Finite Group Databasebeta

← #27456802

Subgroups — #27456802

Order 13568.

728 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
1#20279139C211yesmaximal, characteristic
2#20277724C211yesmaximal, characteristic
3#20276126C211yesmaximal, characteristic, complemented
4#20276092C211yesmaximal, characteristic, complemented
5#20275908C211yesmaximal, characteristic
6#20275475C211yesmaximal, characteristic, complemented
7#20275442C211yesmaximal, characteristic, complemented
8#20275318C211yesmaximal, characteristic
9#20275302C211yesmaximal, characteristic, complemented
10#20275296C211yesmaximal, characteristic, complemented
11#20274981C211yesmaximal, characteristic, complemented
12#20274907C211yesmaximal, characteristic
13#20274806C211yesmaximal, characteristic
14#103×Dic53C211yesmaximal, characteristic, complemented
15C53×#2035C211yesmaximal, characteristic
16C2×#13133210—21no—
17C2×Q8×Dic53C2211yescharacteristic
18C22×C4⋊Dic53C2211yescharacteristic
19C2×#13133148C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133146C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×C22⋊#88322C2211yescharacteristic
26C2×C8×Dic53C2211yescharacteristic
27#16810565—21no—
28#16810551—21no—
29#16810530C2211yescharacteristic
30#16810522—21no—
31#16810522C2211yescharacteristic
32#16810522C2211yescharacteristic
33#16810519C2211yescharacteristic
34#16810509—21no—
35#16810505—21no—
36#16810498—21no—
37#16810492C2211yescharacteristic
38#16810483C2211yescharacteristic
39C4⋊C4×Dic53C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41C22⋊C4×Dic53—21no—
42C4×C22⋊Dic53C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×C22⋊Q8×C53C2211yescharacteristic
53C2×#106×C53C2211yescharacteristic
54C2×#103×C53C2211yescharacteristic
55C2×#103×C53C2211yescharacteristic
56C2×C22⋊C8×C53C2211yescharacteristic
57C53×#380C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C23×Dic53—21no—
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62#13133210—44no—
63Q8×Dic53—22no—
64Q8×Dic53—41no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—22no—
70C2×C4⋊Dic53—41no—
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53D411yescharacteristic
80C2×C22⋊Dic53D411yescharacteristic
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53C2311yescharacteristic
85C2×C4×Dic53C2311yescharacteristic
86#13133148—22no—
87#13133148—22no—
88#13133148—41no—
89#13133146—22no—
90#13133141—22no—
91#13133141—22no—
92C22⋊#88322—22no—
93C8×Dic53—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#13133124D411yescharacteristic
103#13133124D411yescharacteristic
104C22×Q8×C53C2311yescharacteristic
105C23×C4×C53C2311yescharacteristic
106C22×C8×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108C22⋊Q8×C53—22no—
109C22⋊Q8×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×C22⋊C4×C53—21no—
116#106×C53—22no—
117#103×C53—22no—
118#103×C53—41no—
119C22⋊C8×C53—41no—
120#95×C53—21no—
121C24×C53C2×D411yescharacteristic
122C2×Q8×C53—21no—
123C2×Q8×C53—21no—
124C2×Q8×C53—21no—
125C2×Q8×C53—21no—
126C2×Q8×C53—21no—
127C2×Q8×C53#4112yes—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2411yescharacteristic
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—42no—
145C22×Dic53—42no—
146C22×Dic53—81no—
147C22×Dic53C2×D411yescharacteristic
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53C2×D411yescharacteristic
150C22×Dic53C2×D411yescharacteristic
151C2×C8×C53—21no—
152C2×C8×C53—21no—
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C4⋊C4×C53—24no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53#4112yes—
164C22⋊C4×C53—24no—
165C22⋊C4×C53—42no—
166C22⋊Dic53—41no—
167C22⋊Dic53—42no—
168C22⋊Dic53SD1614yes—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—42no—
177C4⋊Dic53—42no—
178C4⋊Dic53—42no—
179C4⋊Dic53—81no—
180C4⋊Dic53#4112yes—
181#11719919—42no—
182#11719919—42no—
183#11719919—42no—
184C4×Dic53—21no—
185C4×Dic53—21no—
186C4×Dic53—21no—
187C4×Dic53—21no—
188C4×Dic53—42no—
189C4×Dic53—42no—
190C4×Dic53—81no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322#4112yes—
197C23×C53—21no—
198C23×C53—21no—
199C23×C53—21no—
200C23×C53—21no—
201C23×C53—21no—
202C23×C53—21no—
203C23×C53C2×SD1612yes—
204C23×C53C22×D411yescharacteristic
205Q8×C53—22no—
206Q8×C53—22no—
207Q8×C53—41no—
208Q8×C53—41no—
209C2×C4×C53—21no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
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×C53C2×#4111yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C2×C4×C53C2×#4111yescharacteristic
231C2×C4×C53C2×#4111yescharacteristic
232C2×C4×C53H(5,𝔽2)11yescharacteristic
233C2×C4×C532−1+411yescharacteristic
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—22no—
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
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—84no—
257C2×Dic53C2×SD1612yes—
258C2×Dic53C2×SD1612yes—
259C8×C53—41no—
260C8×C53—41no—
261#88322—22no—
262#88322—41no—
263#27034—531nomaximal, Sylow
264C22×C53—22no—
265C22×C53—22no—
266C22×C53—22no—
267C22×C53—41no—
268C22×C53—41no—
269C22×C53—41no—
270C22×C53—41no—
271C22×C53#52511yescharacteristic
272C22×C53#54611yescharacteristic
273C22×C53#54711yescharacteristic
274C22×C53C22×SD1611yescharacteristic
275C22×C53#57511yescharacteristic
276C22×C53#57711yescharacteristic
277C22×C53#57811yescharacteristic
278C4×C53—42no—
279C4×C53—41no—
280C4×C53—42no—
281C4×C53—81no—
282C4×C53—81no—
283C4×C53#52911yescharacteristic
284C4×C53#53911yescharacteristic
285C4×C53#55311yescharacteristic
286C4×C53#55611yescharacteristic
287Dic53—22no—
288Dic53—22no—
289Dic53—41no—
290Dic53—41no—
291Dic53—42no—
292Dic53—81no—
293Dic53—81no—
294Dic53—81no—
295#2496—531no—
296#2360—531no—
297#2072—531no—
298#2066—531no—
299#2035—531no—
300#2035D5311yescharacteristic
301#1948—531no—
302#1941—531no—
303#1923—531no—
304#1920—531no—
305#1919—531no—
306#1865—531no—
307#1853—531no—
308#1824—531no—
309C4×#103—531no—
310C2×C53—42no—
311C2×C53#246611yescharacteristic
312C2×C53#310311yescharacteristic
313C2×C53#312011yescharacteristic
314C2×C53#329811yescharacteristic
315C2×C53#331111yescharacteristic
316C2×C53#333511yescharacteristic
317C2×C53#335211yescharacteristic
318C2×#126—1061no—
319C2×C22⋊Q8C2×D5311yescharacteristic
320C2×C4×Q8—531no—
321C22×C4⋊C4—531no—
322C2×#106—531no—
323C2×#106—531no—
324C2×#106—531no—
325C2×#106C2×D5311yescharacteristic
326C2×C4⋊C8—531no—
327C2×#103—531no—
328C2×#103—531no—
329C2×#103C2×D5311yescharacteristic
330C2×#103C2×D5311yescharacteristic
331C2×C22⋊C8—531no—
332C2×C22⋊C8C2×D5311yescharacteristic
333C2×C4×C8—531no—
334#400—1061no—
335#397—1061no—
336#391—531no—
337#389—531no—
338#389—531no—
339#389—531no—
340#389—1061no—
341#387—1061no—
342#387—1061no—
343#385—1061no—
344#384—531no—
345#380—531no—
346#380C2×D5311yescharacteristic
347C4×C4⋊C4—531no—
348C4×C4⋊C4—531no—
349C4×C22⋊C4—531no—
350C4×C22⋊C4—1061no—
351#340—531no—
352#340—531no—
353#340—531no—
354#340—531no—
355#340—531no—
356#340—531no—
357#340—531no—
358#340C2×D5311yescharacteristic
359#336—531no—
360#336—531no—
361C53#2703411yescharacteristic, complemented, Sylow
362C22×Q8C22×D5311yescharacteristic
363C23×C4—1061no—
364C23×C4C22×D5311yescharacteristic
365C22×C8—531no—
366C22×C8—531no—
367C22×C8C22×D5311yescharacteristic
368C22×C8C22×D5311yescharacteristic
369#126—2124no—
370C22⋊Q8—22no—
371C22⋊Q8—22no—
372C4×Q8—1062no—
373C4×Q8—2121no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—531no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—531no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—1062no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4—2121no—
388C2×C4⋊C4C22×D5311yescharacteristic
389C2×C4⋊C4C22×D5311yescharacteristic
390C2×C4⋊C4C22×D5311yescharacteristic
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—531no—
394C2×C22⋊C4—531no—
395C2×C22⋊C4—1061no—
396C2×C42—531no—
397C2×C42—531no—
398C2×C42—1061no—
399C2×C42—1061no—
400C2×C42—1061no—
401#106—22no—
402#106—1062no—
403#106—1062no—
404#106—2121no—
405C4⋊C8—1062no—
406#103—22no—
407#103—41no—
408#103—1062no—
409#103—1062no—
410C22⋊C8—41no—
411C22⋊C8—1062no—
412C4×C8—2121no—
413#95—21no—
414#95—531no—
415#95—531no—
416#95—1061no—
417#95—1061no—
418#95—1061no—
419#95—1061no—
420#95—1061no—
421#95—1061no—
422#95—1061no—
423#95—1061no—
424C24D4×D5311yescharacteristic
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8#1171994712yes—
431C22×C4—21no—
432C22×C4—21no—
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—1061no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—4241no—
450C22×C4—2122no—
451C22×C4—2122no—
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—532no—
461C2×C8—1061no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C4⋊C4—24no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—532no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—4241no—
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—42no—
490C22⋊C4—534no—
491C22⋊C4—2121no—
492C22⋊C4—2122no—
493C42—1061no—
494C42—1061no—
495C42—1061no—
496C42—1061no—
497C42—4241no—
498C42—2122no—
499C42—2122no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23SD16×D5312yes—
507C23C2×D4×D5311yescharacteristic
508Q8—22no—
509Q8—22no—
510Q8—41no—
511Q8—41no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
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—1061no—
534C2×C4—1061no—
535C2×C4—1061no—
536C2×C4—1061no—
537C2×C4—1061no—
538C2×C4—1061no—
539C2×C4—1061no—
540C2×C4—1061no—
541C2×C4—1061no—
542C2×C4—1061no—
543C2×C4—1062no—
544C2×C4—4244no—
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—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4#41×D5311yescharacteristic
557C2×C4#41×D5311yescharacteristic
558C2×C4C2×#1171994711yescharacteristic
559C2×C4C2×#1171994711yescharacteristic
560C2×C4#1313326211yescharacteristic
561C2×C4#1313326411yescharacteristic
562C8—41no—
563C8—41no—
564C8—1062no—
565C8—2121no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22#1681112911yescharacteristic
574C22#1681126211yescharacteristic
575C22#1681127111yescharacteristic
576C22C2×SD16×D5311yescharacteristic
577C22#1681142211yescharacteristic
578C22#1681143211yescharacteristic
579C22#1681143711yescharacteristic
580C4—42no—
581C4—41no—
582C4—42no—
583C4—81no—
584C4—81no—
585C4—1062no—
586C4—1062no—
587C4—4241no—
588C4—4241no—
589C4—4241no—
590C4—2121no—
591C4—2122no—
592C4—2121no—
593C4#1681115011yescharacteristic
594C4#1681121811yescharacteristic
595C4#1681132011yescharacteristic
596C4#1681134211yescharacteristic
597C2—42no—
598C2#2027874411yescharacteristic
599C2#2028474111yescharacteristic
600C2#2028493011yescharacteristic
601C2#2028657511yescharacteristic
602C2#2028674611yescharacteristic
603C2#2028706511yescharacteristic
604C2#2028728311yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2035C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2035D53
hypocenterC53#27034
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2035

Complements

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

NH ≅ G/Ntype
3. #20276126C2semidirect
4. #20276092C2semidirect
6. #20275475C2semidirect
7. #20275442C2semidirect
9. #20275302C2semidirect
10. #20275296C2semidirect
11. #20274981C2semidirect
14. #103×Dic53C2semidirect
361. C53#27034semidirect

Maximal subgroups

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

subgroupindex
#202791392
#202777242
#202761262
#202760922
#202759082
#202754752
#202754422
#202753182
#202753022
#202752962
#202749812
#202749072
#202748062
#103×Dic532
C53×#20352
#2703453