Group #34000203 — subgroups

Subgroups — #34000203

Order 18688.

222 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 86 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	C₄×#14556664	C₂	1	1	yes	maximal, characteristic
2	#22625481	C₂	1	1	yes	maximal, characteristic
3	#22608678	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556664	C₄	1	2	yes	—
5	#17989606	—	2	1	no	—
6	#17989473	—	2	1	no	—
7	C₄×#11924031	—	2	1	no	—
8	#17989450	C₄	1	2	yes	—
9	#17989449	—	2	1	no	—
10	#17988639	—	2	1	no	—
11	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
12	#17988340	C₄	1	1	yes	characteristic
13	#17988337	—	2	1	no	—
14	#335×C₇₃	C₄	1	1	yes	characteristic
15	#14556664	—	2	2	no	—
16	#14556664	C₈	1	4	yes	complemented
17	C₂×#11924031	—	2	2	no	—
18	C₄⋊#10596833	—	2	2	no	—
19	#14556648	—	4	1	no	—
20	C₄×#10596833	—	4	1	no	—
21	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×D₇₃	Q₈	1	1	yes	characteristic
23	C₄²×D₇₃	D₄	1	1	yes	characteristic
24	C₄⋊#10596831	—	2	1	no	—
25	#14556503	—	4	1	no	—
26	C₄×#10596831	D₄	1	1	yes	characteristic
27	C₈×Dic₇₃	—	4	1	no	—
28	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₇₃	D₄	1	1	yes	characteristic
30	C₄⋊C₈×C₇₃	—	2	1	no	—
31	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₂×#10596833	—	4	2	no	—
33	#11924031	—	2	4	no	—
34	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×D₇₃	SD₁₆	1	1	yes	characteristic
36	C₂×C₄×D₇₃	SD₁₆	1	1	yes	characteristic
37	C₂×#10596831	—	4	1	no	—
38	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
40	C₈×D₇₃	—	2	2	no	—
41	C₈×D₇₃	C₂×C₈	1	2	yes	—
42	C₈×D₇₃	M₁₆	1	2	yes	—
43	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₄×Dic₇₃	D₈	1	1	yes	characteristic
45	C₄×Dic₇₃	Q₁₆	1	1	yes	characteristic
46	C₂×C₈×C₇₃	—	4	1	no	—
47	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
48	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×D₇₃	#106	1	1	yes	characteristic
51	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
52	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
53	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
54	C₂×Dic₇₃	#102	1	1	yes	characteristic
55	C₂×Dic₇₃	#103	1	1	yes	characteristic
56	C₂×Dic₇₃	#107	1	1	yes	characteristic
57	C₄×D₇₃	—	2	1	no	—
58	C₄×D₇₃	—	2	1	no	—
59	C₄×D₇₃	—	2	2	no	—
60	C₄×D₇₃	C₄⋊C₈	1	2	yes	—
61	C₄×D₇₃	#108	1	2	yes	—
62	#10596833	—	8	2	no	—
63	C₈×C₇₃	—	2	1	no	—
64	C₈×C₇₃	—	8	1	no	—
65	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
66	C₈×C₇₃	#97	1	1	yes	characteristic
67	#10596831	—	2	1	no	—
68	#10596831	—	8	1	no	—
69	#10596831	C₂²⋊C₈	1	2	yes	—
70	C₂²×C₇₃	#340	1	1	yes	characteristic
71	C₂×D₇₃	—	2	1	no	—
72	C₂×D₇₃	—	2	1	no	—
73	C₂×D₇₃	#334	1	2	yes	—
74	C₄×C₇₃	—	2	1	no	—
75	C₄×C₇₃	#336	1	1	yes	characteristic
76	C₄×C₇₃	#341	1	1	yes	characteristic
77	C₄×C₇₃	#359	1	1	yes	characteristic
78	C₄×C₇₃	#359	1	1	yes	characteristic
79	Dic₇₃	—	2	1	no	—
80	Dic₇₃	#335	1	2	yes	—
81	Dic₇₃	#359	1	2	yes	—
82	#7155	—	73	1	no	maximal, Sylow
83	C₂×C₇₃	#1277	1	1	yes	characteristic
84	C₂×C₇₃	#1382	1	1	yes	characteristic
85	C₂×C₇₃	#1382	1	1	yes	characteristic
86	D₇₃	—	2	2	no	—
87	C₂×C₄×C₁₆	—	73	1	no	—
88	#1564	—	73	1	no	—
89	#1564	—	73	1	no	—
90	C₇₃	#7155	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₁₆	—	73	2	no	—
92	#431	—	146	1	no	—
93	#431	—	146	1	no	—
94	C₂×C₄×C₈	—	73	1	no	—
95	C₄×C₁₆	—	146	1	no	—
96	C₄×C₁₆	—	146	1	no	—
97	#335	—	73	2	no	—
98	#335	—	73	1	no	—
99	#335	#64408	1	1	yes	characteristic
100	#334	—	146	1	no	—
101	#334	—	146	1	no	—
102	C₂²×C₈	—	73	1	no	—
103	C₂²×C₈	—	73	1	no	—
104	C₂×C₄²	—	73	1	no	—
105	C₂×C₁₆	—	73	4	no	—
106	C₂×C₁₆	—	146	2	no	—
107	C₂×C₁₆	—	146	2	no	—
108	C₄⋊C₈	—	2	1	no	—
109	C₄⋊C₈	—	146	2	no	—
110	C₄⋊C₈	—	146	1	no	—
111	C₂²⋊C₈	—	292	1	no	—
112	C₂²⋊C₈	—	292	1	no	—
113	C₄×C₈	—	292	1	no	—
114	C₄×C₈	—	292	1	no	—
115	C₄×C₈	—	73	1	no	—
116	C₄×C₈	—	73	1	no	—
117	C₄×C₈	—	73	1	no	—
118	C₄×C₈	C₂×#64408	1	1	yes	characteristic
119	C₂²×C₄	—	73	1	no	—
120	C₂²×C₄	—	73	1	no	—
121	C₂²×C₄	—	73	1	no	—
122	C₂×C₈	—	4	1	no	—
123	C₂×C₈	—	292	2	no	—
124	C₂×C₈	—	292	1	no	—
125	C₂×C₈	—	73	1	no	—
126	C₂×C₈	—	73	2	no	—
127	C₂×C₈	—	73	2	no	—
128	C₂×C₈	—	73	1	no	—
129	C₂×C₈	—	146	2	no	—
130	C₂×C₈	C₄×#64408	1	1	yes	characteristic
131	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
132	C₄²	—	73	1	no	—
133	C₄²	—	73	1	no	—
134	C₄²	—	73	1	no	—
135	C₄²	C₂²⋊#64408	1	1	yes	characteristic
136	C₁₆	—	146	4	no	—
137	C₂³	—	73	1	no	—
138	C₂×C₄	—	73	1	no	—
139	C₂×C₄	—	73	1	no	—
140	C₂×C₄	—	73	2	no	—
141	C₂×C₄	—	73	2	no	—
142	C₂×C₄	—	73	1	no	—
143	C₂×C₄	—	146	2	no	—
144	C₂×C₄	—	146	1	no	—
145	C₂×C₄	—	146	1	no	—
146	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
147	C₂×C₄	#14556653	1	1	yes	characteristic
148	C₂×C₄	#14556654	1	1	yes	characteristic
149	C₈	—	2	1	no	—
150	C₈	—	8	1	no	—
151	C₈	—	584	2	no	—
152	C₈	—	584	1	no	—
153	C₈	—	73	2	no	—
154	C₈	—	146	1	no	—
155	C₈	C₈×#64408	1	1	yes	characteristic
156	C₈	#14556645	1	1	yes	characteristic
157	C₂²	—	73	2	no	—
158	C₂²	—	146	1	no	—
159	C₂²	—	146	1	no	—
160	C₂²	#17989460	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	73	2	no	—
163	C₄	—	73	2	no	—
164	C₄	—	146	1	no	—
165	C₄	#17989452	1	1	yes	characteristic
166	C₄	#17989462	1	1	yes	characteristic
167	C₄	#17989501	1	1	yes	characteristic
168	C₄	#17989502	1	1	yes	characteristic
169	C₂	—	146	2	no	—
170	C₂	#22624760	1	1	yes	characteristic
171	C₂	#22624992	1	1	yes	characteristic
172	C₂	#22624994	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14556664

C₂

yes

maximal, characteristic

#22625481

C₂

yes

maximal, characteristic

#22608678

C₂

yes

maximal, characteristic

C₂×#14556664

C₄

yes

—

—

—

—

C₄

yes

—

#17989449

—

#17988639

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988340

C₄

yes

characteristic

#17988337

—

#335×C₇₃

C₄

yes

characteristic

#14556664

—

#14556664

C₈

yes

—

—

—

—

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₂×C₈×D₇₃

Q₈

yes

characteristic

C₄²×D₇₃

D₄

yes

—

—

D₄

yes

characteristic

C₈×Dic₇₃

—

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

D₄

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×#10596833

—

#11924031

—

C₂×C₄×D₇₃

C₄⋊C₄

yes

characteristic

C₂×C₄×D₇₃

SD₁₆

yes

characteristic

C₂×C₄×D₇₃

SD₁₆

yes

characteristic

C₂×#10596831

—

C₂×#10596831

C₂²⋊C₄

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

C₂×C₈

yes

—

C₈×D₇₃

M₁₆

yes

—

C₄×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₇₃

D₈

yes

characteristic

C₄×Dic₇₃

Q₁₆

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₄²×C₇₃

C₂²⋊C₄

yes

C₂²×D₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₄⋊C₈

yes

—

C₄×D₇₃

#108

yes

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

C₄×C₈

yes

C₈×C₇₃

yes

—

—

C₂²⋊C₈

yes

—

C₂²×C₇₃

#340

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

#334

yes

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

#335

yes

—

Dic₇₃

#359

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

C₂×C₄×C₁₆

—

#1564

—

#1564

—

C₇₃

#7155

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

#431

—

146

—

#431

—

146

—

C₂×C₄×C₈

—

C₄×C₁₆

—

146

—

C₄×C₁₆

—

146

—

—

—

yes

100

—

146

—

101

#334

—

146

—

102

C₂²×C₈

—

103

C₂²×C₈

—

104

C₂×C₄²

—

105

C₂×C₁₆

—

106

C₂×C₁₆

—

146

—

107

C₂×C₁₆

—

146

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

146

—

110

C₄⋊C₈

—

146

—

111

C₂²⋊C₈

—

292

—

112

C₂²⋊C₈

—

292

—

113

C₄×C₈

—

292

—

114

C₄×C₈

—

292

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

C₂×#64408

yes

characteristic

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂×C₈

—

123

C₂×C₈

—

292

—

124

C₂×C₈

—

292

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

146

—

130

C₂×C₈

C₄×#64408

yes

characteristic

131

C₂×C₈

C₄⋊#64408

yes

characteristic

132

C₄²

—

133

C₄²

—

134

C₄²

—

135

C₄²

C₂²⋊#64408

yes

characteristic

136

C₁₆

—

146

—

137

C₂³

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

146

—

144

C₂×C₄

—

146

—

145

C₂×C₄

—

146

—

146

C₂×C₄

(C₂×#64408)⋊C₄

yes

characteristic

147

C₂×C₄

#14556653

yes

characteristic

148

C₂×C₄

#14556654

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

—

584

—

152

C₈

—

584

—

153

C₈

—

154

C₈

—

146

—

155

C₈

C₈×#64408

yes

characteristic

156

C₈

#14556645

yes

characteristic

157

C₂²

—

158

C₂²

—

146

—

159

C₂²

—

146

—

160

C₂²

#17989460

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

146

—

165

C₄

#17989452

yes

characteristic

166

C₄

#17989462

yes

characteristic

167

C₄

#17989501

yes

characteristic

168

C₄

#17989502

yes

characteristic

169

C₂

—

146

—

170

C₂

#22624760

yes

characteristic

171

C₂

#22624992

yes

characteristic

172

C₂

#22624994

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556653
commutator	C₈×C₇₃	C₄×C₈
Frattini	C₄×C₈	C₂×#64408
Fitting	#335×C₇₃	C₄
socle	C₂²×C₇₃	#340
prime radical	C₂²×D₇₃	#106
cosocle	C₄×C₈×D₇₃	C₂²
hypercenter	#335	#64408
hypocenter	C₇₃	#7155
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #335

subgroup

G/H

center

C₂×C₄

#14556653

commutator

C₈×C₇₃

C₄×C₈

Frattini

C₄×C₈

C₂×#64408

Fitting

#335×C₇₃

C₄

C₂²×C₇₃

C₂²×D₇₃

C₄×C₈×D₇₃

C₂²

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #335

N	H ≅ G/N	type
16. #14556664	C₈	semidirect
90. C₇₃	#7155	semidirect

H ≅ G/N

type

16. #14556664

C₈

semidirect

90. C₇₃

#7155

semidirect

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

subgroup	index
C₄×#14556664	2
#22625481	2
#22608678	2
#7155	73

subgroup

index

Subgroups — #34000203

Conjugacy classes of subgroups

Complements

Maximal subgroups