Finite Group Databasebeta

← #27456852

Subgroups — #27456852

Order 13568.

700 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279115C211yesmaximal, characteristic
2#20277718C211yesmaximal, characteristic
3#20276135C211yesmaximal, characteristic, complemented
4#20276095C211yesmaximal, characteristic, complemented
5#20275919C211yesmaximal, characteristic
6#20275912C211yesmaximal, characteristic
7#20275485C211yesmaximal, characteristic, complemented
8#20275443C211yesmaximal, characteristic, complemented
9#20275308C211yesmaximal, characteristic, complemented
10#20275301C211yesmaximal, characteristic, complemented
11#20274986C211yesmaximal, characteristic, complemented
12#20274910C211yesmaximal, characteristic
13#20274802C211yesmaximal, characteristic
14C4×#13133142C211yesmaximal, characteristic, complemented
15C53×#1923C211yesmaximal, characteristic
16C2×#13133196C2211yescharacteristic
17C2×C4×Dic106C2211yescharacteristic
18C22×C4⋊Dic53C2211yescharacteristic
19C2×#13133151C2211yescharacteristic
20C2×#13133149C2211yescharacteristic
21C2×#13133149C2211yescharacteristic
22C2×#13133149C2211yescharacteristic
23C2×#13133142C2211yescharacteristic
24C2×#13133142C2211yescharacteristic
25C2×#13133142C2211yescharacteristic
26C2×#13133142C2211yescharacteristic
27C53⋊#401—21no—
28#16810551—21no—
29#16810529C2211yescharacteristic
30#16810521C2211yescharacteristic
31#16810521C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810520—21no—
34#16810517—21no—
35#16810510—21no—
36#16810505—21no—
37#16810495—21no—
38#16810489C2211yescharacteristic
39#16810483C2211yescharacteristic
40#16810483C2211yescharacteristic
41C4×C4⋊Dic53C2211yescharacteristic
42C4×C4⋊Dic53C2211yescharacteristic
43C22⋊C4×Dic53—21no—
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52C2×C4⋊C8×C53C2211yescharacteristic
53C2×C22⋊C8×C53C2211yescharacteristic
54C2×C22⋊C8×C53C2211yescharacteristic
55C2×C4×C8×C53C2211yescharacteristic
56C4×C22⋊C4×C53C2211yescharacteristic
57C53×#336C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C23×Dic53—21no—
61#13133196—22no—
62#13133196—22no—
63C4×Dic106—22no—
64C4×Dic106—41no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53—22no—
67C2×C4⋊Dic53—41no—
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86#13133151—22no—
87#13133149—22no—
88#13133149—22no—
89#13133149—41no—
90#13133142—22no—
91#13133142—22no—
92#13133142—22no—
93#13133142—41no—
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—
103#13133124—21no—
104#13133124—21no—
105C23×C4×C53C2311yescharacteristic
106C22×C8×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C2×C22⋊C4×C53D411yescharacteristic
111C2×C22⋊C4×C53D411yescharacteristic
112C2×C42×C53C2311yescharacteristic
113C2×C42×C53C2311yescharacteristic
114C4⋊C8×C53—22no—
115C22⋊C8×C53—22no—
116C22⋊C8×C53—41no—
117C4×C8×C53—41no—
118#95×C53D411yescharacteristic
119#95×C53D411yescharacteristic
120C24×C53C2×D411yescharacteristic
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2411yescharacteristic
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—42no—
142C22×Dic53—42no—
143C22×Dic53—81no—
144C2×Dic106—21no—
145C2×Dic106—21no—
146C2×Dic106—21no—
147C2×Dic106—21no—
148C2×Dic106—21no—
149C2×Dic106#4112yes—
150C2×C8×C53—21no—
151C2×C8×C53—21no—
152C2×C8×C53—21no—
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#4112yes—
162C22⋊C4×C53—22no—
163C22⋊C4×C53—22no—
164C42×C53—21no—
165C42×C53—21no—
166C42×C53—21no—
167C42×C53—21no—
168C22⋊Dic53—24no—
169C22⋊Dic53—42no—
170C22⋊Dic53—81no—
171C4⋊Dic53—21no—
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—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—42no—
184C4⋊Dic53#4112yes—
185C4⋊Dic53#4112yes—
186#11719919—24no—
187#11719919—42no—
188#11719919—42no—
189#11719919—81no—
190#11719919—81no—
191C4×Dic53—42no—
192C4×Dic53—81no—
193C4×Dic53—81no—
194C23×C53—21no—
195C23×C53—21no—
196C23×C53—21no—
197C23×C53—21no—
198C23×C53—21no—
199C23×C53—21no—
200C23×C53#13512yes—
201C23×C53C22×D411yescharacteristic
202C2×C4×C53—21no—
203C2×C4×C53—21no—
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—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53#13712yes—
221C2×C4×C53C2×#4111yescharacteristic
222C2×C4×C53C2×#4111yescharacteristic
223C2×C4×C53C2×#4111yescharacteristic
224C2×C4×C53C2×#4111yescharacteristic
225C2×C4×C53H(5,𝔽2)11yescharacteristic
226C2×C4×C532−1+411yescharacteristic
227C2×Dic53—22no—
228C2×Dic53—22no—
229C2×Dic53—22no—
230C2×Dic53—41no—
231C2×Dic53—41no—
232C2×Dic53—41no—
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—84no—
252Dic106—22no—
253Dic106—22no—
254Dic106—41no—
255Dic106—41no—
256C8×C53—22no—
257C8×C53—41no—
258C8×C53—41no—
259C8×C53—41no—
260#27037—531nomaximal, Sylow
261C22×C53—22no—
262C22×C53—22no—
263C22×C53—22no—
264C22×C53—41no—
265C22×C53—41no—
266C22×C53—41no—
267C22×C53—41no—
268C22×C53#52511yescharacteristic
269C22×C53#54611yescharacteristic
270C22×C53#54711yescharacteristic
271C22×C53C2×#13511yescharacteristic
272C22×C53C2×#13711yescharacteristic
273C22×C53#57711yescharacteristic
274C22×C53#57711yescharacteristic
275C4×C53—22no—
276C4×C53—41no—
277C4×C53—41no—
278C4×C53—41no—
279C4×C53—41no—
280C4×C53#52911yescharacteristic
281C4×C53#53911yescharacteristic
282C4×C53#55311yescharacteristic
283C4×C53#55611yescharacteristic
284Dic53—42no—
285Dic53—42no—
286Dic53—42no—
287Dic53—81no—
288Dic53—81no—
289Dic53—81no—
290Dic53—81no—
291Dic53—81no—
292#2494—531no—
293#2360—531no—
294#2074—531no—
295#2066—531no—
296#2036—531no—
297#2035—531no—
298#1949—531no—
299#1941—531no—
300#1923D5311yescharacteristic
301#1921—531no—
302#1920—531no—
303#1865—531no—
304#1853—531no—
305#1823—531no—
306C4×#103—531no—
307C2×C53—42no—
308C2×C53#246611yescharacteristic
309C2×C53#310811yescharacteristic
310C2×C53#312511yescharacteristic
311C2×C53#329611yescharacteristic
312C2×C53#332011yescharacteristic
313C2×C53#333711yescharacteristic
314C2×C53#335611yescharacteristic
315C2×C22⋊Q8—531no—
316C2×C4×Q8—531no—
317C22×C4⋊C4—531no—
318C2×#107—531no—
319C2×#106—531no—
320C2×#106—531no—
321C2×#106—531no—
322C2×C4⋊C8C2×D5311yescharacteristic
323C2×#103—531no—
324C2×#103—531no—
325C2×#103—531no—
326C2×#103—531no—
327C2×C22⋊C8C2×D5311yescharacteristic
328C2×C22⋊C8C2×D5311yescharacteristic
329C2×C4×C8C2×D5311yescharacteristic
330#401—1061no—
331#397—1061no—
332#391—531no—
333#389—531no—
334#389—531no—
335#389—531no—
336#389—1061no—
337#388—1061no—
338#387—1061no—
339#387—1061no—
340#385—1061no—
341#382—531no—
342#380—531no—
343#380—531no—
344C4×C4⋊C4—531no—
345C4×C4⋊C4—531no—
346C4×C22⋊C4—1061no—
347C4×C22⋊C4C2×D5311yescharacteristic
348#340—531no—
349#340—531no—
350#340—531no—
351#340—531no—
352#340—531no—
353#340—531no—
354#340—531no—
355#340—531no—
356#336C2×D5311yescharacteristic
357#336C2×D5311yescharacteristic
358C53#2703711yescharacteristic, complemented, Sylow
359C22×Q8—531no—
360C23×C4—1061no—
361C23×C4C22×D5311yescharacteristic
362C22×C8C22×D5311yescharacteristic
363C22×C8C22×D5311yescharacteristic
364C22×C8C22×D5311yescharacteristic
365C22×C8C22×D5311yescharacteristic
366C22⋊Q8—1062no—
367C22⋊Q8—1062no—
368C4×Q8—1062no—
369C4×Q8—2121no—
370C2×C4⋊C4—531no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—531no—
373C2×C4⋊C4—531no—
374C2×C4⋊C4—531no—
375C2×C4⋊C4—531no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—1062no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4—2121no—
385C2×C22⋊C4—1061no—
386C2×C22⋊C4—1061no—
387C2×C22⋊C4—1061no—
388C2×C22⋊C4D21211yescharacteristic
389C2×C22⋊C4D21211yescharacteristic
390C2×C42—1061no—
391C2×C42—1061no—
392C2×C42—1061no—
393C2×C42C22×D5311yescharacteristic
394C2×C42C22×D5311yescharacteristic
395#107—1062no—
396#106—1062no—
397#106—1062no—
398#106—2121no—
399C4⋊C8—22no—
400#103—1062no—
401#103—1062no—
402#103—1062no—
403#103—2121no—
404C22⋊C8—22no—
405C22⋊C8—41no—
406C4×C8—41no—
407#95—1061no—
408#95—1061no—
409#95—1061no—
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#95D21211yescharacteristic
419#95D21211yescharacteristic
420C24C2×D21211yescharacteristic
421C2×Q8—532no—
422C2×Q8—1061no—
423C2×Q8—1061no—
424C2×Q8—1061no—
425C2×Q8—1061no—
426C2×Q8—1061no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—1061no—
434C22×C4—1061no—
435C22×C4—1061no—
436C22×C4—1061no—
437C22×C4—1061no—
438C22×C4—1061no—
439C22×C4—1061no—
440C22×C4—1061no—
441C22×C4—4241no—
442C22×C4—2122no—
443C22×C4—2122no—
444C22×C4C2×D21211yescharacteristic
445C22×C4C2×D21211yescharacteristic
446C22×C4C2×D21211yescharacteristic
447C22×C4C2×D21211yescharacteristic
448C22×C4C2×D21211yescharacteristic
449C22×C4C23×D5311yescharacteristic
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8#1171994712yes—
462C4⋊C4—532no—
463C4⋊C4—532no—
464C4⋊C4—1064no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—4241no—
478C4⋊C4—4241no—
479C4⋊C4—2122no—
480C4⋊C4—2122no—
481C4⋊C4—2122no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—1064no—
485C22⋊C4—4241no—
486C22⋊C4—2122no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—4241no—
492C42—4241no—
493C42—2122no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23#1313323912yes—
501C23C22×D21211yescharacteristic
502Q8—1062no—
503Q8—1062no—
504Q8—2121no—
505Q8—2121no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
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—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—1062no—
525C2×C4—1062no—
526C2×C4—1062no—
527C2×C4—4244no—
528C2×C4—2121no—
529C2×C4—2121no—
530C2×C4—2121no—
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#1313325012yes—
550C2×C4#41×D5311yescharacteristic
551C2×C4#41×D5311yescharacteristic
552C2×C4#41×D5311yescharacteristic
553C2×C4C2×#1171994711yescharacteristic
554C2×C4#1313326211yescharacteristic
555C2×C4#1313326411yescharacteristic
556C8—22no—
557C8—41no—
558C8—41no—
559C8—41no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22#1681113511yescharacteristic
568C22#1681126811yescharacteristic
569C22#1681127711yescharacteristic
570C22C2×#1313323911yescharacteristic
571C22C2×#1313325011yescharacteristic
572C22#1681143511yescharacteristic
573C22#1681143511yescharacteristic
574C4—22no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—4241no—
580C4—4241no—
581C4—4241no—
582C4—4241no—
583C4—4241no—
584C4—2122no—
585C4—2122no—
586C4—2122no—
587C4#1681115211yescharacteristic
588C4#1681121611yescharacteristic
589C4#1681131611yescharacteristic
590C4#1681134011yescharacteristic
591C2—42no—
592C2#2027874911yescharacteristic
593C2#2028479711yescharacteristic
594C2#2028499411yescharacteristic
595C2#2028655211yescharacteristic
596C2#2028688811yescharacteristic
597C2#2028709611yescharacteristic
598C2#2028733111yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1923C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1923D53
hypocenterC53#27037
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1923

Complements

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

NH ≅ G/Ntype
3. #20276135C2semidirect
4. #20276095C2semidirect
7. #20275485C2semidirect
8. #20275443C2semidirect
9. #20275308C2semidirect
10. #20275301C2semidirect
11. #20274986C2semidirect
14. C4×#13133142C2semidirect
358. C53#27037semidirect

Maximal subgroups

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

subgroupindex
#202791152
#202777182
#202761352
#202760952
#202759192
#202759122
#202754852
#202754432
#202753082
#202753012
#202749862
#202749102
#202748022
C4×#131331422
C53×#19232
#2703753