Finite Group Databasebeta

← #27410941

Subgroups — #27410941

Order 13568.

2170 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133248C211yesmaximal, characteristic, complemented
2C2×#16811134C211yesmaximal, characteristic
3C2×#16810622C211yesmaximal, characteristic, complemented
4C2×#16810632C211yesmaximal, characteristic, complemented
5C2×#16810587C211yesmaximal, characteristic, complemented
6#20275723C214yesmaximal, complemented
7#20275716C214yesmaximal, complemented
8#20274434C211yesmaximal, characteristic, complemented
9C22×C22⋊Q8×C53C211yesmaximal, characteristic, complemented
10C22×#11719945C2211yescharacteristic
11C2×#13133248—24no—
12C2×#13133248—24no—
13C2×#13133248—24no—
14C22×#11719925C2212yescomplemented
15C22×#11719924C2212yescomplemented
16C22×#11719917C2211yescharacteristic
17C22×#11719917C2211yescharacteristic
18#16811134—24no—
19#16811134—24no—
20C2×#13133200—21no—
21C2×#13133196C2212yes—
22C2×#13133191C2212yes—
23C2×#13133186—21no—
24C2×C4×C22⋊D53—22no—
25C2×#13133176C2211yescharacteristic
26#16810632—24no—
27#16810622—24no—
28C2×#13133141C2214yescomplemented
29C2×#13133140C2212yescomplemented
30C2×#13133137C2212yescomplemented
31#16810587—24no—
32C2×C22⋊#88322C2214yes—
33#16810385C2214yescomplemented
34C23×Q8×C53C2211yescharacteristic
35C24×C4×C53C2211yescharacteristic
36C2×C22⋊Q8×C53—24no—
37C2×C22⋊Q8×C53—22no—
38C2×C22⋊Q8×C53—24no—
39C2×C22⋊Q8×C53C2214yes—
40C22×C4⋊C4×C53—21no—
41C22×C4⋊C4×C53C2211yescharacteristic
42C22×C22⋊C4×C53—21no—
43C2×#11719945—22no—
44C2×#11719945—22no—
45C2×#11719945—22no—
46C2×#11719945—22no—
47C2×#11719945—22no—
48C2×#11719945D414yes—
49C22×Dic106C2311yescharacteristic
50C22×D212C2311yescharacteristic
51C22×C22⋊D53—22no—
52C22×C22⋊D53—21no—
53C22×C4×D53—21no—
54#13133248—232no—
55C2×#11719925—24no—
56C2×#11719925—24no—
57C2×#11719925—24no—
58C2×#11719924—24no—
59C2×#11719924—24no—
60C2×#11719924—24no—
61C2×#11719917—22no—
62C2×#11719917—22no—
63C2×#11719917—22no—
64C2×#11719917—22no—
65C2×#11719917—22no—
66C2×#11719917D414yescomplemented
67C22×#88322C2312yes—
68C22×#88322C2312yes—
69#13133200—28no—
70#13133196—24no—
71#13133196—24no—
72#13133191—24no—
73#13133191—24no—
74#13133186—28no—
75C4×C22⋊D53—216no—
76#13133176—22no—
77#13133176D414yes—
78C2×C4⋊Dic53C2312yes—
79C2×#11719919—22no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—22no—
82C2×#11719921—22no—
83C2×C4×Dic53—21no—
84#13133141—28no—
85#13133140—24no—
86#13133137—24no—
87C22⋊#88322—28no—
88C25×C53D411yescharacteristic
89C22×Q8×C53—22no—
90C22×Q8×C53—21no—
91C22×Q8×C53—22no—
92C22×Q8×C53—22no—
93C22×Q8×C53—22no—
94C22×Q8×C53—22no—
95C22×Q8×C53D414yescomplemented
96C22×Q8×C53C2312yes—
97C23×C4×C53—21no—
98C23×C4×C53—21no—
99C23×C4×C53—22no—
100C23×C4×C53—22no—
101C23×C4×C53—22no—
102C23×C4×C53—22no—
103C23×C4×C53—22no—
104C23×C4×C53—22no—
105C23×C4×C53D411yescharacteristic
106C23×C4×C53D412yes—
107C23×C4×C53C2312yes—
108C23×C4×C53C2311yescharacteristic
109C22⋊Q8×C53—216no—
110C22⋊Q8×C53—28no—
111C22⋊Q8×C53—28no—
112C2×C4⋊C4×C53—28no—
113C2×C4⋊C4×C53—24no—
114C2×C4⋊C4×C53—22no—
115C2×C4⋊C4×C53—21no—
116C2×C4⋊C4×C53D414yescomplemented
117C2×C4⋊C4×C53C2312yes—
118C2×C22⋊C4×C53—28no—
119C2×C22⋊C4×C53—24no—
120C24×C53—22no—
121C24×C53—24no—
122C24×C53—22no—
123C24×C53—22no—
124C24×C53—42no—
125C24×C53C2×D412yes—
126C24×C53C2×D411yescharacteristic
127C23×D53—21no—
128C2×Q8×C53—24no—
129C2×Q8×C53—22no—
130C2×Q8×C53—28no—
131C2×Q8×C53—28no—
132C2×Q8×C53—22no—
133C2×Q8×C53—22no—
134C2×Q8×C53—22no—
135C2×Q8×C53—22no—
136C2×Q8×C53—42no—
137C2×Q8×C53—42no—
138C2×Q8×C53—44no—
139C2×Q8×C53—44no—
140C2×Q8×C53C2×D414yes—
141C22×C4×C53—24no—
142C22×C4×C53—24no—
143C22×C4×C53—22no—
144C22×C4×C53—22no—
145C22×C4×C53—22no—
146C22×C4×C53—22no—
147C22×C4×C53—22no—
148C22×C4×C53—24no—
149C22×C4×C53—24no—
150C22×C4×C53—24no—
151C22×C4×C53—22no—
152C22×C4×C53—22no—
153C22×C4×C53—22no—
154C22×C4×C53—22no—
155C22×C4×C53—22no—
156C22×C4×C53—22no—
157C22×C4×C53—44no—
158C22×C4×C53—42no—
159C22×C4×C53—42no—
160C22×C4×C53—44no—
161C22×C4×C53—44no—
162C22×C4×C53—44no—
163C22×C4×C53—44no—
164C22×C4×C53C2×D412yes—
165C22×C4×C53C2×D411yescharacteristic
166C22×C4×C53C2×D412yes—
167C22×C4×C53C2×D412yes—
168C22×C4×C53C2×D412yes—
169C22×C4×C53C2×D412yes—
170C22×C4×C53C2×D412yes—
171C22×C4×C53#4112yes—
172C22×C4×C53C2411yescharacteristic
173C2×C22⋊D53—28no—
174C2×C22⋊D53—24no—
175C2×C22⋊D53—44no—
176C2×C22⋊D53—42no—
177C2×C22⋊D53—42no—
178C2×C22⋊D53—44no—
179C22×Dic53—22no—
180C22×Dic53—21no—
181#11719945—28no—
182#11719945—28no—
183#11719945—44no—
184#11719945—44no—
185C2×D212—21no—
186C2×D212—21no—
187C2×D212—21no—
188C2×D212—21no—
189C2×D212—21no—
190C2×D212C2×D412yes—
191C2×C4×D53—24no—
192C2×C4×D53—42no—
193C2×C4×D53—42no—
194C2×Dic106—21no—
195C2×Dic106—21no—
196C2×Dic106—21no—
197C2×Dic106—21no—
198C2×Dic106—21no—
199C2×Dic106C2×D412yes—
200C4⋊C4×C53—216no—
201C4⋊C4×C53—24no—
202C4⋊C4×C53—22no—
203C4⋊C4×C53C2×D414yes—
204C22⋊C4×C53—216no—
205C22⋊Dic53—28no—
206C22⋊Dic53—24no—
207#11719925—216no—
208#11719924—216no—
209#11719921—28no—
210C4⋊Dic53—22no—
211C4⋊Dic53C2×D414yes—
212#11719919—28no—
213C4×Dic53—24no—
214#11719917—28no—
215#11719917—44no—
216C2×#88322—22no—
217C2×#88322—22no—
218C2×#88322—22no—
219C2×#88322—22no—
220C2×#88322—22no—
221C2×#88322C2×D414yes—
222C23×C53—22no—
223C23×C53—22no—
224C23×C53—28no—
225C23×C53—22no—
226C23×C53—22no—
227C23×C53—22no—
228C23×C53—22no—
229C23×C53—22no—
230C23×C53—22no—
231C23×C53—48no—
232C23×C53—48no—
233C23×C53—44no—
234C23×C53—44no—
235C23×C53#12012yes—
236C23×C53#12012yes—
237C23×C53#12112yes—
238C23×C53#13714yescomplemented
239C23×C53C22×D411yescharacteristic
240C22×D53—22no—
241C22×D53—41no—
242C22×D53—41no—
243C22×D53—42no—
244C22×D53—42no—
245Q8×C53—24no—
246Q8×C53—24no—
247Q8×C53—42no—
248Q8×C53—42no—
249Q8×C53—48no—
250C2×C4×C53—22no—
251C2×C4×C53—22no—
252C2×C4×C53—24no—
253C2×C4×C53—24no—
254C2×C4×C53—22no—
255C2×C4×C53—48no—
256C2×C4×C53—42no—
257C2×C4×C53—42no—
258C2×C4×C53—44no—
259C2×C4×C53—44no—
260C2×C4×C53—48no—
261C2×C4×C53—42no—
262C2×C4×C53—42no—
263C2×C4×C53—42no—
264C2×C4×C53—42no—
265C2×C4×C53—48no—
266C2×C4×C53#12014yes—
267C2×C4×C53#12012yes—
268C2×C4×C53#12012yes—
269C2×C4×C53#12114yes—
270C2×C4×C53#12114yes—
271C2×C4×C53#12112yes—
272C2×C4×C53#12714yes—
273C2×C4×C53C22×D411yescharacteristic
274C2×C4×C53C22×D411yescharacteristic
275C2×C4×C53C22×D411yescharacteristic
276C2×C4×C53C22×D411yescharacteristic
277C2×C4×C53C22×D411yescharacteristic
278C2×C4×C53C2×#4111yescharacteristic
279C22⋊D53—416no—
280C22⋊D53—48no—
281C2×Dic53—22no—
282C2×Dic53—24no—
283C2×Dic53—41no—
284C2×Dic53—41no—
285C2×Dic53—42no—
286C2×Dic53—42no—
287D212—22no—
288D212—22no—
289D212—41no—
290D212—41no—
291C4×D53—48no—
292Dic106—22no—
293Dic106—22no—
294Dic106—41no—
295Dic106—41no—
296#88322—24no—
297#88322—42no—
298#23392—531nomaximal, Sylow
299C22×C53—22no—
300C22×C53—22no—
301C22×C53—24no—
302C22×C53—24no—
303C22×C53—24no—
304C22×C53—24no—
305C22×C53—24no—
306C22×C53—416no—
307C22×C53—42no—
308C22×C53—42no—
309C22×C53—42no—
310C22×C53—42no—
311C22×C53#39214yes—
312C22×C53C2×#12011yescharacteristic
313C22×C53C2×#12011yescharacteristic
314C22×C53C2×#12111yescharacteristic
315C22×C53C2×#13712yes—
316C22×C53#57512yes—
317C2×D53—42no—
318C2×D53—44no—
319C2×D53—81no—
320C2×D53—81no—
321C2×D53—82no—
322C4×C53—42no—
323C4×C53—44no—
324C4×C53—44no—
325C4×C53#39214yes—
326C4×C53C2×#12011yescharacteristic
327C4×C53C2×#12111yescharacteristic
328C4×C53C2×#12111yescharacteristic
329C4×C53C2×#12711yescharacteristic
330Dic53—42no—
331Dic53—44no—
332C22×#137—531no—
333C2×#525—531no—
334C22×C22⋊Q8D5311yescharacteristic, complemented
335C2×#417—531no—
336C2×#419—531no—
337C2×#407—531no—
338#2001—534no—
339#2000—534no—
340#1736—531no—
341C2×C53—24no—
342C2×C53—44no—
343C2×C53C2×#39211yescharacteristic
344C2×C53#300411yescharacteristic
345C2×C53#300511yescharacteristic
346C2×C53#300712yes—
347C2×C53#305612yes—
348D53—82no—
349C22×#41—531no—
350C23×Q8C2×D5311yescharacteristic
351C24×C4C2×D5311yescharacteristic
352C2×#137—1064no—
353C2×#137—1064no—
354C2×#137—1064no—
355C22×Q16—532no—
356C22×SD16—532no—
357C22×M16—531no—
358C22×M16—531no—
359#525—1064no—
360#525—1064no—
361C2×#123—1061no—
362C2×C22⋊Q8—24no—
363C2×C22⋊Q8—22no—
364C2×C22⋊Q8—24no—
365C2×C22⋊Q8—532no—
366C2×C22⋊Q8C2×D5314yes—
367C2×#121—532no—
368C2×#120—1061no—
369C2×C4×D4—1062no—
370C2×#117—531no—
371C22×C4⋊C4—21no—
372C22×C4⋊C4C2×D5311yescharacteristic
373C22×C22⋊C4—21no—
374#419—1064no—
375#417—1064no—
376C2×#103—534no—
377C2×#103—532no—
378C2×#102—532no—
379#407—1064no—
380C2×C22⋊C8—534no—
381#340—534no—
382C53#2339211yescharacteristic, complemented, Sylow
383C25C22⋊D5311yescharacteristic
384C2×#41—534no—
385C2×#41—1062no—
386C2×#41—1062no—
387C2×#41—1062no—
388C2×#41—1062no—
389C2×#41—1062no—
390C22×Q8—22no—
391C22×Q8—21no—
392C22×Q8—22no—
393C22×Q8—22no—
394C22×Q8—22no—
395C22×Q8—22no—
396C22×Q8—531no—
397C22×Q8C22⋊D5314yescomplemented
398C22×Q8C22×D5312yes—
399C22×D4—531no—
400C22×D4—1062no—
401C22×D4—1061no—
402C23×C4—21no—
403C23×C4—21no—
404C23×C4—22no—
405C23×C4—22no—
406C23×C4—22no—
407C23×C4—22no—
408C23×C4—22no—
409C23×C4—22no—
410C23×C4—1061no—
411C23×C4C22⋊D5311yescharacteristic
412C23×C4C22⋊D5312yes—
413C23×C4C22×D5312yes—
414C23×C4C22×D5311yescharacteristic
415#137—10632no—
416C2×Q16—1064no—
417C2×Q16—1064no—
418C2×Q16—1064no—
419C2×SD16—1064no—
420C2×SD16—1064no—
421C2×SD16—1064no—
422C2×M16—534no—
423C2×M16—1062no—
424C2×M16—1062no—
425C2×M16—1062no—
426C2×M16—1062no—
427C2×M16—1062no—
428C22×C8—532no—
429C22×C8—532no—
430#123—1068no—
431C22⋊Q8—216no—
432C22⋊Q8—28no—
433C22⋊Q8—28no—
434C22⋊Q8—1064no—
435C22⋊Q8—1064no—
436#121—1064no—
437#121—1064no—
438#120—1068no—
439C4×D4—10616no—
440#117—534no—
441#117—1062no—
442C2×C4⋊C4—28no—
443C2×C4⋊C4—24no—
444C2×C4⋊C4—21no—
445C2×C4⋊C4—22no—
446C2×C4⋊C4—532no—
447C2×C4⋊C4—1062no—
448C2×C4⋊C4C22⋊D5314yescomplemented
449C2×C4⋊C4C22×D5312yes—
450C2×C22⋊C4—28no—
451C2×C22⋊C4—24no—
452C2×C22⋊C4—1061no—
453C2×C22⋊C4—1062no—
454C2×C22⋊C4—1062no—
455C2×C42—1061no—
456#103—1068no—
457#103—1064no—
458#102—1064no—
459C22⋊C8—1068no—
460C24—22no—
461C24—24no—
462C24—22no—
463C24—22no—
464C24—42no—
465C24—1061no—
466C24C2×C22⋊D5312yes—
467C24C2×C22⋊D5311yescharacteristic
468#41—1068no—
469#41—1068no—
470#41—2124no—
471#41—2124no—
472C2×Q8—24no—
473C2×Q8—22no—
474C2×Q8—28no—
475C2×Q8—28no—
476C2×Q8—22no—
477C2×Q8—22no—
478C2×Q8—22no—
479C2×Q8—22no—
480C2×Q8—42no—
481C2×Q8—44no—
482C2×Q8—42no—
483C2×Q8—44no—
484C2×Q8—532no—
485C2×Q8—1061no—
486C2×Q8—1061no—
487C2×Q8—1061no—
488C2×Q8—1061no—
489C2×Q8—1061no—
490C2×Q8C2×C22⋊D5314yes—
491C2×D4—532no—
492C2×D4—1068no—
493C2×D4—1064no—
494C2×D4—1061no—
495C2×D4—1061no—
496C2×D4—1061no—
497C2×D4—1061no—
498C2×D4—1061no—
499C2×D4—2124no—
500C2×D4—2122no—
501C2×D4—2122no—
502C2×D4—2124no—
503C22×C4—22no—
504C22×C4—24no—
505C22×C4—24no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—24no—
511C22×C4—24no—
512C22×C4—24no—
513C22×C4—22no—
514C22×C4—22no—
515C22×C4—22no—
516C22×C4—22no—
517C22×C4—22no—
518C22×C4—22no—
519C22×C4—44no—
520C22×C4—42no—
521C22×C4—42no—
522C22×C4—44no—
523C22×C4—44no—
524C22×C4—44no—
525C22×C4—44no—
526C22×C4—1062no—
527C22×C4—1061no—
528C22×C4—1064no—
529C22×C4—2122no—
530C22×C4—2122no—
531C22×C4D4×D5312yes—
532C22×C4D4×D5312yes—
533C22×C4D4×D5312yes—
534C22×C4#1171994712yes—
535C22×C4C2×C22⋊D5312yes—
536C22×C4C2×C22⋊D5311yescharacteristic
537C22×C4C2×C22⋊D5312yes—
538C22×C4C2×C22⋊D5312yes—
539C22×C4C23×D5311yescharacteristic
540Q16—10616no—
541SD16—10616no—
542M16—1068no—
543M16—2124no—
544C2×C8—534no—
545C2×C8—1062no—
546C2×C8—1062no—
547C2×C8—1062no—
548C2×C8—1062no—
549C2×C8—1062no—
550C4⋊C4—216no—
551C4⋊C4—24no—
552C4⋊C4—22no—
553C4⋊C4—534no—
554C4⋊C4—1068no—
555C4⋊C4—1062no—
556C4⋊C4C2×C22⋊D5314yes—
557C22⋊C4—216no—
558C22⋊C4—1068no—
559C22⋊C4—1068no—
560C22⋊C4—1064no—
561C42—1064no—
562C23—22no—
563C23—22no—
564C23—28no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—48no—
572C23—48no—
573C23—44no—
574C23—44no—
575C23—1062no—
576C23—2121no—
577C23—2121no—
578C23—2122no—
579C23—2122no—
580C23#1313318512yes—
581C23#1313318612yes—
582C23#1313318912yes—
583C23#1313324814yescomplemented
584C23C22×C22⋊D5311yescharacteristic
585Q8—24no—
586Q8—24no—
587Q8—48no—
588Q8—42no—
589Q8—42no—
590Q8—1062no—
591Q8—1062no—
592Q8—2121no—
593Q8—2121no—
594D4—1062no—
595D4—1062no—
596D4—21216no—
597D4—2128no—
598D4—2121no—
599D4—2121no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—48no—
606C2×C4—44no—
607C2×C4—48no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—1064no—
617C2×C4—1062no—
618C2×C4—2121no—
619C2×C4—2121no—
620C2×C4—2122no—
621C2×C4—2122no—
622C2×C4—2128no—
623C2×C4#1313318512yes—
624C2×C4#1313318514yes—
625C2×C4#1313318612yes—
626C2×C4#1313318814yes—
627C2×C4#1313318912yes—
628C2×C4#1313318914yes—
629C2×C4#1313321314yes—
630C2×C4C2×D4×D5311yescharacteristic
631C2×C4C2×D4×D5311yescharacteristic
632C2×C4C2×D4×D5311yescharacteristic
633C2×C4C22×C22⋊D5311yescharacteristic
634C2×C4C22×C22⋊D5311yescharacteristic
635C2×C4C2×#1171994711yescharacteristic
636C8—1064no—
637C8—2122no—
638C22—22no—
639C22—22no—
640C22—24no—
641C22—24no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—416no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—4241no—
651C22—4241no—
652C22—4242no—
653C22—2124no—
654C22—2122no—
655C22#1681053114yes—
656C22C2×#1313318511yescharacteristic
657C22C2×#1313318611yescharacteristic
658C22C2×#1313318911yescharacteristic
659C22C2×#1313324812yes—
660C22#1681142112yes—
661C4—44no—
662C4—42no—
663C4—44no—
664C4—2124no—
665C4—2122no—
666C4#1681053114yes—
667C4C2×#1313318511yescharacteristic
668C4C2×#1313318811yescharacteristic
669C4C2×#1313318911yescharacteristic
670C4C2×#1313321311yescharacteristic
671C2—24no—
672C2—44no—
673C2—4242no—
674C2C2×#1681053111yescharacteristic
675C2#2028394911yescharacteristic
676C2#2028395711yescharacteristic
677C2#2028397312yes—
678C2#2028434112yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC22×C22⋊Q8×C53C2
socleC23×C53C22×D4
prime radicalC2×#16811134C2
hypercenterC22×C22⋊Q8D53
hypocenterC53#23392
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#13133248C2semidirect
3. C2×#16810622C2semidirect
4. C2×#16810632C2semidirect
5. C2×#16810587C2semidirect
6. #20275723C2semidirect
7. #20275716C2semidirect
8. #20274434C2semidirect
9. C22×C22⋊Q8×C53C2semidirect
14. C22×#11719925C22semidirect
15. C22×#11719924C22semidirect
28. C2×#13133141C22semidirect
29. C2×#13133140C22semidirect
30. C2×#13133137C22semidirect
33. #16810385C22semidirect
66. C2×#11719917D4semidirect
95. C22×Q8×C53D4semidirect
116. C2×C4⋊C4×C53D4semidirect
238. C23×C53#137semidirect
334. C22×C22⋊Q8D53semidirect
382. C53#23392semidirect
397. C22×Q8C22⋊D53semidirect
448. C2×C4⋊C4C22⋊D53semidirect
583. C23#13133248semidirect

Maximal subgroups

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

subgroupindex
C22×#131332482
C2×#168111342
C2×#168106222
C2×#168106322
C2×#168105872
#202757232
#202757162
#202744342
C22×C22⋊Q8×C532
#2339253