Finite Group Databasebeta

← #34111723

Subgroups — #34111723

Order 20096.

488 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097803C211yesmaximal, characteristic, complemented
2C2×#19097794C211yesmaximal, characteristic, complemented
3#23757945C211yesmaximal, characteristic, complemented
4#23757935C211yesmaximal, characteristic, complemented
5#23757662C211yesmaximal, characteristic, complemented
6#23757527C211yesmaximal, characteristic, complemented
7#23757503C211yesmaximal, characteristic, complemented
8#23757495C211yesmaximal, characteristic, complemented
9#23757473C211yesmaximal, characteristic, complemented
10#23757438C211yesmaximal, characteristic, complemented
11#23757421C211yesmaximal, characteristic, complemented
12#23757421C211yesmaximal, characteristic, complemented
13#23757413C211yesmaximal, characteristic, complemented
14#23757346C211yesmaximal, characteristic, complemented
15#441×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×#14578955C2211yescharacteristic
18C2×#14578955C2211yescharacteristic
19C2×Q8×D157C2211yescharacteristic
20#19097803—24no—
21#19097794—24no—
22C2×#14578944C2211yescharacteristic
23C2×#14578943C2211yescharacteristic
24C2×#14578943C2211yescharacteristic
25C2×#14578941C2211yescharacteristic
26C2×#14578941C2211yescharacteristic
27C2×#14578941C2211yescharacteristic
28C2×D1256C2211yescharacteristic
29C2×D1256C2211yescharacteristic
30C2×#14578937C2211yescharacteristic
31C2×C8×D157C2211yescharacteristic
32#19097768—21no—
33#19097768C2211yescharacteristic
34#19097759—21no—
35#19097759C2211yescharacteristic
36#19097751—21no—
37#19097751C2211yescharacteristic
38#19097746—21no—
39#19097746—21no—
40#19097746—21no—
41#19097746C2211yescharacteristic
42#19097746C2211yescharacteristic
43#19097746C2211yescharacteristic
44C4×D628—21no—
45C4×D628—21no—
46C4×D628C2211yescharacteristic
47C4⋊#11929434C2211yescharacteristic
48#19097694C2211yescharacteristic
49#19097691C2211yescharacteristic
50#19097691C2211yescharacteristic
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—22no—
63#14578955—22no—
64#14578955—41no—
65#14578955—41no—
66#14578955—41no—
67C2×Dic314—21no—
68C2×Dic314C2311yescharacteristic
69Q8×D157—22no—
70Q8×D157—41no—
71C2×D628—21no—
72C2×D628—21no—
73C2×D628—21no—
74C2×D628—21no—
75C2×D628—21no—
76C2×D628—21no—
77C2×D628—21no—
78C2×D628—21no—
79C2×D628D411yescharacteristic
80C2×D628C2311yescharacteristic
81C2×D628C2311yescharacteristic
82C2×D628C2311yescharacteristic
83C2×C4×D157—21no—
84C2×C4×D157—21no—
85C2×C4×D157—21no—
86C2×C4×D157—21no—
87C2×C4×D157C2311yescharacteristic
88C2×C4×D157C2311yescharacteristic
89#14578944—22no—
90#14578943—22no—
91#14578943—41no—
92#14578941—22no—
93#14578941—22no—
94#14578941—22no—
95D1256—22no—
96D1256—41no—
97#14578937—22no—
98C2×#11929434C2311yescharacteristic
99C2×#11929434C2311yescharacteristic
100C8×D157—22no—
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×D157—21no—
125C22×D157—21no—
126C22×D157C2×D411yescharacteristic
127C22×D157C2×D411yescharacteristic
128Dic314—21no—
129Dic314—21no—
130Dic314—42no—
131D628—21no—
132D628—21no—
133D628—21no—
134D628—21no—
135D628—21no—
136D628—21no—
137D628—41no—
138D628—42no—
139D628—42no—
140D628—42no—
141D628—42no—
142D628—42no—
143D628—81no—
144D628—81no—
145D628—81no—
146C2×Dic157—21no—
147C2×Dic157C2×D411yescharacteristic
148C2×Dic157C2×D411yescharacteristic
149C4×D157—21no—
150C4×D157—21no—
151C4×D157—21no—
152C4×D157—21no—
153C4×D157—42no—
154C4×D157—42no—
155C4×D157—42no—
156C4×D157—42no—
157#11929434—21no—
158#11929434—21no—
159Q8×C157—21no—
160Q8×C157—21no—
161Q8×C157C2×D412yes—
162Q8×C157C2×D412yes—
163C2×C4×C157—21no—
164C2×C4×C157—21no—
165C2×C4×C157—21no—
166C2×C4×C157—21no—
167C2×C4×C157C2×D411yescharacteristic
168C2×C4×C157C2×D411yescharacteristic
169C2×C4×C157C2411yescharacteristic
170C8×C157—21no—
171C8×C157C2×D412yes—
172C22×C157C22×D411yescharacteristic
173C2×D157—21no—
174C2×D157—21no—
175C2×D157—21no—
176C2×D157—21no—
177C2×D157—21no—
178C2×D157—21no—
179C2×D157—41no—
180C2×D157—41no—
181C2×D157—41no—
182C2×D157—41no—
183C2×D157—41no—
184C2×D157—41no—
185C2×D157—41no—
186C2×D157—41no—
187C2×D157—41no—
188C4×C157—22no—
189C4×C157—22no—
190C4×C157—21no—
191C4×C157—21no—
192C4×C157—41no—
193C4×C157—41no—
194C4×C157C22×D411yescharacteristic
195C4×C157H(5,𝔽2)11yescharacteristic
196Dic157—21no—
197Dic157—21no—
198Dic157—41no—
199C2×C157D4211yescharacteristic
200C2×C157#57511yescharacteristic
201C2×C157#57711yescharacteristic
202D157—41no—
203D157—41no—
204D157—81no—
205D157—81no—
206D157—81no—
207C157#327111yescharacteristic, complemented, Sylow
208#3271—1571nomaximal, Sylow
209C2×#137—1571no—
210C2×#135—1571no—
211#550—1571no—
212#548—1571no—
213#496—1571no—
214#469—1571no—
215#465—1571no—
216#464—1571no—
217#460—1571no—
218#452—1571no—
219#449—1571no—
220#449—1571no—
221#448—1571no—
222#441D15711yescharacteristic, complemented
223#435—1571no—
224C2×#41—1571no—
225C2×#41—1571no—
226C2×#41—1571no—
227C22×Q8—1571no—
228#137—3144no—
229#135—3144no—
230C2×Q16—1571no—
231C2×Q16C2×D15711yescharacteristic
232C2×SD16—1571no—
233C2×SD16—1571no—
234C2×SD16—1571no—
235C2×SD16—1571no—
236C2×SD16—1571no—
237C2×D8—1571no—
238C2×D8—1571no—
239C2×M16—1571no—
240C22×C8—1571no—
241#127—3141no—
242#127—1571no—
243#124—3141no—
244#124—1571no—
245C22⋊Q8—3141no—
246C22⋊Q8—1571no—
247#121—3141no—
248#121—3141no—
249#121—3141no—
250#121—1571no—
251#121—1571no—
252#121—1571no—
253C4×Q8C2×D15711yescharacteristic
254C4×Q8C2×D15711yescharacteristic
255C4×D4—3141no—
256C4×D4—3141no—
257C4×D4—1571no—
258#106C2×D15711yescharacteristic
259C4⋊C8—1571no—
260#103—1571no—
261#103C2×D15711yescharacteristic
262#103C2×D15711yescharacteristic
263#102—1571no—
264#102—1571no—
265#102—1571no—
266C22⋊C8—1571no—
267C22⋊C8—1571no—
268#97C2×D15711yescharacteristic
269#41—3142no—
270#41—3142no—
271#41—3142no—
272#41—6281no—
273#41—6281no—
274#41—6281no—
275C2×Q8—3141no—
276C2×Q8—3142no—
277C2×Q8—6281no—
278C2×Q8—1571no—
279C2×Q8C22×D15711yescharacteristic
280C2×Q8C22×D15711yescharacteristic
281C2×D4—3141no—
282C2×D4—3141no—
283C2×D4—3141no—
284C2×D4—3141no—
285C2×D4—3141no—
286C2×D4—3141no—
287C2×D4—3141no—
288C2×D4—3141no—
289C2×D4—1571no—
290C2×D4—1571no—
291C2×D4—1571no—
292C2×D4—1571no—
293C22×C4—3141no—
294C22×C4—3141no—
295C22×C4—3141no—
296C22×C4—3141no—
297C22×C4—1571no—
298C22×C4—1571no—
299Q16—3142no—
300Q16D62814yescomplemented
301SD16—3142no—
302SD16—3142no—
303SD16—3142no—
304SD16—3142no—
305SD16—6281no—
306D8—3142no—
307D8—6281no—
308M16—3142no—
309C2×C8—3142no—
310C2×C8—1571no—
311C2×C8—1571no—
312C2×C8C22×D15711yescharacteristic
313C2×C8C22×D15711yescharacteristic
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—3141no—
317C4⋊C4—1571no—
318C4⋊C4C22×D15711yescharacteristic
319C4⋊C4C22×D15711yescharacteristic
320C22⋊C4—3141no—
321C22⋊C4—3141no—
322C22⋊C4—3141no—
323C22⋊C4—3141no—
324C22⋊C4—3141no—
325C22⋊C4—3141no—
326C22⋊C4—1571no—
327C22⋊C4—1571no—
328C42—21no—
329C42—21no—
330C42C22×D15711yescharacteristic
331C23—3141no—
332C23—3141no—
333C23—3141no—
334C23—1571no—
335C23—1571no—
336Q8—21no—
337Q8—21no—
338Q8—3141no—
339Q8—3141no—
340Q8—6282no—
341Q8C2×D62812yes—
342Q8C2×D62812yes—
343D4—3141no—
344D4—3141no—
345D4—3141no—
346D4—3141no—
347D4—3141no—
348D4—3141no—
349D4—6281no—
350D4—6282no—
351D4—6282no—
352D4—6282no—
353D4—6282no—
354D4—6282no—
355D4—12561no—
356D4—12561no—
357D4—12561no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—3141no—
363C2×C4—3141no—
364C2×C4—3141no—
365C2×C4—3141no—
366C2×C4—3141no—
367C2×C4—6282no—
368C2×C4—6282no—
369C2×C4—6282no—
370C2×C4—6282no—
371C2×C4—1571no—
372C2×C4—1571no—
373C2×C4D4×D15711yescharacteristic
374C2×C4D4×D15711yescharacteristic
375C2×C4C23×D15711yescharacteristic
376C8—21no—
377C8—3141no—
378C8—3141no—
379C8C2×D62812yes—
380C22—3141no—
381C22—3141no—
382C22—3141no—
383C22—3141no—
384C22—3141no—
385C22—3141no—
386C22—6281no—
387C22—6281no—
388C22—6281no—
389C22—6281no—
390C22—6281no—
391C22—6281no—
392C22—6281no—
393C22—6281no—
394C22—6281no—
395C22C2×D4×D15711yescharacteristic
396C4—21no—
397C4—22no—
398C4—22no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—3141no—
403C4—3141no—
404C4—6281no—
405C4C22×D62811yescharacteristic
406C4#1909781711yescharacteristic
407C2—6281no—
408C2—6281no—
409C2—12561no—
410C2—12561no—
411C2—12561no—
412C2D4×D62811yescharacteristic
413C2#2375807211yescharacteristic
414C2#2375808211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#441×C157C2
socleC22×C157C22×D4
hypercenter#441D157
hypocenterC157#3271
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×#19097794C2semidirect
3. #23757945C2semidirect
4. #23757935C2semidirect
5. #23757662C2semidirect
6. #23757527C2semidirect
7. #23757503C2semidirect
8. #23757495C2semidirect
9. #23757473C2semidirect
10. #23757438C2semidirect
11. #23757421C2semidirect
12. #23757421C2semidirect
13. #23757413C2semidirect
14. #23757346C2semidirect
15. #441×C157C2semidirect
113. Q16×C157D4semidirect
207. C157#3271semidirect
222. #441D157semidirect
300. Q16D628semidirect

Maximal subgroups

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

subgroupindex
C2×#190978032
C2×#190977942
#237579452
#237579352
#237576622
#237575272
#237575032
#237574952
#237574732
#237574382
#237574212
#237574132
#237573462
#441×C1572
#3271157