Finite Group Databasebeta

← #27469745

Subgroups — #27469745

Order 13568.

670 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20280545C211yesmaximal, characteristic
2#20276282C211yesmaximal, characteristic, complemented
3#20276282C211yesmaximal, characteristic, complemented
4#20276277C211yesmaximal, characteristic, complemented
5#20276272C211yesmaximal, characteristic, complemented
6#20275683C211yesmaximal, characteristic, complemented
7#20275679C211yesmaximal, characteristic, complemented
8#20274848C211yesmaximal, characteristic, complemented
9#20274842C211yesmaximal, characteristic, complemented
10#20274810C211yesmaximal, characteristic
11#20274810C211yesmaximal, characteristic
12#20274805C211yesmaximal, characteristic
13#20274801C211yesmaximal, characteristic
14#20274481C211yesmaximal, characteristic
15C53×#2317C211yesmaximal, characteristic
16C2×#13133209C2211yescharacteristic
17C2×C22⋊#88322C2211yescharacteristic
18C2×C22⋊#88322C2211yescharacteristic
19C2×#13133128C2211yescharacteristic
20C2×C4×#88322C2211yescharacteristic
21#16810556—21no—
22#16810556—21no—
23#16810546—21no—
24#16810546—21no—
25#16810521C2211yescharacteristic
26#16810521C2211yescharacteristic
27#16810493C2211yescharacteristic
28#16810483C2211yescharacteristic
29#16810483C2211yescharacteristic
30#16810385C2211yescharacteristic
31#16810385C2211yescharacteristic
32#16810385C2211yescharacteristic
33#16810385C2211yescharacteristic
34#16810385C2211yescharacteristic
35#16810385C2211yescharacteristic
36#16810385C2211yescharacteristic
37#16810385C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×#117×C53C2211yescharacteristic
49C2×#117×C53C2211yescharacteristic
50C22×C4⋊C4×C53C2211yescharacteristic
51C53×#389—21no—
52C53×#389—21no—
53C53×#385—21no—
54C53×#385—21no—
55C53×#384C2211yescharacteristic
56C53×#384C2211yescharacteristic
57C53×#382C2211yescharacteristic
58C53×#382C2211yescharacteristic
59C22×#88322C2311yescharacteristic
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63#13133209—22no—
64#13133209—22no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×#11719919—21no—
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×C22⋊Dic53—21no—
76C2×C22⋊Dic53—21no—
77C22⋊#88322—22no—
78C22⋊#88322—22no—
79#13133128—22no—
80C4×#88322—22no—
81#13133124—21no—
82#13133124—21no—
83#13133124—21no—
84#13133124—21no—
85C23×C4×C53—21no—
86C23×C4×C53C2311yescharacteristic
87#117×C53—22no—
88#117×C53—22no—
89#117×C53—22no—
90#117×C53—22no—
91C2×C4⋊C4×C53—21no—
92C2×C4⋊C4×C53—21no—
93C2×C4⋊C4×C53—21no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53—22no—
97C2×C4⋊C4×C53—22no—
98C2×C4⋊C4×C53D411yescharacteristic
99C2×C4⋊C4×C53D411yescharacteristic
100C2×C4⋊C4×C53C2311yescharacteristic
101C2×C4⋊C4×C53C2311yescharacteristic
102C2×C4⋊C4×C53C2311yescharacteristic
103C2×C4⋊C4×C53C2311yescharacteristic
104C2×C22⋊C4×C53—21no—
105C2×C22⋊C4×C53D411yescharacteristic
106C2×C22⋊C4×C53D411yescharacteristic
107C2×C42×C53—21no—
108C2×C42×C53C2311yescharacteristic
109C2×C42×C53C2311yescharacteristic
110#95×C53—21no—
111#95×C53—21no—
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—42no—
124C22×C4×C53—81no—
125C22×C4×C53—81no—
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53#4112yes—
132C22×C4×C53#4112yes—
133C22×C4×C53C2411yescharacteristic
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C4⋊C4×C53—22no—
139C4⋊C4×C53—22no—
140C4⋊C4×C53—21no—
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—42no—
149C4⋊C4×C53—81no—
150C4⋊C4×C53—81no—
151C4⋊C4×C53—81no—
152C4⋊C4×C53—81no—
153C22⋊C4×C53—22no—
154C22⋊C4×C53—22no—
155C22⋊C4×C53—42no—
156C42×C53—21no—
157C42×C53—21no—
158C42×C53—42no—
159C42×C53#4112yes—
160C42×C53#4112yes—
161C22⋊Dic53—81no—
162C22⋊Dic53—81no—
163C4⋊Dic53—21no—
164C4⋊Dic53—21no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—81no—
172C4⋊Dic53—81no—
173#11719919—42no—
174#11719919—42no—
175#11719919—81no—
176#11719919—81no—
177C2×#88322—21no—
178C2×#88322—21no—
179C2×#88322—21no—
180C2×#88322—21no—
181C2×#88322—21no—
182C2×#88322—21no—
183C2×#88322—21no—
184C2×#88322—21no—
185C2×#88322—21no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C23×C53—21no—
190C23×C53—21no—
191C23×C53—21no—
192C23×C53—21no—
193C23×C53—21no—
194C23×C53—21no—
195C23×C53#12412yes—
196C23×C53C22×D411yescharacteristic
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—22no—
204C2×C4×C53—22no—
205C2×C4×C53—22no—
206C2×C4×C53—22no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—41no—
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—84no—
226C2×C4×C53#12312yes—
227C2×C4×C53#12312yes—
228C2×C4×C53#12412yes—
229C2×C4×C53C2×#4111yescharacteristic
230C2×C4×C53C2×#4111yescharacteristic
231C2×C4×C53C2×#4111yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×C4×C532−1+411yescharacteristic
234C2×C4×C532−1+411yescharacteristic
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—41no—
246C2×Dic53—41no—
247#88322—41no—
248#88322—41no—
249#88322—41no—
250#88322—41no—
251#27970—531nomaximal, Sylow
252C22×C53—22no—
253C22×C53—22no—
254C22×C53—22no—
255C22×C53—41no—
256C22×C53—41no—
257C22×C53—41no—
258C22×C53—41no—
259C22×C53C2×#12311yescharacteristic
260C22×C53C2×#12411yescharacteristic
261C22×C53#53611yescharacteristic
262C22×C53#57611yescharacteristic
263C22×C53#57711yescharacteristic
264C22×C53#57711yescharacteristic
265C22×C53#57811yescharacteristic
266C4×C53—22no—
267C4×C53—22no—
268C4×C53—22no—
269C4×C53—41no—
270C4×C53—41no—
271C4×C53—81no—
272C4×C53—81no—
273C4×C53—81no—
274C4×C53—81no—
275C4×C53#55211yescharacteristic
276C4×C53#55211yescharacteristic
277C4×C53#55211yescharacteristic
278C4×C53#55211yescharacteristic
279Dic53—81no—
280Dic53—81no—
281Dic53—81no—
282Dic53—81no—
283#2629—531no—
284#2317D5311yescharacteristic
285#2101—531no—
286#2101—531no—
287#2100—531no—
288#2099—531no—
289#1992—531no—
290#1991—531no—
291#1836—531no—
292#1834—531no—
293#1825—531no—
294#1825—531no—
295#1824—531no—
296#1823—531no—
297#1750—531no—
298C2×C53—42no—
299C2×C53#250311yescharacteristic
300C2×C53#309811yescharacteristic
301C2×C53#309911yescharacteristic
302C2×C53#313611yescharacteristic
303C2×C53#313711yescharacteristic
304C2×C53#324111yescharacteristic
305C2×C53#324211yescharacteristic
306C2×#125—531no—
307C2×#117C2×D5311yescharacteristic
308C2×#117C2×D5311yescharacteristic
309C22×C4⋊C4C2×D5311yescharacteristic
310C2×C22⋊C8—531no—
311C2×C22⋊C8—531no—
312C2×#97—531no—
313C2×C4×C8—531no—
314#398—1061no—
315#398—1061no—
316#396—1061no—
317#396—1061no—
318#389—21no—
319#389—21no—
320#389—531no—
321#389—531no—
322#385—21no—
323#385—21no—
324#384—531no—
325#384C2×D5311yescharacteristic
326#384C2×D5311yescharacteristic
327#382C2×D5311yescharacteristic
328#382C2×D5311yescharacteristic
329#380—531no—
330#380—531no—
331#340—531no—
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#340—531no—
343#340—531no—
344#340—531no—
345#340—531no—
346#340—531no—
347#336—531no—
348#336—531no—
349C53#2797011yescharacteristic, complemented, Sylow
350C23×C4—21no—
351C23×C4C22×D5311yescharacteristic
352C22×C8—531no—
353C22×C8—531no—
354C22×C8—531no—
355C22×C8—531no—
356#125—1062no—
357#125—1062no—
358#117—22no—
359#117—22no—
360#117—22no—
361#117—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—531no—
370C2×C4⋊C4—531no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—531no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4—1061no—
377C2×C4⋊C4—1061no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4C22⋊D5311yescharacteristic
380C2×C4⋊C4C22⋊D5311yescharacteristic
381C2×C4⋊C4C22×D5311yescharacteristic
382C2×C4⋊C4C22×D5311yescharacteristic
383C2×C4⋊C4C22×D5311yescharacteristic
384C2×C4⋊C4C22×D5311yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—1061no—
387C2×C22⋊C4—1061no—
388C2×C22⋊C4C22⋊D5311yescharacteristic
389C2×C22⋊C4C22⋊D5311yescharacteristic
390C2×C42—21no—
391C2×C42C22×D5311yescharacteristic
392C2×C42C22×D5311yescharacteristic
393C22⋊C8—1062no—
394C22⋊C8—1062no—
395#97—1062no—
396C4×C8—1062no—
397#95—21no—
398#95—21no—
399#95—21no—
400#95—21no—
401#95—1061no—
402#95—1061no—
403#95—1061no—
404#95—1061no—
405C24C2×C22⋊D5311yescharacteristic
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—42no—
415C22×C4—81no—
416C22×C4—81no—
417C22×C4—1061no—
418C22×C4—1061no—
419C22×C4—1061no—
420C22×C4—1061no—
421C22×C4#1171994712yes—
422C22×C4#1171994712yes—
423C22×C4C2×C22⋊D5311yescharacteristic
424C22×C4C2×C22⋊D5311yescharacteristic
425C22×C4C2×C22⋊D5311yescharacteristic
426C22×C4C2×C22⋊D5311yescharacteristic
427C22×C4C2×C22⋊D5311yescharacteristic
428C22×C4C23×D5311yescharacteristic
429C2×C8—1061no—
430C2×C8—1061no—
431C2×C8—1061no—
432C2×C8—1061no—
433C2×C8—1061no—
434C2×C8—1061no—
435C2×C8—1061no—
436C2×C8—1061no—
437C2×C8—1061no—
438C2×C8—1061no—
439C2×C8—1061no—
440C2×C8—1061no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—42no—
452C4⋊C4—81no—
453C4⋊C4—81no—
454C4⋊C4—81no—
455C4⋊C4—81no—
456C4⋊C4—1061no—
457C4⋊C4—1061no—
458C4⋊C4—1061no—
459C4⋊C4—1061no—
460C4⋊C4—1061no—
461C4⋊C4—1061no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—4241no—
465C4⋊C4—4241no—
466C4⋊C4—4241no—
467C4⋊C4—4241no—
468C4⋊C4—2122no—
469C4⋊C4—2122no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—42no—
473C22⋊C4—4241no—
474C22⋊C4—4241no—
475C42—21no—
476C42—21no—
477C42—42no—
478C42#1171994512yes—
479C42#1171994512yes—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23#1313320412yes—
487C23C22×C22⋊D5311yescharacteristic
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
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—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—84no—
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—2121no—
529C2×C4#1313320012yes—
530C2×C4#1313320012yes—
531C2×C4#1313320412yes—
532C2×C4C2×#1171994711yescharacteristic
533C2×C4C2×#1171994711yescharacteristic
534C2×C4C2×#1171994511yescharacteristic
535C2×C4C2×#1171994511yescharacteristic
536C2×C4#1313326511yescharacteristic
537C2×C4#1313326511yescharacteristic
538C8—2121no—
539C8—2121no—
540C8—2121no—
541C8—2121no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22C2×#1313320011yescharacteristic
550C22C2×#1313320411yescharacteristic
551C22#1681119711yescharacteristic
552C22#1681142611yescharacteristic
553C22#1681143111yescharacteristic
554C22#1681143411yescharacteristic
555C22#1681143811yescharacteristic
556C4—22no—
557C4—22no—
558C4—22no—
559C4—41no—
560C4—41no—
561C4—81no—
562C4—81no—
563C4—81no—
564C4—81no—
565C4—4241no—
566C4—4241no—
567C4—4241no—
568C4—4241no—
569C4#1681130811yescharacteristic
570C4#1681130811yescharacteristic
571C4#1681130811yescharacteristic
572C4#1681130811yescharacteristic
573C2—42no—
574C2#2027919611yescharacteristic
575C2#2028469311yescharacteristic
576C2#2028470411yescharacteristic
577C2#2028506911yescharacteristic
578C2#2028508011yescharacteristic
579C2#2028596811yescharacteristic
580C2#2028598311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2317C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2317D53
hypocenterC53#27970
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
2. #20276282C2semidirect
3. #20276282C2semidirect
4. #20276277C2semidirect
5. #20276272C2semidirect
6. #20275683C2semidirect
7. #20275679C2semidirect
8. #20274848C2semidirect
9. #20274842C2semidirect
349. C53#27970semidirect

Maximal subgroups

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

subgroupindex
#202805452
#202762822
#202762772
#202762722
#202756832
#202756792
#202748482
#202748422
#202748102
#202748052
#202748012
#202744812
C53×#23172
#2797053