Finite Group Databasebeta

← #28485754

Subgroups — #28485754

Order 15104.

704 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432545C211yesmaximal, characteristic, complemented
2#21424036C211yesmaximal, characteristic, complemented
3#21422664C211yesmaximal, characteristic, complemented
4#21422456C211yesmaximal, characteristic, complemented
5#21422001C211yesmaximal, characteristic, complemented
6#21421983C211yesmaximal, characteristic, complemented
7C59×#3006C211yesmaximal, characteristic
8C2×#14273734—21no—
9C2×#14273651—21no—
10C2×#14273648—21no—
11#16837369—22no—
12#16837369—21no—
13#16837366—21no—
14#16837366C2211yescharacteristic
15C2×C22⋊(C59⋊C8)—21no—
16C2×C22⋊(C59⋊C8)C2211yescharacteristic
17C2×C22⋊(C59⋊C8)C2211yescharacteristic
18#16837262C2211yescharacteristic
19#16837136—21no—
20#16837136—21no—
21#16837126—22no—
22C22×#41×C59C2211yescharacteristic
23C2×AGL(1,ℤ/8)×C59C411yescharacteristic, complemented
24C2×AGL(1,ℤ/8)×C59C411yescharacteristic, complemented
25C2×#121×C59C2211yescharacteristic
26C59×#449—21no—
27C59×#449—21no—
28C59×#448—21no—
29C59×#447—21no—
30C59×#417C411yescharacteristic, complemented
31C59×#417C411yescharacteristic, complemented
32C59×#407C2211yescharacteristic
33#14273734—42no—
34#14273734—42no—
35C2×#11745664—22no—
36C2×#11745664—21no—
37C22×C59⋊C8—22no—
38C22×C59⋊C8—21no—
39C22×C59⋊C8—21no—
40C2×C4⋊Dic59—21no—
41C2×C22⋊Dic59—21no—
42#14273651—42no—
43#14273648—42no—
44C22⋊(C59⋊C8)—22no—
45C22⋊(C59⋊C8)—22no—
46C22⋊(C59⋊C8)—21no—
47C22⋊(C59⋊C8)—21no—
48C22⋊(C59⋊C8)—21no—
49C22⋊(C59⋊C8)—21no—
50C22⋊(C59⋊C8)D412yes—
51C22⋊(C59⋊C8)D411yescharacteristic
52C22⋊(C59⋊C8)D411yescharacteristic
53C59⋊#95—21no—
54C2×#41×C59—21no—
55C2×#41×C59—21no—
56C2×#41×C59—42no—
57C2×#41×C59—42no—
58C2×#41×C59C2×C411yescharacteristic
59C2×#41×C59C2×C411yescharacteristic
60C2×#41×C59D411yescharacteristic
61C2×#41×C59D411yescharacteristic
62C22×Q8×C59D411yescharacteristic
63C22×D4×C59—21no—
64C22×D4×C59—21no—
65C22×D4×C59C2×C411yescharacteristic
66C22×D4×C59D411yescharacteristic
67C23×C4×C59—21no—
68C23×C4×C59C2311yescharacteristic
69AGL(1,ℤ/8)×C59—42no—
70AGL(1,ℤ/8)×C59—42no—
71C2×SD16×C59—21no—
72C2×SD16×C59—21no—
73C2×D8×C59—21no—
74C2×D8×C59—21no—
75C2×M16×C59C2×C411yescharacteristic
76C2×M16×C59C2×C411yescharacteristic
77#121×C59—21no—
78#121×C59—21no—
79#121×C59—21no—
80#121×C59—21no—
81C2×C4⋊C4×C59C2×C411yescharacteristic
82C2×C22⋊C4×C59—21no—
83#103×C59—21no—
84#103×C59—21no—
85#102×C59—21no—
86#102×C59—21no—
87C22⋊C8×C59—21no—
88C22⋊C8×C59—21no—
89C24×C59—21no—
90C24×C59—21no—
91C24×C59#4111yescharacteristic
92#41×C59—22no—
93#41×C59—22no—
94#41×C59—22no—
95#41×C59—22no—
96#41×C59—41no—
97#41×C59—41no—
98#41×C59—41no—
99#41×C59—41no—
100#41×C59—44no—
101#41×C59—44no—
102C2×Q8×C59—21no—
103C2×Q8×C59—22no—
104C2×Q8×C59—41no—
105C2×Q8×C59C22⋊C411yescharacteristic
106C2×Q8×C59C22⋊C411yescharacteristic
107C2×D4×C59—21no—
108C2×D4×C59—22no—
109C2×D4×C59—21no—
110C2×D4×C59—21no—
111C2×D4×C59—21no—
112C2×D4×C59—21no—
113C2×D4×C59—21no—
114C2×D4×C59—21no—
115C2×D4×C59—41no—
116C2×D4×C59—41no—
117C2×D4×C59—41no—
118C2×D4×C59—44no—
119C2×D4×C59—41no—
120C2×D4×C59—41no—
121C2×D4×C59—81no—
122C2×D4×C59—81no—
123C2×D4×C59C22⋊C411yescharacteristic
124C2×D4×C59C22⋊C411yescharacteristic
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—41no—
132C22×C4×C59—41no—
133C22×C4×C59—44no—
134C22×C4×C59C22×C411yescharacteristic
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59#4111yescharacteristic
140C22×Dic59—41no—
141SD16×C59—42no—
142SD16×C59—42no—
143D8×C59—42no—
144D8×C59—42no—
145M16×C59—41no—
146M16×C59—41no—
147C2×C8×C59—21no—
148C2×C8×C59—21no—
149C4⋊C4×C59—21no—
150C4⋊C4×C59—21no—
151C22⋊C4×C59—41no—
152C22⋊C4×C59—41no—
153C22⋊Dic59—81no—
154C4⋊Dic59—41no—
155C4⋊Dic59—41no—
156#11745664—44no—
157#11745664—41no—
158#11745664—41no—
159C2×C59⋊C8—21no—
160C2×C59⋊C8—22no—
161C2×C59⋊C8—21no—
162C2×C59⋊C8—44no—
163C2×C59⋊C8—41no—
164C2×C59⋊C8—41no—
165C2×C59⋊C8—41no—
166C2×C59⋊C8—41no—
167C2×C59⋊C8—41no—
168C23×C59—21no—
169C23×C59—21no—
170C23×C59—21no—
171C23×C59—21no—
172C23×C59—41no—
173C23×C59—41no—
174C23×C59—42no—
175C23×C59—42no—
176C23×C59—41no—
177C23×C59—41no—
178C23×C59—41no—
179C23×C59—81no—
180C23×C59—81no—
181C23×C59—81no—
182C23×C59C4×D411yescharacteristic
183C23×C59#12111yescharacteristic
184C23×C59#12311yescharacteristic
185Q8×C59—21no—
186Q8×C59—21no—
187Q8×C59—21no—
188Q8×C59—42no—
189Q8×C59#9912yes—
190D4×C59—24no—
191D4×C59—21no—
192D4×C59—21no—
193D4×C59—21no—
194D4×C59—42no—
195D4×C59—42no—
196D4×C59—42no—
197D4×C59—42no—
198D4×C59—41no—
199D4×C59—41no—
200D4×C59—81no—
201D4×C59—82no—
202D4×C59—81no—
203D4×C59—81no—
204D4×C59#9912yes—
205C2×C4×C59—24no—
206C2×C4×C59—21no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—41no—
215C2×C4×C59—41no—
216C2×C4×C59—41no—
217C2×C4×C59—42no—
218C2×C4×C59—42no—
219C2×C4×C59—42no—
220C2×C4×C59—42no—
221C2×C4×C59—41no—
222C2×C4×C59—81no—
223C2×C4×C59C2×C22⋊C411yescharacteristic
224C2×C4×C59C4×D411yescharacteristic
225C2×C4×C59#12011yescharacteristic
226C2×C4×C59#12111yescharacteristic
227C2×Dic59—81no—
228C2×Dic59—81no—
229C2×Dic59—81no—
230C8×C59—41no—
231C8×C59—41no—
232C59⋊C8—41no—
233C59⋊C8—42no—
234C59⋊C8—41no—
235C59⋊C8—81no—
236#9347—591nomaximal, Sylow
237C22×C59—21no—
238C22×C59—21no—
239C22×C59—21no—
240C22×C59—21no—
241C22×C59—21no—
242C22×C59—22no—
243C22×C59—21no—
244C22×C59—21no—
245C22×C59—21no—
246C22×C59—41no—
247C22×C59—41no—
248C22×C59—41no—
249C22×C59—41no—
250C22×C59—41no—
251C22×C59—81no—
252C22×C59—81no—
253C22×C59—82no—
254C22×C59—82no—
255C22×C59—81no—
256C22×C59—81no—
257C22×C59—81no—
258C22×C59—162no—
259C22×C59#38611yescharacteristic
260C22×C59#47111yescharacteristic
261C22×C59#47211yescharacteristic
262C4×C59—21no—
263C4×C59—21no—
264C4×C59—22no—
265C4×C59—21no—
266C4×C59—41no—
267C4×C59—81no—
268C4×C59C2×#9911yescharacteristic
269C4×C59C2×#9911yescharacteristic
270Dic59—161no—
271#3006D5911yescharacteristic
272#2880—591no—
273#1788—591no—
274#1519—591no—
275#1471—591no—
276#1305—591no—
277#1298—591no—
278C2×C59—21no—
279C2×C59—21no—
280C2×C59—41no—
281C2×C59—41no—
282C2×C59—42no—
283C2×C59—81no—
284C2×C59—161no—
285C2×C59#188111yescharacteristic
286C2×C59#189811yescharacteristic
287C2×C59#190711yescharacteristic
288C22×#41C2×D5911yescharacteristic
289C2×AGL(1,ℤ/8)Dic5911yescharacteristic, complemented
290C2×AGL(1,ℤ/8)Dic5911yescharacteristic, complemented
291C2×#131—1181no—
292C2×#121C2×D5911yescharacteristic
293#449—21no—
294#449—21no—
295#448—21no—
296#447—21no—
297#417Dic5911yescharacteristic, complemented
298#417Dic5911yescharacteristic, complemented
299C2×#103—1181no—
300C2×#102—1181no—
301#408—1182no—
302#408—1181no—
303#407—591no—
304#407—1181no—
305#407C2×D5911yescharacteristic
306C2×C22⋊C8—591no—
307C2×C22⋊C8—591no—
308C2×C22⋊C8—1181no—
309#380—591no—
310#327—1181no—
311#327—1181no—
312#323—1182no—
313C59#934711yescharacteristic, complemented, Sylow
314C2×#41—21no—
315C2×#41—21no—
316C2×#41—42no—
317C2×#41—42no—
318C2×#41C2×Dic5911yescharacteristic
319C2×#41C2×Dic5911yescharacteristic
320C2×#41C22⋊D5911yescharacteristic
321C2×#41C22⋊D5911yescharacteristic
322C22×Q8C22⋊D5911yescharacteristic
323C22×D4—21no—
324C22×D4—21no—
325C22×D4C2×Dic5911yescharacteristic
326C22×D4C22⋊D5911yescharacteristic
327C23×C4—21no—
328C23×C4C22×D5911yescharacteristic
329AGL(1,ℤ/8)—42no—
330AGL(1,ℤ/8)—42no—
331C2×SD16—21no—
332C2×SD16—21no—
333C2×D8—21no—
334C2×D8—21no—
335#131—2362no—
336#131—2362no—
337C2×M16—1182no—
338C2×M16—1181no—
339C2×M16C2×Dic5911yescharacteristic
340C2×M16C2×Dic5911yescharacteristic
341C22×C8—1182no—
342C22×C8—1181no—
343C22×C8—1181no—
344#121—21no—
345#121—21no—
346#121—21no—
347#121—21no—
348C2×C4⋊C4—1181no—
349C2×C4⋊C4C2×Dic5911yescharacteristic
350C2×C22⋊C4—21no—
351C2×C22⋊C4—1181no—
352#103—21no—
353#103—21no—
354#103—2362no—
355#102—21no—
356#102—21no—
357#102—2362no—
358C22⋊C8—21no—
359C22⋊C8—21no—
360C22⋊C8—592no—
361C22⋊C8—591no—
362C22⋊C8—591no—
363C22⋊C8—1182no—
364C22⋊C8—1182no—
365C22⋊C8—1181no—
366C22⋊C8—1181no—
367C22⋊C8—1181no—
368C22⋊C8—1181no—
369#95—1181no—
370C24—21no—
371C24—21no—
372C24#1174568711yescharacteristic
373#41—22no—
374#41—22no—
375#41—22no—
376#41—22no—
377#41—41no—
378#41—41no—
379#41—41no—
380#41—41no—
381#41—44no—
382#41—44no—
383C2×Q8—21no—
384C2×Q8—22no—
385C2×Q8—41no—
386C2×Q8C22⋊Dic5911yescharacteristic
387C2×Q8C22⋊Dic5911yescharacteristic
388C2×D4—21no—
389C2×D4—22no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—44no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—81no—
403C2×D4—81no—
404C2×D4C22⋊Dic5911yescharacteristic
405C2×D4C22⋊Dic5911yescharacteristic
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—44no—
415C22×C4—2361no—
416C22×C4D4×D5911yescharacteristic
417C22×C4D4×D5911yescharacteristic
418C22×C4#1174568711yescharacteristic
419C22×C4C22×Dic5911yescharacteristic
420C22×C4C2×C22⋊D5911yescharacteristic
421C22×C4C2×C22⋊D5911yescharacteristic
422SD16—42no—
423SD16—42no—
424D8—42no—
425D8—42no—
426M16—41no—
427M16—41no—
428M16—2364no—
429M16—2361no—
430M16—2361no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—1181no—
434C2×C8—1182no—
435C2×C8—1181no—
436C2×C8—2361no—
437C2×C8—2361no—
438C2×C8—2361no—
439C2×C8—2364no—
440C2×C8—2361no—
441C2×C8—2361no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—2361no—
445C4⋊C4—2361no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C22⋊C4—4721no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—41no—
454C23—41no—
455C23—42no—
456C23—42no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—81no—
461C23—81no—
462C23—81no—
463C23D4×Dic5911yescharacteristic
464C23#1427369911yescharacteristic
465C23#1427370811yescharacteristic
466Q8—21no—
467Q8—21no—
468Q8—21no—
469Q8—42no—
470Q8#1427364212yes—
471D4—24no—
472D4—21no—
473D4—21no—
474D4—21no—
475D4—42no—
476D4—42no—
477D4—42no—
478D4—42no—
479D4—41no—
480D4—41no—
481D4—81no—
482D4—82no—
483D4—81no—
484D4—81no—
485D4#1427364212yes—
486C2×C4—24no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—81no—
504C2×C4—4721no—
505C2×C4—4721no—
506C2×C4—4721no—
507C2×C4C2×C22⋊Dic5911yescharacteristic
508C2×C4D4×Dic5911yescharacteristic
509C2×C4#1427369511yescharacteristic
510C2×C4#1427369911yescharacteristic
511C8—41no—
512C8—41no—
513C8—4721no—
514C8—2361no—
515C8—2362no—
516C8—2361no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—22no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—41no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—81no—
532C22—81no—
533C22—82no—
534C22—82no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—162no—
539C22#1683728211yescharacteristic
540C22#1683766311yescharacteristic
541C22#1683767011yescharacteristic
542C4—21no—
543C4—21no—
544C4—22no—
545C4—21no—
546C4—41no—
547C4—81no—
548C4—9441no—
549C4C2×#1427364211yescharacteristic
550C4C2×#1427364211yescharacteristic
551C2—21no—
552C2—21no—
553C2—41no—
554C2—41no—
555C2—42no—
556C2—81no—
557C2—161no—
558C2#2142446011yescharacteristic
559C2#2142455011yescharacteristic
560C2#2142460611yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
Fitting, prime radicalC59×#3006C2
socleC22×C59#386
cosocleC23×C4×C59C23
hypercenter#3006D59
hypocenterC59#9347
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3006

Complements

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

NH ≅ G/Ntype
1. #21432545C2semidirect
2. #21424036C2semidirect
3. #21422664C2semidirect
4. #21422456C2semidirect
5. #21422001C2semidirect
6. #21421983C2semidirect
23. C2×AGL(1,ℤ/8)×C59C4semidirect
24. C2×AGL(1,ℤ/8)×C59C4semidirect
30. C59×#417C4semidirect
31. C59×#417C4semidirect
289. C2×AGL(1,ℤ/8)Dic59semidirect
290. C2×AGL(1,ℤ/8)Dic59semidirect
297. #417Dic59semidirect
298. #417Dic59semidirect
313. C59#9347semidirect

Maximal subgroups

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

subgroupindex
#214325452
#214240362
#214226642
#214224562
#214220012
#214219832
C59×#30062
#934759