Finite Group Databasebeta

← #34111708

Subgroups — #34111708

Order 20096.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097803C211yesmaximal, characteristic, complemented
2C2×Q16×D157C211yesmaximal, characteristic, complemented
3Q8×D628C211yesmaximal, characteristic, complemented
4#23757935C211yesmaximal, characteristic
5#23757667C211yesmaximal, characteristic, complemented
6#23757532C211yesmaximal, characteristic, complemented
7#23757503C211yesmaximal, characteristic, complemented
8#23757495C211yesmaximal, characteristic, complemented
9#23757483C211yesmaximal, characteristic, complemented
10#23757438C211yesmaximal, characteristic, complemented
11#23757430C211yesmaximal, characteristic, complemented
12#23757430C211yesmaximal, characteristic, complemented
13#23757413C211yesmaximal, characteristic, complemented
14#23757346C211yesmaximal, characteristic, complemented
15#441×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×Q8×D157C2211yescharacteristic
18C2×Q8×D157C2211yescharacteristic
19C2×Q8×D157C2211yescharacteristic
20#19097803—24no—
21C2×Dic628C2211yescharacteristic
22C2×Dic628C2211yescharacteristic
23C2×#14578944C2211yescharacteristic
24C2×#14578944C2211yescharacteristic
25C2×#14578944C2211yescharacteristic
26Q16×D157—24no—
27C2×#14578943C2211yescharacteristic
28C2×#14578943C2211yescharacteristic
29C2×#14578941C2211yescharacteristic
30C2×#14578937C2211yescharacteristic
31C2×C8×D157C2211yescharacteristic
32#19097771—21no—
33#19097771C2211yescharacteristic
34#19097759—21no—
35#19097759C2211yescharacteristic
36#19097751—21no—
37#19097751—21no—
38#19097751—21no—
39#19097751C2211yescharacteristic
40#19097751C2211yescharacteristic
41#19097751C2211yescharacteristic
42#19097746—21no—
43#19097746C2211yescharacteristic
44C4×D628—21no—
45C4×D628—21no—
46C4×D628C2211yescharacteristic
47C4⋊#11929434C2211yescharacteristic
48#19097694C2211yescharacteristic
49#19097694C2211yescharacteristic
50#19097694C2211yescharacteristic
51#19097691C2211yescharacteristic
52#19097683C2211yescharacteristic
53#19097683C2211yescharacteristic
54C2×Q16×C157C2211yescharacteristic
55C4×Q8×C157C2211yescharacteristic
56C4×Q8×C157C2211yescharacteristic
57#106×C157C2211yescharacteristic
58#103×C157C2211yescharacteristic
59#103×C157C2211yescharacteristic
60#97×C157C2211yescharacteristic
61#14578955—22no—
62#14578955—41no—
63C2×Dic314—21no—
64C2×Dic314—21no—
65C2×Dic314—21no—
66C2×Dic314C2311yescharacteristic
67C2×Dic314C2311yescharacteristic
68C2×Dic314C2311yescharacteristic
69Q8×D157—22no—
70Q8×D157—22no—
71Q8×D157—22no—
72Q8×D157—41no—
73Q8×D157—41no—
74Q8×D157—41no—
75C2×D628—21no—
76C2×D628—21no—
77C2×D628D411yescharacteristic
78C2×D628C2311yescharacteristic
79C2×C4×D157—21no—
80C2×C4×D157—21no—
81C2×C4×D157—21no—
82C2×C4×D157—21no—
83C2×C4×D157C2311yescharacteristic
84C2×C4×D157C2311yescharacteristic
85Dic628—22no—
86Dic628—41no—
87#14578944—22no—
88#14578944—22no—
89#14578944—22no—
90#14578943—22no—
91#14578943—41no—
92#14578941—22no—
93#14578937—22no—
94C2×#11929434C2311yescharacteristic
95C2×#11929434C2311yescharacteristic
96C8×D157—22no—
97C4⋊Dic157—21no—
98C4⋊Dic157—21no—
99C4⋊Dic157—21no—
100C4⋊Dic157—21no—
101C4⋊Dic157—21no—
102C4⋊Dic157D411yescharacteristic
103#14578931—21no—
104#14578931—21no—
105#14578931—21no—
106#14578931—21no—
107#14578931—21no—
108#14578931—21no—
109#14578931D411yescharacteristic
110#14578931D411yescharacteristic
111C2×Q8×C157C2311yescharacteristic
112C2×Q8×C157C2311yescharacteristic
113Q16×C157D414yescomplemented
114C2×C8×C157C2311yescharacteristic
115C2×C8×C157C2311yescharacteristic
116C4⋊C4×C157—21no—
117C4⋊C4×C157—21no—
118C4⋊C4×C157C2311yescharacteristic
119C4⋊C4×C157C2311yescharacteristic
120C42×C157—21no—
121C42×C157—21no—
122C42×C157C2311yescharacteristic
123C22×D157—21no—
124C22×D157C2×D411yescharacteristic
125C22×D157C2×D411yescharacteristic
126Dic314—21no—
127Dic314—21no—
128Dic314—21no—
129Dic314—21no—
130Dic314—21no—
131Dic314—21no—
132Dic314—42no—
133Dic314—42no—
134Dic314—42no—
135D628—22no—
136D628—21no—
137D628—21no—
138D628—42no—
139D628—42no—
140C2×Dic157—21no—
141C2×Dic157—21no—
142C2×Dic157—21no—
143C2×Dic157C2×D411yescharacteristic
144C2×Dic157C2×D411yescharacteristic
145C4×D157—21no—
146C4×D157—21no—
147C4×D157—21no—
148C4×D157—21no—
149C4×D157—42no—
150C4×D157—42no—
151C4×D157—42no—
152C4×D157—42no—
153#11929434—21no—
154#11929434—21no—
155Q8×C157—21no—
156Q8×C157—21no—
157Q8×C157C2×D412yes—
158Q8×C157C2×D412yes—
159C2×C4×C157—21no—
160C2×C4×C157—21no—
161C2×C4×C157—21no—
162C2×C4×C157—21no—
163C2×C4×C157C2×D411yescharacteristic
164C2×C4×C157C2×D411yescharacteristic
165C2×C4×C157C2411yescharacteristic
166C8×C157—21no—
167C8×C157C2×D412yes—
168C22×C157C22×D411yescharacteristic
169C2×D157—21no—
170C2×D157—21no—
171C2×D157—21no—
172C2×D157—21no—
173C2×D157—21no—
174C2×D157—41no—
175C2×D157—41no—
176C2×D157—41no—
177C2×D157#13712yescomplemented
178C4×C157—22no—
179C4×C157—22no—
180C4×C157—21no—
181C4×C157—21no—
182C4×C157—41no—
183C4×C157—41no—
184C4×C157C22×D411yescharacteristic
185C4×C157H(5,𝔽2)11yescharacteristic
186Dic157—21no—
187Dic157—21no—
188Dic157—41no—
189Dic157—41no—
190Dic157—41no—
191C2×C157D4211yescharacteristic
192C2×C157C2×#13711yescharacteristic
193C2×C157#57811yescharacteristic
194D157—22no—
195D157—41no—
196D157—81no—
197C157#327011yescharacteristic, complemented, Sylow
198#3270—1571nomaximal, Sylow
199C2×#137—1571no—
200C22×Q16—1571no—
201D4×Q8—1571no—
202#548—1571no—
203#497—1571no—
204#470—1571no—
205#465—1571no—
206#464—1571no—
207#462—1571no—
208#452—1571no—
209#451—1571no—
210#451—1571no—
211#448—1571no—
212#441D15711yescharacteristic, complemented
213#435—1571no—
214C2×#41—1571no—
215C22×Q8—1571no—
216C22×Q8—1571no—
217C22×Q8—1571no—
218#137—3144no—
219C2×Q16—3144no—
220C2×Q16—1571no—
221C2×Q16—1571no—
222C2×Q16—1571no—
223C2×Q16—1571no—
224C2×Q16—1571no—
225C2×Q16C2×D15711yescharacteristic
226C2×SD16—1571no—
227C2×SD16—1571no—
228C2×SD16—1571no—
229C2×M16—1571no—
230C22×C8—1571no—
231C4⋊Q8—3141no—
232C4⋊Q8—1571no—
233#124—3141no—
234#124—1571no—
235C22⋊Q8—3141no—
236C22⋊Q8—3141no—
237C22⋊Q8—3141no—
238C22⋊Q8—1571no—
239C22⋊Q8—1571no—
240C22⋊Q8—1571no—
241#121—3141no—
242#121—1571no—
243C4×Q8C2×D15711yescharacteristic
244C4×Q8C2×D15711yescharacteristic
245C4×D4—3141no—
246C4×D4—3141no—
247C4×D4—1571no—
248#106C2×D15711yescharacteristic
249C4⋊C8—1571no—
250#103—1571no—
251#103—1571no—
252#103—1571no—
253#103C2×D15711yescharacteristic
254#103C2×D15711yescharacteristic
255#102—1571no—
256C22⋊C8—1571no—
257C22⋊C8—1571no—
258#97C2×D15711yescharacteristic
259#41—3142no—
260#41—6281no—
261C2×Q8—3141no—
262C2×Q8—3141no—
263C2×Q8—3141no—
264C2×Q8—3142no—
265C2×Q8—3142no—
266C2×Q8—3142no—
267C2×Q8—6281no—
268C2×Q8—6281no—
269C2×Q8—6281no—
270C2×Q8—1571no—
271C2×Q8—1571no—
272C2×Q8—1571no—
273C2×Q8C22×D15711yescharacteristic
274C2×Q8C22×D15711yescharacteristic
275C2×D4—3141no—
276C2×D4—3141no—
277C2×D4—1571no—
278C2×D4—1571no—
279C22×C4—3141no—
280C22×C4—3141no—
281C22×C4—3141no—
282C22×C4—3141no—
283C22×C4—1571no—
284C22×C4—1571no—
285Q16—3142no—
286Q16—3142no—
287Q16—3142no—
288Q16—3142no—
289Q16—6281no—
290Q16D62814yescomplemented
291SD16—3142no—
292SD16—3142no—
293SD16—6281no—
294M16—3142no—
295C2×C8—3142no—
296C2×C8—1571no—
297C2×C8—1571no—
298C2×C8C22×D15711yescharacteristic
299C2×C8C22×D15711yescharacteristic
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—3141no—
303C4⋊C4—3141no—
304C4⋊C4—3141no—
305C4⋊C4—3141no—
306C4⋊C4—3141no—
307C4⋊C4—1571no—
308C4⋊C4C22×D15711yescharacteristic
309C4⋊C4C22×D15711yescharacteristic
310C22⋊C4—3141no—
311C22⋊C4—3141no—
312C22⋊C4—3141no—
313C22⋊C4—3141no—
314C22⋊C4—3141no—
315C22⋊C4—3141no—
316C22⋊C4—1571no—
317C22⋊C4—1571no—
318C42—21no—
319C42—21no—
320C42C22×D15711yescharacteristic
321C23—3141no—
322C23—1571no—
323C23—1571no—
324Q8—21no—
325Q8—21no—
326Q8—3141no—
327Q8—3141no—
328Q8—3141no—
329Q8—3141no—
330Q8—3141no—
331Q8—3141no—
332Q8—6282no—
333Q8—6282no—
334Q8—6282no—
335Q8C2×D62812yes—
336Q8C2×D62812yes—
337D4—3142no—
338D4—3141no—
339D4—3141no—
340D4—6282no—
341D4—6282no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—3141no—
347C2×C4—3141no—
348C2×C4—3141no—
349C2×C4—3141no—
350C2×C4—3141no—
351C2×C4—3141no—
352C2×C4—3141no—
353C2×C4—6282no—
354C2×C4—6282no—
355C2×C4—6282no—
356C2×C4—6282no—
357C2×C4—1571no—
358C2×C4—1571no—
359C2×C4D4×D15711yescharacteristic
360C2×C4D4×D15711yescharacteristic
361C2×C4C23×D15711yescharacteristic
362C8—21no—
363C8—3141no—
364C8—3141no—
365C8C2×D62812yes—
366C22—3141no—
367C22—3141no—
368C22—3141no—
369C22—3141no—
370C22—3141no—
371C22—6281no—
372C22—6281no—
373C22—6281no—
374C22—1572no—
375C22C2×D4×D15711yescharacteristic
376C4—21no—
377C4—22no—
378C4—22no—
379C4—21no—
380C4—41no—
381C4—41no—
382C4—3141no—
383C4—3141no—
384C4—6281no—
385C4—6281no—
386C4—6281no—
387C4C22×D62811yescharacteristic
388C4#1909781711yescharacteristic
389C2—3142no—
390C2—6281no—
391C2—12561no—
392C2D4×D62811yescharacteristic
393C2#137×D15711yescharacteristic
394C2#2375808911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#441×C157C2
socleC22×C157C22×D4
prime radical#23757935C2
hypercenter#441D157
hypocenterC157#3270
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
1. C2×#19097803C2semidirect
2. C2×Q16×D157C2semidirect
3. Q8×D628C2semidirect
5. #23757667C2semidirect
6. #23757532C2semidirect
7. #23757503C2semidirect
8. #23757495C2semidirect
9. #23757483C2semidirect
10. #23757438C2semidirect
11. #23757430C2semidirect
12. #23757430C2semidirect
13. #23757413C2semidirect
14. #23757346C2semidirect
15. #441×C157C2semidirect
113. Q16×C157D4semidirect
177. C2×D157#137semidirect
197. C157#3270semidirect
212. #441D157semidirect
290. Q16D628semidirect

Maximal subgroups

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

subgroupindex
C2×#190978032
C2×Q16×D1572
Q8×D6282
#237579352
#237576672
#237575322
#237575032
#237574952
#237574832
#237574382
#237574302
#237574132
#237573462
#441×C1572
#3270157