Finite Group Databasebeta

← #27486598

Subgroups — #27486598

Order 13568.

774 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277286C211yesmaximal, characteristic, complemented
2#20277180C211yesmaximal, characteristic
3#20276189C211yesmaximal, characteristic, complemented
4#20276121C211yesmaximal, characteristic
5#20276047C211yesmaximal, characteristic, complemented
6#20275849C211yesmaximal, characteristic
7#20275345C211yesmaximal, characteristic
8#20275344C211yesmaximal, characteristic
9#20275310C211yesmaximal, characteristic, complemented
10#20275309C211yesmaximal, characteristic, complemented
11#20275294C211yesmaximal, characteristic, complemented
12#20275290C211yesmaximal, characteristic, complemented
13#20274962C211yesmaximal, characteristic
14#20274857C211yesmaximal, characteristic, complemented
15C53×#1862C211yesmaximal, characteristic
16C2×#13133216C2211yescharacteristic
17C2×#13133182—21no—
18C2×#13133176—21no—
19C2×#13133150C2211yescharacteristic
20C2×#13133149C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133138C2211yescharacteristic
23C2×#13133138C2211yescharacteristic
24#16810527—21no—
25#16810526C2211yescharacteristic
26#16810526C2211yescharacteristic
27#16810522C2211yescharacteristic
28#16810522C2211yescharacteristic
29#16810521C2211yescharacteristic
30#16810518—21no—
31#16810516—21no—
32#16810516—21no—
33#16810492C2211yescharacteristic
34C4⋊C4×Dic53C2211yescharacteristic
35C4⋊C4×Dic53C2211yescharacteristic
36C4×C4⋊Dic53C2211yescharacteristic
37#16810474—21no—
38C4×C22⋊Dic53—21no—
39C42×Dic53C2211yescharacteristic
40#16810386C2211yescharacteristic
41#16810386C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810378C2211yescharacteristic
48#16810377C2211yescharacteristic
49#16810377C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×C4×D4×C53C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic
54C2×#102×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C53×#389C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×Dic106—21no—
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62#13133216—22no—
63#13133216—22no—
64#13133182—44no—
65#13133176—44no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×C22⋊Dic53—21no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53D411yescharacteristic
84C2×C4×Dic53D411yescharacteristic
85C2×C4×Dic53D411yescharacteristic
86C2×C4×Dic53D411yescharacteristic
87C2×C4×Dic53C2311yescharacteristic
88C2×C4×Dic53C2311yescharacteristic
89#13133150—22no—
90#13133149—22no—
91#13133148—22no—
92#13133138—22no—
93#13133138—22no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100C22×D4×C53C2311yescharacteristic
101C23×C4×C53—21no—
102C22×C8×C53C2311yescharacteristic
103C22×C8×C53C2311yescharacteristic
104C4×D4×C53—22no—
105C4×D4×C53—22no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C22⋊C4×C53—21no—
111C2×C42×C53C2311yescharacteristic
112#107×C53—22no—
113#102×C53—22no—
114#102×C53—22no—
115#95×C53—21no—
116C24×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53—21no—
122C2×D4×C53#4112yes—
123C22×C4×C53—24no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—42no—
128C22×C4×C53—42no—
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2411yescharacteristic
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53C2×D411yescharacteristic
137C22×Dic53C2×D411yescharacteristic
138C22×Dic53C2×D411yescharacteristic
139C22×Dic53C2×D411yescharacteristic
140C2×Dic106—24no—
141C2×Dic106—42no—
142C2×Dic106—42no—
143C2×C8×C53—21no—
144C2×C8×C53—21no—
145C2×C8×C53—21no—
146C2×C8×C53—21no—
147C2×C8×C53—21no—
148C2×C8×C53—21no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—42no—
154C4⋊C4×C53#4112yes—
155C4⋊C4×C53#4112yes—
156C22⋊C4×C53—24no—
157C42×C53—21no—
158C42×C53—21no—
159C22⋊Dic53—42no—
160C22⋊Dic53—42no—
161C22⋊Dic53—42no—
162C22⋊Dic53—42no—
163C4⋊Dic53—21no—
164C4⋊Dic53—21no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—42no—
171C4⋊Dic53—42no—
172C4⋊Dic53#4112yes—
173#11719919—24no—
174#11719919—42no—
175#11719919—42no—
176#11719919—42no—
177C4×Dic53—22no—
178C4×Dic53—21no—
179C4×Dic53—21no—
180C4×Dic53—21no—
181C4×Dic53—21no—
182C4×Dic53—22no—
183C4×Dic53—22no—
184C4×Dic53—22no—
185C4×Dic53—42no—
186C4×Dic53—42no—
187C4×Dic53—42no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C23×C53—22no—
195C23×C53—41no—
196C23×C53—41no—
197C23×C53—41no—
198C23×C53—41no—
199C23×C53—41no—
200C23×C53—41no—
201C23×C53C22×D411yescharacteristic
202D4×C53—22no—
203D4×C53—22no—
204D4×C53—41no—
205D4×C53—41no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—48no—
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×C53AGL(1,ℤ/8)12yes—
222C2×C4×C53C2×#4111yescharacteristic
223C2×C4×C53C2×#4111yescharacteristic
224C2×C4×C53C2×#4111yescharacteristic
225C2×C4×C53C2×#4111yescharacteristic
226C2×C4×C532−1+411yescharacteristic
227C2×C4×C532−1+411yescharacteristic
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—22no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
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#13512yes—
250C2×Dic53AGL(1,ℤ/8)12yes—
251C2×Dic53#13712yes—
252Dic106—48no—
253C8×C53—41no—
254C8×C53—41no—
255#88322—41no—
256#88322—41no—
257#29389—531nomaximal, Sylow
258C22×C53—42no—
259C22×C53—42no—
260C22×C53—42no—
261C22×C53—81no—
262C22×C53—81no—
263C22×C53—81no—
264C22×C53—81no—
265C22×C53#52511yescharacteristic
266C22×C53#52511yescharacteristic
267C22×C53#53611yescharacteristic
268C22×C53C2×#13511yescharacteristic
269C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C53C2×#13711yescharacteristic
272C4×C53—22no—
273C4×C53—42no—
274C4×C53—41no—
275C4×C53—42no—
276C4×C53—81no—
277C4×C53#55211yescharacteristic
278C4×C53#55211yescharacteristic
279C4×C53#55611yescharacteristic
280C4×C53#55611yescharacteristic
281Dic53—22no—
282Dic53—22no—
283Dic53—22no—
284Dic53—42no—
285Dic53—41no—
286Dic53—81no—
287Dic53—81no—
288Dic53—81no—
289C4×C4⋊Q8—531no—
290#2293—531no—
291#2083—531no—
292#2071—531no—
293#2058—531no—
294#2023—531no—
295#1926—531no—
296#1926—531no—
297#1921—531no—
298#1921—531no—
299#1919—531no—
300#1919—531no—
301#1862—531no—
302#1862D5311yescharacteristic
303#1839—531no—
304C2×C53—82no—
305C2×C53#259911yescharacteristic
306C2×C53#310511yescharacteristic
307C2×C53#310711yescharacteristic
308C2×C53#310911yescharacteristic
309C2×C53#311211yescharacteristic
310C2×C53#322811yescharacteristic
311C2×C53#323911yescharacteristic
312C2×C4⋊Q8—531no—
313C2×C4×Q8—1061no—
314C2×C4×D4C2×D5311yescharacteristic
315C2×#117—1061no—
316C2×#107—531no—
317C2×#107C2×D5311yescharacteristic
318C2×#106—531no—
319C2×#106—531no—
320C2×#102—531no—
321C2×#102—531no—
322C2×#102C2×D5311yescharacteristic
323C2×#102C2×D5311yescharacteristic
324#391—1061no—
325#390—531no—
326#390—531no—
327#389—531no—
328#389—531no—
329#389—531no—
330#389—1061no—
331#389C2×D5311yescharacteristic
332#388—1061no—
333#388—1061no—
334#384—531no—
335C4×C4⋊C4—531no—
336C4×C4⋊C4—531no—
337C4×C4⋊C4—531no—
338C4×C4⋊C4—1061no—
339C4×C22⋊C4—1061no—
340C43—531no—
341#340—531no—
342#340—531no—
343#340—531no—
344#340—531no—
345#340—531no—
346#340—531no—
347#340—531no—
348#340C2×D5311yescharacteristic
349#336—531no—
350#336—531no—
351#336—531no—
352#336—531no—
353#336—531no—
354#336C2×D5311yescharacteristic
355C53#2938911yescharacteristic, complemented, Sylow
356C22×Q8—1061no—
357C22×D4C22×D5311yescharacteristic
358C23×C4—21no—
359C22×C8—531no—
360C22×C8—531no—
361C22×C8C22×D5311yescharacteristic
362C22×C8C22×D5311yescharacteristic
363C4⋊Q8—1062no—
364C4⋊Q8—1062no—
365C4×Q8—2124no—
366C4×D4—22no—
367C4×D4—22no—
368#117—2124no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—531no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—531no—
373C2×C4⋊C4—531no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4—1061no—
377C2×C4⋊C4—1061no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4C22×D5311yescharacteristic
381C2×C4⋊C4C22×D5311yescharacteristic
382C2×C4⋊C4C22×D5311yescharacteristic
383C2×C22⋊C4—21no—
384C2×C22⋊C4—1061no—
385C2×C22⋊C4—1061no—
386C2×C22⋊C4—1061no—
387C2×C22⋊C4—1061no—
388C2×C42—531no—
389C2×C42—531no—
390C2×C42—531no—
391C2×C42—531no—
392C2×C42—531no—
393C2×C42—531no—
394C2×C42—1061no—
395C2×C42—1061no—
396C2×C42—1061no—
397C2×C42C22×D5311yescharacteristic
398#107—22no—
399#107—1062no—
400#106—1062no—
401#106—1062no—
402#102—22no—
403#102—22no—
404#102—1062no—
405#102—1062no—
406#95—21no—
407#95—1061no—
408#95—1061no—
409#95—1061no—
410#95—1061no—
411#95—1061no—
412#95—1061no—
413C24—21no—
414C2×Q8—1064no—
415C2×Q8—2122no—
416C2×Q8—2122no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4#1171994712yes—
423C22×C4—24no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—531no—
430C22×C4—531no—
431C22×C4—531no—
432C22×C4—531no—
433C22×C4—1061no—
434C22×C4—1061no—
435C22×C4—1061no—
436C22×C4—1061no—
437C22×C4D4×D5311yescharacteristic
438C22×C4D4×D5311yescharacteristic
439C22×C4C23×D5311yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—1061no—
447C2×C8—1061no—
448C2×C8—1061no—
449C2×C8—1061no—
450C2×C8—1061no—
451C2×C8—1061no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—42no—
457C4⋊C4—532no—
458C4⋊C4—1064no—
459C4⋊C4—1061no—
460C4⋊C4—1061no—
461C4⋊C4—1061no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—2122no—
467C4⋊C4—2122no—
468C4⋊C4—2122no—
469C4⋊C4—2122no—
470C4⋊C4—2122no—
471C4⋊C4#1171994512yes—
472C4⋊C4#1171994712yes—
473C22⋊C4—24no—
474C22⋊C4—2122no—
475C22⋊C4—2122no—
476C22⋊C4—2122no—
477C22⋊C4—2122no—
478C42—21no—
479C42—21no—
480C42—1062no—
481C42—1062no—
482C42—1062no—
483C42—1062no—
484C42—1061no—
485C42—1061no—
486C42—1061no—
487C42—1061no—
488C42—2122no—
489C42—2122no—
490C42—2122no—
491C23—22no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23C2×D4×D5311yescharacteristic
499Q8—2128no—
500D4—22no—
501D4—22no—
502D4—41no—
503D4—41no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—48no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—532no—
520C2×C4—532no—
521C2×C4—532no—
522C2×C4—1061no—
523C2×C4—1061no—
524C2×C4—1061no—
525C2×C4—1061no—
526C2×C4—1061no—
527C2×C4—1061no—
528C2×C4—1062no—
529C2×C4—1061no—
530C2×C4—1061no—
531C2×C4—1061no—
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#1313324112yes—
544C2×C4#41×D5311yescharacteristic
545C2×C4C2×#1171994711yescharacteristic
546C2×C4C2×#1171994711yescharacteristic
547C2×C4C2×#1171994511yescharacteristic
548C2×C4#1313326411yescharacteristic
549C2×C4#1313326511yescharacteristic
550C8—41no—
551C8—41no—
552C8—2121no—
553C8—2121no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22—81no—
561C22#1681112911yescharacteristic
562C22#1681113211yescharacteristic
563C22#1681119611yescharacteristic
564C22#135×D5311yescharacteristic
565C22AGL(1,ℤ/8)×D5311yescharacteristic
566C22C2×#1313324111yescharacteristic
567C22#137×D5311yescharacteristic
568C4—22no—
569C4—42no—
570C4—41no—
571C4—42no—
572C4—81no—
573C4—1062no—
574C4—1062no—
575C4—1062no—
576C4—4241no—
577C4—4241no—
578C4—4241no—
579C4—2122no—
580C4—2121no—
581C4#1681130811yescharacteristic
582C4#1681131011yescharacteristic
583C4#1681134311yescharacteristic
584C4#1681134411yescharacteristic
585C2—82no—
586C2#2028028811yescharacteristic
587C2#2028476011yescharacteristic
588C2#2028477911yescharacteristic
589C2#2028480111yescharacteristic
590C2#2028483511yescharacteristic
591C2#2028585011yescharacteristic
592C2#2028594411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1862C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#1862D53
hypocenterC53#29389
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1862

Complements

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

NH ≅ G/Ntype
1. #20277286C2semidirect
3. #20276189C2semidirect
5. #20276047C2semidirect
9. #20275310C2semidirect
10. #20275309C2semidirect
11. #20275294C2semidirect
12. #20275290C2semidirect
14. #20274857C2semidirect
355. C53#29389semidirect

Maximal subgroups

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

subgroupindex
#202772862
#202771802
#202761892
#202761212
#202760472
#202758492
#202753452
#202753442
#202753102
#202753092
#202752942
#202752902
#202749622
#202748572
C53×#18622
#2938953