Group #34000433 — subgroups

Subgroups — #34000433

Order 18688.

214 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 62 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	#22625852	C₂	1	1	yes	maximal, characteristic
2	#22625494	C₂	1	1	yes	maximal, characteristic
3	#22611732	C₂	1	1	yes	maximal, characteristic
4	#17989609	—	2	1	no	—
5	#17989533	—	2	1	no	—
6	#17989461	—	2	1	no	—
7	#17989461	C₄	1	2	yes	—
8	#17989451	C₄	1	2	yes	complemented
9	#17989449	—	2	1	no	—
10	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
11	#17988405	—	2	1	no	—
12	#17988366	—	2	1	no	—
13	#17988363	C₄	1	1	yes	characteristic, complemented
14	#346×C₇₃	C₄	1	1	yes	characteristic, complemented
15	C₂×C₄⋊#64408	—	2	2	no	—
16	C₂×C₄⋊#64408	—	4	1	no	—
17	C₂×C₄⋊#64408	—	4	1	no	—
18	C₄⋊#10596833	—	2	2	no	—
19	#14556648	—	4	1	no	—
20	#14556646	—	4	1	no	—
21	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
23	C₄²×D₇₃	D₄	1	1	yes	characteristic
24	C₂×#11924003	—	2	1	no	—
25	C₄×#10596831	Q₈	1	1	yes	characteristic
26	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₇₃	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₇₃	—	2	1	no	—
29	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×#64408	—	4	2	no	—
31	C₂×#10596833	—	4	2	no	—
32	C₄⋊#64408	—	4	2	no	—
33	C₄⋊#64408	—	4	2	no	—
34	C₄⋊#64408	—	8	2	no	—
35	C₄⋊#64408	—	8	2	no	—
36	C₂×C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
38	C₂×C₄×D₇₃	SD₁₆	1	1	yes	characteristic
39	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
40	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
41	C₈×D₇₃	—	4	1	no	—
42	C₈×D₇₃	—	4	1	no	—
43	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₄×Dic₇₃	SD₁₆	1	1	yes	characteristic
45	C₄×Dic₇₃	Q₁₆	1	1	yes	characteristic
46	#11924003	—	4	1	no	—
47	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
48	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
50	C₁₆×C₇₃	—	4	1	no	—
51	C₂²×D₇₃	#102	1	1	yes	characteristic
52	C₂×#64408	—	8	2	no	—
53	C₂×#64408	—	8	2	no	—
54	C₂×#64408	—	8	2	no	—
55	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
57	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
58	C₂×Dic₇₃	#102	1	1	yes	characteristic
59	C₂×Dic₇₃	#103	1	1	yes	characteristic
60	C₂×Dic₇₃	#103	1	1	yes	characteristic
61	C₄×D₇₃	—	2	1	no	—
62	C₄×D₇₃	—	2	1	no	—
63	C₄×D₇₃	—	2	1	no	—
64	C₄×D₇₃	—	2	1	no	—
65	C₄×D₇₃	—	2	2	no	—
66	#10596833	—	8	2	no	—
67	C₈×C₇₃	—	2	1	no	—
68	C₈×C₇₃	—	2	1	no	—
69	#10596831	—	2	1	no	—
70	#10596831	—	2	1	no	—
71	C₂²×C₇₃	#340	1	1	yes	characteristic
72	C₂×D₇₃	—	2	1	no	—
73	C₂×D₇₃	—	2	1	no	—
74	C₂×D₇₃	#361	1	1	yes	characteristic
75	C₂×D₇₃	#362	1	1	yes	characteristic
76	#64408	—	16	2	no	—
77	C₄×C₇₃	—	2	1	no	—
78	C₄×C₇₃	—	2	1	no	—
79	C₄×C₇₃	H(3,ℤ/4)	1	1	yes	characteristic
80	C₄×C₇₃	#344	1	1	yes	characteristic
81	Dic₇₃	—	2	1	no	—
82	Dic₇₃	—	2	1	no	—
83	Dic₇₃	#360	1	2	yes	—
84	#7222	—	73	1	no	maximal, Sylow
85	C₂×C₇₃	#1286	1	1	yes	characteristic
86	C₂×C₇₃	#1386	1	1	yes	characteristic
87	C₂×C₇₃	#1387	1	1	yes	characteristic
88	D₇₃	—	2	1	no	—
89	D₇₃	—	2	1	no	—
90	#2163	—	73	1	no	—
91	#1834	—	73	1	no	—
92	#1568	—	73	1	no	—
93	C₇₃	#7222	1	1	yes	characteristic, complemented, Sylow
94	#432	—	146	1	no	—
95	C₂×C₄×C₈	—	73	1	no	—
96	#384	—	146	1	no	—
97	C₄⋊C₁₆	—	146	1	no	—
98	C₂²⋊C₁₆	—	146	1	no	—
99	#346	—	73	1	no	—
100	#346	#64408	1	1	yes	characteristic, complemented
101	#340	—	73	2	no	—
102	#340	—	146	1	no	—
103	#335	—	73	2	no	—
104	#334	—	146	1	no	—
105	C₂²×C₈	—	73	1	no	—
106	C₂²×C₈	—	73	1	no	—
107	C₂×C₄⋊C₄	—	292	1	no	—
108	C₂×C₄⋊C₄	—	292	1	no	—
109	C₂×C₄⋊C₄	—	146	2	no	—
110	C₂×C₄²	—	73	1	no	—
111	C₂×C₁₆	—	2	1	no	—
112	C₂×C₁₆	—	146	1	no	—
113	C₄⋊C₈	—	146	2	no	—
114	C₂²⋊C₈	—	292	1	no	—
115	#97	—	292	1	no	—
116	C₄×C₈	—	73	1	no	—
117	C₄×C₈	—	73	1	no	—
118	C₄×C₈	—	73	1	no	—
119	C₄×C₈	C₂×#64408	1	1	yes	characteristic
120	C₂²×C₄	—	292	2	no	—
121	C₂²×C₄	—	73	1	no	—
122	C₂²×C₄	—	73	1	no	—
123	C₂²×C₄	—	73	1	no	—
124	C₂×C₈	—	292	1	no	—
125	C₂×C₈	—	292	1	no	—
126	C₂×C₈	—	292	2	no	—
127	C₂×C₈	—	73	1	no	—
128	C₂×C₈	—	73	1	no	—
129	C₂×C₈	C₄×#64408	1	1	yes	characteristic
130	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
131	C₄⋊C₄	—	292	2	no	—
132	C₄⋊C₄	—	292	2	no	—
133	C₄⋊C₄	—	584	2	no	—
134	C₄⋊C₄	—	584	2	no	—
135	C₄²	—	73	1	no	—
136	C₄²	—	73	1	no	—
137	C₄²	—	73	1	no	—
138	C₄²	C₄⋊#64408	1	1	yes	characteristic
139	C₁₆	—	4	1	no	—
140	C₁₆	—	292	1	no	—
141	C₂³	—	73	1	no	—
142	C₂×C₄	—	584	2	no	—
143	C₂×C₄	—	584	2	no	—
144	C₂×C₄	—	584	2	no	—
145	C₂×C₄	—	73	1	no	—
146	C₂×C₄	—	73	1	no	—
147	C₂×C₄	—	73	1	no	—
148	C₂×C₄	—	146	2	no	—
149	C₂×C₄	—	146	1	no	—
150	C₂×C₄	—	146	1	no	—
151	C₂×C₄	—	146	1	no	—
152	C₂×C₄	—	146	1	no	—
153	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
154	C₂×C₄	#14556659	1	1	yes	characteristic
155	C₂×C₄	#14556660	1	1	yes	characteristic
156	C₈	—	2	1	no	—
157	C₈	—	2	1	no	—
158	C₈	—	584	2	no	—
159	C₈	—	146	1	no	—
160	C₈	—	146	1	no	—
161	C₂²	—	73	1	no	—
162	C₂²	—	73	1	no	—
163	C₂²	—	146	1	no	—
164	C₂²	—	146	1	no	—
165	C₂²	#17989461	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	73	2	no	—
169	C₄	—	1168	2	no	—
170	C₄	—	146	1	no	—
171	C₄	—	146	1	no	—
172	C₄	#17989454	1	1	yes	characteristic
173	C₄	#17989469	1	1	yes	characteristic
174	C₂	—	146	1	no	—
175	C₂	—	146	1	no	—
176	C₂	#22624785	1	1	yes	characteristic
177	C₂	#22625003	1	1	yes	characteristic
178	C₂	#22625006	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22625852

C₂

yes

maximal, characteristic

#22625494

C₂

yes

maximal, characteristic

#22611732

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

—

#17989451

C₄

yes

complemented

#17989449

—

C₄×C₈×D₇₃

C₂²

yes

—

—

C₄

yes

characteristic, complemented

#346×C₇₃

C₄

yes

characteristic, complemented

C₂×C₄⋊#64408

—

C₂×C₄⋊#64408

—

C₂×C₄⋊#64408

—

C₄⋊#10596833

—

#14556648

—

#14556646

—

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₂×C₈×D₇₃

D₄

yes

characteristic

C₄²×D₇₃

D₄

yes

characteristic

C₂×#11924003

—

C₄×#10596831

Q₈

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×#64408

—

—

—

—

—

—

C₂×C₄×D₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₇₃

D₈

yes

characteristic

C₂×C₄×D₇₃

SD₁₆

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₄×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₇₃

SD₁₆

yes

characteristic

C₄×Dic₇₃

Q₁₆

yes

characteristic

#11924003

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₄²×C₇₃

C₄⋊C₄

yes

characteristic

C₁₆×C₇₃

—

C₂²×D₇₃

yes

—

—

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

#340

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

yes

C₂×D₇₃

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#360

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

D₇₃

—

#2163

—

#1834

—

#1568

—

C₇₃

#7222

yes

characteristic, complemented, Sylow

#432

—

146

—

C₂×C₄×C₈

—

#384

—

146

—

C₄⋊C₁₆

—

146

—

C₂²⋊C₁₆

—

146

—

#346

—

100

#346

#64408

yes

characteristic, complemented

101

#340

—

102

#340

—

146

—

103

#335

—

104

#334

—

146

—

105

C₂²×C₈

—

106

C₂²×C₈

—

107

C₂×C₄⋊C₄

—

292

—

108

C₂×C₄⋊C₄

—

292

—

109

C₂×C₄⋊C₄

—

146

—

110

C₂×C₄²

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

146

—

113

C₄⋊C₈

—

146

—

114

C₂²⋊C₈

—

292

—

115

#97

—

292

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₄×C₈

C₂×#64408

yes

characteristic

120

C₂²×C₄

—

292

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂×C₈

—

292

—

125

C₂×C₈

—

292

—

126

C₂×C₈

—

292

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

C₄×#64408

yes

characteristic

130

C₂×C₈

C₂²⋊#64408

yes

characteristic

131

C₄⋊C₄

—

292

—

132

C₄⋊C₄

—

292

—

133

C₄⋊C₄

—

584

—

134

C₄⋊C₄

—

584

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

C₄⋊#64408

yes

characteristic

139

C₁₆

—

140

C₁₆

—

292

—

141

C₂³

—

142

C₂×C₄

—

584

—

143

C₂×C₄

—

584

—

144

C₂×C₄

—

584

—

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

—

146

—

153

C₂×C₄

(C₂×#64408)⋊C₄

yes

characteristic

154

C₂×C₄

#14556659

yes

characteristic

155

C₂×C₄

#14556660

yes

characteristic

156

C₈

—

157

C₈

—

158

C₈

—

584

—

159

C₈

—

146

—

160

C₈

—

146

—

161

C₂²

—

162

C₂²

—

163

C₂²

—

146

—

164

C₂²

—

146

—

165

C₂²

#17989461

yes

characteristic

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

1168

—

170

C₄

—

146

—

171

C₄

—

146

—

172

C₄

#17989454

yes

characteristic

173

C₄

#17989469

yes

characteristic

174

C₂

—

146

—

175

C₂

—

146

—

176

C₂

#22624785

yes

characteristic

177

C₂

#22625003

yes

characteristic

178

C₂

#22625006

yes

characteristic

	subgroup	G/H
center	C₂²	#17989461
commutator	C₂×C₈×C₇₃	C₄²
Frattini	C₄×C₈	C₂×#64408
Fitting	#346×C₇₃	C₄
socle	C₂²×C₇₃	#340
prime radical	C₂²×D₇₃	#102
cosocle	C₄×C₈×D₇₃	C₂²
hypercenter	#346	#64408
hypocenter	C₇₃	#7222
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

subgroup

G/H

center

C₂²

#17989461

commutator

C₂×C₈×C₇₃

C₄²

Frattini

C₄×C₈

C₂×#64408

Fitting

#346×C₇₃

C₄

C₂²×C₇₃

C₂²×D₇₃

C₄×C₈×D₇₃

C₂²

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

N	H ≅ G/N	type
8. #17989451	C₄	semidirect
13. #17988363	C₄	semidirect
14. #346×C₇₃	C₄	semidirect
93. C₇₃	#7222	semidirect
100. #346	#64408	semidirect

H ≅ G/N

type

8. #17989451

C₄

semidirect

13. #17988363

C₄

semidirect

14. #346×C₇₃

C₄

semidirect

93. C₇₃

#7222

semidirect

100. #346

#64408

semidirect

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

subgroup	index
#22625852	2
#22625494	2
#22611732	2
#7222	73

subgroup

index

Subgroups — #34000433

Conjugacy classes of subgroups

Complements

Maximal subgroups