Finite Group Databasebeta

← #28743503

Subgroups — #28743503

Order 15104.

584 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433675C211yesmaximal, characteristic
2#427×D59C211yesmaximal, characteristic, complemented
3#21426307C211yesmaximal, characteristic, complemented
4#21426306C211yesmaximal, characteristic, complemented
5#21426305C211yesmaximal, characteristic, complemented
6#21426304C211yesmaximal, characteristic, complemented
7#21426286C211yesmaximal, characteristic, complemented
8#21426285C211yesmaximal, characteristic, complemented
9#21426284C211yesmaximal, characteristic, complemented
10#21425883C211yesmaximal, characteristic, complemented
11#21425881C211yesmaximal, characteristic, complemented
12#21425805C211yesmaximal, characteristic, complemented
13#21425805C211yesmaximal, characteristic, complemented
14#21425705C211yesmaximal, characteristic, complemented
15C59×#2234C211yesmaximal, characteristic, complemented
16C2×#14273746C2211yescharacteristic
17C22×C8×D59C2211yescharacteristic
18#16837912—21no—
19#117×D59—21no—
20C2×#14273666C2211yescharacteristic
21C2×#14273666C2211yescharacteristic
22#16837644C2211yescharacteristic
23#16837643C2211yescharacteristic
24#16837635C2211yescharacteristic
25#16837630—21no—
26#16837629—21no—
27#16837440C2211yescharacteristic
28#16837439C2211yescharacteristic
29#16837439C2211yescharacteristic
30#16837439C2211yescharacteristic
31#16837438—21no—
32#16837437—21no—
33#107×D59C2211yescharacteristic
34#107×D59C2211yescharacteristic
35#106×D59—21no—
36#16837395—21no—
37#16837230C2211yescharacteristic, complemented
38#16837229C2211yescharacteristic, complemented
39#16837229C2211yescharacteristic, complemented
40#16837228C2211yescharacteristic, complemented
41#16837227C2211yescharacteristic, complemented
42#16837227C2211yescharacteristic, complemented
43#16837207C2211yescharacteristic, complemented
44#16837207C2211yescharacteristic, complemented
45#16837205C2211yescharacteristic, complemented
46#16837205C2211yescharacteristic, complemented
47#16837203C2211yescharacteristic, complemented
48#16837203C2211yescharacteristic, complemented
49C22⋊(C59⋊C16)C2211yescharacteristic
50(C2×D59)⋊C16C2211yescharacteristic
51(C2×D59)⋊C16C2211yescharacteristic
52C59×#466C2211yescharacteristic
53C59×#427C2211yescharacteristic
54C59×#367C2211yescharacteristic, complemented
55C59×#366C2211yescharacteristic, complemented
56C59×#357C2211yescharacteristic, complemented
57C59×#357C2211yescharacteristic, complemented
58C59×C22⋊C16C2211yescharacteristic
59C2×#11745687—21no—
60C22×C4×D59D411yescharacteristic
61#14273746—22no—
62#14273746—22no—
63C2×Dic236C2311yescharacteristic
64C2×#11745670—21no—
65C22×C59⋊C8D411yescharacteristic
66C2×C8×D59—22no—
67C2×C8×D59—22no—
68C2×C8×D59D411yescharacteristic
69C2×C8×D59D411yescharacteristic
70C2×C8×D59C2311yescharacteristic
71C2×C8×D59C2311yescharacteristic
72#14273709—41no—
73#14273707—21no—
74#14273706—21no—
75#14273698—21no—
76#14273695—41no—
77D4×Dic59—41no—
78(C22⋊D59)⋊C4—41no—
79#14273686—21no—
80#14273685—21no—
81#14273685—21no—
82#14273685—21no—
83#14273684—41no—
84#14273683—41no—
85C4⋊C4×D59—21no—
86C4⋊C4×D59—21no—
87C22⋊C4×D59—41no—
88C42×D59—41no—
89#14273666—22no—
90#14273666—22no—
91C2×C59⋊C16C2311yescharacteristic
92C2×C59⋊C16C2311yescharacteristic
93#14273661C2311yescharacteristic
94#14273661C2311yescharacteristic
95#14273661C2311yescharacteristic
96#14273660—21no—
97#14273660—21no—
98#14273659—21no—
99#14273658—21no—
100#14273658—21no—
101#14273658—21no—
102#14273652—21no—
103#14273650—21no—
104#14273648—21no—
105C2×D8×C59C2311yescharacteristic
106C22×C8×C59C2311yescharacteristic
107#121×C59—21no—
108#117×C59—21no—
109C2×C16×C59C2311yescharacteristic
110C2×C16×C59C2311yescharacteristic
111#107×C59C2311yescharacteristic
112#107×C59C2311yescharacteristic
113#107×C59C2311yescharacteristic
114#106×C59—21no—
115#102×C59—21no—
116C23×D59D811yescharacteristic
117C2×D4×C59—21no—
118C2×D4×C59—41no—
119C22×C4×C59C2×D411yescharacteristic
120C2×C22⋊D59—41no—
121C2×C22⋊D59—41no—
122C22×Dic59—41no—
123C22×Dic59D811yescharacteristic
124#11745687—42no—
125#11745687—42no—
126C2×C4×D59—22no—
127C2×C4×D59—22no—
128C2×C4×D59—41no—
129C2×C4×D59—41no—
130C2×C4×D59D811yescharacteristic
131C2×C4×D59D811yescharacteristic
132C2×C4×D59C2×D411yescharacteristic
133C2×C4×D59C2×D411yescharacteristic
134C2×Dic118—21no—
135D8×C59—21no—
136D8×C59—21no—
137C2×C8×C59—21no—
138C2×C8×C59—21no—
139C2×C8×C59C2×D411yescharacteristic
140C2×C8×C59C2411yescharacteristic
141C4⋊C4×C59—21no—
142C4⋊C4×C59—21no—
143C4⋊C4×C59—21no—
144C22⋊C4×C59—41no—
145C22⋊C4×C59—41no—
146C42×C59—41no—
147C22⋊Dic59—41no—
148C22⋊Dic59—41no—
149C22⋊Dic59—41no—
150#11745670—42no—
151(C2×D59)⋊C4—41no—
152(C2×D59)⋊C4—41no—
153(C2×D59)⋊C4—41no—
154(C2×D59)⋊C4—41no—
155C4⋊Dic59—21no—
156C4⋊Dic59—21no—
157C4⋊Dic59—21no—
158C4⋊Dic59—41no—
159Dic59⋊C4—41no—
160Dic59⋊C4—41no—
161Dic59⋊C4—41no—
162C4×Dic59—41no—
163C4×Dic59—41no—
164C4×Dic59—41no—
165C4×Dic59—41no—
166C2×C59⋊C8—22no—
167C2×C59⋊C8C2×D411yescharacteristic
168C2×C59⋊C8C2×D411yescharacteristic
169Dic236—21no—
170Dic236—21no—
171C8×D59—22no—
172C8×D59—21no—
173C8×D59—21no—
174C8×D59—22no—
175C8×D59#4112yes—
176C8×D59#4112yes—
177C16×C59—21no—
178C16×C59—21no—
179C59⋊C16—21no—
180C59⋊C16—21no—
181C23×C59—41no—
182C23×C59C2×D811yescharacteristic
183C22×D59—22no—
184C22×D59—41no—
185C22×D59—41no—
186C22×D59C2×D811yescharacteristic
187C22×D59C2×D811yescharacteristic
188D4×C59—41no—
189D4×C59—41no—
190D4×C59—82no—
191C2×C4×C59—21no—
192C2×C4×C59—21no—
193C2×C4×C59—41no—
194C2×C4×C59—41no—
195C2×C4×C59—41no—
196C2×C4×C59C2×D811yescharacteristic
197C2×C4×C59C22×D411yescharacteristic
198C22⋊D59—82no—
199C22⋊D59—82no—
200C2×Dic59—22no—
201C2×Dic59—41no—
202C2×Dic59—41no—
203C2×Dic59—41no—
204C2×Dic59—41no—
205C2×Dic59—82no—
206C2×Dic59C2×D811yescharacteristic
207C2×Dic59C2×D811yescharacteristic
208C4×D59—22no—
209C4×D59—21no—
210C4×D59—21no—
211C4×D59—22no—
212C4×D59—82no—
213C4×D59—82no—
214C4×D59#12312yes—
215C4×D59#13712yes—
216Dic118—41no—
217Dic118—41no—
218C8×C59—21no—
219C8×C59C2×#4111yescharacteristic
220C8×C59C2×#4111yescharacteristic
221C59⋊C8—21no—
222C59⋊C8#12312yes—
223#33647—591nomaximal, Sylow
224C22×C59—21no—
225C22×C59—21no—
226C22×C59—21no—
227C22×C59—81no—
228C22×C59—81no—
229C22×C59—81no—
230C22×C59C22×D811yescharacteristic
231C2×D59—21no—
232C2×D59—21no—
233C2×D59—21no—
234C2×D59—21no—
235C2×D59—21no—
236C2×D59—44no—
237C2×D59#48012yes—
238C4×C59—21no—
239C4×C59—81no—
240C4×C59—81no—
241C4×C59—81no—
242C4×C59C2×#12311yescharacteristic
243C4×C59C2×#13711yescharacteristic
244Dic59—21no—
245Dic59—81no—
246Dic59—81no—
247Dic59—81no—
248Dic59—81no—
249Dic59#48012yes—
250#3050—591no—
251C2×#427—591no—
252#2237—591no—
253#2237—591no—
254#2237—591no—
255#2237—591no—
256#2234—591no—
257#2234—591no—
258#2234—591no—
259#2234D5911yescharacteristic, complemented
260#2159—591no—
261#2159—591no—
262#2139—591no—
263#2139—591no—
264#2112—591no—
265C2×C59—41no—
266C2×C59—161no—
267C2×C59C2×#48011yescharacteristic
268C2×C59#341411yescharacteristic
269C2×C59#341411yescharacteristic
270D59—22no—
271D59—41no—
272C2×#135—591no—
273C23×C8—591no—
274#525—1181no—
275C2×#117—1181no—
276C2×M32—591no—
277C2×M32—591no—
278#467—591no—
279#467—591no—
280#466—591no—
281#466C2×D5911yescharacteristic
282#465—1181no—
283#465—1181no—
284#427—591no—
285#427—591no—
286#427—591no—
287#427—591no—
288#427—1181no—
289#427—1181no—
290#427C2×D5911yescharacteristic
291C2×#107—591no—
292C2×#107—591no—
293C2×#106—1181no—
294#416—1181no—
295#367—591no—
296#367—591no—
297#367—591no—
298#367C2×D5911yescharacteristic, complemented
299#366—591no—
300#366—591no—
301#366—591no—
302#366C2×D5911yescharacteristic, complemented
303#358—591no—
304#358—591no—
305#358—591no—
306#358—591no—
307#357—591no—
308#357—591no—
309#357C2×D5911yescharacteristic, complemented
310#357C2×D5911yescharacteristic, complemented
311C22⋊C16—591no—
312C22⋊C16—591no—
313C22⋊C16—591no—
314C22⋊C16C2×D5911yescharacteristic
315C59#3364711yescharacteristic, complemented, Sylow
316C2×#41—1181no—
317C23×C4—591no—
318#135—1182no—
319#135—1182no—
320C2×Q16—591no—
321C2×SD16—1181no—
322C2×D8C22×D5911yescharacteristic
323C22×C8—591no—
324C22×C8—591no—
325C22×C8—591no—
326C22×C8—591no—
327C22×C8—591no—
328C22×C8—1182no—
329C22×C8—1182no—
330C22×C8C22×D5911yescharacteristic
331#123—2361no—
332C22⋊Q8—1181no—
333C22⋊Q8—1181no—
334#121—21no—
335#121—1181no—
336#120—2361no—
337C4×D4—2361no—
338C4×D4—2361no—
339#117—21no—
340#117—1181no—
341#117—1181no—
342#117—1181no—
343#117—1181no—
344#117—2361no—
345#117—2361no—
346C2×C4⋊C4—1181no—
347C2×C4⋊C4—1181no—
348C2×C22⋊C4—2361no—
349C2×C42—2361no—
350M32—1182no—
351M32—1182no—
352C2×C16—591no—
353C2×C16—591no—
354C2×C16C22×D5911yescharacteristic
355C2×C16C22×D5911yescharacteristic
356#107—591no—
357#107—591no—
358#107—591no—
359#107—1181no—
360#107—1181no—
361#107C22×D5911yescharacteristic
362#107C22×D5911yescharacteristic
363#107C22×D5911yescharacteristic
364#106—21no—
365#106—1181no—
366#106—1181no—
367#106—1181no—
368#106—1181no—
369#103—1181no—
370#103—1181no—
371#102—21no—
372#102—1181no—
373C24—591no—
374#41—2362no—
375#41—2362no—
376C2×Q8—1181no—
377C2×D4—21no—
378C2×D4—41no—
379C2×D4—2361no—
380C2×D4—2361no—
381C22×C4—591no—
382C22×C4—591no—
383C22×C4—591no—
384C22×C4—591no—
385C22×C4—591no—
386C22×C4—1182no—
387C22×C4—1182no—
388C22×C4—2361no—
389C22×C4—2361no—
390C22×C4—2361no—
391C22×C4D4×D5911yescharacteristic
392Q16—1181no—
393Q16—1181no—
394SD16—2362no—
395D8—21no—
396D8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—591no—
400C2×C8—592no—
401C2×C8—592no—
402C2×C8—591no—
403C2×C8—1182no—
404C2×C8—1182no—
405C2×C8—1182no—
406C2×C8—1181no—
407C2×C8—1181no—
408C2×C8D4×D5911yescharacteristic
409C2×C8C23×D5911yescharacteristic
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—1181no—
414C4⋊C4—1181no—
415C4⋊C4—1181no—
416C4⋊C4—2361no—
417C4⋊C4—2361no—
418C4⋊C4—2361no—
419C4⋊C4—2361no—
420C22⋊C4—41no—
421C22⋊C4—41no—
422C22⋊C4—2361no—
423C22⋊C4—2361no—
424C22⋊C4—2361no—
425C22⋊C4—2361no—
426C22⋊C4—2361no—
427C22⋊C4—2361no—
428C22⋊C4—2361no—
429C42—41no—
430C42—2361no—
431C42—2361no—
432C42—2361no—
433C42—2361no—
434C16—21no—
435C16—21no—
436C16—1181no—
437C16—1181no—
438C23—41no—
439C23—591no—
440C23—591no—
441C23—1182no—
442C23—2361no—
443C23—2361no—
444C23D8×D5911yescharacteristic
445Q8—2361no—
446Q8—2361no—
447D4—41no—
448D4—41no—
449D4—82no—
450D4—4722no—
451D4—4722no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—591no—
458C2×C4—592no—
459C2×C4—592no—
460C2×C4—591no—
461C2×C4—1182no—
462C2×C4—1182no—
463C2×C4—1182no—
464C2×C4—1181no—
465C2×C4—1181no—
466C2×C4—4722no—
467C2×C4—4722no—
468C2×C4—4722no—
469C2×C4—2361no—
470C2×C4—2361no—
471C2×C4—2361no—
472C2×C4—2361no—
473C2×C4D8×D5911yescharacteristic
474C2×C4C2×D4×D5911yescharacteristic
475C8—21no—
476C8—592no—
477C8—1181no—
478C8#41×D5911yescharacteristic
479C8#41×D5911yescharacteristic
480C22—21no—
481C22—21no—
482C22—21no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22—592no—
487C22—1181no—
488C22—1181no—
489C22—1181no—
490C22—1181no—
491C22—1181no—
492C22—2364no—
493C22C2×D8×D5911yescharacteristic
494C4—21no—
495C4—81no—
496C4—81no—
497C4—81no—
498C4—592no—
499C4—1181no—
500C4—4721no—
501C4—4721no—
502C4—4721no—
503C4—4721no—
504C4#123×D5911yescharacteristic
505C4#137×D5911yescharacteristic
506C2—41no—
507C2—161no—
508C2—1182no—
509C2—2361no—
510C2#480×D5911yescharacteristic
511C2#2143722811yescharacteristic
512C2#2143722911yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2234C2
socleC22×C59C22×D8
prime radical#21433675C2
hypercenter#2234D59
hypocenterC59#33647
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2234

Complements

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

NH ≅ G/Ntype
2. #427×D59C2semidirect
3. #21426307C2semidirect
4. #21426306C2semidirect
5. #21426305C2semidirect
6. #21426304C2semidirect
7. #21426286C2semidirect
8. #21426285C2semidirect
9. #21426284C2semidirect
10. #21425883C2semidirect
11. #21425881C2semidirect
12. #21425805C2semidirect
13. #21425805C2semidirect
14. #21425705C2semidirect
15. C59×#2234C2semidirect
37. #16837230C22semidirect
38. #16837229C22semidirect
39. #16837229C22semidirect
40. #16837228C22semidirect
41. #16837227C22semidirect
42. #16837227C22semidirect
43. #16837207C22semidirect
44. #16837207C22semidirect
45. #16837205C22semidirect
46. #16837205C22semidirect
47. #16837203C22semidirect
48. #16837203C22semidirect
54. C59×#367C22semidirect
55. C59×#366C22semidirect
56. C59×#357C22semidirect
57. C59×#357C22semidirect
259. #2234D59semidirect
298. #367C2×D59semidirect
302. #366C2×D59semidirect
309. #357C2×D59semidirect
310. #357C2×D59semidirect
315. C59#33647semidirect

Maximal subgroups

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

subgroupindex
#214336752
#427×D592
#214263072
#214263062
#214263052
#214263042
#214262862
#214262852
#214262842
#214258832
#214258812
#214258052
#214257052
C59×#22342
#3364759