Finite Group Databasebeta

← #28647500

Subgroups — #28647500

Order 15104.

774 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837802C211yesmaximal, characteristic, complemented
2C2×#16837442C211yesmaximal, characteristic
3#21427814C211yesmaximal, characteristic
4#21427810C211yesmaximal, characteristic
5#21425582C211yesmaximal, characteristic, complemented
6#21425575C211yesmaximal, characteristic, complemented
7#21425437C211yesmaximal, characteristic, complemented
8#21425436C211yesmaximal, characteristic, complemented
9#21424833C211yesmaximal, characteristic, complemented
10#21424823C211yesmaximal, characteristic, complemented
11#21424297C211yesmaximal, characteristic
12#21424261C211yesmaximal, characteristic, complemented
13#21424127C211yesmaximal, characteristic
14#21423882C211yesmaximal, characteristic
15C59×#1853C211yesmaximal, characteristic
16C22×#11745664C2211yescharacteristic
17C2×#14273718C2211yescharacteristic
18C2×#14273718C2211yescharacteristic
19C2×#14273686C2211yescharacteristic
20#16837802—24no—
21#16837442—24no—
22C2×#14273661C2211yescharacteristic
23C2×#14273660C2211yescharacteristic
24C2×#14273660C2211yescharacteristic
25C2×#14273660C2211yescharacteristic
26C2×#14273659C2211yescharacteristic
27C2×#14273658C2211yescharacteristic
28C2×#14273658C2211yescharacteristic
29C2×#14273658C2211yescharacteristic
30C2×#14273637C2211yescharacteristic
31C2×C8×Dic59C2211yescharacteristic
32#16837341—21no—
33#16837338—21no—
34#16837335—21no—
35#16837325—21no—
36#16837301C2211yescharacteristic
37#16837301C2211yescharacteristic
38#16837299—21no—
39#16837298C2211yescharacteristic
40#16837298C2211yescharacteristic
41#16837289—21no—
42#16837275—21no—
43#16837271C2211yescharacteristic
44#16837267C2211yescharacteristic
45#16837262—21no—
46#16837262C2211yescharacteristic
47C4×C22⋊Dic59C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837156C2211yescharacteristic
53#16837156C2211yescharacteristic
54C22×C4⋊C4×C59C2211yescharacteristic
55C2×#106×C59C2211yescharacteristic
56C2×#106×C59C2211yescharacteristic
57C2×C22⋊C8×C59C2211yescharacteristic
58C59×#380C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C59×#340C2211yescharacteristic
61C2×#11745664—22no—
62C2×#11745664—22no—
63C2×#11745664Q814yes—
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66#14273718—22no—
67#14273718—22no—
68#14273718—41no—
69#14273718—41no—
70#14273686—22no—
71#14273686—41no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59—21no—
74C2×C4⋊Dic59—21no—
75C2×C4⋊Dic59—21no—
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—21no—
82C2×Dic59⋊C4—21no—
83C2×Dic59⋊C4—21no—
84C2×Dic59⋊C4—21no—
85C2×Dic59⋊C4—21no—
86C2×C22⋊Dic59—21no—
87C2×C22⋊Dic59—21no—
88C2×C22⋊Dic59D411yescharacteristic
89C2×C22⋊Dic59D411yescharacteristic
90C2×C4×Dic59—21no—
91C2×C4×Dic59C2311yescharacteristic
92C2×C4×Dic59C2311yescharacteristic
93#14273661—41no—
94#14273660—22no—
95#14273660—22no—
96#14273660D414yes—
97#14273659—22no—
98#14273658—22no—
99#14273658—22no—
100#14273658—22no—
101#14273637—22no—
102C8×Dic59—22no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C59⋊#95—21no—
110C59⋊#95D411yescharacteristic
111C59⋊#95D411yescharacteristic
112C23×C4×C59—21no—
113C23×C4×C59C2311yescharacteristic
114C22×C8×C59C2311yescharacteristic
115C22×C8×C59C2311yescharacteristic
116C2×C4⋊C4×C59—22no—
117C2×C4⋊C4×C59—21no—
118C2×C4⋊C4×C59—41no—
119C2×C4⋊C4×C59C2311yescharacteristic
120C2×C4⋊C4×C59C2311yescharacteristic
121C2×C4⋊C4×C59C2311yescharacteristic
122C2×C4⋊C4×C59C2311yescharacteristic
123C2×C22⋊C4×C59—21no—
124#106×C59—22no—
125#106×C59—41no—
126C22⋊C8×C59Q814yes—
127#95×C59—21no—
128C24×C59C2×D411yescharacteristic
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—42no—
138C22×C4×C59—42no—
139C22×C4×C59—81no—
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×Q812yes—
142C22×C4×C59C2×Q812yes—
143C22×C4×C59C2411yescharacteristic
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—21no—
147C22×Dic59—21no—
148C22×Dic59C2×D411yescharacteristic
149C22×Dic59C2×D411yescharacteristic
150C22×Dic59C2×D411yescharacteristic
151C22×Dic59C2×D411yescharacteristic
152C2×C8×C59—21no—
153C2×C8×C59—21no—
154C2×C8×C59—21no—
155C2×C8×C59—21no—
156C2×C8×C59C2×Q812yes—
157C2×C8×C59C2×Q812yes—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—42no—
166C4⋊C4×C59C2×D412yes—
167C22⋊C4×C59—81no—
168C22⋊Dic59—41no—
169C22⋊Dic59—41no—
170C22⋊Dic59—42no—
171C22⋊Dic59—42no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—42no—
180C4⋊Dic59—42no—
181C4⋊Dic59—81no—
182C4⋊Dic59—81no—
183C4⋊Dic59C2×D412yes—
184Dic59⋊C4—42no—
185Dic59⋊C4—42no—
186Dic59⋊C4—42no—
187Dic59⋊C4—42no—
188Dic59⋊C4—81no—
189Dic59⋊C4—81no—
190C4×Dic59—21no—
191C4×Dic59—21no—
192C4×Dic59—21no—
193C4×Dic59—21no—
194C4×Dic59—42no—
195#11745664—28no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8C2×D412yes—
199C2×C59⋊C8C2×Q812yes—
200C2×C59⋊C8C2×Q812yes—
201C2×C59⋊C8#4112yes—
202C23×C59—21no—
203C23×C59—21no—
204C23×C59—21no—
205C23×C59—21no—
206C23×C59—21no—
207C23×C59—21no—
208C23×C59C4⋊Q812yes—
209C23×C59C22×D411yescharacteristic
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—22no—
213C2×C4×C59—22no—
214C2×C4×C59—22no—
215C2×C4×C59—22no—
216C2×C4×C59—22no—
217C2×C4×C59—41no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—84no—
229C2×C4×C59C4⋊Q812yes—
230C2×C4×C59C22×D411yescharacteristic
231C2×C4×C59C22×Q811yescharacteristic
232C2×C4×C59C22×Q811yescharacteristic
233C2×C4×C59C2×#4111yescharacteristic
234C2×C4×C59H(5,𝔽2)11yescharacteristic
235C2×C4×C592−1+411yescharacteristic
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—22no—
239C2×Dic59—21no—
240C2×Dic59—21no—
241C2×Dic59—21no—
242C2×Dic59—21no—
243C2×Dic59—21no—
244C2×Dic59—21no—
245C2×Dic59—21no—
246C2×Dic59—21no—
247C2×Dic59—21no—
248C2×Dic59—21no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C8×C59—22no—
261C8×C59—22no—
262C59⋊C8—22no—
263C59⋊C8C22⋊Q814yes—
264#26075—591nomaximal, Sylow
265C22×C59—22no—
266C22×C59—22no—
267C22×C59—22no—
268C22×C59—41no—
269C22×C59—41no—
270C22×C59—41no—
271C22×C59—41no—
272C22×C59C2×C4⋊Q811yescharacteristic
273C22×C59D4211yescharacteristic
274C22×C59#54711yescharacteristic
275C22×C59#57511yescharacteristic
276C22×C59#57511yescharacteristic
277C22×C59#57711yescharacteristic
278C22×C59#57711yescharacteristic
279C4×C59—22no—
280C4×C59—42no—
281C4×C59—81no—
282C4×C59—81no—
283C4×C59—81no—
284C4×C59C2×C22⋊Q811yescharacteristic
285C4×C59#54411yescharacteristic
286C4×C59D4×Q811yescharacteristic
287C4×C59#55411yescharacteristic
288Dic59—41no—
289Dic59—41no—
290Dic59—41no—
291Dic59—42no—
292Dic59—41no—
293Dic59—81no—
294Dic59—81no—
295Dic59—81no—
296C2×#499—591no—
297C2×#428—591no—
298#2442—591no—
299#2441—591no—
300#2085—591no—
301#2084—591no—
302#2059—591no—
303#2059—591no—
304#1943—591no—
305#1941—591no—
306#1854—591no—
307#1853D5911yescharacteristic
308#1847—591no—
309#1809—591no—
310#1753—591no—
311C2×C59—42no—
312C2×C59#244011yescharacteristic
313C2×C59#316311yescharacteristic
314C2×C59#316611yescharacteristic
315C2×C59#326711yescharacteristic
316C2×C59#327111yescharacteristic
317C2×C59#335111yescharacteristic
318C2×C59#335111yescharacteristic
319C22×M16—591no—
320C2×#125—591no—
321C2×#125—591no—
322C2×#117—591no—
323C22×C4⋊C4C2×D5911yescharacteristic
324#499—1184no—
325#428—1184no—
326C2×#107—591no—
327C2×#107—591no—
328C2×#107—591no—
329C2×#107—591no—
330C2×#106—591no—
331C2×#106—591no—
332C2×#106—591no—
333C2×#106—591no—
334C2×#106C2×D5911yescharacteristic
335C2×#106C2×D5911yescharacteristic
336C2×C22⋊C8C2×D5911yescharacteristic
337C2×#97—591no—
338C2×C4×C8—591no—
339#400—1181no—
340#399—1181no—
341#398—1181no—
342#396—1181no—
343#389—591no—
344#389—591no—
345#389—591no—
346#389—591no—
347#389—1181no—
348#387—1181no—
349#385—1181no—
350#384—591no—
351#382—591no—
352#380—591no—
353#380—1181no—
354#380C2×D5911yescharacteristic
355C4×C22⋊C4—591no—
356#340—591no—
357#340—591no—
358#340—591no—
359#340—591no—
360#340C2×D5911yescharacteristic
361#340C2×D5911yescharacteristic
362#336—591no—
363#336—591no—
364C59#2607511yescharacteristic, complemented, Sylow
365C23×C4—21no—
366C23×C4C22×D5911yescharacteristic
367C2×M16—594no—
368C2×M16—1182no—
369C2×M16—1182no—
370C22×C8—591no—
371C22×C8—591no—
372C22×C8C22×D5911yescharacteristic
373C22×C8C22×D5911yescharacteristic
374#125—1182no—
375#125—1182no—
376#125—2361no—
377#125—2361no—
378#117—1182no—
379#117—2361no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—41no—
383C2×C4⋊C4—591no—
384C2×C4⋊C4—591no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—1181no—
388C2×C4⋊C4—1181no—
389C2×C4⋊C4—1181no—
390C2×C4⋊C4—1181no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4—1181no—
395C2×C4⋊C4—1181no—
396C2×C4⋊C4—1181no—
397C2×C4⋊C4C22×D5911yescharacteristic
398C2×C4⋊C4C22×D5911yescharacteristic
399C2×C4⋊C4C22×D5911yescharacteristic
400C2×C4⋊C4C22×D5911yescharacteristic
401C2×C22⋊C4—21no—
402C2×C22⋊C4—591no—
403C2×C22⋊C4—591no—
404C2×C22⋊C4—1181no—
405C2×C22⋊C4—1181no—
406C2×C42—591no—
407C2×C42—591no—
408C2×C42—1181no—
409#107—594no—
410#107—1182no—
411#107—1182no—
412#107—2361no—
413#106—22no—
414#106—41no—
415#106—1182no—
416#106—1182no—
417#106—1182no—
418#106—1182no—
419C22⋊C8Dic11814yes—
420#97—1182no—
421C4×C8—1182no—
422#95—21no—
423#95—591no—
424#95—591no—
425#95—1181no—
426#95—1181no—
427#95—1181no—
428#95—1181no—
429#95—1181no—
430#95—1181no—
431#95—1181no—
432C24D4×D5911yescharacteristic
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—81no—
444C22×C4—591no—
445C22×C4—591no—
446C22×C4—591no—
447C22×C4—591no—
448C22×C4—1181no—
449C22×C4—1181no—
450C22×C4—1181no—
451C22×C4—1181no—
452C22×C4C2×Dic11812yes—
453C22×C4D4×D5911yescharacteristic
454C22×C4Q8×D5912yes—
455C22×C4C23×D5911yescharacteristic
456M16—1188no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—592no—
462C2×C8—592no—
463C2×C8—592no—
464C2×C8—592no—
465C2×C8—1181no—
466C2×C8—1181no—
467C2×C8C2×Dic11812yes—
468C2×C8C2×Dic11812yes—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—42no—
477C4⋊C4—592no—
478C4⋊C4—1181no—
479C4⋊C4—1181no—
480C4⋊C4—1181no—
481C4⋊C4—1181no—
482C4⋊C4—1181no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C4⋊C4—4721no—
486C4⋊C4—4721no—
487C4⋊C4—4721no—
488C4⋊C4—4721no—
489C4⋊C4—2362no—
490C4⋊C4—2362no—
491C4⋊C4—2362no—
492C4⋊C4—2362no—
493C4⋊C4—2362no—
494C4⋊C4—2362no—
495C4⋊C4D4×D5912yes—
496C22⋊C4—81no—
497C22⋊C4—2361no—
498C22⋊C4—2361no—
499C22⋊C4—2362no—
500C22⋊C4—2362no—
501C42—1181no—
502C42—1181no—
503C42—1181no—
504C42—1181no—
505C42—2362no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23#1427372612yes—
513C23C2×D4×D5911yescharacteristic
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—84no—
533C2×C4—1181no—
534C2×C4—1181no—
535C2×C4—1181no—
536C2×C4—1181no—
537C2×C4—1181no—
538C2×C4—1181no—
539C2×C4—1181no—
540C2×C4—1181no—
541C2×C4—1181no—
542C2×C4—1181no—
543C2×C4—1182no—
544C2×C4—1181no—
545C2×C4—1181no—
546C2×C4—2361no—
547C2×C4—2361no—
548C2×C4—2361no—
549C2×C4—2361no—
550C2×C4—2361no—
551C2×C4—2361no—
552C2×C4—2361no—
553C2×C4—2361no—
554C2×C4—2361no—
555C2×C4—2361no—
556C2×C4—2361no—
557C2×C4#1427372612yes—
558C2×C4C2×D4×D5911yescharacteristic
559C2×C4C2×Q8×D5911yescharacteristic
560C2×C4C22×Dic11811yescharacteristic
561C2×C4#41×D5911yescharacteristic
562C2×C4#1427377211yescharacteristic
563C2×C4#1427377411yescharacteristic
564C8—22no—
565C8—22no—
566C8—594no—
567C8—1182no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22C2×#1427372611yescharacteristic
576C22#1683803411yescharacteristic
577C22#1683805011yescharacteristic
578C22#1683819811yescharacteristic
579C22#1683820111yescharacteristic
580C22#1683820811yescharacteristic
581C22#1683820811yescharacteristic
582C4—22no—
583C4—42no—
584C4—81no—
585C4—81no—
586C4—81no—
587C4—4721no—
588C4—4721no—
589C4—4721no—
590C4—2361no—
591C4—2361no—
592C4—2362no—
593C4—2361no—
594C4—2361no—
595C4C22⋊Q8×D5911yescharacteristic
596C4#1683803111yescharacteristic
597C4D4×Dic11811yescharacteristic
598C4#1683811111yescharacteristic
599C2—42no—
600C2#2142779711yescharacteristic
601C2#2143462411yescharacteristic
602C2#2143464611yescharacteristic
603C2#2143561911yescharacteristic
604C2#2143568411yescharacteristic
605C2#2143665911yescharacteristic
606C2#2143666011yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#1853C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#1853D59
hypocenterC59#26075
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
1. C2×#16837802C2semidirect
5. #21425582C2semidirect
6. #21425575C2semidirect
7. #21425437C2semidirect
8. #21425436C2semidirect
9. #21424833C2semidirect
10. #21424823C2semidirect
12. #21424261C2semidirect
364. C59#26075semidirect

Maximal subgroups

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

subgroupindex
C2×#168378022
C2×#168374422
#214278142
#214278102
#214255822
#214255752
#214254372
#214254362
#214248332
#214248232
#214242972
#214242612
#214241272
#214238822
C59×#18532
#2607559