Finite Group Databasebeta

← #27394412

Subgroups — #27394412

Order 13568.

1010 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810586C211yesmaximal, characteristic
2C2×#16810585C211yesmaximal, characteristic
3#407×D53C211yesmaximal, characteristic
4#20274922C211yesmaximal, characteristic, complemented
5#20274921C212yesmaximal, complemented
6#20274920C211yesmaximal, characteristic, complemented
7#20274919C211yesmaximal, characteristic, complemented
8#20274918C212yesmaximal, complemented
9#20274756C212yesmaximal, complemented
10#20274754C212yesmaximal, complemented
11C53×#1855C211yesmaximal, characteristic, complemented
12C23×C4×D53C2211yescharacteristic
13C22×#11719917C2211yescharacteristic
14C22×#11719912C2212yes—
15C2×M16×D53C2211yescharacteristic
16C2×M16×D53C2211yescharacteristic
17C2×#13133153C2212yescomplemented
18C2×#13133152C2212yescomplemented
19C2×#13133152C2212yescomplemented
20#16810587C2211yescharacteristic
21#16810586—21no—
22#16810586—21no—
23#16810586C2211yescharacteristic
24#16810586C2211yescharacteristic
25#16810586C2212yes—
26#16810586C2212yes—
27#16810585—21no—
28#16810585—21no—
29#16810585C412yes—
30#16810585C412yes—
31#16810585C412yes—
32#16810585C412yes—
33C2×C22⋊#88322C2211yescharacteristic
34C2×#13133129—21no—
35C2×#13133129—21no—
36C2×#13133129C2212yes—
37C22⋊C8×D53—21no—
38C22⋊C8×D53—21no—
39#16810388C2212yescomplemented
40#16810387C2212yescomplemented
41#16810387C2212yescomplemented
42C22×M16×C53C2211yescharacteristic
43C2×#108×C53C2212yescomplemented
44C53×#407C2211yescharacteristic
45C2×C22⋊C8×C53C2211yescharacteristic
46C53×#341C2212yescomplemented
47C24×D53Q811yescharacteristic
48C23×Dic53Q811yescharacteristic
49C22×C4×D53—42no—
50C22×C4×D53—42no—
51C22×C4×D53C2×C411yescharacteristic
52C22×C4×D53C2×C411yescharacteristic
53C22×C4×D53C2×C411yescharacteristic
54C22×C4×D53C2×C411yescharacteristic
55C22×C4×D53D411yescharacteristic
56C22×C4×D53D411yescharacteristic
57C22×C4×D53D411yescharacteristic
58C22×C4×D53D411yescharacteristic
59C22×C4×D53Q811yescharacteristic
60C22×C4×D53Q811yescharacteristic
61C22×C4×D53C2311yescharacteristic
62C22×C4×D53C2311yescharacteristic
63C2×#11719917—21no—
64C2×#11719917—42no—
65C2×#11719917C2312yes—
66C2×#11719917C2311yescharacteristic
67C2×#11719917C2311yescharacteristic
68C2×#11719912—21no—
69C2×#11719912—21no—
70C2×#11719912—42no—
71C2×#11719912—42no—
72C2×#11719912C2×C414yes—
73C2×#11719912C2×C414yes—
74M16×D53—42no—
75M16×D53—42no—
76C22×#88322—21no—
77C22×#88322C2312yes—
78C2×C8×D53—21no—
79C2×C8×D53—21no—
80#13133153—42no—
81#13133152—42no—
82#13133152—42no—
83C22⋊#88322—21no—
84C22⋊#88322—21no—
85C22⋊#88322C2×C412yes—
86C22⋊#88322C2×C412yes—
87#13133129—21no—
88#13133129—21no—
89#13133129—21no—
90#13133129—21no—
91#13133129—22no—
92#13133129—22no—
93#13133129—22no—
94#13133129—22no—
95#13133129C2×C414yes—
96#13133129C2×C414yes—
97C23×C4×C53C2311yescharacteristic
98C2×M16×C53—21no—
99C2×M16×C53—42no—
100C2×M16×C53C2312yes—
101C2×M16×C53C2311yescharacteristic
102C2×M16×C53C2311yescharacteristic
103C22×C8×C53—21no—
104C22×C8×C53C2312yes—
105#108×C53—42no—
106C22⋊C8×C53—21no—
107C22⋊C8×C53—21no—
108C22⋊C8×C53C2×C412yes—
109C22⋊C8×C53C2×C412yes—
110C24×C53C2×Q811yescharacteristic
111C23×D53—42no—
112C23×D53—42no—
113C23×D53—42no—
114C23×D53C4⋊C411yescharacteristic
115C23×D53C4⋊C411yescharacteristic
116C23×D53C4⋊C411yescharacteristic
117C23×D53C4⋊C411yescharacteristic
118C23×D53C2×Q811yescharacteristic
119C23×D53C2×Q811yescharacteristic
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53C22×C411yescharacteristic
123C22×C4×C53C22×C411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×Q811yescharacteristic
127C22×C4×C53C2411yescharacteristic
128C22×Dic53—42no—
129C22×Dic53C4⋊C411yescharacteristic
130C22×Dic53C4⋊C411yescharacteristic
131C22×Dic53C4⋊C411yescharacteristic
132C22×Dic53C4⋊C411yescharacteristic
133C22×Dic53C2×Q811yescharacteristic
134C22×Dic53C2×Q811yescharacteristic
135C2×C4×D53—22no—
136C2×C4×D53—22no—
137C2×C4×D53—22no—
138C2×C4×D53—22no—
139C2×C4×D53—42no—
140C2×C4×D53—42no—
141C2×C4×D53—42no—
142C2×C4×D53—42no—
143C2×C4×D53—41no—
144C2×C4×D53—41no—
145C2×C4×D53—41no—
146C2×C4×D53—41no—
147C2×C4×D53—41no—
148C2×C4×D53—41no—
149C2×C4×D53—41no—
150C2×C4×D53—41no—
151C2×C4×D53—42no—
152C2×C4×D53—42no—
153C2×C4×D53C4⋊C411yescharacteristic
154C2×C4×D53C4⋊C411yescharacteristic
155C2×C4×D53C4⋊C411yescharacteristic
156C2×C4×D53C4⋊C411yescharacteristic
157C2×C4×D53C4⋊C411yescharacteristic
158C2×C4×D53C4⋊C411yescharacteristic
159C2×C4×D53C4⋊C411yescharacteristic
160C2×C4×D53C4⋊C411yescharacteristic
161C2×C4×D53C22×C411yescharacteristic
162C2×C4×D53C22×C411yescharacteristic
163C2×C4×D53C22×C411yescharacteristic
164C2×C4×D53C22×C411yescharacteristic
165C2×C4×D53C2×D411yescharacteristic
166C2×C4×D53C2×D411yescharacteristic
167C2×C4×D53C2×D411yescharacteristic
168C2×C4×D53C2×D411yescharacteristic
169M16×C53—22no—
170M16×C53—22no—
171M16×C53—22no—
172M16×C53—22no—
173M16×C53—42no—
174C2×C8×C53—22no—
175C2×C8×C53—22no—
176C2×C8×C53—21no—
177C2×C8×C53—21no—
178C2×C8×C53—22no—
179C2×C8×C53—42no—
180C2×C8×C53C22×C414yes—
181#11719917—22no—
182#11719917—22no—
183#11719917—22no—
184#11719917—22no—
185#11719917—42no—
186C2×#88322—22no—
187C2×#88322—22no—
188C2×#88322—22no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—42no—
192C2×#88322C22×C414yes—
193#11719912—44no—
194#11719912—44no—
195#11719912—42no—
196#11719912—42no—
197C8×D53—42no—
198C8×D53—42no—
199C23×C53—22no—
200C23×C53—22no—
201C23×C53—41no—
202C23×C53C2×C4⋊C411yescharacteristic
203C23×C53C2×C4⋊C411yescharacteristic
204C23×C53C22×Q811yescharacteristic
205C22×D53—22no—
206C22×D53—22no—
207C22×D53—22no—
208C22×D53—22no—
209C22×D53—22no—
210C22×D53—22no—
211C22×D53—22no—
212C22×D53—41no—
213C22×D53—41no—
214C22×D53—41no—
215C22×D53—41no—
216C22×D53—41no—
217C22×D53—41no—
218C22×D53—41no—
219C22×D53—41no—
220C22×D53—41no—
221C22×D53—41no—
222C22×D53—41no—
223C22×D53—416no—
224C22×D53C2×C4⋊C411yescharacteristic
225C22×D53C2×C4⋊C411yescharacteristic
226C22×D53C2×C4⋊C411yescharacteristic
227C22×D53C2×C4⋊C411yescharacteristic
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53C2×C4⋊C411yescharacteristic
236C2×C4×C53C2×C4⋊C411yescharacteristic
237C2×C4×C53C23×C411yescharacteristic
238C2×C4×C53C22×D411yescharacteristic
239C2×C4×C53H(5,𝔽2)12yes—
240C2×C4×C53H(5,𝔽2)12yes—
241C2×Dic53—22no—
242C2×Dic53—22no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53C2×C4⋊C411yescharacteristic
248C2×Dic53C2×C4⋊C411yescharacteristic
249C2×Dic53C2×C4⋊C411yescharacteristic
250C2×Dic53C2×C4⋊C411yescharacteristic
251C4×D53—22no—
252C4×D53—22no—
253C4×D53—22no—
254C4×D53—22no—
255C4×D53—21no—
256C4×D53—21no—
257C4×D53—21no—
258C4×D53—21no—
259C4×D53—21no—
260C4×D53—21no—
261C4×D53—21no—
262C4×D53—21no—
263C4×D53—41no—
264C4×D53—41no—
265C4×D53—41no—
266C4×D53—41no—
267C4×D53—41no—
268C4×D53—41no—
269C4×D53—41no—
270C4×D53—41no—
271C8×C53—24no—
272C8×C53—41no—
273C8×C53—41no—
274C8×C53—42no—
275#88322—24no—
276#88322—41no—
277#88322—41no—
278#88322—42no—
279#21949—531nomaximal, Sylow
280C22×C53—21no—
281C22×C53—21no—
282C22×C53—21no—
283C22×C53—21no—
284C22×C53—21no—
285C22×C53—21no—
286C22×C53—21no—
287C22×C53—44no—
288C22×C53#43012yes—
289C22×C53C22×C4⋊C411yescharacteristic
290C22×C53#54312yes—
291C2×D53—21no—
292C2×D53—21no—
293C2×D53—21no—
294C2×D53—21no—
295C2×D53—21no—
296C2×D53—21no—
297C2×D53—21no—
298C2×D53—21no—
299C2×D53—21no—
300C2×D53—21no—
301C2×D53—21no—
302C2×D53—44no—
303C2×D53—44no—
304C2×D53—44no—
305C2×D53—44no—
306C2×D53—44no—
307C2×D53—44no—
308C2×D53#43012yes—
309C4×C53—21no—
310C4×C53—21no—
311C4×C53#51812yes—
312C4×C53#53411yescharacteristic
313C4×C53#53411yescharacteristic
314Dic53—21no—
315Dic53—21no—
316Dic53—21no—
317Dic53—21no—
318C2×#407—531no—
319C2×#407—531no—
320C2×#407—531no—
321#1855—532no—
322#1855—531no—
323#1855—531no—
324#1855—532no—
325#1855—531no—
326#1855D5311yescharacteristic, complemented
327#1811—532no—
328#1811—532no—
329C2×C53—22no—
330C2×C53—22no—
331C2×C53—41no—
332C2×C53#231511yescharacteristic
333C2×C53C2×#43011yescharacteristic
334C2×C53C2×#43011yescharacteristic
335D53—22no—
336D53—41no—
337D53—41no—
338D53—41no—
339C24×C4—531no—
340C22×M16—532no—
341C22×M16—531no—
342C22×M16—531no—
343C22×M16—531no—
344C22×M16C2×D5311yescharacteristic
345C2×#108—532no—
346C2×#108—532no—
347C2×#108—532no—
348C2×#108C2×D5312yescomplemented
349#407—532no—
350#407—532no—
351#407—532no—
352#407—532no—
353#407—531no—
354#407—531no—
355#407—531no—
356#407—532no—
357#407—532no—
358#407—1061no—
359#407—1061no—
360#407—1061no—
361#407—1061no—
362#407C2×D5311yescharacteristic
363C2×C22⋊C8—531no—
364C2×C22⋊C8—532no—
365C2×C22⋊C8—1061no—
366C2×C22⋊C8—1061no—
367C2×C22⋊C8—1061no—
368C2×C22⋊C8—1061no—
369C2×C22⋊C8C2×D5311yescharacteristic
370#341—532no—
371#341—532no—
372#341—532no—
373#341C2×D5312yescomplemented
374C53#2194911yescharacteristic, complemented, Sylow
375C25—531no—
376C23×C4—531no—
377C23×C4—531no—
378C23×C4—531no—
379C23×C4—531no—
380C23×C4—531no—
381C23×C4—531no—
382C23×C4—531no—
383C23×C4—531no—
384C23×C4—531no—
385C23×C4—531no—
386C23×C4—531no—
387C23×C4—531no—
388C23×C4—531no—
389C23×C4—2122no—
390C23×C4—2122no—
391C23×C4C22×D5311yescharacteristic
392C2×M16—21no—
393C2×M16—42no—
394C2×M16—534no—
395C2×M16—534no—
396C2×M16—531no—
397C2×M16—532no—
398C2×M16—531no—
399C2×M16—1061no—
400C2×M16—1061no—
401C2×M16—1061no—
402C2×M16—2122no—
403C2×M16—2122no—
404C2×M16—2122no—
405C2×M16—2122no—
406C2×M16—2122no—
407C2×M16C22×D5311yescharacteristic
408C2×M16C22×D5311yescharacteristic
409C2×M16C22×D5312yes—
410C22×C8—21no—
411C22×C8—532no—
412C22×C8—1061no—
413C22×C8—1061no—
414C22×C8—1061no—
415C22×C8C22×D5312yes—
416#108—42no—
417#108—2122no—
418#108—2122no—
419#108—2122no—
420C22⋊C8—21no—
421C22⋊C8—21no—
422C22⋊C8—532no—
423C22⋊C8—532no—
424C22⋊C8—534no—
425C22⋊C8—534no—
426C22⋊C8—1061no—
427C22⋊C8—1061no—
428C22⋊C8—1061no—
429C22⋊C8—1061no—
430C22⋊C8—1062no—
431C22⋊C8—1062no—
432C22⋊C8—1062no—
433C22⋊C8—1062no—
434C22⋊C8—1061no—
435C22⋊C8—1061no—
436C22⋊C8C4×D5312yes—
437C22⋊C8C4×D5312yes—
438C24—531no—
439C24—531no—
440C24—531no—
441C24—531no—
442C24—531no—
443C24—531no—
444C24—2122no—
445C24—2122no—
446C24—2122no—
447C24Q8×D5311yescharacteristic
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—531no—
451C22×C4—531no—
452C22×C4—531no—
453C22×C4—531no—
454C22×C4—531no—
455C22×C4—531no—
456C22×C4—531no—
457C22×C4—531no—
458C22×C4—531no—
459C22×C4—531no—
460C22×C4—531no—
461C22×C4—531no—
462C22×C4—531no—
463C22×C4—531no—
464C22×C4—531no—
465C22×C4—531no—
466C22×C4—531no—
467C22×C4—531no—
468C22×C4—531no—
469C22×C4—531no—
470C22×C4—531no—
471C22×C4—531no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—2122no—
477C22×C4—2122no—
478C22×C4—2122no—
479C22×C4—2122no—
480C22×C4—2121no—
481C22×C4—2121no—
482C22×C4—2121no—
483C22×C4—2121no—
484C22×C4—2121no—
485C22×C4—2121no—
486C22×C4—2121no—
487C22×C4—2121no—
488C22×C4—2122no—
489C22×C4—2122no—
490C22×C4—2122no—
491C22×C4C2×C4×D5311yescharacteristic
492C22×C4C2×C4×D5311yescharacteristic
493C22×C4D4×D5311yescharacteristic
494C22×C4D4×D5311yescharacteristic
495C22×C4Q8×D5311yescharacteristic
496C22×C4C23×D5311yescharacteristic
497M16—22no—
498M16—22no—
499M16—22no—
500M16—22no—
501M16—42no—
502M16—1062no—
503M16—1062no—
504M16—1062no—
505M16—1062no—
506M16—2124no—
507M16—2124no—
508M16—2122no—
509M16—2122no—
510M16—2122no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—22no—
516C2×C8—42no—
517C2×C8—534no—
518C2×C8—1062no—
519C2×C8—1062no—
520C2×C8—1061no—
521C2×C8—1062no—
522C2×C8—1061no—
523C2×C8—2122no—
524C2×C8—2122no—
525C2×C8—2122no—
526C2×C8C2×C4×D5314yes—
527C23—22no—
528C23—22no—
529C23—41no—
530C23—531no—
531C23—531no—
532C23—531no—
533C23—531no—
534C23—1062no—
535C23—1062no—
536C23—1062no—
537C23—1062no—
538C23—1062no—
539C23—1062no—
540C23—1062no—
541C23—2121no—
542C23—2121no—
543C23—2121no—
544C23—2121no—
545C23—2121no—
546C23—2121no—
547C23—2121no—
548C23—2121no—
549C23—2121no—
550C23—2121no—
551C23—2121no—
552C23—21216no—
553C23C4⋊C4×D5311yescharacteristic
554C23C4⋊C4×D5311yescharacteristic
555C23C2×Q8×D5311yescharacteristic
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—531no—
564C2×C4—531no—
565C2×C4—531no—
566C2×C4—531no—
567C2×C4—1062no—
568C2×C4—1062no—
569C2×C4—1062no—
570C2×C4—1062no—
571C2×C4—1062no—
572C2×C4—1062no—
573C2×C4—1061no—
574C2×C4—1061no—
575C2×C4—1061no—
576C2×C4—1061no—
577C2×C4—1061no—
578C2×C4—1061no—
579C2×C4—1061no—
580C2×C4—1061no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4C4⋊C4×D5311yescharacteristic
594C2×C4C4⋊C4×D5311yescharacteristic
595C2×C4C22×C4×D5311yescharacteristic
596C2×C4C2×D4×D5311yescharacteristic
597C2×C4#1313326212yes—
598C2×C4#1313326312yes—
599C8—24no—
600C8—42no—
601C8—41no—
602C8—41no—
603C8—1064no—
604C8—2121no—
605C8—2121no—
606C8—2122no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—44no—
615C22—532no—
616C22—1061no—
617C22—1061no—
618C22—1061no—
619C22—1061no—
620C22—1061no—
621C22—1061no—
622C22—1061no—
623C22—1061no—
624C22—1061no—
625C22—1061no—
626C22—1061no—
627C22—2124no—
628C22—2124no—
629C22—2124no—
630C22—2124no—
631C22—2124no—
632C22—2124no—
633C22#1681067012yes—
634C22C2×C4⋊C4×D5311yescharacteristic
635C22#1681124812yes—
636C4—21no—
637C4—21no—
638C4—1061no—
639C4—1061no—
640C4—1061no—
641C4—1061no—
642C4#1681109512yes—
643C4#1681118311yescharacteristic
644C4#1681118411yescharacteristic
645C2—22no—
646C2—22no—
647C2—41no—
648C2—1062no—
649C2—2121no—
650C2—2121no—
651C2—2121no—
652C2#2027732611yescharacteristic
653C2#430×D5311yescharacteristic
654C2C2×#1681067011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1855C2
socleC22×C53C22×C4⋊C4
prime radicalC24×D53Q8
cosocleC22×C4×C53C24
hypercenter#1855D53
hypocenterC53#21949
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1855

Complements

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

NH ≅ G/Ntype
4. #20274922C2semidirect
5. #20274921C2semidirect
6. #20274920C2semidirect
7. #20274919C2semidirect
8. #20274918C2semidirect
9. #20274756C2semidirect
10. #20274754C2semidirect
11. C53×#1855C2semidirect
17. C2×#13133153C22semidirect
18. C2×#13133152C22semidirect
19. C2×#13133152C22semidirect
39. #16810388C22semidirect
40. #16810387C22semidirect
41. #16810387C22semidirect
43. C2×#108×C53C22semidirect
46. C53×#341C22semidirect
326. #1855D53semidirect
348. C2×#108C2×D53semidirect
373. #341C2×D53semidirect
374. C53#21949semidirect

Maximal subgroups

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

subgroupindex
C2×#168105862
C2×#168105852
#407×D532
#202749222
#202749212
#202749202
#202749192
#202749182
#202747562
#202747542
C53×#18552
#2194953