Finite Group Databasebeta

← #27283233

Subgroups — #27283233

Order 13568.

650 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283018C211yesmaximal, characteristic
2#20274693C211yesmaximal, characteristic, complemented
3C2×#16810360C211yesmaximal, characteristic, complemented
4C2×#16810358C211yesmaximal, characteristic
5C2×#16810356C211yesmaximal, characteristic
6#20272940C211yesmaximal, characteristic, complemented
7C2×C53×#328C211yesmaximal, characteristic, complemented
8C2×#13133206C411yescharacteristic, complemented
9C2×#13133191C2211yescharacteristic
10C22×C4×Dic53C2211yescharacteristic
11C2×#13133146—21no—
12C2×C22⋊#88322C2211yescharacteristic
13#16810530C411yescharacteristic, complemented
14#16810524C411yescharacteristic, complemented
15#16810513—21no—
16#16810513—21no—
17#16810500—21no—
18#16810500—21no—
19#16810488C411yescharacteristic, complemented
20C2×#13133124—21no—
21C2×#13133124—21no—
22C2×#13133124C2211yescharacteristic
23#16810377—21no—
24#16810360—22no—
25#16810358—22no—
26#16810356—22no—
27C2×C22⋊Q8×C53C2211yescharacteristic
28C2×C22⋊C8×C53C2211yescharacteristic
29C2×#95×C53C2211yescharacteristic
30C53×#328—22no—
31C22×D212C2×C411yescharacteristic
32C22×C22⋊D53—21no—
33C23×Dic53—21no—
34C23×Dic53D411yescharacteristic
35C23×Dic53D411yescharacteristic
36C22×#88322—21no—
37#13133206—42no—
38#13133191—21no—
39#13133191—21no—
40#13133191—21no—
41#13133191—21no—
42C2×C4⋊Dic53C2×C411yescharacteristic
43C2×#11719921—21no—
44C2×#11719921—21no—
45C2×#11719921—21no—
46C2×#11719921—21no—
47C2×#11719921—21no—
48C2×C4×Dic53—42no—
49C2×C4×Dic53C2×C411yescharacteristic
50C2×C4×Dic53C2×C411yescharacteristic
51C2×C4×Dic53D411yescharacteristic
52C2×C4×Dic53D411yescharacteristic
53#13133146—42no—
54C22⋊#88322—21no—
55C22⋊#88322—21no—
56#13133124—21no—
57#13133124—21no—
58#13133124—21no—
59#13133124—21no—
60#13133124—21no—
61#13133124—42no—
62#13133124—42no—
63#13133124D412yes—
64C22×Q8×C53C2×C411yescharacteristic
65C23×C4×C53—21no—
66C23×C4×C53C2311yescharacteristic
67C22×C8×C53—21no—
68C22⋊Q8×C53—21no—
69C22⋊Q8×C53—21no—
70C22⋊Q8×C53—21no—
71C22⋊Q8×C53—21no—
72C2×C4⋊C4×C53—21no—
73C2×C4⋊C4×C53C2×C411yescharacteristic
74C2×C22⋊C4×C53—21no—
75C22⋊C8×C53—21no—
76C22⋊C8×C53—21no—
77#95×C53—21no—
78#95×C53D412yes—
79C24×C53C2×D411yescharacteristic
80C23×D53—21no—
81C2×Q8×C53—21no—
82C2×Q8×C53—21no—
83C2×Q8×C53—41no—
84C2×Q8×C53—41no—
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—22no—
88C22×C4×C53—41no—
89C22×C4×C53—41no—
90C22×C4×C53—41no—
91C22×C4×C53—41no—
92C22×C4×C53—41no—
93C22×C4×C53—41no—
94C22×C4×C53—41no—
95C22×C4×C53—41no—
96C22×C4×C53C22×C411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53#4111yescharacteristic
101C22×C4×C53#4111yescharacteristic
102C2×C22⋊D53—41no—
103C2×C22⋊D53—41no—
104C2×C22⋊D53—41no—
105C2×C22⋊D53—41no—
106C2×C22⋊D53—41no—
107C2×C22⋊D53—41no—
108C22×Dic53—21no—
109C22×Dic53—21no—
110C22×Dic53—21no—
111C22×Dic53—21no—
112C22×Dic53—22no—
113C22×Dic53—21no—
114C22×Dic53—22no—
115C22×Dic53—41no—
116C22×Dic53—41no—
117C22×Dic53—41no—
118C22×Dic53—41no—
119C22×Dic53—41no—
120C22×Dic53—41no—
121C22×Dic53—41no—
122C22×Dic53—41no—
123C22×Dic53—41no—
124C22×Dic53C22⋊C411yescharacteristic
125C22×Dic53C22⋊C411yescharacteristic
126C22×Dic53C22⋊C411yescharacteristic
127C22×Dic53C22⋊C411yescharacteristic
128C2×D212—21no—
129C2×D212—21no—
130C2×D212—41no—
131C2×D212—41no—
132C2×C8×C53—41no—
133C2×C8×C53—41no—
134C2×C8×C53—41no—
135C4⋊C4×C53—21no—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—41no—
138C4⋊C4×C53—41no—
139C22⋊C4×C53—41no—
140C22⋊C4×C53—41no—
141#11719921—42no—
142#11719921—42no—
143#11719921—42no—
144#11719921—41no—
145#11719921—41no—
146#11719921—42no—
147C4⋊Dic53—21no—
148C4⋊Dic53—21no—
149C4×Dic53—22no—
150C4×Dic53—22no—
151C4×Dic53—41no—
152C4×Dic53—41no—
153C2×#88322—41no—
154C2×#88322—41no—
155C2×#88322—41no—
156C23×C53—21no—
157C23×C53—21no—
158C23×C53—21no—
159C23×C53—21no—
160C23×C53C2×C22⋊C411yescharacteristic
161C23×C53#12111yescharacteristic
162C23×C53#12111yescharacteristic
163C23×C53#13511yescharacteristic
164C23×C53#13511yescharacteristic
165C23×C53AGL(1,ℤ/8)11yescharacteristic
166C23×C53#13711yescharacteristic
167C22×D53—41no—
168C22×D53—41no—
169C22×D53—41no—
170C22×D53—41no—
171C22×D53—41no—
172C22×D53—41no—
173C22×D53—41no—
174Q8×C53—81no—
175Q8×C53—81no—
176C2×C4×C53—21no—
177C2×C4×C53—21no—
178C2×C4×C53—21no—
179C2×C4×C53—21no—
180C2×C4×C53—41no—
181C2×C4×C53—41no—
182C2×C4×C53—41no—
183C2×C4×C53—41no—
184C2×C4×C53—42no—
185C2×C4×C53—42no—
186C2×C4×C53—41no—
187C2×C4×C53—41no—
188C2×C4×C53—41no—
189C2×C4×C53—41no—
190C2×C4×C53—41no—
191C2×C4×C53—41no—
192C2×C4×C53—42no—
193C2×C4×C53—81no—
194C2×C4×C53—81no—
195C2×C4×C53—81no—
196C2×C4×C53—81no—
197C2×C4×C53C4×D411yescharacteristic
198C2×C4×C53C4×D411yescharacteristic
199C2×C4×C53#12411yescharacteristic
200C2×C4×C53#12711yescharacteristic
201C22⋊D53—81no—
202C22⋊D53—81no—
203C22⋊D53—81no—
204C22⋊D53—81no—
205C2×Dic53—21no—
206C2×Dic53—21no—
207C2×Dic53—21no—
208C2×Dic53—21no—
209C2×Dic53—21no—
210C2×Dic53—21no—
211C2×Dic53—21no—
212C2×Dic53—21no—
213C2×Dic53—21no—
214C2×Dic53—21no—
215C2×Dic53—21no—
216C2×Dic53—42no—
217C2×Dic53—42no—
218C2×Dic53—42no—
219C2×Dic53—42no—
220C2×Dic53—42no—
221C2×Dic53—42no—
222C2×Dic53—41no—
223C2×Dic53—41no—
224C2×Dic53—41no—
225C2×Dic53—42no—
226C2×Dic53—81no—
227C2×Dic53—81no—
228C2×Dic53—81no—
229C2×Dic53—81no—
230C2×Dic53#9912yes—
231D212—81no—
232D212—81no—
233C8×C53—81no—
234#88322—81no—
235#9716—531nomaximal, Sylow
236C22×C53—21no—
237C22×C53—21no—
238C22×C53—21no—
239C22×C53—21no—
240C22×C53—21no—
241C22×C53—21no—
242C22×C53—21no—
243C22×C53—21no—
244C22×C53—21no—
245C22×C53—21no—
246C22×C53—21no—
247C22×C53—21no—
248C22×C53#39011yescharacteristic
249C22×C53C2×#9911yescharacteristic
250C22×C53#41011yescharacteristic
251C22×C53#41611yescharacteristic
252C22×C53#41711yescharacteristic
253C22×C53#42111yescharacteristic
254C22×C53#42111yescharacteristic
255C22×C53#46311yescharacteristic
256C22×C53#46311yescharacteristic
257C22×C53#46411yescharacteristic
258C22×C53#46411yescharacteristic
259C2×D53—81no—
260C2×D53—81no—
261C2×D53—81no—
262C2×D53—81no—
263C2×D53—81no—
264C2×D53—81no—
265C2×D53—81no—
266C4×C53—21no—
267C4×C53—21no—
268C4×C53—21no—
269C4×C53—21no—
270C4×C53—82no—
271C4×C53—81no—
272C4×C53—81no—
273Dic53—22no—
274Dic53—41no—
275Dic53—41no—
276Dic53—41no—
277Dic53—82no—
278Dic53—81no—
279#2850—531no—
280#1797—531no—
281C2×#328—531no—
282C2×#328D5311yescharacteristic, complemented
283C2×#327—531no—
284C2×#327—531no—
285#1434—531no—
286C2×C53—21no—
287C2×C53—21no—
288C2×C53—21no—
289C2×C53—21no—
290C2×C53#151211yescharacteristic
291C2×C53#151311yescharacteristic
292C2×C53#151611yescharacteristic
293C2×C53#152111yescharacteristic
294C2×C53#196311yescharacteristic
295C2×C53#196411yescharacteristic
296C2×C53#198311yescharacteristic
297D53—161no—
298C2×#124—531no—
299C2×C22⋊Q8C2×D5311yescharacteristic
300C2×#121—531no—
301C22×C42—531no—
302C2×C4⋊C8—1061no—
303C2×C22⋊C8—531no—
304C2×C22⋊C8C2×D5311yescharacteristic
305#391—531no—
306#390—531no—
307#388—1061no—
308#388—1061no—
309#386—1061no—
310#386—1061no—
311#382—531no—
312C2×#95—531no—
313C2×#95—1061no—
314C2×#95—1061no—
315C2×#95C2×D5311yescharacteristic
316#336—1061no—
317#328—22no—
318#328—1062no—
319#327—1062no—
320#327—1062no—
321C53#971611yescharacteristic, complemented, Sylow
322C22×Q8C4×D5311yescharacteristic
323C22×D4—531no—
324C22×D4—1061no—
325C23×C4—21no—
326C23×C4—531no—
327C23×C4—531no—
328C23×C4—1061no—
329C23×C4C22×D5311yescharacteristic
330C22×C8—21no—
331C22×C8—1061no—
332#124—2122no—
333C22⋊Q8—21no—
334C22⋊Q8—21no—
335C22⋊Q8—21no—
336C22⋊Q8—21no—
337#121—1061no—
338#121—1061no—
339#121—1061no—
340#121—1061no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—531no—
343C2×C4⋊C4C4×D5311yescharacteristic
344C2×C22⋊C4—21no—
345C2×C22⋊C4—1061no—
346C2×C22⋊C4—1061no—
347C2×C22⋊C4—1061no—
348C2×C22⋊C4—1061no—
349C2×C22⋊C4—1061no—
350C2×C42—531no—
351C2×C42—531no—
352C2×C42—531no—
353C2×C42—531no—
354C2×C42—2122no—
355C4⋊C8—2122no—
356C22⋊C8—21no—
357C22⋊C8—21no—
358C22⋊C8—1061no—
359C22⋊C8—1061no—
360#95—21no—
361#95—532no—
362#95—1061no—
363#95—1061no—
364#95—1061no—
365#95—1061no—
366#95—1061no—
367#95—2122no—
368#95—2122no—
369#95C22⋊D5312yes—
370C24—1061no—
371C24D4×D5311yescharacteristic
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—41no—
375C2×Q8—41no—
376C2×D4—1061no—
377C2×D4—1061no—
378C2×D4—2121no—
379C2×D4—2121no—
380C2×D4—2121no—
381C2×D4—2121no—
382C2×D4—2121no—
383C2×D4—2121no—
384C2×D4—2121no—
385C2×D4—2121no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—531no—
398C22×C4—531no—
399C22×C4—531no—
400C22×C4—531no—
401C22×C4—1062no—
402C22×C4—1061no—
403C22×C4—1061no—
404C22×C4—1061no—
405C22×C4—1061no—
406C22×C4—1061no—
407C22×C4—1062no—
408C22×C4—2121no—
409C22×C4—2121no—
410C22×C4—2121no—
411C22×C4—2121no—
412C22×C4—2121no—
413C22×C4—2121no—
414C22×C4—2121no—
415C22×C4—2121no—
416C22×C4—2121no—
417C22×C4C2×C4×D5311yescharacteristic
418C22×C4#1171994511yescharacteristic
419C22×C4D4×D5311yescharacteristic
420C22×C4D4×D5311yescharacteristic
421C22×C4#1171994711yescharacteristic
422C22×C4C2×C22⋊D5311yescharacteristic
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—2121no—
427C2×C8—2121no—
428C2×C8—2121no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—1061no—
434C4⋊C4—1061no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—2122no—
438C22⋊C4—2122no—
439C22⋊C4—2122no—
440C22⋊C4—2122no—
441C22⋊C4—2121no—
442C22⋊C4—2121no—
443C42—1062no—
444C42—1062no—
445C42—2121no—
446C42—2121no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—2121no—
452C23—2121no—
453C23—2121no—
454C23—2121no—
455C23—2121no—
456C23—2121no—
457C23—2121no—
458C23C22⋊C4×D5311yescharacteristic
459C23#1313318911yescharacteristic
460C23#1313319011yescharacteristic
461C23#1313323811yescharacteristic
462C23#1313324011yescharacteristic
463C23#1313324411yescharacteristic
464C23#1313324911yescharacteristic
465Q8—81no—
466Q8—81no—
467D4—4241no—
468D4—4241no—
469D4—4241no—
470D4—4241no—
471D4—4241no—
472D4—4241no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—532no—
495C2×C4—1061no—
496C2×C4—1061no—
497C2×C4—1061no—
498C2×C4—1061no—
499C2×C4—1061no—
500C2×C4—1061no—
501C2×C4—1061no—
502C2×C4—1061no—
503C2×C4—1061no—
504C2×C4—1061no—
505C2×C4—1061no—
506C2×C4—4241no—
507C2×C4—4241no—
508C2×C4—4241no—
509C2×C4—4241no—
510C2×C4—2122no—
511C2×C4—2122no—
512C2×C4—2122no—
513C2×C4—2122no—
514C2×C4—2121no—
515C2×C4—2122no—
516C2×C4—2121no—
517C2×C4—2121no—
518C2×C4—2122no—
519C2×C4—2122no—
520C2×C4#1313317711yescharacteristic
521C2×C4C4×C22⋊D5311yescharacteristic
522C2×C4#1313320311yescharacteristic
523C2×C4#1313321311yescharacteristic
524C8—81no—
525C8—4241no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—4241no—
539C22—4241no—
540C22—4241no—
541C22—4241no—
542C22—4241no—
543C22—4241no—
544C22—4241no—
545C22#1681052311yescharacteristic
546C22#99×D5311yescharacteristic
547C22#1681059511yescharacteristic
548C22#1681061811yescharacteristic
549C22#1681062311yescharacteristic
550C22#1681063811yescharacteristic
551C22#1681063811yescharacteristic
552C22#1681084011yescharacteristic
553C22#1681084111yescharacteristic
554C22#1681084711yescharacteristic
555C22#1681084811yescharacteristic
556C4—21no—
557C4—21no—
558C4—21no—
559C4—21no—
560C4—82no—
561C4—81no—
562C4—81no—
563C4—1062no—
564C4—4242no—
565C4—4241no—
566C4—2121no—
567C4—2121no—
568C4—2121no—
569C2—21no—
570C2—21no—
571C2—21no—
572C2—21no—
573C2—8481no—
574C2#2027325211yescharacteristic
575C2#2027325711yescharacteristic
576C2#2027327011yescharacteristic
577C2#2027329111yescharacteristic
578C2#2027554511yescharacteristic
579C2#2027554911yescharacteristic
580C2#2027564411yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#328C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133191C22
cosocleC23×C4×C53C23
hypercenterC2×#328D53
hypocenterC53#9716
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
2. #20274693C2semidirect
3. C2×#16810360C2semidirect
6. #20272940C2semidirect
7. C2×C53×#328C2semidirect
8. C2×#13133206C4semidirect
13. #16810530C4semidirect
14. #16810524C4semidirect
19. #16810488C4semidirect
282. C2×#328D53semidirect
321. C53#9716semidirect

Maximal subgroups

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

subgroupindex
#202830182
#202746932
C2×#168103602
C2×#168103582
C2×#168103562
#202729402
C2×C53×#3282
#971653