Finite Group Databasebeta

← #28485012

Subgroups — #28485012

Order 15104.

712 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432545C211yesmaximal, characteristic, complemented
2#21424040C211yesmaximal, characteristic, complemented
3#21422612C211yesmaximal, characteristic, complemented
4#21422456C211yesmaximal, characteristic, complemented
5#21421973C211yesmaximal, characteristic, complemented
6#21421943C211yesmaximal, characteristic, complemented
7C2×C59×#449C211yesmaximal, characteristic
8C2×#14273734—21no—
9C2×#14273654—21no—
10C2×#14273654—21no—
11#16837369—22no—
12#16837369—21no—
13#16837366—21no—
14#16837366C2211yescharacteristic
15#16837366C2211yescharacteristic
16C2×C22⋊(C59⋊C8)—21no—
17C2×C22⋊(C59⋊C8)C2211yescharacteristic
18C4×C22⋊Dic59C2211yescharacteristic
19#16837136—22no—
20#16837126—21no—
21#16837126—21no—
22C22×#41×C59C2211yescharacteristic
23C22×SD16×C59C411yescharacteristic, complemented
24C22×D8×C59C411yescharacteristic, complemented
25C2×#121×C59C2211yescharacteristic
26C59×#449—21no—
27C59×#449—21no—
28C59×#449—21no—
29C59×#449—21no—
30C2×#103×C59C411yescharacteristic, complemented
31C2×#102×C59C411yescharacteristic, complemented
32C2×C22⋊C8×C59C2211yescharacteristic
33#14273734—42no—
34#14273734—42no—
35C2×#11745664—22no—
36C2×#11745664—21no—
37C2×#11745664—21no—
38C22×C59⋊C8—22no—
39C22×C59⋊C8—21no—
40C2×C22⋊Dic59—21no—
41C2×C4×Dic59—21no—
42#14273654—42no—
43#14273654—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)D411yescharacteristic
51C22⋊(C59⋊C8)D411yescharacteristic
52C22⋊(C59⋊C8)D412yes—
53C59⋊#95—21no—
54C2×#41×C59—21no—
55C2×#41×C59—21no—
56C2×#41×C59—42no—
57C2×#41×C59—42no—
58C2×#41×C59D411yescharacteristic
59C2×#41×C59D411yescharacteristic
60C2×#41×C59D411yescharacteristic
61C2×#41×C59D411yescharacteristic
62C22×Q8×C59C2×C411yescharacteristic
63C22×D4×C59—21no—
64C22×D4×C59—21no—
65C22×D4×C59C2×C411yescharacteristic
66C22×D4×C59C2×C411yescharacteristic
67C23×C4×C59—21no—
68C23×C4×C59C2311yescharacteristic
69C2×SD16×C59—21no—
70C2×SD16×C59—21no—
71C2×SD16×C59—42no—
72C2×D8×C59—21no—
73C2×D8×C59—21no—
74C2×D8×C59—42no—
75C22×C8×C59C2×C411yescharacteristic
76C22×C8×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—82no—
122C2×D4×C59C22⋊C411yescharacteristic
123C2×D4×C59C22⋊C411yescharacteristic
124C22×C4×C59—21no—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—41no—
131C22×C4×C59—41no—
132C22×C4×C59—44no—
133C22×C4×C59C22×C411yescharacteristic
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59#4111yescharacteristic
139C22×Dic59—41no—
140SD16×C59—42no—
141SD16×C59—42no—
142D8×C59—42no—
143D8×C59—42no—
144C2×C8×C59—21no—
145C2×C8×C59—21no—
146C2×C8×C59—41no—
147C2×C8×C59—41no—
148C4⋊C4×C59—21no—
149C4⋊C4×C59—21no—
150C22⋊C4×C59—41no—
151C22⋊C4×C59—41no—
152C22⋊Dic59—22no—
153C22⋊Dic59—41no—
154C4×Dic59—41no—
155C4×Dic59—41no—
156#11745664—44no—
157#11745664—41no—
158#11745664—41no—
159#11745664—41no—
160#11745664—41no—
161C2×C59⋊C8—21no—
162C2×C59⋊C8—22no—
163C2×C59⋊C8—21no—
164C2×C59⋊C8—44no—
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—41no—
176C23×C59—41no—
177C23×C59—41no—
178C23×C59—41no—
179C23×C59—41no—
180C23×C59—81no—
181C23×C59—81no—
182C23×C59—81no—
183C23×C59C4×D411yescharacteristic
184C23×C59#12111yescharacteristic
185C23×C59#12311yescharacteristic
186Q8×C59—21no—
187Q8×C59—21no—
188Q8×C59—21no—
189Q8×C59—42no—
190Q8×C59#9912yes—
191D4×C59—24no—
192D4×C59—21no—
193D4×C59—21no—
194D4×C59—21no—
195D4×C59—42no—
196D4×C59—42no—
197D4×C59—42no—
198D4×C59—42no—
199D4×C59—41no—
200D4×C59—41no—
201D4×C59—81no—
202D4×C59—82no—
203D4×C59—82no—
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—42no—
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#9241—591nomaximal, Sylow
237C22×C59—21no—
238C22×C59—21no—
239C22×C59—21no—
240C22×C59—21no—
241C22×C59—21no—
242C22×C59—21no—
243C22×C59—22no—
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—161no—
259C22×C59—161no—
260C22×C59#38611yescharacteristic
261C22×C59#44311yescharacteristic
262C22×C59#44411yescharacteristic
263C4×C59—21no—
264C4×C59—21no—
265C4×C59—22no—
266C4×C59—21no—
267C4×C59—41no—
268C4×C59—81no—
269C4×C59C2×#9911yescharacteristic
270C4×C59C2×#9911yescharacteristic
271Dic59—82no—
272C2×#449D5911yescharacteristic
273#2880—591no—
274#1789—591no—
275#1507—591no—
276#1471—591no—
277#1293—591no—
278#1282—591no—
279C2×C59—21no—
280C2×C59—21no—
281C2×C59—41no—
282C2×C59—41no—
283C2×C59—42no—
284C2×C59—81no—
285C2×C59—161no—
286C2×C59#187611yescharacteristic
287C2×C59#189811yescharacteristic
288C2×C59#190311yescharacteristic
289C22×#41C2×D5911yescharacteristic
290C22×SD16Dic5911yescharacteristic, complemented
291C22×D8Dic5911yescharacteristic, complemented
292C2×#131—1181no—
293C2×#121C2×D5911yescharacteristic
294#449—21no—
295#449—21no—
296#449—21no—
297#449—21no—
298C2×#104—1181no—
299C2×#104—1181no—
300C2×#103Dic5911yescharacteristic, complemented
301C2×#102Dic5911yescharacteristic, complemented
302#408—1182no—
303#408—1181no—
304#407—591no—
305#407—591no—
306#407—1181no—
307C2×C22⋊C8—591no—
308C2×C22⋊C8—1181no—
309C2×C22⋊C8C2×D5911yescharacteristic
310C4×C22⋊C4—591no—
311#327—1182no—
312#323—1181no—
313#323—1181no—
314C59#924111yescharacteristic, complemented, Sylow
315C2×#41—21no—
316C2×#41—21no—
317C2×#41—42no—
318C2×#41—42no—
319C2×#41C22⋊D5911yescharacteristic
320C2×#41C22⋊D5911yescharacteristic
321C2×#41C22⋊D5911yescharacteristic
322C2×#41C22⋊D5911yescharacteristic
323C22×Q8C2×Dic5911yescharacteristic
324C22×D4—21no—
325C22×D4—21no—
326C22×D4C2×Dic5911yescharacteristic
327C22×D4C2×Dic5911yescharacteristic
328C23×C4—21no—
329C23×C4C22×D5911yescharacteristic
330C2×SD16—21no—
331C2×SD16—21no—
332C2×SD16—42no—
333C2×D8—21no—
334C2×D8—21no—
335C2×D8—42no—
336#131—2362no—
337#131—2362no—
338C2×M16—1182no—
339C2×M16—1181no—
340C2×M16—1181no—
341C22×C8—1182no—
342C22×C8—1181no—
343C22×C8C2×Dic5911yescharacteristic
344C22×C8C2×Dic5911yescharacteristic
345#121—21no—
346#121—21no—
347#121—21no—
348#121—21no—
349C2×C4⋊C4C2×Dic5911yescharacteristic
350C2×C22⋊C4—21no—
351C2×C22⋊C4—1181no—
352C2×C42—1181no—
353#104—2362no—
354#104—2362no—
355#103—21no—
356#103—21no—
357#102—21no—
358#102—21no—
359C22⋊C8—21no—
360C22⋊C8—21no—
361C22⋊C8—591no—
362C22⋊C8—591no—
363C22⋊C8—592no—
364C22⋊C8—1182no—
365C22⋊C8—1182no—
366C22⋊C8—1181no—
367C22⋊C8—1181no—
368C22⋊C8—1181no—
369C22⋊C8—1181no—
370#95—1181no—
371C24—21no—
372C24—21no—
373C24#1174568711yescharacteristic
374#41—22no—
375#41—22no—
376#41—22no—
377#41—22no—
378#41—41no—
379#41—41no—
380#41—41no—
381#41—41no—
382#41—44no—
383#41—44no—
384C2×Q8—21no—
385C2×Q8—22no—
386C2×Q8—41no—
387C2×Q8C22⋊Dic5911yescharacteristic
388C2×Q8C22⋊Dic5911yescharacteristic
389C2×D4—21no—
390C2×D4—22no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—44no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—82no—
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—2361no—
427M16—2361no—
428M16—2364no—
429M16—2361no—
430M16—2361no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—1181no—
436C2×C8—1181no—
437C2×C8—1182no—
438C2×C8—2361no—
439C2×C8—2364no—
440C2×C8—2361no—
441C2×C8—2361no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C22⋊C4—1182no—
447C22⋊C4—2361no—
448C42—2361no—
449C42—2361no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—41no—
455C23—41no—
456C23—42no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—81no—
463C23—81no—
464C23—81no—
465C23D4×Dic5911yescharacteristic
466C23#1427369911yescharacteristic
467C23#1427370811yescharacteristic
468Q8—21no—
469Q8—21no—
470Q8—21no—
471Q8—42no—
472Q8#1427364212yes—
473D4—24no—
474D4—21no—
475D4—21no—
476D4—21no—
477D4—42no—
478D4—42no—
479D4—42no—
480D4—42no—
481D4—41no—
482D4—41no—
483D4—81no—
484D4—82no—
485D4—82no—
486D4#1427364212yes—
487C2×C4—24no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—81no—
505C2×C4—4721no—
506C2×C4—4721no—
507C2×C4—2362no—
508C2×C4C2×C22⋊Dic5911yescharacteristic
509C2×C4D4×Dic5911yescharacteristic
510C2×C4#1427369511yescharacteristic
511C2×C4#1427369911yescharacteristic
512C8—41no—
513C8—41no—
514C8—4721no—
515C8—2361no—
516C8—2361no—
517C8—2362no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—22no—
525C22—21no—
526C22—21no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—81no—
533C22—81no—
534C22—82no—
535C22—82no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—161no—
540C22—161no—
541C22#1683728211yescharacteristic
542C22#1683751911yescharacteristic
543C22#1683752411yescharacteristic
544C4—21no—
545C4—21no—
546C4—22no—
547C4—21no—
548C4—41no—
549C4—81no—
550C4—4722no—
551C4C2×#1427364211yescharacteristic
552C4C2×#1427364211yescharacteristic
553C2—21no—
554C2—21no—
555C2—41no—
556C2—41no—
557C2—42no—
558C2—81no—
559C2—161no—
560C2#2142442511yescharacteristic
561C2#2142455011yescharacteristic
562C2#2142457811yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
Fitting, prime radicalC2×C59×#449C2
socleC22×C59#386
cosocleC23×C4×C59C23
hypercenterC2×#449D59
hypocenterC59#9241
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#449

Complements

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

NH ≅ G/Ntype
1. #21432545C2semidirect
2. #21424040C2semidirect
3. #21422612C2semidirect
4. #21422456C2semidirect
5. #21421973C2semidirect
6. #21421943C2semidirect
23. C22×SD16×C59C4semidirect
24. C22×D8×C59C4semidirect
30. C2×#103×C59C4semidirect
31. C2×#102×C59C4semidirect
290. C22×SD16Dic59semidirect
291. C22×D8Dic59semidirect
300. C2×#103Dic59semidirect
301. C2×#102Dic59semidirect
314. C59#9241semidirect

Maximal subgroups

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

subgroupindex
#214325452
#214240402
#214226122
#214224562
#214219732
#214219432
C2×C59×#4492
#924159