Finite Group Databasebeta

← #27440641

Subgroups — #27440641

Order 13568.

1322 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810982C211yesmaximal, characteristic, complemented
2C2×#16810918C211yesmaximal, characteristic, complemented
3#20279707C211yesmaximal, characteristic, complemented
4#20275915C212yesmaximal, complemented
5#20275894C212yesmaximal, complemented
6#20275648C211yesmaximal, characteristic, complemented
7#20275514C211yesmaximal, characteristic, complemented
8#20274655C212yesmaximal, complemented
9#20274650C212yesmaximal, complemented
10#20274545C211yesmaximal, characteristic, complemented
11C2×C53×#548C211yesmaximal, characteristic
12C2×#13133217C2211yescharacteristic
13C2×#13133204—21no—
14#16810982—24no—
15#16810918—24no—
16C2×#13133150C2212yes—
17C2×C4⋊#88322C2211yescharacteristic
18C2×#13133141C2212yes—
19C2×#13133141C2212yes—
20C2×#13133141C2212yes—
21C2×#13133141C2212yes—
22C2×#13133138C2212yes—
23C2×#13133138C2212yes—
24C2×C22⋊#88322C2212yes—
25C2×C22⋊#88322C2212yes—
26C2×#13133128C2211yescharacteristic
27#16810548—21no—
28#16810548—21no—
29#16810528—21no—
30#16810483C2212yes—
31#16810483C2212yes—
32C4×C4⋊Dic53C2211yescharacteristic
33#16810385C2212yes—
34C22×#41×C53C2211yescharacteristic
35C23×Q8×C53C2211yescharacteristic
36C53×#548—24no—
37C53×#548—24no—
38C2×#124×C53—21no—
39C2×#124×C53C2211yescharacteristic
40C2×C22⋊Q8×C53—21no—
41C2×C22⋊Q8×C53C2211yescharacteristic
42C2×#121×C53—21no—
43C2×#121×C53C2211yescharacteristic
44C2×C4×Q8×C53C2211yescharacteristic
45C2×C4×D4×C53—21no—
46C2×C4×D4×C53C2211yescharacteristic
47C22×Dic106—21no—
48C22×#88322C2312yes—
49C22×#88322C2312yes—
50#13133217—22no—
51#13133217—22no—
52#13133204—44no—
53C2×C4⋊Dic53—21no—
54C2×C4⋊Dic53C2312yes—
55C2×C4⋊Dic53C2312yes—
56C2×C22⋊Dic53—22no—
57C2×C22⋊Dic53—22no—
58C2×C4×Dic53—21no—
59#13133150—24no—
60C4⋊#88322—22no—
61#13133141—24no—
62#13133141—24no—
63#13133141—24no—
64#13133141—42no—
65#13133138—24no—
66#13133138—42no—
67C22⋊#88322—42no—
68C22⋊#88322—42no—
69#13133128—22no—
70#13133124—22no—
71#13133124—22no—
72#13133124—21no—
73C2×#41×C53—22no—
74C2×#41×C53—22no—
75C2×#41×C53—24no—
76C2×#41×C53—42no—
77C22×Q8×C53—22no—
78C22×Q8×C53—22no—
79C22×Q8×C53—21no—
80C22×Q8×C53—24no—
81C22×Q8×C53—42no—
82C22×Q8×C53C2311yescharacteristic
83C22×Q8×C53C2311yescharacteristic
84C22×D4×C53—21no—
85C22×D4×C53—21no—
86C22×D4×C53D411yescharacteristic
87C22×D4×C53C2311yescharacteristic
88C23×C4×C53—21no—
89C23×C4×C53—21no—
90C23×C4×C53C2311yescharacteristic
91C23×C4×C53C2311yescharacteristic
92#124×C53—28no—
93#124×C53—22no—
94#124×C53D414yes—
95C22⋊Q8×C53—28no—
96C22⋊Q8×C53—22no—
97C22⋊Q8×C53—22no—
98#121×C53—28no—
99#121×C53—22no—
100#121×C53—22no—
101C4×Q8×C53—22no—
102C4×Q8×C53D414yes—
103C4×D4×C53—28no—
104C4×D4×C53—22no—
105C4×D4×C53—22no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53D411yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C22⋊C4×C53—21no—
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×C53—21no—
117C2×C42×C53C2311yescharacteristic
118C24×C53—21no—
119C24×C53C2×D411yescharacteristic
120C24×C53C2×D411yescharacteristic
121#41×C53—48no—
122#41×C53—44no—
123#41×C53—44no—
124C2×Q8×C53—24no—
125C2×Q8×C53—21no—
126C2×Q8×C53—21no—
127C2×Q8×C53—22no—
128C2×Q8×C53—22no—
129C2×Q8×C53—22no—
130C2×Q8×C53—44no—
131C2×Q8×C53—48no—
132C2×Q8×C53—44no—
133C2×Q8×C53—44no—
134C2×Q8×C53C2×D412yes—
135C2×Q8×C53C2×D412yes—
136C2×Q8×C53#4114yes—
137C2×D4×C53—24no—
138C2×D4×C53—24no—
139C2×D4×C53—22no—
140C2×D4×C53—22no—
141C2×D4×C53—21no—
142C2×D4×C53—22no—
143C2×D4×C53—22no—
144C2×D4×C53—41no—
145C2×D4×C53—44no—
146C2×D4×C53—42no—
147C2×D4×C53—42no—
148C2×D4×C53C2×D412yes—
149C22×C4×C53—21no—
150C22×C4×C53—24no—
151C22×C4×C53—24no—
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53—21no—
155C22×C4×C53—21no—
156C22×C4×C53—21no—
157C22×C4×C53—21no—
158C22×C4×C53—22no—
159C22×C4×C53—22no—
160C22×C4×C53—22no—
161C22×C4×C53—22no—
162C22×C4×C53—44no—
163C22×C4×C53—44no—
164C22×C4×C53C2×D411yescharacteristic
165C22×C4×C53C2×D411yescharacteristic
166C22×C4×C53C2×D411yescharacteristic
167C22×C4×C53C2×D411yescharacteristic
168C22×C4×C53C2411yescharacteristic
169C22×Dic53—22no—
170C22×Dic53—22no—
171C2×Dic106—42no—
172C2×Dic106—42no—
173C2×Dic106—42no—
174C4⋊C4×C53—24no—
175C4⋊C4×C53—24no—
176C4⋊C4×C53—22no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53C2×D412yes—
179C22⋊C4×C53—24no—
180C22⋊C4×C53—24no—
181C22⋊C4×C53—24no—
182C22⋊C4×C53—24no—
183C22⋊C4×C53—22no—
184C22⋊C4×C53—22no—
185C42×C53—24no—
186C42×C53C2×D412yes—
187C42×C53C2×D412yes—
188C22⋊Dic53—44no—
189C22⋊Dic53—44no—
190C4⋊Dic53—22no—
191C4⋊Dic53—22no—
192C4⋊Dic53—22no—
193C4⋊Dic53—22no—
194C4⋊Dic53—42no—
195C4×Dic53—42no—
196C2×#88322—22no—
197C2×#88322—22no—
198C2×#88322—22no—
199C2×#88322—22no—
200C2×#88322—22no—
201C2×#88322—22no—
202C23×C53—21no—
203C23×C53—22no—
204C23×C53—21no—
205C23×C53—22no—
206C23×C53—21no—
207C23×C53—22no—
208C23×C53—22no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—22no—
212C23×C53—42no—
213C23×C53—41no—
214C23×C53—41no—
215C23×C53—42no—
216C23×C53AGL(1,ℤ/8)12yes—
217C23×C53C22×D411yescharacteristic
218Q8×C53—22no—
219Q8×C53—24no—
220Q8×C53—24no—
221Q8×C53—48no—
222Q8×C53—41no—
223Q8×C53—41no—
224Q8×C53#12414yescomplemented
225D4×C53—24no—
226D4×C53—44no—
227D4×C53—48no—
228D4×C53—48no—
229D4×C53—41no—
230D4×C53—41no—
231C2×C4×C53—22no—
232C2×C4×C53—24no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—22no—
236C2×C4×C53—21no—
237C2×C4×C53—22no—
238C2×C4×C53—21no—
239C2×C4×C53—22no—
240C2×C4×C53—22no—
241C2×C4×C53—22no—
242C2×C4×C53—48no—
243C2×C4×C53—48no—
244C2×C4×C53—42no—
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—42no—
252C2×C4×C53—42no—
253C2×C4×C53#12012yes—
254C2×C4×C53#12012yes—
255C2×C4×C53C22×D411yescharacteristic
256C2×C4×C53C22×D411yescharacteristic
257C2×C4×C53C2×#4112yes—
258C2×C4×C53H(5,𝔽2)12yes—
259C2×Dic53—42no—
260C2×Dic53—42no—
261C2×Dic53—42no—
262C2×Dic53—42no—
263C2×Dic53—42no—
264C2×Dic53—42no—
265Dic106—84no—
266#88322—42no—
267#88322—42no—
268#25896—531nomaximal, Sylow
269C22×C53—22no—
270C22×C53—24no—
271C22×C53—41no—
272C22×C53—41no—
273C22×C53—42no—
274C22×C53—42no—
275C22×C53—41no—
276C22×C53—41no—
277C22×C53—42no—
278C22×C53—41no—
279C22×C53—42no—
280C22×C53—42no—
281C22×C53—42no—
282C22×C53—82no—
283C22×C53—82no—
284C22×C53C2×#12011yescharacteristic
285C22×C53#54612yes—
286C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
287C22×C53#57511yescharacteristic
288C22×C53#57812yes—
289C4×C53—24no—
290C4×C53—22no—
291C4×C53—22no—
292C4×C53—41no—
293C4×C53—41no—
294C4×C53—42no—
295C4×C53—42no—
296C4×C53C2×#12411yescharacteristic
297C4×C53#53511yescharacteristic
298C4×C53#54612yes—
299Dic53—82no—
300Dic53—82no—
301C2×#548D5311yescharacteristic
302C2×#489—531no—
303C2×#477—531no—
304#2547—531no—
305#2036—532no—
306#2033—532no—
307#1983—531no—
308#1955—531no—
309#1788—532no—
310#1787—532no—
311#1763—531no—
312C2×C53—42no—
313C2×C53—81no—
314C2×C53—82no—
315C2×C53#243511yescharacteristic
316C2×C53#300611yescharacteristic
317C2×C53#301111yescharacteristic
318C2×C53#330812yes—
319C2×C53#331312yes—
320C22×#41C2×D5311yescharacteristic
321C23×Q8C2×D5311yescharacteristic
322#548—24no—
323#548—24no—
324C2×C4⋊Q8—531no—
325C2×#124—21no—
326C2×#124—1061no—
327C2×#124C2×D5311yescharacteristic
328C2×C22⋊Q8—21no—
329C2×C22⋊Q8C2×D5311yescharacteristic
330C2×#121—21no—
331C2×#121C2×D5311yescharacteristic
332C2×C4×Q8C2×D5311yescharacteristic
333C2×C4×D4—21no—
334C2×C4×D4C2×D5311yescharacteristic
335#489—1064no—
336#477—1064no—
337C2×#107—532no—
338C2×C4⋊C8—531no—
339C2×#103—532no—
340C2×#103—532no—
341C2×#103—532no—
342C2×#103—532no—
343C2×#102—532no—
344C2×#102—532no—
345C2×C22⋊C8—532no—
346C2×C22⋊C8—532no—
347C2×#97—531no—
348#397—1061no—
349#397—1061no—
350#391—1061no—
351#380—532no—
352#380—532no—
353C4×C4⋊C4—531no—
354#340—532no—
355C53#2589611yescharacteristic, complemented, Sylow
356C2×#41—22no—
357C2×#41—22no—
358C2×#41—24no—
359C2×#41—42no—
360C22×Q8—22no—
361C22×Q8—22no—
362C22×Q8—21no—
363C22×Q8—24no—
364C22×Q8—42no—
365C22×Q8—1061no—
366C22×Q8C22×D5311yescharacteristic
367C22×Q8C22×D5311yescharacteristic
368C22×D4—21no—
369C22×D4—21no—
370C22×D4C22⋊D5311yescharacteristic
371C22×D4C22×D5311yescharacteristic
372C23×C4—21no—
373C23×C4—21no—
374C23×C4C22×D5311yescharacteristic
375C23×C4C22×D5311yescharacteristic
376C22×C8—532no—
377C22×C8—532no—
378C4⋊Q8—1062no—
379C4⋊Q8—1062no—
380#124—28no—
381#124—22no—
382#124—2124no—
383#124C22⋊D5314yes—
384C22⋊Q8—28no—
385C22⋊Q8—22no—
386C22⋊Q8—22no—
387#121—28no—
388#121—22no—
389#121—22no—
390C4×Q8—22no—
391C4×Q8C22⋊D5314yes—
392C4×D4—28no—
393C4×D4—22no—
394C4×D4—22no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—532no—
398C2×C4⋊C4—532no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4C22⋊D5311yescharacteristic
401C2×C4⋊C4C22×D5311yescharacteristic
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—1062no—
407C2×C22⋊C4—1062no—
408C2×C22⋊C4C22⋊D5311yescharacteristic
409C2×C22⋊C4C22⋊D5311yescharacteristic
410C2×C42—21no—
411C2×C42—1061no—
412C2×C42C22×D5311yescharacteristic
413#107—1064no—
414C4⋊C8—1062no—
415#103—1064no—
416#103—1064no—
417#103—1064no—
418#103—2122no—
419#102—1064no—
420#102—2122no—
421C22⋊C8—2122no—
422C22⋊C8—2122no—
423#97—1062no—
424#95—1061no—
425#95—1062no—
426#95—1062no—
427C24—21no—
428C24C2×C22⋊D5311yescharacteristic
429C24C2×C22⋊D5311yescharacteristic
430#41—48no—
431#41—44no—
432#41—44no—
433C2×Q8—24no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—44no—
440C2×Q8—48no—
441C2×Q8—44no—
442C2×Q8—44no—
443C2×Q8—2122no—
444C2×Q8—2122no—
445C2×Q8—2122no—
446C2×Q8#1171994714yes—
447C2×Q8C2×C22⋊D5312yes—
448C2×Q8C2×C22⋊D5312yes—
449C2×D4—24no—
450C2×D4—24no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—21no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—41no—
457C2×D4—44no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4C2×C22⋊D5312yes—
461C22×C4—21no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4C2×C22⋊D5311yescharacteristic
479C22×C4C2×C22⋊D5311yescharacteristic
480C22×C4C2×C22⋊D5311yescharacteristic
481C22×C4C2×C22⋊D5311yescharacteristic
482C22×C4C23×D5311yescharacteristic
483C2×C8—1062no—
484C2×C8—1062no—
485C2×C8—1062no—
486C2×C8—1062no—
487C2×C8—1062no—
488C2×C8—1062no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—1062no—
494C4⋊C4—1062no—
495C4⋊C4—1062no—
496C4⋊C4—1062no—
497C4⋊C4—2122no—
498C4⋊C4C2×C22⋊D5312yes—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—2124no—
506C22⋊C4—2124no—
507C42—24no—
508C42—2122no—
509C42C2×C22⋊D5312yes—
510C42C2×C22⋊D5312yes—
511C23—22no—
512C23—21no—
513C23—21no—
514C23—22no—
515C23—22no—
516C23—21no—
517C23—22no—
518C23—21no—
519C23—21no—
520C23—22no—
521C23—42no—
522C23—41no—
523C23—41no—
524C23—42no—
525C23#1313324212yes—
526C23C22×C22⋊D5311yescharacteristic
527Q8—24no—
528Q8—22no—
529Q8—24no—
530Q8—48no—
531Q8—41no—
532Q8—41no—
533Q8—4244no—
534Q8#1313320414yescomplemented
535D4—24no—
536D4—44no—
537D4—48no—
538D4—48no—
539D4—41no—
540D4—41no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—48no—
553C2×C4—48no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—2122no—
564C2×C4—2122no—
565C2×C4—2122no—
566C2×C4—2122no—
567C2×C4—2122no—
568C2×C4—2122no—
569C2×C4#1313318612yes—
570C2×C4#1313318612yes—
571C2×C4C22×C22⋊D5311yescharacteristic
572C2×C4C22×C22⋊D5311yescharacteristic
573C2×C4C2×#1171994712yes—
574C2×C4#1313326212yes—
575C8—2122no—
576C8—2122no—
577C22—22no—
578C22—24no—
579C22—42no—
580C22—41no—
581C22—41no—
582C22—42no—
583C22—41no—
584C22—42no—
585C22—42no—
586C22—41no—
587C22—41no—
588C22—42no—
589C22—42no—
590C22—82no—
591C22—82no—
592C22C2×#1313318611yescharacteristic
593C22#1681126712yes—
594C22C2×#1313324211yescharacteristic
595C22#1681142111yescharacteristic
596C22#1681143812yes—
597C4—22no—
598C4—24no—
599C4—22no—
600C4—42no—
601C4—41no—
602C4—41no—
603C4—42no—
604C4—4242no—
605C4—4242no—
606C4C2×#1313320411yescharacteristic
607C4#1681119011yescharacteristic
608C4#1681126712yes—
609C2—42no—
610C2—81no—
611C2—82no—
612C2#2027836211yescharacteristic
613C2#2028396711yescharacteristic
614C2#2028401011yescharacteristic
615C2#2028670612yes—
616C2#2028678112yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#548C2
socleC23×C53C22×D4
hypercenterC2×#548D53
hypocenterC53#25896
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#548

Complements

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

NH ≅ G/Ntype
1. C2×#16810982C2semidirect
2. C2×#16810918C2semidirect
3. #20279707C2semidirect
4. #20275915C2semidirect
5. #20275894C2semidirect
6. #20275648C2semidirect
7. #20275514C2semidirect
8. #20274655C2semidirect
9. #20274650C2semidirect
10. #20274545C2semidirect
224. Q8×C53#124semidirect
355. C53#25896semidirect
534. Q8#13133204semidirect

Maximal subgroups

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

subgroupindex
C2×#168109822
C2×#168109182
#202797072
#202759152
#202758942
#202756482
#202755142
#202746552
#202746502
#202745452
C2×C53×#5482
#2589653