Finite Group Databasebeta

← #34110062

Subgroups — #34110062

Order 20096.

458 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 129 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#23757785C211yesmaximal, characteristic
2C2×C4⋊C4×D157C211yesmaximal, characteristic, complemented
3#23757610C212yesmaximal, complemented
4#23757605C212yesmaximal, complemented
5#23757593C212yesmaximal, complemented
6#23757588C212yesmaximal, complemented
7#23757316C211yesmaximal, characteristic, complemented
8#23757275C211yesmaximal, characteristic, complemented
9#23757272C211yesmaximal, characteristic, complemented
10#23757236C211yesmaximal, characteristic, complemented
11#428×C157C211yesmaximal, characteristic, complemented
12C2×#14578954C2211yescharacteristic
13C22×C4×D157—21no—
14C22×C4×D157C2211yescharacteristic
15C2×#14578938C2211yescharacteristic
16#19097755—21no—
17#19097751C2211yescharacteristic
18#19097751C2211yescharacteristic
19#19097746C2211yescharacteristic
20#19097746C2211yescharacteristic
21#19097741—21no—
22C4×D628—21no—
23C4×D628—21no—
24C2×C4⋊Dic157C2211yescharacteristic
25C2×#14578933—21no—
26C4⋊C4×D157—21no—
27C4⋊C4×D157—21no—
28C4⋊C4×D157C2212yes—
29C4⋊C4×D157C2212yes—
30#19097704C2212yescomplemented
31#19097702C2212yescomplemented
32#19097696C2212yescomplemented
33#19097694C2212yescomplemented
34#19097693C2212yescomplemented
35#19097691C2212yescomplemented
36#19097683C2212yes—
37#19097683C2212yes—
38C2×M16×C157C2211yescharacteristic
39#117×C157C2211yescharacteristic
40C2×C4⋊C4×C157C2211yescharacteristic
41#107×C157C2212yescomplemented
42#106×C157C2212yescomplemented
43C23×D157D411yescharacteristic
44#14578954—22no—
45#14578954—22no—
46C2×Dic314C2311yescharacteristic
47C2×D628—21no—
48C2×D628—21no—
49C2×D628C2311yescharacteristic
50C2×C22⋊D157—21no—
51C22×Dic157—21no—
52C22×Dic157D411yescharacteristic
53C2×C4×D157—22no—
54C2×C4×D157—22no—
55C2×C4×D157—21no—
56C2×C4×D157—22no—
57C2×C4×D157—22no—
58C2×C4×D157—44no—
59C2×C4×D157D411yescharacteristic
60C2×C4×D157D411yescharacteristic
61C2×C4×D157C2311yescharacteristic
62C2×C4×D157C2311yescharacteristic
63#14578938—22no—
64C2×#11929434C2312yes—
65C4⋊Dic157—21no—
66C4⋊Dic157—21no—
67C4⋊Dic157—21no—
68C4⋊Dic157C2312yes—
69#14578933—22no—
70#14578933—22no—
71#14578931—21no—
72#14578931—21no—
73#14578931—21no—
74#14578931—21no—
75C22×C4×C157—21no—
76C22×C4×C157C2311yescharacteristic
77M16×C157—22no—
78C2×C8×C157C2312yes—
79C4⋊C4×C157—21no—
80C4⋊C4×C157C2311yescharacteristic
81C4⋊C4×C157C2312yes—
82C4⋊C4×C157C2311yescharacteristic
83C22⋊C4×C157—21no—
84C42×C157—21no—
85C22×D157—22no—
86C22×D157—22no—
87C22×D157—22no—
88C22×D157—21no—
89C22×D157C2×D411yescharacteristic
90C22×D157C2×D411yescharacteristic
91Dic314—21no—
92Dic314—21no—
93D628—21no—
94D628—21no—
95D628—42no—
96D628—42no—
97C22⋊D157—42no—
98C2×Dic157—22no—
99C2×Dic157—22no—
100C2×Dic157—21no—
101C2×Dic157—42no—
102C2×Dic157C2×D411yescharacteristic
103C2×Dic157C2×D411yescharacteristic
104C4×D157—22no—
105C4×D157—22no—
106C4×D157—21no—
107C4×D157—21no—
108C4×D157—21no—
109C4×D157—21no—
110C4×D157—44no—
111C4×D157—44no—
112C4×D157—42no—
113#11929434—22no—
114C23×C157C2×D411yescharacteristic
115C2×C4×C157—22no—
116C2×C4×C157—21no—
117C2×C4×C157—21no—
118C2×C4×C157—42no—
119C2×C4×C157C2×D411yescharacteristic
120C2×C4×C157#4112yes—
121C2×C4×C157#4112yes—
122C2×C4×C157C2411yescharacteristic
123C8×C157—22no—
124C22×C157—21no—
125C22×C157—21no—
126C22×C157#12312yes—
127C22×C157C22×D411yescharacteristic
128C2×D157—28no—
129C2×D157—21no—
130C2×D157—21no—
131C2×D157—21no—
132C2×D157—21no—
133C2×D157—41no—
134C2×D157—41no—
135C2×D157—41no—
136C2×D157AGL(1,ℤ/8)12yes—
137C2×D157#13712yes—
138C4×C157—42no—
139C4×C157—41no—
140C4×C157—41no—
141C4×C157#12312yes—
142C4×C157C2×#4111yescharacteristic
143C4×C157C2×#4111yescharacteristic
144Dic157—21no—
145Dic157—21no—
146Dic157—41no—
147Dic157—42no—
148C2×C157—22no—
149C2×C157C2×#12311yescharacteristic
150C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
151C2×C157C2×#13711yescharacteristic
152D157—22no—
153D157—22no—
154D157—81no—
155C157#309211yescharacteristic, complemented, Sylow
156#3092—1571nomaximal, Sylow
157#525—1571no—
158C22×C4⋊C4—1571no—
159#484—1572no—
160#483—1572no—
161#481—1572no—
162#480—1572no—
163#428—1571no—
164#428D15711yescharacteristic, complemented
165#417—1571no—
166#417—1571no—
167#407—1571no—
168C2×#41—1571no—
169C23×C4—3141no—
170C23×C4—1571no—
171C2×M16—1571no—
172C2×M16C2×D15711yescharacteristic
173#123—3141no—
174C22⋊Q8—1571no—
175C22⋊Q8—1571no—
176#121—1571no—
177#121—1571no—
178#120—3141no—
179C4×D4—3141no—
180C4×D4—3141no—
181#117C2×D15711yescharacteristic
182C2×C4⋊C4—3141no—
183C2×C4⋊C4—3141no—
184C2×C4⋊C4—3141no—
185C2×C4⋊C4—1571no—
186C2×C4⋊C4—1572no—
187C2×C4⋊C4—1572no—
188C2×C4⋊C4C2×D15711yescharacteristic
189#107—1572no—
190#107C2×D15712yescomplemented
191#106—1572no—
192#106C2×D15712yescomplemented
193#103—1572no—
194#103—1572no—
195#102—1572no—
196#102—1572no—
197C22⋊C8—1572no—
198C22⋊C8—1572no—
199C24—1571no—
200#41—3142no—
201#41—3142no—
202C2×Q8—1571no—
203C2×D4—3141no—
204C2×D4—3141no—
205C2×D4—3141no—
206C2×D4—1571no—
207C22×C4—21no—
208C22×C4—3142no—
209C22×C4—3142no—
210C22×C4—3142no—
211C22×C4—3142no—
212C22×C4—3141no—
213C22×C4—3141no—
214C22×C4—6284no—
215C22×C4—1571no—
216C22×C4—1571no—
217C22×C4—1571no—
218C22×C4—1571no—
219C22×C4—1571no—
220C22×C4C22×D15711yescharacteristic
221M16—22no—
222M16—3142no—
223C2×C8—1572no—
224C2×C8C22×D15712yes—
225C4⋊C4—21no—
226C4⋊C4—3142no—
227C4⋊C4—3142no—
228C4⋊C4—3141no—
229C4⋊C4—3141no—
230C4⋊C4—3141no—
231C4⋊C4—1572no—
232C4⋊C4C22×D15712yes—
233C4⋊C4C22×D15711yescharacteristic
234C4⋊C4C22×D15711yescharacteristic
235C22⋊C4—21no—
236C22⋊C4—3141no—
237C22⋊C4—3141no—
238C22⋊C4—3141no—
239C22⋊C4—3141no—
240C42—21no—
241C23—3142no—
242C23—3142no—
243C23—3142no—
244C23—3141no—
245C23—1571no—
246C23—1571no—
247C23D4×D15711yescharacteristic
248Q8—3141no—
249Q8—3141no—
250D4—3141no—
251D4—3141no—
252D4—6282no—
253D4—6282no—
254D4—6282no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—3142no—
260C2×C4—3142no—
261C2×C4—3142no—
262C2×C4—3141no—
263C2×C4—3141no—
264C2×C4—3141no—
265C2×C4—3141no—
266C2×C4—3141no—
267C2×C4—3142no—
268C2×C4—6284no—
269C2×C4—6284no—
270C2×C4—6282no—
271C2×C4—6282no—
272C2×C4—1571no—
273C2×C4—1571no—
274C2×C4D4×D15711yescharacteristic
275C2×C4#1457895412yes—
276C2×C4#1457895512yes—
277C2×C4C23×D15711yescharacteristic
278C8—22no—
279C8—3142no—
280C22—21no—
281C22—21no—
282C22—3148no—
283C22—3141no—
284C22—3141no—
285C22—3141no—
286C22—3141no—
287C22—6281no—
288C22—6281no—
289C22—6281no—
290C22—1572no—
291C22—1572no—
292C22#1909775612yes—
293C22C2×D4×D15711yescharacteristic
294C4—42no—
295C4—41no—
296C4—41no—
297C4—3141no—
298C4—3141no—
299C4—6281no—
300C4—6282no—
301C4#1909775612yes—
302C4C2×#1457895411yescharacteristic
303C4C2×#1457895511yescharacteristic
304C2—22no—
305C2—3142no—
306C2—3142no—
307C2—12561no—
308C2C2×#1909775611yescharacteristic
309C2AGL(1,ℤ/8)×D15711yescharacteristic
310C2#137×D15711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#428×C157C2
socleC22×C157C22×D4
prime radical#23757785C2
hypercenter#428D157
hypocenterC157#3092
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D157C2semidirect
3. #23757610C2semidirect
4. #23757605C2semidirect
5. #23757593C2semidirect
6. #23757588C2semidirect
7. #23757316C2semidirect
8. #23757275C2semidirect
9. #23757272C2semidirect
10. #23757236C2semidirect
11. #428×C157C2semidirect
30. #19097704C22semidirect
31. #19097702C22semidirect
32. #19097696C22semidirect
33. #19097694C22semidirect
34. #19097693C22semidirect
35. #19097691C22semidirect
41. #107×C157C22semidirect
42. #106×C157C22semidirect
155. C157#3092semidirect
164. #428D157semidirect
190. #107C2×D157semidirect
192. #106C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237577852
C2×C4⋊C4×D1572
#237576102
#237576052
#237575932
#237575882
#237573162
#237572752
#237572722
#237572362
#428×C1572
#3092157