Finite Group Databasebeta

← #27368427

Subgroups — #27368427

Order 13568.

578 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285456C211yesmaximal, characteristic
2#20284845C211yesmaximal, characteristic
3Q8×#11719917C211yesmaximal, characteristic
4#20274299C211yesmaximal, characteristic, complemented
5#20274267C211yesmaximal, characteristic, complemented
6#20274223C211yesmaximal, characteristic, complemented
7#20274207C211yesmaximal, characteristic, complemented
8#20274175C211yesmaximal, characteristic, complemented
9#20274143C211yesmaximal, characteristic, complemented
10#20274099C211yesmaximal, characteristic, complemented
11#20274083C211yesmaximal, characteristic, complemented
12#20273480C211yesmaximal, characteristic
13#20273452C211yesmaximal, characteristic
14#20273445C211yesmaximal, characteristic
15C53×#3078C211yesmaximal, characteristic
16#16811180C2211yescharacteristic
17#16811019C2211yescharacteristic
18#16811002C2211yescharacteristic
19#16810975C2211yescharacteristic
20#16810973C2211yescharacteristic
21#16810953—21no—
22#16810908C2211yescharacteristic
23#16810908C2211yescharacteristic
24#16810879—21no—
25#16810879—21no—
26#16810833C2211yescharacteristic
27#16810833C2211yescharacteristic
28#16810779—21no—
29#16810754—21no—
30#16810754—21no—
31#16810754C2211yescharacteristic
32#16810754C2211yescharacteristic
33Q8×#88322C2211yescharacteristic
34Q8×#88322C2211yescharacteristic
35#16810728—21no—
36#16810728—21no—
37#16810728—21no—
38#16810728—21no—
39#16810680C2211yescharacteristic
40#16810647—21no—
41#16810647C2211yescharacteristic
42C4×#11719917—21no—
43C4×#11719917C2211yescharacteristic
44#16810375C2211yescharacteristic
45#16810375C2211yescharacteristic
46#16810373C2211yescharacteristic
47#16810373C2211yescharacteristic
48#16810372C2211yescharacteristic
49#16810372C2211yescharacteristic
50#16810370C2211yescharacteristic
51#16810370C2211yescharacteristic
52#16810369C2211yescharacteristic
53#16810369C2211yescharacteristic
54#16810368C2211yescharacteristic
55#16810368C2211yescharacteristic
56C53×#533C2211yescharacteristic
57C2×C4×Q8×C53C2211yescharacteristic
58C53×#479—21no—
59C53×#479—21no—
60C53×#477C2211yescharacteristic
61C53×#477C2211yescharacteristic
62C53×#475C2211yescharacteristic
63C53×#475C2211yescharacteristic
64C53×#428—21no—
65C53×#423C2211yescharacteristic
66C53×#419—21no—
67C2×#11719925—21no—
68C2×#11719925—21no—
69C2×#11719925—21no—
70C2×#11719925—21no—
71C2×#11719917—21no—
72C2×#11719917D411yescharacteristic
73C2×#11719917D411yescharacteristic
74#13133217C2311yescharacteristic
75#13133217C2311yescharacteristic
76#13133209—21no—
77#13133196—21no—
78#13133196—21no—
79C4×Dic106—21no—
80C4×Dic106—21no—
81#13133176—21no—
82#13133148—21no—
83#13133148—21no—
84#13133148—21no—
85#13133148—21no—
86C4⋊#88322—21no—
87C4⋊#88322—21no—
88C4⋊#88322—21no—
89C4⋊#88322—21no—
90C4⋊#88322—21no—
91C4⋊#88322C2311yescharacteristic
92C4⋊#88322C2311yescharacteristic
93C4⋊#88322C2311yescharacteristic
94C4⋊#88322C2311yescharacteristic
95#13133141—21no—
96#13133141—21no—
97#13133141—21no—
98#13133141—21no—
99#13133140—21no—
100#13133140—21no—
101#13133140—21no—
102#13133140—21no—
103C22⋊#88322—21no—
104#13133128—21no—
105#13133128—21no—
106#13133128—21no—
107#13133128D411yescharacteristic
108#13133128D411yescharacteristic
109C4×#88322—21no—
110C4×#88322—21no—
111C4×#88322C2311yescharacteristic
112C4×#88322C2311yescharacteristic
113C22×Q8×C53D411yescharacteristic
114C2×M16×C53—21no—
115C4⋊Q8×C53C2311yescharacteristic
116C4⋊Q8×C53C2311yescharacteristic
117#125×C53—21no—
118C22⋊Q8×C53—21no—
119C22⋊Q8×C53—21no—
120C4×Q8×C53—21no—
121C4×Q8×C53—21no—
122C4×Q8×C53—21no—
123C4×Q8×C53—21no—
124C4×Q8×C53D411yescharacteristic
125C4×Q8×C53D411yescharacteristic
126C4×Q8×C53C2311yescharacteristic
127C4×Q8×C53C2311yescharacteristic
128#117×C53—21no—
129C2×C4⋊C4×C53—21no—
130C2×C4⋊C4×C53D411yescharacteristic
131C2×C42×C53—21no—
132C2×C42×C53C2311yescharacteristic
133#107×C53—21no—
134#107×C53—21no—
135#106×C53—21no—
136#106×C53—21no—
137C4⋊C8×C53—21no—
138C4⋊C8×C53C2311yescharacteristic
139C4⋊C8×C53C2311yescharacteristic
140#103×C53—21no—
141#103×C53—21no—
142#103×C53—21no—
143#103×C53—21no—
144C2×Q8×C53—21no—
145C2×Q8×C53—22no—
146C2×Q8×C53—21no—
147C2×Q8×C53—21no—
148C2×Q8×C53—41no—
149C2×Q8×C53C2×D411yescharacteristic
150C2×Q8×C53C2×D411yescharacteristic
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53#4111yescharacteristic
156C2×Dic106—21no—
157C2×Dic106—21no—
158M16×C53—42no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—41no—
171C4⋊C4×C53—41no—
172C4⋊C4×C53C2×D411yescharacteristic
173C4⋊C4×C53C2×D411yescharacteristic
174C22⋊C4×C53—41no—
175C42×C53—21no—
176C42×C53—21no—
177C42×C53—21no—
178C42×C53—21no—
179C42×C53—41no—
180C42×C53C2×D411yescharacteristic
181C42×C53C2×D411yescharacteristic
182C42×C53#4111yescharacteristic
183C42×C53C2411yescharacteristic
184C22⋊Dic53—41no—
185#11719925—42no—
186#11719925—42no—
187#11719925—42no—
188#11719925—42no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193#11719919—41no—
194#11719919—41no—
195C4×Dic53—41no—
196#11719917—22no—
197#11719917—41no—
198#11719917—42no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322C2×D411yescharacteristic
204C2×#88322C2×D411yescharacteristic
205C2×#88322C2×D411yescharacteristic
206C2×#88322C2×D411yescharacteristic
207C23×C53#12111yescharacteristic
208Q8×C53—21no—
209Q8×C53—21no—
210Q8×C53—21no—
211Q8×C53—42no—
212Q8×C53—41no—
213Q8×C53—41no—
214Q8×C53—81no—
215Q8×C53#13712yes—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—41no—
225C2×C4×C53—42no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—81no—
229C2×C4×C53#12111yescharacteristic
230C2×C4×C53#12111yescharacteristic
231C2×C4×C53#12111yescharacteristic
232C2×C4×C53C22×D411yescharacteristic
233C2×C4×C53C22×D411yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×Dic53—41no—
236C2×Dic53—41no—
237Dic106—41no—
238Dic106—41no—
239Dic106—41no—
240Dic106—41no—
241C8×C53—41no—
242C8×C53—41no—
243#88322—21no—
244#88322—21no—
245#88322—21no—
246#88322—41no—
247#88322—41no—
248#88322—41no—
249#88322—41no—
250#88322#13712yes—
251#19311—531nomaximal, Sylow
252C22×C53—21no—
253C22×C53—21no—
254C22×C53—21no—
255C22×C53C2×#12111yescharacteristic
256C4×C53—21no—
257C4×C53—21no—
258C4×C53—22no—
259C4×C53—21no—
260C4×C53—21no—
261C4×C53—41no—
262C4×C53—41no—
263C4×C53—41no—
264C4×C53—81no—
265C4×C53—81no—
266C4×C53C2×#13711yescharacteristic
267C4×C53C2×#13711yescharacteristic
268C4×C53#57511yescharacteristic
269C4×C53#57511yescharacteristic
270Dic53—81no—
271Dic53—81no—
272#3185—531no—
273#3113—531no—
274#3078D5311yescharacteristic
275Q8×M16—531no—
276#1704—531no—
277#1700—531no—
278#1692—531no—
279#1690—531no—
280#1684—531no—
281#1680—531no—
282#1672—531no—
283#1670—531no—
284#1569—531no—
285#1561—531no—
286#1559—531no—
287C2×C53—41no—
288C2×C53#303711yescharacteristic
289C2×C53#305611yescharacteristic
290C2×C53#306911yescharacteristic
291#533—531no—
292#533C2×D5311yescharacteristic
293C2×C4×Q8C2×D5311yescharacteristic
294#498—531no—
295#494—531no—
296#487—531no—
297#487—531no—
298#483—1061no—
299#479—21no—
300#479—21no—
301#477C2×D5311yescharacteristic
302#477C2×D5311yescharacteristic
303#475—531no—
304#475—531no—
305#475C2×D5311yescharacteristic
306#475C2×D5311yescharacteristic
307#470—1061no—
308#470—1061no—
309#462—531no—
310#462—531no—
311#451—1061no—
312#446—531no—
313#446—531no—
314#446—1061no—
315#446—1061no—
316C8×Q8—531no—
317C8×Q8—531no—
318#441—1061no—
319#441—1061no—
320#441—1061no—
321#441—1061no—
322#432—531no—
323#428—21no—
324#423—531no—
325#423—1061no—
326#423C2×D5311yescharacteristic
327#419—21no—
328C4×M16—531no—
329C4×M16—1061no—
330#335—531no—
331#335—531no—
332#334—531no—
333#334—531no—
334#333—531no—
335#333—531no—
336#332—531no—
337#332—531no—
338#332—531no—
339#332—531no—
340#332—531no—
341#332—531no—
342C53#1931111yescharacteristic, complemented, Sylow
343C22×Q8C22⋊D5311yescharacteristic
344C2×Q16—1061no—
345C2×Q16—1061no—
346C2×Q16—1061no—
347C2×Q16—1061no—
348C2×M16—21no—
349C2×M16—531no—
350C2×M16—531no—
351C2×M16—1061no—
352C4⋊Q8—531no—
353C4⋊Q8—531no—
354C4⋊Q8C22×D5311yescharacteristic
355C4⋊Q8C22×D5311yescharacteristic
356#125—21no—
357#125—1061no—
358C22⋊Q8—21no—
359C22⋊Q8—21no—
360C22⋊Q8—1061no—
361C22⋊Q8—1061no—
362C4×Q8—21no—
363C4×Q8—21no—
364C4×Q8—21no—
365C4×Q8—21no—
366C4×Q8—1061no—
367C4×Q8—1061no—
368C4×Q8C22⋊D5311yescharacteristic
369C4×Q8C22⋊D5311yescharacteristic
370C4×Q8C22×D5311yescharacteristic
371C4×Q8C22×D5311yescharacteristic
372#117—21no—
373#117—1061no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4C22⋊D5311yescharacteristic
376C2×C42—21no—
377C2×C42C22×D5311yescharacteristic
378#107—21no—
379#107—21no—
380#106—21no—
381#106—21no—
382#106—1061no—
383#106—1061no—
384#106—1061no—
385#106—1061no—
386C4⋊C8—21no—
387C4⋊C8—531no—
388C4⋊C8—531no—
389C4⋊C8—531no—
390C4⋊C8—531no—
391C4⋊C8—1061no—
392C4⋊C8—1061no—
393C4⋊C8—1061no—
394C4⋊C8—1061no—
395C4⋊C8—1061no—
396C4⋊C8C22×D5311yescharacteristic
397C4⋊C8C22×D5311yescharacteristic
398#103—21no—
399#103—21no—
400#103—21no—
401#103—21no—
402#103—1061no—
403#103—1061no—
404#103—1061no—
405#103—1061no—
406#103—1061no—
407#103—1061no—
408#103—1061no—
409#103—1061no—
410C22⋊C8—1061no—
411#97—531no—
412#97—531no—
413#97—1061no—
414#97—1061no—
415#97—1061no—
416C4×C8—531no—
417C4×C8—531no—
418C4×C8—1061no—
419C4×C8—1061no—
420C2×Q8—21no—
421C2×Q8—22no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—41no—
425C2×Q8—1061no—
426C2×Q8—1061no—
427C2×Q8C2×C22⋊D5311yescharacteristic
428C2×Q8C2×C22⋊D5311yescharacteristic
429C22×C4—21no—
430C22×C4—21no—
431C22×C4D4×D5311yescharacteristic
432C22×C4#1171994711yescharacteristic
433C22×C4C2×C22⋊D5311yescharacteristic
434Q16—2122no—
435Q16—2122no—
436Q16—2122no—
437Q16—2122no—
438M16—42no—
439M16—1062no—
440M16—2121no—
441M16—2122no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—531no—
445C2×C8—531no—
446C2×C8—531no—
447C2×C8—531no—
448C2×C8—1061no—
449C2×C8—1061no—
450C2×C8—1061no—
451C2×C8—1061no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—2121no—
468C4⋊C4—2121no—
469C4⋊C4C2×C22⋊D5311yescharacteristic
470C4⋊C4C2×C22⋊D5311yescharacteristic
471C22⋊C4—41no—
472C22⋊C4—2121no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—41no—
478C42—2121no—
479C42D4×D5311yescharacteristic
480C42#1171994711yescharacteristic
481C42C2×C22⋊D5311yescharacteristic
482C42C23×D5311yescharacteristic
483C23#1313318911yescharacteristic
484Q8—21no—
485Q8—21no—
486Q8—21no—
487Q8—42no—
488Q8—41no—
489Q8—41no—
490Q8—81no—
491Q8—2121no—
492Q8—2121no—
493Q8—2121no—
494Q8—2121no—
495Q8#1313324812yes—
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—42no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—81no—
509C2×C4—2121no—
510C2×C4—2121no—
511C2×C4#1313318911yescharacteristic
512C2×C4#1313318911yescharacteristic
513C2×C4#1313318911yescharacteristic
514C2×C4C2×D4×D5311yescharacteristic
515C2×C4C22×C22⋊D5311yescharacteristic
516C2×C4C2×#1171994711yescharacteristic
517C8—41no—
518C8—41no—
519C8—532no—
520C8—1061no—
521C8—1061no—
522C8—1061no—
523C8—2121no—
524C8—2121no—
525C8—2121no—
526C8—2121no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22C2×#1313318911yescharacteristic
531C4—21no—
532C4—21no—
533C4—21no—
534C4—21no—
535C4—22no—
536C4—41no—
537C4—41no—
538C4—41no—
539C4—81no—
540C4—81no—
541C4—4241no—
542C4—4241no—
543C4#137×D5311yescharacteristic
544C4C2×#1313324811yescharacteristic
545C4#1681142111yescharacteristic
546C4#1681142211yescharacteristic
547C2—41no—
548C2#2028418411yescharacteristic
549C2#2028434111yescharacteristic
550C2#2028445411yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3078C2
socleC22×C53C2×#121
prime radicalC23×C53#121
hypercenter#3078D53
hypocenterC53#19311
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3078

Complements

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

NH ≅ G/Ntype
4. #20274299C2semidirect
5. #20274267C2semidirect
6. #20274223C2semidirect
7. #20274207C2semidirect
8. #20274175C2semidirect
9. #20274143C2semidirect
10. #20274099C2semidirect
11. #20274083C2semidirect
342. C53#19311semidirect

Maximal subgroups

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

subgroupindex
#202854562
#202848452
Q8×#117199172
#202742992
#202742672
#202742232
#202742072
#202741752
#202741432
#202740992
#202740832
#202734802
#202734522
#202734452
C53×#30782
#1931153