Finite Group Databasebeta

← #27468222

Subgroups — #27468222

Order 13568.

634 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279128C211yesmaximal, characteristic
2#20279111C211yesmaximal, characteristic
3#20276286C211yesmaximal, characteristic, complemented
4#20276285C211yesmaximal, characteristic, complemented
5#20276143C211yesmaximal, characteristic
6#20275442C211yesmaximal, characteristic, complemented
7#20275442C211yesmaximal, characteristic, complemented
8#20275306C211yesmaximal, characteristic, complemented
9#20275288C211yesmaximal, characteristic, complemented
10#20274878C211yesmaximal, characteristic, complemented
11#20274738C211yesmaximal, characteristic
12#20274738C211yesmaximal, characteristic
13#20274603C211yesmaximal, characteristic, complemented
14#20274489C211yesmaximal, characteristic
15C53×#2075C211yesmaximal, characteristic
16C23×#88322C2211yescharacteristic
17C2×#13133151C2211yescharacteristic
18C2×#13133148C2211yescharacteristic
19C2×#13133148C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×C8×Dic53C2211yescharacteristic
23C2×C8×Dic53C2211yescharacteristic
24C53⋊#401—21no—
25#16810563—21no—
26#16810551—21no—
27#16810551—21no—
28#16810522C2211yescharacteristic
29#16810522C2211yescharacteristic
30#16810519C2211yescharacteristic
31#16810519C2211yescharacteristic
32#16810516—21no—
33#16810516—21no—
34#16810508—21no—
35#16810490—21no—
36#16810488C2211yescharacteristic
37#16810488C2211yescharacteristic
38#16810483C2211yescharacteristic
39#16810483C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41C4⋊C4×Dic53C2211yescharacteristic
42C4×C22⋊Dic53C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#107×C53C2211yescharacteristic
53C2×C22⋊C8×C53C2211yescharacteristic
54C53×#380C2211yescharacteristic
55C53×#380C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×#88322—22no—
60C22×#88322—22no—
61C22×#88322—22no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53C2311yescharacteristic
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×#11719919—21no—
71C2×#11719919—21no—
72C2×C22⋊Dic53—21no—
73C2×C22⋊Dic53—21no—
74C2×C22⋊Dic53D411yescharacteristic
75C2×C22⋊Dic53D411yescharacteristic
76C2×C4×Dic53—21no—
77C2×C4×Dic53—21no—
78C2×C4×Dic53C2311yescharacteristic
79C2×C4×Dic53C2311yescharacteristic
80#13133151—22no—
81#13133148—22no—
82#13133148—22no—
83#13133148—22no—
84#13133148—22no—
85C8×Dic53—22no—
86C8×Dic53—22no—
87#13133124—21no—
88#13133124—21no—
89#13133124—21no—
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#13133124D411yescharacteristic
100#13133124D411yescharacteristic
101C23×C4×C53C2311yescharacteristic
102C22×C8×C53C2311yescharacteristic
103C22×C8×C53C2311yescharacteristic
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C4⋊C4×C53C2311yescharacteristic
106C2×C4⋊C4×C53C2311yescharacteristic
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C22⋊C4×C53—21no—
109C2×C22⋊C4×C53—21no—
110#107×C53—22no—
111C22⋊C8×C53—22no—
112#95×C53—21no—
113#95×C53—21no—
114C24×C53C2×D411yescharacteristic
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2411yescharacteristic
127C22×Dic53—21no—
128C22×Dic53—21no—
129C22×Dic53—21no—
130C22×Dic53—21no—
131C22×Dic53C2×D411yescharacteristic
132C22×Dic53C2×D411yescharacteristic
133C22×Dic53C2×D411yescharacteristic
134C22×Dic53C2×D411yescharacteristic
135C2×C8×C53—21no—
136C2×C8×C53—21no—
137C2×C8×C53—21no—
138C2×C8×C53—21no—
139C2×C8×C53—21no—
140C2×C8×C53#4112yes—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—21no—
149C22⋊C4×C53—42no—
150C22⋊C4×C53—42no—
151C22⋊Dic53—22no—
152C22⋊Dic53—22no—
153C22⋊Dic53—81no—
154C22⋊Dic53—81no—
155C4⋊Dic53—21no—
156C4⋊Dic53—21no—
157C4⋊Dic53—21no—
158C4⋊Dic53—21no—
159C4⋊Dic53—21no—
160C4⋊Dic53—21no—
161C4⋊Dic53—21no—
162C4⋊Dic53—21no—
163C4⋊Dic53—42no—
164C4⋊Dic53—81no—
165#11719919—81no—
166#11719919—81no—
167C4×Dic53—21no—
168C4×Dic53—21no—
169C4×Dic53—21no—
170C4×Dic53—21no—
171C4×Dic53—42no—
172C4×Dic53—81no—
173C2×#88322—21no—
174C2×#88322—21no—
175C2×#88322—21no—
176C2×#88322—44no—
177C2×#88322#4112yes—
178C2×#88322#4112yes—
179C2×#88322#4112yes—
180C23×C53—21no—
181C23×C53—21no—
182C23×C53—21no—
183C23×C53—21no—
184C23×C53—21no—
185C23×C53—21no—
186C23×C53#13512yes—
187C23×C53C22×D411yescharacteristic
188C2×C4×C53—21no—
189C2×C4×C53—21no—
190C2×C4×C53—21no—
191C2×C4×C53—41no—
192C2×C4×C53—41no—
193C2×C4×C53—41no—
194C2×C4×C53—41no—
195C2×C4×C53—41no—
196C2×C4×C53—41no—
197C2×C4×C53—41no—
198C2×C4×C53—41no—
199C2×C4×C53—41no—
200C2×C4×C53—41no—
201C2×C4×C53—41no—
202C2×C4×C53—41no—
203C2×C4×C53—41no—
204C2×C4×C53—41no—
205C2×C4×C53—41no—
206C2×C4×C53—41no—
207C2×C4×C53C2×#4111yescharacteristic
208C2×C4×C53C2×#4111yescharacteristic
209C2×C4×C53C2×#4111yescharacteristic
210C2×C4×C53C2×#4111yescharacteristic
211C2×C4×C532−1+411yescharacteristic
212C2×C4×C532−1+411yescharacteristic
213C2×Dic53—21no—
214C2×Dic53—21no—
215C2×Dic53—21no—
216C2×Dic53—21no—
217C2×Dic53—21no—
218C2×Dic53—21no—
219C2×Dic53—21no—
220C2×Dic53—21no—
221C2×Dic53—21no—
222C2×Dic53—21no—
223C2×Dic53—21no—
224C2×Dic53—41no—
225C2×Dic53—41no—
226C2×Dic53—41no—
227C2×Dic53—41no—
228C2×Dic53—41no—
229C2×Dic53—41no—
230C2×Dic53—41no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53#13512yes—
237C8×C53—22no—
238C8×C53—41no—
239#88322—41no—
240#88322#12614yes—
241#27836—531nomaximal, Sylow
242C22×C53—22no—
243C22×C53—22no—
244C22×C53—22no—
245C22×C53—41no—
246C22×C53—41no—
247C22×C53—41no—
248C22×C53—41no—
249C22×C53#53211yescharacteristic
250C22×C53#54711yescharacteristic
251C22×C53#54711yescharacteristic
252C22×C53C2×#13511yescharacteristic
253C22×C53C2×#13511yescharacteristic
254C22×C53#57711yescharacteristic
255C22×C53#57711yescharacteristic
256C4×C53—41no—
257C4×C53—81no—
258C4×C53—81no—
259C4×C53—81no—
260C4×C53—81no—
261C4×C53C2×#12611yescharacteristic
262C4×C53#54111yescharacteristic
263C4×C53#55211yescharacteristic
264C4×C53#55611yescharacteristic
265Dic53—22no—
266Dic53—41no—
267Dic53—41no—
268Dic53—41no—
269Dic53—81no—
270Dic53—81no—
271Dic53—81no—
272Dic53—81no—
273#2495—531no—
274#2494—531no—
275#2101—531no—
276#2101—531no—
277#2076—531no—
278#2075D5311yescharacteristic
279#1941—531no—
280#1941—531no—
281#1921—531no—
282#1918—531no—
283#1846—531no—
284#1807—531no—
285#1807—531no—
286C4×#106—531no—
287C8×C22⋊C4—531no—
288C2×C53—42no—
289C2×C53#249611yescharacteristic
290C2×C53#316811yescharacteristic
291C2×C53#317311yescharacteristic
292C2×C53#333711yescharacteristic
293C2×C53#333711yescharacteristic
294C2×C53#333711yescharacteristic
295C2×C53#334011yescharacteristic
296C23×C8—531no—
297C2×#107—531no—
298C2×#107C2×D5311yescharacteristic
299C2×#106—531no—
300C2×#106—531no—
301C2×#106—531no—
302C2×#106—531no—
303C2×C22⋊C8C2×D5311yescharacteristic
304C2×C4×C8—531no—
305C2×C4×C8—531no—
306#401—1061no—
307#400—1061no—
308#397—1061no—
309#397—1061no—
310#389—531no—
311#389—531no—
312#389—531no—
313#389—531no—
314#388—1061no—
315#388—1061no—
316#387—1061no—
317#383—1061no—
318#382—531no—
319#382—531no—
320#380—531no—
321#380—531no—
322#380C2×D5311yescharacteristic
323#380C2×D5311yescharacteristic
324C4×C4⋊C4—531no—
325C4×C4⋊C4—531no—
326C4×C22⋊C4—531no—
327#340—531no—
328#340—531no—
329#340—531no—
330#340—531no—
331#340—531no—
332#340—531no—
333#340—531no—
334#340C2×D5311yescharacteristic
335#340C2×D5311yescharacteristic
336#340C2×D5311yescharacteristic
337#336—531no—
338#336—531no—
339C53#2783611yescharacteristic, complemented, Sylow
340C23×C4C22×D5311yescharacteristic
341C22×C8—531no—
342C22×C8—531no—
343C22×C8—1062no—
344C22×C8—1062no—
345C22×C8—1062no—
346C22×C8C22×D5311yescharacteristic
347C22×C8C22×D5311yescharacteristic
348C2×C4⋊C4—531no—
349C2×C4⋊C4—531no—
350C2×C4⋊C4—531no—
351C2×C4⋊C4—531no—
352C2×C4⋊C4—1061no—
353C2×C4⋊C4—1061no—
354C2×C4⋊C4—1061no—
355C2×C4⋊C4—1061no—
356C2×C4⋊C4C22×D5311yescharacteristic
357C2×C4⋊C4C22×D5311yescharacteristic
358C2×C4⋊C4C22×D5311yescharacteristic
359C2×C4⋊C4C22×D5311yescharacteristic
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—531no—
363C2×C22⋊C4—531no—
364C2×C22⋊C4—1061no—
365C2×C22⋊C4—1061no—
366C2×C42—531no—
367C2×C42—531no—
368C2×C42—1061no—
369C2×C42—1061no—
370#107—22no—
371#107—1062no—
372#106—1062no—
373#106—1062no—
374#106—1062no—
375#106—1062no—
376C22⋊C8—22no—
377C4×C8—1062no—
378C4×C8—1062no—
379#95—21no—
380#95—21no—
381#95—531no—
382#95—531no—
383#95—1061no—
384#95—1061no—
385#95—1061no—
386#95—1061no—
387#95—1061no—
388#95—1061no—
389#95—1061no—
390#95—1061no—
391#95—1061no—
392#95—1061no—
393#95—1061no—
394#95—1061no—
395C24D4×D5311yescharacteristic
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—531no—
407C22×C4—531no—
408C22×C4—531no—
409C22×C4—531no—
410C22×C4—1061no—
411C22×C4—1061no—
412C22×C4—1061no—
413C22×C4—1061no—
414C22×C4D4×D5311yescharacteristic
415C22×C4C23×D5311yescharacteristic
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—532no—
422C2×C8—532no—
423C2×C8—532no—
424C2×C8—1061no—
425C2×C8—1061no—
426C2×C8—1061no—
427C2×C8—2124no—
428C2×C8#1171994712yes—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—1061no—
438C4⋊C4—1061no—
439C4⋊C4—1061no—
440C4⋊C4—1061no—
441C4⋊C4—1061no—
442C4⋊C4—1061no—
443C4⋊C4—1061no—
444C4⋊C4—1061no—
445C4⋊C4—4241no—
446C4⋊C4—4241no—
447C4⋊C4—4241no—
448C4⋊C4—2122no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—1062no—
452C22⋊C4—1062no—
453C22⋊C4—4241no—
454C22⋊C4—4241no—
455C42—1061no—
456C42—1061no—
457C42—1061no—
458C42—1061no—
459C42—4241no—
460C42—2122no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23#1313323612yes—
468C23C2×D4×D5311yescharacteristic
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—532no—
489C2×C4—1061no—
490C2×C4—1061no—
491C2×C4—1061no—
492C2×C4—1061no—
493C2×C4—1061no—
494C2×C4—1061no—
495C2×C4—1061no—
496C2×C4—1061no—
497C2×C4—1061no—
498C2×C4—1061no—
499C2×C4—1061no—
500C2×C4—2121no—
501C2×C4—2121no—
502C2×C4—2121no—
503C2×C4—2121no—
504C2×C4—2121no—
505C2×C4—2121no—
506C2×C4—2121no—
507C2×C4—2121no—
508C2×C4—2121no—
509C2×C4—2121no—
510C2×C4—2121no—
511C2×C4—2121no—
512C2×C4#41×D5311yescharacteristic
513C2×C4#41×D5311yescharacteristic
514C2×C4#41×D5311yescharacteristic
515C2×C4C2×#1171994711yescharacteristic
516C2×C4#1313326411yescharacteristic
517C2×C4#1313326411yescharacteristic
518C8—22no—
519C8—41no—
520C8—534no—
521C8—2121no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—41no—
529C22#1681116811yescharacteristic
530C22#1681127111yescharacteristic
531C22#1681127111yescharacteristic
532C22#135×D5311yescharacteristic
533C22C2×#1313323611yescharacteristic
534C22#1681143211yescharacteristic
535C22#1681143211yescharacteristic
536C4—41no—
537C4—81no—
538C4—81no—
539C4—81no—
540C4—81no—
541C4—1062no—
542C4—4241no—
543C4—4241no—
544C4—4241no—
545C4—4241no—
546C4—2121no—
547C4—2121no—
548C4—2121no—
549C4#126×D5311yescharacteristic
550C4#1681123011yescharacteristic
551C4#1681131011yescharacteristic
552C4#1681134211yescharacteristic
553C2—42no—
554C2#2027913011yescharacteristic
555C2#2028528011yescharacteristic
556C2#2028533311yescharacteristic
557C2#2028709711yescharacteristic
558C2#2028709811yescharacteristic
559C2#2028709811yescharacteristic
560C2#2028712111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2075C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2075D53
hypocenterC53#27836
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2075

Complements

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

NH ≅ G/Ntype
3. #20276286C2semidirect
4. #20276285C2semidirect
6. #20275442C2semidirect
7. #20275442C2semidirect
8. #20275306C2semidirect
9. #20275288C2semidirect
10. #20274878C2semidirect
13. #20274603C2semidirect
339. C53#27836semidirect

Maximal subgroups

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

subgroupindex
#202791282
#202791112
#202762862
#202762852
#202761432
#202754422
#202753062
#202752882
#202748782
#202747382
#202746032
#202744892
C53×#20752
#2783653