Finite Group Databasebeta

← #27534432

Subgroups — #27534432

Order 13568.

870 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283074C211yesmaximal, characteristic, complemented
2C4×D4×Dic53C211yesmaximal, characteristic
3#20275472C211yesmaximal, characteristic, complemented
4#20275470C211yesmaximal, characteristic, complemented
5#20275288C211yesmaximal, characteristic, complemented
6#20275287C211yesmaximal, characteristic, complemented
7#20275284C211yesmaximal, characteristic, complemented
8#20275283C211yesmaximal, characteristic, complemented
9#20274964C211yesmaximal, characteristic
10#20274962C211yesmaximal, characteristic
11#20274960C211yesmaximal, characteristic
12#20274830C211yesmaximal, characteristic, complemented
13#20274713C211yesmaximal, characteristic
14#20274713C211yesmaximal, characteristic
15C53×#1862C211yesmaximal, characteristic
16C2×#13133207—21no—
17C2×D4×Dic53C2211yescharacteristic
18C2×D4×Dic53C2211yescharacteristic
19C22×C4×Dic53—21no—
20C2×#13133151C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133146C2211yescharacteristic
24C2×#13133146C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27#16810522C2211yescharacteristic
28#16810521C2211yescharacteristic
29#16810520—21no—
30#16810519C2211yescharacteristic
31#16810518—21no—
32#16810492C2211yescharacteristic
33#16810492C2211yescharacteristic
34#16810487—21no—
35C4⋊C4×Dic53C2211yescharacteristic
36C4⋊C4×Dic53C2211yescharacteristic
37C4×C4⋊Dic53C2211yescharacteristic
38C22⋊C4×Dic53—21no—
39C22⋊C4×Dic53—21no—
40C4×C22⋊Dic53—21no—
41C42×Dic53C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810378C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×C4×D4×C53C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic
54C2×#102×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C53×#389C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C23×Dic53—21no—
60C23×Dic53—21no—
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63#13133207—44no—
64D4×Dic53—22no—
65D4×Dic53—22no—
66D4×Dic53—22no—
67D4×Dic53Q814yes—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—44no—
85C2×C4×Dic53D411yescharacteristic
86C2×C4×Dic53D411yescharacteristic
87C2×C4×Dic53D411yescharacteristic
88C2×C4×Dic53D411yescharacteristic
89C2×C4×Dic53C2311yescharacteristic
90C2×C4×Dic53C2311yescharacteristic
91#13133151—22no—
92#13133150—22no—
93#13133150—22no—
94#13133146—22no—
95#13133146—22no—
96#13133138—22no—
97#13133138—22no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104C22×D4×C53C2311yescharacteristic
105C23×C4×C53—21no—
106C22×C8×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108C4×D4×C53—22no—
109C4×D4×C53—22no—
110C2×C4⋊C4×C53—21no—
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C22⋊C4×C53—21no—
115C2×C42×C53C2311yescharacteristic
116#107×C53—22no—
117#102×C53—22no—
118#102×C53—22no—
119#95×C53—21no—
120C24×C53—21no—
121C2×D4×C53—21no—
122C2×D4×C53—21no—
123C2×D4×C53—21no—
124C2×D4×C53C2×Q812yes—
125C2×D4×C53#4112yes—
126C2×D4×C53#4112yes—
127C22×C4×C53—24no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—42no—
132C22×C4×C53—42no—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2411yescharacteristic
136C22×Dic53—24no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—24no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—42no—
143C22×Dic53—42no—
144C22×Dic53—42no—
145C22×Dic53—42no—
146C22×Dic53C2×D411yescharacteristic
147C22×Dic53C2×D411yescharacteristic
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53C2×D411yescharacteristic
150C2×C8×C53—21no—
151C2×C8×C53—21no—
152C2×C8×C53—21no—
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×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—42no—
162C4⋊C4×C53#4112yes—
163C22⋊C4×C53—24no—
164C42×C53—21no—
165C42×C53—21no—
166C22⋊Dic53—24no—
167C22⋊Dic53—24no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—42no—
175C4⋊Dic53C2×Q812yes—
176C4⋊Dic53#4112yes—
177#11719919—42no—
178#11719919—42no—
179#11719919—42no—
180#11719919—42no—
181#11719919—42no—
182#11719919—42no—
183C4×Dic53—21no—
184C4×Dic53—21no—
185C4×Dic53—22no—
186C4×Dic53—41no—
187C4×Dic53—41no—
188C4×Dic53—42no—
189C4×Dic53—42no—
190C4×Dic53—42no—
191C4×Dic53D814yes—
192C4×Dic53C2×Q812yes—
193C4×Dic53#4112yes—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C23×C53—22no—
201C23×C53—22no—
202C23×C53—22no—
203C23×C53—41no—
204C23×C53—41no—
205C23×C53—41no—
206C23×C53—41no—
207C23×C53C22×D411yescharacteristic
208D4×C53—22no—
209D4×C53—22no—
210D4×C53—22no—
211D4×C53#12514yescomplemented
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—22no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—48no—
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×C53C2×D812yes—
228C2×C4×C53C22×Q811yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C2×C4×C53C2×#4111yescharacteristic
231C2×C4×C53C2×#4111yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—22no—
239C2×Dic53—22no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—48no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—48no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53C22⋊Q812yes—
256C2×Dic53C22⋊Q812yes—
257C2×Dic53C2×D812yes—
258C2×Dic53C2×D812yes—
259C2×Dic53#13712yes—
260C8×C53—41no—
261C8×C53—41no—
262#88322—41no—
263#88322—41no—
264#33066—531nomaximal, Sylow
265C22×C53—24no—
266C22×C53—42no—
267C22×C53—42no—
268C22×C53—42no—
269C22×C53—42no—
270C22×C53—42no—
271C22×C53—42no—
272C22×C53C2×C22⋊Q811yescharacteristic
273C22×C53#52511yescharacteristic
274C22×C53#52511yescharacteristic
275C22×C53C22×D811yescharacteristic
276C22×C53C2×#13711yescharacteristic
277C22×C53#57511yescharacteristic
278C22×C53#57511yescharacteristic
279C4×C53—22no—
280C4×C53—42no—
281C4×C53—41no—
282C4×C53—81no—
283C4×C53—81no—
284C4×C53C2×#12511yescharacteristic
285C4×C53#52911yescharacteristic
286C4×C53#52911yescharacteristic
287C4×C53#53311yescharacteristic
288Dic53—22no—
289Dic53—22no—
290Dic53—42no—
291Dic53—41no—
292Dic53—42no—
293Dic53—81no—
294Dic53—81no—
295Dic53#47414yes—
296#2859—531no—
297C42×D4—531no—
298#1947—531no—
299#1947—531no—
300#1918—531no—
301#1918—531no—
302#1918—531no—
303#1918—531no—
304#1862—531no—
305#1862—531no—
306#1862—531no—
307#1862D5311yescharacteristic
308#1831—531no—
309#1802—531no—
310#1802—531no—
311C2×C53—44no—
312C2×C53#285111yescharacteristic
313C2×C53C2×#47411yescharacteristic
314C2×C53#308011yescharacteristic
315C2×C53#310011yescharacteristic
316C2×C53#310011yescharacteristic
317C2×C53#311611yescharacteristic
318C2×C53#311611yescharacteristic
319C2×#125—1061no—
320C2×C4×D4—531no—
321C2×C4×D4—531no—
322C2×C4×D4C2×D5311yescharacteristic
323C22×C42—1061no—
324C2×#107—531no—
325C2×#107—531no—
326C2×#107—531no—
327C2×#107C2×D5311yescharacteristic
328C2×C4⋊C8—531no—
329C2×C4⋊C8—531no—
330C2×#102—531no—
331C2×#102—531no—
332C2×#102C2×D5311yescharacteristic
333C2×#102C2×D5311yescharacteristic
334#389—531no—
335#389—531no—
336#389—531no—
337#389—1061no—
338#389—1061no—
339#389C2×D5311yescharacteristic
340#384—531no—
341#384—531no—
342#382—1061no—
343C4×C4⋊C4—531no—
344C4×C4⋊C4—531no—
345C4×C4⋊C4—531no—
346C4×C22⋊C4—1061no—
347C4×C22⋊C4—1061no—
348C4×C22⋊C4—1061no—
349C43—531no—
350#340—531no—
351#340—531no—
352#340—531no—
353#340—531no—
354#340—531no—
355#340—531no—
356#340—531no—
357#340C2×D5311yescharacteristic
358#336—531no—
359#336—531no—
360#336—531no—
361#336C2×D5311yescharacteristic
362C53#3306611yescharacteristic, complemented, Sylow
363C22×D4C22×D5311yescharacteristic
364C23×C4—21no—
365C23×C4—1061no—
366C23×C4—1061no—
367C22×C8—531no—
368C22×C8—531no—
369C22×C8C22×D5311yescharacteristic
370C22×C8C22×D5311yescharacteristic
371#125—2124no—
372C4×D4—22no—
373C4×D4—22no—
374C4×D4—534no—
375C4×D4—1062no—
376C4×D4—1062no—
377C4×D4—1062no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—531no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—531no—
383C2×C4⋊C4—1061no—
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⋊C4C22×D5311yescharacteristic
391C2×C4⋊C4C22×D5311yescharacteristic
392C2×C4⋊C4C22×D5311yescharacteristic
393C2×C22⋊C4—21no—
394C2×C22⋊C4—1061no—
395C2×C22⋊C4—1061no—
396C2×C42—531no—
397C2×C42—531no—
398C2×C42—531no—
399C2×C42—531no—
400C2×C42—531no—
401C2×C42—531no—
402C2×C42—1061no—
403C2×C42—1061no—
404C2×C42—1061no—
405C2×C42—2124no—
406C2×C42C22×D5311yescharacteristic
407#107—22no—
408#107—1062no—
409#107—1062no—
410#107—1062no—
411C4⋊C8—1062no—
412C4⋊C8—1062no—
413#102—22no—
414#102—22no—
415#102—1062no—
416#102—1062no—
417#95—21no—
418#95—1061no—
419#95—1061no—
420#95—1061no—
421#95—1061no—
422#95—1061no—
423#95—1061no—
424C24—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4#1171994512yes—
429C2×D4#1171994712yes—
430C2×D4Q8×D5312yes—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—531no—
438C22×C4—531no—
439C22×C4—531no—
440C22×C4—531no—
441C22×C4—1064no—
442C22×C4—1064no—
443C22×C4—1061no—
444C22×C4—1061no—
445C22×C4—1061no—
446C22×C4—1061no—
447C22×C4—2122no—
448C22×C4—2122no—
449C22×C4—2122no—
450C22×C4—2122no—
451C22×C4D4×D5311yescharacteristic
452C22×C4D4×D5311yescharacteristic
453C22×C4C23×D5311yescharacteristic
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—1061no—
461C2×C8—1061no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—42no—
472C4⋊C4—532no—
473C4⋊C4—532no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—2122no—
481C4⋊C4—2122no—
482C4⋊C4—2122no—
483C4⋊C4—2122no—
484C4⋊C4—2122no—
485C4⋊C4—2122no—
486C4⋊C4—2122no—
487C4⋊C4#1171994712yes—
488C22⋊C4—24no—
489C22⋊C4—1064no—
490C22⋊C4—1064no—
491C42—21no—
492C42—21no—
493C42—534no—
494C42—532no—
495C42—532no—
496C42—1062no—
497C42—1061no—
498C42—1061no—
499C42—2121no—
500C42—2121no—
501C42—2122no—
502C42—2122no—
503C42—2122no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23C2×D4×D5311yescharacteristic
512D4—22no—
513D4—22no—
514D4—22no—
515D4#1313320714yescomplemented
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—48no—
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—532no—
532C2×C4—532no—
533C2×C4—532no—
534C2×C4—532no—
535C2×C4—532no—
536C2×C4—1061no—
537C2×C4—1061no—
538C2×C4—1061no—
539C2×C4—1061no—
540C2×C4—1062no—
541C2×C4—1061no—
542C2×C4—1062no—
543C2×C4—1061no—
544C2×C4—1061no—
545C2×C4—2128no—
546C2×C4—2128no—
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×C4D8×D5312yes—
558C2×C4C2×Q8×D5311yescharacteristic
559C2×C4#41×D5311yescharacteristic
560C2×C4#41×D5311yescharacteristic
561C2×C4C2×#1171994711yescharacteristic
562C2×C4C2×#1171994711yescharacteristic
563C2×C4C2×#1171994511yescharacteristic
564C8—41no—
565C8—41no—
566C8—2121no—
567C8—2121no—
568C22—24no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22C22⋊Q8×D5311yescharacteristic
576C22#1681112911yescharacteristic
577C22#1681113211yescharacteristic
578C22C2×D8×D5311yescharacteristic
579C22#137×D5311yescharacteristic
580C22#1681141911yescharacteristic
581C22#1681142011yescharacteristic
582C4—22no—
583C4—42no—
584C4—41no—
585C4—81no—
586C4—81no—
587C4—534no—
588C4—1062no—
589C4—1062no—
590C4—4241no—
591C4—4241no—
592C4—2122no—
593C4—2122no—
594C4—2121no—
595C4C2×#1313320711yescharacteristic
596C4#1681115011yescharacteristic
597C4#1681115311yescharacteristic
598C4#1681117511yescharacteristic
599C2—44no—
600C2#2028302111yescharacteristic
601C2#474×D5311yescharacteristic
602C2#2028453211yescharacteristic
603C2#2028471011yescharacteristic
604C2#2028471311yescharacteristic
605C2#2028487011yescharacteristic
606C2#2028487311yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1862C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#1862D53
hypocenterC53#33066
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1862

Complements

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

NH ≅ G/Ntype
1. #20283074C2semidirect
3. #20275472C2semidirect
4. #20275470C2semidirect
5. #20275288C2semidirect
6. #20275287C2semidirect
7. #20275284C2semidirect
8. #20275283C2semidirect
12. #20274830C2semidirect
211. D4×C53#125semidirect
362. C53#33066semidirect
515. D4#13133207semidirect

Maximal subgroups

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

subgroupindex
#202830742
C4×D4×Dic532
#202754722
#202754702
#202752882
#202752872
#202752842
#202752832
#202749642
#202749622
#202749602
#202748302
#202747132
C53×#18622
#3306653