Group #33005642 — subgroups

Subgroups — #33005642

Order 18688.

318 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 99 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	#22620455	C₂	1	1	yes	maximal, characteristic
2	#22611705	C₂	1	2	yes	maximal
3	#22608032	C₂	1	2	yes	maximal
4	#22608011	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1946	C₂	1	1	yes	maximal, characteristic
6	#17989144	—	2	1	no	—
7	C₂×#14556531	C₂²	1	2	yes	—
8	#17988996	—	2	4	no	—
9	#17988625	—	2	1	no	—
10	#17988625	C₂²	1	2	yes	—
11	#17988411	—	2	4	no	—
12	C₂×#108×C₇₃	C₂²	1	1	yes	characteristic
13	#427×C₇₃	C₂²	1	1	yes	characteristic
14	#424×C₇₃	—	2	1	no	—
15	C₂×#97×C₇₃	C₂²	1	1	yes	characteristic
16	#339×C₇₃	—	2	1	no	—
17	#14556591	—	2	2	no	—
18	#14556583	—	2	2	no	—
19	#14556570	—	4	2	no	—
20	C₄×Dic₁₄₆	—	4	2	no	—
21	#14556550	—	2	2	no	—
22	#14556531	—	2	2	no	—
23	#14556531	—	2	2	no	—
24	C₂×#11924003	Q₈	1	4	yes	complemented
25	#14556525	—	2	2	no	—
26	#14556525	—	2	2	no	—
27	#14556523	—	2	2	no	—
28	#14556523	D₄	1	4	yes	—
29	C₂×M₁₆×C₇₃	—	2	1	no	—
30	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
32	#117×C₇₃	—	2	1	no	—
33	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
34	#108×C₇₃	—	2	2	no	—
35	#107×C₇₃	—	2	1	no	—
36	#106×C₇₃	Q₈	1	2	yes	complemented
37	C₄⋊C₈×C₇₃	—	4	2	no	—
38	#97×C₇₃	—	2	2	no	—
39	C₂×Dic₁₄₆	—	4	2	no	—
40	C₄⋊Dic₇₃	—	2	4	no	—
41	C₄⋊Dic₇₃	—	4	2	no	—
42	#11924007	—	4	4	no	—
43	C₂²⋊Dic₇₃	—	4	2	no	—
44	C₄×Dic₇₃	—	4	2	no	—
45	#11924003	—	2	4	no	—
46	C₂²×C₄×C₇₃	—	2	1	no	—
47	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
48	M₁₆×C₇₃	—	4	1	no	—
49	M₁₆×C₇₃	—	4	1	no	—
50	C₂×C₈×C₇₃	—	2	1	no	—
51	C₂×C₈×C₇₃	—	2	1	no	—
52	C₂×C₈×C₇₃	—	4	1	no	—
53	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
54	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
55	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
56	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
58	C₄⋊C₄×C₇₃	—	2	2	no	—
59	C₂²⋊C₄×C₇₃	—	4	1	no	—
60	C₄²×C₇₃	—	2	1	no	—
61	C₄²×C₇₃	—	4	1	no	—
62	C₄²×C₇₃	SD₁₆	1	2	yes	—
63	C₂³×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₂×C₄×C₇₃	—	2	2	no	—
65	C₂×C₄×C₇₃	—	4	2	no	—
66	C₂×C₄×C₇₃	—	8	1	no	—
67	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
68	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
73	C₂×Dic₇₃	—	4	4	no	—
74	Dic₁₄₆	—	8	4	no	—
75	C₈×C₇₃	—	2	2	no	—
76	C₈×C₇₃	—	8	1	no	—
77	C₈×C₇₃	C₂²⋊Q₈	1	2	yes	—
78	C₈×C₇₃	C₄⋊Q₈	1	2	yes	—
79	C₂²×C₇₃	—	4	1	no	—
80	C₂²×C₇₃	#450	1	1	yes	characteristic
81	C₂²×C₇₃	#476	1	1	yes	characteristic
82	C₂²×C₇₃	#501	1	1	yes	characteristic
83	C₄×C₇₃	—	4	2	no	—
84	C₄×C₇₃	—	8	2	no	—
85	C₄×C₇₃	#395	1	1	yes	characteristic
86	C₄×C₇₃	#448	1	1	yes	characteristic
87	C₄×C₇₃	#475	1	1	yes	characteristic
88	C₄×C₇₃	#498	1	1	yes	characteristic
89	Dic₇₃	—	8	4	no	—
90	#13459	—	73	1	no	maximal, Sylow
91	C₂×C₇₃	—	2	1	no	—
92	C₂×C₇₃	—	2	1	no	—
93	C₂×C₇₃	—	2	1	no	—
94	C₂×C₇₃	#2058	1	1	yes	characteristic
95	#3162	—	73	1	no	—
96	#2155	—	73	2	no	—
97	#1946	D₇₃	1	1	yes	characteristic
98	#1384	—	73	2	no	—
99	#1375	—	73	1	no	—
100	C₇₃	#13459	1	1	yes	characteristic, complemented, Sylow
101	#533	—	146	1	no	—
102	C₂×M₃₂	—	73	2	no	—
103	#501	—	146	4	no	—
104	C₂×#108	C₂×D₇₃	1	1	yes	characteristic
105	#427	—	73	2	no	—
106	#427	—	146	1	no	—
107	#427	C₂×D₇₃	1	1	yes	characteristic
108	#424	—	2	1	no	—
109	C₂×#97	C₂×D₇₃	1	1	yes	characteristic
110	#365	—	146	4	no	—
111	#339	—	2	1	no	—
112	C₂×M₁₆	—	2	1	no	—
113	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
114	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
115	C₄⋊Q₈	—	146	2	no	—
116	#125	—	146	2	no	—
117	C₂²⋊Q₈	—	292	2	no	—
118	C₄×Q₈	—	292	2	no	—
119	#117	—	2	1	no	—
120	#117	—	146	2	no	—
121	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
122	M₃₂	—	146	2	no	—
123	M₃₂	—	146	2	no	—
124	C₂×C₁₆	—	73	4	no	—
125	#108	—	2	2	no	—
126	#107	—	2	1	no	—
127	#107	—	146	2	no	—
128	#107	—	146	2	no	—
129	#106	—	73	4	no	—
130	#106	—	146	2	no	—
131	#106	Dic₁₄₆	1	2	yes	complemented
132	C₄⋊C₈	—	4	2	no	—
133	#97	—	2	2	no	—
134	C₂×Q₈	—	292	2	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
137	M₁₆	—	4	1	no	—
138	M₁₆	—	4	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	4	1	no	—
142	C₂×C₈	#11924016	1	2	yes	—
143	C₂×C₈	D₄×D₇₃	1	2	yes	—
144	C₂×C₈	Q₈×D₇₃	1	2	yes	—
145	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
146	C₂×C₈	#11924028	1	1	yes	characteristic
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	292	4	no	—
149	C₄⋊C₄	—	292	2	no	—
150	C₄⋊C₄	—	146	4	no	—
151	C₂²⋊C₄	—	4	1	no	—
152	C₂²⋊C₄	—	292	2	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	4	1	no	—
155	C₄²	—	292	2	no	—
156	C₄²	#11924015	1	2	yes	—
157	C₁₆	—	146	4	no	—
158	C₂³	#14556616	1	1	yes	characteristic
159	Q₈	—	584	4	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	2	no	—
162	C₂×C₄	—	8	1	no	—
163	C₂×C₄	—	292	4	no	—
164	C₂×C₄	#14556559	1	1	yes	characteristic
165	C₂×C₄	#14556570	1	1	yes	characteristic
166	C₂×C₄	#14556589	1	1	yes	characteristic
167	C₂×C₄	C₂×#11924015	1	1	yes	characteristic
168	C₂×C₄	C₂×#11924016	1	1	yes	characteristic
169	C₂×C₄	#14556621	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	8	1	no	—
172	C₈	#14556569	1	2	yes	—
173	C₈	#14556590	1	2	yes	—
174	C₂²	—	4	1	no	—
175	C₂²	#17988737	1	1	yes	characteristic
176	C₂²	#17988877	1	1	yes	characteristic
177	C₂²	#17988997	1	1	yes	characteristic
178	C₄	—	4	2	no	—
179	C₄	—	8	2	no	—
180	C₄	—	584	4	no	—
181	C₄	#17988507	1	1	yes	characteristic
182	C₄	#17988725	1	1	yes	characteristic
183	C₄	#17988872	1	1	yes	characteristic
184	C₄	#17988983	1	1	yes	characteristic
185	C₂	—	2	1	no	—
186	C₂	—	2	1	no	—
187	C₂	—	2	1	no	—
188	C₂	#22611264	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22620455

C₂

yes

maximal, characteristic

#22611705

C₂

yes

maximal

#22608032

C₂

yes

maximal

#22608011

C₂

yes

maximal, characteristic

C₇₃×#1946

C₂

yes

maximal, characteristic

#17989144

—

C₂×#14556531

C₂²

yes

—

#17988996

—

#17988625

—

#17988625

C₂²

yes

—

#17988411

—

C₂×#108×C₇₃

C₂²

yes

characteristic

#427×C₇₃

C₂²

yes

characteristic

#424×C₇₃

—

C₂×#97×C₇₃

C₂²

yes

characteristic

#339×C₇₃

—

#14556591

—

#14556583

—

#14556570

—

C₄×Dic₁₄₆

—

—

—

—

Q₈

yes

—

—

—

D₄

yes

—

C₂×M₁₆×C₇₃

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

C₂³

yes

characteristic

#117×C₇₃

—

C₂×C₄²×C₇₃

D₄

yes

characteristic

#108×C₇₃

—

#107×C₇₃

—

#106×C₇₃

Q₈

yes

complemented

C₄⋊C₈×C₇₃

—

#97×C₇₃

—

C₂×Dic₁₄₆

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

#11924003

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

SD₁₆

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₂×C₈×C₇₃

C₂×Q₈

yes

—

C₂×C₈×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₃

#41

yes

characteristic

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

SD₁₆

yes

—

C₂³×C₇₃

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#120

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₃

#137

yes

characteristic

C₂×Dic₇₃

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

C₂²⋊Q₈

yes

—

C₈×C₇₃

C₄⋊Q₈

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

#13459

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

—

—

D₇₃

yes

characteristic

#1384

—

#1375

—

100

C₇₃

#13459

yes

characteristic, complemented, Sylow

101

#533

—

146

—

102

C₂×M₃₂

—

103

#501

—

146

—

104

C₂×#108

C₂×D₇₃

yes

characteristic

105

#427

—

106

#427

—

146

—

107

#427

C₂×D₇₃

yes

characteristic

108

#424

—

109

C₂×#97

C₂×D₇₃

yes

characteristic

110

#365

—

146

—

111

#339

—

112

C₂×M₁₆

—

113

C₂²×C₈

C₂²⋊D₇₃

yes

characteristic

114

C₂²×C₈

C₂²×D₇₃

yes

characteristic

115

C₄⋊Q₈

—

146

—

116

#125

—

146

—

117

C₂²⋊Q₈

—

292

—

118

C₄×Q₈

—

292

—

119

#117

—

120

#117

—

146

—

121

C₂×C₄²

C₂²⋊D₇₃

yes

characteristic

122

M₃₂

—

146

—

123

M₃₂

—

146

—

124

C₂×C₁₆

—

125

#108

—

126

#107

—

127

#107

—

146

—

128

#107

—

146

—

129

#106

—

130

#106

—

146

—

131

#106

Dic₁₄₆

yes

complemented

132

C₄⋊C₈

—

133

#97

—

134

C₂×Q₈

—

292

—

135

C₂²×C₄

—

136

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

137

M₁₆

—

138

M₁₆

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

#11924016

yes

—

143

C₂×C₈

D₄×D₇₃

yes

—

144

C₂×C₈

Q₈×D₇₃

yes

—

145

C₂×C₈

C₂×Dic₁₄₆

yes

characteristic

146

C₂×C₈

#11924028

yes

characteristic

147

C₄⋊C₄

—

148

C₄⋊C₄

—

292

—

149

C₄⋊C₄

—

292

—

150

C₄⋊C₄

—

146

—

151

C₂²⋊C₄

—

152

C₂²⋊C₄

—

292

—

153

C₄²

—

154

C₄²

—

155

C₄²

—

292

—

156

C₄²

#11924015

yes

—

157

C₁₆

—

146

—

158

C₂³

#14556616

yes

characteristic

159

Q₈

—

584

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

292

—

164

C₂×C₄

#14556559

yes

characteristic

165

C₂×C₄

#14556570

yes

characteristic

166

C₂×C₄

#14556589

yes

characteristic

167

C₂×C₄

C₂×#11924015

yes

characteristic

168

C₂×C₄

C₂×#11924016

yes

characteristic

169

C₂×C₄

#14556621

yes

characteristic

170

C₈

—

171

C₈

—

172

C₈

#14556569

yes

—

173

C₈

#14556590

yes

—

174

C₂²

—

175

C₂²

#17988737

yes

characteristic

176

C₂²

#17988877

yes

characteristic

177

C₂²

#17988997

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

—

584

—

181

C₄

#17988507

yes

characteristic

182

C₄

#17988725

yes

characteristic

183

C₄

#17988872

yes

characteristic

184

C₄

#17988983

yes

characteristic

185

C₂

—

186

C₂

—

187

C₂

—

188

C₂

#22611264

yes

characteristic

	subgroup	G/H
center	C₄	#17988983
commutator, cosocle	C₂²×C₈×C₇₃	C₂³
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1946	C₂
socle	C₂×C₇₃	#2058
prime radical	C₂³×C₇₃	AGL(1,ℤ/8)
hypercenter	#1946	D₇₃
hypocenter	C₇₃	#13459
LCS	G ▷ C₂²×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1946

subgroup

G/H

center

C₄

#17988983

commutator, cosocle

C₂²×C₈×C₇₃

C₂³

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₇₃×#1946

C₂

C₂×C₇₃

C₂³×C₇₃

D₇₃

C₇₃

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

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1946

N	H ≅ G/N	type
24. C₂×#11924003	Q₈	semidirect
36. #106×C₇₃	Q₈	semidirect
100. C₇₃	#13459	semidirect
131. #106	Dic₁₄₆	semidirect

H ≅ G/N

type

24. C₂×#11924003

Q₈

semidirect

36. #106×C₇₃

Q₈

semidirect

100. C₇₃

#13459

semidirect

131. #106

Dic₁₄₆

semidirect

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

subgroup	index
#22620455	2
#22611705	2
#22608032	2
#22608011	2
C₇₃×#1946	2
#13459	73

subgroup

index

C₇₃×#1946

Subgroups — #33005642

Conjugacy classes of subgroups

Complements

Maximal subgroups