Finite Group Databasebeta

← #27393342

Subgroups — #27393342

Order 13568.

1442 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811134C211yesmaximal, characteristic
2C2×#16810681C212yesmaximal, complemented
3#20275511C212yesmaximal, complemented
4#20275511C212yesmaximal, complemented
5#20275507C212yesmaximal, complemented
6#20275507C212yesmaximal, complemented
7#20274626C212yesmaximal, complemented
8#20274625C211yesmaximal, characteristic, complemented
9C53×#1779C211yesmaximal, characteristic, complemented
10C22×#11719945C2211yescharacteristic
11#16811134—28no—
12C2×#13133200C412yes—
13C2×#13133196C411yescharacteristic
14C2×#13133196C411yescharacteristic
15C2×#13133191C411yescharacteristic
16C2×#13133191C411yescharacteristic
17C2×#13133186C412yes—
18C2×C4×C22⋊D53C2212yes—
19C2×C4×C22⋊D53C2212yes—
20C2×#13133176C2211yescharacteristic
21#16810681—28no—
22C2×#13133146C2212yescomplemented
23C2×#13133146C2212yescomplemented
24C2×C22⋊#88322C2212yes—
25C2×C22⋊#88322C2211yescharacteristic
26C2×C22⋊#88322C2212yes—
27C2×C22⋊#88322C2211yescharacteristic
28C2×#13133129C2212yescomplemented
29C2×#13133129C2212yescomplemented
30C2×#13133129C2212yescomplemented
31C2×#13133129C2212yescomplemented
32C2×#13133127C2212yescomplemented
33C2×C8×Dic53C2212yescomplemented
34C24×C4×C53C2211yescharacteristic
35C2×C22⋊C8×C53C2212yes—
36C2×C22⋊C8×C53C2211yescharacteristic
37C2×C22⋊C8×C53C2212yes—
38C2×C22⋊C8×C53C2211yescharacteristic
39C2×#11719945—24no—
40C2×#11719945—41no—
41C2×#11719945—41no—
42C2×#11719945—42no—
43C22×Dic106C2×C411yescharacteristic
44C22×D212C2×C411yescharacteristic
45C22×C22⋊D53C2×C412yes—
46C22×C22⋊D53C2×C412yes—
47C22×C22⋊D53C2×C412yes—
48C22×C4×D53C2312yes—
49C22×#88322C2312yes—
50C22×#88322C2312yes—
51#13133200—28no—
52#13133196—24no—
53#13133196—24no—
54#13133191—24no—
55#13133191—24no—
56#13133186—28no—
57C4×C22⋊D53—24no—
58C4×C22⋊D53—24no—
59C4×C22⋊D53—24no—
60C4×C22⋊D53—24no—
61#13133176D418yes—
62C2×C4⋊Dic53C2×C411yescharacteristic
63C2×C4⋊Dic53C2×C411yescharacteristic
64C2×#11719919C2×C412yes—
65C2×#11719919C2×C412yes—
66C2×C22⋊Dic53C2×C412yes—
67C2×C22⋊Dic53C2×C412yes—
68C2×C22⋊Dic53C2×C412yes—
69C2×#11719921C2×C412yes—
70C2×#11719921C2×C412yes—
71C2×C4×Dic53C2312yes—
72#13133146—24no—
73#13133146—24no—
74C22⋊#88322—24no—
75C22⋊#88322—41no—
76C22⋊#88322—42no—
77C22⋊#88322—41no—
78#13133129—42no—
79#13133129—42no—
80#13133129—42no—
81#13133129—42no—
82#13133127—24no—
83C8×Dic53—24no—
84C25×C53C2×C411yescharacteristic
85C23×C4×C53—24no—
86C23×C4×C53—42no—
87C23×C4×C53—41no—
88C23×C4×C53—41no—
89C23×C4×C53C2×C411yescharacteristic
90C23×C4×C53C2×C411yescharacteristic
91C23×C4×C53C2×C411yescharacteristic
92C23×C4×C53C2311yescharacteristic
93C23×C4×C53C2311yescharacteristic
94C23×C4×C53C2311yescharacteristic
95C22×C8×C53C2312yes—
96C22×C8×C53C2312yes—
97C22⋊C8×C53—24no—
98C22⋊C8×C53—41no—
99C22⋊C8×C53—42no—
100C22⋊C8×C53—41no—
101C24×C53—22no—
102C24×C53—22no—
103C24×C53—22no—
104C24×C53—41no—
105C24×C53—41no—
106C24×C53—41no—
107C24×C53—41no—
108C24×C53C22×C411yescharacteristic
109C24×C53C22×C411yescharacteristic
110C24×C53C22×C411yescharacteristic
111C23×D53C22×C412yes—
112C22×C4×C53—21no—
113C22×C4×C53—21no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—22no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—22no—
126C22×C4×C53—41no—
127C22×C4×C53—41no—
128C22×C4×C53—41no—
129C22×C4×C53—41no—
130C22×C4×C53—41no—
131C22×C4×C53—41no—
132C22×C4×C53—48no—
133C22×C4×C53—48no—
134C22×C4×C53C22×C411yescharacteristic
135C22×C4×C53C22×C411yescharacteristic
136C22×C4×C53C22×C411yescharacteristic
137C22×C4×C53C2×D414yes—
138C22×C4×C53C2411yescharacteristic
139C2×C22⋊D53—24no—
140C2×C22⋊D53—24no—
141C2×C22⋊D53—24no—
142C2×C22⋊D53—24no—
143C2×C22⋊D53—24no—
144C2×C22⋊D53—42no—
145C2×C22⋊D53—42no—
146C2×C22⋊D53—42no—
147C2×C22⋊D53—42no—
148C22×Dic53C22×C412yes—
149C22×Dic53C22×C412yes—
150C22×Dic53C22×C412yes—
151#11719945—48no—
152#11719945—48no—
153C2×D212—22no—
154C2×D212—41no—
155C2×D212—41no—
156C2×C4×D53—22no—
157C2×C4×D53—22no—
158C2×C4×D53—22no—
159C2×C4×D53—22no—
160C2×C4×D53—22no—
161C2×C4×D53—22no—
162C2×Dic106—22no—
163C2×Dic106—41no—
164C2×Dic106—41no—
165C2×C8×C53—22no—
166C2×C8×C53—22no—
167C2×C8×C53—22no—
168C2×C8×C53—22no—
169C2×C8×C53—22no—
170C2×C8×C53—22no—
171C22⋊Dic53—24no—
172C22⋊Dic53—24no—
173C22⋊Dic53C22⋊C418yes—
174#11719921—24no—
175#11719921—24no—
176C4⋊Dic53C22⋊C414yes—
177C4⋊Dic53C22⋊C414yes—
178#11719919—24no—
179#11719919—24no—
180C4×Dic53C2×D414yes—
181C4×Dic53C2×D414yes—
182C2×#88322—22no—
183C2×#88322—22no—
184C2×#88322—22no—
185C2×#88322—22no—
186C2×#88322—22no—
187C2×#88322—22no—
188C23×C53—21no—
189C23×C53—21no—
190C23×C53—21no—
191C23×C53—21no—
192C23×C53—21no—
193C23×C53—21no—
194C23×C53—21no—
195C23×C53—21no—
196C23×C53—21no—
197C23×C53—21no—
198C23×C53—21no—
199C23×C53—21no—
200C23×C53—21no—
201C23×C53—21no—
202C23×C53—21no—
203C23×C53—21no—
204C23×C53—21no—
205C23×C53—21no—
206C23×C53—44no—
207C23×C53—44no—
208C23×C53—44no—
209C23×C53—44no—
210C23×C53—44no—
211C23×C53—44no—
212C23×C53—44no—
213C23×C53C2×C22⋊C412yes—
214C23×C53#13112yes—
215C23×C53#13112yes—
216C23×C53C23×C411yescharacteristic
217C22×D53—22no—
218C22×D53—22no—
219C22×D53—22no—
220C22×D53—22no—
221C22×D53—22no—
222C22×D53—22no—
223C22×D53—22no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—24no—
227C2×C4×C53—24no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—44no—
235C2×C4×C53—44no—
236C2×C4×C53—44no—
237C2×C4×C53—42no—
238C2×C4×C53—42no—
239C2×C4×C53—42no—
240C2×C4×C53—42no—
241C2×C4×C53C2×C22⋊C412yes—
242C2×C4×C53C22×D412yes—
243C2×C4×C53H(5,𝔽2)11yescharacteristic
244C2×C4×C53H(5,𝔽2)11yescharacteristic
245C2×C4×C53H(5,𝔽2)12yes—
246C22⋊D53—48no—
247C22⋊D53—48no—
248C22⋊D53—48no—
249C2×Dic53—22no—
250C2×Dic53—22no—
251C2×Dic53—22no—
252C2×Dic53—22no—
253C2×Dic53—22no—
254C2×Dic53—22no—
255C2×Dic53—22no—
256C2×Dic53C2×C22⋊C414yes—
257C2×Dic53C2×C22⋊C414yes—
258D212—44no—
259C4×D53—42no—
260C4×D53—42no—
261C4×D53—42no—
262C4×D53—42no—
263Dic106—44no—
264C8×C53—42no—
265C8×C53—42no—
266#88322—42no—
267#88322—42no—
268#21792—531nomaximal, Sylow
269C22×C53—22no—
270C22×C53—22no—
271C22×C53—22no—
272C22×C53—22no—
273C22×C53—22no—
274C22×C53—22no—
275C22×C53—22no—
276C22×C53—22no—
277C22×C53—22no—
278C22×C53—416no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53—41no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53—41no—
289C22×C53—41no—
290C22×C53—41no—
291C22×C53C22×C22⋊C411yescharacteristic
292C22×C53#51811yescharacteristic
293C22×C53#51811yescharacteristic
294C22×C53C2×#13111yescharacteristic
295C22×C53C2×#13111yescharacteristic
296C22×C53#56811yescharacteristic
297C22×C53#56811yescharacteristic
298C2×D53—42no—
299C2×D53—42no—
300C2×D53—42no—
301C2×D53—42no—
302C2×D53—42no—
303C2×D53—42no—
304C2×D53—42no—
305C4×C53—22no—
306C4×C53—41no—
307C4×C53—41no—
308C4×C53#53512yes—
309C4×C53#53512yes—
310Dic53—24no—
311Dic53—24no—
312Dic53—42no—
313C2×#525—531no—
314C2×#433—532no—
315#1955—532no—
316#1955—532no—
317#1954—532no—
318#1954—532no—
319#1780—532no—
320#1779—531no—
321#1779D5311yescharacteristic, complemented
322C2×C53—42no—
323C2×C53—42no—
324C2×C53—42no—
325C2×C53#230911yescharacteristic
326C2×C53#288011yescharacteristic
327C2×C53#288011yescharacteristic
328C2×C53#297611yescharacteristic
329C2×C53#297611yescharacteristic
330C2×C53#297711yescharacteristic
331C2×C53#297811yescharacteristic
332D53—82no—
333C22×#41—531no—
334C24×C4C2×D5311yescharacteristic
335#525—1068no—
336C2×#123—532no—
337C2×C22⋊Q8—531no—
338C2×C22⋊Q8—531no—
339C2×#121—531no—
340C2×#121—531no—
341C2×#120—532no—
342C2×C4×D4—532no—
343C2×C4×D4—532no—
344C2×#117—531no—
345#433—1068no—
346C2×C4⋊C8—532no—
347C2×C4⋊C8—532no—
348C2×C22⋊C8—532no—
349C2×C22⋊C8—531no—
350C2×C22⋊C8—532no—
351C2×C22⋊C8—531no—
352C2×C22⋊C8—532no—
353C2×C22⋊C8—532no—
354C2×C22⋊C8—532no—
355C2×C22⋊C8—532no—
356C2×C22⋊C8C2×D5312yes—
357C2×C22⋊C8C2×D5311yescharacteristic
358C2×C22⋊C8C2×D5312yes—
359C2×C22⋊C8C2×D5311yescharacteristic
360C2×#97—532no—
361C2×C4×C8—532no—
362C53#2179211yescharacteristic, complemented, Sylow
363C25C4×D5311yescharacteristic
364C2×#41—1064no—
365C2×#41—2121no—
366C2×#41—2121no—
367C2×#41—2122no—
368C22×Q8—531no—
369C22×D4—532no—
370C22×D4—532no—
371C22×D4—532no—
372C22×D4—531no—
373C23×C4—24no—
374C23×C4—42no—
375C23×C4—41no—
376C23×C4—41no—
377C23×C4—532no—
378C23×C4C4×D5311yescharacteristic
379C23×C4C4×D5311yescharacteristic
380C23×C4C4×D5311yescharacteristic
381C23×C4C22×D5311yescharacteristic
382C23×C4C22×D5311yescharacteristic
383C23×C4C22×D5311yescharacteristic
384C22×C8—532no—
385C22×C8—532no—
386C22×C8C22×D5312yes—
387C22×C8C22×D5312yes—
388#123—1068no—
389C22⋊Q8—1064no—
390C22⋊Q8—1064no—
391#121—1064no—
392#121—1064no—
393#120—1068no—
394C4×D4—1064no—
395C4×D4—1064no—
396C4×D4—1064no—
397C4×D4—1064no—
398#117—538no—
399C2×C4⋊C4—531no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—532no—
402C2×C4⋊C4—532no—
403C2×C22⋊C4—532no—
404C2×C22⋊C4—532no—
405C2×C22⋊C4—532no—
406C2×C22⋊C4—532no—
407C2×C22⋊C4—532no—
408C2×C42—532no—
409C4⋊C8—1064no—
410C4⋊C8—1064no—
411C22⋊C8—24no—
412C22⋊C8—41no—
413C22⋊C8—42no—
414C22⋊C8—41no—
415C22⋊C8—1064no—
416C22⋊C8—2122no—
417C22⋊C8—2122no—
418C22⋊C8—2122no—
419C22⋊C8—2122no—
420C22⋊C8—2121no—
421C22⋊C8—2122no—
422C22⋊C8—2121no—
423#97—1064no—
424C4×C8—1064no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—41no—
429C24—41no—
430C24—41no—
431C24—41no—
432C24—532no—
433C24C2×C4×D5311yescharacteristic
434C24C2×C4×D5311yescharacteristic
435C24C2×C4×D5311yescharacteristic
436#41—2128no—
437#41—2128no—
438C2×Q8—1062no—
439C2×Q8—2121no—
440C2×Q8—2121no—
441C2×D4—1064no—
442C2×D4—1064no—
443C2×D4—1064no—
444C2×D4—1064no—
445C2×D4—1064no—
446C2×D4—1062no—
447C2×D4—2121no—
448C2×D4—2121no—
449C2×D4—2122no—
450C2×D4—2122no—
451C2×D4—2122no—
452C2×D4—2122no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—532no—
476C22×C4—532no—
477C22×C4—532no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4C2×C4×D5311yescharacteristic
485C22×C4C2×C4×D5311yescharacteristic
486C22×C4C2×C4×D5311yescharacteristic
487C22×C4D4×D5314yes—
488C22×C4C23×D5311yescharacteristic
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—1062no—
496C2×C8—1062no—
497C2×C8—1062no—
498C2×C8—1062no—
499C2×C8—1062no—
500C2×C8—1062no—
501C4⋊C4—534no—
502C4⋊C4—534no—
503C4⋊C4—1064no—
504C4⋊C4—1064no—
505C22⋊C4—538no—
506C22⋊C4—1064no—
507C22⋊C4—1064no—
508C22⋊C4—1064no—
509C22⋊C4—1064no—
510C42—534no—
511C42—534no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—1062no—
538C23—1062no—
539C23—1062no—
540C23—1062no—
541C23—1062no—
542C23—1062no—
543C23—1062no—
544C23C22⋊C4×D5312yes—
545C23#1313322312yes—
546C23#1313322512yes—
547C23C22×C4×D5311yescharacteristic
548Q8—2124no—
549D4—2124no—
550D4—2128no—
551D4—2128no—
552D4—2128no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—534no—
571C2×C4—534no—
572C2×C4—1062no—
573C2×C4—1062no—
574C2×C4—1062no—
575C2×C4—1062no—
576C2×C4—1062no—
577C2×C4—1062no—
578C2×C4—1062no—
579C2×C4—2122no—
580C2×C4—2122no—
581C2×C4—2122no—
582C2×C4—2122no—
583C2×C4C22⋊C4×D5312yes—
584C2×C4C2×D4×D5312yes—
585C2×C4#1313326211yescharacteristic
586C2×C4#1313326211yescharacteristic
587C2×C4#1313326212yes—
588C8—42no—
589C8—42no—
590C8—2122no—
591C8—2122no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—416no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—2122no—
615C22—2122no—
616C22—2122no—
617C22—2122no—
618C22—2122no—
619C22—2122no—
620C22—2122no—
621C22C2×C22⋊C4×D5311yescharacteristic
622C22#1681109611yescharacteristic
623C22#1681109611yescharacteristic
624C22C2×#1313322311yescharacteristic
625C22C2×#1313322511yescharacteristic
626C22#1681138911yescharacteristic
627C22#1681138911yescharacteristic
628C4—22no—
629C4—41no—
630C4—41no—
631C4—1064no—
632C4—1064no—
633C4—2122no—
634C4#1681118812yes—
635C4#1681118812yes—
636C2—42no—
637C2—42no—
638C2—42no—
639C2—4242no—
640C2#2027729611yescharacteristic
641C2#2028316111yescharacteristic
642C2#2028316511yescharacteristic
643C2#2028376011yescharacteristic
644C2#2028376511yescharacteristic
645C2#2028376911yescharacteristic
646C2#2028377511yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D53
commutator, socleC23×C53C23×C4
FittingC53×#1779C2
prime radicalC2×#16811134C2
cosocleC22×C4×C53C24
hypercenter#1779D53
hypocenterC53#21792
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
2. C2×#16810681C2semidirect
3. #20275511C2semidirect
4. #20275511C2semidirect
5. #20275507C2semidirect
6. #20275507C2semidirect
7. #20274626C2semidirect
8. #20274625C2semidirect
9. C53×#1779C2semidirect
22. C2×#13133146C22semidirect
23. C2×#13133146C22semidirect
28. C2×#13133129C22semidirect
29. C2×#13133129C22semidirect
30. C2×#13133129C22semidirect
31. C2×#13133129C22semidirect
32. C2×#13133127C22semidirect
33. C2×C8×Dic53C22semidirect
321. #1779D53semidirect
362. C53#21792semidirect

Maximal subgroups

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

subgroupindex
C2×#168111342
C2×#168106812
#202755112
#202755072
#202746262
#202746252
C53×#17792
#2179253