Finite Group Databasebeta

← #27484286

Subgroups — #27484286

Order 13568.

720 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
1#20280549C211yesmaximal, characteristic
2#20277393C211yesmaximal, characteristic
3#20276275C211yesmaximal, characteristic, complemented
4#20276250C211yesmaximal, characteristic, complemented
5#20276055C211yesmaximal, characteristic, complemented
6#20275865C211yesmaximal, characteristic, complemented
7#20275305C211yesmaximal, characteristic, complemented
8#20275288C211yesmaximal, characteristic, complemented
9#20275148C211yesmaximal, characteristic
10#20275136C211yesmaximal, characteristic
11#20274980C211yesmaximal, characteristic, complemented
12#20274970C211yesmaximal, characteristic, complemented
13#20274742C211yesmaximal, characteristic
14#20274413C211yesmaximal, characteristic
15C53×#1809C211yesmaximal, characteristic
16C23×#88322C2211yescharacteristic
17C2×#13133208C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C2×#13133182C2211yescharacteristic
20C2×#13133176C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133142C2211yescharacteristic
24C2×#13133142C2211yescharacteristic
25C2×#13133140C2211yescharacteristic
26C2×#13133140C2211yescharacteristic
27#16810557—21no—
28#16810555—21no—
29#16810546—21no—
30#16810546—21no—
31#16810528C2211yescharacteristic
32#16810522C2211yescharacteristic
33#16810520—21no—
34#16810519C2211yescharacteristic
35#16810519C2211yescharacteristic
36#16810517—21no—
37#16810511—21no—
38#16810510—21no—
39#16810492C2211yescharacteristic
40#16810483C2211yescharacteristic
41C4⋊C4×Dic53C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810377C2211yescharacteristic
49#16810377C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C22×M16×C53C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic
54C2×#106×C53C2211yescharacteristic
55C53×#380C2211yescharacteristic
56C53×#380C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C22×#88322—22no—
61C22×#88322—22no—
62C22×#88322—41no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65#13133208—41no—
66#13133208—41no—
67#13133196—22no—
68#13133196Q814yes—
69#13133182—41no—
70#13133182—41no—
71#13133176—41no—
72#13133176—41no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53D411yescharacteristic
76C2×C4⋊Dic53D411yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×#11719919—21no—
87C2×C22⋊Dic53—21no—
88C2×C22⋊Dic53—21no—
89C2×C22⋊Dic53D411yescharacteristic
90C2×C22⋊Dic53D411yescharacteristic
91C2×C4×Dic53—21no—
92C2×C4×Dic53—21no—
93C2×C4×Dic53C2311yescharacteristic
94C2×C4×Dic53C2311yescharacteristic
95#13133150—22no—
96#13133150—41no—
97#13133142—22no—
98#13133142—41no—
99#13133140—22no—
100#13133140—41no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108C23×C4×C53C2311yescharacteristic
109C2×M16×C53—22no—
110C2×M16×C53—22no—
111C2×M16×C53—41no—
112C22×C8×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C22⋊C4×C53—21no—
119C2×C22⋊C4×C53—21no—
120#107×C53—22no—
121#106×C53—41no—
122#95×C53—21no—
123#95×C53—21no—
124C24×C53C2×D411yescharacteristic
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×Q812yes—
135C22×C4×C53#4112yes—
136C22×C4×C53C2411yescharacteristic
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53C2×D411yescharacteristic
142C22×Dic53C2×D411yescharacteristic
143C22×Dic53C2×D411yescharacteristic
144C22×Dic53C2×D411yescharacteristic
145C2×Dic106—21no—
146C2×Dic106—21no—
147C2×Dic106—21no—
148C2×Dic106—21no—
149C2×Dic106—21no—
150C2×Dic106C2×Q812yes—
151M16×C53—44no—
152C2×C8×C53—21no—
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×C53#4112yes—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C22⋊C4×C53—81no—
167C22⋊C4×C53—81no—
168C22⋊Dic53—24no—
169C22⋊Dic53—41no—
170C22⋊Dic53—41no—
171C22⋊Dic53—81no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—41no—
178C4⋊Dic53—41no—
179C4⋊Dic53—42no—
180C4⋊Dic53—81no—
181C4⋊Dic53C2×Q812yes—
182#11719919—24no—
183#11719919—42no—
184#11719919—42no—
185#11719919—81no—
186#11719919—81no—
187#11719919—81no—
188#11719919—81no—
189C4×Dic53—21no—
190C4×Dic53—21no—
191C4×Dic53—21no—
192C4×Dic53—21no—
193C4×Dic53—42no—
194C4×Dic53—81no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—44no—
201C2×#88322#4112yes—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53C22⋊Q812yes—
209C23×C53C22×D411yescharacteristic
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53C22⋊Q812yes—
229C2×C4×C53C22×Q811yescharacteristic
230C2×C4×C53C2×#4111yescharacteristic
231C2×C4×C53C2×#4111yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×C4×C53H(5,𝔽2)11yescharacteristic
234C2×C4×C53H(5,𝔽2)11yescharacteristic
235C2×Dic53—22no—
236C2×Dic53—22no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259Dic106—22no—
260Dic106—22no—
261Dic106—41no—
262Dic106—41no—
263C8×C53—22no—
264C8×C53—41no—
265#88322—22no—
266#88322—41no—
267#29173—531nomaximal, Sylow
268C22×C53—22no—
269C22×C53—22no—
270C22×C53—22no—
271C22×C53—41no—
272C22×C53—41no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53C2×C22⋊Q811yescharacteristic
276C22×C53#53211yescharacteristic
277C22×C53#53411yescharacteristic
278C22×C53#57611yescharacteristic
279C22×C53#57711yescharacteristic
280C22×C53#57811yescharacteristic
281C22×C53#57811yescharacteristic
282C4×C53—22no—
283C4×C53—81no—
284C4×C53—81no—
285C4×C53—81no—
286C4×C53—81no—
287C4×C53#55111yescharacteristic
288C4×C53#55111yescharacteristic
289C4×C53#55411yescharacteristic
290C4×C53#55411yescharacteristic
291Dic53—42no—
292Dic53—42no—
293Dic53—41no—
294Dic53—41no—
295Dic53—41no—
296Dic53—41no—
297Dic53—81no—
298Dic53—81no—
299#2629—531no—
300#2323—531no—
301#2099—531no—
302#2095—531no—
303#2059—531no—
304#2026—531no—
305#1921—531no—
306#1918—531no—
307#1894—531no—
308#1892—531no—
309#1865—531no—
310#1863—531no—
311#1809D5311yescharacteristic
312#1808—531no—
313#1729—531no—
314C2×C53—42no—
315C2×C53#256411yescharacteristic
316C2×C53#308211yescharacteristic
317C2×C53#308311yescharacteristic
318C2×C53#317711yescharacteristic
319C2×C53#317911yescharacteristic
320C2×C53#320111yescharacteristic
321C2×C53#320311yescharacteristic
322C22×M16C2×D5311yescharacteristic
323C23×C8—531no—
324C2×#125—531no—
325C2×C22⋊Q8—531no—
326C2×C4×Q8—531no—
327C2×#117—531no—
328C2×#107—531no—
329C2×#107—531no—
330C2×#107C2×D5311yescharacteristic
331C2×#106C2×D5311yescharacteristic
332C2×#103—531no—
333C2×#103—531no—
334C2×#103—531no—
335C2×#103—531no—
336#398—1061no—
337#398—1061no—
338#396—1061no—
339#396—1061no—
340#391—531no—
341#389—531no—
342#389—531no—
343#389—531no—
344#389—1061no—
345#388—1061no—
346#388—1061no—
347#387—1061no—
348#384—531no—
349#380—531no—
350#380C2×D5311yescharacteristic
351#380C2×D5311yescharacteristic
352C4×C4⋊C4—531no—
353#340—531no—
354#340—531no—
355#340—531no—
356#340—531no—
357#340—531no—
358#340—531no—
359#340C2×D5311yescharacteristic
360#340C2×D5311yescharacteristic
361#336—531no—
362#336—531no—
363#336—531no—
364#336—531no—
365C53#2917311yescharacteristic, complemented, Sylow
366C22×Q8—531no—
367C23×C4C22×D5311yescharacteristic
368C2×M16—22no—
369C2×M16—22no—
370C2×M16—41no—
371C22×C8—531no—
372C22×C8—531no—
373C22×C8—1062no—
374C22×C8—1062no—
375C22×C8—2121no—
376C22×C8C22×D5311yescharacteristic
377C22×C8C22×D5311yescharacteristic
378#125—2121no—
379#125—2121no—
380C22⋊Q8—534no—
381C22⋊Q8—1062no—
382C4×Q8—2121no—
383C4×Q8—2121no—
384#117—2121no—
385#117—2121no—
386C2×C4⋊C4—531no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4C22×D5311yescharacteristic
401C2×C4⋊C4C22×D5311yescharacteristic
402C2×C4⋊C4C22×D5311yescharacteristic
403C2×C4⋊C4C22×D5311yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—531no—
407C2×C22⋊C4—531no—
408C2×C22⋊C4—1061no—
409C2×C22⋊C4—1061no—
410C2×C42—531no—
411C2×C42—531no—
412C2×C42—1061no—
413C2×C42—1061no—
414#107—22no—
415#107—1062no—
416#107—2121no—
417#106—41no—
418#103—1062no—
419#103—1062no—
420#103—2121no—
421#103—2121no—
422#95—21no—
423#95—21no—
424#95—1061no—
425#95—1061no—
426#95—1061no—
427#95—1061no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431C24D4×D5311yescharacteristic
432C2×Q8—532no—
433C2×Q8—1061no—
434C2×Q8—1061no—
435C2×Q8—1061no—
436C2×Q8—1061no—
437C2×Q8—1061no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—531no—
447C22×C4—531no—
448C22×C4—531no—
449C22×C4—531no—
450C22×C4—1061no—
451C22×C4—1061no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4#1171994512yes—
455C22×C4D4×D5311yescharacteristic
456C22×C4Q8×D5312yes—
457C22×C4C23×D5311yescharacteristic
458M16—44no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—532no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8—1061no—
470C2×C8—2124no—
471C2×C8#1171994712yes—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—532no—
481C4⋊C4—1064no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—4241no—
488C4⋊C4—4241no—
489C4⋊C4—4241no—
490C4⋊C4—4241no—
491C4⋊C4—4241no—
492C4⋊C4—2121no—
493C4⋊C4—2121no—
494C4⋊C4—2122no—
495C4⋊C4—2122no—
496C4⋊C4—2122no—
497C22⋊C4—81no—
498C22⋊C4—81no—
499C22⋊C4—1064no—
500C22⋊C4—4241no—
501C22⋊C4—2121no—
502C22⋊C4—2121no—
503C42—1061no—
504C42—1061no—
505C42—1061no—
506C42—1061no—
507C42—4241no—
508C42—2122no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23#1313319512yes—
516C23C2×D4×D5311yescharacteristic
517Q8—1062no—
518Q8—1062no—
519Q8—2121no—
520Q8—2121no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—1062no—
540C2×C4—1062no—
541C2×C4—1061no—
542C2×C4—1061no—
543C2×C4—1061no—
544C2×C4—1061no—
545C2×C4—1061no—
546C2×C4—1061no—
547C2×C4—1061no—
548C2×C4—1061no—
549C2×C4—1061no—
550C2×C4—1061no—
551C2×C4—1061no—
552C2×C4—1061no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4#1313319512yes—
564C2×C4C2×Q8×D5311yescharacteristic
565C2×C4#41×D5311yescharacteristic
566C2×C4C2×#1171994711yescharacteristic
567C2×C4C2×#1171994511yescharacteristic
568C2×C4#1313326211yescharacteristic
569C2×C4#1313326311yescharacteristic
570C8—22no—
571C8—41no—
572C8—1062no—
573C8—2121no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22C2×#1313319511yescharacteristic
582C22#1681116811yescharacteristic
583C22#1681118111yescharacteristic
584C22#1681142711yescharacteristic
585C22#1681143311yescharacteristic
586C22#1681143911yescharacteristic
587C22#1681143911yescharacteristic
588C4—22no—
589C4—81no—
590C4—81no—
591C4—81no—
592C4—81no—
593C4—4241no—
594C4—4241no—
595C4—2122no—
596C4—2122no—
597C4—2121no—
598C4—2121no—
599C4—2121no—
600C4—2121no—
601C4#1681129711yescharacteristic
602C4#1681129911yescharacteristic
603C4#1681132711yescharacteristic
604C4#1681133111yescharacteristic
605C2—42no—
606C2#2027989311yescharacteristic
607C2#2028455511yescharacteristic
608C2#2028456511yescharacteristic
609C2#2028536911yescharacteristic
610C2#2028539911yescharacteristic
611C2#2028559211yescharacteristic
612C2#2028561311yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1809C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1809D53
hypocenterC53#29173
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1809

Complements

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

NH ≅ G/Ntype
3. #20276275C2semidirect
4. #20276250C2semidirect
5. #20276055C2semidirect
6. #20275865C2semidirect
7. #20275305C2semidirect
8. #20275288C2semidirect
11. #20274980C2semidirect
12. #20274970C2semidirect
365. C53#29173semidirect

Maximal subgroups

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

subgroupindex
#202805492
#202773932
#202762752
#202762502
#202760552
#202758652
#202753052
#202752882
#202751482
#202751362
#202749802
#202749702
#202747422
#202744132
C53×#18092
#2917353