Group #34000668 — subgroups

Subgroups — #34000668

Order 18688.

216 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 68 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×#14556662	C₂	1	1	yes	maximal, characteristic
2	#22625489	C₂	1	1	yes	maximal, characteristic
3	#22611729	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556662	—	2	1	no	—
5	#17989591	—	2	1	no	—
6	C₄×#11924038	—	2	1	no	—
7	#17989463	C₄	1	2	yes	—
8	#17989450	—	2	1	no	—
9	#17989450	C₄	1	2	yes	—
10	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
11	C₄⋊#11924003	C₄	1	1	yes	characteristic
12	#17988366	—	2	1	no	—
13	C₁₆×Dic₇₃	—	2	1	no	—
14	C₄⋊C₁₆×C₇₃	C₄	1	1	yes	characteristic
15	C₂×#11924038	—	2	2	no	—
16	C₂×#11924038	—	4	1	no	—
17	#14556662	—	2	2	no	—
18	#14556662	—	2	2	no	—
19	C₄⋊#10596833	—	2	2	no	—
20	C₄⋊#10596833	—	4	1	no	—
21	#14556646	—	4	1	no	—
22	C₄×#10596833	—	4	1	no	—
23	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
25	C₄²×D₇₃	D₄	1	1	yes	characteristic
26	C₂×#11924003	—	2	1	no	—
27	C₄×#10596831	Q₈	1	1	yes	characteristic
28	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₇₃	D₄	1	1	yes	characteristic
30	C₂×C₁₆×C₇₃	—	2	1	no	—
31	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
32	#11924038	—	2	4	no	—
33	#11924038	—	4	2	no	—
34	#11924038	—	8	2	no	—
35	C₂×#10596833	—	4	2	no	—
36	C₂×#10596833	—	4	2	no	—
37	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×D₇₃	SD₁₆	1	1	yes	characteristic
39	C₂×C₄×D₇₃	Q₁₆	1	1	yes	characteristic
40	C₂×#10596831	C₄⋊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	—	4	1	no	—
45	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₄×Dic₇₃	D₈	1	1	yes	characteristic
47	C₄×Dic₇₃	SD₁₆	1	1	yes	characteristic
48	#11924003	—	4	1	no	—
49	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
50	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
52	C₁₆×C₇₃	—	4	1	no	—
53	C₂²×D₇₃	#103	1	1	yes	characteristic
54	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
55	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
57	C₂×Dic₇₃	#102	1	1	yes	characteristic
58	C₂×Dic₇₃	#102	1	1	yes	characteristic
59	C₂×Dic₇₃	#103	1	1	yes	characteristic
60	C₄×D₇₃	—	2	1	no	—
61	C₄×D₇₃	—	2	1	no	—
62	C₄×D₇₃	—	2	1	no	—
63	C₄×D₇₃	—	2	1	no	—
64	C₄×D₇₃	—	2	1	no	—
65	C₄×D₇₃	—	2	1	no	—
66	#10596833	—	4	4	no	—
67	#10596833	—	8	2	no	—
68	C₈×C₇₃	—	2	1	no	—
69	C₈×C₇₃	#104	1	1	yes	characteristic
70	C₈×C₇₃	#104	1	1	yes	characteristic
71	#10596831	—	2	1	no	—
72	#10596831	—	2	1	no	—
73	C₂²×C₇₃	#340	1	1	yes	characteristic
74	C₂×D₇₃	—	2	1	no	—
75	C₂×D₇₃	—	2	1	no	—
76	C₂×D₇₃	—	2	1	no	—
77	C₄×C₇₃	—	2	1	no	—
78	C₄×C₇₃	#339	1	1	yes	characteristic
79	C₄×C₇₃	#344	1	1	yes	characteristic
80	C₄×C₇₃	#368	1	1	yes	characteristic
81	C₄×C₇₃	#368	1	1	yes	characteristic
82	Dic₇₃	—	2	1	no	—
83	Dic₇₃	—	2	1	no	—
84	Dic₇₃	#361	1	1	yes	characteristic
85	Dic₇₃	#362	1	1	yes	characteristic
86	#7298	—	73	1	no	maximal, Sylow
87	C₂×C₇₃	#1303	1	1	yes	characteristic
88	C₂×C₇₃	#1381	1	1	yes	characteristic
89	C₂×C₇₃	#1387	1	1	yes	characteristic
90	D₇₃	—	4	1	no	—
91	#2162	—	73	1	no	—
92	C₄×#108	—	73	1	no	—
93	#1567	—	73	1	no	—
94	C₇₃	#7298	1	1	yes	characteristic, complemented, Sylow
95	C₂×#108	—	146	1	no	—
96	#423	—	146	1	no	—
97	C₄×M₁₆	—	146	1	no	—
98	C₂×C₄×C₈	—	73	1	no	—
99	C₄⋊C₁₆	—	73	1	no	—
100	C₄⋊C₁₆	#64408	1	1	yes	characteristic
101	C₂²⋊C₁₆	—	146	1	no	—
102	C₄×C₁₆	—	146	1	no	—
103	#341	—	73	2	no	—
104	#335	—	73	2	no	—
105	#335	—	146	1	no	—
106	C₂×M₁₆	—	292	1	no	—
107	C₂×M₁₆	—	146	2	no	—
108	C₂²×C₈	—	73	1	no	—
109	C₂²×C₈	—	73	1	no	—
110	C₂×C₄²	—	73	1	no	—
111	C₂×C₁₆	—	2	1	no	—
112	C₂×C₁₆	—	146	1	no	—
113	#108	—	146	2	no	—
114	#108	—	146	2	no	—
115	C₄⋊C₈	—	292	1	no	—
116	C₄⋊C₈	—	146	2	no	—
117	#97	—	292	1	no	—
118	C₄×C₈	—	292	1	no	—
119	C₄×C₈	—	73	1	no	—
120	C₄×C₈	—	73	1	no	—
121	C₄×C₈	—	73	1	no	—
122	C₄×C₈	C₂×#64408	1	1	yes	characteristic
123	C₂²×C₄	—	73	1	no	—
124	C₂²×C₄	—	73	1	no	—
125	C₂²×C₄	—	73	1	no	—
126	M₁₆	—	292	2	no	—
127	M₁₆	—	584	2	no	—
128	M₁₆	—	146	4	no	—
129	C₂×C₈	—	292	1	no	—
130	C₂×C₈	—	292	2	no	—
131	C₂×C₈	—	292	2	no	—
132	C₂×C₈	—	73	1	no	—
133	C₂×C₈	—	73	1	no	—
134	C₂×C₈	—	146	1	no	—
135	C₂×C₈	—	146	1	no	—
136	C₂×C₈	C₄×#64408	1	1	yes	characteristic
137	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
138	C₄²	—	73	1	no	—
139	C₄²	—	73	1	no	—
140	C₄²	—	73	1	no	—
141	C₄²	C₄⋊#64408	1	1	yes	characteristic
142	C₁₆	—	4	1	no	—
143	C₁₆	—	292	1	no	—
144	C₂³	—	73	1	no	—
145	C₂×C₄	—	73	1	no	—
146	C₂×C₄	—	73	1	no	—
147	C₂×C₄	—	73	1	no	—
148	C₂×C₄	—	146	1	no	—
149	C₂×C₄	—	146	1	no	—
150	C₂×C₄	—	146	1	no	—
151	C₂×C₄	—	146	1	no	—
152	C₂×C₄	—	146	1	no	—
153	C₂×C₄	—	146	1	no	—
154	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
155	C₂×C₄	#14556659	1	1	yes	characteristic
156	C₂×C₄	#14556660	1	1	yes	characteristic
157	C₈	—	2	1	no	—
158	C₈	—	292	4	no	—
159	C₈	—	584	2	no	—
160	C₈	—	146	1	no	—
161	C₈	—	146	1	no	—
162	C₈	#14556655	1	1	yes	characteristic
163	C₈	#14556656	1	1	yes	characteristic
164	C₂²	—	146	1	no	—
165	C₂²	—	146	1	no	—
166	C₂²	—	146	1	no	—
167	C₂²	#17989461	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	73	1	no	—
170	C₄	—	73	1	no	—
171	C₄	—	146	1	no	—
172	C₄	—	146	1	no	—
173	C₄	#17989457	1	1	yes	characteristic
174	C₄	#17989470	1	1	yes	characteristic
175	C₄	#17989515	1	1	yes	characteristic
176	C₄	#17989516	1	1	yes	characteristic
177	C₂	—	292	1	no	—
178	C₂	#22624828	1	1	yes	characteristic
179	C₂	#22624991	1	1	yes	characteristic
180	C₂	#22625006	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14556662

C₂

yes

maximal, characteristic

#22625489

C₂

yes

maximal, characteristic

#22611729

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

—

#17989450

—

#17989450

C₄

yes

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

C₄⋊#11924003

C₄

yes

characteristic

#17988366

—

C₁₆×Dic₇₃

—

C₄⋊C₁₆×C₇₃

C₄

yes

—

—

—

—

—

—

—

—

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₂×C₈×D₇₃

D₄

yes

characteristic

C₄²×D₇₃

D₄

yes

characteristic

C₂×#11924003

—

C₄×#10596831

Q₈

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

D₄

yes

characteristic

C₂×C₁₆×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

C₄⋊C₄

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₈×D₇₃

—

C₄×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₇₃

D₈

yes

characteristic

C₄×Dic₇₃

SD₁₆

yes

characteristic

#11924003

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₄²×C₇₃

C₄⋊C₄

yes

characteristic

C₁₆×C₇₃

—

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

—

C₄×D₇₃

—

#10596833

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

yes

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

yes

Dic₇₃

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#2162

—

C₄×#108

—

#1567

—

C₇₃

#7298

yes

characteristic, complemented, Sylow

C₂×#108

—

146

—

#423

—

146

—

C₄×M₁₆

—

146

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

100

C₄⋊C₁₆

#64408

yes

characteristic

101

C₂²⋊C₁₆

—

146

—

102

C₄×C₁₆

—

146

—

103

#341

—

104

#335

—

105

#335

—

146

—

106

C₂×M₁₆

—

292

—

107

C₂×M₁₆

—

146

—

108

C₂²×C₈

—

109

C₂²×C₈

—

110

C₂×C₄²

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

146

—

113

#108

—

146

—

114

#108

—

146

—

115

C₄⋊C₈

—

292

—

116

C₄⋊C₈

—

146

—

117

#97

—

292

—

118

C₄×C₈

—

292

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

—

122

C₄×C₈

C₂×#64408

yes

characteristic

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

M₁₆

—

292

—

127

M₁₆

—

584

—

128

M₁₆

—

146

—

129

C₂×C₈

—

292

—

130

C₂×C₈

—

292

—

131

C₂×C₈

—

292

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

146

—

135

C₂×C₈

—

146

—

136

C₂×C₈

C₄×#64408

yes

characteristic

137

C₂×C₈

C₂²⋊#64408

yes

characteristic

138

C₄²

—

139

C₄²

—

140

C₄²

—

141

C₄²

C₄⋊#64408

yes

characteristic

142

C₁₆

—

143

C₁₆

—

292

—

144

C₂³

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

146

—

149

C₂×C₄

—

146

—

150

C₂×C₄

—

146

—

151

C₂×C₄

—

146

—

152

C₂×C₄

—

146

—

153

C₂×C₄

—

146

—

154

C₂×C₄

(C₂×#64408)⋊C₄

yes

characteristic

155

C₂×C₄

#14556659

yes

characteristic

156

C₂×C₄

#14556660

yes

characteristic

157

C₈

—

158

C₈

—

292

—

159

C₈

—

584

—

160

C₈

—

146

—

161

C₈

—

146

—

162

C₈

#14556655

yes

characteristic

163

C₈

#14556656

yes

characteristic

164

C₂²

—

146

—

165

C₂²

—

146

—

166

C₂²

—

146

—

167

C₂²

#17989461

yes

characteristic

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

146

—

172

C₄

—

146

—

173

C₄

#17989457

yes

characteristic

174

C₄

#17989470

yes

characteristic

175

C₄

#17989515

yes

characteristic

176

C₄

#17989516

yes

characteristic

177

C₂

—

292

—

178

C₂

#22624828

yes

characteristic

179

C₂

#22624991

yes

characteristic

180

C₂

#22625006

yes

characteristic

	subgroup	G/H
center	C₂²	#17989461
commutator	C₂×C₈×C₇₃	C₄²
Frattini	C₄×C₈	C₂×#64408
Fitting	C₄⋊C₁₆×C₇₃	C₄
socle	C₂²×C₇₃	#340
prime radical	C₂²×D₇₃	#103
cosocle	C₄×C₈×D₇₃	C₂²
hypercenter	C₄⋊C₁₆	#64408
hypocenter	C₇₃	#7298
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂²

#17989461

commutator

C₂×C₈×C₇₃

C₄²

Frattini

C₄×C₈

C₂×#64408

Fitting

C₄⋊C₁₆×C₇₃

C₄

C₂²×C₇₃

C₂²×D₇₃

C₄×C₈×D₇₃

C₂²

C₄⋊C₁₆

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
94. C₇₃	#7298	semidirect

H ≅ G/N

type

94. C₇₃

#7298

semidirect

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

subgroup	index
C₄×#14556662	2
#22625489	2
#22611729	2
#7298	73

subgroup

index

Subgroups — #34000668

Conjugacy classes of subgroups

Complements

Maximal subgroups