Finite Group Databasebeta

← #27469772

Subgroups — #27469772

Order 13568.

678 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20280539C211yesmaximal, characteristic
2#20277336C211yesmaximal, characteristic
3#20276288C211yesmaximal, characteristic, complemented
4#20276286C211yesmaximal, characteristic, complemented
5#20276281C211yesmaximal, characteristic, complemented
6#20276276C211yesmaximal, characteristic, complemented
7#20275688C211yesmaximal, characteristic, complemented
8#20275675C211yesmaximal, characteristic, complemented
9#20274845C211yesmaximal, characteristic, complemented
10#20274809C211yesmaximal, characteristic
11#20274808C211yesmaximal, characteristic
12#20274804C211yesmaximal, characteristic
13#20274800C211yesmaximal, characteristic
14#20274484C211yesmaximal, characteristic
15C53×#1836C211yesmaximal, characteristic, complemented
16C2×#13133217C2211yescharacteristic
17C2×#13133175C2211yescharacteristic
18C2×#13133174C2211yescharacteristic
19C2×C4⋊C4×D53C2211yescharacteristic
20C2×#13133129C2211yescharacteristic
21C2×#13133129C2211yescharacteristic
22C2×C4×#88322C2211yescharacteristic
23#16810563—21no—
24#16810560—21no—
25#16810544—21no—
26#16810536—21no—
27#16810522C2211yescharacteristic
28#16810522C2211yescharacteristic
29#16810518—21no—
30#16810518—21no—
31#16810494—21no—
32#16810494—21no—
33#16810492C2211yescharacteristic
34#16810489C2211yescharacteristic
35#16810488C2211yescharacteristic
36#16810480C2211yescharacteristic
37#16810480C2211yescharacteristic
38#16810386C2211yescharacteristic
39#16810386C2211yescharacteristic
40#16810386C2211yescharacteristic
41#16810386C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#97×C53C2211yescharacteristic
53C53×#384C2211yescharacteristic
54C53×#382C2211yescharacteristic
55C53×#340C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×Dic106—21no—
60C22×C4×D53—21no—
61C22×C4×D53C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133217—22no—
65#13133217—22no—
66#13133175—22no—
67#13133175—22no—
68#13133174—22no—
69#13133174—22no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919D411yescharacteristic
81C2×#11719919D411yescharacteristic
82C4⋊C4×D53—22no—
83C4⋊C4×D53—22no—
84C2×#11719921—21no—
85C2×#11719921—21no—
86C2×#11719921—21no—
87C2×#11719921D411yescharacteristic
88C2×#11719921D411yescharacteristic
89C2×C4×Dic53—21no—
90C2×C4×Dic53C2311yescharacteristic
91#13133129—22no—
92#13133129—22no—
93C4×#88322—22no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102C22×C8×C53C2311yescharacteristic
103C22×C8×C53C2311yescharacteristic
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53C2311yescharacteristic
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C42×C53C2311yescharacteristic
111#97×C53—22no—
112#95×C53—21no—
113#95×C53—21no—
114C23×D53C2×D411yescharacteristic
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2411yescharacteristic
122C22×Dic53—21no—
123C22×Dic53—21no—
124C22×Dic53—21no—
125C22×Dic53—21no—
126C22×Dic53C2×D411yescharacteristic
127C22×Dic53C2×D411yescharacteristic
128C22×Dic53C2×D411yescharacteristic
129C2×C4×D53—21no—
130C2×C4×D53—21no—
131C2×C4×D53—21no—
132C2×C4×D53—21no—
133C2×C4×D53—42no—
134C2×C4×D53—81no—
135C2×C4×D53—81no—
136C2×C4×D53#4112yes—
137C2×C4×D53#4112yes—
138C2×Dic106—42no—
139C2×Dic106—81no—
140C2×Dic106—81no—
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—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—81no—
156C4⋊C4×C53—81no—
157C42×C53#4112yes—
158C42×C53#4112yes—
159#11719921—22no—
160#11719921—22no—
161#11719921—42no—
162#11719921—81no—
163#11719921—81no—
164C4⋊Dic53—21no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—42no—
173C4⋊Dic53—81no—
174C4⋊Dic53—81no—
175#11719919—22no—
176#11719919—22no—
177#11719919—42no—
178#11719919—81no—
179#11719919—81no—
180C4×Dic53—21no—
181C4×Dic53—21no—
182C4×Dic53—42no—
183C2×#88322—21no—
184C2×#88322—21no—
185C2×#88322—21no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C23×C53C22×D411yescharacteristic
190C22×D53—21no—
191C22×D53—21no—
192C22×D53—21no—
193C22×D53—21no—
194C22×D53—21no—
195C22×D53—21no—
196C22×D53#12412yes—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—41no—
202C2×C4×C53—41no—
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×C53#12312yes—
215C2×C4×C53C2×#4111yescharacteristic
216C2×C4×C53C2×#4111yescharacteristic
217C2×C4×C53C2×#4111yescharacteristic
218C2×C4×C53C2×#4111yescharacteristic
219C2×C4×C532−1+411yescharacteristic
220C2×C4×C532−1+411yescharacteristic
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—41no—
230C2×Dic53—41no—
231C2×Dic53—41no—
232C2×Dic53—41no—
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#12412yes—
242C4×D53—22no—
243C4×D53—22no—
244C4×D53—22no—
245C4×D53—22no—
246C4×D53—84no—
247Dic106—84no—
248C8×C53—41no—
249C8×C53—41no—
250#88322—41no—
251#88322—41no—
252#27971—531nomaximal, Sylow
253C22×C53C2×#12311yescharacteristic
254C22×C53C2×#12411yescharacteristic
255C22×C53#53611yescharacteristic
256C22×C53#57611yescharacteristic
257C22×C53#57711yescharacteristic
258C22×C53#57711yescharacteristic
259C22×C53#57811yescharacteristic
260C2×D53—22no—
261C2×D53—22no—
262C2×D53—22no—
263C2×D53—41no—
264C2×D53—41no—
265C2×D53—41no—
266C2×D53—41no—
267C4×C53—22no—
268C4×C53—22no—
269C4×C53—81no—
270C4×C53—81no—
271C4×C53—81no—
272C4×C53—81no—
273C4×C53#55211yescharacteristic
274C4×C53#55211yescharacteristic
275C4×C53#55211yescharacteristic
276C4×C53#55211yescharacteristic
277Dic53—22no—
278Dic53—41no—
279Dic53—41no—
280Dic53—81no—
281Dic53—81no—
282Dic53—81no—
283Dic53—81no—
284#2628—531no—
285#2317—531no—
286#2101—531no—
287#2101—531no—
288#2100—531no—
289#2099—531no—
290#1993—531no—
291#1990—531no—
292#1836D5311yescharacteristic, complemented
293#1835—531no—
294#1825—531no—
295#1825—531no—
296#1824—531no—
297#1823—531no—
298#1750—531no—
299C2×C53#250311yescharacteristic
300C2×C53#309611yescharacteristic
301C2×C53#309711yescharacteristic
302C2×C53#313611yescharacteristic
303C2×C53#313711yescharacteristic
304C2×C53#324311yescharacteristic
305C2×C53#324411yescharacteristic
306D53—42no—
307C2×C4⋊Q8—531no—
308C2×#117—531no—
309C2×#117—531no—
310C22×C4⋊C4—531no—
311C2×C22⋊C8—531no—
312C2×C22⋊C8—531no—
313C2×#97C2×D5311yescharacteristic
314C2×C4×C8—531no—
315#400—1061no—
316#399—1061no—
317#395—1061no—
318#393—1061no—
319#389—531no—
320#389—531no—
321#389—1061no—
322#389—1061no—
323#385—1061no—
324#385—1061no—
325#384—531no—
326#384C2×D5311yescharacteristic
327#382—531no—
328#382—531no—
329#382C2×D5311yescharacteristic
330#380—531no—
331#380—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#340C2×D5311yescharacteristic
345#340C2×D5311yescharacteristic
346#340C2×D5311yescharacteristic
347#340C2×D5311yescharacteristic
348#336—531no—
349#336—531no—
350C53#2797111yescharacteristic, complemented, Sylow
351C22×Q8—1061no—
352C23×C4—531no—
353C23×C4—1061no—
354C22×C8—531no—
355C22×C8—531no—
356C22×C8C22×D5311yescharacteristic
357C22×C8C22×D5311yescharacteristic
358C4⋊Q8—1062no—
359C4⋊Q8—1062no—
360#117—1062no—
361#117—1062no—
362#117—1062no—
363#117—1062no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—531no—
367C2×C4⋊C4—531no—
368C2×C4⋊C4—531no—
369C2×C4⋊C4—531no—
370C2×C4⋊C4—531no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—1062no—
373C2×C4⋊C4—1062no—
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×C4⋊C4C22×D5311yescharacteristic
384C2×C22⋊C4—531no—
385C2×C22⋊C4—531no—
386C2×C22⋊C4—1061no—
387C2×C22⋊C4—1061no—
388C2×C22⋊C4—1061no—
389C2×C42—531no—
390C2×C42—1061no—
391C2×C42C22×D5311yescharacteristic
392C22⋊C8—1062no—
393C22⋊C8—1062no—
394#97—22no—
395C4×C8—1062no—
396#95—21no—
397#95—21no—
398#95—1061no—
399#95—1061no—
400#95—1061no—
401#95—1061no—
402#95—1061no—
403#95—1061no—
404#95—1061no—
405#95—1061no—
406C24—531no—
407C2×Q8—4241no—
408C2×Q8—4241no—
409C2×Q8—2122no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—531no—
415C22×C4—532no—
416C22×C4—532no—
417C22×C4—531no—
418C22×C4—531no—
419C22×C4—1061no—
420C22×C4—1061no—
421C22×C4—1061no—
422C22×C4—1061no—
423C22×C4—1061no—
424C22×C4—1061no—
425C22×C4—1061no—
426C22×C4—1061no—
427C22×C4—4241no—
428C22×C4—4241no—
429C22×C4—2122no—
430C22×C4D4×D5311yescharacteristic
431C22×C4D4×D5311yescharacteristic
432C22×C4C23×D5311yescharacteristic
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—1061no—
440C2×C8—1061no—
441C2×C8—1061no—
442C2×C8—1061no—
443C2×C8—1061no—
444C2×C8—1061no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—81no—
454C4⋊C4—81no—
455C4⋊C4—1062no—
456C4⋊C4—1062no—
457C4⋊C4—1061no—
458C4⋊C4—1061no—
459C4⋊C4—1061no—
460C4⋊C4—1061no—
461C4⋊C4—1061no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—4241no—
466C4⋊C4—4241no—
467C4⋊C4—4241no—
468C4⋊C4—4241no—
469C4⋊C4—2122no—
470C4⋊C4—2122no—
471C22⋊C4—1062no—
472C22⋊C4—1062no—
473C22⋊C4—4241no—
474C22⋊C4—4241no—
475C22⋊C4—2122no—
476C42—1061no—
477C42—1061no—
478C42—2122no—
479C42#1171994512yes—
480C42#1171994912yes—
481C23—532no—
482C23—1061no—
483C23—1061no—
484C23—1061no—
485C23—1061no—
486C23—1061no—
487C23—1061no—
488C23C2×D4×D5311yescharacteristic
489Q8—4244no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
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—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—532no—
507C2×C4—1061no—
508C2×C4—1061no—
509C2×C4—1062no—
510C2×C4—1062no—
511C2×C4—1062no—
512C2×C4—1062no—
513C2×C4—1061no—
514C2×C4—1061no—
515C2×C4—1061no—
516C2×C4—1061no—
517C2×C4—1061no—
518C2×C4—1061no—
519C2×C4—4244no—
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—2121no—
530C2×C4—2121no—
531C2×C4—2121no—
532C2×C4#1313320212yes—
533C2×C4#1313320212yes—
534C2×C4#41×D5311yescharacteristic
535C2×C4#41×D5311yescharacteristic
536C2×C4C2×#1171994511yescharacteristic
537C2×C4C2×#1171994911yescharacteristic
538C2×C4#1313326411yescharacteristic
539C2×C4#1313326411yescharacteristic
540C8—41no—
541C8—41no—
542C8—2121no—
543C8—2121no—
544C22—1062no—
545C22—1062no—
546C22—1062no—
547C22—2121no—
548C22—2121no—
549C22—2121no—
550C22—2121no—
551C22C2×#1313320211yescharacteristic
552C22#124×D5311yescharacteristic
553C22#1681119311yescharacteristic
554C22#1681142711yescharacteristic
555C22#1681143211yescharacteristic
556C22#1681143311yescharacteristic
557C22#1681143911yescharacteristic
558C4—22no—
559C4—22no—
560C4—81no—
561C4—81no—
562C4—81no—
563C4—81no—
564C4—1062no—
565C4—4241no—
566C4—4241no—
567C4—4241no—
568C4—4241no—
569C4—2121no—
570C4—2121no—
571C4#1681131111yescharacteristic
572C4#1681131111yescharacteristic
573C4#1681131211yescharacteristic
574C4#1681131211yescharacteristic
575C2—2122no—
576C2#2027919711yescharacteristic
577C2#2028467511yescharacteristic
578C2#2028468611yescharacteristic
579C2#2028507011yescharacteristic
580C2#2028508111yescharacteristic
581C2#2028599311yescharacteristic
582C2#2028600811yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1836C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#1836D53
hypocenterC53#27971
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1836

Complements

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

NH ≅ G/Ntype
3. #20276288C2semidirect
4. #20276286C2semidirect
5. #20276281C2semidirect
6. #20276276C2semidirect
7. #20275688C2semidirect
8. #20275675C2semidirect
9. #20274845C2semidirect
15. C53×#1836C2semidirect
292. #1836D53semidirect
350. C53#27971semidirect

Maximal subgroups

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

subgroupindex
#202805392
#202773362
#202762882
#202762862
#202762812
#202762762
#202756882
#202756752
#202748452
#202748092
#202748082
#202748042
#202748002
#202744842
C53×#18362
#2797153