Finite Group Databasebeta

← #27468229

Subgroups — #27468229

Order 13568.

634 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279126C211yesmaximal, characteristic
2#20279115C211yesmaximal, characteristic
3#20276284C211yesmaximal, characteristic, complemented
4#20276284C211yesmaximal, characteristic, complemented
5#20276143C211yesmaximal, characteristic
6#20276138C211yesmaximal, characteristic
7#20275443C211yesmaximal, characteristic, complemented
8#20275443C211yesmaximal, characteristic, complemented
9#20275308C211yesmaximal, characteristic, complemented
10#20275287C211yesmaximal, characteristic, complemented
11#20274879C211yesmaximal, characteristic, complemented
12#20274740C211yesmaximal, characteristic
13#20274740C211yesmaximal, characteristic
14C4×#13133149C211yesmaximal, characteristic, complemented
15C8×C22⋊C4×C53C211yesmaximal, characteristic
16C2×#13133151C2211yescharacteristic
17C2×#13133151C2211yescharacteristic
18C2×#13133149C2211yescharacteristic
19C2×#13133149C2211yescharacteristic
20C2×#13133149C2211yescharacteristic
21C2×#13133149C2211yescharacteristic
22C53⋊#401—21no—
23#16810564—21no—
24#16810551—21no—
25#16810551—21no—
26#16810521C2211yescharacteristic
27#16810521C2211yescharacteristic
28#16810521C2211yescharacteristic
29#16810521C2211yescharacteristic
30#16810517—21no—
31#16810511—21no—
32#16810505—21no—
33#16810490—21no—
34#16810489C2211yescharacteristic
35#16810489C2211yescharacteristic
36#16810483C2211yescharacteristic
37#16810483C2211yescharacteristic
38#16810483C2211yescharacteristic
39#16810483C2211yescharacteristic
40C4×C4⋊Dic53C2211yescharacteristic
41C4×C4⋊Dic53C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52C23×C8×C53C2211yescharacteristic
53C2×C22⋊C8×C53C2211yescharacteristic
54C2×C4×C8×C53C2211yescharacteristic
55C2×C4×C8×C53C2211yescharacteristic
56C4×C22⋊C4×C53C2211yescharacteristic
57C53×#336C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C2×C4⋊Dic53C2311yescharacteristic
60C2×C4⋊Dic53C2311yescharacteristic
61C2×C4⋊Dic53C2311yescharacteristic
62C2×C4⋊Dic53C2311yescharacteristic
63C2×C4⋊Dic53C2311yescharacteristic
64C2×C4⋊Dic53C2311yescharacteristic
65C2×C4⋊Dic53C2311yescharacteristic
66C2×C4⋊Dic53C2311yescharacteristic
67C2×#11719919—21no—
68C2×#11719919—21no—
69C2×#11719919—21no—
70C2×#11719919—21no—
71C2×C22⋊Dic53—21no—
72C2×C22⋊Dic53—21no—
73C2×C22⋊Dic53—21no—
74C2×C22⋊Dic53—21no—
75C2×C4×Dic53—21no—
76C2×C4×Dic53—21no—
77#13133151—22no—
78#13133151—22no—
79#13133149—22no—
80#13133149—22no—
81#13133149—22no—
82#13133149—22no—
83#13133124—21no—
84#13133124—21no—
85#13133124—21no—
86#13133124—21no—
87#13133124—21no—
88#13133124—21no—
89#13133124—21no—
90#13133124—21no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97C23×C4×C53C2311yescharacteristic
98C22×C8×C53—22no—
99C22×C8×C53—22no—
100C22×C8×C53—22no—
101C22×C8×C53C2311yescharacteristic
102C22×C8×C53C2311yescharacteristic
103C22×C8×C53C2311yescharacteristic
104C22×C8×C53C2311yescharacteristic
105C2×C22⋊C4×C53D411yescharacteristic
106C2×C22⋊C4×C53D411yescharacteristic
107C2×C42×C53C2311yescharacteristic
108C2×C42×C53C2311yescharacteristic
109C22⋊C8×C53—22no—
110C4×C8×C53—22no—
111C4×C8×C53—22no—
112#95×C53D411yescharacteristic
113#95×C53D411yescharacteristic
114C24×C53C2×D411yescharacteristic
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2411yescharacteristic
127C22×Dic53—21no—
128C22×Dic53—21no—
129C22×Dic53—21no—
130C22×Dic53—21no—
131C22×Dic53—21no—
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—21no—
135C2×C8×C53—21no—
136C2×C8×C53—21no—
137C2×C8×C53—21no—
138C2×C8×C53—21no—
139C2×C8×C53—21no—
140C2×C8×C53—21no—
141C2×C8×C53—21no—
142C2×C8×C53—21no—
143C2×C8×C53—44no—
144C2×C8×C53#4112yes—
145C2×C8×C53#4112yes—
146C2×C8×C53#4112yes—
147C2×C8×C53#4112yes—
148C22⋊C4×C53—22no—
149C22⋊C4×C53—22no—
150C42×C53—21no—
151C42×C53—21no—
152C42×C53—21no—
153C42×C53—21no—
154C22⋊Dic53—42no—
155C22⋊Dic53—42no—
156C22⋊Dic53—81no—
157C22⋊Dic53—81no—
158C4⋊Dic53—21no—
159C4⋊Dic53—21no—
160C4⋊Dic53—21no—
161C4⋊Dic53—21no—
162C4⋊Dic53—21no—
163C4⋊Dic53—21no—
164C4⋊Dic53—21no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174#11719919—42no—
175#11719919—81no—
176#11719919—81no—
177#11719919—81no—
178C4×Dic53—42no—
179C4×Dic53—81no—
180C23×C53—21no—
181C23×C53—21no—
182C23×C53—21no—
183C23×C53—21no—
184C23×C53—21no—
185C23×C53—21no—
186C23×C53#13512yes—
187C23×C53C22×D411yescharacteristic
188C2×C4×C53—21no—
189C2×C4×C53—21no—
190C2×C4×C53—21no—
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—21no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—41no—
203C2×C4×C53—41no—
204C2×C4×C53—41no—
205C2×C4×C53—41no—
206C2×C4×C53#13512yes—
207C2×C4×C53C2×#4111yescharacteristic
208C2×C4×C53C2×#4111yescharacteristic
209C2×C4×C53C2×#4111yescharacteristic
210C2×C4×C53C2×#4111yescharacteristic
211C2×C4×C532−1+411yescharacteristic
212C2×C4×C532−1+411yescharacteristic
213C2×Dic53—41no—
214C2×Dic53—41no—
215C2×Dic53—41no—
216C2×Dic53—41no—
217C2×Dic53—41no—
218C2×Dic53—41no—
219C2×Dic53—41no—
220C2×Dic53—41no—
221C2×Dic53—41no—
222C2×Dic53—41no—
223C2×Dic53—41no—
224C2×Dic53—41no—
225C2×Dic53—41no—
226C2×Dic53—41no—
227C2×Dic53—41no—
228C2×Dic53—41no—
229C2×Dic53—41no—
230C2×Dic53—41no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C8×C53—22no—
238C8×C53—41no—
239C8×C53—41no—
240C8×C53#12614yes—
241#27836—531nomaximal, Sylow
242C22×C53—22no—
243C22×C53—22no—
244C22×C53—22no—
245C22×C53—41no—
246C22×C53—41no—
247C22×C53—41no—
248C22×C53—41no—
249C22×C53#53211yescharacteristic
250C22×C53#54711yescharacteristic
251C22×C53#54711yescharacteristic
252C22×C53C2×#13511yescharacteristic
253C22×C53C2×#13511yescharacteristic
254C22×C53#57711yescharacteristic
255C22×C53#57711yescharacteristic
256C4×C53—22no—
257C4×C53—41no—
258C4×C53—41no—
259C4×C53—41no—
260C4×C53—41no—
261C4×C53C2×#12611yescharacteristic
262C4×C53#54111yescharacteristic
263C4×C53#55211yescharacteristic
264C4×C53#55611yescharacteristic
265Dic53—81no—
266Dic53—81no—
267Dic53—81no—
268Dic53—81no—
269Dic53—81no—
270Dic53—81no—
271Dic53—81no—
272Dic53—81no—
273#2495—531no—
274#2494—531no—
275#2101—531no—
276#2101—531no—
277#2076—531no—
278#2075—531no—
279#1941—531no—
280#1941—531no—
281#1921—531no—
282#1918—531no—
283#1846—531no—
284#1807—531no—
285#1807—531no—
286C4×#106—531no—
287C8×C22⋊C4D5311yescharacteristic
288C2×C53—42no—
289C2×C53#249611yescharacteristic
290C2×C53#316811yescharacteristic
291C2×C53#317311yescharacteristic
292C2×C53#333711yescharacteristic
293C2×C53#333711yescharacteristic
294C2×C53#333711yescharacteristic
295C2×C53#334011yescharacteristic
296C23×C8C2×D5311yescharacteristic
297C2×#107—531no—
298C2×#107—531no—
299C2×#106—531no—
300C2×#106—531no—
301C2×#106—531no—
302C2×#106—531no—
303C2×C22⋊C8C2×D5311yescharacteristic
304C2×C4×C8C2×D5311yescharacteristic
305C2×C4×C8C2×D5311yescharacteristic
306#401—1061no—
307#400—1061no—
308#397—1061no—
309#397—1061no—
310#389—531no—
311#389—531no—
312#389—531no—
313#389—531no—
314#388—1061no—
315#388—1061no—
316#387—1061no—
317#383—1061no—
318#382—531no—
319#382—531no—
320#380—531no—
321#380—531no—
322#380—531no—
323#380—531no—
324C4×C4⋊C4—531no—
325C4×C4⋊C4—531no—
326C4×C22⋊C4C2×D5311yescharacteristic
327#340—531no—
328#340—531no—
329#340—531no—
330#340—531no—
331#340—531no—
332#340—531no—
333#340—531no—
334#340—531no—
335#340—531no—
336#340—531no—
337#336C2×D5311yescharacteristic
338#336C2×D5311yescharacteristic
339C53#2783611yescharacteristic, complemented, Sylow
340C23×C4C22×D5311yescharacteristic
341C22×C8—22no—
342C22×C8—22no—
343C22×C8—22no—
344C22×C8C22×D5311yescharacteristic
345C22×C8C22×D5311yescharacteristic
346C22×C8C22×D5311yescharacteristic
347C22×C8C22×D5311yescharacteristic
348C2×C4⋊C4—531no—
349C2×C4⋊C4—531no—
350C2×C4⋊C4—531no—
351C2×C4⋊C4—531no—
352C2×C4⋊C4—531no—
353C2×C4⋊C4—531no—
354C2×C4⋊C4—531no—
355C2×C4⋊C4—531no—
356C2×C4⋊C4—1061no—
357C2×C4⋊C4—1061no—
358C2×C4⋊C4—1061no—
359C2×C4⋊C4—1061no—
360C2×C22⋊C4—1061no—
361C2×C22⋊C4—1061no—
362C2×C22⋊C4—1061no—
363C2×C22⋊C4—1061no—
364C2×C22⋊C4D21211yescharacteristic
365C2×C22⋊C4D21211yescharacteristic
366C2×C42—1061no—
367C2×C42—1061no—
368C2×C42C22×D5311yescharacteristic
369C2×C42C22×D5311yescharacteristic
370#107—1062no—
371#107—1062no—
372#106—1062no—
373#106—1062no—
374#106—1062no—
375#106—1062no—
376C22⋊C8—22no—
377C4×C8—22no—
378C4×C8—22no—
379#95—1061no—
380#95—1061no—
381#95—1061no—
382#95—1061no—
383#95—1061no—
384#95—1061no—
385#95—1061no—
386#95—1061no—
387#95—1061no—
388#95—1061no—
389#95—1061no—
390#95—1061no—
391#95—1061no—
392#95—1061no—
393#95D21211yescharacteristic
394#95D21211yescharacteristic
395C24C2×D21211yescharacteristic
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—1061no—
403C22×C4—1061no—
404C22×C4—1061no—
405C22×C4—1061no—
406C22×C4—1061no—
407C22×C4—1061no—
408C22×C4—1061no—
409C22×C4—1061no—
410C22×C4C2×D21211yescharacteristic
411C22×C4C2×D21211yescharacteristic
412C22×C4C2×D21211yescharacteristic
413C22×C4C2×D21211yescharacteristic
414C22×C4C2×D21211yescharacteristic
415C22×C4C23×D5311yescharacteristic
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—44no—
425C2×C8#1171994512yes—
426C2×C8#1171994512yes—
427C2×C8#1171994712yes—
428C2×C8#1171994712yes—
429C4⋊C4—1061no—
430C4⋊C4—1061no—
431C4⋊C4—1061no—
432C4⋊C4—1061no—
433C4⋊C4—1061no—
434C4⋊C4—1061no—
435C4⋊C4—1061no—
436C4⋊C4—1061no—
437C4⋊C4—1061no—
438C4⋊C4—1061no—
439C4⋊C4—1061no—
440C4⋊C4—1061no—
441C4⋊C4—1061no—
442C4⋊C4—1061no—
443C4⋊C4—1061no—
444C4⋊C4—1061no—
445C4⋊C4—4241no—
446C4⋊C4—4241no—
447C4⋊C4—4241no—
448C4⋊C4—2122no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—4241no—
452C22⋊C4—4241no—
453C22⋊C4—2122no—
454C22⋊C4—2122no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—4241no—
460C42—2122no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23#1313323912yes—
468C23C22×D21211yescharacteristic
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—2121no—
488C2×C4—2121no—
489C2×C4—2121no—
490C2×C4—2121no—
491C2×C4—2121no—
492C2×C4—2121no—
493C2×C4—2121no—
494C2×C4—2121no—
495C2×C4—2121no—
496C2×C4—2121no—
497C2×C4—2121no—
498C2×C4—2121no—
499C2×C4—2121no—
500C2×C4—2121no—
501C2×C4—2121no—
502C2×C4—2121no—
503C2×C4—2121no—
504C2×C4—2121no—
505C2×C4—2121no—
506C2×C4—2121no—
507C2×C4—2121no—
508C2×C4—2121no—
509C2×C4—2121no—
510C2×C4—2121no—
511C2×C4#1313323912yes—
512C2×C4C2×#1171994711yescharacteristic
513C2×C4C2×#1171994711yescharacteristic
514C2×C4C2×#1171994511yescharacteristic
515C2×C4C2×#1171994511yescharacteristic
516C2×C4#1313326411yescharacteristic
517C2×C4#1313326411yescharacteristic
518C8—22no—
519C8—41no—
520C8—41no—
521C8#1313321014yes—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—41no—
529C22#1681117311yescharacteristic
530C22#1681127711yescharacteristic
531C22#1681127711yescharacteristic
532C22C2×#1313323911yescharacteristic
533C22C2×#1313323911yescharacteristic
534C22#1681143511yescharacteristic
535C22#1681143511yescharacteristic
536C4—22no—
537C4—41no—
538C4—41no—
539C4—41no—
540C4—41no—
541C4—4241no—
542C4—4241no—
543C4—4241no—
544C4—4241no—
545C4—4241no—
546C4—4241no—
547C4—4241no—
548C4—4241no—
549C4C2×#1313321011yescharacteristic
550C4#1681123511yescharacteristic
551C4#1681131311yescharacteristic
552C4#1681134311yescharacteristic
553C2—42no—
554C2#2027913711yescharacteristic
555C2#2028528511yescharacteristic
556C2#2028533211yescharacteristic
557C2#2028709611yescharacteristic
558C2#2028709611yescharacteristic
559C2#2028709611yescharacteristic
560C2#2028712911yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC8×C22⋊C4×C53C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenterC8×C22⋊C4D53
hypocenterC53#27836
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #20276284C2semidirect
4. #20276284C2semidirect
7. #20275443C2semidirect
8. #20275443C2semidirect
9. #20275308C2semidirect
10. #20275287C2semidirect
11. #20274879C2semidirect
14. C4×#13133149C2semidirect
339. C53#27836semidirect

Maximal subgroups

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

subgroupindex
#202791262
#202791152
#202762842
#202761432
#202761382
#202754432
#202753082
#202752872
#202748792
#202747402
C4×#131331492
C8×C22⋊C4×C532
#2783653