Finite Group Databasebeta

← #28745662

Subgroups — #28745662

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#21436444C211yesmaximal, characteristic, complemented
2#21435783C211yesmaximal, characteristic, complemented
3#21426333C211yesmaximal, characteristic, complemented
4#21426294C211yesmaximal, characteristic, complemented
5#21426282C211yesmaximal, characteristic, complemented
6#21426244C211yesmaximal, characteristic, complemented
7#21426199C211yesmaximal, characteristic, complemented
8#21426192C211yesmaximal, characteristic, complemented
9#21426152C211yesmaximal, characteristic, complemented
10#21426021C211yesmaximal, characteristic, complemented
11#21426010C211yesmaximal, characteristic, complemented
12#21425973C211yesmaximal, characteristic, complemented
13#21425809C211yesmaximal, characteristic, complemented
14C2×#16837204C211yesmaximal, characteristic, complemented
15C59×#2168C211yesmaximal, characteristic, complemented
16C2×#14273749C2211yescharacteristic
17C22×D472C2211yescharacteristic
18#16838056—21no—
19#16838048—21no—
20C2×#14273672C2211yescharacteristic, complemented
21C2×D944C2211yescharacteristic
22C2×D944C2211yescharacteristic
23#16837807C2211yescharacteristic, complemented
24#16837786C2211yescharacteristic
25#16837716—21no—
26#16837716—21no—
27#16837683—21no—
28#16837643C2211yescharacteristic
29#16837638C2211yescharacteristic
30#16837638C2211yescharacteristic
31#16837638C2211yescharacteristic, complemented
32#16837618—21no—
33#16837555—21no—
34#16837549—21no—
35C4×D472C2211yescharacteristic
36C4×D472C2211yescharacteristic, complemented
37C2×#14273661C2211yescharacteristic
38C2×#14273661C2211yescharacteristic, complemented
39#16837230C2211yescharacteristic, complemented
40#16837230C2211yescharacteristic, complemented
41#16837230C2211yescharacteristic, complemented
42#16837207C2211yescharacteristic, complemented
43#16837207C2211yescharacteristic, complemented
44#16837207C2211yescharacteristic, complemented
45#16837204C2211yescharacteristic
46#16837204C2211yescharacteristic
47#16837204C2211yescharacteristic
48#16837204C2211yescharacteristic
49#16837204C2211yescharacteristic
50#16837204C2211yescharacteristic
51#16837204C2211yescharacteristic, complemented
52C2×M32×C59C2211yescharacteristic, complemented
53C22×C16×C59C2211yescharacteristic
54C8×D4×C59C2211yescharacteristic
55C59×C4⋊C16C2211yescharacteristic, complemented
56C59×C22⋊C16C2211yescharacteristic
57C59×C22⋊C16C2211yescharacteristic, complemented
58C59×#346C2211yescharacteristic, complemented
59C2×#11745685—21no—
60C2×#11745687—41no—
61C22×D236—21no—
62#14273749—22no—
63#14273749—41no—
64C2×Dic236C2311yescharacteristic
65C2×#11745660—21no—
66C2×#11745660—21no—
67C2×D472—22no—
68C2×D472—21no—
69C2×D472—41no—
70C2×D472C2311yescharacteristic
71C2×D472C2311yescharacteristic
72C2×D472C2311yescharacteristic
73#14273727—21no—
74#14273715—21no—
75#14273712—41no—
76#14273711—41no—
77#14273711—41no—
78#14273707—41no—
79#14273706—21no—
80#14273701—21no—
81#14273701—21no—
82#14273701—21no—
83#14273700—41no—
84#14273695—41no—
85C4×D236—21no—
86C4×D236—21no—
87C2×C4⋊Dic59—21no—
88C2×C4⋊Dic59—21no—
89C22⋊C4×D59—41no—
90#14273672—41no—
91D944—22no—
92D944—41no—
93#14273661—21no—
94#14273661—21no—
95#14273661—21no—
96#14273661C2311yescharacteristic
97#14273661C2311yescharacteristic
98#14273661C2311yescharacteristic
99#14273661C2311yescharacteristic
100#14273652—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—
120C2×D4×C59D811yescharacteristic
121C22×C4×C59C2×D411yescharacteristic
122C22×C4×C59C2×D411yescharacteristic
123C2×C22⋊D59—41no—
124C2×C22⋊D59—41no—
125C2×C22⋊D59—41no—
126C2×C22⋊D59—41no—
127C22×Dic59—41no—
128C22×Dic59—41no—
129#11745687—84no—
130#11745685—42no—
131#11745685—81no—
132C2×D236—21no—
133C2×D236—21no—
134C2×D236—21no—
135C2×D236—42no—
136C2×D236—41no—
137C2×D236—81no—
138C2×C4×D59—41no—
139C2×C4×D59—41no—
140C2×C4×D59—41no—
141C2×Dic118—21no—
142C2×Dic118—41no—
143C2×C8×C59—21no—
144C2×C8×C59—21no—
145C2×C8×C59—21no—
146C2×C8×C59—21no—
147C2×C8×C59C2×D411yescharacteristic
148C2×C8×C59C2×D411yescharacteristic
149C2×C8×C59C2×D411yescharacteristic
150C2×C8×C59C2411yescharacteristic
151C4⋊C4×C59D811yescharacteristic
152C22⋊C4×C59D811yescharacteristic
153C22⋊C4×C59D811yescharacteristic
154C42×C59C2×D411yescharacteristic
155C22⋊Dic59—41no—
156(C2×D59)⋊C4—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—
168C4⋊Dic59—41no—
169C4⋊Dic59—41no—
170Dic59⋊C4—41no—
171Dic236—21no—
172Dic236—21no—
173D472—21no—
174D472—21no—
175D472—21no—
176D472—21no—
177D472—21no—
178D472—42no—
179D472#4112yescomplemented
180#11745660—42no—
181#11745660—81no—
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—82no—
192C22×D59—82no—
193C22×D59—161no—
194D4×C59—22no—
195C2×C4×C59—21no—
196C2×C4×C59—21no—
197C2×C4×C59—21no—
198C2×C4×C59—21no—
199C2×C4×C59C2×D811yescharacteristic
200C2×C4×C59C2×D811yescharacteristic
201C2×C4×C59C2×D811yescharacteristic
202C2×C4×C59C2×D811yescharacteristic
203C2×C4×C59C22×D411yescharacteristic
204C22⋊D59—82no—
205C22⋊D59—82no—
206C22⋊D59—82no—
207C22⋊D59—161no—
208C2×Dic59—41no—
209C2×Dic59—41no—
210C2×Dic59—41no—
211C2×Dic59—41no—
212C2×Dic59—41no—
213C2×Dic59—82no—
214C2×Dic59—82no—
215D236—22no—
216D236—41no—
217D236—41no—
218D236—41no—
219D236—41no—
220D236—41no—
221D236—82no—
222C4×D59—82no—
223C4×D59—82no—
224C4×D59—161no—
225Dic118—41no—
226Dic118—41no—
227Dic118—82no—
228C8×C59—21no—
229C8×C59—21no—
230C8×C59—21no—
231C8×C59C2×#4111yescharacteristic
232C8×C59H(5,𝔽2)11yescharacteristic
233#33873—591nomaximal, Sylow
234C22×C59—21no—
235C22×C59—21no—
236C22×C59—21no—
237C22×C59—21no—
238C22×C59—21no—
239C22×C59#50912yes—
240C22×C59C22×D811yescharacteristic
241C2×D59—42no—
242C2×D59—81no—
243C2×D59—81no—
244C2×D59—81no—
245C2×D59—81no—
246C2×D59—81no—
247C2×D59—81no—
248C2×D59—81no—
249C2×D59—81no—
250C2×D59—164no—
251C4×C59—21no—
252C4×C59—21no—
253C4×C59—21no—
254C4×C59—21no—
255C4×C59#54611yescharacteristic
256C4×C59#57711yescharacteristic
257Dic59—81no—
258Dic59—81no—
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#2186—591no—
272#2184—591no—
273#2177—591no—
274#2168D5911yescharacteristic, complemented
275#2139—591no—
276C2×#357—591no—
277C2×C59—22no—
278C2×C59—41no—
279C2×C59#329411yescharacteristic
280C2×C59C2×#50911yescharacteristic
281C2×C59#341611yescharacteristic
282D59—82no—
283D59—161no—
284D59—161no—
285C2×#135—591no—
286C22×D8—591no—
287#547—1181no—
288#546—1181no—
289C2×SD32—591no—
290C2×D16—591no—
291C2×D16—591no—
292C2×M32C2×D5911yescharacteristic, complemented
293C22×C16C2×D5911yescharacteristic
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#466—591no—
303#463—1181no—
304#450—1181no—
305#449—1181no—
306C4×D8—591no—
307C4×D8—591no—
308C8×D4C2×D5911yescharacteristic
309C2×#107—591no—
310C2×#107—591no—
311#367—591no—
312#367—591no—
313#367—591no—
314C4⋊C16C2×D5911yescharacteristic, complemented
315#358—591no—
316#358—591no—
317#358—591no—
318#357—591no—
319#357—591no—
320#357—591no—
321#357—591no—
322#357—591no—
323#357—591no—
324#357—591no—
325C22⋊C16C2×D5911yescharacteristic
326C22⋊C16C2×D5911yescharacteristic, complemented
327#346C2×D5911yescharacteristic, complemented
328C59#3387311yescharacteristic, complemented, Sylow
329C2×#41—1181no—
330C2×#41—2361no—
331C22×D4—1181no—
332#135—1182no—
333#135—2361no—
334C2×Q16—591no—
335C2×SD16—1181no—
336C2×SD16—1181no—
337C2×D8—591no—
338C2×D8—591no—
339C2×D8—591no—
340C2×D8—1182no—
341C2×D8—1181no—
342C2×D8—2361no—
343C22×C8C22×D5911yescharacteristic
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—1181no—
353#121—1181no—
354#121—1181no—
355#121—2361no—
356#120—2361no—
357C4×D4—1181no—
358C4×D4—1181no—
359C4×D4D23611yescharacteristic
360C2×C4⋊C4—1181no—
361C2×C4⋊C4—1181no—
362C2×C22⋊C4—2361no—
363SD32—2361no—
364D16—1182no—
365D16—2361no—
366M32—41no—
367C2×C16—41no—
368C2×C16C22×D5911yescharacteristic
369C2×C16C22×D5911yescharacteristic
370C2×C16C22×D5911yescharacteristic
371C2×C16C22×D5911yescharacteristic
372#107—591no—
373#107—591no—
374#107—591no—
375#107—591no—
376#107—1181no—
377#107—1181no—
378#107—1181no—
379C4⋊C8D23611yescharacteristic
380#103—1181no—
381#102—1181no—
382#102—1181no—
383#102—1181no—
384#102—1181no—
385#102—1181no—
386C22⋊C8D23611yescharacteristic
387C22⋊C8D23611yescharacteristic
388C4×C8C22×D5911yescharacteristic
389C24—2361no—
390#41—4721no—
391#41—4724no—
392#41—2362no—
393C2×Q8—1181no—
394C2×Q8—2361no—
395C2×D4—1181no—
396C2×D4—1181no—
397C2×D4—1181no—
398C2×D4—4721no—
399C2×D4—2362no—
400C2×D4—2361no—
401C2×D4—2361no—
402C2×D4—2361no—
403C2×D4—2361no—
404C2×D4—2361no—
405C2×D4D47211yescharacteristic
406C22×C4—2361no—
407C22×C4—2361no—
408C22×C4—2361no—
409C22×C4—2361no—
410C22×C4—2361no—
411C22×C4C2×D23611yescharacteristic
412C22×C4C2×D23611yescharacteristic
413Q16—1181no—
414Q16—1181no—
415SD16—4721no—
416SD16—2362no—
417D8—592no—
418D8—1181no—
419D8—1181no—
420D8—1181no—
421D8—1181no—
422D8—1181no—
423D8—2362no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8C2×D23611yescharacteristic
429C2×C8C2×D23611yescharacteristic
430C2×C8C2×D23611yescharacteristic
431C2×C8C23×D5911yescharacteristic
432C4⋊C4—1181no—
433C4⋊C4—1181no—
434C4⋊C4—1181no—
435C4⋊C4—1181no—
436C4⋊C4—2361no—
437C4⋊C4—2361no—
438C4⋊C4—2361no—
439C4⋊C4—2361no—
440C4⋊C4D47211yescharacteristic
441C22⋊C4—2361no—
442C22⋊C4—2361no—
443C22⋊C4—2361no—
444C22⋊C4—2361no—
445C22⋊C4—2361no—
446C22⋊C4—2361no—
447C22⋊C4—2361no—
448C22⋊C4—2361no—
449C22⋊C4D47211yescharacteristic
450C22⋊C4D47211yescharacteristic
451C42C2×D23611yescharacteristic
452C16—21no—
453C16—21no—
454C16—21no—
455C16—21no—
456C23—9441no—
457C23—4722no—
458C23—4722no—
459C23—2361no—
460C23—2361no—
461C23—2361no—
462C23C2×D47211yescharacteristic
463C23C2×D47211yescharacteristic
464Q8—4722no—
465Q8—2361no—
466Q8—2361no—
467D4—22no—
468D4—1182no—
469D4—9441no—
470D4—4722no—
471D4—4722no—
472D4—4722no—
473D4—4722no—
474D4—2361no—
475D4—2361no—
476D4—2361no—
477D4—2361no—
478D4—2361no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—9441no—
484C2×C4—4722no—
485C2×C4—4722no—
486C2×C4—4722no—
487C2×C4—4722no—
488C2×C4—2361no—
489C2×C4—2361no—
490C2×C4—2361no—
491C2×C4—2361no—
492C2×C4—2361no—
493C2×C4C2×D47211yescharacteristic
494C2×C4C2×D47211yescharacteristic
495C2×C4C2×D47211yescharacteristic
496C2×C4C2×D47211yescharacteristic
497C2×C4C22×D23611yescharacteristic
498C8—21no—
499C8—21no—
500C8—21no—
501C8#41×D5911yescharacteristic
502C8#1427377211yescharacteristic
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—9444no—
509C22—4721no—
510C22—4721no—
511C22—4721no—
512C22—4721no—
513C22—4721no—
514C22—4721no—
515C22—4721no—
516C22—4721no—
517C22—2362no—
518C22#1683784012yes—
519C22C22×D47211yescharacteristic
520C4—21no—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—4721no—
525C4—4721no—
526C4—4721no—
527C4—4721no—
528C4—4721no—
529C4#1683804711yescharacteristic
530C4#1683821411yescharacteristic
531C2—22no—
532C2—41no—
533C2—9441no—
534C2—9441no—
535C2—4722no—
536C2#2143591011yescharacteristic
537C2C2×#1683784011yescharacteristic
538C2#2143724411yescharacteristic
subgroupG/H
centerC22C22×D472
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2168C2
socleC22×C59C22×D8
hypercenter#2168D59
hypocenterC59#33873
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. #21436444C2semidirect
2. #21435783C2semidirect
3. #21426333C2semidirect
4. #21426294C2semidirect
5. #21426282C2semidirect
6. #21426244C2semidirect
7. #21426199C2semidirect
8. #21426192C2semidirect
9. #21426152C2semidirect
10. #21426021C2semidirect
11. #21426010C2semidirect
12. #21425973C2semidirect
13. #21425809C2semidirect
14. C2×#16837204C2semidirect
15. C59×#2168C2semidirect
20. C2×#14273672C22semidirect
23. #16837807C22semidirect
31. #16837638C22semidirect
36. C4×D472C22semidirect
38. C2×#14273661C22semidirect
39. #16837230C22semidirect
40. #16837230C22semidirect
41. #16837230C22semidirect
42. #16837207C22semidirect
43. #16837207C22semidirect
44. #16837207C22semidirect
51. #16837204C22semidirect
52. C2×M32×C59C22semidirect
55. C59×C4⋊C16C22semidirect
57. C59×C22⋊C16C22semidirect
58. C59×#346C22semidirect
179. D472#41semidirect
274. #2168D59semidirect
292. C2×M32C2×D59semidirect
314. C4⋊C16C2×D59semidirect
326. C22⋊C16C2×D59semidirect
327. #346C2×D59semidirect
328. C59#33873semidirect

Maximal subgroups

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

subgroupindex
#214364442
#214357832
#214263332
#214262942
#214262822
#214262442
#214261992
#214261922
#214261522
#214260212
#214260102
#214259732
#214258092
C2×#168372042
C59×#21682
#3387359