Finite Group Databasebeta

← #27280415

Subgroups — #27280415

Order 13568.

712 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283164C211yesmaximal, characteristic, complemented
2#20274659C211yesmaximal, characteristic, complemented
3#20273231C211yesmaximal, characteristic, complemented
4#20273075C211yesmaximal, characteristic, complemented
5#20272592C211yesmaximal, characteristic, complemented
6#20272562C211yesmaximal, characteristic, complemented
7C2×C53×#449C211yesmaximal, characteristic
8C2×#13133224—21no—
9C2×#13133144—21no—
10C2×#13133144—21no—
11#16810590—22no—
12#16810590—21no—
13#16810587—21no—
14#16810587C2211yescharacteristic
15#16810587C2211yescharacteristic
16C2×C22⋊#88322—21no—
17C2×C22⋊#88322C2211yescharacteristic
18C4×C22⋊Dic53C2211yescharacteristic
19#16810357—22no—
20#16810347—21no—
21#16810347—21no—
22C22×#41×C53C2211yescharacteristic
23C22×SD16×C53C411yescharacteristic, complemented
24C22×D8×C53C411yescharacteristic, complemented
25C2×#121×C53C2211yescharacteristic
26C53×#449—21no—
27C53×#449—21no—
28C53×#449—21no—
29C53×#449—21no—
30C2×#103×C53C411yescharacteristic, complemented
31C2×#102×C53C411yescharacteristic, complemented
32C2×C22⋊C8×C53C2211yescharacteristic
33#13133224—42no—
34#13133224—42no—
35C2×#11719917—22no—
36C2×#11719917—21no—
37C2×#11719917—21no—
38C22×#88322—22no—
39C22×#88322—21no—
40C2×C22⋊Dic53—21no—
41C2×C4×Dic53—21no—
42#13133144—42no—
43#13133144—42no—
44C22⋊#88322—22no—
45C22⋊#88322—22no—
46C22⋊#88322—21no—
47C22⋊#88322—21no—
48C22⋊#88322—21no—
49C22⋊#88322—21no—
50C22⋊#88322D411yescharacteristic
51C22⋊#88322D411yescharacteristic
52C22⋊#88322D412yes—
53#13133124—21no—
54C2×#41×C53—21no—
55C2×#41×C53—21no—
56C2×#41×C53—42no—
57C2×#41×C53—42no—
58C2×#41×C53D411yescharacteristic
59C2×#41×C53D411yescharacteristic
60C2×#41×C53D411yescharacteristic
61C2×#41×C53D411yescharacteristic
62C22×Q8×C53C2×C411yescharacteristic
63C22×D4×C53—21no—
64C22×D4×C53—21no—
65C22×D4×C53C2×C411yescharacteristic
66C22×D4×C53C2×C411yescharacteristic
67C23×C4×C53—21no—
68C23×C4×C53C2311yescharacteristic
69C2×SD16×C53—21no—
70C2×SD16×C53—21no—
71C2×SD16×C53—42no—
72C2×D8×C53—21no—
73C2×D8×C53—21no—
74C2×D8×C53—42no—
75C22×C8×C53C2×C411yescharacteristic
76C22×C8×C53C2×C411yescharacteristic
77#121×C53—21no—
78#121×C53—21no—
79#121×C53—21no—
80#121×C53—21no—
81C2×C4⋊C4×C53C2×C411yescharacteristic
82C2×C22⋊C4×C53—21no—
83#103×C53—21no—
84#103×C53—21no—
85#102×C53—21no—
86#102×C53—21no—
87C22⋊C8×C53—21no—
88C22⋊C8×C53—21no—
89C24×C53—21no—
90C24×C53—21no—
91C24×C53#4111yescharacteristic
92#41×C53—22no—
93#41×C53—22no—
94#41×C53—22no—
95#41×C53—22no—
96#41×C53—41no—
97#41×C53—41no—
98#41×C53—41no—
99#41×C53—41no—
100#41×C53—44no—
101#41×C53—44no—
102C2×Q8×C53—21no—
103C2×Q8×C53—22no—
104C2×Q8×C53—41no—
105C2×Q8×C53C22⋊C411yescharacteristic
106C2×Q8×C53C22⋊C411yescharacteristic
107C2×D4×C53—21no—
108C2×D4×C53—22no—
109C2×D4×C53—21no—
110C2×D4×C53—21no—
111C2×D4×C53—21no—
112C2×D4×C53—21no—
113C2×D4×C53—21no—
114C2×D4×C53—21no—
115C2×D4×C53—41no—
116C2×D4×C53—41no—
117C2×D4×C53—41no—
118C2×D4×C53—44no—
119C2×D4×C53—41no—
120C2×D4×C53—41no—
121C2×D4×C53—82no—
122C2×D4×C53C22⋊C411yescharacteristic
123C2×D4×C53C22⋊C411yescharacteristic
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—41no—
131C22×C4×C53—41no—
132C22×C4×C53—44no—
133C22×C4×C53C22×C411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53#4111yescharacteristic
139C22×Dic53—41no—
140SD16×C53—42no—
141SD16×C53—42no—
142D8×C53—42no—
143D8×C53—42no—
144C2×C8×C53—21no—
145C2×C8×C53—21no—
146C2×C8×C53—41no—
147C2×C8×C53—41no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—21no—
150C22⋊C4×C53—41no—
151C22⋊C4×C53—41no—
152C22⋊Dic53—22no—
153C22⋊Dic53—41no—
154C4×Dic53—41no—
155C4×Dic53—41no—
156#11719917—44no—
157#11719917—41no—
158#11719917—41no—
159#11719917—41no—
160#11719917—41no—
161C2×#88322—21no—
162C2×#88322—22no—
163C2×#88322—21no—
164C2×#88322—44no—
165C2×#88322—41no—
166C2×#88322—41no—
167C2×#88322—41no—
168C23×C53—21no—
169C23×C53—21no—
170C23×C53—21no—
171C23×C53—21no—
172C23×C53—41no—
173C23×C53—41no—
174C23×C53—42no—
175C23×C53—41no—
176C23×C53—41no—
177C23×C53—41no—
178C23×C53—41no—
179C23×C53—41no—
180C23×C53—81no—
181C23×C53—81no—
182C23×C53—81no—
183C23×C53C4×D411yescharacteristic
184C23×C53#12111yescharacteristic
185C23×C53#12311yescharacteristic
186Q8×C53—21no—
187Q8×C53—21no—
188Q8×C53—21no—
189Q8×C53—42no—
190Q8×C53#9912yes—
191D4×C53—24no—
192D4×C53—21no—
193D4×C53—21no—
194D4×C53—21no—
195D4×C53—42no—
196D4×C53—42no—
197D4×C53—42no—
198D4×C53—42no—
199D4×C53—41no—
200D4×C53—41no—
201D4×C53—81no—
202D4×C53—82no—
203D4×C53—82no—
204D4×C53#9912yes—
205C2×C4×C53—24no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53—42no—
221C2×C4×C53—41no—
222C2×C4×C53—81no—
223C2×C4×C53C2×C22⋊C411yescharacteristic
224C2×C4×C53C4×D411yescharacteristic
225C2×C4×C53#12011yescharacteristic
226C2×C4×C53#12111yescharacteristic
227C2×Dic53—42no—
228C2×Dic53—81no—
229C2×Dic53—81no—
230C8×C53—41no—
231C8×C53—41no—
232#88322—41no—
233#88322—42no—
234#88322—41no—
235#88322—81no—
236#9241—531nomaximal, Sylow
237C22×C53—21no—
238C22×C53—21no—
239C22×C53—21no—
240C22×C53—21no—
241C22×C53—21no—
242C22×C53—21no—
243C22×C53—22no—
244C22×C53—21no—
245C22×C53—21no—
246C22×C53—41no—
247C22×C53—41no—
248C22×C53—41no—
249C22×C53—41no—
250C22×C53—41no—
251C22×C53—81no—
252C22×C53—81no—
253C22×C53—82no—
254C22×C53—82no—
255C22×C53—81no—
256C22×C53—81no—
257C22×C53—81no—
258C22×C53—161no—
259C22×C53—161no—
260C22×C53#38611yescharacteristic
261C22×C53#44311yescharacteristic
262C22×C53#44411yescharacteristic
263C4×C53—21no—
264C4×C53—21no—
265C4×C53—22no—
266C4×C53—21no—
267C4×C53—41no—
268C4×C53—81no—
269C4×C53C2×#9911yescharacteristic
270C4×C53C2×#9911yescharacteristic
271Dic53—82no—
272C2×#449D5311yescharacteristic
273#2880—531no—
274#1789—531no—
275#1507—531no—
276#1471—531no—
277#1293—531no—
278#1282—531no—
279C2×C53—21no—
280C2×C53—21no—
281C2×C53—41no—
282C2×C53—41no—
283C2×C53—42no—
284C2×C53—81no—
285C2×C53—161no—
286C2×C53#187611yescharacteristic
287C2×C53#189811yescharacteristic
288C2×C53#190311yescharacteristic
289C22×#41C2×D5311yescharacteristic
290C22×SD16Dic5311yescharacteristic, complemented
291C22×D8Dic5311yescharacteristic, complemented
292C2×#131—1061no—
293C2×#121C2×D5311yescharacteristic
294#449—21no—
295#449—21no—
296#449—21no—
297#449—21no—
298C2×#104—1061no—
299C2×#104—1061no—
300C2×#103Dic5311yescharacteristic, complemented
301C2×#102Dic5311yescharacteristic, complemented
302#408—1062no—
303#408—1061no—
304#407—531no—
305#407—531no—
306#407—1061no—
307C2×C22⋊C8—531no—
308C2×C22⋊C8—1061no—
309C2×C22⋊C8C2×D5311yescharacteristic
310C4×C22⋊C4—531no—
311#327—1062no—
312#323—1061no—
313#323—1061no—
314C53#924111yescharacteristic, complemented, Sylow
315C2×#41—21no—
316C2×#41—21no—
317C2×#41—42no—
318C2×#41—42no—
319C2×#41C22⋊D5311yescharacteristic
320C2×#41C22⋊D5311yescharacteristic
321C2×#41C22⋊D5311yescharacteristic
322C2×#41C22⋊D5311yescharacteristic
323C22×Q8C2×Dic5311yescharacteristic
324C22×D4—21no—
325C22×D4—21no—
326C22×D4C2×Dic5311yescharacteristic
327C22×D4C2×Dic5311yescharacteristic
328C23×C4—21no—
329C23×C4C22×D5311yescharacteristic
330C2×SD16—21no—
331C2×SD16—21no—
332C2×SD16—42no—
333C2×D8—21no—
334C2×D8—21no—
335C2×D8—42no—
336#131—2122no—
337#131—2122no—
338C2×M16—1062no—
339C2×M16—1061no—
340C2×M16—1061no—
341C22×C8—1062no—
342C22×C8—1061no—
343C22×C8C2×Dic5311yescharacteristic
344C22×C8C2×Dic5311yescharacteristic
345#121—21no—
346#121—21no—
347#121—21no—
348#121—21no—
349C2×C4⋊C4C2×Dic5311yescharacteristic
350C2×C22⋊C4—21no—
351C2×C22⋊C4—1061no—
352C2×C42—1061no—
353#104—2122no—
354#104—2122no—
355#103—21no—
356#103—21no—
357#102—21no—
358#102—21no—
359C22⋊C8—21no—
360C22⋊C8—21no—
361C22⋊C8—531no—
362C22⋊C8—531no—
363C22⋊C8—532no—
364C22⋊C8—1062no—
365C22⋊C8—1062no—
366C22⋊C8—1061no—
367C22⋊C8—1061no—
368C22⋊C8—1061no—
369C22⋊C8—1061no—
370#95—1061no—
371C24—21no—
372C24—21no—
373C24#1171994711yescharacteristic
374#41—22no—
375#41—22no—
376#41—22no—
377#41—22no—
378#41—41no—
379#41—41no—
380#41—41no—
381#41—41no—
382#41—44no—
383#41—44no—
384C2×Q8—21no—
385C2×Q8—22no—
386C2×Q8—41no—
387C2×Q8C22⋊Dic5311yescharacteristic
388C2×Q8C22⋊Dic5311yescharacteristic
389C2×D4—21no—
390C2×D4—22no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—44no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—82no—
404C2×D4C22⋊Dic5311yescharacteristic
405C2×D4C22⋊Dic5311yescharacteristic
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—44no—
415C22×C4—2121no—
416C22×C4D4×D5311yescharacteristic
417C22×C4D4×D5311yescharacteristic
418C22×C4#1171994711yescharacteristic
419C22×C4C22×Dic5311yescharacteristic
420C22×C4C2×C22⋊D5311yescharacteristic
421C22×C4C2×C22⋊D5311yescharacteristic
422SD16—42no—
423SD16—42no—
424D8—42no—
425D8—42no—
426M16—2121no—
427M16—2121no—
428M16—2124no—
429M16—2121no—
430M16—2121no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—1061no—
436C2×C8—1061no—
437C2×C8—1062no—
438C2×C8—2121no—
439C2×C8—2124no—
440C2×C8—2121no—
441C2×C8—2121no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C22⋊C4—1062no—
447C22⋊C4—2121no—
448C42—2121no—
449C42—2121no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—41no—
455C23—41no—
456C23—42no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—81no—
463C23—81no—
464C23—81no—
465C23D4×Dic5311yescharacteristic
466C23#1313318911yescharacteristic
467C23#1313319811yescharacteristic
468Q8—21no—
469Q8—21no—
470Q8—21no—
471Q8—42no—
472Q8#1313313212yes—
473D4—24no—
474D4—21no—
475D4—21no—
476D4—21no—
477D4—42no—
478D4—42no—
479D4—42no—
480D4—42no—
481D4—41no—
482D4—41no—
483D4—81no—
484D4—82no—
485D4—82no—
486D4#1313313212yes—
487C2×C4—24no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—81no—
505C2×C4—4241no—
506C2×C4—4241no—
507C2×C4—2122no—
508C2×C4C2×C22⋊Dic5311yescharacteristic
509C2×C4D4×Dic5311yescharacteristic
510C2×C4#1313318511yescharacteristic
511C2×C4#1313318911yescharacteristic
512C8—41no—
513C8—41no—
514C8—4241no—
515C8—2121no—
516C8—2121no—
517C8—2122no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—22no—
525C22—21no—
526C22—21no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—81no—
533C22—81no—
534C22—82no—
535C22—82no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—161no—
540C22—161no—
541C22#1681050311yescharacteristic
542C22#1681074011yescharacteristic
543C22#1681074511yescharacteristic
544C4—21no—
545C4—21no—
546C4—22no—
547C4—21no—
548C4—41no—
549C4—81no—
550C4—4242no—
551C4C2×#1313313211yescharacteristic
552C4C2×#1313313211yescharacteristic
553C2—21no—
554C2—21no—
555C2—41no—
556C2—41no—
557C2—42no—
558C2—81no—
559C2—161no—
560C2#2027504411yescharacteristic
561C2#2027516911yescharacteristic
562C2#2027519711yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
Fitting, prime radicalC2×C53×#449C2
socleC22×C53#386
cosocleC23×C4×C53C23
hypercenterC2×#449D53
hypocenterC53#9241
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#449

Complements

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

NH ≅ G/Ntype
1. #20283164C2semidirect
2. #20274659C2semidirect
3. #20273231C2semidirect
4. #20273075C2semidirect
5. #20272592C2semidirect
6. #20272562C2semidirect
23. C22×SD16×C53C4semidirect
24. C22×D8×C53C4semidirect
30. C2×#103×C53C4semidirect
31. C2×#102×C53C4semidirect
290. C22×SD16Dic53semidirect
291. C22×D8Dic53semidirect
300. C2×#103Dic53semidirect
301. C2×#102Dic53semidirect
314. C53#9241semidirect

Maximal subgroups

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

subgroupindex
#202831642
#202746592
#202732312
#202730752
#202725922
#202725622
C2×C53×#4492
#924153