Group #34032984 — subgroups

Subgroups — #34032984

Order 18688.

222 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 75 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	#22627129	C₂	1	1	yes	maximal, characteristic
2	C₂×#17989690	C₂	1	1	yes	maximal, characteristic
3	#22625132	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11924040	C₄	1	2	yes	—
5	#17989690	—	2	1	no	—
6	#17989690	—	2	1	no	—
7	#17989690	—	2	1	no	—
8	C₂²⋊#11924040	C₄	1	2	yes	—
9	C₄×#11924040	—	2	1	no	—
10	C₄×#11924037	C₄	1	1	yes	characteristic
11	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
12	#17989429	—	2	1	no	—
13	C₈×#10596833	—	2	1	no	—
14	#17988637	C₄	1	1	yes	characteristic
15	C₂×#11924040	—	2	2	no	—
16	C₂×#11924040	—	2	2	no	—
17	C₂×#11924040	—	4	2	no	—
18	C₂×#11924037	C₈	1	2	yes	—
19	C₂²×#10596833	C₂×C₄	1	1	yes	characteristic
20	C₂²×#10596833	C₂×C₄	1	1	yes	characteristic
21	#14556646	—	2	1	no	—
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	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
28	#14556520	—	2	1	no	—
29	C₂²⋊#10596831	C₈	1	1	yes	characteristic
30	C₈×Dic₇₃	—	2	1	no	—
31	C₂²⋊C₈×C₇₃	C₈	1	1	yes	characteristic
32	#11924040	—	4	2	no	—
33	#11924040	—	4	2	no	—
34	#11924037	—	2	2	no	—
35	#11924037	C₁₆	1	4	yes	complemented
36	C₂×#10596833	—	2	2	no	—
37	C₂×#10596833	—	2	1	no	—
38	C₂×#10596833	—	2	1	no	—
39	C₂×#10596833	—	4	1	no	—
40	C₂×#10596833	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×#10596833	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×#10596833	C₄⋊C₄	1	1	yes	characteristic
43	C₂×#10596833	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×Dic₇₃	C₄²	1	1	yes	characteristic
45	C₂²×Dic₇₃	C₂×C₈	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	M₁₆	1	1	yes	characteristic
50	C₄×Dic₇₃	M₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
52	C₂×C₈×C₇₃	—	2	1	no	—
53	C₂³×C₇₃	C₄×C₈	1	1	yes	characteristic
54	C₂×C₄×C₇₃	—	4	1	no	—
55	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
57	C₂×Dic₇₃	—	2	1	no	—
58	C₂×Dic₇₃	—	2	1	no	—
59	C₂×Dic₇₃	#95	1	1	yes	characteristic
60	C₂×Dic₇₃	#97	1	1	yes	characteristic
61	C₂×Dic₇₃	C₂²⋊C₈	1	1	yes	characteristic
62	C₂×Dic₇₃	C₄⋊C₈	1	1	yes	characteristic
63	C₂×Dic₇₃	C₂×C₁₆	1	2	yes	—
64	C₂×Dic₇₃	M₃₂	1	2	yes	—
65	#10596833	—	2	1	no	—
66	#10596833	—	2	1	no	—
67	#10596833	—	2	1	no	—
68	#10596833	—	2	4	no	—
69	#10596833	#100	1	1	yes	characteristic
70	#10596833	#101	1	1	yes	characteristic
71	C₈×C₇₃	—	4	1	no	—
72	#10596831	—	4	1	no	—
73	C₂²×C₇₃	—	2	1	no	—
74	C₂²×C₇₃	—	2	1	no	—
75	C₂²×C₇₃	#336	1	1	yes	characteristic
76	C₂²×C₇₃	C₄×C₁₆	1	1	yes	characteristic
77	C₂²×C₇₃	#346	1	1	yes	characteristic
78	C₄×C₇₃	—	2	1	no	—
79	C₄×C₇₃	—	2	1	no	—
80	Dic₇₃	—	2	1	no	—
81	Dic₇₃	#343	1	1	yes	characteristic
82	Dic₇₃	#344	1	1	yes	characteristic
83	Dic₇₃	C₂²⋊C₁₆	1	2	yes	—
84	Dic₇₃	C₄⋊C₁₆	1	2	yes	—
85	#7209	—	73	1	no	maximal, Sylow
86	C₂×C₇₃	—	2	1	no	—
87	C₂×C₇₃	—	2	1	no	—
88	C₂×C₇₃	#1283	1	1	yes	characteristic
89	C₂×C₇₃	#1374	1	1	yes	characteristic
90	C₂×C₇₃	#1378	1	1	yes	characteristic
91	#2162	—	73	1	no	—
92	C₂×C₄⋊C₁₆	—	73	1	no	—
93	C₈×M₁₆	—	73	1	no	—
94	C₇₃	#7209	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₁₆	—	73	2	no	—
96	#431	—	73	1	no	—
97	C₄×M₁₆	—	73	1	no	—
98	C₂×C₄×C₈	—	73	1	no	—
99	C₄⋊C₁₆	—	146	1	no	—
100	C₄⋊C₁₆	—	146	1	no	—
101	C₄⋊C₁₆	—	146	1	no	—
102	C₂²⋊C₁₆	—	73	2	no	—
103	C₄×C₁₆	—	146	1	no	—
104	#322	—	146	1	no	—
105	C₈²	—	146	1	no	—
106	C₂×M₁₆	—	73	2	no	—
107	C₂²×C₈	—	73	1	no	—
108	C₂²×C₈	—	73	1	no	—
109	C₂×C₄²	—	73	1	no	—
110	C₂×C₁₆	—	292	2	no	—
111	C₂×C₁₆	—	146	2	no	—
112	C₂×C₁₆	—	146	2	no	—
113	C₄⋊C₈	—	146	1	no	—
114	C₂²⋊C₈	—	73	1	no	—
115	C₂²⋊C₈	#10596842	1	1	yes	characteristic
116	#97	—	146	1	no	—
117	C₄×C₈	—	73	1	no	—
118	C₄×C₈	—	73	1	no	—
119	C₄×C₈	—	73	1	no	—
120	C₄×C₈	—	73	1	no	—
121	C₄×C₈	—	146	1	no	—
122	C₄×C₈	—	146	1	no	—
123	C₂²×C₄	—	73	1	no	—
124	C₂²×C₄	—	73	1	no	—
125	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
126	M₁₆	—	73	4	no	—
127	M₁₆	—	146	2	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	292	1	no	—
130	C₂×C₈	—	73	1	no	—
131	C₂×C₈	—	73	1	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	—	146	1	no	—
137	C₂×C₈	—	146	2	no	—
138	C₄²	—	73	1	no	—
139	C₄²	—	73	1	no	—
140	C₄²	—	73	1	no	—
141	C₄²	—	73	1	no	—
142	C₁₆	—	292	2	no	—
143	C₁₆	—	292	2	no	—
144	C₂³	C₄×#10596842	1	1	yes	characteristic
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	73	1	no	—
147	C₂×C₄	—	73	1	no	—
148	C₂×C₄	—	73	1	no	—
149	C₂×C₄	—	73	2	no	—
150	C₂×C₄	—	73	2	no	—
151	C₂×C₄	—	73	1	no	—
152	C₂×C₄	—	146	1	no	—
153	C₂×C₄	—	146	1	no	—
154	C₂×C₄	C₂²⋊#10596842	1	1	yes	characteristic
155	C₂×C₄	C₄⋊#10596842	1	1	yes	characteristic
156	C₈	—	4	1	no	—
157	C₈	—	292	1	no	—
158	C₈	—	73	1	no	—
159	C₈	—	73	1	no	—
160	C₈	—	146	1	no	—
161	C₈	—	146	1	no	—
162	C₈	—	146	1	no	—
163	C₈	—	146	4	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	(C₂×#10596842)⋊C₄	1	1	yes	characteristic
167	C₂²	#64408⋊C₁₆	1	1	yes	characteristic
168	C₂²	#17989682	1	1	yes	characteristic
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	73	1	no	—
172	C₄	—	73	1	no	—
173	C₄	—	73	2	no	—
174	C₄	—	73	2	no	—
175	C₄	—	146	1	no	—
176	C₂	—	2	1	no	—
177	C₂	—	2	1	no	—
178	C₂	#22626943	1	1	yes	characteristic
179	C₂	#22627002	1	1	yes	characteristic
180	C₂	#22627009	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22627129

C₂

yes

maximal, characteristic

C₂×#17989690

C₂

yes

maximal, characteristic

#22625132

C₂

yes

maximal, characteristic

C₂²×#11924040

C₄

yes

—

—

—

—

C₄

yes

—

C₄×#11924040

—

C₄×#11924037

C₄

yes

characteristic

C₂×C₄×#10596833

C₂²

yes

—

—

C₄

yes

—

—

—

C₈

yes

—

C₂²×#10596833

C₂×C₄

yes

characteristic

C₂²×#10596833

C₂×C₄

yes

—

—

D₄

yes

characteristic

C₄×#10596833

D₄

yes

characteristic

C₄×#10596833

D₄

yes

characteristic

C₄×#10596833

Q₈

yes

characteristic

C₂×C₄×Dic₇₃

C₂×C₄

yes

characteristic

#14556520

—

C₂²⋊#10596831

C₈

yes

characteristic

C₈×Dic₇₃

—

C₂²⋊C₈×C₇₃

C₈

yes

—

—

—

C₁₆

yes

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#10596833

C₂²⋊C₄

yes

characteristic

C₂×#10596833

C₄⋊C₄

yes

characteristic

C₂×#10596833

C₄⋊C₄

yes

characteristic

C₂²×Dic₇₃

C₄²

yes

characteristic

C₂²×Dic₇₃

C₂×C₈

yes

characteristic

C₂×#10596831

—

C₄×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₇₃

C₄⋊C₄

yes

characteristic

C₄×Dic₇₃

M₁₆

yes

characteristic

C₄×Dic₇₃

M₁₆

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

—

C₂³×C₇₃

C₄×C₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₃

C₄⋊C₈

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

C₂²⋊C₈

yes

characteristic

C₂×Dic₇₃

C₄⋊C₈

yes

characteristic

C₂×Dic₇₃

C₂×C₁₆

yes

—

C₂×Dic₇₃

M₃₂

yes

—

—

—

—

—

yes

yes

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#336

yes

characteristic

C₂²×C₇₃

C₄×C₁₆

yes

characteristic

C₂²×C₇₃

#346

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

Dic₇₃

—

Dic₇₃

yes

Dic₇₃

yes

Dic₇₃

C₂²⋊C₁₆

yes

—

Dic₇₃

C₄⋊C₁₆

yes

—

#7209

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₄⋊C₁₆

—

C₈×M₁₆

—

C₇₃

#7209

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

#431

—

C₄×M₁₆

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

146

—

100

C₄⋊C₁₆

—

146

—

101

C₄⋊C₁₆

—

146

—

102

C₂²⋊C₁₆

—

103

C₄×C₁₆

—

146

—

104

#322

—

146

—

105

C₈²

—

146

—

106

C₂×M₁₆

—

107

C₂²×C₈

—

108

C₂²×C₈

—

109

C₂×C₄²

—

110

C₂×C₁₆

—

292

—

111

C₂×C₁₆

—

146

—

112

C₂×C₁₆

—

146

—

113

C₄⋊C₈

—

146

—

114

C₂²⋊C₈

—

115

C₂²⋊C₈

#10596842

yes

characteristic

116

#97

—

146

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

—

146

—

122

C₄×C₈

—

146

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

C₂×#10596842

yes

characteristic

126

M₁₆

—

127

M₁₆

—

146

—

128

C₂×C₈

—

129

C₂×C₈

—

292

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

146

—

135

C₂×C₈

—

146

—

136

C₂×C₈

—

146

—

137

C₂×C₈

—

146

—

138

C₄²

—

139

C₄²

—

140

C₄²

—

141

C₄²

—

142

C₁₆

—

292

—

143

C₁₆

—

292

—

144

C₂³

C₄×#10596842

yes

characteristic

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

146

—

153

C₂×C₄

—

146

—

154

C₂×C₄

C₂²⋊#10596842

yes

characteristic

155

C₂×C₄

C₄⋊#10596842

yes

characteristic

156

C₈

—

157

C₈

—

292

—

158

C₈

—

159

C₈

—

160

C₈

—

146

—

161

C₈

—

146

—

162

C₈

—

146

—

163

C₈

—

146

—

164

C₂²

—

165

C₂²

—

166

C₂²

(C₂×#10596842)⋊C₄

yes

characteristic

167

C₂²

#64408⋊C₁₆

yes

characteristic

168

C₂²

#17989682

yes

characteristic

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

146

—

176

C₂

—

177

C₂

—

178

C₂

#22626943

yes

characteristic

179

C₂

#22627002

yes

characteristic

180

C₂

#22627009

yes

characteristic

	subgroup	G/H
center	C₂²	(C₂×#10596842)⋊C₄
commutator	C₂²×C₇₃	C₄×C₁₆
Frattini	C₂²×C₄	C₂×#10596842
Fitting	C₂²⋊C₈×C₇₃	C₈
socle	C₂²×C₇₃	#336
prime radical	C₂³×C₇₃	C₄×C₈
cosocle	C₂×C₄×#10596833	C₂²
hypercenter	C₂²⋊C₈	#10596842
hypocenter	C₇₃	#7209
LCS	G ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

subgroup

G/H

center

C₂²

(C₂×#10596842)⋊C₄

commutator

C₂²×C₇₃

C₄×C₁₆

Frattini

C₂²×C₄

C₂×#10596842

Fitting

C₂²⋊C₈×C₇₃

C₈

socle

C₂²×C₇₃

#336

prime radical

C₂³×C₇₃

C₄×C₈

C₂²

C₂²⋊C₈

C₇₃

G ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

N	H ≅ G/N	type
35. #11924037	C₁₆	semidirect
94. C₇₃	#7209	semidirect

H ≅ G/N

type

35. #11924037

C₁₆

semidirect

94. C₇₃

#7209

semidirect

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

subgroup	index
#22627129	2
C₂×#17989690	2
#22625132	2
#7209	73

subgroup

index

Subgroups — #34032984

Conjugacy classes of subgroups

Complements

Maximal subgroups