Finite Group Databasebeta

← #27465592

Subgroups — #27465592

Order 13568.

726 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13133196C211yesmaximal, characteristic
2#20277113C211yesmaximal, characteristic
3#20276148C211yesmaximal, characteristic
4#20276143C211yesmaximal, characteristic
5#20275919C211yesmaximal, characteristic
6#20275912C211yesmaximal, characteristic
7#20275485C211yesmaximal, characteristic, complemented
8#20275485C211yesmaximal, characteristic, complemented
9#20275357C211yesmaximal, characteristic, complemented
10#20275350C211yesmaximal, characteristic, complemented
11#20275294C211yesmaximal, characteristic, complemented
12#20275287C211yesmaximal, characteristic, complemented
13#20274731C211yesmaximal, characteristic, complemented
14#20274731C211yesmaximal, characteristic, complemented
15C53×#1797C211yesmaximal, characteristic
16C2×#13133196C2211yescharacteristic
17C2×C4×Dic106C2211yescharacteristic
18C2×#13133151C2211yescharacteristic
19C2×#13133151C2211yescharacteristic
20C2×#13133149C2211yescharacteristic
21C2×#13133149C2211yescharacteristic
22C2×#13133142C2211yescharacteristic
23C2×#13133142C2211yescharacteristic
24C2×#13133142C2211yescharacteristic
25C2×#13133142C2211yescharacteristic
26#16810529C2211yescharacteristic
27#16810521C2211yescharacteristic
28#16810507—21no—
29#16810496—21no—
30#16810491—21no—
31#16810489C2211yescharacteristic
32#16810489C2211yescharacteristic
33#16810486—21no—
34#16810483C2211yescharacteristic
35#16810483C2211yescharacteristic
36#16810483C2211yescharacteristic
37C4×#11719919—21no—
38C4×C4⋊Dic53C2211yescharacteristic
39C4×C4⋊Dic53C2211yescharacteristic
40C4×C4⋊Dic53C2211yescharacteristic
41C4×C22⋊Dic53—21no—
42C4×C22⋊Dic53—21no—
43#16810469—21no—
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52C22×C42×C53C2211yescharacteristic
53C2×C4⋊C8×C53C2211yescharacteristic
54C2×C4⋊C8×C53C2211yescharacteristic
55C2×C22⋊C8×C53C2211yescharacteristic
56C2×C22⋊C8×C53C2211yescharacteristic
57C53×#336C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60#13133196—22no—
61#13133196—22no—
62C4×Dic106—22no—
63C4×Dic106—22no—
64C2×C4⋊Dic53C2311yescharacteristic
65C2×C4⋊Dic53C2311yescharacteristic
66C2×C4⋊Dic53C2311yescharacteristic
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×#11719919—21no—
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×C22⋊Dic53—21no—
75C2×C22⋊Dic53—21no—
76C2×C22⋊Dic53—21no—
77C2×C22⋊Dic53—21no—
78C2×C4×Dic53—21no—
79C2×C4×Dic53—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53—21no—
82#13133151—22no—
83#13133151—41no—
84#13133149—22no—
85#13133149—41no—
86#13133142—22no—
87#13133142—22no—
88#13133142—41no—
89#13133142—41no—
90#13133124—21no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103C23×C4×C53D411yescharacteristic
104C23×C4×C53D411yescharacteristic
105C23×C4×C53C2311yescharacteristic
106C22×C8×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C2×C42×C53—22no—
111C2×C42×C53—22no—
112C2×C42×C53D411yescharacteristic
113C2×C42×C53D411yescharacteristic
114C2×C42×C53C2311yescharacteristic
115C2×C42×C53C2311yescharacteristic
116C4⋊C8×C53—22no—
117C4⋊C8×C53—22no—
118C22⋊C8×C53—41no—
119C22⋊C8×C53—41no—
120C24×C53C2×D411yescharacteristic
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
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—41no—
132C22×C4×C53—41no—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2411yescharacteristic
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C2×Dic106—21no—
148C2×Dic106—21no—
149C2×Dic106—21no—
150C2×Dic106—21no—
151C2×Dic106—21no—
152C2×Dic106#4112yes—
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—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C42×C53—22no—
166C42×C53—22no—
167C42×C53—21no—
168C42×C53—21no—
169C42×C53#4112yes—
170C42×C53#4112yes—
171C22⋊Dic53—24no—
172C22⋊Dic53—42no—
173C22⋊Dic53—42no—
174C22⋊Dic53—42no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
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#4112yes—
189#11719919—24no—
190#11719919—42no—
191#11719919—42no—
192C4×Dic53—42no—
193C4×Dic53—42no—
194C4×Dic53—42no—
195C4×Dic53—42no—
196C23×C53—21no—
197C23×C53—21no—
198C23×C53—21no—
199C23×C53—21no—
200C23×C53—21no—
201C23×C53—21no—
202C23×C53#13712yes—
203C23×C53C22×D411yescharacteristic
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
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—44no—
217C2×C4×C53—44no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53#12312yes—
223C2×C4×C53#12312yes—
224C2×C4×C53#13512yes—
225C2×C4×C53C2×#4111yescharacteristic
226C2×C4×C53C2×#4111yescharacteristic
227C2×C4×C53C2×#4111yescharacteristic
228C2×C4×C53C2×#4111yescharacteristic
229C2×C4×C53H(5,𝔽2)11yescharacteristic
230C2×C4×C532−1+411yescharacteristic
231C2×Dic53—22no—
232C2×Dic53—22no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
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—
255Dic106—22no—
256Dic106—22no—
257Dic106—41no—
258Dic106—41no—
259C8×C53—41no—
260C8×C53—41no—
261C8×C53—41no—
262C8×C53—41no—
263#27635—531nomaximal, Sylow
264C22×C53—22no—
265C22×C53—22no—
266C22×C53—22no—
267C22×C53—41no—
268C22×C53—41no—
269C22×C53—41no—
270C22×C53—41no—
271C22×C53C2×#12311yescharacteristic
272C22×C53#52511yescharacteristic
273C22×C53#53711yescharacteristic
274C22×C53C2×#13511yescharacteristic
275C22×C53C2×#13711yescharacteristic
276C22×C53#57511yescharacteristic
277C22×C53#57511yescharacteristic
278C4×C53—22no—
279C4×C53—22no—
280C4×C53—22no—
281C4×C53—41no—
282C4×C53—41no—
283C4×C53#55211yescharacteristic
284C4×C53#55311yescharacteristic
285C4×C53#55511yescharacteristic
286C4×C53#55611yescharacteristic
287Dic53—42no—
288Dic53—42no—
289Dic53—81no—
290Dic53—81no—
291Dic53—81no—
292Dic53—81no—
293Dic53—81no—
294Dic53—81no—
295C4×C22⋊Q8—531no—
296#2283—531no—
297#2077—531no—
298#2076—531no—
299#2036—531no—
300#2035—531no—
301#1949—531no—
302#1949—531no—
303#1928—531no—
304#1927—531no—
305#1919—531no—
306#1918—531no—
307#1805—531no—
308#1805—531no—
309#1797D5311yescharacteristic
310C2×C53—42no—
311C2×C53#248511yescharacteristic
312C2×C53#309111yescharacteristic
313C2×C53#309511yescharacteristic
314C2×C53#310211yescharacteristic
315C2×C53#311711yescharacteristic
316C2×C53#324811yescharacteristic
317C2×C53#325611yescharacteristic
318C2×C22⋊Q8—531no—
319C2×C4×Q8—531no—
320C22×C42C2×D5311yescharacteristic
321C2×#107—531no—
322C2×#107—531no—
323C2×#106—531no—
324C2×#106—531no—
325C2×C4⋊C8C2×D5311yescharacteristic
326C2×C4⋊C8C2×D5311yescharacteristic
327C2×#103—531no—
328C2×#103—531no—
329C2×#103—531no—
330C2×#103—531no—
331C2×C22⋊C8C2×D5311yescharacteristic
332C2×C22⋊C8C2×D5311yescharacteristic
333#391—531no—
334#389—531no—
335#387—1061no—
336#385—1061no—
337#383—1061no—
338#382—531no—
339#382—531no—
340#381—1061no—
341#380—531no—
342#380—531no—
343#380—531no—
344C4×C4⋊C4—531no—
345C4×C4⋊C4—531no—
346C4×C4⋊C4—531no—
347C4×C4⋊C4—1061no—
348C4×C22⋊C4—1061no—
349C4×C22⋊C4—1061no—
350#376—1061no—
351#340—531no—
352#340—531no—
353#340—531no—
354#340—531no—
355#340—531no—
356#340—531no—
357#340—531no—
358#340—531no—
359#336C2×D5311yescharacteristic
360#336C2×D5311yescharacteristic
361C53#2763511yescharacteristic, complemented, Sylow
362C22×Q8—531no—
363C23×C4D21211yescharacteristic
364C23×C4D21211yescharacteristic
365C23×C4C22×D5311yescharacteristic
366C22×C8C22×D5311yescharacteristic
367C22×C8C22×D5311yescharacteristic
368C22×C8C22×D5311yescharacteristic
369C22×C8C22×D5311yescharacteristic
370C22⋊Q8—1062no—
371C22⋊Q8—1062no—
372C4×Q8—1062no—
373C4×Q8—1062no—
374C2×C4⋊C4—531no—
375C2×C4⋊C4—531no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—531no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—531no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4—1061no—
384C2×C22⋊C4—1061no—
385C2×C22⋊C4—1061no—
386C2×C22⋊C4—1061no—
387C2×C22⋊C4—1061no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—1061no—
391C2×C42—1061no—
392C2×C42—1061no—
393C2×C42—1061no—
394C2×C42D21211yescharacteristic
395C2×C42D21211yescharacteristic
396C2×C42C22×D5311yescharacteristic
397C2×C42C22×D5311yescharacteristic
398#107—1062no—
399#107—2121no—
400#106—1062no—
401#106—2121no—
402C4⋊C8—22no—
403C4⋊C8—22no—
404#103—1062no—
405#103—1062no—
406#103—2121no—
407#103—2121no—
408C22⋊C8—41no—
409C22⋊C8—41no—
410#95—1061no—
411#95—1061no—
412#95—1061no—
413#95—1061no—
414#95—1061no—
415#95—1061no—
416#95—1061no—
417#95—1061no—
418#95—1061no—
419#95—1061no—
420#95—1061no—
421#95—1061no—
422#95—1061no—
423C24C2×D21211yescharacteristic
424C2×Q8—532no—
425C2×Q8—1061no—
426C2×Q8—1061no—
427C2×Q8—1061no—
428C2×Q8—1061no—
429C2×Q8—1061no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—41no—
441C22×C4—41no—
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×C4C23×D5311yescharacteristic
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
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—
468C4⋊C4—532no—
469C4⋊C4—1064no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4—2122no—
484C4⋊C4—2122no—
485C22⋊C4—1064no—
486C22⋊C4—2122no—
487C22⋊C4—2122no—
488C22⋊C4—2122no—
489C42—22no—
490C42—22no—
491C42—21no—
492C42—21no—
493C42—2122no—
494C42—2122no—
495C42—2122no—
496C42—2122no—
497C42#1171994712yes—
498C42#1171994712yes—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23#1313325012yes—
506C23C22×D21211yescharacteristic
507Q8—1062no—
508Q8—1062no—
509Q8—2121no—
510Q8—2121no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
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—44no—
524C2×C4—44no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—1062no—
530C2×C4—1062no—
531C2×C4—2121no—
532C2×C4—2121no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4—2121no—
536C2×C4—2121no—
537C2×C4—2121no—
538C2×C4—2121no—
539C2×C4—2121no—
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×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4#1313320112yes—
554C2×C4#1313320112yes—
555C2×C4#1313323912yes—
556C2×C4#41×D5311yescharacteristic
557C2×C4#41×D5311yescharacteristic
558C2×C4C2×#1171994711yescharacteristic
559C2×C4C2×#1171994711yescharacteristic
560C2×C4#1313326211yescharacteristic
561C2×C4#1313326511yescharacteristic
562C8—41no—
563C8—41no—
564C8—41no—
565C8—41no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22C2×#1313320111yescharacteristic
574C22#1681113511yescharacteristic
575C22#1681120411yescharacteristic
576C22C2×#1313323911yescharacteristic
577C22C2×#1313325011yescharacteristic
578C22#1681142311yescharacteristic
579C22#1681142311yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—41no—
584C4—41no—
585C4—4241no—
586C4—4241no—
587C4—4241no—
588C4—4241no—
589C4—4241no—
590C4—4241no—
591C4—2122no—
592C4—2122no—
593C4#1681130711yescharacteristic
594C4#1681132011yescharacteristic
595C4#1681133711yescharacteristic
596C4#1681134411yescharacteristic
597C2—42no—
598C2#2027901311yescharacteristic
599C2#2028463011yescharacteristic
600C2#2028466411yescharacteristic
601C2#2028473511yescharacteristic
602C2#2028488811yescharacteristic
603C2#2028605311yescharacteristic
604C2#2028611611yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1797C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1797D53
hypocenterC53#27635
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
7. #20275485C2semidirect
8. #20275485C2semidirect
9. #20275357C2semidirect
10. #20275350C2semidirect
11. #20275294C2semidirect
12. #20275287C2semidirect
13. #20274731C2semidirect
14. #20274731C2semidirect
361. C53#27635semidirect

Maximal subgroups

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

subgroupindex
C4×#131331962
#202771132
#202761482
#202761432
#202759192
#202759122
#202754852
#202753572
#202753502
#202752942
#202752872
#202747312
C53×#17972
#2763553