Group #32990973 — subgroups

Subgroups — #32990973

Order 18688.

320 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 97 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	#22610871	C₂	1	1	yes	maximal, characteristic
2	#22608707	C₂	1	2	yes	maximal
3	#22608359	C₂	1	2	yes	maximal
4	#22607762	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1849	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₄₆	C₂²	1	1	yes	characteristic
7	#17988640	C₂²	1	2	yes	—
8	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
9	#17988341	—	2	2	no	—
10	#17988339	—	2	4	no	—
11	#17988317	—	2	4	no	—
12	C₂×#108×C₇₃	—	2	2	no	—
13	#423×C₇₃	C₂²	1	2	yes	—
14	C₂×#97×C₇₃	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₄₆	Q₈	1	1	yes	characteristic
16	C₂²×#10596831	—	2	1	no	—
17	C₄×Dic₁₄₆	—	2	2	no	—
18	C₄×Dic₁₄₆	—	2	2	no	—
19	C₂×C₄⋊Dic₇₃	Q₈	1	1	yes	characteristic
20	C₂×#11924007	—	2	1	no	—
21	C₂×C₄×Dic₇₃	—	2	1	no	—
22	C₄⋊#10596831	—	2	2	no	—
23	C₄⋊#10596831	—	2	2	no	—
24	C₂²⋊#10596831	—	2	2	no	—
25	C₄×#10596831	D₄	1	4	yes	—
26	C₂×M₁₆×C₇₃	—	2	2	no	—
27	C₂²×C₈×C₇₃	D₄	1	2	yes	—
28	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
29	#108×C₇₃	—	4	4	no	—
30	C₄⋊C₈×C₇₃	—	2	2	no	—
31	C₄⋊C₈×C₇₃	—	2	2	no	—
32	#97×C₇₃	—	2	2	no	—
33	C₂×Dic₁₄₆	—	2	2	no	—
34	C₂×Dic₁₄₆	—	4	2	no	—
35	C₂²×Dic₇₃	—	2	1	no	—
36	C₂²×Dic₇₃	—	2	1	no	—
37	C₂×#10596831	—	2	4	no	—
38	C₂×#10596831	—	4	1	no	—
39	C₂×#10596831	—	4	2	no	—
40	C₄⋊Dic₇₃	—	2	2	no	—
41	#11924007	—	4	2	no	—
42	C₄×Dic₇₃	—	4	1	no	—
43	C₄×Dic₇₃	—	4	1	no	—
44	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
47	M₁₆×C₇₃	—	4	4	no	—
48	C₂×C₈×C₇₃	—	2	2	no	—
49	C₂×C₈×C₇₃	—	2	2	no	—
50	C₂×C₈×C₇₃	—	4	2	no	—
51	C₂×C₈×C₇₃	Q₁₆	1	4	yes	—
52	C₄²×C₇₃	C₂×D₄	1	2	yes	—
53	C₄²×C₇₃	#41	1	2	yes	—
54	C₂³×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
55	C₂×C₄×C₇₃	—	4	1	no	—
56	C₂×C₄×C₇₃	—	4	1	no	—
57	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
58	C₂×C₄×C₇₃	#123	1	1	yes	characteristic
59	C₂×C₄×C₇₃	#123	1	2	yes	—
60	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
61	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
62	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
63	C₂×Dic₇₃	—	2	2	no	—
64	C₂×Dic₇₃	—	4	1	no	—
65	C₂×Dic₇₃	—	4	1	no	—
66	C₂×Dic₇₃	—	4	2	no	—
67	C₂×Dic₇₃	—	4	1	no	—
68	Dic₁₄₆	—	2	4	no	—
69	Dic₁₄₆	—	4	2	no	—
70	C₈×C₇₃	—	2	4	no	—
71	C₈×C₇₃	—	8	2	no	—
72	#10596831	—	4	4	no	—
73	#10596831	—	8	1	no	—
74	C₂²×C₇₃	—	2	1	no	—
75	C₂²×C₇₃	—	2	1	no	—
76	C₂²×C₇₃	#399	1	1	yes	characteristic
77	C₂²×C₇₃	#477	1	2	yes	—
78	C₄×C₇₃	—	2	2	no	—
79	C₄×C₇₃	—	2	2	no	—
80	C₄×C₇₃	#484	1	2	yes	—
81	C₄×C₇₃	#494	1	1	yes	characteristic
82	C₄×C₇₃	#496	1	1	yes	characteristic
83	Dic₇₃	—	4	2	no	—
84	Dic₇₃	—	8	1	no	—
85	#10662	—	73	1	no	maximal, Sylow
86	C₂×C₇₃	—	2	2	no	—
87	C₂×C₇₃	#2091	1	1	yes	characteristic
88	C₂×C₇₃	#2095	1	1	yes	characteristic
89	C₂×C₇₃	#2098	1	1	yes	characteristic
90	#1985	—	73	1	no	—
91	#1849	D₇₃	1	1	yes	characteristic
92	#1571	—	73	2	no	—
93	#1487	—	73	2	no	—
94	#1275	—	73	1	no	—
95	C₇₃	#10662	1	1	yes	characteristic, complemented, Sylow
96	C₂×C₄×Q₈	—	73	1	no	—
97	#431	—	73	2	no	—
98	C₂×#108	—	2	2	no	—
99	#423	C₂×D₇₃	1	2	yes	—
100	C₂×C₄⋊C₈	—	73	1	no	—
101	C₂×#97	C₂×D₇₃	1	1	yes	characteristic
102	#336	—	146	2	no	—
103	#335	—	146	4	no	—
104	Q₈⋊C₈	—	146	4	no	—
105	C₂²×Q₈	—	73	1	no	—
106	C₂×M₁₆	—	2	2	no	—
107	C₂²×C₈	—	146	1	no	—
108	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
109	C₄×Q₈	—	146	2	no	—
110	C₄×Q₈	—	146	2	no	—
111	C₂×C₄⋊C₄	—	73	1	no	—
112	C₂×C₄⋊C₄	—	146	1	no	—
113	C₂×C₄²	—	146	1	no	—
114	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
115	#108	—	4	4	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	146	2	no	—
119	C₄⋊C₈	—	146	2	no	—
120	C₂²⋊C₈	—	146	2	no	—
121	#97	—	2	2	no	—
122	C₄×C₈	—	73	4	no	—
123	C₂×Q₈	—	292	2	no	—
124	C₂×Q₈	—	146	2	no	—
125	C₂²×C₄	—	146	1	no	—
126	C₂²×C₄	—	146	1	no	—
127	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
128	C₂²×C₄	Q₈×D₇₃	1	1	yes	characteristic
129	C₂²×C₄	#11924029	1	1	yes	characteristic
130	M₁₆	—	4	4	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	—	292	1	no	—
135	C₂×C₈	—	292	2	no	—
136	C₂×C₈	—	146	4	no	—
137	C₂×C₈	#11924018	1	4	yes	—
138	C₄⋊C₄	—	292	2	no	—
139	C₄⋊C₄	—	146	2	no	—
140	C₄²	—	292	1	no	—
141	C₄²	—	292	1	no	—
142	C₄²	D₄×D₇₃	1	2	yes	—
143	C₄²	#11924028	1	2	yes	—
144	C₂³	#14556567	1	1	yes	characteristic
145	Q₈	—	292	2	no	—
146	Q₈	—	146	4	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	292	1	no	—
150	C₂×C₄	—	292	1	no	—
151	C₂×C₄	—	292	2	no	—
152	C₂×C₄	—	292	1	no	—
153	C₂×C₄	—	146	2	no	—
154	C₂×C₄	#14556569	1	2	yes	—
155	C₂×C₄	#14556574	1	1	yes	characteristic
156	C₂×C₄	#14556576	1	2	yes	—
157	C₂×C₄	#14556587	1	1	yes	characteristic
158	C₂×C₄	C₂×#11924018	1	2	yes	—
159	C₂×C₄	#14556617	1	2	yes	—
160	C₈	—	2	4	no	—
161	C₈	—	8	2	no	—
162	C₈	—	292	4	no	—
163	C₈	—	584	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	#17988522	1	1	yes	characteristic
167	C₂²	#17988879	1	2	yes	—
168	C₄	—	2	2	no	—
169	C₄	—	2	2	no	—
170	C₄	—	292	2	no	—
171	C₄	—	584	1	no	—
172	C₄	#17988922	1	2	yes	—
173	C₄	#17988966	1	1	yes	characteristic
174	C₄	#17988974	1	1	yes	characteristic
175	C₂	—	2	2	no	—
176	C₂	#22611446	1	1	yes	characteristic
177	C₂	#22611470	1	1	yes	characteristic
178	C₂	#22611483	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22610871

C₂

yes

maximal, characteristic

#22608707

C₂

yes

maximal

#22608359

C₂

yes

maximal

#22607762

C₂

yes

maximal, characteristic

C₇₃×#1849

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₄₆

C₂²

yes

characteristic

#17988640

C₂²

yes

—

C₂×C₄⋊#10596831

C₂²

yes

—

—

—

C₂×#108×C₇₃

—

#423×C₇₃

C₂²

yes

—

C₂×#97×C₇₃

C₂²

yes

characteristic

C₂²×Dic₁₄₆

Q₈

yes

characteristic

C₂²×#10596831

—

C₄×Dic₁₄₆

—

C₄×Dic₁₄₆

—

C₂×C₄⋊Dic₇₃

Q₈

yes

characteristic

C₂×#11924007

—

C₂×C₄×Dic₇₃

—

—

—

—

D₄

yes

—

C₂×M₁₆×C₇₃

—

C₂²×C₈×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

#108×C₇₃

—

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

—

#97×C₇₃

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

#11924007

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₇₃

#41

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

Q₁₆

yes

—

C₄²×C₇₃

C₂×D₄

yes

—

C₄²×C₇₃

#41

yes

—

C₂³×C₇₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₃

#123

yes

characteristic

C₂×C₄×C₇₃

#123

yes

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₁₆

yes

—

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#399

yes

characteristic

C₂²×C₇₃

#477

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#484

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

#10662

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

yes

—

—

—

C₇₃

#10662

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

#431

—

C₂×#108

—

#423

C₂×D₇₃

yes

—

100

C₂×C₄⋊C₈

—

101

C₂×#97

C₂×D₇₃

yes

characteristic

102

#336

—

146

—

103

#335

—

146

—

104

Q₈⋊C₈

—

146

—

105

C₂²×Q₈

—

106

C₂×M₁₆

—

107

C₂²×C₈

—

146

—

108

C₂²×C₈

C₂²⋊D₇₃

yes

—

109

C₄×Q₈

—

146

—

110

C₄×Q₈

—

146

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

146

—

113

C₂×C₄²

—

146

—

114

C₂×C₄²

C₂²×D₇₃

yes

characteristic

115

#108

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

146

—

119

C₄⋊C₈

—

146

—

120

C₂²⋊C₈

—

146

—

121

#97

—

122

C₄×C₈

—

123

C₂×Q₈

—

292

—

124

C₂×Q₈

—

146

—

125

C₂²×C₄

—

146

—

126

C₂²×C₄

—

146

—

127

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

128

C₂²×C₄

Q₈×D₇₃

yes

characteristic

129

C₂²×C₄

#11924029

yes

characteristic

130

M₁₆

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

292

—

135

C₂×C₈

—

292

—

136

C₂×C₈

—

146

—

137

C₂×C₈

#11924018

yes

—

138

C₄⋊C₄

—

292

—

139

C₄⋊C₄

—

146

—

140

C₄²

—

292

—

141

C₄²

—

292

—

142

C₄²

D₄×D₇₃

yes

—

143

C₄²

#11924028

yes

—

144

C₂³

#14556567

yes

characteristic

145

Q₈

—

292

—

146

Q₈

—

146

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

292

—

150

C₂×C₄

—

292

—

151

C₂×C₄

—

292

—

152

C₂×C₄

—

292

—

153

C₂×C₄

—

146

—

154

C₂×C₄

#14556569

yes

—

155

C₂×C₄

#14556574

yes

characteristic

156

C₂×C₄

#14556576

yes

—

157

C₂×C₄

#14556587

yes

characteristic

158

C₂×C₄

C₂×#11924018

yes

—

159

C₂×C₄

#14556617

yes

—

160

C₈

—

161

C₈

—

162

C₈

—

292

—

163

C₈

—

584

—

164

C₂²

—

165

C₂²

—

166

C₂²

#17988522

yes

characteristic

167

C₂²

#17988879

yes

—

168

C₄

—

169

C₄

—

170

C₄

—

292

—

171

C₄

—

584

—

172

C₄

#17988922

yes

—

173

C₄

#17988966

yes

characteristic

174

C₄

#17988974

yes

characteristic

175

C₂

—

176

C₂

#22611446

yes

characteristic

177

C₂

#22611470

yes

characteristic

178

C₂

#22611483

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556587
commutator, cosocle	C₂×C₄²×C₇₃	C₂³
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1849	C₂
socle	C₂²×C₇₃	#399
prime radical	C₂³×C₇₃	C₂²⋊Q₈
hypercenter	#1849	D₇₃
hypocenter	C₇₃	#10662
LCS	G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1849

subgroup

G/H

center

C₂×C₄

#14556587

commutator, cosocle

C₂×C₄²×C₇₃

C₂³

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1849

C₂

socle

C₂²×C₇₃

#399

prime radical

C₂³×C₇₃

C₂²⋊Q₈

D₇₃

C₇₃

G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1849

N	H ≅ G/N	type
95. C₇₃	#10662	semidirect

H ≅ G/N

type

95. C₇₃

#10662

semidirect

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

subgroup	index
#22610871	2
#22608707	2
#22608359	2
#22607762	2
C₇₃×#1849	2
#10662	73

subgroup

index

C₇₃×#1849

Subgroups — #32990973

Conjugacy classes of subgroups

Complements

Maximal subgroups