Finite Group Databasebeta

← #27396334

Subgroups — #27396334

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×#16810922C211yesmaximal, characteristic
2#20277455C211yesmaximal, characteristic, complemented
3#20277354C211yesmaximal, characteristic
4#20275669C211yesmaximal, characteristic, complemented
5#20275470C211yesmaximal, characteristic, complemented
6#20275445C211yesmaximal, characteristic, complemented
7#20275412C211yesmaximal, characteristic
8#20275397C211yesmaximal, characteristic
9#20275334C211yesmaximal, characteristic
10#20275333C211yesmaximal, characteristic
11#20275310C211yesmaximal, characteristic, complemented
12#20275295C211yesmaximal, characteristic, complemented
13#20274714C211yesmaximal, characteristic
14#20274575C211yesmaximal, characteristic, complemented
15C53×#1944C211yesmaximal, characteristic, complemented
16C2×#13133208C2211yescharacteristic
17C2×#13133179C2211yescharacteristic
18#16810922C418yescomplemented
19C2×#13133150C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133146C2211yescharacteristic
22C2×#13133146C2211yescharacteristic
23C2×#13133139C2211yescharacteristic
24C2×#13133139C2211yescharacteristic
25C2×#13133137C2211yescharacteristic
26C2×#13133137C2211yescharacteristic
27C2×C4×#88322C2211yescharacteristic
28#16810525C2211yescharacteristic
29#16810524C2211yescharacteristic
30#16810522C2211yescharacteristic
31#16810521C2211yescharacteristic
32#16810519—21no—
33#16810518—21no—
34#16810515—21no—
35#16810513—21no—
36#16810508—21no—
37#16810505—21no—
38#16810492C2211yescharacteristic
39#16810485—21no—
40#16810480—21no—
41C4⋊C4×Dic53C2211yescharacteristic
42C4⋊C4×Dic53C2211yescharacteristic
43C4⋊C4×Dic53C2211yescharacteristic
44#16810469C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic
54C2×#106×C53C2211yescharacteristic
55C2×#97×C53C2211yescharacteristic
56C53×#389C2211yescharacteristic
57C53×#389C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C22×D212C2311yescharacteristic
61C22×C4×D53—21no—
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133208—22no—
65#13133208C2×C414yes—
66#13133179—22no—
67#13133179C2×C414yes—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719921—21no—
78C2×#11719921—21no—
79C2×#11719921—21no—
80C2×#11719921—21no—
81C2×#11719921—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53C2311yescharacteristic
86C2×C4×Dic53C2311yescharacteristic
87#13133150—41no—
88#13133150C2×C414yes—
89#13133146—22no—
90#13133146C2×C414yes—
91#13133139—41no—
92#13133139C2×C414yes—
93#13133137—41no—
94#13133137C2×C414yes—
95C4×#88322—41no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124D411yescharacteristic
102#13133124D411yescharacteristic
103#13133124D411yescharacteristic
104#13133124D411yescharacteristic
105C22×C8×C53C2311yescharacteristic
106C22×C8×C53C2311yescharacteristic
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C42×C53C2311yescharacteristic
114#107×C53—41no—
115#106×C53C2×C414yes—
116#97×C53—41no—
117#95×C53—21no—
118#95×C53—21no—
119C23×D53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2411yescharacteristic
127C22×Dic53—21no—
128C22×Dic53—21no—
129C22×Dic53—21no—
130C22×Dic53C2×D411yescharacteristic
131C22×Dic53C2×D411yescharacteristic
132C22×Dic53C2×D411yescharacteristic
133C22×Dic53C2×D411yescharacteristic
134C2×D212—21no—
135C2×D212—21no—
136C2×D212—21no—
137C2×D212—21no—
138C2×D212—21no—
139C2×D212C22×C412yes—
140C2×C4×D53—24no—
141C2×C4×D53—81no—
142C2×C4×D53—81no—
143C2×C8×C53—21no—
144C2×C8×C53—21no—
145C2×C8×C53—21no—
146C2×C8×C53—21no—
147C2×C8×C53—21no—
148C2×C8×C53C22×C412yes—
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—
157C4⋊C4×C53C22×C412yes—
158C4⋊C4×C53C22×C412yes—
159C42×C53—21no—
160C42×C53—21no—
161#11719921—24no—
162#11719921—81no—
163#11719921—81no—
164#11719921—81no—
165#11719921—81no—
166C4⋊Dic53—21no—
167C4⋊Dic53—24no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—81no—
173C4⋊Dic53C22×C412yes—
174#11719919—24no—
175#11719919—81no—
176#11719919—81no—
177#11719919—81no—
178C4×Dic53—21no—
179C4×Dic53—21no—
180C4×Dic53—81no—
181C4×Dic53—81no—
182C4×Dic53—81no—
183C4×Dic53C22×C412yes—
184C4×Dic53#4112yes—
185C2×#88322—21no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322C22×C412yes—
191C23×C53C22×D411yescharacteristic
192C22×D53—22no—
193C22×D53—41no—
194C22×D53—41no—
195C22×D53—41no—
196C22×D53—41no—
197C22×D53—41no—
198C22×D53—41no—
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—21no—
206C2×C4×C53—21no—
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—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53C23×C411yescharacteristic
218C2×C4×C53C2×#4111yescharacteristic
219C2×C4×C53H(5,𝔽2)11yescharacteristic
220C2×C4×C532−1+411yescharacteristic
221C2×C4×C532−1+411yescharacteristic
222C2×C4×C532−1+411yescharacteristic
223C2×Dic53—21no—
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—21no—
227C2×Dic53—22no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
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×Dic53C4×D412yes—
243C2×Dic53C4×D412yes—
244D212—22no—
245D212—22no—
246D212—41no—
247D212—41no—
248C4×D53—84no—
249C8×C53—22no—
250C8×C53—41no—
251#88322—22no—
252#88322—41no—
253#22126—531nomaximal, Sylow
254C22×C53C2×C4×D411yescharacteristic
255C22×C53#53611yescharacteristic
256C22×C53#53711yescharacteristic
257C22×C53#57611yescharacteristic
258C22×C53#57611yescharacteristic
259C22×C53#57711yescharacteristic
260C22×C53#57811yescharacteristic
261C2×D53—42no—
262C2×D53—42no—
263C2×D53—42no—
264C2×D53—81no—
265C2×D53—81no—
266C2×D53—81no—
267C2×D53—81no—
268C4×C53—42no—
269C4×C53—42no—
270C4×C53—41no—
271C4×C53—41no—
272C4×C53—81no—
273C4×C53—81no—
274C4×C53#51911yescharacteristic
275C4×C53#52011yescharacteristic
276C4×C53#54111yescharacteristic
277C4×C53#54211yescharacteristic
278Dic53—22no—
279Dic53—22no—
280Dic53—42no—
281Dic53—41no—
282Dic53—41no—
283Dic53—81no—
284Dic53—81no—
285C2×#478—531no—
286#2330—531no—
287#2319—531no—
288#1989—531no—
289#1947—531no—
290#1944D5311yescharacteristic, complemented
291#1942—531no—
292#1936—531no—
293#1934—531no—
294#1925—531no—
295#1925—531no—
296#1921—531no—
297#1919—531no—
298#1802—531no—
299#1768—531no—
300C2×C53#232311yescharacteristic
301C2×C53#294611yescharacteristic
302C2×C53#295111yescharacteristic
303C2×C53#324111yescharacteristic
304C2×C53#324311yescharacteristic
305C2×C53#325811yescharacteristic
306C2×C53#325911yescharacteristic
307D53—82no—
308C2×#125—531no—
309C2×C4×D4—531no—
310#478—538no—
311C2×#107—531no—
312C2×#107—531no—
313C2×#107C2×D5311yescharacteristic
314C2×#106C2×D5311yescharacteristic
315C2×C4⋊C8—531no—
316C2×C4⋊C8—531no—
317C2×#102—531no—
318C2×#102—531no—
319C2×#102—531no—
320C2×#102—531no—
321C2×#97C2×D5311yescharacteristic
322C2×C4×C8—531no—
323#390—531no—
324#390—531no—
325#389—531no—
326#389—531no—
327#389—1061no—
328#389—1061no—
329#389C2×D5311yescharacteristic
330#389C2×D5311yescharacteristic
331#388—1061no—
332#388—1061no—
333#387—1061no—
334#387—1061no—
335#384—531no—
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×D4—531no—
354C23×C4—1061no—
355C22×C8—531no—
356C22×C8—531no—
357C22×C8C22×D5311yescharacteristic
358C22×C8C22×D5311yescharacteristic
359#125—534no—
360#125—1062no—
361C4×D4—534no—
362C4×D4—1062no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—531no—
366C2×C4⋊C4—531no—
367C2×C4⋊C4—531no—
368C2×C4⋊C4—1061no—
369C2×C4⋊C4—1061no—
370C2×C4⋊C4—1061no—
371C2×C4⋊C4—1061no—
372C2×C4⋊C4—1061no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4C22×D5311yescharacteristic
375C2×C4⋊C4C22×D5311yescharacteristic
376C2×C4⋊C4C22×D5311yescharacteristic
377C2×C4⋊C4C22×D5311yescharacteristic
378C2×C22⋊C4—1061no—
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—41no—
390#107—534no—
391#107—2121no—
392#106C4×D5314yes—
393C4⋊C8—534no—
394C4⋊C8—1062no—
395#102—534no—
396#102—534no—
397#102—2121no—
398#102—2121no—
399#97—41no—
400C4×C8—2121no—
401#95—21no—
402#95—21no—
403#95—531no—
404#95—531no—
405#95—531no—
406#95—531no—
407#95—1061no—
408#95—1061no—
409#95—1061no—
410#95—1061no—
411#95—1061no—
412C24—1061no—
413C2×D4—532no—
414C2×D4—1061no—
415C2×D4—1061no—
416C2×D4—1061no—
417C2×D4—1061no—
418C2×D4—1061no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—531no—
424C22×C4—531no—
425C22×C4—531no—
426C22×C4—531no—
427C22×C4—1064no—
428C22×C4—1061no—
429C22×C4—1061no—
430C22×C4—1061no—
431C22×C4—4241no—
432C22×C4—4241no—
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—532no—
442C2×C8—1061no—
443C2×C8—1061no—
444C2×C8—1061no—
445C2×C8—1061no—
446C2×C8—1061no—
447C2×C8C2×C4×D5312yes—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—81no—
455C4⋊C4—81no—
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—4241no—
465C4⋊C4—4241no—
466C4⋊C4—4241no—
467C4⋊C4—4241no—
468C4⋊C4C2×C4×D5312yes—
469C4⋊C4C2×C4×D5312yes—
470C22⋊C4—1064no—
471C22⋊C4—4241no—
472C22⋊C4—4241no—
473C22⋊C4—4241no—
474C22⋊C4—4241no—
475C42—21no—
476C42—21no—
477C42—532no—
478C42—532no—
479C42—1061no—
480C42—1061no—
481C42—4241no—
482C42—4241no—
483C42—4241no—
484C23—1062no—
485C23—2121no—
486C23—2121no—
487C23—2121no—
488C23—2121no—
489C23—2121no—
490C23—2121no—
491C23C2×D4×D5311yescharacteristic
492D4—1062no—
493D4—1062no—
494D4—2121no—
495D4—2121no—
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—532no—
515C2×C4—532no—
516C2×C4—1061no—
517C2×C4—1061no—
518C2×C4—1061no—
519C2×C4—1061no—
520C2×C4—1062no—
521C2×C4—1061no—
522C2×C4—1061no—
523C2×C4—1061no—
524C2×C4—1061no—
525C2×C4—1061no—
526C2×C4—1061no—
527C2×C4—4244no—
528C2×C4—2121no—
529C2×C4—2121no—
530C2×C4—2121no—
531C2×C4—2121no—
532C2×C4—2121no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4—2121no—
536C2×C4C22×C4×D5311yescharacteristic
537C2×C4#41×D5311yescharacteristic
538C2×C4#1313326211yescharacteristic
539C2×C4#1313326411yescharacteristic
540C2×C4#1313326511yescharacteristic
541C2×C4#1313326511yescharacteristic
542C8—22no—
543C8—41no—
544C8—1062no—
545C8—2121no—
546C22—4241no—
547C22—4241no—
548C22—4241no—
549C22—4241no—
550C22—2122no—
551C22—2122no—
552C22—2122no—
553C22C4×D4×D5311yescharacteristic
554C22#1681119811yescharacteristic
555C22#1681120011yescharacteristic
556C22#1681142711yescharacteristic
557C22#1681142711yescharacteristic
558C22#1681143211yescharacteristic
559C22#1681143911yescharacteristic
560C4—42no—
561C4—41no—
562C4—42no—
563C4—41no—
564C4—81no—
565C4—81no—
566C4—1062no—
567C4—1062no—
568C4—4241no—
569C4—4241no—
570C4—2121no—
571C4—2121no—
572C4—2122no—
573C4#1681109911yescharacteristic
574C4#1681110411yescharacteristic
575C4#1681123211yescharacteristic
576C4#1681124311yescharacteristic
577C2—4242no—
578C2#2027739211yescharacteristic
579C2#2028354611yescharacteristic
580C2#2028358711yescharacteristic
581C2#2028596911yescharacteristic
582C2#2028599911yescharacteristic
583C2#2028613611yescharacteristic
584C2#2028615111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1944C2
socleC23×C53C22×D4
prime radicalC22×D212C23
cosocleC22×C4×C53C24
hypercenter#1944D53
hypocenterC53#22126
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1944

Complements

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

NH ≅ G/Ntype
2. #20277455C2semidirect
4. #20275669C2semidirect
5. #20275470C2semidirect
6. #20275445C2semidirect
11. #20275310C2semidirect
12. #20275295C2semidirect
14. #20274575C2semidirect
15. C53×#1944C2semidirect
18. #16810922C4semidirect
290. #1944D53semidirect
352. C53#22126semidirect

Maximal subgroups

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

subgroupindex
C2×#168109222
#202774552
#202773542
#202756692
#202754702
#202754452
#202754122
#202753972
#202753342
#202753332
#202753102
#202752952
#202747142
#202745752
C53×#19442
#2212653