Finite Group Databasebeta

← #27465442

Subgroups — #27465442

Order 13568.

670 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277324C211yesmaximal, characteristic
2#20276278C211yesmaximal, characteristic, complemented
3#20276273C211yesmaximal, characteristic, complemented
4#20276196C211yesmaximal, characteristic, complemented
5#20276196C211yesmaximal, characteristic, complemented
6#20275304C211yesmaximal, characteristic, complemented
7#20275297C211yesmaximal, characteristic, complemented
8#20275290C211yesmaximal, characteristic, complemented
9#20275283C211yesmaximal, characteristic, complemented
10#20274908C211yesmaximal, characteristic
11#20274903C211yesmaximal, characteristic
12#20274810C211yesmaximal, characteristic
13#20274810C211yesmaximal, characteristic
14#20274786C211yesmaximal, characteristic
15C53×#2316C211yesmaximal, characteristic
16C2×#13133150C2211yescharacteristic
17C2×#13133150C2211yescharacteristic
18C2×#13133148C2211yescharacteristic
19C2×#13133148C2211yescharacteristic
20C2×C22⋊#88322C2211yescharacteristic
21C2×C22⋊#88322C2211yescharacteristic
22#16810516—21no—
23#16810516—21no—
24#16810509—21no—
25#16810509—21no—
26#16810493C2211yescharacteristic
27#16810489C2211yescharacteristic
28#16810483C2211yescharacteristic
29#16810483C2211yescharacteristic
30C4×C4⋊Dic53C2211yescharacteristic
31C4×C4⋊Dic53C2211yescharacteristic
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#16810378C2211yescharacteristic
45#16810378C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×#117×C53C2211yescharacteristic
49C2×#117×C53C2211yescharacteristic
50C22×C4⋊C4×C53C2211yescharacteristic
51C53×#389—21no—
52C53×#389—21no—
53C53×#389C2211yescharacteristic
54C53×#389C2211yescharacteristic
55C53×#389C2211yescharacteristic
56C53×#389C2211yescharacteristic
57C53×#380—21no—
58C53×#380—21no—
59C22×#88322C2311yescharacteristic
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C2×C4⋊Dic53—21no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53C2311yescharacteristic
66C2×C4⋊Dic53C2311yescharacteristic
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C22⋊Dic53—21no—
70C2×C22⋊Dic53—21no—
71C2×C4×Dic53—21no—
72C2×C4×Dic53—21no—
73#13133150—22no—
74#13133150—22no—
75#13133148—22no—
76#13133148—22no—
77C22⋊#88322—22no—
78C22⋊#88322—22no—
79#13133124—21no—
80#13133124—21no—
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—22no—
92C2×C4⋊C4×C53—22no—
93C2×C4⋊C4×C53—21no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53—21no—
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—24no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
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—24no—
139C4⋊C4×C53—22no—
140C4⋊C4×C53—22no—
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—81no—
148C4⋊C4×C53—81no—
149C4⋊C4×C53—81no—
150C4⋊C4×C53—81no—
151C4⋊C4×C53#4112yes—
152C4⋊C4×C53#4112yes—
153C22⋊C4×C53—22no—
154C22⋊C4×C53—22no—
155C22⋊C4×C53—42no—
156C42×C53—21no—
157C42×C53—21no—
158C42×C53—21no—
159C42×C53—21no—
160C42×C53—42no—
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—
173C4×Dic53—81no—
174C4×Dic53—81no—
175C2×#88322—21no—
176C2×#88322—21no—
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—
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—21no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
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—21no—
214C2×C4×C53—21no—
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×C53C2×#4111yescharacteristic
228C2×C4×C53C2×#4111yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C2×C4×C53C2×#4111yescharacteristic
231C2×C4×C532−1+411yescharacteristic
232C2×C4×C532−1+411yescharacteristic
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—
245#88322—41no—
246#88322—41no—
247#88322—41no—
248#88322—41no—
249#27623—531nomaximal, Sylow
250C22×C53—22no—
251C22×C53—22no—
252C22×C53—22no—
253C22×C53—41no—
254C22×C53—41no—
255C22×C53—41no—
256C22×C53—41no—
257C22×C53C2×#12311yescharacteristic
258C22×C53#52511yescharacteristic
259C22×C53#53611yescharacteristic
260C22×C53#57611yescharacteristic
261C22×C53#57711yescharacteristic
262C22×C53#57711yescharacteristic
263C22×C53#57811yescharacteristic
264C4×C53—22no—
265C4×C53—41no—
266C4×C53—41no—
267C4×C53—41no—
268C4×C53—41no—
269C4×C53—42no—
270C4×C53—42no—
271C4×C53—81no—
272C4×C53—81no—
273C4×C53#55211yescharacteristic
274C4×C53#55211yescharacteristic
275C4×C53#55211yescharacteristic
276C4×C53#55211yescharacteristic
277Dic53—81no—
278Dic53—81no—
279Dic53—81no—
280Dic53—81no—
281#2316D5311yescharacteristic
282#2314—531no—
283#2100—531no—
284#2099—531no—
285#2085—531no—
286#2085—531no—
287#1921—531no—
288#1920—531no—
289#1919—531no—
290#1918—531no—
291#1853—531no—
292#1852—531no—
293#1825—531no—
294#1825—531no—
295#1818—531no—
296C2×C53—42no—
297C2×C53#248411yescharacteristic
298C2×C53#309811yescharacteristic
299C2×C53#309911yescharacteristic
300C2×C53#311811yescharacteristic
301C2×C53#312111yescharacteristic
302C2×C53#324111yescharacteristic
303C2×C53#324211yescharacteristic
304C2×#117C2×D5311yescharacteristic
305C2×#117C2×D5311yescharacteristic
306C22×C4⋊C4C2×D5311yescharacteristic
307C2×#107—531no—
308C2×#107—531no—
309C2×#106—531no—
310C2×#106—531no—
311C2×C22⋊C8—531no—
312C2×C22⋊C8—531no—
313#389—21no—
314#389—21no—
315#389C2×D5311yescharacteristic
316#389C2×D5311yescharacteristic
317#389C2×D5311yescharacteristic
318#389C2×D5311yescharacteristic
319#388—1061no—
320#388—1061no—
321#387—1061no—
322#387—1061no—
323#384—531no—
324#382—531no—
325#380—21no—
326#380—21no—
327#380—531no—
328#380—531no—
329C4×C4⋊C4—531no—
330C4×C4⋊C4—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#336—531no—
344#336—531no—
345#336—531no—
346#336—531no—
347C53#2762311yescharacteristic, complemented, Sylow
348C23×C4—21no—
349C23×C4C22×D5311yescharacteristic
350C22×C8—531no—
351C22×C8—531no—
352C22×C8—531no—
353C22×C8—531no—
354#117—22no—
355#117—22no—
356#117—22no—
357#117—22no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—531no—
366C2×C4⋊C4—531no—
367C2×C4⋊C4—531no—
368C2×C4⋊C4—531no—
369C2×C4⋊C4—1061no—
370C2×C4⋊C4—1061no—
371C2×C4⋊C4C22⋊D5311yescharacteristic
372C2×C4⋊C4C22⋊D5311yescharacteristic
373C2×C4⋊C4C22×D5311yescharacteristic
374C2×C4⋊C4C22×D5311yescharacteristic
375C2×C4⋊C4C22×D5311yescharacteristic
376C2×C4⋊C4C22×D5311yescharacteristic
377C2×C22⋊C4—21no—
378C2×C22⋊C4—1061no—
379C2×C22⋊C4—1061no—
380C2×C22⋊C4C22⋊D5311yescharacteristic
381C2×C22⋊C4C22⋊D5311yescharacteristic
382C2×C42—21no—
383C2×C42—1061no—
384C2×C42—1061no—
385C2×C42C22×D5311yescharacteristic
386C2×C42C22×D5311yescharacteristic
387#107—1062no—
388#107—1062no—
389#106—1062no—
390#106—1062no—
391C22⋊C8—1062no—
392C22⋊C8—1062no—
393#95—21no—
394#95—21no—
395#95—21no—
396#95—21no—
397#95—1061no—
398#95—1061no—
399#95—1061no—
400#95—1061no—
401#95—1061no—
402#95—1061no—
403C24C2×C22⋊D5311yescharacteristic
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—81no—
414C22×C4—81no—
415C22×C4—1061no—
416C22×C4—1061no—
417C22×C4—1061no—
418C22×C4—1061no—
419C22×C4#1171994712yes—
420C22×C4#1171994712yes—
421C22×C4C2×C22⋊D5311yescharacteristic
422C22×C4C2×C22⋊D5311yescharacteristic
423C22×C4C2×C22⋊D5311yescharacteristic
424C22×C4C2×C22⋊D5311yescharacteristic
425C22×C4C2×C22⋊D5311yescharacteristic
426C22×C4C23×D5311yescharacteristic
427C2×C8—1061no—
428C2×C8—1061no—
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—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—81no—
449C4⋊C4—81no—
450C4⋊C4—81no—
451C4⋊C4—81no—
452C4⋊C4—1061no—
453C4⋊C4—1061no—
454C4⋊C4—1061no—
455C4⋊C4—1061no—
456C4⋊C4—1061no—
457C4⋊C4—1061no—
458C4⋊C4—1061no—
459C4⋊C4—1061no—
460C4⋊C4—4241no—
461C4⋊C4—4241no—
462C4⋊C4#1171994512yes—
463C4⋊C4#1171994512yes—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—42no—
467C22⋊C4—4241no—
468C22⋊C4—4241no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—42no—
474C42—4241no—
475C42—4241no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23#1313319812yes—
483C23C22×C22⋊D5311yescharacteristic
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
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—84no—
515C2×C4—2121no—
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#1313319812yes—
528C2×C4C2×#1171994711yescharacteristic
529C2×C4C2×#1171994711yescharacteristic
530C2×C4C2×#1171994511yescharacteristic
531C2×C4C2×#1171994511yescharacteristic
532C2×C4#1313326511yescharacteristic
533C2×C4#1313326511yescharacteristic
534C8—2121no—
535C8—2121no—
536C8—2121no—
537C8—2121no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22C2×#1313319811yescharacteristic
546C22#1681113411yescharacteristic
547C22#1681119711yescharacteristic
548C22#1681142611yescharacteristic
549C22#1681143111yescharacteristic
550C22#1681143411yescharacteristic
551C22#1681143811yescharacteristic
552C4—22no—
553C4—41no—
554C4—41no—
555C4—41no—
556C4—41no—
557C4—42no—
558C4—42no—
559C4—81no—
560C4—81no—
561C4—4241no—
562C4—4241no—
563C4—4241no—
564C4—4241no—
565C4#1681130811yescharacteristic
566C4#1681130811yescharacteristic
567C4#1681130811yescharacteristic
568C4#1681130811yescharacteristic
569C2—42no—
570C2#2027900211yescharacteristic
571C2#2028468911yescharacteristic
572C2#2028470011yescharacteristic
573C2#2028489911yescharacteristic
574C2#2028494411yescharacteristic
575C2#2028596811yescharacteristic
576C2#2028598311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2316C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2316D53
hypocenterC53#27623
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2316

Complements

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

NH ≅ G/Ntype
2. #20276278C2semidirect
3. #20276273C2semidirect
4. #20276196C2semidirect
5. #20276196C2semidirect
6. #20275304C2semidirect
7. #20275297C2semidirect
8. #20275290C2semidirect
9. #20275283C2semidirect
347. C53#27623semidirect

Maximal subgroups

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

subgroupindex
#202773242
#202762782
#202762732
#202761962
#202753042
#202752972
#202752902
#202752832
#202749082
#202749032
#202748102
#202747862
C53×#23162
#2762353