Finite Group Databasebeta

← #28318180

Subgroups — #28318180

Order 13568.

1444 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811609C211yesmaximal, characteristic
2C2×#16811602C211yesmaximal, characteristic
3#20291278C212yesmaximal, complemented
4#20290471C214yesmaximal, complemented
5#20290470C214yesmaximal, complemented
6#20290425C212yesmaximal, complemented
7C2×#16811389C211yesmaximal, characteristic
8C2×D53⋊M16C2212yes—
9C2×#13133296C2212yes—
10#16811609C412yescomplemented
11#16811609C412yescomplemented
12#16811609C2212yes—
13#16811609C2212yes—
14C2×#13133278C412yescomplemented
15C2×#13133277C412yescomplemented
16#16811602C412yes—
17#16811602C412yes—
18#16811602C2212yes—
19#16811602C2212yes—
20C2×#13133276C412yes—
21C2×#13133275C412yes—
22#16811589—24no—
23#16811586C2218yescomplemented
24M16×#6157—24no—
25#16811505C2218yescomplemented
26C22×#11719945C2211yescharacteristic
27#16811389—24no—
28#16811389—24no—
29C2×#13133225C412yescomplemented
30C2×#13133225C2212yes—
31C2×#13133223C412yescomplemented
32C2×#13133223C2212yes—
33C22×#11719917C411yescharacteristic, complemented
34C2×M16×D53C411yescharacteristic, complemented
35C2×M16×D53C411yescharacteristic, complemented
36C2×M16×D53C2211yescharacteristic
37C2×M16×D53C2211yescharacteristic
38C22×M16×C53C411yescharacteristic, complemented
39C2×#11719936—22no—
40D53⋊M16—22no—
41D53⋊M16—22no—
42D53⋊M16C2×C414yes—
43D53⋊M16C2314yes—
44C2×#11719940C2×C414yes—
45C2×D53⋊C8—22no—
46#13133296—22no—
47#13133296—22no—
48#13133296C2×C414yes—
49#13133296C2314yes—
50C2×C4⋊#6157—22no—
51C2×C22⋊#6157C2×C414yes—
52C2×C4×#6157—22no—
53#13133278C2×C414yescomplemented
54#13133278C2×C414yescomplemented
55#13133277C2×C414yescomplemented
56#13133277C2×C414yescomplemented
57#13133276C2×C414yes—
58#13133276C2×C414yes—
59#13133275C2×C414yes—
60#13133275C2×C414yes—
61#13133271—28no—
62C8×#6157—28no—
63C2×#11719945—22no—
64C2×#11719945—22no—
65C2×#11719945—22no—
66C2×#11719945—22no—
67C2×#11719945C2×C411yescharacteristic
68C2×#11719945C2×C411yescharacteristic
69C2×#11719945C2×C411yescharacteristic
70C2×#11719945C2×C411yescharacteristic
71C2×#11719945C2×C411yescharacteristic
72C2×#11719945C2×C411yescharacteristic
73C2×#11719945C2311yescharacteristic
74C2×#11719945C2311yescharacteristic
75C22×Dic106C2×C411yescharacteristic
76C22×D212C2×C411yescharacteristic
77C22×C22⋊D53C2×C411yescharacteristic
78C22×C22⋊D53C2×C411yescharacteristic
79C22×C4×D53C2×C411yescharacteristic
80C22×C4×D53C2311yescharacteristic
81#13133225—28no—
82#13133225—28no—
83#13133223—28no—
84#13133223—28no—
85C2×#11719917—22no—
86C2×#11719917—22no—
87C2×#11719917C2×C411yescharacteristic
88C2×#11719917C2×C411yescharacteristic
89C2×#11719917C2×C412yescomplemented
90C2×#11719912C2×C412yes—
91C2×#11719912C2×C412yescomplemented
92C2×#11719912C2×C412yescomplemented
93C2×#11719912C2312yes—
94M16×D53—22no—
95M16×D53—22no—
96M16×D53—22no—
97M16×D53—22no—
98M16×D53—22no—
99M16×D53—22no—
100M16×D53D414yes—
101M16×D53D414yes—
102C22×#88322C2×C412yescomplemented
103C2×C8×D53C2×C412yes—
104C2×C8×D53C2×C412yescomplemented
105C2×C8×D53C2×C412yescomplemented
106C2×C8×D53C2312yes—
107C23×C4×C53C2×C411yescharacteristic
108C2×M16×C53—22no—
109C2×M16×C53—22no—
110C2×M16×C53C2×C411yescharacteristic
111C2×M16×C53C2×C411yescharacteristic
112C2×M16×C53C2×C412yescomplemented
113C22×C8×C53C2×C412yescomplemented
114C24×C53C4211yescharacteristic
115C23×D53C4211yescharacteristic
116C23×D53C22×C411yescharacteristic
117C22×#6157—24no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53C4211yescharacteristic
123C22×C4×C53C4211yescharacteristic
124C22×C4×C53C4211yescharacteristic
125C22×C4×C53C22×C411yescharacteristic
126C22×C4×C53C22×C411yescharacteristic
127C22×C4×C53C22×C411yescharacteristic
128C2×C22⋊D53—22no—
129C2×C22⋊D53—22no—
130C2×C22⋊D53—22no—
131C2×C22⋊D53—22no—
132C2×C22⋊D53C4211yescharacteristic
133C2×C22⋊D53C4211yescharacteristic
134C2×C22⋊D53C4211yescharacteristic
135C2×C22⋊D53C4211yescharacteristic
136C2×C22⋊D53C22×C411yescharacteristic
137C2×C22⋊D53C22×C411yescharacteristic
138C2×C22⋊D53C22×C411yescharacteristic
139C2×C22⋊D53C22×C411yescharacteristic
140C22×Dic53C4211yescharacteristic
141C22×Dic53C22×C411yescharacteristic
142#11719945—22no—
143#11719945—22no—
144#11719945—22no—
145#11719945—22no—
146#11719945—22no—
147#11719945—22no—
148#11719945—22no—
149#11719945—22no—
150#11719945—22no—
151#11719945—22no—
152#11719945—22no—
153#11719945—22no—
154#11719945—22no—
155#11719945—22no—
156#11719945—22no—
157#11719945—22no—
158C2×D212—22no—
159C2×D212—22no—
160C2×D212C4211yescharacteristic
161C2×D212C4211yescharacteristic
162C2×D212C22×C411yescharacteristic
163C2×D212C22×C411yescharacteristic
164C2×C4×D53—21no—
165C2×C4×D53—21no—
166C2×C4×D53—21no—
167C2×C4×D53—21no—
168C2×C4×D53C4211yescharacteristic
169C2×C4×D53C4211yescharacteristic
170C2×C4×D53C22×C411yescharacteristic
171C2×C4×D53C22×C411yescharacteristic
172C2×C4×D53C22×C411yescharacteristic
173C2×C4×D53C22×C411yescharacteristic
174C2×C4×D53C22×C411yescharacteristic
175C2×C4×D53C2×D412yes—
176C2×C4×D53C2×D412yes—
177C2×C4×D53#4112yes—
178C2×C4×D53#4112yes—
179C2×C4×D53C2411yescharacteristic
180C2×Dic106—22no—
181C2×Dic106—22no—
182C2×Dic106C4211yescharacteristic
183C2×Dic106C4211yescharacteristic
184C2×Dic106C22×C411yescharacteristic
185C2×Dic106C22×C411yescharacteristic
186C22⋊#6157—24no—
187C22⋊#6157C22×C418yes—
188#11719940—24no—
189#11719940C22×C418yes—
190C2×#88324—24no—
191C4⋊#6157—24no—
192C4⋊#6157—24no—
193C4×#6157—24no—
194C4×#6157—24no—
195#11719936—24no—
196#11719936—24no—
197D53⋊C8—24no—
198D53⋊C8—24no—
199M16×C53—21no—
200M16×C53—21no—
201M16×C53—24no—
202M16×C53C22⋊C412yes—
203M16×C53C22⋊C412yes—
204C2×C8×C53—24no—
205C2×C8×C53C22×C412yes—
206C2×C8×C53C22×C412yes—
207#11719917—21no—
208#11719917—21no—
209#11719917—24no—
210#11719917C22⋊C412yes—
211#11719917C22⋊C412yes—
212C2×#88322—24no—
213C2×#88322C22×C412yes—
214C2×#88322C22×C412yes—
215#11719912—24no—
216#11719912—24no—
217#11719912—24no—
218#11719912C22⋊C418yes—
219C8×D53—24no—
220C8×D53—24no—
221C8×D53—24no—
222C8×D53C2×D418yes—
223C23×C53—21no—
224C23×C53—21no—
225C23×C53—81no—
226C23×C53C2×C4211yescharacteristic
227C23×C53C2×C4211yescharacteristic
228C23×C53C2×C4211yescharacteristic
229C22×D53—21no—
230C22×D53—21no—
231C22×D53—81no—
232C22×D53—81no—
233C22×D53C2×C4211yescharacteristic
234C22×D53C2×C4211yescharacteristic
235C22×D53C2×C4211yescharacteristic
236C22×D53C4×D412yes—
237C22×D53C4×D412yes—
238C22×D53C23×C411yescharacteristic
239C2×#6157—28no—
240C2×#6157—84no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—21no—
249C2×C4×C53—21no—
250C2×C4×C53—21no—
251C2×C4×C53C2×C4211yescharacteristic
252C2×C4×C53C2×C4211yescharacteristic
253C2×C4×C53C2×C4211yescharacteristic
254C2×C4×C53C2×C22⋊C411yescharacteristic
255C2×C4×C53C2×C22⋊C411yescharacteristic
256C2×C4×C53#11711yescharacteristic
257C2×C4×C53#11711yescharacteristic
258C2×C4×C53C23×C411yescharacteristic
259C22⋊D53—22no—
260C22⋊D53—22no—
261C22⋊D53—22no—
262C22⋊D53—22no—
263C22⋊D53—22no—
264C22⋊D53—22no—
265C22⋊D53—22no—
266C22⋊D53—22no—
267C2×Dic53—21no—
268C2×Dic53—21no—
269C2×Dic53C2×C4211yescharacteristic
270C2×Dic53C2×C4211yescharacteristic
271C2×Dic53C2×C4211yescharacteristic
272C2×Dic53C4×D412yes—
273C2×Dic53C4×D412yes—
274C2×Dic53C23×C411yescharacteristic
275D212—22no—
276D212—22no—
277D212—22no—
278D212—22no—
279C4×D53—21no—
280C4×D53—21no—
281C4×D53—21no—
282C4×D53—21no—
283C4×D53—21no—
284C4×D53—21no—
285C4×D53—21no—
286C4×D53—21no—
287C4×D53C2×C22⋊C412yes—
288C4×D53#11712yes—
289C4×D53C4×D412yes—
290C4×D53C4×D412yes—
291C4×D53C4×D412yes—
292C4×D53C4×D412yes—
293C4×D53C22×D412yes—
294C4×D53C2×#4112yes—
295Dic106—22no—
296Dic106—22no—
297Dic106—22no—
298Dic106—22no—
299#88324—28no—
300C8×C53—22no—
301C8×C53C2×C22⋊C414yes—
302#88322—22no—
303#88322C2×C22⋊C414yes—
304#20401—531nomaximal, Sylow
305C22×C53—21no—
306C22×C53—21no—
307C22×C53—41no—
308C22×C53—41no—
309C22×C53—41no—
310C22×C53—81no—
311C22×C53—81no—
312C22×C53C4×C22⋊C411yescharacteristic
313C22×C53C4×C22⋊C411yescharacteristic
314C22×C53C22×C4211yescharacteristic
315C2×D53—21no—
316C2×D53—21no—
317C2×D53—42no—
318C2×D53—41no—
319C2×D53—42no—
320C2×D53—41no—
321C2×D53—41no—
322C2×D53—41no—
323C2×D53—81no—
324C2×D53—81no—
325C2×D53C4×C22⋊C412yes—
326C2×D53C2×C4×D412yes—
327#6157—48no—
328C4×C53—21no—
329C4×C53—21no—
330C4×C53C4×C22⋊C411yescharacteristic
331C4×C53C4×C22⋊C411yescharacteristic
332C4×C53C22×C22⋊C411yescharacteristic
333C4×C53C2×#11711yescharacteristic
334Dic53—21no—
335Dic53—21no—
336Dic53C4×C22⋊C412yes—
337Dic53C2×C4×D412yes—
338C2×#568—531no—
339C2×#413—531no—
340C2×#410—531no—
341#2873—532no—
342#1757—534no—
343#1757—534no—
344C2×#344—532no—
345C2×C53—21no—
346C2×C53—21no—
347C2×C53—21no—
348C2×C53—41no—
349C2×C53—41no—
350C2×C53C2×C4×C22⋊C411yescharacteristic
351D53—22no—
352D53—22no—
353D53—41no—
354D53—41no—
355C22×#41—531no—
356#568—1064no—
357#568—1064no—
358C2×#131—532no—
359C2×#131—532no—
360C2×#131—532no—
361C2×#131—532no—
362C22×M16—531no—
363C22×M16—531no—
364C22×M16—531no—
365C22×M16—531no—
366C22×M16—531no—
367C22×M16—532no—
368C22×M16#615711yescharacteristic, complemented
369C2×#117—532no—
370#413—532no—
371#413—532no—
372#413—532no—
373#413—532no—
374C2×#101—532no—
375C2×#100—532no—
376#410—532no—
377#410—532no—
378#410—532no—
379#410—532no—
380C2×#99—532no—
381C2×#99—532no—
382#405—538no—
383#405—1064no—
384C4×M16—1064no—
385#344—538no—
386C53#2040111yescharacteristic, complemented, Sylow
387C2×#41—531no—
388C2×#41—531no—
389C2×#41—531no—
390C2×#41—531no—
391C2×#41—531no—
392C2×#41—531no—
393C2×#41—531no—
394C2×#41—531no—
395C2×#41—1062no—
396C2×#41—1062no—
397C2×#41—1062no—
398C2×#41—1062no—
399C22×Q8—531no—
400C22×D4—531no—
401C22×D4—531no—
402C22×D4—531no—
403C23×C4—531no—
404C23×C4—531no—
405C23×C4C2×#615711yescharacteristic
406#131—1068no—
407#131—1068no—
408#131—1068no—
409#131—1068no—
410C2×M16—22no—
411C2×M16—22no—
412C2×M16—531no—
413C2×M16—534no—
414C2×M16—534no—
415C2×M16—531no—
416C2×M16—532no—
417C2×M16—532no—
418C2×M16—532no—
419C2×M16—532no—
420C2×M16—532no—
421C2×M16—534no—
422C2×M16—534no—
423C2×M16—534no—
424C2×M16—1062no—
425C2×M16—1062no—
426C2×M16—1062no—
427C2×M16—1062no—
428C2×M16—1062no—
429C2×M16—1062no—
430C2×M16—1062no—
431C2×M16—1062no—
432C2×M16—1062no—
433C2×M16—1062no—
434C2×M16—1062no—
435C2×M16C2×#615711yescharacteristic
436C2×M16C2×#615711yescharacteristic
437C2×M16C2×#615712yescomplemented
438C22×C8—532no—
439C22×C8—532no—
440C22×C8—532no—
441C22×C8—532no—
442C22×C8—532no—
443C22×C8—1062no—
444C22×C8C2×#615712yescomplemented
445#117—534no—
446#117—534no—
447#117—1062no—
448#117—1062no—
449C2×C4⋊C4—1062no—
450C2×C22⋊C4—534no—
451C2×C42—1062no—
452#101—534no—
453#101—534no—
454#100—534no—
455#100—534no—
456#99—534no—
457#99—534no—
458#99—534no—
459#99—534no—
460#97—1068no—
461C4×C8—1068no—
462C24—531no—
463C24—531no—
464C24C4×#615711yescharacteristic
465#41—1062no—
466#41—1062no—
467#41—1062no—
468#41—1062no—
469#41—1062no—
470#41—1062no—
471#41—1062no—
472#41—1062no—
473#41—1062no—
474#41—1062no—
475#41—1062no—
476#41—1062no—
477#41—1062no—
478#41—1062no—
479#41—1062no—
480#41—1062no—
481C2×Q8—531no—
482C2×Q8—531no—
483C2×Q8—531no—
484C2×Q8—531no—
485C2×Q8—1062no—
486C2×Q8—1062no—
487C2×D4—531no—
488C2×D4—531no—
489C2×D4—531no—
490C2×D4—531no—
491C2×D4—531no—
492C2×D4—531no—
493C2×D4—531no—
494C2×D4—531no—
495C2×D4—531no—
496C2×D4—531no—
497C2×D4—531no—
498C2×D4—531no—
499C2×D4—1062no—
500C2×D4—1062no—
501C2×D4—1062no—
502C2×D4—1062no—
503C2×D4—1062no—
504C2×D4—1062no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—531no—
510C22×C4—531no—
511C22×C4—531no—
512C22×C4—532no—
513C22×C4—532no—
514C22×C4—532no—
515C22×C4—532no—
516C22×C4—531no—
517C22×C4—531no—
518C22×C4—531no—
519C22×C4—531no—
520C22×C4—531no—
521C22×C4—531no—
522C22×C4—531no—
523C22×C4—1061no—
524C22×C4—1061no—
525C22×C4—1061no—
526C22×C4—1061no—
527C22×C4—1064no—
528C22×C4C4×#615711yescharacteristic
529C22×C4C4×#615711yescharacteristic
530C22×C4C4×#615711yescharacteristic
531C22×C4C22×#615711yescharacteristic
532C22×C4C22×#615711yescharacteristic
533C22×C4C22×#615711yescharacteristic
534M16—21no—
535M16—21no—
536M16—24no—
537M16—532no—
538M16—532no—
539M16—538no—
540M16—538no—
541M16—1064no—
542M16—1064no—
543M16—1064no—
544M16—1064no—
545M16—1064no—
546M16—1064no—
547M16—1061no—
548M16—1061no—
549M16—1064no—
550M16C22⋊#615712yes—
551M16C22⋊#615712yes—
552C2×C8—24no—
553C2×C8—532no—
554C2×C8—538no—
555C2×C8—532no—
556C2×C8—1064no—
557C2×C8—1064no—
558C2×C8—1064no—
559C2×C8—1064no—
560C2×C8—1064no—
561C2×C8—1064no—
562C2×C8—1064no—
563C2×C8C22×#615712yes—
564C2×C8C22×#615712yes—
565C4⋊C4—1064no—
566C4⋊C4—1064no—
567C22⋊C4—538no—
568C22⋊C4—1064no—
569C42—1064no—
570C42—1064no—
571C23—21no—
572C23—21no—
573C23—81no—
574C23—532no—
575C23—532no—
576C23—531no—
577C23—531no—
578C23—531no—
579C23—531no—
580C23—1061no—
581C23—1061no—
582C23—4241no—
583C23—4241no—
584C23C2×C4×#615711yescharacteristic
585C23C2×C4×#615711yescharacteristic
586C23C2×C4×#615711yescharacteristic
587Q8—1062no—
588Q8—1062no—
589Q8—1062no—
590Q8—1062no—
591D4—1062no—
592D4—1062no—
593D4—1062no—
594D4—1062no—
595D4—1062no—
596D4—1062no—
597D4—1062no—
598D4—1062no—
599D4—1062no—
600D4—1062no—
601D4—1062no—
602D4—1062no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—532no—
614C2×C4—532no—
615C2×C4—531no—
616C2×C4—531no—
617C2×C4—532no—
618C2×C4—532no—
619C2×C4—532no—
620C2×C4—532no—
621C2×C4—532no—
622C2×C4—532no—
623C2×C4—532no—
624C2×C4—532no—
625C2×C4—531no—
626C2×C4—531no—
627C2×C4—1061no—
628C2×C4—1061no—
629C2×C4—1061no—
630C2×C4—1061no—
631C2×C4—1061no—
632C2×C4—1061no—
633C2×C4—1061no—
634C2×C4—1061no—
635C2×C4—1061no—
636C2×C4—1061no—
637C2×C4—1068no—
638C2×C4—4244no—
639C2×C4C2×C4×#615711yescharacteristic
640C2×C4C2×C4×#615711yescharacteristic
641C2×C4C2×C4×#615711yescharacteristic
642C2×C4C2×C22⋊#615711yescharacteristic
643C2×C4C2×C22⋊#615711yescharacteristic
644C2×C4#1313329611yescharacteristic
645C2×C4#1313329611yescharacteristic
646C2×C4C23×#615711yescharacteristic
647C8—22no—
648C8—534no—
649C8—1068no—
650C8—1062no—
651C8C2×C22⋊#615714yes—
652C22—21no—
653C22—21no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22—81no—
658C22—81no—
659C22—532no—
660C22—532no—
661C22—1061no—
662C22—1061no—
663C22—4241no—
664C22—4241no—
665C22—2122no—
666C22—2121no—
667C22—2121no—
668C22—2122no—
669C22—2121no—
670C22—2121no—
671C22C4×C22⋊#615711yescharacteristic
672C22C4×C22⋊#615711yescharacteristic
673C22C22×C4×#615711yescharacteristic
674C4—21no—
675C4—21no—
676C4—532no—
677C4—532no—
678C4—1061no—
679C4—1061no—
680C4—2128no—
681C4C4×C22⋊#615711yescharacteristic
682C4C4×C22⋊#615711yescharacteristic
683C4C22×C22⋊#615711yescharacteristic
684C4C2×#1313329611yescharacteristic
685C2—21no—
686C2—21no—
687C2—21no—
688C2—41no—
689C2—41no—
690C2—1062no—
691C2—1062no—
692C2—2121no—
693C2—2121no—
694C2C2×C4×C22⋊#615711yescharacteristic
subgroupG/H
centerC4C22×C22⋊#6157
commutatorC22×C53C22×C42
FrattiniC2×C4C23×#6157
FittingC22×M16×C53C4
socleC2×C53C2×C4×C22⋊C4
prime radicalC22×#11719945C22
cosocleC2×C4×D53C24
hypercenterC22×M16#6157
hypocenterC53#20401
LCSG ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×M16 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
3. #20291278C2semidirect
4. #20290471C2semidirect
5. #20290470C2semidirect
6. #20290425C2semidirect
10. #16811609C4semidirect
11. #16811609C4semidirect
14. C2×#13133278C4semidirect
15. C2×#13133277C4semidirect
23. #16811586C22semidirect
25. #16811505C22semidirect
29. C2×#13133225C4semidirect
31. C2×#13133223C4semidirect
33. C22×#11719917C4semidirect
34. C2×M16×D53C4semidirect
35. C2×M16×D53C4semidirect
38. C22×M16×C53C4semidirect
53. #13133278C2×C4semidirect
54. #13133278C2×C4semidirect
55. #13133277C2×C4semidirect
56. #13133277C2×C4semidirect
89. C2×#11719917C2×C4semidirect
91. C2×#11719912C2×C4semidirect
92. C2×#11719912C2×C4semidirect
102. C22×#88322C2×C4semidirect
104. C2×C8×D53C2×C4semidirect
105. C2×C8×D53C2×C4semidirect
112. C2×M16×C53C2×C4semidirect
113. C22×C8×C53C2×C4semidirect
368. C22×M16#6157semidirect
386. C53#20401semidirect
437. C2×M16C2×#6157semidirect
444. C22×C8C2×#6157semidirect

Maximal subgroups

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

subgroupindex
C2×#168116092
C2×#168116022
#202912782
#202904712
#202904702
#202904252
C2×#168113892
#2040153