Finite Group Databasebeta

← #28746049

Subgroups — #28746049

Order 15104.

516 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21436485C211yesmaximal, characteristic, complemented
2#21436441C211yesmaximal, characteristic
3#21426377C211yesmaximal, characteristic, complemented
4#21426292C211yesmaximal, characteristic, complemented
5#21426283C211yesmaximal, characteristic, complemented
6#21426282C211yesmaximal, characteristic, complemented
7#21426242C211yesmaximal, characteristic, complemented
8#21426236C211yesmaximal, characteristic, complemented
9#21426195C211yesmaximal, characteristic, complemented
10#21426194C211yesmaximal, characteristic, complemented
11#21425945C211yesmaximal, characteristic, complemented
12(C22⋊D59)⋊C16C211yesmaximal, characteristic, complemented
13C2×#16837229C211yesmaximal, characteristic, complemented
14#367×D59C211yesmaximal, characteristic, complemented
15C59×#2235C211yesmaximal, characteristic, complemented
16#16838110—21no—
17#16838053—21no—
18C2×#14273670C2211yescharacteristic
19C22×C59⋊C16C2211yescharacteristic
20C2×C16×D59C2211yescharacteristic
21#16837806C2211yescharacteristic
22#16837806C2211yescharacteristic
23#16837736—21no—
24#16837735—21no—
25#16837717—21no—
26#16837715—21no—
27#16837695—21no—
28#16837681—21no—
29#16837640C2211yescharacteristic
30#16837639C2211yescharacteristic
31#16837638C2211yescharacteristic
32Q16×Dic59C2211yescharacteristic
33#16837484C2211yescharacteristic
34C8×C22⋊D59C2211yescharacteristic
35C2×#14273661C2211yescharacteristic
36#107×D59C2211yescharacteristic
37#107×D59C2211yescharacteristic
38#16837230C2211yescharacteristic
39#16837230C2211yescharacteristic
40#16837229C2211yescharacteristic
41#16837229C2211yescharacteristic
42#16837229C2211yescharacteristic
43#16837229C2211yescharacteristic
44#16837229C2211yescharacteristic
45Dic59⋊C16C2211yescharacteristic
46#16837206C2211yescharacteristic
47#16837204C2211yescharacteristic
48#16837204C2211yescharacteristic
49#16837202C2211yescharacteristic
50(C2×D59)⋊C16C2211yescharacteristic
51C16×Dic59C2211yescharacteristic
52C59×#467C2211yescharacteristic
53C2×#107×C59C2211yescharacteristic
54C59×#367C2211yescharacteristic
55C59×#367C2211yescharacteristic
56C59×#358C2211yescharacteristic
57C59×#358C2211yescharacteristic
58C59×C22⋊C16C2211yescharacteristic
59C2×#11745685—41no—
60C2×D472C2311yescharacteristic
61C2×C8×D59C2311yescharacteristic
62#14273726—21no—
63#14273726—21no—
64#14273717—41no—
65#14273712—41no—
66#14273710—41no—
67#14273707—41no—
68#14273705—41no—
69#14273704—41no—
70#14273703—21no—
71#14273702—21no—
72#14273701—21no—
73Q8×Dic59—21no—
74#14273689—21no—
75C4×C22⋊D59D411yescharacteristic
76(C22⋊D59)⋊C4—41no—
77C2×C4⋊Dic59—21no—
78C4⋊C4×D59—21no—
79C4⋊C4×D59—21no—
80#14273670—22no—
81C2×C59⋊C16—22no—
82C2×C59⋊C16C2311yescharacteristic
83C2×C59⋊C16C2311yescharacteristic
84C16×D59—22no—
85#14273661—21no—
86#14273661C2311yescharacteristic
87#14273661C2311yescharacteristic
88#14273661C2311yescharacteristic
89#14273660—21no—
90#14273660—21no—
91#14273660—21no—
92#14273660—21no—
93Dic59⋊C8D411yescharacteristic
94#14273651—21no—
95#14273651—21no—
96#14273649—21no—
97#14273647—21no—
98#14273647—21no—
99C22⋊(C59⋊C8)D411yescharacteristic
100(C2×D59)⋊C8D411yescharacteristic
101C8×Dic59C2311yescharacteristic
102C2×Q16×C59C2311yescharacteristic
103C22×C8×C59C2311yescharacteristic
104C22⋊Q8×C59—21no—
105C2×C4⋊C4×C59—21no—
106C2×C16×C59C2311yescharacteristic
107C2×C16×C59C2311yescharacteristic
108#107×C59—21no—
109#107×C59C2311yescharacteristic
110#107×C59C2311yescharacteristic
111#107×C59C2311yescharacteristic
112#103×C59—21no—
113C2×Q8×C59—21no—
114C22×C4×C59—41no—
115C22×C4×C59C2×D411yescharacteristic
116C2×C22⋊D59—41no—
117C2×C22⋊D59D811yescharacteristic
118C22×Dic59—41no—
119#11745685—84no—
120C2×D236—21no—
121C2×D236—41no—
122C2×C4×D59—41no—
123C2×C4×D59—41no—
124C2×C4×D59—41no—
125C2×C4×D59C2×D411yescharacteristic
126C2×Dic118—41no—
127C2×Dic118—41no—
128Q16×C59—21no—
129Q16×C59—21no—
130C2×C8×C59—21no—
131C2×C8×C59—21no—
132C2×C8×C59C2×D411yescharacteristic
133C2×C8×C59C2411yescharacteristic
134C4⋊C4×C59—21no—
135C4⋊C4×C59—21no—
136C4⋊C4×C59—21no—
137C4⋊C4×C59—41no—
138C4⋊C4×C59—41no—
139C22⋊C4×C59—41no—
140C22⋊Dic59D811yescharacteristic
141(C2×D59)⋊C4—41no—
142(C2×D59)⋊C4—41no—
143(C2×D59)⋊C4—41no—
144(C2×D59)⋊C4—41no—
145(C2×D59)⋊C4D811yescharacteristic
146C4⋊Dic59—21no—
147C4⋊Dic59—21no—
148C4⋊Dic59—21no—
149C4⋊Dic59—41no—
150C4⋊Dic59—41no—
151Dic59⋊C4—41no—
152Dic59⋊C4—41no—
153Dic59⋊C4—41no—
154Dic59⋊C4—41no—
155Dic59⋊C4—41no—
156Dic59⋊C4D811yescharacteristic
157C4×Dic59—41no—
158C4×Dic59C2×D411yescharacteristic
159C2×C59⋊C8C2×D411yescharacteristic
160C2×C59⋊C8C2×D411yescharacteristic
161D472—21no—
162D472—21no—
163C8×D59—21no—
164C8×D59—21no—
165C16×C59—21no—
166C16×C59—21no—
167C59⋊C16—21no—
168C59⋊C16#4112yescomplemented
169C23×C59C2×D811yescharacteristic
170C22×D59—41no—
171C22×D59C2×D811yescharacteristic
172Q8×C59—41no—
173Q8×C59—41no—
174C2×C4×C59—21no—
175C2×C4×C59—21no—
176C2×C4×C59—41no—
177C2×C4×C59—41no—
178C2×C4×C59—41no—
179C2×C4×C59—41no—
180C2×C4×C59—82no—
181C2×C4×C59C2×D811yescharacteristic
182C2×C4×C59C22×D411yescharacteristic
183C22⋊D59—82no—
184C22⋊D59SD3214yescomplemented
185C2×Dic59—41no—
186C2×Dic59—41no—
187C2×Dic59—41no—
188C2×Dic59—161no—
189C2×Dic59C2×D811yescharacteristic
190C2×Dic59C2×D811yescharacteristic
191C2×Dic59C2×D811yescharacteristic
192D236—41no—
193D236—41no—
194D236—82no—
195C4×D59—21no—
196C4×D59—21no—
197C4×D59—82no—
198C4×D59—82no—
199C4×D59—161no—
200Dic118—82no—
201Dic118—161no—
202C8×C59—21no—
203C8×C59C2×#4111yescharacteristic
204C8×C592−1+411yescharacteristic
205C59⋊C8—21no—
206C59⋊C8—21no—
207#33902—591nomaximal, Sylow
208C22×C59—21no—
209C22×C59—21no—
210C22×C59C2×SD3212yes—
211C22×C59C22×D811yescharacteristic
212C2×D59—21no—
213C2×D59—21no—
214C2×D59—81no—
215C2×D59—81no—
216C2×D59—81no—
217C2×D59C2×SD3212yes—
218C4×C59—21no—
219C4×C59—81no—
220C4×C59—81no—
221C4×C59—81no—
222C4×C59—81no—
223C4×C59#54711yescharacteristic
224C4×C59#57811yescharacteristic
225Dic59—21no—
226Dic59—21no—
227Dic59—81no—
228Dic59—81no—
229Dic59—81no—
230Dic59C2×SD3212yes—
231#3338—591no—
232#3334—591no—
233#2254—591no—
234#2235—591no—
235#2235D5911yescharacteristic, complemented
236#2233—591no—
237#2233—591no—
238#2226—591no—
239#2225—591no—
240#2217—591no—
241#2217—591no—
242C4×SD32—591no—
243D4×C16—591no—
244C2×#367—591no—
245C2×#367—591no—
246C2×C59—22no—
247C2×C59#333411yescharacteristic
248C2×C59C22×SD3211yescharacteristic
249C2×C59#341611yescharacteristic
250D59—22no—
251D59—161no—
252#554—1181no—
253#547—1181no—
254C2×SD32—591no—
255C22×C16—591no—
256C22×C16—591no—
257#500—591no—
258#500—591no—
259#484—1181no—
260#484—1181no—
261#480—1181no—
262#480—1181no—
263#477—1181no—
264#474—1181no—
265#467—591no—
266#467C2×D5911yescharacteristic
267#466—591no—
268#466—591no—
269C4×Q16—591no—
270C4×D8—591no—
271C8×D4—591no—
272C2×#107—591no—
273C2×#107—591no—
274C2×#107—591no—
275C2×#107C2×D5911yescharacteristic
276#367—591no—
277#367—591no—
278#367—591no—
279#367—591no—
280#367—591no—
281#367—591no—
282#367—591no—
283#367C2×D5911yescharacteristic
284#367C2×D5911yescharacteristic
285C4⋊C16—591no—
286#358—591no—
287#358C2×D5911yescharacteristic
288#358C2×D5911yescharacteristic
289#357—591no—
290#357—591no—
291#357—591no—
292C22⋊C16—591no—
293C22⋊C16C2×D5911yescharacteristic
294C4×C16—591no—
295C59#3390211yescharacteristic, complemented, Sylow
296C2×#41—2361no—
297C2×Q16C22×D5911yescharacteristic
298C2×D8—591no—
299C22×C8—591no—
300C22×C8C22×D5911yescharacteristic
301C4⋊Q8—1181no—
302C4⋊Q8—1181no—
303#125—2361no—
304#123—2361no—
305#123—2361no—
306C22⋊Q8—21no—
307C22⋊Q8—1181no—
308C22⋊Q8—2361no—
309C22⋊Q8—2361no—
310C22⋊Q8—2361no—
311#121—1181no—
312#121—1181no—
313C4×Q8—1181no—
314C4×D4—591no—
315C4×D4—1181no—
316C4×D4—2361no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—1181no—
319C2×C4⋊C4—1181no—
320C2×C4⋊C4—1181no—
321SD32—1182no—
322C2×C16—591no—
323C2×C16—591no—
324C2×C16—1182no—
325C2×C16—1182no—
326C2×C16C22×D5911yescharacteristic
327C2×C16C22×D5911yescharacteristic
328#107—21no—
329#107—591no—
330#107—591no—
331#107—591no—
332#107—1181no—
333#107—1181no—
334#107—1181no—
335#107—1181no—
336#107—1181no—
337#107C22×D5911yescharacteristic
338#107C22×D5911yescharacteristic
339#107C22×D5911yescharacteristic
340C4⋊C8—591no—
341#103—21no—
342#103—1181no—
343#103—1181no—
344#102—1181no—
345#102—1181no—
346#102—1181no—
347C22⋊C8—591no—
348C22⋊C8—591no—
349C4×C8—591no—
350#41—4724no—
351C2×Q8—21no—
352C2×Q8—2361no—
353C2×Q8—2361no—
354C2×D4—591no—
355C2×D4—1181no—
356C2×D4—2361no—
357C2×D4—2361no—
358C22×C4—41no—
359C22×C4—591no—
360C22×C4—2361no—
361C22×C4—2361no—
362C22×C4—2361no—
363C22×C4—2361no—
364C22×C4D4×D5911yescharacteristic
365Q16—21no—
366Q16—21no—
367D8—1181no—
368D8—1181no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—591no—
372C2×C8—591no—
373C2×C8—1181no—
374C2×C8—1181no—
375C2×C8D4×D5911yescharacteristic
376C2×C8C23×D5911yescharacteristic
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—591no—
383C4⋊C4—1181no—
384C4⋊C4—1181no—
385C4⋊C4—1181no—
386C4⋊C4—2361no—
387C4⋊C4—2361no—
388C4⋊C4—2361no—
389C4⋊C4—2361no—
390C4⋊C4—2361no—
391C4⋊C4—2361no—
392C4⋊C4—2361no—
393C22⋊C4—41no—
394C22⋊C4—591no—
395C22⋊C4—591no—
396C22⋊C4—2361no—
397C22⋊C4—2361no—
398C22⋊C4—2361no—
399C22⋊C4—2361no—
400C42—591no—
401C42—2361no—
402C16—21no—
403C16—21no—
404C16—592no—
405C16—1181no—
406C23—591no—
407C23—2361no—
408C23D8×D5911yescharacteristic
409Q8—41no—
410Q8—41no—
411Q8—9441no—
412Q8—4722no—
413D4—594no—
414D4—4722no—
415D4—4722no—
416D4—2361no—
417D4—2361no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—82no—
425C2×C4—591no—
426C2×C4—591no—
427C2×C4—591no—
428C2×C4—1181no—
429C2×C4—1181no—
430C2×C4—9441no—
431C2×C4—9441no—
432C2×C4—4722no—
433C2×C4—4722no—
434C2×C4—2361no—
435C2×C4—2361no—
436C2×C4—2361no—
437C2×C4D8×D5911yescharacteristic
438C2×C4C2×D4×D5911yescharacteristic
439C8—21no—
440C8—1181no—
441C8—1181no—
442C8#41×D5911yescharacteristic
443C8#1427377411yescharacteristic
444C22—21no—
445C22—21no—
446C22—592no—
447C22—1181no—
448C22—1181no—
449C22—4721no—
450C22—4721no—
451C22—4721no—
452C22SD32×D5912yes—
453C22C2×D8×D5911yescharacteristic
454C4—21no—
455C4—81no—
456C4—81no—
457C4—81no—
458C4—81no—
459C4—592no—
460C4—1181no—
461C4—1181no—
462C4—4721no—
463C4—4721no—
464C4—4721no—
465C4#1683805011yescharacteristic
466C4#1683821811yescharacteristic
467C2—22no—
468C2—1182no—
469C2—9441no—
470C2#2143643811yescharacteristic
471C2C2×SD32×D5911yescharacteristic
472C2#2143724111yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2235C2
socleC22×C59C22×D8
prime radical#21436441C2
hypercenter#2235D59
hypocenterC59#33902
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2235

Complements

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

NH ≅ G/Ntype
1. #21436485C2semidirect
3. #21426377C2semidirect
4. #21426292C2semidirect
5. #21426283C2semidirect
6. #21426282C2semidirect
7. #21426242C2semidirect
8. #21426236C2semidirect
9. #21426195C2semidirect
10. #21426194C2semidirect
11. #21425945C2semidirect
12. (C22⋊D59)⋊C16C2semidirect
13. C2×#16837229C2semidirect
14. #367×D59C2semidirect
15. C59×#2235C2semidirect
168. C59⋊C16#41semidirect
184. C22⋊D59SD32semidirect
235. #2235D59semidirect
295. C59#33902semidirect

Maximal subgroups

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

subgroupindex
#214364852
#214364412
#214263772
#214262922
#214262832
#214262822
#214262422
#214262362
#214261952
#214261942
#214259452
(C22⋊D59)⋊C162
C2×#168372292
#367×D592
C59×#22352
#3390259