Finite Group Databasebeta

← #27396335

Subgroups — #27396335

Order 13568.

732 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810923C211yesmaximal, characteristic
2#20277454C211yesmaximal, characteristic, complemented
3#20277358C211yesmaximal, characteristic
4#20275670C211yesmaximal, characteristic, complemented
5#20275471C211yesmaximal, characteristic, complemented
6#20275457C211yesmaximal, characteristic, complemented
7#20275447C211yesmaximal, characteristic, complemented
8#20275409C211yesmaximal, characteristic
9#20275395C211yesmaximal, characteristic
10#20275338C211yesmaximal, characteristic
11#20275338C211yesmaximal, characteristic
12#20275310C211yesmaximal, characteristic, complemented
13#20275296C211yesmaximal, characteristic, complemented
14#20274574C211yesmaximal, characteristic, complemented
15C53×#1802C211yesmaximal, characteristic
16C2×#13133209C2211yescharacteristic
17#16810923C418yescomplemented
18C2×#13133150C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×C4⋊#88322C2211yescharacteristic
23C2×#13133138C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×#13133127C2211yescharacteristic
26C2×C8×Dic53C2211yescharacteristic
27#16810526C2211yescharacteristic
28#16810526C2211yescharacteristic
29#16810522C2211yescharacteristic
30#16810522C2211yescharacteristic
31#16810519C2211yescharacteristic
32#16810519C2211yescharacteristic
33#16810518—21no—
34#16810518—21no—
35#16810516—21no—
36#16810516—21no—
37#16810504—21no—
38#16810504—21no—
39#16810486—21no—
40#16810483—21no—
41C4⋊C4×Dic53C2211yescharacteristic
42C4⋊C4×Dic53C2211yescharacteristic
43C4×C4⋊Dic53C2211yescharacteristic
44#16810469C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×C4×D4×C53C2211yescharacteristic
54C2×C4⋊C8×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C2×#102×C53C2211yescharacteristic
57C53×#384C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62#13133209—22no—
63#13133209C2×C414yes—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53C2311yescharacteristic
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×#11719919—21no—
71C2×#11719919—21no—
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×C22⋊Dic53—21no—
75C2×C22⋊Dic53—21no—
76C2×C22⋊Dic53—21no—
77C2×C22⋊Dic53—21no—
78C2×C4×Dic53—21no—
79C2×C4×Dic53—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53C2311yescharacteristic
82C2×C4×Dic53C2311yescharacteristic
83#13133150—41no—
84#13133150—41no—
85#13133150C2×C414yes—
86#13133148C2×C414yes—
87C4⋊#88322C2×C414yes—
88#13133138C2×C414yes—
89#13133138C2×C414yes—
90#13133127—41no—
91C8×Dic53—41no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124D411yescharacteristic
100#13133124D411yescharacteristic
101#13133124D411yescharacteristic
102#13133124D411yescharacteristic
103C22×D4×C53C2311yescharacteristic
104C23×C4×C53—21no—
105C22×C8×C53C2311yescharacteristic
106C22×C8×C53C2311yescharacteristic
107C4×D4×C53—22no—
108C4×D4×C53C2×C414yes—
109C2×C4⋊C4×C53—21no—
110C2×C4⋊C4×C53—21no—
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C22⋊C4×C53—21no—
115C2×C42×C53C2311yescharacteristic
116C4⋊C8×C53—22no—
117#102×C53—41no—
118#102×C53—41no—
119C24×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53—21no—
122C2×D4×C53—21no—
123C2×D4×C53—21no—
124C2×D4×C53—21no—
125C2×D4×C53C22×C412yes—
126C22×C4×C53—21no—
127C22×C4×C53—24no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—81no—
131C22×C4×C53—81no—
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2411yescharacteristic
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53C2×D411yescharacteristic
140C22×Dic53C2×D411yescharacteristic
141C22×Dic53C2×D411yescharacteristic
142C22×Dic53C2×D411yescharacteristic
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—21no—
154C4⋊C4×C53—81no—
155C4⋊C4×C53—81no—
156C4⋊C4×C53C22×C412yes—
157C22⋊C4×C53—24no—
158C42×C53C22×C412yes—
159C42×C53#4112yes—
160C22⋊Dic53—81no—
161C22⋊Dic53—81no—
162C22⋊Dic53—81no—
163C22⋊Dic53—81no—
164C4⋊Dic53—21no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—81no—
171C4⋊Dic53—81no—
172C4⋊Dic53C22×C412yes—
173C4⋊Dic53C22×C412yes—
174#11719919—24no—
175#11719919—24no—
176#11719919—81no—
177#11719919—81no—
178C4×Dic53—21no—
179C4×Dic53—21no—
180C4×Dic53—21no—
181C4×Dic53—21no—
182C4×Dic53—81no—
183C4×Dic53—81no—
184C4×Dic53—81no—
185C2×#88322—21no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322C22×C412yes—
190C2×#88322C22×C412yes—
191C23×C53—22no—
192C23×C53—41no—
193C23×C53—41no—
194C23×C53—41no—
195C23×C53—41no—
196C23×C53—41no—
197C23×C53—41no—
198C23×C53C22×D411yescharacteristic
199D4×C53—22no—
200D4×C53—22no—
201D4×C53—41no—
202D4×C53—41no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—22no—
208C2×C4×C53—41no—
209C2×C4×C53—41no—
210C2×C4×C53—41no—
211C2×C4×C53—41no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—84no—
217C2×C4×C53C4×D412yes—
218C2×C4×C53C4×D412yes—
219C2×C4×C53C23×C411yescharacteristic
220C2×C4×C53C2×#4111yescharacteristic
221C2×C4×C53H(5,𝔽2)11yescharacteristic
222C2×C4×C532−1+411yescharacteristic
223C2×C4×C532−1+411yescharacteristic
224C2×C4×C532−1+411yescharacteristic
225C2×Dic53—22no—
226C2×Dic53—22no—
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
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—
249C8×C53—41no—
250C8×C53—41no—
251#88322—22no—
252#88322—22no—
253#22126—531nomaximal, Sylow
254C22×C53—42no—
255C22×C53—42no—
256C22×C53—42no—
257C22×C53—81no—
258C22×C53—81no—
259C22×C53—81no—
260C22×C53—81no—
261C22×C53C2×C4×D411yescharacteristic
262C22×C53#53611yescharacteristic
263C22×C53#53711yescharacteristic
264C22×C53#57611yescharacteristic
265C22×C53#57611yescharacteristic
266C22×C53#57711yescharacteristic
267C22×C53#57811yescharacteristic
268C4×C53—22no—
269C4×C53—22no—
270C4×C53—42no—
271C4×C53—81no—
272C4×C53—81no—
273C4×C53#51911yescharacteristic
274C4×C53#52011yescharacteristic
275C4×C53#54111yescharacteristic
276C4×C53#54211yescharacteristic
277Dic53—42no—
278Dic53—42no—
279Dic53—41no—
280Dic53—41no—
281Dic53—41no—
282Dic53—41no—
283Dic53—81no—
284Dic53—81no—
285C2×#478—531no—
286#2330—531no—
287#2319—531no—
288#1989—531no—
289#1947—531no—
290#1944—531no—
291#1942—531no—
292#1936—531no—
293#1934—531no—
294#1925—531no—
295#1925—531no—
296#1921—531no—
297#1919—531no—
298#1802D5311yescharacteristic
299#1768—531no—
300C2×C53—82no—
301C2×C53#232311yescharacteristic
302C2×C53#294611yescharacteristic
303C2×C53#295111yescharacteristic
304C2×C53#324111yescharacteristic
305C2×C53#324311yescharacteristic
306C2×C53#325811yescharacteristic
307C2×C53#325911yescharacteristic
308C2×#125—531no—
309C2×C4×D4C2×D5311yescharacteristic
310#478—538no—
311C2×#107—531no—
312C2×#107—531no—
313C2×#107—531no—
314C2×#106—531no—
315C2×C4⋊C8—531no—
316C2×C4⋊C8C2×D5311yescharacteristic
317C2×#102—531no—
318C2×#102—531no—
319C2×#102C2×D5311yescharacteristic
320C2×#102C2×D5311yescharacteristic
321C2×#97—531no—
322C2×C4×C8—531no—
323#390—531no—
324#390—531no—
325#389—531no—
326#389—531no—
327#389—531no—
328#389—531no—
329#389—1061no—
330#389—1061no—
331#388—1061no—
332#388—1061no—
333#387—1061no—
334#387—1061no—
335#384C2×D5311yescharacteristic
336#381—1061no—
337#380—1061no—
338C4×C4⋊C4—531no—
339C4×C4⋊C4—531no—
340C4×C4⋊C4—531no—
341#376—531no—
342#340—531no—
343#340—531no—
344#340—531no—
345#340—531no—
346#340—531no—
347#340—531no—
348#340C2×D5311yescharacteristic
349#340C2×D5311yescharacteristic
350#336—531no—
351#336—531no—
352C53#2212611yescharacteristic, complemented, Sylow
353C22×D4C22×D5311yescharacteristic
354C23×C4—21no—
355C22×C8—531no—
356C22×C8—531no—
357C22×C8C22×D5311yescharacteristic
358C22×C8C22×D5311yescharacteristic
359#125—534no—
360#125—1062no—
361C4×D4—22no—
362C4×D4C4×D5314yes—
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⋊C4—1061no—
372C2×C4⋊C4—1061no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4C22×D5311yescharacteristic
376C2×C4⋊C4C22×D5311yescharacteristic
377C2×C4⋊C4C22×D5311yescharacteristic
378C2×C22⋊C4—21no—
379C2×C22⋊C4—1061no—
380C2×C22⋊C4—1061no—
381C2×C22⋊C4—1061no—
382C2×C22⋊C4—1061no—
383C2×C42—531no—
384C2×C42—531no—
385C2×C42—1061no—
386C2×C42—1061no—
387C2×C42—1061no—
388C2×C42C22×D5311yescharacteristic
389#107—534no—
390#107—2121no—
391#107—2121no—
392#106—534no—
393C4⋊C8—22no—
394C4⋊C8—534no—
395#102—41no—
396#102—41no—
397#102—534no—
398#102—534no—
399#97—2121no—
400C4×C8—2121no—
401#95—531no—
402#95—531no—
403#95—531no—
404#95—531no—
405#95—1061no—
406#95—1061no—
407#95—1061no—
408#95—1061no—
409#95—1061no—
410#95—1061no—
411#95—1061no—
412C24—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4C2×C4×D5312yes—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—81no—
424C22×C4—81no—
425C22×C4—531no—
426C22×C4—531no—
427C22×C4—531no—
428C22×C4—531no—
429C22×C4—1061no—
430C22×C4—1061no—
431C22×C4—1061no—
432C22×C4—1061no—
433C22×C4D4×D5311yescharacteristic
434C22×C4D4×D5311yescharacteristic
435C22×C4C23×D5311yescharacteristic
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—532no—
443C2×C8—532no—
444C2×C8—1061no—
445C2×C8—1061no—
446C2×C8—1061no—
447C2×C8—1061no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—81no—
454C4⋊C4—81no—
455C4⋊C4—532no—
456C4⋊C4—532no—
457C4⋊C4—1064no—
458C4⋊C4—1064no—
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⋊C4C2×C4×D5312yes—
470C22⋊C4—24no—
471C22⋊C4—4241no—
472C22⋊C4—4241no—
473C22⋊C4—4241no—
474C22⋊C4—4241no—
475C42—1061no—
476C42—1061no—
477C42—1061no—
478C42—1061no—
479C42—4241no—
480C42—4241no—
481C42—4241no—
482C42C2×C4×D5312yes—
483C42#1171994712yes—
484C23—22no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23C2×D4×D5311yescharacteristic
492D4—22no—
493D4—22no—
494D4—41no—
495D4—41no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—84no—
510C2×C4—1062no—
511C2×C4—1062no—
512C2×C4—1061no—
513C2×C4—1061no—
514C2×C4—1061no—
515C2×C4—1061no—
516C2×C4—1061no—
517C2×C4—1061no—
518C2×C4—1061no—
519C2×C4—1061no—
520C2×C4—1061no—
521C2×C4—1061no—
522C2×C4—1061no—
523C2×C4—1061no—
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—2121no—
533C2×C4—2121no—
534C2×C4#1313317712yes—
535C2×C4#1313317712yes—
536C2×C4C22×C4×D5311yescharacteristic
537C2×C4C2×#1171994711yescharacteristic
538C2×C4#1313326211yescharacteristic
539C2×C4#1313326411yescharacteristic
540C2×C4#1313326411yescharacteristic
541C2×C4#1313326511yescharacteristic
542C8—41no—
543C8—41no—
544C8—1062no—
545C8—1062no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22C2×#1313317711yescharacteristic
554C22#1681119611yescharacteristic
555C22#1681120211yescharacteristic
556C22#1681142411yescharacteristic
557C22#1681142711yescharacteristic
558C22#1681142911yescharacteristic
559C22#1681143911yescharacteristic
560C4—22no—
561C4—22no—
562C4—42no—
563C4—81no—
564C4—81no—
565C4—4241no—
566C4—4241no—
567C4—2121no—
568C4—2121no—
569C4—2122no—
570C4—2121no—
571C4—2122no—
572C4—2121no—
573C4#1681110011yescharacteristic
574C4#1681110411yescharacteristic
575C4#1681123311yescharacteristic
576C4#1681124011yescharacteristic
577C2—82no—
578C2#2027739311yescharacteristic
579C2#2028354111yescharacteristic
580C2#2028358611yescharacteristic
581C2#2028596711yescharacteristic
582C2#2028599511yescharacteristic
583C2#2028613711yescharacteristic
584C2#2028615211yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1802C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
cosocleC22×C4×C53C24
hypercenter#1802D53
hypocenterC53#22126
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. #20277454C2semidirect
4. #20275670C2semidirect
5. #20275471C2semidirect
6. #20275457C2semidirect
7. #20275447C2semidirect
12. #20275310C2semidirect
13. #20275296C2semidirect
14. #20274574C2semidirect
17. #16810923C4semidirect
352. C53#22126semidirect

Maximal subgroups

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

subgroupindex
C2×#168109232
#202774542
#202773582
#202756702
#202754712
#202754572
#202754472
#202754092
#202753952
#202753382
#202753102
#202752962
#202745742
C53×#18022
#2212653