Finite Group Databasebeta

← #27442092

Subgroups — #27442092

Order 13568.

732 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20278431C211yesmaximal, characteristic
2#20277580C211yesmaximal, characteristic
3#20276277C211yesmaximal, characteristic, complemented
4#20276214C211yesmaximal, characteristic, complemented
5#20276203C211yesmaximal, characteristic, complemented
6#20276055C211yesmaximal, characteristic, complemented
7#20275671C211yesmaximal, characteristic, complemented
8#20275306C211yesmaximal, characteristic, complemented
9#20275286C211yesmaximal, characteristic, complemented
10#20274913C211yesmaximal, characteristic
11#20274896C211yesmaximal, characteristic
12#20274804C211yesmaximal, characteristic
13#20274745C211yesmaximal, characteristic
14#20274433C211yesmaximal, characteristic
15C53×#1948C211yesmaximal, characteristic, complemented
16C22×#11719912C2211yescharacteristic
17C2×#13133208C2211yescharacteristic
18C2×#13133175C2211yescharacteristic
19C2×#13133174—21no—
20C2×C4⋊C4×D53C2211yescharacteristic
21C2×#13133151C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133146C2211yescharacteristic
24C2×#13133129C2211yescharacteristic
25#16810558—21no—
26#16810545—21no—
27#16810521C2211yescharacteristic
28#16810521C2211yescharacteristic
29#16810519C2211yescharacteristic
30#16810519C2211yescharacteristic
31#16810518—21no—
32#16810509—21no—
33#16810507—21no—
34#16810505—21no—
35#16810485—21no—
36#16810480C2211yescharacteristic
37#16810480C2211yescharacteristic
38C4⋊C4×Dic53C2211yescharacteristic
39C4×#11719921C2211yescharacteristic
40#16810386C2211yescharacteristic
41#16810386C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810378C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#106×C53C2211yescharacteristic
53C2×#106×C53C2211yescharacteristic
54C2×C4⋊C8×C53C2211yescharacteristic
55C53×#384C2211yescharacteristic
56C4×C4⋊C4×C53C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×C4×D53—21no—
60C22×C4×D53C2311yescharacteristic
61C2×#11719912—22no—
62C2×#11719912—22no—
63C2×#11719912—41no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133208—22no—
67#13133208—41no—
68#13133175—22no—
69#13133175—41no—
70#13133174—44no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C4⋊C4×D53—22no—
83C4⋊C4×D53Q814yes—
84C2×#11719921—21no—
85C2×#11719921—21no—
86C2×#11719921—21no—
87C2×#11719921D411yescharacteristic
88C2×#11719921D411yescharacteristic
89C2×C4×Dic53—21no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53C2311yescharacteristic
92#13133151—22no—
93#13133150—22no—
94#13133146—41no—
95#13133129—22no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124D411yescharacteristic
104#13133124D411yescharacteristic
105C22×C8×C53C2311yescharacteristic
106C22×C8×C53C2311yescharacteristic
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C42×C53—21no—
114C2×C42×C53C2311yescharacteristic
115#106×C53—41no—
116#106×C53—41no—
117C4⋊C8×C53—22no—
118#95×C53—21no—
119C23×D53C2×D411yescharacteristic
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53C2×D411yescharacteristic
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
134C2×C4×D53—21no—
135C2×C4×D53—21no—
136C2×C4×D53—21no—
137C2×C4×D53—21no—
138C2×C4×D53—24no—
139C2×C4×D53—42no—
140C2×C4×D53—42no—
141C2×C4×D53C2×Q812yes—
142C2×C4×D53#4112yes—
143C2×C8×C53—21no—
144C2×C8×C53—21no—
145C2×C8×C53—21no—
146C2×C8×C53—21no—
147C2×C8×C53—21no—
148C2×C8×C53#4112yes—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—42no—
157C4⋊C4×C53—81no—
158C4⋊C4×C53C2×Q812yes—
159C42×C53—21no—
160C42×C53—21no—
161C42×C53—42no—
162#11719921—22no—
163#11719921—41no—
164#11719921—42no—
165#11719921—42no—
166#11719921—81no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—42no—
174C4⋊Dic53C2×Q812yes—
175C4⋊Dic53#4112yes—
176#11719919—24no—
177#11719919—42no—
178#11719919—42no—
179#11719919—42no—
180#11719919—42no—
181#11719919—81no—
182C4×Dic53—21no—
183C4×Dic53—21no—
184C4×Dic53—42no—
185C4×Dic53—42no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192#11719912—44no—
193C23×C53C22×D411yescharacteristic
194C22×D53—21no—
195C22×D53—21no—
196C22×D53—21no—
197C22×D53—21no—
198C22×D53—21no—
199C22×D53—21no—
200C22×D53C22⋊Q812yes—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—22no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—41no—
208C2×C4×C53—41no—
209C2×C4×C53—41no—
210C2×C4×C53—41no—
211C2×C4×C53—41no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53AGL(1,ℤ/8)12yes—
219C2×C4×C53C22×Q811yescharacteristic
220C2×C4×C53C2×#4111yescharacteristic
221C2×C4×C53C2×#4111yescharacteristic
222C2×C4×C53C2×#4111yescharacteristic
223C2×C4×C53H(5,𝔽2)11yescharacteristic
224C2×C4×C532−1+411yescharacteristic
225C2×Dic53—21no—
226C2×Dic53—21no—
227C2×Dic53—22no—
228C2×Dic53—22no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
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×Dic53C22⋊Q812yes—
246C4×D53—22no—
247C4×D53—22no—
248C4×D53—22no—
249C4×D53—22no—
250C4×D53—48no—
251C8×C53—22no—
252C8×C53—41no—
253#88322—41no—
254#88322—41no—
255#26014—531nomaximal, Sylow
256C22×C53C2×C22⋊Q811yescharacteristic
257C22×C53#54611yescharacteristic
258C22×C53#54711yescharacteristic
259C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
260C22×C53#57511yescharacteristic
261C22×C53#57711yescharacteristic
262C22×C53#57811yescharacteristic
263C2×D53—22no—
264C2×D53—22no—
265C2×D53—22no—
266C2×D53—41no—
267C2×D53—41no—
268C2×D53—41no—
269C2×D53—41no—
270C4×C53—22no—
271C4×C53—42no—
272C4×C53—41no—
273C4×C53—81no—
274C4×C53—81no—
275C4×C53—81no—
276C4×C53#53311yescharacteristic
277C4×C53#54411yescharacteristic
278C4×C53#55111yescharacteristic
279C4×C53#55211yescharacteristic
280Dic53—22no—
281Dic53—41no—
282Dic53—42no—
283Dic53—42no—
284Dic53—41no—
285Dic53—81no—
286Dic53—81no—
287#2441—531no—
288#2344—531no—
289#2100—531no—
290#2088—531no—
291#2086—531no—
292#2059—531no—
293#1989—531no—
294#1948D5311yescharacteristic, complemented
295#1921—531no—
296#1918—531no—
297#1854—531no—
298#1851—531no—
299#1824—531no—
300#1809—531no—
301#1736—531no—
302C2×C53#243811yescharacteristic
303C2×C53#307711yescharacteristic
304C2×C53#308211yescharacteristic
305C2×C53#331711yescharacteristic
306C2×C53#333011yescharacteristic
307C2×C53#334511yescharacteristic
308C2×C53#335211yescharacteristic
309D53—42no—
310C22×M16—531no—
311C2×#125—531no—
312C2×#117—531no—
313C2×#117—1061no—
314C22×C4⋊C4—531no—
315C2×#107—531no—
316C2×#107—531no—
317C2×#106C2×D5311yescharacteristic
318C2×#106C2×D5311yescharacteristic
319C2×C4⋊C8—531no—
320C2×C4⋊C8C2×D5311yescharacteristic
321C2×C22⋊C8—531no—
322#399—1061no—
323#396—1061no—
324#389—531no—
325#389—531no—
326#389—531no—
327#389—531no—
328#389—1061no—
329#387—1061no—
330#387—1061no—
331#387—1061no—
332#384C2×D5311yescharacteristic
333#381—1061no—
334#380—531no—
335#380—531no—
336C4×C4⋊C4—531no—
337C4×C4⋊C4C2×D5311yescharacteristic
338C4×C22⋊C4—531no—
339#340—531no—
340#340—531no—
341#340—531no—
342#340—531no—
343#340—531no—
344#340—531no—
345#340—531no—
346#340—531no—
347#340—531no—
348#340—531no—
349#340C2×D5311yescharacteristic
350#340C2×D5311yescharacteristic
351#336—531no—
352#336—531no—
353C53#2601411yescharacteristic, complemented, Sylow
354C23×C4—531no—
355C23×C4—1061no—
356C2×M16—1062no—
357C2×M16—1062no—
358C2×M16—2121no—
359C22×C8—531no—
360C22×C8—531no—
361C22×C8C22×D5311yescharacteristic
362C22×C8C22×D5311yescharacteristic
363#125—1062no—
364#125—2121no—
365#117—1062no—
366#117—2124no—
367#117—2121no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—531no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—534no—
373C2×C4⋊C4—531no—
374C2×C4⋊C4—531no—
375C2×C4⋊C4—1062no—
376C2×C4⋊C4—1061no—
377C2×C4⋊C4—1061no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4C22×D5311yescharacteristic
384C2×C4⋊C4C22×D5311yescharacteristic
385C2×C4⋊C4C22×D5311yescharacteristic
386C2×C4⋊C4C22×D5311yescharacteristic
387C2×C22⋊C4—531no—
388C2×C22⋊C4—531no—
389C2×C22⋊C4—1061no—
390C2×C22⋊C4—1061no—
391C2×C22⋊C4—1061no—
392C2×C42—21no—
393C2×C42—531no—
394C2×C42—1061no—
395C2×C42—1061no—
396C2×C42C22×D5311yescharacteristic
397#107—1062no—
398#107—1062no—
399#106—41no—
400#106—41no—
401C4⋊C8—22no—
402C4⋊C8—2121no—
403C22⋊C8—1062no—
404#95—21no—
405#95—531no—
406#95—531no—
407#95—1061no—
408#95—1061no—
409#95—1061no—
410#95—1061no—
411#95—1061no—
412#95—1061no—
413#95—1061no—
414C24—531no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—531no—
420C22×C4—532no—
421C22×C4—532no—
422C22×C4—531no—
423C22×C4—531no—
424C22×C4—1064no—
425C22×C4—1061no—
426C22×C4—1061no—
427C22×C4—1061no—
428C22×C4—1061no—
429C22×C4—1061no—
430C22×C4—1061no—
431C22×C4—1061no—
432C22×C4—1061no—
433C22×C4—2122no—
434C22×C4—2122no—
435C22×C4D4×D5311yescharacteristic
436C22×C4D4×D5311yescharacteristic
437C22×C4C23×D5311yescharacteristic
438M16—2124no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—1061no—
445C2×C8—1061no—
446C2×C8—1061no—
447C2×C8—1061no—
448C2×C8—1061no—
449C2×C8—1061no—
450C2×C8#1171994512yes—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—42no—
459C4⋊C4—81no—
460C4⋊C4—532no—
461C4⋊C4—532no—
462C4⋊C4—1064no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—4241no—
470C4⋊C4—2122no—
471C4⋊C4—2122no—
472C4⋊C4—2122no—
473C4⋊C4—2122no—
474C4⋊C4—2122no—
475C4⋊C4Q8×D5312yes—
476C22⋊C4—1062no—
477C22⋊C4—4241no—
478C22⋊C4—2121no—
479C22⋊C4—2122no—
480C22⋊C4—2122no—
481C42—21no—
482C42—21no—
483C42—42no—
484C42—1061no—
485C42—1061no—
486C42—2122no—
487C42—2122no—
488C23—532no—
489C23—1061no—
490C23—1061no—
491C23—1061no—
492C23—1061no—
493C23—1061no—
494C23—1061no—
495C23C2×D4×D5311yescharacteristic
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—532no—
514C2×C4—1061no—
515C2×C4—1061no—
516C2×C4—1062no—
517C2×C4—1062no—
518C2×C4—1062no—
519C2×C4—1062no—
520C2×C4—1061no—
521C2×C4—1061no—
522C2×C4—1062no—
523C2×C4—1062no—
524C2×C4—1061no—
525C2×C4—1061no—
526C2×C4—1061no—
527C2×C4—1061no—
528C2×C4—2128no—
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#1313324412yes—
540C2×C4C2×Q8×D5311yescharacteristic
541C2×C4#41×D5311yescharacteristic
542C2×C4#41×D5311yescharacteristic
543C2×C4C2×#1171994511yescharacteristic
544C2×C4#1313326211yescharacteristic
545C2×C4#1313326511yescharacteristic
546C8—22no—
547C8—41no—
548C8—2121no—
549C8—2121no—
550C22—1062no—
551C22—1062no—
552C22—1062no—
553C22—2121no—
554C22—2121no—
555C22—2121no—
556C22—2121no—
557C22C22⋊Q8×D5311yescharacteristic
558C22#1681127011yescharacteristic
559C22#1681127411yescharacteristic
560C22C2×#1313324411yescharacteristic
561C22#1681141911yescharacteristic
562C22#1681142911yescharacteristic
563C22#1681143911yescharacteristic
564C4—22no—
565C4—41no—
566C4—42no—
567C4—81no—
568C4—81no—
569C4—81no—
570C4—1062no—
571C4—4241no—
572C4—4241no—
573C4—2121no—
574C4—2122no—
575C4—2122no—
576C4—2121no—
577C4#1681117711yescharacteristic
578C4#1681125011yescharacteristic
579C4#1681130211yescharacteristic
580C4#1681130511yescharacteristic
581C2—2122no—
582C2#2027839811yescharacteristic
583C2#2028451311yescharacteristic
584C2#2028455011yescharacteristic
585C2#2028684811yescharacteristic
586C2#2028702311yescharacteristic
587C2#2028718111yescharacteristic
588C2#2028728611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1948C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#1948D53
hypocenterC53#26014
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
3. #20276277C2semidirect
4. #20276214C2semidirect
5. #20276203C2semidirect
6. #20276055C2semidirect
7. #20275671C2semidirect
8. #20275306C2semidirect
9. #20275286C2semidirect
15. C53×#1948C2semidirect
294. #1948D53semidirect
353. C53#26014semidirect

Maximal subgroups

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

subgroupindex
#202784312
#202775802
#202762772
#202762142
#202762032
#202760552
#202756712
#202753062
#202752862
#202749132
#202748962
#202748042
#202747452
#202744332
C53×#19482
#2601453