Finite Group Databasebeta

← #28745735

Subgroups — #28745735

Order 15104.

562 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21436527C211yesmaximal, characteristic, complemented
2#21426332C211yesmaximal, characteristic, complemented
3#21426301C211yesmaximal, characteristic, complemented
4#21426284C211yesmaximal, characteristic, complemented
5#21426243C211yesmaximal, characteristic, complemented
6#21426202C211yesmaximal, characteristic, complemented
7#21426190C211yesmaximal, characteristic, complemented
8#21426155C211yesmaximal, characteristic, complemented
9#21426036C211yesmaximal, characteristic, complemented
10#21426025C211yesmaximal, characteristic, complemented
11#21425960C211yesmaximal, characteristic, complemented
12#21425922C211yesmaximal, characteristic, complemented
13#21425812C211yesmaximal, characteristic, complemented
14#21425803C211yesmaximal, characteristic, complemented
15C59×#3280C211yesmaximal, characteristic
16#16838122—21no—
17C2×#14273673C2211yescharacteristic, complemented
18C2×#14273671C2211yescharacteristic
19C2×#14273671C2211yescharacteristic
20C2×#14273667C2211yescharacteristic
21C22×C59⋊C16C2211yescharacteristic, complemented
22#16837807C2211yescharacteristic, complemented
23#16837744—21no—
24#16837744—21no—
25#16837688—21no—
26#16837643C2211yescharacteristic
27#16837643C2211yescharacteristic, complemented
28C4×Dic236C2211yescharacteristic, complemented
29#16837440C2211yescharacteristic
30#16837440C2211yescharacteristic, complemented
31#16837229C2211yescharacteristic, complemented
32#16837229C2211yescharacteristic, complemented
33#16837227C2211yescharacteristic, complemented
34C4⋊(C59⋊C16)C2211yescharacteristic, complemented
35#16837206C2211yescharacteristic, complemented
36#16837206C2211yescharacteristic, complemented
37#16837206C2211yescharacteristic, complemented
38#16837205C2211yescharacteristic, complemented
39#16837203C2211yescharacteristic
40#16837203C2211yescharacteristic
41#16837203C2211yescharacteristic
42#16837203C2211yescharacteristic
43#16837203C2211yescharacteristic
44#16837203C2211yescharacteristic
45C22⋊(C59⋊C16)C2211yescharacteristic
46C22⋊(C59⋊C16)C2211yescharacteristic, complemented
47#16837178C2211yescharacteristic, complemented
48C2×#135×C59C2211yescharacteristic
49C22×D8×C59C2211yescharacteristic
50C59×#546—21no—
51C59×#495C2211yescharacteristic
52C59×#467C2211yescharacteristic
53C59×#466C2211yescharacteristic
54C59×#463—21no—
55C59×#450—21no—
56C59×#449—21no—
57C4×D8×C59C2211yescharacteristic
58C8×D4×C59C2211yescharacteristic
59C2×Dic236C2311yescharacteristic
60#14273727—21no—
61#14273720—41no—
62#14273720—41no—
63#14273714—41no—
64#14273706—21no—
65#14273706—21no—
66C4×Dic118—21no—
67#14273692—41no—
68#14273686—21no—
69#14273686—21no—
70#14273673—41no—
71#14273671—22no—
72#14273671—41no—
73#14273667—41no—
74C2×C59⋊C16—41no—
75C2×C59⋊C16C2311yescharacteristic
76C2×C59⋊C16C2311yescharacteristic
77C2×C59⋊C16C2311yescharacteristic
78C2×C59⋊C16C2311yescharacteristic
79#14273661—21no—
80#14273661C2311yescharacteristic
81#14273661C2311yescharacteristic
82#14273661C2311yescharacteristic
83#14273659—21no—
84#14273659—21no—
85#14273652—21no—
86#14273652—21no—
87#14273652—21no—
88C2×#41×C59—21no—
89C22×D4×C59—21no—
90#135×C59—22no—
91#135×C59—41no—
92C2×Q16×C59C2311yescharacteristic
93C2×SD16×C59—21no—
94C2×SD16×C59—21no—
95C2×D8×C59—22no—
96C2×D8×C59—21no—
97C2×D8×C59—41no—
98C2×D8×C59C2311yescharacteristic
99C2×D8×C59C2311yescharacteristic
100C22×C8×C59C2311yescharacteristic
101C22×C8×C59C2311yescharacteristic
102#124×C59—21no—
103#123×C59—41no—
104C22⋊Q8×C59—21no—
105#121×C59—21no—
106#121×C59—41no—
107#120×C59—41no—
108C4×D4×C59—21no—
109C4×D4×C59D411yescharacteristic
110C2×C22⋊C4×C59—41no—
111#107×C59C2311yescharacteristic
112C4⋊C8×C59D411yescharacteristic
113#103×C59—21no—
114#102×C59—21no—
115#102×C59—21no—
116C22⋊C8×C59D411yescharacteristic
117C22⋊C8×C59D411yescharacteristic
118C4×C8×C59C2311yescharacteristic
119C24×C59—41no—
120#41×C59—42no—
121#41×C59—81no—
122C2×Q8×C59—21no—
123C2×D4×C59—21no—
124C2×D4×C59—21no—
125C2×D4×C59—42no—
126C2×D4×C59—41no—
127C2×D4×C59—41no—
128C2×D4×C59—41no—
129C2×D4×C59—81no—
130C2×D4×C59D811yescharacteristic
131C22×C4×C59—41no—
132C22×C4×C59—41no—
133C22×C4×C59C2×D411yescharacteristic
134C22×C4×C59C2×D411yescharacteristic
135C2×Dic118—21no—
136C2×Dic118—41no—
137Q16×C59—21no—
138Q16×C59—21no—
139SD16×C59—42no—
140SD16×C59—81no—
141D8×C59—21no—
142D8×C59—21no—
143D8×C59—21no—
144D8×C59—42no—
145D8×C59#4112yes—
146C2×C8×C59—21no—
147C2×C8×C59—21no—
148C2×C8×C59—21no—
149C2×C8×C59—21no—
150C2×C8×C59C2×D411yescharacteristic
151C2×C8×C59C2×D411yescharacteristic
152C2×C8×C59C2×D411yescharacteristic
153C2×C8×C59C2411yescharacteristic
154C4⋊C4×C59—21no—
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×C59—41no—
162C22⋊C4×C59D811yescharacteristic
163C22⋊C4×C59D811yescharacteristic
164C42×C59C2×D411yescharacteristic
165C22⋊Dic59—41no—
166C22⋊Dic59—41no—
167C22⋊Dic59—41no—
168C22⋊Dic59—41no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—41no—
173Dic59⋊C4—41no—
174Dic59⋊C4—41no—
175Dic59⋊C4—41no—
176C4×Dic59—41no—
177C4×Dic59—41no—
178C4×Dic59—41no—
179Dic236—21no—
180Dic236—21no—
181C59⋊C16—21no—
182C59⋊C16—21no—
183C59⋊C16—21no—
184C59⋊C16—21no—
185C23×C59—41no—
186C23×C59—41no—
187C23×C59—82no—
188C23×C59—82no—
189C23×C59—161no—
190C23×C59C2×D811yescharacteristic
191C23×C59C2×D811yescharacteristic
192Q8×C59—41no—
193Q8×C59—41no—
194D4×C59—22no—
195D4×C59—41no—
196D4×C59—41no—
197D4×C59—41no—
198D4×C59—41no—
199D4×C59—82no—
200D4×C59—82no—
201D4×C59—161no—
202C2×C4×C59—21no—
203C2×C4×C59—21no—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—41no—
207C2×C4×C59—41no—
208C2×C4×C59—82no—
209C2×C4×C59—161no—
210C2×C4×C59C2×D811yescharacteristic
211C2×C4×C59C2×D811yescharacteristic
212C2×C4×C59C2×D811yescharacteristic
213C2×C4×C59C2×D811yescharacteristic
214C2×C4×C59C22×D411yescharacteristic
215C2×Dic59—41no—
216C2×Dic59—41no—
217C2×Dic59—41no—
218C2×Dic59—41no—
219Dic118—41no—
220Dic118—41no—
221Dic118—82no—
222C8×C59—21no—
223C8×C59—21no—
224C8×C59—21no—
225C8×C59C2×#4111yescharacteristic
226C8×C59H(5,𝔽2)11yescharacteristic
227#33878—591nomaximal, Sylow
228C22×C59—21no—
229C22×C59—21no—
230C22×C59—21no—
231C22×C59—21no—
232C22×C59—21no—
233C22×C59—21no—
234C22×C59—42no—
235C22×C59—81no—
236C22×C59—81no—
237C22×C59—81no—
238C22×C59—81no—
239C22×C59—81no—
240C22×C59—164no—
241C22×C59C22×D811yescharacteristic
242C4×C59—21no—
243C4×C59—21no—
244C4×C59—21no—
245C4×C59—21no—
246C4×C59—81no—
247C4×C59—81no—
248C4×C59#54611yescharacteristic
249C4×C59#57711yescharacteristic
250Dic59—81no—
251Dic59—81no—
252Dic59—81no—
253Dic59—81no—
254#3342—591no—
255#3280D5911yescharacteristic
256#2243—591no—
257#2237—591no—
258#2234—591no—
259#2226—591no—
260#2219—591no—
261#2217—591no—
262#2211—591no—
263#2189—591no—
264#2187—591no—
265#2175—591no—
266#2168—591no—
267#2140—591no—
268#2138—591no—
269C2×C59—41no—
270C2×C59—41no—
271C2×C59—82no—
272C2×C59—161no—
273C2×C59#329411yescharacteristic
274C2×C59#341411yescharacteristic
275C2×C59#341611yescharacteristic
276C2×#135C2×D5911yescharacteristic
277C22×D8C2×D5911yescharacteristic
278#556—1181no—
279#546—21no—
280C2×Q32—591no—
281C2×SD32—591no—
282C2×SD32—591no—
283C2×M32—591no—
284C22×C16—591no—
285#500—591no—
286#495C2×D5911yescharacteristic
287#485—1181no—
288#485—1181no—
289#475—1181no—
290#467—591no—
291#467—591no—
292#467C2×D5911yescharacteristic
293#466C2×D5911yescharacteristic
294#463—21no—
295#450—21no—
296#449—21no—
297C4×Q16—591no—
298C4×D8C2×D5911yescharacteristic
299C8×D4C2×D5911yescharacteristic
300#427—591no—
301#427—591no—
302#367—591no—
303#367—591no—
304#366—591no—
305C4⋊C16—591no—
306#358—591no—
307#358—591no—
308#358—591no—
309#358—591no—
310#357—591no—
311#357—591no—
312#357—591no—
313#357—591no—
314#357—591no—
315#357—591no—
316C22⋊C16—591no—
317C22⋊C16—591no—
318#346—591no—
319C59#3387811yescharacteristic, complemented, Sylow
320C2×#41—21no—
321C22×D4—21no—
322#135—22no—
323#135—41no—
324C2×Q16—591no—
325C2×Q16C22×D5911yescharacteristic
326C2×SD16—21no—
327C2×SD16—21no—
328C2×D8—22no—
329C2×D8—21no—
330C2×D8—41no—
331C2×D8C22×D5911yescharacteristic
332C2×D8C22×D5911yescharacteristic
333C22×C8C22×D5911yescharacteristic
334C22×C8C22×D5911yescharacteristic
335C4⋊Q8—1181no—
336#126—2361no—
337#126—2361no—
338#124—21no—
339#124—2361no—
340#123—41no—
341C22⋊Q8—21no—
342C22⋊Q8—1181no—
343C22⋊Q8—1181no—
344#121—21no—
345#121—41no—
346#120—41no—
347C4×Q8—1181no—
348C4×Q8—2361no—
349C4×D4—21no—
350C4×D4C22⋊D5911yescharacteristic
351#117—1181no—
352#117—1181no—
353C2×C22⋊C4—41no—
354Q32—2361no—
355SD32—1182no—
356SD32—2361no—
357M32—2361no—
358C2×C16—591no—
359C2×C16—591no—
360C2×C16—591no—
361C2×C16—591no—
362C2×C16—2361no—
363#107—591no—
364#107—591no—
365#107—591no—
366#107—1181no—
367#107C22×D5911yescharacteristic
368#106—1181no—
369#106—1181no—
370C4⋊C8C22⋊D5911yescharacteristic
371#103—21no—
372#103—1181no—
373#103—1181no—
374#103—1181no—
375#102—21no—
376#102—21no—
377C22⋊C8C22⋊D5911yescharacteristic
378C22⋊C8C22⋊D5911yescharacteristic
379C4×C8C22×D5911yescharacteristic
380C24—41no—
381#41—42no—
382#41—81no—
383C2×Q8—21no—
384C2×Q8—1181no—
385C2×Q8—2361no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—42no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—81no—
393C2×D4#1174566911yescharacteristic
394C22×C4—41no—
395C22×C4—41no—
396C22×C4C2×C22⋊D5911yescharacteristic
397C22×C4C2×C22⋊D5911yescharacteristic
398Q16—21no—
399Q16—21no—
400Q16—1181no—
401Q16—1181no—
402SD16—42no—
403SD16—81no—
404D8—21no—
405D8—21no—
406D8—21no—
407D8—42no—
408D8#1174568712yes—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8C2×C22⋊D5911yescharacteristic
414C2×C8C2×C22⋊D5911yescharacteristic
415C2×C8C2×C22⋊D5911yescharacteristic
416C2×C8C23×D5911yescharacteristic
417C4⋊C4—21no—
418C4⋊C4—41no—
419C4⋊C4—1181no—
420C4⋊C4—1181no—
421C4⋊C4—1181no—
422C4⋊C4—2361no—
423C4⋊C4—2361no—
424C4⋊C4—2361no—
425C4⋊C4—2361no—
426C4⋊C4#1174566911yescharacteristic
427C22⋊C4—41no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—2361no—
433C22⋊C4—2361no—
434C22⋊C4—2361no—
435C22⋊C4—2361no—
436C22⋊C4#1174566911yescharacteristic
437C22⋊C4#1174566911yescharacteristic
438C42—2361no—
439C42—2361no—
440C42—2361no—
441C42C2×C22⋊D5911yescharacteristic
442C16—1181no—
443C16—1181no—
444C16—1181no—
445C16—1181no—
446C23—41no—
447C23—41no—
448C23—82no—
449C23—82no—
450C23—161no—
451C23C2×#1174566911yescharacteristic
452C23C2×#1174566911yescharacteristic
453Q8—41no—
454Q8—41no—
455Q8—4722no—
456Q8—2361no—
457Q8—2361no—
458D4—22no—
459D4—41no—
460D4—41no—
461D4—41no—
462D4—41no—
463D4—82no—
464D4—82no—
465D4—161no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—82no—
473C2×C4—161no—
474C2×C4—2361no—
475C2×C4—2361no—
476C2×C4—2361no—
477C2×C4—2361no—
478C2×C4C2×#1174566911yescharacteristic
479C2×C4C2×#1174566911yescharacteristic
480C2×C4C2×#1174566911yescharacteristic
481C2×C4C2×#1174566911yescharacteristic
482C2×C4C22×C22⋊D5911yescharacteristic
483C8—21no—
484C8—21no—
485C8—21no—
486C8C2×#1174568711yescharacteristic
487C8#1427377211yescharacteristic
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—42no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22—81no—
499C22—81no—
500C22—164no—
501C22C22×#1174566911yescharacteristic
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—81no—
507C4—81no—
508C4—4721no—
509C4—4721no—
510C4—4721no—
511C4—4721no—
512C4#1683804611yescharacteristic
513C4#1683821311yescharacteristic
514C2—41no—
515C2—41no—
516C2—82no—
517C2—161no—
518C2#2143590911yescharacteristic
519C2#2143723011yescharacteristic
520C2#2143724311yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
Fitting, prime radicalC59×#3280C2
socleC22×C59C22×D8
hypercenter#3280D59
hypocenterC59#33878
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3280

Complements

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

NH ≅ G/Ntype
1. #21436527C2semidirect
2. #21426332C2semidirect
3. #21426301C2semidirect
4. #21426284C2semidirect
5. #21426243C2semidirect
6. #21426202C2semidirect
7. #21426190C2semidirect
8. #21426155C2semidirect
9. #21426036C2semidirect
10. #21426025C2semidirect
11. #21425960C2semidirect
12. #21425922C2semidirect
13. #21425812C2semidirect
14. #21425803C2semidirect
17. C2×#14273673C22semidirect
21. C22×C59⋊C16C22semidirect
22. #16837807C22semidirect
27. #16837643C22semidirect
28. C4×Dic236C22semidirect
30. #16837440C22semidirect
31. #16837229C22semidirect
32. #16837229C22semidirect
33. #16837227C22semidirect
34. C4⋊(C59⋊C16)C22semidirect
35. #16837206C22semidirect
36. #16837206C22semidirect
37. #16837206C22semidirect
38. #16837205C22semidirect
46. C22⋊(C59⋊C16)C22semidirect
47. #16837178C22semidirect
319. C59#33878semidirect

Maximal subgroups

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

subgroupindex
#214365272
#214263322
#214263012
#214262842
#214262432
#214262022
#214261902
#214261552
#214260362
#214260252
#214259602
#214259222
#214258122
#214258032
C59×#32802
#3387859