Group #34005430 — subgroups

Subgroups — #34005430

Order 18688.

322 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 112 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	#22625848	C₂	1	1	yes	maximal, characteristic
2	#22625846	C₂	1	1	yes	maximal, characteristic
3	#22624761	C₂	1	4	yes	maximal
4	#22609675	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#10596833	C₂²	1	2	yes	—
6	C₂×C₄⋊#10596833	C₂²	1	2	yes	—
7	#17989453	—	2	2	no	—
8	#17989453	—	2	2	no	—
9	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
10	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
11	#17988342	C₄	1	1	yes	characteristic
12	#17988341	C₄	1	2	yes	—
13	#336×C₇₃	C₄	1	1	yes	characteristic
14	C₂²×#10596833	—	2	2	no	—
15	C₂²×#10596833	—	2	2	no	—
16	C₄⋊#10596833	—	2	4	no	—
17	C₄⋊#10596833	—	2	4	no	—
18	C₂²×#10596831	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₇₃	D₄	1	2	yes	—
21	C₂×C₄×Dic₇₃	Q₈	1	2	yes	—
22	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
23	C₈×Dic₇₃	—	2	2	no	—
24	C₈×Dic₇₃	—	2	2	no	—
25	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂×#10596833	—	4	2	no	—
28	C₂×#10596833	—	4	2	no	—
29	C₂×#10596833	—	4	4	no	—
30	C₂×#10596833	—	4	4	no	—
31	C₂²×Dic₇₃	C₄⋊C₄	1	2	yes	—
32	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
33	C₂²×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
34	C₂×#10596831	—	2	2	no	—
35	C₂×#10596831	—	2	2	no	—
36	C₂×#10596831	—	2	2	no	—
37	C₄×Dic₇₃	—	2	1	no	—
38	C₄×Dic₇₃	—	2	1	no	—
39	C₄×Dic₇₃	—	2	4	no	—
40	C₄×Dic₇₃	—	4	2	no	—
41	C₄×Dic₇₃	#41	1	2	yes	—
42	C₄×Dic₇₃	#41	1	2	yes	—
43	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₇₃	—	2	2	no	—
46	C₂×C₈×C₇₃	—	2	2	no	—
47	C₂×C₈×C₇₃	—	2	2	no	—
48	C₄²×C₇₃	—	2	2	no	—
49	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₇₃	—	2	2	no	—
51	C₂×C₄×C₇₃	—	2	2	no	—
52	C₂×C₄×C₇₃	—	2	2	no	—
53	C₂×C₄×C₇₃	#117	1	2	yes	—
54	C₂×C₄×C₇₃	#131	1	2	yes	—
55	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
57	C₂×Dic₇₃	—	2	2	no	—
58	C₂×Dic₇₃	—	2	2	no	—
59	C₂×Dic₇₃	—	2	2	no	—
60	C₂×Dic₇₃	—	2	2	no	—
61	C₂×Dic₇₃	#124	1	1	yes	characteristic
62	C₂×Dic₇₃	#124	1	1	yes	characteristic
63	C₂×Dic₇₃	#125	1	2	yes	—
64	C₂×Dic₇₃	#135	1	4	yes	—
65	#10596833	—	8	2	no	—
66	#10596833	—	8	2	no	—
67	C₈×C₇₃	—	4	2	no	—
68	#10596831	—	4	2	no	—
69	C₂²×C₇₃	#382	1	1	yes	characteristic
70	C₂²×C₇₃	#424	1	1	yes	characteristic
71	C₂²×C₇₃	#424	1	1	yes	characteristic
72	C₂²×C₇₃	#427	1	2	yes	—
73	C₂²×C₇₃	#430	1	2	yes	—
74	C₄×C₇₃	—	4	2	no	—
75	C₄×C₇₃	#433	1	2	yes	—
76	C₄×C₇₃	#433	1	2	yes	—
77	Dic₇₃	—	2	2	no	—
78	Dic₇₃	—	4	2	no	—
79	Dic₇₃	#488	1	2	yes	—
80	Dic₇₃	#488	1	2	yes	—
81	#8788	—	73	1	no	maximal, Sylow
82	C₂×C₇₃	#1572	1	2	yes	—
83	C₂×C₇₃	#1572	1	2	yes	—
84	C₂×C₇₃	#1832	1	1	yes	characteristic
85	C₂×C₇₃	#1835	1	1	yes	characteristic
86	C₂×C₇₃	#1838	1	1	yes	characteristic
87	#1831	—	73	1	no	—
88	#1831	—	73	1	no	—
89	#1744	—	73	1	no	—
90	#1277	—	73	4	no	—
91	C₇₃	#8788	1	1	yes	characteristic, complemented, Sylow
92	C₂×C₄⋊C₈	—	73	2	no	—
93	C₂×C₄⋊C₈	—	73	2	no	—
94	C₂×C₄×C₈	—	73	2	no	—
95	C₄³	—	73	1	no	—
96	#336	—	73	2	no	—
97	#336	—	73	1	no	—
98	#336	—	146	2	no	—
99	#336	—	146	2	no	—
100	#336	#64408	1	1	yes	characteristic
101	C₂²×C₈	—	73	2	no	—
102	C₂²×C₈	—	146	2	no	—
103	C₂²×C₈	—	146	2	no	—
104	C₂²×C₈	C₂×#64408	1	2	yes	—
105	C₂×C₄²	—	73	2	no	—
106	C₂×C₄²	—	73	1	no	—
107	C₂×C₄²	—	73	2	no	—
108	C₂×C₄²	—	73	1	no	—
109	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
110	C₄⋊C₈	—	146	4	no	—
111	C₄⋊C₈	—	146	4	no	—
112	C₄×C₈	—	146	2	no	—
113	C₄×C₈	—	146	2	no	—
114	C₂²×C₄	—	73	1	no	—
115	C₂²×C₄	—	73	1	no	—
116	C₂²×C₄	—	73	2	no	—
117	C₂²×C₄	C₄⋊#64408	1	2	yes	—
118	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	292	2	no	—
123	C₂×C₈	—	292	2	no	—
124	C₂×C₈	—	292	4	no	—
125	C₂×C₈	—	292	4	no	—
126	C₂×C₈	—	146	2	no	—
127	C₂×C₈	—	146	2	no	—
128	C₂×C₈	—	146	2	no	—
129	C₄²	—	2	2	no	—
130	C₄²	—	292	2	no	—
131	C₄²	—	73	2	no	—
132	C₄²	—	73	2	no	—
133	C₄²	—	146	4	no	—
134	C₄²	—	146	1	no	—
135	C₄²	—	146	1	no	—
136	C₂³	C₂×C₄⋊#64408	1	1	yes	characteristic
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	73	1	no	—
141	C₂×C₄	—	73	1	no	—
142	C₂×C₄	—	73	4	no	—
143	C₂×C₄	—	73	2	no	—
144	C₂×C₄	—	146	2	no	—
145	C₂×C₄	—	146	2	no	—
146	C₂×C₄	—	146	2	no	—
147	C₂×C₄	—	146	2	no	—
148	C₂×C₄	#14556670	1	2	yes	—
149	C₂×C₄	#14556678	1	1	yes	characteristic
150	C₂×C₄	#14556678	1	2	yes	—
151	C₂×C₄	#14556679	1	1	yes	characteristic
152	C₈	—	4	2	no	—
153	C₈	—	292	2	no	—
154	C₈	—	584	2	no	—
155	C₈	—	584	2	no	—
156	C₂²	#17989531	1	1	yes	characteristic
157	C₂²	#17989593	1	1	yes	characteristic
158	C₂²	#17989594	1	1	yes	characteristic
159	C₂²	#17989597	1	2	yes	—
160	C₂²	#17989602	1	2	yes	—
161	C₄	—	4	2	no	—
162	C₄	—	292	2	no	—
163	C₄	—	73	2	no	—
164	C₄	—	73	2	no	—
165	C₄	—	146	2	no	—
166	C₄	#17989611	1	2	yes	—
167	C₄	#17989612	1	2	yes	—
168	C₂	#22625502	1	2	yes	—
169	C₂	#22625503	1	2	yes	—
170	C₂	#22625849	1	1	yes	characteristic
171	C₂	#22625853	1	1	yes	characteristic
172	C₂	#22625857	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22625848

C₂

yes

maximal, characteristic

#22625846

C₂

yes

maximal, characteristic

#22624761

C₂

yes

maximal

#22609675

C₂

yes

maximal, characteristic

C₂×C₄⋊#10596833

C₂²

yes

—

C₂×C₄⋊#10596833

C₂²

yes

—

#17989453

—

#17989453

—

C₂×C₈×Dic₇₃

C₂²

yes

—

C₄²×Dic₇₃

C₂²

yes

characteristic

#17988342

C₄

yes

characteristic

#17988341

C₄

yes

—

#336×C₇₃

C₄

yes

—

—

—

—

C₂×C₄

yes

—

C₂×C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₇₃

D₄

yes

—

C₂×C₄×Dic₇₃

Q₈

yes

—

C₂×C₄×Dic₇₃

C₂³

yes

characteristic

C₈×Dic₇₃

—

C₈×Dic₇₃

—

C₂²×C₈×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂×C₄

yes

—

—

—

—

C₂²×Dic₇₃

C₄⋊C₄

yes

—

C₂²×Dic₇₃

C₂×D₄

yes

characteristic

C₂²×Dic₇₃

C₂×Q₈

yes

—

—

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

#41

yes

—

C₄×Dic₇₃

#41

yes

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#117

yes

—

C₂×C₄×C₇₃

#131

yes

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

—

C₂×Dic₇₃

#135

yes

—

#10596833

—

#10596833

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₂²×C₇₃

#430

yes

—

C₄×C₇₃

—

C₄×C₇₃

#433

yes

—

C₄×C₇₃

#433

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#488

yes

—

Dic₇₃

#488

yes

—

#8788

—

maximal, Sylow

C₂×C₇₃

#1572

yes

—

C₂×C₇₃

#1572

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

—

—

C₇₃

#8788

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₄³

—

#336

—

#336

—

#336

—

146

—

#336

—

146

—

100

#336

#64408

yes

characteristic

101

C₂²×C₈

—

102

C₂²×C₈

—

146

—

103

C₂²×C₈

—

146

—

104

C₂²×C₈

C₂×#64408

yes

—

105

C₂×C₄²

—

106

C₂×C₄²

—

107

C₂×C₄²

—

108

C₂×C₄²

—

109

C₂×C₄²

C₂×#64408

yes

characteristic

110

C₄⋊C₈

—

146

—

111

C₄⋊C₈

—

146

—

112

C₄×C₈

—

146

—

113

C₄×C₈

—

146

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

C₄⋊#64408

yes

—

118

C₂²×C₄

C₂²×#64408

yes

characteristic

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

292

—

123

C₂×C₈

—

292

—

124

C₂×C₈

—

292

—

125

C₂×C₈

—

292

—

126

C₂×C₈

—

146

—

127

C₂×C₈

—

146

—

128

C₂×C₈

—

146

—

129

C₄²

—

130

C₄²

—

292

—

131

C₄²

—

132

C₄²

—

133

C₄²

—

146

—

134

C₄²

—

146

—

135

C₄²

—

146

—

136

C₂³

C₂×C₄⋊#64408

yes

characteristic

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

146

—

145

C₂×C₄

—

146

—

146

C₂×C₄

—

146

—

147

C₂×C₄

—

146

—

148

C₂×C₄

#14556670

yes

—

149

C₂×C₄

#14556678

yes

characteristic

150

C₂×C₄

#14556678

yes

—

151

C₂×C₄

#14556679

yes

characteristic

152

C₈

—

153

C₈

—

292

—

154

C₈

—

584

—

155

C₈

—

584

—

156

C₂²

#17989531

yes

characteristic

157

C₂²

#17989593

yes

characteristic

158

C₂²

#17989594

yes

characteristic

159

C₂²

#17989597

yes

—

160

C₂²

#17989602

yes

—

161

C₄

—

162

C₄

—

292

—

163

C₄

—

164

C₄

—

165

C₄

—

146

—

166

C₄

#17989611

yes

—

167

C₄

#17989612

yes

—

168

C₂

#22625502

yes

—

169

C₂

#22625503

yes

—

170

C₂

#22625849

yes

characteristic

171

C₂

#22625853

yes

characteristic

172

C₂

#22625857

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#64408
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#64408
Fitting	#336×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	#336	#64408
hypocenter	C₇₃	#8788
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×C₄⋊#64408

commutator

C₂²×C₄×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×#64408

Fitting

#336×C₇₃

C₄

socle, prime radical

C₂³×C₇₃

C₂×C₄⋊C₄

cosocle

C₂×C₄×Dic₇₃

C₂³

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
91. C₇₃	#8788	semidirect

H ≅ G/N

type

91. C₇₃

#8788

semidirect

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

subgroup	index
#22625848	2
#22625846	2
#22624761	2
#22609675	2
#8788	73

subgroup

index

Subgroups — #34005430

Conjugacy classes of subgroups

Complements

Maximal subgroups