Group #32977005 — subgroups

Subgroups — #32977005

Order 18688.

368 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 133 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	#22618866	C₂	1	1	yes	maximal, characteristic
2	#22608526	C₂	1	1	yes	maximal, characteristic
3	#22608221	C₂	1	2	yes	maximal
4	#22607822	C₂	1	2	yes	maximal
5	C₇₃×#1544	C₂	1	1	yes	maximal, characteristic
6	#17988718	—	2	2	no	—
7	Q₈×#10596831	—	2	2	no	—
8	#17988614	C₄	1	2	yes	—
9	#17988611	C₂²	1	1	yes	characteristic
10	#17988543	C₄	1	2	yes	—
11	C₄×#11924012	C₂²	1	2	yes	—
12	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
13	#17988329	C₄	1	4	yes	—
14	#17988309	—	2	2	no	—
15	#17988308	—	2	2	no	—
16	#533×C₇₃	C₂²	1	1	yes	characteristic
17	#431×C₇₃	C₂²	1	2	yes	—
18	#332×C₇₃	C₄	1	4	yes	—
19	C₂×#11924012	—	2	2	no	—
20	C₂×#11924012	C₂×C₄	1	2	yes	—
21	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
22	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
23	C₄⋊#10596831	—	2	4	no	—
24	C₄⋊#10596831	—	2	2	no	—
25	#14556502	—	2	2	no	—
26	#14556502	C₂×C₄	1	4	yes	—
27	C₄×#10596831	—	2	2	no	—
28	C₄×#10596831	—	2	2	no	—
29	C₄×#10596831	D₄	1	2	yes	—
30	C₄×#10596831	D₄	1	2	yes	—
31	C₄⋊Q₈×C₇₃	C₂×C₄	1	2	yes	—
32	#125×C₇₃	C₂×C₄	1	2	yes	—
33	C₂²⋊Q₈×C₇₃	—	2	2	no	—
34	C₄×Q₈×C₇₃	—	2	2	no	—
35	#117×C₇₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₇₃	C₂×C₄	1	4	yes	—
38	C₂²⋊C₈×C₇₃	—	2	2	no	—
39	C₄×C₈×C₇₃	—	2	2	no	—
40	#11924012	—	4	4	no	—
41	#11924012	—	4	2	no	—
42	C₂×#10596831	—	2	2	no	—
43	C₂×#10596831	—	2	2	no	—
44	C₂×#10596831	—	2	4	no	—
45	C₂×#10596831	—	4	1	no	—
46	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
48	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×Q₈×C₇₃	—	2	2	no	—
50	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₇₃	#41	1	2	yes	—
52	C₂×C₈×C₇₃	—	2	4	no	—
53	C₄⋊C₄×C₇₃	—	2	4	no	—
54	C₄⋊C₄×C₇₃	—	2	2	no	—
55	C₄⋊C₄×C₇₃	C₄²	1	4	yes	—
56	C₂²⋊C₄×C₇₃	—	2	2	no	—
57	C₄²×C₇₃	—	2	2	no	—
58	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₇₃	C₂×D₄	1	2	yes	—
61	C₂³×C₇₃	#117	1	1	yes	characteristic
62	Q₈×C₇₃	—	4	4	no	—
63	C₂×C₄×C₇₃	—	2	1	no	—
64	C₂×C₄×C₇₃	—	2	1	no	—
65	C₂×C₄×C₇₃	—	2	4	no	—
66	C₂×C₄×C₇₃	—	2	4	no	—
67	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
70	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
71	C₈×C₇₃	—	4	4	no	—
72	#10596831	—	2	1	no	—
73	#10596831	—	2	2	no	—
74	#10596831	—	4	4	no	—
75	#10596831	—	4	2	no	—
76	#10596831	#101	1	2	yes	—
77	C₂²×C₇₃	—	2	1	no	—
78	C₂²×C₇₃	—	2	2	no	—
79	C₂²×C₇₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
80	C₄×C₇₃	—	2	1	no	—
81	C₄×C₇₃	—	2	2	no	—
82	C₄×C₇₃	—	4	4	no	—
83	C₄×C₇₃	C₂×#101	1	1	yes	characteristic
84	C₄×C₇₃	#413	1	1	yes	characteristic
85	C₄×C₇₃	#444	1	4	yes	—
86	#8244	—	73	1	no	maximal, Sylow
87	C₂×C₇₃	—	4	1	no	—
88	C₂×C₇₃	C₄×#101	1	1	yes	characteristic
89	C₂×C₇₃	#1765	1	2	yes	—
90	#2960	—	73	1	no	—
91	#1544	D₇₃	1	1	yes	characteristic
92	#1526	—	73	1	no	—
93	#1451	—	73	2	no	—
94	#1299	—	73	2	no	—
95	C₇₃	#8244	1	1	yes	characteristic, complemented, Sylow
96	#533	C₂×D₇₃	1	1	yes	characteristic
97	#446	—	146	2	no	—
98	C₈×Q₈	—	146	2	no	—
99	#431	C₂×D₇₃	1	2	yes	—
100	#424	—	73	2	no	—
101	#423	—	73	1	no	—
102	#405	—	73	2	no	—
103	C₄×M₁₆	—	73	2	no	—
104	C₂×C₄×C₈	—	73	1	no	—
105	#332	Dic₇₃	1	4	yes	—
106	#330	—	73	4	no	—
107	#322	—	146	2	no	—
108	#322	—	146	2	no	—
109	C₂×M₁₆	—	73	2	no	—
110	C₂×M₁₆	—	146	2	no	—
111	C₂²×C₈	—	73	1	no	—
112	C₂²×C₈	—	73	1	no	—
113	C₄⋊Q₈	C₂×Dic₇₃	1	2	yes	—
114	#125	C₄×D₇₃	1	2	yes	—
115	C₂²⋊Q₈	—	2	2	no	—
116	C₄×Q₈	—	2	2	no	—
117	#117	C₄×D₇₃	1	2	yes	—
118	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
119	C₄⋊C₈	—	146	4	no	—
120	C₄⋊C₈	—	146	2	no	—
121	C₄⋊C₈	C₂×Dic₇₃	1	4	yes	—
122	C₂²⋊C₈	—	2	2	no	—
123	#97	—	73	4	no	—
124	#97	—	146	2	no	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	—	73	2	no	—
127	C₄×C₈	—	73	2	no	—
128	C₄×C₈	—	146	2	no	—
129	C₄×C₈	—	146	2	no	—
130	C₂×Q₈	—	2	2	no	—
131	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
132	C₂²×C₄	#11924027	1	2	yes	—
133	M₁₆	—	292	4	no	—
134	M₁₆	—	292	2	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	292	1	no	—
137	C₂×C₈	—	73	1	no	—
138	C₂×C₈	—	73	1	no	—
139	C₂×C₈	—	73	2	no	—
140	C₂×C₈	—	146	2	no	—
141	C₂×C₈	—	146	4	no	—
142	C₂×C₈	—	146	2	no	—
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	C₄×Dic₇₃	1	4	yes	—
146	C₂²⋊C₄	—	2	2	no	—
147	C₄²	—	2	2	no	—
148	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
149	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
150	C₄²	D₄×D₇₃	1	2	yes	—
151	C₂³	#14556547	1	1	yes	characteristic
152	Q₈	—	4	4	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
158	C₂×C₄	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
159	C₂×C₄	#14556551	1	2	yes	—
160	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
161	C₈	—	4	4	no	—
162	C₈	—	292	4	no	—
163	C₈	—	292	2	no	—
164	C₈	—	73	2	no	—
165	C₈	—	146	1	no	—
166	C₈	—	146	2	no	—
167	C₂²	—	2	1	no	—
168	C₂²	—	2	2	no	—
169	C₂²	C₂²⋊C₄×Dic₇₃	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	2	2	no	—
172	C₄	—	4	4	no	—
173	C₄	#101×D₇₃	1	1	yes	characteristic
174	C₄	#17988570	1	1	yes	characteristic
175	C₄	#17988709	1	4	yes	—
176	C₂	—	4	1	no	—
177	C₂	#101×Dic₇₃	1	1	yes	characteristic
178	C₂	#22609778	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22618866

C₂

yes

maximal, characteristic

#22608526

C₂

yes

maximal, characteristic

#22608221

C₂

yes

maximal

#22607822

C₂

yes

maximal

C₇₃×#1544

C₂

yes

maximal, characteristic

#17988718

—

Q₈×#10596831

—

#17988614

C₄

yes

—

#17988611

C₂²

yes

characteristic

#17988543

C₄

yes

—

C₄×#11924012

C₂²

yes

—

C₂×C₄×#10596831

C₂²

yes

characteristic

#17988329

C₄

yes

—

#17988309

—

#17988308

—

#533×C₇₃

C₂²

yes

characteristic

#431×C₇₃

C₂²

yes

—

#332×C₇₃

C₄

yes

—

C₂×#11924012

—

C₂×#11924012

C₂×C₄

yes

—

C₂²×#10596831

C₂×C₄

yes

characteristic

C₂²×#10596831

C₂×C₄

yes

—

—

—

C₂×C₄

yes

—

C₄×#10596831

—

C₄×#10596831

—

C₄×#10596831

D₄

yes

—

C₄×#10596831

D₄

yes

—

C₄⋊Q₈×C₇₃

C₂×C₄

yes

—

#125×C₇₃

C₂×C₄

yes

—

C₂²⋊Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

#117×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

C₂×C₄

yes

—

C₂²⋊C₈×C₇₃

—

C₄×C₈×C₇₃

—

—

—

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#10596831

C₂²⋊C₄

yes

—

C₂×#10596831

C₂²⋊C₄

yes

characteristic

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₃

#41

yes

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₄²

yes

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

—

C₂³×C₇₃

#117

yes

characteristic

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₄×D₄

yes

—

C₂×C₄×C₇₃

C₄×D₄

yes

—

C₈×C₇₃

—

—

—

—

—

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₄×C₂²⋊C₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

#8244

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

C₄×#101

yes

characteristic

C₂×C₇₃

#1765

yes

—

#2960

—

#1544

D₇₃

yes

—

—

—

C₇₃

#8244

yes

characteristic, complemented, Sylow

#533

C₂×D₇₃

yes

characteristic

#446

—

146

—

C₈×Q₈

—

146

—

#431

C₂×D₇₃

yes

—

100

#424

—

101

#423

—

102

#405

—

103

C₄×M₁₆

—

104

C₂×C₄×C₈

—

105

#332

Dic₇₃

yes

—

106

#330

—

107

#322

—

146

—

108

#322

—

146

—

109

C₂×M₁₆

—

110

C₂×M₁₆

—

146

—

111

C₂²×C₈

—

112

C₂²×C₈

—

113

C₄⋊Q₈

C₂×Dic₇₃

yes

—

114

#125

C₄×D₇₃

yes

—

115

C₂²⋊Q₈

—

116

C₄×Q₈

—

117

#117

C₄×D₇₃

yes

—

118

C₂×C₄²

C₂²×D₇₃

yes

characteristic

119

C₄⋊C₈

—

146

—

120

C₄⋊C₈

—

146

—

121

C₄⋊C₈

C₂×Dic₇₃

yes

—

122

C₂²⋊C₈

—

123

#97

—

124

#97

—

146

—

125

C₄×C₈

—

126

C₄×C₈

—

127

C₄×C₈

—

128

C₄×C₈

—

146

—

129

C₄×C₈

—

146

—

130

C₂×Q₈

—

131

C₂²×C₄

C₂×C₄×D₇₃

yes

characteristic

132

C₂²×C₄

#11924027

yes

—

133

M₁₆

—

292

—

134

M₁₆

—

292

—

135

C₂×C₈

—

136

C₂×C₈

—

292

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

146

—

141

C₂×C₈

—

146

—

142

C₂×C₈

—

146

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

C₄×Dic₇₃

yes

—

146

C₂²⋊C₄

—

147

C₄²

—

148

C₄²

C₂×C₄×D₇₃

yes

characteristic

149

C₄²

C₂²×Dic₇₃

yes

characteristic

150

C₄²

D₄×D₇₃

yes

—

151

C₂³

#14556547

yes

characteristic

152

Q₈

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

C₂×C₄×Dic₇₃

yes

characteristic

158

C₂×C₄

C₂²⋊C₄×D₇₃

yes

characteristic

159

C₂×C₄

#14556551

yes

—

160

C₂×C₄

D₄×Dic₇₃

yes

—

161

C₈

—

162

C₈

—

292

—

163

C₈

—

292

—

164

C₈

—

165

C₈

—

146

—

166

C₈

—

146

—

167

C₂²

—

168

C₂²

—

169

C₂²

C₂²⋊C₄×Dic₇₃

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

#101×D₇₃

yes

characteristic

174

C₄

#17988570

yes

characteristic

175

C₄

#17988709

yes

—

176

C₂

—

177

C₂

#101×Dic₇₃

yes

characteristic

178

C₂

#22609778

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1544	C₂
socle	C₂²×C₇₃	C₄×C₂²⋊C₄
prime radical	C₂³×C₇₃	#117
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1544	D₇₃
hypocenter	C₇₃	#8244
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1544

subgroup

G/H

center

C₂×C₄

C₂²⋊C₄×D₇₃

commutator

C₂×C₄×C₇₃

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1544

C₂

socle

C₂²×C₇₃

C₄×C₂²⋊C₄

prime radical

C₂³×C₇₃

#117

cosocle

C₂×C₄²×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1544

N	H ≅ G/N	type
95. C₇₃	#8244	semidirect

H ≅ G/N

type

95. C₇₃

#8244

semidirect

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

subgroup	index
#22618866	2
#22608526	2
#22608221	2
#22607822	2
C₇₃×#1544	2
#8244	73

subgroup

index

C₇₃×#1544

Subgroups — #32977005

Conjugacy classes of subgroups

Complements

Maximal subgroups