Finite Group Databasebeta

← #27449518

Subgroups — #27449518

Order 13568.

716 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277784C211yesmaximal, characteristic
2#20276266C211yesmaximal, characteristic, complemented
3#20276239C211yesmaximal, characteristic
4#20276107C211yesmaximal, characteristic, complemented
5#20276100C211yesmaximal, characteristic
6#20275647C211yesmaximal, characteristic
7#20275529C211yesmaximal, characteristic
8#20275471C211yesmaximal, characteristic, complemented
9#20275439C211yesmaximal, characteristic, complemented
10#20274968C211yesmaximal, characteristic, complemented
11#20274961C211yesmaximal, characteristic
12#20274727C211yesmaximal, characteristic, complemented
13#20274722C211yesmaximal, characteristic
14#20274613C211yesmaximal, characteristic, complemented
15C53×#2571C211yesmaximal, characteristic, complemented
16C2×#13133205C2211yescharacteristic
17C2×C4×Dic106C2211yescharacteristic
18C2×C4×D212C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×C4⋊#88322C2211yescharacteristic
25C2×C4⋊#88322C2211yescharacteristic
26C2×#13133140C2211yescharacteristic
27C2×#13133140C2211yescharacteristic
28C2×#13133140C2211yescharacteristic
29C2×#13133140C2211yescharacteristic
30C2×#13133137C2211yescharacteristic
31C2×#13133137C2211yescharacteristic
32C2×#13133137C2211yescharacteristic
33C2×#13133137C2211yescharacteristic
34C2×#13133128C2211yescharacteristic
35#16810529C2211yescharacteristic
36#16810525C2211yescharacteristic
37#16810513—21no—
38#16810512—21no—
39#16810494—21no—
40C4×C4⋊Dic53C2211yescharacteristic
41#16810470—21no—
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C53×#400—21no—
49C53×#400—21no—
50C53×#389—21no—
51C53×#389C2211yescharacteristic
52C53×#389C2211yescharacteristic
53C53×#384C2211yescharacteristic
54C53×#382—21no—
55C53×#382C2211yescharacteristic
56C53×#382C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C4×C4⋊C4×C53C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C22×D212C2311yescharacteristic
61C22×C4×D53—21no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133205—22no—
67#13133205Q814yes—
68C4×Dic106—22no—
69C4×Dic106—41no—
70C4×D212—22no—
71C4×D212—41no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×#11719919—21no—
75C2×#11719921—21no—
76C2×#11719921—21no—
77C2×#11719921—21no—
78C2×#11719921—21no—
79C2×#11719921—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53—21no—
82#13133150—22no—
83#13133150—41no—
84#13133150—41no—
85#13133148—22no—
86C4⋊#88322—22no—
87C4⋊#88322—41no—
88C4⋊#88322—41no—
89#13133140—22no—
90#13133140—41no—
91#13133140—41no—
92#13133140—41no—
93#13133137—22no—
94#13133137—41no—
95#13133137—41no—
96#13133137—41no—
97#13133128—22no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53D411yescharacteristic
106C2×C4⋊C4×C53D411yescharacteristic
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C42×C53—21no—
112C2×C42×C53C2311yescharacteristic
113C2×C42×C53C2311yescharacteristic
114C2×C42×C53C2311yescharacteristic
115#95×C53—21no—
116#95×C53—21no—
117#95×C53—21no—
118#95×C53—21no—
119#95×C53—21no—
120#95×C53—21no—
121#95×C53—21no—
122#95×C53D411yescharacteristic
123#95×C53D411yescharacteristic
124C23×D53—21no—
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×D411yescharacteristic
135C22×C4×C53C2411yescharacteristic
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C2×D212—21no—
140C2×D212—21no—
141C2×D212—21no—
142C2×D212—21no—
143C2×D212—21no—
144C2×D212C2×Q812yes—
145C2×C4×D53—42no—
146C2×C4×D53—42no—
147C2×C4×D53—81no—
148C2×Dic106—21no—
149C2×Dic106—21no—
150C2×Dic106—21no—
151C2×Dic106—21no—
152C2×Dic106—21no—
153C2×Dic106C2×Q812yes—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—41no—
163C4⋊C4×C53—41no—
164C4⋊C4×C53—42no—
165C4⋊C4×C53—81no—
166C4⋊C4×C53—81no—
167C4⋊C4×C53—81no—
168C42×C53—21no—
169C42×C53—21no—
170C42×C53—21no—
171C42×C53—21no—
172C42×C53—42no—
173C42×C53C2×Q812yes—
174C42×C53#4112yes—
175#11719921—24no—
176#11719921—24no—
177#11719921—42no—
178#11719921—42no—
179#11719921—81no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53#4112yes—
184#11719919—42no—
185C4×Dic53—42no—
186C4×Dic53—81no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322#4112yes—
199C23×C53C22×D411yescharacteristic
200C22×D53—22no—
201C22×D53—41no—
202C22×D53—41no—
203C22×D53—41no—
204C22×D53—41no—
205C22×D53—41no—
206C22×D53—41no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
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—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53C22⋊Q812yes—
236C2×C4×C53C22⋊Q812yes—
237C2×C4×C53C22×Q811yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×C4×C53H(5,𝔽2)11yescharacteristic
242C2×C4×C53H(5,𝔽2)11yescharacteristic
243C2×Dic53—22no—
244C2×Dic53—22no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252D212—22no—
253D212—22no—
254D212—41no—
255D212—41no—
256C4×D53—84no—
257Dic106—22no—
258Dic106—22no—
259Dic106—41no—
260Dic106—41no—
261#88322—22no—
262#88322—41no—
263#88322—41no—
264#88322—41no—
265#26545—531nomaximal, Sylow
266C22×C53C2×C22⋊Q811yescharacteristic
267C22×C53#54611yescharacteristic
268C22×C53#55011yescharacteristic
269C22×C53#57511yescharacteristic
270C22×C53#57511yescharacteristic
271C22×C53#57611yescharacteristic
272C22×C53#57811yescharacteristic
273C2×D53—42no—
274C2×D53—42no—
275C2×D53—42no—
276C2×D53—81no—
277C2×D53—81no—
278C2×D53—81no—
279C2×D53—81no—
280C4×C53—22no—
281C4×C53—22no—
282C4×C53—41no—
283C4×C53—41no—
284C4×C53—41no—
285C4×C53—41no—
286C4×C53—81no—
287C4×C53—81no—
288C4×C53—81no—
289C4×C53—81no—
290C4×C53#52911yescharacteristic
291C4×C53#54011yescharacteristic
292C4×C53#55411yescharacteristic
293C4×C53#55411yescharacteristic
294Dic53—42no—
295Dic53—42no—
296Dic53—81no—
297#2571D5311yescharacteristic, complemented
298#2367—531no—
299#2097—531no—
300#2092—531no—
301#2068—531no—
302#2067—531no—
303#1983—531no—
304#1959—531no—
305#1947—531no—
306#1940—531no—
307#1863—531no—
308#1862—531no—
309#1805—531no—
310#1804—531no—
311#1776—531no—
312C2×C53#245411yescharacteristic
313C2×C53#308011yescharacteristic
314C2×C53#308311yescharacteristic
315C2×C53#331311yescharacteristic
316C2×C53#331911yescharacteristic
317C2×C53#340211yescharacteristic
318C2×C53#340311yescharacteristic
319D53—82no—
320C2×#124—531no—
321C2×C4×Q8—531no—
322C2×C4×D4—531no—
323C2×#107—531no—
324C2×#107—531no—
325C2×#107—531no—
326C2×#106—531no—
327C2×C4⋊C8—531no—
328C2×C4⋊C8—531no—
329C2×C4⋊C8—531no—
330C2×#103—531no—
331C2×#103—531no—
332C2×#103—531no—
333C2×#103—531no—
334C2×#102—531no—
335C2×#102—531no—
336C2×#102—531no—
337C2×#102—531no—
338C2×#97—531no—
339#400—21no—
340#400—21no—
341#391—531no—
342#390—531no—
343#389—21no—
344#389C2×D5311yescharacteristic
345#389C2×D5311yescharacteristic
346#388—1061no—
347#388—1061no—
348#385—1061no—
349#384C2×D5311yescharacteristic
350#382—21no—
351#382C2×D5311yescharacteristic
352#382C2×D5311yescharacteristic
353C4×C4⋊C4—531no—
354C4×C4⋊C4C2×D5311yescharacteristic
355C4×C4⋊C4C2×D5311yescharacteristic
356C4×C22⋊C4—1061no—
357#340—531no—
358#340—531no—
359#340—531no—
360#340—531no—
361#336—531no—
362#336—531no—
363C53#2654511yescharacteristic, complemented, Sylow
364C22×Q8—531no—
365C22×D4—531no—
366C23×C4—1061no—
367C22×C8—531no—
368C22×C8—531no—
369C22×C8—531no—
370C22×C8—531no—
371#124—534no—
372#124—1062no—
373C4×Q8—1062no—
374C4×Q8—2121no—
375C4×D4—1062no—
376C4×D4—2121no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—531no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4C22⋊D5311yescharacteristic
385C2×C4⋊C4C22⋊D5311yescharacteristic
386C2×C4⋊C4C22×D5311yescharacteristic
387C2×C4⋊C4C22×D5311yescharacteristic
388C2×C4⋊C4C22×D5311yescharacteristic
389C2×C4⋊C4C22×D5311yescharacteristic
390C2×C22⋊C4—1061no—
391C2×C22⋊C4—1061no—
392C2×C22⋊C4—1061no—
393C2×C22⋊C4—1061no—
394C2×C22⋊C4—1061no—
395C2×C42—21no—
396C2×C42—1061no—
397C2×C42—1061no—
398C2×C42C22×D5311yescharacteristic
399C2×C42C22×D5311yescharacteristic
400C2×C42C22×D5311yescharacteristic
401#107—1062no—
402#107—2121no—
403#107—2121no—
404#106—1062no—
405C4⋊C8—1062no—
406C4⋊C8—2121no—
407C4⋊C8—2121no—
408#103—1062no—
409#103—2121no—
410#103—2121no—
411#103—2121no—
412#102—1062no—
413#102—2121no—
414#102—2121no—
415#102—2121no—
416#97—1062no—
417#95—21no—
418#95—21no—
419#95—21no—
420#95—21no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—1061no—
425#95—1061no—
426#95—1061no—
427#95C22⋊D5311yescharacteristic
428#95C22⋊D5311yescharacteristic
429C24—1061no—
430C2×Q8—532no—
431C2×Q8—1061no—
432C2×Q8—1061no—
433C2×Q8—1061no—
434C2×Q8—1061no—
435C2×Q8—1061no—
436C2×D4—532no—
437C2×D4—1061no—
438C2×D4—1061no—
439C2×D4—1061no—
440C2×D4—1061no—
441C2×D4—1061no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—4241no—
450C22×C4—2122no—
451C22×C4—2122no—
452C22×C4C2×C22⋊D5311yescharacteristic
453C22×C4C2×C22⋊D5311yescharacteristic
454C22×C4C2×C22⋊D5311yescharacteristic
455C22×C4C2×C22⋊D5311yescharacteristic
456C22×C4C2×C22⋊D5311yescharacteristic
457C22×C4C2×C22⋊D5311yescharacteristic
458C22×C4C23×D5311yescharacteristic
459C2×C8—532no—
460C2×C8—1061no—
461C2×C8—1061no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8—1061no—
470C2×C8—1061no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—42no—
482C4⋊C4—81no—
483C4⋊C4—81no—
484C4⋊C4—81no—
485C4⋊C4—532no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—2122no—
490C22⋊C4—1064no—
491C22⋊C4—1064no—
492C22⋊C4—4241no—
493C22⋊C4—2122no—
494C22⋊C4—2122no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—42no—
500C42—4241no—
501C42—2122no—
502C42Q8×D5312yes—
503C42#1171994912yes—
504C23—1062no—
505C23—2121no—
506C23—2121no—
507C23—2121no—
508C23—2121no—
509C23—2121no—
510C23—2121no—
511C23C22×C22⋊D5311yescharacteristic
512Q8—1062no—
513Q8—1062no—
514Q8—2121no—
515Q8—2121no—
516D4—1062no—
517D4—1062no—
518D4—2121no—
519D4—2121no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
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—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—1062no—
549C2×C4—1062no—
550C2×C4—4244no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4#1313319312yes—
559C2×C4#1313319312yes—
560C2×C4C2×Q8×D5311yescharacteristic
561C2×C4#41×D5311yescharacteristic
562C2×C4#41×D5311yescharacteristic
563C2×C4C2×#1171994911yescharacteristic
564C2×C4#1313326311yescharacteristic
565C2×C4#1313326311yescharacteristic
566C8—1062no—
567C8—2121no—
568C8—2121no—
569C8—2121no—
570C22—4241no—
571C22—4241no—
572C22—4241no—
573C22—4241no—
574C22—2122no—
575C22—2122no—
576C22—2122no—
577C22C2×#1313319311yescharacteristic
578C22#1681126111yescharacteristic
579C22#1681129211yescharacteristic
580C22#1681142111yescharacteristic
581C22#1681142111yescharacteristic
582C22#1681142611yescharacteristic
583C22#1681143811yescharacteristic
584C4—22no—
585C4—22no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—41no—
590C4—81no—
591C4—81no—
592C4—81no—
593C4—81no—
594C4—4241no—
595C4—2122no—
596C4—2122no—
597C4#1681115811yescharacteristic
598C4#1681122911yescharacteristic
599C4#1681132911yescharacteristic
600C4#1681132911yescharacteristic
601C2—4242no—
602C2#2027858311yescharacteristic
603C2#2028453311yescharacteristic
604C2#2028456211yescharacteristic
605C2#2028677511yescharacteristic
606C2#2028686511yescharacteristic
607C2#2028779011yescharacteristic
608C2#2028779911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2571C2
socleC23×C53C22×D4
prime radicalC22×D212C23
hypercenter#2571D53
hypocenterC53#26545
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2571

Complements

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

NH ≅ G/Ntype
2. #20276266C2semidirect
4. #20276107C2semidirect
8. #20275471C2semidirect
9. #20275439C2semidirect
10. #20274968C2semidirect
12. #20274727C2semidirect
14. #20274613C2semidirect
15. C53×#2571C2semidirect
297. #2571D53semidirect
363. C53#26545semidirect

Maximal subgroups

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

subgroupindex
#202777842
#202762662
#202762392
#202761072
#202761002
#202756472
#202755292
#202754712
#202754392
#202749682
#202749612
#202747272
#202747222
#202746132
C53×#25712
#2654553