Group #33005874 — subgroups

Subgroups — #33005874

Order 18688.

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

normal subgroups only

This group has 292 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	#22611726	C₂	1	1	yes	maximal, characteristic, complemented
2	#22609969	C₂	1	2	yes	maximal
3	#22608043	C₂	1	2	yes	maximal, complemented
4	#22608035	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#2112	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556527	C₂²	1	2	yes	—
7	C₂×#14556527	C₂²	1	2	yes	—
8	#17988551	—	2	2	no	—
9	#17988417	—	2	2	no	—
10	#17988352	—	2	2	no	—
11	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
12	C₂×M₃₂×C₇₃	C₂²	1	1	yes	characteristic
13	C₂×M₃₂×C₇₃	C₂²	1	1	yes	characteristic
14	C₂²⋊C₁₆×C₇₃	—	2	2	no	—
15	C₂×#11924012	—	2	2	no	—
16	C₂×#11924012	—	2	2	no	—
17	#14556527	—	2	2	no	—
18	#14556527	—	2	2	no	—
19	#14556527	—	2	2	no	—
20	#14556527	—	2	2	no	—
21	C₂²⋊#10596831	—	4	4	no	—
22	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₇₃	—	2	1	no	—
24	C₂²×C₈×C₇₃	—	2	1	no	—
25	C₂²×C₈×C₇₃	—	2	2	no	—
26	C₂²×C₈×C₇₃	—	2	2	no	—
27	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
29	M₃₂×C₇₃	—	2	2	no	—
30	M₃₂×C₇₃	—	2	2	no	—
31	C₂×C₁₆×C₇₃	—	2	1	no	—
32	C₂×C₁₆×C₇₃	D₄	1	2	yes	—
33	#11924012	—	4	2	no	—
34	#11924012	—	4	2	no	—
35	#11924012	—	4	2	no	—
36	#11924012	—	4	2	no	—
37	C₂×#10596831	—	4	2	no	—
38	C₂×#10596831	—	4	2	no	—
39	C₂⁴×C₇₃	Q₁₆	1	1	yes	characteristic
40	C₂²×C₄×C₇₃	—	2	1	no	—
41	C₂²×C₄×C₇₃	—	2	2	no	—
42	C₂²×C₄×C₇₃	—	2	1	no	—
43	C₂²×C₄×C₇₃	—	2	2	no	—
44	C₂²×C₄×C₇₃	Q₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₇₃	—	2	1	no	—
47	C₂×C₈×C₇₃	—	2	2	no	—
48	C₂×C₈×C₇₃	—	4	1	no	—
49	C₂×C₈×C₇₃	—	4	1	no	—
50	C₂×C₈×C₇₃	—	4	2	no	—
51	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₇₃	#41	1	2	yes	—
53	C₂×C₈×C₇₃	#41	1	2	yes	—
54	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
55	C₁₆×C₇₃	—	2	2	no	—
56	C₁₆×C₇₃	—	4	1	no	—
57	C₂³×C₇₃	—	2	1	no	—
58	C₂³×C₇₃	—	2	2	no	—
59	C₂³×C₇₃	—	4	2	no	—
60	C₂³×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
61	C₂×C₄×C₇₃	—	2	1	no	—
62	C₂×C₄×C₇₃	—	2	2	no	—
63	C₂×C₄×C₇₃	—	4	1	no	—
64	C₂×C₄×C₇₃	—	4	1	no	—
65	C₂×C₄×C₇₃	—	4	2	no	—
66	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
67	C₂×C₄×C₇₃	#123	1	2	yes	—
68	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
69	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
70	C₈×C₇₃	—	4	1	no	—
71	C₈×C₇₃	#123	1	1	yes	characteristic
72	C₈×C₇₃	#124	1	1	yes	characteristic
73	C₈×C₇₃	#126	1	2	yes	—
74	#10596831	—	8	2	no	—
75	#10596831	—	8	2	no	—
76	C₂²×C₇₃	—	2	2	no	—
77	C₂²×C₇₃	—	2	4	no	—
78	C₂²×C₇₃	—	4	1	no	—
79	C₂²×C₇₃	—	4	1	no	—
80	C₂²×C₇₃	—	4	2	no	—
81	C₂²×C₇₃	—	4	1	no	—
82	C₂²×C₇₃	#467	1	1	yes	characteristic
83	C₂²×C₇₃	#484	1	2	yes	—
84	C₄×C₇₃	—	4	1	no	—
85	C₄×C₇₃	#397	1	1	yes	characteristic
86	C₄×C₇₃	#469	1	1	yes	characteristic
87	C₄×C₇₃	#482	1	2	yes	—
88	#13498	—	73	1	no	maximal, Sylow
89	C₂×C₇₃	—	2	1	no	—
90	C₂×C₇₃	—	2	2	no	—
91	C₂×C₇₃	—	4	2	no	—
92	C₂×C₇₃	#2076	1	1	yes	characteristic
93	#2161	—	73	1	no	—
94	#2112	D₇₃	1	1	yes	characteristic
95	#1810	—	73	2	no	—
96	#1388	—	73	2	no	—
97	#1385	—	73	1	no	—
98	C₇₃	#13498	1	1	yes	characteristic, complemented, Sylow
99	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
100	C₂×M₃₂	C₂×D₇₃	1	1	yes	characteristic
101	C₂×M₃₂	C₂×D₇₃	1	1	yes	characteristic
102	C₂×#108	—	73	2	no	—
103	C₂×#108	—	73	2	no	—
104	#407	—	146	2	no	—
105	#368	—	146	2	no	—
106	C₂²⋊C₁₆	—	2	2	no	—
107	#341	—	146	2	no	—
108	C₂³×C₄	D₂₉₂	1	1	yes	characteristic
109	C₂×M₁₆	—	146	2	no	—
110	C₂×M₁₆	—	146	2	no	—
111	C₂²×C₈	—	2	1	no	—
112	C₂²×C₈	—	2	2	no	—
113	C₂²×C₈	—	2	2	no	—
114	C₂²×C₈	—	2	1	no	—
115	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
117	M₃₂	—	2	2	no	—
118	M₃₂	—	2	2	no	—
119	C₂×C₁₆	—	2	1	no	—
120	C₂×C₁₆	C₂²⋊D₇₃	1	2	yes	—
121	#108	—	146	2	no	—
122	#108	—	146	2	no	—
123	#108	—	146	2	no	—
124	#108	—	146	2	no	—
125	C₂²⋊C₈	—	292	4	no	—
126	C₂⁴	Dic₂₉₂	1	1	yes	characteristic
127	C₂²×C₄	—	2	1	no	—
128	C₂²×C₄	—	2	2	no	—
129	C₂²×C₄	—	2	1	no	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	Dic₂₉₂	1	1	yes	characteristic
132	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
133	M₁₆	—	292	2	no	—
134	M₁₆	—	292	2	no	—
135	M₁₆	—	292	2	no	—
136	M₁₆	—	292	2	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	4	1	no	—
140	C₂×C₈	—	4	1	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	292	2	no	—
143	C₂×C₈	—	292	2	no	—
144	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
145	C₂×C₈	#11924027	1	2	yes	—
146	C₂×C₈	#11924027	1	2	yes	—
147	C₂×C₈	#11924028	1	1	yes	characteristic
148	C₁₆	—	2	2	no	—
149	C₁₆	—	4	1	no	—
150	C₂³	—	2	2	no	—
151	C₂³	—	2	1	no	—
152	C₂³	—	4	2	no	—
153	C₂³	C₂×Dic₂₉₂	1	1	yes	characteristic
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	#14556565	1	1	yes	characteristic
160	C₂×C₄	#14556575	1	2	yes	—
161	C₂×C₄	#14556613	1	1	yes	characteristic
162	C₂×C₄	#14556619	1	2	yes	—
163	C₈	—	4	1	no	—
164	C₈	—	584	2	no	—
165	C₈	—	584	2	no	—
166	C₈	#14556572	1	1	yes	characteristic
167	C₈	#14556578	1	1	yes	characteristic
168	C₈	#14556584	1	2	yes	—
169	C₂²	—	2	4	no	—
170	C₂²	—	2	2	no	—
171	C₂²	—	4	1	no	—
172	C₂²	—	4	2	no	—
173	C₂²	—	4	1	no	—
174	C₂²	—	4	1	no	—
175	C₂²	#17988828	1	1	yes	characteristic
176	C₂²	#17988923	1	2	yes	—
177	C₄	—	4	1	no	—
178	C₄	#17988512	1	1	yes	characteristic
179	C₄	#17988840	1	1	yes	characteristic
180	C₄	#17988912	1	2	yes	—
181	C₂	—	2	2	no	—
182	C₂	—	2	1	no	—
183	C₂	—	4	2	no	—
184	C₂	#22611359	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22611726

C₂

yes

maximal, characteristic, complemented

#22609969

C₂

yes

maximal

#22608043

C₂

yes

maximal, complemented

#22608035

C₂

yes

maximal, characteristic, complemented

C₇₃×#2112

C₂

yes

maximal, characteristic

C₂×#14556527

C₂²

yes

—

C₂×#14556527

C₂²

yes

—

#17988551

—

#17988417

—

#17988352

—

C₂³×C₈×C₇₃

C₂²

yes

characteristic

C₂×M₃₂×C₇₃

C₂²

yes

characteristic

C₂×M₃₂×C₇₃

C₂²

yes

characteristic

C₂²⋊C₁₆×C₇₃

—

—

—

—

—

—

—

—

C₂³×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

—

C₂²×C₈×C₇₃

—

C₂²×C₈×C₇₃

—

C₂²×C₈×C₇₃

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

C₂³

yes

characteristic

M₃₂×C₇₃

—

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

—

—

—

—

—

—

—

C₂⁴×C₇₃

Q₁₆

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

Q₁₆

yes

characteristic

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×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

—

C₁₆×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#121

yes

characteristic

C₂×C₄×C₇₃

#123

yes

—

C₂×C₄×C₇₃

#135

yes

characteristic

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₈×C₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

C₈×C₇₃

yes

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#467

yes

characteristic

C₂²×C₇₃

#484

yes

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

#13498

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

—

D₇₃

yes

—

—

—

C₇₃

#13498

yes

characteristic, complemented, Sylow

C₂³×C₈

C₂×D₇₃

yes

characteristic

100

C₂×M₃₂

C₂×D₇₃

yes

characteristic

101

C₂×M₃₂

C₂×D₇₃

yes

characteristic

102

C₂×#108

—

103

C₂×#108

—

104

#407

—

146

—

105

#368

—

146

—

106

C₂²⋊C₁₆

—

107

#341

—

146

—

108

C₂³×C₄

D₂₉₂

yes

characteristic

109

C₂×M₁₆

—

146

—

110

C₂×M₁₆

—

146

—

111

C₂²×C₈

—

112

C₂²×C₈

—

113

C₂²×C₈

—

114

C₂²×C₈

—

115

C₂²×C₈

D₂₉₂

yes

characteristic

116

C₂²×C₈

C₂²×D₇₃

yes

characteristic

117

M₃₂

—

118

M₃₂

—

119

C₂×C₁₆

—

120

C₂×C₁₆

C₂²⋊D₇₃

yes

—

121

#108

—

146

—

122

#108

—

146

—

123

#108

—

146

—

124

#108

—

146

—

125

C₂²⋊C₈

—

292

—

126

C₂⁴

Dic₂₉₂

yes

characteristic

127

C₂²×C₄

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

Dic₂₉₂

yes

characteristic

132

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

133

M₁₆

—

292

—

134

M₁₆

—

292

—

135

M₁₆

—

292

—

136

M₁₆

—

292

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

292

—

143

C₂×C₈

—

292

—

144

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

145

C₂×C₈

#11924027

yes

—

146

C₂×C₈

#11924027

yes

—

147

C₂×C₈

#11924028

yes

characteristic

148

C₁₆

—

149

C₁₆

—

150

C₂³

—

151

C₂³

—

152

C₂³

—

153

C₂³

C₂×Dic₂₉₂

yes

characteristic

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

#14556565

yes

characteristic

160

C₂×C₄

#14556575

yes

—

161

C₂×C₄

#14556613

yes

characteristic

162

C₂×C₄

#14556619

yes

—

163

C₈

—

164

C₈

—

584

—

165

C₈

—

584

—

166

C₈

#14556572

yes

characteristic

167

C₈

#14556578

yes

characteristic

168

C₈

#14556584

yes

—

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

#17988828

yes

characteristic

176

C₂²

#17988923

yes

—

177

C₄

—

178

C₄

#17988512

yes

characteristic

179

C₄

#17988840

yes

characteristic

180

C₄

#17988912

yes

—

181

C₂

—

182

C₂

—

183

C₂

—

184

C₂

#22611359

yes

characteristic

	subgroup	G/H
center	C₄	#17988840
commutator, cosocle	C₂²×C₈×C₇₃	C₂³
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#2112	C₂
socle	C₂×C₇₃	#2076
prime radical	C₂⁴×C₇₃	Q₁₆
hypercenter	#2112	D₇₃
hypocenter	C₇₃	#13498
LCS	G ▷ C₂²×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ #2112

subgroup

G/H

center

C₄

#17988840

commutator, cosocle

C₂²×C₈×C₇₃

C₂³

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₇₃×#2112

C₂

socle

C₂×C₇₃

#2076

prime radical

C₂⁴×C₇₃

Q₁₆

D₇₃

C₇₃

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

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ #2112

N	H ≅ G/N	type
1. #22611726	C₂	semidirect
3. #22608043	C₂	semidirect
4. #22608035	C₂	semidirect
98. C₇₃	#13498	semidirect

H ≅ G/N

type

1. #22611726

C₂

semidirect

3. #22608043

C₂

semidirect

4. #22608035

C₂

semidirect

98. C₇₃

#13498

semidirect

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

subgroup	index
#22611726	2
#22609969	2
#22608043	2
#22608035	2
C₇₃×#2112	2
#13498	73

subgroup

index

C₇₃×#2112

Subgroups — #33005874

Conjugacy classes of subgroups

Complements

Maximal subgroups