Group #32973251 — subgroups

Subgroups — #32973251

Order 18688.

238 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 91 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	#22611734	C₂	1	1	yes	maximal, characteristic
2	#22608687	C₂	1	1	yes	maximal, characteristic
3	C₄×#106×C₇₃	C₂	1	1	yes	maximal, characteristic
4	#17988611	—	2	1	no	—
5	C₄⋊#11924003	C₄	1	2	yes	complemented
6	C₂²⋊#11924003	—	2	1	no	—
7	#17988363	—	2	1	no	—
8	#17988338	—	2	1	no	—
9	#17988338	C₄	1	2	yes	complemented
10	C₂×#106×C₇₃	—	2	1	no	—
11	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₇₃	—	2	1	no	—
13	#340×C₇₃	C₄	1	2	yes	—
14	C₂×#11924012	—	4	1	no	—
15	C₂×#11924003	—	2	2	no	—
16	C₄⋊#10596831	—	2	2	no	—
17	C₄⋊#10596831	—	4	1	no	—
18	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
21	C₂×C₄²×C₇₃	—	4	1	no	—
22	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
23	#106×C₇₃	—	2	4	no	—
24	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
27	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
28	#95×C₇₃	—	4	1	no	—
29	#11924012	—	8	2	no	—
30	C₂×#10596831	—	4	2	no	—
31	#11924003	—	4	2	no	—
32	C₂²×C₄×C₇₃	—	4	2	no	—
33	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
35	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
36	C₂×C₈×C₇₃	—	2	2	no	—
37	C₂×C₈×C₇₃	—	4	1	no	—
38	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
42	C₄⋊C₄×C₇₃	—	2	4	no	—
43	C₄⋊C₄×C₇₃	—	4	2	no	—
44	C₄²×C₇₃	—	8	2	no	—
45	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
47	C₄²×C₇₃	SD₁₆	1	1	yes	characteristic
48	C₄²×C₇₃	Q₁₆	1	1	yes	characteristic
49	C₂³×C₇₃	#102	1	1	yes	characteristic
50	C₂×C₄×C₇₃	—	2	2	no	—
51	C₂×C₄×C₇₃	—	2	1	no	—
52	C₂×C₄×C₇₃	—	2	1	no	—
53	C₂×C₄×C₇₃	—	2	2	no	—
54	C₂×C₄×C₇₃	—	4	4	no	—
55	C₂×C₄×C₇₃	—	8	2	no	—
56	C₂×C₄×C₇₃	—	8	2	no	—
57	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
58	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
59	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
60	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
61	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
62	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
63	C₈×C₇₃	—	2	1	no	—
64	C₈×C₇₃	—	2	1	no	—
65	C₈×C₇₃	—	2	1	no	—
66	C₈×C₇₃	#104	1	2	yes	—
67	#10596831	—	8	2	no	—
68	C₂²×C₇₃	—	2	1	no	—
69	C₂²×C₇₃	—	2	1	no	—
70	C₂²×C₇₃	#340	1	1	yes	characteristic
71	C₂²×C₇₃	#360	1	2	yes	—
72	C₄×C₇₃	—	2	1	no	—
73	C₄×C₇₃	—	2	1	no	—
74	C₄×C₇₃	—	2	1	no	—
75	C₄×C₇₃	—	2	1	no	—
76	C₄×C₇₃	—	8	4	no	—
77	C₄×C₇₃	#339	1	1	yes	characteristic
78	C₄×C₇₃	#344	1	1	yes	characteristic
79	C₄×C₇₃	#365	1	2	yes	—
80	#7302	—	73	1	no	maximal, Sylow
81	C₂×C₇₃	—	2	2	no	—
82	C₂×C₇₃	#1303	1	1	yes	characteristic
83	C₂×C₇₃	#1383	1	1	yes	characteristic
84	C₂×C₇₃	#1386	1	1	yes	characteristic
85	#2163	—	73	1	no	—
86	C₄×#106	D₇₃	1	1	yes	characteristic
87	#1566	—	73	1	no	—
88	C₇₃	#7302	1	1	yes	characteristic, complemented, Sylow
89	C₂×#106	—	2	1	no	—
90	#423	—	146	1	no	—
91	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
92	C₄×C₄⋊C₄	—	2	1	no	—
93	C₄⋊C₁₆	—	73	2	no	—
94	C₂²⋊C₁₆	—	146	1	no	—
95	#346	—	146	1	no	—
96	#340	Dic₇₃	1	2	yes	—
97	#334	—	73	2	no	—
98	#334	—	146	1	no	—
99	C₂×M₁₆	—	292	1	no	—
100	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
101	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
102	C₂×C₄⋊C₄	—	2	2	no	—
103	C₂×C₄²	—	4	1	no	—
104	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
105	C₂×C₁₆	—	146	2	no	—
106	#106	—	2	4	no	—
107	C₄⋊C₈	—	292	1	no	—
108	C₄⋊C₈	—	146	2	no	—
109	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
110	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
111	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
112	C₄×C₈	D₂₉₂	1	1	yes	characteristic
113	#95	—	4	1	no	—
114	C₂²×C₄	—	4	2	no	—
115	C₂²×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
116	C₂²×C₄	#11924013	1	1	yes	characteristic
117	C₂²×C₄	#11924016	1	1	yes	characteristic
118	M₁₆	—	584	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	4	1	no	—
121	C₂×C₈	—	292	2	no	—
122	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
123	C₂×C₈	#11924005	1	1	yes	characteristic
124	C₂×C₈	#11924007	1	1	yes	characteristic
125	C₂×C₈	C₄⋊Dic₇₃	1	1	yes	characteristic
126	C₄⋊C₄	—	2	4	no	—
127	C₄⋊C₄	—	4	2	no	—
128	C₄²	—	8	2	no	—
129	C₄²	#11924005	1	1	yes	characteristic
130	C₄²	#11924007	1	1	yes	characteristic
131	C₄²	#11924015	1	1	yes	characteristic
132	C₄²	#11924018	1	1	yes	characteristic
133	C₁₆	—	292	2	no	—
134	C₂³	#14556512	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	4	4	no	—
140	C₂×C₄	—	8	2	no	—
141	C₂×C₄	—	8	2	no	—
142	C₂×C₄	#14556498	1	1	yes	characteristic
143	C₂×C₄	#14556511	1	1	yes	characteristic
144	C₂×C₄	#14556514	1	1	yes	characteristic
145	C₂×C₄	#14556515	1	1	yes	characteristic
146	C₂×C₄	#14556522	1	1	yes	characteristic
147	C₂×C₄	#14556524	1	1	yes	characteristic
148	C₈	—	2	1	no	—
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	—	584	2	no	—
152	C₈	#14556519	1	2	yes	—
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	#17988349	1	1	yes	characteristic
156	C₂²	#17988397	1	2	yes	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	—	8	4	no	—
162	C₄	#17988347	1	1	yes	characteristic
163	C₄	#17988357	1	1	yes	characteristic
164	C₄	#17988409	1	2	yes	—
165	C₂	—	2	2	no	—
166	C₂	#22607834	1	1	yes	characteristic
167	C₂	#22608029	1	1	yes	characteristic
168	C₂	#22608036	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22611734

C₂

yes

maximal, characteristic

#22608687

C₂

yes

maximal, characteristic

C₄×#106×C₇₃

C₂

yes

maximal, characteristic

#17988611

—

C₄⋊#11924003

C₄

yes

—

—

—

C₄

yes

complemented

C₂×#106×C₇₃

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

C₄×C₄⋊C₄×C₇₃

—

#340×C₇₃

C₄

yes

—

—

—

—

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

—

C₂×C₄²×C₇₃

D₄

yes

characteristic

#106×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

Q₈

yes

characteristic

#95×C₇₃

—

#11924012

—

C₂×#10596831

—

#11924003

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

D₈

yes

characteristic

C₂²×C₄×C₇₃

SD₁₆

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂²⋊C₄

yes

characteristic

C₄²×C₇₃

C₄⋊C₄

yes

characteristic

C₄²×C₇₃

SD₁₆

yes

characteristic

C₄²×C₇₃

Q₁₆

yes

characteristic

C₂³×C₇₃

#102

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

#104

yes

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#340

yes

characteristic

C₂²×C₇₃

#360

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

#7302

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

yes

characteristic

#1566

—

C₇₃

#7302

yes

characteristic, complemented, Sylow

C₂×#106

—

#423

—

146

—

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

C₄×C₄⋊C₄

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

146

—

#346

—

146

—

#340

Dic₇₃

yes

—

#334

—

#334

—

146

—

C₂×M₁₆

—

292

—

100

C₂²×C₈

C₂×Dic₇₃

yes

characteristic

101

C₂²×C₈

C₂²⋊D₇₃

yes

characteristic

102

C₂×C₄⋊C₄

—

103

C₂×C₄²

—

104

C₂×C₄²

C₂²⋊D₇₃

yes

characteristic

105

C₂×C₁₆

—

146

—

106

#106

—

107

C₄⋊C₈

—

292

—

108

C₄⋊C₈

—

146

—

109

C₄×C₈

Dic₁₄₆

yes

characteristic

110

C₄×C₈

C₄×D₇₃

yes

characteristic

111

C₄×C₈

C₄×D₇₃

yes

characteristic

112

C₄×C₈

D₂₉₂

yes

characteristic

113

#95

—

114

C₂²×C₄

—

115

C₂²×C₄

C₂²⋊Dic₇₃

yes

characteristic

116

C₂²×C₄

#11924013

yes

characteristic

117

C₂²×C₄

#11924016

yes

characteristic

118

M₁₆

—

584

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

292

—

122

C₂×C₈

C₄×Dic₇₃

yes

characteristic

123

C₂×C₈

#11924005

yes

characteristic

124

C₂×C₈

#11924007

yes

characteristic

125

C₂×C₈

C₄⋊Dic₇₃

yes

characteristic

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄²

—

129

C₄²

#11924005

yes

characteristic

130

C₄²

#11924007

yes

characteristic

131

C₄²

#11924015

yes

characteristic

132

C₄²

#11924018

yes

characteristic

133

C₁₆

—

292

—

134

C₂³

#14556512

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

#14556498

yes

characteristic

143

C₂×C₄

#14556511

yes

characteristic

144

C₂×C₄

#14556514

yes

characteristic

145

C₂×C₄

#14556515

yes

characteristic

146

C₂×C₄

#14556522

yes

characteristic

147

C₂×C₄

#14556524

yes

characteristic

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

584

—

152

C₈

#14556519

yes

—

153

C₂²

—

154

C₂²

—

155

C₂²

#17988349

yes

characteristic

156

C₂²

#17988397

yes

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

#17988347

yes

characteristic

163

C₄

#17988357

yes

characteristic

164

C₄

#17988409

yes

—

165

C₂

—

166

C₂

#22607834

yes

characteristic

167

C₂

#22608029

yes

characteristic

168

C₂

#22608036

yes

characteristic

	subgroup	G/H
center	C₂²	#17988349
commutator	C₂×C₈×C₇₃	C₄²
Frattini	C₂×C₄×C₈	C₂×D₇₃
Fitting	C₄×#106×C₇₃	C₂
socle	C₂²×C₇₃	#340
prime radical	C₂³×C₇₃	#102
cosocle	C₂×C₄×C₈×C₇₃	C₂²
hypercenter	C₄×#106	D₇₃
hypocenter	C₇₃	#7302
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#106

subgroup

G/H

center

C₂²

#17988349

commutator

C₂×C₈×C₇₃

C₄²

Frattini

C₂×C₄×C₈

C₂×D₇₃

Fitting

C₄×#106×C₇₃

C₂

C₂²×C₇₃

C₂³×C₇₃

C₂×C₄×C₈×C₇₃

C₂²

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#106

N	H ≅ G/N	type
5. C₄⋊#11924003	C₄	semidirect
9. #17988338	C₄	semidirect
88. C₇₃	#7302	semidirect

H ≅ G/N

type

5. C₄⋊#11924003

C₄

semidirect

9. #17988338

C₄

semidirect

88. C₇₃

#7302

semidirect

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

subgroup	index
#22611734	2
#22608687	2
C₄×#106×C₇₃	2
#7302	73

subgroup

index

#22611734

#22608687

C₄×#106×C₇₃

#7302

Subgroups — #32973251

Conjugacy classes of subgroups

Complements

Maximal subgroups