Finite Group Databasebeta

← #27388992

Subgroups — #27388992

Order 13568.

826 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810681C211yesmaximal, characteristic
2#20277322C211yesmaximal, characteristic
3#20277072C211yesmaximal, characteristic
4#20275480C211yesmaximal, characteristic, complemented
5#20275480C211yesmaximal, characteristic, complemented
6#20275477C211yesmaximal, characteristic, complemented
7#20275472C211yesmaximal, characteristic, complemented
8#20274916C211yesmaximal, characteristic
9#20274912C211yesmaximal, characteristic
10#20274908C211yesmaximal, characteristic
11#20274614C211yesmaximal, characteristic, complemented
12#20274611C211yesmaximal, characteristic, complemented
13#107×Dic53C211yesmaximal, characteristic, complemented
14#20274603C211yesmaximal, characteristic, complemented
15C53×#1852C211yesmaximal, characteristic
16C2×#13133176C2211yescharacteristic
17C2×#13133176C2211yescharacteristic
18#16810681C418yescomplemented
19C2×#13133151C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133149C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×#13133146C2211yescharacteristic
26C2×#13133146C2211yescharacteristic
27C2×C22⋊#88322C2211yescharacteristic
28C2×#13133127C2211yescharacteristic
29C2×C8×Dic53C2211yescharacteristic
30#16810517—21no—
31#16810511—21no—
32#16810508—21no—
33#16810506—21no—
34#16810492C2211yescharacteristic
35#16810488C2211yescharacteristic
36#16810483C2211yescharacteristic
37C4⋊C4×Dic53—21no—
38C4⋊C4×Dic53—21no—
39C4⋊C4×Dic53C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41C4⋊C4×Dic53C2211yescharacteristic
42C4⋊C4×Dic53C2211yescharacteristic
43C4⋊C4×Dic53C2211yescharacteristic
44C4⋊C4×Dic53C2211yescharacteristic
45C4×C22⋊Dic53—21no—
46C4×C22⋊Dic53—21no—
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53C22×C4⋊C4×C53C2211yescharacteristic
54C2×#107×C53C2211yescharacteristic
55C2×#107×C53C2211yescharacteristic
56C2×C22⋊C8×C53C2211yescharacteristic
57C53×#380C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62#13133176—22no—
63#13133176—22no—
64#13133176C2×C414yes—
65#13133176Q814yes—
66C2×C4⋊Dic53D411yescharacteristic
67C2×C4⋊Dic53D411yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×C22⋊Dic53—21no—
75C2×C22⋊Dic53—21no—
76C2×C22⋊Dic53D411yescharacteristic
77C2×C22⋊Dic53D411yescharacteristic
78C2×C4×Dic53—21no—
79C2×C4×Dic53—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53C2311yescharacteristic
86C2×C4×Dic53C2311yescharacteristic
87#13133151—22no—
88#13133150—22no—
89#13133149—22no—
90#13133148—22no—
91#13133148—22no—
92#13133148—22no—
93#13133146C2×C414yescomplemented
94#13133146C2×C414yescomplemented
95C22⋊#88322C2×C414yes—
96#13133127C2×C414yescomplemented
97C8×Dic53C2×C414yescomplemented
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107C23×C4×C53—21no—
108C23×C4×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C2×C4⋊C4×C53—22no—
112C2×C4⋊C4×C53—22no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C22⋊C4×C53—21no—
119#107×C53—22no—
120#107×C53—22no—
121C22⋊C8×C53C2×C414yes—
122#95×C53—21no—
123C24×C53C2×D411yescharacteristic
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—24no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—81no—
134C22×C4×C53—81no—
135C22×C4×C53C22×C412yes—
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×Q812yes—
138C22×C4×C53C2411yescharacteristic
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53C2×D411yescharacteristic
144C22×Dic53C2×D411yescharacteristic
145C22×Dic53C2×D411yescharacteristic
146C22×Dic53C2×D411yescharacteristic
147C2×C8×C53—21no—
148C2×C8×C53—21no—
149C2×C8×C53—21no—
150C2×C8×C53—21no—
151C2×C8×C53C22×C412yes—
152C2×C8×C53C22×C412yes—
153C4⋊C4×C53—24no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53#4112yes—
161C4⋊C4×C53#4112yes—
162C22⋊C4×C53—81no—
163C22⋊Dic53—24no—
164C22⋊Dic53—81no—
165C22⋊Dic53C4⋊C414yes—
166C22⋊Dic53C4⋊C414yes—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53C4⋊C414yes—
174C4⋊Dic53C4⋊C414yes—
175C4⋊Dic53#4112yes—
176C4⋊Dic53#4112yes—
177#11719919—81no—
178#11719919—81no—
179C4×Dic53—24no—
180C4×Dic53—81no—
181C4×Dic53—81no—
182C4×Dic53—81no—
183C4×Dic53—81no—
184C4×Dic53—81no—
185C4×Dic53—81no—
186C4×Dic53C22×C412yes—
187C4×Dic53C22×C412yes—
188C4×Dic53C2×Q812yes—
189C4×Dic53C2×Q812yes—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322C22×C412yes—
195C2×#88322C22×C412yes—
196C23×C53—21no—
197C23×C53—21no—
198C23×C53—21no—
199C23×C53—21no—
200C23×C53—21no—
201C23×C53—21no—
202C23×C53C2×C4⋊C412yes—
203C23×C53C22×D411yescharacteristic
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—84no—
223C2×C4×C53C2×C4⋊C412yes—
224C2×C4×C53C23×C411yescharacteristic
225C2×C4×C53C22×Q811yescharacteristic
226C2×C4×C53C2×#4111yescharacteristic
227C2×C4×C53C2×#4111yescharacteristic
228C2×C4×C53C2×#4111yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—22no—
239C2×Dic53—22no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53C2×C4⋊C412yes—
251C2×Dic53C2×C4⋊C412yes—
252C2×Dic53C2×C4⋊C412yes—
253C2×Dic53C2×C4⋊C412yes—
254C8×C53—22no—
255C8×C53—22no—
256#88322—22no—
257#88322—22no—
258#21367—531nomaximal, Sylow
259C22×C53—22no—
260C22×C53—22no—
261C22×C53—22no—
262C22×C53—41no—
263C22×C53—41no—
264C22×C53—41no—
265C22×C53—41no—
266C22×C53C22×C4⋊C411yescharacteristic
267C22×C53#52511yescharacteristic
268C22×C53#52511yescharacteristic
269C22×C53#57611yescharacteristic
270C22×C53#57711yescharacteristic
271C22×C53#57711yescharacteristic
272C22×C53#57811yescharacteristic
273C4×C53—22no—
274C4×C53—42no—
275C4×C53—42no—
276C4×C53—81no—
277C4×C53—81no—
278C4×C53C4×#4111yescharacteristic
279C4×C53C4×#4111yescharacteristic
280C4×C53#53311yescharacteristic
281C4×C53#53311yescharacteristic
282Dic53—22no—
283Dic53—22no—
284Dic53—22no—
285Dic53—22no—
286Dic53—42no—
287Dic53—42no—
288Dic53—81no—
289Dic53—81no—
290C2×#433—531no—
291#2314—531no—
292#2274—531no—
293#1949—531no—
294#1949—531no—
295#1948—531no—
296#1947—531no—
297#1854—531no—
298#1854—531no—
299#1853—531no—
300#1852D5311yescharacteristic
301#1776—531no—
302#1776—531no—
303C4×#107—531no—
304C4×#106—531no—
305C2×C53—42no—
306C2×C53#229611yescharacteristic
307C2×C53#291211yescharacteristic
308C2×C53#291311yescharacteristic
309C2×C53#311811yescharacteristic
310C2×C53#311911yescharacteristic
311C2×C53#312011yescharacteristic
312C2×C53#312111yescharacteristic
313C2×#117—531no—
314C2×#117—531no—
315C22×C4⋊C4C2×D5311yescharacteristic
316#433—538no—
317C2×#107—531no—
318C2×#107—531no—
319C2×#107C2×D5311yescharacteristic
320C2×#107C2×D5311yescharacteristic
321C2×#106—531no—
322C2×#106—531no—
323C2×#106—531no—
324C2×#106—531no—
325C2×C4⋊C8—531no—
326C2×C4⋊C8—531no—
327C2×C22⋊C8—531no—
328C2×C22⋊C8C2×D5311yescharacteristic
329C2×#97—531no—
330C2×C4×C8—531no—
331#388—1061no—
332#388—1061no—
333#387—1061no—
334#387—1061no—
335#384—531no—
336#382—531no—
337#380—531no—
338#380C2×D5311yescharacteristic
339C4×C4⋊C4—531no—
340C4×C4⋊C4—531no—
341C4×C4⋊C4—531no—
342C4×C4⋊C4—531no—
343C4×C4⋊C4—531no—
344C4×C4⋊C4—531no—
345C4×C4⋊C4—1061no—
346C4×C4⋊C4—1061no—
347C4×C22⋊C4—1061no—
348C4×C22⋊C4—1061no—
349#340—531no—
350#340—531no—
351#340—531no—
352#340—531no—
353#340—531no—
354#340—531no—
355#340C2×D5311yescharacteristic
356#340C2×D5311yescharacteristic
357C53#2136711yescharacteristic, complemented, Sylow
358C23×C4—21no—
359C23×C4C22×D5311yescharacteristic
360C22×C8—531no—
361C22×C8—531no—
362C22×C8C22×D5311yescharacteristic
363C22×C8C22×D5311yescharacteristic
364#117—534no—
365#117—534no—
366#117—1062no—
367#117—1062no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—531no—
373C2×C4⋊C4—531no—
374C2×C4⋊C4—531no—
375C2×C4⋊C4—531no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—1061no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4C22×D5311yescharacteristic
380C2×C4⋊C4C22×D5311yescharacteristic
381C2×C4⋊C4C22×D5311yescharacteristic
382C2×C4⋊C4C22×D5311yescharacteristic
383C2×C22⋊C4—21no—
384C2×C22⋊C4—531no—
385C2×C22⋊C4—531no—
386C2×C22⋊C4—1061no—
387C2×C22⋊C4—1061no—
388C2×C42—531no—
389C2×C42—531no—
390C2×C42—1061no—
391C2×C42—1061no—
392C2×C42—1061no—
393C2×C42—1061no—
394C2×C42—1061no—
395C2×C42—1061no—
396C2×C42—1061no—
397#107—22no—
398#107—22no—
399#107—1062no—
400#107—1062no—
401#106—1062no—
402#106—1062no—
403#106—1062no—
404#106—1062no—
405C4⋊C8—534no—
406C4⋊C8—534no—
407C22⋊C8—534no—
408C22⋊C8C4×D5314yes—
409#97—534no—
410C4×C8—534no—
411#95—21no—
412#95—1061no—
413#95—1061no—
414#95—1061no—
415#95—1061no—
416#95—1061no—
417#95—1061no—
418#95—1061no—
419#95—1061no—
420#95—1061no—
421C24D4×D5311yescharacteristic
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—81no—
432C22×C4—81no—
433C22×C4—531no—
434C22×C4—531no—
435C22×C4—531no—
436C22×C4—531no—
437C22×C4—1061no—
438C22×C4—1061no—
439C22×C4—1061no—
440C22×C4—1061no—
441C22×C4C2×C4×D5312yes—
442C22×C4D4×D5311yescharacteristic
443C22×C4Q8×D5312yes—
444C22×C4C23×D5311yescharacteristic
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—532no—
450C2×C8—532no—
451C2×C8—1061no—
452C2×C8—1061no—
453C2×C8—1061no—
454C2×C8—1061no—
455C2×C8C2×C4×D5312yes—
456C2×C8C2×C4×D5312yes—
457C4⋊C4—24no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—534no—
465C4⋊C4—534no—
466C4⋊C4—532no—
467C4⋊C4—532no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—4241no—
475C4⋊C4—4241no—
476C4⋊C4#1171994712yes—
477C4⋊C4#1171994712yes—
478C22⋊C4—81no—
479C22⋊C4—534no—
480C22⋊C4—534no—
481C22⋊C4—1064no—
482C22⋊C4—4241no—
483C42—532no—
484C42—532no—
485C42—532no—
486C42—532no—
487C42—1064no—
488C42—4241no—
489C42—4241no—
490C42—4241no—
491C42—4241no—
492C42—4241no—
493C42—4241no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23C4⋊C4×D5312yes—
501C23C2×D4×D5311yescharacteristic
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—84no—
521C2×C4—532no—
522C2×C4—532no—
523C2×C4—532no—
524C2×C4—532no—
525C2×C4—1061no—
526C2×C4—1061no—
527C2×C4—1061no—
528C2×C4—1061no—
529C2×C4—1061no—
530C2×C4—1061no—
531C2×C4—1061no—
532C2×C4—1061no—
533C2×C4—1062no—
534C2×C4—1062no—
535C2×C4—2121no—
536C2×C4—2121no—
537C2×C4—2121no—
538C2×C4—2121no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4—2121no—
543C2×C4—2121no—
544C2×C4—2121no—
545C2×C4C4⋊C4×D5312yes—
546C2×C4C22×C4×D5311yescharacteristic
547C2×C4C2×Q8×D5311yescharacteristic
548C2×C4#41×D5311yescharacteristic
549C2×C4#41×D5311yescharacteristic
550C2×C4C2×#1171994711yescharacteristic
551C2×C4C2×#1171994711yescharacteristic
552C8—22no—
553C8—22no—
554C8—1062no—
555C8—1062no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22C2×C4⋊C4×D5311yescharacteristic
564C22#1681112911yescharacteristic
565C22#1681113111yescharacteristic
566C22#1681142411yescharacteristic
567C22#1681143211yescharacteristic
568C22#1681143311yescharacteristic
569C22#1681143611yescharacteristic
570C4—22no—
571C4—42no—
572C4—42no—
573C4—81no—
574C4—81no—
575C4—1062no—
576C4—1062no—
577C4—1062no—
578C4—1062no—
579C4—4241no—
580C4—4241no—
581C4—2122no—
582C4—2122no—
583C4C4×#1171994711yescharacteristic
584C4C4×#1171994711yescharacteristic
585C4#1681117511yescharacteristic
586C4#1681117511yescharacteristic
587C2—42no—
588C2#2027719411yescharacteristic
589C2#2028330911yescharacteristic
590C2#2028331111yescharacteristic
591C2#2028489211yescharacteristic
592C2#2028491711yescharacteristic
593C2#2028492411yescharacteristic
594C2#2028494511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1852C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
cosocleC22×C4×C53C24
hypercenter#1852D53
hypocenterC53#21367
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
4. #20275480C2semidirect
5. #20275480C2semidirect
6. #20275477C2semidirect
7. #20275472C2semidirect
11. #20274614C2semidirect
12. #20274611C2semidirect
13. #107×Dic53C2semidirect
14. #20274603C2semidirect
18. #16810681C4semidirect
93. #13133146C2×C4semidirect
94. #13133146C2×C4semidirect
96. #13133127C2×C4semidirect
97. C8×Dic53C2×C4semidirect
357. C53#21367semidirect

Maximal subgroups

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

subgroupindex
C2×#168106812
#202773222
#202770722
#202754802
#202754772
#202754722
#202749162
#202749122
#202749082
#202746142
#202746112
#107×Dic532
#202746032
C53×#18522
#2136753