Group #34000234 — subgroups

Subgroups — #34000234

Order 18688.

222 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 70 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	#22626295	C₂	1	1	yes	maximal, characteristic
2	C₈×#11924038	C₂	1	1	yes	maximal, characteristic
3	#346×D₇₃	C₂	1	1	yes	maximal, characteristic
4	#17989606	C₄	1	1	yes	characteristic
5	C₄×#11924038	C₄	1	1	yes	characteristic
6	C₄⋊#11924031	C₄	1	2	yes	—
7	#17989482	—	2	1	no	—
8	#17989476	—	2	1	no	—
9	#17989429	—	2	1	no	—
10	C₈×#10596833	—	2	1	no	—
11	C₂×C₁₆×D₇₃	—	2	1	no	—
12	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
13	#17988363	C₄	1	1	yes	characteristic
14	#17988362	—	2	1	no	—
15	#346×C₇₃	C₄	1	1	yes	characteristic
16	C₂×#11924038	—	2	1	no	—
17	C₂×#11924031	—	2	2	no	—
18	C₄⋊#10596833	—	2	1	no	—
19	#14556648	C₈	1	2	yes	—
20	#14556646	C₈	1	2	yes	—
21	C₄×#10596833	—	2	1	no	—
22	C₄×#10596833	—	2	1	no	—
23	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
24	C₂×C₈×D₇₃	Q₈	1	1	yes	characteristic
25	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂×#11924003	—	2	1	no	—
27	C₁₆×D₇₃	—	2	4	no	—
28	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₇₃	D₄	1	1	yes	characteristic
30	C₈×Dic₇₃	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₇₃	—	2	1	no	—
32	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
33	#11924038	—	2	4	no	—
34	C₂×#10596833	—	2	2	no	—
35	C₂×#10596833	—	2	2	no	—
36	#11924031	—	4	2	no	—
37	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
39	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
40	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
42	C₈×D₇₃	—	2	1	no	—
43	C₈×D₇₃	—	2	1	no	—
44	C₈×D₇₃	—	2	2	no	—
45	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
47	C₄×Dic₇₃	M₁₆	1	1	yes	characteristic
48	#11924003	—	2	2	no	—
49	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₇₃	C₄²	1	1	yes	characteristic
52	C₁₆×C₇₃	—	2	2	no	—
53	C₂²×D₇₃	C₄⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
55	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#97	1	1	yes	characteristic
57	C₂×Dic₇₃	C₂²⋊C₈	1	1	yes	characteristic
58	C₂×Dic₇₃	C₂²⋊C₈	1	1	yes	characteristic
59	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
60	C₄×D₇₃	—	2	2	no	—
61	C₄×D₇₃	—	2	1	no	—
62	C₄×D₇₃	—	2	1	no	—
63	C₄×D₇₃	—	4	1	no	—
64	#10596833	—	2	4	no	—
65	#10596833	—	4	2	no	—
66	C₈×C₇₃	—	2	1	no	—
67	C₈×C₇₃	#108	1	1	yes	characteristic
68	C₈×C₇₃	#108	1	1	yes	characteristic
69	#10596831	—	2	1	no	—
70	#10596831	#104	1	2	yes	—
71	C₂²×C₇₃	#336	1	1	yes	characteristic
72	C₂×D₇₃	—	2	1	no	—
73	C₂×D₇₃	—	2	1	no	—
74	C₂×D₇₃	#364	1	2	yes	—
75	C₄×C₇₃	—	2	1	no	—
76	C₄×C₇₃	—	2	1	no	—
77	C₄×C₇₃	#339	1	1	yes	characteristic
78	C₄×C₇₃	#341	1	1	yes	characteristic
79	Dic₇₃	—	2	1	no	—
80	Dic₇₃	—	2	1	no	—
81	Dic₇₃	—	2	1	no	—
82	#7165	—	73	1	no	maximal, Sylow
83	C₂×C₇₃	#1278	1	1	yes	characteristic
84	C₂×C₇₃	#1376	1	1	yes	characteristic
85	C₂×C₇₃	#1378	1	1	yes	characteristic
86	D₇₃	—	2	2	no	—
87	#2163	—	73	1	no	—
88	C₂×#346	—	73	1	no	—
89	C₈×M₁₆	—	73	1	no	—
90	C₇₃	#7165	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₁₆	—	146	1	no	—
92	#431	—	73	1	no	—
93	C₄×M₁₆	—	73	1	no	—
94	C₂×C₄×C₈	—	73	1	no	—
95	C₄⋊C₁₆	—	73	2	no	—
96	C₂²⋊C₁₆	—	146	1	no	—
97	#346	—	73	1	no	—
98	#346	—	146	1	no	—
99	#346	—	146	1	no	—
100	#346	#64408	1	1	yes	characteristic
101	#322	—	146	1	no	—
102	C₈²	—	146	1	no	—
103	C₂×M₁₆	—	146	1	no	—
104	C₂²×C₈	—	73	1	no	—
105	C₂²×C₈	—	73	1	no	—
106	C₂×C₄²	—	73	1	no	—
107	C₂×C₁₆	—	2	1	no	—
108	C₂×C₁₆	—	146	4	no	—
109	C₂×C₁₆	—	146	2	no	—
110	C₂×C₁₆	—	146	1	no	—
111	C₄⋊C₈	—	146	1	no	—
112	C₂²⋊C₈	—	73	2	no	—
113	#97	—	73	2	no	—
114	C₄×C₈	—	73	1	no	—
115	C₄×C₈	—	73	1	no	—
116	C₄×C₈	—	73	1	no	—
117	C₄×C₈	—	146	1	no	—
118	C₄×C₈	—	146	1	no	—
119	C₄×C₈	C₂×#64408	1	1	yes	characteristic
120	C₂²×C₄	—	73	1	no	—
121	C₂²×C₄	—	73	1	no	—
122	C₂²×C₄	—	73	1	no	—
123	M₁₆	—	146	4	no	—
124	C₂×C₈	—	73	1	no	—
125	C₂×C₈	—	73	1	no	—
126	C₂×C₈	—	146	2	no	—
127	C₂×C₈	—	146	2	no	—
128	C₂×C₈	—	146	2	no	—
129	C₂×C₈	—	146	1	no	—
130	C₂×C₈	—	146	1	no	—
131	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
132	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
133	C₄²	—	73	1	no	—
134	C₄²	—	73	1	no	—
135	C₄²	—	73	1	no	—
136	C₄²	C₄×#64408	1	1	yes	characteristic
137	C₁₆	—	2	2	no	—
138	C₁₆	—	292	2	no	—
139	C₁₆	—	146	2	no	—
140	C₂³	—	73	1	no	—
141	C₂×C₄	—	292	1	no	—
142	C₂×C₄	—	73	1	no	—
143	C₂×C₄	—	73	1	no	—
144	C₂×C₄	—	73	1	no	—
145	C₂×C₄	—	146	2	no	—
146	C₂×C₄	—	146	1	no	—
147	C₂×C₄	—	146	1	no	—
148	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
149	C₂×C₄	C₈×#64408	1	1	yes	characteristic
150	C₂×C₄	#14556645	1	1	yes	characteristic
151	C₈	—	2	1	no	—
152	C₈	—	292	2	no	—
153	C₈	—	73	2	no	—
154	C₈	—	146	4	no	—
155	C₈	—	146	1	no	—
156	C₈	#14556661	1	1	yes	characteristic
157	C₈	#14556662	1	1	yes	characteristic
158	C₂²	—	73	2	no	—
159	C₂²	—	146	1	no	—
160	C₂²	—	146	1	no	—
161	C₂²	#17989452	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	146	1	no	—
165	C₄	—	146	1	no	—
166	C₄	—	146	1	no	—
167	C₄	#17989458	1	1	yes	characteristic
168	C₄	#17989463	1	1	yes	characteristic
169	C₂	—	146	2	no	—
170	C₂	#22624763	1	1	yes	characteristic
171	C₂	#22624974	1	1	yes	characteristic
172	C₂	#22624983	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626295

C₂

yes

maximal, characteristic

C₈×#11924038

C₂

yes

maximal, characteristic

#346×D₇₃

C₂

yes

maximal, characteristic

#17989606

C₄

yes

characteristic

C₄×#11924038

C₄

yes

characteristic

C₄⋊#11924031

C₄

yes

—

—

—

—

—

C₂×C₁₆×D₇₃

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988363

C₄

yes

characteristic

#17988362

—

#346×C₇₃

C₄

yes

—

—

—

C₈

yes

—

#14556646

C₈

yes

—

C₄×#10596833

—

C₄×#10596833

—

C₂×C₈×D₇₃

D₄

yes

characteristic

C₂×C₈×D₇₃

Q₈

yes

characteristic

C₄²×D₇₃

C₂×C₄

yes

characteristic

C₂×#11924003

—

C₁₆×D₇₃

—

C₄×#10596831

C₂×C₄

yes

characteristic

C₈×Dic₇₃

D₄

yes

characteristic

C₈×Dic₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

—

—

—

C₂×C₄×D₇₃

C₄⋊C₄

yes

characteristic

C₂×C₄×D₇₃

C₂×C₈

yes

characteristic

C₂×C₄×D₇₃

M₁₆

yes

characteristic

C₂×#10596831

C₂²⋊C₄

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₈×D₇₃

—

C₄×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₇₃

C₂×C₈

yes

characteristic

C₄×Dic₇₃

M₁₆

yes

characteristic

#11924003

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₄²×C₇₃

C₄²

yes

characteristic

C₁₆×C₇₃

—

C₂²×D₇₃

C₄⋊C₈

yes

characteristic

C₂×C₄×C₇₃

#95

yes

characteristic

C₂×C₄×C₇₃

C₄×C₈

yes

characteristic

C₂×C₄×C₇₃

#97

yes

characteristic

C₂×Dic₇₃

C₂²⋊C₈

yes

characteristic

C₂×Dic₇₃

C₂²⋊C₈

yes

characteristic

C₂×Dic₇₃

C₄⋊C₈

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

#10596833

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

—

yes

—

C₂²×C₇₃

#336

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

#364

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#2163

—

C₂×#346

—

C₈×M₁₆

—

C₇₃

#7165

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

146

—

#431

—

C₄×M₁₆

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

146

—

#346

—

#346

—

146

—

#346

—

146

—

100

yes

101

—

146

—

102

C₈²

—

146

—

103

C₂×M₁₆

—

146

—

104

C₂²×C₈

—

105

C₂²×C₈

—

106

C₂×C₄²

—

107

C₂×C₁₆

—

108

C₂×C₁₆

—

146

—

109

C₂×C₁₆

—

146

—

110

C₂×C₁₆

—

146

—

111

C₄⋊C₈

—

146

—

112

C₂²⋊C₈

—

113

#97

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

—

146

—

118

C₄×C₈

—

146

—

119

C₄×C₈

C₂×#64408

yes

characteristic

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

M₁₆

—

146

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

146

—

127

C₂×C₈

—

146

—

128

C₂×C₈

—

146

—

129

C₂×C₈

—

146

—

130

C₂×C₈

—

146

—

131

C₂×C₈

C₂²⋊#64408

yes

characteristic

132

C₂×C₈

C₄⋊#64408

yes

characteristic

133

C₄²

—

134

C₄²

—

135

C₄²

—

136

C₄²

C₄×#64408

yes

characteristic

137

C₁₆

—

138

C₁₆

—

292

—

139

C₁₆

—

146

—

140

C₂³

—

141

C₂×C₄

—

292

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

—

146

C₂×C₄

—

146

—

147

C₂×C₄

—

146

—

148

C₂×C₄

(C₂×#64408)⋊C₄

yes

characteristic

149

C₂×C₄

C₈×#64408

yes

characteristic

150

C₂×C₄

#14556645

yes

characteristic

151

C₈

—

152

C₈

—

292

—

153

C₈

—

154

C₈

—

146

—

155

C₈

—

146

—

156

C₈

#14556661

yes

characteristic

157

C₈

#14556662

yes

characteristic

158

C₂²

—

159

C₂²

—

146

—

160

C₂²

—

146

—

161

C₂²

#17989452

yes

characteristic

162

C₄

—

163

C₄

—

164

C₄

—

146

—

165

C₄

—

146

—

166

C₄

—

146

—

167

C₄

#17989458

yes

characteristic

168

C₄

#17989463

yes

characteristic

169

C₂

—

146

—

170

C₂

#22624763

yes

characteristic

171

C₂

#22624974

yes

characteristic

172

C₂

#22624983

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×#64408)⋊C₄
commutator	C₂×C₄×C₇₃	C₄×C₈
Frattini	C₄×C₈	C₂×#64408
Fitting	#346×C₇₃	C₄
socle	C₂²×C₇₃	#336
prime radical	C₂²×D₇₃	C₄⋊C₈
cosocle	C₄×C₈×D₇₃	C₂²
hypercenter	#346	#64408
hypocenter	C₇₃	#7165
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346

subgroup

G/H

center

C₂×C₄

(C₂×#64408)⋊C₄

commutator

C₂×C₄×C₇₃

C₄×C₈

Frattini

C₄×C₈

C₂×#64408

Fitting

#346×C₇₃

C₄

socle

C₂²×C₇₃

#336

prime radical

C₂²×D₇₃

C₄⋊C₈

cosocle

C₄×C₈×D₇₃

C₂²

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346

N	H ≅ G/N	type
90. C₇₃	#7165	semidirect

H ≅ G/N

type

90. C₇₃

#7165

semidirect

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

subgroup	index
#22626295	2
C₈×#11924038	2
#346×D₇₃	2
#7165	73

subgroup

index

#22626295

C₈×#11924038

#346×D₇₃

#7165

Subgroups — #34000234

Conjugacy classes of subgroups

Complements

Maximal subgroups