Finite Group Databasebeta

← #28745543

Subgroups — #28745543

Order 15104.

580 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21435783C211yesmaximal, characteristic, complemented
2#21426330C211yesmaximal, characteristic, complemented
3#21426298C211yesmaximal, characteristic, complemented
4#21426287C211yesmaximal, characteristic, complemented
5#21426250C211yesmaximal, characteristic, complemented
6#21426190C211yesmaximal, characteristic, complemented
7#21426180C211yesmaximal, characteristic, complemented
8#21426151C211yesmaximal, characteristic, complemented
9#21426017C211yesmaximal, characteristic, complemented
10#21426007C211yesmaximal, characteristic, complemented
11C4×#14273668C211yesmaximal, characteristic, complemented
12D4×C59⋊C16C211yesmaximal, characteristic, complemented
13#21425801C211yesmaximal, characteristic, complemented
14C2×#16837202C211yesmaximal, characteristic, complemented
15C59×#3339C211yesmaximal, characteristic, complemented
16C2×#14273749C2211yescharacteristic
17C22×D472C2211yescharacteristic
18#16838048—21no—
19C2×#14273671C2211yescharacteristic, complemented
20C2×#14273668C2211yescharacteristic
21C2×#14273668C2211yescharacteristic
22C22×C59⋊C16C2211yescharacteristic
23C22×C59⋊C16C2211yescharacteristic, complemented
24#16837786C2211yescharacteristic
25#16837643C2211yescharacteristic
26#16837638C2211yescharacteristic
27#16837618—21no—
28#16837555—21no—
29#16837549—21no—
30C4×D472C2211yescharacteristic
31#16837229C2211yescharacteristic, complemented
32#16837227C2211yescharacteristic, complemented
33#16837227C2211yescharacteristic, complemented
34C4⋊(C59⋊C16)C2211yescharacteristic, complemented
35#16837205C2211yescharacteristic, complemented
36#16837205C2211yescharacteristic, complemented
37#16837205C2211yescharacteristic, complemented
38#16837203C2211yescharacteristic, complemented
39#16837202C2211yescharacteristic
40#16837202C2211yescharacteristic
41#16837202C2211yescharacteristic
42#16837202C2211yescharacteristic
43#16837202C2211yescharacteristic
44#16837202C2211yescharacteristic
45C22⋊(C59⋊C16)C2211yescharacteristic
46C22⋊(C59⋊C16)C2211yescharacteristic, complemented
47C4×C59⋊C16C2211yescharacteristic, complemented
48C59×#553—21no—
49C59×#499C2211yescharacteristic, complemented
50C59×#482—21no—
51C59×#481—21no—
52C59×#478—21no—
53C59×#466C2211yescharacteristic
54C59×#466C2211yescharacteristic, complemented
55C4×D8×C59C2211yescharacteristic, complemented
56C8×D4×C59C2211yescharacteristic
57C59×#427C2211yescharacteristic, complemented
58C2×#107×C59C2211yescharacteristic
59C2×#11745685—21no—
60C22×D236—21no—
61#14273749—22no—
62#14273749—41no—
63C2×Dic236C2311yescharacteristic
64C2×#11745660—21no—
65C2×#11745660—21no—
66C2×D472—22no—
67C2×D472—21no—
68C2×D472—41no—
69C2×D472C2311yescharacteristic
70C2×D472C2311yescharacteristic
71#14273715—21no—
72#14273712—41no—
73#14273706—21no—
74#14273701—21no—
75#14273700—41no—
76#14273695—41no—
77C4×D236—21no—
78C22⋊C4×D59—41no—
79#14273671—41no—
80#14273668—22no—
81#14273668—41no—
82C2×C59⋊C16—41no—
83C2×C59⋊C16—41no—
84C2×C59⋊C16C2311yescharacteristic
85C2×C59⋊C16C2311yescharacteristic
86C2×C59⋊C16C2311yescharacteristic
87C2×C59⋊C16C2311yescharacteristic
88#14273661C2311yescharacteristic
89#14273652—21no—
90#14273649—21no—
91#14273649—21no—
92C2×D8×C59C2311yescharacteristic
93C22×C8×C59C2311yescharacteristic
94C22×C8×C59C2311yescharacteristic
95#126×C59—41no—
96#125×C59—21no—
97#123×C59—41no—
98#121×C59—21no—
99#121×C59—21no—
100#121×C59—41no—
101C4×D4×C59—21no—
102C4×D4×C59—41no—
103C4×D4×C59D411yescharacteristic
104#117×C59—21no—
105C2×C4⋊C4×C59—21no—
106#107×C59—21no—
107#107×C59C2311yescharacteristic
108#107×C59C2311yescharacteristic
109#107×C59C2311yescharacteristic
110#106×C59—21no—
111#106×C59—21no—
112C4⋊C8×C59D411yescharacteristic
113#102×C59—21no—
114#102×C59—21no—
115#102×C59—21no—
116C22⋊C8×C59D411yescharacteristic
117C22⋊C8×C59D411yescharacteristic
118C4×C8×C59C2311yescharacteristic
119C23×D59—41no—
120C2×D4×C59—21no—
121C2×D4×C59—41no—
122C2×D4×C59—41no—
123C2×D4×C59D811yescharacteristic
124C22×C4×C59—41no—
125C22×C4×C59—41no—
126C22×C4×C59C2×D411yescharacteristic
127C22×C4×C59C2×D411yescharacteristic
128C2×C22⋊D59—41no—
129C2×C22⋊D59—41no—
130#11745685—42no—
131#11745685—81no—
132C2×D236—21no—
133C2×D236—21no—
134C2×D236—42no—
135C2×D236—41no—
136C2×D236—81no—
137C2×C4×D59—41no—
138C2×C4×D59—41no—
139C2×Dic118—21no—
140D8×C59—21no—
141D8×C59—21no—
142C2×C8×C59—21no—
143C2×C8×C59—21no—
144C2×C8×C59—21no—
145C2×C8×C59—21no—
146C2×C8×C59C2×D411yescharacteristic
147C2×C8×C59C2×D411yescharacteristic
148C2×C8×C59C2×D411yescharacteristic
149C2×C8×C59C2411yescharacteristic
150C4⋊C4×C59—21no—
151C4⋊C4×C59—21no—
152C4⋊C4×C59—21no—
153C4⋊C4×C59—41no—
154C4⋊C4×C59—41no—
155C4⋊C4×C59—41no—
156C4⋊C4×C59D811yescharacteristic
157C22⋊C4×C59—41no—
158C22⋊C4×C59—41no—
159C22⋊C4×C59—41no—
160C22⋊C4×C59—41no—
161C22⋊C4×C59D811yescharacteristic
162C22⋊C4×C59D811yescharacteristic
163C42×C59—41no—
164C42×C59C2×D411yescharacteristic
165C22⋊Dic59—41no—
166(C2×D59)⋊C4—41no—
167(C2×D59)⋊C4—41no—
168(C2×D59)⋊C4—41no—
169(C2×D59)⋊C4—41no—
170C4⋊Dic59—21no—
171Dic59⋊C4—41no—
172Dic236—21no—
173Dic236—21no—
174D472—21no—
175D472—21no—
176D472—21no—
177D472—42no—
178D472#4112yes—
179#11745660—42no—
180#11745660—81no—
181C59⋊C16—21no—
182C59⋊C16—21no—
183C59⋊C16—21no—
184C59⋊C16—21no—
185C23×C59—41no—
186C23×C59C2×D811yescharacteristic
187C23×C59C2×D811yescharacteristic
188C22×D59—41no—
189C22×D59—41no—
190C22×D59—82no—
191C22×D59—82no—
192C22×D59—161no—
193D4×C59—22no—
194D4×C59—41no—
195D4×C59—41no—
196D4×C59—82no—
197D4×C59—161no—
198C2×C4×C59—21no—
199C2×C4×C59—21no—
200C2×C4×C59—21no—
201C2×C4×C59—21no—
202C2×C4×C59—41no—
203C2×C4×C59—41no—
204C2×C4×C59—41no—
205C2×C4×C59—161no—
206C2×C4×C59—161no—
207C2×C4×C59C2×D811yescharacteristic
208C2×C4×C59C2×D811yescharacteristic
209C2×C4×C59C2×D811yescharacteristic
210C2×C4×C59C2×D811yescharacteristic
211C2×C4×C59C22×D411yescharacteristic
212C22⋊D59—82no—
213C22⋊D59—161no—
214C2×Dic59—41no—
215C2×Dic59—41no—
216D236—22no—
217D236—41no—
218D236—41no—
219D236—41no—
220D236—82no—
221C4×D59—82no—
222C4×D59—161no—
223Dic118—41no—
224Dic118—41no—
225C8×C59—21no—
226C8×C59—21no—
227C8×C59—21no—
228C8×C59C2×#4111yescharacteristic
229C8×C59H(5,𝔽2)11yescharacteristic
230#33864—591nomaximal, Sylow
231C22×C59—21no—
232C22×C59—21no—
233C22×C59—21no—
234C22×C59—21no—
235C22×C59—21no—
236C22×C59—81no—
237C22×C59—81no—
238C22×C59—81no—
239C22×C59#50812yes—
240C22×C59C22×D811yescharacteristic
241C2×D59—42no—
242C2×D59—81no—
243C2×D59—81no—
244C2×D59—81no—
245C2×D59—81no—
246C2×D59—81no—
247C2×D59—164no—
248C4×C59—21no—
249C4×C59—21no—
250C4×C59—21no—
251C4×C59—21no—
252C4×C59—81no—
253C4×C59—81no—
254C4×C59—81no—
255C4×C59#54611yescharacteristic
256C4×C59#57711yescharacteristic
257Dic59—81no—
258Dic59—81no—
259#3339D5911yescharacteristic, complemented
260#3280—591no—
261#2242—591no—
262#2236—591no—
263#2234—591no—
264#2228—591no—
265#2217—591no—
266#2215—591no—
267#2210—591no—
268#2186—591no—
269#2184—591no—
270C4×D16—591no—
271D4×C16—591no—
272#2138—591no—
273C2×#357—591no—
274C2×C59—22no—
275C2×C59—41no—
276C2×C59—161no—
277C2×C59#329411yescharacteristic
278C2×C59C2×#50811yescharacteristic
279C2×C59#341511yescharacteristic
280D59—82no—
281D59—161no—
282C2×#135—591no—
283C22×D8—591no—
284#553—21no—
285#546—1181no—
286C2×SD32—591no—
287C2×D16—591no—
288C2×D16—591no—
289C22×C16—591no—
290C22×C16—591no—
291#499C2×D5911yescharacteristic, complemented
292#495—591no—
293#482—21no—
294#481—21no—
295#478—21no—
296#467—591no—
297#466—591no—
298#466C2×D5911yescharacteristic
299#466C2×D5911yescharacteristic, complemented
300#463—1181no—
301#450—1181no—
302#449—1181no—
303C4×D8—591no—
304C4×D8C2×D5911yescharacteristic, complemented
305C8×D4C2×D5911yescharacteristic
306#427C2×D5911yescharacteristic, complemented
307C2×#107C2×D5911yescharacteristic
308#367—591no—
309#366—591no—
310#366—591no—
311C4⋊C16—591no—
312#358—591no—
313#358—591no—
314#358—591no—
315#357—591no—
316#357—591no—
317#357—591no—
318#357—591no—
319#357—591no—
320#357—591no—
321#357—591no—
322C22⋊C16—591no—
323C22⋊C16—591no—
324C4×C16—591no—
325C59#3386411yescharacteristic, complemented, Sylow
326C2×#41—1181no—
327C22×D4—1181no—
328#135—1182no—
329#135—2361no—
330C2×Q16—591no—
331C2×SD16—1181no—
332C2×SD16—1181no—
333C2×D8—591no—
334C2×D8—591no—
335C2×D8—1182no—
336C2×D8—1181no—
337C2×D8—2361no—
338C2×D8C22×D5911yescharacteristic
339C22×C8C22×D5911yescharacteristic
340C22×C8C22×D5911yescharacteristic
341#126—41no—
342#125—21no—
343#124—1181no—
344#123—41no—
345#123—2361no—
346C22⋊Q8—1181no—
347#121—21no—
348#121—21no—
349#121—41no—
350#121—1181no—
351#121—2361no—
352#120—2361no—
353C4×D4—21no—
354C4×D4—41no—
355C4×D4—1181no—
356C4×D4C22⋊D5911yescharacteristic
357#117—21no—
358C2×C4⋊C4—21no—
359C2×C22⋊C4—2361no—
360SD32—2361no—
361D16—1182no—
362D16—2361no—
363C2×C16—591no—
364C2×C16—591no—
365C2×C16—591no—
366C2×C16—591no—
367C2×C16—2361no—
368C2×C16—2361no—
369#107—21no—
370#107—591no—
371#107C22×D5911yescharacteristic
372#107C22×D5911yescharacteristic
373#107C22×D5911yescharacteristic
374#106—21no—
375#106—21no—
376C4⋊C8C22⋊D5911yescharacteristic
377#103—1181no—
378#102—21no—
379#102—21no—
380#102—21no—
381#102—1181no—
382#102—1181no—
383C22⋊C8C22⋊D5911yescharacteristic
384C22⋊C8C22⋊D5911yescharacteristic
385C4×C8C22×D5911yescharacteristic
386C24—2361no—
387#41—4721no—
388#41—2362no—
389C2×Q8—1181no—
390C2×D4—21no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4—1181no—
394C2×D4—1181no—
395C2×D4—4721no—
396C2×D4—2362no—
397C2×D4—2361no—
398C2×D4—2361no—
399C2×D4—2361no—
400C2×D4#1174566911yescharacteristic
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—2361no—
404C22×C4—2361no—
405C22×C4C2×C22⋊D5911yescharacteristic
406C22×C4C2×C22⋊D5911yescharacteristic
407Q16—1181no—
408Q16—1181no—
409SD16—4721no—
410SD16—2362no—
411D8—21no—
412D8—21no—
413D8—592no—
414D8—1181no—
415D8—1181no—
416D8—1181no—
417D8—2362no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8C2×C22⋊D5911yescharacteristic
423C2×C8C2×C22⋊D5911yescharacteristic
424C2×C8C2×C22⋊D5911yescharacteristic
425C2×C8C23×D5911yescharacteristic
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—1181no—
433C4⋊C4—2361no—
434C4⋊C4#1174566911yescharacteristic
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—2361no—
440C22⋊C4—2361no—
441C22⋊C4—2361no—
442C22⋊C4—2361no—
443C22⋊C4—2361no—
444C22⋊C4#1174566911yescharacteristic
445C22⋊C4#1174566911yescharacteristic
446C42—41no—
447C42C2×C22⋊D5911yescharacteristic
448C16—1181no—
449C16—1181no—
450C16—1181no—
451C16—1181no—
452C23—41no—
453C23—9441no—
454C23—4722no—
455C23—4722no—
456C23—2361no—
457C23—2361no—
458C23C2×#1174566911yescharacteristic
459C23C2×#1174566911yescharacteristic
460Q8—2361no—
461Q8—2361no—
462D4—22no—
463D4—41no—
464D4—41no—
465D4—82no—
466D4—161no—
467D4—1182no—
468D4—9441no—
469D4—4722no—
470D4—4722no—
471D4—2361no—
472D4—2361no—
473D4—2361no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—161no—
482C2×C4—161no—
483C2×C4—9441no—
484C2×C4—4722no—
485C2×C4—2361no—
486C2×C4—2361no—
487C2×C4C2×#1174566911yescharacteristic
488C2×C4C2×#1174566911yescharacteristic
489C2×C4C2×#1174566911yescharacteristic
490C2×C4C2×#1174566911yescharacteristic
491C2×C4C22×C22⋊D5911yescharacteristic
492C8—21no—
493C8—21no—
494C8—21no—
495C8#41×D5911yescharacteristic
496C8#1427377311yescharacteristic
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22—9444no—
506C22—4721no—
507C22—4721no—
508C22—4721no—
509C22—4721no—
510C22—4721no—
511C22—2362no—
512C22#1683783212yes—
513C22C22×#1174566911yescharacteristic
514C4—21no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—81no—
519C4—81no—
520C4—81no—
521C4—4721no—
522C4—4721no—
523C4#1683804011yescharacteristic
524C4#1683821011yescharacteristic
525C2—22no—
526C2—41no—
527C2—161no—
528C2—9441no—
529C2—4722no—
530C2#2143590311yescharacteristic
531C2C2×#1683783211yescharacteristic
532C2#2143723511yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3339C2
socleC22×C59C22×D8
hypercenter#3339D59
hypocenterC59#33864
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3339

Complements

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

NH ≅ G/Ntype
1. #21435783C2semidirect
2. #21426330C2semidirect
3. #21426298C2semidirect
4. #21426287C2semidirect
5. #21426250C2semidirect
6. #21426190C2semidirect
7. #21426180C2semidirect
8. #21426151C2semidirect
9. #21426017C2semidirect
10. #21426007C2semidirect
11. C4×#14273668C2semidirect
12. D4×C59⋊C16C2semidirect
13. #21425801C2semidirect
14. C2×#16837202C2semidirect
15. C59×#3339C2semidirect
19. C2×#14273671C22semidirect
23. C22×C59⋊C16C22semidirect
31. #16837229C22semidirect
32. #16837227C22semidirect
33. #16837227C22semidirect
34. C4⋊(C59⋊C16)C22semidirect
35. #16837205C22semidirect
36. #16837205C22semidirect
37. #16837205C22semidirect
38. #16837203C22semidirect
46. C22⋊(C59⋊C16)C22semidirect
47. C4×C59⋊C16C22semidirect
49. C59×#499C22semidirect
54. C59×#466C22semidirect
55. C4×D8×C59C22semidirect
57. C59×#427C22semidirect
259. #3339D59semidirect
291. #499C2×D59semidirect
299. #466C2×D59semidirect
304. C4×D8C2×D59semidirect
306. #427C2×D59semidirect
325. C59#33864semidirect

Maximal subgroups

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

subgroupindex
#214357832
#214263302
#214262982
#214262872
#214262502
#214261902
#214261802
#214261512
#214260172
#214260072
C4×#142736682
D4×C59⋊C162
#214258012
C2×#168372022
C59×#33392
#3386459