Group #33005859 — subgroups

Subgroups — #33005859

Order 18688.

284 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 87 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×#17988413	C₂	1	1	yes	maximal, characteristic, complemented
2	#22611544	C₂	1	1	yes	maximal, characteristic
3	#22609992	C₂	1	2	yes	maximal
4	#22608041	C₂	1	2	yes	maximal, complemented
5	C₇₃×#1381	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14556597	C₂²	1	1	yes	characteristic
7	C₂×#14556531	C₂²	1	1	yes	characteristic
8	C₂×#14556525	C₂²	1	2	yes	—
9	#17988589	—	2	2	no	—
10	#17988413	—	2	2	no	—
11	#17988366	—	2	2	no	—
12	#17988355	—	2	2	no	—
13	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
14	C₂×#108×C₇₃	C₂²	1	2	yes	—
15	C₂×#11924028	D₄	1	1	yes	characteristic
16	#14556597	—	2	2	no	—
17	#14556597	—	2	2	no	—
18	C₂×#11924012	D₄	1	1	yes	characteristic
19	C₂×#11924011	—	2	1	no	—
20	C₂×C₈×D₇₃	—	2	1	no	—
21	C₂×C₄⋊Dic₇₃	—	2	2	no	—
22	#14556531	—	2	2	no	—
23	C₂×#11924003	—	2	1	no	—
24	#14556525	—	2	2	no	—
25	#14556525	—	2	2	no	—
26	#14556516	—	4	2	no	—
27	#14556513	—	4	2	no	—
28	C₂×M₁₆×C₇₃	—	2	2	no	—
29	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
30	C₂×C₁₆×C₇₃	—	2	2	no	—
31	C₂×C₁₆×C₇₃	D₄	1	2	yes	—
32	#108×C₇₃	—	2	2	no	—
33	#108×C₇₃	—	2	2	no	—
34	#11924028	—	2	2	no	—
35	#11924028	—	2	2	no	—
36	C₂×Dic₁₄₆	D₈	1	1	yes	characteristic
37	C₂×D₂₉₂	D₈	1	1	yes	characteristic
38	C₂×C₂²⋊D₇₃	—	2	1	no	—
39	C₂²×Dic₇₃	—	4	2	no	—
40	C₂×C₄×D₇₃	—	2	1	no	—
41	#11924012	—	2	2	no	—
42	#11924011	—	4	2	no	—
43	C₂×#10596831	—	2	1	no	—
44	C₈×D₇₃	—	4	1	no	—
45	C₈×D₇₃	—	4	1	no	—
46	C₄⋊Dic₇₃	—	4	2	no	—
47	C₄⋊Dic₇₃	—	4	2	no	—
48	#11924003	—	4	1	no	—
49	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
50	M₁₆×C₇₃	—	4	2	no	—
51	M₁₆×C₇₃	—	4	2	no	—
52	C₂×C₈×C₇₃	—	4	2	no	—
53	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₇₃	#41	1	2	yes	—
55	C₂×C₈×C₇₃	#41	1	2	yes	—
56	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
57	C₁₆×C₇₃	—	2	2	no	—
58	C₂³×C₇₃	#135	1	1	yes	characteristic
59	C₂²×D₇₃	—	2	1	no	—
60	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
61	C₂×C₄×C₇₃	#123	1	2	yes	—
62	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₇₃	#137	1	2	yes	—
64	C₂²⋊D₇₃	—	4	2	no	—
65	C₂×Dic₇₃	—	2	1	no	—
66	C₂×Dic₇₃	—	8	2	no	—
67	C₂×Dic₇₃	—	16	2	no	—
68	D₂₉₂	—	2	2	no	—
69	C₄×D₇₃	—	4	1	no	—
70	C₄×D₇₃	—	4	1	no	—
71	Dic₁₄₆	—	2	2	no	—
72	C₈×C₇₃	—	8	2	no	—
73	C₈×C₇₃	#123	1	1	yes	characteristic
74	C₈×C₇₃	#124	1	1	yes	characteristic
75	C₈×C₇₃	#126	1	2	yes	—
76	#10596831	—	4	1	no	—
77	C₂²×C₇₃	—	2	1	no	—
78	C₂²×C₇₃	—	2	1	no	—
79	C₂²×C₇₃	#466	1	1	yes	characteristic
80	C₂²×C₇₃	#485	1	2	yes	—
81	C₂×D₇₃	—	4	1	no	—
82	C₂×D₇₃	—	4	1	no	—
83	C₂×D₇₃	—	4	1	no	—
84	C₄×C₇₃	#397	1	1	yes	characteristic
85	C₄×C₇₃	#470	1	1	yes	characteristic
86	C₄×C₇₃	#480	1	2	yes	—
87	Dic₇₃	—	4	1	no	—
88	Dic₇₃	—	16	2	no	—
89	#13495	—	73	1	no	maximal, Sylow
90	C₂×C₇₃	—	2	2	no	—
91	C₂×C₇₃	#2075	1	1	yes	characteristic
92	C₂×C₇₃	#2221	1	1	yes	characteristic
93	C₂×C₇₃	#2222	1	1	yes	characteristic
94	D₇₃	—	8	1	no	—
95	C₂×#366	—	73	1	no	—
96	#2113	—	73	1	no	—
97	#1816	—	73	2	no	—
98	#1387	—	73	2	no	—
99	#1381	D₇₃	1	1	yes	characteristic, complemented
100	C₇₃	#13495	1	1	yes	characteristic, complemented, Sylow
101	C₂×#131	—	73	1	no	—
102	C₂×M₃₂	—	73	1	no	—
103	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
104	C₂×#108	C₂×D₇₃	1	2	yes	—
105	C₂×#107	—	73	2	no	—
106	#417	—	146	2	no	—
107	#366	—	146	2	no	—
108	C₂²⋊C₁₆	—	146	2	no	—
109	#343	—	146	2	no	—
110	C₂×#41	—	73	1	no	—
111	#131	—	146	2	no	—
112	#131	—	146	2	no	—
113	C₂×M₁₆	—	2	2	no	—
114	C₂×M₁₆	—	73	1	no	—
115	C₂×M₁₆	—	146	1	no	—
116	C₂²×C₈	—	146	1	no	—
117	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
118	C₂×C₄⋊C₄	—	146	2	no	—
119	M₃₂	—	146	2	no	—
120	C₂×C₁₆	—	2	2	no	—
121	C₂×C₁₆	—	146	1	no	—
122	C₂×C₁₆	D₂₉₂	1	2	yes	—
123	#108	—	2	2	no	—
124	#108	—	2	2	no	—
125	#107	—	146	2	no	—
126	#107	—	146	2	no	—
127	#103	—	292	2	no	—
128	#102	—	292	2	no	—
129	#41	—	146	2	no	—
130	#41	—	146	2	no	—
131	C₂×Q₈	—	73	1	no	—
132	C₂×D₄	—	73	1	no	—
133	C₂×D₄	—	146	1	no	—
134	C₂²×C₄	—	292	2	no	—
135	C₂²×C₄	—	146	1	no	—
136	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
137	M₁₆	—	4	2	no	—
138	M₁₆	—	4	2	no	—
139	M₁₆	—	292	2	no	—
140	M₁₆	—	146	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	292	1	no	—
143	C₂×C₈	—	292	1	no	—
144	C₂×C₈	—	146	1	no	—
145	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
146	C₂×C₈	#11924027	1	2	yes	—
147	C₂×C₈	#11924028	1	2	yes	—
148	C₂×C₈	#11924029	1	1	yes	characteristic
149	C₄⋊C₄	—	292	2	no	—
150	C₄⋊C₄	—	292	2	no	—
151	C₁₆	—	2	2	no	—
152	C₁₆	—	292	1	no	—
153	C₂³	—	146	1	no	—
154	C₂³	#14556614	1	1	yes	characteristic
155	Q₈	—	146	2	no	—
156	D₄	—	292	2	no	—
157	D₄	—	146	2	no	—
158	C₂×C₄	—	292	1	no	—
159	C₂×C₄	—	292	1	no	—
160	C₂×C₄	—	584	2	no	—
161	C₂×C₄	—	1168	2	no	—
162	C₂×C₄	—	146	1	no	—
163	C₂×C₄	#14556566	1	1	yes	characteristic
164	C₂×C₄	#14556573	1	2	yes	—
165	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
166	C₂×C₄	#14556620	1	2	yes	—
167	C₈	—	8	2	no	—
168	C₈	—	292	1	no	—
169	C₈	#14556575	1	1	yes	characteristic
170	C₈	#14556579	1	1	yes	characteristic
171	C₈	#14556586	1	2	yes	—
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	—	292	1	no	—
175	C₂²	—	292	1	no	—
176	C₂²	—	292	1	no	—
177	C₂²	#17988824	1	1	yes	characteristic
178	C₂²	#17988927	1	2	yes	—
179	C₄	—	292	1	no	—
180	C₄	—	1168	2	no	—
181	C₄	#17988514	1	1	yes	characteristic
182	C₄	#17988846	1	1	yes	characteristic
183	C₄	#17988899	1	2	yes	—
184	C₂	—	2	2	no	—
185	C₂	—	584	1	no	—
186	C₂	#22611356	1	1	yes	characteristic
187	C₂	#22612052	1	1	yes	characteristic
188	C₂	#22612057	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988413

C₂

yes

maximal, characteristic, complemented

#22611544

C₂

yes

maximal, characteristic

#22609992

C₂

yes

maximal

#22608041

C₂

yes

maximal, complemented

C₇₃×#1381

C₂

yes

maximal, characteristic, complemented

C₂×#14556597

C₂²

yes

characteristic

C₂×#14556531

C₂²

yes

characteristic

C₂×#14556525

C₂²

yes

—

—

—

—

—

C₂²×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×#108×C₇₃

C₂²

yes

—

C₂×#11924028

D₄

yes

—

—

D₄

yes

characteristic

C₂×#11924011

—

C₂×C₈×D₇₃

—

C₂×C₄⋊Dic₇₃

—

—

—

—

—

—

—

C₂×M₁₆×C₇₃

—

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

—

#108×C₇₃

—

#108×C₇₃

—

#11924028

—

#11924028

—

C₂×Dic₁₄₆

D₈

yes

characteristic

C₂×D₂₉₂

D₈

yes

characteristic

C₂×C₂²⋊D₇₃

—

C₂²×Dic₇₃

—

C₂×C₄×D₇₃

—

#11924012

—

#11924011

—

C₂×#10596831

—

C₈×D₇₃

—

C₈×D₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924003

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

#41

yes

—

C₂×C₈×C₇₃

#41

yes

—

C₂×C₈×C₇₃

#41

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

#135

yes

characteristic

C₂²×D₇₃

—

C₂×C₄×C₇₃

#121

yes

characteristic

C₂×C₄×C₇₃

#123

yes

—

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

#137

yes

—

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

C₈×C₇₃

yes

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#466

yes

characteristic

C₂²×C₇₃

#485

yes

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

Dic₇₃

—

#13495

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

—

—

—

—

D₇₃

yes

characteristic, complemented

100

C₇₃

#13495

yes

characteristic, complemented, Sylow

101

C₂×#131

—

102

C₂×M₃₂

—

103

C₂²×C₁₆

C₂×D₇₃

yes

characteristic

104

C₂×#108

C₂×D₇₃

yes

—

105

C₂×#107

—

106

#417

—

146

—

107

#366

—

146

—

108

C₂²⋊C₁₆

—

146

—

109

#343

—

146

—

110

C₂×#41

—

111

#131

—

146

—

112

#131

—

146

—

113

C₂×M₁₆

—

114

C₂×M₁₆

—

115

C₂×M₁₆

—

146

—

116

C₂²×C₈

—

146

—

117

C₂²×C₈

C₂²×D₇₃

yes

characteristic

118

C₂×C₄⋊C₄

—

146

—

119

M₃₂

—

146

—

120

C₂×C₁₆

—

121

C₂×C₁₆

—

146

—

122

C₂×C₁₆

D₂₉₂

yes

—

123

#108

—

124

#108

—

125

#107

—

146

—

126

#107

—

146

—

127

#103

—

292

—

128

#102

—

292

—

129

#41

—

146

—

130

#41

—

146

—

131

C₂×Q₈

—

132

C₂×D₄

—

133

C₂×D₄

—

146

—

134

C₂²×C₄

—

292

—

135

C₂²×C₄

—

146

—

136

C₂²×C₄

D₄×D₇₃

yes

characteristic

137

M₁₆

—

138

M₁₆

—

139

M₁₆

—

292

—

140

M₁₆

—

146

—

141

C₂×C₈

—

142

C₂×C₈

—

292

—

143

C₂×C₈

—

292

—

144

C₂×C₈

—

146

—

145

C₂×C₈

C₂×D₂₉₂

yes

characteristic

146

C₂×C₈

#11924027

yes

—

147

C₂×C₈

#11924028

yes

—

148

C₂×C₈

#11924029

yes

characteristic

149

C₄⋊C₄

—

292

—

150

C₄⋊C₄

—

292

—

151

C₁₆

—

152

C₁₆

—

292

—

153

C₂³

—

146

—

154

C₂³

#14556614

yes

characteristic

155

Q₈

—

146

—

156

D₄

—

292

—

157

D₄

—

146

—

158

C₂×C₄

—

292

—

159

C₂×C₄

—

292

—

160

C₂×C₄

—

584

—

161

C₂×C₄

—

1168

—

162

C₂×C₄

—

146

—

163

C₂×C₄

#14556566

yes

characteristic

164

C₂×C₄

#14556573

yes

—

165

C₂×C₄

D₈×D₇₃

yes

characteristic

166

C₂×C₄

#14556620

yes

—

167

C₈

—

168

C₈

—

292

—

169

C₈

#14556575

yes

characteristic

170

C₈

#14556579

yes

characteristic

171

C₈

#14556586

yes

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

292

—

175

C₂²

—

292

—

176

C₂²

—

292

—

177

C₂²

#17988824

yes

characteristic

178

C₂²

#17988927

yes

—

179

C₄

—

292

—

180

C₄

—

1168

—

181

C₄

#17988514

yes

characteristic

182

C₄

#17988846

yes

characteristic

183

C₄

#17988899

yes

—

184

C₂

—

185

C₂

—

584

—

186

C₂

#22611356

yes

characteristic

187

C₂

#22612052

yes

characteristic

188

C₂

#22612057

yes

characteristic

	subgroup	G/H
center	C₂²	#17988824
commutator, cosocle	C₂²×C₈×C₇₃	C₂³
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1381	C₂
socle	C₂²×C₇₃	#466
prime radical	C₂×#11924028	D₄
hypercenter	#1381	D₇₃
hypocenter	C₇₃	#13495
LCS	G ▷ C₂²×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂²

#17988824

commutator, cosocle

C₂²×C₈×C₇₃

C₂³

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₇₃×#1381

C₂

C₂²×C₇₃

D₄

D₇₃

C₇₃

G ▷ C₂²×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
1. C₂×#17988413	C₂	semidirect
4. #22608041	C₂	semidirect
5. C₇₃×#1381	C₂	semidirect
99. #1381	D₇₃	semidirect
100. C₇₃	#13495	semidirect

H ≅ G/N

type

1. C₂×#17988413

C₂

semidirect

4. #22608041

C₂

semidirect

5. C₇₃×#1381

C₂

semidirect

99. #1381

D₇₃

semidirect

100. C₇₃

#13495

semidirect

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

subgroup	index
C₂×#17988413	2
#22611544	2
#22609992	2
#22608041	2
C₇₃×#1381	2
#13495	73

subgroup

index

C₇₃×#1381

Subgroups — #33005859

Conjugacy classes of subgroups

Complements

Maximal subgroups