Finite Group Databasebeta

← #28746054

Subgroups — #28746054

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#21436486C211yesmaximal, characteristic, complemented
2#21436440C211yesmaximal, characteristic
3#21426377C211yesmaximal, characteristic, complemented
4#21426295C211yesmaximal, characteristic, complemented
5#21426294C211yesmaximal, characteristic, complemented
6#21426280C211yesmaximal, characteristic, complemented
7#21426241C211yesmaximal, characteristic, complemented
8#21426237C211yesmaximal, characteristic, complemented
9#21426191C211yesmaximal, characteristic, complemented
10#21426190C211yesmaximal, characteristic, complemented
11#21425951C211yesmaximal, characteristic, complemented
12(C22⋊D59)⋊C16C211yesmaximal, characteristic, complemented
13C2×#16837229C211yesmaximal, characteristic, complemented
14#367×D59C211yesmaximal, characteristic, complemented
15C59×#2233C211yesmaximal, characteristic, complemented
16#16838107—21no—
17#16838052—21no—
18C2×#14273671C2211yescharacteristic
19C22×C59⋊C16C2211yescharacteristic
20C2×C16×D59C2211yescharacteristic
21#16837806C2211yescharacteristic
22#16837806C2211yescharacteristic
23#16837739—21no—
24#16837737—21no—
25#16837714—21no—
26#16837713—21no—
27#16837696—21no—
28#16837680—21no—
29#16837644C2211yescharacteristic
30#16837643C2211yescharacteristic
31#16837635C2211yescharacteristic
32#16837496C2211yescharacteristic
33D8×Dic59C2211yescharacteristic
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#16837207C2211yescharacteristic
47#16837207C2211yescharacteristic
48#16837205C2211yescharacteristic
49#16837203C2211yescharacteristic
50(C2×D59)⋊C16C2211yescharacteristic
51C16×Dic59C2211yescharacteristic
52C59×#466C2211yescharacteristic
53C2×#107×C59C2211yescharacteristic
54C59×#367C2211yescharacteristic
55C59×#367C2211yescharacteristic
56C59×#357C2211yescharacteristic
57C59×#357C2211yescharacteristic
58C59×C22⋊C16C2211yescharacteristic
59C2×#11745687—41no—
60C2×Dic236C2311yescharacteristic
61C2×C8×D59C2311yescharacteristic
62#14273726—21no—
63#14273726—21no—
64#14273717—41no—
65#14273709—41no—
66#14273708—41no—
67#14273707—21no—
68#14273706—21no—
69#14273706—41no—
70#14273705—41no—
71#14273704—41no—
72#14273698—21no—
73#14273692—21no—
74D4×Dic59—21no—
75C4×C22⋊D59—41no—
76C4×C22⋊D59D411yescharacteristic
77C2×C4⋊Dic59—21no—
78C4⋊C4×D59—21no—
79C4⋊C4×D59—21no—
80#14273671—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#14273652—21no—
95#14273650—21no—
96#14273650—21no—
97#14273648—21no—
98#14273648—21no—
99C22⋊(C59⋊C8)D411yescharacteristic
100(C2×D59)⋊C8D411yescharacteristic
101C8×Dic59C2311yescharacteristic
102C2×D8×C59C2311yescharacteristic
103C22×C8×C59C2311yescharacteristic
104#121×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#102×C59—21no—
113C2×D4×C59—21no—
114C2×D4×C59—41no—
115C22×C4×C59—41no—
116C22×C4×C59C2×D411yescharacteristic
117C2×C22⋊D59—41no—
118C2×C22⋊D59D811yescharacteristic
119C22×Dic59—41no—
120C22×Dic59—41no—
121#11745687—84no—
122C2×C4×D59—41no—
123C2×C4×D59—41no—
124C2×C4×D59C2×D411yescharacteristic
125C2×Dic118—21no—
126C2×Dic118—41no—
127C2×Dic118—41no—
128D8×C59—21no—
129D8×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—
138C22⋊C4×C59—41no—
139C22⋊Dic59—41no—
140C22⋊Dic59—41no—
141C22⋊Dic59—41no—
142C22⋊Dic59D811yescharacteristic
143(C2×D59)⋊C4—41no—
144(C2×D59)⋊C4D811yescharacteristic
145C4⋊Dic59—21no—
146C4⋊Dic59—21no—
147C4⋊Dic59—21no—
148C4⋊Dic59—41no—
149C4⋊Dic59—41no—
150Dic59⋊C4—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
161Dic236—21no—
162Dic236—21no—
163C8×D59—21no—
164C8×D59—21no—
165C16×C59—21no—
166C16×C59—21no—
167C59⋊C16—21no—
168C59⋊C16#4112yescomplemented
169C23×C59—41no—
170C23×C59C2×D811yescharacteristic
171C22×D59C2×D811yescharacteristic
172D4×C59—41no—
173D4×C59—41no—
174D4×C59—82no—
175C2×C4×C59—21no—
176C2×C4×C59—21no—
177C2×C4×C59—41no—
178C2×C4×C59—41no—
179C2×C4×C59—41no—
180C2×C4×C59—161no—
181C2×C4×C59C2×D811yescharacteristic
182C2×C4×C59C22×D411yescharacteristic
183C22⋊D59—82no—
184C22⋊D59SD3214yescomplemented
185C2×Dic59—41no—
186C2×Dic59—41no—
187C2×Dic59—41no—
188C2×Dic59—41no—
189C2×Dic59—82no—
190C2×Dic59—82no—
191C2×Dic59C2×D811yescharacteristic
192C2×Dic59C2×D811yescharacteristic
193C2×Dic59C2×D811yescharacteristic
194C4×D59—21no—
195C4×D59—21no—
196C4×D59—82no—
197C4×D59—161no—
198Dic118—41no—
199Dic118—41no—
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×C59—81no—
211C22×C59—81no—
212C22×C59—81no—
213C22×C59C2×SD3212yes—
214C22×C59C22×D811yescharacteristic
215C2×D59—21no—
216C2×D59—21no—
217C2×D59C2×SD3212yes—
218C4×C59—21no—
219C4×C59—81no—
220C4×C59—81no—
221C4×C59—81no—
222C4×C59#54711yescharacteristic
223C4×C59#57811yescharacteristic
224Dic59—21no—
225Dic59—21no—
226Dic59—81no—
227Dic59—81no—
228Dic59—81no—
229Dic59—81no—
230Dic59C2×SD3212yes—
231#3338—591no—
232#3334—591no—
233#2254—591no—
234#2235—591no—
235#2235—591no—
236#2233—591no—
237#2233D5911yescharacteristic, complemented
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—161no—
248C2×C59#333411yescharacteristic
249C2×C59C22×SD3211yescharacteristic
250C2×C59#341611yescharacteristic
251D59—22no—
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#467—591no—
267#466—591no—
268#466C2×D5911yescharacteristic
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#358—591no—
288#358—591no—
289#357—591no—
290#357C2×D5911yescharacteristic
291#357C2×D5911yescharacteristic
292C22⋊C16—591no—
293C22⋊C16C2×D5911yescharacteristic
294C4×C16—591no—
295C59#3390211yescharacteristic, complemented, Sylow
296C2×#41—2361no—
297C2×Q16—591no—
298C2×D8C22×D5911yescharacteristic
299C22×C8—591no—
300C22×C8C22×D5911yescharacteristic
301C4⋊Q8—1181no—
302C4⋊Q8—1181no—
303#125—2361no—
304#123—2361no—
305#123—2361no—
306C22⋊Q8—1181no—
307C22⋊Q8—1181no—
308C22⋊Q8—2361no—
309C22⋊Q8—2361no—
310C22⋊Q8—2361no—
311#121—21no—
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—1181no—
342#103—1181no—
343#103—1181no—
344#102—21no—
345#102—1181no—
346#102—1181no—
347C22⋊C8—591no—
348C22⋊C8—591no—
349C4×C8—591no—
350#41—4724no—
351C2×Q8—1181no—
352C2×Q8—2361no—
353C2×Q8—2361no—
354C2×D4—21no—
355C2×D4—41no—
356C2×D4—591no—
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—1181no—
366Q16—1181no—
367D8—21no—
368D8—21no—
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—591no—
382C4⋊C4—1181no—
383C4⋊C4—1181no—
384C4⋊C4—1181no—
385C4⋊C4—2361no—
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—41no—
407C23—591no—
408C23D8×D5911yescharacteristic
409Q8—9441no—
410Q8—4722no—
411Q8—2361no—
412Q8—2361no—
413D4—41no—
414D4—41no—
415D4—82no—
416D4—594no—
417D4—4722no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—161no—
424C2×C4—591no—
425C2×C4—591no—
426C2×C4—591no—
427C2×C4—1181no—
428C2×C4—1181no—
429C2×C4—9441no—
430C2×C4—4722no—
431C2×C4—4722no—
432C2×C4—4722no—
433C2×C4—2361no—
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—81no—
447C22—81no—
448C22—81no—
449C22—592no—
450C22—1181no—
451C22—1181no—
452C22SD32×D5912yes—
453C22C2×D8×D5911yescharacteristic
454C4—21no—
455C4—81no—
456C4—81no—
457C4—81no—
458C4—592no—
459C4—1181no—
460C4—1181no—
461C4—4721no—
462C4—4721no—
463C4—4721no—
464C4—4721no—
465C4#1683805011yescharacteristic
466C4#1683821811yescharacteristic
467C2—22no—
468C2—161no—
469C2—1182no—
470C2#2143643811yescharacteristic
471C2C2×SD32×D5911yescharacteristic
472C2#2143724211yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2233C2
socleC22×C59C22×D8
prime radical#21436440C2
hypercenter#2233D59
hypocenterC59#33902
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2233

Complements

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

NH ≅ G/Ntype
1. #21436486C2semidirect
3. #21426377C2semidirect
4. #21426295C2semidirect
5. #21426294C2semidirect
6. #21426280C2semidirect
7. #21426241C2semidirect
8. #21426237C2semidirect
9. #21426191C2semidirect
10. #21426190C2semidirect
11. #21425951C2semidirect
12. (C22⋊D59)⋊C16C2semidirect
13. C2×#16837229C2semidirect
14. #367×D59C2semidirect
15. C59×#2233C2semidirect
168. C59⋊C16#41semidirect
184. C22⋊D59SD32semidirect
237. #2233D59semidirect
295. C59#33902semidirect

Maximal subgroups

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

subgroupindex
#214364862
#214364402
#214263772
#214262952
#214262942
#214262802
#214262412
#214262372
#214261912
#214261902
#214259512
(C22⋊D59)⋊C162
C2×#168372292
#367×D592
C59×#22332
#3390259