Finite Group Databasebeta

← #27351648

Subgroups — #27351648

Order 13568.

1046 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
1#20288747C211yesmaximal, characteristic
2#20277358C211yesmaximal, characteristic, complemented
3#20273284C212yesmaximal, complemented
4#20273272C212yesmaximal, complemented
5#20273259C212yesmaximal, complemented
6#20273245C212yesmaximal, complemented
7#20273241C211yesmaximal, characteristic, complemented
8#20273224C211yesmaximal, characteristic, complemented
9#20273211C211yesmaximal, characteristic, complemented
10#20273202C211yesmaximal, characteristic, complemented
11C53×#2974C211yesmaximal, characteristic, complemented
12#16811338C411yescharacteristic, complemented
13#16811330C411yescharacteristic, complemented
14#16811321—21no—
15#16811319C412yescomplemented
16#16811294C411yescharacteristic, complemented
17#16811269C412yescomplemented
18#16811263—22no—
19D4×D212C411yescharacteristic, complemented
20D4×C22⋊D53—21no—
21#16811224—21no—
22#16811206—22no—
23#16811173C2211yescharacteristic
24#16811157C2211yescharacteristic
25#16811134C2212yes—
26C2×#13133191C2212yes—
27#16811095—21no—
28#16810526—21no—
29#16810522—21no—
30#16810516—22no—
31#16810486C2212yescomplemented
32#16810483C2212yescomplemented
33C4×C4⋊Dic53C2211yescharacteristic, complemented
34#16810469C2211yescharacteristic, complemented
35#16810361C2211yescharacteristic, complemented
36#16810361C2211yescharacteristic, complemented
37#16810361C2212yescomplemented
38#16810358C2211yescharacteristic
39#16810358C2211yescharacteristic
40#16810358C2212yes—
41#16810358C2212yescomplemented
42#16810358C2212yescomplemented
43#16810358C2212yescomplemented
44#16810358C2212yescomplemented
45C2×C4×D4×C53C2211yescharacteristic
46C53×#436—21no—
47C53×#435—22no—
48C8×D4×C53—21no—
49C53×#432C2211yescharacteristic, complemented
50C53×#431C2211yescharacteristic, complemented
51C53×#407C2212yescomplemented
52C2×C22⋊C8×C53C2212yescomplemented
53C2×#11719945—21no—
54C2×#11719945C2×C412yes—
55C22×D212C2×C412yes—
56C22×C22⋊D53—22no—
57C2×D4×D53—21no—
58#13133217C2×C411yescharacteristic
59#13133214C2×C411yescharacteristic
60#13133213—21no—
61#13133212—22no—
62#13133211—21no—
63#13133209C2×C411yescharacteristic
64#13133208—21no—
65#13133205—21no—
66#13133205C2×C411yescharacteristic
67#13133203—22no—
68#13133202—22no—
69#13133200—22no—
70#13133200—21no—
71#13133199—22no—
72#13133196C2×C412yes—
73#13133196C2311yescharacteristic
74#13133194—22no—
75#13133192—21no—
76#13133191—22no—
77#13133191—21no—
78#13133191—22no—
79#13133191C2×C412yes—
80#13133191C2×C412yes—
81#13133191C2×C412yes—
82#13133191C2311yescharacteristic
83#13133191C2312yes—
84#13133190—22no—
85#13133190—22no—
86#13133189—22no—
87#13133188—21no—
88#13133187—22no—
89#13133186—22no—
90#13133185—22no—
91C4×Dic106C2×C411yescharacteristic
92C4×D212—21no—
93C4×D212C2×C412yes—
94C4×D212C2×C411yescharacteristic
95D4×Dic53—21no—
96#13133179—21no—
97C4×C22⋊D53—22no—
98C4×C22⋊D53—21no—
99C4×C22⋊D53—22no—
100#13133177—22no—
101#13133176C2×C412yes—
102#13133175—21no—
103#13133174—21no—
104C2×C4⋊Dic53—21no—
105C2×C4⋊Dic53—22no—
106C2×C4⋊Dic53C2×C412yes—
107C2×C22⋊Dic53—22no—
108C2×C22⋊Dic53—22no—
109C2×#11719921—22no—
110C22⋊C4×D53—22no—
111C2×C4×Dic53—21no—
112C2×C4×Dic53—21no—
113#13133124—21no—
114#13133124—22no—
115#13133124C2311yescharacteristic
116#13133124C2311yescharacteristic
117#13133124C2312yes—
118C22×D4×C53D411yescharacteristic
119C23×C4×C53C2312yes—
120C2×M16×C53—22no—
121C22×C8×C53—22no—
122C4×D4×C53C2×C412yes—
123C4×D4×C53C2×C411yescharacteristic
124C4×D4×C53C2×C411yescharacteristic
125C4×D4×C53D411yescharacteristic
126C4×D4×C53D412yes—
127C4×D4×C53D411yescharacteristic
128C2×C4⋊C4×C53D411yescharacteristic
129C2×C22⋊C4×C53D412yes—
130C2×C42×C53C2311yescharacteristic
131C4⋊C8×C53—21no—
132C4⋊C8×C53—21no—
133C22⋊C8×C53—22no—
134C22⋊C8×C53—22no—
135C22⋊C8×C53C2311yescharacteristic
136C22⋊C8×C53C2312yes—
137C22⋊C8×C53C2311yescharacteristic
138#97×C53—21no—
139C4×C8×C53—21no—
140C24×C53C2×D412yes—
141C23×D53—22no—
142C2×D4×C53—42no—
143C2×D4×C53C22⋊C411yescharacteristic
144C2×D4×C53C22⋊C411yescharacteristic
145C2×D4×C53C22⋊C412yes—
146C22×C4×C53—82no—
147C22×C4×C53C22×C412yes—
148C22×C4×C53C22×C412yes—
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D412yes—
152C22×C4×C53C2×D412yes—
153C22×C4×C53C2×D412yes—
154C22×C4×C53C2411yescharacteristic
155C2×C22⋊D53—22no—
156C2×C22⋊D53—22no—
157C2×C22⋊D53—21no—
158C2×C22⋊D53—22no—
159C2×C22⋊D53—22no—
160C2×C22⋊D53—22no—
161C2×C22⋊D53—82no—
162C2×C22⋊D53—82no—
163C22×Dic53—22no—
164C22×Dic53—21no—
165C22×Dic53—21no—
166C22×Dic53—22no—
167D4×D53—44no—
168#11719945—82no—
169#11719945—81no—
170#11719945—81no—
171C2×D212—21no—
172C2×D212—21no—
173C2×D212—22no—
174C2×D212—82no—
175C2×D212C22×C411yescharacteristic
176C2×D212C22×C412yes—
177C2×C4×D53—21no—
178C2×C4×D53—22no—
179C2×C4×D53—22no—
180C2×Dic106—21no—
181C2×Dic106C22×C411yescharacteristic
182M16×C53—82no—
183C2×C8×C53—21no—
184C2×C8×C53—21no—
185C2×C8×C53—22no—
186C2×C8×C53—82no—
187C4⋊C4×C53C22⋊C411yescharacteristic
188C4⋊C4×C53C22⋊C411yescharacteristic
189C4⋊C4×C53C22⋊C412yes—
190C22⋊C4×C53C22⋊C412yes—
191C22⋊C4×C53C22⋊C412yes—
192C22⋊C4×C53C22⋊C412yes—
193C22⋊C4×C53C22⋊C412yes—
194C42×C53C22×C411yescharacteristic
195C42×C53C22×C411yescharacteristic
196C42×C53C2×D411yescharacteristic
197C42×C53C2×D411yescharacteristic
198C22⋊Dic53—22no—
199C22⋊Dic53—22no—
200C22⋊Dic53—21no—
201C22⋊Dic53—82no—
202C22⋊Dic53—82no—
203#11719921—21no—
204#11719921—22no—
205#11719921—22no—
206#11719921—21no—
207#11719921—22no—
208#11719921—22no—
209#11719921—21no—
210#11719921—21no—
211#11719921—22no—
212C4⋊Dic53—22no—
213C4⋊Dic53—21no—
214C4⋊Dic53—42no—
215C4⋊Dic53—82no—
216C4⋊Dic53C22×C411yescharacteristic
217C4⋊Dic53C22×C412yes—
218C4⋊Dic53C22×C411yescharacteristic
219#11719919—21no—
220#11719919—21no—
221#11719919—22no—
222#11719919—21no—
223#11719919—21no—
224C4×Dic53—21no—
225C4×Dic53—22no—
226C4×Dic53—42no—
227C4×Dic53—81no—
228C23×C53—22no—
229C23×C53—22no—
230C23×C53—42no—
231C23×C53—42no—
232C23×C53C2×C22⋊C412yes—
233C23×C53C2×C22⋊C412yes—
234C23×C53C22×D411yescharacteristic
235C22×D53—22no—
236C22×D53—21no—
237C22×D53—44no—
238C22×D53—82no—
239C22×D53—82no—
240D4×C53—22no—
241D4×C53—22no—
242D4×C53—42no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—22no—
246C2×C4×C53—42no—
247C2×C4×C53—42no—
248C2×C4×C53—41no—
249C2×C4×C53—42no—
250C2×C4×C53—42no—
251C2×C4×C53—42no—
252C2×C4×C53C2×C22⋊C411yescharacteristic
253C2×C4×C53C2×C22⋊C411yescharacteristic
254C2×C4×C53C2×C22⋊C412yes—
255C2×C4×C53C2×C22⋊C412yes—
256C2×C4×C53C2×C22⋊C411yescharacteristic
257C2×C4×C53C2×C22⋊C411yescharacteristic
258C2×C4×C53C23×C411yescharacteristic
259C2×C4×C53C22×D411yescharacteristic
260C22⋊D53—44no—
261C22⋊D53—42no—
262C22⋊D53—41no—
263C22⋊D53—42no—
264C22⋊D53—41no—
265C22⋊D53—82no—
266C22⋊D53—82no—
267C22⋊D53—82no—
268C2×Dic53—22no—
269C2×Dic53—21no—
270C2×Dic53—21no—
271C2×Dic53—21no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—42no—
275C2×Dic53—41no—
276C2×Dic53—42no—
277C2×Dic53—41no—
278C2×Dic53—42no—
279C2×Dic53—41no—
280C2×Dic53—41no—
281C2×Dic53—82no—
282D212—41no—
283D212—42no—
284D212—42no—
285D212—82no—
286D212—81no—
287C4×D53—42no—
288C4×D53—82no—
289C4×D53—82no—
290Dic106—41no—
291Dic106—81no—
292C8×C53—41no—
293C8×C53—42no—
294C8×C53—41no—
295#17599—531nomaximal, Sylow
296C22×C53—21no—
297C22×C53—21no—
298C22×C53—22no—
299C22×C53—22no—
300C22×C53—22no—
301C22×C53—22no—
302C22×C53—22no—
303C22×C53—22no—
304C22×C53—82no—
305C22×C53—82no—
306C22×C53C22×C22⋊C411yescharacteristic
307C22×C53#57611yescharacteristic
308C22×C53#57711yescharacteristic
309C2×D53—42no—
310C2×D53—41no—
311C2×D53—42no—
312C2×D53—41no—
313C2×D53—41no—
314C2×D53—42no—
315C2×D53—164no—
316C4×C53—21no—
317C4×C53—21no—
318C4×C53—22no—
319C4×C53—22no—
320C4×C53—21no—
321C4×C53—21no—
322C4×C53—21no—
323C4×C53#42112yes—
324Dic53—41no—
325Dic53—42no—
326Dic53—41no—
327Dic53—41no—
328Dic53—81no—
329Dic53—82no—
330Dic53—81no—
331#3519—531no—
332#2974D5311yescharacteristic, complemented
333#2319—531no—
334#1519—532no—
335#1516—532no—
336#1513—532no—
337#1510—532no—
338#1509—531no—
339#1505—531no—
340#1502—531no—
341#1500—531no—
342C2×C53—21no—
343C2×C53—21no—
344C2×C53—42no—
345C2×C53—42no—
346C2×C53#288311yescharacteristic
347C2×C53#289711yescharacteristic
348C2×C53C2×#42111yescharacteristic
349D53—82no—
350D53—81no—
351#555—531no—
352#554—531no—
353#553—532no—
354#553—1061no—
355#550—531no—
356#546—532no—
357#546—1062no—
358D42—531no—
359D42—1061no—
360#540—1061no—
361#538—1062no—
362#532—531no—
363#529—531no—
364#525—532no—
365C2×#121—532no—
366#518—1061no—
367C2×C4×D4C2×D5311yescharacteristic
368#436—21no—
369#435—22no—
370C8×D4—21no—
371#432C2×D5311yescharacteristic, complemented
372#431C2×D5311yescharacteristic, complemented
373#407C2×D5312yescomplemented
374C2×C22⋊C8C2×D5312yescomplemented
375#390—1061no—
376#389—1061no—
377#388—1062no—
378#381—532no—
379#380—532no—
380C4×C4⋊C4—531no—
381#376—531no—
382#328—531no—
383#328—531no—
384#328—532no—
385#327—531no—
386#327—531no—
387#327—532no—
388#327—532no—
389#327—532no—
390#327—532no—
391#327—532no—
392C53#1759911yescharacteristic, complemented, Sylow
393C2×#41—532no—
394C2×#41—1061no—
395C22×D4—532no—
396C22×D4—1062no—
397C22×D4—1061no—
398C22×D4D21211yescharacteristic
399C23×C4C22×D5312yes—
400C2×M16—22no—
401C22×C8—22no—
402C4⋊Q8—531no—
403#127—531no—
404#127—1061no—
405#126—1062no—
406#126—1061no—
407#125—531no—
408#125—1061no—
409#124—531no—
410#124—1062no—
411#124—1061no—
412#123—1061no—
413#123—1062no—
414#123—1062no—
415#123—1062no—
416C22⋊Q8—532no—
417C22⋊Q8—531no—
418C22⋊Q8—1062no—
419#121—532no—
420#121—532no—
421#121—532no—
422#121—531no—
423#121—532no—
424#121—1062no—
425#121—1061no—
426#121—1062no—
427#121—1061no—
428#121—1062no—
429#121—1061no—
430#121—1062no—
431#121—1062no—
432#120—1062no—
433#120—1062no—
434#120—1062no—
435C4×Q8—531no—
436C4×D4—531no—
437C4×D4—532no—
438C4×D4—1061no—
439C4×D4—1062no—
440C4×D4—1061no—
441C4×D4—1062no—
442C4×D4—1061no—
443C4×D4—1062no—
444C4×D4—1061no—
445C4×D4C4×D5311yescharacteristic
446C4×D4C4×D5312yes—
447C4×D4C4×D5311yescharacteristic
448C4×D4C22⋊D5311yescharacteristic
449C4×D4C22⋊D5312yes—
450C4×D4C22⋊D5311yescharacteristic
451#117—532no—
452#117—1061no—
453#117—1061no—
454C2×C4⋊C4—532no—
455C2×C4⋊C4—1062no—
456C2×C4⋊C4—1061no—
457C2×C4⋊C4D21211yescharacteristic
458C2×C22⋊C4—1062no—
459C2×C22⋊C4—1062no—
460C2×C22⋊C4—1062no—
461C2×C22⋊C4—1062no—
462C2×C22⋊C4D21212yes—
463C2×C42—1061no—
464C2×C42—1061no—
465C2×C42C22×D5311yescharacteristic
466C4⋊C8—21no—
467C4⋊C8—21no—
468C22⋊C8—22no—
469C22⋊C8—22no—
470C22⋊C8C22×D5311yescharacteristic
471C22⋊C8C22×D5312yes—
472C22⋊C8C22×D5311yescharacteristic
473#97—21no—
474C4×C8—21no—
475#95—531no—
476#95—531no—
477#95—532no—
478#95—1061no—
479#95—1062no—
480C24—1062no—
481C24C2×D21212yes—
482#41—4241no—
483#41—4241no—
484#41—4242no—
485C2×Q8—531no—
486C2×Q8—1061no—
487C2×D4—42no—
488C2×D4—531no—
489C2×D4—532no—
490C2×D4—1062no—
491C2×D4—1062no—
492C2×D4—1062no—
493C2×D4—1062no—
494C2×D4—1062no—
495C2×D4—1061no—
496C2×D4—1061no—
497C2×D4—1062no—
498C2×D4—1061no—
499C2×D4—4242no—
500C2×D4—4242no—
501C2×D4—4242no—
502C2×D4—2124no—
503C2×D4#1171992111yescharacteristic
504C2×D4#1171992111yescharacteristic
505C2×D4#1171992112yes—
506C22×C4—82no—
507C22×C4—1062no—
508C22×C4—1062no—
509C22×C4—1062no—
510C22×C4—1061no—
511C22×C4—1061no—
512C22×C4—1061no—
513C22×C4—1062no—
514C22×C4C2×C4×D5312yes—
515C22×C4C2×C4×D5312yes—
516C22×C4C2×D21211yescharacteristic
517C22×C4C2×D21211yescharacteristic
518C22×C4C2×D21212yes—
519C22×C4C2×C22⋊D5312yes—
520C22×C4C2×C22⋊D5312yes—
521C22×C4C23×D5311yescharacteristic
522M16—82no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—22no—
526C2×C8—82no—
527C4⋊C4—531no—
528C4⋊C4—532no—
529C4⋊C4—531no—
530C4⋊C4—1062no—
531C4⋊C4—1061no—
532C4⋊C4—1061no—
533C4⋊C4—1061no—
534C4⋊C4—1062no—
535C4⋊C4—1061no—
536C4⋊C4—1061no—
537C4⋊C4—4242no—
538C4⋊C4—2122no—
539C4⋊C4#1171992111yescharacteristic
540C4⋊C4#1171992111yescharacteristic
541C4⋊C4#1171992112yes—
542C22⋊C4—1062no—
543C22⋊C4—1062no—
544C22⋊C4—1061no—
545C22⋊C4—1062no—
546C22⋊C4—1061no—
547C22⋊C4—1062no—
548C22⋊C4—1061no—
549C22⋊C4—1062no—
550C22⋊C4—1061no—
551C22⋊C4—1062no—
552C22⋊C4—1061no—
553C22⋊C4—1062no—
554C22⋊C4—4242no—
555C22⋊C4—4242no—
556C22⋊C4#1171992112yes—
557C22⋊C4#1171992112yes—
558C22⋊C4#1171992112yes—
559C22⋊C4#1171992112yes—
560C42—1062no—
561C42—1061no—
562C42—4241no—
563C42—2122no—
564C42C2×C4×D5311yescharacteristic
565C42C2×C4×D5311yescharacteristic
566C42C2×C22⋊D5311yescharacteristic
567C42C2×C22⋊D5311yescharacteristic
568C23—22no—
569C23—22no—
570C23—42no—
571C23—42no—
572C23—1061no—
573C23—1062no—
574C23—4242no—
575C23—4242no—
576C23—2124no—
577C23C2×#1171992112yes—
578C23C2×#1171992112yes—
579C23C22×D21211yescharacteristic
580Q8—4241no—
581Q8—2121no—
582D4—22no—
583D4—22no—
584D4—42no—
585D4—4242no—
586D4—4241no—
587D4—4242no—
588D4—4242no—
589D4—4242no—
590D4—2121no—
591D4—2122no—
592D4—2124no—
593D4—2121no—
594D4—2122no—
595D4—2121no—
596D4—2122no—
597D4—2122no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—1061no—
608C2×C4—1062no—
609C2×C4—1061no—
610C2×C4—1061no—
611C2×C4—4242no—
612C2×C4—4242no—
613C2×C4—4242no—
614C2×C4—2121no—
615C2×C4—2121no—
616C2×C4—2121no—
617C2×C4—2122no—
618C2×C4—2121no—
619C2×C4—2122no—
620C2×C4—2122no—
621C2×C4—2121no—
622C2×C4—2121no—
623C2×C4—2122no—
624C2×C4C2×#1171992112yes—
625C2×C4C2×#1171992111yescharacteristic
626C2×C4C2×#1171992111yescharacteristic
627C2×C4C2×#1171992111yescharacteristic
628C2×C4C2×#1171992112yes—
629C2×C4C2×#1171992111yescharacteristic
630C2×C4C22×C4×D5311yescharacteristic
631C2×C4C22×C22⋊D5311yescharacteristic
632C8—41no—
633C8—42no—
634C8—41no—
635C22—21no—
636C22—21no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—82no—
644C22—82no—
645C22—8484no—
646C22—2122no—
647C22—2121no—
648C22—2122no—
649C22—2121no—
650C22—2121no—
651C22—2122no—
652C22C22×#1171992111yescharacteristic
653C22#1681142811yescharacteristic
654C22#1681143511yescharacteristic
655C4—21no—
656C4—21no—
657C4—22no—
658C4—22no—
659C4—21no—
660C4—21no—
661C4—21no—
662C4—4241no—
663C4—4242no—
664C4—4241no—
665C4—2121no—
666C4—2121no—
667C4—2122no—
668C4—2121no—
669C4#1681063912yes—
670C2—21no—
671C2—21no—
672C2—42no—
673C2—42no—
674C2—4241no—
675C2—4242no—
676C2#2028317511yescharacteristic
677C2#2028324211yescharacteristic
678C2C2×#1681063911yescharacteristic
subgroupG/H
centerC22C22×#11719921
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#2974C2
socleC22×C53C22×C22⋊C4
prime radical#20288747C2
cosocleC22×C4×C53C24
hypercenter#2974D53
hypocenterC53#17599
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2974

Complements

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

NH ≅ G/Ntype
2. #20277358C2semidirect
3. #20273284C2semidirect
4. #20273272C2semidirect
5. #20273259C2semidirect
6. #20273245C2semidirect
7. #20273241C2semidirect
8. #20273224C2semidirect
9. #20273211C2semidirect
10. #20273202C2semidirect
11. C53×#2974C2semidirect
12. #16811338C4semidirect
13. #16811330C4semidirect
15. #16811319C4semidirect
16. #16811294C4semidirect
17. #16811269C4semidirect
19. D4×D212C4semidirect
31. #16810486C22semidirect
32. #16810483C22semidirect
33. C4×C4⋊Dic53C22semidirect
34. #16810469C22semidirect
35. #16810361C22semidirect
36. #16810361C22semidirect
37. #16810361C22semidirect
41. #16810358C22semidirect
42. #16810358C22semidirect
43. #16810358C22semidirect
44. #16810358C22semidirect
49. C53×#432C22semidirect
50. C53×#431C22semidirect
51. C53×#407C22semidirect
52. C2×C22⋊C8×C53C22semidirect
332. #2974D53semidirect
371. #432C2×D53semidirect
372. #431C2×D53semidirect
373. #407C2×D53semidirect
374. C2×C22⋊C8C2×D53semidirect
392. C53#17599semidirect

Maximal subgroups

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

subgroupindex
#202887472
#202773582
#202732842
#202732722
#202732592
#202732452
#202732412
#202732242
#202732112
#202732022
C53×#29742
#1759953