Finite Group Databasebeta

← #27442953

Subgroups — #27442953

Order 13568.

818 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811032C211yesmaximal, characteristic, complemented
2C2×#16810662C211yesmaximal, characteristic
3#20278430C211yesmaximal, characteristic
4#20278419C211yesmaximal, characteristic
5#20276195C211yesmaximal, characteristic, complemented
6#20276190C211yesmaximal, characteristic, complemented
7#20276053C211yesmaximal, characteristic, complemented
8#20276048C211yesmaximal, characteristic, complemented
9#20275446C211yesmaximal, characteristic, complemented
10#20274917C211yesmaximal, characteristic
11#20274909C211yesmaximal, characteristic
12#20274870C211yesmaximal, characteristic, complemented
13#20274749C211yesmaximal, characteristic
14#20274500C211yesmaximal, characteristic
15C53×#1943C211yesmaximal, characteristic, complemented
16C22×#11719912C2211yescharacteristic
17C2×#13133217C2211yescharacteristic
18C2×#13133209C2211yescharacteristic
19C2×#13133195—21no—
20C2×#13133175C2211yescharacteristic
21C2×C4⋊C4×D53C2211yescharacteristic
22#16811032—24no—
23#16810662—24no—
24C2×#13133151C2211yescharacteristic
25C2×#13133151C2211yescharacteristic
26C2×#13133150C2211yescharacteristic
27C2×#13133149C2211yescharacteristic
28C2×#13133149C2211yescharacteristic
29C2×#13133148C2211yescharacteristic
30C2×#13133148C2211yescharacteristic
31C2×#13133148C2211yescharacteristic
32C2×#13133129C2211yescharacteristic
33C2×C8×Dic53C2211yescharacteristic
34#16810558—21no—
35#16810557—21no—
36#16810547—21no—
37#16810543—21no—
38#16810519C2211yescharacteristic
39#16810519C2211yescharacteristic
40#16810518—21no—
41#16810505—21no—
42#16810497—21no—
43#16810492C2211yescharacteristic
44#16810492C2211yescharacteristic
45#16810480C2211yescharacteristic
46#16810480C2211yescharacteristic
47C4×#11719921C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810378C2211yescharacteristic
53#16810377C2211yescharacteristic
54C2×#107×C53C2211yescharacteristic
55C2×#106×C53C2211yescharacteristic
56C2×#97×C53C2211yescharacteristic
57C53×#389C2211yescharacteristic
58C53×#389C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C22×Dic106—21no—
62C22×C4×D53—21no—
63C22×C4×D53C2311yescharacteristic
64C2×#11719912—22no—
65C2×#11719912—22no—
66C2×#11719912Q814yescomplemented
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69#13133217—22no—
70#13133217—41no—
71#13133209—22no—
72#13133209—41no—
73#13133195—44no—
74#13133175—22no—
75#13133175—41no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×#11719919—21no—
87C2×#11719919—21no—
88C2×#11719919—21no—
89C2×#11719919—21no—
90C4⋊C4×D53—22no—
91C4⋊C4×D53—41no—
92C2×#11719921—21no—
93C2×#11719921—21no—
94C2×#11719921—21no—
95C2×#11719921D411yescharacteristic
96C2×#11719921D411yescharacteristic
97C2×C4×Dic53—21no—
98C2×C4×Dic53C2311yescharacteristic
99#13133151—22no—
100#13133151D414yes—
101#13133150—22no—
102#13133149—22no—
103#13133149—22no—
104#13133148—22no—
105#13133148—41no—
106#13133148—41no—
107#13133129Q814yescomplemented
108C8×Dic53—22no—
109#13133124—21no—
110#13133124—21no—
111#13133124—21no—
112#13133124—21no—
113#13133124D411yescharacteristic
114#13133124D411yescharacteristic
115C22×C8×C53C2311yescharacteristic
116C22×C8×C53C2311yescharacteristic
117C2×C4⋊C4×C53—21no—
118C2×C4⋊C4×C53—21no—
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C4⋊C4×C53C2311yescharacteristic
122C2×C4⋊C4×C53C2311yescharacteristic
123C2×C42×C53C2311yescharacteristic
124#107×C53—22no—
125#106×C53—22no—
126#97×C53—22no—
127#95×C53—21no—
128#95×C53—21no—
129C23×D53C2×D411yescharacteristic
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2411yescharacteristic
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53C2×D411yescharacteristic
142C22×Dic53C2×D411yescharacteristic
143C22×Dic53C2×D411yescharacteristic
144C2×C4×D53—21no—
145C2×C4×D53—21no—
146C2×C4×D53—21no—
147C2×C4×D53—21no—
148C2×C4×D53—42no—
149C2×C4×D53—42no—
150C2×C4×D53—81no—
151C2×C4×D53C2×Q812yes—
152C2×C4×D53C2×Q812yes—
153C2×Dic106—42no—
154C2×Dic106—42no—
155C2×Dic106—81no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53C2×D412yes—
159C2×C8×C53C2×Q812yes—
160C2×C8×C53C2×Q812yes—
161C2×C8×C53#4112yes—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—21no—
170C4⋊C4×C53—42no—
171C4⋊C4×C53—81no—
172C42×C53—21no—
173C42×C53—21no—
174#11719921—22no—
175#11719921—22no—
176#11719921—42no—
177#11719921—42no—
178#11719921—81no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—42no—
186C4⋊Dic53C2×D412yes—
187C4⋊Dic53C2×D412yes—
188#11719919—42no—
189#11719919—42no—
190#11719919—42no—
191#11719919—42no—
192#11719919—42no—
193#11719919—42no—
194#11719919—42no—
195#11719919—81no—
196#11719919—81no—
197C4×Dic53—21no—
198C4×Dic53—21no—
199C4×Dic53—42no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322C2×Q812yes—
205C2×#88322C2×Q812yes—
206#11719912—28no—
207C23×C53C22×D411yescharacteristic
208C22×D53—21no—
209C22×D53—21no—
210C22×D53—21no—
211C22×D53—21no—
212C22×D53—21no—
213C22×D53—21no—
214C22×D53C4⋊Q812yes—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53C22×D411yescharacteristic
234C2×C4×C53C22×Q811yescharacteristic
235C2×C4×C53C22×Q811yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53H(5,𝔽2)11yescharacteristic
238C2×C4×C532−1+411yescharacteristic
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—22no—
246C2×Dic53—22no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53C4⋊Q812yes—
258C2×Dic53AGL(1,ℤ/8)12yes—
259C2×Dic53#13712yes—
260C4×D53—22no—
261C4×D53—22no—
262C4×D53—22no—
263C4×D53—22no—
264C4×D53—84no—
265Dic106—84no—
266C8×C53—22no—
267C8×C53C22⋊Q814yescomplemented
268#88322—22no—
269#88322—22no—
270#26078—531nomaximal, Sylow
271C22×C53C2×C4⋊Q811yescharacteristic
272C22×C53D4211yescharacteristic
273C22×C53#54711yescharacteristic
274C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
275C22×C53C2×#13711yescharacteristic
276C22×C53#57711yescharacteristic
277C22×C53#57711yescharacteristic
278C2×D53—22no—
279C2×D53—22no—
280C2×D53—22no—
281C2×D53—41no—
282C2×D53—41no—
283C2×D53—41no—
284C2×D53—41no—
285C4×C53—41no—
286C4×C53—41no—
287C4×C53—81no—
288C4×C53—81no—
289C4×C53—81no—
290C4×C53—81no—
291C4×C53C2×C22⋊Q811yescharacteristic
292C4×C53#54411yescharacteristic
293C4×C53D4×Q811yescharacteristic
294C4×C53#55411yescharacteristic
295Dic53—22no—
296Dic53—22no—
297Dic53—22no—
298Dic53—42no—
299Dic53—42no—
300Dic53—81no—
301Dic53—81no—
302C2×#501—531no—
303C2×#428—531no—
304#2441—531no—
305#2440—531no—
306#2084—531no—
307#2083—531no—
308#2059—531no—
309#2058—531no—
310#1943D5311yescharacteristic, complemented
311#1942—531no—
312#1854—531no—
313#1853—531no—
314#1844—531no—
315#1809—531no—
316#1753—531no—
317C2×C53#244011yescharacteristic
318C2×C53#316311yescharacteristic
319C2×C53#316511yescharacteristic
320C2×C53#327211yescharacteristic
321C2×C53#327811yescharacteristic
322C2×C53#334411yescharacteristic
323C2×C53#334511yescharacteristic
324D53—42no—
325C22×M16—531no—
326C2×C4⋊Q8—531no—
327C2×#125—531no—
328C2×C22⋊Q8—1061no—
329C2×#117—531no—
330C22×C4⋊C4—531no—
331#501—1064no—
332#428—1064no—
333C2×#107—531no—
334C2×#107—531no—
335C2×#107—531no—
336C2×#107C2×D5311yescharacteristic
337C2×#106—531no—
338C2×#106—531no—
339C2×#106—531no—
340C2×#106—531no—
341C2×#106—531no—
342C2×#106C2×D5311yescharacteristic
343C2×C22⋊C8—531no—
344C2×#97C2×D5311yescharacteristic
345C2×C4×C8—531no—
346#399—1061no—
347#398—1061no—
348#396—1061no—
349#395—1061no—
350#389—531no—
351#389—531no—
352#389—1061no—
353#389C2×D5311yescharacteristic
354#389C2×D5311yescharacteristic
355#387—1061no—
356#385—1061no—
357#384—531no—
358#384—531no—
359#380—531no—
360#380—531no—
361C4×C22⋊C4—531no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340C2×D5311yescharacteristic
367#340C2×D5311yescharacteristic
368#336—531no—
369#336—531no—
370C53#2607811yescharacteristic, complemented, Sylow
371C22×Q8—1061no—
372C23×C4—531no—
373C23×C4—1061no—
374C2×M16—534no—
375C2×M16—1062no—
376C2×M16—1062no—
377C22×C8—531no—
378C22×C8—531no—
379C22×C8C22×D5311yescharacteristic
380C22×C8C22×D5311yescharacteristic
381C4⋊Q8—1062no—
382C4⋊Q8—2121no—
383#125—1062no—
384#125—2121no—
385C22⋊Q8—2124no—
386#117—1062no—
387#117—2121no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—1062no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4—1061no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4—2121no—
406C2×C4⋊C4C22×D5311yescharacteristic
407C2×C4⋊C4C22×D5311yescharacteristic
408C2×C4⋊C4C22×D5311yescharacteristic
409C2×C4⋊C4C22×D5311yescharacteristic
410C2×C22⋊C4—531no—
411C2×C22⋊C4—531no—
412C2×C22⋊C4—1061no—
413C2×C22⋊C4—1061no—
414C2×C22⋊C4—1061no—
415C2×C42—531no—
416C2×C42—1061no—
417C2×C42C22×D5311yescharacteristic
418#107—22no—
419#107—534no—
420#107—1062no—
421#107—1062no—
422#106—22no—
423#106—1062no—
424#106—1062no—
425#106—1062no—
426#106—2121no—
427#106—2121no—
428C22⋊C8—534no—
429#97—22no—
430C4×C8—1062no—
431#95—21no—
432#95—21no—
433#95—531no—
434#95—531no—
435#95—1061no—
436#95—1061no—
437#95—1061no—
438#95—1061no—
439C24—531no—
440C2×Q8—4241no—
441C2×Q8—2122no—
442C2×Q8—2122no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—531no—
448C22×C4—531no—
449C22×C4—531no—
450C22×C4—532no—
451C22×C4—532no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—4241no—
461C22×C4—2122no—
462C22×C4—2122no—
463C22×C4D4×D5311yescharacteristic
464C22×C4D4×D5311yescharacteristic
465C22×C4C23×D5311yescharacteristic
466M16—1068no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—532no—
470C2×C8—532no—
471C2×C8—1061no—
472C2×C8—1061no—
473C2×C8—1061no—
474C2×C8—1061no—
475C2×C8#1171994512yes—
476C2×C8D4×D5312yes—
477C2×C8Q8×D5312yes—
478C2×C8Q8×D5312yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—42no—
488C4⋊C4—81no—
489C4⋊C4—532no—
490C4⋊C4—532no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—4241no—
498C4⋊C4—4241no—
499C4⋊C4—2122no—
500C4⋊C4—2122no—
501C4⋊C4—2122no—
502C4⋊C4—2122no—
503C4⋊C4—2122no—
504C4⋊C4—2122no—
505C4⋊C4—2122no—
506C4⋊C4—2122no—
507C22⋊C4—1062no—
508C22⋊C4—1062no—
509C22⋊C4—4241no—
510C22⋊C4—2122no—
511C22⋊C4—2122no—
512C42—21no—
513C42—21no—
514C42—1061no—
515C42—1061no—
516C42—2122no—
517C23—532no—
518C23—1061no—
519C23—1061no—
520C23—1061no—
521C23—1061no—
522C23—1061no—
523C23—1061no—
524C23C2×D4×D5311yescharacteristic
525Q8—4244no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—532no—
545C2×C4—532no—
546C2×C4—532no—
547C2×C4—1061no—
548C2×C4—1061no—
549C2×C4—1061no—
550C2×C4—1061no—
551C2×C4—1061no—
552C2×C4—1061no—
553C2×C4—1062no—
554C2×C4—1062no—
555C2×C4—1062no—
556C2×C4—1062no—
557C2×C4—1062no—
558C2×C4—1062no—
559C2×C4—4244no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4C2×D4×D5311yescharacteristic
571C2×C4C2×Q8×D5311yescharacteristic
572C2×C4C2×Q8×D5311yescharacteristic
573C2×C4C2×#1171994511yescharacteristic
574C2×C4#1313326311yescharacteristic
575C2×C4#1313326411yescharacteristic
576C8—22no—
577C8—1062no—
578C8—1062no—
579C8#1313319514yescomplemented
580C22—1062no—
581C22—1062no—
582C22—1062no—
583C22—2121no—
584C22—2121no—
585C22—2121no—
586C22—2121no—
587C22C4⋊Q8×D5311yescharacteristic
588C22#1681125711yescharacteristic
589C22#1681127811yescharacteristic
590C22AGL(1,ℤ/8)×D5311yescharacteristic
591C22#137×D5311yescharacteristic
592C22#1681143211yescharacteristic
593C22#1681143311yescharacteristic
594C4—41no—
595C4—41no—
596C4—81no—
597C4—81no—
598C4—81no—
599C4—81no—
600C4—1062no—
601C4—1062no—
602C4—1062no—
603C4—4241no—
604C4—4241no—
605C4—2122no—
606C4—2122no—
607C4C2×#1313319511yescharacteristic
608C4#1681125311yescharacteristic
609C4#1681128611yescharacteristic
610C4#1681133311yescharacteristic
611C2—2122no—
612C2#2027841911yescharacteristic
613C2#2028524211yescharacteristic
614C2#2028526011yescharacteristic
615C2#2028631211yescharacteristic
616C2#2028637811yescharacteristic
617C2#2028717511yescharacteristic
618C2#2028719011yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1943C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#1943D53
hypocenterC53#26078
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1943

Complements

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

NH ≅ G/Ntype
1. C2×#16811032C2semidirect
5. #20276195C2semidirect
6. #20276190C2semidirect
7. #20276053C2semidirect
8. #20276048C2semidirect
9. #20275446C2semidirect
12. #20274870C2semidirect
15. C53×#1943C2semidirect
66. C2×#11719912Q8semidirect
107. #13133129Q8semidirect
267. C8×C53C22⋊Q8semidirect
310. #1943D53semidirect
370. C53#26078semidirect
579. C8#13133195semidirect

Maximal subgroups

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

subgroupindex
C2×#168110322
C2×#168106622
#202784302
#202784192
#202761952
#202761902
#202760532
#202760482
#202754462
#202749172
#202749092
#202748702
#202747492
#202745002
C53×#19432
#2607853