Group #34000664 — subgroups

Subgroups — #34000664

Order 18688.

208 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 64 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	#22626290	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556662	C₂	1	1	yes	maximal, characteristic
3	#22608693	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556662	—	2	1	no	—
5	C₄×#11924038	—	2	1	no	—
6	#17989508	C₄	1	2	yes	—
7	#17989482	—	2	1	no	—
8	C₄×#11924031	—	2	1	no	—
9	#17989463	C₄	1	2	yes	—
10	#17988643	—	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	C₂×#11924038	—	2	2	no	—
16	C₂×#11924031	—	2	2	no	—
17	#14556662	—	2	4	no	—
18	#14556646	—	4	1	no	—
19	C₄×#10596833	—	4	1	no	—
20	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
22	C₄²×D₇₃	D₄	1	1	yes	characteristic
23	C₄⋊#10596831	—	2	1	no	—
24	#14556503	—	4	1	no	—
25	#14556501	—	4	1	no	—
26	C₄×#10596831	D₄	1	1	yes	characteristic
27	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
28	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
29	C₄⋊C₈×C₇₃	—	2	1	no	—
30	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
31	#11924038	—	2	4	no	—
32	#11924038	—	4	2	no	—
33	C₂×#10596833	—	4	2	no	—
34	#11924031	—	4	2	no	—
35	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₄×D₇₃	SD₁₆	1	1	yes	characteristic
37	C₂×C₄×D₇₃	Q₁₆	1	1	yes	characteristic
38	C₂×#10596831	—	4	1	no	—
39	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
41	C₈×D₇₃	—	2	2	no	—
42	C₈×D₇₃	—	4	1	no	—
43	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
44	C₄×Dic₇₃	D₈	1	1	yes	characteristic
45	C₄×Dic₇₃	SD₁₆	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	#103	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	#106	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	1	no	—
60	C₄×D₇₃	—	2	1	no	—
61	C₄×D₇₃	—	4	1	no	—
62	#10596833	—	4	4	no	—
63	C₈×C₇₃	—	2	1	no	—
64	C₈×C₇₃	—	2	1	no	—
65	C₈×C₇₃	—	8	1	no	—
66	#10596831	—	2	1	no	—
67	#10596831	—	8	1	no	—
68	#10596831	#104	1	2	yes	—
69	C₂²×C₇₃	#340	1	1	yes	characteristic
70	C₂×D₇₃	—	2	1	no	—
71	C₂×D₇₃	—	2	1	no	—
72	C₂×D₇₃	—	2	1	no	—
73	C₄×C₇₃	—	2	1	no	—
74	C₄×C₇₃	#339	1	1	yes	characteristic
75	C₄×C₇₃	#344	1	1	yes	characteristic
76	C₄×C₇₃	#359	1	1	yes	characteristic
77	C₄×C₇₃	#359	1	1	yes	characteristic
78	Dic₇₃	—	2	1	no	—
79	Dic₇₃	—	2	1	no	—
80	Dic₇₃	—	2	1	no	—
81	#7297	—	73	1	no	maximal, Sylow
82	C₂×C₇₃	#1303	1	1	yes	characteristic
83	C₂×C₇₃	#1382	1	1	yes	characteristic
84	C₂×C₇₃	#1388	1	1	yes	characteristic
85	D₇₃	—	4	1	no	—
86	#2162	—	73	1	no	—
87	C₄×#108	—	73	1	no	—
88	#1568	—	73	1	no	—
89	C₇₃	#7297	1	1	yes	characteristic, complemented, Sylow
90	#432	—	146	1	no	—
91	C₂×#108	—	146	1	no	—
92	C₄×M₁₆	—	146	1	no	—
93	C₂×C₄×C₈	—	73	1	no	—
94	C₄⋊C₁₆	—	73	2	no	—
95	C₂²⋊C₁₆	—	146	1	no	—
96	C₄×C₁₆	—	146	1	no	—
97	#341	—	73	2	no	—
98	#335	—	73	1	no	—
99	#335	#64408	1	1	yes	characteristic
100	#334	—	146	1	no	—
101	C₂×M₁₆	—	146	2	no	—
102	C₂²×C₈	—	73	1	no	—
103	C₂²×C₈	—	73	1	no	—
104	C₂×C₄²	—	73	1	no	—
105	C₂×C₁₆	—	146	2	no	—
106	#108	—	146	4	no	—
107	C₄⋊C₈	—	2	1	no	—
108	C₄⋊C₈	—	146	1	no	—
109	C₂²⋊C₈	—	292	1	no	—
110	#97	—	292	1	no	—
111	#97	—	292	1	no	—
112	C₄×C₈	—	292	1	no	—
113	C₄×C₈	—	73	1	no	—
114	C₄×C₈	—	73	1	no	—
115	C₄×C₈	—	73	1	no	—
116	C₄×C₈	C₂×#64408	1	1	yes	characteristic
117	C₂²×C₄	—	73	1	no	—
118	C₂²×C₄	—	73	1	no	—
119	C₂²×C₄	—	73	1	no	—
120	M₁₆	—	292	2	no	—
121	M₁₆	—	146	4	no	—
122	C₂×C₈	—	4	1	no	—
123	C₂×C₈	—	292	1	no	—
124	C₂×C₈	—	292	2	no	—
125	C₂×C₈	—	292	1	no	—
126	C₂×C₈	—	73	1	no	—
127	C₂×C₈	—	73	1	no	—
128	C₂×C₈	—	146	2	no	—
129	C₂×C₈	C₄×#64408	1	1	yes	characteristic
130	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
131	C₄²	—	73	1	no	—
132	C₄²	—	73	1	no	—
133	C₄²	—	73	1	no	—
134	C₄²	C₂²⋊#64408	1	1	yes	characteristic
135	C₁₆	—	292	2	no	—
136	C₂³	—	73	1	no	—
137	C₂×C₄	—	292	1	no	—
138	C₂×C₄	—	73	1	no	—
139	C₂×C₄	—	73	1	no	—
140	C₂×C₄	—	73	1	no	—
141	C₂×C₄	—	146	1	no	—
142	C₂×C₄	—	146	1	no	—
143	C₂×C₄	—	146	1	no	—
144	C₂×C₄	—	146	1	no	—
145	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
146	C₂×C₄	#14556653	1	1	yes	characteristic
147	C₂×C₄	#14556654	1	1	yes	characteristic
148	C₈	—	2	1	no	—
149	C₈	—	2	1	no	—
150	C₈	—	8	1	no	—
151	C₈	—	292	4	no	—
152	C₈	—	584	1	no	—
153	C₈	—	73	2	no	—
154	C₈	—	146	1	no	—
155	C₂²	—	146	1	no	—
156	C₂²	—	146	1	no	—
157	C₂²	—	146	1	no	—
158	C₂²	#17989460	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	146	1	no	—
161	C₄	—	146	1	no	—
162	C₄	—	146	1	no	—
163	C₄	#17989459	1	1	yes	characteristic
164	C₄	#17989469	1	1	yes	characteristic
165	C₄	#17989501	1	1	yes	characteristic
166	C₄	#17989502	1	1	yes	characteristic
167	C₂	—	292	1	no	—
168	C₂	#22624827	1	1	yes	characteristic
169	C₂	#22624992	1	1	yes	characteristic
170	C₂	#22625008	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626290

C₂

yes

maximal, characteristic

C₄×#14556662

C₂

yes

maximal, characteristic

#22608693

C₂

yes

maximal, characteristic

C₂×#14556662

—

C₄×#11924038

—

#17989508

C₄

yes

—

#17989482

—

C₄×#11924031

—

#17989463

C₄

yes

—

#17988643

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988340

C₄

yes

characteristic

#17988337

—

#335×C₇₃

C₄

yes

—

—

—

—

—

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₂×C₈×D₇₃

D₄

yes

characteristic

C₄²×D₇₃

D₄

yes

—

—

—

D₄

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

Q₈

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

—

—

—

C₂×C₄×D₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₇₃

SD₁₆

yes

characteristic

C₂×C₄×D₇₃

Q₁₆

yes

characteristic

C₂×#10596831

—

C₂×#10596831

C₂²⋊C₄

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₄×Dic₇₃

C₄⋊C₄

yes

characteristic

C₄×Dic₇₃

D₈

yes

characteristic

C₄×Dic₇₃

SD₁₆

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×D₇₃

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

—

—

yes

—

C₂²×C₇₃

#340

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#2162

—

C₄×#108

—

#1568

—

C₇₃

#7297

yes

characteristic, complemented, Sylow

#432

—

146

—

C₂×#108

—

146

—

C₄×M₁₆

—

146

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

146

—

C₄×C₁₆

—

146

—

—

—

yes

100

—

146

—

101

C₂×M₁₆

—

146

—

102

C₂²×C₈

—

103

C₂²×C₈

—

104

C₂×C₄²

—

105

C₂×C₁₆

—

146

—

106

#108

—

146

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

146

—

109

C₂²⋊C₈

—

292

—

110

#97

—

292

—

111

#97

—

292

—

112

C₄×C₈

—

292

—

113

C₄×C₈

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₄×C₈

C₂×#64408

yes

characteristic

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

M₁₆

—

292

—

121

M₁₆

—

146

—

122

C₂×C₈

—

123

C₂×C₈

—

292

—

124

C₂×C₈

—

292

—

125

C₂×C₈

—

292

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

146

—

129

C₂×C₈

C₄×#64408

yes

characteristic

130

C₂×C₈

C₄⋊#64408

yes

characteristic

131

C₄²

—

132

C₄²

—

133

C₄²

—

134

C₄²

C₂²⋊#64408

yes

characteristic

135

C₁₆

—

292

—

136

C₂³

—

137

C₂×C₄

—

292

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

146

—

142

C₂×C₄

—

146

—

143

C₂×C₄

—

146

—

144

C₂×C₄

—

146

—

145

C₂×C₄

(C₂×#64408)⋊C₄

yes

characteristic

146

C₂×C₄

#14556653

yes

characteristic

147

C₂×C₄

#14556654

yes

characteristic

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

292

—

152

C₈

—

584

—

153

C₈

—

154

C₈

—

146

—

155

C₂²

—

146

—

156

C₂²

—

146

—

157

C₂²

—

146

—

158

C₂²

#17989460

yes

characteristic

159

C₄

—

160

C₄

—

146

—

161

C₄

—

146

—

162

C₄

—

146

—

163

C₄

#17989459

yes

characteristic

164

C₄

#17989469

yes

characteristic

165

C₄

#17989501

yes

characteristic

166

C₄

#17989502

yes

characteristic

167

C₂

—

292

—

168

C₂

#22624827

yes

characteristic

169

C₂

#22624992

yes

characteristic

170

C₂

#22625008

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556653
commutator	C₂×C₈×C₇₃	C₄²
Frattini	C₄×C₈	C₂×#64408
Fitting	#335×C₇₃	C₄
socle	C₂²×C₇₃	#340
prime radical	C₂²×D₇₃	#103
cosocle	C₄×C₈×D₇₃	C₂²
hypercenter	#335	#64408
hypocenter	C₇₃	#7297
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #335

subgroup

G/H

center

C₂×C₄

#14556653

commutator

C₂×C₈×C₇₃

C₄²

Frattini

C₄×C₈

C₂×#64408

Fitting

#335×C₇₃

C₄

C₂²×C₇₃

C₂²×D₇₃

C₄×C₈×D₇₃

C₂²

C₇₃

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

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #335

N	H ≅ G/N	type
89. C₇₃	#7297	semidirect

H ≅ G/N

type

89. C₇₃

#7297

semidirect

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

subgroup	index
#22626290	2
C₄×#14556662	2
#22608693	2
#7297	73

subgroup

index

Subgroups — #34000664

Conjugacy classes of subgroups

Complements

Maximal subgroups