Finite Group Databasebeta

← #27287381

Subgroups — #27287381

Order 13568.

632 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#20280457C211yesmaximal, characteristic
2#20275032C211yesmaximal, characteristic
3#20274903C211yesmaximal, characteristic, complemented
4C2×#16810356C211yesmaximal, characteristic
5C2×#16810349C211yesmaximal, characteristic, complemented
6#20272594C211yesmaximal, characteristic, complemented
7C53×#1441C211yesmaximal, characteristic, complemented
8C2×#13133200—21no—
9C2×#13133191C2211yescharacteristic
10C2×#13133150—21no—
11C2×#13133137—21no—
12C2×C22⋊#88322C2211yescharacteristic
13C2×C22⋊#88322C2211yescharacteristic
14#16810553—21no—
15#16810545—21no—
16#16810545—21no—
17#16810540—21no—
18#16810502—21no—
19#16810483C2211yescharacteristic
20#16810385—21no—
21#16810385—21no—
22#16810356—22no—
23#16810349—22no—
24C22×C4⋊C4×C53C2211yescharacteristic
25C22×C4⋊C4×C53C2211yescharacteristic
26C2×#95×C53—21no—
27C2×#95×C53—21no—
28C2×#95×C53C2211yescharacteristic
29C22×D212D411yescharacteristic
30C22×C22⋊D53—21no—
31C22×#88322—21no—
32C22×#88322—21no—
33#13133200—42no—
34#13133200—42no—
35#13133191—21no—
36#13133191—21no—
37#13133191—21no—
38#13133191—21no—
39C2×C4⋊Dic53D411yescharacteristic
40C2×C4⋊Dic53D411yescharacteristic
41C2×C4⋊Dic53D411yescharacteristic
42C2×#11719919—21no—
43C2×#11719919—21no—
44C2×C22⋊Dic53—21no—
45C2×#11719921—21no—
46C2×#11719921—21no—
47C2×#11719921—21no—
48C2×#11719921—21no—
49C2×#11719921—21no—
50#13133150—42no—
51#13133137—42no—
52C22⋊#88322—21no—
53C22⋊#88322—21no—
54C22⋊#88322—21no—
55C22⋊#88322D412yes—
56#13133124—21no—
57#13133124—21no—
58#13133124—21no—
59C23×C4×C53—21no—
60C23×C4×C53—21no—
61C23×C4×C53D411yescharacteristic
62C23×C4×C53D411yescharacteristic
63C23×C4×C53C2311yescharacteristic
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53—21no—
67C2×C4⋊C4×C53—21no—
68C2×C4⋊C4×C53—21no—
69C2×C4⋊C4×C53—21no—
70C2×C4⋊C4×C53—21no—
71C2×C4⋊C4×C53—21no—
72C2×C4⋊C4×C53—42no—
73#95×C53—21no—
74#95×C53—21no—
75#95×C53—42no—
76#95×C53—42no—
77C24×C53C2×D411yescharacteristic
78C23×D53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—22no—
86C22×C4×C53—22no—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
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×C53—41no—
97C22×C4×C53—41no—
98C22×C4×C53—41no—
99C22×C4×C53—41no—
100C22×C4×C53—41no—
101C22×C4×C53—41no—
102C22×C4×C53—41no—
103C22×C4×C53—41no—
104C22×C4×C53—41no—
105C22×C4×C53—41no—
106C22×C4×C53D812yes—
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53#4111yescharacteristic
111C22×C4×C53#4111yescharacteristic
112C22×C4×C53#4111yescharacteristic
113C2×C22⋊D53—41no—
114C2×C22⋊D53—41no—
115C2×C22⋊D53—41no—
116C2×C22⋊D53—41no—
117C2×C22⋊D53—41no—
118C2×C22⋊D53—41no—
119C22×Dic53—21no—
120C22×Dic53—21no—
121C22×Dic53—21no—
122C2×D212—21no—
123C2×D212—21no—
124C2×D212—41no—
125C2×D212—41no—
126C4⋊C4×C53—22no—
127C4⋊C4×C53—22no—
128C4⋊C4×C53—41no—
129C4⋊C4×C53—41no—
130C4⋊C4×C53—41no—
131C4⋊C4×C53—41no—
132C4⋊C4×C53—41no—
133C4⋊C4×C53—41no—
134C22⋊Dic53—41no—
135C22⋊Dic53—41no—
136#11719921—42no—
137#11719921—42no—
138#11719921—41no—
139#11719921—41no—
140#11719921—81no—
141#11719921—81no—
142C4⋊Dic53—21no—
143C4⋊Dic53—21no—
144C4⋊Dic53—41no—
145C4⋊Dic53—41no—
146#11719919—42no—
147#11719919—42no—
148C2×#88322—22no—
149C2×#88322—41no—
150C2×#88322—41no—
151C2×#88322—41no—
152C2×#88322—41no—
153C2×#88322—41no—
154C23×C53—21no—
155C23×C53—21no—
156C23×C53—21no—
157C23×C53—21no—
158C23×C53#12011yescharacteristic
159C23×C53#12111yescharacteristic
160C23×C53#12311yescharacteristic
161C23×C53C2×D811yescharacteristic
162C23×C53#13511yescharacteristic
163C23×C53AGL(1,ℤ/8)11yescharacteristic
164C23×C53#13711yescharacteristic
165C22×D53—41no—
166C22×D53—41no—
167C22×D53—41no—
168C22×D53—41no—
169C22×D53—41no—
170C22×D53—41no—
171C22×D53—41no—
172C2×C4×C53—21no—
173C2×C4×C53—21no—
174C2×C4×C53—21no—
175C2×C4×C53—21no—
176C2×C4×C53—22no—
177C2×C4×C53—22no—
178C2×C4×C53—22no—
179C2×C4×C53—41no—
180C2×C4×C53—41no—
181C2×C4×C53—41no—
182C2×C4×C53—41no—
183C2×C4×C53—41no—
184C2×C4×C53—41no—
185C2×C4×C53—41no—
186C2×C4×C53—41no—
187C2×C4×C53—42no—
188C2×C4×C53—42no—
189C2×C4×C53—42no—
190C2×C4×C53—42no—
191C2×C4×C53—42no—
192C2×C4×C53—42no—
193C2×C4×C53—41no—
194C2×C4×C53—42no—
195C2×C4×C53—41no—
196C2×C4×C53—41no—
197C2×C4×C53—81no—
198C2×C4×C53—81no—
199C2×C4×C53—81no—
200C2×C4×C53—81no—
201C2×C4×C53—81no—
202C2×C4×C53—81no—
203C2×C4×C53—81no—
204C2×C4×C53—81no—
205C2×C4×C53—81no—
206C2×C4×C53#12111yescharacteristic
207C2×C4×C53#12111yescharacteristic
208C2×C4×C53#12311yescharacteristic
209C2×C4×C53#12411yescharacteristic
210C22⋊D53—81no—
211C22⋊D53—81no—
212C22⋊D53—81no—
213C22⋊D53—81no—
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—
223D212—42no—
224D212—81no—
225#88322—42no—
226#88322—81no—
227#10397—531nomaximal, Sylow
228C22×C53—21no—
229C22×C53—21no—
230C22×C53—21no—
231C22×C53—21no—
232C22×C53—21no—
233C22×C53—21no—
234C22×C53—21no—
235C22×C53—21no—
236C22×C53—21no—
237C22×C53—21no—
238C22×C53—21no—
239C22×C53—21no—
240C22×C53#39411yescharacteristic
241C22×C53#44711yescharacteristic
242C22×C53#45211yescharacteristic
243C22×C53#45411yescharacteristic
244C22×C53#45611yescharacteristic
245C22×C53B(4,𝔽2)11yescharacteristic
246C22×C53#45811yescharacteristic
247C22×C53#46611yescharacteristic
248C22×C53#46811yescharacteristic
249C22×C53#48011yescharacteristic
250C22×C53#48211yescharacteristic
251C2×D53—81no—
252C2×D53—81no—
253C2×D53—81no—
254C2×D53—81no—
255C2×D53—81no—
256C2×D53—81no—
257C2×D53—81no—
258C4×C53—21no—
259C4×C53—21no—
260C4×C53—21no—
261C4×C53—21no—
262C4×C53—42no—
263C4×C53—82no—
264C4×C53—82no—
265C4×C53—81no—
266Dic53—81no—
267Dic53—81no—
268Dic53—81no—
269#2618—531no—
270#1874—531no—
271#1852—531no—
272C2×#327—531no—
273C2×#324—531no—
274#1441D5311yescharacteristic, complemented
275#1294—531no—
276C2×C53—21no—
277C2×C53—21no—
278C2×C53—21no—
279C2×C53—21no—
280C2×C53#159811yescharacteristic
281C2×C53#160311yescharacteristic
282C2×C53#161011yescharacteristic
283C2×C53#161511yescharacteristic
284C2×C53#203311yescharacteristic
285C2×C53#203811yescharacteristic
286C2×C53#204411yescharacteristic
287D53—161no—
288C2×#123—1061no—
289C2×#121—531no—
290C22×C4⋊C4C2×D5311yescharacteristic
291C22×C4⋊C4C2×D5311yescharacteristic
292C2×#107—1061no—
293C2×#102—1061no—
294C2×C22⋊C8—531no—
295C2×C22⋊C8—531no—
296#398—1061no—
297#396—1061no—
298#396—1061no—
299#394—1061no—
300#386—1061no—
301#380—531no—
302C2×#95—21no—
303C2×#95—21no—
304C2×#95C2×D5311yescharacteristic
305#340—1061no—
306#340—1061no—
307#327—1062no—
308#324—1062no—
309C53#1039711yescharacteristic, complemented, Sylow
310C22×D4—531no—
311C22×D4—1061no—
312C23×C4—21no—
313C23×C4—21no—
314C23×C4C22⋊D5311yescharacteristic
315C23×C4C22⋊D5311yescharacteristic
316C23×C4C22×D5311yescharacteristic
317C22×C8—1061no—
318C22×C8—1061no—
319#123—2122no—
320#123—2122no—
321#121—1061no—
322#121—1061no—
323#121—1061no—
324#121—1061no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—42no—
334C2×C4⋊C4—531no—
335C2×C4⋊C4—531no—
336C2×C4⋊C4—531no—
337C2×C4⋊C4—1061no—
338C2×C4⋊C4—1061no—
339C2×C22⋊C4—1061no—
340C2×C22⋊C4—1061no—
341C2×C22⋊C4—1061no—
342C2×C22⋊C4—1061no—
343C2×C22⋊C4—1061no—
344C2×C22⋊C4—1061no—
345#107—2122no—
346#102—2122no—
347C22⋊C8—532no—
348C22⋊C8—1061no—
349C22⋊C8—1061no—
350C22⋊C8—1061no—
351#95—21no—
352#95—21no—
353#95—42no—
354#95—42no—
355#95—1061no—
356#95—1061no—
357#95—1061no—
358C24—1061no—
359C24C2×C22⋊D5311yescharacteristic
360C2×D4—1061no—
361C2×D4—1061no—
362C2×D4—2121no—
363C2×D4—2121no—
364C2×D4—2121no—
365C2×D4—2121no—
366C2×D4—2121no—
367C2×D4—2121no—
368C2×D4—2121no—
369C2×D4—2121no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
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—1061no—
398C22×C4—1061no—
399C22×C4—1061no—
400C22×C4#1171992212yes—
401C22×C4#1171994511yescharacteristic
402C22×C4#1171994511yescharacteristic
403C22×C4D4×D5311yescharacteristic
404C22×C4D4×D5311yescharacteristic
405C22×C4D4×D5311yescharacteristic
406C22×C4#1171994711yescharacteristic
407C2×C8—1062no—
408C2×C8—2121no—
409C2×C8—2121no—
410C2×C8—2121no—
411C2×C8—2121no—
412C2×C8—2121no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—1061no—
422C4⋊C4—1061no—
423C4⋊C4—2121no—
424C4⋊C4—2121no—
425C4⋊C4—2122no—
426C4⋊C4—2122no—
427C22⋊C4—4241no—
428C22⋊C4—4241no—
429C22⋊C4—2122no—
430C22⋊C4—2122no—
431C22⋊C4—2121no—
432C22⋊C4—2121no—
433C22⋊C4—2121no—
434C22⋊C4—2121no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—2121no—
440C23—2121no—
441C23—2121no—
442C23—2121no—
443C23—2121no—
444C23—2121no—
445C23—2121no—
446C23#1313318511yescharacteristic
447C23#1313318911yescharacteristic
448C23#1313320011yescharacteristic
449C23C2×#1171992211yescharacteristic
450C23#1313323711yescharacteristic
451C23#1313324211yescharacteristic
452C23#1313324811yescharacteristic
453D4—4241no—
454D4—4241no—
455D4—4241no—
456D4—4241no—
457D4—4241no—
458D4—2122no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
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#1313319011yescharacteristic
503C2×C4#1313319011yescharacteristic
504C2×C4#1313319911yescharacteristic
505C2×C4#1313320311yescharacteristic
506C8—4241no—
507C8—2122no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—4241no—
521C22—4241no—
522C22—4241no—
523C22—4241no—
524C22—4241no—
525C22—4241no—
526C22—4241no—
527C22#1681053811yescharacteristic
528C22#1681075611yescharacteristic
529C22#1681078311yescharacteristic
530C22#1681079511yescharacteristic
531C22#1681080511yescharacteristic
532C22#1681080911yescharacteristic
533C22#1681081211yescharacteristic
534C22#1681085711yescharacteristic
535C22#1681087111yescharacteristic
536C22#1681093611yescharacteristic
537C22#1681094711yescharacteristic
538C4—21no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—42no—
543C4—82no—
544C4—82no—
545C4—81no—
546C4—4241no—
547C4—4241no—
548C4—4241no—
549C2—21no—
550C2—21no—
551C2—21no—
552C2—21no—
553C2—8481no—
554C2#2027364911yescharacteristic
555C2#2027367611yescharacteristic
556C2#2027371711yescharacteristic
557C2#2027374411yescharacteristic
558C2#2027589711yescharacteristic
559C2#2027593211yescharacteristic
560C2#2027596411yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#1441C2
socleC23×C53#120
prime radicalC2×#13133191C22
hypercenter#1441D53
hypocenterC53#10397
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1441

Complements

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

NH ≅ G/Ntype
3. #20274903C2semidirect
5. C2×#16810349C2semidirect
6. #20272594C2semidirect
7. C53×#1441C2semidirect
274. #1441D53semidirect
309. C53#10397semidirect

Maximal subgroups

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

subgroupindex
#202804572
#202750322
#202749032
C2×#168103562
C2×#168103492
#202725942
C53×#14412
#1039753