Group #34032985 — subgroups

Subgroups — #34032985

Order 18688.

206 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 65 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	#22627134	C₂	1	1	yes	maximal, characteristic
2	#22627117	C₂	1	1	yes	maximal, characteristic
3	#22625143	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556686	C₄	1	2	yes	—
5	#17989690	—	2	1	no	—
6	C₄⋊#11924040	—	2	1	no	—
7	C₄⋊#11924040	—	2	1	no	—
8	#17989684	C₄	1	2	yes	—
9	#17989681	—	2	1	no	—
10	C₄×#11924038	C₄	1	1	yes	characteristic
11	C₄×#11924036	C₂²	1	1	yes	characteristic
12	#17989429	—	2	1	no	—
13	#17989428	—	2	1	no	—
14	#17988639	C₄	1	1	yes	characteristic
15	#14556686	—	4	2	no	—
16	C₂×#11924040	—	2	2	no	—
17	C₂×#11924040	—	2	2	no	—
18	C₂×#11924038	—	2	1	no	—
19	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
20	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
21	#14556646	C₈	1	2	yes	—
22	C₄×#10596833	—	2	1	no	—
23	C₄×#10596833	D₄	1	1	yes	characteristic
24	C₄×#10596833	D₄	1	1	yes	characteristic
25	C₄×#10596833	D₄	1	1	yes	characteristic
26	C₄×#10596833	Q₈	1	1	yes	characteristic
27	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
28	C₄⋊#10596831	C₈	1	1	yes	characteristic
29	#14556503	—	2	1	no	—
30	C₈×Dic₇₃	—	2	1	no	—
31	C₄⋊C₈×C₇₃	C₈	1	1	yes	characteristic
32	#11924040	—	4	2	no	—
33	#11924040	—	4	2	no	—
34	#11924038	—	4	2	no	—
35	#11924036	—	2	1	no	—
36	#11924036	—	2	1	no	—
37	#11924036	—	4	1	no	—
38	C₂×#10596833	—	2	2	no	—
39	C₂×#10596833	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×#10596833	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×#10596833	C₄⋊C₄	1	1	yes	characteristic
42	C₂×#10596833	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×D₇₃	C₄²	1	1	yes	characteristic
44	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
45	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
46	C₂×#10596831	—	2	1	no	—
47	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
49	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
50	C₂×C₈×C₇₃	—	2	1	no	—
51	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
52	C₂²×D₇₃	#97	1	1	yes	characteristic
53	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
54	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
55	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
56	C₂×Dic₇₃	#95	1	1	yes	characteristic
57	C₂×Dic₇₃	C₂²⋊C₈	1	1	yes	characteristic
58	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
59	C₄×D₇₃	—	2	2	no	—
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	#10596833	—	2	1	no	—
65	#10596833	—	2	1	no	—
66	#10596833	—	2	1	no	—
67	#10596833	—	4	2	no	—
68	#10596833	#100	1	1	yes	characteristic
69	#10596833	#101	1	1	yes	characteristic
70	C₈×C₇₃	—	4	1	no	—
71	#10596831	—	4	1	no	—
72	C₂²×C₇₃	#336	1	1	yes	characteristic
73	C₂×D₇₃	—	2	1	no	—
74	C₂×D₇₃	—	2	1	no	—
75	C₂×D₇₃	#347	1	2	yes	—
76	C₄×C₇₃	—	2	1	no	—
77	C₄×C₇₃	—	2	1	no	—
78	C₄×C₇₃	#350	1	1	yes	characteristic
79	C₄×C₇₃	#350	1	1	yes	characteristic
80	Dic₇₃	—	2	1	no	—
81	Dic₇₃	—	2	1	no	—
82	Dic₇₃	#343	1	1	yes	characteristic
83	Dic₇₃	#344	1	1	yes	characteristic
84	#7210	—	73	1	no	maximal, Sylow
85	C₂×C₇₃	#1283	1	1	yes	characteristic
86	C₂×C₇₃	#1375	1	1	yes	characteristic
87	C₂×C₇₃	#1379	1	1	yes	characteristic
88	D₇₃	—	2	2	no	—
89	#2163	—	73	1	no	—
90	C₂×C₄⋊C₁₆	—	73	1	no	—
91	#1451	—	73	1	no	—
92	C₇₃	#7210	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₁₆	—	73	2	no	—
94	#431	—	73	1	no	—
95	C₄×M₁₆	—	73	1	no	—
96	C₂×C₄×C₈	—	73	1	no	—
97	C₄⋊C₁₆	—	146	1	no	—
98	C₄⋊C₁₆	—	146	1	no	—
99	C₄⋊C₁₆	—	146	1	no	—
100	C₂²⋊C₁₆	—	73	2	no	—
101	#346	—	146	1	no	—
102	#322	—	146	1	no	—
103	#322	—	146	1	no	—
104	C₂×M₁₆	—	146	1	no	—
105	C₂²×C₈	—	73	1	no	—
106	C₂²×C₈	—	73	1	no	—
107	C₂×C₄²	—	73	1	no	—
108	C₂×C₁₆	—	292	2	no	—
109	C₂×C₁₆	—	146	2	no	—
110	C₂×C₁₆	—	146	2	no	—
111	C₄⋊C₈	—	73	1	no	—
112	C₄⋊C₈	#10596842	1	1	yes	characteristic
113	C₂²⋊C₈	—	146	1	no	—
114	#97	—	73	2	no	—
115	C₄×C₈	—	73	1	no	—
116	C₄×C₈	—	73	1	no	—
117	C₄×C₈	—	73	1	no	—
118	C₄×C₈	—	73	1	no	—
119	C₄×C₈	—	146	1	no	—
120	C₄×C₈	—	146	1	no	—
121	C₂²×C₄	—	73	1	no	—
122	C₂²×C₄	—	73	1	no	—
123	C₂²×C₄	—	73	1	no	—
124	M₁₆	—	292	2	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	292	1	no	—
127	C₂×C₈	—	73	1	no	—
128	C₂×C₈	—	73	1	no	—
129	C₂×C₈	—	73	1	no	—
130	C₂×C₈	—	73	1	no	—
131	C₂×C₈	—	146	2	no	—
132	C₂×C₈	—	146	1	no	—
133	C₂×C₈	—	146	1	no	—
134	C₂×C₈	—	146	1	no	—
135	C₄²	—	73	1	no	—
136	C₄²	—	73	1	no	—
137	C₄²	—	73	1	no	—
138	C₄²	C₂×#10596842	1	1	yes	characteristic
139	C₁₆	—	292	2	no	—
140	C₁₆	—	292	2	no	—
141	C₂³	—	73	1	no	—
142	C₂×C₄	—	73	1	no	—
143	C₂×C₄	—	73	1	no	—
144	C₂×C₄	—	73	1	no	—
145	C₂×C₄	—	146	1	no	—
146	C₂×C₄	—	146	1	no	—
147	C₂×C₄	—	146	2	no	—
148	C₂×C₄	—	146	1	no	—
149	C₂×C₄	—	146	1	no	—
150	C₂×C₄	C₄×#10596842	1	1	yes	characteristic
151	C₂×C₄	C₂²⋊#10596842	1	1	yes	characteristic
152	C₂×C₄	C₄⋊#10596842	1	1	yes	characteristic
153	C₈	—	4	1	no	—
154	C₈	—	292	2	no	—
155	C₈	—	292	1	no	—
156	C₈	—	73	1	no	—
157	C₈	—	73	1	no	—
158	C₈	—	146	1	no	—
159	C₈	—	146	1	no	—
160	C₈	—	146	1	no	—
161	C₂²	—	73	2	no	—
162	C₂²	—	146	1	no	—
163	C₂²	—	146	1	no	—
164	C₂²	(C₂×#10596842)⋊C₄	1	1	yes	characteristic
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	—	73	1	no	—
168	C₄	—	73	1	no	—
169	C₄	—	146	1	no	—
170	C₄	—	146	1	no	—
171	C₄	#17989687	1	1	yes	characteristic
172	C₄	#17989688	1	1	yes	characteristic
173	C₂	—	146	2	no	—
174	C₂	#22626943	1	1	yes	characteristic
175	C₂	#22627004	1	1	yes	characteristic
176	C₂	#22627011	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22627134

C₂

yes

maximal, characteristic

#22627117

C₂

yes

maximal, characteristic

#22625143

C₂

yes

maximal, characteristic

C₂×#14556686

C₄

yes

—

—

—

—

C₄

yes

—

#17989681

—

C₄×#11924038

C₄

yes

characteristic

C₄×#11924036

C₂²

yes

—

—

C₄

yes

—

—

—

—

C₂×C₄

yes

characteristic

C₂×#11924036

C₂×C₄

yes

characteristic

#14556646

C₈

yes

—

C₄×#10596833

—

C₄×#10596833

D₄

yes

characteristic

C₄×#10596833

D₄

yes

characteristic

C₄×#10596833

D₄

yes

characteristic

C₄×#10596833

Q₈

yes

characteristic

C₄²×D₇₃

C₂×C₄

yes

characteristic

C₄⋊#10596831

C₈

yes

characteristic

#14556503

—

C₈×Dic₇₃

—

C₄⋊C₈×C₇₃

C₈

yes

—

—

—

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#10596833

C₂²⋊C₄

yes

characteristic

C₂×#10596833

C₄⋊C₄

yes

characteristic

C₂×#10596833

C₄⋊C₄

yes

characteristic

C₂×C₄×D₇₃

C₄²

yes

characteristic

C₂×C₄×D₇₃

M₁₆

yes

characteristic

C₂×C₄×D₇₃

M₁₆

yes

characteristic

C₂×#10596831

—

C₄×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₇₃

C₄⋊C₄

yes

characteristic

C₄×Dic₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

—

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₂²×D₇₃

#97

yes

characteristic

C₂×C₄×C₇₃

C₄×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₃

C₄⋊C₈

yes

characteristic

C₂×Dic₇₃

#95

yes

characteristic

C₂×Dic₇₃

C₂²⋊C₈

yes

characteristic

C₂×Dic₇₃

C₄⋊C₈

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

—

—

—

—

yes

yes

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

#336

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

#347

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

yes

Dic₇₃

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#2163

—

C₂×C₄⋊C₁₆

—

#1451

—

C₇₃

#7210

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

#431

—

C₄×M₁₆

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

146

—

C₄⋊C₁₆

—

146

—

C₄⋊C₁₆

—

146

—

100

C₂²⋊C₁₆

—

101

#346

—

146

—

102

#322

—

146

—

103

#322

—

146

—

104

C₂×M₁₆

—

146

—

105

C₂²×C₈

—

106

C₂²×C₈

—

107

C₂×C₄²

—

108

C₂×C₁₆

—

292

—

109

C₂×C₁₆

—

146

—

110

C₂×C₁₆

—

146

—

111

C₄⋊C₈

—

112

C₄⋊C₈

#10596842

yes

characteristic

113

C₂²⋊C₈

—

146

—

114

#97

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₄×C₈

—

146

—

120

C₄×C₈

—

146

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

M₁₆

—

292

—

125

C₂×C₈

—

126

C₂×C₈

—

292

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

146

—

132

C₂×C₈

—

146

—

133

C₂×C₈

—

146

—

134

C₂×C₈

—

146

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

C₂×#10596842

yes

characteristic

139

C₁₆

—

292

—

140

C₁₆

—

292

—

141

C₂³

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

—

146

C₂×C₄

—

146

—

147

C₂×C₄

—

146

—

148

C₂×C₄

—

146

—

149

C₂×C₄

—

146

—

150

C₂×C₄

C₄×#10596842

yes

characteristic

151

C₂×C₄

C₂²⋊#10596842

yes

characteristic

152

C₂×C₄

C₄⋊#10596842

yes

characteristic

153

C₈

—

154

C₈

—

292

—

155

C₈

—

292

—

156

C₈

—

157

C₈

—

158

C₈

—

146

—

159

C₈

—

146

—

160

C₈

—

146

—

161

C₂²

—

162

C₂²

—

146

—

163

C₂²

—

146

—

164

C₂²

(C₂×#10596842)⋊C₄

yes

characteristic

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

146

—

170

C₄

—

146

—

171

C₄

#17989687

yes

characteristic

172

C₄

#17989688

yes

characteristic

173

C₂

—

146

—

174

C₂

#22626943

yes

characteristic

175

C₂

#22627004

yes

characteristic

176

C₂

#22627011

yes

characteristic

	subgroup	G/H
center	C₂²	(C₂×#10596842)⋊C₄
commutator	C₂×C₄×C₇₃	C₄×C₈
Frattini	C₄²	C₂×#10596842
Fitting	C₄⋊C₈×C₇₃	C₈
socle	C₂²×C₇₃	#336
prime radical	C₂²×D₇₃	#97
cosocle	C₄×#11924036	C₂²
hypercenter	C₄⋊C₈	#10596842
hypocenter	C₇₃	#7210
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

subgroup

G/H

center

C₂²

(C₂×#10596842)⋊C₄

commutator

C₂×C₄×C₇₃

C₄×C₈

Frattini

C₄²

C₂×#10596842

Fitting

C₄⋊C₈×C₇₃

C₈

C₂²×C₇₃

C₂²×D₇₃

C₂²

C₄⋊C₈

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

N	H ≅ G/N	type
92. C₇₃	#7210	semidirect

H ≅ G/N

type

92. C₇₃

#7210

semidirect

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

subgroup	index
#22627134	2
#22627117	2
#22625143	2
#7210	73

subgroup

index

Subgroups — #34032985

Conjugacy classes of subgroups

Complements

Maximal subgroups