Finite Group Databasebeta

← #27419205

Subgroups — #27419205

Order 13568.

1482 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811173C211yesmaximal, characteristic
2C2×#16810845C212yesmaximal, complemented
3C2×#16810783C212yesmaximal, complemented
4#20275961C211yesmaximal, characteristic, complemented
5#20275949C211yesmaximal, characteristic, complemented
6#20275814C211yesmaximal, characteristic, complemented
7#20275802C211yesmaximal, characteristic, complemented
8#20275647C212yesmaximal, complemented
9#20274695C211yesmaximal, characteristic, complemented
10#20274653C212yesmaximal, complemented
11C4×C22⋊Q8×C53C211yesmaximal, characteristic, complemented
12C22×#11719945C2212yes—
13C22×#11719925C2212yes—
14C22×#11719924C2212yes—
15#16811173—28no—
16C2×#13133217C2211yescharacteristic
17C2×#13133214C2211yescharacteristic
18C2×#13133205C2212yes—
19C2×#13133191—21no—
20C2×#13133191—21no—
21C2×C4×D212—21no—
22C2×C4×D212—21no—
23#16810845—28no—
24#16810783—28no—
25C2×C4⋊#88322C2212yes—
26C2×#13133140C2212yes—
27C2×#13133140C2212yes—
28C2×#13133140C2212yes—
29C2×#13133137C2212yes—
30C2×#13133137C2212yes—
31C2×#13133137C2212yes—
32C2×C22⋊#88322C2212yes—
33#16810378C2211yescharacteristic
34#16810378C2211yescharacteristic
35C2×C22⋊Q8×C53C2211yescharacteristic
36C2×C4×Q8×C53C2211yescharacteristic
37C22×C42×C53C2211yescharacteristic
38C53×#391C2211yescharacteristic
39C53×#389C2211yescharacteristic
40C53×#387—21no—
41C53×#385—21no—
42C53×#380C2211yescharacteristic
43C4×C4⋊C4×C53—21no—
44C4×C4⋊C4×C53C2211yescharacteristic
45C4×C22⋊C4×C53—21no—
46C2×#11719945—24no—
47C2×#11719945—24no—
48C2×#11719945—24no—
49C2×#11719945—24no—
50C2×#11719945—42no—
51C2×#11719945—42no—
52C22×Dic106C2312yes—
53C22×D212—21no—
54C22×D212—21no—
55C22×D212C2312yes—
56C22×C22⋊D53—22no—
57C22×C4×D53—22no—
58C2×#11719925—24no—
59C2×#11719925—24no—
60C2×#11719925—42no—
61C2×#11719924—24no—
62C2×#11719924—24no—
63C2×#11719924—42no—
64C22×#88322C2312yes—
65C22×#88322C2312yes—
66#13133217—24no—
67#13133214—24no—
68#13133205—24no—
69#13133205D418yes—
70#13133191—28no—
71#13133191—28no—
72C4×D212—28no—
73C4×D212—28no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×#11719921—22no—
77C2×#11719921—22no—
78C4⋊#88322—24no—
79#13133140—24no—
80#13133140—24no—
81#13133140—42no—
82#13133137—24no—
83#13133137—24no—
84#13133137—42no—
85C22⋊#88322—24no—
86C22×Q8×C53C2311yescharacteristic
87C23×C4×C53D411yescharacteristic
88C23×C4×C53D411yescharacteristic
89C23×C4×C53C2311yescharacteristic
90C22⋊Q8×C53—22no—
91C22⋊Q8×C53D414yescomplemented
92C4×Q8×C53—24no—
93C2×C4⋊C4×C53—21no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53C2311yescharacteristic
97C2×C4⋊C4×C53C2311yescharacteristic
98C2×C4⋊C4×C53C2311yescharacteristic
99C2×C22⋊C4×C53—21no—
100C2×C22⋊C4×C53—21no—
101C2×C42×C53—24no—
102C2×C42×C53—21no—
103C2×C42×C53—21no—
104C2×C42×C53D411yescharacteristic
105C2×C42×C53D411yescharacteristic
106C2×C42×C53C2311yescharacteristic
107C2×C42×C53C2311yescharacteristic
108#95×C53—21no—
109#95×C53—21no—
110#95×C53—21no—
111#95×C53—21no—
112#95×C53—21no—
113C24×C53C2×D411yescharacteristic
114C23×D53—22no—
115C2×Q8×C53—21no—
116C2×Q8×C53—22no—
117C2×Q8×C53—22no—
118C2×Q8×C53C2×D412yes—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—41no—
130C22×C4×C53—41no—
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D412yes—
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53#4112yes—
138C22×C4×C53C2411yescharacteristic
139C2×C22⋊D53—28no—
140C2×C22⋊D53—44no—
141C2×C22⋊D53—44no—
142C22×Dic53—22no—
143#11719945—48no—
144#11719945—48no—
145#11719945—48no—
146#11719945—48no—
147C2×D212—22no—
148C2×D212—22no—
149C2×D212—22no—
150C2×D212—22no—
151C2×D212—24no—
152C2×D212—24no—
153C2×D212—22no—
154C2×D212—42no—
155C2×D212—42no—
156C2×D212—42no—
157C2×D212—42no—
158C2×D212C2×D414yes—
159C2×C4×D53—28no—
160C2×C4×D53—44no—
161C2×C4×D53—44no—
162C2×Dic106—22no—
163C2×Dic106—22no—
164C2×Dic106—22no—
165C2×Dic106—22no—
166C2×Dic106—22no—
167C2×Dic106C2×D414yes—
168C4⋊C4×C53—24no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—22no—
171C4⋊C4×C53—22no—
172C4⋊C4×C53—42no—
173C4⋊C4×C53—42no—
174C4⋊C4×C53C2×D412yes—
175C22⋊C4×C53—24no—
176C22⋊C4×C53—42no—
177C42×C53—22no—
178C42×C53—22no—
179C42×C53—22no—
180C42×C53—42no—
181C42×C53—42no—
182C42×C53C2×D414yes—
183#11719925—48no—
184#11719924—48no—
185#11719921—28no—
186#11719921—28no—
187C4⋊Dic53—24no—
188C4⋊Dic53—24no—
189C2×#88322—22no—
190C2×#88322—22no—
191C2×#88322—22no—
192C2×#88322—22no—
193C2×#88322—22no—
194C2×#88322—22no—
195C23×C53—21no—
196C23×C53—21no—
197C23×C53—21no—
198C23×C53—21no—
199C23×C53—21no—
200C23×C53—21no—
201C23×C53#12112yes—
202C23×C53C22×D411yescharacteristic
203C22×D53—24no—
204C22×D53—42no—
205C22×D53—42no—
206C22×D53—42no—
207C22×D53—42no—
208C22×D53—42no—
209C22×D53—42no—
210Q8×C53—24no—
211Q8×C53—42no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—24no—
223C2×C4×C53—24no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—44no—
227C2×C4×C53—44no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53#12012yes—
239C2×C4×C53#12012yes—
240C2×C4×C53#12112yes—
241C2×C4×C53#13512yes—
242C2×C4×C53#13712yes—
243C2×C4×C53C22×D412yes—
244C2×C4×C53C22×D411yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246C2×C4×C53H(5,𝔽2)12yes—
247C22⋊D53—416no—
248C2×Dic53—24no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251D212—24no—
252D212—24no—
253D212—48no—
254D212—48no—
255D212—42no—
256D212—42no—
257C4×D53—416no—
258Dic106—24no—
259Dic106—24no—
260Dic106—42no—
261Dic106—42no—
262#88322—42no—
263#88322—42no—
264#24216—531nomaximal, Sylow
265C22×C53—22no—
266C22×C53—22no—
267C22×C53—22no—
268C22×C53—41no—
269C22×C53—41no—
270C22×C53—41no—
271C22×C53—41no—
272C22×C53C2×#12011yescharacteristic
273C22×C53C2×#12111yescharacteristic
274C22×C53#53511yescharacteristic
275C22×C53C2×#13511yescharacteristic
276C22×C53C2×#13711yescharacteristic
277C22×C53#57511yescharacteristic
278C22×C53#57511yescharacteristic
279C2×D53—44no—
280C2×D53—44no—
281C2×D53—44no—
282C2×D53—82no—
283C2×D53—82no—
284C2×D53—82no—
285C2×D53—82no—
286C4×C53—22no—
287C4×C53—22no—
288C4×C53—22no—
289C4×C53—22no—
290C4×C53—22no—
291C4×C53—42no—
292C4×C53—42no—
293C4×C53—81no—
294C4×C53—81no—
295C4×C53D4212yes—
296C4×C53#54612yes—
297Dic53—44no—
298C2×#532—531no—
299C2×#464—532no—
300C2×#452—532no—
301C4×C22⋊Q8D5311yescharacteristic, complemented
302#2043—531no—
303#2041—531no—
304#2016—531no—
305#2014—531no—
306#1983—532no—
307#1797—531no—
308#1788—532no—
309C2×C53—42no—
310C2×C53#240811yescharacteristic
311C2×C53#300211yescharacteristic
312C2×C53#300711yescharacteristic
313C2×C53#303511yescharacteristic
314C2×C53#303711yescharacteristic
315C2×C53#321011yescharacteristic
316C2×C53#321911yescharacteristic
317D53—84no—
318C22×#41—532no—
319C22×Q16—532no—
320C22×SD16—532no—
321#532—1068no—
322C2×C4⋊Q8—531no—
323C2×#127—531no—
324C2×#124—532no—
325C2×C22⋊Q8C2×D5311yescharacteristic
326C2×#121—1061no—
327C2×#121—1061no—
328C2×C4×Q8C2×D5311yescharacteristic
329C2×C4×D4—1061no—
330C2×C4×D4—1061no—
331C22×C42C2×D5311yescharacteristic
332#464—1068no—
333#452—1068no—
334C2×C4⋊C8—532no—
335C2×#103—532no—
336C2×#103—532no—
337C2×#103—532no—
338C2×#102—532no—
339C2×#102—532no—
340C2×#102—532no—
341C2×C22⋊C8—532no—
342#391C2×D5311yescharacteristic
343#389C2×D5311yescharacteristic
344#387—21no—
345#385—21no—
346#380C2×D5311yescharacteristic
347C4×C4⋊C4—21no—
348C4×C4⋊C4C2×D5311yescharacteristic
349C4×C22⋊C4—21no—
350#336—531no—
351#336—531no—
352C53#2421611yescharacteristic, complemented, Sylow
353C2×#41—1064no—
354C2×#41—1064no—
355C2×#41—1064no—
356C2×#41—1064no—
357C2×#41—2122no—
358C2×#41—2122no—
359C22×Q8—532no—
360C22×Q8C22×D5311yescharacteristic
361C22×D4—532no—
362C22×D4—1062no—
363C22×D4—1061no—
364C22×D4—1061no—
365C23×C4—1062no—
366C23×C4C22⋊D5311yescharacteristic
367C23×C4C22⋊D5311yescharacteristic
368C23×C4C22×D5311yescharacteristic
369C2×Q16—1064no—
370C2×Q16—1064no—
371C2×Q16—2122no—
372C2×SD16—1064no—
373C2×SD16—1064no—
374C2×SD16—2122no—
375C22×C8—532no—
376C22×C8—532no—
377C4⋊Q8—1064no—
378#127—1064no—
379#124—538no—
380#124—1064no—
381C22⋊Q8—22no—
382C22⋊Q8D21214yescomplemented
383#121—1068no—
384#121—1068no—
385C4×Q8—24no—
386C4×D4—1068no—
387C4×D4—1068no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4C22×D5311yescharacteristic
394C2×C4⋊C4C22×D5311yescharacteristic
395C2×C4⋊C4C22×D5311yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—1062no—
399C2×C22⋊C4—1062no—
400C2×C42—24no—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42C22⋊D5311yescharacteristic
404C2×C42C22⋊D5311yescharacteristic
405C2×C42C22×D5311yescharacteristic
406C2×C42C22×D5311yescharacteristic
407C4⋊C8—1064no—
408#103—1064no—
409#103—1064no—
410#103—2122no—
411#102—1064no—
412#102—1064no—
413#102—2122no—
414C22⋊C8—1064no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—21no—
419#95—21no—
420C24—1062no—
421C24C2×C22⋊D5311yescharacteristic
422#41—2128no—
423#41—2128no—
424#41—2128no—
425#41—2128no—
426C2×Q8—21no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—534no—
430C2×Q8—1062no—
431C2×Q8—1062no—
432C2×Q8—1062no—
433C2×Q8—1062no—
434C2×Q8—1062no—
435C2×Q8C2×D21212yes—
436C2×D4—534no—
437C2×D4—1064no—
438C2×D4—1064no—
439C2×D4—1068no—
440C2×D4—1062no—
441C2×D4—1062no—
442C2×D4—1062no—
443C2×D4—1062no—
444C2×D4—1062no—
445C2×D4—2122no—
446C2×D4—2122no—
447C2×D4—2124no—
448C2×D4—2124no—
449C2×D4—2122no—
450C2×D4—2122no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—1068no—
464C22×C4—1062no—
465C22×C4—2124no—
466C22×C4—2124no—
467C22×C4C2×D21212yes—
468C22×C4#1171994512yes—
469C22×C4C2×C22⋊D5311yescharacteristic
470C22×C4C2×C22⋊D5311yescharacteristic
471C22×C4C2×C22⋊D5311yescharacteristic
472C22×C4C2×C22⋊D5311yescharacteristic
473C22×C4C2×C22⋊D5311yescharacteristic
474C22×C4C23×D5311yescharacteristic
475Q16—2128no—
476SD16—2128no—
477C2×C8—1062no—
478C2×C8—1062no—
479C2×C8—1062no—
480C2×C8—1062no—
481C2×C8—1062no—
482C2×C8—1062no—
483C4⋊C4—24no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—1064no—
490C4⋊C4—1064no—
491C4⋊C4C2×D21212yes—
492C22⋊C4—24no—
493C22⋊C4—42no—
494C22⋊C4—1068no—
495C22⋊C4—1068no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—42no—
500C42—42no—
501C42D4×D5314yes—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—1064no—
509C23—2122no—
510C23—2122no—
511C23—2122no—
512C23—2122no—
513C23—2122no—
514C23—2122no—
515C23#1313319112yes—
516C23C22×C22⋊D5311yescharacteristic
517Q8—24no—
518Q8—42no—
519Q8—1064no—
520Q8—1064no—
521Q8—2122no—
522Q8—2122no—
523D4—1064no—
524D4—1064no—
525D4—2128no—
526D4—2128no—
527D4—21216no—
528D4—2122no—
529D4—2122no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—1064no—
557C2×C4—21216no—
558C2×C4—2122no—
559C2×C4—2122no—
560C2×C4#1313318512yes—
561C2×C4#1313318512yes—
562C2×C4#1313319112yes—
563C2×C4#1313323712yes—
564C2×C4#1313324812yes—
565C2×C4C2×D4×D5312yes—
566C2×C4C22×D21211yescharacteristic
567C2×C4C2×#1171994511yescharacteristic
568C2×C4#1313326312yes—
569C8—2122no—
570C8—2122no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—4242no—
579C22—4242no—
580C22—4242no—
581C22—4242no—
582C22—2124no—
583C22—2124no—
584C22—2124no—
585C22C2×#1313318511yescharacteristic
586C22C2×#1313319111yescharacteristic
587C22#1681118711yescharacteristic
588C22C2×#1313323711yescharacteristic
589C22C2×#1313324811yescharacteristic
590C22#1681142111yescharacteristic
591C22#1681142111yescharacteristic
592C4—22no—
593C4—22no—
594C4—22no—
595C4—22no—
596C4—22no—
597C4—42no—
598C4—42no—
599C4—81no—
600C4—81no—
601C4—2124no—
602C4D4×D21212yes—
603C4#1681126912yes—
604C2—42no—
605C2—4244no—
606C2#2027807111yescharacteristic
607C2#2028393111yescharacteristic
608C2#2028397311yescharacteristic
609C2#2028417011yescharacteristic
610C2#2028418711yescharacteristic
611C2#2028566711yescharacteristic
612C2#2028574811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC4×C22⋊Q8×C53C2
socleC23×C53C22×D4
prime radicalC2×#16811173C2
hypercenterC4×C22⋊Q8D53
hypocenterC53#24216
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#16810845C2semidirect
3. C2×#16810783C2semidirect
4. #20275961C2semidirect
5. #20275949C2semidirect
6. #20275814C2semidirect
7. #20275802C2semidirect
8. #20275647C2semidirect
9. #20274695C2semidirect
10. #20274653C2semidirect
11. C4×C22⋊Q8×C53C2semidirect
91. C22⋊Q8×C53D4semidirect
301. C4×C22⋊Q8D53semidirect
352. C53#24216semidirect
382. C22⋊Q8D212semidirect

Maximal subgroups

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

subgroupindex
C2×#168111732
C2×#168108452
C2×#168107832
#202759612
#202759492
#202758142
#202758022
#202756472
#202746952
#202746532
C4×C22⋊Q8×C532
#2421653