄

yes

—

C₂²×#10596831

C₂³

yes

characteristic

C₂×C₄×Dic₇₃

C₂×C₄

yes

—

—

—

C₈×Dic₇₃

D₄

yes

—

C₈×Dic₇₃

Q₈

yes

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

#117×C₇₃

C₂×C₄

yes

—

#107×C₇₃

C₈

yes

—

#106×C₇₃

C₈

yes

—

#11924031

—

#11924031

—

#11924031

—

C₂²×Dic₇₃

—

—

—

—

C₂×D₄

yes

characteristic

C₂×#10596831

C₂×D₄

yes

characteristic

C₂×#10596831

C₂×Q₈

yes

characteristic

C₂×#10596831

C₂×Q₈

yes

yes

yes

C₄×Dic₇₃

—

C₄×Dic₇₃

C₄⋊C₄

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

M₁₆

yes

—

C₄⋊C₄×C₇₃

C₂×C₈

yes

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

#131

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₄×D₄

yes

characteristic

C₂×C₄×C₇₃

C₄×Q₈

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₃

#131

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

C₄⋊C₈

yes

—

C₈×C₇₃

C₄⋊C₈

yes

—

—

yes

C₂²⋊Q₈

yes

yes

C₄⋊Q₈

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

#424

yes

characteristic

C₂²×C₇₃

C₈×D₄

yes

characteristic

C₂²×C₇₃

#446

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

#389

yes

characteristic

C₄×C₇₃

C₂×C₄⋊C₈

yes

characteristic

C₄×C₇₃

#436

yes

characteristic

C₄×C₇₃

C₈×Q₈

yes

characteristic

Dic₇₃

—

#13162

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1939

yes

characteristic

C₂×C₇₃

#2171

yes

—

—

—

—

—

—

C₇₃

#13162

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

100

C₂²×C₁₆

—

101

C₂²×C₁₆

—

102

C₂×#108

—

103

yes

104

—

105

C₂×C₄×C₈

—

106

#364

—

146

—

107

C₄⋊C₁₆

—

146

—

108

#346

—

146

—

109

#339

—

110

C₂×M₁₆

—

111

C₂²×C₈

—

112

C₂²×C₈

yes

113

yes

—

114

C₂×C₄²

—

115

M₃₂

—

146

—

116

M₃₂

—

146

—

117

C₂×C₁₆

—

118

C₂×C₁₆

—

119

C₂×C₁₆

—

146

—

120

C₂×C₁₆

—

146

—

121

C₂×C₁₆

—

146

—

122

C₂×C₁₆

—

146

—

123

C₂×C₁₆

—

146

—

124

#108

—

146

—

125

#107

#10596833

yes

—

126

#106

#10596833

yes

—

127

#97

—

146

—

128

C₄×C₈

—

129

C₄×C₈

—

130

C₄×C₈

—

146

—

131

C₂²×C₄

—

146

—

132

C₂²×C₄

C₂²×#64408

yes

characteristic

133

M₁₆

—

146

—

134

M₁₆

—

146

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

146

—

142

C₂×C₈

C₄⋊#64408

yes

—

143

C₂×C₈

C₂×#10596833

yes

characteristic

144

C₂×C₈

C₂×#10596833

yes

characteristic

145

C₂×C₈

#11924038

yes

—

146

C₄⋊C₄

C₂×#10596833

yes

—

147

C₂²⋊C₄

—

148

C₄²

—

149

C₄²

—

150

C₄²

—

146

—

151

C₁₆

—

292

—

152

C₁₆

—

146

—

153

C₁₆

—

146

—

154

C₂³

#14556678

yes

characteristic

155

C₂×C₄

—

156

C₂×C₄

—

292

—

157

C₂×C₄

—

146

—

158

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

159

C₂×C₄

D₄×#64408

yes

characteristic

160

C₂×C₄

Q₈×#64408

yes

characteristic

161

C₂×C₄

C₂²×#10596833

yes

characteristic

162

C₂×C₄

C₂×#11924038

yes

characteristic

163

C₂×C₄

#14556679

yes

characteristic

164

C₈

—

292

—

165

C₈

—

166

C₈

—

167

C₈

—

168

C₈

—

169

C₈

C₄⋊#10596833

yes

—

170

C₈

C₄⋊#10596833

yes

—

171

C₂²

—

172

C₂²

#17989594

yes

characteristic

173

C₂²

D₄×#10596833

yes

characteristic

174

C₂²

#17989637

yes

characteristic

175

C₄

—

176

C₄

—

292

—

177

C₄

#17989538

yes

characteristic

178

C₄

C₂×C₄⋊#10596833

yes

characteristic

179

C₄

#17989621

yes

characteristic

180

C₄

Q₈×#10596833

yes

characteristic

181

C₂

—

182

C₂

—

183

C₂

#22626028

yes

characteristic

184

C₂

#22626321

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×#64408
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	#427×C₇₃	C₄
socle	C₂²×C₇₃	C₈×D₄
prime radical	C₂³×C₇₃	#131
cosocle	C₂²×#10596831	C₂³
hypercenter	#427	#64408
hypocenter	C₇₃	#13162
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×#64408

commutator

C₂×C₄×C₇₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#64408

Fitting

#427×C₇₃

C₄

socle

C₂²×C₇₃

C₈×D₄

C₂³×C₇₃

C₂³

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
98. C₇₃	#13162	semidirect

H ≅ G/N

type

98. C₇₃

#13162

semidirect

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

subgroup	index
C₂×#17989509	2
#22626266	2
C₂×#17989476	2
#22624985	2
#22609812	2
#13162	73

subgroup

index

Subgroups — #34013873

Conjugacy classes of subgroups

Complements

Maximal subgroups