Finite Group Databasebeta

← #28647506

Subgroups — #28647506

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×#16837801C211yesmaximal, characteristic, complemented
2C2×#16837445C211yesmaximal, characteristic
3#21427820C211yesmaximal, characteristic
4#21425577C211yesmaximal, characteristic, complemented
5#21425573C211yesmaximal, characteristic, complemented
6#21425434C211yesmaximal, characteristic, complemented
7#21425431C211yesmaximal, characteristic, complemented
8#21424831C211yesmaximal, characteristic, complemented
9#21424821C211yesmaximal, characteristic, complemented
10#21424295C211yesmaximal, characteristic
11#21424288C211yesmaximal, characteristic
12#21424262C211yesmaximal, characteristic, complemented
13#21424127C211yesmaximal, characteristic
14#21423880C211yesmaximal, characteristic
15C59×#2441C211yesmaximal, characteristic
16C22×#11745664C2211yescharacteristic
17C2×#14273719C2211yescharacteristic
18C22×C4⋊Dic59C2211yescharacteristic
19#16837801—24no—
20#16837445—24no—
21C2×#14273660C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273660C2211yescharacteristic
24C2×#14273660C2211yescharacteristic
25C2×#14273658C2211yescharacteristic
26C2×#14273658C2211yescharacteristic
27C2×#14273658C2211yescharacteristic
28C2×#14273658C2211yescharacteristic
29C2×#14273658C2211yescharacteristic
30C2×#14273658C2211yescharacteristic
31C2×C22⋊(C59⋊C8)C2211yescharacteristic
32C2×#14273638C2211yescharacteristic
33C2×C4×C59⋊C8C2211yescharacteristic
34#16837344—21no—
35#16837336—21no—
36#16837300C2211yescharacteristic
37#16837300C2211yescharacteristic
38#16837274—21no—
39#16837268C2211yescharacteristic
40#16837262C2211yescharacteristic
41#16837260—21no—
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×#125×C59C2211yescharacteristic
51C2×#117×C59C2211yescharacteristic
52C59×#399—21no—
53C59×#396—21no—
54C59×#389—21no—
55C59×#389C2211yescharacteristic
56C59×#389C2211yescharacteristic
57C59×#387—21no—
58C59×#384C2211yescharacteristic
59C59×#380C2211yescharacteristic
60C4×C22⋊C4×C59C2211yescharacteristic
61C23×Dic59—21no—
62C2×#11745664—22no—
63C2×#11745664—22no—
64C2×#11745664Q814yes—
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69#14273719—22no—
70#14273719—41no—
71C2×C4⋊Dic59—22no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59—41no—
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—21no—
82C2×C22⋊Dic59—21no—
83#14273660—22no—
84#14273660—22no—
85#14273660—41no—
86#14273660D414yes—
87#14273658—22no—
88#14273658—22no—
89#14273658—22no—
90#14273658—22no—
91#14273658—22no—
92#14273658—41no—
93C22⋊(C59⋊C8)Q814yes—
94#14273638—22no—
95C4×C59⋊C8—22no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C23×C4×C59C2311yescharacteristic
102#125×C59—22no—
103#125×C59—41no—
104#117×C59—22no—
105#117×C59—41no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59—21no—
108C2×C4⋊C4×C59—21no—
109C2×C4⋊C4×C59—21no—
110C2×C4⋊C4×C59—21no—
111C2×C4⋊C4×C59—21no—
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C4⋊C4×C59C2311yescharacteristic
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C22⋊C4×C59—21no—
117C2×C22⋊C4×C59—21no—
118C2×C22⋊C4×C59D411yescharacteristic
119C2×C22⋊C4×C59D411yescharacteristic
120C2×C42×C59—21no—
121C2×C42×C59C2311yescharacteristic
122C2×C42×C59C2311yescharacteristic
123#95×C59—21no—
124#95×C59—21no—
125#95×C59—21no—
126#95×C59D411yescharacteristic
127#95×C59D411yescharacteristic
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×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×Q812yes—
143C22×C4×C59C2×Q812yes—
144C22×C4×C59C2411yescharacteristic
145C22×Dic59—21no—
146C22×Dic59—21no—
147C22×Dic59—21no—
148C22×Dic59—21no—
149C22×Dic59—42no—
150C22×Dic59—42no—
151C22×Dic59—81no—
152C4⋊C4×C59—21no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—21no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—21no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—42no—
160C4⋊C4×C59—42no—
161C4⋊C4×C59—42no—
162C4⋊C4×C59—42no—
163C4⋊C4×C59—81no—
164C4⋊C4×C59—81no—
165C4⋊C4×C59C2×D412yes—
166C22⋊C4×C59—41no—
167C22⋊C4×C59—41no—
168C22⋊C4×C59—42no—
169C22⋊C4×C59—81no—
170C42×C59—21no—
171C42×C59—21no—
172C42×C59—21no—
173C42×C59—21no—
174C42×C59—42no—
175C22⋊Dic59—42no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—42no—
184C4⋊Dic59C2×D412yes—
185Dic59⋊C4—42no—
186Dic59⋊C4—42no—
187Dic59⋊C4—81no—
188Dic59⋊C4—81no—
189#11745664—28no—
190C2×C59⋊C8—21no—
191C2×C59⋊C8—21no—
192C2×C59⋊C8—21no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8C2×D412yes—
197C2×C59⋊C8C2×Q812yes—
198C2×C59⋊C8C2×Q812yes—
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—21no—
217C2×C4×C59—21no—
218C2×C4×C59—22no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59—41no—
236C2×C4×C59—41no—
237C2×C4×C59—41no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59C4⋊Q812yes—
241C2×C4×C59C22×D411yescharacteristic
242C2×C4×C59C22×Q811yescharacteristic
243C2×C4×C59C22×Q811yescharacteristic
244C2×C4×C59C2×#4111yescharacteristic
245C2×C4×C59H(5,𝔽2)11yescharacteristic
246C2×C4×C592−1+411yescharacteristic
247C2×Dic59—22no—
248C2×Dic59—41no—
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—84no—
260C59⋊C8—22no—
261C59⋊C8—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—41no—
281C4×C59—41no—
282C4×C59—41no—
283C4×C59—42no—
284C4×C59—41no—
285C4×C59—81no—
286C4×C59—81no—
287C4×C59—81no—
288C4×C59C2×C22⋊Q811yescharacteristic
289C4×C59#54411yescharacteristic
290C4×C59D4×Q811yescharacteristic
291C4×C59#55411yescharacteristic
292Dic59—42no—
293Dic59—81no—
294Dic59—81no—
295Dic59—81no—
296C2×#499—591no—
297C2×#428—591no—
298#2442—591no—
299#2441D5911yescharacteristic
300#2085—591no—
301#2084—591no—
302#2059—591no—
303#2059—591no—
304#1943—591no—
305#1941—591no—
306#1854—591no—
307#1853—591no—
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×#125C2×D5911yescharacteristic
322C2×#117C2×D5911yescharacteristic
323C22×C4⋊C4—591no—
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×#106—591no—
335C2×#106—591no—
336C2×C22⋊C8—591no—
337C2×#97—591no—
338C2×C4×C8—591no—
339#400—1181no—
340#399—21no—
341#398—1181no—
342#396—21no—
343#389—21no—
344#389—591no—
345#389—591no—
346#389C2×D5911yescharacteristic
347#389C2×D5911yescharacteristic
348#387—21no—
349#385—1181no—
350#384C2×D5911yescharacteristic
351#382—591no—
352#380—591no—
353#380—1181no—
354#380C2×D5911yescharacteristic
355C4×C22⋊C4C2×D5911yescharacteristic
356#340—591no—
357#340—591no—
358#340—591no—
359#340—591no—
360#340—591no—
361#340—591no—
362#336—591no—
363#336—591no—
364C59#2607511yescharacteristic, complemented, Sylow
365C23×C4—1181no—
366C23×C4C22×D5911yescharacteristic
367C2×M16—594no—
368C2×M16—1182no—
369C2×M16—1182no—
370C22×C8—591no—
371C22×C8—591no—
372C22×C8—591no—
373C22×C8—591no—
374#125—22no—
375#125—41no—
376#125—1182no—
377#125—2361no—
378#117—22no—
379#117—41no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—591no—
390C2×C4⋊C4—1182no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4—1181no—
395C2×C4⋊C4—1181no—
396C2×C4⋊C4—2361no—
397C2×C4⋊C4C22×D5911yescharacteristic
398C2×C4⋊C4C22×D5911yescharacteristic
399C2×C4⋊C4C22×D5911yescharacteristic
400C2×C4⋊C4C22×D5911yescharacteristic
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—1181no—
404C2×C22⋊C4C22⋊D5911yescharacteristic
405C2×C22⋊C4C22⋊D5911yescharacteristic
406C2×C42—21no—
407C2×C42C22×D5911yescharacteristic
408C2×C42C22×D5911yescharacteristic
409#107—594no—
410#107—1182no—
411#107—1182no—
412#107—2361no—
413#106—1182no—
414#106—1182no—
415#106—1182no—
416#106—1182no—
417#106—1182no—
418#106—2361no—
419C22⋊C8—594no—
420#97—1182no—
421C4×C8—1182no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—1181no—
426#95—1181no—
427#95—1181no—
428#95—1181no—
429#95—1181no—
430#95C22⋊D5911yescharacteristic
431#95C22⋊D5911yescharacteristic
432C24C2×C22⋊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—1181no—
442C22×C4—1181no—
443C22×C4—1181no—
444C22×C4—1181no—
445C22×C4—4721no—
446C22×C4—2362no—
447C22×C4—2362no—
448C22×C4Q8×D5912yes—
449C22×C4Q8×D5912yes—
450C22×C4C2×C22⋊D5911yescharacteristic
451C22×C4C2×C22⋊D5911yescharacteristic
452C22×C4C2×C22⋊D5911yescharacteristic
453C22×C4C2×C22⋊D5911yescharacteristic
454C22×C4C2×C22⋊D5911yescharacteristic
455C22×C4C23×D5911yescharacteristic
456M16—1188no—
457C2×C8—592no—
458C2×C8—592no—
459C2×C8—592no—
460C2×C8—592no—
461C2×C8—592no—
462C2×C8—592no—
463C2×C8—1181no—
464C2×C8—1181no—
465C2×C8—1181no—
466C2×C8—1181no—
467C2×C8—1181no—
468C2×C8—1181no—
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—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—81no—
481C4⋊C4—81no—
482C4⋊C4—592no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C4⋊C4—1181no—
486C4⋊C4—1181no—
487C4⋊C4—1181no—
488C4⋊C4—1181no—
489C4⋊C4—1181no—
490C4⋊C4—4721no—
491C4⋊C4—4721no—
492C4⋊C4—2362no—
493C4⋊C4—2362no—
494C4⋊C4—2362no—
495C4⋊C4D4×D5912yes—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C22⋊C4—42no—
499C22⋊C4—81no—
500C22⋊C4—2362no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—42no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23#1427372512yes—
513C23C22×C22⋊D5911yescharacteristic
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—1182no—
545C2×C4—4724no—
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#1427372512yes—
558C2×C4C2×D4×D5911yescharacteristic
559C2×C4C2×Q8×D5911yescharacteristic
560C2×C4C2×Q8×D5911yescharacteristic
561C2×C4#41×D5911yescharacteristic
562C2×C4#1427377211yescharacteristic
563C2×C4#1427377411yescharacteristic
564C8—594no—
565C8—1182no—
566C8—1182no—
567C8—1182no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22C2×#1427372511yescharacteristic
576C22D4×C22⋊D5911yescharacteristic
577C22#1683805111yescharacteristic
578C22#1683820011yescharacteristic
579C22#1683820011yescharacteristic
580C22#1683821011yescharacteristic
581C22#1683821311yescharacteristic
582C4—22no—
583C4—41no—
584C4—41no—
585C4—42no—
586C4—41no—
587C4—41no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4—4721no—
592C4—4721no—
593C4—4721no—
594C4—2362no—
595C4C22⋊Q8×D5911yescharacteristic
596C4#1683803111yescharacteristic
597C4#1683806511yescharacteristic
598C4#1683810611yescharacteristic
599C2—42no—
600C2#2142779511yescharacteristic
601C2#2143462211yescharacteristic
602C2#2143464911yescharacteristic
603C2#2143562211yescharacteristic
604C2#2143567711yescharacteristic
605C2#2143665011yescharacteristic
606C2#2143665611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2441C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#2441D59
hypocenterC59#26075
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2441

Complements

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

NH ≅ G/Ntype
1. C2×#16837801C2semidirect
4. #21425577C2semidirect
5. #21425573C2semidirect
6. #21425434C2semidirect
7. #21425431C2semidirect
8. #21424831C2semidirect
9. #21424821C2semidirect
12. #21424262C2semidirect
364. C59#26075semidirect

Maximal subgroups

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

subgroupindex
C2×#168378012
C2×#168374452
#214278202
#214255772
#214255732
#214254342
#214254312
#214248312
#214248212
#214242952
#214242882
#214242622
#214241272
#214238802
C59×#24412
#2607559