Finite Group Databasebeta

← #28661880

Subgroups — #28661880

Order 15104.

762 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21428682C211yesmaximal, characteristic
2#21425639C211yesmaximal, characteristic, complemented
3#21425411C211yesmaximal, characteristic
4#21425339C211yesmaximal, characteristic, complemented
5#21425179C211yesmaximal, characteristic
6#21424953C211yesmaximal, characteristic
7#21424727C211yesmaximal, characteristic, complemented
8#21424723C211yesmaximal, characteristic
9#21424674C211yesmaximal, characteristic, complemented
10#21424655C211yesmaximal, characteristic, complemented
11#21424451C211yesmaximal, characteristic
12#21424451C211yesmaximal, characteristic
13#21424101C211yesmaximal, characteristic, complemented
14C4×#14273651C211yesmaximal, characteristic, complemented
15C59×#2365C211yesmaximal, characteristic, complemented
16C22×#11745671C2211yescharacteristic
17C22×#11745671C2211yescharacteristic
18C2×#14273722—21no—
19C2×#14273658C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×C4⋊(C59⋊C8)C2211yescharacteristic
22C2×#14273651C2211yescharacteristic
23C2×#14273651C2211yescharacteristic
24C2×#14273651C2211yescharacteristic
25C2×#14273651C2211yescharacteristic
26C2×#14273651C2211yescharacteristic
27C2×#14273651C2211yescharacteristic
28C2×#14273647C2211yescharacteristic
29C2×#14273647C2211yescharacteristic
30C2×C4×C59⋊C8C2211yescharacteristic
31#16837329—21no—
32#16837324—21no—
33#16837304C2211yescharacteristic
34#16837304C2211yescharacteristic
35#16837304C2211yescharacteristic
36#16837300C2211yescharacteristic
37#16837294—21no—
38#16837272C2211yescharacteristic
39C4×C4⋊Dic59C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837157C2211yescharacteristic
45#16837157C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C2×C4⋊Q8×C59C2211yescharacteristic
49C2×C4×Q8×C59C2211yescharacteristic
50C2×C4×Q8×C59C2211yescharacteristic
51C59×#391C2211yescharacteristic
52C59×#391C2211yescharacteristic
53C59×#389—21no—
54C59×#387—21no—
55C59×#387—21no—
56C4×C4⋊C4×C59—21no—
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C4⋊C4×C59C2211yescharacteristic
59C22×D236C2311yescharacteristic
60C2×#11745671—22no—
61C2×#11745671—22no—
62C2×#11745671—22no—
63C2×#11745671—41no—
64C2×#11745671—41no—
65C2×#11745671—41no—
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69C22×C59⋊C8C2311yescharacteristic
70#14273722—44no—
71C2×C4⋊Dic59—21no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×Dic59⋊C4—21no—
77C2×(C2×D59)⋊C4—21no—
78C2×(C2×D59)⋊C4—21no—
79C2×(C2×D59)⋊C4—21no—
80C2×(C2×D59)⋊C4—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×C4×Dic59—21no—
84#14273658—22no—
85#14273658—41no—
86C4⋊(C59⋊C8)—41no—
87#14273651—22no—
88#14273651—22no—
89#14273651—22no—
90#14273651—41no—
91#14273651—41no—
92#14273651—41no—
93#14273647—22no—
94#14273647—41no—
95C4×C59⋊C8—41no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C22×Q8×C59C2311yescharacteristic
99C22×Q8×C59C2311yescharacteristic
100C4⋊Q8×C59—22no—
101C4⋊Q8×C59—22no—
102C4×Q8×C59—22no—
103C4×Q8×C59—22no—
104C4×Q8×C59—41no—
105C4×Q8×C59—41no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59—21no—
108C2×C4⋊C4×C59—21no—
109C2×C4⋊C4×C59—21no—
110C2×C4⋊C4×C59D411yescharacteristic
111C2×C4⋊C4×C59D411yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C4⋊C4×C59C2311yescharacteristic
114C2×C42×C59—21no—
115C2×C42×C59—21no—
116C2×C42×C59—21no—
117C2×C42×C59C2311yescharacteristic
118C2×C42×C59C2311yescharacteristic
119C2×C42×C59C2311yescharacteristic
120#95×C59—21no—
121#95×C59—21no—
122#95×C59—21no—
123#95×C59—21no—
124#95×C59—21no—
125#95×C59D411yescharacteristic
126#95×C59D411yescharacteristic
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—21no—
134C2×Q8×C59—21no—
135C2×Q8×C59—21no—
136C2×Q8×C59—21no—
137C2×Q8×C59—21no—
138C2×Q8×C59#4112yes—
139C2×Q8×C59#4112yes—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—21no—
143C22×C4×C59—21no—
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2411yescharacteristic
151C22×Dic59—21no—
152C22×Dic59—21no—
153C22×Dic59—21no—
154C2×D236—21no—
155C2×D236—21no—
156C2×D236—21no—
157C2×D236—21no—
158C2×D236—21no—
159C2×D236—21no—
160C4⋊C4×C59—24no—
161C4⋊C4×C59—24no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—41no—
166C4⋊C4×C59—42no—
167C4⋊C4×C59—42no—
168C4⋊C4×C59SD1614yescomplemented
169C4⋊C4×C59#4112yes—
170C42×C59—21no—
171C42×C59—21no—
172C42×C59—21no—
173C42×C59—21no—
174C42×C59—21no—
175C42×C59—21no—
176C42×C59—42no—
177C42×C59—42no—
178C42×C59—42no—
179#11745671—44no—
180#11745671—44no—
181(C2×D59)⋊C4—42no—
182(C2×D59)⋊C4—42no—
183(C2×D59)⋊C4—42no—
184(C2×D59)⋊C4—42no—
185(C2×D59)⋊C4—81no—
186(C2×D59)⋊C4—81no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59—42no—
194C4⋊Dic59—81no—
195Dic59⋊C4—42no—
196C4×Dic59—42no—
197C2×C59⋊C8—21no—
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#4112yes—
209C23×C59C22×D411yescharacteristic
210C22×D59—41no—
211C22×D59—41no—
212C22×D59—41no—
213C22×D59—41no—
214C22×D59—41no—
215C22×D59—41no—
216C22×D59—41no—
217Q8×C59—22no—
218Q8×C59—22no—
219Q8×C59—22no—
220Q8×C59—22no—
221Q8×C59—41no—
222Q8×C59—41no—
223Q8×C59—41no—
224Q8×C59—41no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—22no—
233C2×C4×C59—22no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—22no—
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—41no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59—41no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59C2×SD1612yes—
253C2×C4×C59C2×SD1612yes—
254C2×C4×C59C2×SD1612yes—
255C2×C4×C59C2×#4111yescharacteristic
256C2×C4×C59C2×#4111yescharacteristic
257C2×C4×C59C2×#4111yescharacteristic
258C2×C4×C59C2×#4111yescharacteristic
259C2×C4×C59H(5,𝔽2)11yescharacteristic
260C2×C4×C59H(5,𝔽2)11yescharacteristic
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270D236—41no—
271D236—41no—
272D236—41no—
273D236—41no—
274C59⋊C8—22no—
275C59⋊C8—41no—
276C59⋊C8—41no—
277C59⋊C8—41no—
278#27066—591nomaximal, Sylow
279C22×C59#52511yescharacteristic
280C22×C59#54611yescharacteristic
281C22×C59#55011yescharacteristic
282C22×C59C22×SD1611yescharacteristic
283C22×C59#57511yescharacteristic
284C22×C59#57711yescharacteristic
285C22×C59#57811yescharacteristic
286C2×D59—81no—
287C2×D59—81no—
288C2×D59—81no—
289C2×D59—81no—
290C2×D59—81no—
291C2×D59—81no—
292C2×D59—81no—
293C4×C59—22no—
294C4×C59—22no—
295C4×C59—22no—
296C4×C59—41no—
297C4×C59—42no—
298C4×C59—42no—
299C4×C59—41no—
300C4×C59—41no—
301C4×C59—42no—
302C4×C59—81no—
303C4×C59#52911yescharacteristic
304C4×C59#53811yescharacteristic
305C4×C59#55111yescharacteristic
306C4×C59#55311yescharacteristic
307Dic59—81no—
308Dic59—81no—
309Dic59—81no—
310#2512—591no—
311#2365D5911yescharacteristic, complemented
312#2096—591no—
313#2055—591no—
314#2043—591no—
315#2013—591no—
316#1968—591no—
317#1927—591no—
318#1926—591no—
319#1919—591no—
320#1916—591no—
321#1880—591no—
322#1880—591no—
323#1803—591no—
324C4×#103—591no—
325C2×C59#246711yescharacteristic
326C2×C59#310311yescharacteristic
327C2×C59#312111yescharacteristic
328C2×C59#329811yescharacteristic
329C2×C59#330711yescharacteristic
330C2×C59#339211yescharacteristic
331C2×C59#340211yescharacteristic
332D59—161no—
333C22×SD16—591no—
334C22×SD16—591no—
335C2×C4⋊Q8C2×D5911yescharacteristic
336C2×#126—1181no—
337C2×C4×Q8C2×D5911yescharacteristic
338C2×C4×Q8C2×D5911yescharacteristic
339C2×#106—591no—
340C2×#106—591no—
341C2×C4⋊C8—591no—
342C2×#103—591no—
343C2×#103—591no—
344C2×#103—591no—
345C2×#103—591no—
346C2×#103—591no—
347C2×#103—591no—
348C2×#102—591no—
349C2×#102—591no—
350C2×C4×C8—591no—
351#397—1181no—
352#396—1181no—
353#391C2×D5911yescharacteristic
354#391C2×D5911yescharacteristic
355#390—591no—
356#390—591no—
357#390—591no—
358#389—21no—
359#389—591no—
360#388—1181no—
361#387—21no—
362#387—21no—
363#384—591no—
364C4×C4⋊C4—21no—
365C4×C4⋊C4—591no—
366C4×C4⋊C4C2×D5911yescharacteristic
367C4×C4⋊C4C2×D5911yescharacteristic
368#340—591no—
369#340—591no—
370#340—591no—
371#340—591no—
372#336—591no—
373#336—591no—
374#336—591no—
375#336—591no—
376C59#2706611yescharacteristic, complemented, Sylow
377C22×Q8C22×D5911yescharacteristic
378C22×Q8C22×D5911yescharacteristic
379C22×D4—591no—
380C2×SD16—1182no—
381C2×SD16—1182no—
382C2×SD16—1182no—
383C2×SD16—2361no—
384C2×SD16—2361no—
385C2×SD16—2361no—
386C22×C8—591no—
387C22×C8—591no—
388C22×C8—591no—
389C22×C8—591no—
390C4⋊Q8—22no—
391C4⋊Q8—22no—
392#126—2364no—
393C4×Q8—22no—
394C4×Q8—22no—
395C4×Q8—41no—
396C4×Q8—41no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—591no—
402C2×C4⋊C4—591no—
403C2×C4⋊C4—591no—
404C2×C4⋊C4—1181no—
405C2×C4⋊C4—1181no—
406C2×C4⋊C4—1181no—
407C2×C4⋊C4C22⋊D5911yescharacteristic
408C2×C4⋊C4C22⋊D5911yescharacteristic
409C2×C4⋊C4C22×D5911yescharacteristic
410C2×C4⋊C4C22×D5911yescharacteristic
411C2×C22⋊C4—1181no—
412C2×C22⋊C4—1181no—
413C2×C22⋊C4—1181no—
414C2×C22⋊C4—1181no—
415C2×C22⋊C4—1181no—
416C2×C22⋊C4—1181no—
417C2×C42—21no—
418C2×C42—21no—
419C2×C42—21no—
420C2×C42—1181no—
421C2×C42C22×D5911yescharacteristic
422C2×C42C22×D5911yescharacteristic
423C2×C42C22×D5911yescharacteristic
424#106—1182no—
425#106—2361no—
426C4⋊C8—2361no—
427#103—1182no—
428#103—1182no—
429#103—1182no—
430#103—2361no—
431#103—2361no—
432#103—2361no—
433#102—1182no—
434#102—2361no—
435C4×C8—2361no—
436#95—21no—
437#95—21no—
438#95—21no—
439#95—21no—
440#95—21no—
441#95—1181no—
442#95—1181no—
443#95C22⋊D5911yescharacteristic
444#95C22⋊D5911yescharacteristic
445C24—1181no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8#1174568512yes—
457C2×Q8#1174568512yes—
458C2×D4—1181no—
459C2×D4—1181no—
460C2×D4—1181no—
461C2×D4—1181no—
462C2×D4—1181no—
463C2×D4—1181no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—1181no—
469C22×C4—1181no—
470C22×C4—1181no—
471C22×C4C2×C22⋊D5911yescharacteristic
472C22×C4C2×C22⋊D5911yescharacteristic
473C22×C4C2×C22⋊D5911yescharacteristic
474C22×C4C2×C22⋊D5911yescharacteristic
475C22×C4C2×C22⋊D5911yescharacteristic
476C22×C4C2×C22⋊D5911yescharacteristic
477C22×C4C23×D5911yescharacteristic
478SD16—2364no—
479SD16—2364no—
480C2×C8—592no—
481C2×C8—1181no—
482C2×C8—1181no—
483C2×C8—1181no—
484C2×C8—1181no—
485C2×C8—1181no—
486C2×C8—1181no—
487C2×C8—1181no—
488C2×C8—1181no—
489C2×C8—1181no—
490C2×C8—1181no—
491C2×C8—1181no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—41no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—1181no—
501C4⋊C4—1181no—
502C4⋊C4—1181no—
503C4⋊C4—1181no—
504C4⋊C4—1181no—
505C4⋊C4—1181no—
506C4⋊C4—4721no—
507C4⋊C4—2362no—
508C4⋊C4—2362no—
509C4⋊C4#1174567114yescomplemented
510C4⋊C4#1174568712yes—
511C22⋊C4—4721no—
512C22⋊C4—4721no—
513C22⋊C4—2362no—
514C22⋊C4—2362no—
515C22⋊C4—2362no—
516C22⋊C4—2362no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—42no—
524C42—42no—
525C42—42no—
526C42—2362no—
527C23—2361no—
528C23—2361no—
529C23—2361no—
530C23—2361no—
531C23—2361no—
532C23—2361no—
533C23—2361no—
534C23C22×C22⋊D5911yescharacteristic
535Q8—22no—
536Q8—22no—
537Q8—22no—
538Q8—22no—
539Q8—41no—
540Q8—41no—
541Q8—41no—
542Q8—41no—
543D4—2361no—
544D4—2361no—
545D4—2361no—
546D4—2361no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—2361no—
575C2×C4—2361no—
576C2×C4—2361no—
577C2×C4—2361no—
578C2×C4—2361no—
579C2×C4—2361no—
580C2×C4—2361no—
581C2×C4—2361no—
582C2×C4—2361no—
583C2×C4C2×#1174567112yes—
584C2×C4C2×#1174567112yes—
585C2×C4C2×#1174567112yes—
586C2×C4#41×D5911yescharacteristic
587C2×C4C2×#1174568711yescharacteristic
588C2×C4C2×#1174568511yescharacteristic
589C2×C4C2×#1174568511yescharacteristic
590C2×C4#1427377211yescharacteristic
591C2×C4#1427377311yescharacteristic
592C8—1182no—
593C8—2361no—
594C8—2361no—
595C8—2361no—
596C22—4721no—
597C22—4721no—
598C22—4721no—
599C22—4721no—
600C22—4721no—
601C22—4721no—
602C22—4721no—
603C22#1683791311yescharacteristic
604C22#1683804611yescharacteristic
605C22#1683807111yescharacteristic
606C22C22×#1174567111yescharacteristic
607C22#1683820011yescharacteristic
608C22#1683821011yescharacteristic
609C22#1683821711yescharacteristic
610C4—22no—
611C4—22no—
612C4—22no—
613C4—41no—
614C4—41no—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—41no—
619C4—81no—
620C4—4721no—
621C4—4721no—
622C4—4721no—
623C4#1683793211yescharacteristic
624C4#1683798511yescharacteristic
625C4#1683808111yescharacteristic
626C4#1683809811yescharacteristic
627C2—9441no—
628C2#2142814111yescharacteristic
629C2#2143412711yescharacteristic
630C2#2143432511yescharacteristic
631C2#2143596111yescharacteristic
632C2#2143607211yescharacteristic
633C2#2143707311yescharacteristic
634C2#2143717111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2365C2
socleC23×C59C22×D4
prime radicalC22×D236C23
hypercenter#2365D59
hypocenterC59#27066
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2365

Complements

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

NH ≅ G/Ntype
2. #21425639C2semidirect
4. #21425339C2semidirect
7. #21424727C2semidirect
9. #21424674C2semidirect
10. #21424655C2semidirect
13. #21424101C2semidirect
14. C4×#14273651C2semidirect
15. C59×#2365C2semidirect
168. C4⋊C4×C59SD16semidirect
311. #2365D59semidirect
376. C59#27066semidirect
509. C4⋊C4#11745671semidirect

Maximal subgroups

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

subgroupindex
#214286822
#214256392
#214254112
#214253392
#214251792
#214249532
#214247272
#214247232
#214246742
#214246552
#214244512
#214241012
C4×#142736512
C59×#23652
#2706659