Finite Group Databasebeta

← #34110059

Subgroups — #34110059

Order 20096.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757784C211yesmaximal, characteristic
2#23757608C212yesmaximal, complemented
3#23757603C212yesmaximal, complemented
4#23757591C212yesmaximal, complemented
5#23757586C212yesmaximal, complemented
6#23757313C212yesmaximal, complemented
7#23757272C212yesmaximal, complemented
8#23757237C211yesmaximal, characteristic, complemented
9C22×C4⋊C4×C157C211yesmaximal, characteristic, complemented
10C2×#14578954C2211yescharacteristic
11C2×#14578938C2212yes—
12#19097754—21no—
13#19097750C2211yescharacteristic
14#19097750C2211yescharacteristic
15#19097745C2211yescharacteristic
16#19097745C2211yescharacteristic
17#19097740—21no—
18C4×C22⋊D157—21no—
19C4×C22⋊D157—21no—
20#19097730C2211yescharacteristic
21#19097704C2214yescomplemented
22#19097702C2214yescomplemented
23#19097694C2214yescomplemented
24#19097691C2214yescomplemented
25C22⋊#11929434C2212yes—
26C22⋊#11929434C2212yes—
27C23×C4×C157—21no—
28C23×C4×C157C2211yescharacteristic
29C2×C4⋊C4×C157—21no—
30C2×C4⋊C4×C157—21no—
31C2×C4⋊C4×C157—21no—
32C2×C4⋊C4×C157C2212yes—
33C2×C4⋊C4×C157C2212yes—
34C2×C4⋊C4×C157C2212yes—
35#14578954—24no—
36C2×Dic314C2311yescharacteristic
37C2×D628C2311yescharacteristic
38C2×C22⋊D157—21no—
39C2×C22⋊D157—21no—
40C2×C22⋊D157—21no—
41C2×C4×D157—21no—
42#14578938—24no—
43C2×#11929434C2314yes—
44C4⋊Dic157C2311yescharacteristic
45C4⋊Dic157C2311yescharacteristic
46#14578933—21no—
47#14578933—21no—
48C22⋊Dic157—21no—
49C22⋊Dic157—21no—
50C22⋊Dic157—21no—
51#14578931—21no—
52#14578931—21no—
53C4×Dic157—21no—
54C24×C157D411yescharacteristic
55C22×C4×C157—24no—
56C22×C4×C157—22no—
57C22×C4×C157—22no—
58C22×C4×C157—22no—
59C22×C4×C157—44no—
60C22×C4×C157D411yescharacteristic
61C22×C4×C157D411yescharacteristic
62C22×C4×C157D411yescharacteristic
63C22×C4×C157C2311yescharacteristic
64C22×C4×C157C2311yescharacteristic
65C22×C4×C157C2311yescharacteristic
66C4⋊C4×C157—22no—
67C4⋊C4×C157—22no—
68C4⋊C4×C157—22no—
69C4⋊C4×C157C2314yes—
70C22×D157—21no—
71Dic314—22no—
72D628—22no—
73C22⋊D157—42no—
74C22⋊D157—42no—
75C22⋊D157—42no—
76C2×Dic157—21no—
77C2×Dic157—21no—
78C2×Dic157—21no—
79C4×D157—42no—
80#11929434—24no—
81C23×C157—22no—
82C23×C157—22no—
83C23×C157—22no—
84C23×C157C2×D411yescharacteristic
85C23×C157C2×D411yescharacteristic
86C23×C157C2×D411yescharacteristic
87C2×C4×C157—22no—
88C2×C4×C157—22no—
89C2×C4×C157—22no—
90C2×C4×C157—22no—
91C2×C4×C157—22no—
92C2×C4×C157—24no—
93C2×C4×C157—44no—
94C2×C4×C157—44no—
95C2×C4×C157—44no—
96C2×C4×C157C2×D411yescharacteristic
97C2×C4×C157C2×D411yescharacteristic
98C2×C4×C157C2×D411yescharacteristic
99C2×C4×C157#4114yes—
100C2×C4×C157C2411yescharacteristic
101C22×C157—28no—
102C22×C157—21no—
103C22×C157—21no—
104C22×C157—21no—
105C22×C157—21no—
106C22×C157—21no—
107C22×C157—21no—
108C22×C157#12312yes—
109C22×C157AGL(1,ℤ/8)12yes—
110C22×C157#13712yes—
111C22×C157C22×D411yescharacteristic
112C2×D157—41no—
113C2×D157—41no—
114C2×D157—41no—
115C4×C157—21no—
116C4×C157—21no—
117C4×C157—44no—
118C4×C157#12312yes—
119C4×C157C2×#4112yes—
120Dic157—41no—
121Dic157—41no—
122Dic157—41no—
123C2×C157—22no—
124C2×C157—22no—
125C2×C157—22no—
126C2×C157C2×#12311yescharacteristic
127C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
128C2×C157C2×#13711yescharacteristic
129D157—81no—
130C157#309211yescharacteristic, complemented, Sylow
131#3092—1571nomaximal, Sylow
132#525—1571no—
133C22×C4⋊C4D15711yescharacteristic, complemented
134#484—1572no—
135#483—1572no—
136#481—1572no—
137#480—1572no—
138#428—1572no—
139#417—1572no—
140#407—1571no—
141C2×#41—1571no—
142C23×C4—21no—
143C23×C4C2×D15711yescharacteristic
144C2×M16—1572no—
145#123—3141no—
146C22⋊Q8—1571no—
147C22⋊Q8—1571no—
148#121—1571no—
149#121—1571no—
150#120—3141no—
151C4×D4—3141no—
152C4×D4—3141no—
153#117—1571no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4C2×D15712yes—
158C2×C4⋊C4C2×D15712yes—
159C2×C4⋊C4C2×D15712yes—
160#107—1574no—
161#106—1574no—
162#103—1574no—
163#102—1574no—
164C22⋊C8—1572no—
165C22⋊C8—1572no—
166C24C22⋊D15711yescharacteristic
167#41—3144no—
168C2×Q8—1571no—
169C2×D4—3141no—
170C2×D4—3141no—
171C2×D4—3141no—
172C2×D4—1571no—
173C22×C4—24no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—44no—
178C22×C4—3141no—
179C22×C4C22⋊D15711yescharacteristic
180C22×C4C22⋊D15711yescharacteristic
181C22×C4C22⋊D15711yescharacteristic
182C22×C4C22×D15711yescharacteristic
183C22×C4C22×D15711yescharacteristic
184C22×C4C22×D15711yescharacteristic
185M16—3144no—
186C2×C8—1574no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—3141no—
191C4⋊C4—3141no—
192C4⋊C4—1571no—
193C4⋊C4—1571no—
194C4⋊C4C22×D15714yes—
195C22⋊C4—3141no—
196C22⋊C4—3141no—
197C22⋊C4—3141no—
198C22⋊C4—3141no—
199C22⋊C4—3141no—
200C42—3141no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—3141no—
205C23C2×C22⋊D15711yescharacteristic
206C23C2×C22⋊D15711yescharacteristic
207C23C2×C22⋊D15711yescharacteristic
208Q8—3142no—
209D4—3142no—
210D4—6282no—
211D4—6282no—
212D4—6282no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—3141no—
223C2×C4—3141no—
224C2×C4—3141no—
225C2×C4—6282no—
226C2×C4C2×C22⋊D15711yescharacteristic
227C2×C4C2×C22⋊D15711yescharacteristic
228C2×C4C2×C22⋊D15711yescharacteristic
229C2×C4#1457895414yes—
230C2×C4C23×D15711yescharacteristic
231C8—3144no—
232C22—28no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—6281no—
240C22—6281no—
241C22—6281no—
242C22#1909775412yes—
243C22#1909779712yes—
244C22#1909780212yes—
245C22C22×C22⋊D15711yescharacteristic
246C4—21no—
247C4—21no—
248C4—44no—
249C4—6281no—
250C4—6281no—
251C4—6281no—
252C4#1909775412yes—
253C4C2×#1457895412yes—
254C2—22no—
255C2—22no—
256C2—22no—
257C2—12561no—
258C2C2×#1909775411yescharacteristic
259C2C2×#1909779711yescharacteristic
260C2C2×#1909780211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC22×C4⋊C4×C157C2
socleC22×C157C22×D4
prime radical#23757784C2
hypercenterC22×C4⋊C4D157
hypocenterC157#3092
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #23757608C2semidirect
3. #23757603C2semidirect
4. #23757591C2semidirect
5. #23757586C2semidirect
6. #23757313C2semidirect
7. #23757272C2semidirect
8. #23757237C2semidirect
9. C22×C4⋊C4×C157C2semidirect
21. #19097704C22semidirect
22. #19097702C22semidirect
23. #19097694C22semidirect
24. #19097691C22semidirect
130. C157#3092semidirect
133. C22×C4⋊C4D157semidirect

Maximal subgroups

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

subgroupindex
#237577842
#237576082
#237576032
#237575912
#237575862
#237573132
#237572722
#237572372
C22×C4⋊C4×C1572
#3092157