Finite Group Databasebeta

← #27311312

Subgroups — #27311312

Order 13568.

1542 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133182C211yesmaximal, characteristic
2C2×#16810490C211yesmaximal, characteristic
3C2×#16810488C211yesmaximal, characteristic
4#20272985C212yesmaximal
5#20272981C212yesmaximal
6#20272969C211yesmaximal, characteristic
7#20272968C211yesmaximal, characteristic
8#20272965C211yesmaximal, characteristic
9#20272941C212yesmaximal
10#20272940C212yesmaximal
11C53×#1441C211yesmaximal, characteristic
12C23×Dic106C2211yescharacteristic
13C2×#13133182—24no—
14C2×#13133182—24no—
15C2×#13133182C418yescomplemented
16C22×C4⋊Dic53C2211yescharacteristic
17C22×#11719919C2211yescharacteristic
18C22×#11719919C2211yescharacteristic
19C22×C4×Dic53C2211yescharacteristic
20C22×C4×Dic53C2211yescharacteristic
21C22×C4×Dic53C2211yescharacteristic
22#16810490—24no—
23#16810488—24no—
24C2×#13133124C2212yes—
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2211yescharacteristic
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2212yes—
34C2×#13133124C2212yes—
35C22×C4⋊C4×C53C2211yescharacteristic
36C22×C4⋊C4×C53C2211yescharacteristic
37C2×#95×C53C2212yes—
38C2×#95×C53C2211yescharacteristic
39C2×#95×C53C2212yes—
40C22×Dic106—22no—
41C22×Dic106—22no—
42C22×Dic106—22no—
43C22×Dic106—22no—
44C22×Dic106—22no—
45C22×Dic106—22no—
46C22×Dic106C2×C414yes—
47C23×Dic53C2312yes—
48C23×Dic53C2312yes—
49C23×Dic53C2311yescharacteristic
50C23×Dic53C2311yescharacteristic
51C23×Dic53C2311yescharacteristic
52C23×Dic53C2311yescharacteristic
53#13133182—232no—
54C2×C4⋊Dic53—22no—
55C2×C4⋊Dic53—42no—
56C2×#11719919—22no—
57C2×#11719919—22no—
58C2×#11719919—22no—
59C2×#11719919—22no—
60C2×#11719919C2×C414yes—
61C2×#11719919C2×C414yes—
62C2×C4×Dic53—22no—
63C2×C4×Dic53—22no—
64C2×C4×Dic53—22no—
65C2×C4×Dic53—22no—
66C2×C4×Dic53C2×C414yes—
67C2×C4×Dic53C2×C414yes—
68C2×C4×Dic53C2×C414yes—
69C2×C4×Dic53D414yes—
70C2×C4×Dic53D414yes—
71#13133124—22no—
72#13133124—24no—
73#13133124—24no—
74#13133124—24no—
75#13133124—24no—
76#13133124—24no—
77#13133124—24no—
78#13133124—24no—
79#13133124—42no—
80#13133124—42no—
81#13133124—42no—
82C23×C4×C53C2312yes—
83C23×C4×C53C2312yes—
84C23×C4×C53C2311yescharacteristic
85C23×C4×C53C2311yescharacteristic
86C23×C4×C53C2311yescharacteristic
87C2×C4⋊C4×C53—22no—
88C2×C4⋊C4×C53—22no—
89C2×C4⋊C4×C53—22no—
90C2×C4⋊C4×C53—42no—
91C2×C4⋊C4×C53C2×C414yes—
92#95×C53—22no—
93#95×C53—24no—
94#95×C53—42no—
95C24×C53C2411yescharacteristic
96C22×C4×C53—21no—
97C22×C4×C53—21no—
98C22×C4×C53—21no—
99C22×C4×C53—21no—
100C22×C4×C53—22no—
101C22×C4×C53—22no—
102C22×C4×C53—21no—
103C22×C4×C53—21no—
104C22×C4×C53—22no—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53C22×C412yes—
121C22×C4×C53C22×C412yes—
122C22×C4×C53C22×C412yes—
123C22×C4×C53C2×D412yes—
124C22×C4×C53C2×D412yes—
125C22×C4×C53#4112yes—
126C22×C4×C53#4112yes—
127C22×Dic53—21no—
128C22×Dic53—21no—
129C22×Dic53—21no—
130C22×Dic53—21no—
131C22×Dic53—21no—
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53C22×C412yes—
156C22×Dic53C22×C412yes—
157C22×Dic53C22×C412yes—
158C22×Dic53C22×C412yes—
159C22×Dic53C2×D412yes—
160C22×Dic53C2×D412yes—
161C22×Dic53C2×D412yes—
162C22×Dic53C2×D412yes—
163C22×Dic53#4112yes—
164C22×Dic53#4112yes—
165C2×Dic106—28no—
166C2×Dic106—28no—
167C2×Dic106—28no—
168C2×Dic106—28no—
169C2×Dic106—28no—
170C2×Dic106—28no—
171C2×Dic106—28no—
172C4⋊C4×C53—28no—
173C4⋊C4×C53—44no—
174C4⋊Dic53—44no—
175#11719919—28no—
176#11719919—28no—
177C4×Dic53—28no—
178C4×Dic53—28no—
179C4×Dic53C22⋊C4116yes—
180C23×C53C23×C411yescharacteristic
181C23×C53C22×D411yescharacteristic
182C23×C53C22×D411yescharacteristic
183C23×C53C2×#4111yescharacteristic
184C23×C53C2×#4111yescharacteristic
185C23×C53C2×#4111yescharacteristic
186C23×C53C2×#4111yescharacteristic
187C23×C53H(5,𝔽2)12yes—
188C23×C532−1+412yes—
189C23×C532−1+412yes—
190C23×C532−1+412yes—
191C2×C4×C53—22no—
192C2×C4×C53—22no—
193C2×C4×C53—24no—
194C2×C4×C53—22no—
195C2×C4×C53—22no—
196C2×C4×C53—22no—
197C2×C4×C53—22no—
198C2×C4×C53—22no—
199C2×C4×C53—22no—
200C2×C4×C53—24no—
201C2×C4×C53—22no—
202C2×C4×C53—24no—
203C2×C4×C53—41no—
204C2×C4×C53—41no—
205C2×C4×C53—42no—
206C2×C4×C53—42no—
207C2×C4×C53—42no—
208C2×C4×C53—42no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53—42no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53C2×C22⋊C414yes—
221C2×C4×C53#11714yes—
222C2×Dic53—22no—
223C2×Dic53—22no—
224C2×Dic53—22no—
225C2×Dic53—22no—
226C2×Dic53—24no—
227C2×Dic53—22no—
228C2×Dic53—22no—
229C2×Dic53—22no—
230C2×Dic53—22no—
231C2×Dic53—22no—
232C2×Dic53—22no—
233C2×Dic53—22no—
234C2×Dic53—22no—
235C2×Dic53—22no—
236C2×Dic53—24no—
237C2×Dic53—22no—
238C2×Dic53—22no—
239C2×Dic53—24no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—42no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247C2×Dic53—42no—
248C2×Dic53—42no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257C2×Dic53C2×C22⋊C414yes—
258C2×Dic53C2×C22⋊C414yes—
259C2×Dic53#11714yes—
260Dic106—232no—
261#14346—531nomaximal, Sylow
262C22×C53C22×C22⋊C411yescharacteristic
263C22×C53C2×#11711yescharacteristic
264C22×C53C2×#11711yescharacteristic
265C22×C53#51912yes—
266C22×C53#52012yes—
267C22×C53#52511yescharacteristic
268C22×C53#52511yescharacteristic
269C22×C53#52511yescharacteristic
270C22×C53#52511yescharacteristic
271C22×C53#53511yescharacteristic
272C22×C53#53611yescharacteristic
273C22×C53#53611yescharacteristic
274C22×C53#53612yes—
275C22×C53#53611yescharacteristic
276C22×C53#53712yes—
277C22×C53#54112yes—
278C22×C53#54112yes—
279C22×C53#54112yes—
280C22×C53#54112yes—
281C22×C53#54212yes—
282C22×C53#54212yes—
283C22×C53#54212yes—
284C22×C53#54212yes—
285C4×C53—24no—
286C4×C53—24no—
287C4×C53—42no—
288C4×C53—82no—
289C4×C53—82no—
290Dic53—24no—
291Dic53—24no—
292Dic53—24no—
293Dic53—42no—
294Dic53—82no—
295Dic53—82no—
296C22×C4×Q8—531no—
297C2×#383—531no—
298C2×#382—531no—
299#1446—532no—
300#1445—532no—
301#1441—531no—
302#1441—531no—
303#1441—531no—
304#1441D5311yescharacteristic
305#1434—532no—
306#1434—532no—
307C2×C53#228011yescharacteristic
308C2×C53#231011yescharacteristic
309C2×C53#231111yescharacteristic
310C2×C53#232012yes—
311C2×C53#232012yes—
312C2×C53#263211yescharacteristic
313C2×C53#263211yescharacteristic
314C2×C53#263411yescharacteristic
315C2×C53#263411yescharacteristic
316C2×C53#263712yes—
317C2×C53#263712yes—
318C23×Q8—531no—
319C2×C4×Q8—538no—
320C2×C4×Q8—1064no—
321C2×C4×Q8—1064no—
322C22×C4⋊C4—531no—
323C22×C4⋊C4—531no—
324C22×C4⋊C4—531no—
325C22×C4⋊C4C2×D5311yescharacteristic
326C22×C4⋊C4C2×D5311yescharacteristic
327C22×C42—531no—
328C22×C42—531no—
329C22×C42—531no—
330#383—1064no—
331#382—1064no—
332C2×#95—532no—
333C2×#95—532no—
334C2×#95—532no—
335C2×#95—532no—
336C2×#95—532no—
337C2×#95—531no—
338C2×#95—532no—
339C2×#95—532no—
340C2×#95—532no—
341C2×#95—532no—
342C2×#95—532no—
343C2×#95C2×D5312yes—
344C2×#95C2×D5311yescharacteristic
345C2×#95C2×D5312yes—
346C53#1434611yescharacteristic, complemented, Sylow
347C22×Q8—534no—
348C22×Q8—1062no—
349C22×Q8—1062no—
350C22×Q8—1062no—
351C22×Q8—1062no—
352C22×Q8—1062no—
353C22×Q8—1062no—
354C23×C4—531no—
355C23×C4—531no—
356C23×C4—531no—
357C23×C4—531no—
358C23×C4—532no—
359C23×C4—532no—
360C23×C4C22×D5312yes—
361C23×C4C22×D5312yes—
362C23×C4C22×D5311yescharacteristic
363C23×C4C22×D5311yescharacteristic
364C23×C4C22×D5311yescharacteristic
365C4×Q8—10632no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—42no—
370C2×C4⋊C4—534no—
371C2×C4⋊C4—534no—
372C2×C4⋊C4—1062no—
373C2×C4⋊C4—1062no—
374C2×C4⋊C4—1062no—
375C2×C4⋊C4—1062no—
376C2×C4⋊C4—1062no—
377C2×C4⋊C4—2122no—
378C2×C4⋊C4C4×D5314yes—
379C2×C42—534no—
380C2×C42—534no—
381C2×C42—534no—
382C2×C42—534no—
383C2×C42—534no—
384C2×C42—1062no—
385C2×C42—1062no—
386C2×C42—1062no—
387C2×C42—1062no—
388#95—22no—
389#95—24no—
390#95—42no—
391#95—1064no—
392#95—1064no—
393#95—1064no—
394#95—1064no—
395#95—1064no—
396#95—1064no—
397#95—1062no—
398#95—1064no—
399#95—2122no—
400#95—2122no—
401#95—2122no—
402C24C23×D5311yescharacteristic
403C2×Q8—1068no—
404C2×Q8—1068no—
405C2×Q8—1068no—
406C2×Q8—1068no—
407C2×Q8—1068no—
408C2×Q8—1068no—
409C2×Q8—1068no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—532no—
435C22×C4—532no—
436C22×C4—532no—
437C22×C4—532no—
438C22×C4—532no—
439C22×C4—532no—
440C22×C4—532no—
441C22×C4—532no—
442C22×C4—532no—
443C22×C4—532no—
444C22×C4—1061no—
445C22×C4—1061no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—1061no—
450C22×C4—1061no—
451C22×C4—1061no—
452C22×C4—1062no—
453C22×C4—1062no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—1062no—
457C22×C4—1062no—
458C22×C4—1062no—
459C22×C4—1062no—
460C22×C4—1062no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4—1062no—
464C22×C4—1062no—
465C22×C4—1062no—
466C22×C4—1062no—
467C22×C4—1062no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4C2×C4×D5312yes—
473C22×C4C2×C4×D5312yes—
474C22×C4C2×C4×D5312yes—
475C22×C4D4×D5312yes—
476C22×C4D4×D5312yes—
477C22×C4#1171994712yes—
478C22×C4#1171994912yes—
479C4⋊C4—28no—
480C4⋊C4—44no—
481C4⋊C4—1068no—
482C4⋊C4—1068no—
483C4⋊C4—2124no—
484C42—5316no—
485C42—1068no—
486C42—1068no—
487C23C22×C4×D5311yescharacteristic
488C23C2×D4×D5311yescharacteristic
489C23C2×D4×D5311yescharacteristic
490C23#41×D5311yescharacteristic
491C23#41×D5311yescharacteristic
492C23C2×#1171994711yescharacteristic
493C23C2×#1171994911yescharacteristic
494C23#1313326212yes—
495C23#1313326412yes—
496C23#1313326412yes—
497C23#1313326512yes—
498Q8—10632no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—534no—
529C2×C4—534no—
530C2×C4—534no—
531C2×C4—1062no—
532C2×C4—1062no—
533C2×C4—1062no—
534C2×C4—1062no—
535C2×C4—1064no—
536C2×C4—1062no—
537C2×C4—1062no—
538C2×C4—1062no—
539C2×C4—1062no—
540C2×C4—1062no—
541C2×C4—1062no—
542C2×C4—1062no—
543C2×C4—1062no—
544C2×C4—1062no—
545C2×C4—1064no—
546C2×C4—1062no—
547C2×C4—1062no—
548C2×C4—1064no—
549C2×C4—2122no—
550C2×C4—2122no—
551C2×C4—2122no—
552C2×C4—2122no—
553C2×C4—2122no—
554C2×C4—2122no—
555C2×C4—2122no—
556C2×C4—2122no—
557C2×C4—2122no—
558C2×C4—2122no—
559C2×C4—2122no—
560C2×C4—2122no—
561C2×C4—2122no—
562C2×C4—2122no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2122no—
566C2×C4C22⋊C4×D5314yes—
567C2×C4#1313317514yes—
568C22C2×C22⋊C4×D5311yescharacteristic
569C22#117×D5311yescharacteristic
570C22C2×#1313317511yescharacteristic
571C22#1681109912yes—
572C22#1681110212yes—
573C22#1681113311yescharacteristic
574C22#1681113311yescharacteristic
575C22#1681113611yescharacteristic
576C22#1681113611yescharacteristic
577C22#1681118811yescharacteristic
578C22#1681119311yescharacteristic
579C22#1681119511yescharacteristic
580C22#1681119612yes—
581C22#1681119811yescharacteristic
582C22#1681120012yes—
583C22#1681123012yes—
584C22#1681123212yes—
585C22#1681123312yes—
586C22#1681123612yes—
587C22#1681124012yes—
588C22#1681124112yes—
589C22#1681124312yes—
590C22#1681124712yes—
591C4—24no—
592C4—24no—
593C4—42no—
594C4—82no—
595C4—82no—
596C4—1064no—
597C4—1064no—
598C4—1064no—
599C4—4242no—
600C4—4242no—
601C4—2122no—
602C2#2027709711yescharacteristic
603C2#2027729811yescharacteristic
604C2#2027730211yescharacteristic
605C2#2027736212yes—
606C2#2027736412yes—
607C2#2028058011yescharacteristic
608C2#2028058111yescharacteristic
609C2#2028059611yescharacteristic
610C2#2028059711yescharacteristic
611C2#2028062812yes—
612C2#2028063012yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1441C2
socle, prime radical, cosocleC24×C53C24
hypercenter#1441D53
hypocenterC53#14346
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
15. C2×#13133182C4semidirect
346. C53#14346semidirect

Maximal subgroups

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

subgroupindex
C22×#131331822
C2×#168104902
C2×#168104882
#202729852
#202729812
#202729692
#202729682
#202729652
#202729412
#202729402
C53×#14412
#1434653