Finite Group Databasebeta

← #28745846

Subgroups — #28745846

Order 15104.

622 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21436501C211yesmaximal, characteristic, complemented
2#21435770C211yesmaximal, characteristic, complemented
3#21426336C211yesmaximal, characteristic, complemented
4#21426290C211yesmaximal, characteristic, complemented
5#21426279C211yesmaximal, characteristic, complemented
6#21426256C211yesmaximal, characteristic, complemented
7#21426198C211yesmaximal, characteristic, complemented
8#21426129C211yesmaximal, characteristic, complemented
9#21426007C211yesmaximal, characteristic, complemented
10#21426006C211yesmaximal, characteristic, complemented
11#21425969C211yesmaximal, characteristic, complemented
12#21425921C211yesmaximal, characteristic, complemented
13#21425810C211yesmaximal, characteristic, complemented
14C2×#16837204C211yesmaximal, characteristic, complemented
15C59×#2215C211yesmaximal, characteristic, complemented
16C22×D472C2211yescharacteristic
17C2×D8×D59C2211yescharacteristic
18#16838104—21no—
19#16838036—21no—
20C2×#14273668C2211yescharacteristic
21C2×#14273668C2211yescharacteristic, complemented
22C2×#14273666C2211yescharacteristic, complemented
23#16837803C2211yescharacteristic, complemented
24#16837777C2211yescharacteristic
25#16837724—21no—
26#16837718—21no—
27#16837700—21no—
28#16837639C2211yescharacteristic
29#16837635C2211yescharacteristic, complemented
30#16837596—21no—
31#16837537—21no—
32#16837535—21no—
33D8×Dic59C2211yescharacteristic, complemented
34#16837484C2211yescharacteristic
35C8×C22⋊D59C2211yescharacteristic
36#16837439C2211yescharacteristic, complemented
37C2×#14273661C2211yescharacteristic, complemented
38#16837229C2211yescharacteristic, complemented
39#16837229C2211yescharacteristic, 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
51#16837177C2211yescharacteristic, complemented
52C2×D16×C59C2211yescharacteristic, complemented
53C22×C16×C59C2211yescharacteristic, complemented
54C59×#466C2211yescharacteristic
55C59×#466C2211yescharacteristic, complemented
56C59×#366C2211yescharacteristic, complemented
57C59×#357C2211yescharacteristic, complemented
58C59×#357C2211yescharacteristic, complemented
59C22×D236—21no—
60C2×D4×D59—21no—
61C2×D4×D59—41no—
62C2×D472—22no—
63C2×D472—21no—
64C2×D472—41no—
65C2×D472C2311yescharacteristic
66C2×D472C2311yescharacteristic
67C2×#11745669—21no—
68C2×#11745669—21no—
69D8×D59—22no—
70D8×D59—41no—
71C2×C8×D59C2311yescharacteristic
72#14273723—21no—
73#14273720—41no—
74#14273718—21no—
75#14273711—41no—
76#14273702—21no—
77#14273700—41no—
78#14273699—41no—
79#14273698—21no—
80#14273697—41no—
81#14273695—41no—
82D4×Dic59—21no—
83D4×Dic59—41no—
84#14273689—21no—
85C4×C22⋊D59D411yescharacteristic
86#14273685—21no—
87C2×C4⋊Dic59—21no—
88#14273668—22no—
89#14273668—41no—
90#14273666—41no—
91C2×C59⋊C16C2311yescharacteristic
92C2×C59⋊C16C2311yescharacteristic
93#14273661—21no—
94#14273661C2311yescharacteristic
95#14273661C2311yescharacteristic
96#14273658—21no—
97#14273658—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—
133D4×D59—42no—
134D4×D59—81no—
135D4×D59—84no—
136C2×D236—21no—
137C2×D236—21no—
138C2×D236—42no—
139C2×D236—41no—
140C2×D236—41no—
141C2×D236—81no—
142C2×C4×D59—41no—
143C2×C4×D59C2×D411yescharacteristic
144D8×C59—21no—
145D8×C59—21no—
146D8×C59—21no—
147D8×C59—21no—
148C2×C8×C59—21no—
149C2×C8×C59—21no—
150C2×C8×C59C2×D411yescharacteristic
151C2×C8×C59C2411yescharacteristic
152C4⋊C4×C59—21no—
153C4⋊C4×C59—21no—
154C22⋊C4×C59—41no—
155C22⋊C4×C59—41no—
156C22⋊Dic59—41no—
157C22⋊Dic59—41no—
158C22⋊Dic59D811yescharacteristic
159#11745669—42no—
160#11745669—81no—
161(C2×D59)⋊C4—41no—
162(C2×D59)⋊C4—41no—
163(C2×D59)⋊C4—41no—
164(C2×D59)⋊C4D811yescharacteristic
165C4⋊Dic59—21no—
166C4⋊Dic59—21no—
167C4⋊Dic59—41no—
168C4⋊Dic59—41no—
169Dic59⋊C4—41no—
170Dic59⋊C4D811yescharacteristic
171C4×Dic59—41no—
172C4×Dic59C2×D411yescharacteristic
173C2×C59⋊C8C2×D411yescharacteristic
174C2×C59⋊C8C2×D411yescharacteristic
175D472—21no—
176D472—21no—
177D472—21no—
178D472—42no—
179D472#4112yes—
180C8×D59—21no—
181C8×D59—21no—
182C16×C59—21no—
183C16×C59—21no—
184C59⋊C16—21no—
185C59⋊C16—21no—
186C23×C59—41no—
187C23×C59—41no—
188C23×C59C2×D811yescharacteristic
189C22×D59—41no—
190C22×D59—41no—
191C22×D59—82no—
192C22×D59—82no—
193C22×D59—82no—
194C22×D59—82no—
195C22×D59—161no—
196C22×D59—161no—
197C22×D59C2×D811yescharacteristic
198D4×C59—41no—
199D4×C59—41no—
200D4×C59—41no—
201D4×C59—41no—
202D4×C59—82no—
203D4×C59—82no—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—41no—
207C2×C4×C59—41no—
208C2×C4×C59C2×D811yescharacteristic
209C2×C4×C59C22×D411yescharacteristic
210C22⋊D59—22no—
211C22⋊D59—82no—
212C22⋊D59—82no—
213C22⋊D59—161no—
214C22⋊D59—161no—
215C2×Dic59—41no—
216C2×Dic59—41no—
217C2×Dic59—161no—
218C2×Dic59—161no—
219C2×Dic59C2×D811yescharacteristic
220C2×Dic59C2×D811yescharacteristic
221C2×Dic59C2×D811yescharacteristic
222D236—22no—
223D236—41no—
224D236—41no—
225D236—41no—
226D236—82no—
227D236—82no—
228C4×D59—21no—
229C4×D59—21no—
230C4×D59—82no—
231C8×C59—21no—
232C8×C59C2×#4111yescharacteristic
233C8×C59H(5,𝔽2)11yescharacteristic
234C59⋊C8—21no—
235C59⋊C8—21no—
236#33885—591nomaximal, Sylow
237C22×C59—21no—
238C22×C59—21no—
239C22×C59—81no—
240C22×C59—81no—
241C22×C59—81no—
242C22×C59—81no—
243C22×C59—81no—
244C22×C59—81no—
245C22×C59#50912yes—
246C22×C59C22×D811yescharacteristic
247C2×D59—21no—
248C2×D59—21no—
249C2×D59—21no—
250C2×D59—42no—
251C2×D59—81no—
252C2×D59—81no—
253C2×D59—81no—
254C2×D59—81no—
255C2×D59—81no—
256C2×D59—164no—
257C2×D59—164no—
258C4×C59—21no—
259C4×C59—81no—
260C4×C59—81no—
261C4×C59#54611yescharacteristic
262C4×C59#57711yescharacteristic
263Dic59—21no—
264Dic59—21no—
265Dic59—21no—
266Dic59—81no—
267Dic59—81no—
268#3339—591no—
269D4×D8—591no—
270#2244—591no—
271#2234—591no—
272#2233—591no—
273#2228—591no—
274#2218—591no—
275#2215D5911yescharacteristic, complemented
276#2206—591no—
277#2184—591no—
278#2184—591no—
279#2177—591no—
280#2168—591no—
281#2140—591no—
282C2×#357—591no—
283C2×C59—22no—
284C2×C59—161no—
285C2×C59—161no—
286C2×C59#329411yescharacteristic
287C2×C59C2×#50911yescharacteristic
288C2×C59#341511yescharacteristic
289D59—41no—
290D59—82no—
291D59—161no—
292C22×D8—591no—
293C22×D8—591no—
294#553—1181no—
295D42—1181no—
296C2×D16—591no—
297C2×D16—591no—
298C2×D16C2×D5911yescharacteristic, complemented
299C2×M32—591no—
300C22×C16C2×D5911yescharacteristic, complemented
301#499—591no—
302#493—591no—
303#482—1181no—
304#481—1181no—
305#478—1181no—
306#466—591no—
307#466—591no—
308#466C2×D5911yescharacteristic
309#466C2×D5911yescharacteristic, complemented
310#459—1181no—
311#447—1181no—
312#447—1181no—
313C4×D8—591no—
314C4×D8—591no—
315C8×D4—591no—
316#427—591no—
317C2×#107—591no—
318#367—591no—
319#367—591no—
320#366C2×D5911yescharacteristic, complemented
321C4⋊C16—591no—
322#357—591no—
323#357—591no—
324#357—591no—
325#357—591no—
326#357—591no—
327#357—591no—
328#357—591no—
329#357—591no—
330#357C2×D5911yescharacteristic, complemented
331#357C2×D5911yescharacteristic, complemented
332C22⋊C16—591no—
333C22⋊C16—591no—
334#346—591no—
335C59#3388511yescharacteristic, complemented, Sylow
336C22×D4—1181no—
337C22×D4—1181no—
338C22×D4—2361no—
339C2×D8—591no—
340C2×D8—591no—
341C2×D8—1182no—
342C2×D8—1181no—
343C2×D8—1181no—
344C2×D8—1182no—
345C2×D8—1181no—
346C2×D8—2361no—
347C2×D8—2361no—
348C2×D8C22×D5911yescharacteristic
349C2×D8C22×D5911yescharacteristic
350C22×C8—591no—
351C22×C8C22×D5911yescharacteristic
352#127—1181no—
353#126—2361no—
354#125—1181no—
355#123—2361no—
356#121—21no—
357#121—21no—
358#121—1181no—
359#121—1181no—
360#121—2361no—
361#121—2361no—
362#120—2361no—
363#120—2361no—
364C4×D4—591no—
365C4×D4—1181no—
366C4×D4—1181no—
367C4×D4—2361no—
368#117—1181no—
369C2×C4⋊C4—1181no—
370D16—41no—
371D16—1182no—
372D16—2361no—
373M32—2361no—
374C2×C16—41no—
375C2×C16—591no—
376C2×C16—591no—
377C2×C16C22×D5911yescharacteristic
378C2×C16C22×D5911yescharacteristic
379#107—591no—
380#107—591no—
381#107—1181no—
382#107C22×D5911yescharacteristic
383#107C22×D5911yescharacteristic
384#106—1181no—
385#106—1181no—
386C4⋊C8—591no—
387#102—21no—
388#102—21no—
389#102—1181no—
390#102—1181no—
391#102—1181no—
392#102—1181no—
393C22⋊C8—591no—
394C22⋊C8—591no—
395C4×C8—591no—
396C24—2361no—
397C24—2361no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—591no—
403C2×D4—1181no—
404C2×D4—1181no—
405C2×D4—4721no—
406C2×D4—4721no—
407C2×D4—4724no—
408C2×D4—2362no—
409C2×D4—2361no—
410C2×D4—2361no—
411C2×D4—2361no—
412C2×D4—2361no—
413C2×D4—2362no—
414C2×D4—2361no—
415C2×D4—2361no—
416C22×C4—591no—
417C22×C4—2361no—
418C22×C4—2361no—
419C22×C4—2361no—
420C22×C4D4×D5911yescharacteristic
421D8—21no—
422D8—21no—
423D8—21no—
424D8—21no—
425D8—592no—
426D8—1181no—
427D8—1181no—
428D8—1181no—
429D8—4721no—
430D8—2362no—
431D8—2362no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—591no—
435C2×C8—591no—
436C2×C8—1181no—
437C2×C8—1181no—
438C2×C8D4×D5911yescharacteristic
439C2×C8C23×D5911yescharacteristic
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—591no—
443C4⋊C4—1181no—
444C4⋊C4—1181no—
445C4⋊C4—2361no—
446C4⋊C4—2361no—
447C4⋊C4—2361no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—591no—
451C22⋊C4—591no—
452C22⋊C4—2361no—
453C22⋊C4—2361no—
454C22⋊C4—2361no—
455C22⋊C4—2361no—
456C22⋊C4—2361no—
457C42—591no—
458C42—2361no—
459C16—21no—
460C16—21no—
461C16—1181no—
462C16—1181no—
463C23—41no—
464C23—41no—
465C23—591no—
466C23—9441no—
467C23—9441no—
468C23—4722no—
469C23—4722no—
470C23—4722no—
471C23—4722no—
472C23—2361no—
473C23—2361no—
474C23D8×D5911yescharacteristic
475D4—41no—
476D4—41no—
477D4—41no—
478D4—41no—
479D4—82no—
480D4—82no—
481D4—1182no—
482D4—1182no—
483D4—9441no—
484D4—9441no—
485D4—4722no—
486D4—4722no—
487D4—4722no—
488D4—4722no—
489D4—2361no—
490D4—2361no—
491D4—2361no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—591no—
497C2×C4—591no—
498C2×C4—591no—
499C2×C4—1181no—
500C2×C4—1181no—
501C2×C4—9441no—
502C2×C4—9441no—
503C2×C4—4722no—
504C2×C4—2361no—
505C2×C4—2361no—
506C2×C4D8×D5911yescharacteristic
507C2×C4C2×D4×D5911yescharacteristic
508C8—21no—
509C8—1181no—
510C8—1181no—
511C8#41×D5911yescharacteristic
512C8#1427377211yescharacteristic
513C22—21no—
514C22—21no—
515C22—81no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22—81no—
520C22—81no—
521C22—1181no—
522C22—1181no—
523C22—1181no—
524C22—9444no—
525C22—9444no—
526C22—4721no—
527C22—4721no—
528C22—4721no—
529C22—4721no—
530C22—4721no—
531C22—2362no—
532C22#1683783612yes—
533C22C2×D8×D5911yescharacteristic
534C4—21no—
535C4—81no—
536C4—81no—
537C4—1181no—
538C4—1181no—
539C4—1181no—
540C4—4721no—
541C4—4721no—
542C4#1683804111yescharacteristic
543C4#1683820911yescharacteristic
544C2—22no—
545C2—161no—
546C2—161no—
547C2—9441no—
548C2—4722no—
549C2—2361no—
550C2#2143590411yescharacteristic
551C2C2×#1683783611yescharacteristic
552C2#2143723411yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2215C2
socleC22×C59C22×D8
hypercenter#2215D59
hypocenterC59#33885
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. #21436501C2semidirect
2. #21435770C2semidirect
3. #21426336C2semidirect
4. #21426290C2semidirect
5. #21426279C2semidirect
6. #21426256C2semidirect
7. #21426198C2semidirect
8. #21426129C2semidirect
9. #21426007C2semidirect
10. #21426006C2semidirect
11. #21425969C2semidirect
12. #21425921C2semidirect
13. #21425810C2semidirect
14. C2×#16837204C2semidirect
15. C59×#2215C2semidirect
21. C2×#14273668C22semidirect
22. C2×#14273666C22semidirect
23. #16837803C22semidirect
29. #16837635C22semidirect
33. D8×Dic59C22semidirect
36. #16837439C22semidirect
37. C2×#14273661C22semidirect
38. #16837229C22semidirect
39. #16837229C22semidirect
40. Dic59⋊C16C22semidirect
43. #16837204C22semidirect
44. #16837204C22semidirect
45. #16837203C22semidirect
46. #16837203C22semidirect
48. #16837202C22semidirect
49. C22⋊(C59⋊C16)C22semidirect
50. (C2×D59)⋊C16C22semidirect
51. #16837177C22semidirect
52. C2×D16×C59C22semidirect
53. C22×C16×C59C22semidirect
55. C59×#466C22semidirect
56. C59×#366C22semidirect
57. C59×#357C22semidirect
58. C59×#357C22semidirect
275. #2215D59semidirect
298. C2×D16C2×D59semidirect
300. C22×C16C2×D59semidirect
309. #466C2×D59semidirect
320. #366C2×D59semidirect
330. #357C2×D59semidirect
331. #357C2×D59semidirect
335. C59#33885semidirect

Maximal subgroups

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

subgroupindex
#214365012
#214357702
#214263362
#214262902
#214262792
#214262562
#214261982
#214261292
#214260072
#214260062
#214259692
#214259212
#214258102
C2×#168372042
C59×#22152
#3388559