Finite Group Databasebeta

← #34108990

Subgroups — #34108990

Order 20096.

492 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D157C211yesmaximal, characteristic
2#23757749C211yesmaximal, characteristic, complemented
3#23757745C211yesmaximal, characteristic
4#23757376C211yesmaximal, characteristic, complemented
5#23757373C211yesmaximal, characteristic, complemented
6#23757371C211yesmaximal, characteristic, complemented
7#23757370C211yesmaximal, characteristic, complemented
8C4×#14578943C211yesmaximal, characteristic, complemented
9SD16×Dic157C211yesmaximal, characteristic, complemented
10#23757360C211yesmaximal, characteristic, complemented
11#23757308C211yesmaximal, characteristic, complemented
12#23757280C211yesmaximal, characteristic, complemented
13#23757276C211yesmaximal, characteristic
14#23757227C211yesmaximal, characteristic, complemented
15C4×SD16×C157C211yesmaximal, characteristic, complemented
16C2×Q8×D157C2211yescharacteristic
17C2×D4×D157C2211yescharacteristic
18C2×#14578943C2211yescharacteristic
19C2×#14578942C2211yescharacteristic
20C2×#14578941C2211yescharacteristic
21SD16×D157C418yescomplemented
22C2×#14578937C2211yescharacteristic
23C2×C8×D157C2211yescharacteristic
24Q8×Dic157C2211yescharacteristic, complemented
25C4×Dic314—21no—
26C4×Dic314C2211yescharacteristic, complemented
27#19097736—21no—
28#19097736C2211yescharacteristic, complemented
29C4×D628C2211yescharacteristic
30D4×Dic157C2211yescharacteristic
31#19097733C2211yescharacteristic
32C4×C22⋊D157—21no—
33#19097731—21no—
34#19097729—21no—
35#19097729C2211yescharacteristic
36#19097729C2211yescharacteristic
37#19097728—21no—
38#19097728C2211yescharacteristic
39C2×#14578931—21no—
40C22⋊C4×D157—21no—
41#19097704C2211yescharacteristic, complemented
42#19097704C2211yescharacteristic, complemented
43#19097703C2211yescharacteristic, complemented
44#19097696C2211yescharacteristic, complemented
45#19097695C2211yescharacteristic, complemented
46#19097694C2211yescharacteristic, complemented
47#19097693C2211yescharacteristic
48#19097692C2211yescharacteristic
49#19097691C2211yescharacteristic
50#19097682C2211yescharacteristic, complemented
51#19097681C2211yescharacteristic, complemented
52C8×Dic157C2211yescharacteristic, complemented
53C2×SD16×C157C2211yescharacteristic
54C4×Q8×C157C2211yescharacteristic, complemented
55C4×D4×C157C2211yescharacteristic
56#106×C157C2211yescharacteristic, complemented
57#103×C157C2211yescharacteristic, complemented
58#102×C157C2211yescharacteristic
59C4×C8×C157C2211yescharacteristic, complemented
60C23×D157—21no—
61C2×Dic314—21no—
62C2×Dic314C2311yescharacteristic
63Q8×D157—22no—
64Q8×D157C2×C414yescomplemented
65C2×D628C2311yescharacteristic
66C2×C22⋊D157—21no—
67D4×D157—22no—
68D4×D157C2×C414yes—
69C22×Dic157—21no—
70C2×C4×D157—21no—
71C2×C4×D157—21no—
72C2×C4×D157—21no—
73C2×C4×D157C2311yescharacteristic
74#14578943C2×C414yescomplemented
75#14578942C2×C414yescomplemented
76#14578941C2×C414yescomplemented
77#14578937—22no—
78C2×#11929434C2311yescharacteristic
79C2×#11929434C2311yescharacteristic
80C8×D157C2×C414yescomplemented
81C4⋊Dic157—21no—
82C4⋊Dic157C2311yescharacteristic
83C4⋊Dic157C2311yescharacteristic
84#14578933—21no—
85#14578933—21no—
86#14578933D411yescharacteristic
87#14578933D411yescharacteristic
88C22⋊Dic157—21no—
89#14578931—21no—
90#14578931—21no—
91#14578931—21no—
92#14578931—21no—
93#14578931D411yescharacteristic
94#14578931D411yescharacteristic
95C4×Dic157—21no—
96C4×Dic157—21no—
97C4×Dic157—21no—
98C4×Dic157—21no—
99C4×Dic157—21no—
100C4×Dic157C2311yescharacteristic
101C2×Q8×C157C2311yescharacteristic
102C2×D4×C157C2311yescharacteristic
103C22×C4×C157—21no—
104SD16×C157C2×C414yescomplemented
105C2×C8×C157C2311yescharacteristic
106C2×C8×C157C2311yescharacteristic
107C4⋊C4×C157—21no—
108C4⋊C4×C157C2311yescharacteristic
109C4⋊C4×C157C2311yescharacteristic
110C22⋊C4×C157—21no—
111C42×C157—21no—
112C42×C157C2311yescharacteristic
113C22×D157—21no—
114C22×D157—24no—
115C22×D157—81no—
116C22×D157—81no—
117C22×D157C2×D411yescharacteristic
118Dic314—21no—
119Dic314—24no—
120Dic314C22×C412yes—
121D628—21no—
122D628C22×C412yes—
123C22⋊D157—24no—
124C2×Dic157—21no—
125C2×Dic157—21no—
126C2×Dic157—21no—
127C2×Dic157—81no—
128C2×Dic157C2×D411yescharacteristic
129C2×Dic157C2×D411yescharacteristic
130C2×Dic157C2×D411yescharacteristic
131C4×D157—24no—
132C4×D157—81no—
133C4×D157—81no—
134C4×D157C22×C412yes—
135C4×D157#4112yes—
136#11929434—21no—
137#11929434C22×C412yes—
138C23×C157—21no—
139Q8×C157—21no—
140Q8×C157C22×C412yes—
141D4×C157—21no—
142D4×C157C22×C412yes—
143C2×C4×C157—21no—
144C2×C4×C157—21no—
145C2×C4×C157—21no—
146C2×C4×C157—81no—
147C2×C4×C157C2×D411yescharacteristic
148C2×C4×C157C2×D411yescharacteristic
149C2×C4×C157C2411yescharacteristic
150C8×C157—21no—
151C8×C157C22×C412yes—
152C22×C157—22no—
153C22×C157—41no—
154C22×C157—41no—
155C22×C157C22×D411yescharacteristic
156C2×D157—21no—
157C2×D157—21no—
158C2×D157—22no—
159C2×D157—41no—
160C2×D157—41no—
161C2×D157—84no—
162C2×D157C4×D412yes—
163C4×C157—22no—
164C4×C157—21no—
165C4×C157—21no—
166C4×C157—41no—
167C4×C157—41no—
168C4×C157C23×C411yescharacteristic
169C4×C157C2×#4111yescharacteristic
170Dic157—22no—
171Dic157—21no—
172Dic157—21no—
173Dic157—41no—
174Dic157—41no—
175Dic157C4×D412yes—
176C2×C157—42no—
177C2×C157C2×C4×D411yescharacteristic
178C2×C157#57711yescharacteristic
179C2×C157#57711yescharacteristic
180D157—22no—
181D157—42no—
182C157#294911yescharacteristic, complemented, Sylow
183#2949—1571nomaximal, Sylow
184C22×SD16—1571no—
185#519—1571no—
186#518—1571no—
187#440—1571no—
188#440—1571no—
189#440—1571no—
190#440—1571no—
191C4×SD16—1571no—
192C4×SD16—1571no—
193C4×SD16—1571no—
194C4×SD16D15711yescharacteristic, complemented
195#427—1571no—
196#419—1571no—
197#418—1571no—
198#405—1571no—
199C22×Q8—1571no—
200C22×D4—1571no—
201C2×SD16—1571no—
202C2×SD16—1571no—
203C2×SD16—1578no—
204C2×SD16—1571no—
205C2×SD16C2×D15711yescharacteristic
206C2×M16—1571no—
207C22×C8—1571no—
208C4×Q8—3141no—
209C4×Q8—3141no—
210C4×Q8—1571no—
211C4×Q8—1571no—
212C4×Q8—1571no—
213C4×Q8C2×D15711yescharacteristic, complemented
214C4×D4—3141no—
215C4×D4—3141no—
216C4×D4—1571no—
217C4×D4—1571no—
218C4×D4—1571no—
219C4×D4C2×D15711yescharacteristic
220#117—3141no—
221#117—3141no—
222#117—1571no—
223#117—1571no—
224#117—1571no—
225C2×C22⋊C4—3141no—
226C2×C22⋊C4—3141no—
227#107—1571no—
228#107—1571no—
229#106—1571no—
230#106C2×D15711yescharacteristic, complemented
231#103—1571no—
232#103—1571no—
233#103—1571no—
234#103C2×D15711yescharacteristic, complemented
235#102—1571no—
236#102—1571no—
237#102—1571no—
238#102C2×D15711yescharacteristic
239#97—1571no—
240#97—1571no—
241C4×C8—1571no—
242C4×C8C2×D15711yescharacteristic, complemented
243C24—3141no—
244C2×Q8—3141no—
245C2×Q8—3142no—
246C2×Q8—1574no—
247C2×Q8—1571no—
248C2×Q8C22×D15711yescharacteristic
249C2×D4—3142no—
250C2×D4—3141no—
251C2×D4—1574no—
252C2×D4—1571no—
253C2×D4C22×D15711yescharacteristic
254C22×C4—21no—
255C22×C4—3141no—
256C22×C4—3141no—
257C22×C4—3141no—
258C22×C4—3141no—
259C22×C4—1571no—
260SD16—1574no—
261SD16—1574no—
262SD16—1574no—
263SD16C4×D15714yescomplemented
264M16—3142no—
265C2×C8—1571no—
266C2×C8—1571no—
267C2×C8—1574no—
268C2×C8C22×D15711yescharacteristic
269C2×C8C22×D15711yescharacteristic
270C4⋊C4—21no—
271C4⋊C4—3141no—
272C4⋊C4—3141no—
273C4⋊C4—3141no—
274C4⋊C4—1571no—
275C4⋊C4—1571no—
276C4⋊C4—1571no—
277C4⋊C4—1571no—
278C4⋊C4C22×D15711yescharacteristic
279C4⋊C4C22×D15711yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—3141no—
282C22⋊C4—3141no—
283C22⋊C4—3141no—
284C22⋊C4—3141no—
285C22⋊C4—3141no—
286C22⋊C4—1571no—
287C22⋊C4—1571no—
288C42—21no—
289C42—3141no—
290C42—3141no—
291C42—3141no—
292C42—3141no—
293C42—3141no—
294C42—1571no—
295C42C22×D15711yescharacteristic
296C23—21no—
297C23—3144no—
298C23—3141no—
299C23—1571no—
300C23—12561no—
301C23—12561no—
302Q8—21no—
303Q8—3144no—
304Q8—3141no—
305Q8—1572no—
306Q8C2×C4×D15712yes—
307D4—21no—
308D4—3144no—
309D4—3141no—
310D4—1572no—
311D4C2×C4×D15712yes—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—81no—
316C2×C4—3144no—
317C2×C4—3141no—
318C2×C4—3141no—
319C2×C4—3141no—
320C2×C4—1571no—
321C2×C4—1571no—
322C2×C4—1571no—
323C2×C4—1572no—
324C2×C4—1572no—
325C2×C4—12561no—
326C2×C4—12561no—
327C2×C4—12561no—
328C2×C4D4×D15711yescharacteristic
329C2×C4D4×D15711yescharacteristic
330C2×C4C23×D15711yescharacteristic
331C8—21no—
332C8—3141no—
333C8—1572no—
334C8C2×C4×D15712yes—
335C22—22no—
336C22—41no—
337C22—41no—
338C22—3141no—
339C22—3141no—
340C22—3142no—
341C22—6281no—
342C22—6281no—
343C22—1572no—
344C22—12564no—
345C22C2×D4×D15711yescharacteristic
346C4—21no—
347C4—22no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—3141no—
352C4—3142no—
353C4—3141no—
354C4—6281no—
355C4—6281no—
356C4—1572no—
357C4C22×C4×D15711yescharacteristic
358C4#41×D15711yescharacteristic
359C2—42no—
360C2—3142no—
361C2—6282no—
362C2C4×D4×D15711yescharacteristic
363C2#2375807911yescharacteristic
364C2#2375808011yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutatorC4×C157C23×C4
FrattiniC2×C4C23×D157
FittingC4×SD16×C157C2
socleC22×C157C22×D4
prime radicalC2×D4×D157C22
cosocleC2×C4×C157C24
hypercenterC4×SD16D157
hypocenterC157#2949
LCSG ▷ C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
2. #23757749C2semidirect
4. #23757376C2semidirect
5. #23757373C2semidirect
6. #23757371C2semidirect
7. #23757370C2semidirect
8. C4×#14578943C2semidirect
9. SD16×Dic157C2semidirect
10. #23757360C2semidirect
11. #23757308C2semidirect
12. #23757280C2semidirect
14. #23757227C2semidirect
15. C4×SD16×C157C2semidirect
21. SD16×D157C4semidirect
24. Q8×Dic157C22semidirect
26. C4×Dic314C22semidirect
28. #19097736C22semidirect
41. #19097704C22semidirect
42. #19097704C22semidirect
43. #19097703C22semidirect
44. #19097696C22semidirect
45. #19097695C22semidirect
46. #19097694C22semidirect
50. #19097682C22semidirect
51. #19097681C22semidirect
52. C8×Dic157C22semidirect
54. C4×Q8×C157C22semidirect
56. #106×C157C22semidirect
57. #103×C157C22semidirect
59. C4×C8×C157C22semidirect
64. Q8×D157C2×C4semidirect
74. #14578943C2×C4semidirect
75. #14578942C2×C4semidirect
76. #14578941C2×C4semidirect
80. C8×D157C2×C4semidirect
104. SD16×C157C2×C4semidirect
182. C157#2949semidirect
194. C4×SD16D157semidirect
213. C4×Q8C2×D157semidirect
230. #106C2×D157semidirect
234. #103C2×D157semidirect
242. C4×C8C2×D157semidirect
263. SD16C4×D157semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1572
#237577492
#237577452
#237573762
#237573732
#237573712
#237573702
C4×#145789432
SD16×Dic1572
#237573602
#237573082
#237572802
#237572762
#237572272
C4×SD16×C1572
#2949157