Finite Group Databasebeta

← #27487203

Subgroups — #27487203

Order 13568.

724 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277192C211yesmaximal, characteristic, complemented
2#20276216C211yesmaximal, characteristic, complemented
3#20276093C211yesmaximal, characteristic, complemented
4#20276089C211yesmaximal, characteristic
5#20276032C211yesmaximal, characteristic
6#20275568C211yesmaximal, characteristic
7#20275440C211yesmaximal, characteristic, complemented
8#20275401C211yesmaximal, characteristic
9#20275390C211yesmaximal, characteristic, complemented
10#20275339C211yesmaximal, characteristic
11#20275290C211yesmaximal, characteristic, complemented
12#20275073C211yesmaximal, characteristic
13#20274982C211yesmaximal, characteristic, complemented
14C4×C4⋊#88322C211yesmaximal, characteristic, complemented
15C53×#2860C211yesmaximal, characteristic
16C22×#11719923C2211yescharacteristic
17C2×C4×Dic106C2211yescharacteristic
18C2×#13133148C2211yescharacteristic
19C2×#13133148C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22C2×C4⋊#88322C2211yescharacteristic
23C2×#13133140C2211yescharacteristic
24C2×#13133140C2211yescharacteristic
25C2×#13133140C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×#13133138C2211yescharacteristic
29C2×C4×#88322C2211yescharacteristic
30C2×C4×#88322C2211yescharacteristic
31#16810529C2211yescharacteristic
32#16810529C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810521C2211yescharacteristic
35#16810507—21no—
36#16810507—21no—
37C4×#11719919—21no—
38C4×C4⋊Dic53C2211yescharacteristic
39#16810469—21no—
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×#126×C53—21no—
49C53×#390C2211yescharacteristic
50C53×#390C2211yescharacteristic
51C53×#390C2211yescharacteristic
52C53×#389C2211yescharacteristic
53C53×#389C2211yescharacteristic
54C53×#389C2211yescharacteristic
55C53×#388—21no—
56C53×#388—21no—
57C53×#388—21no—
58C43×C53C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C2×#11719923—22no—
61C2×#11719923—22no—
62C2×#11719923—41no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C4×Dic106—22no—
68C4×Dic106—22no—
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×C4×Dic53—21no—
76C2×C4×Dic53—21no—
77#13133148—22no—
78#13133148—22no—
79#13133148—41no—
80C4⋊#88322—22no—
81C4⋊#88322—22no—
82#13133140—22no—
83#13133140—22no—
84#13133140—41no—
85#13133138—22no—
86#13133138—22no—
87#13133138—41no—
88C4×#88322—41no—
89C4×#88322—41no—
90#13133124—21no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97C22×D4×C53C2311yescharacteristic
98#126×C53—44no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53C2311yescharacteristic
103C2×C4⋊C4×C53C2311yescharacteristic
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C22⋊C4×C53—21no—
106C2×C22⋊C4×C53—21no—
107C2×C22⋊C4×C53—21no—
108C2×C22⋊C4×C53—21no—
109C2×C22⋊C4×C53—21no—
110C2×C22⋊C4×C53—21no—
111C2×C42×C53D411yescharacteristic
112C2×C42×C53D411yescharacteristic
113C2×C42×C53D411yescharacteristic
114C2×C42×C53D411yescharacteristic
115C2×C42×C53C2311yescharacteristic
116C2×C42×C53C2311yescharacteristic
117C2×C42×C53C2311yescharacteristic
118#95×C53—21no—
119#95×C53—21no—
120#95×C53—21no—
121C24×C53—21no—
122C2×D4×C53—21no—
123C2×D4×C53—21no—
124C2×D4×C53—21no—
125C2×D4×C53—21no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2411yescharacteristic
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C2×Dic106—21no—
143C2×Dic106—21no—
144C2×Dic106—21no—
145C2×Dic106—21no—
146C2×Dic106—21no—
147C2×Dic106#4112yes—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—42no—
155C4⋊C4×C53—42no—
156C4⋊C4×C53—42no—
157C22⋊C4×C53—42no—
158C22⋊C4×C53—42no—
159C22⋊C4×C53—42no—
160C22⋊C4×C53—42no—
161C22⋊C4×C53—42no—
162C22⋊C4×C53—42no—
163C42×C53—22no—
164C42×C53—21no—
165C42×C53—21no—
166C42×C53—21no—
167C42×C53—21no—
168C42×C53—21no—
169C42×C53—21no—
170C42×C53—41no—
171C42×C53—41no—
172C42×C53SD1614yescomplemented
173#11719923—44no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53#4112yes—
180#11719919—24no—
181#11719919—42no—
182#11719919—42no—
183C4×Dic53—24no—
184C4×Dic53—42no—
185C2×#88322—21no—
186C2×#88322—21no—
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#4112yes—
196C2×#88322#4112yes—
197C23×C53—41no—
198C23×C53—41no—
199C23×C53—41no—
200C23×C53—41no—
201C23×C53—41no—
202C23×C53—41no—
203C23×C53—41no—
204C23×C53C22×D411yescharacteristic
205D4×C53—41no—
206D4×C53—41no—
207D4×C53—41no—
208D4×C53—41no—
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×C53C2×SD1612yes—
233C2×C4×C53C2×SD1612yes—
234C2×C4×C53C2×SD1612yes—
235C2×C4×C53#13512yes—
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53H(5,𝔽2)11yescharacteristic
241C2×C4×C532−1+411yescharacteristic
242C2×Dic53—22no—
243C2×Dic53—22no—
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—
254Dic106—22no—
255Dic106—22no—
256Dic106—41no—
257Dic106—41no—
258#88322—22no—
259#88322—22no—
260#88322—41no—
261#88322—41no—
262#29434—531nomaximal, Sylow
263C22×C53—81no—
264C22×C53—81no—
265C22×C53—81no—
266C22×C53—81no—
267C22×C53—81no—
268C22×C53—81no—
269C22×C53—81no—
270C22×C53#52511yescharacteristic
271C22×C53#53211yescharacteristic
272C22×C53#53711yescharacteristic
273C22×C53C22×SD1611yescharacteristic
274C22×C53C2×#13511yescharacteristic
275C22×C53#57511yescharacteristic
276C22×C53#57511yescharacteristic
277C4×C53—22no—
278C4×C53—22no—
279C4×C53—22no—
280C4×C53—22no—
281C4×C53—41no—
282C4×C53—41no—
283C4×C53—81no—
284C4×C53—81no—
285C4×C53—81no—
286C4×C53#55111yescharacteristic
287C4×C53#55211yescharacteristic
288C4×C53#55311yescharacteristic
289C4×C53#55611yescharacteristic
290Dic53—42no—
291Dic53—42no—
292Dic53—81no—
293Dic53—81no—
294#2860D5311yescharacteristic
295#2295—531no—
296#2088—531no—
297#2066—531no—
298#2065—531no—
299#2055—531no—
300#1968—531no—
301#1941—531no—
302#1934—531no—
303#1933—531no—
304#1926—531no—
305#1919—531no—
306#1880—531no—
307#1865—531no—
308C4×C4⋊C8—531no—
309C2×C53—161no—
310C2×C53#260111yescharacteristic
311C2×C53#310211yescharacteristic
312C2×C53#310311yescharacteristic
313C2×C53#316711yescharacteristic
314C2×C53#318311yescharacteristic
315C2×C53#324611yescharacteristic
316C2×C53#325011yescharacteristic
317C22×SD16—531no—
318C2×#126—21no—
319C2×C4×Q8—531no—
320C2×#106—531no—
321C2×#106—531no—
322C2×#106—531no—
323C2×C4⋊C8—531no—
324C2×C4⋊C8—531no—
325C2×#103—531no—
326C2×#103—531no—
327C2×#103—531no—
328C2×#102—531no—
329C2×#102—531no—
330C2×#102—531no—
331C2×C4×C8—531no—
332C2×C4×C8—531no—
333#391—531no—
334#391—531no—
335#390C2×D5311yescharacteristic
336#390C2×D5311yescharacteristic
337#390C2×D5311yescharacteristic
338#389—531no—
339#389—531no—
340#389C2×D5311yescharacteristic
341#389C2×D5311yescharacteristic
342#389C2×D5311yescharacteristic
343#388—21no—
344#388—21no—
345#388—21no—
346#387—1061no—
347#387—1061no—
348C4×C4⋊C4—531no—
349C4×C4⋊C4—1061no—
350#376—1061no—
351C43C2×D5311yescharacteristic
352#340—531no—
353#340—531no—
354#340—531no—
355#340—531no—
356#340—531no—
357#340—531no—
358#336—531no—
359#336—531no—
360C53#2943411yescharacteristic, complemented, Sylow
361C22×Q8—531no—
362C22×D4C22×D5311yescharacteristic
363C2×SD16—1062no—
364C2×SD16—1062no—
365C2×SD16—2121no—
366C22×C8—531no—
367C22×C8—531no—
368C22×C8—531no—
369C22×C8—531no—
370#126—44no—
371C4×Q8—1062no—
372C4×Q8—1062no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—531no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4C22×D5311yescharacteristic
383C2×C4⋊C4C22×D5311yescharacteristic
384C2×C4⋊C4C22×D5311yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C42—1061no—
392C2×C42—1061no—
393C2×C42C22⋊D5311yescharacteristic
394C2×C42C22⋊D5311yescharacteristic
395C2×C42C22⋊D5311yescharacteristic
396C2×C42C22⋊D5311yescharacteristic
397C2×C42C22×D5311yescharacteristic
398C2×C42C22×D5311yescharacteristic
399C2×C42C22×D5311yescharacteristic
400#106—1062no—
401#106—1062no—
402#106—2121no—
403C4⋊C8—1062no—
404C4⋊C8—1062no—
405#103—1062no—
406#103—1062no—
407#103—2121no—
408#102—1062no—
409#102—1062no—
410#102—2121no—
411C4×C8—2121no—
412C4×C8—2121no—
413#95—21no—
414#95—21no—
415#95—21no—
416#95—1061no—
417#95—1061no—
418#95—1061no—
419#95—1061no—
420#95—1061no—
421#95—1061no—
422#95—1061no—
423C24—21no—
424C2×Q8—532no—
425C2×Q8—1061no—
426C2×Q8—1061no—
427C2×Q8—1061no—
428C2×Q8—1061no—
429C2×Q8—1061no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—1061no—
440C22×C4—1061no—
441C22×C4—1061no—
442C22×C4—1061no—
443C22×C4C2×C22⋊D5311yescharacteristic
444C22×C4C2×C22⋊D5311yescharacteristic
445C22×C4C2×C22⋊D5311yescharacteristic
446C22×C4C2×C22⋊D5311yescharacteristic
447C22×C4C2×C22⋊D5311yescharacteristic
448C22×C4C2×C22⋊D5311yescharacteristic
449C22×C4C23×D5311yescharacteristic
450SD16—2124no—
451C2×C8—532no—
452C2×C8—532no—
453C2×C8—1061no—
454C2×C8—1061no—
455C2×C8—1061no—
456C2×C8—1061no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8—1061no—
462C2×C8—1061no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—532no—
473C4⋊C4—1064no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4—2122no—
480C4⋊C4—2122no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C42—22no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—41no—
495C42—41no—
496C42—1064no—
497C42—2122no—
498C42#1171992314yescomplemented
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23C22×C22⋊D5311yescharacteristic
507Q8—1062no—
508Q8—1062no—
509Q8—2121no—
510Q8—2121no—
511D4—41no—
512D4—41no—
513D4—41no—
514D4—41no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
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—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—1062no—
539C2×C4—1062no—
540C2×C4—2121no—
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×C4C2×#1171992312yes—
551C2×C4C2×#1171992312yes—
552C2×C4C2×#1171992312yes—
553C2×C4#1313323712yes—
554C2×C4#41×D5311yescharacteristic
555C2×C4#41×D5311yescharacteristic
556C2×C4#41×D5311yescharacteristic
557C2×C4#41×D5311yescharacteristic
558C2×C4#1313326311yescharacteristic
559C2×C4#1313326411yescharacteristic
560C8—1062no—
561C8—1062no—
562C8—2121no—
563C8—2121no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22#1681113011yescharacteristic
572C22#1681116911yescharacteristic
573C22#1681120111yescharacteristic
574C22C22×#1171992311yescharacteristic
575C22C2×#1313323711yescharacteristic
576C22#1681142111yescharacteristic
577C22#1681142111yescharacteristic
578C4—22no—
579C4—22no—
580C4—22no—
581C4—22no—
582C4—41no—
583C4—41no—
584C4—81no—
585C4—81no—
586C4—81no—
587C4—4241no—
588C4—4241no—
589C4—2122no—
590C4—2122no—
591C4#1681129811yescharacteristic
592C4#1681130911yescharacteristic
593C4#1681132411yescharacteristic
594C4#1681134011yescharacteristic
595C2—161no—
596C2#2028031411yescharacteristic
597C2#2028472711yescharacteristic
598C2#2028474211yescharacteristic
599C2#2028527011yescharacteristic
600C2#2028543811yescharacteristic
601C2#2028603111yescharacteristic
602C2#2028606211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2860C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2860D53
hypocenterC53#29434
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C43 ◁ #2860

Complements

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

NH ≅ G/Ntype
1. #20277192C2semidirect
2. #20276216C2semidirect
3. #20276093C2semidirect
7. #20275440C2semidirect
9. #20275390C2semidirect
11. #20275290C2semidirect
13. #20274982C2semidirect
14. C4×C4⋊#88322C2semidirect
172. C42×C53SD16semidirect
360. C53#29434semidirect
498. C42#11719923semidirect

Maximal subgroups

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

subgroupindex
#202771922
#202762162
#202760932
#202760892
#202760322
#202755682
#202754402
#202754012
#202753902
#202753392
#202752902
#202750732
#202749822
C4×C4⋊#883222
C53×#28602
#2943453