Finite Group Databasebeta

← #28745842

Subgroups — #28745842

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#21436499C211yesmaximal, characteristic, complemented
2D4×D472C211yesmaximal, characteristic, complemented
3#21426338C211yesmaximal, characteristic, complemented
4#21426288C211yesmaximal, characteristic, complemented
5#21426282C211yesmaximal, characteristic, complemented
6#21426254C211yesmaximal, characteristic, complemented
7#21426199C211yesmaximal, characteristic, complemented
8#21426182C211yesmaximal, characteristic, complemented
9#21426132C211yesmaximal, characteristic, complemented
10#21426010C211yesmaximal, characteristic, complemented
11#21426010C211yesmaximal, characteristic, complemented
12#21425973C211yesmaximal, characteristic, complemented
13#21425813C211yesmaximal, characteristic, complemented
14C2×#16837204C211yesmaximal, characteristic, complemented
15C59×#2168C211yesmaximal, characteristic, complemented
16C22×D472C2211yescharacteristic
17C22×D472C2211yescharacteristic
18#16838098—21no—
19D4×D236—21no—
20C2×D944C2211yescharacteristic
21C2×D944C2211yescharacteristic, complemented
22C2×D944C2211yescharacteristic, complemented
23#16837804C2211yescharacteristic, complemented
24#16837779C2211yescharacteristic
25#16837727—21no—
26#16837721—21no—
27#16837703—21no—
28#16837638C2211yescharacteristic
29#16837638C2211yescharacteristic
30#16837638C2211yescharacteristic, complemented
31#16837638C2211yescharacteristic, complemented
32#16837598—21no—
33#16837538—21no—
34#16837538—21no—
35C4×D472C2211yescharacteristic
36C4×D472C2211yescharacteristic, complemented
37#16837440C2211yescharacteristic, complemented
38C2×#14273661C2211yescharacteristic, complemented
39#16837230C2211yescharacteristic, complemented
40#16837230C2211yescharacteristic, complemented
41#16837228C2211yescharacteristic, complemented
42#16837204C2211yescharacteristic
43#16837204C2211yescharacteristic
44#16837204C2211yescharacteristic
45#16837204C2211yescharacteristic, complemented
46#16837204C2211yescharacteristic, complemented
47#16837204C2211yescharacteristic, complemented
48#16837204C2211yescharacteristic, complemented
49#16837204C2211yescharacteristic, complemented
50#16837204C2211yescharacteristic, complemented
51#16837204C2211yescharacteristic, complemented
52C2×M32×C59C2211yescharacteristic, complemented
53C22×C16×C59C2211yescharacteristic, complemented
54C8×D4×C59C2211yescharacteristic
55C59×C4⋊C16C2211yescharacteristic, complemented
56C59×C22⋊C16C2211yescharacteristic, complemented
57C59×C22⋊C16C2211yescharacteristic, complemented
58C59×#346C2211yescharacteristic, complemented
59C22×D236—21no—
60C22×D236—21no—
61C2×D4×D59—41no—
62C2×D472—22no—
63C2×D472—22no—
64C2×D472—21no—
65C2×D472—21no—
66C2×D472—21no—
67C2×D472—41no—
68C2×D472—41no—
69C2×D472C2311yescharacteristic
70C2×D472C2311yescharacteristic
71C2×D472C2311yescharacteristic
72C2×D472C2311yescharacteristic
73#14273724—21no—
74#14273722—41no—
75#14273719—21no—
76#14273709—41no—
77#14273702—41no—
78#14273701—21no—
79#14273701—21no—
80#14273701—21no—
81#14273701—21no—
82#14273699—41no—
83#14273697—41no—
84#14273697—41no—
85C4×D236—21no—
86C4×D236—21no—
87(C22⋊D59)⋊C4—41no—
88#14273686—21no—
89C2×C4⋊Dic59—21no—
90D944—22no—
91D944—41no—
92D944—41no—
93#14273661—21no—
94#14273661C2311yescharacteristic
95#14273661C2311yescharacteristic
96#14273661C2311yescharacteristic
97#14273661C2311yescharacteristic
98#14273659—21no—
99#14273659—21no—
100#14273649—21no—
101#14273649—21no—
102#14273649—21no—
103#14273649—21no—
104#14273649—21no—
105#14273649—21no—
106C22×C8×C59C2311yescharacteristic
107C22×C8×C59C2311yescharacteristic
108C4×D4×C59D411yescharacteristic
109M32×C59—41no—
110C2×C16×C59—41no—
111C2×C16×C59C2311yescharacteristic
112C2×C16×C59C2311yescharacteristic
113C2×C16×C59C2311yescharacteristic
114C2×C16×C59C2311yescharacteristic
115C4⋊C8×C59D411yescharacteristic
116C22⋊C8×C59D411yescharacteristic
117C22⋊C8×C59D411yescharacteristic
118C4×C8×C59C2311yescharacteristic
119C23×D59—41no—
120C23×D59—41no—
121C2×D4×C59D811yescharacteristic
122C22×C4×C59C2×D411yescharacteristic
123C22×C4×C59C2×D411yescharacteristic
124C2×C22⋊D59—41no—
125C2×C22⋊D59—41no—
126C2×C22⋊D59—41no—
127C2×C22⋊D59—41no—
128C22×Dic59—41no—
129D4×D59—84no—
130C2×D236—21no—
131C2×D236—21no—
132C2×D236—21no—
133C2×D236—21no—
134C2×D236—42no—
135C2×D236—42no—
136C2×D236—41no—
137C2×D236—41no—
138C2×D236—41no—
139C2×D236—41no—
140C2×D236—81no—
141C2×D236—81no—
142C2×C4×D59—41no—
143C2×C4×D59—41no—
144C2×C8×C59—21no—
145C2×C8×C59—21no—
146C2×C8×C59—21no—
147C2×C8×C59—21no—
148C2×C8×C59C2×D411yescharacteristic
149C2×C8×C59C2×D411yescharacteristic
150C2×C8×C59C2×D411yescharacteristic
151C2×C8×C59C2411yescharacteristic
152C4⋊C4×C59D811yescharacteristic
153C22⋊C4×C59D811yescharacteristic
154C22⋊C4×C59D811yescharacteristic
155C42×C59C2×D411yescharacteristic
156C22⋊Dic59—41no—
157(C2×D59)⋊C4—41no—
158(C2×D59)⋊C4—41no—
159(C2×D59)⋊C4—41no—
160(C2×D59)⋊C4—41no—
161(C2×D59)⋊C4—41no—
162(C2×D59)⋊C4—41no—
163C4⋊Dic59—21no—
164C4⋊Dic59—21no—
165C4⋊Dic59—21no—
166C4⋊Dic59—21no—
167C4⋊Dic59—41no—
168Dic59⋊C4—41no—
169Dic59⋊C4—41no—
170C4×Dic59—41no—
171D472—21no—
172D472—21no—
173D472—21no—
174D472—21no—
175D472—21no—
176D472—21no—
177D472—21no—
178D472—42no—
179D472—42no—
180D472—81no—
181D472#4112yes—
182C16×C59—21no—
183C16×C59—21no—
184C16×C59—21no—
185C16×C59—21no—
186C23×C59C2×D811yescharacteristic
187C23×C59C2×D811yescharacteristic
188C22×D59—41no—
189C22×D59—41no—
190C22×D59—41no—
191C22×D59—41no—
192C22×D59—82no—
193C22×D59—82no—
194C22×D59—82no—
195C22×D59—82no—
196C22×D59—161no—
197C22×D59—161no—
198D4×C59—22no—
199C2×C4×C59—21no—
200C2×C4×C59—21no—
201C2×C4×C59—21no—
202C2×C4×C59—21no—
203C2×C4×C59C2×D811yescharacteristic
204C2×C4×C59C2×D811yescharacteristic
205C2×C4×C59C2×D811yescharacteristic
206C2×C4×C59C2×D811yescharacteristic
207C2×C4×C59C22×D411yescharacteristic
208C22⋊D59—82no—
209C22⋊D59—82no—
210C22⋊D59—82no—
211C22⋊D59—161no—
212C2×Dic59—41no—
213C2×Dic59—41no—
214C2×Dic59—41no—
215C2×Dic59—41no—
216C2×Dic59—161no—
217D236—22no—
218D236—41no—
219D236—41no—
220D236—41no—
221D236—41no—
222D236—41no—
223D236—41no—
224D236—41no—
225D236—82no—
226D236—82no—
227D236—82no—
228D236—161no—
229C4×D59—82no—
230C4×D59—161no—
231C8×C59—21no—
232C8×C59—21no—
233C8×C59—21no—
234C8×C59C2×#4111yescharacteristic
235C8×C59H(5,𝔽2)11yescharacteristic
236#33885—591nomaximal, Sylow
237C22×C59—21no—
238C22×C59—21no—
239C22×C59—21no—
240C22×C59—21no—
241C22×C59—21no—
242C22×C59#50912yes—
243C22×C59C22×D811yescharacteristic
244C2×D59—42no—
245C2×D59—81no—
246C2×D59—81no—
247C2×D59—81no—
248C2×D59—81no—
249C2×D59—81no—
250C2×D59—81no—
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—21no—
260C4×C59—21no—
261C4×C59—21no—
262C4×C59#54611yescharacteristic
263C4×C59#57711yescharacteristic
264Dic59—81no—
265Dic59—81no—
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#2215—591no—
276#2206—591no—
277#2184—591no—
278#2184—591no—
279#2177—591no—
280#2168D5911yescharacteristic, complemented
281#2140—591no—
282C2×#357—591no—
283C2×C59—22no—
284C2×C59—41no—
285C2×C59#329411yescharacteristic
286C2×C59C2×#50911yescharacteristic
287C2×C59#341511yescharacteristic
288D59—82no—
289D59—161no—
290D59—161no—
291D59—161no—
292C22×D8—591no—
293C22×D8—591no—
294#553—1181no—
295D42—1181no—
296C2×D16—591no—
297C2×D16—591no—
298C2×D16—591no—
299C2×M32C2×D5911yescharacteristic, complemented
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#466—591no—
309#466—591no—
310#459—1181no—
311#447—1181no—
312#447—1181no—
313C4×D8—591no—
314C4×D8—591no—
315C8×D4C2×D5911yescharacteristic
316#427—591no—
317C2×#107—591no—
318#367—591no—
319#367—591no—
320#366—591no—
321C4⋊C16C2×D5911yescharacteristic, complemented
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#357—591no—
331#357—591no—
332C22⋊C16C2×D5911yescharacteristic, complemented
333C22⋊C16C2×D5911yescharacteristic, complemented
334#346C2×D5911yescharacteristic, complemented
335C59#3388511yescharacteristic, complemented, Sylow
336C22×D4—1181no—
337C22×D4—1181no—
338C22×D4—2361no—
339C2×D8—591no—
340C2×D8—591no—
341C2×D8—591no—
342C2×D8—591no—
343C2×D8—1182no—
344C2×D8—1182no—
345C2×D8—1181no—
346C2×D8—1181no—
347C2×D8—1181no—
348C2×D8—2361no—
349C2×D8—2361no—
350C22×C8C22×D5911yescharacteristic
351C22×C8C22×D5911yescharacteristic
352#127—1181no—
353#126—2361no—
354#125—1181no—
355#123—2361no—
356#121—1181no—
357#121—1181no—
358#121—1181no—
359#121—1181no—
360#121—2361no—
361#121—2361no—
362#120—2361no—
363#120—2361no—
364C4×D4—1181no—
365C4×D4—1181no—
366C4×D4—2361no—
367C4×D4D23611yescharacteristic
368#117—1181no—
369C2×C4⋊C4—1181no—
370D16—1182no—
371D16—2361no—
372D16—2361no—
373M32—41no—
374C2×C16—41no—
375C2×C16C22×D5911yescharacteristic
376C2×C16C22×D5911yescharacteristic
377C2×C16C22×D5911yescharacteristic
378C2×C16C22×D5911yescharacteristic
379#107—591no—
380#107—591no—
381#107—591no—
382#107—591no—
383#107—1181no—
384#106—1181no—
385#106—1181no—
386C4⋊C8D23611yescharacteristic
387#102—1181no—
388#102—1181no—
389#102—1181no—
390#102—1181no—
391#102—1181no—
392#102—1181no—
393C22⋊C8D23611yescharacteristic
394C22⋊C8D23611yescharacteristic
395C4×C8C22×D5911yescharacteristic
396C24—2361no—
397C24—2361no—
398C2×D4—1181no—
399C2×D4—1181no—
400C2×D4—1181no—
401C2×D4—1181no—
402C2×D4—4721no—
403C2×D4—4721no—
404C2×D4—4724no—
405C2×D4—2362no—
406C2×D4—2362no—
407C2×D4—2361no—
408C2×D4—2361no—
409C2×D4—2361no—
410C2×D4—2361no—
411C2×D4—2361no—
412C2×D4—2361no—
413C2×D4—2361no—
414C2×D4—2361no—
415C2×D4D47211yescharacteristic
416C22×C4—2361no—
417C22×C4—2361no—
418C22×C4—2361no—
419C22×C4C2×D23611yescharacteristic
420C22×C4C2×D23611yescharacteristic
421D8—592no—
422D8—1181no—
423D8—1181no—
424D8—1181no—
425D8—1181no—
426D8—1181no—
427D8—1181no—
428D8—1181no—
429D8—4721no—
430D8—2362no—
431D8—2362no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8C2×D23611yescharacteristic
437C2×C8C2×D23611yescharacteristic
438C2×C8C2×D23611yescharacteristic
439C2×C8C23×D5911yescharacteristic
440C4⋊C4—1181no—
441C4⋊C4—1181no—
442C4⋊C4—1181no—
443C4⋊C4—1181no—
444C4⋊C4—2361no—
445C4⋊C4—2361no—
446C4⋊C4—2361no—
447C4⋊C4D47211yescharacteristic
448C22⋊C4—2361no—
449C22⋊C4—2361no—
450C22⋊C4—2361no—
451C22⋊C4—2361no—
452C22⋊C4—2361no—
453C22⋊C4—2361no—
454C22⋊C4—2361no—
455C22⋊C4D47211yescharacteristic
456C22⋊C4D47211yescharacteristic
457C42—2361no—
458C42C2×D23611yescharacteristic
459C16—21no—
460C16—21no—
461C16—21no—
462C16—21no—
463C23—9441no—
464C23—9441no—
465C23—4722no—
466C23—4722no—
467C23—4722no—
468C23—4722no—
469C23—2361no—
470C23—2361no—
471C23—2361no—
472C23—2361no—
473C23C2×D47211yescharacteristic
474C23C2×D47211yescharacteristic
475D4—22no—
476D4—1182no—
477D4—9441no—
478D4—9441no—
479D4—4722no—
480D4—4722no—
481D4—4722no—
482D4—4722no—
483D4—4722no—
484D4—4722no—
485D4—2361no—
486D4—2361no—
487D4—2361no—
488D4—2361no—
489D4—2361no—
490D4—2361no—
491D4—2361no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—9441no—
497C2×C4—9441no—
498C2×C4—4722no—
499C2×C4—2361no—
500C2×C4—2361no—
501C2×C4—2361no—
502C2×C4—2361no—
503C2×C4C2×D47211yescharacteristic
504C2×C4C2×D47211yescharacteristic
505C2×C4C2×D47211yescharacteristic
506C2×C4C2×D47211yescharacteristic
507C2×C4C22×D23611yescharacteristic
508C8—21no—
509C8—21no—
510C8—21no—
511C8#41×D5911yescharacteristic
512C8#1427377211yescharacteristic
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—9444no—
519C22—9444no—
520C22—4721no—
521C22—4721no—
522C22—4721no—
523C22—4721no—
524C22—4721no—
525C22—4721no—
526C22—4721no—
527C22—4721no—
528C22—4721no—
529C22—4721no—
530C22—4721no—
531C22—2362no—
532C22#1683784012yes—
533C22C22×D47211yescharacteristic
534C4—21no—
535C4—21no—
536C4—21no—
537C4—21no—
538C4—4721no—
539C4—4721no—
540C4—4721no—
541C4—4721no—
542C4#1683804711yescharacteristic
543C4#1683821411yescharacteristic
544C2—22no—
545C2—41no—
546C2—9441no—
547C2—9441no—
548C2—9441no—
549C2—4722no—
550C2#2143591011yescharacteristic
551C2C2×#1683784011yescharacteristic
552C2#2143723711yescharacteristic
subgroupG/H
centerC22C22×D472
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2168C2
socleC22×C59C22×D8
hypercenter#2168D59
hypocenterC59#33885
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #2168

Complements

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

NH ≅ G/Ntype
1. #21436499C2semidirect
2. D4×D472C2semidirect
3. #21426338C2semidirect
4. #21426288C2semidirect
5. #21426282C2semidirect
6. #21426254C2semidirect
7. #21426199C2semidirect
8. #21426182C2semidirect
9. #21426132C2semidirect
10. #21426010C2semidirect
11. #21426010C2semidirect
12. #21425973C2semidirect
13. #21425813C2semidirect
14. C2×#16837204C2semidirect
15. C59×#2168C2semidirect
21. C2×D944C22semidirect
22. C2×D944C22semidirect
23. #16837804C22semidirect
30. #16837638C22semidirect
31. #16837638C22semidirect
36. C4×D472C22semidirect
37. #16837440C22semidirect
38. C2×#14273661C22semidirect
39. #16837230C22semidirect
40. #16837230C22semidirect
41. #16837228C22semidirect
45. #16837204C22semidirect
46. #16837204C22semidirect
47. #16837204C22semidirect
48. #16837204C22semidirect
49. #16837204C22semidirect
50. #16837204C22semidirect
51. #16837204C22semidirect
52. C2×M32×C59C22semidirect
53. C22×C16×C59C22semidirect
55. C59×C4⋊C16C22semidirect
56. C59×C22⋊C16C22semidirect
57. C59×C22⋊C16C22semidirect
58. C59×#346C22semidirect
280. #2168D59semidirect
299. C2×M32C2×D59semidirect
300. C22×C16C2×D59semidirect
321. C4⋊C16C2×D59semidirect
332. C22⋊C16C2×D59semidirect
333. C22⋊C16C2×D59semidirect
334. #346C2×D59semidirect
335. C59#33885semidirect

Maximal subgroups

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

subgroupindex
#214364992
D4×D4722
#214263382
#214262882
#214262822
#214262542
#214261992
#214261822
#214261322
#214260102
#214259732
#214258132
C2×#168372042
C59×#21682
#3388559