Finite Group Databasebeta

← #27469616

Subgroups — #27469616

Order 13568.

770 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283065C211yesmaximal, characteristic
2#20277118C211yesmaximal, characteristic
3#20276195C211yesmaximal, characteristic, complemented
4#20276194C211yesmaximal, characteristic, complemented
5#20276190C211yesmaximal, characteristic, complemented
6#20276189C211yesmaximal, characteristic, complemented
7#20275686C211yesmaximal, characteristic, complemented
8#20275686C211yesmaximal, characteristic, complemented
9#20274916C211yesmaximal, characteristic
10#20274914C211yesmaximal, characteristic
11#20274912C211yesmaximal, characteristic
12#20274882C211yesmaximal, characteristic, complemented
13#20274880C211yesmaximal, characteristic, complemented
14#20274479C211yesmaximal, characteristic
15C53×#1854C211yesmaximal, characteristic
16C2×#13133216C2211yescharacteristic
17C2×#13133194—21no—
18C2×#13133176C2211yescharacteristic
19C22×C4×Dic53C2211yescharacteristic
20C2×#13133151C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×C22⋊#88322C2211yescharacteristic
27C2×#13133127C2211yescharacteristic
28C2×#13133127C2211yescharacteristic
29#16810527—21no—
30#16810522C2211yescharacteristic
31#16810520—21no—
32#16810519C2211yescharacteristic
33#16810492C2211yescharacteristic
34#16810492C2211yescharacteristic
35#16810492C2211yescharacteristic
36#16810488C2211yescharacteristic
37#16810488C2211yescharacteristic
38#16810483C2211yescharacteristic
39#16810481—21no—
40#16810481—21no—
41#16810481—21no—
42C4×#11719919—21no—
43C4×#11719919—21no—
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#117×C53C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic
54C2×#106×C53C2211yescharacteristic
55C2×C22⋊C8×C53C2211yescharacteristic
56C53×#380C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×Dic106—21no—
60C23×Dic53D411yescharacteristic
61C23×Dic53D411yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133216—22no—
65#13133216—22no—
66#13133194—44no—
67#13133176—41no—
68#13133176—41no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×#11719919—21no—
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×C4×Dic53—22no—
85C2×C4×Dic53—22no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53D411yescharacteristic
88C2×C4×Dic53D411yescharacteristic
89C2×C4×Dic53C2311yescharacteristic
90C2×C4×Dic53C2311yescharacteristic
91#13133151—41no—
92#13133150—41no—
93#13133150—41no—
94#13133149—41no—
95#13133148—41no—
96#13133148—41no—
97C22⋊#88322Q814yescomplemented
98#13133127—22no—
99#13133127—22no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107C23×C4×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110#117×C53—41no—
111#117×C53—41no—
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C22⋊C4×C53—21no—
117C2×C22⋊C4×C53—21no—
118C2×C42×C53—21no—
119#107×C53—41no—
120#106×C53—41no—
121C22⋊C8×C53Q814yescomplemented
122#95×C53—21no—
123C24×C53C2×D411yescharacteristic
124C22×C4×C53—21no—
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×C53C2×D411yescharacteristic
133C22×C4×C53C2×Q812yes—
134C22×C4×C53C2×Q812yes—
135C22×C4×C53C2411yescharacteristic
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—41no—
145C22×Dic53—41no—
146C22×Dic53C2×D411yescharacteristic
147C22×Dic53C2×D411yescharacteristic
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53C2×D411yescharacteristic
150C2×Dic106—42no—
151C2×Dic106—42no—
152C2×Dic106—42no—
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×C53C2×Q812yes—
158C2×C8×C53C2×Q812yes—
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—
166C4⋊C4×C53—21no—
167C22⋊C4×C53—42no—
168C22⋊C4×C53—42no—
169C42×C53—42no—
170C22⋊Dic53—42no—
171C22⋊Dic53—42no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—42no—
181#11719919—42no—
182#11719919—42no—
183#11719919—42no—
184#11719919—42no—
185#11719919—42no—
186#11719919—42no—
187#11719919—42no—
188#11719919—42no—
189C4×Dic53—21no—
190C4×Dic53—21no—
191C4×Dic53—22no—
192C4×Dic53—22no—
193C4×Dic53—42no—
194C4×Dic53#4112yes—
195C4×Dic53#4112yes—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322C2×Q812yes—
201C2×#88322C2×Q812yes—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
208C23×C53C4⋊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×C53C4⋊Q812yes—
229C2×C4×C53C22×Q811yescharacteristic
230C2×C4×C53C22×Q811yescharacteristic
231C2×C4×C53C2×#4111yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×C4×C53H(5,𝔽2)11yescharacteristic
234C2×C4×C53H(5,𝔽2)11yescharacteristic
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—44no—
244C2×Dic53—44no—
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#12312yes—
258C2×Dic53#12312yes—
259C2×Dic53AGL(1,ℤ/8)12yes—
260C2×Dic53#13712yes—
261Dic106—84no—
262C8×C53—22no—
263C8×C53—22no—
264#88322—22no—
265#88322—22no—
266#27955—531nomaximal, Sylow
267C22×C53—22no—
268C22×C53—22no—
269C22×C53—22no—
270C22×C53—41no—
271C22×C53—41no—
272C22×C53—41no—
273C22×C53—41no—
274C22×C53C2×#12311yescharacteristic
275C22×C53C2×C4⋊Q811yescharacteristic
276C22×C53#53511yescharacteristic
277C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
278C22×C53C2×#13711yescharacteristic
279C22×C53#57511yescharacteristic
280C22×C53#57511yescharacteristic
281C4×C53—22no—
282C4×C53—81no—
283C4×C53—81no—
284C4×C53—81no—
285C4×C53—81no—
286C4×C53#55411yescharacteristic
287C4×C53#55411yescharacteristic
288C4×C53#55411yescharacteristic
289C4×C53#55411yescharacteristic
290Dic53—22no—
291Dic53—22no—
292Dic53—22no—
293Dic53—22no—
294Dic53—81no—
295Dic53—81no—
296Dic53—81no—
297Dic53—81no—
298#2857—531no—
299#2285—531no—
300#2084—531no—
301#2084—531no—
302#2083—531no—
303#2083—531no—
304#1993—531no—
305#1993—531no—
306#1854—531no—
307#1854—531no—
308#1854—531no—
309#1854D5311yescharacteristic
310#1847—531no—
311#1847—531no—
312#1749—531no—
313C2×C53—42no—
314C2×C53#250211yescharacteristic
315C2×C53#309411yescharacteristic
316C2×C53#309511yescharacteristic
317C2×C53#316211yescharacteristic
318C2×C53#316511yescharacteristic
319C2×C53#321411yescharacteristic
320C2×C53#321611yescharacteristic
321C2×C4⋊Q8—531no—
322C2×C22⋊Q8—1061no—
323C2×#117—531no—
324C2×#117C2×D5311yescharacteristic
325C22×C42—531no—
326C2×#107—531no—
327C2×#107—531no—
328C2×#107—531no—
329C2×#107C2×D5311yescharacteristic
330C2×#106—531no—
331C2×#106—531no—
332C2×#106—531no—
333C2×#106C2×D5311yescharacteristic
334C2×C22⋊C8—531no—
335C2×C22⋊C8C2×D5311yescharacteristic
336C2×#97—531no—
337C2×#97—531no—
338#391—1061no—
339#389—531no—
340#389—531no—
341#389—1061no—
342#384—531no—
343#384—531no—
344#384—531no—
345#382—531no—
346#382—531no—
347#380—531no—
348#380—1061no—
349#380—1061no—
350#380—1061no—
351#380C2×D5311yescharacteristic
352C4×C4⋊C4—1061no—
353C4×C4⋊C4—1061no—
354#340—531no—
355#340—531no—
356#340—531no—
357#340—531no—
358#340—531no—
359#340—531no—
360#340C2×D5311yescharacteristic
361#340C2×D5311yescharacteristic
362#336—531no—
363#336—531no—
364C53#2795511yescharacteristic, complemented, Sylow
365C22×Q8—1061no—
366C23×C4—531no—
367C23×C4—531no—
368C23×C4C22×D5311yescharacteristic
369C22×C8—531no—
370C22×C8—531no—
371C22×C8C22×D5311yescharacteristic
372C22×C8C22×D5311yescharacteristic
373C4⋊Q8—1062no—
374C4⋊Q8—1062no—
375C22⋊Q8—2124no—
376#117—41no—
377#117—41no—
378#117—2121no—
379#117—2121no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—531no—
383C2×C4⋊C4—531no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4—1061no—
389C2×C4⋊C4—1061no—
390C2×C4⋊C4—1061no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4C22×D5311yescharacteristic
394C2×C4⋊C4C22×D5311yescharacteristic
395C2×C4⋊C4C22×D5311yescharacteristic
396C2×C4⋊C4C22×D5311yescharacteristic
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—1061no—
400C2×C22⋊C4—1061no—
401C2×C42—21no—
402C2×C42—531no—
403C2×C42—531no—
404C2×C42—531no—
405C2×C42—531no—
406C2×C42—1062no—
407C2×C42—1062no—
408C2×C42—1061no—
409#107—41no—
410#107—2121no—
411#107—2121no—
412#107—2121no—
413#106—41no—
414#106—2121no—
415#106—2121no—
416#106—2121no—
417C22⋊C8—534no—
418C22⋊C8Dic10614yescomplemented
419#97—1062no—
420#97—1062no—
421#95—21no—
422#95—1061no—
423#95—1061no—
424#95—1061no—
425#95—1061no—
426#95—1061no—
427#95—1061no—
428#95—1061no—
429C24D4×D5311yescharacteristic
430C2×Q8—2122no—
431C2×Q8—2122no—
432C2×Q8—2122no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—531no—
442C22×C4—531no—
443C22×C4—531no—
444C22×C4—531no—
445C22×C4—1062no—
446C22×C4—1062no—
447C22×C4—1062no—
448C22×C4—1062no—
449C22×C4—1061no—
450C22×C4—1061no—
451C22×C4—1061no—
452C22×C4—1061no—
453C22×C4—2121no—
454C22×C4—2121no—
455C22×C4C2×Dic10612yes—
456C22×C4D4×D5311yescharacteristic
457C22×C4Q8×D5312yes—
458C22×C4C23×D5311yescharacteristic
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—532no—
464C2×C8—532no—
465C2×C8—1061no—
466C2×C8—1061no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8C2×Dic10612yes—
470C2×C8C2×Dic10612yes—
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—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—2122no—
488C4⋊C4—2122no—
489C4⋊C4—2122no—
490C4⋊C4—2122no—
491C4⋊C4—2122no—
492C4⋊C4—2122no—
493C4⋊C4—2122no—
494C4⋊C4—2122no—
495C4⋊C4—2122no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—2122no—
499C22⋊C4—2122no—
500C42—42no—
501C42—532no—
502C42—532no—
503C42—1062no—
504C42—1062no—
505C42—1061no—
506C42—1061no—
507C42—2122no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23#1313321612yes—
515C23C2×D4×D5311yescharacteristic
516Q8—4244no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
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—532no—
536C2×C4—532no—
537C2×C4—532no—
538C2×C4—532no—
539C2×C4—1061no—
540C2×C4—1061no—
541C2×C4—1061no—
542C2×C4—1061no—
543C2×C4—1061no—
544C2×C4—1061no—
545C2×C4—1061no—
546C2×C4—1061no—
547C2×C4—2124no—
548C2×C4—2124no—
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#1313321612yes—
562C2×C4C2×Q8×D5311yescharacteristic
563C2×C4C22×Dic10611yescharacteristic
564C2×C4#41×D5311yescharacteristic
565C2×C4#41×D5311yescharacteristic
566C2×C4#1313326211yescharacteristic
567C2×C4#1313326211yescharacteristic
568C8—22no—
569C8—22no—
570C8—1062no—
571C8—1062no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22#123×D5311yescharacteristic
580C22C2×#1313321611yescharacteristic
581C22#1681118811yescharacteristic
582C22AGL(1,ℤ/8)×D5311yescharacteristic
583C22#137×D5311yescharacteristic
584C22#1681142011yescharacteristic
585C22#1681142011yescharacteristic
586C4—22no—
587C4—81no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4—1062no—
592C4—1062no—
593C4—1062no—
594C4—1062no—
595C4—4241no—
596C4—4241no—
597C4—4241no—
598C4—4241no—
599C4#1681132711yescharacteristic
600C4#1681132711yescharacteristic
601C4#1681133211yescharacteristic
602C4#1681133211yescharacteristic
603C2—42no—
604C2#2027919111yescharacteristic
605C2#2028465411yescharacteristic
606C2#2028466211yescharacteristic
607C2#2028523811yescharacteristic
608C2#2028525911yescharacteristic
609C2#2028570711yescharacteristic
610C2#2028572311yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1854C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1854D53
hypocenterC53#27955
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
3. #20276195C2semidirect
4. #20276194C2semidirect
5. #20276190C2semidirect
6. #20276189C2semidirect
7. #20275686C2semidirect
8. #20275686C2semidirect
12. #20274882C2semidirect
13. #20274880C2semidirect
97. C22⋊#88322Q8semidirect
121. C22⋊C8×C53Q8semidirect
364. C53#27955semidirect
418. C22⋊C8Dic106semidirect

Maximal subgroups

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

subgroupindex
#202830652
#202771182
#202761952
#202761942
#202761902
#202761892
#202756862
#202749162
#202749142
#202749122
#202748822
#202748802
#202744792
C53×#18542
#2795553