Finite Group Databasebeta

← #28745653

Subgroups — #28745653

Order 15104.

602 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21436440C211yesmaximal, characteristic, complemented
2#21435775C211yesmaximal, characteristic, complemented
3#21426334C211yesmaximal, characteristic, complemented
4#21426292C211yesmaximal, characteristic, complemented
5#21426282C211yesmaximal, characteristic, complemented
6#21426241C211yesmaximal, characteristic, complemented
7#21426197C211yesmaximal, characteristic, complemented
8#21426189C211yesmaximal, characteristic, complemented
9#21426153C211yesmaximal, characteristic, complemented
10#21426006C211yesmaximal, characteristic, complemented
11#21425972C211yesmaximal, characteristic, complemented
12#21425924C211yesmaximal, characteristic, complemented
13#21425808C211yesmaximal, characteristic, complemented
14#357×D59C211yesmaximal, characteristic, complemented
15C59×#2186C211yesmaximal, characteristic, complemented
16C2×D8×D59C2211yescharacteristic
17#16838052—21no—
18#16838040—21no—
19C2×#14273668C2211yescharacteristic
20C2×#14273667C2211yescharacteristic, complemented
21C2×C16×D59C2211yescharacteristic
22#16837806C2211yescharacteristic, complemented
23#16837787C2211yescharacteristic
24#16837714—21no—
25#16837713—21no—
26#16837680—21no—
27#16837640C2211yescharacteristic
28#16837638C2211yescharacteristic
29#16837635C2211yescharacteristic
30#16837619—21no—
31#16837551—21no—
32#16837547—21no—
33D8×Dic59C2211yescharacteristic
34D8×Dic59C2211yescharacteristic, complemented
35C8×C22⋊D59C2211yescharacteristic
36C2×#14273661C2211yescharacteristic, complemented
37#107×D59C2211yescharacteristic
38#16837230C2211yescharacteristic, complemented
39#16837230C2211yescharacteristic, complemented
40#16837229C2211yescharacteristic, complemented
41Dic59⋊C16C2211yescharacteristic, complemented
42#16837206C2211yescharacteristic, complemented
43#16837206C2211yescharacteristic, complemented
44#16837204C2211yescharacteristic
45#16837204C2211yescharacteristic
46#16837203C2211yescharacteristic
47#16837203C2211yescharacteristic
48#16837203C2211yescharacteristic, complemented
49#16837202C2211yescharacteristic
50(C2×D59)⋊C16C2211yescharacteristic
51#16837177C2211yescharacteristic, complemented
52C2×#135×C59C2211yescharacteristic
53C2×SD32×C59C2211yescharacteristic, complemented
54C2×D16×C59C2211yescharacteristic
55C59×#466C2211yescharacteristic, complemented
56C59×#358C2211yescharacteristic, complemented
57C59×#357C2211yescharacteristic
58C59×C22⋊C16C2211yescharacteristic, complemented
59C2×#11745687—41no—
60C2×D4×D59—21no—
61C2×#11745671—21no—
62C2×D472C2311yescharacteristic
63C2×#11745669—21no—
64D8×D59—22no—
65D8×D59—41no—
66C2×C8×D59C2311yescharacteristic
67#14273726—21no—
68#14273716—21no—
69#14273710—41no—
70#14273709—41no—
71#14273708—41no—
72#14273704—41no—
73#14273703—21no—
74#14273701—21no—
75#14273699—41no—
76#14273698—21no—
77#14273695—41no—
78D4×Dic59—21no—
79D4×Dic59—21no—
80C4×C22⋊D59D411yescharacteristic
81C2×C4⋊Dic59—21no—
82C4⋊C4×D59—21no—
83C2×(C2×D59)⋊C4—41no—
84#14273668—41no—
85#14273667—41no—
86C2×C59⋊C16C2311yescharacteristic
87C2×C59⋊C16C2311yescharacteristic
88C16×D59—41no—
89#14273661—21no—
90#14273661C2311yescharacteristic
91#14273661C2311yescharacteristic
92#14273661C2311yescharacteristic
93#14273660—21no—
94#14273660—21no—
95Dic59⋊C8D411yescharacteristic
96#14273651—21no—
97#14273649—21no—
98#14273648—21no—
99#14273648—21no—
100#14273648—21no—
101C22⋊(C59⋊C8)D411yescharacteristic
102(C2×D59)⋊C8D411yescharacteristic
103C8×Dic59C2311yescharacteristic
104C2×#41×C59—21no—
105#135×C59—22no—
106#135×C59—41no—
107C2×Q16×C59C2311yescharacteristic
108C2×SD16×C59—21no—
109C2×D8×C59C2311yescharacteristic
110C2×D8×C59C2311yescharacteristic
111C22×C8×C59C2311yescharacteristic
112#121×C59—21no—
113SD32×C59—41no—
114D16×C59—22no—
115C2×C16×C59C2311yescharacteristic
116C2×C16×C59C2311yescharacteristic
117#107×C59C2311yescharacteristic
118#102×C59—21no—
119C23×D59—41no—
120#41×C59—42no—
121#41×C59—81no—
122C2×Q8×C59—21no—
123C2×D4×C59—21no—
124C2×D4×C59—21no—
125C2×D4×C59—41no—
126C2×D4×C59—41no—
127C22×C4×C59—41no—
128C22×C4×C59C2×D411yescharacteristic
129C2×C22⋊D59—41no—
130C2×C22⋊D59—41no—
131C2×C22⋊D59—41no—
132C2×C22⋊D59D811yescharacteristic
133C22×Dic59—41no—
134C22×Dic59—41no—
135C22×Dic59—41no—
136#11745687—84no—
137D4×D59—42no—
138D4×D59—81no—
139C2×D236—21no—
140C2×C4×D59—41no—
141C2×C4×D59C2×D411yescharacteristic
142C2×Dic118—41no—
143Q16×C59—21no—
144Q16×C59—21no—
145SD16×C59—42no—
146D8×C59—21no—
147D8×C59—21no—
148D8×C59—21no—
149D8×C59#4112yescomplemented
150C2×C8×C59—21no—
151C2×C8×C59—21no—
152C2×C8×C59C2×D411yescharacteristic
153C2×C8×C59C2411yescharacteristic
154C4⋊C4×C59—21no—
155C22⋊C4×C59—41no—
156C22⋊Dic59—41no—
157C22⋊Dic59—41no—
158C22⋊Dic59—41no—
159C22⋊Dic59D811yescharacteristic
160#11745671—81no—
161#11745669—42no—
162(C2×D59)⋊C4—41no—
163(C2×D59)⋊C4—41no—
164(C2×D59)⋊C4—41no—
165(C2×D59)⋊C4—41no—
166(C2×D59)⋊C4D811yescharacteristic
167C4⋊Dic59—21no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—41no—
171Dic59⋊C4—41no—
172Dic59⋊C4—41no—
173Dic59⋊C4—41no—
174Dic59⋊C4D811yescharacteristic
175C4×Dic59C2×D411yescharacteristic
176C2×C59⋊C8C2×D411yescharacteristic
177C2×C59⋊C8C2×D411yescharacteristic
178D472—21no—
179D472—21no—
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—82no—
191C22×D59—82no—
192C22×D59—161no—
193C22×D59C2×D811yescharacteristic
194Q8×C59—41no—
195Q8×C59—41no—
196D4×C59—22no—
197D4×C59—41no—
198D4×C59—41no—
199D4×C59—41no—
200D4×C59—82no—
201D4×C59—82no—
202C2×C4×C59—21no—
203C2×C4×C59—21no—
204C2×C4×C59—41no—
205C2×C4×C59—41no—
206C2×C4×C59—82no—
207C2×C4×C59C2×D811yescharacteristic
208C2×C4×C59C22×D411yescharacteristic
209C22⋊D59—22no—
210C22⋊D59—82no—
211C22⋊D59—82no—
212C22⋊D59—161no—
213C2×Dic59—41no—
214C2×Dic59—41no—
215C2×Dic59—41no—
216C2×Dic59—82no—
217C2×Dic59—82no—
218C2×Dic59—161no—
219C2×Dic59C2×D811yescharacteristic
220C2×Dic59C2×D811yescharacteristic
221C2×Dic59C2×D811yescharacteristic
222D236—41no—
223D236—41no—
224C4×D59—21no—
225C4×D59—21no—
226C4×D59—82no—
227Dic118—82no—
228C8×C59—21no—
229C8×C59C2×#4111yescharacteristic
230C8×C59H(5,𝔽2)11yescharacteristic
231C59⋊C8—21no—
232C59⋊C8—21no—
233#33873—591nomaximal, Sylow
234C22×C59—21no—
235C22×C59—21no—
236C22×C59—21no—
237C22×C59—42no—
238C22×C59—81no—
239C22×C59—81no—
240C22×C59—81no—
241C22×C59—81no—
242C22×C59—81no—
243C22×C59C22×D811yescharacteristic
244C2×D59—21no—
245C2×D59—21no—
246C2×D59—81no—
247C2×D59—81no—
248C2×D59—81no—
249C2×D59—164no—
250C2×D59#50912yes—
251C4×C59—21no—
252C4×C59—81no—
253C4×C59—81no—
254C4×C59#54611yescharacteristic
255C4×C59#57711yescharacteristic
256Dic59—21no—
257Dic59—21no—
258Dic59—21no—
259Dic59—81no—
260Dic59—81no—
261Dic59—81no—
262#3334—591no—
263#3280—591no—
264#2243—591no—
265#2235—591no—
266#2233—591no—
267#2226—591no—
268#2218—591no—
269#2217—591no—
270#2210—591no—
271#2186D5911yescharacteristic, complemented
272#2184—591no—
273#2177—591no—
274#2168—591no—
275#2139—591no—
276C2×#357—591no—
277C2×C59—41no—
278C2×C59—82no—
279C2×C59—161no—
280C2×C59#329411yescharacteristic
281C2×C59C2×#50911yescharacteristic
282C2×C59#341611yescharacteristic
283D59—22no—
284D59—161no—
285C2×#135C2×D5911yescharacteristic
286C22×D8—591no—
287#547—1181no—
288#546—1181no—
289C2×SD32C2×D5911yescharacteristic, complemented
290C2×D16—591no—
291C2×D16C2×D5911yescharacteristic
292C2×M32—591no—
293C22×C16—591no—
294#500—591no—
295#495—591no—
296#480—1181no—
297#480—1181no—
298#474—1181no—
299#467—591no—
300#466—591no—
301#466—591no—
302#466C2×D5911yescharacteristic, complemented
303#463—1181no—
304#450—1181no—
305#449—1181no—
306C4×D8—591no—
307C4×D8—591no—
308C8×D4—591no—
309C2×#107—591no—
310C2×#107—591no—
311#367—591no—
312#367—591no—
313#367—591no—
314C4⋊C16—591no—
315#358—591no—
316#358—591no—
317#358C2×D5911yescharacteristic, complemented
318#357—591no—
319#357—591no—
320#357—591no—
321#357—591no—
322#357—591no—
323#357—591no—
324#357C2×D5911yescharacteristic
325C22⋊C16—591no—
326C22⋊C16C2×D5911yescharacteristic, complemented
327#346—591no—
328C59#3387311yescharacteristic, complemented, Sylow
329C2×#41—21no—
330C2×#41—2361no—
331C22×D4—1181no—
332#135—22no—
333#135—41no—
334C2×Q16C22×D5911yescharacteristic
335C2×SD16—21no—
336C2×SD16—1181no—
337C2×D8—591no—
338C2×D8—1181no—
339C2×D8—1182no—
340C2×D8—2361no—
341C2×D8C22×D5911yescharacteristic
342C2×D8C22×D5911yescharacteristic
343C22×C8—591no—
344C22×C8C22×D5911yescharacteristic
345C4⋊Q8—1181no—
346#124—1181no—
347#123—2361no—
348#123—2361no—
349#123—2361no—
350C22⋊Q8—1181no—
351C22⋊Q8—2361no—
352#121—21no—
353#121—1181no—
354#121—1181no—
355#121—2361no—
356#120—2361no—
357C4×D4—591no—
358C4×D4—1181no—
359C4×D4—1181no—
360C2×C4⋊C4—1181no—
361C2×C4⋊C4—1181no—
362C2×C22⋊C4—2361no—
363SD32—41no—
364D16—22no—
365D16—2361no—
366M32—2361no—
367C2×C16—591no—
368C2×C16—591no—
369C2×C16—2361no—
370C2×C16C22×D5911yescharacteristic
371C2×C16C22×D5911yescharacteristic
372#107—591no—
373#107—591no—
374#107—591no—
375#107—1181no—
376#107—1181no—
377#107—1181no—
378#107C22×D5911yescharacteristic
379C4⋊C8—591no—
380#103—1181no—
381#102—21no—
382#102—1181no—
383#102—1181no—
384#102—1181no—
385#102—1181no—
386C22⋊C8—591no—
387C22⋊C8—591no—
388C4×C8—591no—
389C24—2361no—
390#41—42no—
391#41—81no—
392#41—4724no—
393C2×Q8—21no—
394C2×Q8—2361no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—591no—
400C2×D4—1181no—
401C2×D4—4721no—
402C2×D4—2361no—
403C2×D4—2362no—
404C2×D4—2361no—
405C2×D4—2361no—
406C22×C4—41no—
407C22×C4—591no—
408C22×C4—2361no—
409C22×C4—2361no—
410C22×C4—2361no—
411C22×C4—2361no—
412C22×C4D4×D5911yescharacteristic
413Q16—21no—
414Q16—21no—
415SD16—42no—
416SD16—4721no—
417D8—21no—
418D8—21no—
419D8—21no—
420D8—1181no—
421D8—1181no—
422D8—2362no—
423D8#1174568712yescomplemented
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—591no—
427C2×C8—591no—
428C2×C8—1181no—
429C2×C8—1181no—
430C2×C8D4×D5911yescharacteristic
431C2×C8C23×D5911yescharacteristic
432C4⋊C4—21no—
433C4⋊C4—591no—
434C4⋊C4—1181no—
435C4⋊C4—1181no—
436C4⋊C4—1181no—
437C4⋊C4—2361no—
438C4⋊C4—2361no—
439C4⋊C4—2361no—
440C4⋊C4—2361no—
441C22⋊C4—41no—
442C22⋊C4—591no—
443C22⋊C4—591no—
444C22⋊C4—2361no—
445C22⋊C4—2361no—
446C22⋊C4—2361no—
447C22⋊C4—2361no—
448C22⋊C4—2361no—
449C22⋊C4—2361no—
450C22⋊C4—2361no—
451C42—591no—
452C16—21no—
453C16—21no—
454C16—1181no—
455C16—1181no—
456C23—41no—
457C23—41no—
458C23—591no—
459C23—9441no—
460C23—4722no—
461C23—4722no—
462C23—2361no—
463C23D8×D5911yescharacteristic
464Q8—41no—
465Q8—41no—
466Q8—4722no—
467D4—22no—
468D4—41no—
469D4—41no—
470D4—41no—
471D4—82no—
472D4—82no—
473D4—1182no—
474D4—9441no—
475D4—4722no—
476D4—4722no—
477D4—2361no—
478D4—2361no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—82no—
484C2×C4—591no—
485C2×C4—591no—
486C2×C4—591no—
487C2×C4—1181no—
488C2×C4—1181no—
489C2×C4—9441no—
490C2×C4—4722no—
491C2×C4—4722no—
492C2×C4—4722no—
493C2×C4—2361no—
494C2×C4—2361no—
495C2×C4—2361no—
496C2×C4D8×D5911yescharacteristic
497C2×C4C2×D4×D5911yescharacteristic
498C8—21no—
499C8—1181no—
500C8—1181no—
501C8C2×#1174568711yescharacteristic
502C8#1427377211yescharacteristic
503C22—21no—
504C22—21no—
505C22—21no—
506C22—42no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—592no—
513C22—1181no—
514C22—1181no—
515C22—9444no—
516C22—4721no—
517C22—4721no—
518C22—4721no—
519C22C2×D8×D5911yescharacteristic
520C4—21no—
521C4—81no—
522C4—81no—
523C4—1181no—
524C4—1181no—
525C4—1181no—
526C4—4721no—
527C4—4721no—
528C4—4721no—
529C4#1683803911yescharacteristic
530C4#1683821211yescharacteristic
531C2—41no—
532C2—82no—
533C2—161no—
534C2—1182no—
535C2—9441no—
536C2#2143590211yescharacteristic
537C2#509×D5911yescharacteristic
538C2#2143723911yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2186C2
socleC22×C59C22×D8
hypercenter#2186D59
hypocenterC59#33873
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2186

Complements

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

NH ≅ G/Ntype
1. #21436440C2semidirect
2. #21435775C2semidirect
3. #21426334C2semidirect
4. #21426292C2semidirect
5. #21426282C2semidirect
6. #21426241C2semidirect
7. #21426197C2semidirect
8. #21426189C2semidirect
9. #21426153C2semidirect
10. #21426006C2semidirect
11. #21425972C2semidirect
12. #21425924C2semidirect
13. #21425808C2semidirect
14. #357×D59C2semidirect
15. C59×#2186C2semidirect
20. C2×#14273667C22semidirect
22. #16837806C22semidirect
34. D8×Dic59C22semidirect
36. C2×#14273661C22semidirect
38. #16837230C22semidirect
39. #16837230C22semidirect
40. #16837229C22semidirect
41. Dic59⋊C16C22semidirect
42. #16837206C22semidirect
43. #16837206C22semidirect
48. #16837203C22semidirect
51. #16837177C22semidirect
53. C2×SD32×C59C22semidirect
55. C59×#466C22semidirect
56. C59×#358C22semidirect
58. C59×C22⋊C16C22semidirect
149. D8×C59#41semidirect
271. #2186D59semidirect
289. C2×SD32C2×D59semidirect
302. #466C2×D59semidirect
317. #358C2×D59semidirect
326. C22⋊C16C2×D59semidirect
328. C59#33873semidirect
423. D8#11745687semidirect

Maximal subgroups

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

subgroupindex
#214364402
#214357752
#214263342
#214262922
#214262822
#214262412
#214261972
#214261892
#214261532
#214260062
#214259722
#214259242
#214258082
#357×D592
C59×#21862
#3387359