Finite Group Databasebeta

← #27351801

Subgroups — #27351801

Order 13568.

1070 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20288511C211yesmaximal, characteristic
2#20283737C211yesmaximal, characteristic
3#20277622C211yesmaximal, characteristic
4#20273288C211yesmaximal, characteristic, complemented
5#20273287C211yesmaximal, characteristic, complemented
6#20273285C212yesmaximal, complemented
7#20273277C212yesmaximal, complemented
8#20273212C212yesmaximal, complemented
9C2×#16810359C211yesmaximal, characteristic, complemented
10#328×D53C212yesmaximal, complemented
11C2×C53×#328C211yesmaximal, characteristic, complemented
12#16811331C412yescomplemented
13#16811328—22no—
14#16811328C412yescomplemented
15#16811327—22no—
16Q8×C22⋊D53C412yescomplemented
17#16811286C412yescomplemented
18#16811252—21no—
19#16811249—21no—
20#16811248—21no—
21#16811175C2212yes—
22C2×#13133196C2211yescharacteristic
23C22⋊Q8×D53C2212yes—
24#16811096—21no—
25C2×C4×C22⋊D53—21no—
26C2×C4×C22⋊D53C2211yescharacteristic
27C8×C22⋊D53—22no—
28#16810686—22no—
29#16810673C2212yes—
30C2×C22⋊#88322C2211yescharacteristic
31C22⋊C8×D53C2212yes—
32#16810520—21no—
33#16810497—21no—
34#16810496—21no—
35#16810487C2211yescharacteristic
36#16810483C2211yescharacteristic
37#16810482C2211yescharacteristic
38#16810474C2211yescharacteristic
39#95×D53C2211yescharacteristic
40#16810361C2212yes—
41#16810361C2212yes—
42#16810360C2212yes—
43#16810360C2212yes—
44#16810359C2212yes—
45#16810359C2212yes—
46C2×C22⋊Q8×C53C2211yescharacteristic
47C2×C22⋊C8×C53C2211yescharacteristic
48C2×#95×C53C2211yescharacteristic
49C53×#328C2212yes—
50C53×#328C2212yes—
51C22×Dic106C2×C411yescharacteristic, complemented
52C2×Q8×D53C2×C412yescomplemented
53C22×C22⋊D53D411yescharacteristic
54C22×C4×D53—21no—
55C22×C4×D53C2311yescharacteristic
56C22×#88322—21no—
57C2×C8×D53—22no—
58#13133216—22no—
59#13133216C2×C412yescomplemented
60#13133215—22no—
61#13133215C2×C412yescomplemented
62#13133207—22no—
63#13133207—22no—
64#13133207—22no—
65#13133197—22no—
66#13133197C2×C412yescomplemented
67#13133196—22no—
68#13133196—22no—
69#13133196—21no—
70#13133196—21no—
71#13133196C2×C412yescomplemented
72#13133196C2312yes—
73#13133195—22no—
74#13133195—22no—
75#13133195—22no—
76#13133194—22no—
77#13133194—22no—
78#13133194—22no—
79#13133194—22no—
80#13133193—22no—
81#13133193C2×C412yescomplemented
82Q8×Dic53C2×C412yescomplemented
83#13133182—22no—
84#13133182C2×C412yescomplemented
85C4×C22⋊D53—22no—
86C4×C22⋊D53—22no—
87C4×C22⋊D53—44no—
88C4×C22⋊D53C2×C412yes—
89C4×C22⋊D53C2×C412yes—
90C4×C22⋊D53D412yes—
91C4×C22⋊D53D412yes—
92#13133177—22no—
93#13133177—22no—
94#13133173—22no—
95C2×C4⋊Dic53—21no—
96C2×C4⋊Dic53C2×C411yescharacteristic, complemented
97C2×#11719919—21no—
98C2×#11719919—21no—
99C2×#11719919—21no—
100C2×#11719919—21no—
101C2×#11719919D411yescharacteristic
102C4⋊C4×D53—22no—
103C4⋊C4×D53C2×C412yescomplemented
104C2×C22⋊Dic53—21no—
105C2×C22⋊Dic53—21no—
106C2×C22⋊Dic53D411yescharacteristic
107C2×#11719921—21no—
108C2×#11719921D411yescharacteristic
109C22⋊C4×D53—22no—
110C2×C4×Dic53—21no—
111C2×C4×Dic53C2311yescharacteristic
112#13133146—22no—
113C22⋊#88322—21no—
114C22⋊#88322C2312yes—
115#13133129—22no—
116C8×Dic53—22no—
117#13133124—21no—
118#13133124—21no—
119#13133124C2311yescharacteristic
120#13133124C2311yescharacteristic
121C22×Q8×C53C2×C411yescharacteristic, complemented
122C23×C4×C53—21no—
123C23×C4×C53C2311yescharacteristic
124C22×C8×C53—21no—
125C22⋊Q8×C53—21no—
126C22⋊Q8×C53—21no—
127C22⋊Q8×C53C2×C412yescomplemented
128C22⋊Q8×C53C2312yes—
129C2×C4⋊C4×C53—21no—
130C2×C4⋊C4×C53C2×C411yescharacteristic, complemented
131C2×C22⋊C4×C53—21no—
132C22⋊C8×C53—21no—
133C22⋊C8×C53C2312yes—
134#95×C53—21no—
135#95×C53C2311yescharacteristic
136#95×C53C2311yescharacteristic
137C24×C53C2×D411yescharacteristic
138C23×D53C2×D411yescharacteristic
139C2×Q8×C53—21no—
140C2×Q8×C53—81no—
141C2×Q8×C53C22×C412yes—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—42no—
147C22×C4×C53—42no—
148C22×C4×C53—81no—
149C22×C4×C53—81no—
150C22×C4×C53C22×C411yescharacteristic
151C22×C4×C53C22×C411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2411yescharacteristic
156C2×C22⋊D53C22⋊C412yes—
157C2×C22⋊D53C22⋊C412yes—
158C2×C22⋊D53SD1614yes—
159C2×C22⋊D53Q1614yes—
160C22×Dic53—21no—
161C22×Dic53—21no—
162C22×Dic53—21no—
163C22×Dic53—21no—
164C22×Dic53C2×D411yescharacteristic
165C22×Dic53C2×D411yescharacteristic
166C22×Dic53C2×D411yescharacteristic
167Q8×D53—82no—
168C2×C4×D53—22no—
169C2×C4×D53—22no—
170C2×C4×D53—42no—
171C2×C4×D53—42no—
172C2×C4×D53—81no—
173C2×C4×D53—81no—
174C2×C4×D53C22×C412yes—
175C2×C4×D53C2×D412yes—
176C2×Dic106—21no—
177C2×Dic106—22no—
178C2×Dic106—22no—
179C2×Dic106—81no—
180C2×Dic106C22×C412yes—
181C2×C8×C53—22no—
182C2×C8×C53—81no—
183C4⋊C4×C53—22no—
184C4⋊C4×C53—21no—
185C4⋊C4×C53—22no—
186C4⋊C4×C53C22×C412yes—
187C22⋊C4×C53—22no—
188C22⋊C4×C53—22no—
189C22⋊Dic53—22no—
190C22⋊Dic53—22no—
191C22⋊Dic53—42no—
192C22⋊Dic53C22⋊C412yes—
193C22⋊Dic53C22⋊C412yes—
194#11719921—22no—
195#11719921—22no—
196#11719921—42no—
197#11719921C22⋊C412yes—
198#11719921C22⋊C412yes—
199C4⋊Dic53—21no—
200C4⋊Dic53—22no—
201C4⋊Dic53—22no—
202C4⋊Dic53—81no—
203C4⋊Dic53C22×C412yes—
204#11719919—22no—
205#11719919—22no—
206#11719919—22no—
207#11719919—22no—
208#11719919—22no—
209#11719919—22no—
210#11719919—22no—
211#11719919—22no—
212#11719919—42no—
213#11719919—42no—
214#11719919—81no—
215#11719919C22⋊C412yes—
216#11719919C22⋊C412yes—
217C4×Dic53—22no—
218C4×Dic53—22no—
219C4×Dic53—42no—
220C4×Dic53C22×C412yes—
221C4×Dic53C2×D412yes—
222C2×#88322—22no—
223C2×#88322—81no—
224C8×D53—82no—
225C23×C53—41no—
226C23×C53—41no—
227C23×C53C2×C22⋊C411yescharacteristic
228C23×C53C2×C22⋊C411yescharacteristic
229C23×C53C2×SD1612yes—
230C23×C53C2×Q1612yes—
231C23×C53C22×D411yescharacteristic
232C22×D53—41no—
233C22×D53—41no—
234C22×D53C2×C22⋊C412yes—
235C22×D53C2×SD1612yes—
236C22×D53C2×Q1612yes—
237Q8×C53—42no—
238Q8×C53—81no—
239C2×C4×C53—22no—
240C2×C4×C53—22no—
241C2×C4×C53—22no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—81no—
254C2×C4×C53—81no—
255C2×C4×C53—84no—
256C2×C4×C53C2×C22⋊C411yescharacteristic
257C2×C4×C53C2×C22⋊C411yescharacteristic
258C2×C4×C53C23×C411yescharacteristic
259C2×C4×C53C22×D411yescharacteristic
260C22⋊D53#10318yes—
261C22⋊D53#10318yes—
262C2×Dic53—22no—
263C2×Dic53—22no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—81no—
273C2×Dic53—81no—
274C2×Dic53C2×C22⋊C412yes—
275C2×Dic53C2×C22⋊C412yes—
276C2×Dic53C2×C22⋊C412yes—
277C2×Dic53C2×SD1612yes—
278C2×Dic53C2×Q1612yes—
279C4×D53—42no—
280C4×D53—42no—
281C4×D53—82no—
282C4×D53—82no—
283C4×D53—84no—
284Dic106—42no—
285Dic106—82no—
286Dic106—82no—
287Dic106—81no—
288C8×C53—42no—
289#88322—42no—
290#17614—531nomaximal, Sylow
291C22×C53—21no—
292C22×C53—21no—
293C22×C53—21no—
294C22×C53—21no—
295C22×C53—21no—
296C22×C53—21no—
297C22×C53—42no—
298C22×C53—42no—
299C22×C53C2×#10312yes—
300C22×C53C2×#10312yes—
301C22×C53C22×C22⋊C411yescharacteristic
302C22×C53C22×SD1611yescharacteristic
303C22×C53C22×Q1611yescharacteristic
304C2×D53—22no—
305C2×D53—22no—
306C2×D53—42no—
307C2×D53—42no—
308C2×D53C2×#10314yes—
309C4×C53—21no—
310C4×C53—21no—
311C4×C53—21no—
312C4×C53—21no—
313C4×C53—42no—
314C4×C53—42no—
315C4×C53—42no—
316C4×C53—81no—
317Dic53—22no—
318Dic53—22no—
319Dic53—42no—
320Dic53—42no—
321Dic53—81no—
322Dic53—81no—
323Dic53C2×#10314yes—
324#3501—531no—
325#2972—531no—
326#2349—531no—
327#1520—531no—
328#1520—532no—
329#1520—531no—
330#1518—532no—
331#1503—532no—
332C2×#328—531no—
333C2×#328—532no—
334C2×#328D5311yescharacteristic, complemented
335C2×C53—21no—
336C2×C53—21no—
337C2×C53—22no—
338C2×C53—22no—
339C2×C53#288311yescharacteristic
340C2×C53C22×#10311yescharacteristic
341C2×C53#290111yescharacteristic
342D53—24no—
343#554—532no—
344#554—532no—
345#554—1062no—
346#554—1062no—
347D4×Q8—532no—
348D4×Q8—532no—
349#544—1061no—
350#543—1061no—
351#543—1061no—
352#533—532no—
353C2×C22⋊Q8—531no—
354C2×C22⋊Q8—532no—
355C2×C22⋊Q8C2×D5311yescharacteristic
356#518—1061no—
357C2×C4×D4—531no—
358C2×C4×D4—1061no—
359C8×D4—1062no—
360C8×D4—1062no—
361#431—532no—
362C2×C22⋊C8—531no—
363C2×C22⋊C8—532no—
364C2×C22⋊C8C2×D5311yescharacteristic
365#389—1061no—
366#385—1061no—
367#385—1061no—
368#382—531no—
369#380—531no—
370#380—531no—
371C4×C4⋊C4—531no—
372C2×#95—531no—
373C2×#95C2×D5311yescharacteristic
374#328—532no—
375#328—532no—
376#328—532no—
377#328—532no—
378#328—532no—
379#328—532no—
380#328C2×D5312yes—
381#328C2×D5312yes—
382C53#1761411yescharacteristic, complemented, Sylow
383C22×Q8—532no—
384C22×Q8—531no—
385C22×Q8C4×D5311yescharacteristic, complemented
386C22×D4—531no—
387C23×C4—21no—
388C23×C4—531no—
389C23×C4—1061no—
390C23×C4C22×D5311yescharacteristic
391C22×C8—21no—
392C22×C8—1062no—
393C22×C8—1061no—
394C4⋊Q8—532no—
395C4⋊Q8—532no—
396C4⋊Q8—1062no—
397C4⋊Q8—1062no—
398#125—1062no—
399#125—1062no—
400#125—1062no—
401C22⋊Q8—21no—
402C22⋊Q8—21no—
403C22⋊Q8—532no—
404C22⋊Q8—532no—
405C22⋊Q8—532no—
406C22⋊Q8—532no—
407C22⋊Q8—1062no—
408C22⋊Q8—1062no—
409C22⋊Q8—1062no—
410C22⋊Q8—1062no—
411C22⋊Q8—1062no—
412C22⋊Q8—1062no—
413C22⋊Q8—1062no—
414C22⋊Q8—1062no—
415C22⋊Q8—1062no—
416C22⋊Q8—1062no—
417C22⋊Q8—1062no—
418C22⋊Q8—1061no—
419C22⋊Q8—1061no—
420C22⋊Q8C4×D5312yescomplemented
421C22⋊Q8C22×D5312yes—
422C4×Q8—532no—
423C4×Q8—532no—
424C4×Q8—1062no—
425C4×D4—532no—
426C4×D4—532no—
427C4×D4—532no—
428C4×D4—532no—
429C4×D4—1062no—
430C4×D4—1062no—
431C4×D4—1062no—
432C4×D4—1062no—
433C4×D4—2124no—
434#117—1062no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—531no—
437C2×C4⋊C4—531no—
438C2×C4⋊C4—532no—
439C2×C4⋊C4—1062no—
440C2×C4⋊C4—1061no—
441C2×C4⋊C4—1061no—
442C2×C4⋊C4—1061no—
443C2×C4⋊C4—1061no—
444C2×C4⋊C4—1061no—
445C2×C4⋊C4C4×D5311yescharacteristic, complemented
446C2×C22⋊C4—21no—
447C2×C22⋊C4—531no—
448C2×C22⋊C4—531no—
449C2×C22⋊C4—1062no—
450C2×C22⋊C4—1061no—
451C2×C22⋊C4—1061no—
452C2×C22⋊C4—1061no—
453C2×C42—531no—
454C2×C42—1061no—
455C4⋊C8—1062no—
456C22⋊C8—21no—
457C22⋊C8—532no—
458C22⋊C8—1062no—
459C22⋊C8—1061no—
460C22⋊C8C22×D5312yes—
461C4×C8—1062no—
462#95—21no—
463#95—531no—
464#95—531no—
465#95—1061no—
466#95—1061no—
467#95C22×D5311yescharacteristic
468#95C22×D5311yescharacteristic
469C24—531no—
470C24D4×D5311yescharacteristic
471C2×Q8—21no—
472C2×Q8—81no—
473C2×Q8—532no—
474C2×Q8—1062no—
475C2×Q8—1062no—
476C2×Q8—1061no—
477C2×Q8—4241no—
478C2×Q8—4242no—
479C2×Q8C2×C4×D5312yes—
480C2×D4—532no—
481C2×D4—532no—
482C2×D4—534no—
483C2×D4—534no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—81no—
491C22×C4—81no—
492C22×C4—531no—
493C22×C4—531no—
494C22×C4—531no—
495C22×C4—532no—
496C22×C4—532no—
497C22×C4—1062no—
498C22×C4—1062no—
499C22×C4—1061no—
500C22×C4—1061no—
501C22×C4—1061no—
502C22×C4—1061no—
503C22×C4—4241no—
504C22×C4—4241no—
505C22×C4—2122no—
506C22×C4—2122no—
507C22×C4C2×C4×D5311yescharacteristic
508C22×C4C2×C4×D5311yescharacteristic
509C22×C4D4×D5311yescharacteristic
510C22×C4D4×D5311yescharacteristic
511C22×C4D4×D5311yescharacteristic
512C22×C4C23×D5311yescharacteristic
513C2×C8—22no—
514C2×C8—81no—
515C2×C8—1062no—
516C2×C8—4242no—
517C2×C8—4241no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—22no—
521C4⋊C4—532no—
522C4⋊C4—532no—
523C4⋊C4—532no—
524C4⋊C4—1062no—
525C4⋊C4—1062no—
526C4⋊C4—1062no—
527C4⋊C4—1062no—
528C4⋊C4—1062no—
529C4⋊C4—1062no—
530C4⋊C4—1062no—
531C4⋊C4—1062no—
532C4⋊C4—1061no—
533C4⋊C4—1062no—
534C4⋊C4—1062no—
535C4⋊C4—4241no—
536C4⋊C4—4241no—
537C4⋊C4—2122no—
538C4⋊C4—2122no—
539C4⋊C4C2×C4×D5312yes—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—532no—
543C22⋊C4—532no—
544C22⋊C4—532no—
545C22⋊C4—532no—
546C22⋊C4—1062no—
547C22⋊C4—1062no—
548C22⋊C4—1062no—
549C22⋊C4—1062no—
550C22⋊C4—2122no—
551C22⋊C4—2122no—
552C42—532no—
553C42—532no—
554C42—1062no—
555C42—1062no—
556C42—2122no—
557C23—41no—
558C23—41no—
559C23—532no—
560C23—532no—
561C23—532no—
562C23—2121no—
563C23—2121no—
564C23C22⋊C4×D5311yescharacteristic
565C23C22⋊C4×D5311yescharacteristic
566C23SD16×D5312yes—
567C23Q16×D5312yes—
568C23C2×D4×D5311yescharacteristic
569Q8—42no—
570Q8—81no—
571Q8—4241no—
572Q8—4242no—
573Q8—4242no—
574Q8—2122no—
575D4—538no—
576D4—538no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—84no—
594C2×C4—532no—
595C2×C4—532no—
596C2×C4—532no—
597C2×C4—532no—
598C2×C4—532no—
599C2×C4—1062no—
600C2×C4—1062no—
601C2×C4—4244no—
602C2×C4—4242no—
603C2×C4—4242no—
604C2×C4—4241no—
605C2×C4—4241no—
606C2×C4—2122no—
607C2×C4—2122no—
608C2×C4—2121no—
609C2×C4—2121no—
610C2×C4—2121no—
611C2×C4—2121no—
612C2×C4—2121no—
613C2×C4—2121no—
614C2×C4—2121no—
615C2×C4—2121no—
616C2×C4C22⋊C4×D5311yescharacteristic
617C2×C4C22⋊C4×D5311yescharacteristic
618C2×C4C22×C4×D5311yescharacteristic
619C2×C4C2×D4×D5311yescharacteristic
620C8—42no—
621C8—2122no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—42no—
629C22—42no—
630C22—534no—
631C22—1062no—
632C22—1062no—
633C22—2122no—
634C22—2122no—
635C22#103×D5312yes—
636C22#103×D5312yes—
637C22C2×C22⋊C4×D5311yescharacteristic
638C22C2×SD16×D5311yescharacteristic
639C22C2×Q16×D5311yescharacteristic
640C4—21no—
641C4—21no—
642C4—21no—
643C4—21no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—81no—
648C4—534no—
649C4—1062no—
650C4—1062no—
651C4—4241no—
652C4—4241no—
653C4—2122no—
654C4—2122no—
655C2—21no—
656C2—21no—
657C2—22no—
658C2—22no—
659C2—1064no—
660C2#2028317311yescharacteristic
661C2C2×#103×D5311yescharacteristic
662C2#2028325911yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#328C2
socleC22×C53C22×C22⋊C4
prime radicalC22×C22⋊D53D4
cosocleC22×C4×C53C24
hypercenterC2×#328D53
hypocenterC53#17614
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
4. #20273288C2semidirect
5. #20273287C2semidirect
6. #20273285C2semidirect
7. #20273277C2semidirect
8. #20273212C2semidirect
9. C2×#16810359C2semidirect
10. #328×D53C2semidirect
11. C2×C53×#328C2semidirect
12. #16811331C4semidirect
14. #16811328C4semidirect
16. Q8×C22⋊D53C4semidirect
17. #16811286C4semidirect
51. C22×Dic106C2×C4semidirect
52. C2×Q8×D53C2×C4semidirect
59. #13133216C2×C4semidirect
61. #13133215C2×C4semidirect
66. #13133197C2×C4semidirect
71. #13133196C2×C4semidirect
81. #13133193C2×C4semidirect
82. Q8×Dic53C2×C4semidirect
84. #13133182C2×C4semidirect
96. C2×C4⋊Dic53C2×C4semidirect
103. C4⋊C4×D53C2×C4semidirect
121. C22×Q8×C53C2×C4semidirect
127. C22⋊Q8×C53C2×C4semidirect
130. C2×C4⋊C4×C53C2×C4semidirect
334. C2×#328D53semidirect
382. C53#17614semidirect
385. C22×Q8C4×D53semidirect
420. C22⋊Q8C4×D53semidirect
445. C2×C4⋊C4C4×D53semidirect

Maximal subgroups

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

subgroupindex
#202885112
#202837372
#202776222
#202732882
#202732872
#202732852
#202732772
#202732122
C2×#168103592
#328×D532
C2×C53×#3282
#1761453