Finite Group Databasebeta

← #27441196

Subgroups — #27441196

Order 13568.

958 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810932C211yesmaximal, characteristic, complemented
2C2×#16810619C211yesmaximal, characteristic
3#20278429C211yesmaximal, characteristic, complemented
4#20277556C211yesmaximal, characteristic
5#20276246C211yesmaximal, characteristic, complemented
6#20276101C211yesmaximal, characteristic, complemented
7#20276085C211yesmaximal, characteristic, complemented
8#20275859C211yesmaximal, characteristic, complemented
9#20274985C211yesmaximal, characteristic, complemented
10#20274981C211yesmaximal, characteristic, complemented
11#20274916C211yesmaximal, characteristic, complemented
12#20274892C211yesmaximal, characteristic, complemented
13#20274738C211yesmaximal, characteristic, complemented
14#20274718C211yesmaximal, characteristic, complemented
15C53×#1788C211yesmaximal, characteristic
16C23×#88322C2211yescharacteristic
17C2×#13133208C2211yescharacteristic
18C2×Q8×Dic53C2211yescharacteristic
19C2×Q8×Dic53C2211yescharacteristic
20C2×#13133176C2211yescharacteristic
21#16810932—24no—
22C2×#13133151C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×#13133146C2211yescharacteristic
27C2×#13133146C2211yescharacteristic
28#16810619—24no—
29C2×#13133141C2211yescharacteristic
30C2×#13133141C2211yescharacteristic
31C2×#13133138C2211yescharacteristic
32C2×#13133138C2211yescharacteristic
33#16810559—21no—
34#16810546—21no—
35#16810526—21no—
36#16810526—21no—
37#16810526C2211yescharacteristic
38#16810526C2211yescharacteristic
39#16810522C2211yescharacteristic
40#16810520—21no—
41#16810519C2211yescharacteristic
42#16810510—21no—
43#16810492C2211yescharacteristic
44#16810483—21no—
45#16810483C2211yescharacteristic
46C4×C22⋊Dic53—21no—
47C4×C22⋊Dic53C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810377C2211yescharacteristic
53#16810377C2211yescharacteristic
54C22×#41×C53C2211yescharacteristic
55C2×#103×C53C2211yescharacteristic
56C2×#103×C53C2211yescharacteristic
57C2×#102×C53C2211yescharacteristic
58C2×#102×C53C2211yescharacteristic
59C2×C22⋊C8×C53C2211yescharacteristic
60C53×#380C2211yescharacteristic
61C22×#88322—22no—
62C22×#88322—22no—
63C22×#88322—41no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133208—22no—
67#13133208—41no—
68Q8×Dic53—22no—
69Q8×Dic53—22no—
70Q8×Dic53—41no—
71Q8×Dic53—41no—
72#13133176—22no—
73#13133176—41no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53—21no—
88C2×C22⋊Dic53—21no—
89C2×C22⋊Dic53—21no—
90C2×C22⋊Dic53—21no—
91C2×C22⋊Dic53—21no—
92C2×C22⋊Dic53D411yescharacteristic
93C2×C22⋊Dic53D411yescharacteristic
94C2×C4×Dic53—21no—
95C2×C4×Dic53—21no—
96C2×C4×Dic53—21no—
97C2×C4×Dic53C2311yescharacteristic
98C2×C4×Dic53C2311yescharacteristic
99#13133151—22no—
100#13133149—41no—
101#13133148—22no—
102#13133148—41no—
103#13133146—22no—
104#13133146—41no—
105#13133141—22no—
106#13133141D414yes—
107#13133138—22no—
108#13133138—22no—
109#13133124—21no—
110#13133124—21no—
111#13133124—21no—
112#13133124D411yescharacteristic
113#13133124D411yescharacteristic
114C2×#41×C53—22no—
115C2×#41×C53—22no—
116C2×#41×C53—22no—
117C2×#41×C53—41no—
118C2×#41×C53—41no—
119C2×#41×C53Q814yes—
120C22×Q8×C53C2311yescharacteristic
121C22×D4×C53—21no—
122C22×D4×C53C2311yescharacteristic
123C23×C4×C53—21no—
124C23×C4×C53C2311yescharacteristic
125C22×C8×C53C2311yescharacteristic
126C22×C8×C53C2311yescharacteristic
127C2×C4⋊C4×C53C2311yescharacteristic
128C2×C4⋊C4×C53C2311yescharacteristic
129C2×C22⋊C4×C53—21no—
130#103×C53—22no—
131#103×C53—22no—
132#102×C53—41no—
133#102×C53—41no—
134C22⋊C8×C53—22no—
135#95×C53—21no—
136C24×C53—21no—
137C24×C53C2×D411yescharacteristic
138#41×C53—28no—
139#41×C53—44no—
140#41×C53—44no—
141#41×C53—44no—
142C2×Q8×C53—21no—
143C2×Q8×C53—21no—
144C2×Q8×C53—21no—
145C2×Q8×C53C2×D412yes—
146C2×Q8×C53C2×Q812yes—
147C2×Q8×C53#4112yes—
148C2×D4×C53—24no—
149C2×D4×C53—21no—
150C2×D4×C53—21no—
151C2×D4×C53—21no—
152C2×D4×C53—21no—
153C2×D4×C53—21no—
154C2×D4×C53—42no—
155C2×D4×C53—42no—
156C2×D4×C53C2×Q812yes—
157C22×C4×C53—21no—
158C22×C4×C53—21no—
159C22×C4×C53—21no—
160C22×C4×C53—21no—
161C22×C4×C53—21no—
162C22×C4×C53—24no—
163C22×C4×C53—21no—
164C22×C4×C53—21no—
165C22×C4×C53—42no—
166C22×C4×C53—42no—
167C22×C4×C53C2×D411yescharacteristic
168C22×C4×C53C2×Q812yes—
169C22×C4×C53#4112yes—
170C22×C4×C53C2411yescharacteristic
171C22×Dic53—21no—
172C22×Dic53—21no—
173C22×Dic53—21no—
174C22×Dic53—21no—
175C22×Dic53C2×D411yescharacteristic
176C22×Dic53C2×D411yescharacteristic
177C22×Dic53C2×D411yescharacteristic
178C22×Dic53C2×D411yescharacteristic
179C2×C8×C53—21no—
180C2×C8×C53—21no—
181C2×C8×C53—21no—
182C2×C8×C53—21no—
183C2×C8×C53—21no—
184C2×C8×C53—21no—
185C4⋊C4×C53—21no—
186C4⋊C4×C53—21no—
187C4⋊C4×C53—21no—
188C4⋊C4×C53—21no—
189C22⋊C4×C53—81no—
190C22⋊Dic53—22no—
191C22⋊Dic53—41no—
192C22⋊Dic53—42no—
193C22⋊Dic53—42no—
194C22⋊Dic53—42no—
195C22⋊Dic53—81no—
196C22⋊Dic53—81no—
197C22⋊Dic53—81no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53—21no—
203C4⋊Dic53—21no—
204C4⋊Dic53—21no—
205C4⋊Dic53—42no—
206C4⋊Dic53—42no—
207C4⋊Dic53—42no—
208C4⋊Dic53—81no—
209C4⋊Dic53C2×D412yes—
210#11719919—42no—
211#11719919—42no—
212#11719919—42no—
213#11719919—81no—
214C4×Dic53—21no—
215C4×Dic53—21no—
216C4×Dic53—21no—
217C4×Dic53—21no—
218C4×Dic53—42no—
219C4×Dic53—42no—
220C4×Dic53—42no—
221C2×#88322—21no—
222C2×#88322—21no—
223C2×#88322—21no—
224C2×#88322—21no—
225C2×#88322—21no—
226C2×#88322—44no—
227C2×#88322C2×D412yes—
228C23×C53—21no—
229C23×C53—21no—
230C23×C53—21no—
231C23×C53—21no—
232C23×C53—21no—
233C23×C53—21no—
234C23×C53—22no—
235C23×C53—41no—
236C23×C53—41no—
237C23×C53—41no—
238C23×C53—41no—
239C23×C53—41no—
240C23×C53—41no—
241C23×C53C22⋊Q812yes—
242C23×C53C22×D411yescharacteristic
243Q8×C53—22no—
244Q8×C53—22no—
245Q8×C53—22no—
246Q8×C53C22⋊Q814yes—
247D4×C53—22no—
248D4×C53—22no—
249D4×C53—48no—
250D4×C53—41no—
251D4×C53—41no—
252C2×C4×C53—22no—
253C2×C4×C53—21no—
254C2×C4×C53—22no—
255C2×C4×C53—21no—
256C2×C4×C53—22no—
257C2×C4×C53—22no—
258C2×C4×C53—22no—
259C2×C4×C53—22no—
260C2×C4×C53—22no—
261C2×C4×C53—48no—
262C2×C4×C53—41no—
263C2×C4×C53—41no—
264C2×C4×C53—41no—
265C2×C4×C53—41no—
266C2×C4×C53—41no—
267C2×C4×C53—41no—
268C2×C4×C53C22⋊Q812yes—
269C2×C4×C53C22×D411yescharacteristic
270C2×C4×C53C22×Q811yescharacteristic
271C2×C4×C53C2×#4111yescharacteristic
272C2×C4×C53C2×#4111yescharacteristic
273C2×C4×C53H(5,𝔽2)11yescharacteristic
274C2×C4×C53H(5,𝔽2)11yescharacteristic
275C2×Dic53—21no—
276C2×Dic53—21no—
277C2×Dic53—22no—
278C2×Dic53—21no—
279C2×Dic53—21no—
280C2×Dic53—21no—
281C2×Dic53—21no—
282C2×Dic53—21no—
283C2×Dic53—21no—
284C2×Dic53—21no—
285C2×Dic53—21no—
286C2×Dic53—21no—
287C2×Dic53—41no—
288C2×Dic53—41no—
289C2×Dic53—41no—
290C2×Dic53—41no—
291C2×Dic53—41no—
292C2×Dic53—41no—
293C2×Dic53—41no—
294C2×Dic53—41no—
295C2×Dic53—41no—
296C2×Dic53—41no—
297C2×Dic53—41no—
298C2×Dic53#13512yes—
299C8×C53—41no—
300C8×C53—41no—
301#88322—22no—
302#88322—41no—
303#25953—531nomaximal, Sylow
304C22×C53—22no—
305C22×C53—22no—
306C22×C53—22no—
307C22×C53—41no—
308C22×C53—41no—
309C22×C53—41no—
310C22×C53—41no—
311C22×C53—42no—
312C22×C53—42no—
313C22×C53—42no—
314C22×C53—81no—
315C22×C53—81no—
316C22×C53—81no—
317C22×C53—81no—
318C22×C53C2×C22⋊Q811yescharacteristic
319C22×C53D4211yescharacteristic
320C22×C53#54611yescharacteristic
321C22×C53C2×#13511yescharacteristic
322C22×C53#57511yescharacteristic
323C22×C53#57711yescharacteristic
324C22×C53#57711yescharacteristic
325C4×C53—22no—
326C4×C53—24no—
327C4×C53—81no—
328C4×C53—81no—
329C4×C53C2×C22⋊Q811yescharacteristic
330C4×C53#54311yescharacteristic
331C4×C53#54611yescharacteristic
332C4×C53#55111yescharacteristic
333Dic53—22no—
334Dic53—41no—
335Dic53—41no—
336Dic53—42no—
337Dic53—41no—
338Dic53—81no—
339Dic53—81no—
340Dic53—81no—
341C2×#479—531no—
342C2×#416—531no—
343#2441—531no—
344#2341—531no—
345#2094—531no—
346#2067—531no—
347#2065—531no—
348#2025—531no—
349#1865—531no—
350#1865—531no—
351#1854—531no—
352#1850—531no—
353#1807—531no—
354#1803—531no—
355#1788D5311yescharacteristic
356C2×C53—42no—
357C2×C53—82no—
358C2×C53#243711yescharacteristic
359C2×C53#307611yescharacteristic
360C2×C53#308411yescharacteristic
361C2×C53#327111yescharacteristic
362C2×C53#328411yescharacteristic
363C2×C53#330111yescharacteristic
364C2×C53#331711yescharacteristic
365C22×#41C2×D5311yescharacteristic
366C23×C8—531no—
367C2×#125—531no—
368C2×C4×Q8—531no—
369C2×C4×Q8—531no—
370C2×#117—531no—
371#479—1064no—
372C2×#107—531no—
373C2×#106—531no—
374C2×#106—531no—
375C2×#106—531no—
376C2×C4⋊C8—531no—
377C2×C4⋊C8—531no—
378#416—1064no—
379C2×#103—531no—
380C2×#103—531no—
381C2×#103C2×D5311yescharacteristic
382C2×#103C2×D5311yescharacteristic
383C2×#102—531no—
384C2×#102—531no—
385C2×#102C2×D5311yescharacteristic
386C2×#102C2×D5311yescharacteristic
387C2×C22⋊C8C2×D5311yescharacteristic
388#399—1061no—
389#396—1061no—
390#390—531no—
391#390—531no—
392#390—1061no—
393#390—1061no—
394#389—531no—
395#389—531no—
396#389—1061no—
397#387—1061no—
398#384—531no—
399#380—531no—
400#380—1061no—
401#380C2×D5311yescharacteristic
402C4×C22⋊C4—531no—
403C4×C22⋊C4—1061no—
404#340—531no—
405#340—531no—
406#340—531no—
407#340—531no—
408#336—531no—
409#336—531no—
410C53#2595311yescharacteristic, complemented, Sylow
411C2×#41—22no—
412C2×#41—22no—
413C2×#41—22no—
414C2×#41—41no—
415C2×#41—41no—
416C2×#41Dic10614yes—
417C22×Q8C22×D5311yescharacteristic
418C22×D4—21no—
419C22×D4C22×D5311yescharacteristic
420C23×C4—21no—
421C23×C4C22×D5311yescharacteristic
422C22×C8—531no—
423C22×C8—531no—
424C22×C8—1062no—
425C22×C8—1062no—
426C22×C8—2121no—
427C22×C8C22×D5311yescharacteristic
428C22×C8C22×D5311yescharacteristic
429#125—1062no—
430#125—2121no—
431C4×Q8—1062no—
432C4×Q8—1062no—
433C4×Q8—2121no—
434C4×Q8—2121no—
435#117—1062no—
436#117—2121no—
437C2×C4⋊C4—531no—
438C2×C4⋊C4—531no—
439C2×C4⋊C4—531no—
440C2×C4⋊C4—531no—
441C2×C4⋊C4—1061no—
442C2×C4⋊C4—1061no—
443C2×C4⋊C4—1061no—
444C2×C4⋊C4—1061no—
445C2×C4⋊C4—1061no—
446C2×C4⋊C4—1061no—
447C2×C4⋊C4—1061no—
448C2×C4⋊C4—1061no—
449C2×C4⋊C4C22×D5311yescharacteristic
450C2×C4⋊C4C22×D5311yescharacteristic
451C2×C22⋊C4—21no—
452C2×C22⋊C4—531no—
453C2×C22⋊C4—531no—
454C2×C22⋊C4—1061no—
455C2×C22⋊C4—1061no—
456C2×C22⋊C4—1061no—
457C2×C22⋊C4—1061no—
458C2×C22⋊C4—1061no—
459C2×C22⋊C4—1061no—
460C2×C42—531no—
461C2×C42—531no—
462C2×C42—1061no—
463C2×C42—1061no—
464C2×C42—1061no—
465#107—1062no—
466#106—1062no—
467#106—2121no—
468#106—2121no—
469C4⋊C8—1062no—
470C4⋊C8—2121no—
471#103—22no—
472#103—22no—
473#103—534no—
474#103—1062no—
475#102—41no—
476#102—41no—
477#102—1062no—
478#102—1062no—
479C22⋊C8—22no—
480#95—21no—
481#95—531no—
482#95—531no—
483#95—1061no—
484#95—1061no—
485#95—1061no—
486C24—21no—
487C24D4×D5311yescharacteristic
488#41—28no—
489#41—44no—
490#41—44no—
491#41—44no—
492C2×Q8—21no—
493C2×Q8—21no—
494C2×Q8—21no—
495C2×Q8C2×Dic10612yes—
496C2×Q8D4×D5312yes—
497C2×Q8#1171994712yes—
498C2×D4—24no—
499C2×D4—21no—
500C2×D4—21no—
501C2×D4—21no—
502C2×D4—21no—
503C2×D4—21no—
504C2×D4—42no—
505C2×D4—42no—
506C2×D4C2×Dic10612yes—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—24no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—42no—
516C22×C4—42no—
517C22×C4—531no—
518C22×C4—531no—
519C22×C4—531no—
520C22×C4—531no—
521C22×C4—1061no—
522C22×C4—1061no—
523C22×C4—1061no—
524C22×C4—1061no—
525C22×C4C2×Dic10612yes—
526C22×C4D4×D5311yescharacteristic
527C22×C4#1171994712yes—
528C22×C4C23×D5311yescharacteristic
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—532no—
536C2×C8—1061no—
537C2×C8—1061no—
538C2×C8—1061no—
539C2×C8—1061no—
540C2×C8—1061no—
541C2×C8—2124no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—532no—
547C4⋊C4—1061no—
548C4⋊C4—1061no—
549C4⋊C4—1061no—
550C4⋊C4—1061no—
551C4⋊C4—1061no—
552C4⋊C4—1061no—
553C4⋊C4—1061no—
554C4⋊C4—4241no—
555C4⋊C4—4241no—
556C4⋊C4—2122no—
557C4⋊C4—2122no—
558C4⋊C4—2122no—
559C4⋊C4—2122no—
560C4⋊C4—2122no—
561C4⋊C4—2122no—
562C22⋊C4—81no—
563C22⋊C4—1062no—
564C22⋊C4—4241no—
565C22⋊C4—4241no—
566C22⋊C4—4241no—
567C22⋊C4—2122no—
568C22⋊C4—2121no—
569C22⋊C4—2122no—
570C22⋊C4—2122no—
571C42—1061no—
572C42—1061no—
573C42—1061no—
574C42—1061no—
575C42—2122no—
576C42—2122no—
577C42—2122no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—22no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23#1313319412yes—
592C23C2×D4×D5311yescharacteristic
593Q8—22no—
594Q8—22no—
595Q8—22no—
596Q8#1313319414yes—
597D4—22no—
598D4—22no—
599D4—48no—
600D4—41no—
601D4—41no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—48no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—532no—
619C2×C4—1061no—
620C2×C4—1061no—
621C2×C4—1061no—
622C2×C4—1061no—
623C2×C4—1061no—
624C2×C4—1061no—
625C2×C4—1061no—
626C2×C4—1061no—
627C2×C4—1061no—
628C2×C4—1061no—
629C2×C4—1062no—
630C2×C4—1061no—
631C2×C4—2121no—
632C2×C4—2121no—
633C2×C4—2121no—
634C2×C4—2121no—
635C2×C4—2121no—
636C2×C4—2121no—
637C2×C4—2121no—
638C2×C4—2121no—
639C2×C4—2121no—
640C2×C4—2121no—
641C2×C4—2121no—
642C2×C4#1313319412yes—
643C2×C4C2×D4×D5311yescharacteristic
644C2×C4C22×Dic10611yescharacteristic
645C2×C4C2×#1171994711yescharacteristic
646C2×C4C2×#1171994711yescharacteristic
647C2×C4#1313326211yescharacteristic
648C2×C4#1313326211yescharacteristic
649C8—41no—
650C8—41no—
651C8—1062no—
652C8—2121no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22—41no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22—81no—
667C22C2×#1313319411yescharacteristic
668C22#1681125511yescharacteristic
669C22#1681126011yescharacteristic
670C22#135×D5311yescharacteristic
671C22#1681142011yescharacteristic
672C22#1681143211yescharacteristic
673C22#1681143311yescharacteristic
674C4—22no—
675C4—24no—
676C4—81no—
677C4—81no—
678C4—1062no—
679C4—4241no—
680C4—4241no—
681C4—4241no—
682C4—2121no—
683C4—2121no—
684C4—2121no—
685C4—2122no—
686C4C2×#1313319411yescharacteristic
687C4#1681124911yescharacteristic
688C4#1681126011yescharacteristic
689C4#1681130411yescharacteristic
690C2—42no—
691C2—82no—
692C2#2027838111yescharacteristic
693C2#2028449911yescharacteristic
694C2#2028458111yescharacteristic
695C2#2028629311yescharacteristic
696C2#2028643811yescharacteristic
697C2#2028661011yescharacteristic
698C2#2028684511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1788C2
socleC23×C53C22×D4
prime radicalC22×#41×C53C22
hypercenter#1788D53
hypocenterC53#25953
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1788

Complements

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

NH ≅ G/Ntype
1. C2×#16810932C2semidirect
3. #20278429C2semidirect
5. #20276246C2semidirect
6. #20276101C2semidirect
7. #20276085C2semidirect
8. #20275859C2semidirect
9. #20274985C2semidirect
10. #20274981C2semidirect
11. #20274916C2semidirect
12. #20274892C2semidirect
13. #20274738C2semidirect
14. #20274718C2semidirect
410. C53#25953semidirect

Maximal subgroups

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

subgroupindex
C2×#168109322
C2×#168106192
#202784292
#202775562
#202762462
#202761012
#202760852
#202758592
#202749852
#202749812
#202749162
#202748922
#202747382
#202747182
C53×#17882
#2595353