Finite Group Databasebeta

← #27412788

Subgroups — #27412788

Order 13568.

1178 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810666C211yesmaximal, characteristic, complemented
2C2×#16810646C211yesmaximal, characteristic, complemented
3C2×#16810625C211yesmaximal, characteristic
4C2×#16810622C211yesmaximal, characteristic
5#20277203C211yesmaximal, characteristic
6C2×#16810519C211yesmaximal, characteristic, complemented
7#20276111C212yesmaximal, complemented
8#20276104C212yesmaximal, complemented
9#20276097C212yesmaximal, complemented
10#20276090C212yesmaximal, complemented
11C53×#1736C211yesmaximal, characteristic, complemented
12C22×#11719945C2211yescharacteristic
13C22×#11719917C2211yescharacteristic
14C22×C4⋊Dic53C2211yescharacteristic
15C22×#11719919—21no—
16C22×C4×Dic53C2211yescharacteristic
17#16810666—24no—
18C2×#13133150C2212yes—
19C2×#13133148C2212yes—
20#16810646—24no—
21C2×#13133146C2212yes—
22C2×#13133146C2212yes—
23#16810625—24no—
24#16810622—24no—
25C2×#13133142C2212yes—
26C2×#13133140C2212yes—
27C2×#13133139C2212yes—
28C2×#13133137C2212yes—
29#16810528C2211yescharacteristic
30#16810528C2211yescharacteristic
31#16810524C2211yescharacteristic
32#16810524C2211yescharacteristic
33#16810519—21no—
34#16810519—21no—
35#16810519C2212yes—
36#16810519C2212yes—
37#16810500—21no—
38#16810500—21no—
39C4×#11719921—21no—
40C4×#11719921—21no—
41C2×#13133124—21no—
42C22×M16×C53C2211yescharacteristic
43C2×#117×C53C2211yescharacteristic
44C22×C4⋊C4×C53C2211yescharacteristic
45C53×#340C2212yes—
46C53×#340C2212yes—
47C2×#11719945—22no—
48C2×#11719945—22no—
49C2×#11719945—22no—
50C2×#11719945—22no—
51C2×#11719945—22no—
52C2×#11719945Q814yes—
53C22×Dic106C2311yescharacteristic
54C22×D212C2311yescharacteristic
55C22×C22⋊D53—21no—
56C23×Dic53—21no—
57C23×Dic53D411yescharacteristic
58C23×Dic53D411yescharacteristic
59C22×C4×D53—21no—
60C2×#11719917—22no—
61C2×#11719917—22no—
62C2×#11719917D414yes—
63C22×#88322C2312yes—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53—22no—
66C2×C4⋊Dic53D414yes—
67C2×C4⋊Dic53C2312yes—
68C2×#11719919—22no—
69C2×#11719919—22no—
70C2×#11719919—44no—
71C2×#11719921—21no—
72C2×#11719921—21no—
73C2×#11719921—21no—
74C2×#11719921—21no—
75C2×C4×Dic53—22no—
76C2×C4×Dic53—22no—
77C2×C4×Dic53D411yescharacteristic
78C2×C4×Dic53D411yescharacteristic
79C2×C4×Dic53C2311yescharacteristic
80C2×C4×Dic53C2311yescharacteristic
81#13133150—24no—
82#13133148—24no—
83#13133146—24no—
84#13133146—24no—
85#13133142—24no—
86#13133140—24no—
87#13133139—24no—
88#13133137—24no—
89#13133124—21no—
90#13133124—21no—
91#13133124—21no—
92#13133124—21no—
93#13133124—22no—
94#13133124—22no—
95C23×C4×C53—21no—
96C23×C4×C53C2311yescharacteristic
97C2×M16×C53—22no—
98C2×M16×C53—22no—
99C2×M16×C53—22no—
100C22×C8×C53C2312yes—
101#117×C53—22no—
102#117×C53—22no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—22no—
105C2×C4⋊C4×C53—22no—
106C2×C4⋊C4×C53C2312yes—
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C4⋊C4×C53C2311yescharacteristic
109C2×C22⋊C4×C53—21no—
110C2×C42×C53—21no—
111C24×C53C2×D411yescharacteristic
112C23×D53—21no—
113C22×C4×C53—21no—
114C22×C4×C53—21no—
115C22×C4×C53—22no—
116C22×C4×C53—42no—
117C22×C4×C53—42no—
118C22×C4×C53—42no—
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53C2×D412yes—
121C22×C4×C53C2×D412yes—
122C22×C4×C53C2×Q812yes—
123C22×C4×C53#4112yes—
124C22×C4×C53#4112yes—
125C22×C4×C53#4112yes—
126C22×C4×C53C2411yescharacteristic
127C2×C22⋊D53—24no—
128C2×C22⋊D53—42no—
129C2×C22⋊D53—42no—
130C22×Dic53—22no—
131C22×Dic53—22no—
132C22×Dic53—24no—
133C22×Dic53—21no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—41no—
138C22×Dic53—41no—
139C22×Dic53—42no—
140C22×Dic53—42no—
141C22×Dic53C2×D411yescharacteristic
142C22×Dic53C2×D411yescharacteristic
143C22×Dic53C2×D411yescharacteristic
144C22×Dic53C2×D411yescharacteristic
145#11719945—28no—
146#11719945—28no—
147#11719945—44no—
148#11719945—44no—
149C2×D212—21no—
150C2×D212—21no—
151C2×D212—21no—
152C2×D212—21no—
153C2×D212—21no—
154C2×D212C2×Q812yes—
155C2×C4×D53—24no—
156C2×C4×D53—42no—
157C2×C4×D53—42no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106—21no—
162C2×Dic106—21no—
163C2×Dic106C2×Q812yes—
164M16×C53—44no—
165C2×C8×C53—22no—
166C2×C8×C53—22no—
167C2×C8×C53—22no—
168C4⋊C4×C53—22no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—22no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—42no—
175C22⋊C4×C53—42no—
176C42×C53—42no—
177#11719921—42no—
178#11719921—42no—
179#11719921—42no—
180#11719921—42no—
181C4⋊Dic53—24no—
182C4⋊Dic53—22no—
183C4⋊Dic53C2×D414yes—
184#11719919—44no—
185#11719919—44no—
186C4×Dic53—22no—
187C4×Dic53—22no—
188C4×Dic53—21no—
189C4×Dic53—21no—
190C4×Dic53—21no—
191C4×Dic53—21no—
192#11719917—28no—
193C2×#88322—22no—
194C2×#88322—22no—
195C2×#88322C2×D414yes—
196C23×C53—21no—
197C23×C53—21no—
198C23×C53—21no—
199C23×C53—21no—
200C23×C53#12112yes—
201C23×C53#12112yes—
202C23×C53C22⋊Q812yes—
203C23×C53C22×D411yescharacteristic
204C22×D53—22no—
205C22×D53—41no—
206C22×D53—41no—
207C22×D53—41no—
208C22×D53—41no—
209C22×D53—41no—
210C22×D53—41no—
211C2×C4×C53—42no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—42no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—42no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—84no—
221C2×C4×C53#12114yes—
222C2×C4×C53#12112yes—
223C2×C4×C53#12112yes—
224C2×C4×C53C22⋊Q814yes—
225C2×C4×C53C22⋊Q814yes—
226C2×C4×C53C22⋊Q812yes—
227C2×C4×C53#12614yes—
228C2×C4×C53C22×D411yescharacteristic
229C2×C4×C53C22×D411yescharacteristic
230C2×C4×C53C22×Q811yescharacteristic
231C2×C4×C53C2×#4111yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C22⋊D53—48no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—24no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—22no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—44no—
248C2×Dic53—44no—
249C2×Dic53—48no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53AGL(1,ℤ/8)12yes—
255C2×Dic53#13712yes—
256D212—22no—
257D212—22no—
258D212—41no—
259D212—41no—
260C4×D53—48no—
261Dic106—22no—
262Dic106—22no—
263Dic106—41no—
264Dic106—41no—
265C8×C53—42no—
266#88322—24no—
267#23617—531nomaximal, Sylow
268C22×C53—22no—
269C22×C53—22no—
270C22×C53—22no—
271C22×C53—22no—
272C22×C53—22no—
273C22×C53—22no—
274C22×C53#39614yes—
275C22×C53C2×#12111yescharacteristic
276C22×C53C2×#12111yescharacteristic
277C22×C53C2×C22⋊Q811yescharacteristic
278C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C53C2×#13711yescharacteristic
280C22×C53#57511yescharacteristic
281C22×C53#57511yescharacteristic
282C2×D53—42no—
283C2×D53—42no—
284C2×D53—42no—
285C2×D53—81no—
286C2×D53—81no—
287C2×D53—81no—
288C2×D53—81no—
289C4×C53—81no—
290C4×C53—81no—
291C4×C53—82no—
292C4×C53#39614yes—
293C4×C53C2×#12111yescharacteristic
294C4×C53C2×C22⋊Q811yescharacteristic
295C4×C53C2×C22⋊Q811yescharacteristic
296C4×C53C2×#12611yescharacteristic
297Dic53—22no—
298Dic53—22no—
299Dic53—42no—
300Dic53—44no—
301Dic53—41no—
302Dic53—41no—
303C2×#428—531no—
304C2×#423—531no—
305C2×#417—531no—
306C2×#417—531no—
307#2297—531no—
308C2×#389—531no—
309#2068—532no—
310#2067—532no—
311#2066—532no—
312#2065—532no—
313#1736D5311yescharacteristic, complemented
314C2×C53—24no—
315C2×C53C2×#39611yescharacteristic
316C2×C53#303611yescharacteristic
317C2×C53#303711yescharacteristic
318C2×C53#305511yescharacteristic
319C2×C53#305611yescharacteristic
320C2×C53#307911yescharacteristic
321C2×C53#308011yescharacteristic
322D53—82no—
323C22×#41—531no—
324C22×M16—531no—
325C22×M16C2×D5311yescharacteristic
326C2×#117C2×D5311yescharacteristic
327C22×C4⋊C4—531no—
328C22×C4⋊C4—1061no—
329C22×C4⋊C4C2×D5311yescharacteristic
330C22×C42—531no—
331#428—1064no—
332C2×#107—532no—
333C2×#106—532no—
334#423—1064no—
335C2×C4⋊C8—532no—
336C2×C4⋊C8—532no—
337#417—1064no—
338#417—1064no—
339C2×#103—532no—
340C2×#103—532no—
341C2×#102—532no—
342C2×#102—532no—
343#391—531no—
344#391—531no—
345#390—531no—
346#390—531no—
347#389—532no—
348#389—532no—
349#389—1061no—
350#389—1061no—
351#386—1061no—
352#386—1061no—
353C4×C22⋊C4—1061no—
354C4×C22⋊C4—1061no—
355C2×#95—1061no—
356#340C2×D5312yes—
357#340C2×D5312yes—
358C53#2361711yescharacteristic, complemented, Sylow
359C2×#41—534no—
360C2×#41—1062no—
361C2×#41—1062no—
362C2×#41—1062no—
363C2×#41—1062no—
364C2×#41—1062no—
365C22×Q8—531no—
366C22×D4—531no—
367C22×D4—1061no—
368C23×C4—21no—
369C23×C4—531no—
370C23×C4—531no—
371C23×C4—1061no—
372C23×C4—1061no—
373C23×C4C22×D5311yescharacteristic
374C2×M16—22no—
375C2×M16—22no—
376C2×M16—22no—
377C2×M16—534no—
378C2×M16—1062no—
379C2×M16—1062no—
380C22×C8—532no—
381C22×C8C22×D5312yes—
382#117—22no—
383#117—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—534no—
388C2×C4⋊C4—532no—
389C2×C4⋊C4—1062no—
390C2×C4⋊C4—1061no—
391C2×C4⋊C4—1062no—
392C2×C4⋊C4—1062no—
393C2×C4⋊C4—2124no—
394C2×C4⋊C4C22×D5312yes—
395C2×C4⋊C4C22×D5311yescharacteristic
396C2×C4⋊C4C22×D5311yescharacteristic
397C2×C22⋊C4—21no—
398C2×C22⋊C4—1061no—
399C2×C22⋊C4—1061no—
400C2×C22⋊C4—1061no—
401C2×C22⋊C4—1061no—
402C2×C42—21no—
403C2×C42—531no—
404C2×C42—531no—
405C2×C42—531no—
406C2×C42—531no—
407C2×C42—1062no—
408C2×C42—1062no—
409#107—1064no—
410#106—1064no—
411C4⋊C8—1064no—
412C4⋊C8—1064no—
413#103—1064no—
414#103—1064no—
415#102—1064no—
416#102—1064no—
417#95—1062no—
418#95—1062no—
419#95—1061no—
420#95—1061no—
421#95—1061no—
422#95—1061no—
423C24—1061no—
424C24D4×D5311yescharacteristic
425#41—1068no—
426#41—1068no—
427#41—2124no—
428#41—2124no—
429C2×Q8—532no—
430C2×Q8—1061no—
431C2×Q8—1061no—
432C2×Q8—1061no—
433C2×Q8—1061no—
434C2×Q8—1061no—
435C2×D4—532no—
436C2×D4—1064no—
437C2×D4—1061no—
438C2×D4—1061no—
439C2×D4—1061no—
440C2×D4—1061no—
441C2×D4—1061no—
442C2×D4—2122no—
443C2×D4—2122no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—531no—
451C22×C4—531no—
452C22×C4—531no—
453C22×C4—531no—
454C22×C4—1062no—
455C22×C4—1062no—
456C22×C4—1062no—
457C22×C4—1062no—
458C22×C4—1064no—
459C22×C4—1061no—
460C22×C4—1062no—
461C22×C4—1064no—
462C22×C4—2121no—
463C22×C4—2121no—
464C22×C4—2122no—
465C22×C4—2122no—
466C22×C4—2122no—
467C22×C4—2122no—
468C22×C4#1171994512yes—
469C22×C4#1171994512yes—
470C22×C4#1171994512yes—
471C22×C4D4×D5311yescharacteristic
472C22×C4D4×D5312yes—
473C22×C4D4×D5312yes—
474C22×C4Q8×D5312yes—
475C22×C4C23×D5311yescharacteristic
476M16—44no—
477M16—1068no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—534no—
482C2×C8—1062no—
483C2×C8—1062no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—42no—
491C4⋊C4—534no—
492C4⋊C4—1064no—
493C4⋊C4—1062no—
494C4⋊C4—2124no—
495C4⋊C4—2124no—
496C22⋊C4—42no—
497C22⋊C4—2122no—
498C22⋊C4—2122no—
499C22⋊C4—2122no—
500C22⋊C4—2122no—
501C42—42no—
502C42—1062no—
503C42—1062no—
504C42—1061no—
505C42—1061no—
506C42—1061no—
507C42—1061no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—1062no—
513C23—2121no—
514C23—2121no—
515C23—2121no—
516C23—2121no—
517C23—2121no—
518C23—2121no—
519C23#1313319012yes—
520C23#1313319012yes—
521C23#1313319512yes—
522C23C2×D4×D5311yescharacteristic
523Q8—1062no—
524Q8—1062no—
525Q8—2121no—
526Q8—2121no—
527D4—1062no—
528D4—1062no—
529D4—2128no—
530D4—2121no—
531D4—2121no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—84no—
542C2×C4—532no—
543C2×C4—532no—
544C2×C4—1061no—
545C2×C4—1061no—
546C2×C4—1061no—
547C2×C4—1061no—
548C2×C4—1061no—
549C2×C4—1061no—
550C2×C4—1064no—
551C2×C4—1062no—
552C2×C4—1061no—
553C2×C4—1061no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—2124no—
557C2×C4—2124no—
558C2×C4—2128no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4—2122no—
562C2×C4—2122no—
563C2×C4—2128no—
564C2×C4#1313319012yes—
565C2×C4#1313319012yes—
566C2×C4#1313319014yes—
567C2×C4#1313319512yes—
568C2×C4#1313319514yes—
569C2×C4#1313319514yes—
570C2×C4#1313321114yes—
571C2×C4C2×D4×D5311yescharacteristic
572C2×C4C2×D4×D5311yescharacteristic
573C2×C4C2×Q8×D5311yescharacteristic
574C2×C4C2×#1171994511yescharacteristic
575C2×C4C2×#1171994511yescharacteristic
576C2×C4C2×#1171994511yescharacteristic
577C8—42no—
578C8—1064no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—4241no—
586C22—4241no—
587C22—4241no—
588C22—4241no—
589C22—2122no—
590C22—2122no—
591C22—2122no—
592C22#1681054714yes—
593C22C2×#1313319011yescharacteristic
594C22C2×#1313319011yescharacteristic
595C22C2×#1313319511yescharacteristic
596C22AGL(1,ℤ/8)×D5311yescharacteristic
597C22#137×D5311yescharacteristic
598C22#1681141911yescharacteristic
599C22#1681142211yescharacteristic
600C4—81no—
601C4—81no—
602C4—82no—
603C4—1062no—
604C4—1062no—
605C4—2124no—
606C4—2121no—
607C4—2122no—
608C4—2121no—
609C4#1681054714yes—
610C4C2×#1313319011yescharacteristic
611C4C2×#1313319511yescharacteristic
612C4C2×#1313319511yescharacteristic
613C4C2×#1313321111yescharacteristic
614C2—24no—
615C2—4242no—
616C2C2×#1681054711yescharacteristic
617C2#2028417811yescharacteristic
618C2#2028418611yescharacteristic
619C2#2028433211yescharacteristic
620C2#2028434311yescharacteristic
621C2#2028452811yescharacteristic
622C2#2028453611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1736C2
socleC23×C53C22×D4
prime radicalC22×#11719945C22
hypercenter#1736D53
hypocenterC53#23617
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
1. C2×#16810666C2semidirect
2. C2×#16810646C2semidirect
6. C2×#16810519C2semidirect
7. #20276111C2semidirect
8. #20276104C2semidirect
9. #20276097C2semidirect
10. #20276090C2semidirect
11. C53×#1736C2semidirect
313. #1736D53semidirect
358. C53#23617semidirect

Maximal subgroups

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

subgroupindex
C2×#168106662
C2×#168106462
C2×#168106252
C2×#168106222
#202772032
C2×#168105192
#202761112
#202761042
#202760972
#202760902
C53×#17362
#2361753