Finite Group Databasebeta

← #28745507

Subgroups — #28745507

Order 15104.

654 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21436440C211yesmaximal, characteristic, complemented
2#21435770C211yesmaximal, characteristic, complemented
3#21426318C211yesmaximal, characteristic, complemented
4#21426278C211yesmaximal, characteristic, complemented
5#21426274C211yesmaximal, characteristic, complemented
6#21426232C211yesmaximal, characteristic, complemented
7#21426181C211yesmaximal, characteristic, complemented
8#21426129C211yesmaximal, characteristic, complemented
9#21426007C211yesmaximal, characteristic, complemented
10#21426006C211yesmaximal, characteristic, complemented
11#21425940C211yesmaximal, characteristic, complemented
12C16×C22⋊D59C211yesmaximal, characteristic, complemented
13C2×#16837204C211yesmaximal, characteristic, complemented
14#357×D59C211yesmaximal, characteristic, complemented
15C59×#2215C211yesmaximal, characteristic, complemented
16C22×D472C2211yescharacteristic
17C2×D8×D59C2211yescharacteristic
18#16838052—21no—
19#16838036—21no—
20C2×#14273668C2211yescharacteristic
21C2×#14273668C2211yescharacteristic, complemented
22C2×C16×D59C2211yescharacteristic, complemented
23#16837806C2211yescharacteristic, complemented
24#16837777C2211yescharacteristic
25#16837714—21no—
26#16837713—21no—
27#16837680—21no—
28#16837639C2211yescharacteristic
29#16837635C2211yescharacteristic, complemented
30#16837596—21no—
31#16837537—21no—
32#16837535—21no—
33D8×Dic59C2211yescharacteristic, complemented
34#16837484C2211yescharacteristic
35C8×C22⋊D59C2211yescharacteristic
36C2×#14273661C2211yescharacteristic, complemented
37#107×D59C2211yescharacteristic, complemented
38#16837227C2211yescharacteristic, complemented
39#16837227C2211yescharacteristic, complemented
40Dic59⋊C16C2211yescharacteristic, complemented
41#16837204C2211yescharacteristic
42#16837204C2211yescharacteristic
43#16837204C2211yescharacteristic, complemented
44#16837204C2211yescharacteristic, complemented
45#16837203C2211yescharacteristic, complemented
46#16837203C2211yescharacteristic, complemented
47#16837202C2211yescharacteristic
48#16837202C2211yescharacteristic, complemented
49C22⋊(C59⋊C16)C2211yescharacteristic, complemented
50(C2×D59)⋊C16C2211yescharacteristic, complemented
51C16×Dic59C2211yescharacteristic, complemented
52C2×D16×C59C2211yescharacteristic, complemented
53C22×C16×C59C2211yescharacteristic, complemented
54C59×#466C2211yescharacteristic
55C59×#466C2211yescharacteristic, complemented
56C59×#366C2211yescharacteristic, complemented
57C59×#357C2211yescharacteristic, complemented
58C59×#357C2211yescharacteristic, complemented
59C2×#11745687—41no—
60C22×D236—21no—
61C2×D4×D59—21no—
62C2×D4×D59—41no—
63C2×D472—22no—
64C2×D472—21no—
65C2×D472—41no—
66C2×D472C2311yescharacteristic
67C2×D472C2311yescharacteristic
68C2×#11745669—21no—
69C2×#11745669—21no—
70D8×D59—22no—
71D8×D59—41no—
72C2×C8×D59C2311yescharacteristic
73#14273726—21no—
74#14273723—21no—
75#14273709—41no—
76#14273708—41no—
77#14273704—41no—
78#14273702—21no—
79#14273700—41no—
80#14273698—21no—
81#14273697—41no—
82#14273695—41no—
83D4×Dic59—21no—
84#14273689—21no—
85C4×C22⋊D59D411yescharacteristic
86C2×C4⋊Dic59—21no—
87C4⋊C4×D59—21no—
88#14273668—22no—
89#14273668—41no—
90C2×C59⋊C16C2311yescharacteristic
91C2×C59⋊C16C2311yescharacteristic
92C16×D59—41no—
93#14273661—21no—
94#14273661C2311yescharacteristic
95#14273661C2311yescharacteristic
96#14273660—21no—
97#14273660—21no—
98Dic59⋊C8D411yescharacteristic
99#14273648—21no—
100#14273648—21no—
101#14273647—21no—
102#14273647—21no—
103C22⋊(C59⋊C8)D411yescharacteristic
104(C2×D59)⋊C8D411yescharacteristic
105C8×Dic59C2311yescharacteristic
106C2×D8×C59C2311yescharacteristic
107C2×D8×C59C2311yescharacteristic
108C22×C8×C59C2311yescharacteristic
109#121×C59—21no—
110#121×C59—21no—
111D16×C59—41no—
112C2×C16×C59—41no—
113C2×C16×C59C2311yescharacteristic
114C2×C16×C59C2311yescharacteristic
115#107×C59C2311yescharacteristic
116#107×C59C2311yescharacteristic
117#102×C59—21no—
118#102×C59—21no—
119C23×D59—41no—
120C23×D59—41no—
121C2×D4×C59—21no—
122C2×D4×C59—21no—
123C2×D4×C59—41no—
124C2×D4×C59—41no—
125C22×C4×C59C2×D411yescharacteristic
126C2×C22⋊D59—41no—
127C2×C22⋊D59—41no—
128C2×C22⋊D59—41no—
129C2×C22⋊D59—41no—
130C2×C22⋊D59D811yescharacteristic
131C22×Dic59—41no—
132C22×Dic59—41no—
133#11745687—84no—
134D4×D59—42no—
135D4×D59—81no—
136D4×D59—84no—
137C2×D236—21no—
138C2×D236—21no—
139C2×D236—42no—
140C2×D236—41no—
141C2×D236—41no—
142C2×D236—81no—
143C2×C4×D59—41no—
144C2×C4×D59—41no—
145C2×C4×D59C2×D411yescharacteristic
146C2×Dic118—41no—
147D8×C59—21no—
148D8×C59—21no—
149D8×C59—21no—
150D8×C59—21no—
151C2×C8×C59—21no—
152C2×C8×C59—21no—
153C2×C8×C59C2×D411yescharacteristic
154C2×C8×C59C2411yescharacteristic
155C4⋊C4×C59—21no—
156C4⋊C4×C59—21no—
157C22⋊C4×C59—41no—
158C22⋊C4×C59—41no—
159C22⋊Dic59—41no—
160C22⋊Dic59—41no—
161C22⋊Dic59D811yescharacteristic
162#11745669—42no—
163#11745669—81no—
164(C2×D59)⋊C4—41no—
165(C2×D59)⋊C4—41no—
166(C2×D59)⋊C4D811yescharacteristic
167C4⋊Dic59—21no—
168C4⋊Dic59—21no—
169C4⋊Dic59—41no—
170Dic59⋊C4—41no—
171Dic59⋊C4—41no—
172Dic59⋊C4D811yescharacteristic
173C4×Dic59C2×D411yescharacteristic
174C2×C59⋊C8C2×D411yescharacteristic
175C2×C59⋊C8C2×D411yescharacteristic
176D472—21no—
177D472—21no—
178D472—21no—
179D472—42no—
180D472#4112yescomplemented
181C8×D59—21no—
182C8×D59—21no—
183C16×C59—21no—
184C16×C59—21no—
185C59⋊C16—21no—
186C59⋊C16—21no—
187C23×C59—41no—
188C23×C59—41no—
189C23×C59C2×D811yescharacteristic
190C22×D59—41no—
191C22×D59—41no—
192C22×D59—82no—
193C22×D59—82no—
194C22×D59—82no—
195C22×D59—82no—
196C22×D59—161no—
197C22×D59—161no—
198C22×D59C2×D811yescharacteristic
199D4×C59—41no—
200D4×C59—41no—
201D4×C59—41no—
202D4×C59—41no—
203D4×C59—82no—
204D4×C59—82no—
205C2×C4×C59—21no—
206C2×C4×C59—21no—
207C2×C4×C59—41no—
208C2×C4×C59—41no—
209C2×C4×C59C2×D811yescharacteristic
210C2×C4×C59C22×D411yescharacteristic
211C22⋊D59—82no—
212C22⋊D59—82no—
213C22⋊D59—82no—
214C22⋊D59—161no—
215C22⋊D59D1614yescomplemented
216C2×Dic59—41no—
217C2×Dic59—41no—
218C2×Dic59—82no—
219C2×Dic59—82no—
220C2×Dic59C2×D811yescharacteristic
221C2×Dic59C2×D811yescharacteristic
222C2×Dic59C2×D811yescharacteristic
223D236—22no—
224D236—41no—
225D236—41no—
226D236—41no—
227D236—82no—
228D236—82no—
229C4×D59—21no—
230C4×D59—21no—
231C4×D59—82no—
232C4×D59—82no—
233Dic118—82no—
234C8×C59—21no—
235C8×C59C2×#4111yescharacteristic
236C8×C59H(5,𝔽2)11yescharacteristic
237C59⋊C8—21no—
238C59⋊C8—21no—
239#33861—591nomaximal, Sylow
240C22×C59—21no—
241C22×C59—21no—
242C22×C59—81no—
243C22×C59—81no—
244C22×C59—81no—
245C22×C59—81no—
246C22×C59—81no—
247C22×C59—81no—
248C22×C59C2×D1612yes—
249C22×C59C22×D811yescharacteristic
250C2×D59—21no—
251C2×D59—21no—
252C2×D59—42no—
253C2×D59—81no—
254C2×D59—81no—
255C2×D59—81no—
256C2×D59—81no—
257C2×D59—81no—
258C2×D59—164no—
259C2×D59—164no—
260C2×D59C2×D1612yes—
261C4×C59—21no—
262C4×C59—81no—
263C4×C59—81no—
264C4×C59#54611yescharacteristic
265C4×C59#57711yescharacteristic
266Dic59—21no—
267Dic59—21no—
268Dic59—81no—
269Dic59—81no—
270Dic59C2×D1612yes—
271#3334—591no—
272D4×D8—591no—
273#2240—591no—
274#2232—591no—
275#2232—591no—
276#2224—591no—
277#2215—591no—
278#2215D5911yescharacteristic, complemented
279#2206—591no—
280#2184—591no—
281#2184—591no—
282C4×D16—591no—
283D4×C16—591no—
284C2×#357—591no—
285C2×#357—591no—
286C2×C59—22no—
287C2×C59—161no—
288C2×C59—161no—
289C2×C59#329411yescharacteristic
290C2×C59C22×D1611yescharacteristic
291C2×C59#341611yescharacteristic
292D59—22no—
293D59—82no—
294D59—161no—
295C22×D8—591no—
296C22×D8—591no—
297#547—1181no—
298D42—1181no—
299C2×D16—591no—
300C2×D16—591no—
301C2×D16C2×D5911yescharacteristic, complemented
302C22×C16—591no—
303C22×C16C2×D5911yescharacteristic, complemented
304#500—591no—
305#493—591no—
306#480—1181no—
307#480—1181no—
308#474—1181no—
309#466—591no—
310#466—591no—
311#466C2×D5911yescharacteristic
312#466C2×D5911yescharacteristic, complemented
313#459—1181no—
314#447—1181no—
315#447—1181no—
316C4×D8—591no—
317C4×D8—591no—
318C8×D4—591no—
319C2×#107—591no—
320C2×#107—591no—
321#366—591no—
322#366—591no—
323#366C2×D5911yescharacteristic, complemented
324C4⋊C16—591no—
325#357—591no—
326#357—591no—
327#357—591no—
328#357—591no—
329#357—591no—
330#357—591no—
331#357—591no—
332#357—591no—
333#357C2×D5911yescharacteristic, complemented
334#357C2×D5911yescharacteristic, complemented
335C22⋊C16—591no—
336C22⋊C16—591no—
337C4×C16—591no—
338C59#3386111yescharacteristic, complemented, Sylow
339C2×#41—2361no—
340C22×D4—1181no—
341C22×D4—1181no—
342C22×D4—2361no—
343C2×D8—591no—
344C2×D8—591no—
345C2×D8—1182no—
346C2×D8—1181no—
347C2×D8—1181no—
348C2×D8—1182no—
349C2×D8—1181no—
350C2×D8—2361no—
351C2×D8—2361no—
352C2×D8C22×D5911yescharacteristic
353C2×D8C22×D5911yescharacteristic
354C22×C8—591no—
355C22×C8C22×D5911yescharacteristic
356C4⋊Q8—1181no—
357#127—1181no—
358#123—2361no—
359#123—2361no—
360C22⋊Q8—2361no—
361#121—21no—
362#121—21no—
363#121—1181no—
364#121—1181no—
365#121—2361no—
366#120—2361no—
367#120—2361no—
368C4×D4—591no—
369C4×D4—1181no—
370C4×D4—1181no—
371C2×C4⋊C4—1181no—
372C2×C4⋊C4—1181no—
373D16—41no—
374D16—1182no—
375D16—2361no—
376C2×C16—41no—
377C2×C16—591no—
378C2×C16—591no—
379C2×C16—2361no—
380C2×C16C22×D5911yescharacteristic
381C2×C16C22×D5911yescharacteristic
382#107—591no—
383#107—591no—
384#107—1181no—
385#107—1181no—
386#107—1181no—
387#107C22×D5911yescharacteristic
388#107C22×D5911yescharacteristic
389C4⋊C8—591no—
390#102—21no—
391#102—21no—
392#102—1181no—
393#102—1181no—
394#102—1181no—
395#102—1181no—
396C22⋊C8—591no—
397C22⋊C8—591no—
398C4×C8—591no—
399C24—2361no—
400C24—2361no—
401#41—4724no—
402C2×Q8—2361no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—591no—
408C2×D4—1181no—
409C2×D4—1181no—
410C2×D4—4721no—
411C2×D4—4721no—
412C2×D4—4724no—
413C2×D4—2362no—
414C2×D4—2361no—
415C2×D4—2361no—
416C2×D4—2361no—
417C2×D4—2361no—
418C2×D4—2362no—
419C2×D4—2361no—
420C2×D4—2361no—
421C22×C4—591no—
422C22×C4—2361no—
423C22×C4—2361no—
424C22×C4—2361no—
425C22×C4—2361no—
426C22×C4D4×D5911yescharacteristic
427D8—21no—
428D8—21no—
429D8—21no—
430D8—21no—
431D8—592no—
432D8—1181no—
433D8—1181no—
434D8—1181no—
435D8—4721no—
436D8—2362no—
437D8—2362no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—591no—
441C2×C8—591no—
442C2×C8—1181no—
443C2×C8—1181no—
444C2×C8D4×D5911yescharacteristic
445C2×C8C23×D5911yescharacteristic
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—591no—
449C4⋊C4—1181no—
450C4⋊C4—1181no—
451C4⋊C4—2361no—
452C4⋊C4—2361no—
453C4⋊C4—2361no—
454C22⋊C4—41no—
455C22⋊C4—41no—
456C22⋊C4—591no—
457C22⋊C4—591no—
458C22⋊C4—2361no—
459C22⋊C4—2361no—
460C22⋊C4—2361no—
461C22⋊C4—2361no—
462C42—591no—
463C16—21no—
464C16—21no—
465C16—1181no—
466C16—1181no—
467C23—41no—
468C23—41no—
469C23—591no—
470C23—9441no—
471C23—9441no—
472C23—4722no—
473C23—4722no—
474C23—4722no—
475C23—4722no—
476C23—2361no—
477C23—2361no—
478C23D8×D5911yescharacteristic
479Q8—4722no—
480D4—41no—
481D4—41no—
482D4—41no—
483D4—41no—
484D4—82no—
485D4—82no—
486D4—594no—
487D4—1182no—
488D4—9441no—
489D4—4722no—
490D4—4722no—
491D4—4722no—
492D4—4722no—
493D4—4722no—
494D4—2361no—
495D4—2361no—
496D4—2361no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—591no—
502C2×C4—591no—
503C2×C4—591no—
504C2×C4—1181no—
505C2×C4—1181no—
506C2×C4—4722no—
507C2×C4—4722no—
508C2×C4—4722no—
509C2×C4—4722no—
510C2×C4—2361no—
511C2×C4—2361no—
512C2×C4D8×D5911yescharacteristic
513C2×C4C2×D4×D5911yescharacteristic
514C8—21no—
515C8—1181no—
516C8—1181no—
517C8#41×D5911yescharacteristic
518C8#1427377211yescharacteristic
519C22—21no—
520C22—21no—
521C22—81no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22—81no—
527C22—592no—
528C22—1181no—
529C22—1181no—
530C22—9444no—
531C22—9444no—
532C22—4721no—
533C22—4721no—
534C22—4721no—
535C22—4721no—
536C22—4721no—
537C22—2362no—
538C22D16×D5912yes—
539C22C2×D8×D5911yescharacteristic
540C4—21no—
541C4—81no—
542C4—81no—
543C4—592no—
544C4—1181no—
545C4—1181no—
546C4—4721no—
547C4—4721no—
548C4#1683804111yescharacteristic
549C4#1683820911yescharacteristic
550C2—22no—
551C2—161no—
552C2—161no—
553C2—1182no—
554C2—9441no—
555C2—4722no—
556C2#2143590411yescharacteristic
557C2C2×D16×D5911yescharacteristic
558C2#2143723911yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2215C2
socleC22×C59C22×D8
hypercenter#2215D59
hypocenterC59#33861
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2215

Complements

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

NH ≅ G/Ntype
1. #21436440C2semidirect
2. #21435770C2semidirect
3. #21426318C2semidirect
4. #21426278C2semidirect
5. #21426274C2semidirect
6. #21426232C2semidirect
7. #21426181C2semidirect
8. #21426129C2semidirect
9. #21426007C2semidirect
10. #21426006C2semidirect
11. #21425940C2semidirect
12. C16×C22⋊D59C2semidirect
13. C2×#16837204C2semidirect
14. #357×D59C2semidirect
15. C59×#2215C2semidirect
21. C2×#14273668C22semidirect
22. C2×C16×D59C22semidirect
23. #16837806C22semidirect
29. #16837635C22semidirect
33. D8×Dic59C22semidirect
36. C2×#14273661C22semidirect
37. #107×D59C22semidirect
38. #16837227C22semidirect
39. #16837227C22semidirect
40. Dic59⋊C16C22semidirect
43. #16837204C22semidirect
44. #16837204C22semidirect
45. #16837203C22semidirect
46. #16837203C22semidirect
48. #16837202C22semidirect
49. C22⋊(C59⋊C16)C22semidirect
50. (C2×D59)⋊C16C22semidirect
51. C16×Dic59C22semidirect
52. C2×D16×C59C22semidirect
53. C22×C16×C59C22semidirect
55. C59×#466C22semidirect
56. C59×#366C22semidirect
57. C59×#357C22semidirect
58. C59×#357C22semidirect
180. D472#41semidirect
215. C22⋊D59D16semidirect
278. #2215D59semidirect
301. C2×D16C2×D59semidirect
303. C22×C16C2×D59semidirect
312. #466C2×D59semidirect
323. #366C2×D59semidirect
333. #357C2×D59semidirect
334. #357C2×D59semidirect
338. C59#33861semidirect

Maximal subgroups

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

subgroupindex
#214364402
#214357702
#214263182
#214262782
#214262742
#214262322
#214261812
#214261292
#214260072
#214260062
#214259402
C16×C22⋊D592
C2×#168372042
#357×D592
C59×#22152
#3386159