Finite Group Databasebeta

← #27434167

Subgroups — #27434167

Order 13568.

840 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810982C211yesmaximal, characteristic
2C2×#16810842C211yesmaximal, characteristic
3#20279923C211yesmaximal, characteristic, complemented
4#20276123C211yesmaximal, characteristic, complemented
5#20276118C211yesmaximal, characteristic
6#20276037C211yesmaximal, characteristic, complemented
7#20276032C211yesmaximal, characteristic
8#20275667C211yesmaximal, characteristic, complemented
9#20275439C211yesmaximal, characteristic, complemented
10#20275056C211yesmaximal, characteristic, complemented
11#20275049C211yesmaximal, characteristic
12#20274720C211yesmaximal, characteristic, complemented
13#20274715C211yesmaximal, characteristic
14#20274548C211yesmaximal, characteristic, complemented
15C53×#2367C211yesmaximal, characteristic
16C22×#11719925C2211yescharacteristic
17C22×#11719923C2211yescharacteristic
18C2×#13133217C2211yescharacteristic
19C2×C4×Dic106C2211yescharacteristic
20#16810982—24no—
21#16810842—24no—
22C2×C4⋊#88322C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×C4⋊#88322C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×#13133140C2211yescharacteristic
29C2×#13133140C2211yescharacteristic
30C2×#13133138C2211yescharacteristic
31C2×#13133138C2211yescharacteristic
32C2×#13133138C2211yescharacteristic
33C2×#13133128C2211yescharacteristic
34#16810544—21no—
35#16810544—21no—
36#16810529C2211yescharacteristic
37#16810529C2211yescharacteristic
38#16810522—21no—
39#16810522—21no—
40#16810493C2211yescharacteristic
41#16810493C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×#124×C53C2211yescharacteristic
51C2×C4×Q8×C53C2211yescharacteristic
52C2×C4×D4×C53C2211yescharacteristic
53C53×#391C2211yescharacteristic
54C53×#390C2211yescharacteristic
55C53×#388—21no—
56C53×#388—21no—
57C53×#385—21no—
58C4×C4⋊C4×C53C2211yescharacteristic
59C4×C4⋊C4×C53C2211yescharacteristic
60C4×C22⋊C4×C53—21no—
61C22×Dic106—21no—
62C22×Dic106C2311yescharacteristic
63C2×#11719925—22no—
64C2×#11719925—22no—
65C2×#11719925—41no—
66C2×#11719923—22no—
67C2×#11719923—22no—
68C2×#11719923—41no—
69C22×#88322C2311yescharacteristic
70C22×#88322C2311yescharacteristic
71C22×#88322C2311yescharacteristic
72C22×#88322C2311yescharacteristic
73#13133217—22no—
74#13133217—22no—
75C4×Dic106—22no—
76C4×Dic106—22no—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53—21no—
79C2×C4⋊Dic53—21no—
80C2×C4⋊Dic53—21no—
81C2×C4⋊Dic53—21no—
82C2×C4⋊Dic53C2311yescharacteristic
83C2×C4⋊Dic53C2311yescharacteristic
84C2×C4⋊Dic53C2311yescharacteristic
85C2×#11719919—21no—
86C2×C4×Dic53—21no—
87C4⋊#88322—22no—
88C4⋊#88322—41no—
89C4⋊#88322—41no—
90#13133141—22no—
91#13133141—22no—
92#13133141—22no—
93#13133140—22no—
94#13133140—22no—
95#13133138—22no—
96#13133138—22no—
97#13133138—22no—
98#13133128—22no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103C22×Q8×C53C2311yescharacteristic
104C22×D4×C53C2311yescharacteristic
105C23×C4×C53—21no—
106#124×C53—22no—
107#124×C53D414yes—
108C4×Q8×C53—22no—
109C4×Q8×C53—41no—
110C4×D4×C53—22no—
111C4×D4×C53—41no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53D411yescharacteristic
114C2×C4⋊C4×C53D411yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C22⋊C4×C53—21no—
118C2×C22⋊C4×C53—21no—
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53—21no—
121C2×C22⋊C4×C53—21no—
122C2×C42×C53—21no—
123C2×C42×C53—21no—
124C2×C42×C53C2311yescharacteristic
125C2×C42×C53C2311yescharacteristic
126C2×C42×C53C2311yescharacteristic
127#95×C53—21no—
128#95×C53—21no—
129#95×C53—21no—
130#95×C53D411yescharacteristic
131#95×C53D411yescharacteristic
132C24×C53—21no—
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53—21no—
137C2×Q8×C53—21no—
138C2×Q8×C53C2×D412yes—
139C2×D4×C53—21no—
140C2×D4×C53—21no—
141C2×D4×C53—21no—
142C2×D4×C53—21no—
143C2×D4×C53—21no—
144C2×D4×C53C2×D412yes—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—42no—
149C22×C4×C53—42no—
150C22×C4×C53—81no—
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2411yescharacteristic
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53—21no—
161C22×Dic53—21no—
162C2×Dic106—21no—
163C2×Dic106—21no—
164C2×Dic106—21no—
165C2×Dic106—21no—
166C2×Dic106—21no—
167C2×Dic106—42no—
168C2×Dic106—81no—
169C2×Dic106—81no—
170C2×Dic106#4112yes—
171C4⋊C4×C53—22no—
172C4⋊C4×C53—22no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—42no—
177C4⋊C4×C53#4112yes—
178C22⋊C4×C53—24no—
179C22⋊C4×C53—24no—
180C22⋊C4×C53—42no—
181C22⋊C4×C53—42no—
182C22⋊C4×C53—81no—
183C42×C53—21no—
184C42×C53—21no—
185C42×C53—21no—
186C42×C53—21no—
187C42×C53—42no—
188C42×C53—81no—
189C42×C53C2×D412yes—
190C42×C53#4112yes—
191#11719925—44no—
192#11719923—44no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—42no—
200C4⋊Dic53—81no—
201C4⋊Dic53—81no—
202C4⋊Dic53—81no—
203C4⋊Dic53—81no—
204#11719919—42no—
205C4×Dic53—42no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322—21no—
215C2×#88322—21no—
216C2×#88322—21no—
217C2×#88322—21no—
218C23×C53—22no—
219C23×C53—41no—
220C23×C53—41no—
221C23×C53—41no—
222C23×C53—41no—
223C23×C53—41no—
224C23×C53—41no—
225C23×C53C22×D411yescharacteristic
226Q8×C53—22no—
227Q8×C53—22no—
228Q8×C53—41no—
229Q8×C53—41no—
230D4×C53—22no—
231D4×C53—22no—
232D4×C53—41no—
233D4×C53—41no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—22no—
243C2×C4×C53—22no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—21no—
249C2×C4×C53—21no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—41no—
257C2×C4×C53—84no—
258C2×C4×C53#12112yes—
259C2×C4×C53#12112yes—
260C2×C4×C53#13712yes—
261C2×C4×C53#13712yes—
262C2×C4×C53C22×D411yescharacteristic
263C2×C4×C53C2×#4111yescharacteristic
264C2×C4×C53C2×#4111yescharacteristic
265C2×C4×C53C2×#4111yescharacteristic
266C2×C4×C53H(5,𝔽2)11yescharacteristic
267C2×C4×C532−1+411yescharacteristic
268C2×Dic53—22no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280Dic106—22no—
281Dic106—22no—
282Dic106—41no—
283Dic106—41no—
284Dic106—84no—
285#88322—41no—
286#88322—41no—
287#88322—41no—
288#88322—41no—
289#25336—531nomaximal, Sylow
290C22×C53—42no—
291C22×C53—42no—
292C22×C53—42no—
293C22×C53—81no—
294C22×C53—81no—
295C22×C53—81no—
296C22×C53—81no—
297C22×C53C2×#12111yescharacteristic
298C22×C53#54611yescharacteristic
299C22×C53#54811yescharacteristic
300C22×C53C2×#13711yescharacteristic
301C22×C53C2×#13711yescharacteristic
302C22×C53#57611yescharacteristic
303C22×C53#57711yescharacteristic
304C4×C53—22no—
305C4×C53—22no—
306C4×C53—22no—
307C4×C53—22no—
308C4×C53—42no—
309C4×C53—41no—
310C4×C53—42no—
311C4×C53—41no—
312C4×C53—81no—
313C4×C53#52911yescharacteristic
314C4×C53#54211yescharacteristic
315C4×C53#54611yescharacteristic
316C4×C53#54711yescharacteristic
317Dic53—42no—
318Dic53—81no—
319Dic53—81no—
320Dic53—81no—
321C2×#489—531no—
322C2×#464—531no—
323#2567—531no—
324#2367D5311yescharacteristic
325#2072—531no—
326#2071—531no—
327#2056—531no—
328#2055—531no—
329#1988—531no—
330#1940—531no—
331#1878—531no—
332#1877—531no—
333#1803—531no—
334#1802—531no—
335#1763—531no—
336C2×C53—82no—
337C2×C53#242711yescharacteristic
338C2×C53#303911yescharacteristic
339C2×C53#304511yescharacteristic
340C2×C53#331411yescharacteristic
341C2×C53#331611yescharacteristic
342C2×C53#336911yescharacteristic
343C2×C53#337511yescharacteristic
344C22×Q16—531no—
345C22×SD16—531no—
346C2×C4⋊Q8—531no—
347C2×#124C2×D5311yescharacteristic
348C2×C4×Q8—531no—
349C2×C4×Q8C2×D5311yescharacteristic
350C2×C4×D4C2×D5311yescharacteristic
351#489—1064no—
352#464—1064no—
353C2×C4⋊C8—531no—
354C2×C4⋊C8—531no—
355C2×C4⋊C8—531no—
356C2×#103—531no—
357C2×#103—531no—
358C2×#103—531no—
359C2×#103—531no—
360C2×#103—531no—
361C2×#102—531no—
362C2×#102—531no—
363C2×#102—531no—
364C2×#97—531no—
365#395—1061no—
366#395—1061no—
367#391—531no—
368#391—531no—
369#391C2×D5311yescharacteristic
370#390C2×D5311yescharacteristic
371#389—1061no—
372#389—1061no—
373#388—21no—
374#388—21no—
375#385—21no—
376#384—531no—
377#384—531no—
378C4×C4⋊C4C2×D5311yescharacteristic
379C4×C4⋊C4C2×D5311yescharacteristic
380C4×C22⋊C4—21no—
381#340—531no—
382#340—531no—
383#340—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387#336—531no—
388#336—531no—
389C53#2533611yescharacteristic, complemented, Sylow
390C22×Q8—531no—
391C22×Q8—1061no—
392C22×Q8C22×D5311yescharacteristic
393C22×D4C22×D5311yescharacteristic
394C23×C4—21no—
395C2×Q16—1062no—
396C2×Q16—1062no—
397C2×Q16—2121no—
398C2×SD16—1062no—
399C2×SD16—1062no—
400C2×SD16—2121no—
401C22×C8—531no—
402C22×C8—531no—
403C22×C8—531no—
404C22×C8—531no—
405C4⋊Q8—1062no—
406C4⋊Q8—1062no—
407#124—22no—
408#124D21214yes—
409C4×Q8—22no—
410C4×Q8—41no—
411C4×Q8—1062no—
412C4×Q8—1062no—
413C4×D4—22no—
414C4×D4—41no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—531no—
417C2×C4⋊C4—531no—
418C2×C4⋊C4—531no—
419C2×C4⋊C4—1061no—
420C2×C4⋊C4—1061no—
421C2×C4⋊C4—1061no—
422C2×C4⋊C4—1061no—
423C2×C4⋊C4—1061no—
424C2×C4⋊C4—1061no—
425C2×C4⋊C4C22⋊D5311yescharacteristic
426C2×C4⋊C4C22⋊D5311yescharacteristic
427C2×C4⋊C4C22×D5311yescharacteristic
428C2×C4⋊C4C22×D5311yescharacteristic
429C2×C22⋊C4—21no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C42—21no—
435C2×C42—21no—
436C2×C42—1061no—
437C2×C42C22×D5311yescharacteristic
438C2×C42C22×D5311yescharacteristic
439C2×C42C22×D5311yescharacteristic
440C4⋊C8—1062no—
441C4⋊C8—2121no—
442C4⋊C8—2121no—
443#103—1062no—
444#103—1062no—
445#103—1062no—
446#103—1062no—
447#103—1062no—
448#102—1062no—
449#102—1062no—
450#102—1062no—
451#97—1062no—
452#95—21no—
453#95—21no—
454#95—21no—
455#95—1061no—
456#95—1061no—
457#95—1061no—
458#95—1061no—
459#95C22⋊D5311yescharacteristic
460#95C22⋊D5311yescharacteristic
461C24—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—532no—
468C2×Q8—1061no—
469C2×Q8—1061no—
470C2×Q8—1061no—
471C2×Q8—1061no—
472C2×Q8—1061no—
473C2×Q8—4241no—
474C2×Q8—4241no—
475C2×Q8—2122no—
476C2×Q8C2×D21212yes—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4C2×D21212yes—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—81no—
489C22×C4—1061no—
490C22×C4—1061no—
491C22×C4—1061no—
492C22×C4—1061no—
493C22×C4C2×C22⋊D5311yescharacteristic
494C22×C4C2×C22⋊D5311yescharacteristic
495C22×C4C2×C22⋊D5311yescharacteristic
496C22×C4C2×C22⋊D5311yescharacteristic
497C22×C4C2×C22⋊D5311yescharacteristic
498C22×C4C2×C22⋊D5311yescharacteristic
499C22×C4C23×D5311yescharacteristic
500Q16—2124no—
501SD16—2124no—
502C2×C8—1061no—
503C2×C8—1061no—
504C2×C8—1061no—
505C2×C8—1061no—
506C2×C8—1061no—
507C2×C8—1061no—
508C2×C8—1061no—
509C2×C8—1061no—
510C2×C8—1061no—
511C2×C8—1061no—
512C2×C8—1061no—
513C2×C8—1061no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—42no—
520C4⋊C4—1061no—
521C4⋊C4—1061no—
522C4⋊C4—1061no—
523C4⋊C4—1061no—
524C4⋊C4—1061no—
525C4⋊C4—1061no—
526C4⋊C4—4241no—
527C4⋊C4—4241no—
528C4⋊C4—4241no—
529C4⋊C4—4241no—
530C4⋊C4—2122no—
531C4⋊C4—2122no—
532C4⋊C4#1171994712yes—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C22⋊C4—81no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—42no—
543C42—81no—
544C42—2122no—
545C42C2×D21212yes—
546C42#1171994512yes—
547C23—22no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23C22×C22⋊D5311yescharacteristic
555Q8—22no—
556Q8—22no—
557Q8—41no—
558Q8—41no—
559Q8—1062no—
560Q8—1062no—
561Q8—4244no—
562Q8—2121no—
563Q8—2121no—
564D4—22no—
565D4—22no—
566D4—41no—
567D4—41no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—84no—
592C2×C4—1062no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4—2121no—
604C2×C4#1313319112yes—
605C2×C4#1313319112yes—
606C2×C4#1313324712yes—
607C2×C4#1313324712yes—
608C2×C4C22×D21211yescharacteristic
609C2×C4#41×D5311yescharacteristic
610C2×C4C2×#1171994711yescharacteristic
611C2×C4C2×#1171994511yescharacteristic
612C2×C4#1313326211yescharacteristic
613C2×C4#1313326411yescharacteristic
614C8—2121no—
615C8—2121no—
616C8—2121no—
617C8—2121no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22C2×#1313319111yescharacteristic
626C22#1681126711yescharacteristic
627C22#1681128011yescharacteristic
628C22C2×#1313324711yescharacteristic
629C22C2×#1313324711yescharacteristic
630C22#1681142611yescharacteristic
631C22#1681143111yescharacteristic
632C4—22no—
633C4—22no—
634C4—22no—
635C4—22no—
636C4—42no—
637C4—41no—
638C4—42no—
639C4—41no—
640C4—81no—
641C4—4241no—
642C4—4241no—
643C4—4241no—
644C4—2122no—
645C4#1681115511yescharacteristic
646C4#1681123811yescharacteristic
647C4#1681126811yescharacteristic
648C4#1681127711yescharacteristic
649C2—82no—
650C2#2027828611yescharacteristic
651C2#2028420211yescharacteristic
652C2#2028425411yescharacteristic
653C2#2028679611yescharacteristic
654C2#2028682611yescharacteristic
655C2#2028745111yescharacteristic
656C2#2028752511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2367C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2367D53
hypocenterC53#25336
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2367

Complements

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

NH ≅ G/Ntype
3. #20279923C2semidirect
4. #20276123C2semidirect
6. #20276037C2semidirect
8. #20275667C2semidirect
9. #20275439C2semidirect
10. #20275056C2semidirect
12. #20274720C2semidirect
14. #20274548C2semidirect
389. C53#25336semidirect

Maximal subgroups

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

subgroupindex
C2×#168109822
C2×#168108422
#202799232
#202761232
#202761182
#202760372
#202760322
#202756672
#202754392
#202750562
#202750492
#202747202
#202747152
#202745482
C53×#23672
#2533653