Finite Group Databasebeta

← #28742030

Subgroups — #28742030

Order 15104.

696 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837839C211yesmaximal, characteristic
2#21432971C211yesmaximal, characteristic, complemented
3#21432943C211yesmaximal, characteristic, complemented
4#21425973C211yesmaximal, characteristic, complemented
5#21425969C211yesmaximal, characteristic, complemented
6#21425963C211yesmaximal, characteristic, complemented
7SD32×Dic59C211yesmaximal, characteristic, complemented
8C4×#14273670C211yesmaximal, characteristic, complemented
9#21425942C211yesmaximal, characteristic, complemented
10C4×#14273668C211yesmaximal, characteristic, complemented
11#21425881C211yesmaximal, characteristic, complemented
12#21425802C211yesmaximal, characteristic, complemented
13#357×D59C211yesmaximal, characteristic, complemented
14#21425694C211yesmaximal, characteristic, complemented
15C59×#2175C211yesmaximal, characteristic, complemented
16C2×#14273750C2211yescharacteristic
17C2×D8×D59C2211yescharacteristic
18#16837884—21no—
19#16837874—21no—
20#16837839C418yescomplemented
21C2×#14273670C2211yescharacteristic
22C2×D944C2211yescharacteristic
23C2×#14273668C2211yescharacteristic
24C2×#14273666C2211yescharacteristic
25C2×C16×D59C2211yescharacteristic
26#16837512—21no—
27#16837511—21no—
28#16837505—21no—
29#16837500—21no—
30Q16×Dic59C2211yescharacteristic, complemented
31C4×D472C2211yescharacteristic
32C4×D472C2211yescharacteristic, complemented
33D8×Dic59C2211yescharacteristic
34#16837484C2211yescharacteristic
35#16837484C2211yescharacteristic, complemented
36#16837439C2211yescharacteristic
37#107×D59C2211yescharacteristic
38#16837405—21no—
39#16837402—21no—
40#16837356C2211yescharacteristic
41#16837230C2211yescharacteristic, complemented
42#16837229C2211yescharacteristic, complemented
43#16837227C2211yescharacteristic, complemented
44#16837206C2211yescharacteristic, complemented
45#16837204C2211yescharacteristic
46#16837204C2211yescharacteristic, complemented
47#16837203C2211yescharacteristic
48#16837202C2211yescharacteristic
49#16837202C2211yescharacteristic, complemented
50#16837177C2211yescharacteristic, complemented
51C4×C59⋊C16C2211yescharacteristic, complemented
52C16×Dic59C2211yescharacteristic, complemented
53C2×SD32×C59C2211yescharacteristic
54C4×Q16×C59C2211yescharacteristic, complemented
55C4×D8×C59C2211yescharacteristic
56C59×#366C2211yescharacteristic, complemented
57C59×#358C2211yescharacteristic, complemented
58C59×#357C2211yescharacteristic
59C59×#346C2211yescharacteristic, complemented
60C2×#11745689—21no—
61C2×D4×D59—21no—
62#14273750—22no—
63#14273750C2×C414yescomplemented
64C2×#11745671—21no—
65C2×D472C2311yescharacteristic
66C2×D472C2311yescharacteristic
67C2×#11745669—21no—
68D8×D59—22no—
69D8×D59C2×C414yescomplemented
70C2×#11745659D411yescharacteristic
71C2×C8×D59C2311yescharacteristic
72Q8×Dic59—21no—
73C4×D236—21no—
74C4×D236—21no—
75C4×D236—41no—
76D4×Dic59—21no—
77#14273689—21no—
78#14273689—21no—
79#14273689—41no—
80C4×C22⋊D59—41no—
81(C22⋊D59)⋊C4—41no—
82#14273685—21no—
83#14273684—41no—
84#14273684D411yescharacteristic
85C4⋊C4×D59—21no—
86C4⋊C4×D59—41no—
87C2×(C2×D59)⋊C4—41no—
88C22⋊C4×D59—41no—
89#14273670C2×C414yescomplemented
90D944C2×C414yescomplemented
91#14273668C2×C414yescomplemented
92#14273666C2×C414yescomplemented
93C2×C59⋊C16C2311yescharacteristic
94C2×C59⋊C16C2311yescharacteristic
95C16×D59—22no—
96#14273661C2311yescharacteristic
97#14273661C2311yescharacteristic
98#14273660—21no—
99#14273658—21no—
100#14273651—21no—
101#14273649—21no—
102#14273649—21no—
103#14273648—21no—
104#14273647—21no—
105#14273647—21no—
106#14273638D411yescharacteristic
107#14273637D411yescharacteristic
108C8×Dic59C2311yescharacteristic
109C2×Q16×C59C2311yescharacteristic
110C2×D8×C59C2311yescharacteristic
111C4×Q8×C59—21no—
112C4×D4×C59—21no—
113SD32×C59C2×C414yescomplemented
114C2×C16×C59C2311yescharacteristic
115C2×C16×C59C2311yescharacteristic
116#107×C59C2311yescharacteristic
117#107×C59C2311yescharacteristic
118#103×C59—21no—
119#102×C59—21no—
120C4×C8×C59C2311yescharacteristic
121C23×D59—41no—
122C2×Q8×C59—21no—
123C2×D4×C59—21no—
124C22×C4×C59—41no—
125C2×C22⋊D59—41no—
126C22×Dic59—41no—
127#11745689—24no—
128#11745689—42no—
129D4×D59—24no—
130D4×D59—42no—
131C2×D236—21no—
132C2×D236—21no—
133C2×D236—41no—
134C2×C4×D59—41no—
135C2×C4×D59—41no—
136C2×C4×D59—41no—
137C2×C4×D59—41no—
138C2×C4×D59—41no—
139C2×C4×D59—41no—
140C2×C4×D59C2×D411yescharacteristic
141Q16×C59—21no—
142Q16×C59C22×C412yes—
143D8×C59—21no—
144D8×C59C22×C412yes—
145C2×C8×C59C2×D411yescharacteristic
146C2×C8×C59C2411yescharacteristic
147C4⋊C4×C59—21no—
148C4⋊C4×C59—21no—
149C4⋊C4×C59—41no—
150C22⋊C4×C59—41no—
151C42×C59—41no—
152C42×C59C2×D411yescharacteristic
153C22⋊Dic59—41no—
154#11745671—24no—
155#11745669—24no—
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
162(C2×D59)⋊C4D811yescharacteristic
163C4⋊Dic59—21no—
164C4⋊Dic59—21no—
165C4⋊Dic59—41no—
166Dic59⋊C4—41no—
167Dic59⋊C4D811yescharacteristic
168Dic59⋊C4D811yescharacteristic
169C4×Dic59—41no—
170C4×Dic59—41no—
171C4×Dic59C2×D411yescharacteristic
172C2×C59⋊C8C2×D411yescharacteristic
173C2×C59⋊C8C2×D411yescharacteristic
174D472—21no—
175D472—21no—
176D472C22×C412yes—
177D472C22×C412yes—
178#11745659—22no—
179C8×D59C22×C412yes—
180C8×D59#4112yes—
181C16×C59—21no—
182C16×C59C22×C412yes—
183C59⋊C16—21no—
184C59⋊C16C22×C412yes—
185C23×C59—41no—
186C22×D59—41no—
187C22×D59—41no—
188C22×D59—44no—
189C22×D59—161no—
190C22×D59—161no—
191C22×D59C2×D811yescharacteristic
192Q8×C59—22no—
193Q8×C59—41no—
194D4×C59—22no—
195D4×C59—41no—
196C2×C4×C59—41no—
197C2×C4×C59—41no—
198C2×C4×C59—41no—
199C2×C4×C59—161no—
200C2×C4×C59C2×D811yescharacteristic
201C2×C4×C59C2×D811yescharacteristic
202C2×C4×C59C22×D411yescharacteristic
203C22⋊D59—44no—
204C2×Dic59—41no—
205C2×Dic59—41no—
206C2×Dic59—161no—
207C2×Dic59C2×D811yescharacteristic
208C2×Dic59C2×D811yescharacteristic
209C2×Dic59C2×D811yescharacteristic
210D236—22no—
211D236—22no—
212D236—44no—
213D236—41no—
214D236—41no—
215C4×D59—44no—
216C4×D59—161no—
217C4×D59—161no—
218C4×D59—161no—
219C4×D59—161no—
220C4×D59—161no—
221C4×D59C4×D412yes—
222C4×D59#13512yes—
223C8×C59—21no—
224C8×C59C23×C411yescharacteristic
225C8×C59C2×#4111yescharacteristic
226C59⋊C8—21no—
227C59⋊C8C4×D412yes—
228#33454—591nomaximal, Sylow
229C22×C59—42no—
230C22×C59—81no—
231C22×C59—81no—
232C22×C59C22×D811yescharacteristic
233C2×D59—21no—
234C2×D59—21no—
235C2×D59—42no—
236C2×D59—42no—
237C2×D59—81no—
238C2×D59—81no—
239C2×D59—81no—
240C2×D59—81no—
241C2×D59—164no—
242C2×D59C4×D812yes—
243C4×C59—21no—
244C4×C59—21no—
245C4×C59—42no—
246C4×C59—81no—
247C4×C59—81no—
248C4×C59C2×C4×D411yescharacteristic
249C4×C59C2×#13511yescharacteristic
250Dic59—21no—
251Dic59—21no—
252Dic59—81no—
253Dic59—81no—
254Dic59C4×D812yes—
255C2×#509—591no—
256#2951—591no—
257#2948—591no—
258#2177—591no—
259#2177—591no—
260#2175—591no—
261#2175D5911yescharacteristic, complemented
262C4×SD32—591no—
263C4×SD32—591no—
264C4×D16—591no—
265C4×D16—591no—
266#2159—591no—
267#2138—591no—
268C2×#357—591no—
269#2108—591no—
270C2×C59—82no—
271C2×C59C2×C4×D811yescharacteristic
272C2×C59#341511yescharacteristic
273C2×C59#341611yescharacteristic
274D59—22no—
275D59—82no—
276D59—82no—
277C2×#135—591no—
278C22×D8—591no—
279#520—1181no—
280#518—1181no—
281#509—598no—
282C2×SD32—591no—
283C2×SD32C2×D5911yescharacteristic
284C2×D16—591no—
285C2×D16—591no—
286C2×M32—591no—
287C22×C16—591no—
288#442—1181no—
289#442—1181no—
290#440—1181no—
291#440—1181no—
292C4×Q16—591no—
293C4×Q16C2×D5911yescharacteristic, complemented
294C4×D8—591no—
295C4×D8—591no—
296C4×D8—591no—
297C4×D8—591no—
298C4×D8—591no—
299C4×D8C2×D5911yescharacteristic
300#427—591no—
301C2×#107—591no—
302#418—1181no—
303#417—1181no—
304#405—591no—
305#367—591no—
306#367—591no—
307#366—591no—
308#366C2×D5911yescharacteristic, complemented
309#358—591no—
310#358C2×D5911yescharacteristic, complemented
311#357—591no—
312#357—591no—
313#357—591no—
314#357—591no—
315#357—591no—
316#357C2×D5911yescharacteristic
317#346—591no—
318#346C2×D5911yescharacteristic, complemented
319C4×C16—591no—
320C4×C16—591no—
321C59#3345411yescharacteristic, complemented, Sylow
322C2×#41—1181no—
323C22×D4—1181no—
324#135—594no—
325#135—1182no—
326C2×Q16C22×D5911yescharacteristic
327C2×SD16—1181no—
328C2×D8—594no—
329C2×D8—591no—
330C2×D8—591no—
331C2×D8—1182no—
332C2×D8—1181no—
333C2×D8C22×D5911yescharacteristic
334C2×M16—591no—
335C22×C8—591no—
336C4×Q8—21no—
337C4×Q8—1181no—
338C4×D4—21no—
339C4×D4—1181no—
340C4×D4—1181no—
341C4×D4—1181no—
342C4×D4—1181no—
343C4×D4—1181no—
344C4×D4—2361no—
345C4×D4—2361no—
346C4×D4—2361no—
347C4×D4—2361no—
348#117—591no—
349#117—1181no—
350#117—2361no—
351C2×C4⋊C4—1181no—
352C2×C4⋊C4—2361no—
353C2×C22⋊C4—2361no—
354C2×C22⋊C4—2361no—
355SD32—594no—
356SD32C4×D5914yescomplemented
357D16—594no—
358D16—594no—
359M32—594no—
360C2×C16—591no—
361C2×C16—591no—
362C2×C16—1182no—
363C2×C16C22×D5911yescharacteristic
364C2×C16C22×D5911yescharacteristic
365#107—591no—
366#107—591no—
367#107—1181no—
368#107C22×D5911yescharacteristic
369#107C22×D5911yescharacteristic
370#106—1181no—
371#103—21no—
372#103—1181no—
373#102—21no—
374#102—1181no—
375#102—1181no—
376#102—1181no—
377#102—1181no—
378#102—1181no—
379#97—591no—
380#97—591no—
381C4×C8—591no—
382C4×C8C22×D5911yescharacteristic
383C24—2361no—
384#41—1184no—
385#41—2362no—
386C2×Q8—21no—
387C2×D4—21no—
388C2×D4—1184no—
389C2×D4—1181no—
390C2×D4—1181no—
391C2×D4—2361no—
392C2×D4—2362no—
393C2×D4—2361no—
394C22×C4—41no—
395C22×C4—591no—
396C22×C4—2361no—
397C22×C4—2361no—
398C22×C4—2361no—
399C22×C4—2361no—
400C22×C4—2361no—
401C22×C4—2361no—
402C22×C4—2361no—
403Q16—21no—
404Q16C2×C4×D5912yes—
405SD16—1184no—
406D8—21no—
407D8—592no—
408D8—592no—
409D8—1184no—
410D8—1181no—
411D8—1181no—
412D8C2×C4×D5912yes—
413M16—1182no—
414C2×C8—591no—
415C2×C8—592no—
416C2×C8—592no—
417C2×C8—591no—
418C2×C8D4×D5911yescharacteristic
419C2×C8C23×D5911yescharacteristic
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—41no—
423C4⋊C4—591no—
424C4⋊C4—591no—
425C4⋊C4—1181no—
426C4⋊C4—1181no—
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—
438C42—41no—
439C42—591no—
440C42—2361no—
441C42—2361no—
442C42D4×D5911yescharacteristic
443C16—21no—
444C16—592no—
445C16—1181no—
446C16C2×C4×D5912yes—
447C23—41no—
448C23—591no—
449C23—9441no—
450C23—9441no—
451C23—2364no—
452C23—2361no—
453C23—2361no—
454Q8—22no—
455Q8—41no—
456D4—22no—
457D4—41no—
458D4—1182no—
459D4—1182no—
460D4—2364no—
461D4—2364no—
462D4—2361no—
463D4—2361no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—161no—
468C2×C4—591no—
469C2×C4—592no—
470C2×C4—592no—
471C2×C4—591no—
472C2×C4—591no—
473C2×C4—9441no—
474C2×C4—9441no—
475C2×C4—9441no—
476C2×C4—9441no—
477C2×C4—9441no—
478C2×C4—9441no—
479C2×C4—2364no—
480C2×C4—2361no—
481C2×C4—2361no—
482C2×C4D8×D5911yescharacteristic
483C2×C4D8×D5911yescharacteristic
484C2×C4C2×D4×D5911yescharacteristic
485C8—21no—
486C8—592no—
487C8—1181no—
488C8C22×C4×D5911yescharacteristic
489C8#41×D5911yescharacteristic
490C22—42no—
491C22—81no—
492C22—81no—
493C22—592no—
494C22—1181no—
495C22—1181no—
496C22—9444no—
497C22—4721no—
498C22—4721no—
499C22—4721no—
500C22—4721no—
501C22—2362no—
502C22—2362no—
503C22C2×D8×D5911yescharacteristic
504C4—21no—
505C4—21no—
506C4—42no—
507C4—81no—
508C4—81no—
509C4—592no—
510C4—1181no—
511C4—1181no—
512C4—4721no—
513C4—4721no—
514C4C4×D4×D5911yescharacteristic
515C4#135×D5911yescharacteristic
516C2—82no—
517C2—1182no—
518C2—4722no—
519C2—4722no—
520C2C4×D8×D5911yescharacteristic
521C2#2143723611yescharacteristic
522C2#2143724111yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutatorC8×C59C23×C4
FrattiniC2×C8C23×D59
FittingC59×#2175C2
socleC22×C59C22×D8
prime radicalC2×#16837839C2
cosocleC2×C8×C59C24
hypercenter#2175D59
hypocenterC59#33454
LCSG ▷ C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2175

Complements

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

NH ≅ G/Ntype
2. #21432971C2semidirect
3. #21432943C2semidirect
4. #21425973C2semidirect
5. #21425969C2semidirect
6. #21425963C2semidirect
7. SD32×Dic59C2semidirect
8. C4×#14273670C2semidirect
9. #21425942C2semidirect
10. C4×#14273668C2semidirect
11. #21425881C2semidirect
12. #21425802C2semidirect
13. #357×D59C2semidirect
14. #21425694C2semidirect
15. C59×#2175C2semidirect
20. #16837839C4semidirect
30. Q16×Dic59C22semidirect
32. C4×D472C22semidirect
35. #16837484C22semidirect
41. #16837230C22semidirect
42. #16837229C22semidirect
43. #16837227C22semidirect
44. #16837206C22semidirect
46. #16837204C22semidirect
49. #16837202C22semidirect
50. #16837177C22semidirect
51. C4×C59⋊C16C22semidirect
52. C16×Dic59C22semidirect
54. C4×Q16×C59C22semidirect
56. C59×#366C22semidirect
57. C59×#358C22semidirect
59. C59×#346C22semidirect
63. #14273750C2×C4semidirect
69. D8×D59C2×C4semidirect
89. #14273670C2×C4semidirect
90. D944C2×C4semidirect
91. #14273668C2×C4semidirect
92. #14273666C2×C4semidirect
113. SD32×C59C2×C4semidirect
261. #2175D59semidirect
293. C4×Q16C2×D59semidirect
308. #366C2×D59semidirect
310. #358C2×D59semidirect
318. #346C2×D59semidirect
321. C59#33454semidirect
356. SD32C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168378392
#214329712
#214329432
#214259732
#214259692
#214259632
SD32×Dic592
C4×#142736702
#214259422
C4×#142736682
#214258812
#214258022
#357×D592
#214256942
C59×#21752
#3345459