Finite Group Databasebeta

← #28319944

Subgroups — #28319944

Order 13568.

790 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291358C211yesmaximal, characteristic
2#20291280C211yesmaximal, characteristic
3#20290783C212yesmaximal, complemented
4#20290782C212yesmaximal, complemented
5#20290781C212yesmaximal, complemented
6#20290780C212yesmaximal, complemented
7C2×#16811499C211yesmaximal, characteristic
8#20290395C211yesmaximal, characteristic
9#20290395C211yesmaximal, characteristic
10#20290380C211yesmaximal, characteristic
11#427×D53C211yesmaximal, characteristic
12C22×#11719936C2211yescharacteristic
13C22×#11719936C2211yescharacteristic
14C2×D53⋊M16C2211yescharacteristic
15C2×#13133296C2211yescharacteristic
16#16811630C2212yes—
17#16811630C2212yes—
18#16811629—22no—
19#16811628—21no—
20#16811627—21no—
21#16811588C2212yes—
22#16811588C2212yes—
23#16811587—22no—
24C4×#11719936—21no—
25C4×#11719936—21no—
26#16811499C2212yes—
27#16811499C2212yes—
28#16811493C2212yes—
29#16811493C2212yes—
30#16811493C2212yes—
31#16811493C2212yes—
32(C2×#6157)⋊C8C2212yes—
33(C2×#6157)⋊C8C2212yes—
34C22×C8×D53C2211yescharacteristic
35#117×D53C2211yescharacteristic
36#117×D53C2211yescharacteristic
37#16810661C411yescharacteristic, complemented
38#16810660C2211yescharacteristic
39#16810660C2211yescharacteristic
40#16810659C411yescharacteristic, complemented
41#16810659C411yescharacteristic, complemented
42#16810658C2211yescharacteristic
43#16810658C2211yescharacteristic
44#107×D53C411yescharacteristic, complemented
45#107×D53C411yescharacteristic, complemented
46#106×D53C411yescharacteristic, complemented
47#106×D53C411yescharacteristic, complemented
48C53×#427C411yescharacteristic, complemented
49C2×#11719936—21no—
50C2×#11719936—22no—
51C2×#11719936—22no—
52C2×#11719936—22no—
53C2×#11719936—21no—
54C2×#11719936D411yescharacteristic
55C2×#11719936D411yescharacteristic
56C2×#11719936Q814yes—
57C2×#11719936C2312yes—
58C2×#11719936C2312yes—
59D53⋊M16—22no—
60D53⋊M16—22no—
61C2×#11719940D412yes—
62C2×D53⋊C8C2312yes—
63C22×#88324—21no—
64C22×#88324—21no—
65#13133296—22no—
66#13133296—22no—
67C2×C4⋊#6157—21no—
68C2×C22⋊#6157—21no—
69C2×C4×#6157C2312yes—
70#13133284—22no—
71#13133284—22no—
72C4⋊#88324—22no—
73C4⋊#88324—22no—
74#13133272—22no—
75#13133272—22no—
76C4×#88324—22no—
77C4×#88324—22no—
78C22×C4×D53C2311yescharacteristic
79C22×#88322C2×C411yescharacteristic
80C2×C8×D53—22no—
81C2×C8×D53—22no—
82C2×C8×D53C2×C411yescharacteristic
83C2×C8×D53C2×C411yescharacteristic
84C2×C8×D53C2311yescharacteristic
85C2×C8×D53C2311yescharacteristic
86#13133176C2×C411yescharacteristic
87#13133176C2×C411yescharacteristic
88#13133175C2311yescharacteristic
89#13133175C2311yescharacteristic
90#13133175C2311yescharacteristic
91#13133175C2311yescharacteristic
92#13133174—21no—
93#13133174—21no—
94#13133173—21no—
95#13133173—21no—
96C4⋊C4×D53C2×C411yescharacteristic
97C4⋊C4×D53C2×C411yescharacteristic
98C4⋊C4×D53C2×C411yescharacteristic
99C4⋊C4×D53C2×C411yescharacteristic
100C22⋊C4×D53—21no—
101C22⋊C4×D53—21no—
102C42×D53—21no—
103C42×D53—21no—
104#13133151C2×C411yescharacteristic
105#13133151C2×C411yescharacteristic
106#13133150C2×C411yescharacteristic
107#13133150C2×C411yescharacteristic
108#13133150C2×C411yescharacteristic
109#13133150C2×C411yescharacteristic
110#13133149C2×C411yescharacteristic
111#13133149C2×C411yescharacteristic
112#13133148C2×C411yescharacteristic
113#13133148C2×C411yescharacteristic
114#13133148C2×C411yescharacteristic
115#13133148C2×C411yescharacteristic
116C22×C8×C53C2×C411yescharacteristic
117#117×C53C2×C411yescharacteristic
118#117×C53C2×C411yescharacteristic
119#107×C53C2×C411yescharacteristic
120#107×C53C2×C411yescharacteristic
121#106×C53C2×C411yescharacteristic
122#106×C53C2×C411yescharacteristic
123C23×D53#4111yescharacteristic
124C22×#6157—22no—
125C22×C4×C53C22×C411yescharacteristic
126C22×Dic53C2×D411yescharacteristic
127C2×C4×D53—21no—
128C2×C4×D53—21no—
129C2×C4×D53—21no—
130C2×C4×D53—21no—
131C2×C4×D53C22×C411yescharacteristic
132C2×C4×D53C2×D411yescharacteristic
133C2×C4×D53C2×Q812yes—
134C2×C4×D53#4112yes—
135C2×C4×D53#4112yes—
136C2×C4×D53#4112yes—
137C2×C4×D53#4111yescharacteristic
138C2×C4×D53C2411yescharacteristic
139C22⋊#6157—42no—
140#11719940—24no—
141C2×#88324—24no—
142C2×#88324—22no—
143C2×#88324—22no—
144C2×#88324—42no—
145C2×#88324C2×D412yes—
146C2×#88324C2×D412yes—
147C4⋊#6157—42no—
148C4×#6157—22no—
149C4×#6157—22no—
150#11719936—24no—
151#11719936—22no—
152#11719936—22no—
153#11719936—22no—
154#11719936—22no—
155#11719936—22no—
156#11719936—42no—
157#11719936C2×Q814yes—
158D53⋊C8—22no—
159D53⋊C8—22no—
160C2×C8×C53—22no—
161C2×C8×C53C22×C411yescharacteristic
162C2×C8×C53C22×C411yescharacteristic
163C4⋊C4×C53C22×C411yescharacteristic
164C4⋊C4×C53C22×C411yescharacteristic
165C4⋊C4×C53C22×C411yescharacteristic
166C4⋊C4×C53C22×C411yescharacteristic
167C22⋊C4×C53—21no—
168C22⋊C4×C53—21no—
169C42×C53—21no—
170C42×C53—21no—
171C22⋊Dic53—21no—
172C22⋊Dic53—21no—
173#11719921—21no—
174#11719921—21no—
175#11719921—21no—
176#11719921—21no—
177C4⋊Dic53C22×C411yescharacteristic
178C4⋊Dic53C22×C411yescharacteristic
179C4⋊Dic53C22×C411yescharacteristic
180C4⋊Dic53C22×C411yescharacteristic
181#11719919—21no—
182#11719919—21no—
183#11719919—21no—
184#11719919—21no—
185C4×Dic53—21no—
186C4×Dic53—21no—
187C4×Dic53—21no—
188C4×Dic53—21no—
189C4×Dic53—21no—
190C4×Dic53—21no—
191C2×#88322—22no—
192C2×#88322C22×C411yescharacteristic
193C2×#88322C22×C411yescharacteristic
194C8×D53—22no—
195C8×D53—22no—
196C8×D53—21no—
197C8×D53—21no—
198C8×D53—21no—
199C8×D53—21no—
200C23×C53C4×D411yescharacteristic
201C22×D53—41no—
202C22×D53—41no—
203C22×D53C4×D411yescharacteristic
204C22×D53#12512yes—
205C22×D53#12612yes—
206C22×D53C2×#4111yescharacteristic
207C2×#6157—42no—
208C2×#6157—42no—
209C2×#6157—42no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53#11712yes—
215C2×C4×C53C4×D411yescharacteristic
216C2×C4×C53C4×Q812yes—
217C2×C4×C53C23×C411yescharacteristic
218C2×Dic53—21no—
219C2×Dic53—21no—
220C2×Dic53—21no—
221C2×Dic53—21no—
222C2×Dic53C4×D411yescharacteristic
223C2×Dic53C22⋊Q812yes—
224C2×Dic53#12312yes—
225C2×Dic53C22×D411yescharacteristic
226C4×D53—81no—
227C4×D53—81no—
228C4×D53—81no—
229C4×D53—81no—
230C4×D53#11712yes—
231C4×D53C4×Q812yes—
232C4×D53C22⋊Q812yes—
233C4×D53#12312yes—
234C4×D53#12512yes—
235C4×D53#12612yes—
236C4×D53C22×Q811yescharacteristic
237C4×D53C2×#4111yescharacteristic
238C4×D53C2×#4111yescharacteristic
239C4×D53C2×#4111yescharacteristic
240#88324—24no—
241#88324—22no—
242#88324—22no—
243#88324—42no—
244C8×C53—21no—
245C8×C53—21no—
246#88322—21no—
247#88322—21no—
248#21043—531nomaximal, Sylow
249C22×C53—21no—
250C22×C53—21no—
251C22×C53#38712yes—
252C22×C53C2×C4×D411yescharacteristic
253C2×D53—21no—
254C2×D53—21no—
255C2×D53—21no—
256C2×D53—21no—
257C2×D53—42no—
258C2×D53—42no—
259C2×D53#38712yes—
260C2×D53C2×#12511yescharacteristic
261C2×D53C2×#12611yescharacteristic
262#6157—82no—
263C4×C53—41no—
264C4×C53—41no—
265C4×C53—41no—
266C4×C53—41no—
267C4×C53#38712yes—
268C4×C53C2×#11711yescharacteristic
269C4×C53C2×C4×Q811yescharacteristic
270Dic53—41no—
271Dic53—41no—
272Dic53—41no—
273Dic53—41no—
274Dic53#38712yes—
275Dic53C2×C22⋊Q811yescharacteristic
276Dic53C2×#12311yescharacteristic
277C2×#427—531no—
278#2905—531no—
279#2873—531no—
280#1922—532no—
281#1922—532no—
282#1922—532no—
283#1922—532no—
284C2×#341—531no—
285#1733—531no—
286#1733—531no—
287#1729—531no—
288C2×C53—22no—
289C2×C53C2×#38711yescharacteristic
290C2×C53#295711yescharacteristic
291C2×C53#295711yescharacteristic
292D53—21no—
293D53—21no—
294D53—22no—
295C22×M16—531no—
296C22×M16—531no—
297C22×M16—531no—
298C23×C8—531no—
299C2×#117—531no—
300C2×#117—531no—
301C2×#117—531no—
302#427—531no—
303#427—531no—
304#427—531no—
305#427—531no—
306#427—531no—
307#427—531no—
308#427—531no—
309#427#615711yescharacteristic, complemented
310C2×#107—531no—
311C2×#107—531no—
312C2×#106—531no—
313C2×#106—531no—
314#424—532no—
315#424—532no—
316#424—1062no—
317#423—1061no—
318#423—1061no—
319#405—532no—
320#405—532no—
321#405—1062no—
322C4×M16—1061no—
323C4×M16—1061no—
324#341—532no—
325#341—532no—
326#339—532no—
327#339—532no—
328#339—532no—
329#339—532no—
330#336—532no—
331#336—532no—
332C53#2104311yescharacteristic, complemented, Sylow
333C23×C4—531no—
334C2×M16—531no—
335C2×M16—531no—
336C2×M16—532no—
337C2×M16—532no—
338C2×M16—534no—
339C2×M16—532no—
340C2×M16—1061no—
341C2×M16—1062no—
342C2×M16—1062no—
343C2×M16—1062no—
344C2×M16—1061no—
345C2×M16—1062no—
346C2×M16—1062no—
347C22×C8—532no—
348C22×C8—531no—
349C22×C8—531no—
350C22×C8—531no—
351C22×C8—531no—
352C22×C8—531no—
353C22×C8—1062no—
354C22×C8—1062no—
355C22×C8—1061no—
356C22×C8—1061no—
357C22×C8C2×#615711yescharacteristic
358#117—531no—
359#117—531no—
360#117—531no—
361#117—531no—
362#117—531no—
363#117—531no—
364#117—1061no—
365#117—1061no—
366#117—1061no—
367#117—1061no—
368#117—1062no—
369#117—1062no—
370#117C2×#615711yescharacteristic
371#117C2×#615711yescharacteristic
372C2×C4⋊C4—531no—
373C2×C4⋊C4—531no—
374C2×C4⋊C4—531no—
375C2×C4⋊C4—531no—
376C2×C4⋊C4—1061no—
377C2×C22⋊C4—1061no—
378C2×C22⋊C4—1061no—
379C2×C22⋊C4—1061no—
380C2×C42—532no—
381C2×C42—1061no—
382C2×C42—1061no—
383#107—531no—
384#107—531no—
385#107—531no—
386#107—531no—
387#107—531no—
388#107—531no—
389#107C2×#615711yescharacteristic
390#107C2×#615711yescharacteristic
391#106—531no—
392#106—531no—
393#106—531no—
394#106—531no—
395#106—531no—
396#106—531no—
397#106C2×#615711yescharacteristic
398#106C2×#615711yescharacteristic
399C4⋊C8—1062no—
400C4⋊C8—1062no—
401C4⋊C8—1062no—
402C4⋊C8—1062no—
403#97—1062no—
404#97—1062no—
405C4×C8—1062no—
406C4×C8—1062no—
407C24—531no—
408C22×C4—531no—
409C22×C4—532no—
410C22×C4—532no—
411C22×C4—531no—
412C22×C4—531no—
413C22×C4—532no—
414C22×C4—532no—
415C22×C4—531no—
416C22×C4—531no—
417C22×C4—1061no—
418C22×C4—1061no—
419C22×C4—1061no—
420C22×C4—1061no—
421C22×C4—1062no—
422C22×C4C22×#615711yescharacteristic
423M16—534no—
424M16—1062no—
425M16—1062no—
426M16—1064no—
427M16—1062no—
428M16—1064no—
429M16—1062no—
430M16—1062no—
431M16—2122no—
432C2×C8—22no—
433C2×C8—532no—
434C2×C8—532no—
435C2×C8—531no—
436C2×C8—531no—
437C2×C8—1062no—
438C2×C8—1062no—
439C2×C8—1061no—
440C2×C8—1061no—
441C2×C8—1061no—
442C2×C8—1061no—
443C2×C8—1062no—
444C2×C8—1064no—
445C2×C8—1062no—
446C2×C8—1062no—
447C2×C8—1062no—
448C2×C8—1062no—
449C2×C8—2122no—
450C2×C8C22×#615711yescharacteristic
451C2×C8C22×#615711yescharacteristic
452C4⋊C4—531no—
453C4⋊C4—531no—
454C4⋊C4—531no—
455C4⋊C4—531no—
456C4⋊C4—1061no—
457C4⋊C4—1061no—
458C4⋊C4—1061no—
459C4⋊C4—1061no—
460C4⋊C4—2122no—
461C4⋊C4C22×#615711yescharacteristic
462C4⋊C4C22×#615711yescharacteristic
463C4⋊C4C22×#615711yescharacteristic
464C4⋊C4C22×#615711yescharacteristic
465C22⋊C4—21no—
466C22⋊C4—21no—
467C22⋊C4—1061no—
468C22⋊C4—1061no—
469C22⋊C4—1061no—
470C22⋊C4—1061no—
471C22⋊C4—1061no—
472C22⋊C4—1061no—
473C22⋊C4—2122no—
474C42—21no—
475C42—21no—
476C42—1061no—
477C42—1061no—
478C42—1061no—
479C42—1061no—
480C42—1062no—
481C42—1062no—
482C42—1061no—
483C42—1061no—
484C23—531no—
485C23—531no—
486C23—532no—
487C23—532no—
488C23—2121no—
489C23—2121no—
490C23D4×#615711yescharacteristic
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—532no—
496C2×C4—532no—
497C2×C4—531no—
498C2×C4—531no—
499C2×C4—532no—
500C2×C4—532no—
501C2×C4—532no—
502C2×C4—532no—
503C2×C4—532no—
504C2×C4—531no—
505C2×C4—531no—
506C2×C4—532no—
507C2×C4—531no—
508C2×C4—531no—
509C2×C4—1061no—
510C2×C4—1061no—
511C2×C4—1061no—
512C2×C4—1061no—
513C2×C4—4241no—
514C2×C4—4241no—
515C2×C4—4241no—
516C2×C4—4241no—
517C2×C4—2122no—
518C2×C4—2122no—
519C2×C4—2122no—
520C2×C4#1313329612yes—
521C2×C4D4×#615711yescharacteristic
522C2×C4Q8×#615712yes—
523C2×C4C23×#615711yescharacteristic
524C8—21no—
525C8—21no—
526C8—1062no—
527C8—1062no—
528C8—1064no—
529C8—1061no—
530C8—1061no—
531C8—2122no—
532C22—21no—
533C22—21no—
534C22—531no—
535C22—531no—
536C22—532no—
537C22—1061no—
538C22—1061no—
539C22—1061no—
540C22—1061no—
541C22—2122no—
542C22—2122no—
543C22#1681157212yes—
544C22C2×D4×#615711yescharacteristic
545C4—41no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—531no—
550C4—531no—
551C4—532no—
552C4—4242no—
553C4—2121no—
554C4—2121no—
555C4—2121no—
556C4—2121no—
557C4#1681157212yes—
558C4C2×#1313329611yescharacteristic
559C4C2×Q8×#615711yescharacteristic
560C2—22no—
561C2—1061no—
562C2—1061no—
563C2—1062no—
564C2C2×#1681157211yescharacteristic
565C2#2029149211yescharacteristic
566C2#2029149411yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC53×#427C4
socleC22×C53C2×C4×D4
prime radicalC23×D53#41
cosocleC2×C4×D53C24
hypercenter#427#6157
hypocenterC53#21043
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
3. #20290783C2semidirect
4. #20290782C2semidirect
5. #20290781C2semidirect
6. #20290780C2semidirect
37. #16810661C4semidirect
40. #16810659C4semidirect
41. #16810659C4semidirect
44. #107×D53C4semidirect
45. #107×D53C4semidirect
46. #106×D53C4semidirect
47. #106×D53C4semidirect
48. C53×#427C4semidirect
309. #427#6157semidirect
332. C53#21043semidirect

Maximal subgroups

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

subgroupindex
#202913582
#202912802
#202907832
#202907822
#202907812
#202907802
C2×#168114992
#202903952
#202903802
#427×D532
#2104353