Finite Group Databasebeta

← #27417312

Subgroups — #27417312

Order 13568.

1338 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810951C211yesmaximal, characteristic, complemented
2C2×#16810624C211yesmaximal, characteristic, complemented
3#20279032C211yesmaximal, characteristic, complemented
4#20275958C212yesmaximal, complemented
5#20275908C212yesmaximal, complemented
6#20275149C211yesmaximal, characteristic, complemented
7#20275013C211yesmaximal, characteristic, complemented
8#20274976C212yesmaximal, complemented
9#20274655C211yesmaximal, characteristic, complemented
10#20274655C212yesmaximal, complemented
11C2×C53×#547C211yesmaximal, characteristic
12C22×#11719917C2211yescharacteristic
13C22×C4⋊Dic53C2211yescharacteristic
14#16810951—24no—
15C2×#13133148C2212yes—
16#16810624—24no—
17C2×#13133141C2212yes—
18C2×#13133141C2212yes—
19C2×#13133141C2212yes—
20C2×#13133141C2212yes—
21C2×#13133138C2212yes—
22C2×#13133138C2212yes—
23C2×C22⋊#88322C2211yescharacteristic
24C2×C22⋊#88322C2212yes—
25#16810562—21no—
26#16810559—21no—
27#16810521C2212yes—
28#16810503—21no—
29#16810495—21no—
30#16810483C2212yes—
31#16810483C2211yescharacteristic
32#16810385C2212yes—
33#16810378C2212yes—
34C22×#41×C53C2211yescharacteristic
35C22×#41×C53C2211yescharacteristic
36C53×#547—24no—
37C53×#547—24no—
38C2×C4⋊Q8×C53C2211yescharacteristic
39C2×#123×C53—21no—
40C2×#123×C53—21no—
41C2×C22⋊Q8×C53C2211yescharacteristic
42C2×C22⋊Q8×C53C2211yescharacteristic
43C2×#121×C53—21no—
44C2×C4×D4×C53C2211yescharacteristic
45C2×C4×D4×C53C2211yescharacteristic
46C22×C4⋊C4×C53—21no—
47C23×Dic53—21no—
48C2×#11719917—22no—
49C2×#11719917—42no—
50C22×#88322C2312yes—
51C22×#88322C2312yes—
52C2×C4⋊Dic53—21no—
53C2×C4⋊Dic53—22no—
54C2×C4⋊Dic53—22no—
55C2×C4⋊Dic53C2312yes—
56C2×C4⋊Dic53C2312yes—
57C2×#11719919—22no—
58C2×C22⋊Dic53—21no—
59C2×C22⋊Dic53—22no—
60#13133148—24no—
61#13133141—24no—
62#13133141—24no—
63#13133141—42no—
64#13133141—42no—
65#13133138—24no—
66#13133138—24no—
67C22⋊#88322—22no—
68C22⋊#88322—42no—
69#13133124—22no—
70#13133124—21no—
71#13133124—22no—
72C2×#41×C53—22no—
73C2×#41×C53—22no—
74C2×#41×C53—22no—
75C2×#41×C53—24no—
76C2×#41×C53—24no—
77C2×#41×C53—42no—
78C2×#41×C53—42no—
79C2×#41×C53D414yes—
80C22×Q8×C53C2311yescharacteristic
81C22×Q8×C53C2311yescharacteristic
82C22×D4×C53—21no—
83C22×D4×C53—21no—
84C22×D4×C53D411yescharacteristic
85C22×D4×C53C2311yescharacteristic
86C23×C4×C53—21no—
87C23×C4×C53—21no—
88C23×C4×C53—21no—
89C23×C4×C53C2311yescharacteristic
90C23×C4×C53C2311yescharacteristic
91C4⋊Q8×C53—22no—
92C4⋊Q8×C53—22no—
93#123×C53—28no—
94#123×C53—28no—
95C22⋊Q8×C53—22no—
96C22⋊Q8×C53—22no—
97C22⋊Q8×C53—22no—
98C22⋊Q8×C53D414yes—
99#121×C53—28no—
100C4×D4×C53—22no—
101C4×D4×C53—22no—
102C4×D4×C53—22no—
103C4×D4×C53—22no—
104C2×C4⋊C4×C53—28no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53D411yescharacteristic
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C22⋊C4×C53—21no—
112C2×C22⋊C4×C53—21no—
113C2×C22⋊C4×C53—21no—
114C2×C22⋊C4×C53D411yescharacteristic
115C2×C22⋊C4×C53D411yescharacteristic
116C2×C42×C53C2311yescharacteristic
117C24×C53—21no—
118C24×C53C2×D411yescharacteristic
119C24×C53C2×D411yescharacteristic
120#41×C53—28no—
121#41×C53—44no—
122#41×C53—48no—
123#41×C53—48no—
124#41×C53—44no—
125#41×C53—44no—
126C2×Q8×C53—21no—
127C2×Q8×C53—21no—
128C2×Q8×C53—22no—
129C2×Q8×C53—22no—
130C2×Q8×C53—22no—
131C2×Q8×C53—22no—
132C2×Q8×C53C2×D412yes—
133C2×Q8×C53C2×D412yes—
134C2×D4×C53—24no—
135C2×D4×C53—22no—
136C2×D4×C53—24no—
137C2×D4×C53—22no—
138C2×D4×C53—21no—
139C2×D4×C53—22no—
140C2×D4×C53—41no—
141C2×D4×C53—44no—
142C2×D4×C53—44no—
143C2×D4×C53C2×D412yes—
144C2×D4×C53#4114yes—
145C22×C4×C53—22no—
146C22×C4×C53—22no—
147C22×C4×C53—24no—
148C22×C4×C53—24no—
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53—24no—
153C22×C4×C53—21no—
154C22×C4×C53—21no—
155C22×C4×C53—22no—
156C22×C4×C53—22no—
157C22×C4×C53—44no—
158C22×C4×C53—44no—
159C22×C4×C53—42no—
160C22×C4×C53—42no—
161C22×C4×C53C2×D411yescharacteristic
162C22×C4×C53C2×D411yescharacteristic
163C22×C4×C53C2×D411yescharacteristic
164C22×C4×C53C2×D412yes—
165C22×C4×C53C2×D412yes—
166C22×C4×C53C2×D411yescharacteristic
167C22×C4×C53C2411yescharacteristic
168C22×Dic53—22no—
169C22×Dic53—22no—
170C22×Dic53—42no—
171C22×Dic53—82no—
172C4⋊C4×C53—24no—
173C4⋊C4×C53—24no—
174C4⋊C4×C53—24no—
175C4⋊C4×C53—22no—
176C4⋊C4×C53—24no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53C2×D412yes—
179C22⋊C4×C53—24no—
180C22⋊C4×C53—24no—
181C22⋊C4×C53—24no—
182C22⋊C4×C53—22no—
183C22⋊C4×C53—22no—
184C42×C53—21no—
185C42×C53—21no—
186C22⋊Dic53—42no—
187C22⋊Dic53—82no—
188C4⋊Dic53—22no—
189C4⋊Dic53—22no—
190C4⋊Dic53—22no—
191C4⋊Dic53—22no—
192C4⋊Dic53—42no—
193#11719919—82no—
194#11719917—44no—
195C2×#88322—22no—
196C2×#88322—22no—
197C2×#88322—22no—
198C2×#88322—22no—
199C2×#88322—22no—
200C2×#88322—22no—
201C23×C53—22no—
202C23×C53—22no—
203C23×C53—24no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—22no—
207C23×C53—21no—
208C23×C53—22no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—22no—
212C23×C53—41no—
213C23×C53—41no—
214C23×C53—42no—
215C23×C53#12012yes—
216C23×C53C22×D411yescharacteristic
217Q8×C53—22no—
218Q8×C53—22no—
219Q8×C53—24no—
220Q8×C53—41no—
221Q8×C53—41no—
222Q8×C53—42no—
223D4×C53—22no—
224D4×C53—24no—
225D4×C53—44no—
226D4×C53—48no—
227D4×C53—48no—
228D4×C53#12314yes—
229C2×C4×C53—21no—
230C2×C4×C53—22no—
231C2×C4×C53—21no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—24no—
235C2×C4×C53—21no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—21no—
240C2×C4×C53—22no—
241C2×C4×C53—22no—
242C2×C4×C53—48no—
243C2×C4×C53—48no—
244C2×C4×C53—48no—
245C2×C4×C53—42no—
246C2×C4×C53—42no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—42no—
252C2×C4×C53#12012yes—
253C2×C4×C53#13712yes—
254C2×C4×C53C22×D411yescharacteristic
255C2×C4×C53C22×D411yescharacteristic
256C2×C4×C53C2×#4112yes—
257C2×C4×C53H(5,𝔽2)12yes—
258C2×Dic53—42no—
259C2×Dic53—42no—
260C2×Dic53—42no—
261C2×Dic53—42no—
262C2×Dic53—42no—
263C2×Dic53—42no—
264C2×Dic53—84no—
265#88322—42no—
266#88322—42no—
267#24055—531nomaximal, Sylow
268C22×C53—22no—
269C22×C53—24no—
270C22×C53—22no—
271C22×C53—22no—
272C22×C53—42no—
273C22×C53—44no—
274C22×C53—42no—
275C22×C53—42no—
276C22×C53—42no—
277C22×C53—44no—
278C22×C53—42no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53—41no—
282C22×C53—42no—
283C22×C53C2×#12011yescharacteristic
284C22×C53#54612yes—
285C22×C53C2×#13711yescharacteristic
286C22×C53#57511yescharacteristic
287C22×C53#57712yes—
288C4×C53—22no—
289C4×C53—22no—
290C4×C53—41no—
291C4×C53—42no—
292C4×C53—42no—
293C4×C53—42no—
294C4×C53—41no—
295C4×C53C2×#12311yescharacteristic
296C4×C53#53411yescharacteristic
297C4×C53#54612yes—
298Dic53—82no—
299Dic53—82no—
300C2×#547D5311yescharacteristic
301C2×#483—531no—
302C2×#417—531no—
303#2487—531no—
304#2043—532no—
305#2035—532no—
306#1894—531no—
307#1870—531no—
308#1864—532no—
309#1788—532no—
310#1788—531no—
311C2×C53—42no—
312C2×C53—44no—
313C2×C53—81no—
314C2×C53#240511yescharacteristic
315C2×C53#300711yescharacteristic
316C2×C53#301011yescharacteristic
317C2×C53#330712yes—
318C2×C53#333112yes—
319C22×#41C2×D5311yescharacteristic
320C22×#41C2×D5311yescharacteristic
321C22×M16—531no—
322#547—24no—
323#547—24no—
324C2×C4⋊Q8C2×D5311yescharacteristic
325C2×#123—21no—
326C2×#123—21no—
327C2×C22⋊Q8C2×D5311yescharacteristic
328C2×C22⋊Q8C2×D5311yescharacteristic
329C2×#121—21no—
330C2×C4×D4C2×D5311yescharacteristic
331C2×C4×D4C2×D5311yescharacteristic
332C22×C4⋊C4—21no—
333C22×C4⋊C4—531no—
334#483—1064no—
335C2×#106—532no—
336#417—1064no—
337C2×#103—532no—
338C2×#103—532no—
339C2×#103—532no—
340C2×#103—532no—
341C2×#102—532no—
342C2×#102—532no—
343C2×C22⋊C8—531no—
344C2×C22⋊C8—532no—
345#400—1061no—
346#399—1061no—
347#389—532no—
348#386—1061no—
349#385—1061no—
350#380—531no—
351#380—532no—
352#340—532no—
353#336—532no—
354C53#2405511yescharacteristic, complemented, Sylow
355C2×#41—22no—
356C2×#41—22no—
357C2×#41—22no—
358C2×#41—24no—
359C2×#41—24no—
360C2×#41—42no—
361C2×#41—42no—
362C2×#41C22⋊D5314yes—
363C22×Q8C22×D5311yescharacteristic
364C22×Q8C22×D5311yescharacteristic
365C22×D4—21no—
366C22×D4—21no—
367C22×D4C22⋊D5311yescharacteristic
368C22×D4C22×D5311yescharacteristic
369C23×C4—21no—
370C23×C4—21no—
371C23×C4—21no—
372C23×C4—1061no—
373C23×C4C22×D5311yescharacteristic
374C23×C4C22×D5311yescharacteristic
375C2×M16—1062no—
376C2×M16—2122no—
377C22×C8—532no—
378C22×C8—532no—
379C4⋊Q8—22no—
380C4⋊Q8—22no—
381#123—28no—
382#123—28no—
383C22⋊Q8—22no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386C22⋊Q8C22⋊D5314yes—
387#121—28no—
388C4×D4—22no—
389C4×D4—22no—
390C4×D4—22no—
391C4×D4—22no—
392C2×C4⋊C4—28no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—532no—
398C2×C4⋊C4—532no—
399C2×C4⋊C4—1062no—
400C2×C4⋊C4—1062no—
401C2×C4⋊C4—1062no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4C22⋊D5311yescharacteristic
404C2×C4⋊C4C22×D5311yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4—1062no—
409C2×C22⋊C4—1061no—
410C2×C22⋊C4C22⋊D5311yescharacteristic
411C2×C22⋊C4C22⋊D5311yescharacteristic
412C2×C42C22×D5311yescharacteristic
413#106—1064no—
414#103—1064no—
415#103—1064no—
416#103—2122no—
417#103—2122no—
418#102—1064no—
419#102—1064no—
420C22⋊C8—1062no—
421C22⋊C8—2122no—
422#95—1062no—
423#95—1061no—
424#95—1062no—
425C24—21no—
426C24C2×C22⋊D5311yescharacteristic
427C24C2×C22⋊D5311yescharacteristic
428#41—28no—
429#41—44no—
430#41—48no—
431#41—48no—
432#41—44no—
433#41—44no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8C2×C22⋊D5312yes—
441C2×Q8C2×C22⋊D5312yes—
442C2×D4—24no—
443C2×D4—22no—
444C2×D4—24no—
445C2×D4—22no—
446C2×D4—21no—
447C2×D4—22no—
448C2×D4—41no—
449C2×D4—44no—
450C2×D4—44no—
451C2×D4#1171994714yes—
452C2×D4C2×C22⋊D5312yes—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—24no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—4242no—
472C22×C4—2122no—
473C22×C4C2×C22⋊D5311yescharacteristic
474C22×C4C2×C22⋊D5311yescharacteristic
475C22×C4C2×C22⋊D5311yescharacteristic
476C22×C4C2×C22⋊D5312yes—
477C22×C4C2×C22⋊D5312yes—
478C22×C4C2×C22⋊D5311yescharacteristic
479C22×C4C23×D5311yescharacteristic
480M16—2124no—
481C2×C8—1062no—
482C2×C8—1062no—
483C2×C8—1062no—
484C2×C8—1062no—
485C2×C8—1062no—
486C2×C8—1062no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—21no—
493C4⋊C4—1062no—
494C4⋊C4—1062no—
495C4⋊C4—1062no—
496C4⋊C4—1062no—
497C4⋊C4—4242no—
498C4⋊C4—2122no—
499C4⋊C4C2×C22⋊D5312yes—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—4242no—
506C22⋊C4—2122no—
507C42—21no—
508C42—21no—
509C23—22no—
510C23—24no—
511C23—22no—
512C23—21no—
513C23—21no—
514C23—22no—
515C23—22no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—22no—
520C23—41no—
521C23—41no—
522C23—42no—
523C23#1313318612yes—
524C23C22×C22⋊D5311yescharacteristic
525Q8—24no—
526Q8—22no—
527Q8—22no—
528Q8—42no—
529Q8—41no—
530Q8—41no—
531D4—24no—
532D4—22no—
533D4—44no—
534D4—48no—
535D4—48no—
536D4#1313319814yes—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—48no—
551C2×C4—48no—
552C2×C4—48no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—4244no—
561C2×C4—2122no—
562C2×C4—2122no—
563C2×C4—2122no—
564C2×C4—2122no—
565C2×C4—2122no—
566C2×C4—2122no—
567C2×C4#1313318612yes—
568C2×C4#1313324712yes—
569C2×C4C22×C22⋊D5311yescharacteristic
570C2×C4C22×C22⋊D5311yescharacteristic
571C2×C4C2×#1171994712yes—
572C2×C4#1313326212yes—
573C8—2122no—
574C8—2122no—
575C22—22no—
576C22—24no—
577C22—22no—
578C22—22no—
579C22—42no—
580C22—44no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—44no—
585C22—41no—
586C22—42no—
587C22—41no—
588C22—41no—
589C22—42no—
590C22C2×#1313318611yescharacteristic
591C22#1681126712yes—
592C22C2×#1313324711yescharacteristic
593C22#1681142111yescharacteristic
594C22#1681143112yes—
595C4—22no—
596C4—22no—
597C4—41no—
598C4—42no—
599C4—42no—
600C4—42no—
601C4—41no—
602C4—4242no—
603C4—4242no—
604C4C2×#1313319811yescharacteristic
605C4#1681118211yescharacteristic
606C4#1681126712yes—
607C2—42no—
608C2—44no—
609C2—81no—
610C2#2027803811yescharacteristic
611C2#2028397511yescharacteristic
612C2#2028400111yescharacteristic
613C2#2028669112yes—
614C2#2028703112yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#547C2
socleC23×C53C22×D4
hypercenterC2×#547D53
hypocenterC53#24055
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×#16810951C2semidirect
2. C2×#16810624C2semidirect
3. #20279032C2semidirect
4. #20275958C2semidirect
5. #20275908C2semidirect
6. #20275149C2semidirect
7. #20275013C2semidirect
8. #20274976C2semidirect
9. #20274655C2semidirect
10. #20274655C2semidirect
354. C53#24055semidirect

Maximal subgroups

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

subgroupindex
C2×#168109512
C2×#168106242
#202790322
#202759582
#202759082
#202751492
#202750132
#202749762
#202746552
C2×C53×#5472
#2405553