Finite Group Databasebeta

← #28745527

Subgroups — #28745527

Order 15104.

568 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21436459C211yesmaximal, characteristic, complemented
2#21435775C211yesmaximal, characteristic
3#21426329C211yesmaximal, characteristic, complemented
4#21426292C211yesmaximal, characteristic, complemented
5#21426288C211yesmaximal, characteristic, complemented
6#21426251C211yesmaximal, characteristic, complemented
7#21426194C211yesmaximal, characteristic, complemented
8#21426184C211yesmaximal, characteristic, complemented
9#21426160C211yesmaximal, characteristic, complemented
10#21426023C211yesmaximal, characteristic, complemented
11SD32×Dic59C211yesmaximal, characteristic, complemented
12(C22⋊D59)⋊C16C211yesmaximal, characteristic, complemented
13#21425803C211yesmaximal, characteristic, complemented
14#357×D59C211yesmaximal, characteristic, complemented
15C59×#2189C211yesmaximal, characteristic, complemented
16C2×D8×D59C2211yescharacteristic
17#16838086—21no—
18#16838040—21no—
19C2×#14273670C2211yescharacteristic
20C22×C59⋊C16C2211yescharacteristic, complemented
21C2×C16×D59C2211yescharacteristic
22#16837803C2211yescharacteristic, complemented
23#16837787C2211yescharacteristic
24#16837740—21no—
25#16837731—21no—
26#16837707—21no—
27#16837640C2211yescharacteristic
28#16837640C2211yescharacteristic
29#16837638C2211yescharacteristic
30#16837619—21no—
31#16837551—21no—
32#16837547—21no—
33Q16×Dic59C2211yescharacteristic, complemented
34D8×Dic59C2211yescharacteristic
35C8×C22⋊D59C2211yescharacteristic
36#16837440C2211yescharacteristic, complemented
37#107×D59C2211yescharacteristic
38#16837230C2211yescharacteristic, complemented
39#16837229C2211yescharacteristic, complemented
40#16837228C2211yescharacteristic, complemented
41Dic59⋊C16C2211yescharacteristic, complemented
42#16837206C2211yescharacteristic, complemented
43#16837206C2211yescharacteristic, complemented
44#16837206C2211yescharacteristic, complemented
45#16837204C2211yescharacteristic
46#16837204C2211yescharacteristic
47#16837203C2211yescharacteristic
48#16837203C2211yescharacteristic
49#16837202C2211yescharacteristic
50(C2×D59)⋊C16C2211yescharacteristic
51C16×Dic59C2211yescharacteristic, complemented
52C2×#135×C59C2211yescharacteristic
53C2×Q32×C59C2211yescharacteristic, complemented
54C2×SD32×C59C2211yescharacteristic
55C59×#467C2211yescharacteristic, complemented
56C59×#358C2211yescharacteristic, complemented
57C59×#357C2211yescharacteristic
58C59×C22⋊C16C2211yescharacteristic, complemented
59C2×D4×D59—21no—
60C2×#11745671—21no—
61C2×D472C2311yescharacteristic
62C2×#11745669—21no—
63D8×D59—22no—
64D8×D59—41no—
65C2×C8×D59C2311yescharacteristic
66#14273720—41no—
67#14273718—21no—
68#14273717—41no—
69#14273716—21no—
70#14273710—41no—
71#14273709—41no—
72#14273703—21no—
73#14273703—21no—
74#14273701—21no—
75#14273699—41no—
76#14273695—41no—
77Q8×Dic59—21no—
78D4×Dic59—21no—
79C4×C22⋊D59D411yescharacteristic
80#14273686—21no—
81#14273685—41no—
82C4⋊C4×D59—21no—
83C2×(C2×D59)⋊C4—41no—
84#14273670—41no—
85C2×C59⋊C16—41no—
86C2×C59⋊C16C2311yescharacteristic
87C2×C59⋊C16C2311yescharacteristic
88C16×D59—41no—
89#14273661C2311yescharacteristic
90#14273661C2311yescharacteristic
91#14273661C2311yescharacteristic
92#14273660—21no—
93#14273659—21no—
94#14273658—21no—
95Dic59⋊C8D411yescharacteristic
96#14273651—21no—
97#14273651—21no—
98#14273651—21no—
99#14273649—21no—
100#14273648—21no—
101C22⋊(C59⋊C8)D411yescharacteristic
102(C2×D59)⋊C8D411yescharacteristic
103C8×Dic59C2311yescharacteristic
104C2×#41×C59—21no—
105#135×C59—22no—
106#135×C59—41no—
107C2×Q16×C59C2311yescharacteristic
108C2×Q16×C59C2311yescharacteristic
109C2×SD16×C59—21no—
110C2×D8×C59C2311yescharacteristic
111C22×C8×C59C2311yescharacteristic
112C22⋊Q8×C59—21no—
113Q32×C59—41no—
114SD32×C59—22no—
115C2×C16×C59C2311yescharacteristic
116C2×C16×C59C2311yescharacteristic
117#107×C59C2311yescharacteristic
118#103×C59—21no—
119C23×D59—41no—
120#41×C59—42no—
121#41×C59—81no—
122C2×Q8×C59—21no—
123C2×Q8×C59—21no—
124C2×D4×C59—21no—
125C2×D4×C59—41no—
126C22×C4×C59—41no—
127C22×C4×C59C2×D411yescharacteristic
128C2×C22⋊D59—41no—
129C2×C22⋊D59—41no—
130C2×C22⋊D59D811yescharacteristic
131C22×Dic59—41no—
132D4×D59—42no—
133D4×D59—81no—
134C2×D236—21no—
135C2×C4×D59—41no—
136C2×C4×D59C2×D411yescharacteristic
137Q16×C59—21no—
138Q16×C59—21no—
139Q16×C59—21no—
140Q16×C59—21no—
141SD16×C59—42no—
142D8×C59—21no—
143D8×C59#4112yes—
144C2×C8×C59—21no—
145C2×C8×C59—21no—
146C2×C8×C59C2×D411yescharacteristic
147C2×C8×C59C2411yescharacteristic
148C4⋊C4×C59—21no—
149C4⋊C4×C59—41no—
150C22⋊C4×C59—41no—
151C22⋊Dic59—41no—
152C22⋊Dic59—41no—
153C22⋊Dic59D811yescharacteristic
154#11745671—81no—
155#11745669—42no—
156(C2×D59)⋊C4—41no—
157(C2×D59)⋊C4—41no—
158(C2×D59)⋊C4—41no—
159(C2×D59)⋊C4—41no—
160(C2×D59)⋊C4—41no—
161(C2×D59)⋊C4D811yescharacteristic
162C4⋊Dic59—21no—
163C4⋊Dic59—21no—
164C4⋊Dic59—21no—
165C4⋊Dic59—41no—
166C4⋊Dic59—41no—
167Dic59⋊C4—41no—
168Dic59⋊C4—41no—
169Dic59⋊C4—41no—
170Dic59⋊C4—41no—
171Dic59⋊C4D811yescharacteristic
172C4×Dic59—41no—
173C4×Dic59—41no—
174C4×Dic59C2×D411yescharacteristic
175C2×C59⋊C8C2×D411yescharacteristic
176C2×C59⋊C8C2×D411yescharacteristic
177D472—21no—
178D472—21no—
179C8×D59—21no—
180C8×D59—21no—
181C16×C59—21no—
182C16×C59—21no—
183C59⋊C16—21no—
184C59⋊C16—21no—
185C23×C59—41no—
186C23×C59C2×D811yescharacteristic
187C22×D59—41no—
188C22×D59—82no—
189C22×D59—82no—
190C22×D59—161no—
191C22×D59C2×D811yescharacteristic
192Q8×C59—41no—
193Q8×C59—41no—
194Q8×C59—41no—
195Q8×C59—41no—
196D4×C59—22no—
197D4×C59—41no—
198D4×C59—82no—
199C2×C4×C59—21no—
200C2×C4×C59—21no—
201C2×C4×C59—41no—
202C2×C4×C59—41no—
203C2×C4×C59—41no—
204C2×C4×C59—82no—
205C2×C4×C59C2×D811yescharacteristic
206C2×C4×C59C22×D411yescharacteristic
207C22⋊D59—22no—
208C22⋊D59—82no—
209C22⋊D59—161no—
210C2×Dic59—41no—
211C2×Dic59—41no—
212C2×Dic59—41no—
213C2×Dic59—161no—
214C2×Dic59C2×D811yescharacteristic
215C2×Dic59C2×D811yescharacteristic
216C2×Dic59C2×D811yescharacteristic
217D236—41no—
218D236—41no—
219C4×D59—21no—
220C4×D59—21no—
221C4×D59—82no—
222C8×C59—21no—
223C8×C59C2×#4111yescharacteristic
224C8×C59H(5,𝔽2)11yescharacteristic
225C59⋊C8—21no—
226C59⋊C8—21no—
227#33863—591nomaximal, Sylow
228C22×C59—21no—
229C22×C59—21no—
230C22×C59—21no—
231C22×C59—42no—
232C22×C59—81no—
233C22×C59—81no—
234C22×C59C22×D811yescharacteristic
235C2×D59—21no—
236C2×D59—21no—
237C2×D59—81no—
238C2×D59—81no—
239C2×D59—81no—
240C2×D59—164no—
241C2×D59#50812yes—
242C4×C59—21no—
243C4×C59—81no—
244C4×C59—81no—
245C4×C59—81no—
246C4×C59#54611yescharacteristic
247C4×C59#57711yescharacteristic
248Dic59—21no—
249Dic59—21no—
250Dic59—21no—
251Dic59—81no—
252Dic59—81no—
253Dic59—81no—
254#3336—591no—
255#3280—591no—
256#2242—591no—
257#2235—591no—
258#2234—591no—
259#2228—591no—
260#2217—591no—
261#2216—591no—
262#2211—591no—
263#2189D5911yescharacteristic, complemented
264#2187—591no—
265C4×SD32—591no—
266D4×C16—591no—
267#2138—591no—
268C2×#357—591no—
269C2×C59—41no—
270C2×C59—82no—
271C2×C59#329411yescharacteristic
272C2×C59C2×#50811yescharacteristic
273C2×C59#341611yescharacteristic
274D59—22no—
275D59—161no—
276C2×#135C2×D5911yescharacteristic
277C22×D8—591no—
278#552—1181no—
279#546—1181no—
280C2×Q32C2×D5911yescharacteristic, complemented
281C2×SD32—591no—
282C2×SD32C2×D5911yescharacteristic
283C22×C16—591no—
284C22×C16—591no—
285#499—591no—
286#495—591no—
287#485—1181no—
288#483—1181no—
289#479—1181no—
290#467—591no—
291#467—591no—
292#467C2×D5911yescharacteristic, complemented
293#466—591no—
294#463—1181no—
295#450—1181no—
296#449—1181no—
297C4×Q16—591no—
298C4×D8—591no—
299C8×D4—591no—
300#427—591no—
301C2×#107—591no—
302#367—591no—
303#367—591no—
304#366—591no—
305C4⋊C16—591no—
306#358—591no—
307#358—591no—
308#358—591no—
309#358C2×D5911yescharacteristic, complemented
310#357—591no—
311#357—591no—
312#357—591no—
313#357—591no—
314#357—591no—
315#357C2×D5911yescharacteristic
316C22⋊C16—591no—
317C22⋊C16C2×D5911yescharacteristic, complemented
318C4×C16—591no—
319C59#3386311yescharacteristic, complemented, Sylow
320C2×#41—21no—
321C22×D4—1181no—
322#135—22no—
323#135—41no—
324C2×Q16C22×D5911yescharacteristic
325C2×Q16C22×D5911yescharacteristic
326C2×SD16—21no—
327C2×SD16—1181no—
328C2×D8—591no—
329C2×D8—1181no—
330C2×D8—1182no—
331C2×D8—2361no—
332C2×D8C22×D5911yescharacteristic
333C22×C8—591no—
334C22×C8C22×D5911yescharacteristic
335#126—2361no—
336#125—1181no—
337#125—2361no—
338#124—1181no—
339#123—2361no—
340#123—2361no—
341C22⋊Q8—21no—
342C22⋊Q8—1181no—
343C22⋊Q8—1181no—
344#121—1181no—
345#121—2361no—
346#120—2361no—
347C4×Q8—1181no—
348C4×D4—591no—
349C4×D4—1181no—
350#117—1181no—
351#117—2361no—
352C2×C4⋊C4—1181no—
353C2×C22⋊C4—2361no—
354Q32—41no—
355SD32—22no—
356SD32—2361no—
357C2×C16—591no—
358C2×C16—591no—
359C2×C16—2361no—
360C2×C16—2361no—
361C2×C16C22×D5911yescharacteristic
362C2×C16C22×D5911yescharacteristic
363#107—591no—
364#107—591no—
365#107—591no—
366#107—1181no—
367#107C22×D5911yescharacteristic
368#106—1181no—
369#106—1181no—
370C4⋊C8—591no—
371#103—21no—
372#103—1181no—
373#103—1181no—
374#103—1181no—
375#102—1181no—
376#102—1181no—
377C22⋊C8—591no—
378C22⋊C8—591no—
379C4×C8—591no—
380C24—2361no—
381#41—42no—
382#41—81no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×D4—21no—
386C2×D4—41no—
387C2×D4—591no—
388C2×D4—1181no—
389C2×D4—4721no—
390C2×D4—2361no—
391C2×D4—2362no—
392C2×D4—2361no—
393C22×C4—41no—
394C22×C4—591no—
395C22×C4—2361no—
396C22×C4—2361no—
397C22×C4D4×D5911yescharacteristic
398Q16—21no—
399Q16—21no—
400Q16—21no—
401Q16—21no—
402SD16—42no—
403SD16—4721no—
404D8—21no—
405D8—1181no—
406D8—1181no—
407D8—2362no—
408D8#1174568712yes—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—591no—
412C2×C8—591no—
413C2×C8—1181no—
414C2×C8—1181no—
415C2×C8D4×D5911yescharacteristic
416C2×C8C23×D5911yescharacteristic
417C4⋊C4—21no—
418C4⋊C4—41no—
419C4⋊C4—591no—
420C4⋊C4—1181no—
421C4⋊C4—1181no—
422C4⋊C4—1181no—
423C4⋊C4—2361no—
424C4⋊C4—2361no—
425C4⋊C4—2361no—
426C4⋊C4—2361no—
427C4⋊C4—2361no—
428C4⋊C4—2361no—
429C22⋊C4—41no—
430C22⋊C4—591no—
431C22⋊C4—591no—
432C22⋊C4—2361no—
433C22⋊C4—2361no—
434C22⋊C4—2361no—
435C22⋊C4—2361no—
436C22⋊C4—2361no—
437C22⋊C4—2361no—
438C22⋊C4—2361no—
439C42—591no—
440C42—2361no—
441C42—2361no—
442C16—21no—
443C16—21no—
444C16—1181no—
445C16—1181no—
446C23—41no—
447C23—591no—
448C23—9441no—
449C23—4722no—
450C23—4722no—
451C23—2361no—
452C23D8×D5911yescharacteristic
453Q8—41no—
454Q8—41no—
455Q8—41no—
456Q8—41no—
457D4—22no—
458D4—41no—
459D4—82no—
460D4—1182no—
461D4—9441no—
462D4—4722no—
463D4—2361no—
464D4—2361no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—82no—
471C2×C4—591no—
472C2×C4—591no—
473C2×C4—591no—
474C2×C4—1181no—
475C2×C4—1181no—
476C2×C4—9441no—
477C2×C4—4722no—
478C2×C4—2361no—
479C2×C4—2361no—
480C2×C4—2361no—
481C2×C4D8×D5911yescharacteristic
482C2×C4C2×D4×D5911yescharacteristic
483C8—21no—
484C8—1181no—
485C8—1181no—
486C8C2×#1174568711yescharacteristic
487C8#1427377211yescharacteristic
488C22—21no—
489C22—21no—
490C22—21no—
491C22—42no—
492C22—81no—
493C22—81no—
494C22—592no—
495C22—1181no—
496C22—1181no—
497C22—9444no—
498C22—4721no—
499C22—4721no—
500C22—4721no—
501C22C2×D8×D5911yescharacteristic
502C4—21no—
503C4—81no—
504C4—81no—
505C4—81no—
506C4—1181no—
507C4—1181no—
508C4—1181no—
509C4—4721no—
510C4—4721no—
511C4—4721no—
512C4#1683803911yescharacteristic
513C4#1683821211yescharacteristic
514C2—41no—
515C2—82no—
516C2—1182no—
517C2—9441no—
518C2#2143590211yescharacteristic
519C2#508×D5911yescharacteristic
520C2#2143724111yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2189C2
socleC22×C59C22×D8
prime radical#21435775C2
hypercenter#2189D59
hypocenterC59#33863
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2189

Complements

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

NH ≅ G/Ntype
1. #21436459C2semidirect
3. #21426329C2semidirect
4. #21426292C2semidirect
5. #21426288C2semidirect
6. #21426251C2semidirect
7. #21426194C2semidirect
8. #21426184C2semidirect
9. #21426160C2semidirect
10. #21426023C2semidirect
11. SD32×Dic59C2semidirect
12. (C22⋊D59)⋊C16C2semidirect
13. #21425803C2semidirect
14. #357×D59C2semidirect
15. C59×#2189C2semidirect
20. C22×C59⋊C16C22semidirect
22. #16837803C22semidirect
33. Q16×Dic59C22semidirect
36. #16837440C22semidirect
38. #16837230C22semidirect
39. #16837229C22semidirect
40. #16837228C22semidirect
41. Dic59⋊C16C22semidirect
42. #16837206C22semidirect
43. #16837206C22semidirect
44. #16837206C22semidirect
51. C16×Dic59C22semidirect
53. C2×Q32×C59C22semidirect
55. C59×#467C22semidirect
56. C59×#358C22semidirect
58. C59×C22⋊C16C22semidirect
263. #2189D59semidirect
280. C2×Q32C2×D59semidirect
292. #467C2×D59semidirect
309. #358C2×D59semidirect
317. C22⋊C16C2×D59semidirect
319. C59#33863semidirect

Maximal subgroups

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

subgroupindex
#214364592
#214357752
#214263292
#214262922
#214262882
#214262512
#214261942
#214261842
#214261602
#214260232
SD32×Dic592
(C22⋊D59)⋊C162
#214258032
#357×D592
C59×#21892
#3386359