Group #33002176 — subgroups

Subgroups — #33002176

Order 18688.

354 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 111 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×#17989005	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988416	C₂	1	2	yes	maximal, complemented
3	#22609978	C₂	1	2	yes	maximal
4	#22608035	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#1385	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14556597	C₂²	1	1	yes	characteristic
7	#17989005	—	2	4	no	—
8	C₂×#14556530	C₄	1	2	yes	—
9	C₂²×#11924003	C₂²	1	1	yes	characteristic
10	C₂×C₁₆×D₇₃	C₄	1	2	yes	—
11	C₂×#14556527	C₂²	1	2	yes	—
12	#17988569	—	2	2	no	—
13	#17988416	—	2	4	no	—
14	#17988345	—	2	2	no	—
15	C₂×M₃₂×C₇₃	C₂²	1	1	yes	characteristic
16	C₂×#108×C₇₃	C₂²	1	2	yes	—
17	C₂×#11924028	D₄	1	1	yes	characteristic
18	#14556597	—	2	2	no	—
19	#14556597	—	2	2	no	—
20	C₂×#11924012	—	2	2	no	—
21	C₂×#11924012	D₄	1	1	yes	characteristic
22	C₂×#11924011	Q₈	1	2	yes	—
23	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
24	#14556530	—	2	4	no	—
25	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
26	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
27	C₂×#11924003	D₄	1	4	yes	—
28	C₁₆×D₇₃	—	2	4	no	—
29	#14556527	—	2	2	no	—
30	#14556527	—	2	2	no	—
31	#14556509	—	4	2	no	—
32	#14556507	—	4	2	no	—
33	C₂×M₁₆×C₇₃	—	2	2	no	—
34	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
35	M₃₂×C₇₃	—	2	2	no	—
36	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
37	#108×C₇₃	—	2	2	no	—
38	#108×C₇₃	—	2	2	no	—
39	#11924028	—	2	2	no	—
40	#11924028	—	2	2	no	—
41	C₂×Dic₁₄₆	D₈	1	1	yes	characteristic
42	C₂×D₂₉₂	D₈	1	1	yes	characteristic
43	C₂×C₂²⋊D₇₃	Q₁₆	1	2	yes	—
44	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
45	#11924012	—	2	2	no	—
46	#11924012	—	4	2	no	—
47	#11924012	—	4	2	no	—
48	#11924011	—	2	4	no	—
49	C₂×#10596831	—	4	2	no	—
50	C₂×#10596831	C₄⋊C₄	1	2	yes	—
51	C₈×D₇₃	—	2	2	no	—
52	C₈×D₇₃	—	2	2	no	—
53	#11924003	C₂²⋊C₄	1	2	yes	—
54	#11924003	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
56	M₁₆×C₇₃	—	4	2	no	—
57	M₁₆×C₇₃	—	4	2	no	—
58	C₂×C₈×C₇₃	—	4	2	no	—
59	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₇₃	#41	1	2	yes	—
63	C₁₆×C₇₃	—	2	2	no	—
64	C₂³×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
65	C₂²×D₇₃	#107	1	2	yes	—
66	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₇₃	#121	1	2	yes	—
68	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
69	C₂×C₄×C₇₃	#135	1	2	yes	—
70	C₂²⋊D₇₃	—	2	4	no	—
71	C₂×Dic₇₃	#107	1	2	yes	—
72	D₂₉₂	—	2	2	no	—
73	C₄×D₇₃	—	2	2	no	—
74	C₄×D₇₃	—	2	2	no	—
75	Dic₁₄₆	—	2	2	no	—
76	C₈×C₇₃	—	8	2	no	—
77	C₈×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₈×C₇₃	#117	1	1	yes	characteristic
79	C₈×C₇₃	C₂²⋊Q₈	1	2	yes	—
80	#10596831	—	2	2	no	—
81	#10596831	—	8	2	no	—
82	C₂²×C₇₃	—	4	1	no	—
83	C₂²×C₇₃	C₂×#107	1	1	yes	characteristic
84	C₂²×C₇₃	#467	1	2	yes	—
85	C₂×D₇₃	—	2	2	no	—
86	C₂×D₇₃	—	4	2	no	—
87	C₄×C₇₃	#380	1	1	yes	characteristic
88	C₄×C₇₃	#427	1	1	yes	characteristic
89	C₄×C₇₃	#466	1	2	yes	—
90	Dic₇₃	—	2	2	no	—
91	#12840	—	73	1	no	maximal, Sylow
92	C₂×C₇₃	—	2	1	no	—
93	C₂×C₇₃	—	2	2	no	—
94	C₂×C₇₃	#1808	1	1	yes	characteristic
95	D₇₃	—	4	2	no	—
96	C₂×#504	—	73	1	no	—
97	C₂×#368	—	73	2	no	—
98	#1813	—	73	2	no	—
99	#1385	—	73	1	no	—
100	#1385	D₇₃	1	1	yes	characteristic, complemented
101	C₇₃	#12840	1	1	yes	characteristic, complemented, Sylow
102	C₂×#131	—	73	1	no	—
103	#504	—	146	4	no	—
104	C₂×M₃₂	—	73	2	no	—
105	C₂×M₃₂	C₂×D₇₃	1	1	yes	characteristic
106	C₂²×C₁₆	—	73	1	no	—
107	C₂²×C₁₆	—	73	2	no	—
108	C₂×#108	—	73	2	no	—
109	C₂×#108	C₂×D₇₃	1	2	yes	—
110	#413	—	146	2	no	—
111	#368	—	146	4	no	—
112	#338	—	146	2	no	—
113	C₂×#41	—	73	1	no	—
114	#131	—	146	2	no	—
115	#131	—	146	2	no	—
116	C₂×M₁₆	—	2	2	no	—
117	C₂×M₁₆	—	73	1	no	—
118	C₂×M₁₆	—	73	2	no	—
119	C₂×M₁₆	—	146	2	no	—
120	C₂²×C₈	—	73	2	no	—
121	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
122	M₃₂	—	2	2	no	—
123	M₃₂	—	146	4	no	—
124	C₂×C₁₆	—	73	1	no	—
125	C₂×C₁₆	—	73	1	no	—
126	C₂×C₁₆	—	73	4	no	—
127	C₂×C₁₆	—	146	4	no	—
128	C₂×C₁₆	C₄×D₇₃	1	2	yes	—
129	#108	—	2	2	no	—
130	#108	—	2	2	no	—
131	#108	—	146	2	no	—
132	#108	—	146	2	no	—
133	#101	—	292	2	no	—
134	#100	—	292	2	no	—
135	#41	—	146	2	no	—
136	#41	—	146	2	no	—
137	C₂×Q₈	—	73	1	no	—
138	C₂×D₄	—	73	2	no	—
139	C₂×D₄	—	73	1	no	—
140	C₂²×C₄	—	73	2	no	—
141	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
142	M₁₆	—	4	2	no	—
143	M₁₆	—	4	2	no	—
144	M₁₆	—	292	2	no	—
145	M₁₆	—	292	2	no	—
146	M₁₆	—	146	2	no	—
147	M₁₆	—	146	4	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	292	2	no	—
150	C₂×C₈	—	73	2	no	—
151	C₂×C₈	—	146	2	no	—
152	C₂×C₈	—	146	2	no	—
153	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
154	C₂×C₈	D₄×D₇₃	1	2	yes	—
155	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
156	C₂×C₈	#11924028	1	2	yes	—
157	C₁₆	—	2	2	no	—
158	C₁₆	—	73	2	no	—
159	C₁₆	—	73	2	no	—
160	C₂³	—	73	2	no	—
161	C₂³	Q₁₆×D₇₃	1	1	yes	characteristic
162	Q₈	—	146	2	no	—
163	D₄	—	146	4	no	—
164	D₄	—	146	2	no	—
165	C₂×C₄	—	73	2	no	—
166	C₂×C₄	—	146	2	no	—
167	C₂×C₄	—	146	2	no	—
168	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
169	C₂×C₄	#14556564	1	2	yes	—
170	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
171	C₂×C₄	#14556612	1	2	yes	—
172	C₈	—	8	2	no	—
173	C₈	—	584	2	no	—
174	C₈	—	146	2	no	—
175	C₈	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
176	C₈	#14556548	1	1	yes	characteristic
177	C₈	#14556569	1	2	yes	—
178	C₂²	—	4	1	no	—
179	C₂²	—	292	2	no	—
180	C₂²	—	146	2	no	—
181	C₂²	#107×D₇₃	1	1	yes	characteristic
182	C₂²	#17988827	1	2	yes	—
183	C₄	—	146	2	no	—
184	C₄	#17988446	1	1	yes	characteristic
185	C₄	#17988622	1	1	yes	characteristic
186	C₄	#17988822	1	2	yes	—
187	C₂	—	2	1	no	—
188	C₂	—	2	2	no	—
189	C₂	—	292	2	no	—
190	C₂	#22609959	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17989005

C₂

yes

maximal, characteristic

C₂×#17988416

C₂

yes

maximal, complemented

#22609978

C₂

yes

maximal

#22608035

C₂

yes

maximal, characteristic, complemented

C₇₃×#1385

C₂

yes

maximal, characteristic, complemented

C₂×#14556597

C₂²

yes

characteristic

#17989005

—

C₂×#14556530

C₄

yes

—

C₂²×#11924003

C₂²

yes

characteristic

C₂×C₁₆×D₇₃

C₄

yes

—

C₂×#14556527

C₂²

yes

—

#17988569

—

#17988416

—

#17988345

—

C₂×M₃₂×C₇₃

C₂²

yes

characteristic

C₂×#108×C₇₃

C₂²

yes

—

C₂×#11924028

D₄

yes

—

—

—

D₄

yes

characteristic

C₂×#11924011

Q₈

yes

—

C₂×C₈×D₇₃

C₂×C₄

yes

—

#14556530

—

C₂×#11924003

C₂×C₄

yes

characteristic

C₂×#11924003

C₂×C₄

yes

characteristic

C₂×#11924003

D₄

yes

—

C₁₆×D₇₃

—

—

—

—

—

C₂×M₁₆×C₇₃

—

C₂²×C₈×C₇₃

C₂³

yes

characteristic

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

#108×C₇₃

—

#108×C₇₃

—

#11924028

—

#11924028

—

C₂×Dic₁₄₆

D₈

yes

characteristic

C₂×D₂₉₂

D₈

yes

characteristic

C₂×C₂²⋊D₇₃

Q₁₆

yes

—

C₂×C₄×D₇₃

C₄⋊C₄

yes

—

—

—

—

—

—

C₄⋊C₄

yes

—

C₈×D₇₃

—

C₈×D₇₃

—

#11924003

C₂²⋊C₄

yes

—

#11924003

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₃

#41

yes

—

C₁₆×C₇₃

—

C₂³×C₇₃

C₂×Q₁₆

yes

characteristic

C₂²×D₇₃

#107

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

#121

yes

—

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

#135

yes

—

C₂²⋊D₇₃

—

C₂×Dic₇₃

#107

yes

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₇₃

#117

yes

characteristic

C₈×C₇₃

C₂²⋊Q₈

yes

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×#107

yes

characteristic

C₂²×C₇₃

#467

yes

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

Dic₇₃

—

#12840

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1808

yes

characteristic

D₇₃

—

—

—

—

—

100

#1385

D₇₃

yes

characteristic, complemented

101

C₇₃

#12840

yes

characteristic, complemented, Sylow

102

C₂×#131

—

103

#504

—

146

—

104

C₂×M₃₂

—

105

C₂×M₃₂

C₂×D₇₃

yes

characteristic

106

C₂²×C₁₆

—

107

C₂²×C₁₆

—

108

C₂×#108

—

109

C₂×#108

C₂×D₇₃

yes

—

110

#413

—

146

—

111

#368

—

146

—

112

#338

—

146

—

113

C₂×#41

—

114

#131

—

146

—

115

#131

—

146

—

116

C₂×M₁₆

—

117

C₂×M₁₆

—

118

C₂×M₁₆

—

119

C₂×M₁₆

—

146

—

120

C₂²×C₈

—

121

C₂²×C₈

C₂²×D₇₃

yes

characteristic

122

M₃₂

—

123

M₃₂

—

146

—

124

C₂×C₁₆

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

127

C₂×C₁₆

—

146

—

128

C₂×C₁₆

C₄×D₇₃

yes

—

129

#108

—

130

#108

—

131

#108

—

146

—

132

#108

—

146

—

133

#101

—

292

—

134

#100

—

292

—

135

#41

—

146

—

136

#41

—

146

—

137

C₂×Q₈

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂²×C₄

—

141

C₂²×C₄

D₄×D₇₃

yes

characteristic

142

M₁₆

—

143

M₁₆

—

144

M₁₆

—

292

—

145

M₁₆

—

292

—

146

M₁₆

—

146

—

147

M₁₆

—

146

—

148

C₂×C₈

—

149

C₂×C₈

—

292

—

150

C₂×C₈

—

151

C₂×C₈

—

146

—

152

C₂×C₈

—

146

—

153

C₂×C₈

C₂×C₄×D₇₃

yes

characteristic

154

C₂×C₈

D₄×D₇₃

yes

—

155

C₂×C₈

Q₈×D₇₃

yes

characteristic

156

C₂×C₈

#11924028

yes

—

157

C₁₆

—

158

C₁₆

—

159

C₁₆

—

160

C₂³

—

161

C₂³

Q₁₆×D₇₃

yes

characteristic

162

Q₈

—

146

—

163

D₄

—

146

—

164

D₄

—

146

—

165

C₂×C₄

—

166

C₂×C₄

—

146

—

167

C₂×C₄

—

146

—

168

C₂×C₄

C₄⋊C₄×D₇₃

yes

characteristic

169

C₂×C₄

#14556564

yes

—

170

C₂×C₄

D₈×D₇₃

yes

characteristic

171

C₂×C₄

#14556612

yes

—

172

C₈

—

173

C₈

—

584

—

174

C₈

—

146

—

175

C₈

C₂²⋊C₄×D₇₃

yes

characteristic

176

C₈

#14556548

yes

characteristic

177

C₈

#14556569

yes

—

178

C₂²

—

179

C₂²

—

292

—

180

C₂²

—

146

—

181

C₂²

#107×D₇₃

yes

characteristic

182

C₂²

#17988827

yes

—

183

C₄

—

146

—

184

C₄

#17988446

yes

characteristic

185

C₄

#17988622

yes

characteristic

186

C₄

#17988822

yes

—

187

C₂

—

188

C₂

—

189

C₂

—

292

—

190

C₂

#22609959

yes

characteristic

	subgroup	G/H
center	C₄	#17988622
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1385	C₂
socle	C₂×C₇₃	#1808
prime radical	C₂×#11924028	D₄
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#1385	D₇₃
hypocenter	C₇₃	#12840
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

subgroup

G/H

center

C₄

#17988622

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₇₃×#1385

C₂

C₂×C₇₃

D₄

C₂²×C₈×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

N	H ≅ G/N	type
2. C₂×#17988416	C₂	semidirect
4. #22608035	C₂	semidirect
5. C₇₃×#1385	C₂	semidirect
100. #1385	D₇₃	semidirect
101. C₇₃	#12840	semidirect

H ≅ G/N

type

2. C₂×#17988416

C₂

semidirect

4. #22608035

C₂

semidirect

5. C₇₃×#1385

C₂

semidirect

100. #1385

D₇₃

semidirect

101. C₇₃

#12840

semidirect

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

subgroup	index
C₂×#17989005	2
C₂×#17988416	2
#22609978	2
#22608035	2
C₇₃×#1385	2
#12840	73

subgroup

index

C₇₃×#1385

Subgroups — #33002176

Conjugacy classes of subgroups

Complements

Maximal subgroups