Finite Group Databasebeta

← #27484273

Subgroups — #27484273

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#20280545C211yesmaximal, characteristic
2#20277397C211yesmaximal, characteristic
3#20276279C211yesmaximal, characteristic, complemented
4#20276257C211yesmaximal, characteristic, complemented
5#20276054C211yesmaximal, characteristic, complemented
6#20275867C211yesmaximal, characteristic, complemented
7#20275301C211yesmaximal, characteristic, complemented
8#20275294C211yesmaximal, characteristic, complemented
9#20275148C211yesmaximal, characteristic
10#20275138C211yesmaximal, characteristic
11#20274986C211yesmaximal, characteristic, complemented
12#20274972C211yesmaximal, characteristic, complemented
13#20274748C211yesmaximal, characteristic
14#20274740C211yesmaximal, characteristic
15C53×#1729C211yesmaximal, characteristic
16C2×#13133209C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×C4×Dic106C2211yescharacteristic
19C2×#13133151C2211yescharacteristic
20C2×#13133149C2211yescharacteristic
21C2×#13133149C2211yescharacteristic
22C2×#13133149C2211yescharacteristic
23C2×#13133142C2211yescharacteristic
24C2×#13133142C2211yescharacteristic
25C2×#13133142C2211yescharacteristic
26C2×#13133142C2211yescharacteristic
27#16810556—21no—
28#16810556—21no—
29#16810546—21no—
30#16810546—21no—
31#16810529C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810521C2211yescharacteristic
35#16810518—21no—
36#16810517—21no—
37#16810511—21no—
38#16810504—21no—
39#16810493C2211yescharacteristic
40#16810483C2211yescharacteristic
41#16810483C2211yescharacteristic
42#16810483C2211yescharacteristic
43C4×C4⋊Dic53C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52C22×M16×C53C2211yescharacteristic
53C23×C8×C53C2211yescharacteristic
54C2×#117×C53C2211yescharacteristic
55C53×#336C2211yescharacteristic
56C53×#336C2211yescharacteristic
57C53×#336C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60#13133209—41no—
61#13133209—41no—
62#13133196—22no—
63#13133196Q814yes—
64C4×Dic106—41no—
65C4×Dic106—41no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53—21no—
85C2×C22⋊Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53—21no—
88#13133151—41no—
89#13133149—22no—
90#13133149—22no—
91#13133149—41no—
92#13133142—22no—
93#13133142—22no—
94#13133142—41no—
95#13133142—41no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105C23×C4×C53C2311yescharacteristic
106C2×M16×C53—22no—
107C2×M16×C53—22no—
108C2×M16×C53—41no—
109C22×C8×C53—22no—
110C22×C8×C53—22no—
111C22×C8×C53—41no—
112C22×C8×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115C22×C8×C53C2311yescharacteristic
116#117×C53—41no—
117#117×C53—41no—
118C2×C4⋊C4×C53D411yescharacteristic
119C2×C4⋊C4×C53D411yescharacteristic
120C2×C22⋊C4×C53D411yescharacteristic
121C2×C22⋊C4×C53D411yescharacteristic
122C2×C42×C53C2311yescharacteristic
123C2×C42×C53C2311yescharacteristic
124C24×C53C2×D411yescharacteristic
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
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×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
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—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—44no—
163C2×C8×C53#4112yes—
164C2×C8×C53#4112yes—
165C4⋊C4×C53—41no—
166C4⋊C4×C53—41no—
167C22⋊C4×C53—41no—
168C22⋊C4×C53—41no—
169C42×C53—21no—
170C42×C53—21no—
171C42×C53—21no—
172C42×C53—21no—
173C22⋊Dic53—24no—
174C22⋊Dic53—81no—
175C22⋊Dic53—81no—
176C22⋊Dic53—81no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—81no—
191C4⋊Dic53—81no—
192C4⋊Dic53C2×Q812yes—
193#11719919—24no—
194#11719919—42no—
195#11719919—42no—
196#11719919—42no—
197#11719919—81no—
198#11719919—81no—
199#11719919—81no—
200C4×Dic53—42no—
201C4×Dic53—81no—
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—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
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—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
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—22no—
265C8×C53—41no—
266C8×C53—41no—
267#29172—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#57711yescharacteristic
279C22×C53#57711yescharacteristic
280C22×C53#57711yescharacteristic
281C22×C53#57811yescharacteristic
282C4×C53—22no—
283C4×C53—41no—
284C4×C53—41no—
285C4×C53—41no—
286C4×C53—41no—
287C4×C53#55111yescharacteristic
288C4×C53#55111yescharacteristic
289C4×C53#55411yescharacteristic
290C4×C53#55411yescharacteristic
291Dic53—42no—
292Dic53—42no—
293Dic53—81no—
294Dic53—81no—
295Dic53—81no—
296Dic53—81no—
297Dic53—81no—
298Dic53—81no—
299#2629—531no—
300#2323—531no—
301#2100—531no—
302#2096—531no—
303#2059—531no—
304#2026—531no—
305#1920—531no—
306#1919—531no—
307#1894—531no—
308#1892—531no—
309#1865—531no—
310#1863—531no—
311#1809—531no—
312#1807—531no—
313#1729D5311yescharacteristic
314C2×C53—42no—
315C2×C53#256411yescharacteristic
316C2×C53#308211yescharacteristic
317C2×C53#308411yescharacteristic
318C2×C53#317411yescharacteristic
319C2×C53#317811yescharacteristic
320C2×C53#320111yescharacteristic
321C2×C53#320411yescharacteristic
322C22×M16C2×D5311yescharacteristic
323C23×C8C2×D5311yescharacteristic
324C2×#125—531no—
325C2×C22⋊Q8—531no—
326C2×C4×Q8—531no—
327C2×#117C2×D5311yescharacteristic
328C2×#107—531no—
329C2×#106—531no—
330C2×#106—531no—
331C2×#106—531no—
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#380—531no—
351#380—531no—
352C4×C4⋊C4—531no—
353#340—531no—
354#340—531no—
355#340—531no—
356#340—531no—
357#340—531no—
358#340—531no—
359#340—531no—
360#340—531no—
361#336C2×D5311yescharacteristic
362#336C2×D5311yescharacteristic
363#336C2×D5311yescharacteristic
364#336C2×D5311yescharacteristic
365C53#2917211yescharacteristic, complemented, Sylow
366C22×Q8—531no—
367C23×C4C22×D5311yescharacteristic
368C2×M16—22no—
369C2×M16—22no—
370C2×M16—41no—
371C22×C8—22no—
372C22×C8—22no—
373C22×C8—41no—
374C22×C8C22×D5311yescharacteristic
375C22×C8C22×D5311yescharacteristic
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—41no—
385#117—41no—
386C2×C4⋊C4—531no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
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⋊C4—1061no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4D21211yescharacteristic
403C2×C4⋊C4D21211yescharacteristic
404C2×C22⋊C4—1061no—
405C2×C22⋊C4—1061no—
406C2×C22⋊C4—1061no—
407C2×C22⋊C4—1061no—
408C2×C22⋊C4D21211yescharacteristic
409C2×C22⋊C4D21211yescharacteristic
410C2×C42—1061no—
411C2×C42—1061no—
412C2×C42C22×D5311yescharacteristic
413C2×C42C22×D5311yescharacteristic
414#107—2121no—
415#106—1062no—
416#106—1062no—
417#106—2121no—
418#103—1062no—
419#103—1062no—
420#103—2121no—
421#103—2121no—
422#95—1061no—
423#95—1061no—
424#95—1061no—
425#95—1061no—
426#95—1061no—
427#95—1061no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431C24C2×D21211yescharacteristic
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—1061no—
443C22×C4—1061no—
444C22×C4—1061no—
445C22×C4—1061no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—1061no—
450C22×C4C2×D21211yescharacteristic
451C22×C4C2×D21211yescharacteristic
452C22×C4C2×D21211yescharacteristic
453C22×C4C2×D21211yescharacteristic
454C22×C4C2×D21211yescharacteristic
455C22×C4#1171994712yes—
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—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—44no—
470C2×C8#1171994512yes—
471C2×C8#1171994512yes—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—532no—
475C4⋊C4—1064no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—4241no—
490C4⋊C4—4241no—
491C4⋊C4—4241no—
492C4⋊C4—4241no—
493C4⋊C4—4241no—
494C4⋊C4—2122no—
495C4⋊C4—2122no—
496C4⋊C4—2122no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—1064no—
500C22⋊C4—4241no—
501C22⋊C4—4241no—
502C22⋊C4—4241no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—4241no—
508C42—2122no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23#1313319712yes—
516C23C22×D21211yescharacteristic
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—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—1062no—
540C2×C4—1062no—
541C2×C4—2121no—
542C2×C4—2121no—
543C2×C4—2121no—
544C2×C4—2121no—
545C2×C4—2121no—
546C2×C4—2121no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
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#1313319712yes—
564C2×C4C2×Q8×D5311yescharacteristic
565C2×C4C2×#1171994711yescharacteristic
566C2×C4C2×#1171994511yescharacteristic
567C2×C4C2×#1171994511yescharacteristic
568C2×C4#1313326211yescharacteristic
569C2×C4#1313326211yescharacteristic
570C8—22no—
571C8—22no—
572C8—41no—
573C8—41no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22C2×#1313319711yescharacteristic
582C22#1681117311yescharacteristic
583C22#1681118511yescharacteristic
584C22#1681143511yescharacteristic
585C22#1681143511yescharacteristic
586C22#1681143511yescharacteristic
587C22#1681144011yescharacteristic
588C4—22no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—41no—
593C4—4241no—
594C4—4241no—
595C4—4241no—
596C4—4241no—
597C4—4241no—
598C4—4241no—
599C4—2122no—
600C4—2122no—
601C4#1681129911yescharacteristic
602C4#1681129911yescharacteristic
603C4#1681132811yescharacteristic
604C4#1681132811yescharacteristic
605C2—42no—
606C2#2027989511yescharacteristic
607C2#2028455811yescharacteristic
608C2#2028457911yescharacteristic
609C2#2028534311yescharacteristic
610C2#2028538311yescharacteristic
611C2#2028559911yescharacteristic
612C2#2028562711yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1729C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1729D53
hypocenterC53#29172
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1729

Complements

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

NH ≅ G/Ntype
3. #20276279C2semidirect
4. #20276257C2semidirect
5. #20276054C2semidirect
6. #20275867C2semidirect
7. #20275301C2semidirect
8. #20275294C2semidirect
11. #20274986C2semidirect
12. #20274972C2semidirect
365. C53#29172semidirect

Maximal subgroups

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

subgroupindex
#202805452
#202773972
#202762792
#202762572
#202760542
#202758672
#202753012
#202752942
#202751482
#202751382
#202749862
#202749722
#202747482
#202747402
C53×#17292
#2917253