Group #34034960 — subgroups

Subgroups — #34034960

Order 18688.

526 conjugacy classes of subgroups; only the 257 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 257 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	#22627155	C₂	1	6	yes	maximal, complemented
2	Q₈×#11924040	C₂	1	2	yes	maximal, complemented
3	#22627131	C₂	1	3	yes	maximal
4	#22627130	C₂	1	3	yes	maximal
5	Q₈×#11924036	C₂	1	1	yes	maximal, characteristic
6	#17989690	C₂²	1	6	yes	—
7	C₄⋊#11924040	C₂²	1	6	yes	—
8	#17989684	C₂²	1	1	yes	characteristic
9	#17989684	C₂²	1	3	yes	—
10	C₄×#11924040	C₂²	1	6	yes	—
11	C₄×#11924040	C₂²	1	6	yes	—
12	#17989636	C₄	1	3	yes	—
13	#17989636	C₂²	1	3	yes	—
14	Q₈×#10596833	C₄	1	1	yes	characteristic
15	Q₈×#10596833	C₂²	1	1	yes	characteristic
16	#17989588	C₄	1	3	yes	—
17	C₄×#11924036	C₂²	1	3	yes	—
18	C₄×Q₈×D₇₃	C₄	1	1	yes	characteristic
19	C₂×#11924040	C₂³	1	6	yes	—
20	C₂×#11924040	C₂³	1	2	yes	—
21	C₂×#11924036	C₂×C₄	1	3	yes	—
22	C₂×#11924036	C₂³	1	1	yes	characteristic
23	#14556658	C₂×C₄	1	6	yes	—
24	C₄⋊#10596833	C₂×C₄	1	3	yes	—
25	C₄⋊#10596833	C₂×C₄	1	3	yes	—
26	C₄×#10596833	C₂×C₄	1	3	yes	—
27	C₄×#10596833	C₂×C₄	1	3	yes	—
28	C₄×#10596833	C₂³	1	3	yes	—
29	C₄×#10596833	C₂³	1	3	yes	—
30	C₂×Q₈×D₇₃	C₈	1	1	yes	characteristic
31	Q₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
32	C₄×Dic₁₄₆	C₈	1	3	yes	—
33	#14556556	C₂×C₄	1	3	yes	—
34	C₄⋊C₄×D₇₃	C₈	1	3	yes	—
35	C₄²×D₇₃	C₂×C₄	1	3	yes	—
36	C₄×Q₈×C₇₃	C₈	1	1	yes	characteristic
37	#11924040	—	2	6	no	—
38	#11924040	—	2	2	no	—
39	#11924036	—	2	2	no	—
40	#11924036	—	4	3	no	—
41	C₂×#10596833	C₂²×C₄	1	3	yes	—
42	C₂×#10596833	C₂²×C₄	1	3	yes	—
43	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
44	C₂×#10596833	C₂⁴	1	1	yes	characteristic
45	C₂×Dic₁₄₆	C₂×C₈	1	3	yes	—
46	Q₈×D₇₃	C₁₆	1	8	yes	complemented
47	C₂×C₄×D₇₃	C₂×C₈	1	3	yes	—
48	C₂×C₄×D₇₃	C₂×C₈	1	3	yes	—
49	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
50	C₄⋊Dic₇₃	C₂×C₈	1	3	yes	—
51	#11924007	C₂×C₈	1	6	yes	—
52	C₄×Dic₇₃	C₂×C₈	1	3	yes	—
53	C₄×Dic₇₃	C₂²×C₄	1	3	yes	—
54	C₄×Dic₇₃	C₂²×C₄	1	3	yes	—
55	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
56	C₄⋊C₄×C₇₃	C₂×C₈	1	3	yes	—
57	C₄²×C₇₃	C₂×C₈	1	3	yes	—
58	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
59	Q₈×C₇₃	C₂×C₁₆	1	4	yes	—
60	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
62	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
63	C₂×Dic₇₃	C₂²×C₈	1	3	yes	—
64	C₂×Dic₇₃	C₂²×C₈	1	3	yes	—
65	C₂×Dic₇₃	C₂³×C₄	1	1	yes	characteristic
66	C₄×D₇₃	—	2	1	no	—
67	C₄×D₇₃	—	2	1	no	—
68	C₄×D₇₃	—	4	3	no	—
69	C₄×D₇₃	C₂×C₁₆	1	12	yes	—
70	Dic₁₄₆	C₂×C₁₆	1	12	yes	—
71	#10596833	—	2	1	no	—
72	#10596833	—	2	3	no	—
73	#10596833	—	2	3	no	—
74	#10596833	2₋¹⁺⁴	1	2	yes	—
75	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
76	C₂×D₇₃	—	2	1	no	—
77	C₂×D₇₃	—	2	1	no	—
78	C₂×D₇₃	C₂²×C₁₆	1	2	yes	—
79	C₄×C₇₃	—	2	3	no	—
80	C₄×C₇₃	—	2	1	no	—
81	C₄×C₇₃	C₂²×C₁₆	1	6	yes	—
82	Dic₇₃	—	2	3	no	—
83	Dic₇₃	C₂²×C₁₆	1	6	yes	—
84	Dic₇₃	#519	1	1	yes	characteristic
85	Dic₇₃	#568	1	1	yes	characteristic
86	#33496	—	73	1	no	maximal, Sylow
87	C₂×C₇₃	#2982	1	1	yes	characteristic
88	C₂×C₇₃	C₂³×C₁₆	1	1	yes	characteristic
89	C₂×C₇₃	#3407	1	1	yes	characteristic
90	D₇₃	—	2	2	no	—
91	C₂×C₈×Q₈	—	73	1	no	—
92	Q₈×C₁₆	—	73	2	no	—
93	Q₈×C₁₆	—	73	6	no	—
94	#2162	—	73	3	no	—
95	#2162	—	73	3	no	—
96	C₇₃	#33496	1	1	yes	characteristic, complemented, Sylow
97	C₂×C₄×Q₈	—	73	1	no	—
98	C₈×Q₈	—	73	1	no	—
99	C₈×Q₈	—	73	1	no	—
100	C₈×Q₈	—	73	3	no	—
101	C₈×Q₈	—	73	3	no	—
102	C₂×C₄⋊C₈	—	73	3	no	—
103	C₂×C₄×C₈	—	73	3	no	—
104	C₄⋊C₁₆	—	73	6	no	—
105	C₄⋊C₁₆	—	73	6	no	—
106	C₂²⋊C₁₆	—	73	1	no	—
107	C₂²⋊C₁₆	—	73	3	no	—
108	C₄×C₁₆	—	73	6	no	—
109	C₄×C₁₆	—	73	6	no	—
110	C₂²×Q₈	—	73	1	no	—
111	C₂²×C₈	—	73	3	no	—
112	C₂²×C₈	—	73	1	no	—
113	C₄×Q₈	—	73	1	no	—
114	C₄×Q₈	—	73	3	no	—
115	C₄×Q₈	—	73	3	no	—
116	C₄×Q₈	#10596842	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	73	3	no	—
118	C₂×C₄²	—	73	3	no	—
119	C₂×C₁₆	—	73	6	no	—
120	C₂×C₁₆	—	73	2	no	—
121	C₄⋊C₈	—	73	3	no	—
122	C₄⋊C₈	—	73	3	no	—
123	C₄⋊C₈	—	73	6	no	—
124	C₄×C₈	—	73	3	no	—
125	C₄×C₈	—	73	3	no	—
126	C₄×C₈	—	73	3	no	—
127	C₄×C₈	—	73	3	no	—
128	C₂×Q₈	—	73	8	no	—
129	C₂×Q₈	—	73	3	no	—
130	C₂×Q₈	C₂×#10596842	1	1	yes	characteristic
131	C₂²×C₄	—	73	3	no	—
132	C₂²×C₄	—	73	3	no	—
133	C₂²×C₄	—	73	1	no	—
134	C₂×C₈	—	292	3	no	—
135	C₂×C₈	—	73	1	no	—
136	C₂×C₈	—	73	1	no	—
137	C₂×C₈	—	73	3	no	—
138	C₂×C₈	—	73	3	no	—
139	C₂×C₈	—	146	2	no	—
140	C₄⋊C₄	—	73	6	no	—
141	C₄⋊C₄	—	73	3	no	—
142	C₄⋊C₄	C₂×#10596842	1	3	yes	—
143	C₄²	—	73	3	no	—
144	C₄²	—	73	3	no	—
145	C₄²	—	73	3	no	—
146	C₄²	C₂×#10596842	1	3	yes	—
147	C₁₆	—	146	6	no	—
148	C₁₆	—	146	2	no	—
149	C₂³	—	73	1	no	—
150	Q₈	—	73	12	no	—
151	Q₈	#14556686	1	4	yes	—
152	C₂×C₄	—	292	3	no	—
153	C₂×C₄	—	73	1	no	—
154	C₂×C₄	—	73	3	no	—
155	C₂×C₄	—	73	12	no	—
156	C₂×C₄	—	73	3	no	—
157	C₂×C₄	—	146	1	no	—
158	C₂×C₄	—	146	1	no	—
159	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
160	C₂×C₄	C₂²×#10596842	1	3	yes	—
161	C₂×C₄	C₂²×#10596842	1	3	yes	—
162	C₈	—	73	2	no	—
163	C₈	—	146	3	no	—
164	C₈	—	146	3	no	—
165	C₈	—	146	1	no	—
166	C₂²	—	73	2	no	—
167	C₂²	—	146	1	no	—
168	C₂²	—	146	1	no	—
169	C₂²	C₂³×#10596842	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	2	3	no	—
172	C₄	—	73	1	no	—
173	C₄	—	73	1	no	—
174	C₄	—	73	6	no	—
175	C₄	—	146	3	no	—
176	C₄	C₂×#14556686	1	6	yes	—
177	C₂	—	146	2	no	—
178	C₂	#22627174	1	1	yes	characteristic
179	C₂	C₂²×#14556686	1	1	yes	characteristic
180	C₂	#22627185	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22627155

C₂

yes

maximal, complemented

Q₈×#11924040

C₂

yes

maximal, complemented

#22627131

C₂

yes

maximal

#22627130

C₂

yes

maximal

Q₈×#11924036

C₂

yes

maximal, characteristic

#17989690

C₂²

yes

—

C₄⋊#11924040

C₂²

yes

—

#17989684

C₂²

yes

characteristic

#17989684

C₂²

yes

—

C₄×#11924040

C₂²

yes

—

C₄×#11924040

C₂²

yes

—

#17989636

C₄

yes

—

#17989636

C₂²

yes

—

Q₈×#10596833

C₄

yes

characteristic

Q₈×#10596833

C₂²

yes

characteristic

#17989588

C₄

yes

—

C₄×#11924036

C₂²

yes

—

C₄×Q₈×D₇₃

C₄

yes

characteristic

C₂×#11924040

C₂³

yes

—

C₂×#11924040

C₂³

yes

—

C₂×#11924036

C₂×C₄

yes

—

C₂×#11924036

C₂³

yes

characteristic

#14556658

C₂×C₄

yes

—

C₄⋊#10596833

C₂×C₄

yes

—

C₄⋊#10596833

C₂×C₄

yes

—

C₄×#10596833

C₂×C₄

yes

—

C₄×#10596833

C₂×C₄

yes

—

C₄×#10596833

C₂³

yes

—

C₄×#10596833

C₂³

yes

—

C₂×Q₈×D₇₃

C₈

yes

characteristic

Q₈×Dic₇₃

C₂×C₄

yes

characteristic

C₄×Dic₁₄₆

C₈

yes

—

#14556556

C₂×C₄

yes

—

C₄⋊C₄×D₇₃

C₈

yes

—

C₄²×D₇₃

C₂×C₄

yes

—

C₄×Q₈×C₇₃

C₈

yes

—

—

—

—

C₂²×C₄

yes

—

C₂×#10596833

C₂²×C₄

yes

—

C₂×#10596833

C₂²×C₄

yes

characteristic

C₂×#10596833

C₂⁴

yes

characteristic

C₂×Dic₁₄₆

C₂×C₈

yes

—

Q₈×D₇₃

C₁₆

yes

complemented

C₂×C₄×D₇₃

C₂×C₈

yes

—

C₂×C₄×D₇₃

C₂×C₈

yes

—

C₂×C₄×D₇₃

C₂²×C₄

yes

characteristic

C₄⋊Dic₇₃

C₂×C₈

yes

—

#11924007

C₂×C₈

yes

—

C₄×Dic₇₃

C₂×C₈

yes

—

C₄×Dic₇₃

C₂²×C₄

yes

—

C₄×Dic₇₃

C₂²×C₄

yes

—

C₂×Q₈×C₇₃

C₂×C₈

yes

characteristic

C₄⋊C₄×C₇₃

C₂×C₈

yes

—

C₄²×C₇₃

C₂×C₈

yes

—

C₂²×D₇₃

C₂²×C₈

yes

characteristic

Q₈×C₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₈

yes

—

C₂×C₄×C₇₃

C₂²×C₈

yes

—

C₂×Dic₇₃

C₂²×C₈

yes

—

C₂×Dic₇₃

C₂²×C₈

yes

—

C₂×Dic₇₃

C₂³×C₄

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₂×C₁₆

yes

—

Dic₁₄₆

C₂×C₁₆

yes

—

—

—

—

2₋¹⁺⁴

yes

—

C₂²×C₇₃

C₂³×C₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

C₂²×C₁₆

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂²×C₁₆

yes

—

Dic₇₃

—

Dic₇₃

C₂²×C₁₆

yes

—

Dic₇₃

yes

Dic₇₃

yes

—

C₂×C₇₃

yes

C₂×C₇₃

C₂³×C₁₆

yes

characteristic

C₂×C₇₃

#3407

yes

characteristic

D₇₃

—

C₂×C₈×Q₈

—

Q₈×C₁₆

—

Q₈×C₁₆

—

#2162

—

#2162

—

C₇₃

#33496

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

C₈×Q₈

—

C₈×Q₈

—

100

C₈×Q₈

—

101

C₈×Q₈

—

102

C₂×C₄⋊C₈

—

103

C₂×C₄×C₈

—

104

C₄⋊C₁₆

—

105

C₄⋊C₁₆

—

106

C₂²⋊C₁₆

—

107

C₂²⋊C₁₆

—

108

C₄×C₁₆

—

109

C₄×C₁₆

—

110

C₂²×Q₈

—

111

C₂²×C₈

—

112

C₂²×C₈

—

113

C₄×Q₈

—

114

C₄×Q₈

—

115

C₄×Q₈

—

116

C₄×Q₈

#10596842

yes

characteristic

117

C₂×C₄⋊C₄

—

118

C₂×C₄²

—

119

C₂×C₁₆

—

120

C₂×C₁₆

—

121

C₄⋊C₈

—

122

C₄⋊C₈

—

123

C₄⋊C₈

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₄×C₈

—

127

C₄×C₈

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

C₂×#10596842

yes

characteristic

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂×C₈

—

292

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

146

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

C₂×#10596842

yes

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₄²

C₂×#10596842

yes

—

147

C₁₆

—

146

—

148

C₁₆

—

146

—

149

C₂³

—

150

Q₈

—

151

Q₈

#14556686

yes

—

152

C₂×C₄

—

292

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

146

—

158

C₂×C₄

—

146

—

159

C₂×C₄

C₂²×#10596842

yes

characteristic

160

C₂×C₄

C₂²×#10596842

yes

—

161

C₂×C₄

C₂²×#10596842

yes

—

162

C₈

—

163

C₈

—

146

—

164

C₈

—

146

—

165

C₈

—

146

—

166

C₂²

—

167

C₂²

—

146

—

168

C₂²

—

146

—

169

C₂²

C₂³×#10596842

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

146

—

176

C₄

C₂×#14556686

yes

—

177

C₂

—

146

—

178

C₂

#22627174

yes

characteristic

179

C₂

C₂²×#14556686

yes

characteristic

180

C₂

#22627185

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10596842
commutator	C₂×C₇₃	C₂³×C₁₆
Fitting	C₄×Q₈×C₇₃	C₈
socle	C₂²×C₇₃	C₂³×C₈
prime radical	C₂²×D₇₃	C₂²×C₈
cosocle	C₂×#10596833	C₂⁴
hypercenter	C₄×Q₈	#10596842
hypocenter	C₇₃	#33496
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#10596842

commutator

C₂×C₇₃

C₂³×C₁₆

Fitting

C₄×Q₈×C₇₃

C₈

socle

C₂²×C₇₃

C₂³×C₈

prime radical

C₂²×D₇₃

C₂²×C₈

C₂⁴

C₄×Q₈

C₇₃

G ▷ C₂×C₇₃ ▷ C₇₃

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
1. #22627155	C₂	semidirect
2. Q₈×#11924040	C₂	semidirect
46. Q₈×D₇₃	C₁₆	semidirect
96. C₇₃	#33496	semidirect

H ≅ G/N

type

1. #22627155

C₂

semidirect

2. Q₈×#11924040

C₂

semidirect

46. Q₈×D₇₃

C₁₆

semidirect

96. C₇₃

#33496

semidirect

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

subgroup	index
#22627155	2
Q₈×#11924040	2
#22627131	2
#22627130	2
Q₈×#11924036	2
#33496	73

subgroup

index

Subgroups — #34034960

Conjugacy classes of subgroups

Complements

Maximal subgroups