Group #34032906 — subgroups

Subgroups — #34032906

Order 18688.

238 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 81 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	#22627131	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556686	C₂	1	1	yes	maximal, characteristic
3	#22625133	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556686	—	2	1	no	—
5	C₄⋊#11924040	C₄	1	2	yes	—
6	#17989684	—	2	1	no	—
7	C₄×#11924040	—	2	1	no	—
8	C₄×#11924040	—	2	1	no	—
9	C₄×#11924040	C₄	1	2	yes	—
10	C₄×#11924038	C₄	1	1	yes	characteristic
11	C₄×#11924036	C₂²	1	1	yes	characteristic
12	#17989429	—	2	1	no	—
13	C₈×#10596833	—	2	1	no	—
14	#17988639	C₄	1	1	yes	characteristic
15	#14556686	—	2	4	no	—
16	C₂×#11924040	—	2	2	no	—
17	C₂×#11924040	—	2	2	no	—
18	C₂×#11924038	C₈	1	2	yes	—
19	C₂×#11924036	D₄	1	1	yes	characteristic
20	C₂×#11924036	Q₈	1	1	yes	characteristic
21	#14556646	—	2	1	no	—
22	C₄×#10596833	—	2	1	no	—
23	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
24	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
25	C₄×#10596833	D₄	1	1	yes	characteristic
26	C₄×#10596833	D₄	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	—	2	4	no	—
33	#11924040	—	4	2	no	—
34	#11924038	—	2	2	no	—
35	#11924038	C₁₆	1	4	yes	complemented
36	#11924036	—	2	2	no	—
37	#11924036	—	2	2	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	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
45	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
46	C₂×#10596831	—	2	1	no	—
47	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
48	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₄×Dic₇₃	M₁₆	1	1	yes	characteristic
50	C₂×C₈×C₇₃	—	2	1	no	—
51	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
52	C₂²×D₇₃	C₄⋊C₈	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	#97	1	1	yes	characteristic
58	C₂×Dic₇₃	C₂²⋊C₈	1	1	yes	characteristic
59	C₄×D₇₃	—	2	1	no	—
60	C₄×D₇₃	—	2	1	no	—
61	C₄×D₇₃	—	2	1	no	—
62	C₄×D₇₃	—	2	1	no	—
63	C₄×D₇₃	C₂×C₁₆	1	2	yes	—
64	C₄×D₇₃	M₃₂	1	2	yes	—
65	#10596833	—	2	1	no	—
66	#10596833	—	2	1	no	—
67	#10596833	—	2	4	no	—
68	#10596833	#104	1	2	yes	—
69	#10596833	#108	1	2	yes	—
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	C₄⋊C₁₆	1	2	yes	—
76	C₄×C₇₃	—	2	1	no	—
77	C₄×C₇₃	C₄×C₁₆	1	1	yes	characteristic
78	C₄×C₇₃	#346	1	1	yes	characteristic
79	C₄×C₇₃	#350	1	1	yes	characteristic
80	C₄×C₇₃	#350	1	1	yes	characteristic
81	Dic₇₃	—	2	1	no	—
82	Dic₇₃	#339	1	1	yes	characteristic
83	Dic₇₃	#341	1	1	yes	characteristic
84	Dic₇₃	C₂²⋊C₁₆	1	2	yes	—
85	#7163	—	73	1	no	maximal, Sylow
86	C₂×C₇₃	#1278	1	1	yes	characteristic
87	C₂×C₇₃	#1374	1	1	yes	characteristic
88	C₂×C₇₃	#1379	1	1	yes	characteristic
89	D₇₃	—	2	2	no	—
90	#2162	—	73	1	no	—
91	C₂×C₄×C₁₆	—	73	1	no	—
92	C₈×M₁₆	—	73	1	no	—
93	C₇₃	#7163	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₁₆	—	146	1	no	—
95	#431	—	73	1	no	—
96	C₄×M₁₆	—	73	1	no	—
97	C₂×C₄×C₈	—	73	1	no	—
98	C₄⋊C₁₆	—	73	2	no	—
99	C₂²⋊C₁₆	—	146	1	no	—
100	C₄×C₁₆	—	73	2	no	—
101	C₄×C₁₆	—	146	1	no	—
102	C₄×C₁₆	—	146	1	no	—
103	#322	—	146	1	no	—
104	C₈²	—	146	1	no	—
105	C₂×M₁₆	—	73	2	no	—
106	C₂²×C₈	—	73	1	no	—
107	C₂²×C₈	—	73	1	no	—
108	C₂×C₄²	—	73	1	no	—
109	C₂×C₁₆	—	146	2	no	—
110	C₂×C₁₆	—	146	2	no	—
111	C₂×C₁₆	—	146	4	no	—
112	C₄⋊C₈	—	73	1	no	—
113	C₄⋊C₈	#10596842	1	1	yes	characteristic
114	C₂²⋊C₈	—	146	1	no	—
115	#97	—	146	1	no	—
116	C₄×C₈	—	73	1	no	—
117	C₄×C₈	—	73	1	no	—
118	C₄×C₈	—	73	1	no	—
119	C₄×C₈	—	73	1	no	—
120	C₄×C₈	—	146	1	no	—
121	C₄×C₈	—	146	1	no	—
122	C₂²×C₄	—	73	1	no	—
123	C₂²×C₄	—	73	1	no	—
124	C₂²×C₄	—	73	1	no	—
125	M₁₆	—	73	4	no	—
126	M₁₆	—	146	2	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	73	1	no	—
129	C₂×C₈	—	73	1	no	—
130	C₂×C₈	—	73	1	no	—
131	C₂×C₈	—	73	1	no	—
132	C₂×C₈	—	146	1	no	—
133	C₂×C₈	—	146	2	no	—
134	C₂×C₈	—	146	2	no	—
135	C₂×C₈	—	146	2	no	—
136	C₄²	—	73	1	no	—
137	C₄²	—	73	1	no	—
138	C₄²	—	73	1	no	—
139	C₄²	C₂×#10596842	1	1	yes	characteristic
140	C₁₆	—	292	2	no	—
141	C₁₆	—	146	4	no	—
142	C₂³	—	73	1	no	—
143	C₂×C₄	—	73	1	no	—
144	C₂×C₄	—	73	1	no	—
145	C₂×C₄	—	73	2	no	—
146	C₂×C₄	—	73	2	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	C₄×#10596842	1	1	yes	characteristic
153	C₂×C₄	C₂²⋊#10596842	1	1	yes	characteristic
154	C₂×C₄	C₄⋊#10596842	1	1	yes	characteristic
155	C₈	—	4	1	no	—
156	C₈	—	292	1	no	—
157	C₈	—	73	2	no	—
158	C₈	—	73	2	no	—
159	C₈	—	146	1	no	—
160	C₈	—	146	1	no	—
161	C₈	—	146	4	no	—
162	C₂²	—	73	2	no	—
163	C₂²	—	146	1	no	—
164	C₂²	—	146	1	no	—
165	C₂²	(C₂×#10596842)⋊C₄	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	73	1	no	—
168	C₄	—	73	1	no	—
169	C₄	—	73	2	no	—
170	C₄	—	146	1	no	—
171	C₄	#64408⋊C₁₆	1	1	yes	characteristic
172	C₄	#17989682	1	1	yes	characteristic
173	C₄	#17989687	1	1	yes	characteristic
174	C₄	#17989688	1	1	yes	characteristic
175	C₂	—	146	2	no	—
176	C₂	#22626938	1	1	yes	characteristic
177	C₂	#22627002	1	1	yes	characteristic
178	C₂	#22627011	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22627131

C₂

yes

maximal, characteristic

C₄×#14556686

C₂

yes

maximal, characteristic

#22625133

C₂

yes

maximal, characteristic

C₂×#14556686

—

C₄⋊#11924040

C₄

yes

—

—

—

—

C₄

yes

—

C₄×#11924038

C₄

yes

characteristic

C₄×#11924036

C₂²

yes

—

—

C₄

yes

—

—

—

C₈

yes

—

C₂×#11924036

D₄

yes

characteristic

C₂×#11924036

Q₈

yes

—

—

C₂×C₄

yes

characteristic

C₄×#10596833

C₂×C₄

yes

characteristic

C₄×#10596833

D₄

yes

characteristic

C₄×#10596833

D₄

yes

characteristic

C₄²×D₇₃

C₂×C₄

yes

characteristic

C₄⋊#10596831

C₈

yes

characteristic

#14556503

—

C₈×Dic₇₃

—

C₄⋊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

C₂×C₄×D₇₃

C₄⋊C₄

yes

characteristic

C₂×C₄×D₇₃

C₂×C₈

yes

characteristic

C₂×C₄×D₇₃

M₁₆

yes

characteristic

C₂×#10596831

—

C₄×Dic₇₃

C₄²

yes

characteristic

C₄×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₇₃

M₁₆

yes

characteristic

C₂×C₈×C₇₃

—

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₂²×D₇₃

C₄⋊C₈

yes

characteristic

C₂×C₄×C₇₃

C₄×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₃

C₄⋊C₈

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

C₂²⋊C₈

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₂×C₁₆

yes

—

C₄×D₇₃

M₃₂

yes

—

—

—

—

yes

—

#10596833

#108

yes

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

#336

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

C₄⋊C₁₆

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₄×C₁₆

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

yes

Dic₇₃

yes

Dic₇₃

C₂²⋊C₁₆

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#2162

—

C₂×C₄×C₁₆

—

C₈×M₁₆

—

C₇₃

#7163

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

146

—

#431

—

C₄×M₁₆

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

146

—

100

C₄×C₁₆

—

101

C₄×C₁₆

—

146

—

102

C₄×C₁₆

—

146

—

103

#322

—

146

—

104

C₈²

—

146

—

105

C₂×M₁₆

—

106

C₂²×C₈

—

107

C₂²×C₈

—

108

C₂×C₄²

—

109

C₂×C₁₆

—

146

—

110

C₂×C₁₆

—

146

—

111

C₂×C₁₆

—

146

—

112

C₄⋊C₈

—

113

C₄⋊C₈

#10596842

yes

characteristic

114

C₂²⋊C₈

—

146

—

115

#97

—

146

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₄×C₈

—

146

—

121

C₄×C₈

—

146

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

M₁₆

—

126

M₁₆

—

146

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

146

—

133

C₂×C₈

—

146

—

134

C₂×C₈

—

146

—

135

C₂×C₈

—

146

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₄²

C₂×#10596842

yes

characteristic

140

C₁₆

—

292

—

141

C₁₆

—

146

—

142

C₂³

—

143

C₂×C₄

—

144

C₂×C₄

—

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

C₄×#10596842

yes

characteristic

153

C₂×C₄

C₂²⋊#10596842

yes

characteristic

154

C₂×C₄

C₄⋊#10596842

yes

characteristic

155

C₈

—

156

C₈

—

292

—

157

C₈

—

158

C₈

—

159

C₈

—

146

—

160

C₈

—

146

—

161

C₈

—

146

—

162

C₂²

—

163

C₂²

—

146

—

164

C₂²

—

146

—

165

C₂²

(C₂×#10596842)⋊C₄

yes

characteristic

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

146

—

171

C₄

#64408⋊C₁₆

yes

characteristic

172

C₄

#17989682

yes

characteristic

173

C₄

#17989687

yes

characteristic

174

C₄

#17989688

yes

characteristic

175

C₂

—

146

—

176

C₂

#22626938

yes

characteristic

177

C₂

#22627002

yes

characteristic

178

C₂

#22627011

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊#10596842
commutator	C₄×C₇₃	C₄×C₁₆
Frattini	C₄²	C₂×#10596842
Fitting	C₄⋊C₈×C₇₃	C₈
socle	C₂²×C₇₃	#336
prime radical	C₂²×D₇₃	C₄⋊C₈
cosocle	C₄×#11924036	C₂²
hypercenter	C₄⋊C₈	#10596842
hypocenter	C₇₃	#7163
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

subgroup

G/H

center

C₂×C₄

C₂²⋊#10596842

commutator

C₄×C₇₃

C₄×C₁₆

Frattini

C₄²

C₂×#10596842

Fitting

C₄⋊C₈×C₇₃

C₈

socle

C₂²×C₇₃

#336

prime radical

C₂²×D₇₃

C₄⋊C₈

C₂²

C₄⋊C₈

C₇₃

G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

N	H ≅ G/N	type
35. #11924038	C₁₆	semidirect
93. C₇₃	#7163	semidirect

H ≅ G/N

type

35. #11924038

C₁₆

semidirect

93. C₇₃

#7163

semidirect

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

subgroup	index
#22627131	2
C₄×#14556686	2
#22625133	2
#7163	73

subgroup

index

Subgroups — #34032906

Conjugacy classes of subgroups

Complements

Maximal subgroups