Finite Group Databasebeta

← #28661970

Subgroups — #28661970

Order 15104.

774 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21428693C211yesmaximal, characteristic, complemented
2#21425617C211yesmaximal, characteristic, complemented
3#21425406C211yesmaximal, characteristic
4#21425332C211yesmaximal, characteristic, complemented
5#21425173C211yesmaximal, characteristic
6#21424953C211yesmaximal, characteristic, complemented
7#21424734C211yesmaximal, characteristic, complemented
8#21424716C211yesmaximal, characteristic, complemented
9#21424674C211yesmaximal, characteristic, complemented
10#21424655C211yesmaximal, characteristic, complemented
11#21424451C211yesmaximal, characteristic, complemented
12#21424444C211yesmaximal, characteristic
13#21424111C211yesmaximal, characteristic, complemented
14C4×#14273648C211yesmaximal, characteristic, complemented
15C59×#2367C211yesmaximal, characteristic, complemented
16C22×#11745671C2211yescharacteristic
17C22×#11745669C2211yescharacteristic
18C2×#14273660C2211yescharacteristic
19C2×#14273658C2211yescharacteristic
20C2×C4⋊(C59⋊C8)C2211yescharacteristic
21C2×#14273651C2211yescharacteristic
22C2×#14273651C2211yescharacteristic
23C2×#14273651C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×#14273647C2211yescharacteristic
28C2×#14273647C2211yescharacteristic
29C2×C4×C59⋊C8C2211yescharacteristic
30#16837331—21no—
31#16837328—21no—
32#16837304C2211yescharacteristic
33#16837304C2211yescharacteristic
34#16837304C2211yescharacteristic
35#16837300C2211yescharacteristic
36#16837292—21no—
37#16837291—21no—
38#16837268C2211yescharacteristic
39C4×C4⋊Dic59C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837157C2211yescharacteristic
45#16837157C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C2×#124×C59C2211yescharacteristic
49C2×C4×Q8×C59C2211yescharacteristic
50C2×C4×D4×C59C2211yescharacteristic
51C59×#391C2211yescharacteristic
52C59×#390C2211yescharacteristic
53C59×#388—21no—
54C59×#388—21no—
55C59×#385—21no—
56C4×C4⋊C4×C59C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C22⋊C4×C59—21no—
59C22×D236C2311yescharacteristic
60C2×#11745671—22no—
61C2×#11745671—22no—
62C2×#11745671—41no—
63C2×#11745669—22no—
64C2×#11745669—41no—
65C2×#11745669—41no—
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69C22×C59⋊C8C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×Dic59⋊C4—21no—
74C2×(C2×D59)⋊C4—21no—
75C2×(C2×D59)⋊C4—21no—
76C2×(C2×D59)⋊C4—21no—
77C2×(C2×D59)⋊C4—21no—
78C2×(C2×D59)⋊C4—21no—
79C2×(C2×D59)⋊C4—21no—
80C2×C4×Dic59—21no—
81#14273660—41no—
82#14273658—22no—
83C4⋊(C59⋊C8)—41no—
84#14273651—22no—
85#14273651—41no—
86#14273651—41no—
87#14273648—22no—
88#14273648—22no—
89#14273648—41no—
90#14273647—22no—
91#14273647—41no—
92C4×C59⋊C8—41no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C22×Q8×C59C2311yescharacteristic
98C22×D4×C59C2311yescharacteristic
99C23×C4×C59—21no—
100#124×C59—22no—
101#124×C59—22no—
102C4×Q8×C59—22no—
103C4×Q8×C59—41no—
104C4×D4×C59—22no—
105C4×D4×C59—41no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59D411yescharacteristic
108C2×C4⋊C4×C59D411yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C22⋊C4×C59—21no—
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59—21no—
114C2×C22⋊C4×C59—21no—
115C2×C22⋊C4×C59—21no—
116C2×C42×C59—21no—
117C2×C42×C59—21no—
118C2×C42×C59C2311yescharacteristic
119C2×C42×C59C2311yescharacteristic
120C2×C42×C59C2311yescharacteristic
121#95×C59—21no—
122#95×C59—21no—
123#95×C59—21no—
124#95×C59D411yescharacteristic
125#95×C59D411yescharacteristic
126C24×C59—21no—
127C23×D59—21no—
128C2×Q8×C59—21no—
129C2×Q8×C59—21no—
130C2×Q8×C59—21no—
131C2×Q8×C59—21no—
132C2×Q8×C59—21no—
133C2×Q8×C59#4112yes—
134C2×D4×C59—21no—
135C2×D4×C59—21no—
136C2×D4×C59—21no—
137C2×D4×C59—21no—
138C2×D4×C59—21no—
139C2×D4×C59#4112yes—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—21no—
143C22×C4×C59—42no—
144C22×C4×C59—42no—
145C22×C4×C59—81no—
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2411yescharacteristic
153C22×Dic59—21no—
154C22×Dic59—21no—
155C22×Dic59—21no—
156C2×D236—21no—
157C2×D236—21no—
158C2×D236—21no—
159C2×D236—21no—
160C2×D236—21no—
161C2×D236—21no—
162C4⋊C4×C59—22no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—41no—
167C4⋊C4×C59—42no—
168C4⋊C4×C59#4112yes—
169C22⋊C4×C59—24no—
170C22⋊C4×C59—24no—
171C22⋊C4×C59—42no—
172C22⋊C4×C59—42no—
173C22⋊C4×C59—81no—
174C42×C59—21no—
175C42×C59—21no—
176C42×C59—21no—
177C42×C59—21no—
178C42×C59—21no—
179C42×C59—21no—
180C42×C59—42no—
181C42×C59—81no—
182#11745671—44no—
183#11745669—44no—
184(C2×D59)⋊C4—42no—
185(C2×D59)⋊C4—42no—
186(C2×D59)⋊C4—81no—
187(C2×D59)⋊C4—81no—
188(C2×D59)⋊C4—81no—
189(C2×D59)⋊C4—81no—
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59—21no—
195C4⋊Dic59—21no—
196Dic59⋊C4—81no—
197C4×Dic59—81no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
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#4112yes—
210C23×C59—22no—
211C23×C59—41no—
212C23×C59—41no—
213C23×C59—41no—
214C23×C59—41no—
215C23×C59—41no—
216C23×C59—41no—
217C23×C59C22×D411yescharacteristic
218C22×D59—41no—
219C22×D59—41no—
220C22×D59—41no—
221C22×D59—41no—
222C22×D59—41no—
223C22×D59—41no—
224C22×D59—41no—
225Q8×C59—22no—
226Q8×C59—22no—
227Q8×C59—41no—
228Q8×C59—41no—
229D4×C59—22no—
230D4×C59—22no—
231D4×C59—41no—
232D4×C59—41no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—22no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—22no—
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—21no—
250C2×C4×C59—21no—
251C2×C4×C59—21no—
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—84no—
260C2×C4×C59#13512yes—
261C2×C4×C59C2×#4111yescharacteristic
262C2×C4×C59C2×#4111yescharacteristic
263C2×C4×C59C2×#4111yescharacteristic
264C2×C4×C59C2×#4111yescharacteristic
265C2×C4×C59H(5,𝔽2)11yescharacteristic
266C2×C4×C59H(5,𝔽2)11yescharacteristic
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276D236—41no—
277D236—41no—
278D236—41no—
279D236—41no—
280C59⋊C8—22no—
281C59⋊C8—41no—
282C59⋊C8—41no—
283C59⋊C8—41no—
284#27072—591nomaximal, Sylow
285C22×C59—42no—
286C22×C59—42no—
287C22×C59—42no—
288C22×C59—81no—
289C22×C59—81no—
290C22×C59—81no—
291C22×C59—81no—
292C22×C59#52511yescharacteristic
293C22×C59#54611yescharacteristic
294C22×C59#55011yescharacteristic
295C22×C59C2×#13511yescharacteristic
296C22×C59#57511yescharacteristic
297C22×C59#57611yescharacteristic
298C22×C59#57711yescharacteristic
299C2×D59—81no—
300C2×D59—81no—
301C2×D59—81no—
302C2×D59—81no—
303C2×D59—81no—
304C2×D59—81no—
305C2×D59—81no—
306C4×C59—22no—
307C4×C59—41no—
308C4×C59—41no—
309C4×C59—42no—
310C4×C59—41no—
311C4×C59—42no—
312C4×C59—41no—
313C4×C59—41no—
314C4×C59—81no—
315C4×C59#52911yescharacteristic
316C4×C59#53811yescharacteristic
317C4×C59#55111yescharacteristic
318C4×C59#55311yescharacteristic
319Dic59—81no—
320Dic59—81no—
321Dic59—81no—
322#2513—591no—
323#2367D5911yescharacteristic, complemented
324#2092—591no—
325#2054—591no—
326#2042—591no—
327#2012—591no—
328#1968—591no—
329#1928—591no—
330#1925—591no—
331#1919—591no—
332#1916—591no—
333#1880—591no—
334#1879—591no—
335#1805—591no—
336C4×#102—591no—
337C2×C59—82no—
338C2×C59#246711yescharacteristic
339C2×C59#310211yescharacteristic
340C2×C59#311911yescharacteristic
341C2×C59#329711yescharacteristic
342C2×C59#332111yescharacteristic
343C2×C59#339411yescharacteristic
344C2×C59#340311yescharacteristic
345D59—161no—
346C22×SD16—591no—
347C22×D8—591no—
348C2×#124C2×D5911yescharacteristic
349C2×C4×Q8C2×D5911yescharacteristic
350C2×C4×D4C2×D5911yescharacteristic
351C2×#107—591no—
352C2×#106—591no—
353C2×C4⋊C8—591no—
354C2×#103—591no—
355C2×#103—591no—
356C2×#103—591no—
357C2×#102—591no—
358C2×#102—591no—
359C2×#102—591no—
360C2×#102—591no—
361C2×#102—591no—
362C2×C4×C8—591no—
363#397—1181no—
364#397—1181no—
365#391C2×D5911yescharacteristic
366#390—591no—
367#390—591no—
368#390—591no—
369#390C2×D5911yescharacteristic
370#389—591no—
371#388—21no—
372#388—21no—
373#388—1181no—
374#388—1181no—
375#385—21no—
376#382—591no—
377C4×C4⋊C4—591no—
378C4×C4⋊C4C2×D5911yescharacteristic
379C4×C4⋊C4C2×D5911yescharacteristic
380C4×C22⋊C4—21no—
381#340—591no—
382#340—591no—
383#340—591no—
384#340—591no—
385#336—591no—
386#336—591no—
387#336—591no—
388#336—591no—
389C59#2707211yescharacteristic, complemented, Sylow
390C22×Q8C22×D5911yescharacteristic
391C22×D4—591no—
392C22×D4C22×D5911yescharacteristic
393C23×C4—21no—
394C2×SD16—1182no—
395C2×SD16—1182no—
396C2×SD16—2361no—
397C2×D8—1182no—
398C2×D8—2361no—
399C2×D8—2361no—
400C22×C8—591no—
401C22×C8—591no—
402C22×C8—591no—
403C22×C8—591no—
404#124—22no—
405#124—22no—
406C4×Q8—22no—
407C4×Q8—41no—
408C4×D4—22no—
409C4×D4—41no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—591no—
412C2×C4⋊C4—591no—
413C2×C4⋊C4—591no—
414C2×C4⋊C4—1181no—
415C2×C4⋊C4C22⋊D5911yescharacteristic
416C2×C4⋊C4C22⋊D5911yescharacteristic
417C2×C4⋊C4C22×D5911yescharacteristic
418C2×C4⋊C4C22×D5911yescharacteristic
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—1181no—
425C2×C22⋊C4—1181no—
426C2×C22⋊C4—1181no—
427C2×C22⋊C4—1181no—
428C2×C22⋊C4—1181no—
429C2×C22⋊C4—1181no—
430C2×C42—21no—
431C2×C42—21no—
432C2×C42—1181no—
433C2×C42C22×D5911yescharacteristic
434C2×C42C22×D5911yescharacteristic
435C2×C42C22×D5911yescharacteristic
436#107—2361no—
437#106—1182no—
438C4⋊C8—2361no—
439#103—1182no—
440#103—2361no—
441#103—2361no—
442#102—1182no—
443#102—1182no—
444#102—1182no—
445#102—2361no—
446#102—2361no—
447C4×C8—2361no—
448#95—21no—
449#95—21no—
450#95—21no—
451#95—1181no—
452#95—1181no—
453#95—1181no—
454#95—1181no—
455#95C22⋊D5911yescharacteristic
456#95C22⋊D5911yescharacteristic
457C24—21no—
458C24—1181no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8#1174568512yes—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—1181no—
471C2×D4—1181no—
472C2×D4—1181no—
473C2×D4—1181no—
474C2×D4—1181no—
475C2×D4—1181no—
476C2×D4#1174568512yes—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—81no—
483C22×C4—1181no—
484C22×C4—1181no—
485C22×C4—1181no—
486C22×C4C2×C22⋊D5911yescharacteristic
487C22×C4C2×C22⋊D5911yescharacteristic
488C22×C4C2×C22⋊D5911yescharacteristic
489C22×C4C2×C22⋊D5911yescharacteristic
490C22×C4C2×C22⋊D5911yescharacteristic
491C22×C4C2×C22⋊D5911yescharacteristic
492C22×C4C23×D5911yescharacteristic
493SD16—2364no—
494D8—2364no—
495C2×C8—592no—
496C2×C8—1181no—
497C2×C8—1181no—
498C2×C8—1181no—
499C2×C8—1181no—
500C2×C8—1181no—
501C2×C8—1181no—
502C2×C8—1181no—
503C2×C8—1181no—
504C2×C8—1181no—
505C2×C8—1181no—
506C2×C8—1181no—
507C4⋊C4—22no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—41no—
512C4⋊C4—42no—
513C4⋊C4—1181no—
514C4⋊C4—1181no—
515C4⋊C4—1181no—
516C4⋊C4—1181no—
517C4⋊C4—1181no—
518C4⋊C4—1181no—
519C4⋊C4—4721no—
520C4⋊C4#1174568712yes—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—81no—
526C22⋊C4—4721no—
527C22⋊C4—4721no—
528C22⋊C4—4721no—
529C22⋊C4—4721no—
530C22⋊C4—2362no—
531C22⋊C4—2362no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—42no—
539C42—81no—
540C42—4721no—
541C23—22no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—2361no—
549C23—2361no—
550C23—2361no—
551C23—2361no—
552C23—2361no—
553C23—2361no—
554C23—2361no—
555C23C22×C22⋊D5911yescharacteristic
556Q8—22no—
557Q8—22no—
558Q8—41no—
559Q8—41no—
560D4—22no—
561D4—22no—
562D4—41no—
563D4—41no—
564D4—2361no—
565D4—2361no—
566D4—2361no—
567D4—2361no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
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—22no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—84no—
595C2×C4—2361no—
596C2×C4—2361no—
597C2×C4—2361no—
598C2×C4—2361no—
599C2×C4—2361no—
600C2×C4—2361no—
601C2×C4—2361no—
602C2×C4—2361no—
603C2×C4—2361no—
604C2×C4#1427374712yes—
605C2×C4#41×D5911yescharacteristic
606C2×C4C2×#1174568711yescharacteristic
607C2×C4C2×#1174568511yescharacteristic
608C2×C4C2×#1174568511yescharacteristic
609C2×C4#1427377211yescharacteristic
610C2×C4#1427377311yescharacteristic
611C8—1182no—
612C8—2361no—
613C8—2361no—
614C8—2361no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—4721no—
623C22—4721no—
624C22—4721no—
625C22—4721no—
626C22—4721no—
627C22—4721no—
628C22—4721no—
629C22#1683791311yescharacteristic
630C22#1683804611yescharacteristic
631C22#1683807111yescharacteristic
632C22C2×#1427374711yescharacteristic
633C22#1683820011yescharacteristic
634C22#1683820511yescharacteristic
635C22#1683821011yescharacteristic
636C4—22no—
637C4—41no—
638C4—41no—
639C4—42no—
640C4—41no—
641C4—41no—
642C4—42no—
643C4—41no—
644C4—81no—
645C4—4721no—
646C4—4721no—
647C4—4721no—
648C4#1683793211yescharacteristic
649C4#1683798511yescharacteristic
650C4#1683808111yescharacteristic
651C4#1683809811yescharacteristic
652C2—82no—
653C2—9441no—
654C2#2142814111yescharacteristic
655C2#2143411411yescharacteristic
656C2#2143429511yescharacteristic
657C2#2143594611yescharacteristic
658C2#2143628211yescharacteristic
659C2#2143709211yescharacteristic
660C2#2143718011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2367C2
socleC23×C59C22×D4
prime radicalC22×#11745669C22
hypercenter#2367D59
hypocenterC59#27072
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2367

Complements

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

NH ≅ G/Ntype
1. #21428693C2semidirect
2. #21425617C2semidirect
4. #21425332C2semidirect
6. #21424953C2semidirect
7. #21424734C2semidirect
8. #21424716C2semidirect
9. #21424674C2semidirect
10. #21424655C2semidirect
11. #21424451C2semidirect
13. #21424111C2semidirect
14. C4×#14273648C2semidirect
15. C59×#2367C2semidirect
323. #2367D59semidirect
389. C59#27072semidirect

Maximal subgroups

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

subgroupindex
#214286932
#214256172
#214254062
#214253322
#214251732
#214249532
#214247342
#214247162
#214246742
#214246552
#214244512
#214244442
#214241112
C4×#142736482
C59×#23672
#2707259