Finite Group Databasebeta

← #27385330

Subgroups — #27385330

Order 13568.

1934 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133193C211yesmaximal, characteristic
2C2×#117×D53C211yesmaximal, characteristic
3C2×#16810632C211yesmaximal, characteristic, complemented
4C2×#16810586C211yesmaximal, characteristic
5#20275024C214yesmaximal, complemented
6#20275019C214yesmaximal, complemented
7#20274436C211yesmaximal, characteristic, complemented
8#20274435C211yesmaximal, characteristic, complemented
9C2×C53×#419C211yesmaximal, characteristic, complemented
10C23×C4×D53C2211yescharacteristic
11C22×#11719917C2211yescharacteristic
12C2×#13133193—24no—
13C2×#13133193—21no—
14C2×#13133193—21no—
15C2×#13133193C418yescomplemented
16C2×#13133193C2212yes—
17C2×#13133193C2212yes—
18C2×#13133176C2211yescharacteristic
19C2×#13133175C2212yes—
20C2×#13133174—21no—
21C2×#13133173—21no—
22#117×D53—24no—
23#117×D53—24no—
24C22×C4⋊Dic53C2211yescharacteristic
25C22×#11719919—21no—
26C2×C4⋊C4×D53C2212yes—
27C22×#11719921—21no—
28C2×C22⋊C4×D53—21no—
29C2×C42×D53—21no—
30#16810632—24no—
31C2×#13133141C2214yescomplemented
32#16810586—24no—
33C2×#13133129C2212yes—
34C2×#13133129C2212yes—
35#16810386C2214yescomplemented
36#16810385C2214yescomplemented
37C23×Q8×C53C2211yescharacteristic
38C22×M16×C53C2211yescharacteristic
39C2×#117×C53C2211yescharacteristic
40C53×#419—24no—
41C2×#103×C53C2214yescomplemented
42C24×D53D411yescharacteristic
43C23×Dic53—21no—
44C23×Dic53D411yescharacteristic
45C22×C4×D53—22no—
46C22×C4×D53—22no—
47C22×C4×D53—22no—
48C22×C4×D53—22no—
49C22×C4×D53—22no—
50C22×C4×D53C2×C414yes—
51C22×C4×D53D411yescharacteristic
52C22×C4×D53D411yescharacteristic
53C22×C4×D53D414yes—
54C22×C4×D53C2311yescharacteristic
55C22×C4×D53C2311yescharacteristic
56C2×#11719917—22no—
57C2×#11719917—22no—
58C2×#11719917—22no—
59C22×#88322C2312yes—
60#13133193—28no—
61#13133193—28no—
62#13133193—24no—
63#13133193—24no—
64#13133193C2×C418yes—
65#13133193C2×C418yescomplemented
66#13133176—22no—
67#13133176D414yes—
68#13133175—24no—
69#13133175—24no—
70#13133174—28no—
71#13133173—28no—
72C2×C4⋊Dic53—22no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53C2×C414yescomplemented
75C2×C4⋊Dic53C2312yes—
76C2×C4⋊Dic53C2312yes—
77C2×#11719919—28no—
78C2×#11719919—22no—
79C2×#11719919—22no—
80C2×#11719919—22no—
81C4⋊C4×D53—24no—
82C4⋊C4×D53—24no—
83C2×C22⋊Dic53—21no—
84C2×#11719921—28no—
85C2×#11719921—22no—
86C2×#11719921—22no—
87C2×#11719921—22no—
88C22⋊C4×D53—28no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53—22no—
91C42×D53—28no—
92#13133141—28no—
93#13133129—24no—
94#13133129—24no—
95C22×Q8×C53—22no—
96C22×Q8×C53—21no—
97C22×Q8×C53—22no—
98C22×Q8×C53—22no—
99C22×Q8×C53—22no—
100C22×Q8×C53—22no—
101C22×Q8×C53C2×C414yescomplemented
102C22×Q8×C53C2312yes—
103C23×C4×C53—21no—
104C23×C4×C53C2311yescharacteristic
105C2×M16×C53—22no—
106C2×M16×C53—22no—
107C2×M16×C53—22no—
108C22×C8×C53C2312yes—
109#117×C53—22no—
110#117×C53D414yes—
111C2×C4⋊C4×C53C2312yes—
112C2×C22⋊C4×C53—21no—
113C2×C42×C53—21no—
114#103×C53—28no—
115C24×C53C2×D411yescharacteristic
116C23×D53—22no—
117C23×D53—22no—
118C23×D53—24no—
119C23×D53—42no—
120C23×D53C22⋊C414yes—
121C23×D53C2×D411yescharacteristic
122C23×D53C2×D411yescharacteristic
123C2×Q8×C53—24no—
124C2×Q8×C53—22no—
125C2×Q8×C53—28no—
126C2×Q8×C53—28no—
127C2×Q8×C53—22no—
128C2×Q8×C53—22no—
129C2×Q8×C53—22no—
130C2×Q8×C53—22no—
131C2×Q8×C53—42no—
132C2×Q8×C53—44no—
133C2×Q8×C53—44no—
134C2×Q8×C53—42no—
135C2×Q8×C53C22×C414yes—
136C22×C4×C53—24no—
137C22×C4×C53—22no—
138C22×C4×C53—22no—
139C22×C4×C53—42no—
140C22×C4×C53—42no—
141C22×C4×C53C22×C412yes—
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D412yes—
144C22×C4×C53C2×D412yes—
145C22×C4×C53C2×D412yes—
146C22×C4×C53#4112yes—
147C22×C4×C53#4112yes—
148C22×C4×C53C2411yescharacteristic
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—24no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—44no—
155C22×Dic53C22⋊C414yes—
156C22×Dic53C2×D411yescharacteristic
157C22×Dic53C2×D411yescharacteristic
158C2×C4×D53—28no—
159C2×C4×D53—28no—
160C2×C4×D53—22no—
161C2×C4×D53—22no—
162C2×C4×D53—28no—
163C2×C4×D53—28no—
164C2×C4×D53—22no—
165C2×C4×D53—22no—
166C2×C4×D53—21no—
167C2×C4×D53—21no—
168C2×C4×D53—21no—
169C2×C4×D53—21no—
170C2×C4×D53—21no—
171C2×C4×D53—21no—
172C2×C4×D53—21no—
173C2×C4×D53—21no—
174C2×C4×D53—28no—
175C2×C4×D53—44no—
176C2×C4×D53—44no—
177C2×C4×D53C22⋊C414yes—
178C2×C4×D53C22⋊C414yes—
179C2×C4×D53C22×C412yes—
180C2×C4×D53C22×C412yes—
181C2×C4×D53C2×D412yes—
182C2×C4×D53C2×D412yes—
183M16×C53—44no—
184C2×C8×C53—22no—
185C2×C8×C53—22no—
186C2×C8×C53—22no—
187C4⋊C4×C53—22no—
188C4⋊C4×C53C2×D414yes—
189C22⋊C4×C53—24no—
190C42×C53—24no—
191C22⋊Dic53—24no—
192#11719921—28no—
193#11719921—28no—
194#11719921—28no—
195C4⋊Dic53—24no—
196C4⋊Dic53—22no—
197C4⋊Dic53—22no—
198C4⋊Dic53C22×C414yes—
199C4⋊Dic53C2×D414yes—
200#11719919—28no—
201#11719919—28no—
202#11719919—28no—
203C4×Dic53—28no—
204C4×Dic53—24no—
205#11719917—44no—
206C2×#88322—22no—
207C2×#88322—22no—
208C2×#88322—22no—
209C23×C53—22no—
210C23×C53—22no—
211C23×C53C2×C22⋊C412yes—
212C23×C53#12012yes—
213C23×C53#12312yes—
214C23×C53C22×D411yescharacteristic
215C22×D53—21no—
216C22×D53—21no—
217C22×D53—22no—
218C22×D53—21no—
219C22×D53—21no—
220C22×D53—22no—
221C22×D53—22no—
222C22×D53—28no—
223C22×D53—28no—
224C22×D53—44no—
225C22×D53—48no—
226C22×D53—48no—
227C22×D53C2×C22⋊C412yes—
228C22×D53C2×C22⋊C412yes—
229C22×D53#13714yes—
230Q8×C53—24no—
231Q8×C53—24no—
232Q8×C53—42no—
233Q8×C53—42no—
234Q8×C53—48no—
235C2×C4×C53—24no—
236C2×C4×C53—24no—
237C2×C4×C53—42no—
238C2×C4×C53—42no—
239C2×C4×C53—42no—
240C2×C4×C53—42no—
241C2×C4×C53—48no—
242C2×C4×C53C2×C22⋊C412yes—
243C2×C4×C53C4×D414yes—
244C2×C4×C53C4×D414yes—
245C2×C4×C53#12012yes—
246C2×C4×C53#12114yes—
247C2×C4×C53#12114yes—
248C2×C4×C53#12312yes—
249C2×C4×C53C23×C411yescharacteristic
250C2×C4×C53C22×D411yescharacteristic
251C2×C4×C53C22×D411yescharacteristic
252C2×C4×C53C22×D411yescharacteristic
253C2×C4×C53C2×#4111yescharacteristic
254C2×C4×C53C2×#4111yescharacteristic
255C2×Dic53—21no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—28no—
260C2×Dic53—24no—
261C2×Dic53—24no—
262C2×Dic53—48no—
263C2×Dic53—42no—
264C2×Dic53—42no—
265C2×Dic53—44no—
266C2×Dic53C2×C22⋊C412yes—
267C2×Dic53C2×C22⋊C412yes—
268C4×D53—22no—
269C4×D53—22no—
270C4×D53—22no—
271C4×D53—22no—
272C4×D53—22no—
273C4×D53—22no—
274C4×D53—22no—
275C4×D53—22no—
276C4×D53—28no—
277C4×D53—28no—
278C4×D53—416no—
279C8×C53—42no—
280#88322—42no—
281#20911—531nomaximal, Sylow
282C22×C53—24no—
283C22×C53—22no—
284C22×C53—22no—
285C22×C53—24no—
286C22×C53#38614yes—
287C22×C53C22×C22⋊C411yescharacteristic
288C22×C53C2×#12011yescharacteristic
289C22×C53C2×#12311yescharacteristic
290C22×C53C2×#13712yes—
291C22×C53#57512yes—
292C2×D53—24no—
293C2×D53—22no—
294C2×D53—24no—
295C2×D53—22no—
296C2×D53—22no—
297C2×D53—22no—
298C2×D53—22no—
299C2×D53—416no—
300C2×D53—42no—
301C2×D53—42no—
302C2×D53#41914yescomplemented
303C4×C53—44no—
304C4×C53—44no—
305C4×C53#38614yes—
306C4×C53C2×C4×D411yescharacteristic
307C4×C53C2×C4×D411yescharacteristic
308C4×C53C2×#12111yescharacteristic
309C4×C53C2×#12111yescharacteristic
310Dic53—22no—
311Dic53—22no—
312Dic53—44no—
313Dic53—44no—
314C22×C22⋊Q8—531no—
315C22×#117—531no—
316C2×#419—531no—
317C2×#419D5311yescharacteristic, complemented
318C2×#407—531no—
319#1873—534no—
320#1872—534no—
321#1736—531no—
322#1736—531no—
323C2×C53—24no—
324C2×C53C2×#38611yescharacteristic
325C2×C53C2×#41911yescharacteristic
326C2×C53#289611yescharacteristic
327C2×C53#300712yes—
328C2×C53#309512yes—
329D53—24no—
330D53—42no—
331C23×Q8C2×D5311yescharacteristic
332C24×C4—531no—
333C22×M16—531no—
334C22×M16C2×D5311yescharacteristic
335C2×C22⋊Q8—538no—
336C2×C22⋊Q8—532no—
337C2×C22⋊Q8—532no—
338C2×C22⋊Q8—1061no—
339C2×C22⋊Q8—1064no—
340C2×C22⋊Q8—1061no—
341C2×#117—531no—
342C2×#117—532no—
343C2×#117—1061no—
344C2×#117—1061no—
345C2×#117—1064no—
346C2×#117—1064no—
347C2×#117C2×D5311yescharacteristic
348C22×C4⋊C4—531no—
349C22×C4⋊C4—532no—
350C22×C4⋊C4—1061no—
351C22×C22⋊C4—1061no—
352C22×C22⋊C4—1061no—
353C22×C42—1061no—
354#419—24no—
355#419—1064no—
356C2×#103—534no—
357C2×#103C2×D5314yescomplemented
358#407—1064no—
359C2×C22⋊C8—532no—
360C2×C22⋊C8—532no—
361#340—534no—
362#340—534no—
363C53#2091111yescharacteristic, complemented, Sylow
364C25—531no—
365C22×Q8—22no—
366C22×Q8—21no—
367C22×Q8—22no—
368C22×Q8—22no—
369C22×Q8—22no—
370C22×Q8—22no—
371C22×Q8C4×D5314yescomplemented
372C22×Q8C22×D5312yes—
373C23×C4—21no—
374C23×C4—531no—
375C23×C4—534no—
376C23×C4—531no—
377C23×C4—534no—
378C23×C4—531no—
379C23×C4—531no—
380C23×C4—531no—
381C23×C4—1062no—
382C23×C4—1062no—
383C23×C4—1062no—
384C23×C4—1062no—
385C23×C4—1062no—
386C23×C4—1061no—
387C23×C4C22×D5311yescharacteristic
388C2×M16—22no—
389C2×M16—22no—
390C2×M16—22no—
391C2×M16—1062no—
392C2×M16—1062no—
393C2×M16—1062no—
394C22×C8—532no—
395C22×C8C22×D5312yes—
396C22⋊Q8—538no—
397C22⋊Q8—538no—
398C22⋊Q8—1068no—
399C22⋊Q8—1068no—
400C22⋊Q8—1064no—
401C22⋊Q8—1064no—
402#117—22no—
403#117—534no—
404#117—1068no—
405#117—1068no—
406#117—1064no—
407#117—1062no—
408#117—1064no—
409#117C22⋊D5314yes—
410C2×C4⋊C4—534no—
411C2×C4⋊C4—532no—
412C2×C4⋊C4—532no—
413C2×C4⋊C4—1064no—
414C2×C4⋊C4—1062no—
415C2×C4⋊C4—1068no—
416C2×C4⋊C4—1062no—
417C2×C4⋊C4—1062no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4—1062no—
420C2×C4⋊C4—1064no—
421C2×C4⋊C4C22×D5312yes—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1068no—
424C2×C22⋊C4—1062no—
425C2×C22⋊C4—1068no—
426C2×C22⋊C4—1062no—
427C2×C22⋊C4—1062no—
428C2×C22⋊C4—1061no—
429C2×C42—21no—
430C2×C42—1068no—
431C2×C42—1062no—
432C2×C42—1061no—
433#103—28no—
434#103—1068no—
435C22⋊C8—1064no—
436C22⋊C8—1064no—
437C24—531no—
438C24—531no—
439C24—534no—
440C24—1064no—
441C24—1062no—
442C24—1062no—
443C24—2122no—
444C24D4×D5311yescharacteristic
445C2×Q8—24no—
446C2×Q8—22no—
447C2×Q8—28no—
448C2×Q8—28no—
449C2×Q8—22no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×Q8—42no—
454C2×Q8—44no—
455C2×Q8—44no—
456C2×Q8—42no—
457C2×Q8C2×C4×D5314yes—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—534no—
464C22×C4—531no—
465C22×C4—531no—
466C22×C4—534no—
467C22×C4—534no—
468C22×C4—532no—
469C22×C4—532no—
470C22×C4—532no—
471C22×C4—532no—
472C22×C4—1068no—
473C22×C4—1062no—
474C22×C4—1062no—
475C22×C4—1068no—
476C22×C4—1068no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1068no—
480C22×C4—1068no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1061no—
484C22×C4—1061no—
485C22×C4—1061no—
486C22×C4—1061no—
487C22×C4—1061no—
488C22×C4—1061no—
489C22×C4—1061no—
490C22×C4—1061no—
491C22×C4—1064no—
492C22×C4—1062no—
493C22×C4—1062no—
494C22×C4—2124no—
495C22×C4—2124no—
496C22×C4—2124no—
497C22×C4C2×C4×D5312yes—
498C22×C4#1171994512yes—
499C22×C4D4×D5311yescharacteristic
500C22×C4D4×D5312yes—
501C22×C4D4×D5312yes—
502C22×C4#1171994712yes—
503C22×C4C2×C22⋊D5312yes—
504C22×C4C23×D5311yescharacteristic
505M16—44no—
506M16—2124no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—1062no—
511C2×C8—1062no—
512C2×C8—1062no—
513C4⋊C4—22no—
514C4⋊C4—534no—
515C4⋊C4—534no—
516C4⋊C4—1068no—
517C4⋊C4—1068no—
518C4⋊C4—1068no—
519C4⋊C4—1064no—
520C4⋊C4—1062no—
521C4⋊C4—1062no—
522C4⋊C4C2×C22⋊D5314yes—
523C22⋊C4—24no—
524C22⋊C4—1068no—
525C22⋊C4—1068no—
526C22⋊C4—1068no—
527C22⋊C4—1064no—
528C42—24no—
529C42—1068no—
530C42—1064no—
531C23—22no—
532C23—22no—
533C23—532no—
534C23—534no—
535C23—532no—
536C23—1068no—
537C23—1061no—
538C23—1061no—
539C23—1062no—
540C23—1068no—
541C23—1061no—
542C23—1061no—
543C23—1062no—
544C23—1062no—
545C23—2124no—
546C23—2128no—
547C23—2128no—
548C23C22⋊C4×D5312yes—
549C23#1313318512yes—
550C23#1313319912yes—
551C23C2×D4×D5311yescharacteristic
552Q8—24no—
553Q8—24no—
554Q8—48no—
555Q8—42no—
556Q8—42no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—48no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—532no—
565C2×C4—532no—
566C2×C4—1068no—
567C2×C4—1061no—
568C2×C4—1061no—
569C2×C4—1061no—
570C2×C4—1061no—
571C2×C4—1068no—
572C2×C4—1068no—
573C2×C4—1062no—
574C2×C4—1062no—
575C2×C4—1062no—
576C2×C4—1062no—
577C2×C4—1062no—
578C2×C4—1062no—
579C2×C4—1062no—
580C2×C4—1062no—
581C2×C4—1064no—
582C2×C4—1064no—
583C2×C4—21216no—
584C2×C4—2128no—
585C2×C4—2122no—
586C2×C4—2122no—
587C2×C4—2124no—
588C2×C4C22⋊C4×D5312yes—
589C2×C4#1313317714yes—
590C2×C4C4×C22⋊D5314yes—
591C2×C4#1313318512yes—
592C2×C4#1313318814yes—
593C2×C4#1313319014yes—
594C2×C4#1313319912yes—
595C2×C4C22×C4×D5311yescharacteristic
596C2×C4C2×D4×D5311yescharacteristic
597C2×C4C2×D4×D5311yescharacteristic
598C2×C4C22×C22⋊D5311yescharacteristic
599C2×C4C2×#1171994711yescharacteristic
600C2×C4C2×#1171994511yescharacteristic
601C8—42no—
602C8—2122no—
603C22—24no—
604C22—22no—
605C22—22no—
606C22—24no—
607C22—534no—
608C22—1064no—
609C22—1062no—
610C22—1062no—
611C22—1064no—
612C22—1062no—
613C22—1062no—
614C22—1062no—
615C22—21216no—
616C22—2122no—
617C22—2122no—
618C22#1681049914yes—
619C22C2×C22⋊C4×D5311yescharacteristic
620C22C2×#1313318511yescharacteristic
621C22C2×#1313319911yescharacteristic
622C22#137×D5312yes—
623C22#1681142212yes—
624C4—44no—
625C4—44no—
626C4—1062no—
627C4—1062no—
628C4—2124no—
629C4—2124no—
630C4#1681049914yes—
631C4C2×#1313317711yescharacteristic
632C4C2×C4×C22⋊D5311yescharacteristic
633C4C2×#1313318811yescharacteristic
634C4C2×#1313319011yescharacteristic
635C2—24no—
636C2—1064no—
637C2—2122no—
638C2C2×#1681049911yescharacteristic
639C2#419×D5311yescharacteristic
640C2#2028323311yescharacteristic
641C2#2028397112yes—
642C2#2028466112yes—
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#419C2
socleC23×C53C22×D4
prime radicalC24×D53D4
cosocleC22×C4×C53C24
hypercenterC2×#419D53
hypocenterC53#20911
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
3. C2×#16810632C2semidirect
5. #20275024C2semidirect
6. #20275019C2semidirect
7. #20274436C2semidirect
8. #20274435C2semidirect
9. C2×C53×#419C2semidirect
15. C2×#13133193C4semidirect
31. C2×#13133141C22semidirect
35. #16810386C22semidirect
36. #16810385C22semidirect
41. C2×#103×C53C22semidirect
65. #13133193C2×C4semidirect
74. C2×C4⋊Dic53C2×C4semidirect
101. C22×Q8×C53C2×C4semidirect
302. C2×D53#419semidirect
317. C2×#419D53semidirect
357. C2×#103C2×D53semidirect
363. C53#20911semidirect
371. C22×Q8C4×D53semidirect

Maximal subgroups

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

subgroupindex
C22×#131331932
C2×#117×D532
C2×#168106322
C2×#168105862
#202750242
#202750192
#202744362
#202744352
C2×C53×#4192
#2091153