Group #33002295 — subgroups

Subgroups — #33002295

Order 18688.

398 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 123 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×#17988625	C₂	1	1	yes	maximal, characteristic
2	#22611542	C₂	1	1	yes	maximal, characteristic
3	#22608041	C₂	1	4	yes	maximal, complemented
4	C₇₃×#1810	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556550	—	2	1	no	—
6	C₂×#14556531	C₂²	1	2	yes	—
7	#17988625	—	2	2	no	—
8	#17988625	C₄	1	4	yes	—
9	C₂×#14556525	C₂²	1	2	yes	—
10	C₂×#14556523	—	2	1	no	—
11	C₂²⋊#11924003	—	2	2	no	—
12	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
13	C₂×#108×C₇₃	C₂²	1	2	yes	—
14	#407×C₇₃	—	2	1	no	—
15	#341×C₇₃	—	2	1	no	—
16	#14556550	—	2	4	no	—
17	#14556550	—	4	2	no	—
18	C₂×C₄⋊Dic₇₃	—	2	2	no	—
19	C₂×C₂²⋊Dic₇₃	—	4	1	no	—
20	C₂×C₄×Dic₇₃	—	4	1	no	—
21	#14556531	—	2	4	no	—
22	C₂×#11924003	—	2	2	no	—
23	#14556525	—	2	2	no	—
24	#14556525	C₂×C₄	1	4	yes	—
25	#14556523	—	2	4	no	—
26	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
27	C₂×M₁₆×C₇₃	—	2	2	no	—
28	C₂²×C₈×C₇₃	—	2	4	no	—
29	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₃	D₄	1	2	yes	—
32	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
33	#108×C₇₃	—	2	4	no	—
34	C₂²⋊C₈×C₇₃	—	4	2	no	—
35	C₂²×Dic₇₃	—	4	2	no	—
36	C₄⋊Dic₇₃	—	2	4	no	—
37	C₄⋊Dic₇₃	—	4	2	no	—
38	C₂²⋊Dic₇₃	—	4	4	no	—
39	C₄×Dic₇₃	—	4	4	no	—
40	#11924003	—	4	2	no	—
41	C₂⁴×C₇₃	SD₁₆	1	1	yes	characteristic
42	C₂²×C₄×C₇₃	—	2	2	no	—
43	C₂²×C₄×C₇₃	—	2	2	no	—
44	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₇₃	D₈	1	2	yes	—
46	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
48	M₁₆×C₇₃	—	4	2	no	—
49	M₁₆×C₇₃	—	4	2	no	—
50	C₂×C₈×C₇₃	—	2	2	no	—
51	C₂×C₈×C₇₃	—	2	4	no	—
52	C₂×C₈×C₇₃	—	2	4	no	—
53	C₂×C₈×C₇₃	—	4	2	no	—
54	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
55	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₇₃	#41	1	4	yes	—
58	C₂³×C₇₃	—	2	2	no	—
59	C₂³×C₇₃	—	4	2	no	—
60	C₂³×C₇₃	#102	1	2	yes	—
61	C₂³×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₇₃	—	2	2	no	—
63	C₂×C₄×C₇₃	—	2	2	no	—
64	C₂×C₄×C₇₃	—	2	2	no	—
65	C₂×C₄×C₇₃	—	2	4	no	—
66	C₂×C₄×C₇₃	#102	1	2	yes	—
67	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₇₃	#123	1	2	yes	—
69	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
70	C₂×C₄×C₇₃	#137	1	2	yes	—
71	C₂×Dic₇₃	—	4	4	no	—
72	C₂×Dic₇₃	—	8	4	no	—
73	C₈×C₇₃	—	2	2	no	—
74	C₈×C₇₃	—	8	2	no	—
75	C₈×C₇₃	#117	1	2	yes	—
76	C₈×C₇₃	#123	1	2	yes	—
77	C₂²×C₇₃	—	2	2	no	—
78	C₂²×C₇₃	—	2	2	no	—
79	C₂²×C₇₃	—	2	4	no	—
80	C₂²×C₇₃	—	4	4	no	—
81	C₂²×C₇₃	C₂×#102	1	1	yes	characteristic
82	C₂²×C₇₃	#483	1	2	yes	—
83	C₄×C₇₃	—	2	2	no	—
84	C₄×C₇₃	#381	1	1	yes	characteristic
85	C₄×C₇₃	#419	1	1	yes	characteristic
86	C₄×C₇₃	#480	1	2	yes	—
87	Dic₇₃	—	8	4	no	—
88	#12872	—	73	1	no	maximal, Sylow
89	C₂×C₇₃	—	2	2	no	—
90	C₂×C₇₃	—	4	2	no	—
91	C₂×C₇₃	#1823	1	1	yes	characteristic
92	C₂×C₇₃	#2148	1	2	yes	—
93	C₂×#427	—	73	1	no	—
94	#2112	—	73	1	no	—
95	#1810	D₇₃	1	1	yes	characteristic
96	#1387	—	73	4	no	—
97	C₇₃	#12872	1	1	yes	characteristic, complemented, Sylow
98	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
99	C₂×#117	—	146	1	no	—
100	C₂×M₃₂	—	73	2	no	—
101	C₂×#108	C₂×D₇₃	1	2	yes	—
102	#427	—	73	4	no	—
103	#427	—	146	2	no	—
104	C₂×#107	—	73	2	no	—
105	C₂×#106	—	146	1	no	—
106	#407	—	2	1	no	—
107	C₂²⋊C₁₆	—	146	2	no	—
108	#341	—	2	1	no	—
109	C₂³×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
110	C₂×M₁₆	—	2	2	no	—
111	C₂²×C₈	—	2	4	no	—
112	C₂²×C₈	C₄×D₇₃	1	2	yes	—
113	C₂²×C₈	D₂₉₂	1	2	yes	—
114	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
116	#117	—	292	2	no	—
117	#117	—	146	4	no	—
118	C₂×C₄⋊C₄	—	146	2	no	—
119	C₂×C₂²⋊C₄	—	292	1	no	—
120	C₂×C₄²	—	292	1	no	—
121	M₃₂	—	146	4	no	—
122	C₂×C₁₆	—	146	2	no	—
123	#108	—	2	4	no	—
124	#107	—	73	4	no	—
125	#107	—	146	2	no	—
126	#106	—	146	4	no	—
127	C₂²⋊C₈	—	4	2	no	—
128	C₂⁴	#11924015	1	1	yes	characteristic
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	—	292	2	no	—
132	C₂²×C₄	#11924005	1	2	yes	—
133	C₂²×C₄	#11924013	1	2	yes	—
134	C₂²×C₄	#11924015	1	1	yes	characteristic
135	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
136	M₁₆	—	4	2	no	—
137	M₁₆	—	4	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	4	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	#11924005	1	2	yes	—
143	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
144	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
145	C₂×C₈	#11924027	1	4	yes	—
146	C₄⋊C₄	—	292	2	no	—
147	C₄⋊C₄	—	146	4	no	—
148	C₂²⋊C₄	—	292	4	no	—
149	C₄²	—	292	4	no	—
150	C₁₆	—	292	2	no	—
151	C₂³	—	2	2	no	—
152	C₂³	—	4	2	no	—
153	C₂³	#14556511	1	2	yes	—
154	C₂³	C₂×#11924015	1	1	yes	characteristic
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	292	4	no	—
160	C₂×C₄	—	584	4	no	—
161	C₂×C₄	#14556511	1	2	yes	—
162	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
163	C₂×C₄	#14556572	1	2	yes	—
164	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
165	C₂×C₄	#14556621	1	2	yes	—
166	C₈	—	2	2	no	—
167	C₈	—	8	2	no	—
168	C₈	#14556547	1	2	yes	—
169	C₈	#14556575	1	2	yes	—
170	C₂²	—	2	2	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	2	4	no	—
173	C₂²	—	4	4	no	—
174	C₂²	C₂×#14556511	1	1	yes	characteristic
175	C₂²	#17988915	1	2	yes	—
176	C₄	—	2	2	no	—
177	C₄	—	584	4	no	—
178	C₄	#17988448	1	1	yes	characteristic
179	C₄	#17988595	1	1	yes	characteristic
180	C₄	#17988898	1	2	yes	—
181	C₂	—	2	2	no	—
182	C₂	—	4	2	no	—
183	C₂	#22610015	1	1	yes	characteristic
184	C₂	#22611678	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988625

C₂

yes

maximal, characteristic

#22611542

C₂

yes

maximal, characteristic

#22608041

C₂

yes

maximal, complemented

C₇₃×#1810

C₂

yes

maximal, characteristic

C₂×#14556550

—

C₂×#14556531

C₂²

yes

—

#17988625

—

#17988625

C₄

yes

—

C₂×#14556525

C₂²

yes

—

C₂×#14556523

—

C₂²⋊#11924003

—

C₂³×C₈×C₇₃

C₂²

yes

characteristic

C₂×#108×C₇₃

C₂²

yes

—

#407×C₇₃

—

#341×C₇₃

—

#14556550

—

#14556550

—

C₂×C₄⋊Dic₇₃

—

C₂×C₂²⋊Dic₇₃

—

C₂×C₄×Dic₇₃

—

—

—

—

C₂×C₄

yes

—

#14556523

—

C₂³×C₄×C₇₃

D₄

yes

characteristic

C₂×M₁₆×C₇₃

—

C₂²×C₈×C₇₃

—

C₂²×C₈×C₇₃

C₂×C₄

yes

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

D₄

yes

—

C₂²×C₈×C₇₃

C₂³

yes

characteristic

#108×C₇₃

—

C₂²⋊C₈×C₇₃

—

C₂²×Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

#11924003

—

C₂⁴×C₇₃

SD₁₆

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₃

D₈

yes

—

C₂²×C₄×C₇₃

SD₁₆

yes

characteristic

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

—

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

#41

yes

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

#102

yes

—

C₂³×C₇₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#102

yes

—

C₂×C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

#123

yes

—

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

#137

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

#117

yes

—

C₈×C₇₃

#123

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×#102

yes

characteristic

C₂²×C₇₃

#483

yes

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

#12872

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1823

yes

characteristic

C₂×C₇₃

#2148

yes

—

C₂×#427

—

#2112

—

#1810

D₇₃

yes

characteristic

#1387

—

C₇₃

#12872

yes

characteristic, complemented, Sylow

C₂³×C₈

C₂×D₇₃

yes

characteristic

C₂×#117

—

146

—

100

C₂×M₃₂

—

101

C₂×#108

C₂×D₇₃

yes

—

102

#427

—

103

#427

—

146

—

104

C₂×#107

—

105

C₂×#106

—

146

—

106

#407

—

107

C₂²⋊C₁₆

—

146

—

108

#341

—

109

C₂³×C₄

C₂²⋊D₇₃

yes

characteristic

110

C₂×M₁₆

—

111

C₂²×C₈

—

112

C₂²×C₈

C₄×D₇₃

yes

—

113

C₂²×C₈

D₂₉₂

yes

—

114

C₂²×C₈

C₂²⋊D₇₃

yes

characteristic

115

C₂²×C₈

C₂²×D₇₃

yes

characteristic

116

#117

—

292

—

117

#117

—

146

—

118

C₂×C₄⋊C₄

—

146

—

119

C₂×C₂²⋊C₄

—

292

—

120

C₂×C₄²

—

292

—

121

M₃₂

—

146

—

122

C₂×C₁₆

—

146

—

123

#108

—

124

#107

—

125

#107

—

146

—

126

#106

—

146

—

127

C₂²⋊C₈

—

128

C₂⁴

#11924015

yes

characteristic

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

—

292

—

132

C₂²×C₄

#11924005

yes

—

133

C₂²×C₄

#11924013

yes

—

134

C₂²×C₄

#11924015

yes

characteristic

135

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

136

M₁₆

—

137

M₁₆

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

#11924005

yes

—

143

C₂×C₈

C₂×C₄×D₇₃

yes

characteristic

144

C₂×C₈

C₂×D₂₉₂

yes

characteristic

145

C₂×C₈

#11924027

yes

—

146

C₄⋊C₄

—

292

—

147

C₄⋊C₄

—

146

—

148

C₂²⋊C₄

—

292

—

149

C₄²

—

292

—

150

C₁₆

—

292

—

151

C₂³

—

152

C₂³

—

153

C₂³

#14556511

yes

—

154

C₂³

C₂×#11924015

yes

characteristic

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

292

—

160

C₂×C₄

—

584

—

161

C₂×C₄

#14556511

yes

—

162

C₂×C₄

C₂×#11924005

yes

characteristic

163

C₂×C₄

#14556572

yes

—

164

C₂×C₄

C₂×#11924013

yes

characteristic

165

C₂×C₄

#14556621

yes

—

166

C₈

—

167

C₈

—

168

C₈

#14556547

yes

—

169

C₈

#14556575

yes

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

C₂×#14556511

yes

characteristic

175

C₂²

#17988915

yes

—

176

C₄

—

177

C₄

—

584

—

178

C₄

#17988448

yes

characteristic

179

C₄

#17988595

yes

characteristic

180

C₄

#17988898

yes

—

181

C₂

—

182

C₂

—

183

C₂

#22610015

yes

characteristic

184

C₂

#22611678

yes

—

	subgroup	G/H
center	C₂²	C₂×#14556511
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1810	C₂
socle	C₂²×C₇₃	C₂×#102
prime radical	C₂⁴×C₇₃	SD₁₆
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#1810	D₇₃
hypocenter	C₇₃	#12872
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1810

subgroup

G/H

center

C₂²

C₂×#14556511

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₇₃×#1810

C₂

socle

C₂²×C₇₃

C₂×#102

prime radical

C₂⁴×C₇₃

SD₁₆

cosocle

C₂²×C₈×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1810

N	H ≅ G/N	type
3. #22608041	C₂	semidirect
97. C₇₃	#12872	semidirect

H ≅ G/N

type

3. #22608041

C₂

semidirect

97. C₇₃

#12872

semidirect

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

subgroup	index
C₂×#17988625	2
#22611542	2
#22608041	2
C₇₃×#1810	2
#12872	73

subgroup

index

C₇₃×#1810

Subgroups — #33002295

Conjugacy classes of subgroups

Complements

Maximal subgroups