Group #34003354 — subgroups

Subgroups — #34003354

Order 18688.

414 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 144 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	#22625645	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556646	C₂	1	1	yes	maximal, characteristic
3	#22624753	C₂	1	4	yes	maximal
4	#22609675	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556646	C₄	1	4	yes	—
6	C₂×C₄×#10596833	C₂²	1	2	yes	—
7	C₂×C₄×#10596833	C₂²	1	2	yes	—
8	#17989453	C₄	1	4	yes	—
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	C₂×C₄	1	4	yes	—
15	C₂²×#10596833	C₂×C₄	1	4	yes	—
16	#14556646	—	2	8	no	—
17	C₄×#10596833	—	2	4	no	—
18	C₄×#10596833	—	2	4	no	—
19	C₂²×#10596831	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
22	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
24	C₈×Dic₇₃	—	2	4	no	—
25	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂×#10596833	—	2	4	no	—
28	C₂×#10596833	—	2	4	no	—
29	C₂×#10596833	—	2	4	no	—
30	C₂×#10596833	—	2	8	no	—
31	C₂×#10596833	M₁₆	1	8	yes	—
32	C₂²×Dic₇₃	C₄²	1	2	yes	—
33	C₂²×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×#10596831	—	2	2	no	—
36	C₂×#10596831	—	2	2	no	—
37	C₂×#10596831	—	2	2	no	—
38	C₄×Dic₇₃	—	2	1	no	—
39	C₄×Dic₇₃	—	2	1	no	—
40	C₄×Dic₇₃	—	2	4	no	—
41	C₄×Dic₇₃	—	2	4	no	—
42	C₄×Dic₇₃	#41	1	4	yes	—
43	C₂²×C₄×C₇₃	C₄²	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×C₄²	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	#117	1	2	yes	—
61	C₂×Dic₇₃	#117	1	2	yes	—
62	C₂×Dic₇₃	C₂×M₁₆	1	4	yes	—
63	C₂×Dic₇₃	#131	1	4	yes	—
64	#10596833	—	2	8	no	—
65	#10596833	—	4	4	no	—
66	C₈×C₇₃	—	4	2	no	—
67	#10596831	—	4	2	no	—
68	C₂²×C₇₃	#376	1	1	yes	characteristic
69	C₂²×C₇₃	C₄×M₁₆	1	2	yes	—
70	C₂²×C₇₃	#405	1	2	yes	—
71	C₂²×C₇₃	#405	1	1	yes	characteristic
72	C₂²×C₇₃	#405	1	1	yes	characteristic
73	C₄×C₇₃	—	4	2	no	—
74	C₄×C₇₃	#433	1	2	yes	—
75	C₄×C₇₃	#433	1	2	yes	—
76	Dic₇₃	—	4	2	no	—
77	Dic₇₃	#432	1	4	yes	—
78	Dic₇₃	#433	1	4	yes	—
79	#8134	—	73	1	no	maximal, Sylow
80	C₂×C₇₃	#1453	1	2	yes	—
81	C₂×C₇₃	#1453	1	2	yes	—
82	C₂×C₇₃	#1745	1	1	yes	characteristic
83	C₂×C₇₃	#1746	1	1	yes	characteristic
84	C₂×C₇₃	#1746	1	1	yes	characteristic
85	#1744	—	73	1	no	—
86	#1744	—	73	1	no	—
87	C₄×#97	—	73	1	no	—
88	#1273	—	73	4	no	—
89	C₇₃	#8134	1	1	yes	characteristic, complemented, Sylow
90	C₂×#97	—	73	4	no	—
91	C₂×C₄×C₈	—	73	2	no	—
92	C₂×C₄×C₈	—	73	2	no	—
93	C₂×C₄×C₈	—	73	2	no	—
94	C₄³	—	73	1	no	—
95	#336	—	73	4	no	—
96	#336	—	73	2	no	—
97	#336	—	73	1	no	—
98	#336	#64408	1	1	yes	characteristic
99	C₂²×C₈	—	73	4	no	—
100	C₂²×C₈	—	73	4	no	—
101	C₂²×C₈	—	73	2	no	—
102	C₂²×C₈	C₂×#64408	1	2	yes	—
103	C₂×C₄²	—	73	1	no	—
104	C₂×C₄²	—	73	2	no	—
105	C₂×C₄²	—	73	2	no	—
106	C₂×C₄²	—	73	1	no	—
107	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
108	#97	—	146	8	no	—
109	C₄×C₈	—	146	4	no	—
110	C₄×C₈	—	146	4	no	—
111	C₄×C₈	—	146	4	no	—
112	C₂²×C₄	—	73	1	no	—
113	C₂²×C₄	—	73	1	no	—
114	C₂²×C₄	—	73	2	no	—
115	C₂²×C₄	C₄×#64408	1	2	yes	—
116	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	73	8	no	—
121	C₂×C₈	—	146	4	no	—
122	C₂×C₈	—	146	8	no	—
123	C₂×C₈	—	146	2	no	—
124	C₂×C₈	—	146	2	no	—
125	C₂×C₈	—	146	2	no	—
126	C₂×C₈	—	146	4	no	—
127	C₂×C₈	—	146	4	no	—
128	C₄²	—	2	2	no	—
129	C₄²	—	73	4	no	—
130	C₄²	—	146	4	no	—
131	C₄²	—	146	4	no	—
132	C₄²	—	146	1	no	—
133	C₄²	—	146	1	no	—
134	C₂³	C₂×C₄×#64408	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	73	2	no	—
139	C₂×C₄	—	73	2	no	—
140	C₂×C₄	—	73	4	no	—
141	C₂×C₄	—	73	4	no	—
142	C₂×C₄	—	146	2	no	—
143	C₂×C₄	—	146	2	no	—
144	C₂×C₄	—	146	2	no	—
145	C₂×C₄	#14556670	1	2	yes	—
146	C₂×C₄	#14556678	1	1	yes	characteristic
147	C₂×C₄	#14556678	1	2	yes	—
148	C₂×C₄	#14556679	1	1	yes	characteristic
149	C₈	—	4	2	no	—
150	C₈	—	292	4	no	—
151	C₈	—	292	2	no	—
152	C₈	—	146	8	no	—
153	C₂²	#17989523	1	1	yes	characteristic
154	C₂²	M₁₆×#64408	1	2	yes	—
155	C₂²	#17989550	1	2	yes	—
156	C₂²	#17989551	1	1	yes	characteristic
157	C₂²	#17989552	1	1	yes	characteristic
158	C₄	—	4	2	no	—
159	C₄	—	292	2	no	—
160	C₄	—	73	4	no	—
161	C₄	—	73	4	no	—
162	C₄	#17989611	1	2	yes	—
163	C₄	#17989612	1	2	yes	—
164	C₂	#22625148	1	2	yes	—
165	C₂	#22625149	1	2	yes	—
166	C₂	#22625647	1	1	yes	characteristic
167	C₂	#22625649	1	1	yes	characteristic
168	C₂	#22625650	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22625645

C₂

yes

maximal, characteristic

C₄×#14556646

C₂

yes

maximal, characteristic

#22624753

C₂

yes

maximal

#22609675

C₂

yes

maximal, characteristic

C₂×#14556646

C₄

yes

—

C₂×C₄×#10596833

C₂²

yes

—

C₂×C₄×#10596833

C₂²

yes

—

#17989453

C₄

yes

—

C₂×C₈×Dic₇₃

C₂²

yes

—

C₄²×Dic₇₃

C₂²

yes

characteristic

#17988342

C₄

yes

characteristic

#17988341

C₄

yes

—

#336×C₇₃

C₄

yes

characteristic

C₂²×#10596833

C₂×C₄

yes

—

C₂²×#10596833

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

—

C₂×C₄×Dic₇₃

C₂×C₄

yes

—

C₂×C₄×Dic₇₃

C₂×C₄

yes

—

C₂×C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₇₃

C₂³

yes

characteristic

C₈×Dic₇₃

—

C₂²×C₈×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂×C₄

yes

—

—

—

—

M₁₆

yes

—

C₂²×Dic₇₃

C₄²

yes

—

C₂²×Dic₇₃

C₂²×C₄

yes

characteristic

C₂²×Dic₇₃

C₂²×C₄

yes

—

—

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

#41

yes

—

C₂²×C₄×C₇₃

C₄²

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

C₂×C₄²

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

#117

yes

—

C₂×Dic₇₃

#117

yes

—

C₂×Dic₇₃

C₂×M₁₆

yes

—

C₂×Dic₇₃

#131

yes

—

#10596833

—

#10596833

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

#376

yes

characteristic

C₂²×C₇₃

C₄×M₁₆

yes

—

C₂²×C₇₃

#405

yes

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

#433

yes

—

C₄×C₇₃

#433

yes

—

Dic₇₃

—

Dic₇₃

#432

yes

—

Dic₇₃

#433

yes

—

#8134

—

maximal, Sylow

C₂×C₇₃

#1453

yes

—

C₂×C₇₃

#1453

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

C₄×#97

—

#1273

—

C₇₃

#8134

yes

characteristic, complemented, Sylow

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄³

—

—

—

—

yes

C₂²×C₈

—

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂²×C₈

C₂×#64408

yes

—

103

C₂×C₄²

—

104

C₂×C₄²

—

105

C₂×C₄²

—

106

C₂×C₄²

—

107

C₂×C₄²

C₂×#64408

yes

characteristic

108

#97

—

146

—

109

C₄×C₈

—

146

—

110

C₄×C₈

—

146

—

111

C₄×C₈

—

146

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

C₄×#64408

yes

—

116

C₂²×C₄

C₂²×#64408

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

146

—

122

C₂×C₈

—

146

—

123

C₂×C₈

—

146

—

124

C₂×C₈

—

146

—

125

C₂×C₈

—

146

—

126

C₂×C₈

—

146

—

127

C₂×C₈

—

146

—

128

C₄²

—

129

C₄²

—

130

C₄²

—

146

—

131

C₄²

—

146

—

132

C₄²

—

146

—

133

C₄²

—

146

—

134

C₂³

C₂×C₄×#64408

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

—

146

—

143

C₂×C₄

—

146

—

144

C₂×C₄

—

146

—

145

C₂×C₄

#14556670

yes

—

146

C₂×C₄

#14556678

yes

characteristic

147

C₂×C₄

#14556678

yes

—

148

C₂×C₄

#14556679

yes

characteristic

149

C₈

—

150

C₈

—

292

—

151

C₈

—

292

—

152

C₈

—

146

—

153

C₂²

#17989523

yes

characteristic

154

C₂²

M₁₆×#64408

yes

—

155

C₂²

#17989550

yes

—

156

C₂²

#17989551

yes

characteristic

157

C₂²

#17989552

yes

characteristic

158

C₄

—

159

C₄

—

292

—

160

C₄

—

161

C₄

—

162

C₄

#17989611

yes

—

163

C₄

#17989612

yes

—

164

C₂

#22625148

yes

—

165

C₂

#22625149

yes

—

166

C₂

#22625647

yes

characteristic

167

C₂

#22625649

yes

characteristic

168

C₂

#22625650

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#64408
commutator, socle, prime radical	C₂³×C₇₃	C₂×C₄²
Fitting	#336×C₇₃	C₄
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	#336	#64408
hypocenter	C₇₃	#8134
LCS	G ▷ C₂³×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×#64408

commutator, socle, prime radical

C₂³×C₇₃

C₂×C₄²

Fitting

#336×C₇₃

C₄

cosocle

C₂×C₄×Dic₇₃

C₂³

C₇₃

G ▷ C₂³×C₇₃ ▷ C₇₃

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
89. C₇₃	#8134	semidirect

H ≅ G/N

type

89. C₇₃

#8134

semidirect

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

subgroup	index
#22625645	2
C₄×#14556646	2
#22624753	2
#22609675	2
#8134	73

subgroup

index

Subgroups — #34003354

Conjugacy classes of subgroups

Complements

Maximal subgroups