Finite Group Databasebeta

← #27351675

Subgroups — #27351675

Order 13568.

938 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20288800C211yesmaximal, characteristic
2#20283804C211yesmaximal, characteristic, complemented
3#20273290C212yesmaximal, complemented
4#20273274C212yesmaximal, complemented
5#20273261C212yesmaximal, complemented
6#20273251C212yesmaximal, complemented
7#20273238C211yesmaximal, characteristic, complemented
8#20273222C211yesmaximal, characteristic, complemented
9#20273209C211yesmaximal, characteristic, complemented
10#20273199C211yesmaximal, characteristic, complemented
11C53×#2320C211yesmaximal, characteristic, complemented
12#16811358C411yescharacteristic, complemented
13#16811358C412yescomplemented
14Q8×Dic106C411yescharacteristic, complemented
15#16811333C411yescharacteristic, complemented
16#16811333C412yescomplemented
17#16811295C411yescharacteristic, complemented
18#16811284—21no—
19#16811274—22no—
20#16811274—21no—
21#16811245—22no—
22#16811245—21no—
23#16811173C2211yescharacteristic
24#16811173C2212yes—
25#16811157C2211yescharacteristic
26#16811157C2212yes—
27#16811105—21no—
28#16810754—21no—
29#16810754—22no—
30Q8×#88322—21no—
31#16810680C2211yescharacteristic
32#16810680C2212yes—
33#16810676C2211yescharacteristic
34#16810676C2212yes—
35#16810359C2211yescharacteristic
36#16810359C2211yescharacteristic
37#16810359C2212yes—
38#16810356C2211yescharacteristic
39#16810356C2211yescharacteristic
40#16810356C2212yes—
41#16810356C2212yes—
42#16810356C2212yes—
43#16810356C2212yes—
44#16810356C2212yes—
45C2×C4×Q8×C53C2211yescharacteristic
46C53×#391—21no—
47C53×#389—21no—
48C53×#387—22no—
49C53×#383C2212yes—
50C53×#382C2212yes—
51C4×C4⋊C4×C53C2211yescharacteristic
52C53×#376C2211yescharacteristic
53C2×#11719949—21no—
54C2×#11719945—21no—
55C2×#11719945—22no—
56#13133217C2×C411yescharacteristic
57#13133217C2×C412yes—
58#13133216—22no—
59#13133216—21no—
60#13133214C2×C411yescharacteristic
61#13133214C2×C412yes—
62#13133211—22no—
63#13133211—21no—
64#13133209C2×C411yescharacteristic
65#13133209C2×C412yes—
66#13133206—21no—
67#13133206—22no—
68#13133205—22no—
69#13133205—21no—
70#13133205C2×C411yescharacteristic
71#13133205C2×C412yes—
72#13133202—22no—
73#13133202—22no—
74#13133202—22no—
75#13133200—21no—
76#13133200—22no—
77#13133196C2311yescharacteristic
78#13133195—22no—
79#13133195—22no—
80#13133195—22no—
81#13133193—21no—
82#13133193—22no—
83#13133192—22no—
84#13133192—21no—
85#13133191—21no—
86#13133191—22no—
87#13133191C2311yescharacteristic
88#13133191C2312yes—
89Q8×Dic53—21no—
90C4×Dic106C2×C411yescharacteristic
91C4×Dic106C2×C412yes—
92C4×D212—21no—
93C4×D212—22no—
94C4×D212C2×C411yescharacteristic
95C4×D212C2×C412yes—
96C4×D212C2×C412yes—
97C4×D212C2×C412yes—
98C4×D212C2×C412yes—
99#13133179—22no—
100#13133179—21no—
101C4×C22⋊D53—21no—
102C4×C22⋊D53—22no—
103#13133174—21no—
104#13133174—22no—
105C4⋊C4×D53—22no—
106C4⋊C4×D53—21no—
107C4⋊#88322—22no—
108C4⋊#88322—21no—
109C4⋊#88322—21no—
110C4⋊#88322—22no—
111C22⋊#88322C2311yescharacteristic
112C22⋊#88322C2311yescharacteristic
113C22⋊#88322C2312yes—
114#13133128—22no—
115#13133128—21no—
116C4×#88322—21no—
117C4×#88322—22no—
118C22×Q8×C53D411yescharacteristic
119C4×Q8×C53C2×C411yescharacteristic
120C4×Q8×C53C2×C411yescharacteristic
121C4×Q8×C53C2×C412yes—
122C4×Q8×C53D411yescharacteristic
123C4×Q8×C53D412yes—
124C4×Q8×C53D411yescharacteristic
125C2×C4⋊C4×C53—21no—
126C2×C4⋊C4×C53—22no—
127C2×C4⋊C4×C53D412yes—
128C2×C4⋊C4×C53D411yescharacteristic
129C2×C42×C53—21no—
130C2×C42×C53—21no—
131C2×C42×C53C2311yescharacteristic
132C2×C42×C53C2312yes—
133#95×C53—21no—
134#95×C53—22no—
135#95×C53—22no—
136#95×C53—22no—
137#95×C53C2311yescharacteristic
138#95×C53C2311yescharacteristic
139#95×C53C2312yes—
140C2×Q8×C53—42no—
141C2×Q8×C53C22⋊C411yescharacteristic
142C2×Q8×C53C22⋊C411yescharacteristic
143C2×Q8×C53C22⋊C412yes—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—22no—
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D412yes—
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D412yes—
151C22×C4×C53C2411yescharacteristic
152C2×C22⋊D53—21no—
153C2×C22⋊D53—22no—
154#11719949—44no—
155#11719945—81no—
156#11719945—82no—
157#11719945—81no—
158#11719945—82no—
159C2×D212—22no—
160C2×D212—21no—
161C2×D212—21no—
162C2×D212—22no—
163C2×D212C22×C411yescharacteristic
164C2×D212C22×C412yes—
165C2×C4×D53—22no—
166C2×C4×D53—21no—
167C2×C4×D53—22no—
168C2×C4×D53—22no—
169C2×C4×D53—22no—
170C2×Dic106—21no—
171C2×Dic106—22no—
172C2×Dic106C22×C411yescharacteristic
173C4⋊C4×C53—42no—
174C4⋊C4×C53—82no—
175C4⋊C4×C53C22⋊C412yes—
176C4⋊C4×C53C22⋊C411yescharacteristic
177C4⋊C4×C53C22⋊C412yes—
178C4⋊C4×C53C22⋊C411yescharacteristic
179C4⋊C4×C53C22⋊C412yes—
180C4⋊C4×C53C22⋊C412yes—
181C4⋊C4×C53C22⋊C412yes—
182C42×C53—42no—
183C42×C53—81no—
184C42×C53C22×C411yescharacteristic
185C42×C53C22×C412yes—
186C42×C53C22×C412yes—
187C42×C53C22×C411yescharacteristic
188C42×C53C2×D411yescharacteristic
189C42×C53C2×D412yes—
190C42×C53C2×D412yes—
191C42×C53C2×D411yescharacteristic
192C22⋊Dic53—21no—
193#11719921—21no—
194#11719921—22no—
195#11719921—21no—
196#11719921—22no—
197#11719921—22no—
198#11719921—22no—
199#11719921—22no—
200#11719921—21no—
201#11719921—22no—
202#11719921—21no—
203#11719921—22no—
204C4⋊Dic53—22no—
205C4⋊Dic53—21no—
206C4⋊Dic53C22×C411yescharacteristic
207C4⋊Dic53C22×C412yes—
208C4⋊Dic53C22×C411yescharacteristic
209#11719919—21no—
210#11719919—22no—
211#11719919—21no—
212#11719919—22no—
213#11719919—21no—
214#11719919—22no—
215#11719919—21no—
216C4×Dic53—22no—
217C4×Dic53—21no—
218C2×#88322—21no—
219C2×#88322—21no—
220C2×#88322—22no—
221C23×C53C22×D411yescharacteristic
222C22×D53—22no—
223C22×D53—21no—
224Q8×C53—22no—
225Q8×C53—22no—
226Q8×C53—42no—
227C2×C4×C53—21no—
228C2×C4×C53—22no—
229C2×C4×C53—21no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—42no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—42no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—42no—
240C2×C4×C53—42no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53C2×C22⋊C411yescharacteristic
244C2×C4×C53C2×C22⋊C412yes—
245C2×C4×C53C2×C22⋊C411yescharacteristic
246C2×C4×C53C2×C22⋊C412yes—
247C2×C4×C53C2×C22⋊C412yes—
248C2×C4×C53C2×C22⋊C411yescharacteristic
249C2×C4×C53C2×C22⋊C412yes—
250C2×C4×C53C2×C22⋊C411yescharacteristic
251C2×C4×C53C23×C411yescharacteristic
252C2×C4×C53C22×D411yescharacteristic
253C22⋊D53—42no—
254C22⋊D53—41no—
255C22⋊D53—42no—
256C22⋊D53—41no—
257C2×Dic53—22no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261D212—41no—
262D212—42no—
263D212—44no—
264D212—42no—
265D212—81no—
266D212—82no—
267C4×D53—44no—
268C4×D53—42no—
269C4×D53—82no—
270C4×D53—82no—
271C4×D53—82no—
272Dic106—41no—
273Dic106—81no—
274Dic106—82no—
275#88322—41no—
276#88322—41no—
277#88322—42no—
278#17602—531nomaximal, Sylow
279C22×C53—21no—
280C22×C53—21no—
281C22×C53C22×C22⋊C411yescharacteristic
282C22×C53#57611yescharacteristic
283C22×C53#57711yescharacteristic
284C2×D53—42no—
285C2×D53—41no—
286C2×D53—42no—
287C2×D53—41no—
288C2×D53—42no—
289C2×D53—41no—
290C4×C53—21no—
291C4×C53—21no—
292C4×C53—21no—
293C4×C53—22no—
294C4×C53—21no—
295C4×C53—22no—
296C4×C53—22no—
297C4×C53—21no—
298C4×C53—22no—
299C4×C53—81no—
300C4×C53—82no—
301C4×C53—81no—
302C4×C53#42112yes—
303Dic53—41no—
304Dic53—42no—
305Dic53—41no—
306Dic53—41no—
307#3522—531no—
308#2984—531no—
309#2320D5311yescharacteristic, complemented
310#1521—532no—
311#1517—532no—
312#1514—532no—
313#1512—532no—
314#1509—531no—
315#1505—531no—
316#1502—531no—
317#1500—531no—
318C2×C53—21no—
319C2×C53—21no—
320C2×C53#288411yescharacteristic
321C2×C53#289711yescharacteristic
322C2×C53C2×#42111yescharacteristic
323D53—82no—
324D53—81no—
325#559—532no—
326#559—531no—
327Q82—531no—
328#554—531no—
329#554—532no—
330#550—531no—
331#548—1061no—
332#547—1062no—
333#547—1061no—
334#542—1061no—
335#542—1062no—
336#532—531no—
337#532—532no—
338#529—531no—
339#529—532no—
340#520—1061no—
341C2×C4×Q8C2×D5311yescharacteristic
342#446—1061no—
343#446—1062no—
344C8×Q8—1061no—
345#432—531no—
346#432—532no—
347#431—531no—
348#431—532no—
349#391—21no—
350#389—21no—
351#387—22no—
352#383C2×D5312yes—
353#382C2×D5312yes—
354C4×C4⋊C4C2×D5311yescharacteristic
355#376C2×D5311yescharacteristic
356#328—531no—
357#328—531no—
358#328—532no—
359#327—531no—
360#327—531no—
361#327—532no—
362#327—532no—
363#327—532no—
364#327—532no—
365#327—532no—
366C53#1760211yescharacteristic, complemented, Sylow
367C2×#41—1061no—
368C2×#41—1061no—
369C2×#41—1062no—
370C22×Q8C22⋊D5311yescharacteristic
371C4⋊Q8—531no—
372C4⋊Q8—532no—
373C4⋊Q8—1062no—
374C4⋊Q8—1061no—
375#127—531no—
376#127—532no—
377#126—1062no—
378#126—1061no—
379#125—531no—
380#125—532no—
381#124—531no—
382#124—532no—
383#124—1061no—
384#124—1062no—
385#124—1062no—
386#124—1061no—
387#123—1061no—
388#123—1062no—
389#123—1062no—
390#123—1062no—
391#123—1062no—
392C22⋊Q8—531no—
393C22⋊Q8—1061no—
394C22⋊Q8—1062no—
395C22⋊Q8—1062no—
396C22⋊Q8—1062no—
397C22⋊Q8—1062no—
398#121—531no—
399#121—532no—
400#121—1061no—
401#121—1062no—
402#121—1062no—
403#121—1061no—
404C4×Q8—531no—
405C4×Q8—532no—
406C4×Q8—1061no—
407C4×Q8C4×D5311yescharacteristic
408C4×Q8C4×D5311yescharacteristic
409C4×Q8C4×D5312yes—
410C4×Q8D21211yescharacteristic
411C4×Q8D21211yescharacteristic
412C4×Q8D21212yes—
413C4×D4—531no—
414C4×D4—532no—
415C4×D4—532no—
416C4×D4—532no—
417C4×D4—532no—
418C4×D4—1061no—
419C4×D4—1062no—
420C4×D4—1062no—
421C4×D4—1061no—
422C4×D4—1061no—
423C4×D4—1062no—
424#117—1061no—
425#117—1062no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—22no—
428C2×C4⋊C4—1062no—
429C2×C4⋊C4—1061no—
430C2×C4⋊C4C22⋊D5312yes—
431C2×C4⋊C4C22⋊D5311yescharacteristic
432C2×C42—21no—
433C2×C42—21no—
434C2×C42C22×D5311yescharacteristic
435C2×C42C22×D5312yes—
436C4⋊C8—1062no—
437C4⋊C8—1061no—
438C4⋊C8—1061no—
439C4⋊C8—1062no—
440C22⋊C8—531no—
441C22⋊C8—531no—
442C22⋊C8—532no—
443#97—1062no—
444#97—1061no—
445C4×C8—1061no—
446C4×C8—1062no—
447#95—21no—
448#95—22no—
449#95—22no—
450#95—22no—
451#95C22×D5311yescharacteristic
452#95C22×D5311yescharacteristic
453#95C22×D5312yes—
454#41—4241no—
455#41—4242no—
456#41—4241no—
457#41—4242no—
458#41—2124no—
459C2×Q8—42no—
460C2×Q8—531no—
461C2×Q8—1062no—
462C2×Q8—1061no—
463C2×Q8#1171992111yescharacteristic
464C2×Q8#1171992111yescharacteristic
465C2×Q8#1171992112yes—
466C2×D4—531no—
467C2×D4—532no—
468C2×D4—1061no—
469C2×D4—1062no—
470C2×D4—1062no—
471C2×D4—1061no—
472C2×D4—1062no—
473C2×D4—1061no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1061no—
482C22×C4C2×C22⋊D5311yescharacteristic
483C22×C4C2×C22⋊D5312yes—
484C22×C4C2×C22⋊D5311yescharacteristic
485C22×C4C2×C22⋊D5312yes—
486C22×C4C23×D5311yescharacteristic
487C2×C8—1061no—
488C2×C8—1061no—
489C2×C8—1062no—
490C4⋊C4—42no—
491C4⋊C4—82no—
492C4⋊C4—531no—
493C4⋊C4—532no—
494C4⋊C4—531no—
495C4⋊C4—1061no—
496C4⋊C4—1062no—
497C4⋊C4—1062no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1062no—
501C4⋊C4—1062no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4#1171992112yes—
505C4⋊C4#1171992111yescharacteristic
506C4⋊C4#1171992112yes—
507C4⋊C4#1171992111yescharacteristic
508C4⋊C4#1171992112yes—
509C4⋊C4#1171992112yes—
510C4⋊C4#1171992112yes—
511C22⋊C4—1061no—
512C22⋊C4—1061no—
513C22⋊C4—1062no—
514C22⋊C4—1062no—
515C22⋊C4—1061no—
516C22⋊C4—1062no—
517C22⋊C4—1062no—
518C22⋊C4—1062no—
519C22⋊C4—1061no—
520C22⋊C4—1062no—
521C22⋊C4—1061no—
522C22⋊C4—1062no—
523C42—42no—
524C42—81no—
525C42—1062no—
526C42—1061no—
527C42C2×C4×D5311yescharacteristic
528C42C2×C4×D5312yes—
529C42C2×C4×D5312yes—
530C42C2×C4×D5311yescharacteristic
531C42C2×D21211yescharacteristic
532C42C2×D21212yes—
533C42C2×D21212yes—
534C42C2×D21211yescharacteristic
535C23—1062no—
536C23—1061no—
537C23C22×C22⋊D5311yescharacteristic
538Q8—22no—
539Q8—22no—
540Q8—42no—
541Q8—4241no—
542Q8—4242no—
543Q8—2121no—
544D4—4241no—
545D4—4242no—
546D4—2121no—
547D4—2122no—
548D4—2124no—
549D4—2122no—
550D4—2121no—
551D4—2121no—
552D4—2122no—
553D4—2122no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—1061no—
571C2×C4—1062no—
572C2×C4—1061no—
573C2×C4—1061no—
574C2×C4—4242no—
575C2×C4—4242no—
576C2×C4—4242no—
577C2×C4—2124no—
578C2×C4—2122no—
579C2×C4C2×#1171992112yes—
580C2×C4C2×#1171992111yescharacteristic
581C2×C4C2×#1171992112yes—
582C2×C4C2×#1171992111yescharacteristic
583C2×C4C2×#1171992112yes—
584C2×C4C2×#1171992112yes—
585C2×C4C2×#1171992111yescharacteristic
586C2×C4C2×#1171992111yescharacteristic
587C2×C4C22×C4×D5311yescharacteristic
588C2×C4C22×D21211yescharacteristic
589C8—2121no—
590C8—2121no—
591C8—2122no—
592C22—21no—
593C22—21no—
594C22—2122no—
595C22—2121no—
596C22—2122no—
597C22—2121no—
598C22—2122no—
599C22—2121no—
600C22C22×#1171992111yescharacteristic
601C22#1681142611yescharacteristic
602C22#1681143111yescharacteristic
603C4—21no—
604C4—21no—
605C4—21no—
606C4—22no—
607C4—21no—
608C4—22no—
609C4—21no—
610C4—22no—
611C4—22no—
612C4—81no—
613C4—82no—
614C4—81no—
615C4—2121no—
616C4—2122no—
617C4—2121no—
618C4—2121no—
619C4#1681064112yes—
620C2—21no—
621C2—21no—
622C2—4242no—
623C2—4241no—
624C2#2028317811yescharacteristic
625C2#2028323811yescharacteristic
626C2C2×#1681064111yescharacteristic
subgroupG/H
centerC22C22×#11719921
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#2320C2
socleC22×C53C22×C22⋊C4
prime radical#20288800C2
cosocleC22×C4×C53C24
hypercenter#2320D53
hypocenterC53#17602
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2320

Complements

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

NH ≅ G/Ntype
2. #20283804C2semidirect
3. #20273290C2semidirect
4. #20273274C2semidirect
5. #20273261C2semidirect
6. #20273251C2semidirect
7. #20273238C2semidirect
8. #20273222C2semidirect
9. #20273209C2semidirect
10. #20273199C2semidirect
11. C53×#2320C2semidirect
12. #16811358C4semidirect
13. #16811358C4semidirect
14. Q8×Dic106C4semidirect
15. #16811333C4semidirect
16. #16811333C4semidirect
17. #16811295C4semidirect
309. #2320D53semidirect
366. C53#17602semidirect

Maximal subgroups

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

subgroupindex
#202888002
#202838042
#202732902
#202732742
#202732612
#202732512
#202732382
#202732222
#202732092
#202731992
C53×#23202
#1760253