Group #34034223 — subgroups

Subgroups — #34034223

Order 18688.

342 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 125 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	C₂×#17989692	C₂	1	2	yes	maximal
2	C₂×#17989691	C₂	1	2	yes	maximal
3	#22627121	C₂	1	1	yes	maximal, characteristic
4	#22627120	C₂	1	1	yes	maximal, characteristic
5	#22625589	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556688	C₂²	1	2	yes	—
7	C₂×#14556688	C₂²	1	1	yes	characteristic
8	C₂×#14556688	C₂²	1	1	yes	characteristic
9	#17989692	—	2	4	no	—
10	#17989691	—	2	4	no	—
11	#17989690	—	2	2	no	—
12	C₄⋊#11924040	—	2	2	no	—
13	C₂×#14556645	C₄	1	2	yes	—
14	C₄×#11924036	C₂²	1	1	yes	characteristic
15	C₂×C₈×#64408	C₂²	1	2	yes	—
16	#17989523	C₄	1	1	yes	characteristic
17	#97×D₇₃	C₄	1	1	yes	characteristic
18	#14556688	—	2	2	no	—
19	#14556688	—	2	2	no	—
20	#14556688	—	2	2	no	—
21	#14556688	—	2	2	no	—
22	C₂×#11924040	—	2	2	no	—
23	C₂×#11924040	—	2	1	no	—
24	C₂×#11924040	—	2	1	no	—
25	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
26	C₂×#11924036	C₂³	1	1	yes	characteristic
27	C₂×C₄×#64408	C₂×C₄	1	2	yes	—
28	#14556645	—	2	4	no	—
29	C₄×#10596833	D₄	1	2	yes	—
30	C₄×#10596833	Q₈	1	2	yes	—
31	C₈×#64408	D₄	1	4	yes	—
32	C₈×#64408	Q₈	1	4	yes	—
33	(C₂×#64408)⋊C₄	—	2	2	no	—
34	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
35	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
36	#14556502	C₈	1	1	yes	characteristic
37	#14556501	C₈	1	2	yes	—
38	#97×C₇₃	C₈	1	1	yes	characteristic
39	#11924040	—	4	2	no	—
40	#11924040	—	4	1	no	—
41	#11924040	—	4	1	no	—
42	C₂²×#64408	—	2	2	no	—
43	#11924036	—	2	2	no	—
44	#11924036	C₂×D₄	1	2	yes	—
45	#11924036	C₂×Q₈	1	2	yes	—
46	C₂×#10596833	C₄⋊C₄	1	2	yes	—
47	C₂×#10596833	C₂×D₄	1	1	yes	characteristic
48	C₂×#10596833	C₂×Q₈	1	1	yes	characteristic
49	C₄×#64408	—	2	2	no	—
50	C₄×#64408	C₄⋊C₄	1	4	yes	—
51	C₂×C₄×D₇₃	M₁₆	1	2	yes	—
52	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×#10596831	C₂×C₈	1	2	yes	—
54	C₈×D₇₃	C₄⋊C₄	1	4	yes	—
55	C₈×D₇₃	M₁₆	1	4	yes	—
56	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
57	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
58	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
59	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
60	C₂²×D₇₃	C₂×M₁₆	1	1	yes	characteristic
61	C₂×#64408	—	2	4	no	—
62	C₂×#64408	—	4	4	no	—
63	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
64	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
65	C₂×Dic₇₃	C₄⋊C₈	1	2	yes	—
66	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₄×D₇₃	—	4	2	no	—
68	C₄×D₇₃	C₂×C₄⋊C₄	1	2	yes	—
69	C₄×D₇₃	C₂×M₁₆	1	2	yes	—
70	#10596833	—	2	2	no	—
71	#10596833	#127	1	1	yes	characteristic
72	#10596833	C₄⋊Q₈	1	1	yes	characteristic
73	#10596833	C₄⋊Q₈	1	2	yes	—
74	C₈×C₇₃	C₄⋊C₈	1	4	yes	—
75	#10596831	C₄⋊C₈	1	4	yes	—
76	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
77	C₂×D₇₃	—	2	2	no	—
78	C₂×D₇₃	#423	1	2	yes	—
79	#64408	—	4	4	no	—
80	C₄×C₇₃	—	2	2	no	—
81	C₄×C₇₃	C₂×C₄⋊C₈	1	2	yes	—
82	Dic₇₃	—	2	2	no	—
83	Dic₇₃	#384	1	1	yes	characteristic
84	Dic₇₃	#423	1	1	yes	characteristic
85	#12901	—	73	1	no	maximal, Sylow
86	C₂×C₇₃	#1842	1	1	yes	characteristic
87	C₂×C₇₃	#2153	1	2	yes	—
88	D₇₃	—	2	2	no	—
89	C₂×#364	—	73	2	no	—
90	C₂×#364	—	73	2	no	—
91	#2150	—	73	1	no	—
92	#2150	—	73	1	no	—
93	#1726	—	73	1	no	—
94	C₇₃	#12901	1	1	yes	characteristic, complemented, Sylow
95	C₂×M₃₂	—	73	2	no	—
96	C₂×M₃₂	—	73	1	no	—
97	C₂×M₃₂	—	73	1	no	—
98	C₂×#97	—	73	1	no	—
99	C₂×#97	—	73	2	no	—
100	C₂×C₄×C₈	—	73	1	no	—
101	C₂×C₄×C₈	—	73	2	no	—
102	#376	—	73	1	no	—
103	#364	—	146	4	no	—
104	#364	—	146	4	no	—
105	C₄⋊C₁₆	—	146	2	no	—
106	C₄⋊C₁₆	—	146	2	no	—
107	C₂²×C₈	—	73	1	no	—
108	C₂²×C₈	—	73	2	no	—
109	C₂²×C₈	—	73	1	no	—
110	C₂×C₄²	—	73	1	no	—
111	C₂×C₄²	—	73	2	no	—
112	M₃₂	—	146	2	no	—
113	M₃₂	—	146	2	no	—
114	M₃₂	—	146	2	no	—
115	M₃₂	—	146	2	no	—
116	C₂×C₁₆	—	146	2	no	—
117	C₂×C₁₆	—	146	1	no	—
118	C₂×C₁₆	—	146	1	no	—
119	#97	—	73	2	no	—
120	#97	—	73	1	no	—
121	#97	—	146	4	no	—
122	#97	#10596842	1	1	yes	characteristic
123	C₄×C₈	—	73	2	no	—
124	C₄×C₈	—	73	4	no	—
125	C₄×C₈	—	73	4	no	—
126	C₄×C₈	—	73	2	no	—
127	#95	—	146	2	no	—
128	C₂²×C₄	—	73	2	no	—
129	C₂²×C₄	—	73	1	no	—
130	C₂²×C₄	—	146	2	no	—
131	C₂×C₈	—	73	1	no	—
132	C₂×C₈	—	73	1	no	—
133	C₂×C₈	—	73	2	no	—
134	C₂×C₈	—	73	2	no	—
135	C₂×C₈	—	73	2	no	—
136	C₂×C₈	—	73	4	no	—
137	C₂×C₈	—	73	4	no	—
138	C₂×C₈	—	73	2	no	—
139	C₂×C₈	—	146	2	no	—
140	C₂×C₈	C₂×#10596842	1	2	yes	—
141	C₄²	—	73	2	no	—
142	C₄²	—	73	1	no	—
143	C₄²	—	73	4	no	—
144	C₄²	—	146	2	no	—
145	C₄²	C₂×#10596842	1	1	yes	characteristic
146	C₁₆	—	292	2	no	—
147	C₁₆	—	292	1	no	—
148	C₁₆	—	292	1	no	—
149	C₂³	—	73	1	no	—
150	C₂×C₄	—	292	2	no	—
151	C₂×C₄	—	292	4	no	—
152	C₂×C₄	—	73	1	no	—
153	C₂×C₄	—	73	2	no	—
154	C₂×C₄	—	73	2	no	—
155	C₂×C₄	—	73	2	no	—
156	C₂×C₄	—	146	4	no	—
157	C₂×C₄	C₄⋊#10596842	1	2	yes	—
158	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
159	C₈	—	73	1	no	—
160	C₈	—	73	1	no	—
161	C₈	—	73	2	no	—
162	C₈	—	73	4	no	—
163	C₈	—	146	2	no	—
164	C₈	C₄⋊#10596842	1	4	yes	—
165	C₂²	—	73	2	no	—
166	C₂²	—	146	2	no	—
167	C₂²	C₂×C₄⋊#10596842	1	1	yes	characteristic
168	C₄	—	2	2	no	—
169	C₄	—	292	4	no	—
170	C₄	—	73	1	no	—
171	C₄	—	73	1	no	—
172	C₄	—	146	2	no	—
173	C₄	C₂×C₄⋊#10596842	1	2	yes	—
174	C₂	—	146	2	no	—
175	C₂	#22627075	1	1	yes	characteristic
176	C₂	#22627127	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17989692

C₂

yes

maximal

C₂×#17989691

C₂

yes

maximal

#22627121

C₂

yes

maximal, characteristic

#22627120

C₂

yes

maximal, characteristic

#22625589

C₂

yes

maximal, characteristic

C₂×#14556688

C₂²

yes

—

C₂×#14556688

C₂²

yes

characteristic

C₂×#14556688

C₂²

yes

—

—

—

—

C₄

yes

—

C₄×#11924036

C₂²

yes

characteristic

C₂×C₈×#64408

C₂²

yes

—

#17989523

C₄

yes

characteristic

#97×D₇₃

C₄

yes

—

—

—

—

—

—

—

C₂×C₄

yes

characteristic

C₂×#11924036

C₂³

yes

characteristic

C₂×C₄×#64408

C₂×C₄

yes

—

#14556645

—

C₄×#10596833

D₄

yes

—

C₄×#10596833

Q₈

yes

—

C₈×#64408

D₄

yes

—

C₈×#64408

Q₈

yes

—

(C₂×#64408)⋊C₄

—

C₂×C₈×D₇₃

C₂×C₄

yes

—

C₄²×D₇₃

C₂×C₄

yes

characteristic

#14556502

C₈

yes

characteristic

#14556501

C₈

yes

—

#97×C₇₃

C₈

yes

—

—

—

C₂²×#64408

—

#11924036

—

#11924036

C₂×D₄

yes

—

#11924036

C₂×Q₈

yes

—

C₂×#10596833

C₄⋊C₄

yes

—

C₂×#10596833

C₂×D₄

yes

characteristic

C₂×#10596833

C₂×Q₈

yes

characteristic

C₄×#64408

—

C₄×#64408

C₄⋊C₄

yes

—

C₂×C₄×D₇₃

M₁₆

yes

—

C₂×C₄×D₇₃

C₂²×C₄

yes

characteristic

C₂×#10596831

C₂×C₈

yes

—

C₈×D₇₃

C₄⋊C₄

yes

—

C₈×D₇₃

M₁₆

yes

—

C₄×Dic₇₃

C₄⋊C₄

yes

—

C₄×Dic₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂×C₈

yes

—

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₂²×D₇₃

C₂×M₁₆

yes

characteristic

C₂×#64408

—

C₂×#64408

—

C₂×C₄×C₇₃

C₄⋊C₈

yes

—

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×Dic₇₃

C₄⋊C₈

yes

—

C₂×Dic₇₃

C₂×C₄⋊C₄

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

C₂×C₄⋊C₄

yes

—

C₄×D₇₃

C₂×M₁₆

yes

—

—

yes

C₄⋊Q₈

yes

characteristic

#10596833

C₄⋊Q₈

yes

—

C₈×C₇₃

C₄⋊C₈

yes

—

#10596831

C₄⋊C₈

yes

—

C₂²×C₇₃

C₂×C₄⋊C₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

#423

yes

—

#64408

—

C₄×C₇₃

—

C₄×C₇₃

C₂×C₄⋊C₈

yes

—

Dic₇₃

—

Dic₇₃

yes

Dic₇₃

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

—

—

—

—

—

—

C₇₃

#12901

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂×#97

—

C₂×#97

—

100

C₂×C₄×C₈

—

101

C₂×C₄×C₈

—

102

#376

—

103

#364

—

146

—

104

#364

—

146

—

105

C₄⋊C₁₆

—

146

—

106

C₄⋊C₁₆

—

146

—

107

C₂²×C₈

—

108

C₂²×C₈

—

109

C₂²×C₈

—

110

C₂×C₄²

—

111

C₂×C₄²

—

112

M₃₂

—

146

—

113

M₃₂

—

146

—

114

M₃₂

—

146

—

115

M₃₂

—

146

—

116

C₂×C₁₆

—

146

—

117

C₂×C₁₆

—

146

—

118

C₂×C₁₆

—

146

—

119

#97

—

120

#97

—

121

#97

—

146

—

122

#97

#10596842

yes

characteristic

123

C₄×C₈

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₄×C₈

—

127

#95

—

146

—

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

146

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

146

—

140

C₂×C₈

C₂×#10596842

yes

—

141

C₄²

—

142

C₄²

—

143

C₄²

—

144

C₄²

—

146

—

145

C₄²

C₂×#10596842

yes

characteristic

146

C₁₆

—

292

—

147

C₁₆

—

292

—

148

C₁₆

—

292

—

149

C₂³

—

150

C₂×C₄

—

292

—

151

C₂×C₄

—

292

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

146

—

157

C₂×C₄

C₄⋊#10596842

yes

—

158

C₂×C₄

C₂²×#10596842

yes

characteristic

159

C₈

—

160

C₈

—

161

C₈

—

162

C₈

—

163

C₈

—

146

—

164

C₈

C₄⋊#10596842

yes

—

165

C₂²

—

166

C₂²

—

146

—

167

C₂²

C₂×C₄⋊#10596842

yes

characteristic

168

C₄

—

169

C₄

—

292

—

170

C₄

—

171

C₄

—

172

C₄

—

146

—

173

C₄

C₂×C₄⋊#10596842

yes

—

174

C₂

—

146

—

175

C₂

#22627075

yes

characteristic

176

C₂

#22627127

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊#10596842
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂×C₄	C₂²×#10596842
Fitting	#97×C₇₃	C₈
socle	C₂²×C₇₃	C₂×C₄⋊C₈
prime radical	C₂²×D₇₃	C₂×M₁₆
cosocle	C₂×#11924036	C₂³
hypercenter	#97	#10596842
hypocenter	C₇₃	#12901
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

C₂×C₄⋊#10596842

commutator

C₂×C₄×C₇₃

C₂²×C₈

Frattini

C₂×C₄

C₂²×#10596842

Fitting

#97×C₇₃

C₈

socle

C₂²×C₇₃

C₂×C₄⋊C₈

prime radical

C₂²×D₇₃

C₂×M₁₆

C₂³

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
94. C₇₃	#12901	semidirect

H ≅ G/N

type

94. C₇₃

#12901

semidirect

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

subgroup	index
C₂×#17989692	2
C₂×#17989691	2
#22627121	2
#22627120	2
#22625589	2
#12901	73

subgroup

index

Subgroups — #34034223

Conjugacy classes of subgroups

Complements

Maximal subgroups