Group #33005629 — subgroups

Subgroups — #33005629

Order 18688.

356 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 105 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×#17988409	C₂	1	2	yes	maximal
2	#22610679	C₂	1	1	yes	maximal, characteristic
3	#22608038	C₂	1	2	yes	maximal
4	#22608013	C₂	1	1	yes	maximal, characteristic
5	C₂×#498×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556531	C₂²	1	2	yes	—
7	C₂×#14556527	C₂²	1	1	yes	characteristic
8	#17988625	C₂²	1	1	yes	characteristic
9	#17988613	—	2	1	no	—
10	#17988409	—	2	4	no	—
11	#17988347	—	2	1	no	—
12	C₂×C₄⋊Q₈×C₇₃	—	2	1	no	—
13	#498×C₇₃	—	2	4	no	—
14	C₂×#106×C₇₃	—	2	1	no	—
15	C₂×#106×C₇₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×#11924012	—	2	1	no	—
18	#14556550	—	2	1	no	—
19	#14556531	—	2	2	no	—
20	#14556531	Q₈	1	4	yes	complemented
21	C₂×#11924003	—	2	2	no	—
22	#14556527	—	2	2	no	—
23	#14556525	Q₈	1	2	yes	complemented
24	#14556523	—	2	1	no	—
25	#14556520	—	4	2	no	—
26	C₂²×Q₈×C₇₃	—	4	1	no	—
27	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
29	C₄⋊Q₈×C₇₃	—	2	4	no	—
30	C₄⋊Q₈×C₇₃	—	4	2	no	—
31	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₇₃	—	4	1	no	—
33	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
34	#106×C₇₃	—	2	4	no	—
35	#106×C₇₃	—	2	2	no	—
36	#106×C₇₃	D₄	1	4	yes	—
37	C₄×C₈×C₇₃	—	2	2	no	—
38	#11924012	—	4	1	no	—
39	#11924012	—	4	1	no	—
40	C₂×#10596831	—	4	1	no	—
41	C₄⋊Dic₇₃	—	2	2	no	—
42	C₂²⋊Dic₇₃	—	4	1	no	—
43	C₄×Dic₇₃	—	4	1	no	—
44	#11924003	—	2	4	no	—
45	C₂×Q₈×C₇₃	—	4	4	no	—
46	C₂×Q₈×C₇₃	—	8	4	no	—
47	C₂²×C₄×C₇₃	—	2	1	no	—
48	C₂²×C₄×C₇₃	—	4	2	no	—
49	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₇₃	—	2	1	no	—
51	C₂×C₈×C₇₃	—	2	1	no	—
52	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
53	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
54	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
57	C₄⋊C₄×C₇₃	—	2	4	no	—
58	C₄⋊C₄×C₇₃	—	4	4	no	—
59	C₄⋊C₄×C₇₃	—	4	2	no	—
60	C₄²×C₇₃	—	2	1	no	—
61	C₄²×C₇₃	SD₁₆	1	2	yes	—
62	C₂³×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
63	Q₈×C₇₃	—	8	8	no	—
64	C₂×C₄×C₇₃	—	2	4	no	—
65	C₂×C₄×C₇₃	—	2	2	no	—
66	C₂×C₄×C₇₃	—	4	4	no	—
67	C₂×C₄×C₇₃	—	8	4	no	—
68	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
73	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
74	C₂×Dic₇₃	—	4	2	no	—
75	C₈×C₇₃	—	2	2	no	—
76	C₈×C₇₃	C₂²⋊Q₈	1	2	yes	—
77	C₈×C₇₃	C₄⋊Q₈	1	2	yes	—
78	#10596831	—	8	1	no	—
79	C₂²×C₇₃	—	2	2	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	—	2	4	no	—
84	C₄×C₇₃	—	8	4	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	2	no	—
90	#13457	—	73	1	no	maximal, Sylow
91	C₂×C₇₃	—	2	1	no	—
92	C₂×C₇₃	—	2	1	no	—
93	C₂×C₇₃	#2058	1	1	yes	characteristic
94	C₂×C₇₃	#2230	1	2	yes	—
95	C₂×#498	D₇₃	1	1	yes	characteristic
96	C₂×#365	—	73	2	no	—
97	#1945	—	73	1	no	—
98	#1386	—	73	2	no	—
99	#1376	—	73	1	no	—
100	C₇₃	#13457	1	1	yes	characteristic, complemented, Sylow
101	C₂×C₄⋊Q₈	—	2	1	no	—
102	C₂×M₃₂	—	73	2	no	—
103	#498	—	2	4	no	—
104	C₂×#108	—	73	1	no	—
105	#427	—	73	1	no	—
106	C₂×#106	—	2	1	no	—
107	C₂×#106	C₂×D₇₃	1	2	yes	—
108	#424	—	146	1	no	—
109	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
110	#365	—	146	4	no	—
111	#339	—	146	1	no	—
112	C₂²×Q₈	—	4	1	no	—
113	C₂×M₁₆	—	146	1	no	—
114	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
116	C₄⋊Q₈	—	2	4	no	—
117	C₄⋊Q₈	—	4	2	no	—
118	#117	—	146	1	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	4	1	no	—
121	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
122	M₃₂	—	73	4	no	—
123	M₃₂	—	146	2	no	—
124	C₂×C₁₆	—	146	2	no	—
125	#108	—	146	2	no	—
126	#107	—	73	2	no	—
127	#106	—	2	4	no	—
128	#106	—	2	2	no	—
129	#106	—	146	1	no	—
130	#106	C₂²⋊D₇₃	1	4	yes	—
131	C₄⋊C₈	—	292	2	no	—
132	C₄×C₈	—	2	2	no	—
133	C₂×Q₈	—	4	4	no	—
134	C₂×Q₈	—	8	4	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
138	M₁₆	—	292	1	no	—
139	M₁₆	—	292	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	292	1	no	—
143	C₂×C₈	#11924015	1	2	yes	—
144	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
145	C₂×C₈	Q₈×D₇₃	1	2	yes	—
146	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
147	C₂×C₈	#11924029	1	1	yes	characteristic
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	4	4	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	146	2	no	—
152	C₂²⋊C₄	—	292	1	no	—
153	C₄²	—	2	1	no	—
154	C₄²	—	292	1	no	—
155	C₄²	#11924016	1	2	yes	—
156	C₁₆	—	146	4	no	—
157	C₂³	#14556617	1	1	yes	characteristic
158	Q₈	—	8	8	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	8	4	no	—
163	C₂×C₄	—	292	2	no	—
164	C₂×C₄	#14556560	1	1	yes	characteristic
165	C₂×C₄	#14556567	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	#14556622	1	1	yes	characteristic
170	C₈	—	2	2	no	—
171	C₈	—	584	1	no	—
172	C₈	#14556567	1	2	yes	—
173	C₈	#14556589	1	2	yes	—
174	C₂²	—	2	2	no	—
175	C₂²	#17988738	1	1	yes	characteristic
176	C₂²	#17988874	1	1	yes	characteristic
177	C₂²	#17988993	1	1	yes	characteristic
178	C₄	—	2	4	no	—
179	C₄	—	8	4	no	—
180	C₄	—	584	2	no	—
181	C₄	#17988506	1	1	yes	characteristic
182	C₄	#17988726	1	1	yes	characteristic
183	C₄	#17988869	1	1	yes	characteristic
184	C₄	#17988983	1	1	yes	characteristic
185	C₂	—	2	1	no	—
186	C₂	—	2	1	no	—
187	C₂	#22611261	1	1	yes	characteristic
188	C₂	#22612099	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988409

C₂

yes

maximal

#22610679

C₂

yes

maximal, characteristic

#22608038

C₂

yes

maximal

#22608013

C₂

yes

maximal, characteristic

C₂×#498×C₇₃

C₂

yes

maximal, characteristic

C₂×#14556531

C₂²

yes

—

C₂×#14556527

C₂²

yes

characteristic

#17988625

C₂²

yes

—

—

—

C₂×C₄⋊Q₈×C₇₃

—

#498×C₇₃

—

C₂×#106×C₇₃

—

C₂×#106×C₇₃

C₂²

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

—

—

—

Q₈

yes

—

—

Q₈

yes

complemented

#14556523

—

#14556520

—

C₂²×Q₈×C₇₃

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

C₂³

yes

characteristic

C₄⋊Q₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

D₄

yes

characteristic

#106×C₇₃

—

#106×C₇₃

—

#106×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

—

#11924012

—

#11924012

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

#11924003

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

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

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

SD₁₆

yes

—

C₂³×C₇₃

AGL(1,ℤ/8)

yes

characteristic

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

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

—

C₈×C₇₃

—

C₈×C₇₃

C₂²⋊Q₈

yes

—

C₈×C₇₃

C₄⋊Q₈

yes

—

#10596831

—

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

—

#13457

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#2058

yes

characteristic

C₂×C₇₃

#2230

yes

—

C₂×#498

D₇₃

yes

—

—

—

—

100

C₇₃

#13457

yes

characteristic, complemented, Sylow

101

C₂×C₄⋊Q₈

—

102

C₂×M₃₂

—

103

#498

—

104

C₂×#108

—

105

#427

—

106

C₂×#106

—

107

C₂×#106

C₂×D₇₃

yes

—

108

#424

—

146

—

109

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

110

#365

—

146

—

111

#339

—

146

—

112

C₂²×Q₈

—

113

C₂×M₁₆

—

146

—

114

C₂²×C₈

C₂²⋊D₇₃

yes

characteristic

115

C₂²×C₈

C₂²×D₇₃

yes

characteristic

116

C₄⋊Q₈

—

117

C₄⋊Q₈

—

118

#117

—

146

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄²

C₂²⋊D₇₃

yes

characteristic

122

M₃₂

—

123

M₃₂

—

146

—

124

C₂×C₁₆

—

146

—

125

#108

—

146

—

126

#107

—

127

#106

—

128

#106

—

129

#106

—

146

—

130

#106

C₂²⋊D₇₃

yes

—

131

C₄⋊C₈

—

292

—

132

C₄×C₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

138

M₁₆

—

292

—

139

M₁₆

—

292

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

292

—

143

C₂×C₈

#11924015

yes

—

144

C₂×C₈

C₂×C₂²⋊D₇₃

yes

—

145

C₂×C₈

Q₈×D₇₃

yes

—

146

C₂×C₈

Q₈×D₇₃

yes

characteristic

147

C₂×C₈

#11924029

yes

characteristic

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

146

—

152

C₂²⋊C₄

—

292

—

153

C₄²

—

154

C₄²

—

292

—

155

C₄²

#11924016

yes

—

156

C₁₆

—

146

—

157

C₂³

#14556617

yes

characteristic

158

Q₈

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

292

—

164

C₂×C₄

#14556560

yes

characteristic

165

C₂×C₄

#14556567

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

#14556622

yes

characteristic

170

C₈

—

171

C₈

—

584

—

172

C₈

#14556567

yes

—

173

C₈

#14556589

yes

—

174

C₂²

—

175

C₂²

#17988738

yes

characteristic

176

C₂²

#17988874

yes

characteristic

177

C₂²

#17988993

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

—

584

—

181

C₄

#17988506

yes

characteristic

182

C₄

#17988726

yes

characteristic

183

C₄

#17988869

yes

characteristic

184

C₄

#17988983

yes

characteristic

185

C₂

—

186

C₂

—

187

C₂

#22611261

yes

characteristic

188

C₂

#22612099

yes

—

	subgroup	G/H
center	C₂²	#17988874
commutator, cosocle	C₂²×C₈×C₇₃	C₂³
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₂×#498×C₇₃	C₂
socle	C₂²×C₇₃	#476
prime radical	C₂³×C₇₃	AGL(1,ℤ/8)
hypercenter	C₂×#498	D₇₃
hypocenter	C₇₃	#13457
LCS	G ▷ C₂²×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#498

subgroup

G/H

center

C₂²

#17988874

commutator, cosocle

C₂²×C₈×C₇₃

C₂³

Frattini

C₂²×C₈

C₂²×D₇₃

Fitting

C₂×#498×C₇₃

C₂

C₂²×C₇₃

C₂³×C₇₃

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#498

N	H ≅ G/N	type
20. #14556531	Q₈	semidirect
23. #14556525	Q₈	semidirect
100. C₇₃	#13457	semidirect

H ≅ G/N

type

20. #14556531

Q₈

semidirect

23. #14556525

Q₈

semidirect

100. C₇₃

#13457

semidirect

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

subgroup	index
C₂×#17988409	2
#22610679	2
#22608038	2
#22608013	2
C₂×#498×C₇₃	2
#13457	73

subgroup

index

C₂×#498×C₇₃

Subgroups — #33005629

Conjugacy classes of subgroups

Complements

Maximal subgroups