Finite Group Databasebeta

← #28634384

Subgroups — #28634384

Order 15104.

850 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837629C211yesmaximal, characteristic
2C2×C4×#11745670C211yesmaximal, characteristic, complemented
3#21427599C211yesmaximal, characteristic, complemented
4#21425515C211yesmaximal, characteristic, complemented
5#21425509C211yesmaximal, characteristic, complemented
6#21425299C211yesmaximal, characteristic, complemented
7#21425275C211yesmaximal, characteristic
8#21424949C211yesmaximal, characteristic, complemented
9#21424821C211yesmaximal, characteristic, complemented
10#21424349C211yesmaximal, characteristic, complemented
11#21424343C211yesmaximal, characteristic, complemented
12#21424258C211yesmaximal, characteristic, complemented
13#21424119C211yesmaximal, characteristic, complemented
14C22⋊C4×C59⋊C8C211yesmaximal, characteristic, complemented
15C59×#2420C211yesmaximal, characteristic
16C22×#11745670C2211yescharacteristic
17C23×C59⋊C8C2211yescharacteristic
18C2×#14273706C2211yescharacteristic
19C2×C4×Dic118C2211yescharacteristic
20#16837629—24no—
21C4×#11745670—24no—
22C2×#14273660C2211yescharacteristic
23C2×#14273660C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×#14273658C2211yescharacteristic
26C2×#14273658C2211yescharacteristic
27C2×#14273650C2211yescharacteristic
28C2×#14273650C2211yescharacteristic
29C2×#14273650C2211yescharacteristic
30C2×#14273648C2211yescharacteristic
31C2×#14273648C2211yescharacteristic
32C2×#14273648C2211yescharacteristic
33C2×C22⋊(C59⋊C8)C2211yescharacteristic
34C2×C4×C59⋊C8C2211yescharacteristic
35C2×C4×C59⋊C8C2211yescharacteristic
36#16837343—21no—
37#16837330—21no—
38#16837308C2211yescharacteristic
39#16837300C2211yescharacteristic
40#16837296—21no—
41#16837284—21no—
42#16837268C2211yescharacteristic
43#16837262C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×#121×C59C2211yescharacteristic
51C2×C4×D4×C59C2211yescharacteristic
52C59×#397—21no—
53C59×#394—21no—
54C59×#390C2211yescharacteristic
55C59×#389C2211yescharacteristic
56C59×#386—21no—
57C59×#382C2211yescharacteristic
58C59×#381—21no—
59C59×#380C2211yescharacteristic
60C4×C22⋊C4×C59C2211yescharacteristic
61C22×Dic118C2311yescharacteristic
62C2×#11745670—22no—
63C2×#11745670—22no—
64C2×#11745670D414yes—
65C22×C59⋊C8—22no—
66C22×C59⋊C8—22no—
67C22×C59⋊C8—22no—
68C22×C59⋊C8C2311yescharacteristic
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71C22×C59⋊C8C2311yescharacteristic
72#14273706—22no—
73#14273706—41no—
74C4×Dic118—22no—
75C4×Dic118—41no—
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C4⋊Dic59C2311yescharacteristic
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—21no—
82C2×C22⋊Dic59—21no—
83C2×C22⋊Dic59—21no—
84C2×C4×Dic59—21no—
85#14273660—41no—
86#14273660—41no—
87#14273658—22no—
88#14273658—22no—
89#14273658—22no—
90#14273650—22no—
91#14273650—22no—
92#14273650—22no—
93#14273648—22no—
94#14273648—22no—
95#14273648—22no—
96C22⋊(C59⋊C8)—22no—
97C4×C59⋊C8—22no—
98C4×C59⋊C8—22no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C22×D4×C59—21no—
106C22×D4×C59C2311yescharacteristic
107C23×C4×C59—21no—
108C23×C4×C59C2311yescharacteristic
109#121×C59—22no—
110#121×C59—41no—
111C4×D4×C59—22no—
112C4×D4×C59—41no—
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C22⋊C4×C59—21no—
118C2×C22⋊C4×C59—21no—
119C2×C22⋊C4×C59—21no—
120C2×C22⋊C4×C59—21no—
121C2×C22⋊C4×C59—21no—
122C2×C22⋊C4×C59D411yescharacteristic
123C2×C22⋊C4×C59D411yescharacteristic
124C2×C42×C59C2311yescharacteristic
125C2×C42×C59C2311yescharacteristic
126#95×C59—21no—
127#95×C59—21no—
128#95×C59—21no—
129#95×C59—21no—
130#95×C59D411yescharacteristic
131#95×C59D411yescharacteristic
132C24×C59—21no—
133C24×C59C2×D411yescharacteristic
134C2×D4×C59—21no—
135C2×D4×C59—21no—
136C2×D4×C59—21no—
137C2×D4×C59—21no—
138C2×D4×C59—21no—
139C2×D4×C59—42no—
140C2×D4×C59—42no—
141C2×D4×C59—81no—
142C2×D4×C59C2×D412yes—
143C22×C4×C59—21no—
144C22×C4×C59—21no—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59—21no—
149C22×C4×C59—21no—
150C22×C4×C59—21no—
151C22×C4×C59—21no—
152C22×C4×C59—42no—
153C22×C4×C59—42no—
154C22×C4×C59—81no—
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59C2×D411yescharacteristic
158C22×C4×C59C2×D411yescharacteristic
159C22×C4×C59C2×D411yescharacteristic
160C22×C4×C59C2411yescharacteristic
161C22×Dic59—21no—
162C22×Dic59—21no—
163C22×Dic59—21no—
164C22×Dic59—21no—
165C2×Dic118—21no—
166C2×Dic118—21no—
167C2×Dic118—21no—
168C2×Dic118—21no—
169C2×Dic118—21no—
170C2×Dic118C2×D412yes—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—21no—
175C4⋊C4×C59—21no—
176C4⋊C4×C59—21no—
177C4⋊C4×C59—81no—
178C22⋊C4×C59—22no—
179C22⋊C4×C59—22no—
180C22⋊C4×C59—42no—
181C22⋊C4×C59—42no—
182C22⋊C4×C59—81no—
183C22⋊C4×C59—81no—
184C22⋊C4×C59—81no—
185C42×C59—21no—
186C42×C59—21no—
187C42×C59—21no—
188C42×C59—21no—
189C22⋊Dic59—42no—
190C22⋊Dic59—42no—
191#11745670—28no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59—21no—
195C4⋊Dic59—21no—
196C4⋊Dic59—21no—
197C4⋊Dic59—21no—
198Dic59⋊C4—42no—
199Dic59⋊C4—42no—
200Dic59⋊C4—42no—
201C4×Dic59—42no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8—21no—
208C2×C59⋊C8—21no—
209C2×C59⋊C8—21no—
210C2×C59⋊C8—44no—
211C2×C59⋊C8C2×D412yes—
212C2×C59⋊C8#4112yes—
213C2×C59⋊C8#4112yes—
214C2×C59⋊C8#4112yes—
215C23×C59—21no—
216C23×C59—21no—
217C23×C59—21no—
218C23×C59—21no—
219C23×C59—21no—
220C23×C59—21no—
221C23×C59—22no—
222C23×C59—41no—
223C23×C59—41no—
224C23×C59—41no—
225C23×C59—41no—
226C23×C59—41no—
227C23×C59—41no—
228C23×C59#13512yes—
229C23×C59C22×D411yescharacteristic
230D4×C59—22no—
231D4×C59—22no—
232D4×C59—41no—
233D4×C59—41no—
234D4×C59—84no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—21no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—21no—
248C2×C4×C59—21no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59—41no—
253C2×C4×C59—41no—
254C2×C4×C59—41no—
255C2×C4×C59—41no—
256C2×C4×C59—41no—
257C2×C4×C59—41no—
258C2×C4×C59—41no—
259C2×C4×C59—41no—
260C2×C4×C59—41no—
261C2×C4×C59—41no—
262C2×C4×C59—84no—
263C2×C4×C59#13512yes—
264C2×C4×C59C22×D411yescharacteristic
265C2×C4×C59C2×#4111yescharacteristic
266C2×C4×C59C2×#4111yescharacteristic
267C2×C4×C59C2×#4111yescharacteristic
268C2×C4×C59H(5,𝔽2)11yescharacteristic
269C2×C4×C592−1+411yescharacteristic
270C2×Dic59—22no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276C2×Dic59—41no—
277C2×Dic59—41no—
278C2×Dic59—41no—
279C2×Dic59—41no—
280C2×Dic59—41no—
281C2×Dic59—41no—
282Dic118—22no—
283Dic118—22no—
284Dic118—41no—
285Dic118—41no—
286C59⋊C8—22no—
287C59⋊C8—41no—
288C59⋊C8—41no—
289C59⋊C8#12314yes—
290#25031—591nomaximal, Sylow
291C22×C59—22no—
292C22×C59—22no—
293C22×C59—22no—
294C22×C59—41no—
295C22×C59—41no—
296C22×C59—41no—
297C22×C59—41no—
298C22×C59—42no—
299C22×C59—42no—
300C22×C59—42no—
301C22×C59—81no—
302C22×C59—81no—
303C22×C59—81no—
304C22×C59—81no—
305C22×C59#53211yescharacteristic
306C22×C59D4211yescharacteristic
307C22×C59#54711yescharacteristic
308C22×C59C2×#13511yescharacteristic
309C22×C59C2×#13511yescharacteristic
310C22×C59#57711yescharacteristic
311C22×C59#57711yescharacteristic
312C4×C59—22no—
313C4×C59—41no—
314C4×C59—41no—
315C4×C59—41no—
316C4×C59—41no—
317C4×C59—81no—
318C4×C59—81no—
319C4×C59—81no—
320C4×C59C2×#12311yescharacteristic
321C4×C59#54211yescharacteristic
322C4×C59#54711yescharacteristic
323C4×C59#55311yescharacteristic
324Dic59—42no—
325Dic59—81no—
326Dic59—81no—
327Dic59—81no—
328C2×#465—591no—
329C2×C4×SD16—591no—
330#2421—591no—
331#2420D5911yescharacteristic
332#2074—591no—
333#2073—591no—
334#2036—591no—
335#2033—591no—
336#1968—591no—
337#1941—591no—
338#1863—591no—
339#1862—591no—
340#1846—591no—
341#1807—591no—
342C8×C22⋊C4—591no—
343C2×C59—42no—
344C2×C59—82no—
345C2×C59#242211yescharacteristic
346C2×C59#316811yescharacteristic
347C2×C59#317411yescharacteristic
348C2×C59#328211yescharacteristic
349C2×C59#328211yescharacteristic
350C2×C59#333711yescharacteristic
351C2×C59#334011yescharacteristic
352C22×SD16—591no—
353C23×C8—591no—
354C2×C22⋊Q8—591no—
355C2×#121C2×D5911yescharacteristic
356C2×C4×Q8—591no—
357C2×C4×D4C2×D5911yescharacteristic
358#465—1184no—
359C4×SD16—1184no—
360C2×#107—591no—
361C2×#107—591no—
362C2×#106—591no—
363C2×#106—591no—
364C2×#106—591no—
365C2×#103—591no—
366C2×#103—591no—
367C2×#103—591no—
368C2×#102—591no—
369C2×#102—591no—
370C2×#102—591no—
371C2×C22⋊C8—591no—
372C2×C4×C8—591no—
373C2×C4×C8—591no—
374#400—1181no—
375#397—21no—
376#397—1181no—
377#394—21no—
378#391—591no—
379#390C2×D5911yescharacteristic
380#389—591no—
381#389C2×D5911yescharacteristic
382#388—1181no—
383#387—1181no—
384#386—21no—
385#382—591no—
386#382C2×D5911yescharacteristic
387#381—21no—
388#380—591no—
389#380C2×D5911yescharacteristic
390C4×C22⋊C4C2×D5911yescharacteristic
391#340—591no—
392#340—591no—
393#340—591no—
394#340—591no—
395#336—591no—
396#336—591no—
397C59#2503111yescharacteristic, complemented, Sylow
398C22×Q8—591no—
399C22×D4—21no—
400C22×D4C22×D5911yescharacteristic
401C23×C4—21no—
402C23×C4C22×D5911yescharacteristic
403C2×SD16—594no—
404C2×SD16—1182no—
405C2×SD16—1182no—
406C22×C8—591no—
407C22×C8—591no—
408C22×C8—591no—
409C22×C8—591no—
410C22×C8—1182no—
411C22×C8—1182no—
412C22×C8—1182no—
413C22⋊Q8—1182no—
414C22⋊Q8—2361no—
415#121—22no—
416#121—41no—
417C4×Q8—1182no—
418C4×Q8—2361no—
419C4×D4—22no—
420C4×D4—41no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—591no—
423C2×C4⋊C4—591no—
424C2×C4⋊C4—591no—
425C2×C4⋊C4—1181no—
426C2×C4⋊C4—1181no—
427C2×C4⋊C4—1181no—
428C2×C4⋊C4C22×D5911yescharacteristic
429C2×C4⋊C4C22×D5911yescharacteristic
430C2×C4⋊C4C22×D5911yescharacteristic
431C2×C22⋊C4—21no—
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—1181no—
437C2×C22⋊C4—1181no—
438C2×C22⋊C4C22⋊D5911yescharacteristic
439C2×C22⋊C4C22⋊D5911yescharacteristic
440C2×C42—1181no—
441C2×C42C22×D5911yescharacteristic
442C2×C42C22×D5911yescharacteristic
443#107—2361no—
444#107—2361no—
445#106—1182no—
446#106—1182no—
447#106—1182no—
448#103—1182no—
449#103—1182no—
450#103—1182no—
451#102—1182no—
452#102—1182no—
453#102—1182no—
454C22⋊C8—1182no—
455C4×C8—1182no—
456C4×C8—1182no—
457#95—21no—
458#95—21no—
459#95—21no—
460#95—21no—
461#95—1181no—
462#95—1181no—
463#95—1181no—
464#95—1181no—
465#95—1181no—
466#95—1181no—
467#95C22⋊D5911yescharacteristic
468#95C22⋊D5911yescharacteristic
469C24—21no—
470C24C2×C22⋊D5911yescharacteristic
471C2×Q8—592no—
472C2×Q8—1181no—
473C2×Q8—1181no—
474C2×Q8—1181no—
475C2×Q8—1181no—
476C2×Q8—1181no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—42no—
483C2×D4—42no—
484C2×D4—81no—
485C2×D4D4×D5912yes—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—81no—
498C22×C4—1181no—
499C22×C4—1181no—
500C22×C4—1181no—
501C22×C4—1181no—
502C22×C4C2×C22⋊D5911yescharacteristic
503C22×C4C2×C22⋊D5911yescharacteristic
504C22×C4C2×C22⋊D5911yescharacteristic
505C22×C4C2×C22⋊D5911yescharacteristic
506C22×C4C2×C22⋊D5911yescharacteristic
507C22×C4C23×D5911yescharacteristic
508SD16—1188no—
509C2×C8—592no—
510C2×C8—592no—
511C2×C8—592no—
512C2×C8—592no—
513C2×C8—1181no—
514C2×C8—1181no—
515C2×C8—1181no—
516C2×C8—1181no—
517C2×C8—1181no—
518C2×C8—1181no—
519C2×C8—1181no—
520C2×C8—1181no—
521C2×C8—2364no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—81no—
529C4⋊C4—1181no—
530C4⋊C4—1181no—
531C4⋊C4—1181no—
532C4⋊C4—1181no—
533C4⋊C4—1181no—
534C4⋊C4—1181no—
535C4⋊C4—2362no—
536C4⋊C4—2362no—
537C4⋊C4—2362no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—42no—
541C22⋊C4—42no—
542C22⋊C4—81no—
543C22⋊C4—81no—
544C22⋊C4—81no—
545C22⋊C4—2362no—
546C22⋊C4—2362no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—2362no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—22no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23#1427374712yes—
566C23C22×C22⋊D5911yescharacteristic
567Q8—1182no—
568Q8—1182no—
569Q8—2361no—
570Q8—2361no—
571D4—22no—
572D4—22no—
573D4—41no—
574D4—41no—
575D4—84no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—84no—
604C2×C4—1182no—
605C2×C4—2361no—
606C2×C4—2361no—
607C2×C4—2361no—
608C2×C4—2361no—
609C2×C4—2361no—
610C2×C4—2361no—
611C2×C4—2361no—
612C2×C4—2361no—
613C2×C4—2361no—
614C2×C4—2361no—
615C2×C4—2361no—
616C2×C4#1427374712yes—
617C2×C4C2×D4×D5911yescharacteristic
618C2×C4#41×D5911yescharacteristic
619C2×C4#41×D5911yescharacteristic
620C2×C4#41×D5911yescharacteristic
621C2×C4#1427377211yescharacteristic
622C2×C4#1427377411yescharacteristic
623C8—594no—
624C8—1182no—
625C8—2361no—
626C8—2361no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22#1683794811yescharacteristic
642C22D4×C22⋊D5911yescharacteristic
643C22#1683805111yescharacteristic
644C22C2×#1427374711yescharacteristic
645C22C2×#1427374711yescharacteristic
646C22#1683821311yescharacteristic
647C22#1683821311yescharacteristic
648C4—22no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—81no—
654C4—81no—
655C4—81no—
656C4—4721no—
657C4—4721no—
658C4—4721no—
659C4—2362no—
660C4#123×D5911yescharacteristic
661C4#1683802011yescharacteristic
662C4#1683805011yescharacteristic
663C4#1683809711yescharacteristic
664C2—42no—
665C2—82no—
666C2#2142760711yescharacteristic
667C2#2143466011yescharacteristic
668C2#2143471811yescharacteristic
669C2#2143579911yescharacteristic
670C2#2143579911yescharacteristic
671C2#2143646911yescharacteristic
672C2#2143650911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2420C2
socleC23×C59C22×D4
prime radicalC2×#121×C59C22
hypercenter#2420D59
hypocenterC59#25031
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2420

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11745670C2semidirect
3. #21427599C2semidirect
4. #21425515C2semidirect
5. #21425509C2semidirect
6. #21425299C2semidirect
8. #21424949C2semidirect
9. #21424821C2semidirect
10. #21424349C2semidirect
11. #21424343C2semidirect
12. #21424258C2semidirect
13. #21424119C2semidirect
14. C22⋊C4×C59⋊C8C2semidirect
397. C59#25031semidirect

Maximal subgroups

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

subgroupindex
C2×#168376292
C2×C4×#117456702
#214275992
#214255152
#214255092
#214252992
#214252752
#214249492
#214248212
#214243492
#214243432
#214242582
#214241192
C22⋊C4×C59⋊C82
C59×#24202
#2503159