Finite Group Databasebeta

← #28506278

Subgroups — #28506278

Order 15104.

672 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432557C211yesmaximal, characteristic
2#21425771C211yesmaximal, characteristic, complemented
3#21425750C211yesmaximal, characteristic, complemented
4#21424066C211yesmaximal, characteristic
5#21422225C211yesmaximal, characteristic, complemented
6#21422217C211yesmaximal, characteristic, complemented
7C59×#3009C211yesmaximal, characteristic
8#16837877—21no—
9#16837419—21no—
10#16837407—21no—
11#16837386C2211yescharacteristic
12#16837376—21no—
13#16837376C2211yescharacteristic
14#16837376C2211yescharacteristic
15#16837376C2211yescharacteristic
16C2×#14273642—22no—
17C2×#14273642—22no—
18#16837197—22no—
19#16837192—22no—
20C2×H(5,𝔽2)×C59C2211yescharacteristic
21C2×#137×C59C411yescharacteristic, complemented
22C22×SD16×C59C411yescharacteristic, complemented
23C59×#536C2211yescharacteristic
24C59×#452—21no—
25C59×#452—21no—
26C59×#450—21no—
27C59×#450—21no—
28C59×#418C411yescharacteristic, complemented
29C59×#416C411yescharacteristic, complemented
30C59×#408C2211yescharacteristic
31C2×#11745664—21no—
32D4×Dic59—42no—
33D4×Dic59—42no—
34#14273686—21no—
35#14273686—21no—
36#14273686—21no—
37C2×C22⋊Dic59—24no—
38#14273654—42no—
39#14273648—42no—
40#14273646—21no—
41#14273644—21no—
42#14273642—24no—
43#14273642—21no—
44#14273642—21no—
45#14273642—21no—
46#14273642—21no—
47#14273642D412yes—
48#14273642D412yes—
49H(5,𝔽2)×C59—42no—
50H(5,𝔽2)×C59—42no—
51C2×#41×C59—21no—
52C2×#41×C59C2×C411yescharacteristic
53C2×#41×C59D411yescharacteristic
54C2×#41×C59D411yescharacteristic
55C2×#41×C59C2311yescharacteristic
56C22×Q8×C59C2×C411yescharacteristic, complemented
57C22×D4×C59—21no—
58C22×D4×C59—21no—
59C22×D4×C59—21no—
60C22×D4×C59C2×C411yescharacteristic
61C22×D4×C59D411yescharacteristic
62C22×D4×C59D411yescharacteristic
63#137×C59—42no—
64C2×Q16×C59—21no—
65C2×SD16×C59—21no—
66C2×SD16×C59—21no—
67C2×SD16×C59—21no—
68C2×SD16×C59—42no—
69C2×M16×C59C2×C411yescharacteristic, complemented
70C22×C8×C59C2×C411yescharacteristic, complemented
71C4⋊Q8×C59—21no—
72#124×C59—21no—
73#123×C59—21no—
74#123×C59—21no—
75C22⋊Q8×C59—21no—
76C22⋊Q8×C59—21no—
77#117×C59C2×C411yescharacteristic, complemented
78#103×C59—21no—
79#102×C59—21no—
80#102×C59—21no—
81#102×C59—21no—
82C22⋊C8×C59—21no—
83C22⋊C8×C59—21no—
84C24×C59—21no—
85C24×C59—21no—
86C24×C59—21no—
87#41×C59—22no—
88#41×C59—22no—
89#41×C59—41no—
90#41×C59—41no—
91#41×C59—41no—
92#41×C59—41no—
93#41×C59—41no—
94#41×C59—41no—
95#41×C59—44no—
96C2×Q8×C59—21no—
97C2×Q8×C59—21no—
98C2×Q8×C59—41no—
99C2×Q8×C59—41no—
100C2×Q8×C59C22⋊C411yescharacteristic
101C2×Q8×C59#4111yescharacteristic
102C2×D4×C59—22no—
103C2×D4×C59—22no—
104C2×D4×C59—22no—
105C2×D4×C59—22no—
106C2×D4×C59—21no—
107C2×D4×C59—21no—
108C2×D4×C59—21no—
109C2×D4×C59—21no—
110C2×D4×C59—21no—
111C2×D4×C59—21no—
112C2×D4×C59—41no—
113C2×D4×C59—41no—
114C2×D4×C59—44no—
115C2×D4×C59—44no—
116C2×D4×C59—44no—
117C2×D4×C59C22⋊C411yescharacteristic
118C2×D4×C59C22⋊C411yescharacteristic
119C2×D4×C59C22⋊C411yescharacteristic
120C2×D4×C59C2×D411yescharacteristic
121C2×D4×C59C2×D411yescharacteristic
122C2×D4×C59#4111yescharacteristic
123C22×C4×C59—21no—
124C22×C4×C59—21no—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59C22×C411yescharacteristic
128C22×C4×C59C2×D411yescharacteristic
129C22×C4×C59C2×D411yescharacteristic
130C22×Dic59—44no—
131Q16×C59—42no—
132SD16×C59—42no—
133SD16×C59—42no—
134SD16×C59—42no—
135M16×C59—41no—
136C2×C8×C59—21no—
137C2×C8×C59—21no—
138C2×C8×C59—41no—
139C4⋊C4×C59—21no—
140C4⋊C4×C59—41no—
141C4⋊C4×C59—41no—
142C22⋊C4×C59—21no—
143C22⋊C4×C59—41no—
144C22⋊C4×C59—41no—
145C42×C59—21no—
146C22⋊Dic59—22no—
147C22⋊Dic59—22no—
148C22⋊Dic59—44no—
149C22⋊Dic59—41no—
150C4⋊Dic59—41no—
151C4⋊Dic59—41no—
152C4⋊Dic59—41no—
153C4×Dic59—41no—
154C4×Dic59—41no—
155C4×Dic59—41no—
156#11745664—41no—
157#11745664—41no—
158C2×C59⋊C8—41no—
159C23×C59—24no—
160C23×C59—21no—
161C23×C59—21no—
162C23×C59—21no—
163C23×C59—42no—
164C23×C59—42no—
165C23×C59—161no—
166C23×C59—161no—
167C23×C59—161no—
168C23×C59C4×D411yescharacteristic
169C23×C59#12011yescharacteristic
170C23×C59#12111yescharacteristic
171Q8×C59—21no—
172Q8×C59—21no—
173Q8×C59—41no—
174Q8×C59—41no—
175Q8×C59—41no—
176Q8×C59—81no—
177D4×C59—24no—
178D4×C59—21no—
179D4×C59—21no—
180D4×C59—21no—
181D4×C59—21no—
182D4×C59—41no—
183D4×C59—41no—
184D4×C59—41no—
185D4×C59—42no—
186D4×C59—42no—
187D4×C59—42no—
188D4×C59—42no—
189D4×C59—42no—
190D4×C59#9912yes—
191D4×C59#9912yes—
192C2×C4×C59—21no—
193C2×C4×C59—21no—
194C2×C4×C59—21no—
195C2×C4×C59—21no—
196C2×C4×C59—21no—
197C2×C4×C59—21no—
198C2×C4×C59—21no—
199C2×C4×C59—42no—
200C2×C4×C59—42no—
201C2×C4×C59—42no—
202C2×C4×C59—41no—
203C2×C4×C59—41no—
204C2×C4×C59—81no—
205C2×C4×C59C2×C22⋊C411yescharacteristic
206C2×C4×C59C4×D411yescharacteristic
207C2×C4×C59#12111yescharacteristic
208C2×C4×C59#12311yescharacteristic
209C2×Dic59—42no—
210C2×Dic59—42no—
211C2×Dic59—81no—
212C2×Dic59—164no—
213C8×C59—41no—
214C8×C59—41no—
215C59⋊C8—81no—
216#13100—591nomaximal, Sylow
217C22×C59—21no—
218C22×C59—21no—
219C22×C59—21no—
220C22×C59—22no—
221C22×C59—22no—
222C22×C59—41no—
223C22×C59—41no—
224C22×C59—41no—
225C22×C59—41no—
226C22×C59—81no—
227C22×C59—81no—
228C22×C59—81no—
229C22×C59—84no—
230C22×C59—162no—
231C22×C59—162no—
232C22×C59#38611yescharacteristic
233C4×C59—21no—
234C4×C59—21no—
235C4×C59—21no—
236C4×C59—41no—
237C4×C59—81no—
238C4×C59—81no—
239C4×C59C2×#9911yescharacteristic
240C4×C59C2×#9911yescharacteristic
241Dic59—82no—
242Dic59—82no—
243Dic59—161no—
244#3009D5911yescharacteristic
245#2883—591no—
246#2129—591no—
247#2123—591no—
248#1795—591no—
249#1395—591no—
250#1391—591no—
251C2×C59—21no—
252C2×C59—41no—
253C2×C59—41no—
254C2×C59—41no—
255C2×C59—42no—
256C2×C59—42no—
257C2×C59—81no—
258C2×C59#189811yescharacteristic
259C2×H(5,𝔽2)C2×D5911yescharacteristic
260C2×#137Dic5911yescharacteristic, complemented
261C22×SD16Dic5911yescharacteristic, complemented
262#536C2×D5911yescharacteristic
263#518—1181no—
264#452—21no—
265#452—21no—
266#450—21no—
267#450—21no—
268#421—1181no—
269#418—1181no—
270#418Dic5911yescharacteristic, complemented
271#416Dic5911yescharacteristic, complemented
272#413—591no—
273#410—591no—
274#410—591no—
275#410—591no—
276#410—1181no—
277C2×#99—1182no—
278C2×#99—1182no—
279#408C2×D5911yescharacteristic
280#354—1182no—
281#352—1182no—
282C59#1310011yescharacteristic, complemented, Sylow
283H(5,𝔽2)—42no—
284H(5,𝔽2)—42no—
285C2×#41—21no—
286C2×#41C2×Dic5911yescharacteristic
287C2×#41C22⋊D5911yescharacteristic
288C2×#41C22⋊D5911yescharacteristic
289C2×#41C22×D5911yescharacteristic
290C22×Q8C2×Dic5911yescharacteristic, complemented
291C22×D4—21no—
292C22×D4—21no—
293C22×D4—21no—
294C22×D4C2×Dic5911yescharacteristic
295C22×D4C22⋊D5911yescharacteristic
296C22×D4C22⋊D5911yescharacteristic
297#137—42no—
298C2×Q16—21no—
299C2×SD16—21no—
300C2×SD16—21no—
301C2×SD16—21no—
302C2×SD16—42no—
303C2×M16—1181no—
304C2×M16C2×Dic5911yescharacteristic, complemented
305C22×C8C2×Dic5911yescharacteristic, complemented
306C4⋊Q8—21no—
307#124—21no—
308#123—21no—
309#123—21no—
310C22⋊Q8—21no—
311C22⋊Q8—21no—
312C4×D4—2362no—
313C4×D4—2362no—
314#117—1181no—
315#117—1181no—
316#117—1181no—
317#117C2×Dic5911yescharacteristic, complemented
318C2×C22⋊C4—1184no—
319#104—2362no—
320#103—21no—
321#102—21no—
322#102—21no—
323#102—21no—
324#102—2362no—
325#101—1181no—
326#100—1181no—
327#99—592no—
328#99—592no—
329#99—1184no—
330#99—1181no—
331#99—1181no—
332#99—1181no—
333#99—1181no—
334C22⋊C8—21no—
335C22⋊C8—21no—
336C24—21no—
337C24—21no—
338C24—21no—
339#41—22no—
340#41—22no—
341#41—41no—
342#41—41no—
343#41—41no—
344#41—41no—
345#41—41no—
346#41—41no—
347#41—44no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—41no—
351C2×Q8—41no—
352C2×Q8C22⋊Dic5911yescharacteristic
353C2×Q8#1174568711yescharacteristic
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—44no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4C22⋊Dic5911yescharacteristic
370C2×D4C22⋊Dic5911yescharacteristic
371C2×D4C22⋊Dic5911yescharacteristic
372C2×D4D4×D5911yescharacteristic
373C2×D4D4×D5911yescharacteristic
374C2×D4#1174568711yescharacteristic
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—2364no—
380C22×C4C22×Dic5911yescharacteristic
381C22×C4C2×C22⋊D5911yescharacteristic
382C22×C4C2×C22⋊D5911yescharacteristic
383Q16—42no—
384SD16—42no—
385SD16—42no—
386SD16—42no—
387M16—41no—
388M16—2361no—
389M16—2361no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—41no—
393C2×C8—2361no—
394C4⋊C4—21no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—2361no—
398C4⋊C4—2361no—
399C4⋊C4—2361no—
400C22⋊C4—21no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—1182no—
404C22⋊C4—1182no—
405C22⋊C4—2361no—
406C22⋊C4—2364no—
407C42—21no—
408C42—2361no—
409C42—2361no—
410C42—2361no—
411C23—24no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—42no—
416C23—42no—
417C23—161no—
418C23—161no—
419C23—161no—
420C23D4×Dic5911yescharacteristic
421C23#1427369511yescharacteristic
422C23#1427369911yescharacteristic
423Q8—21no—
424Q8—21no—
425Q8—41no—
426Q8—41no—
427Q8—41no—
428Q8—81no—
429D4—24no—
430D4—21no—
431D4—21no—
432D4—21no—
433D4—21no—
434D4—41no—
435D4—41no—
436D4—41no—
437D4—42no—
438D4—42no—
439D4—42no—
440D4—42no—
441D4—42no—
442D4#1427364212yes—
443D4#1427364212yes—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—81no—
457C2×C4—9444no—
458C2×C4—4721no—
459C2×C4—2362no—
460C2×C4—2362no—
461C2×C4C2×C22⋊Dic5911yescharacteristic
462C2×C4D4×Dic5911yescharacteristic
463C2×C4#1427369911yescharacteristic
464C2×C4#1427370811yescharacteristic
465C8—41no—
466C8—41no—
467C8—4721no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—22no—
472C22—22no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22—84no—
481C22—162no—
482C22—162no—
483C22#1683728211yescharacteristic
484C4—21no—
485C4—21no—
486C4—21no—
487C4—41no—
488C4—81no—
489C4—81no—
490C4—9441no—
491C4—4722no—
492C4—4722no—
493C4C2×#1427364211yescharacteristic
494C4C2×#1427364211yescharacteristic
495C2—21no—
496C2—41no—
497C2—41no—
498C2—41no—
499C2—42no—
500C2—42no—
501C2—81no—
502C2#2142455011yescharacteristic
subgroupG/H
centerC2#21424550
commutatorC22×C4×C59C22×C4
FrattiniC2×#41C22×D59
FittingC59×#3009C2
socleC2×C59#1898
prime radicalC2×H(5,𝔽2)×C59C22
cosocleC2×#41×C59C23
hypercenter#3009D59
hypocenterC59#13100
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×H(5,𝔽2) ◁ #3009

Complements

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

NH ≅ G/Ntype
2. #21425771C2semidirect
3. #21425750C2semidirect
5. #21422225C2semidirect
6. #21422217C2semidirect
21. C2×#137×C59C4semidirect
22. C22×SD16×C59C4semidirect
28. C59×#418C4semidirect
29. C59×#416C4semidirect
56. C22×Q8×C59C2×C4semidirect
69. C2×M16×C59C2×C4semidirect
70. C22×C8×C59C2×C4semidirect
77. #117×C59C2×C4semidirect
260. C2×#137Dic59semidirect
261. C22×SD16Dic59semidirect
270. #418Dic59semidirect
271. #416Dic59semidirect
282. C59#13100semidirect
290. C22×Q8C2×Dic59semidirect
304. C2×M16C2×Dic59semidirect
305. C22×C8C2×Dic59semidirect
317. #117C2×Dic59semidirect

Maximal subgroups

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

subgroupindex
#214325572
#214257712
#214257502
#214240662
#214222252
#214222172
C59×#30092
#1310059