Finite Group Databasebeta

← #27437648

Subgroups — #27437648

Order 13568.

1172 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811015C211yesmaximal, characteristic, complemented
2C2×#16810985C211yesmaximal, characteristic, complemented
3#20283016C211yesmaximal, characteristic
4#20275723C212yesmaximal, complemented
5#20275716C212yesmaximal
6#20275689C211yesmaximal, characteristic, complemented
7#20275536C211yesmaximal, characteristic, complemented
8#20275070C212yesmaximal, complemented
9#20275056C212yesmaximal, complemented
10#20274477C211yesmaximal, characteristic, complemented
11C2×C53×#533C211yesmaximal, characteristic, complemented
12C22×#11719925C2212yes—
13C22×#11719924C2212yes—
14C2×#13133205C2211yescharacteristic
15C2×#13133196C2211yescharacteristic
16C2×#13133191C2211yescharacteristic
17#16811015—24no—
18#16810985—24no—
19C2×#13133141C2212yes—
20C2×#13133141C2212yes—
21C2×#13133140C2212yes—
22C2×#13133137C2212yes—
23C2×C22⋊#88322C2212yes—
24C2×#13133128C2211yescharacteristic
25C2×#13133128C2211yescharacteristic
26#16810529C2211yescharacteristic
27#16810525C2211yescharacteristic
28#16810514—21no—
29#16810514—21no—
30#16810502—21no—
31#16810502—21no—
32#16810489C2211yescharacteristic
33#16810385C2212yes—
34#16810385C2212yes—
35#16810378C2212yes—
36C53×#533—24no—
37C53×#533—24no—
38C2×C4⋊Q8×C53C2211yescharacteristic
39C2×C4⋊Q8×C53C2211yescharacteristic
40C2×#125×C53—21no—
41C2×C22⋊Q8×C53—21no—
42C2×C22⋊Q8×C53C2212yes—
43C2×C4×Q8×C53—21no—
44C2×C4×Q8×C53C2212yes—
45C2×#117×C53—21no—
46C22×C42×C53C2211yescharacteristic
47C22×Dic106C2311yescharacteristic
48C22×D212C2311yescharacteristic
49C22×C22⋊D53—21no—
50C2×#11719925—24no—
51C2×#11719925—24no—
52C2×#11719925—42no—
53C2×#11719924—24no—
54C2×#11719924—24no—
55C2×#11719924—42no—
56C22×#88322C2312yes—
57C22×#88322C2312yes—
58#13133205—22no—
59#13133205—22no—
60#13133196—42no—
61#13133191—42no—
62C2×C4⋊Dic53C2311yescharacteristic
63C2×C4⋊Dic53C2311yescharacteristic
64C2×#11719919—21no—
65C2×C22⋊Dic53—21no—
66C2×#11719921—21no—
67C2×#11719921—21no—
68C2×#11719921—21no—
69C2×#11719921—21no—
70C2×#11719921—21no—
71#13133141—42no—
72#13133141—42no—
73#13133140—24no—
74#13133137—24no—
75C22⋊#88322—24no—
76#13133128—22no—
77#13133128—22no—
78#13133124—21no—
79#13133124—21no—
80#13133124—21no—
81#13133124—21no—
82C22×Q8×C53—21no—
83C22×Q8×C53C2312yes—
84C23×C4×C53D411yescharacteristic
85C23×C4×C53D411yescharacteristic
86C23×C4×C53C2311yescharacteristic
87C4⋊Q8×C53—22no—
88C4⋊Q8×C53—22no—
89C4⋊Q8×C53D414yescomplemented
90C4⋊Q8×C53D414yescomplemented
91#125×C53—28no—
92C22⋊Q8×C53—28no—
93C22⋊Q8×C53—24no—
94C22⋊Q8×C53—24no—
95C4×Q8×C53—28no—
96C4×Q8×C53—24no—
97C4×Q8×C53—24no—
98#117×C53—28no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—22no—
103C2×C4⋊C4×C53—22no—
104C2×C4⋊C4×C53C2312yes—
105C2×C22⋊C4×C53—22no—
106C2×C42×C53—22no—
107C2×C42×C53—22no—
108C2×C42×C53—22no—
109C2×C42×C53D411yescharacteristic
110C2×C42×C53D411yescharacteristic
111C2×C42×C53C2311yescharacteristic
112C2×C42×C53C2311yescharacteristic
113C24×C53C2×D411yescharacteristic
114C23×D53—21no—
115C2×Q8×C53—24no—
116C2×Q8×C53—22no—
117C2×Q8×C53—22no—
118C2×Q8×C53—22no—
119C2×Q8×C53—22no—
120C2×Q8×C53—22no—
121C2×Q8×C53—42no—
122C2×Q8×C53—42no—
123C2×Q8×C53C2×D414yes—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—22no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—41no—
135C22×C4×C53—41no—
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2411yescharacteristic
142C2×C22⋊D53—42no—
143C2×C22⋊D53—42no—
144C2×C22⋊D53—42no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C2×D212—21no—
149C2×D212—21no—
150C2×D212—22no—
151C2×D212—22no—
152C2×Dic106—21no—
153C2×Dic106—21no—
154C2×Dic106—22no—
155C2×Dic106—22no—
156C4⋊C4×C53—28no—
157C4⋊C4×C53—28no—
158C4⋊C4×C53—24no—
159C4⋊C4×C53—24no—
160C4⋊C4×C53—24no—
161C4⋊C4×C53—22no—
162C4⋊C4×C53C2×D414yes—
163C22⋊C4×C53—28no—
164C42×C53—28no—
165C42×C53—22no—
166C42×C53—22no—
167C42×C53—21no—
168C42×C53—21no—
169C42×C53C2×D412yes—
170C42×C53C2×D412yes—
171C22⋊Dic53—42no—
172#11719925—48no—
173#11719924—48no—
174#11719921—42no—
175#11719921—42no—
176#11719921—42no—
177#11719921—42no—
178#11719921—42no—
179C4⋊Dic53—22no—
180C4⋊Dic53—22no—
181#11719919—42no—
182C2×#88322—22no—
183C2×#88322—22no—
184C2×#88322—22no—
185C2×#88322—22no—
186C2×#88322—22no—
187C2×#88322#4114yes—
188C23×C53—21no—
189C23×C53—21no—
190C23×C53—21no—
191C23×C53—21no—
192C23×C53—21no—
193C23×C53—21no—
194C23×C53#13712yes—
195C23×C53C22×D411yescharacteristic
196C22×D53—41no—
197C22×D53—41no—
198C22×D53—41no—
199C22×D53—41no—
200C22×D53—41no—
201C22×D53—41no—
202C22×D53—41no—
203Q8×C53—24no—
204Q8×C53—24no—
205Q8×C53—48no—
206Q8×C53—42no—
207Q8×C53—42no—
208C2×C4×C53—28no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—24no—
218C2×C4×C53—24no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—44no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53#12012yes—
231C2×C4×C53#12012yes—
232C2×C4×C53#13712yes—
233C2×C4×C53C22×D411yescharacteristic
234C2×C4×C53C22×D411yescharacteristic
235C2×C4×C53C2×#4112yes—
236C2×C4×C53H(5,𝔽2)12yes—
237C22⋊D53—84no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247D212—42no—
248D212—42no—
249Dic106—42no—
250Dic106—42no—
251#88322—24no—
252#88322—42no—
253#25603—531nomaximal, Sylow
254C22×C53—22no—
255C22×C53—22no—
256C22×C53—22no—
257C22×C53—41no—
258C22×C53—41no—
259C22×C53—41no—
260C22×C53—41no—
261C22×C53C2×#12011yescharacteristic
262C22×C53#53211yescharacteristic
263C22×C53#53511yescharacteristic
264C22×C53C2×#13711yescharacteristic
265C22×C53C2×#13711yescharacteristic
266C22×C53#57511yescharacteristic
267C22×C53#57511yescharacteristic
268C2×D53—81no—
269C2×D53—81no—
270C2×D53—81no—
271C2×D53—81no—
272C2×D53—81no—
273C2×D53—81no—
274C2×D53—81no—
275C4×C53—22no—
276C4×C53—22no—
277C4×C53—22no—
278C4×C53—41no—
279C4×C53—44no—
280C4×C53—41no—
281C4×C53—44no—
282C4×C53#54612yes—
283C4×C53#55012yes—
284Dic53—81no—
285Dic53—81no—
286Dic53—81no—
287C2×#533D5311yescharacteristic, complemented
288C2×#497—531no—
289C2×#489—531no—
290#2850—531no—
291#2001—532no—
292#2000—532no—
293#1993—531no—
294#1960—531no—
295#1880—532no—
296#1878—532no—
297#1749—531no—
298C2×C53—42no—
299C2×C53#243111yescharacteristic
300C2×C53#300411yescharacteristic
301C2×C53#300511yescharacteristic
302C2×C53#317211yescharacteristic
303C2×C53#318111yescharacteristic
304C2×C53#321511yescharacteristic
305C2×C53#321611yescharacteristic
306D53—161no—
307C22×Q16—532no—
308C22×SD16—532no—
309#533—24no—
310#533—24no—
311C2×C4⋊Q8C2×D5311yescharacteristic
312C2×C4⋊Q8C2×D5311yescharacteristic
313C2×#125—21no—
314C2×#124—531no—
315C2×C22⋊Q8—21no—
316C2×C22⋊Q8—531no—
317C2×C22⋊Q8C2×D5312yes—
318C2×#121—531no—
319C2×C4×Q8—21no—
320C2×C4×Q8C2×D5312yes—
321C2×#117—21no—
322C22×C42C2×D5311yescharacteristic
323#497—1064no—
324#489—1064no—
325C2×#103—532no—
326C2×#103—532no—
327C2×#103—532no—
328C2×#102—532no—
329C2×C22⋊C8—532no—
330C2×#97—531no—
331C2×#97—531no—
332#391—531no—
333#390—531no—
334#388—1061no—
335#388—1061no—
336#386—1061no—
337#386—1061no—
338#382—531no—
339#340—532no—
340#340—532no—
341#336—532no—
342C53#2560311yescharacteristic, complemented, Sylow
343C22×Q8—21no—
344C22×Q8—531no—
345C22×Q8C22×D5312yes—
346C22×D4—531no—
347C22×D4—1061no—
348C23×C4C22⋊D5311yescharacteristic
349C23×C4C22⋊D5311yescharacteristic
350C23×C4C22×D5311yescharacteristic
351C2×Q16—1064no—
352C2×Q16—1064no—
353C2×Q16—2122no—
354C2×SD16—1064no—
355C2×SD16—1064no—
356C2×SD16—2122no—
357C22×C8—532no—
358C22×C8—532no—
359C4⋊Q8—22no—
360C4⋊Q8—22no—
361C4⋊Q8C22⋊D5314yescomplemented
362C4⋊Q8C22⋊D5314yescomplemented
363#125—28no—
364#124—1062no—
365#124—1062no—
366C22⋊Q8—28no—
367C22⋊Q8—24no—
368C22⋊Q8—24no—
369C22⋊Q8—2122no—
370#121—2122no—
371C4×Q8—28no—
372C4×Q8—24no—
373C4×Q8—24no—
374#117—28no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4C22×D5312yes—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—1061no—
386C2×C22⋊C4—1061no—
387C2×C22⋊C4—1061no—
388C2×C22⋊C4—1061no—
389C2×C22⋊C4—1061no—
390C2×C22⋊C4—1061no—
391C2×C42—22no—
392C2×C42—22no—
393C2×C42—22no—
394C2×C42C22⋊D5311yescharacteristic
395C2×C42C22⋊D5311yescharacteristic
396C2×C42C22×D5311yescharacteristic
397C2×C42C22×D5311yescharacteristic
398#103—1064no—
399#103—2122no—
400#103—2122no—
401#102—1064no—
402C22⋊C8—1064no—
403#97—1062no—
404#97—1062no—
405#95—1061no—
406#95—1061no—
407#95—1061no—
408#95—1061no—
409C24—1061no—
410C24C2×C22⋊D5311yescharacteristic
411C2×Q8—24no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—1061no—
420C2×Q8—1061no—
421C2×Q8—1062no—
422C2×Q8—1062no—
423C2×Q8C2×C22⋊D5314yes—
424C2×D4—1061no—
425C2×D4—1061no—
426C2×D4—1062no—
427C2×D4—1062no—
428C2×D4—2122no—
429C2×D4—2122no—
430C2×D4—2122no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—1061no—
444C22×C4—1061no—
445C22×C4—1061no—
446C22×C4C2×C22⋊D5311yescharacteristic
447C22×C4C2×C22⋊D5311yescharacteristic
448C22×C4C2×C22⋊D5311yescharacteristic
449C22×C4C2×C22⋊D5311yescharacteristic
450C22×C4C2×C22⋊D5311yescharacteristic
451C22×C4C23×D5311yescharacteristic
452Q16—2128no—
453SD16—2128no—
454C2×C8—534no—
455C2×C8—1062no—
456C2×C8—1062no—
457C2×C8—1062no—
458C2×C8—1062no—
459C2×C8—1062no—
460C4⋊C4—28no—
461C4⋊C4—28no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—1062no—
467C4⋊C4—1062no—
468C4⋊C4—2122no—
469C4⋊C4C2×C22⋊D5314yes—
470C22⋊C4—28no—
471C22⋊C4—2122no—
472C22⋊C4—2122no—
473C22⋊C4—2122no—
474C22⋊C4—2122no—
475C22⋊C4—2122no—
476C22⋊C4—2122no—
477C42—28no—
478C42—22no—
479C42—22no—
480C42—21no—
481C42—21no—
482C42C2×C22⋊D5312yes—
483C42C2×C22⋊D5312yes—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—2121no—
491C23—2121no—
492C23—2121no—
493C23—2121no—
494C23—2121no—
495C23—2121no—
496C23—2121no—
497C23#1313324812yes—
498C23C22×C22⋊D5311yescharacteristic
499Q8—24no—
500Q8—24no—
501Q8—48no—
502Q8—42no—
503Q8—42no—
504Q8—2122no—
505Q8—2122no—
506D4—4244no—
507D4—2122no—
508D4—2122no—
509C2×C4—28no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—2121no—
532C2×C4—2121no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4—2121no—
536C2×C4—2121no—
537C2×C4—2121no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4#1313318612yes—
541C2×C4#1313318612yes—
542C2×C4#1313324812yes—
543C2×C4C22×C22⋊D5311yescharacteristic
544C2×C4C22×C22⋊D5311yescharacteristic
545C2×C4#41×D5312yes—
546C2×C4#1313326312yes—
547C8—1064no—
548C8—2122no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—4241no—
557C22—4241no—
558C22—4241no—
559C22—4241no—
560C22—4241no—
561C22—4241no—
562C22—4241no—
563C22C2×#1313318611yescharacteristic
564C22#1681116911yescharacteristic
565C22#1681118711yescharacteristic
566C22C2×#1313324811yescharacteristic
567C22C2×#1313324811yescharacteristic
568C22#1681142111yescharacteristic
569C22#1681142111yescharacteristic
570C4—22no—
571C4—22no—
572C4—22no—
573C4—41no—
574C4—44no—
575C4—41no—
576C4—44no—
577C4—4241no—
578C4—4241no—
579C4—4241no—
580C4#1681126112yes—
581C4#1681129212yes—
582C2—42no—
583C2—8481no—
584C2#2027833011yescharacteristic
585C2#2028394911yescharacteristic
586C2#2028395711yescharacteristic
587C2#2028531511yescharacteristic
588C2#2028541811yescharacteristic
589C2#2028571411yescharacteristic
590C2#2028572211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#533C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenterC2×#533D53
hypocenterC53#25603
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
1. C2×#16811015C2semidirect
2. C2×#16810985C2semidirect
4. #20275723C2semidirect
6. #20275689C2semidirect
7. #20275536C2semidirect
8. #20275070C2semidirect
9. #20275056C2semidirect
10. #20274477C2semidirect
11. C2×C53×#533C2semidirect
89. C4⋊Q8×C53D4semidirect
90. C4⋊Q8×C53D4semidirect
287. C2×#533D53semidirect
342. C53#25603semidirect
361. C4⋊Q8C22⋊D53semidirect
362. C4⋊Q8C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168110152
C2×#168109852
#202830162
#202757232
#202757162
#202756892
#202755362
#202750702
#202750562
#202744772
C2×C53×#5332
#2560353