Finite Group Databasebeta

← #28591095

Subgroups — #28591095

Order 15104.

1218 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837442C211yesmaximal, characteristic, complemented
2C2×#16837401C211yesmaximal, characteristic
3C2×#16837401C211yesmaximal, characteristic
4#21426586C211yesmaximal, characteristic
5#21424737C212yesmaximal, complemented
6#21424730C212yesmaximal, complemented
7#21424723C212yesmaximal, complemented
8#21424716C212yesmaximal, complemented
9#21423815C211yesmaximal, characteristic, complemented
10#21423790C211yesmaximal, characteristic, complemented
11C2×C4×C4⋊C4×C59C211yesmaximal, characteristic, complemented
12C22×#11745685C2211yescharacteristic
13C22×#11745664C2211yescharacteristic
14C22×#11745664C2211yescharacteristic
15C2×#14273686C2211yescharacteristic
16#16837442—24no—
17C2×#14273660C2212yes—
18C2×#14273658C2212yes—
19#16837401—24no—
20#16837401C418yescomplemented
21C2×#14273650C2212yes—
22C2×#14273650C2212yes—
23C2×#14273647C2212yes—
24C2×#14273647C2212yes—
25#16837308C2211yescharacteristic
26#16837308C2211yescharacteristic
27#16837304C2211yescharacteristic
28#16837304C2211yescharacteristic
29#16837281—21no—
30#16837281—21no—
31C4×(C2×D59)⋊C4—21no—
32C4×(C2×D59)⋊C4—21no—
33#16837164C2212yes—
34#16837164C2212yes—
35#16837157C2212yes—
36#16837157C2212yes—
37C22×C4⋊C4×C59C2211yescharacteristic
38C22×C4⋊C4×C59C2211yescharacteristic
39C22×C42×C59—21no—
40C22×C42×C59C2211yescharacteristic
41C4×C4⋊C4×C59—21no—
42C4×C4⋊C4×C59—21no—
43C4×C4⋊C4×C59C2212yes—
44C4×C4⋊C4×C59C2212yes—
45C2×#95×C59—21no—
46C2×#11745685—22no—
47C2×#11745685—22no—
48C2×#11745685—22no—
49C2×#11745685—22no—
50C2×#11745685—22no—
51C2×#11745685C2×C414yes—
52C22×Dic118C2311yescharacteristic
53C22×D236C2311yescharacteristic
54C22×C22⋊D59—21no—
55C22×C4×D59—21no—
56C2×#11745664—22no—
57C2×#11745664—22no—
58C2×#11745664—22no—
59C2×#11745664—22no—
60C2×#11745664—22no—
61C2×#11745664C2×C414yes—
62C22×C59⋊C8C2312yes—
63C22×C59⋊C8C2312yes—
64#14273686—22no—
65#14273686—22no—
66C2×C4⋊Dic59C2311yescharacteristic
67C2×C4⋊Dic59C2311yescharacteristic
68C2×C22⋊Dic59—21no—
69C2×(C2×D59)⋊C4—21no—
70C2×(C2×D59)⋊C4—21no—
71C2×(C2×D59)⋊C4—21no—
72C2×(C2×D59)⋊C4—21no—
73C2×C4×Dic59—21no—
74#14273660—24no—
75#14273658—24no—
76#14273650—24no—
77#14273650C2×C418yescomplemented
78#14273647—24no—
79#14273647C2×C418yescomplemented
80C59⋊#95—21no—
81C59⋊#95—21no—
82C59⋊#95—21no—
83C59⋊#95—21no—
84C23×C4×C59—21no—
85C23×C4×C59—21no—
86C23×C4×C59D411yescharacteristic
87C23×C4×C59D411yescharacteristic
88C23×C4×C59C2311yescharacteristic
89C2×C4⋊C4×C59—21no—
90C2×C4⋊C4×C59—21no—
91C2×C4⋊C4×C59—22no—
92C2×C4⋊C4×C59—22no—
93C2×C4⋊C4×C59C2×C414yes—
94C2×C4⋊C4×C59D414yes—
95C2×C4⋊C4×C59C2312yes—
96C2×C4⋊C4×C59C2312yes—
97C2×C42×C59—22no—
98C2×C42×C59—22no—
99C2×C42×C59—22no—
100C2×C42×C59—22no—
101C2×C42×C59—44no—
102C2×C42×C59D411yescharacteristic
103C2×C42×C59D411yescharacteristic
104C2×C42×C59C2311yescharacteristic
105C2×C42×C59C2311yescharacteristic
106#95×C59—22no—
107#95×C59—22no—
108C24×C59C2×D411yescharacteristic
109C23×D59—21no—
110C22×C4×C59—22no—
111C22×C4×C59—22no—
112C22×C4×C59—22no—
113C22×C4×C59—24no—
114C22×C4×C59—24no—
115C22×C4×C59—22no—
116C22×C4×C59—22no—
117C22×C4×C59—22no—
118C22×C4×C59—41no—
119C22×C4×C59—41no—
120C22×C4×C59—42no—
121C22×C4×C59—42no—
122C22×C4×C59—42no—
123C22×C4×C59—42no—
124C22×C4×C59C22×C412yes—
125C22×C4×C59C2×D411yescharacteristic
126C22×C4×C59C2×D411yescharacteristic
127C22×C4×C59C2×D411yescharacteristic
128C22×C4×C59C2×D411yescharacteristic
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59C2×D412yes—
131C22×C4×C59#4112yes—
132C22×C4×C59#4112yes—
133C22×C4×C59#4112yes—
134C22×C4×C59#4112yes—
135C22×C4×C59C2411yescharacteristic
136C2×C22⋊D59—24no—
137C2×C22⋊D59—42no—
138C2×C22⋊D59—42no—
139C22×Dic59—21no—
140C22×Dic59—21no—
141C22×Dic59—21no—
142#11745685—28no—
143#11745685—28no—
144#11745685—44no—
145#11745685—44no—
146C2×D236—21no—
147C2×D236—21no—
148C2×D236—21no—
149C2×D236—21no—
150C2×D236—21no—
151C2×D236C22×C412yes—
152C2×C4×D59—24no—
153C2×C4×D59—42no—
154C2×C4×D59—42no—
155C2×Dic118—21no—
156C2×Dic118—21no—
157C2×Dic118—21no—
158C2×Dic118—21no—
159C2×Dic118—21no—
160C2×Dic118C22×C412yes—
161C4⋊C4×C59—24no—
162C4⋊C4×C59—24no—
163C4⋊C4×C59—22no—
164C4⋊C4×C59—22no—
165C4⋊C4×C59C22×C414yes—
166C4⋊C4×C59C2×D414yes—
167C42×C59—22no—
168C42×C59—22no—
169C42×C59—21no—
170C42×C59—21no—
171C42×C59—21no—
172C42×C59—21no—
173C42×C59—44no—
174C42×C59—44no—
175C22⋊Dic59—42no—
176(C2×D59)⋊C4—42no—
177(C2×D59)⋊C4—42no—
178(C2×D59)⋊C4—42no—
179(C2×D59)⋊C4—42no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4×Dic59—42no—
185#11745664—28no—
186#11745664—44no—
187C2×C59⋊C8—22no—
188C2×C59⋊C8—22no—
189C2×C59⋊C8—22no—
190C2×C59⋊C8—22no—
191C2×C59⋊C8—22no—
192C2×C59⋊C8C22×C414yes—
193C23×C59—21no—
194C23×C59—21no—
195C23×C59—21no—
196C23×C59—21no—
197C23×C59C4×D412yes—
198C23×C59#12112yes—
199C23×C59#12312yes—
200C23×C59C22×D411yescharacteristic
201C22×D59—22no—
202C22×D59—41no—
203C22×D59—41no—
204C22×D59—41no—
205C22×D59—41no—
206C22×D59—41no—
207C22×D59—41no—
208C2×C4×C59—28no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—24no—
214C2×C4×C59—24no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—44no—
222C2×C4×C59—48no—
223C2×C4×C59—48no—
224C2×C4×C59—42no—
225C2×C4×C59—42no—
226C2×C4×C59—42no—
227C2×C4×C59—42no—
228C2×C4×C59#11714yes—
229C2×C4×C59C4×D412yes—
230C2×C4×C59C4×D414yes—
231C2×C4×C59#12112yes—
232C2×C4×C59#12312yes—
233C2×C4×C59#12414yes—
234C2×C4×C59#12614yes—
235C2×C4×C59AGL(1,ℤ/8)12yes—
236C2×C4×C59#13712yes—
237C2×C4×C59C23×C411yescharacteristic
238C2×C4×C59C22×D411yescharacteristic
239C2×C4×C59C2×#4111yescharacteristic
240C2×C4×C59C2×#4111yescharacteristic
241C2×C4×C59C2×#4111yescharacteristic
242C2×C4×C59C2×#4111yescharacteristic
243C22⋊D59—48no—
244C2×Dic59—22no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253D236—22no—
254D236—22no—
255D236—41no—
256D236—41no—
257C4×D59—48no—
258Dic118—22no—
259Dic118—22no—
260Dic118—41no—
261Dic118—41no—
262C59⋊C8—24no—
263C59⋊C8—42no—
264#21059—591nomaximal, Sylow
265C22×C59—22no—
266C22×C59—22no—
267C22×C59—22no—
268C22×C59—22no—
269C22×C59—22no—
270C22×C59—22no—
271C22×C59#38814yes—
272C22×C59C2×C4×D411yescharacteristic
273C22×C59C2×#12111yescharacteristic
274C22×C59C2×#12311yescharacteristic
275C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C59C2×#13711yescharacteristic
277C22×C59#57511yescharacteristic
278C22×C59#57511yescharacteristic
279C2×D59—42no—
280C2×D59—42no—
281C2×D59—42no—
282C2×D59—81no—
283C2×D59—81no—
284C2×D59—81no—
285C2×D59—81no—
286C4×C59—22no—
287C4×C59—22no—
288C4×C59—41no—
289C4×C59—41no—
290C4×C59—44no—
291C4×C59—44no—
292C4×C59#38814yes—
293C4×C59C2×#11711yescharacteristic
294C4×C59C2×C4×D411yescharacteristic
295C4×C59C2×#12411yescharacteristic
296C4×C59C2×#12611yescharacteristic
297Dic59—42no—
298Dic59—81no—
299Dic59—81no—
300C2×#428—591no—
301C2×#417—591no—
302C2×#417—591no—
303#2297—591no—
304C2×C4×C4⋊C4D5911yescharacteristic, complemented
305#1928—592no—
306#1927—592no—
307#1926—592no—
308#1925—592no—
309#1736—591no—
310#1728—591no—
311C2×C59—24no—
312C2×C59C2×#38811yescharacteristic
313C2×C59C4×AGL(1,ℤ/8)11yescharacteristic
314C2×C59C4×#13711yescharacteristic
315C2×C59#303611yescharacteristic
316C2×C59#303711yescharacteristic
317C2×C59#309211yescharacteristic
318C2×C59#309311yescharacteristic
319D59—82no—
320C22×#41—591no—
321C22×M16—591no—
322C22×M16—591no—
323C2×#117—591no—
324C22×C4⋊C4C2×D5911yescharacteristic
325C22×C4⋊C4C2×D5911yescharacteristic
326C22×C42—21no—
327C22×C42C2×D5911yescharacteristic
328#428—1184no—
329C2×#107—592no—
330C2×#106—592no—
331#417—598no—
332#417—1184no—
333C2×#103—592no—
334C2×#103—592no—
335C2×#102—592no—
336C2×#102—592no—
337#391—591no—
338#391—591no—
339#390—591no—
340#390—591no—
341#386—1181no—
342#386—1181no—
343C4×C4⋊C4—21no—
344C4×C4⋊C4—21no—
345C4×C4⋊C4C2×D5912yes—
346C4×C4⋊C4C2×D5912yes—
347C4×C22⋊C4—1181no—
348C4×C22⋊C4—1181no—
349C2×#95—21no—
350#340—592no—
351#340—592no—
352#336—592no—
353#336—592no—
354C59#2105911yescharacteristic, complemented, Sylow
355C2×#41—594no—
356C2×#41—1182no—
357C2×#41—1182no—
358C2×#41—1182no—
359C2×#41—1182no—
360C2×#41—1182no—
361C22×Q8—591no—
362C22×D4—591no—
363C22×D4—1181no—
364C23×C4—21no—
365C23×C4—21no—
366C23×C4—1181no—
367C23×C4C22⋊D5911yescharacteristic
368C23×C4C22⋊D5911yescharacteristic
369C23×C4C22×D5911yescharacteristic
370C2×M16—594no—
371C2×M16—1182no—
372C2×M16—1182no—
373C2×M16—1182no—
374C2×M16—1182no—
375C2×M16—1182no—
376C22×C8—592no—
377C22×C8—592no—
378#117—1182no—
379#117—1182no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—591no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4C4×D5914yes—
387C2×C4⋊C4D23614yes—
388C2×C4⋊C4C22×D5912yes—
389C2×C4⋊C4C22×D5912yes—
390C2×C22⋊C4—1181no—
391C2×C22⋊C4—1181no—
392C2×C22⋊C4—1181no—
393C2×C22⋊C4—1181no—
394C2×C22⋊C4—1181no—
395C2×C42—22no—
396C2×C42—22no—
397C2×C42—22no—
398C2×C42—22no—
399C2×C42—44no—
400C2×C42—1181no—
401C2×C42C22⋊D5911yescharacteristic
402C2×C42C22⋊D5911yescharacteristic
403C2×C42C22×D5911yescharacteristic
404C2×C42C22×D5911yescharacteristic
405#107—1184no—
406#106—1184no—
407#103—598no—
408#103—1184no—
409#102—598no—
410#102—1184no—
411#95—22no—
412#95—22no—
413#95—1181no—
414#95—1181no—
415#95—1181no—
416#95—1181no—
417C24—1181no—
418C24C2×C22⋊D5911yescharacteristic
419#41—1188no—
420#41—1188no—
421#41—2364no—
422#41—2364no—
423C2×Q8—592no—
424C2×Q8—1181no—
425C2×Q8—1181no—
426C2×Q8—1181no—
427C2×Q8—1181no—
428C2×Q8—1181no—
429C2×D4—592no—
430C2×D4—1184no—
431C2×D4—1181no—
432C2×D4—1181no—
433C2×D4—1181no—
434C2×D4—1181no—
435C2×D4—1181no—
436C2×D4—2362no—
437C2×D4—2362no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—1181no—
453C22×C4—1184no—
454C22×C4—1181no—
455C22×C4—1181no—
456C22×C4—2362no—
457C22×C4—2362no—
458C22×C4C2×C4×D5912yes—
459C22×C4C2×D23612yes—
460C22×C4#1174568512yes—
461C22×C4#1174568512yes—
462C22×C4#1174568512yes—
463C22×C4#1174568512yes—
464C22×C4C2×C22⋊D5911yescharacteristic
465C22×C4C2×C22⋊D5911yescharacteristic
466C22×C4C2×C22⋊D5911yescharacteristic
467C22×C4C2×C22⋊D5911yescharacteristic
468C22×C4C2×C22⋊D5911yescharacteristic
469C22×C4C23×D5911yescharacteristic
470M16—1188no—
471M16—2364no—
472C2×C8—594no—
473C2×C8—1182no—
474C2×C8—1182no—
475C2×C8—1182no—
476C2×C8—1182no—
477C2×C8—1182no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—1181no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C4⋊C4—1181no—
486C4⋊C4C2×C4×D5914yes—
487C4⋊C4C2×D23614yes—
488C22⋊C4—2362no—
489C22⋊C4—2362no—
490C22⋊C4—2362no—
491C22⋊C4—2362no—
492C22⋊C4—2362no—
493C42—22no—
494C42—22no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—44no—
500C42—44no—
501C42—2362no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—1182no—
507C23—2361no—
508C23—2361no—
509C23—2361no—
510C23—2361no—
511C23—2361no—
512C23—2361no—
513C23C4×C22⋊D5912yes—
514C23#1427370112yes—
515C23#1427371012yes—
516C23C22×C22⋊D5911yescharacteristic
517Q8—1182no—
518Q8—1182no—
519Q8—2361no—
520Q8—2361no—
521D4—1182no—
522D4—1182no—
523D4—2368no—
524D4—2361no—
525D4—2361no—
526C2×C4—28no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—24no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—24no—
539C2×C4—48no—
540C2×C4—44no—
541C2×C4—48no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—1182no—
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—2368no—
556C2×C4#1427368414yes—
557C2×C4C4×C22⋊D5912yes—
558C2×C4C4×D23614yes—
559C2×C4#1427370112yes—
560C2×C4#1427371012yes—
561C2×C4#1427371514yes—
562C2×C4#1427372114yes—
563C2×C4#1427375312yes—
564C2×C4#1427375812yes—
565C2×C4C22×C4×D5911yescharacteristic
566C2×C4C22×D23611yescharacteristic
567C2×C4C2×#1174568511yescharacteristic
568C2×C4C2×#1174568511yescharacteristic
569C2×C4C2×#1174568511yescharacteristic
570C2×C4C2×#1174568511yescharacteristic
571C8—1184no—
572C8—2362no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—4721no—
580C22—4721no—
581C22—4721no—
582C22—4721no—
583C22—2362no—
584C22—2362no—
585C22—2362no—
586C22#1683729314yes—
587C22C2×C4×C22⋊D5911yescharacteristic
588C22C2×#1427370111yescharacteristic
589C22C2×#1427371011yescharacteristic
590C22C2×#1427375311yescharacteristic
591C22C2×#1427375811yescharacteristic
592C22#1683820011yescharacteristic
593C22#1683820011yescharacteristic
594C4—22no—
595C4—22no—
596C4—44no—
597C4—41no—
598C4—41no—
599C4—44no—
600C4—4721no—
601C4—4721no—
602C4—2362no—
603C4#1683729314yes—
604C4C2×#1427368411yescharacteristic
605C4C2×C4×D23611yescharacteristic
606C4C2×#1427371511yescharacteristic
607C4C2×#1427372111yescharacteristic
608C2—24no—
609C2—4722no—
610C2C2×#1683729311yescharacteristic
611C2C4×#1427375311yescharacteristic
612C2C4×#1427375811yescharacteristic
613C2#2143356011yescharacteristic
614C2#2143356811yescharacteristic
615C2#2143401911yescharacteristic
616C2#2143402711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C4×C4⋊C4×C59C2
socleC23×C59C22×D4
prime radicalC22×#11745685C22
cosocleC22×C4×C59C24
hypercenterC2×C4×C4⋊C4D59
hypocenterC59#21059
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#16837442C2semidirect
5. #21424737C2semidirect
6. #21424730C2semidirect
7. #21424723C2semidirect
8. #21424716C2semidirect
9. #21423815C2semidirect
10. #21423790C2semidirect
11. C2×C4×C4⋊C4×C59C2semidirect
20. #16837401C4semidirect
77. #14273650C2×C4semidirect
79. #14273647C2×C4semidirect
304. C2×C4×C4⋊C4D59semidirect
354. C59#21059semidirect

Maximal subgroups

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

subgroupindex
C2×#168374422
C2×#168374012
#214265862
#214247372
#214247302
#214247232
#214247162
#214238152
#214237902
C2×C4×C4⋊C4×C592
#2105959