Finite Group Databasebeta

← #27547267

Subgroups — #27547267

Order 13568.

1998 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811178C211yesmaximal, characteristic
2#20277876C214yesmaximal
3#20277805C211yesmaximal, characteristic, complemented
4#20277773C211yesmaximal, characteristic, complemented
5#20277771C211yesmaximal, characteristic, complemented
6#20277676C212yesmaximal, complemented
7#20277634C212yesmaximal, complemented
8#20277607C212yesmaximal
9#20277596C212yesmaximal
10#20277455C212yesmaximal, complemented
11#20277452C212yesmaximal, complemented
12#20277398C212yesmaximal
13#20277395C212yesmaximal
14#20277360C214yesmaximal, complemented
15#20277119C211yesmaximal, characteristic
16#20277111C211yesmaximal, characteristic
17C53×#2369C211yesmaximal, characteristic, complemented
18#16811178C4116yescomplemented
19C2×#13133217C2211yescharacteristic
20C2×#13133208C2212yes—
21C2×#13133197C2212yes—
22C2×#13133193C2212yes—
23C2×Q8×Dic53C2212yes—
24C2×C4×Dic106C2212yes—
25C2×#13133182C2212yes—
26C2×#13133182C2212yes—
27C2×#13133175C2212yes—
28C2×#13133174C2212yes—
29C2×C42×D53C2211yescharacteristic
30#16810530C2214yes—
31#16810529C2212yes—
32#16810528C2212yes—
33#16810522C2212yes—
34#16810522C2212yes—
35#16810521C2212yes—
36#16810521C2214yes—
37#16810520C2214yes—
38#16810519C2212yes—
39#16810519C2214yes—
40#16810518C2214yes—
41#16810515C2214yes—
42#16810514C2214yes—
43#16810513C2214yes—
44#16810512C2214yes—
45#16810508C2214yes—
46#16810508C2214yes—
47#16810507C2214yes—
48#16810506C2214yes—
49#16810506C2214yes—
50#16810505C2214yes—
51#16810492C2211yescharacteristic
52#16810492C2211yescharacteristic
53#16810490C2214yes—
54#16810490C2212yes—
55#16810488C2211yescharacteristic
56#16810488C2211yescharacteristic
57#16810487C2214yes—
58#16810487C2212yes—
59#16810485C2212yes—
60#16810482C2212yes—
61#16810480C2211yescharacteristic
62#16810480C2211yescharacteristic
63#16810480C2211yescharacteristic
64#16810480C2211yescharacteristic
65C4⋊C4×Dic53C2212yes—
66C4⋊C4×Dic53C2212yes—
67C4⋊C4×Dic53C2212yes—
68C4⋊C4×Dic53C2214yes—
69C4×#11719919C2212yes—
70C4×C4⋊Dic53C2211yescharacteristic
71C4×C4⋊Dic53C2211yescharacteristic
72C4×C4⋊Dic53C2212yes—
73C4×#11719921C2212yes—
74#16810469C2214yes—
75#16810469C2212yes—
76C2×C4⋊Q8×C53C2211yescharacteristic
77C53×#391C2214yes—
78C53×#389C2212yes—
79C53×#389C2212yes—
80C4×C4⋊C4×C53C2211yescharacteristic
81C4×C4⋊C4×C53C2211yescharacteristic
82C4×C4⋊C4×C53C2212yes—
83C4×C4⋊C4×C53C2212yes—
84C22×Dic106C2312yes—
85C22×C4×D53C2311yescharacteristic
86C22×C4×D53C2311yescharacteristic
87C22×C4×D53C2311yescharacteristic
88#13133217C2×C418yes—
89#13133208C2×C4116yes—
90#13133197C2×C4116yes—
91#13133193C2×C4116yes—
92Q8×Dic53C2×C4116yes—
93C4×Dic106—44no—
94#13133182—44no—
95#13133182C2×C4116yes—
96#13133175C2×C4116yes—
97#13133174—44no—
98C2×C4⋊Dic53C2311yescharacteristic
99C2×C4⋊Dic53C2311yescharacteristic
100C2×C4⋊Dic53C2311yescharacteristic
101C2×C4⋊Dic53C2311yescharacteristic
102C2×C4⋊Dic53C2314yes—
103C2×C4⋊Dic53C2312yes—
104C2×C4⋊Dic53C2314yes—
105C2×C4⋊Dic53C2312yes—
106C2×#11719919C2312yes—
107C2×#11719919C2312yes—
108C2×#11719919C2314yes—
109C2×#11719919C2312yes—
110C2×#11719919C2312yes—
111C2×#11719919C2314yes—
112C2×#11719919C2314yes—
113C2×#11719921C2312yes—
114C2×#11719921C2312yes—
115C2×#11719921C2312yes—
116C2×#11719921C2312yes—
117C2×#11719921C2314yes—
118C2×C4×Dic53C2312yes—
119C2×C4×Dic53C2312yes—
120C2×C4×Dic53C2314yes—
121C2×C4×Dic53C2314yes—
122C2×C4×Dic53C2314yes—
123C2×C4×Dic53C2311yescharacteristic
124C2×C4×Dic53C2311yescharacteristic
125C2×C4×Dic53C2311yescharacteristic
126C2×C4×Dic53C2314yes—
127C42×D53C2×C418yes—
128#13133124C2312yes—
129#13133124C2314yes—
130#13133124C2314yes—
131#13133124C2314yes—
132#13133124C2312yes—
133#13133124C2312yes—
134#13133124C2314yes—
135#13133124C2314yes—
136#13133124C2312yes—
137#13133124C2312yes—
138#13133124C2314yes—
139#13133124C2312yes—
140#13133124C2312yes—
141#13133124C2312yes—
142#13133124C2312yes—
143#13133124C2312yes—
144C22×Q8×C53C2312yes—
145C4⋊Q8×C53C2×C418yes—
146C2×C4⋊C4×C53C2311yescharacteristic
147C2×C4⋊C4×C53C2311yescharacteristic
148C2×C4⋊C4×C53C2311yescharacteristic
149C2×C4⋊C4×C53C2311yescharacteristic
150C2×C4⋊C4×C53C2314yes—
151C2×C4⋊C4×C53C2312yes—
152C2×C4⋊C4×C53C2312yes—
153C2×C42×C53C2312yes—
154C2×C42×C53C2312yes—
155C2×C42×C53C2314yes—
156C2×C42×C53C2311yescharacteristic
157#95×C53C2314yes—
158#95×C53C2312yes—
159#95×C53C2314yes—
160#95×C53C2312yes—
161C23×D53C2411yescharacteristic
162C2×Q8×C53—42no—
163C2×Q8×C53—42no—
164C2×Q8×C53C22×C418yes—
165C22×C4×C53C2412yes—
166C22×C4×C53C2412yes—
167C22×C4×C53C2412yes—
168C22×C4×C53C2412yes—
169C22×C4×C53C2411yescharacteristic
170C22×C4×C53C2414yes—
171C22×C4×C53C2411yescharacteristic
172C22×C4×C53C2411yescharacteristic
173C22×Dic53C2412yes—
174C22×Dic53C2412yes—
175C22×Dic53C2412yes—
176C22×Dic53C2412yes—
177C22×Dic53C2411yescharacteristic
178C22×Dic53C2414yes—
179C22×Dic53C2411yescharacteristic
180C22×Dic53C2411yescharacteristic
181C2×C4×D53—22no—
182C2×C4×D53—22no—
183C2×C4×D53—22no—
184C2×C4×D53—22no—
185C2×C4×D53—41no—
186C2×C4×D53—41no—
187C2×C4×D53C22×C414yes—
188C2×C4×D53C22×C414yes—
189C2×C4×D53C22×C414yes—
190C2×Dic106—24no—
191C2×Dic106—24no—
192C2×Dic106C22×C418yes—
193C4⋊C4×C53—22no—
194C4⋊C4×C53—22no—
195C4⋊C4×C53—22no—
196C4⋊C4×C53—22no—
197C4⋊C4×C53—44no—
198C4⋊C4×C53C22×C418yes—
199C4⋊C4×C53C22×C418yes—
200C42×C53—24no—
201C42×C53—24no—
202C42×C53—44no—
203C42×C53C22×C414yes—
204#11719921—42no—
205#11719921—42no—
206#11719921—42no—
207#11719921—42no—
208#11719921C22×C4116yes—
209C4⋊Dic53—22no—
210C4⋊Dic53—22no—
211C4⋊Dic53—22no—
212C4⋊Dic53—22no—
213C4⋊Dic53—44no—
214C4⋊Dic53C22×C418yes—
215C4⋊Dic53C22×C418yes—
216C4⋊Dic53C22×C4116yes—
217#11719919—42no—
218#11719919—42no—
219#11719919—42no—
220#11719919—42no—
221#11719919—44no—
222#11719919—44no—
223#11719919C22×C4116yes—
224C4×Dic53—24no—
225C4×Dic53—24no—
226C4×Dic53—44no—
227C4×Dic53—44no—
228C4×Dic53—44no—
229C4×Dic53C22×C414yes—
230C4×Dic53C22×C414yes—
231C4×Dic53C22×C414yes—
232C4×Dic53C22×C4116yes—
233C23×C53C2511yescharacteristic
234C22×D53—21no—
235C22×D53—21no—
236C22×D53—21no—
237C22×D53—21no—
238C22×D53—21no—
239C22×D53—21no—
240C22×D53C23×C412yes—
241Q8×C53—48no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—24no—
245C2×C4×C53—24no—
246C2×C4×C53—22no—
247C2×C4×C53—22no—
248C2×C4×C53—22no—
249C2×C4×C53—22no—
250C2×C4×C53—22no—
251C2×C4×C53—22no—
252C2×C4×C53—22no—
253C2×C4×C53—22no—
254C2×C4×C53—22no—
255C2×C4×C53—22no—
256C2×C4×C53—22no—
257C2×C4×C53—22no—
258C2×C4×C53C23×C412yes—
259C2×C4×C53C23×C418yes—
260C2×C4×C53C23×C412yes—
261C2×C4×C53C23×C412yes—
262C2×C4×C53H(5,𝔽2)12yes—
263C2×C4×C532−1+412yes—
264C2×C4×C532−1+412yes—
265C2×C4×C532−1+412yes—
266C2×Dic53—24no—
267C2×Dic53—21no—
268C2×Dic53—24no—
269C2×Dic53—21no—
270C2×Dic53—21no—
271C2×Dic53—21no—
272C2×Dic53—21no—
273C2×Dic53—21no—
274C2×Dic53—22no—
275C2×Dic53—22no—
276C2×Dic53—22no—
277C2×Dic53—22no—
278C2×Dic53—22no—
279C2×Dic53—22no—
280C2×Dic53—22no—
281C2×Dic53—22no—
282C2×Dic53—22no—
283C2×Dic53—22no—
284C2×Dic53—22no—
285C2×Dic53—22no—
286C2×Dic53C23×C418yes—
287C2×Dic53C23×C412yes—
288C2×Dic53C23×C412yes—
289C2×Dic53C23×C412yes—
290C4×D53—28no—
291C4×D53—28no—
292C4×D53—44no—
293Dic106—216no—
294#34394—531nomaximal, Sylow
295C22×C53C24×C411yescharacteristic
296C22×C53C2×H(5,𝔽2)11yescharacteristic
297C22×C53C2×2−1+411yescharacteristic
298C22×C53C2×2−1+411yescharacteristic
299C22×C53C2×2−1+411yescharacteristic
300C22×C53#58511yescharacteristic
301C22×C53#58511yescharacteristic
302C2×D53—22no—
303C2×D53—22no—
304C2×D53—22no—
305C2×D53—41no—
306C2×D53—41no—
307C2×D53—41no—
308C2×D53—41no—
309C4×C53—22no—
310C4×C53—28no—
311C4×C53—42no—
312C4×C53—42no—
313C4×C53—42no—
314C4×C53—42no—
315C4×C53#51914yes—
316C4×C53#52014yes—
317Dic53—22no—
318Dic53—28no—
319Dic53—22no—
320Dic53—22no—
321Dic53—42no—
322Dic53—42no—
323Dic53—42no—
324Dic53—42no—
325C2×#533—531no—
326#2378—534no—
327#2369—531no—
328#2369D5311yescharacteristic, complemented
329#2365—531no—
330#2365—531no—
331#2356—532no—
332#2351—532no—
333#2347—532no—
334#2346—532no—
335#2330—532no—
336#2330—532no—
337#2323—532no—
338#2323—532no—
339#2320—534no—
340#2285—531no—
341#2283—531no—
342C2×C53C2×#51911yescharacteristic
343C2×C53C2×#52011yescharacteristic
344C2×C53#342811yescharacteristic
345C2×C53#354711yescharacteristic
346C2×C53#354711yescharacteristic
347C2×C53#355511yescharacteristic
348C2×C53#355511yescharacteristic
349D53—42no—
350#533—5316no—
351C2×C4⋊Q8—531no—
352C2×C4⋊Q8C2×D5311yescharacteristic
353C2×#125—532no—
354C2×C22⋊Q8—532no—
355C2×C22⋊Q8—532no—
356C2×C4×Q8—532no—
357C2×C4×Q8—532no—
358C2×C4×Q8—532no—
359C2×C4×Q8—532no—
360C2×#117—532no—
361C2×#117—532no—
362C22×C42—531no—
363#391—534no—
364#391—532no—
365#391—532no—
366#391C2×D5314yes—
367#389—534no—
368#389—532no—
369#389—534no—
370#389—532no—
371#389—532no—
372#389—534no—
373#389—532no—
374#389—534no—
375#389C2×D5312yes—
376#389C2×D5312yes—
377#388—534no—
378#388—534no—
379#388—534no—
380#388—534no—
381#387—534no—
382#387—534no—
383#387—534no—
384#387—534no—
385#387—534no—
386#387—534no—
387#384—531no—
388#384—531no—
389#383—532no—
390#383—534no—
391#382—531no—
392#382—531no—
393#382—532no—
394#382—534no—
395#381—532no—
396#380—531no—
397#380—531no—
398#380—531no—
399#380—531no—
400#380—532no—
401C4×C4⋊C4—534no—
402C4×C4⋊C4—532no—
403C4×C4⋊C4—532no—
404C4×C4⋊C4—532no—
405C4×C4⋊C4—532no—
406C4×C4⋊C4—532no—
407C4×C4⋊C4—531no—
408C4×C4⋊C4—531no—
409C4×C4⋊C4C2×D5312yes—
410C4×C4⋊C4C2×D5312yes—
411C4×C4⋊C4C2×D5311yescharacteristic
412C4×C4⋊C4C2×D5311yescharacteristic
413C4×C22⋊C4—532no—
414#376—534no—
415#376—532no—
416C53#3439411yescharacteristic, complemented, Sylow
417C22×Q8—532no—
418C22×Q8C22×D5312yes—
419C23×C4—531no—
420C23×C4—531no—
421C23×C4—531no—
422C4⋊Q8—538no—
423C4⋊Q8C4×D5318yes—
424#125—5316no—
425C22⋊Q8—5316no—
426C22⋊Q8—5316no—
427C4×Q8—5316no—
428C4×Q8—5316no—
429C4×Q8—2124no—
430C4×Q8—2124no—
431#117—5316no—
432#117—2124no—
433C2×C4⋊C4—532no—
434C2×C4⋊C4—534no—
435C2×C4⋊C4—534no—
436C2×C4⋊C4—532no—
437C2×C4⋊C4—531no—
438C2×C4⋊C4—531no—
439C2×C4⋊C4—531no—
440C2×C4⋊C4—531no—
441C2×C4⋊C4—532no—
442C2×C4⋊C4—532no—
443C2×C4⋊C4—534no—
444C2×C4⋊C4—534no—
445C2×C4⋊C4—532no—
446C2×C4⋊C4—532no—
447C2×C4⋊C4—534no—
448C2×C4⋊C4C22×D5311yescharacteristic
449C2×C4⋊C4C22×D5311yescharacteristic
450C2×C4⋊C4C22×D5311yescharacteristic
451C2×C4⋊C4C22×D5311yescharacteristic
452C2×C4⋊C4C22×D5314yes—
453C2×C4⋊C4C22×D5312yes—
454C2×C4⋊C4C22×D5312yes—
455C2×C22⋊C4—534no—
456C2×C22⋊C4—532no—
457C2×C22⋊C4—532no—
458C2×C22⋊C4—532no—
459C2×C22⋊C4—532no—
460C2×C42—534no—
461C2×C42—531no—
462C2×C42—531no—
463C2×C42—531no—
464C2×C42—538no—
465C2×C42—532no—
466C2×C42—532no—
467C2×C42—534no—
468C2×C42—534no—
469C2×C42—534no—
470C2×C42C22×D5312yes—
471C2×C42C22×D5312yes—
472C2×C42C22×D5314yes—
473C2×C42C22×D5311yescharacteristic
474#95—532no—
475#95—532no—
476#95—532no—
477#95—532no—
478#95—532no—
479#95—532no—
480#95—534no—
481#95—534no—
482#95—532no—
483#95—532no—
484#95—532no—
485#95—532no—
486#95—534no—
487#95—534no—
488#95—534no—
489#95—534no—
490#95C22×D5312yes—
491#95C22×D5312yes—
492#95C22×D5314yes—
493#95C22×D5314yes—
494C24—531no—
495C2×Q8—42no—
496C2×Q8—42no—
497C2×Q8—538no—
498C2×Q8—1064no—
499C2×Q8—1064no—
500C2×Q8C2×C4×D5318yes—
501C22×C4—531no—
502C22×C4—534no—
503C22×C4—531no—
504C22×C4—531no—
505C22×C4—534no—
506C22×C4—534no—
507C22×C4—534no—
508C22×C4—532no—
509C22×C4—532no—
510C22×C4—532no—
511C22×C4—532no—
512C22×C4—1062no—
513C22×C4—1062no—
514C22×C4—1062no—
515C22×C4—1062no—
516C22×C4—2121no—
517C22×C4—2121no—
518C22×C4C23×D5312yes—
519C22×C4C23×D5312yes—
520C22×C4C23×D5312yes—
521C22×C4C23×D5312yes—
522C22×C4C23×D5311yescharacteristic
523C22×C4C23×D5314yes—
524C22×C4C23×D5311yescharacteristic
525C22×C4C23×D5311yescharacteristic
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—44no—
531C4⋊C4—538no—
532C4⋊C4—5316no—
533C4⋊C4—5316no—
534C4⋊C4—538no—
535C4⋊C4—1062no—
536C4⋊C4—1062no—
537C4⋊C4—1062no—
538C4⋊C4—1062no—
539C4⋊C4—2122no—
540C4⋊C4—2122no—
541C4⋊C4—2124no—
542C4⋊C4—2124no—
543C4⋊C4—2122no—
544C4⋊C4—2122no—
545C4⋊C4—2124no—
546C4⋊C4C2×C4×D5318yes—
547C4⋊C4C2×C4×D5318yes—
548C22⋊C4—5316no—
549C22⋊C4—2122no—
550C22⋊C4—2122no—
551C22⋊C4—2122no—
552C22⋊C4—2122no—
553C42—24no—
554C42—24no—
555C42—44no—
556C42—534no—
557C42—534no—
558C42—534no—
559C42—5316no—
560C42—1064no—
561C42—1064no—
562C42—2124no—
563C42—2124no—
564C42—2124no—
565C42C2×C4×D5314yes—
566C23—532no—
567C23—1061no—
568C23—1061no—
569C23—1061no—
570C23—1061no—
571C23—1061no—
572C23—1061no—
573C23C24×D5311yescharacteristic
574Q8—48no—
575Q8—10616no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—532no—
593C2×C4—532no—
594C2×C4—532no—
595C2×C4—538no—
596C2×C4—1064no—
597C2×C4—1061no—
598C2×C4—1061no—
599C2×C4—1061no—
600C2×C4—1061no—
601C2×C4—1064no—
602C2×C4—1061no—
603C2×C4—1061no—
604C2×C4—1068no—
605C2×C4—1068no—
606C2×C4—1062no—
607C2×C4—1062no—
608C2×C4—1062no—
609C2×C4—1062no—
610C2×C4—1062no—
611C2×C4—1062no—
612C2×C4—1062no—
613C2×C4—1062no—
614C2×C4—1062no—
615C2×C4—1062no—
616C2×C4—1062no—
617C2×C4—1062no—
618C2×C4—2124no—
619C2×C4C22×C4×D5312yes—
620C2×C4C22×C4×D5318yes—
621C2×C4C22×C4×D5312yes—
622C2×C4C22×C4×D5312yes—
623C2×C4#1313326312yes—
624C2×C4#1313326412yes—
625C2×C4#1313326512yes—
626C2×C4#1313326512yes—
627C22—1062no—
628C22—1062no—
629C22—1062no—
630C22—2121no—
631C22—2121no—
632C22—2121no—
633C22—2121no—
634C22C23×C4×D5311yescharacteristic
635C22C2×#1313326311yescharacteristic
636C22C2×#1313326411yescharacteristic
637C22C2×#1313326511yescharacteristic
638C22C2×#1313326511yescharacteristic
639C22#1681145811yescharacteristic
640C22#1681145811yescharacteristic
641C4—22no—
642C4—28no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—1062no—
648C4—1062no—
649C4—1068no—
650C4—1062no—
651C4—2122no—
652C4—2122no—
653C4—2122no—
654C4—2122no—
655C4#1681109914yes—
656C4#1681110514yes—
657C2—2122no—
658C2C2×#1681109911yescharacteristic
659C2C2×#1681110511yescharacteristic
660C2#2028791311yescharacteristic
661C2#2028910511yescharacteristic
662C2#2028910711yescharacteristic
663C2#2028920411yescharacteristic
664C2#2028920611yescharacteristic
subgroupG/H
center, FrattiniC23C24×D53
commutatorC22×C53C24×C4
FittingC53×#2369C2
socle, cosocleC23×C53C25
prime radicalC23×D53C24
hypercenter#2369D53
hypocenterC53#34394
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #2369

Complements

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

NH ≅ G/Ntype
3. #20277805C2semidirect
4. #20277773C2semidirect
5. #20277771C2semidirect
6. #20277676C2semidirect
7. #20277634C2semidirect
10. #20277455C2semidirect
11. #20277452C2semidirect
14. #20277360C2semidirect
17. C53×#2369C2semidirect
18. #16811178C4semidirect
328. #2369D53semidirect
416. C53#34394semidirect

Maximal subgroups

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

subgroupindex
C2×#168111782
#202778762
#202778052
#202777732
#202777712
#202776762
#202776342
#202776072
#202775962
#202774552
#202774522
#202773982
#202773952
#202773602
#202771192
#202771112
C53×#23692
#3439453