Group #34000484 — subgroups

Subgroups — #34000484

Order 18688.

280 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 66 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	#22625797	C₂	1	2	yes	maximal
2	#22610866	C₂	1	1	yes	maximal, characteristic
3	#17989590	—	2	2	no	—
4	C₂×#14556652	—	2	2	no	—
5	#17989467	C₄	1	4	yes	—
6	C₂×C₄×Dic₁₄₆	C₂²	1	1	yes	characteristic
7	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
8	#17988341	—	2	2	no	—
9	C₂×C₄⋊C₈×C₇₃	C₄	1	1	yes	characteristic
10	C₂×#11924037	—	2	4	no	—
11	#14556658	—	4	2	no	—
12	C₄⋊#10596833	—	4	2	no	—
13	#14556652	—	4	2	no	—
14	#14556652	—	4	2	no	—
15	C₂²×Dic₁₄₆	D₄	1	1	yes	characteristic
16	C₂²×#10596831	—	2	1	no	—
17	C₄×Dic₁₄₆	—	4	1	no	—
18	C₄×Dic₁₄₆	—	4	1	no	—
19	C₂×C₄⋊Dic₇₃	Q₈	1	1	yes	characteristic
20	C₂×#11924007	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₇₃	D₄	1	2	yes	—
22	C₄×#10596831	—	4	1	no	—
23	C₂²×C₈×C₇₃	—	2	1	no	—
24	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
25	C₄⋊C₈×C₇₃	—	4	1	no	—
26	#11924037	—	4	4	no	—
27	#11924037	—	4	4	no	—
28	C₂×#10596833	—	4	4	no	—
29	C₂×Dic₁₄₆	—	4	1	no	—
30	C₂×Dic₁₄₆	—	4	1	no	—
31	C₂×Dic₁₄₆	D₈	1	1	yes	characteristic
32	C₂×Dic₁₄₆	SD₁₆	1	2	yes	—
33	C₂×Dic₁₄₆	Q₁₆	1	1	yes	characteristic
34	C₂²×Dic₇₃	C₂²⋊C₄	1	2	yes	—
35	C₂²×Dic₇₃	C₄⋊C₄	1	2	yes	—
36	C₂×#10596831	—	2	2	no	—
37	C₂×#10596831	—	4	1	no	—
38	C₂×#10596831	—	4	1	no	—
39	C₄⋊Dic₇₃	—	4	1	no	—
40	#11924007	—	4	2	no	—
41	C₄×Dic₇₃	—	2	2	no	—
42	C₄×Dic₇₃	—	2	2	no	—
43	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
44	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₇₃	—	4	2	no	—
47	C₂×C₈×C₇₃	—	4	1	no	—
48	C₄²×C₇₃	—	2	1	no	—
49	C₄²×C₇₃	—	2	1	no	—
50	C₂³×C₇₃	#95	1	1	yes	characteristic
51	C₂×C₄×C₇₃	—	2	1	no	—
52	C₂×C₄×C₇₃	—	2	1	no	—
53	C₂×C₄×C₇₃	—	2	1	no	—
54	C₂×C₄×C₇₃	—	2	1	no	—
55	C₂×C₄×C₇₃	—	2	1	no	—
56	C₂×C₄×C₇₃	—	4	1	no	—
57	C₂×C₄×C₇₃	#99	1	1	yes	characteristic
58	C₂×C₄×C₇₃	#99	1	1	yes	characteristic
59	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
60	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
61	C₂×Dic₇₃	—	2	2	no	—
62	C₂×Dic₇₃	—	2	2	no	—
63	C₂×Dic₇₃	—	2	2	no	—
64	C₂×Dic₇₃	—	4	2	no	—
65	C₂×Dic₇₃	#102	1	2	yes	—
66	C₂×Dic₇₃	#103	1	2	yes	—
67	Dic₁₄₆	—	4	1	no	—
68	Dic₁₄₆	—	4	2	no	—
69	Dic₁₄₆	—	4	1	no	—
70	#10596833	—	8	4	no	—
71	C₈×C₇₃	—	8	1	no	—
72	#10596831	—	4	2	no	—
73	C₂²×C₇₃	—	2	1	no	—
74	C₂²×C₇₃	—	2	1	no	—
75	C₂²×C₇₃	H(3,ℤ/4)	1	1	yes	characteristic
76	C₂²×C₇₃	#340	1	1	yes	characteristic
77	C₂²×C₇₃	#342	1	1	yes	characteristic
78	C₄×C₇₃	—	2	1	no	—
79	C₄×C₇₃	—	2	1	no	—
80	C₄×C₇₃	—	2	1	no	—
81	C₄×C₇₃	—	4	1	no	—
82	C₄×C₇₃	#355	1	1	yes	characteristic
83	C₄×C₇₃	#356	1	1	yes	characteristic
84	Dic₇₃	—	2	2	no	—
85	Dic₇₃	—	2	2	no	—
86	Dic₇₃	—	4	2	no	—
87	#7238	—	73	1	no	maximal, Sylow
88	C₂×C₇₃	—	2	1	no	—
89	C₂×C₇₃	—	2	1	no	—
90	C₂×C₇₃	#1289	1	1	yes	characteristic
91	C₂×C₇₃	#1392	1	1	yes	characteristic
92	C₂×C₇₃	#1394	1	1	yes	characteristic
93	#1984	—	73	1	no	—
94	#1801	—	73	2	no	—
95	C₇₃	#7238	1	1	yes	characteristic, complemented, Sylow
96	C₂×C₄×Q₈	—	73	1	no	—
97	#423	—	146	2	no	—
98	C₂×C₄⋊C₈	#64408	1	1	yes	characteristic
99	C₂×#101	—	146	2	no	—
100	C₂×C₄×C₈	—	73	1	no	—
101	#343	—	73	4	no	—
102	#336	—	146	2	no	—
103	C₂²×Q₈	—	73	1	no	—
104	C₂×M₁₆	—	146	4	no	—
105	C₂²×C₈	—	2	1	no	—
106	C₂²×C₈	—	146	1	no	—
107	C₄×Q₈	—	292	1	no	—
108	C₄×Q₈	—	292	1	no	—
109	C₂×C₄⋊C₄	—	73	2	no	—
110	C₂×C₄⋊C₄	—	73	1	no	—
111	C₂×C₄²	—	73	2	no	—
112	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
113	C₄⋊C₈	—	4	1	no	—
114	C₄⋊C₈	—	292	2	no	—
115	C₄⋊C₈	—	292	2	no	—
116	#101	—	292	2	no	—
117	#101	—	292	2	no	—
118	C₄×C₈	—	292	1	no	—
119	C₂×Q₈	—	292	1	no	—
120	C₂×Q₈	—	292	1	no	—
121	C₂×Q₈	—	73	2	no	—
122	C₂×Q₈	—	73	1	no	—
123	C₂×Q₈	—	73	1	no	—
124	C₂²×C₄	—	73	2	no	—
125	C₂²×C₄	—	73	2	no	—
126	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
127	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
128	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
129	M₁₆	—	292	4	no	—
130	M₁₆	—	292	4	no	—
131	C₂×C₈	—	4	2	no	—
132	C₂×C₈	—	4	1	no	—
133	C₂×C₈	—	292	1	no	—
134	C₂×C₈	—	292	1	no	—
135	C₂×C₈	—	292	4	no	—
136	C₂×C₈	—	146	2	no	—
137	C₄⋊C₄	—	292	2	no	—
138	C₄⋊C₄	—	292	1	no	—
139	C₄²	—	2	1	no	—
140	C₄²	—	2	1	no	—
141	C₄²	—	146	2	no	—
142	C₄²	—	146	2	no	—
143	C₂³	(C₂×#64408)⋊C₄	1	1	yes	characteristic
144	Q₈	—	292	2	no	—
145	Q₈	—	292	1	no	—
146	Q₈	—	292	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	—	292	2	no	—
154	C₂×C₄	—	73	2	no	—
155	C₂×C₄	—	73	2	no	—
156	C₂×C₄	—	146	2	no	—
157	C₂×C₄	—	146	2	no	—
158	C₂×C₄	—	146	2	no	—
159	C₂×C₄	#14556649	1	1	yes	characteristic
160	C₂×C₄	#14556650	1	1	yes	characteristic
161	C₂×C₄	#14556659	1	1	yes	characteristic
162	C₂×C₄	#14556660	1	1	yes	characteristic
163	C₈	—	8	1	no	—
164	C₈	—	292	2	no	—
165	C₈	—	584	4	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	1	no	—
168	C₂²	#17989455	1	1	yes	characteristic
169	C₂²	#17989461	1	1	yes	characteristic
170	C₂²	#17989465	1	1	yes	characteristic
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	4	1	no	—
175	C₄	—	292	2	no	—
176	C₄	—	146	2	no	—
177	C₄	—	146	2	no	—
178	C₄	#17989495	1	1	yes	characteristic
179	C₄	#17989497	1	1	yes	characteristic
180	C₂	—	2	1	no	—
181	C₂	—	2	1	no	—
182	C₂	#22624794	1	1	yes	characteristic
183	C₂	#22625023	1	1	yes	characteristic
184	C₂	#22625029	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22625797

C₂

yes

maximal

#22610866

C₂

yes

maximal, characteristic

#17989590

—

C₂×#14556652

—

#17989467

C₄

yes

—

C₂×C₄×Dic₁₄₆

C₂²

yes

characteristic

C₂×C₄×#10596831

C₄

yes

characteristic

#17988341

—

C₂×C₄⋊C₈×C₇₃

C₄

yes

—

—

—

—

—

C₂²×Dic₁₄₆

D₄

yes

characteristic

C₂²×#10596831

—

C₄×Dic₁₄₆

—

C₄×Dic₁₄₆

—

C₂×C₄⋊Dic₇₃

Q₈

yes

characteristic

C₂×#11924007

C₂×C₄

yes

—

C₂×C₄×Dic₇₃

D₄

yes

—

C₄×#10596831

—

C₂²×C₈×C₇₃

—

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₇₃

—

#11924037

—

#11924037

—

C₂×#10596833

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

D₈

yes

characteristic

C₂×Dic₁₄₆

SD₁₆

yes

—

C₂×Dic₁₄₆

Q₁₆

yes

characteristic

C₂²×Dic₇₃

C₂²⋊C₄

yes

—

C₂²×Dic₇₃

C₄⋊C₄

yes

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

#11924007

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

C₄²

yes

characteristic

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

#95

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

#102

yes

—

C₂×Dic₇₃

#103

yes

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

#10596833

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

#7238

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

C₇₃

#7238

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

#423

—

146

—

C₂×C₄⋊C₈

#64408

yes

characteristic

C₂×#101

—

146

—

100

C₂×C₄×C₈

—

101

#343

—

102

#336

—

146

—

103

C₂²×Q₈

—

104

C₂×M₁₆

—

146

—

105

C₂²×C₈

—

106

C₂²×C₈

—

146

—

107

C₄×Q₈

—

292

—

108

C₄×Q₈

—

292

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄²

—

112

C₂×C₄²

C₂×#64408

yes

characteristic

113

C₄⋊C₈

—

114

C₄⋊C₈

—

292

—

115

C₄⋊C₈

—

292

—

116

#101

—

292

—

117

#101

—

292

—

118

C₄×C₈

—

292

—

119

C₂×Q₈

—

292

—

120

C₂×Q₈

—

292

—

121

C₂×Q₈

—

122

C₂×Q₈

—

123

C₂×Q₈

—

124

C₂²×C₄

—

125

C₂²×C₄

—

126

C₂²×C₄

C₄×#64408

yes

characteristic

127

C₂²×C₄

C₂²⋊#64408

yes

characteristic

128

C₂²×C₄

C₄⋊#64408

yes

characteristic

129

M₁₆

—

292

—

130

M₁₆

—

292

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

292

—

134

C₂×C₈

—

292

—

135

C₂×C₈

—

292

—

136

C₂×C₈

—

146

—

137

C₄⋊C₄

—

292

—

138

C₄⋊C₄

—

292

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

146

—

142

C₄²

—

146

—

143

C₂³

(C₂×#64408)⋊C₄

yes

characteristic

144

Q₈

—

292

—

145

Q₈

—

292

—

146

Q₈

—

292

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

292

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

146

—

157

C₂×C₄

—

146

—

158

C₂×C₄

—

146

—

159

C₂×C₄

#14556649

yes

characteristic

160

C₂×C₄

#14556650

yes

characteristic

161

C₂×C₄

#14556659

yes

characteristic

162

C₂×C₄

#14556660

yes

characteristic

163

C₈

—

164

C₈

—

292

—

165

C₈

—

584

—

166

C₂²

—

167

C₂²

—

168

C₂²

#17989455

yes

characteristic

169

C₂²

#17989461

yes

characteristic

170

C₂²

#17989465

yes

characteristic

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

292

—

176

C₄

—

146

—

177

C₄

—

146

—

178

C₄

#17989495

yes

characteristic

179

C₄

#17989497

yes

characteristic

180

C₂

—

181

C₂

—

182

C₂

#22624794

yes

characteristic

183

C₂

#22625023

yes

characteristic

184

C₂

#22625029

yes

characteristic

	subgroup	G/H
center	C₂²	#17989465
commutator	C₂²×C₄×C₇₃	C₄²
Frattini	C₂×C₄²	C₂×#64408
Fitting	C₂×C₄⋊C₈×C₇₃	C₄
socle	C₂²×C₇₃	#342
prime radical	C₂³×C₇₃	#95
cosocle	C₂×C₄×Dic₁₄₆	C₂²
hypercenter	C₂×C₄⋊C₈	#64408
hypocenter	C₇₃	#7238
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂²

#17989465

commutator

C₂²×C₄×C₇₃

C₄²

Frattini

C₂×C₄²

C₂×#64408

Fitting

C₂×C₄⋊C₈×C₇₃

C₄

C₂²×C₇₃

C₂³×C₇₃

C₂×C₄×Dic₁₄₆

C₂²

C₂×C₄⋊C₈

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
95. C₇₃	#7238	semidirect

H ≅ G/N

type

95. C₇₃

#7238

semidirect

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

subgroup	index
#22625797	2
#22610866	2
#7238	73

subgroup

index

#22625797

#22610866

#7238

Subgroups — #34000484

Conjugacy classes of subgroups

Complements

Maximal subgroups