Finite Group Databasebeta

← #28608467

Subgroups — #28608467

Order 15104.

846 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×Dic59C211yesmaximal, characteristic
2#21427052C211yesmaximal, characteristic, complemented
3#21427044C211yesmaximal, characteristic
4#21424979C211yesmaximal, characteristic
5#21424740C211yesmaximal, characteristic, complemented
6#21424734C211yesmaximal, characteristic, complemented
7#21424719C211yesmaximal, characteristic
8#21424713C211yesmaximal, characteristic
9#21424254C211yesmaximal, characteristic, complemented
10#21424099C211yesmaximal, characteristic, complemented
11#21424094C211yesmaximal, characteristic
12#21423995C211yesmaximal, characteristic, complemented
13#21423993C211yesmaximal, characteristic, complemented
14#21423964C211yesmaximal, characteristic, complemented
15C59×#1973C211yesmaximal, characteristic
16C2×Q8×Dic59C2211yescharacteristic
17C2×D4×Dic59C2211yescharacteristic
18C2×#14273683—21no—
19SD16×Dic59C418yescomplemented
20C2×#14273661C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273659C2211yescharacteristic
24C2×Dic59⋊C8C2211yescharacteristic
25C2×#14273651C2211yescharacteristic
26C2×#14273651C2211yescharacteristic
27C2×#14273648C2211yescharacteristic
28C2×#14273648C2211yescharacteristic
29C2×#14273637C2211yescharacteristic
30C2×C8×Dic59C2211yescharacteristic
31#16837309C2211yescharacteristic
32#16837305C2211yescharacteristic
33#16837296—21no—
34#16837290—21no—
35#16837287—21no—
36#16837285—21no—
37#16837271C2211yescharacteristic
38#16837271C2211yescharacteristic
39#16837266—21no—
40#16837260—21no—
41C4⋊C4×Dic59C2211yescharacteristic
42C4⋊C4×Dic59C2211yescharacteristic
43C4×Dic59⋊C4C2211yescharacteristic
44C4×C4⋊Dic59C2211yescharacteristic
45C4×C4⋊Dic59C2211yescharacteristic
46#16837253—21no—
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837157C2211yescharacteristic
52#16837156C2211yescharacteristic
53C22×SD16×C59C2211yescharacteristic
54C2×#107×C59C2211yescharacteristic
55C2×#103×C59C2211yescharacteristic
56C2×#102×C59C2211yescharacteristic
57C2×#97×C59C2211yescharacteristic
58C59×#391C2211yescharacteristic
59C59×#390C2211yescharacteristic
60C23×Dic59—21no—
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63Q8×Dic59—22no—
64Q8×Dic59C2×C414yes—
65D4×Dic59—22no—
66D4×Dic59C2×C414yes—
67#14273683—44no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×Dic59⋊C4—21no—
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4D411yescharacteristic
78C2×Dic59⋊C4D411yescharacteristic
79C2×C22⋊Dic59—21no—
80C2×C22⋊Dic59—21no—
81C2×C22⋊Dic59—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59C2311yescharacteristic
88C2×C4×Dic59C2311yescharacteristic
89#14273661—22no—
90#14273660—41no—
91#14273660—41no—
92#14273659C2×C414yes—
93Dic59⋊C8—22no—
94#14273651—41no—
95#14273651C2×C414yes—
96#14273648—41no—
97#14273648C2×C414yes—
98#14273637—22no—
99C8×Dic59C2×C414yes—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95D411yescharacteristic
107C59⋊#95D411yescharacteristic
108C22×Q8×C59C2311yescharacteristic
109C22×D4×C59C2311yescharacteristic
110C2×SD16×C59—22no—
111C2×SD16×C59—22no—
112C2×SD16×C59C2×C414yes—
113C22×C8×C59C2311yescharacteristic
114C22×C8×C59C2311yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C22⋊C4×C59—21no—
118C2×C22⋊C4×C59—21no—
119C2×C42×C59C2311yescharacteristic
120#107×C59—22no—
121#103×C59—41no—
122#102×C59—41no—
123#97×C59—22no—
124#95×C59—21no—
125#95×C59—21no—
126C24×C59—21no—
127C2×Q8×C59—21no—
128C2×Q8×C59—21no—
129C2×Q8×C59—21no—
130C2×Q8×C59—21no—
131C2×Q8×C59—21no—
132C2×Q8×C59C22×C412yes—
133C2×D4×C59—21no—
134C2×D4×C59—21no—
135C2×D4×C59—21no—
136C2×D4×C59—21no—
137C2×D4×C59—21no—
138C2×D4×C59C22×C412yes—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2411yescharacteristic
145C22×Dic59—21no—
146C22×Dic59—21no—
147C22×Dic59—24no—
148C22×Dic59—21no—
149C22×Dic59—21no—
150C22×Dic59—42no—
151C22×Dic59—42no—
152C22×Dic59C2×D411yescharacteristic
153C22×Dic59C2×D411yescharacteristic
154C22×Dic59C2×D411yescharacteristic
155C22×Dic59C2×D411yescharacteristic
156SD16×C59—28no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59C22×C412yes—
161C2×C8×C59#4112yes—
162C2×C8×C59#4112yes—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C22⋊C4×C59—42no—
168C22⋊C4×C59—81no—
169C42×C59—21no—
170C42×C59—21no—
171C22⋊Dic59—24no—
172C22⋊Dic59—42no—
173C22⋊Dic59—81no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—24no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59C22×C412yes—
182C4⋊Dic59C22×C412yes—
183Dic59⋊C4—22no—
184Dic59⋊C4—22no—
185Dic59⋊C4—42no—
186Dic59⋊C4—42no—
187Dic59⋊C4—81no—
188Dic59⋊C4—81no—
189C4×Dic59—21no—
190C4×Dic59—21no—
191C4×Dic59—24no—
192C4×Dic59—42no—
193C4×Dic59—42no—
194C4×Dic59—81no—
195C4×Dic59—81no—
196C4×Dic59C22×C412yes—
197C4×Dic59#4112yes—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8C22×C412yes—
204C23×C59—22no—
205C23×C59—41no—
206C23×C59—41no—
207C23×C59—41no—
208C23×C59—41no—
209C23×C59—41no—
210C23×C59—41no—
211C23×C59C22×D411yescharacteristic
212Q8×C59—22no—
213Q8×C59—22no—
214Q8×C59—41no—
215Q8×C59—41no—
216D4×C59—22no—
217D4×C59—22no—
218D4×C59—41no—
219D4×C59—41no—
220C2×C4×C59—22no—
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—41no—
228C2×C4×C59—41no—
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×C59C23×C411yescharacteristic
236C2×C4×C59C2×#4111yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C59C2×#4111yescharacteristic
239C2×C4×C59H(5,𝔽2)11yescharacteristic
240C2×C4×C592−1+411yescharacteristic
241C2×Dic59—21no—
242C2×Dic59—21no—
243C2×Dic59—22no—
244C2×Dic59—22no—
245C2×Dic59—21no—
246C2×Dic59—21no—
247C2×Dic59—21no—
248C2×Dic59—21no—
249C2×Dic59—21no—
250C2×Dic59—21no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—48no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59C4×D412yes—
263C2×Dic59C4×D412yes—
264C2×Dic59AGL(1,ℤ/8)12yes—
265C2×Dic59#13712yes—
266C8×C59—41no—
267C8×C59#11714yescomplemented
268C59⋊C8—22no—
269C59⋊C8—41no—
270#22811—591nomaximal, Sylow
271C22×C59—42no—
272C22×C59—42no—
273C22×C59—42no—
274C22×C59—81no—
275C22×C59—81no—
276C22×C59—81no—
277C22×C59—81no—
278C22×C59C2×C4×D411yescharacteristic
279C22×C59#54711yescharacteristic
280C22×C59#55011yescharacteristic
281C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C59C2×#13711yescharacteristic
283C22×C59#57611yescharacteristic
284C22×C59#57811yescharacteristic
285C4×C59—41no—
286C4×C59—41no—
287C4×C59—42no—
288C4×C59—81no—
289C4×C59—81no—
290C4×C59C2×#11711yescharacteristic
291C4×C59#52011yescharacteristic
292C4×C59#55511yescharacteristic
293C4×C59#55611yescharacteristic
294Dic59—22no—
295Dic59—41no—
296Dic59—41no—
297Dic59—42no—
298Dic59—42no—
299Dic59—81no—
300Dic59—81no—
301Dic59#44014yes—
302C2×C4×SD16—591no—
303#2355—591no—
304#2354—591no—
305#1973—591no—
306#1973D5911yescharacteristic
307#1928—591no—
308#1928—591no—
309#1925—591no—
310#1925—591no—
311#1845—591no—
312#1803—591no—
313#1802—591no—
314#1776—591no—
315#1776—591no—
316#1770—591no—
317C2×C59—82no—
318C2×C59#235411yescharacteristic
319C2×C59C2×#44011yescharacteristic
320C2×C59#295111yescharacteristic
321C2×C59#335311yescharacteristic
322C2×C59#335711yescharacteristic
323C2×C59#339811yescharacteristic
324C2×C59#339911yescharacteristic
325C22×SD16C2×D5911yescharacteristic
326C2×C4×Q8—591no—
327C2×C4×D4—591no—
328C2×#117—1181no—
329C4×SD16—598no—
330C2×#107—591no—
331C2×#107—591no—
332C2×#107—591no—
333C2×#107C2×D5911yescharacteristic
334C2×#106—591no—
335C2×C4⋊C8—591no—
336C2×#103—591no—
337C2×#103—591no—
338C2×#103C2×D5911yescharacteristic
339C2×#102—591no—
340C2×#102—591no—
341C2×#102C2×D5911yescharacteristic
342C2×#97—591no—
343C2×#97C2×D5911yescharacteristic
344C2×C4×C8—591no—
345#391—591no—
346#391C2×D5911yescharacteristic
347#390—591no—
348#390C2×D5911yescharacteristic
349#388—1181no—
350#388—1181no—
351#387—1181no—
352#387—1181no—
353#384—591no—
354#384—591no—
355#382—1181no—
356#380—1181no—
357C4×C4⋊C4—591no—
358C4×C4⋊C4—591no—
359C4×C4⋊C4—591no—
360C4×C4⋊C4—591no—
361C4×C4⋊C4—591no—
362C4×C4⋊C4—1181no—
363#340—591no—
364#340—591no—
365#340—591no—
366#340—591no—
367#336—591no—
368#336—591no—
369C59#2281111yescharacteristic, complemented, Sylow
370C22×Q8C22×D5911yescharacteristic
371C22×D4C22×D5911yescharacteristic
372C23×C4—1181no—
373C2×SD16—22no—
374C2×SD16—22no—
375C2×SD16C4×D5914yes—
376C22×C8—591no—
377C22×C8—591no—
378C22×C8C22×D5911yescharacteristic
379C22×C8C22×D5911yescharacteristic
380C4×Q8—594no—
381C4×Q8—1182no—
382C4×D4—594no—
383C4×D4—1182no—
384#117—2364no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—591no—
390C2×C4⋊C4—591no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4—1181no—
395C2×C4⋊C4—1181no—
396C2×C4⋊C4C22×D5911yescharacteristic
397C2×C4⋊C4C22×D5911yescharacteristic
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—1181no—
401C2×C22⋊C4—1181no—
402C2×C22⋊C4—1181no—
403C2×C42—591no—
404C2×C42—591no—
405C2×C42—1181no—
406C2×C42—1181no—
407C2×C42—1181no—
408C2×C42—1181no—
409C2×C42—1181no—
410C2×C42C22×D5911yescharacteristic
411#107—22no—
412#107—1182no—
413#107—2361no—
414#107—2361no—
415#106—594no—
416C4⋊C8—1182no—
417#103—41no—
418#103—594no—
419#103—2361no—
420#102—41no—
421#102—594no—
422#102—2361no—
423#97—22no—
424#97—1182no—
425C4×C8—594no—
426#95—21no—
427#95—21no—
428#95—591no—
429#95—591no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433#95—1181no—
434#95—1181no—
435#95—1181no—
436C24—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C2×C4×D5912yes—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4C2×C4×D5912yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—591no—
453C22×C4—591no—
454C22×C4—591no—
455C22×C4—591no—
456C22×C4—1184no—
457C22×C4—1181no—
458C22×C4—1181no—
459C22×C4—1181no—
460C22×C4—1181no—
461C22×C4—2362no—
462C22×C4—2362no—
463C22×C4D4×D5911yescharacteristic
464C22×C4D4×D5911yescharacteristic
465C22×C4C23×D5911yescharacteristic
466SD16—28no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—592no—
471C2×C8—1181no—
472C2×C8—1181no—
473C2×C8—1181no—
474C2×C8—1181no—
475C2×C8—1181no—
476C2×C8C2×C4×D5912yes—
477C2×C8#1174568712yes—
478C2×C8#1174568712yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—592no—
484C4⋊C4—592no—
485C4⋊C4—1182no—
486C4⋊C4—1182no—
487C4⋊C4—1184no—
488C4⋊C4—1181no—
489C4⋊C4—1181no—
490C4⋊C4—1181no—
491C4⋊C4—1181no—
492C4⋊C4—1181no—
493C4⋊C4—1181no—
494C4⋊C4—4721no—
495C4⋊C4—4721no—
496C4⋊C4—2362no—
497C4⋊C4—2362no—
498C22⋊C4—42no—
499C22⋊C4—81no—
500C22⋊C4—1184no—
501C22⋊C4—4721no—
502C22⋊C4—2362no—
503C42—21no—
504C42—21no—
505C42—592no—
506C42—592no—
507C42—1184no—
508C42—1181no—
509C42—1181no—
510C42—4721no—
511C42—4721no—
512C42—2362no—
513C42—2362no—
514C23—22no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C2×D4×D5911yescharacteristic
522Q8—22no—
523Q8—22no—
524Q8—41no—
525Q8—41no—
526D4—22no—
527D4—22no—
528D4—41no—
529D4—41no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—592no—
546C2×C4—592no—
547C2×C4—592no—
548C2×C4—592no—
549C2×C4—1181no—
550C2×C4—1181no—
551C2×C4—1181no—
552C2×C4—1181no—
553C2×C4—1181no—
554C2×C4—1181no—
555C2×C4—1181no—
556C2×C4—1181no—
557C2×C4—1182no—
558C2×C4—1182no—
559C2×C4—2368no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4—2361no—
568C2×C4—2361no—
569C2×C4—2361no—
570C2×C4C22×C4×D5911yescharacteristic
571C2×C4#41×D5911yescharacteristic
572C2×C4C2×#1174568711yescharacteristic
573C2×C4C2×#1174568711yescharacteristic
574C2×C4#1427377211yescharacteristic
575C2×C4#1427377411yescharacteristic
576C8—41no—
577C8—1182no—
578C8—2361no—
579C8#1427368314yescomplemented
580C22—42no—
581C22—42no—
582C22—42no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22C4×D4×D5911yescharacteristic
588C22#1683805211yescharacteristic
589C22#1683807211yescharacteristic
590C22AGL(1,ℤ/8)×D5911yescharacteristic
591C22#137×D5911yescharacteristic
592C22#1683820611yescharacteristic
593C22#1683821811yescharacteristic
594C4—41no—
595C4—41no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4—594no—
600C4—1182no—
601C4—4721no—
602C4—4721no—
603C4—2361no—
604C4—2362no—
605C4—2361no—
606C4—2362no—
607C4C2×#1427368311yescharacteristic
608C4#1683788111yescharacteristic
609C4#1683811611yescharacteristic
610C4#1683812111yescharacteristic
611C2—82no—
612C2#2142704411yescharacteristic
613C2#440×D5911yescharacteristic
614C2#2143296811yescharacteristic
615C2#2143667511yescharacteristic
616C2#2143671711yescharacteristic
617C2#2143712411yescharacteristic
618C2#2143713511yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1973C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenter#1973D59
hypocenterC59#22811
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1973

Complements

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

NH ≅ G/Ntype
2. #21427052C2semidirect
5. #21424740C2semidirect
6. #21424734C2semidirect
9. #21424254C2semidirect
10. #21424099C2semidirect
12. #21423995C2semidirect
13. #21423993C2semidirect
14. #21423964C2semidirect
19. SD16×Dic59C4semidirect
267. C8×C59#117semidirect
369. C59#22811semidirect
579. C8#14273683semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×Dic592
#214270522
#214270442
#214249792
#214247402
#214247342
#214247192
#214247132
#214242542
#214240992
#214240942
#214239952
#214239932
#214239642
C59×#19732
#2281159