Finite Group Databasebeta

← #27430007

Subgroups — #27430007

Order 13568.

792 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810879C211yesmaximal, characteristic
2C2×#16810728C211yesmaximal, characteristic, complemented
3#20278240C211yesmaximal, characteristic
4#20276124C211yesmaximal, characteristic, complemented
5#20276123C211yesmaximal, characteristic, complemented
6#20275911C211yesmaximal, characteristic
7#20275908C211yesmaximal, characteristic
8#20275575C211yesmaximal, characteristic, complemented
9#20275450C211yesmaximal, characteristic, complemented
10#20274983C211yesmaximal, characteristic, complemented
11#20274982C211yesmaximal, characteristic, complemented
12#20274871C211yesmaximal, characteristic, complemented
13#20274746C211yesmaximal, characteristic
14#20274499C211yesmaximal, characteristic
15C53×#2423C211yesmaximal, characteristic
16C22×#11719925C2211yescharacteristic
17C22×#11719917C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C2×C4×Dic106C2211yescharacteristic
20#16810879—24no—
21#16810728—24no—
22C2×#13133150C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×#13133141C2211yescharacteristic
29C2×#13133141C2211yescharacteristic
30C2×#13133140C2211yescharacteristic
31C2×#13133140C2211yescharacteristic
32C2×#13133140C2211yescharacteristic
33C2×C22⋊#88322C2211yescharacteristic
34C2×#13133128C2211yescharacteristic
35C2×C4×#88322C2211yescharacteristic
36#16810559—21no—
37#16810557—21no—
38#16810529C2211yescharacteristic
39#16810521C2211yescharacteristic
40#16810516—21no—
41#16810506—21no—
42#16810493C2211yescharacteristic
43#16810483C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×C22⋊Q8×C53C2211yescharacteristic
51C2×C4×Q8×C53C2211yescharacteristic
52C53×#399—21no—
53C53×#398—21no—
54C53×#391C2211yescharacteristic
55C53×#389C2211yescharacteristic
56C53×#388—21no—
57C53×#387—21no—
58C53×#384C2211yescharacteristic
59C53×#380C2211yescharacteristic
60C4×C22⋊C4×C53C2211yescharacteristic
61C22×Dic106C2311yescharacteristic
62C2×#11719925—22no—
63C2×#11719925—22no—
64C2×#11719925D414yes—
65C2×#11719917—22no—
66C2×#11719917—22no—
67C2×#11719917—22no—
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70C22×#88322C2311yescharacteristic
71C22×#88322C2311yescharacteristic
72#13133196—22no—
73#13133196—41no—
74C4×Dic106—22no—
75C4×Dic106—41no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×C22⋊Dic53—21no—
85C2×C22⋊Dic53—21no—
86C2×C4×Dic53—21no—
87#13133150—22no—
88#13133148—22no—
89#13133148—22no—
90#13133148—41no—
91#13133148—41no—
92#13133141—22no—
93#13133141—22no—
94#13133141—22no—
95#13133140—22no—
96#13133140—22no—
97#13133140—22no—
98C22⋊#88322—22no—
99#13133128—22no—
100C4×#88322—22no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105C22×Q8×C53C2311yescharacteristic
106C23×C4×C53C2311yescharacteristic
107C22⋊Q8×C53—22no—
108C22⋊Q8×C53—41no—
109C4×Q8×C53—22no—
110C4×Q8×C53—41no—
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53—21no—
115C2×C4⋊C4×C53—21no—
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53—21no—
121C2×C22⋊C4×C53D411yescharacteristic
122C2×C22⋊C4×C53D411yescharacteristic
123C2×C42×C53—21no—
124C2×C42×C53C2311yescharacteristic
125C2×C42×C53C2311yescharacteristic
126#95×C53—21no—
127#95×C53—21no—
128#95×C53—21no—
129#95×C53—21no—
130#95×C53D411yescharacteristic
131#95×C53D411yescharacteristic
132C24×C53C2×D411yescharacteristic
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53—21no—
137C2×Q8×C53—21no—
138C2×Q8×C53C2×D412yes—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2411yescharacteristic
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106—21no—
162C2×Dic106—21no—
163C2×Dic106—21no—
164C2×Dic106C2×D412yes—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—42no—
172C4⋊C4×C53—42no—
173C4⋊C4×C53—42no—
174C4⋊C4×C53—81no—
175C4⋊C4×C53—81no—
176C22⋊C4×C53—22no—
177C22⋊C4×C53—22no—
178C22⋊C4×C53—42no—
179C22⋊C4×C53—81no—
180C42×C53—21no—
181C42×C53—21no—
182C42×C53—21no—
183C42×C53—21no—
184C42×C53—42no—
185C22⋊Dic53—42no—
186C22⋊Dic53—81no—
187#11719925—28no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—42no—
195C4⋊Dic53—81no—
196#11719919—42no—
197#11719919—42no—
198#11719919—81no—
199C4×Dic53—42no—
200#11719917—44no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322C2×D412yes—
210C2×#88322#4112yes—
211C2×#88322#4112yes—
212C2×#88322#4112yes—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53C22×D411yescharacteristic
221Q8×C53—22no—
222Q8×C53—22no—
223Q8×C53—41no—
224Q8×C53—41no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—22no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53#13712yes—
255C2×C4×C53#13712yes—
256C2×C4×C53C22×D411yescharacteristic
257C2×C4×C53C2×#4111yescharacteristic
258C2×C4×C53C2×#4111yescharacteristic
259C2×C4×C53C2×#4111yescharacteristic
260C2×C4×C53H(5,𝔽2)11yescharacteristic
261C2×C4×C532−1+411yescharacteristic
262C2×Dic53—22no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274Dic106—22no—
275Dic106—22no—
276Dic106—41no—
277Dic106—41no—
278#88322—22no—
279#88322—41no—
280#88322—41no—
281#88322#12314yes—
282#25047—531nomaximal, Sylow
283C22×C53—41no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53—41no—
289C22×C53—41no—
290C22×C53#53211yescharacteristic
291C22×C53D4211yescharacteristic
292C22×C53#54711yescharacteristic
293C22×C53C2×#13711yescharacteristic
294C22×C53C2×#13711yescharacteristic
295C22×C53#57711yescharacteristic
296C22×C53#57711yescharacteristic
297C4×C53—22no—
298C4×C53—22no—
299C4×C53—41no—
300C4×C53—42no—
301C4×C53—41no—
302C4×C53—41no—
303C4×C53—81no—
304C4×C53—81no—
305C4×C53—81no—
306C4×C53C2×#12311yescharacteristic
307C4×C53#54211yescharacteristic
308C4×C53#54711yescharacteristic
309C4×C53#55311yescharacteristic
310Dic53—42no—
311Dic53—81no—
312Dic53—81no—
313Dic53—81no—
314C2×#470—531no—
315C2×#441—531no—
316#2423—531no—
317#2423D5311yescharacteristic
318#2072—531no—
319#2072—531no—
320#2035—531no—
321#2035—531no—
322#1969—531no—
323#1943—531no—
324#1865—531no—
325#1865—531no—
326#1844—531no—
327#1809—531no—
328#1753—531no—
329C2×C53—81no—
330C2×C53#242211yescharacteristic
331C2×C53#317311yescharacteristic
332C2×C53#318511yescharacteristic
333C2×C53#327811yescharacteristic
334C2×C53#327811yescharacteristic
335C2×C53#334411yescharacteristic
336C2×C53#334411yescharacteristic
337C22×Q16—531no—
338C22×M16—531no—
339C2×C22⋊Q8—531no—
340C2×C22⋊Q8C2×D5311yescharacteristic
341C2×C4×Q8—531no—
342C2×C4×Q8C2×D5311yescharacteristic
343#470—1064no—
344#441—1064no—
345C2×#107—531no—
346C2×#106—531no—
347C2×#106—531no—
348C2×#106—531no—
349C2×#106—531no—
350C2×#103—531no—
351C2×#103—531no—
352C2×#103—531no—
353C2×#103—531no—
354C2×#103—531no—
355C2×#103—531no—
356C2×C22⋊C8—531no—
357C2×#97—531no—
358C2×C4×C8—531no—
359#399—21no—
360#399—1061no—
361#398—21no—
362#398—1061no—
363#391—531no—
364#391C2×D5311yescharacteristic
365#389—531no—
366#389C2×D5311yescharacteristic
367#388—21no—
368#388—1061no—
369#387—21no—
370#387—1061no—
371#384—531no—
372#384C2×D5311yescharacteristic
373#380—531no—
374#380C2×D5311yescharacteristic
375C4×C22⋊C4C2×D5311yescharacteristic
376#340—531no—
377#340—531no—
378#340—531no—
379#340—531no—
380#336—531no—
381#336—531no—
382C53#2504711yescharacteristic, complemented, Sylow
383C22×Q8—531no—
384C22×Q8C22×D5311yescharacteristic
385C23×C4C22×D5311yescharacteristic
386C2×Q16—534no—
387C2×Q16—1062no—
388C2×Q16—1062no—
389C2×M16—1062no—
390C2×M16—1062no—
391C2×M16—1062no—
392C22×C8—531no—
393C22×C8—531no—
394C22×C8—531no—
395C22×C8—531no—
396C22⋊Q8—22no—
397C22⋊Q8—41no—
398C22⋊Q8—1062no—
399C22⋊Q8—2121no—
400C4×Q8—22no—
401C4×Q8—41no—
402C4×Q8—1062no—
403C4×Q8—2121no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—531no—
410C2×C4⋊C4—531no—
411C2×C4⋊C4—531no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4C22×D5311yescharacteristic
418C2×C4⋊C4C22×D5311yescharacteristic
419C2×C4⋊C4C22×D5311yescharacteristic
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—1061no—
423C2×C22⋊C4—1061no—
424C2×C22⋊C4C22⋊D5311yescharacteristic
425C2×C22⋊C4C22⋊D5311yescharacteristic
426C2×C42—21no—
427C2×C42—1061no—
428C2×C42C22×D5311yescharacteristic
429C2×C42C22×D5311yescharacteristic
430#107—1062no—
431#106—1062no—
432#106—1062no—
433#106—2121no—
434#106—2121no—
435#103—1062no—
436#103—1062no—
437#103—1062no—
438#103—1062no—
439#103—1062no—
440#103—1062no—
441C22⋊C8—1062no—
442#97—1062no—
443C4×C8—1062no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—21no—
448#95—1061no—
449#95—1061no—
450#95—1061no—
451#95—1061no—
452#95C22⋊D5311yescharacteristic
453#95C22⋊D5311yescharacteristic
454C24C2×C22⋊D5311yescharacteristic
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—532no—
461C2×Q8—1061no—
462C2×Q8—1061no—
463C2×Q8—1061no—
464C2×Q8—1061no—
465C2×Q8—1061no—
466C2×Q8D4×D5312yes—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4—1061no—
481C22×C4C2×C22⋊D5311yescharacteristic
482C22×C4C2×C22⋊D5311yescharacteristic
483C22×C4C2×C22⋊D5311yescharacteristic
484C22×C4C2×C22⋊D5311yescharacteristic
485C22×C4C2×C22⋊D5311yescharacteristic
486C22×C4C23×D5311yescharacteristic
487Q16—1068no—
488M16—2124no—
489C2×C8—532no—
490C2×C8—532no—
491C2×C8—532no—
492C2×C8—532no—
493C2×C8—1061no—
494C2×C8—1061no—
495C2×C8—1061no—
496C2×C8—1061no—
497C2×C8—1061no—
498C2×C8—1061no—
499C2×C8—1061no—
500C2×C8—1061no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—81no—
511C4⋊C4—81no—
512C4⋊C4—1061no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—4241no—
519C4⋊C4—4241no—
520C4⋊C4—2122no—
521C4⋊C4—2122no—
522C4⋊C4—2122no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—42no—
526C22⋊C4—81no—
527C22⋊C4—4241no—
528C22⋊C4—2122no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—42no—
534C42—2122no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23C22×C22⋊D5311yescharacteristic
543Q8—22no—
544Q8—22no—
545Q8—41no—
546Q8—41no—
547Q8—1062no—
548Q8—1062no—
549Q8—2121no—
550Q8—2121no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—1062no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4#1313324712yes—
593C2×C4#1313324812yes—
594C2×C4C2×D4×D5311yescharacteristic
595C2×C4#41×D5311yescharacteristic
596C2×C4#41×D5311yescharacteristic
597C2×C4#41×D5311yescharacteristic
598C2×C4#1313326211yescharacteristic
599C2×C4#1313326411yescharacteristic
600C8—534no—
601C8—1062no—
602C8—2121no—
603C8—2121no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22#1681116911yescharacteristic
612C22D4×C22⋊D5311yescharacteristic
613C22#1681127211yescharacteristic
614C22C2×#1313324711yescharacteristic
615C22C2×#1313324811yescharacteristic
616C22#1681143111yescharacteristic
617C22#1681143411yescharacteristic
618C4—22no—
619C4—22no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—42no—
624C4—81no—
625C4—81no—
626C4—81no—
627C4—4241no—
628C4—4241no—
629C4—4241no—
630C4—2122no—
631C4#123×D5311yescharacteristic
632C4#1681124111yescharacteristic
633C4#1681127111yescharacteristic
634C4#1681131811yescharacteristic
635C2—81no—
636C2#2027822611yescharacteristic
637C2#2028532611yescharacteristic
638C2#2028545611yescharacteristic
639C2#2028637711yescharacteristic
640C2#2028638111yescharacteristic
641C2#2028716411yescharacteristic
642C2#2028717011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2423C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2423D53
hypocenterC53#25047
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2423

Complements

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

NH ≅ G/Ntype
2. C2×#16810728C2semidirect
4. #20276124C2semidirect
5. #20276123C2semidirect
8. #20275575C2semidirect
9. #20275450C2semidirect
10. #20274983C2semidirect
11. #20274982C2semidirect
12. #20274871C2semidirect
382. C53#25047semidirect

Maximal subgroups

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

subgroupindex
C2×#168108792
C2×#168107282
#202782402
#202761242
#202761232
#202759112
#202759082
#202755752
#202754502
#202749832
#202749822
#202748712
#202747462
#202744992
C53×#24232
#2504753