Finite Group Databasebeta

← #27306705

Subgroups — #27306705

Order 13568.

1486 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
1C2×C4×#11719921C211yesmaximal, characteristic
2#20272975C212yesmaximal, complemented
3#20272963C212yesmaximal
4#20272951C212yesmaximal
5#20272941C211yesmaximal, characteristic, complemented
6#20272941C211yesmaximal, characteristic, complemented
7#20272940C212yesmaximal, complemented
8#20272936C211yesmaximal, characteristic
9#20272936C211yesmaximal, characteristic
10C4×#13133124C211yesmaximal, characteristic, complemented
11C53×#1434C211yesmaximal, characteristic, complemented
12C23×C4×D53C2211yescharacteristic
13C22×#11719921C2212yes—
14C22×#11719921C2212yes—
15C22×#11719921C2211yescharacteristic
16C22×#11719921C2211yescharacteristic
17C22×C4×Dic53C2212yes—
18C22×C4×Dic53C2211yescharacteristic
19C4×#11719921C418yescomplemented
20C2×#13133124C2211yescharacteristic
21C2×#13133124C2211yescharacteristic
22C2×#13133124C2212yes—
23C2×#13133124C2212yes—
24C2×#13133124C2212yes—
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2211yescharacteristic
30C2×#13133124C2211yescharacteristic
31C22×C42×C53C2211yescharacteristic
32C2×#95×C53C2211yescharacteristic
33C2×#95×C53C2211yescharacteristic
34C2×#95×C53C2212yes—
35C2×#95×C53C2212yes—
36C24×D53C2311yescharacteristic
37C23×Dic53C2311yescharacteristic
38C23×Dic53C2311yescharacteristic
39C23×Dic53C2311yescharacteristic
40C23×Dic53C2312yes—
41C23×Dic53C2312yes—
42C22×C4×D53—22no—
43C22×C4×D53—22no—
44C22×C4×D53—42no—
45C22×C4×D53—42no—
46C22×C4×D53C2×C414yes—
47C2×#11719921—22no—
48C2×#11719921—22no—
49C2×#11719921—24no—
50C2×#11719921—24no—
51C2×#11719921—22no—
52C2×#11719921—22no—
53C2×#11719921—24no—
54C2×#11719921—24no—
55C2×#11719921—42no—
56C2×#11719921—42no—
57C2×#11719921C2×C414yes—
58C2×#11719921C2×C414yes—
59C2×C4×Dic53—24no—
60C2×C4×Dic53—24no—
61C2×C4×Dic53—42no—
62C2×C4×Dic53—42no—
63C2×C4×Dic53C2×C414yes—
64#13133124—22no—
65#13133124—22no—
66#13133124—24no—
67#13133124—24no—
68#13133124—24no—
69#13133124—42no—
70#13133124—42no—
71#13133124—42no—
72#13133124—42no—
73#13133124C2×C414yes—
74#13133124C2×C414yes—
75C23×C4×C53C2312yes—
76C23×C4×C53C2312yes—
77C23×C4×C53C2311yescharacteristic
78C23×C4×C53C2311yescharacteristic
79C23×C4×C53C2311yescharacteristic
80C2×C42×C53—42no—
81C2×C42×C53C2×C414yes—
82#95×C53—22no—
83#95×C53—22no—
84#95×C53—24no—
85#95×C53—42no—
86C24×C53C2411yescharacteristic
87C23×D53—21no—
88C23×D53—21no—
89C23×D53—22no—
90C23×D53—22no—
91C23×D53—21no—
92C23×D53—21no—
93C23×D53—22no—
94C23×D53—22no—
95C23×D53C22×C412yes—
96C23×D53#4112yes—
97C23×D53#4112yes—
98C22×C4×C53—22no—
99C22×C4×C53—22no—
100C22×C4×C53—22no—
101C22×C4×C53—22no—
102C22×C4×C53—22no—
103C22×C4×C53—21no—
104C22×C4×C53—21no—
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×C53#4112yes—
124C22×C4×C53#4112yes—
125C22×C4×C53#4112yes—
126C22×C4×C53#4112yes—
127C22×Dic53—22no—
128C22×Dic53—22no—
129C22×Dic53—22no—
130C22×Dic53—22no—
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—21no—
142C22×Dic53—21no—
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×Dic53C22×C412yes—
152C22×Dic53C22×C412yes—
153C22×Dic53C22×C412yes—
154C22×Dic53#4114yes—
155C22×Dic53#4114yes—
156C2×C4×D53—28no—
157C2×C4×D53—44no—
158C2×C4×D53—44no—
159C2×C4×D53—48no—
160C2×C4×D53—48no—
161C42×C53—44no—
162#11719921—28no—
163#11719921—28no—
164#11719921—48no—
165#11719921—48no—
166C4×Dic53—44no—
167C4×Dic53—48no—
168C23×C53C23×C411yescharacteristic
169C23×C53C2×#4111yescharacteristic
170C23×C53C2×#4111yescharacteristic
171C23×C53C2×#4112yes—
172C23×C53C2×#4112yes—
173C23×C53C2×#4111yescharacteristic
174C23×C53C2×#4111yescharacteristic
175C23×C53C2×#4111yescharacteristic
176C23×C53C2×#4111yescharacteristic
177C23×C53H(5,𝔽2)12yes—
178C23×C53H(5,𝔽2)12yes—
179C22×D53—22no—
180C22×D53—22no—
181C22×D53—22no—
182C22×D53—22no—
183C22×D53—22no—
184C22×D53—22no—
185C22×D53—22no—
186C22×D53—22no—
187C22×D53—22no—
188C22×D53—22no—
189C22×D53—24no—
190C22×D53—24no—
191C22×D53—24no—
192C22×D53—24no—
193C22×D53—42no—
194C22×D53—42no—
195C22×D53—42no—
196C22×D53—42no—
197C22×D53—42no—
198C22×D53—42no—
199C22×D53—42no—
200C22×D53—42no—
201C22×D53#11714yes—
202C2×C4×C53—22no—
203C2×C4×C53—22no—
204C2×C4×C53—22no—
205C2×C4×C53—22no—
206C2×C4×C53—24no—
207C2×C4×C53—24no—
208C2×C4×C53—24no—
209C2×C4×C53—24no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
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×C53—42no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53#11714yes—
230C2×C4×C53#11714yes—
231C2×Dic53—22no—
232C2×Dic53—22no—
233C2×Dic53—22no—
234C2×Dic53—22no—
235C2×Dic53—22no—
236C2×Dic53—22no—
237C2×Dic53—22no—
238C2×Dic53—22no—
239C2×Dic53—22no—
240C2×Dic53—24no—
241C2×Dic53—24no—
242C2×Dic53—24no—
243C2×Dic53—24no—
244C2×Dic53—24no—
245C2×Dic53—24no—
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×Dic53—42no—
258C2×Dic53—42no—
259C2×Dic53—42no—
260C4×D53—416no—
261#13947—531nomaximal, Sylow
262C22×C53C2×#11711yescharacteristic
263C22×C53C2×#11711yescharacteristic
264C22×C53C2×#11711yescharacteristic
265C22×C53C4×#4111yescharacteristic
266C22×C53C4×#4111yescharacteristic
267C22×C53#51811yescharacteristic
268C22×C53#51811yescharacteristic
269C22×C53#52912yes—
270C22×C53#52912yes—
271C22×C53#52911yescharacteristic
272C22×C53#52911yescharacteristic
273C22×C53#52911yescharacteristic
274C22×C53#52911yescharacteristic
275C22×C53#53812yes—
276C22×C53#53812yes—
277C22×C53#55112yes—
278C22×C53#55112yes—
279C22×C53#55112yes—
280C22×C53#55312yes—
281C22×C53#55312yes—
282C22×C53#55312yes—
283C22×C53#55312yes—
284C22×C53#55912yes—
285C2×D53—24no—
286C2×D53—24no—
287C2×D53—24no—
288C2×D53—42no—
289C2×D53—44no—
290C2×D53—44no—
291C2×D53—44no—
292C2×D53—44no—
293C2×D53—42no—
294C2×D53—42no—
295C2×D53—42no—
296C4×C53—24no—
297C4×C53—24no—
298C4×C53—42no—
299C4×C53—82no—
300C4×C53—82no—
301Dic53—42no—
302Dic53—44no—
303Dic53—44no—
304Dic53—42no—
305Dic53—42no—
306C2×C4×C22⋊C4—531no—
307#1443—532no—
308#1440—532no—
309#1437—532no—
310#1434—532no—
311#1434—531no—
312#1434—531no—
313#1434D5311yescharacteristic, complemented
314C22⋊#95—531no—
315C22⋊#95—531no—
316C4×#95—531no—
317C2×C53#228311yescharacteristic
318C2×C53#229811yescharacteristic
319C2×C53#231211yescharacteristic
320C2×C53#235311yescharacteristic
321C2×C53#235311yescharacteristic
322C2×C53#235311yescharacteristic
323C2×C53#235711yescharacteristic
324C2×C53#257612yes—
325C2×C53#257812yes—
326C2×C53#259512yes—
327C2×C53#259712yes—
328D53—44no—
329C24×C4—531no—
330C22×C22⋊C4—531no—
331C22×C22⋊C4—531no—
332C22×C22⋊C4—532no—
333C22×C22⋊C4—532no—
334C22×C42—531no—
335C22×C42—532no—
336C22×C42C2×D5311yescharacteristic
337C4×C22⋊C4—538no—
338C2×#95—531no—
339C2×#95—531no—
340C2×#95—531no—
341C2×#95—531no—
342C2×#95—532no—
343C2×#95—532no—
344C2×#95—532no—
345C2×#95—532no—
346C2×#95—532no—
347C2×#95—532no—
348C2×#95—532no—
349C2×#95C2×D5311yescharacteristic
350C2×#95C2×D5311yescharacteristic
351C2×#95C2×D5312yes—
352C2×#95C2×D5312yes—
353C53#1394711yescharacteristic, complemented, Sylow
354C25—531no—
355C23×C4—531no—
356C23×C4—531no—
357C23×C4—532no—
358C23×C4—532no—
359C23×C4—531no—
360C23×C4—534no—
361C23×C4—1062no—
362C23×C4—1062no—
363C23×C4—2122no—
364C23×C4—2122no—
365C23×C4C22×D5312yes—
366C23×C4C22×D5312yes—
367C23×C4C22×D5311yescharacteristic
368C23×C4C22×D5311yescharacteristic
369C23×C4C22×D5311yescharacteristic
370C2×C22⋊C4—534no—
371C2×C22⋊C4—534no—
372C2×C22⋊C4—1062no—
373C2×C22⋊C4—1062no—
374C2×C22⋊C4—1062no—
375C2×C22⋊C4—1062no—
376C2×C22⋊C4—1064no—
377C2×C22⋊C4—1064no—
378C2×C22⋊C4—1064no—
379C2×C22⋊C4—1064no—
380C2×C22⋊C4—2122no—
381C2×C22⋊C4—2122no—
382C2×C42—42no—
383C2×C42—534no—
384C2×C42—1064no—
385C2×C42—1064no—
386C2×C42—2122no—
387C2×C42—2122no—
388C2×C42C4×D5314yes—
389#95—22no—
390#95—22no—
391#95—24no—
392#95—42no—
393#95—534no—
394#95—534no—
395#95—1064no—
396#95—1064no—
397#95—1064no—
398#95—1062no—
399#95—1062no—
400#95—2122no—
401#95—2122no—
402#95—2122no—
403#95—2122no—
404C24—532no—
405C24—532no—
406C24—532no—
407C24—1062no—
408C24—1062no—
409C24—1061no—
410C24—1061no—
411C24—1061no—
412C24—1061no—
413C24—1062no—
414C24—1062no—
415C24C23×D5311yescharacteristic
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
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—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—532no—
439C22×C4—532no—
440C22×C4—532no—
441C22×C4—534no—
442C22×C4—534no—
443C22×C4—1062no—
444C22×C4—1062no—
445C22×C4—1062no—
446C22×C4—1062no—
447C22×C4—1062no—
448C22×C4—1062no—
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—1061no—
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—1068no—
468C22×C4—2124no—
469C22×C4—2124no—
470C22×C4—2128no—
471C22×C4—2128no—
472C22×C4C2×C4×D5312yes—
473C22×C4C2×C4×D5312yes—
474C22×C4C2×C4×D5312yes—
475C22×C4#1171994512yes—
476C22×C4#1171994512yes—
477C22×C4#1171994712yes—
478C22×C4#1171994912yes—
479C22⋊C4—1068no—
480C22⋊C4—1068no—
481C22⋊C4—2128no—
482C22⋊C4—2128no—
483C42—44no—
484C42—2124no—
485C42—2128no—
486C23—534no—
487C23—1064no—
488C23—1064no—
489C23—1064no—
490C23—1064no—
491C23—1062no—
492C23—1062no—
493C23—1062no—
494C23—1062no—
495C23—1062no—
496C23—1062no—
497C23—1062no—
498C23—1062no—
499C23—1062no—
500C23—1062no—
501C23—2122no—
502C23—2122no—
503C23—2122no—
504C23—2122no—
505C23—2122no—
506C23—2122no—
507C23—2122no—
508C23—2122no—
509C23C22×C4×D5311yescharacteristic
510C23#41×D5311yescharacteristic
511C23#41×D5312yes—
512C23#41×D5311yescharacteristic
513C23#41×D5312yes—
514C23C2×#1171994711yescharacteristic
515C23C2×#1171994511yescharacteristic
516C23C2×#1171994511yescharacteristic
517C23C2×#1171994911yescharacteristic
518C23#1313326212yes—
519C23#1313326312yes—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—1064no—
548C2×C4—1064no—
549C2×C4—1064no—
550C2×C4—1064no—
551C2×C4—1062no—
552C2×C4—1062no—
553C2×C4—1062no—
554C2×C4—1062no—
555C2×C4—1062no—
556C2×C4—1062no—
557C2×C4—1062no—
558C2×C4—1062no—
559C2×C4—1062no—
560C2×C4—1064no—
561C2×C4—1064no—
562C2×C4—21216no—
563C2×C4—2122no—
564C2×C4—2122no—
565C2×C4—2122no—
566C2×C4—2122no—
567C2×C4—2122no—
568C2×C4—2122no—
569C2×C4—2122no—
570C2×C4—2122no—
571C2×C4—2122no—
572C2×C4—2122no—
573C2×C4—2122no—
574C2×C4—2122no—
575C2×C4—2122no—
576C2×C4—2122no—
577C2×C4#1313317414yes—
578C2×C4#1313317514yes—
579C22—1064no—
580C22—1064no—
581C22—1064no—
582C22—2122no—
583C22—2122no—
584C22—2124no—
585C22—2122no—
586C22—2124no—
587C22—2124no—
588C22—2122no—
589C22—2124no—
590C22#117×D5311yescharacteristic
591C22C2×#1313317411yescharacteristic
592C22C2×#1313317511yescharacteristic
593C22#1681109111yescharacteristic
594C22#1681109111yescharacteristic
595C22#1681109611yescharacteristic
596C22#1681109711yescharacteristic
597C22#1681115112yes—
598C22#1681115112yes—
599C22#1681115511yescharacteristic
600C22#1681115511yescharacteristic
601C22#1681116011yescharacteristic
602C22#1681116011yescharacteristic
603C22#1681120612yes—
604C22#1681120612yes—
605C22#1681129812yes—
606C22#1681130012yes—
607C22#1681130112yes—
608C22#1681131612yes—
609C22#1681131812yes—
610C22#1681132412yes—
611C22#1681132612yes—
612C22#1681135512yes—
613C4—24no—
614C4—24no—
615C4—42no—
616C4—82no—
617C4—82no—
618C4—2122no—
619C4—2122no—
620C4—2122no—
621C4—2124no—
622C4—2124no—
623C2—2124no—
624C2#2027711111yescharacteristic
625C2#2027721011yescharacteristic
626C2#2027731011yescharacteristic
627C2#2027764811yescharacteristic
628C2#2027765211yescharacteristic
629C2#2027765711yescharacteristic
630C2#2027769011yescharacteristic
631C2#2028002112yes—
632C2#2028005112yes—
633C2#2028023812yes—
634C2#2028026812yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1434C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenter#1434D53
hypocenterC53#13947
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
2. #20272975C2semidirect
5. #20272941C2semidirect
6. #20272941C2semidirect
7. #20272940C2semidirect
10. C4×#13133124C2semidirect
11. C53×#1434C2semidirect
19. C4×#11719921C4semidirect
313. #1434D53semidirect
353. C53#13947semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117199212
#202729752
#202729632
#202729512
#202729412
#202729402
#202729362
C4×#131331242
C53×#14342
#1394753