Finite Group Databasebeta

← #28745586

Subgroups — #28745586

Order 15104.

546 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21436465C211yesmaximal, characteristic, complemented
2#21426327C211yesmaximal, characteristic, complemented
3#21426301C211yesmaximal, characteristic, complemented
4#21426274C211yesmaximal, characteristic, complemented
5#21426260C211yesmaximal, characteristic, complemented
6#21426190C211yesmaximal, characteristic, complemented
7#21426185C211yesmaximal, characteristic, complemented
8#21426155C211yesmaximal, characteristic, complemented
9#21426036C211yesmaximal, characteristic, complemented
10#21426030C211yesmaximal, characteristic, complemented
11C4×#14273673C211yesmaximal, characteristic, complemented
12D4×C59⋊C16C211yesmaximal, characteristic, complemented
13#21425803C211yesmaximal, characteristic, complemented
14C2×#16837206C211yesmaximal, characteristic, complemented
15C59×#3285C211yesmaximal, characteristic
16#16838092—21no—
17C2×#14273673C2211yescharacteristic
18C2×#14273673C2211yescharacteristic
19C2×#14273671C2211yescharacteristic
20C22×C59⋊C16C2211yescharacteristic
21C22×C59⋊C16C2211yescharacteristic
22#16837804C2211yescharacteristic
23#16837744—21no—
24#16837733—21no—
25#16837710—21no—
26#16837643C2211yescharacteristic
27#16837643C2211yescharacteristic
28C4×Dic236C2211yescharacteristic
29#16837440C2211yescharacteristic
30C2×#14273661C2211yescharacteristic
31#16837229C2211yescharacteristic
32#16837227C2211yescharacteristic
33#16837227C2211yescharacteristic
34C4⋊(C59⋊C16)C2211yescharacteristic
35#16837206C2211yescharacteristic
36#16837206C2211yescharacteristic
37#16837206C2211yescharacteristic
38#16837206C2211yescharacteristic
39#16837206C2211yescharacteristic
40#16837206C2211yescharacteristic
41#16837205C2211yescharacteristic
42#16837203C2211yescharacteristic
43#16837203C2211yescharacteristic
44#16837203C2211yescharacteristic
45C22⋊(C59⋊C16)C2211yescharacteristic
46C22⋊(C59⋊C16)C2211yescharacteristic
47C4×C59⋊C16C2211yescharacteristic
48C2×#135×C59C2211yescharacteristic
49C22×Q16×C59C2211yescharacteristic
50C59×#548—21no—
51C59×#495C2211yescharacteristic
52C59×#467C2211yescharacteristic
53C59×#466C2211yescharacteristic
54C59×#464—21no—
55C59×#452—21no—
56C59×#448—21no—
57C4×Q16×C59C2211yescharacteristic
58C8×D4×C59C2211yescharacteristic
59C2×Dic236C2311yescharacteristic
60#14273720—41no—
61#14273719—21no—
62#14273717—41no—
63#14273708—41no—
64#14273706—21no—
65#14273706—21no—
66C4×Dic118—21no—
67#14273686—21no—
68#14273683—41no—
69C2×C4⋊Dic59—21no—
70#14273673—22no—
71#14273673—41no—
72#14273671—41no—
73C2×C59⋊C16—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×Q8×C59—21no—
90#135×C59—22no—
91#135×C59—41no—
92C2×Q16×C59—22no—
93C2×Q16×C59—21no—
94C2×Q16×C59—41no—
95C2×Q16×C59C2311yescharacteristic
96C2×Q16×C59C2311yescharacteristic
97C2×SD16×C59—21no—
98C2×SD16×C59—21no—
99C2×D8×C59C2311yescharacteristic
100C22×C8×C59C2311yescharacteristic
101C22×C8×C59C2311yescharacteristic
102#124×C59—21no—
103#124×C59—41no—
104C22⋊Q8×C59—21no—
105C22⋊Q8×C59—41no—
106#121×C59—21no—
107#121×C59—41no—
108C4×Q8×C59—21no—
109C4×D4×C59—41no—
110C4×D4×C59D411yescharacteristic
111#107×C59C2311yescharacteristic
112C4⋊C8×C59D411yescharacteristic
113#103×C59—21no—
114#103×C59—21no—
115#102×C59—21no—
116C22⋊C8×C59D411yescharacteristic
117C22⋊C8×C59D411yescharacteristic
118C4×C8×C59C2311yescharacteristic
119#41×C59—42no—
120#41×C59—81no—
121C2×Q8×C59—21no—
122C2×Q8×C59—21no—
123C2×Q8×C59—42no—
124C2×Q8×C59—41no—
125C2×Q8×C59—81no—
126C2×D4×C59—21no—
127C2×D4×C59—41no—
128C2×D4×C59—41no—
129C2×D4×C59D811yescharacteristic
130C22×C4×C59—41no—
131C22×C4×C59—41no—
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2×D411yescharacteristic
134C22×Dic59—41no—
135C2×Dic118—21no—
136Q16×C59—21no—
137Q16×C59—21no—
138Q16×C59—21no—
139Q16×C59—42no—
140Q16×C59#4112yes—
141SD16×C59—42no—
142SD16×C59—81no—
143D8×C59—21no—
144D8×C59—21no—
145C2×C8×C59—21no—
146C2×C8×C59—21no—
147C2×C8×C59—21no—
148C2×C8×C59—21no—
149C2×C8×C59C2×D411yescharacteristic
150C2×C8×C59C2×D411yescharacteristic
151C2×C8×C59C2×D411yescharacteristic
152C2×C8×C59C2411yescharacteristic
153C4⋊C4×C59—21no—
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—
166C22⋊Dic59—41no—
167C22⋊Dic59—41no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—41no—
172Dic59⋊C4—41no—
173Dic59⋊C4—41no—
174Dic59⋊C4—41no—
175Dic59⋊C4—41no—
176Dic59⋊C4—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×C59C2×D811yescharacteristic
187C23×C59C2×D811yescharacteristic
188Q8×C59—22no—
189Q8×C59—41no—
190Q8×C59—41no—
191Q8×C59—41no—
192Q8×C59—82no—
193D4×C59—22no—
194D4×C59—41no—
195D4×C59—41no—
196D4×C59—82no—
197D4×C59—82no—
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—82no—
206C2×C4×C59—82no—
207C2×C4×C59C2×D811yescharacteristic
208C2×C4×C59C2×D811yescharacteristic
209C2×C4×C59C2×D811yescharacteristic
210C2×C4×C59C2×D811yescharacteristic
211C2×C4×C59C22×D411yescharacteristic
212C2×Dic59—41no—
213C2×Dic59—41no—
214C2×Dic59—41no—
215C2×Dic59—41no—
216C2×Dic59—82no—
217Dic118—41no—
218Dic118—41no—
219C8×C59—21no—
220C8×C59—21no—
221C8×C59—21no—
222C8×C59C2×#4111yescharacteristic
223C8×C59H(5,𝔽2)11yescharacteristic
224#33867—591nomaximal, Sylow
225C22×C59—21no—
226C22×C59—21no—
227C22×C59—21no—
228C22×C59—21no—
229C22×C59—21no—
230C22×C59—81no—
231C22×C59—81no—
232C22×C59—81no—
233C22×C59#50812yes—
234C22×C59C22×D811yescharacteristic
235C4×C59—21no—
236C4×C59—21no—
237C4×C59—21no—
238C4×C59—21no—
239C4×C59—42no—
240C4×C59—81no—
241C4×C59—81no—
242C4×C59#54611yescharacteristic
243C4×C59#57711yescharacteristic
244Dic59—81no—
245Dic59—81no—
246Dic59—81no—
247Dic59—81no—
248#3336—591no—
249#3285D5911yescharacteristic
250#2242—591no—
251#2237—591no—
252#2232—591no—
253#2229—591no—
254#2217—591no—
255#2216—591no—
256#2211—591no—
257#2189—591no—
258#2188—591no—
259C4×Q32—591no—
260D4×C16—591no—
261#2138—591no—
262C2×#358—591no—
263C2×C59—22no—
264C2×C59—41no—
265C2×C59—161no—
266C2×C59#329411yescharacteristic
267C2×C59C2×#50811yescharacteristic
268C2×C59#341711yescharacteristic
269C2×#135C2×D5911yescharacteristic
270C22×Q16C2×D5911yescharacteristic
271#552—1181no—
272#548—21no—
273C2×Q32—591no—
274C2×Q32—591no—
275C2×SD32—591no—
276C22×C16—591no—
277C22×C16—591no—
278#499—591no—
279#495C2×D5911yescharacteristic
280#485—1181no—
281#483—1181no—
282#479—1181no—
283#467—591no—
284#467—591no—
285#467C2×D5911yescharacteristic
286#466C2×D5911yescharacteristic
287#464—21no—
288#452—21no—
289#448—21no—
290C4×Q16—591no—
291C4×Q16C2×D5911yescharacteristic
292C8×D4C2×D5911yescharacteristic
293#427—591no—
294C2×#107—591no—
295#367—591no—
296#366—591no—
297#366—591no—
298C4⋊C16—591no—
299#358—591no—
300#358—591no—
301#358—591no—
302#358—591no—
303#358—591no—
304#358—591no—
305#358—591no—
306#357—591no—
307#357—591no—
308#357—591no—
309C22⋊C16—591no—
310C22⋊C16—591no—
311C4×C16—591no—
312C59#3386711yescharacteristic, complemented, Sylow
313C2×#41—21no—
314C22×Q8—21no—
315#135—22no—
316#135—41no—
317C2×Q16—22no—
318C2×Q16—21no—
319C2×Q16—41no—
320C2×Q16—591no—
321C2×Q16C22×D5911yescharacteristic
322C2×Q16C22×D5911yescharacteristic
323C2×SD16—21no—
324C2×SD16—21no—
325C2×D8C22×D5911yescharacteristic
326C22×C8C22×D5911yescharacteristic
327C22×C8C22×D5911yescharacteristic
328#126—2361no—
329#125—1181no—
330#125—2361no—
331#124—21no—
332#124—41no—
333#123—2361no—
334C22⋊Q8—21no—
335C22⋊Q8—41no—
336C22⋊Q8—1181no—
337C22⋊Q8—1181no—
338#121—21no—
339#121—41no—
340C4×Q8—21no—
341C4×Q8—1181no—
342C4×D4—41no—
343C4×D4C22⋊D5911yescharacteristic
344#117—1181no—
345#117—2361no—
346C2×C4⋊C4—1181no—
347Q32—1182no—
348Q32—2361no—
349SD32—2361no—
350C2×C16—591no—
351C2×C16—591no—
352C2×C16—591no—
353C2×C16—591no—
354C2×C16—2361no—
355C2×C16—2361no—
356#107—591no—
357#107—591no—
358#107—591no—
359#107—1181no—
360#107C22×D5911yescharacteristic
361#106—1181no—
362#106—1181no—
363C4⋊C8C22⋊D5911yescharacteristic
364#103—21no—
365#103—21no—
366#103—1181no—
367#103—1181no—
368#103—1181no—
369#102—21no—
370C22⋊C8C22⋊D5911yescharacteristic
371C22⋊C8C22⋊D5911yescharacteristic
372C4×C8C22×D5911yescharacteristic
373#41—42no—
374#41—81no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—42no—
378C2×Q8—41no—
379C2×Q8—81no—
380C2×Q8—1181no—
381C2×D4—21no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4#1174566911yescharacteristic
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—2361no—
388C22×C4C2×C22⋊D5911yescharacteristic
389C22×C4C2×C22⋊D5911yescharacteristic
390Q16—21no—
391Q16—21no—
392Q16—21no—
393Q16—42no—
394Q16—1181no—
395Q16—1181no—
396Q16#1174568712yes—
397SD16—42no—
398SD16—81no—
399D8—21no—
400D8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8C2×C22⋊D5911yescharacteristic
406C2×C8C2×C22⋊D5911yescharacteristic
407C2×C8C2×C22⋊D5911yescharacteristic
408C2×C8C23×D5911yescharacteristic
409C4⋊C4—21no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—1181no—
413C4⋊C4—1181no—
414C4⋊C4—1181no—
415C4⋊C4—2361no—
416C4⋊C4—2361no—
417C4⋊C4—2361no—
418C4⋊C4—2361no—
419C4⋊C4—2361no—
420C4⋊C4—2361no—
421C4⋊C4#1174566911yescharacteristic
422C22⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C22⋊C4—2361no—
427C22⋊C4—2361no—
428C22⋊C4—2361no—
429C22⋊C4#1174566911yescharacteristic
430C22⋊C4#1174566911yescharacteristic
431C42—41no—
432C42—2361no—
433C42—2361no—
434C42C2×C22⋊D5911yescharacteristic
435C16—1181no—
436C16—1181no—
437C16—1181no—
438C16—1181no—
439C23—41no—
440C23C2×#1174566911yescharacteristic
441C23C2×#1174566911yescharacteristic
442Q8—22no—
443Q8—41no—
444Q8—41no—
445Q8—41no—
446Q8—82no—
447Q8—2361no—
448Q8—2361no—
449D4—22no—
450D4—41no—
451D4—41no—
452D4—82no—
453D4—82no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—4722no—
464C2×C4—2361no—
465C2×C4—2361no—
466C2×C4—2361no—
467C2×C4—2361no—
468C2×C4C2×#1174566911yescharacteristic
469C2×C4C2×#1174566911yescharacteristic
470C2×C4C2×#1174566911yescharacteristic
471C2×C4C2×#1174566911yescharacteristic
472C2×C4C22×C22⋊D5911yescharacteristic
473C8—21no—
474C8—21no—
475C8—21no—
476C8C2×#1174568711yescharacteristic
477C8#1427377211yescharacteristic
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22#1683783212yes—
487C22C22×#1174566911yescharacteristic
488C4—21no—
489C4—21no—
490C4—21no—
491C4—21no—
492C4—42no—
493C4—81no—
494C4—81no—
495C4—4721no—
496C4—4721no—
497C4—4721no—
498C4—4721no—
499C4#1683804611yescharacteristic
500C4#1683821311yescharacteristic
501C2—22no—
502C2—41no—
503C2—161no—
504C2#2143590911yescharacteristic
505C2C2×#1683783211yescharacteristic
506C2#2143724711yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
Fitting, prime radicalC59×#3285C2
socleC22×C59C22×D8
hypercenter#3285D59
hypocenterC59#33867
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3285

Complements

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

NH ≅ G/Ntype
1. #21436465C2semidirect
2. #21426327C2semidirect
3. #21426301C2semidirect
4. #21426274C2semidirect
5. #21426260C2semidirect
6. #21426190C2semidirect
7. #21426185C2semidirect
8. #21426155C2semidirect
9. #21426036C2semidirect
10. #21426030C2semidirect
11. C4×#14273673C2semidirect
12. D4×C59⋊C16C2semidirect
13. #21425803C2semidirect
14. C2×#16837206C2semidirect
312. C59#33867semidirect

Maximal subgroups

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

subgroupindex
#214364652
#214263272
#214263012
#214262742
#214262602
#214261902
#214261852
#214261552
#214260362
#214260302
C4×#142736732
D4×C59⋊C162
#214258032
C2×#168372062
C59×#32852
#3386759