Finite Group Databasebeta

← #28608131

Subgroups — #28608131

Order 15104.

866 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837503C211yesmaximal, characteristic
2#21427044C211yesmaximal, characteristic
3#21427014C211yesmaximal, characteristic, complemented
4#21424790C211yesmaximal, characteristic
5#21424782C211yesmaximal, characteristic
6#21424687C211yesmaximal, characteristic, complemented
7#21424668C211yesmaximal, characteristic, complemented
8#21424453C211yesmaximal, characteristic
9#21424228C211yesmaximal, characteristic, complemented
10#21424104C211yesmaximal, characteristic
11#21424099C211yesmaximal, characteristic, complemented
12#21423967C211yesmaximal, characteristic, complemented
13#21423928C211yesmaximal, characteristic, complemented
14C4×#14273651C211yesmaximal, characteristic, complemented
15C59×#1880C211yesmaximal, characteristic
16C2×Q8×Dic59C2211yescharacteristic
17C2×Q8×Dic59C2211yescharacteristic
18C2×D4×Dic59C2211yescharacteristic
19C2×#14273683—21no—
20#16837503C418yescomplemented
21C2×#14273661C2211yescharacteristic
22C2×Dic59⋊C8C2211yescharacteristic
23C2×#14273651C2211yescharacteristic
24C2×#14273651C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×#14273637C2211yescharacteristic
28C2×#14273637C2211yescharacteristic
29C2×C4×C59⋊C8C2211yescharacteristic
30#16837305C2211yescharacteristic
31#16837300—21no—
32#16837300C2211yescharacteristic
33#16837298—21no—
34#16837298C2211yescharacteristic
35#16837296—21no—
36#16837290—21no—
37#16837271C2211yescharacteristic
38#16837267—21no—
39#16837267C2211yescharacteristic
40#16837260—21no—
41C4⋊C4×Dic59C2211yescharacteristic
42C4×Dic59⋊C4—21no—
43C4×Dic59⋊C4C2211yescharacteristic
44C4×C4⋊Dic59C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837156C2211yescharacteristic
53C22×SD16×C59C2211yescharacteristic
54C2×C4×Q8×C59C2211yescharacteristic
55C2×#103×C59C2211yescharacteristic
56C2×#102×C59C2211yescharacteristic
57C59×#390C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#336C2211yescharacteristic
60C23×Dic59—21no—
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63Q8×Dic59—22no—
64Q8×Dic59—22no—
65Q8×Dic59—41no—
66Q8×Dic59C2×C414yes—
67D4×Dic59—22no—
68D4×Dic59C2×C414yes—
69#14273683—44no—
70C2×C4⋊Dic59—21no—
71C2×C4⋊Dic59—21no—
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4D411yescharacteristic
81C2×Dic59⋊C4D411yescharacteristic
82C2×C22⋊Dic59—21no—
83C2×C22⋊Dic59—21no—
84C2×C22⋊Dic59—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59—21no—
89C2×C4×Dic59C2311yescharacteristic
90C2×C4×Dic59C2311yescharacteristic
91#14273661C2×C414yes—
92Dic59⋊C8—22no—
93#14273651—22no—
94#14273651C2×C414yes—
95#14273648—41no—
96#14273648C2×C414yes—
97#14273637—41no—
98#14273637C2×C414yes—
99C4×C59⋊C8—41no—
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
115C4×Q8×C59—22no—
116C4×Q8×C59—41no—
117C2×C4⋊C4×C59—21no—
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C4⋊C4×C59C2311yescharacteristic
120C2×C22⋊C4×C59—21no—
121C2×C22⋊C4×C59—21no—
122C2×C42×C59—21no—
123C2×C42×C59C2311yescharacteristic
124#103×C59—22no—
125#102×C59—41no—
126C24×C59—21no—
127C2×Q8×C59—21no—
128C2×Q8×C59—21no—
129C2×Q8×C59—21no—
130C2×Q8×C59C22×C412yes—
131C2×Q8×C59#4112yes—
132C2×Q8×C59#4112yes—
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×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59C22×C412yes—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—42no—
168C22⋊C4×C59—42no—
169C22⋊C4×C59—81no—
170C42×C59—21no—
171C42×C59—21no—
172C42×C59—42no—
173C22⋊Dic59—24no—
174C22⋊Dic59—42no—
175C22⋊Dic59—81no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—24no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—42no—
184C4⋊Dic59C22×C412yes—
185C4⋊Dic59C22×C412yes—
186Dic59⋊C4—22no—
187Dic59⋊C4—41no—
188Dic59⋊C4—42no—
189Dic59⋊C4—42no—
190Dic59⋊C4—42no—
191Dic59⋊C4—81no—
192C4×Dic59—21no—
193C4×Dic59—21no—
194C4×Dic59—24no—
195C4×Dic59—42no—
196C4×Dic59—42no—
197C4×Dic59—42no—
198C4×Dic59C22×C412yes—
199C4×Dic59#4112yes—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8C22×C412yes—
206C23×C59—22no—
207C23×C59—41no—
208C23×C59—41no—
209C23×C59—41no—
210C23×C59—41no—
211C23×C59—41no—
212C23×C59—41no—
213C23×C59C22×D411yescharacteristic
214Q8×C59—22no—
215Q8×C59—22no—
216Q8×C59—22no—
217Q8×C59#11714yescomplemented
218D4×C59—22no—
219D4×C59—22no—
220D4×C59—41no—
221D4×C59—41no—
222C2×C4×C59—22no—
223C2×C4×C59—22no—
224C2×C4×C59—21no—
225C2×C4×C59—22no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
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×C59AGL(1,ℤ/8)12yes—
237C2×C4×C59C23×C411yescharacteristic
238C2×C4×C59C2×#4111yescharacteristic
239C2×C4×C59C2×#4111yescharacteristic
240C2×C4×C59C2×#4111yescharacteristic
241C2×C4×C59H(5,𝔽2)11yescharacteristic
242C2×C4×C592−1+411yescharacteristic
243C2×Dic59—21no—
244C2×Dic59—21no—
245C2×Dic59—21no—
246C2×Dic59—21no—
247C2×Dic59—22no—
248C2×Dic59—22no—
249C2×Dic59—21no—
250C2×Dic59—21no—
251C2×Dic59—21no—
252C2×Dic59—21no—
253C2×Dic59—21no—
254C2×Dic59—21no—
255C2×Dic59—48no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59C4×D412yes—
267C2×Dic59C4×D412yes—
268C8×C59—22no—
269C8×C59—41no—
270C59⋊C8—22no—
271C59⋊C8—41no—
272#22781—591nomaximal, Sylow
273C22×C59—42no—
274C22×C59—42no—
275C22×C59—42no—
276C22×C59—81no—
277C22×C59—81no—
278C22×C59—81no—
279C22×C59—81no—
280C22×C59C2×C4×D411yescharacteristic
281C22×C59#54611yescharacteristic
282C22×C59#54811yescharacteristic
283C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C59#57511yescharacteristic
285C22×C59#57711yescharacteristic
286C22×C59#57811yescharacteristic
287C4×C59—24no—
288C4×C59—22no—
289C4×C59—41no—
290C4×C59—81no—
291C4×C59—81no—
292C4×C59C2×#11711yescharacteristic
293C4×C59#52011yescharacteristic
294C4×C59#55211yescharacteristic
295C4×C59#55311yescharacteristic
296Dic59—22no—
297Dic59—22no—
298Dic59—42no—
299Dic59—42no—
300Dic59—41no—
301Dic59—41no—
302Dic59—81no—
303Dic59—81no—
304C2×#440—591no—
305#2354—591no—
306#2351—591no—
307#1936—591no—
308#1934—591no—
309#1921—591no—
310#1918—591no—
311#1880—591no—
312#1880D5911yescharacteristic
313#1836—591no—
314#1804—591no—
315#1803—591no—
316#1771—591no—
317#1763—591no—
318C4×#103—591no—
319C2×C59—82no—
320C2×C59#235211yescharacteristic
321C2×C59C4×AGL(1,ℤ/8)11yescharacteristic
322C2×C59#294711yescharacteristic
323C2×C59#330811yescharacteristic
324C2×C59#332111yescharacteristic
325C2×C59#336811yescharacteristic
326C2×C59#337711yescharacteristic
327C22×SD16C2×D5911yescharacteristic
328C2×C4×Q8—591no—
329C2×C4×Q8—591no—
330C2×C4×Q8C2×D5911yescharacteristic
331C2×C4×D4—591no—
332C2×#117—1181no—
333#440—598no—
334C2×#107—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×#97—591no—
344C2×C4×C8—591no—
345#390—591no—
346#390C2×D5911yescharacteristic
347#389—591no—
348#389—591no—
349#389—1181no—
350#389—1181no—
351#388—1181no—
352#388—1181no—
353#384—591no—
354#382—591no—
355#382—1181no—
356#380—1181no—
357C4×C4⋊C4—591no—
358C4×C4⋊C4—591no—
359C4×C4⋊C4—591no—
360C4×C4⋊C4—1181no—
361#340—591no—
362#340—591no—
363#340—591no—
364#340—591no—
365#340—591no—
366#340—591no—
367#340—591no—
368#340C2×D5911yescharacteristic
369#336—591no—
370#336C2×D5911yescharacteristic
371C59#2278111yescharacteristic, complemented, Sylow
372C22×Q8C22×D5911yescharacteristic
373C22×D4C22×D5911yescharacteristic
374C23×C4—1181no—
375C2×SD16—22no—
376C2×SD16—22no—
377C2×SD16C4×D5914yes—
378C22×C8—591no—
379C22×C8—591no—
380C22×C8C22×D5911yescharacteristic
381C22×C8C22×D5911yescharacteristic
382C4×Q8—22no—
383C4×Q8—41no—
384C4×Q8—594no—
385C4×Q8—1182no—
386C4×Q8—1182no—
387C4×Q8—2361no—
388C4×D4—594no—
389C4×D4—1182no—
390#117—2364no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—591no—
393C2×C4⋊C4—591no—
394C2×C4⋊C4—591no—
395C2×C4⋊C4—591no—
396C2×C4⋊C4—591no—
397C2×C4⋊C4—591no—
398C2×C4⋊C4—1181no—
399C2×C4⋊C4—1181no—
400C2×C4⋊C4—1181no—
401C2×C4⋊C4—1181no—
402C2×C4⋊C4—1181no—
403C2×C4⋊C4—1181no—
404C2×C4⋊C4C22×D5911yescharacteristic
405C2×C4⋊C4C22×D5911yescharacteristic
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—1181no—
409C2×C22⋊C4—1181no—
410C2×C22⋊C4—1181no—
411C2×C42—21no—
412C2×C42—591no—
413C2×C42—591no—
414C2×C42—1181no—
415C2×C42—1181no—
416C2×C42—1181no—
417C2×C42—1181no—
418C2×C42C22×D5911yescharacteristic
419#107—594no—
420C4⋊C8—1182no—
421#103—22no—
422#103—594no—
423#103—1182no—
424#102—41no—
425#102—594no—
426#102—2361no—
427#97—594no—
428#97—2361no—
429C4×C8—2361no—
430#95—591no—
431#95—591no—
432#95—1181no—
433#95—1181no—
434#95—1181no—
435#95—1181no—
436#95—1181no—
437#95—1181no—
438C24—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C2×C4×D5912yes—
443C2×Q8#1174568712yes—
444C2×Q8#1174568712yes—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4C2×C4×D5912yes—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—591no—
455C22×C4—591no—
456C22×C4—591no—
457C22×C4—591no—
458C22×C4—1184no—
459C22×C4—1181no—
460C22×C4—1181no—
461C22×C4—1181no—
462C22×C4—1181no—
463C22×C4—2362no—
464C22×C4—2362no—
465C22×C4D4×D5911yescharacteristic
466C22×C4D4×D5911yescharacteristic
467C22×C4C23×D5911yescharacteristic
468SD16—28no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—592no—
475C2×C8—1181no—
476C2×C8—1181no—
477C2×C8—1181no—
478C2×C8—1181no—
479C2×C8—1181no—
480C2×C8C2×C4×D5912yes—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—42no—
486C4⋊C4—592no—
487C4⋊C4—592no—
488C4⋊C4—1184no—
489C4⋊C4—1182no—
490C4⋊C4—1181no—
491C4⋊C4—1181no—
492C4⋊C4—1181no—
493C4⋊C4—1181no—
494C4⋊C4—1181no—
495C4⋊C4—1181no—
496C4⋊C4—4721no—
497C4⋊C4—2361no—
498C4⋊C4—2362no—
499C4⋊C4—2362no—
500C4⋊C4—2362no—
501C4⋊C4—2362no—
502C22⋊C4—42no—
503C22⋊C4—81no—
504C22⋊C4—1184no—
505C22⋊C4—4721no—
506C22⋊C4—2362no—
507C42—21no—
508C42—21no—
509C42—42no—
510C42—592no—
511C42—592no—
512C42—1184no—
513C42—1181no—
514C42—1181no—
515C42—2362no—
516C42—2362no—
517C42—2362no—
518C23—22no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23C2×D4×D5911yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—22no—
529Q8#1427368314yescomplemented
530D4—22no—
531D4—22no—
532D4—41no—
533D4—41no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—592no—
549C2×C4—592no—
550C2×C4—1181no—
551C2×C4—1181no—
552C2×C4—1181no—
553C2×C4—1181no—
554C2×C4—1181no—
555C2×C4—1181no—
556C2×C4—1181no—
557C2×C4—1181no—
558C2×C4—1181no—
559C2×C4—1181no—
560C2×C4—1182no—
561C2×C4—1182no—
562C2×C4—2368no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4—2361no—
568C2×C4—2361no—
569C2×C4—2361no—
570C2×C4—2361no—
571C2×C4—2361no—
572C2×C4—2361no—
573C2×C4#1427375412yes—
574C2×C4C22×C4×D5911yescharacteristic
575C2×C4#41×D5911yescharacteristic
576C2×C4C2×#1174568711yescharacteristic
577C2×C4C2×#1174568711yescharacteristic
578C2×C4#1427377211yescharacteristic
579C2×C4#1427377411yescharacteristic
580C8—22no—
581C8—41no—
582C8—1182no—
583C8—2361no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22C4×D4×D5911yescharacteristic
592C22#1683803911yescharacteristic
593C22#1683805811yescharacteristic
594C22C2×#1427375411yescharacteristic
595C22#1683819911yescharacteristic
596C22#1683821211yescharacteristic
597C22#1683821811yescharacteristic
598C4—24no—
599C4—22no—
600C4—41no—
601C4—81no—
602C4—81no—
603C4—1182no—
604C4—1182no—
605C4—4721no—
606C4—4721no—
607C4—2362no—
608C4—2362no—
609C4—2361no—
610C4—2361no—
611C4C2×#1427368311yescharacteristic
612C4#1683788111yescharacteristic
613C4#1683808911yescharacteristic
614C4#1683809511yescharacteristic
615C2—82no—
616C2#2142702311yescharacteristic
617C2C4×#1427375411yescharacteristic
618C2#2143293611yescharacteristic
619C2#2143609111yescharacteristic
620C2#2143628611yescharacteristic
621C2#2143682911yescharacteristic
622C2#2143694211yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1880C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenter#1880D59
hypocenterC59#22781
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1880

Complements

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

NH ≅ G/Ntype
3. #21427014C2semidirect
6. #21424687C2semidirect
7. #21424668C2semidirect
9. #21424228C2semidirect
11. #21424099C2semidirect
12. #21423967C2semidirect
13. #21423928C2semidirect
14. C4×#14273651C2semidirect
20. #16837503C4semidirect
217. Q8×C59#117semidirect
371. C59#22781semidirect
529. Q8#14273683semidirect

Maximal subgroups

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

subgroupindex
C2×#168375032
#214270442
#214270142
#214247902
#214247822
#214246872
#214246682
#214244532
#214242282
#214241042
#214240992
#214239672
#214239282
C4×#142736512
C59×#18802
#2278159