Finite Group Databasebeta

← #27401787

Subgroups — #27401787

Order 13568.

866 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×#107×D53C211yesmaximal, characteristic
2#20277583C211yesmaximal, characteristic
3#20277580C211yesmaximal, characteristic
4#20275452C211yesmaximal, characteristic, complemented
5#20275450C211yesmaximal, characteristic, complemented
6#20275309C211yesmaximal, characteristic, complemented
7#20275308C211yesmaximal, characteristic, complemented
8#20275306C211yesmaximal, characteristic, complemented
9#20274800C211yesmaximal, characteristic
10#20274800C211yesmaximal, characteristic
11#20274739C211yesmaximal, characteristic
12#20274613C211yesmaximal, characteristic, complemented
13#20274611C211yesmaximal, characteristic, complemented
14#20274490C211yesmaximal, characteristic
15C53×#1921C211yesmaximal, characteristic, complemented
16C22×C8×D53C2211yescharacteristic
17C2×#13133175—21no—
18C2×#13133174—21no—
19C2×C4⋊C4×D53C2211yescharacteristic
20C2×C4⋊C4×D53C2211yescharacteristic
21C2×#13133151C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24#107×D53C418yescomplemented
25C2×#13133148C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×#13133129C2211yescharacteristic
28C2×#13133128C2211yescharacteristic
29C2×#13133127C2211yescharacteristic
30#16810522C2211yescharacteristic
31#16810521C2211yescharacteristic
32#16810519C2211yescharacteristic
33#16810510—21no—
34#16810507—21no—
35#16810505—21no—
36#16810504—21no—
37#16810485—21no—
38#16810485—21no—
39#16810480C2211yescharacteristic
40#16810480C2211yescharacteristic
41C4⋊C4×Dic53C2211yescharacteristic
42C4⋊C4×Dic53C2211yescharacteristic
43C4×C4⋊Dic53C2211yescharacteristic
44C4×#11719921C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic
54C53×#389C2211yescharacteristic
55C4×C4⋊C4×C53C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C22×C4×D53—21no—
61C22×C4×D53—21no—
62C22×C4×D53C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C2×C8×D53—22no—
66C2×C8×D53—22no—
67C2×C8×D53C2×C414yes—
68#13133175—44no—
69#13133174—44no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C4⋊C4×D53—22no—
80C4⋊C4×D53—22no—
81C4⋊C4×D53C2×C414yes—
82C4⋊C4×D53C2×C414yes—
83C2×#11719921—21no—
84C2×#11719921—21no—
85C2×#11719921D411yescharacteristic
86C2×#11719921D411yescharacteristic
87C2×C4×Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53C2311yescharacteristic
91#13133151C2×C414yes—
92#13133150C2×C414yes—
93#13133150C2×C414yes—
94#13133148—22no—
95#13133148—22no—
96#13133129—22no—
97#13133128—22no—
98#13133127—22no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124D411yescharacteristic
108#13133124D411yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C42×C53—21no—
117C2×C42×C53C2311yescharacteristic
118#107×C53C2×C414yes—
119#95×C53—21no—
120#95×C53—21no—
121C23×D53C2×D411yescharacteristic
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2411yescharacteristic
129C22×Dic53—21no—
130C22×Dic53—21no—
131C22×Dic53—21no—
132C22×Dic53—21no—
133C22×Dic53C2×D411yescharacteristic
134C22×Dic53C2×D411yescharacteristic
135C22×Dic53C2×D411yescharacteristic
136C2×C4×D53—21no—
137C2×C4×D53—21no—
138C2×C4×D53—21no—
139C2×C4×D53—21no—
140C2×C4×D53—24no—
141C2×C4×D53—24no—
142C2×C4×D53—42no—
143C2×C4×D53—42no—
144C2×C4×D53—42no—
145C2×C4×D53—42no—
146C2×C4×D53C22×C412yes—
147C2×C4×D53#4112yes—
148C2×C8×C53—21no—
149C2×C8×C53—21no—
150C2×C8×C53—21no—
151C2×C8×C53—21no—
152C2×C8×C53—21no—
153C2×C8×C53C22×C412yes—
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—42no—
161C4⋊C4×C53C22×C412yes—
162C4⋊C4×C53C22×C412yes—
163C42×C53—21no—
164C42×C53—21no—
165C42×C53—42no—
166#11719921—22no—
167#11719921—22no—
168#11719921—42no—
169#11719921—42no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—42no—
177C4⋊Dic53C22×C412yes—
178C4⋊Dic53C22×C412yes—
179#11719919—24no—
180#11719919—24no—
181#11719919—42no—
182#11719919—42no—
183C4×Dic53—21no—
184C4×Dic53—21no—
185C4×Dic53—42no—
186C4×Dic53—42no—
187C4×Dic53—42no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322C22×C412yes—
192C2×#88322#4112yes—
193C2×#88322#4112yes—
194C8×D53—28no—
195C23×C53C22×D411yescharacteristic
196C22×D53—21no—
197C22×D53—21no—
198C22×D53—21no—
199C22×D53—21no—
200C22×D53C4×D412yes—
201C22×D53AGL(1,ℤ/8)12yes—
202C22×D53AGL(1,ℤ/8)12yes—
203C2×C4×C53—22no—
204C2×C4×C53—22no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—41no—
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×C53C23×C411yescharacteristic
222C2×C4×C53C2×#4111yescharacteristic
223C2×C4×C53C2×#4111yescharacteristic
224C2×C4×C53C2×#4111yescharacteristic
225C2×C4×C532−1+411yescharacteristic
226C2×C4×C532−1+411yescharacteristic
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—22no—
230C2×Dic53—22no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53C4×D412yes—
248C4×D53—22no—
249C4×D53—22no—
250C4×D53—22no—
251C4×D53—22no—
252C4×D53—48no—
253C4×D53—48no—
254C8×C53—22no—
255C8×C53—41no—
256#88322—41no—
257#88322#11714yescomplemented
258#22617—531nomaximal, Sylow
259C22×C53C2×C4×D411yescharacteristic
260C22×C53#54711yescharacteristic
261C22×C53#54711yescharacteristic
262C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
263C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
264C22×C53#57811yescharacteristic
265C22×C53#57811yescharacteristic
266C2×D53—22no—
267C2×D53—22no—
268C2×D53—22no—
269C2×D53—22no—
270C2×D53—22no—
271C2×D53—22no—
272C2×D53#44214yes—
273C4×C53—41no—
274C4×C53—41no—
275C4×C53—42no—
276C4×C53—42no—
277C4×C53—81no—
278C4×C53—81no—
279C4×C53C2×#11711yescharacteristic
280C4×C53#51911yescharacteristic
281C4×C53#55211yescharacteristic
282C4×C53#55211yescharacteristic
283Dic53—22no—
284Dic53—41no—
285Dic53—41no—
286Dic53—42no—
287Dic53—42no—
288Dic53—81no—
289Dic53—81no—
290C22×#107—531no—
291#2344—531no—
292#2344—531no—
293#1943—531no—
294#1943—531no—
295#1921—531no—
296#1921—531no—
297#1921—531no—
298#1921D5311yescharacteristic, complemented
299#1823—531no—
300#1823—531no—
301#1807—531no—
302#1776—531no—
303#1776—531no—
304#1752—531no—
305C2×C53#234411yescharacteristic
306C2×C53C2×#44211yescharacteristic
307C2×C53#295011yescharacteristic
308C2×C53#334311yescharacteristic
309C2×C53#334311yescharacteristic
310C2×C53#334311yescharacteristic
311C2×C53#334311yescharacteristic
312D53—24no—
313C23×C8—531no—
314C2×#117—1061no—
315C2×#117—1061no—
316C22×C4⋊C4—531no—
317C22×C4⋊C4—531no—
318C2×#107—531no—
319C2×#107—531no—
320C2×#107—531no—
321C2×#107—538no—
322C2×#107C2×D5311yescharacteristic
323C2×#106—531no—
324C2×#106—531no—
325C2×C22⋊C8—531no—
326C2×#97—531no—
327C2×#97—531no—
328#389—531no—
329#389—531no—
330#389—531no—
331#389C2×D5311yescharacteristic
332#387—1061no—
333#387—1061no—
334#387—1061no—
335#387—1061no—
336#381—1061no—
337#381—1061no—
338#380—531no—
339#380—531no—
340C4×C4⋊C4—531no—
341C4×C4⋊C4—531no—
342C4×C4⋊C4—531no—
343C4×C4⋊C4C2×D5311yescharacteristic
344C4×C22⋊C4—531no—
345#340—531no—
346#340—531no—
347#340—531no—
348#340—531no—
349#340—531no—
350#340—531no—
351#340—531no—
352#340C2×D5311yescharacteristic
353#340C2×D5311yescharacteristic
354#340C2×D5311yescharacteristic
355#336—531no—
356#336C2×D5311yescharacteristic
357C53#2261711yescharacteristic, complemented, Sylow
358C23×C4—531no—
359C23×C4—1061no—
360C23×C4—1061no—
361C22×C8—531no—
362C22×C8—534no—
363C22×C8—531no—
364C22×C8—1062no—
365C22×C8—1062no—
366C22×C8C22×D5311yescharacteristic
367C22×C8C22×D5311yescharacteristic
368#117—2124no—
369#117—2124no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—531no—
373C2×C4⋊C4—534no—
374C2×C4⋊C4—534no—
375C2×C4⋊C4—531no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—1062no—
378C2×C4⋊C4—1062no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4C22×D5311yescharacteristic
385C2×C4⋊C4C22×D5311yescharacteristic
386C2×C4⋊C4C22×D5311yescharacteristic
387C2×C4⋊C4C22×D5311yescharacteristic
388C2×C22⋊C4—531no—
389C2×C22⋊C4—531no—
390C2×C22⋊C4—1061no—
391C2×C22⋊C4—1061no—
392C2×C42—21no—
393C2×C42—531no—
394C2×C42—1061no—
395C2×C42—1061no—
396C2×C42—1061no—
397C2×C42C22×D5311yescharacteristic
398#107—534no—
399#107—534no—
400#107—534no—
401#107C4×D5314yes—
402#106—1062no—
403#106—1062no—
404C22⋊C8—1062no—
405#97—1062no—
406#97—1062no—
407#95—21no—
408#95—21no—
409#95—531no—
410#95—531no—
411#95—1061no—
412#95—1061no—
413#95—1061no—
414#95—1061no—
415#95—1061no—
416#95—1061no—
417#95—1061no—
418#95—1061no—
419C24—531no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—531no—
425C22×C4—532no—
426C22×C4—532no—
427C22×C4—531no—
428C22×C4—531no—
429C22×C4—1064no—
430C22×C4—1064no—
431C22×C4—1061no—
432C22×C4—1061no—
433C22×C4—1061no—
434C22×C4—1061no—
435C22×C4—1061no—
436C22×C4—1061no—
437C22×C4—1061no—
438C22×C4—1061no—
439C22×C4—2122no—
440C22×C4—2122no—
441C22×C4—2122no—
442C22×C4—2122no—
443C22×C4D4×D5311yescharacteristic
444C22×C4D4×D5311yescharacteristic
445C22×C4C23×D5311yescharacteristic
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—532no—
452C2×C8—532no—
453C2×C8—532no—
454C2×C8—1068no—
455C2×C8—1061no—
456C2×C8—1061no—
457C2×C8—1061no—
458C2×C8C2×C4×D5312yes—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—42no—
466C4⋊C4—532no—
467C4⋊C4—532no—
468C4⋊C4—1064no—
469C4⋊C4—1064no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—2122no—
477C4⋊C4—2122no—
478C4⋊C4—2122no—
479C4⋊C4C2×C4×D5312yes—
480C4⋊C4C2×C4×D5312yes—
481C22⋊C4—1062no—
482C22⋊C4—1062no—
483C22⋊C4—2122no—
484C22⋊C4—2122no—
485C42—21no—
486C42—21no—
487C42—42no—
488C42—1061no—
489C42—1061no—
490C42—2122no—
491C42—2122no—
492C42—2122no—
493C23—532no—
494C23—532no—
495C23—532no—
496C23—1061no—
497C23—1061no—
498C23—1061no—
499C23—1061no—
500C23C2×D4×D5311yescharacteristic
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—41no—
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—532no—
520C2×C4—1061no—
521C2×C4—1061no—
522C2×C4—1062no—
523C2×C4—1062no—
524C2×C4—1062no—
525C2×C4—1062no—
526C2×C4—1061no—
527C2×C4—1061no—
528C2×C4—1062no—
529C2×C4—1062no—
530C2×C4—1061no—
531C2×C4—1061no—
532C2×C4—1061no—
533C2×C4—1061no—
534C2×C4—2128no—
535C2×C4—2128no—
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×C4—2121no—
546C2×C4C22×C4×D5311yescharacteristic
547C2×C4#41×D5311yescharacteristic
548C2×C4#41×D5311yescharacteristic
549C2×C4#41×D5311yescharacteristic
550C2×C4#1313326411yescharacteristic
551C2×C4#1313326511yescharacteristic
552C8—22no—
553C8—41no—
554C8—534no—
555C8—2121no—
556C22—534no—
557C22—1062no—
558C22—1062no—
559C22—1062no—
560C22—1062no—
561C22—1062no—
562C22—1062no—
563C22C4×D4×D5311yescharacteristic
564C22#1681127111yescharacteristic
565C22#1681127511yescharacteristic
566C22AGL(1,ℤ/8)×D5311yescharacteristic
567C22AGL(1,ℤ/8)×D5311yescharacteristic
568C22#1681143611yescharacteristic
569C22#1681143711yescharacteristic
570C4—42no—
571C4—42no—
572C4—41no—
573C4—41no—
574C4—81no—
575C4—81no—
576C4—1062no—
577C4—4241no—
578C4—4241no—
579C4—2122no—
580C4—2122no—
581C4—2121no—
582C4—2121no—
583C4#117×D5311yescharacteristic
584C4#1681109911yescharacteristic
585C4#1681130511yescharacteristic
586C4#1681130911yescharacteristic
587C2—1064no—
588C2#2027758111yescharacteristic
589C2#442×D5311yescharacteristic
590C2#2028357811yescharacteristic
591C2#2028714811yescharacteristic
592C2#2028715211yescharacteristic
593C2#2028715811yescharacteristic
594C2#2028716211yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1921C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1921D53
hypocenterC53#22617
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1921

Complements

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

NH ≅ G/Ntype
4. #20275452C2semidirect
5. #20275450C2semidirect
6. #20275309C2semidirect
7. #20275308C2semidirect
8. #20275306C2semidirect
12. #20274613C2semidirect
13. #20274611C2semidirect
15. C53×#1921C2semidirect
24. #107×D53C4semidirect
257. #88322#117semidirect
298. #1921D53semidirect
357. C53#22617semidirect

Maximal subgroups

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

subgroupindex
C2×#107×D532
#202775832
#202775802
#202754522
#202754502
#202753092
#202753082
#202753062
#202748002
#202747392
#202746132
#202746112
#202744902
C53×#19212
#2261753