Finite Group Databasebeta

← #28745580

Subgroups — #28745580

Order 15104.

546 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21435839C211yesmaximal, characteristic
2#21426330C211yesmaximal, characteristic, complemented
3#21426304C211yesmaximal, characteristic, complemented
4#21426276C211yesmaximal, characteristic, complemented
5#21426257C211yesmaximal, characteristic, complemented
6#21426194C211yesmaximal, characteristic, complemented
7#21426185C211yesmaximal, characteristic, complemented
8#21426158C211yesmaximal, characteristic, complemented
9#21426034C211yesmaximal, characteristic, complemented
10#21426029C211yesmaximal, characteristic, complemented
11C4×#14273673C211yesmaximal, characteristic, complemented
12D4×C59⋊C16C211yesmaximal, characteristic, complemented
13#21425801C211yesmaximal, characteristic, complemented
14C2×#16837205C211yesmaximal, characteristic, complemented
15C59×#3336C211yesmaximal, characteristic, complemented
16C2×#14273749C2211yescharacteristic
17C22×Dic236C2211yescharacteristic
18#16838061—21no—
19C2×#14273673C2211yescharacteristic
20C2×#14273673C2211yescharacteristic
21C2×#14273670C2211yescharacteristic
22C22×C59⋊C16C2211yescharacteristic
23C22×C59⋊C16C2211yescharacteristic
24#16837786C2211yescharacteristic
25#16837643C2211yescharacteristic
26#16837638C2211yescharacteristic
27#16837625—21no—
28#16837566—21no—
29#16837543—21no—
30C4×Dic236C2211yescharacteristic
31#16837229C2211yescharacteristic
32#16837227C2211yescharacteristic
33#16837227C2211yescharacteristic
34C4⋊(C59⋊C16)C2211yescharacteristic
35#16837206C2211yescharacteristic
36#16837205C2211yescharacteristic
37#16837205C2211yescharacteristic
38#16837205C2211yescharacteristic
39#16837205C2211yescharacteristic
40#16837205C2211yescharacteristic
41#16837205C2211yescharacteristic
42#16837202C2211yescharacteristic
43#16837202C2211yescharacteristic
44#16837202C2211yescharacteristic
45C22⋊(C59⋊C16)C2211yescharacteristic
46C22⋊(C59⋊C16)C2211yescharacteristic
47C4×C59⋊C16C2211yescharacteristic
48C59×#552—21no—
49C59×#499C2211yescharacteristic
50C59×#485—21no—
51C59×#483—21no—
52C59×#479—21no—
53C59×#467C2211yescharacteristic
54C59×#467C2211yescharacteristic
55C4×Q16×C59C2211yescharacteristic
56C8×D4×C59C2211yescharacteristic
57C59×#427C2211yescharacteristic
58C2×#107×C59C2211yescharacteristic
59C2×#11745685—21no—
60C22×Dic118—21no—
61#14273749—22no—
62#14273749—41no—
63C2×Dic236—22no—
64C2×Dic236—21no—
65C2×Dic236—41no—
66C2×Dic236C2311yescharacteristic
67C2×Dic236C2311yescharacteristic
68C2×#11745660—21no—
69C2×#11745660—21no—
70C2×D472C2311yescharacteristic
71#14273715—21no—
72#14273713—41no—
73#14273706—21no—
74#14273705—41no—
75#14273701—21no—
76#14273699—41no—
77C4×Dic118—21no—
78(C22⋊D59)⋊C4—41no—
79#14273673—22no—
80#14273673—41no—
81#14273670—41no—
82C2×C59⋊C16—41no—
83C2×C59⋊C16—41no—
84C2×C59⋊C16C2311yescharacteristic
85C2×C59⋊C16C2311yescharacteristic
86C2×C59⋊C16C2311yescharacteristic
87C2×C59⋊C16C2311yescharacteristic
88#14273661C2311yescharacteristic
89#14273652—21no—
90#14273652—21no—
91#14273649—21no—
92C2×Q16×C59C2311yescharacteristic
93C22×C8×C59C2311yescharacteristic
94C22×C8×C59C2311yescharacteristic
95#126×C59—41no—
96#125×C59—21no—
97#125×C59—41no—
98#123×C59—41no—
99C22⋊Q8×C59—21no—
100C22⋊Q8×C59—21no—
101C4×Q8×C59—21no—
102C4×D4×C59D411yescharacteristic
103#117×C59—21no—
104#117×C59—41no—
105C2×C4⋊C4×C59—21no—
106#107×C59—21no—
107#107×C59C2311yescharacteristic
108#107×C59C2311yescharacteristic
109#107×C59C2311yescharacteristic
110#106×C59—21no—
111#106×C59—21no—
112C4⋊C8×C59D411yescharacteristic
113#103×C59—21no—
114#103×C59—21no—
115#103×C59—21no—
116C22⋊C8×C59D411yescharacteristic
117C22⋊C8×C59D411yescharacteristic
118C4×C8×C59C2311yescharacteristic
119C2×Q8×C59—21no—
120C2×D4×C59D811yescharacteristic
121C22×C4×C59—41no—
122C22×C4×C59C2×D411yescharacteristic
123C22×C4×C59C2×D411yescharacteristic
124C2×C22⋊D59—41no—
125C2×C22⋊D59—41no—
126C22×Dic59—41no—
127#11745685—42no—
128#11745685—81no—
129C2×D236—21no—
130C2×C4×D59—41no—
131C2×Dic118—21no—
132C2×Dic118—21no—
133C2×Dic118—42no—
134C2×Dic118—41no—
135C2×Dic118—81no—
136Q16×C59—21no—
137Q16×C59—21no—
138C2×C8×C59—21no—
139C2×C8×C59—21no—
140C2×C8×C59—21no—
141C2×C8×C59—21no—
142C2×C8×C59C2×D411yescharacteristic
143C2×C8×C59C2×D411yescharacteristic
144C2×C8×C59C2×D411yescharacteristic
145C2×C8×C59C2411yescharacteristic
146C4⋊C4×C59—21no—
147C4⋊C4×C59—21no—
148C4⋊C4×C59—21no—
149C4⋊C4×C59—41no—
150C4⋊C4×C59—41no—
151C4⋊C4×C59—41no—
152C4⋊C4×C59—41no—
153C4⋊C4×C59—41no—
154C4⋊C4×C59—41no—
155C4⋊C4×C59D811yescharacteristic
156C22⋊C4×C59—41no—
157C22⋊C4×C59—41no—
158C22⋊C4×C59—41no—
159C22⋊C4×C59D811yescharacteristic
160C22⋊C4×C59D811yescharacteristic
161C42×C59—41no—
162C42×C59—41no—
163C42×C59C2×D411yescharacteristic
164C22⋊Dic59—41no—
165(C2×D59)⋊C4—41no—
166(C2×D59)⋊C4—41no—
167(C2×D59)⋊C4—41no—
168C4⋊Dic59—21no—
169Dic59⋊C4—41no—
170Dic59⋊C4—41no—
171C4×Dic59—41no—
172Dic236—21no—
173Dic236—21no—
174Dic236—21no—
175Dic236—42no—
176Dic236#4112yes—
177D472—21no—
178D472—21no—
179#11745660—42no—
180#11745660—81no—
181C59⋊C16—21no—
182C59⋊C16—21no—
183C59⋊C16—21no—
184C59⋊C16—21no—
185C23×C59C2×D811yescharacteristic
186C23×C59C2×D811yescharacteristic
187C22×D59—41no—
188Q8×C59—41no—
189Q8×C59—41no—
190D4×C59—22no—
191C2×C4×C59—21no—
192C2×C4×C59—21no—
193C2×C4×C59—21no—
194C2×C4×C59—21no—
195C2×C4×C59—41no—
196C2×C4×C59—41no—
197C2×C4×C59—41no—
198C2×C4×C59—41no—
199C2×C4×C59—82no—
200C2×C4×C59C2×D811yescharacteristic
201C2×C4×C59C2×D811yescharacteristic
202C2×C4×C59C2×D811yescharacteristic
203C2×C4×C59C2×D811yescharacteristic
204C2×C4×C59C22×D411yescharacteristic
205C22⋊D59—82no—
206C22⋊D59—82no—
207C2×Dic59—41no—
208C2×Dic59—41no—
209C2×Dic59—41no—
210C2×Dic59—82no—
211D236—41no—
212D236—41no—
213C4×D59—82no—
214Dic118—22no—
215Dic118—41no—
216Dic118—41no—
217Dic118—41no—
218Dic118—82no—
219C8×C59—21no—
220C8×C59—21no—
221C8×C59—21no—
222C8×C59C2×#4111yescharacteristic
223C8×C59H(5,𝔽2)11yescharacteristic
224#33867—591nomaximal, Sylow
225C22×C59—21no—
226C22×C59—21no—
227C22×C59—21no—
228C22×C59—21no—
229C22×C59—21no—
230C22×C59#50812yes—
231C22×C59C22×D811yescharacteristic
232C2×D59—81no—
233C2×D59—81no—
234C2×D59—81no—
235C4×C59—21no—
236C4×C59—21no—
237C4×C59—21no—
238C4×C59—21no—
239C4×C59—81no—
240C4×C59—81no—
241C4×C59—81no—
242C4×C59—81no—
243C4×C59#54611yescharacteristic
244C4×C59#57711yescharacteristic
245Dic59—42no—
246Dic59—81no—
247Dic59—81no—
248#3336D5911yescharacteristic, complemented
249#3285—591no—
250#2242—591no—
251#2237—591no—
252#2232—591no—
253#2229—591no—
254#2217—591no—
255#2216—591no—
256#2211—591no—
257#2189—591no—
258#2188—591no—
259C4×Q32—591no—
260D4×C16—591no—
261#2138—591no—
262C2×#358—591no—
263C2×C59—22no—
264C2×C59—41no—
265C2×C59#329411yescharacteristic
266C2×C59C2×#50811yescharacteristic
267C2×C59#341711yescharacteristic
268D59—161no—
269C2×#135—591no—
270C22×Q16—591no—
271#552—21no—
272#548—1181no—
273C2×Q32—591no—
274C2×Q32—591no—
275C2×SD32—591no—
276C22×C16—591no—
277C22×C16—591no—
278#499C2×D5911yescharacteristic
279#495—591no—
280#485—21no—
281#483—21no—
282#479—21no—
283#467—591no—
284#467C2×D5911yescharacteristic
285#467C2×D5911yescharacteristic
286#466—591no—
287#464—1181no—
288#452—1181no—
289#448—1181no—
290C4×Q16—591no—
291C4×Q16C2×D5911yescharacteristic
292C8×D4C2×D5911yescharacteristic
293#427C2×D5911yescharacteristic
294C2×#107C2×D5911yescharacteristic
295#367—591no—
296#366—591no—
297#366—591no—
298C4⋊C16—591no—
299#358—591no—
300#358—591no—
301#358—591no—
302#358—591no—
303#358—591no—
304#358—591no—
305#358—591no—
306#357—591no—
307#357—591no—
308#357—591no—
309C22⋊C16—591no—
310C22⋊C16—591no—
311C4×C16—591no—
312C59#3386711yescharacteristic, complemented, Sylow
313C2×#41—1181no—
314C22×Q8—1181no—
315#135—1182no—
316#135—2361no—
317C2×Q16—591no—
318C2×Q16—591no—
319C2×Q16—1182no—
320C2×Q16—1181no—
321C2×Q16—2361no—
322C2×Q16C22×D5911yescharacteristic
323C2×SD16—1181no—
324C2×SD16—1181no—
325C2×D8—591no—
326C22×C8C22×D5911yescharacteristic
327C22×C8C22×D5911yescharacteristic
328#126—41no—
329#125—21no—
330#125—41no—
331#124—1181no—
332#124—2361no—
333#123—41no—
334C22⋊Q8—21no—
335C22⋊Q8—21no—
336C22⋊Q8—1181no—
337C22⋊Q8—2361no—
338#121—1181no—
339#121—2361no—
340C4×Q8—21no—
341C4×Q8—1181no—
342C4×D4—2361no—
343C4×D4C22⋊D5911yescharacteristic
344#117—21no—
345#117—41no—
346C2×C4⋊C4—21no—
347Q32—1182no—
348Q32—2361no—
349SD32—2361no—
350C2×C16—591no—
351C2×C16—591no—
352C2×C16—591no—
353C2×C16—591no—
354C2×C16—2361no—
355C2×C16—2361no—
356#107—21no—
357#107—591no—
358#107C22×D5911yescharacteristic
359#107C22×D5911yescharacteristic
360#107C22×D5911yescharacteristic
361#106—21no—
362#106—21no—
363C4⋊C8C22⋊D5911yescharacteristic
364#103—21no—
365#103—21no—
366#103—21no—
367#103—1181no—
368#103—1181no—
369#102—1181no—
370C22⋊C8C22⋊D5911yescharacteristic
371C22⋊C8C22⋊D5911yescharacteristic
372C4×C8C22×D5911yescharacteristic
373#41—4721no—
374#41—2362no—
375C2×Q8—21no—
376C2×Q8—1181no—
377C2×Q8—1181no—
378C2×Q8—4721no—
379C2×Q8—2362no—
380C2×Q8—2361no—
381C2×D4—1181no—
382C2×D4—2361no—
383C2×D4—2361no—
384C2×D4#1174566911yescharacteristic
385C22×C4—41no—
386C22×C4—2361no—
387C22×C4—2361no—
388C22×C4C2×C22⋊D5911yescharacteristic
389C22×C4C2×C22⋊D5911yescharacteristic
390Q16—21no—
391Q16—21no—
392Q16—592no—
393Q16—1181no—
394Q16—1181no—
395Q16—1181no—
396Q16—2362no—
397SD16—4721no—
398SD16—2362no—
399D8—1181no—
400D8—1181no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8C2×C22⋊D5911yescharacteristic
406C2×C8C2×C22⋊D5911yescharacteristic
407C2×C8C2×C22⋊D5911yescharacteristic
408C2×C8C23×D5911yescharacteristic
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—1181no—
419C4⋊C4—2361no—
420C4⋊C4—2361no—
421C4⋊C4#1174566911yescharacteristic
422C22⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—2361no—
426C22⋊C4—2361no—
427C22⋊C4—2361no—
428C22⋊C4—2361no—
429C22⋊C4#1174566911yescharacteristic
430C22⋊C4#1174566911yescharacteristic
431C42—41no—
432C42—41no—
433C42—2361no—
434C42C2×C22⋊D5911yescharacteristic
435C16—1181no—
436C16—1181no—
437C16—1181no—
438C16—1181no—
439C23—2361no—
440C23C2×#1174566911yescharacteristic
441C23C2×#1174566911yescharacteristic
442Q8—41no—
443Q8—41no—
444Q8—1182no—
445Q8—4722no—
446Q8—2361no—
447Q8—2361no—
448Q8—2361no—
449D4—22no—
450D4—4722no—
451D4—4722no—
452D4—2361no—
453D4—2361no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—82no—
463C2×C4—4722no—
464C2×C4—4722no—
465C2×C4—2361no—
466C2×C4—2361no—
467C2×C4—2361no—
468C2×C4C2×#1174566911yescharacteristic
469C2×C4C2×#1174566911yescharacteristic
470C2×C4C2×#1174566911yescharacteristic
471C2×C4C2×#1174566911yescharacteristic
472C2×C4C22×C22⋊D5911yescharacteristic
473C8—21no—
474C8—21no—
475C8—21no—
476C8#41×D5911yescharacteristic
477C8#1427377311yescharacteristic
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—4721no—
484C22—4721no—
485C22—4721no—
486C22#1683783212yes—
487C22C22×#1174566911yescharacteristic
488C4—21no—
489C4—21no—
490C4—21no—
491C4—21no—
492C4—81no—
493C4—81no—
494C4—81no—
495C4—81no—
496C4—4721no—
497C4—4721no—
498C4—2362no—
499C4#1683804011yescharacteristic
500C4#1683821011yescharacteristic
501C2—22no—
502C2—41no—
503C2—9441no—
504C2#2143590311yescharacteristic
505C2C2×#1683783211yescharacteristic
506C2#2143724711yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3336C2
socleC22×C59C22×D8
prime radical#21435839C2
hypercenter#3336D59
hypocenterC59#33867
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3336

Complements

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

NH ≅ G/Ntype
2. #21426330C2semidirect
3. #21426304C2semidirect
4. #21426276C2semidirect
5. #21426257C2semidirect
6. #21426194C2semidirect
7. #21426185C2semidirect
8. #21426158C2semidirect
9. #21426034C2semidirect
10. #21426029C2semidirect
11. C4×#14273673C2semidirect
12. D4×C59⋊C16C2semidirect
13. #21425801C2semidirect
14. C2×#16837205C2semidirect
15. C59×#3336C2semidirect
248. #3336D59semidirect
312. C59#33867semidirect

Maximal subgroups

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

subgroupindex
#214358392
#214263302
#214263042
#214262762
#214262572
#214261942
#214261852
#214261582
#214260342
#214260292
C4×#142736732
D4×C59⋊C162
#214258012
C2×#168372052
C59×#33362
#3386759