Finite Group Databasebeta

← #27351712

Subgroups — #27351712

Order 13568.

930 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#20288467C211yesmaximal, characteristic
2#20283373C211yesmaximal, characteristic, complemented
3#20273286C211yesmaximal, characteristic, complemented
4#20273278C212yesmaximal, complemented
5#20273256C211yesmaximal, characteristic
6#20273251C212yesmaximal, complemented
7#20273234C211yesmaximal, characteristic, complemented
8#20273226C212yesmaximal, complemented
9#20273204C211yesmaximal, characteristic
10#20273199C212yesmaximal, complemented
11C53×#2346C211yesmaximal, characteristic, complemented
12#16811341—21no—
13#16811328—21no—
14#16811312—21no—
15#16811308—21no—
16#16811306—22no—
17#16811299—22no—
18#16811251C411yescharacteristic, complemented
19#16811244C412yescomplemented
20#16811235C412yescomplemented
21#16811199C411yescharacteristic, complemented
22#16811180C2212yes—
23#16811157C2212yes—
24#16811134C2211yescharacteristic
25C2×#13133196C2211yescharacteristic
26#16811103C411yescharacteristic, complemented
27#16811100C411yescharacteristic, complemented
28#16810684—22no—
29#16810680C2212yes—
30#16810676C2212yes—
31#16810650—21no—
32#16810587C2211yescharacteristic
33C2×C22⋊#88322C2211yescharacteristic
34#16810579—21no—
35#16810359C2212yes—
36#16810359C2212yes—
37#16810359C2211yescharacteristic
38#16810359C2211yescharacteristic
39#16810359C2212yes—
40#16810359C2212yes—
41#16810359C2212yes—
42#16810356C2211yescharacteristic
43#16810356C2212yes—
44#16810356C2211yescharacteristic
45C2×#117×C53C2211yescharacteristic
46C53×#389—21no—
47C53×#388—22no—
48C53×#382C2212yes—
49C53×#380C2211yescharacteristic
50C4×C4⋊C4×C53—21no—
51C4×C22⋊C4×C53C2211yescharacteristic
52C53×#376C2212yes—
53C2×#11719945C2×C411yescharacteristic
54C22×Dic106C2×C411yescharacteristic
55C2×#11719917—21no—
56C22×#88322—21no—
57#13133217C2×C412yes—
58#13133217C2×C412yes—
59#13133216—21no—
60#13133212—22no—
61#13133211—22no—
62#13133210—22no—
63#13133210—22no—
64#13133209—22no—
65#13133209C2×C412yes—
66#13133208—21no—
67#13133207—22no—
68#13133207—22no—
69#13133205C2×C412yes—
70#13133203—22no—
71#13133201—22no—
72#13133200—22no—
73#13133200—21no—
74#13133199—22no—
75#13133197—21no—
76#13133197—21no—
77#13133196—22no—
78#13133196—21no—
79#13133196—21no—
80#13133196C2×C412yes—
81#13133196C2×C411yescharacteristic
82#13133196C2312yes—
83#13133196C2311yescharacteristic
84#13133195—22no—
85#13133194—22no—
86#13133191C2×C411yescharacteristic
87#13133191C2311yescharacteristic
88#13133186—21no—
89C4×Dic106—22no—
90C4×Dic106C2×C412yes—
91C4×Dic106C2×C412yes—
92C4×Dic106C2×C412yes—
93#13133182—21no—
94#13133182—21no—
95C4×D212C2×C412yes—
96C4×C22⋊D53—22no—
97C4×C22⋊D53—21no—
98C4×C22⋊D53—21no—
99#13133177—22no—
100#13133176—22no—
101#13133176C2×C411yescharacteristic
102#13133175—21no—
103#13133174—22no—
104#13133173—22no—
105C2×C4⋊Dic53C2×C411yescharacteristic
106C2×#11719919—21no—
107C4⋊C4×D53—21no—
108C2×C22⋊Dic53—21no—
109C4⋊#88322—22no—
110C4⋊#88322—22no—
111C22⋊#88322—21no—
112C22⋊#88322—21no—
113C22⋊#88322C2311yescharacteristic
114C22⋊#88322C2311yescharacteristic
115C22⋊#88322C2312yes—
116#13133128—22no—
117C4×#88322—22no—
118C23×C4×C53C2311yescharacteristic
119#117×C53C2×C411yescharacteristic
120#117×C53C2×C411yescharacteristic
121#117×C53C2×C412yes—
122#117×C53D412yes—
123#117×C53D411yescharacteristic
124#117×C53D411yescharacteristic
125C2×C4⋊C4×C53—21no—
126C2×C4⋊C4×C53—22no—
127C2×C4⋊C4×C53D411yescharacteristic
128C2×C4⋊C4×C53D411yescharacteristic
129C2×C22⋊C4×C53—21no—
130C2×C22⋊C4×C53—21no—
131C2×C22⋊C4×C53D412yes—
132C2×C42×C53—22no—
133C2×C42×C53—21no—
134C2×C42×C53C2312yes—
135#95×C53—22no—
136#95×C53—22no—
137#95×C53C2311yescharacteristic
138#95×C53C2312yes—
139#95×C53C2311yescharacteristic
140C24×C53C2×D411yescharacteristic
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—22no—
144C22×C4×C53—42no—
145C22×C4×C53C22×C411yescharacteristic
146C22×C4×C53C22×C411yescharacteristic
147C22×C4×C53C2×D412yes—
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D412yes—
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2411yescharacteristic
153C2×C22⋊D53—21no—
154C2×C22⋊D53—21no—
155C2×C22⋊D53—21no—
156C22×Dic53—21no—
157C22×Dic53—21no—
158#11719945—42no—
159C2×D212C22×C411yescharacteristic
160C2×C4×D53—21no—
161C2×C4×D53—22no—
162C2×Dic106—21no—
163C2×Dic106—22no—
164C2×Dic106—42no—
165C2×Dic106C22×C412yes—
166C2×Dic106C22×C411yescharacteristic
167C4⋊C4×C53—42no—
168C4⋊C4×C53—82no—
169C4⋊C4×C53C22⋊C411yescharacteristic
170C4⋊C4×C53C22⋊C411yescharacteristic
171C4⋊C4×C53C22⋊C412yes—
172C4⋊C4×C53C22⋊C411yescharacteristic
173C4⋊C4×C53C22⋊C411yescharacteristic
174C4⋊C4×C53C22⋊C412yes—
175C22⋊C4×C53—42no—
176C22⋊C4×C53—81no—
177C22⋊C4×C53C22⋊C412yes—
178C22⋊C4×C53C22⋊C412yes—
179C22⋊C4×C53C22⋊C412yes—
180C22⋊C4×C53C22⋊C412yes—
181C42×C53—42no—
182C42×C53—44no—
183C42×C53C22×C412yes—
184C42×C53C22×C412yes—
185C42×C53C2×D412yes—
186C42×C53C2×D412yes—
187C22⋊Dic53—22no—
188C22⋊Dic53—21no—
189C22⋊Dic53—21no—
190C22⋊Dic53—22no—
191C22⋊Dic53—21no—
192#11719921—22no—
193#11719921—22no—
194#11719921—21no—
195#11719921—22no—
196#11719921—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—22no—
199C4⋊Dic53—22no—
200C4⋊Dic53—22no—
201C4⋊Dic53C22×C412yes—
202C4⋊Dic53C22×C411yescharacteristic
203C4⋊Dic53C22×C411yescharacteristic
204#11719919—22no—
205#11719919—22no—
206#11719919—22no—
207#11719919—22no—
208#11719919—22no—
209#11719919—22no—
210#11719919—21no—
211#11719919—22no—
212#11719919—21no—
213C4×Dic53—22no—
214C4×Dic53—22no—
215C4×Dic53—22no—
216C4×Dic53—21no—
217#11719917—42no—
218C2×#88322—21no—
219C2×#88322—22no—
220C2×#88322—21no—
221C2×#88322—42no—
222C23×C53—21no—
223C23×C53—21no—
224C23×C53—41no—
225C23×C53—41no—
226C23×C53C2×C22⋊C411yescharacteristic
227C23×C53C2×C22⋊C411yescharacteristic
228C23×C53C22×D411yescharacteristic
229C22×D53—21no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—42no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—42no—
244C2×C4×C53—41no—
245C2×C4×C53—42no—
246C2×C4×C53—42no—
247C2×C4×C53—41no—
248C2×C4×C53C2×C22⋊C412yes—
249C2×C4×C53C2×C22⋊C412yes—
250C2×C4×C53C2×C22⋊C411yescharacteristic
251C2×C4×C53C2×C22⋊C412yes—
252C2×C4×C53C2×C22⋊C412yes—
253C2×C4×C53C2×C22⋊C411yescharacteristic
254C2×C4×C53C23×C411yescharacteristic
255C2×C4×C53C22×D411yescharacteristic
256C22⋊D53—42no—
257C22⋊D53—41no—
258C22⋊D53—41no—
259C22⋊D53—81no—
260C2×Dic53—21no—
261C2×Dic53—22no—
262C2×Dic53—22no—
263C2×Dic53—21no—
264C2×Dic53—21no—
265C2×Dic53—42no—
266C2×Dic53—81no—
267D212—41no—
268C4×D53—42no—
269C4×D53—82no—
270Dic106—42no—
271Dic106—42no—
272Dic106—41no—
273Dic106—82no—
274#88322—41no—
275#88322—41no—
276#88322—42no—
277#17605—531nomaximal, Sylow
278C22×C53—21no—
279C22×C53—21no—
280C22×C53—21no—
281C22×C53—21no—
282C22×C53—21no—
283C22×C53—21no—
284C22×C53—21no—
285C22×C53—21no—
286C22×C53—81no—
287C22×C53—81no—
288C22×C53C22×C22⋊C411yescharacteristic
289C22×C53#57711yescharacteristic
290C22×C53#57811yescharacteristic
291C2×D53—41no—
292C2×D53—41no—
293C2×D53—41no—
294C4×C53—21no—
295C4×C53—21no—
296C4×C53—21no—
297C4×C53—22no—
298C4×C53—22no—
299C4×C53—22no—
300C4×C53—22no—
301C4×C53—42no—
302C4×C53—81no—
303C4×C53—82no—
304C4×C53#41012yes—
305Dic53—42no—
306Dic53—41no—
307Dic53—41no—
308Dic53—41no—
309Dic53—42no—
310#3495—531no—
311#2924—531no—
312#2346D5311yescharacteristic, complemented
313#1520—531no—
314#1518—532no—
315#1513—531no—
316#1512—532no—
317#1508—531no—
318#1506—532no—
319#1501—531no—
320#1500—532no—
321C2×C53—21no—
322C2×C53—21no—
323C2×C53—41no—
324C2×C53—41no—
325C2×C53C2×#41011yescharacteristic
326C2×C53#289811yescharacteristic
327C2×C53#290111yescharacteristic
328D53—81no—
329#556—1061no—
330#554—1061no—
331#552—1062no—
332#552—1061no—
333#552—1061no—
334#551—1062no—
335#544—531no—
336#542—532no—
337#541—532no—
338#536—531no—
339#533—532no—
340#529—532no—
341#525—531no—
342C2×C22⋊Q8—531no—
343#520—531no—
344#519—531no—
345C2×#117C2×D5311yescharacteristic
346#433—1062no—
347#432—532no—
348#431—532no—
349#424—1061no—
350#407—531no—
351C2×C22⋊C8—531no—
352#405—1061no—
353#389—21no—
354#388—22no—
355#382C2×D5312yes—
356#380C2×D5311yescharacteristic
357C4×C4⋊C4—21no—
358C4×C22⋊C4C2×D5311yescharacteristic
359#376C2×D5312yes—
360#328—532no—
361#328—532no—
362#328—531no—
363#328—531no—
364#328—532no—
365#328—532no—
366#328—532no—
367#327—531no—
368#327—532no—
369#327—531no—
370C53#1760511yescharacteristic, complemented, Sylow
371C2×#41—531no—
372C22×Q8—531no—
373C23×C4C22×D5311yescharacteristic
374C2×M16—1061no—
375C22×C8—1061no—
376C4⋊Q8—532no—
377C4⋊Q8—532no—
378C4⋊Q8—1061no—
379#126—1062no—
380#126—1062no—
381#126—1062no—
382#126—1062no—
383#125—532no—
384#125—1062no—
385#125—1062no—
386#125—1061no—
387#125—1062no—
388#124—532no—
389#124—1062no—
390#123—1061no—
391#123—1062no—
392#123—1062no—
393#123—1062no—
394C22⋊Q8—532no—
395C22⋊Q8—531no—
396C22⋊Q8—532no—
397C22⋊Q8—531no—
398C22⋊Q8—1061no—
399C22⋊Q8—1061no—
400C22⋊Q8—1062no—
401C22⋊Q8—1061no—
402C22⋊Q8—1061no—
403C22⋊Q8—1062no—
404C22⋊Q8—1062no—
405#121—531no—
406#121—531no—
407#120—1061no—
408C4×Q8—532no—
409C4×Q8—532no—
410C4×Q8—532no—
411C4×Q8—1061no—
412C4×Q8—1061no—
413C4×Q8—1062no—
414C4×D4—532no—
415C4×D4—1061no—
416C4×D4—1061no—
417C4×D4—1062no—
418C4×D4—1062no—
419#117—531no—
420#117—1062no—
421#117—1061no—
422#117—1062no—
423#117—1062no—
424#117C4×D5311yescharacteristic
425#117C4×D5311yescharacteristic
426#117C4×D5312yes—
427#117D21211yescharacteristic
428#117D21212yes—
429#117D21211yescharacteristic
430C2×C4⋊C4—21no—
431C2×C4⋊C4—22no—
432C2×C4⋊C4—531no—
433C2×C4⋊C4—1061no—
434C2×C4⋊C4—1061no—
435C2×C4⋊C4C22⋊D5311yescharacteristic
436C2×C4⋊C4C22⋊D5311yescharacteristic
437C2×C22⋊C4—21no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—1061no—
440C2×C22⋊C4C22⋊D5312yes—
441C2×C42—22no—
442C2×C42—21no—
443C2×C42C22×D5312yes—
444C4⋊C8—1062no—
445C4⋊C8—1062no—
446C22⋊C8—531no—
447C22⋊C8—531no—
448C22⋊C8—532no—
449C22⋊C8—1061no—
450C22⋊C8—1061no—
451#97—1062no—
452C4×C8—1062no—
453#95—22no—
454#95—22no—
455#95C22×D5311yescharacteristic
456#95C22×D5312yes—
457#95C22×D5311yescharacteristic
458C24C2×C22⋊D5311yescharacteristic
459#41—2122no—
460C2×Q8—532no—
461C2×Q8—531no—
462C2×Q8—1062no—
463C2×Q8—1061no—
464C2×Q8—2122no—
465C2×D4—531no—
466C2×D4—1061no—
467C2×D4—1061no—
468C2×D4—1061no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—42no—
473C22×C4—1061no—
474C22×C4—1061no—
475C22×C4—1062no—
476C22×C4—1061no—
477C22×C4C2×C4×D5311yescharacteristic
478C22×C4C2×C4×D5311yescharacteristic
479C22×C4C2×D21211yescharacteristic
480C22×C4C2×D21211yescharacteristic
481C22×C4C2×C22⋊D5312yes—
482C22×C4C2×C22⋊D5311yescharacteristic
483C22×C4C2×C22⋊D5312yes—
484C22×C4C23×D5311yescharacteristic
485M16—2122no—
486C2×C8—1061no—
487C2×C8—1062no—
488C2×C8—1061no—
489C2×C8—2122no—
490C4⋊C4—42no—
491C4⋊C4—82no—
492C4⋊C4—532no—
493C4⋊C4—531no—
494C4⋊C4—531no—
495C4⋊C4—1061no—
496C4⋊C4—1062no—
497C4⋊C4—1062no—
498C4⋊C4—1062no—
499C4⋊C4—1062no—
500C4⋊C4—1062no—
501C4⋊C4—1062no—
502C4⋊C4—1062no—
503C4⋊C4—1061no—
504C4⋊C4—1062no—
505C4⋊C4—1062no—
506C4⋊C4—1062no—
507C4⋊C4—1061no—
508C4⋊C4#1171992111yescharacteristic
509C4⋊C4#1171992111yescharacteristic
510C4⋊C4#1171992112yes—
511C4⋊C4#1171992111yescharacteristic
512C4⋊C4#1171992111yescharacteristic
513C4⋊C4#1171992112yes—
514C22⋊C4—42no—
515C22⋊C4—81no—
516C22⋊C4—1062no—
517C22⋊C4—1061no—
518C22⋊C4—1061no—
519C22⋊C4—1062no—
520C22⋊C4—1061no—
521C22⋊C4—1062no—
522C22⋊C4—1062no—
523C22⋊C4—1061no—
524C22⋊C4—1062no—
525C22⋊C4—1061no—
526C22⋊C4#1171992112yes—
527C22⋊C4#1171992112yes—
528C22⋊C4#1171992112yes—
529C22⋊C4#1171992112yes—
530C42—42no—
531C42—44no—
532C42—1061no—
533C42—1062no—
534C42—1062no—
535C42—1062no—
536C42C2×C4×D5312yes—
537C42C2×C4×D5312yes—
538C42C2×D21212yes—
539C42C2×D21212yes—
540C23—21no—
541C23—21no—
542C23—41no—
543C23—41no—
544C23—1061no—
545C23C2×#1171992111yescharacteristic
546C23C2×#1171992111yescharacteristic
547C23C22×C22⋊D5311yescharacteristic
548Q8—4242no—
549Q8—2122no—
550Q8—2121no—
551Q8—2122no—
552D4—4241no—
553D4—2121no—
554D4—2122no—
555D4—2121no—
556D4—2121no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—1061no—
576C2×C4—1061no—
577C2×C4—1062no—
578C2×C4—1061no—
579C2×C4—1062no—
580C2×C4—4242no—
581C2×C4—4241no—
582C2×C4—2122no—
583C2×C4—2122no—
584C2×C4C2×#1171992112yes—
585C2×C4C2×#1171992112yes—
586C2×C4C2×#1171992112yes—
587C2×C4C2×#1171992112yes—
588C2×C4C2×#1171992111yescharacteristic
589C2×C4C2×#1171992111yescharacteristic
590C2×C4C22×C4×D5311yescharacteristic
591C2×C4C22×D21211yescharacteristic
592C8—2121no—
593C8—2121no—
594C8—2122no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—81no—
604C22—81no—
605C22—2121no—
606C22—2121no—
607C22—2121no—
608C22C22×#1171992111yescharacteristic
609C22#1681143411yescharacteristic
610C22#1681143811yescharacteristic
611C4—21no—
612C4—21no—
613C4—21no—
614C4—22no—
615C4—22no—
616C4—22no—
617C4—22no—
618C4—42no—
619C4—81no—
620C4—82no—
621C4—2121no—
622C4—2122no—
623C4—2121no—
624C4—2122no—
625C4—2121no—
626C4#1681059612yes—
627C2—21no—
628C2—21no—
629C2—41no—
630C2—41no—
631C2—4241no—
632C2C2×#1681059611yescharacteristic
633C2#2028324511yescharacteristic
634C2#2028326311yescharacteristic
subgroupG/H
centerC22C22×#11719921
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#2346C2
socleC22×C53C22×C22⋊C4
prime radical#16811134C22
cosocleC22×C4×C53C24
hypercenter#2346D53
hypocenterC53#17605
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×#117 ◁ #2346

Complements

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

NH ≅ G/Ntype
2. #20283373C2semidirect
3. #20273286C2semidirect
4. #20273278C2semidirect
6. #20273251C2semidirect
7. #20273234C2semidirect
8. #20273226C2semidirect
10. #20273199C2semidirect
11. C53×#2346C2semidirect
18. #16811251C4semidirect
19. #16811244C4semidirect
20. #16811235C4semidirect
21. #16811199C4semidirect
26. #16811103C4semidirect
27. #16811100C4semidirect
312. #2346D53semidirect
370. C53#17605semidirect

Maximal subgroups

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

subgroupindex
#202884672
#202833732
#202732862
#202732782
#202732562
#202732512
#202732342
#202732262
#202732042
#202731992
C53×#23462
#1760553