Finite Group Databasebeta

← #27465385

Subgroups — #27465385

Order 13568.

700 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277199C211yesmaximal, characteristic
2#20277110C211yesmaximal, characteristic
3#20276286C211yesmaximal, characteristic, complemented
4#20276286C211yesmaximal, characteristic, complemented
5#20276200C211yesmaximal, characteristic, complemented
6#20276199C211yesmaximal, characteristic, complemented
7#20275309C211yesmaximal, characteristic, complemented
8#20275303C211yesmaximal, characteristic, complemented
9#20275295C211yesmaximal, characteristic, complemented
10#20275289C211yesmaximal, characteristic, complemented
11#20274914C211yesmaximal, characteristic
12#20274811C211yesmaximal, characteristic
13#20274807C211yesmaximal, characteristic
14#20274781C211yesmaximal, characteristic
15C53×#1854C211yesmaximal, characteristic
16C2×#13133176C2211yescharacteristic
17C22×C4×Dic53C2211yescharacteristic
18C2×#13133150C2211yescharacteristic
19C2×#13133148C2211yescharacteristic
20C2×C22⋊#88322C2211yescharacteristic
21#16810522C2211yescharacteristic
22#16810522C2211yescharacteristic
23#16810522C2211yescharacteristic
24#16810522C2211yescharacteristic
25#16810498—21no—
26#16810498—21no—
27#16810490—21no—
28#16810488C2211yescharacteristic
29#16810488C2211yescharacteristic
30#16810484—21no—
31#16810483C2211yescharacteristic
32C4⋊C4×Dic53C2211yescharacteristic
33C4⋊C4×Dic53C2211yescharacteristic
34C4×C4⋊Dic53—21no—
35C4×C4⋊Dic53—21no—
36C22⋊C4×Dic53—21no—
37#16810468—21no—
38#16810386C2211yescharacteristic
39#16810386C2211yescharacteristic
40#16810386C2211yescharacteristic
41#16810386C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810377C2211yescharacteristic
49#16810377C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#117×C53C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic
54C2×#106×C53C2211yescharacteristic
55C2×C22⋊C8×C53C2211yescharacteristic
56C53×#380C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C23×Dic53D411yescharacteristic
60C23×Dic53D411yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63#13133176—22no—
64#13133176—22no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C22⋊Dic53—21no—
74C2×C22⋊Dic53—21no—
75C2×C4×Dic53—21no—
76C2×C4×Dic53—22no—
77C2×C4×Dic53—22no—
78C2×C4×Dic53—21no—
79C2×C4×Dic53—21no—
80C2×C4×Dic53D411yescharacteristic
81C2×C4×Dic53D411yescharacteristic
82C2×C4×Dic53C2311yescharacteristic
83C2×C4×Dic53C2311yescharacteristic
84#13133150—22no—
85#13133148—22no—
86C22⋊#88322—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—
98C23×C4×C53C2311yescharacteristic
99C22×C8×C53C2311yescharacteristic
100C22×C8×C53C2311yescharacteristic
101#117×C53—22no—
102#117×C53—22no—
103C2×C4⋊C4×C53C2311yescharacteristic
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C4⋊C4×C53C2311yescharacteristic
106C2×C4⋊C4×C53C2311yescharacteristic
107C2×C22⋊C4×C53—21no—
108C2×C22⋊C4×C53—21no—
109C2×C42×C53—21no—
110#107×C53—22no—
111#106×C53—22no—
112C22⋊C8×C53—22no—
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×C53C2×D411yescharacteristic
124C22×C4×C53#4112yes—
125C22×C4×C53#4112yes—
126C22×C4×C53C2411yescharacteristic
127C22×Dic53—22no—
128C22×Dic53—22no—
129C22×Dic53—22no—
130C22×Dic53—21no—
131C22×Dic53—21no—
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—41no—
135C22×Dic53—41no—
136C22×Dic53—41no—
137C22×Dic53C2×D411yescharacteristic
138C22×Dic53C2×D411yescharacteristic
139C22×Dic53C2×D411yescharacteristic
140C22×Dic53C2×D411yescharacteristic
141C2×C8×C53—21no—
142C2×C8×C53—21no—
143C2×C8×C53—21no—
144C2×C8×C53—21no—
145C2×C8×C53—21no—
146C2×C8×C53—21no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53#4112yes—
154C4⋊C4×C53#4112yes—
155C22⋊C4×C53—24no—
156C22⋊C4×C53—42no—
157C42×C53—24no—
158C22⋊Dic53—42no—
159C22⋊Dic53—42no—
160C4⋊Dic53—21no—
161C4⋊Dic53—21no—
162C4⋊Dic53—21no—
163C4⋊Dic53—21no—
164C4⋊Dic53—21no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53—42no—
169C4⋊Dic53—42no—
170C4⋊Dic53—42no—
171C4⋊Dic53—42no—
172C4×Dic53—22no—
173C4×Dic53—22no—
174C4×Dic53—21no—
175C4×Dic53—21no—
176C4×Dic53—21no—
177C4×Dic53—21no—
178C4×Dic53—42no—
179C4×Dic53—42no—
180C4×Dic53—42no—
181C2×#88322—21no—
182C2×#88322—21no—
183C2×#88322—21no—
184C2×#88322—21no—
185C2×#88322—21no—
186C2×#88322—21no—
187C23×C53—21no—
188C23×C53—21no—
189C23×C53—21no—
190C23×C53—21no—
191C23×C53—21no—
192C23×C53—21no—
193C23×C53#12312yes—
194C23×C53C22×D411yescharacteristic
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53—22no—
198C2×C4×C53—22no—
199C2×C4×C53—22no—
200C2×C4×C53—22no—
201C2×C4×C53—22no—
202C2×C4×C53—22no—
203C2×C4×C53—41no—
204C2×C4×C53—41no—
205C2×C4×C53—41no—
206C2×C4×C53—41no—
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#12312yes—
214C2×C4×C53C2×#4111yescharacteristic
215C2×C4×C53C2×#4111yescharacteristic
216C2×C4×C53C2×#4111yescharacteristic
217C2×C4×C53C2×#4111yescharacteristic
218C2×C4×C532−1+411yescharacteristic
219C2×C4×C532−1+411yescharacteristic
220C2×Dic53—21no—
221C2×Dic53—21no—
222C2×Dic53—21no—
223C2×Dic53—21no—
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—44no—
232C2×Dic53—44no—
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#13512yes—
246C8×C53—41no—
247C8×C53—41no—
248#88322—41no—
249#88322—41no—
250#27618—531nomaximal, Sylow
251C22×C53—22no—
252C22×C53—22no—
253C22×C53—22no—
254C22×C53—41no—
255C22×C53—41no—
256C22×C53—41no—
257C22×C53—41no—
258C22×C53C2×#12311yescharacteristic
259C22×C53#52511yescharacteristic
260C22×C53#53611yescharacteristic
261C22×C53C2×#13511yescharacteristic
262C22×C53#57511yescharacteristic
263C22×C53#57511yescharacteristic
264C22×C53#57511yescharacteristic
265C4×C53—22no—
266C4×C53—42no—
267C4×C53—42no—
268C4×C53—81no—
269C4×C53—81no—
270C4×C53#55211yescharacteristic
271C4×C53#55211yescharacteristic
272C4×C53#55211yescharacteristic
273C4×C53#55211yescharacteristic
274Dic53—22no—
275Dic53—41no—
276Dic53—41no—
277Dic53—41no—
278Dic53—81no—
279Dic53—81no—
280Dic53—81no—
281Dic53—81no—
282#2296—531no—
283#2283—531no—
284#2101—531no—
285#2101—531no—
286#2085—531no—
287#2085—531no—
288#1921—531no—
289#1920—531no—
290#1919—531no—
291#1918—531no—
292#1854—531no—
293#1854D5311yescharacteristic
294#1825—531no—
295#1825—531no—
296#1817—531no—
297C2×C53—42no—
298C2×C53#248411yescharacteristic
299C2×C53#309111yescharacteristic
300C2×C53#309311yescharacteristic
301C2×C53#310211yescharacteristic
302C2×C53#311111yescharacteristic
303C2×C53#323311yescharacteristic
304C2×C53#324011yescharacteristic
305C2×#117—531no—
306C2×#117C2×D5311yescharacteristic
307C22×C42—531no—
308C2×#107—531no—
309C2×#107C2×D5311yescharacteristic
310C2×#106—531no—
311C2×#106C2×D5311yescharacteristic
312C2×C22⋊C8—531no—
313C2×C22⋊C8C2×D5311yescharacteristic
314#389—531no—
315#389—531no—
316#389—531no—
317#389—531no—
318#385—1061no—
319#385—1061no—
320#383—1061no—
321#382—531no—
322#382—531no—
323#381—1061no—
324#380—531no—
325#380C2×D5311yescharacteristic
326C4×C4⋊C4—531no—
327C4×C4⋊C4—531no—
328C4×C4⋊C4—1061no—
329C4×C4⋊C4—1061no—
330C4×C22⋊C4—1061no—
331#376—1061no—
332#340—531no—
333#340—531no—
334#340—531no—
335#340—531no—
336#340—531no—
337#340—531no—
338#340—531no—
339#340—531no—
340#340—531no—
341#340—531no—
342#340C2×D5311yescharacteristic
343#340C2×D5311yescharacteristic
344#336—531no—
345#336—531no—
346#336—531no—
347#336—531no—
348C53#2761811yescharacteristic, complemented, Sylow
349C23×C4—531no—
350C23×C4—531no—
351C23×C4C22×D5311yescharacteristic
352C22×C8—531no—
353C22×C8—531no—
354C22×C8C22×D5311yescharacteristic
355C22×C8C22×D5311yescharacteristic
356#117—22no—
357#117—22no—
358#117—1062no—
359#117—1062no—
360C2×C4⋊C4—531no—
361C2×C4⋊C4—531no—
362C2×C4⋊C4—531no—
363C2×C4⋊C4—531no—
364C2×C4⋊C4—1061no—
365C2×C4⋊C4—1061no—
366C2×C4⋊C4—1061no—
367C2×C4⋊C4—1061no—
368C2×C4⋊C4C22×D5311yescharacteristic
369C2×C4⋊C4C22×D5311yescharacteristic
370C2×C4⋊C4C22×D5311yescharacteristic
371C2×C4⋊C4C22×D5311yescharacteristic
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—1061no—
375C2×C22⋊C4—1061no—
376C2×C42—21no—
377C2×C42—531no—
378C2×C42—531no—
379C2×C42—531no—
380C2×C42—531no—
381C2×C42—1062no—
382C2×C42—1062no—
383C2×C42—1061no—
384C2×C42—1061no—
385C2×C42—1061no—
386#107—22no—
387#107—1062no—
388#106—22no—
389#106—1062no—
390C22⋊C8—22no—
391C22⋊C8—1062no—
392#95—21no—
393#95—1061no—
394#95—1061no—
395#95—1061no—
396#95—1061no—
397#95—1061no—
398#95—1061no—
399#95—1061no—
400#95—1061no—
401#95—1061no—
402#95—1061no—
403#95—1061no—
404C24D4×D5311yescharacteristic
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—531no—
414C22×C4—531no—
415C22×C4—531no—
416C22×C4—531no—
417C22×C4—1062no—
418C22×C4—1062no—
419C22×C4—1062no—
420C22×C4—1061no—
421C22×C4—1061no—
422C22×C4—1061no—
423C22×C4—1061no—
424C22×C4—2121no—
425C22×C4—2121no—
426C22×C4—2121no—
427C22×C4#1171994512yes—
428C22×C4D4×D5311yescharacteristic
429C22×C4#1171994912yes—
430C22×C4C23×D5311yescharacteristic
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—1061no—
438C2×C8—1061no—
439C2×C8—1061no—
440C2×C8—1061no—
441C2×C8—1061no—
442C2×C8—1061no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—1061no—
450C4⋊C4—1061no—
451C4⋊C4—1061no—
452C4⋊C4—1061no—
453C4⋊C4—1061no—
454C4⋊C4—1061no—
455C4⋊C4—1061no—
456C4⋊C4—1061no—
457C4⋊C4—2122no—
458C4⋊C4—2122no—
459C4⋊C4—2122no—
460C4⋊C4—2122no—
461C4⋊C4#1171994712yes—
462C4⋊C4#1171994712yes—
463C22⋊C4—24no—
464C22⋊C4—42no—
465C22⋊C4—2122no—
466C22⋊C4—2122no—
467C42—24no—
468C42—1062no—
469C42—1062no—
470C42—1061no—
471C42—1061no—
472C42—1061no—
473C42—1061no—
474C42—2122no—
475C42—2122no—
476C42—2122no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23#1313320212yes—
484C23C2×D4×D5311yescharacteristic
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—532no—
504C2×C4—1061no—
505C2×C4—1061no—
506C2×C4—1061no—
507C2×C4—1061no—
508C2×C4—1061no—
509C2×C4—1061no—
510C2×C4—1061no—
511C2×C4—1061no—
512C2×C4—1061no—
513C2×C4—1061no—
514C2×C4—1061no—
515C2×C4—2124no—
516C2×C4—2121no—
517C2×C4—2121no—
518C2×C4—2121no—
519C2×C4—2121no—
520C2×C4—2121no—
521C2×C4—2121no—
522C2×C4—2121no—
523C2×C4—2121no—
524C2×C4—2121no—
525C2×C4—2121no—
526C2×C4—2121no—
527C2×C4—2121no—
528C2×C4—2124no—
529C2×C4#1313320212yes—
530C2×C4C2×#1171994711yescharacteristic
531C2×C4C2×#1171994711yescharacteristic
532C2×C4C2×#1171994511yescharacteristic
533C2×C4C2×#1171994911yescharacteristic
534C2×C4#1313326411yescharacteristic
535C2×C4#1313326411yescharacteristic
536C8—41no—
537C8—41no—
538C8—2121no—
539C8—2121no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22C2×#1313320211yescharacteristic
548C22#1681112911yescharacteristic
549C22#1681119511yescharacteristic
550C22#135×D5311yescharacteristic
551C22#1681141911yescharacteristic
552C22#1681142011yescharacteristic
553C22#1681142211yescharacteristic
554C4—22no—
555C4—42no—
556C4—42no—
557C4—81no—
558C4—81no—
559C4—1062no—
560C4—4241no—
561C4—4241no—
562C4—4241no—
563C4—4241no—
564C4—2121no—
565C4—2121no—
566C4—2121no—
567C4#1681131311yescharacteristic
568C4#1681131311yescharacteristic
569C4#1681131511yescharacteristic
570C4#1681131511yescharacteristic
571C2—42no—
572C2#2027900111yescharacteristic
573C2#2028463211yescharacteristic
574C2#2028464911yescharacteristic
575C2#2028473411yescharacteristic
576C2#2028481711yescharacteristic
577C2#2028589111yescharacteristic
578C2#2028595911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1854C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1854D53
hypocenterC53#27618
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
3. #20276286C2semidirect
4. #20276286C2semidirect
5. #20276200C2semidirect
6. #20276199C2semidirect
7. #20275309C2semidirect
8. #20275303C2semidirect
9. #20275295C2semidirect
10. #20275289C2semidirect
348. C53#27618semidirect

Maximal subgroups

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

subgroupindex
#202771992
#202771102
#202762862
#202762002
#202761992
#202753092
#202753032
#202752952
#202752892
#202749142
#202748112
#202748072
#202747812
C53×#18542
#2761853