Finite Group Databasebeta

← #27436145

Subgroups — #27436145

Order 13568.

790 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810911C211yesmaximal, characteristic
2C2×#16810750C211yesmaximal, characteristic, complemented
3#20280071C211yesmaximal, characteristic
4#20279702C211yesmaximal, characteristic, complemented
5#20276219C211yesmaximal, characteristic, complemented
6#20276213C211yesmaximal, characteristic, complemented
7#20276085C211yesmaximal, characteristic
8#20275686C211yesmaximal, characteristic, complemented
9#20275480C211yesmaximal, characteristic, complemented
10#20275454C211yesmaximal, characteristic, complemented
11#20275440C211yesmaximal, characteristic, complemented
12#20275409C211yesmaximal, characteristic
13#20275401C211yesmaximal, characteristic
14#20274542C211yesmaximal, characteristic
15C53×#2067C211yesmaximal, characteristic
16C2×#13133216—21no—
17C2×#13133216C2211yescharacteristic
18C2×C4×Dic106C2211yescharacteristic
19C2×D4×Dic53C2211yescharacteristic
20#16810911—24no—
21#16810750—24no—
22C2×#13133149C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×#13133146C2211yescharacteristic
26C2×#13133146C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×#13133138C2211yescharacteristic
29C2×#13133127C2211yescharacteristic
30C2×#13133127C2211yescharacteristic
31C2×C4×#88322C2211yescharacteristic
32#16810559—21no—
33#16810557—21no—
34#16810556—21no—
35#16810535—21no—
36#16810527—21no—
37#16810526C2211yescharacteristic
38#16810522C2211yescharacteristic
39#16810522C2211yescharacteristic
40#16810521C2211yescharacteristic
41#16810521C2211yescharacteristic
42#16810519C2211yescharacteristic
43#16810498—21no—
44#16810495—21no—
45#16810488C2211yescharacteristic
46C4⋊C4×Dic53C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53#16810385C2211yescharacteristic
54C2×#107×C53C2211yescharacteristic
55C2×C4⋊C8×C53C2211yescharacteristic
56C2×#102×C53C2211yescharacteristic
57C2×#102×C53C2211yescharacteristic
58C53×#390C2211yescharacteristic
59C53×#389C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C22×Dic106—21no—
62C22×Dic106D411yescharacteristic
63C23×Dic53—21no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133216—22no—
67#13133216—22no—
68#13133216—44no—
69C4×Dic106—22no—
70C4×Dic106—22no—
71D4×Dic53—22no—
72D4×Dic53—22no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53D411yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919D411yescharacteristic
85C2×#11719919D411yescharacteristic
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53—21no—
88C2×C22⋊Dic53—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53C2311yescharacteristic
91C2×C4×Dic53C2311yescharacteristic
92#13133149—22no—
93#13133148—22no—
94#13133148—22no—
95#13133146—22no—
96#13133146D414yes—
97#13133138—22no—
98#13133138Q814yescomplemented
99#13133127—22no—
100#13133127—22no—
101C4×#88322—22no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110C22×D4×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C22⋊C4×C53—21no—
118C2×C22⋊C4×C53—21no—
119C2×C42×C53C2311yescharacteristic
120#107×C53—22no—
121C4⋊C8×C53—22no—
122#102×C53—22no—
123#102×C53—22no—
124#95×C53—21no—
125C24×C53—21no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53C2×Q812yes—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2411yescharacteristic
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—42no—
143C22×Dic53—81no—
144C22×Dic53—81no—
145C22×Dic53C2×D411yescharacteristic
146C22×Dic53C2×D411yescharacteristic
147C22×Dic53C2×D411yescharacteristic
148C22×Dic53C2×D411yescharacteristic
149C2×Dic106—22no—
150C2×Dic106—22no—
151C2×Dic106—41no—
152C2×Dic106—42no—
153C2×Dic106—42no—
154C2×Dic106—42no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53C2×D412yes—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—42no—
167C4⋊C4×C53#4112yes—
168C22⋊C4×C53—42no—
169C22⋊C4×C53—81no—
170C42×C53—21no—
171C42×C53—21no—
172C22⋊Dic53—42no—
173C22⋊Dic53—42no—
174C22⋊Dic53—81no—
175C4⋊Dic53—21no—
176C4⋊Dic53—22no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—42no—
184C4⋊Dic53—81no—
185C4⋊Dic53C2×Q812yes—
186#11719919—22no—
187#11719919—22no—
188#11719919—42no—
189#11719919—42no—
190#11719919—42no—
191#11719919—81no—
192C4×Dic53—21no—
193C4×Dic53—21no—
194C4×Dic53—42no—
195C4×Dic53C2×D412yes—
196C4×Dic53#4112yes—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322C2×D412yes—
202C2×#88322C2×Q812yes—
203C23×C53—22no—
204C23×C53—41no—
205C23×C53—41no—
206C23×C53—41no—
207C23×C53—41no—
208C23×C53—41no—
209C23×C53—41no—
210C23×C53C22×D411yescharacteristic
211D4×C53—22no—
212D4×C53—22no—
213D4×C53—41no—
214D4×C53—41no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—22no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53AGL(1,ℤ/8)12yes—
230C2×C4×C53C22×D411yescharacteristic
231C2×C4×C53C22×Q811yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×C4×C532−1+411yescharacteristic
235C2×C4×C532−1+411yescharacteristic
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—22no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—84no—
259C2×Dic53#12112yes—
260C2×Dic53#12112yes—
261Dic106—44no—
262Dic106—84no—
263C8×C53—22no—
264C8×C53—41no—
265#88322—22no—
266#88322—22no—
267#25485—531nomaximal, Sylow
268C22×C53—42no—
269C22×C53—42no—
270C22×C53—42no—
271C22×C53—81no—
272C22×C53—81no—
273C22×C53—81no—
274C22×C53—81no—
275C22×C53C2×#12111yescharacteristic
276C22×C53#54811yescharacteristic
277C22×C53D4×Q811yescharacteristic
278C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C53#57511yescharacteristic
280C22×C53#57711yescharacteristic
281C22×C53#57711yescharacteristic
282C4×C53—22no—
283C4×C53—42no—
284C4×C53—41no—
285C4×C53—81no—
286C4×C53—81no—
287C4×C53#53311yescharacteristic
288C4×C53#54111yescharacteristic
289C4×C53#54711yescharacteristic
290C4×C53D4×Q811yescharacteristic
291Dic53—22no—
292Dic53—22no—
293Dic53—42no—
294Dic53—41no—
295Dic53—41no—
296Dic53—81no—
297Dic53—81no—
298Dic53—81no—
299C2×#476—531no—
300C2×#446—531no—
301#2580—531no—
302#2546—531no—
303#2088—531no—
304#2088—531no—
305#2067D5311yescharacteristic
306#2065—531no—
307#1993—531no—
308#1949—531no—
309#1943—531no—
310#1941—531no—
311#1936—531no—
312#1934—531no—
313#1762—531no—
314C2×C53—82no—
315C2×C53#242911yescharacteristic
316C2×C53#305511yescharacteristic
317C2×C53#305711yescharacteristic
318C2×C53#337111yescharacteristic
319C2×C53#337311yescharacteristic
320C2×C53#338411yescharacteristic
321C2×C53#338811yescharacteristic
322C2×C4⋊Q8—531no—
323C2×C4⋊Q8—1061no—
324C2×C4×Q8—531no—
325C2×C4×D4—531no—
326#476—1064no—
327#446—1064no—
328C2×#107C2×D5311yescharacteristic
329C2×#106—531no—
330C2×#106—531no—
331C2×#106—531no—
332C2×C4⋊C8—531no—
333C2×C4⋊C8—531no—
334C2×C4⋊C8C2×D5311yescharacteristic
335C2×#102—531no—
336C2×#102—531no—
337C2×#102C2×D5311yescharacteristic
338C2×#102C2×D5311yescharacteristic
339C2×#97—531no—
340C2×#97—531no—
341C2×C4×C8—531no—
342#399—1061no—
343#398—1061no—
344#398—1061no—
345#393—1061no—
346#391—1061no—
347#390—531no—
348#390C2×D5311yescharacteristic
349#389—531no—
350#389—531no—
351#389—531no—
352#389—531no—
353#389—531no—
354#389C2×D5311yescharacteristic
355#385—1061no—
356#385—1061no—
357#382—531no—
358C4×C4⋊C4—531no—
359#340—531no—
360#340—531no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340C2×D5311yescharacteristic
367C53#2548511yescharacteristic, complemented, Sylow
368C22×Q8—531no—
369C22×Q8—1061no—
370C22×D4C22×D5311yescharacteristic
371C23×C4—1061no—
372C22×C8—531no—
373C22×C8—531no—
374C22×C8C22×D5311yescharacteristic
375C22×C8C22×D5311yescharacteristic
376C4⋊Q8—1062no—
377C4⋊Q8—1062no—
378C4⋊Q8—2124no—
379C4×Q8—1062no—
380C4×Q8—1062no—
381C4×D4—1062no—
382C4×D4—1062no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—531no—
385C2×C4⋊C4—531no—
386C2×C4⋊C4—531no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4C22×D5311yescharacteristic
398C2×C4⋊C4C22×D5311yescharacteristic
399C2×C4⋊C4C22×D5311yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—1061no—
403C2×C22⋊C4—1061no—
404C2×C22⋊C4—1061no—
405C2×C42—531no—
406C2×C42—531no—
407C2×C42—1061no—
408C2×C42C22×D5311yescharacteristic
409#107—22no—
410#106—1062no—
411#106—1062no—
412#106—1062no—
413C4⋊C8—22no—
414C4⋊C8—534no—
415C4⋊C8—1062no—
416#102—22no—
417#102—22no—
418#102—534no—
419#102—1062no—
420#97—1062no—
421#97—1062no—
422C4×C8—1062no—
423#95—21no—
424#95—1061no—
425#95—1061no—
426#95—1061no—
427#95—1061no—
428#95—1061no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432C24—21no—
433C2×Q8—1062no—
434C2×Q8—1062no—
435C2×Q8—2121no—
436C2×Q8—2122no—
437C2×Q8—2122no—
438C2×Q8—2122no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4Q8×D5312yes—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—531no—
449C22×C4—531no—
450C22×C4—531no—
451C22×C4—531no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4—1061no—
456C22×C4—4241no—
457C22×C4—4241no—
458C22×C4—2122no—
459C22×C4D4×D5311yescharacteristic
460C22×C4D4×D5311yescharacteristic
461C22×C4C23×D5311yescharacteristic
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—532no—
468C2×C8—532no—
469C2×C8—1061no—
470C2×C8—1061no—
471C2×C8—1061no—
472C2×C8—1061no—
473C2×C8D4×D5312yes—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—42no—
480C4⋊C4—532no—
481C4⋊C4—1062no—
482C4⋊C4—1062no—
483C4⋊C4—1062no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—4241no—
492C4⋊C4—4241no—
493C4⋊C4—2122no—
494C4⋊C4—2122no—
495C4⋊C4—2122no—
496C4⋊C4—2122no—
497C4⋊C4#1171994712yes—
498C22⋊C4—42no—
499C22⋊C4—81no—
500C22⋊C4—4241no—
501C22⋊C4—2122no—
502C22⋊C4—2122no—
503C42—21no—
504C42—21no—
505C42—532no—
506C42—532no—
507C42—1061no—
508C42—1061no—
509C42—2122no—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23C2×D4×D5311yescharacteristic
518Q8—4244no—
519Q8—2124no—
520D4—22no—
521D4—22no—
522D4—41no—
523D4—41no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—532no—
539C2×C4—532no—
540C2×C4—1061no—
541C2×C4—1061no—
542C2×C4—1061no—
543C2×C4—1061no—
544C2×C4—1062no—
545C2×C4—1061no—
546C2×C4—1061no—
547C2×C4—1061no—
548C2×C4—1061no—
549C2×C4—1061no—
550C2×C4—1061no—
551C2×C4—4244no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4#1313324112yes—
564C2×C4C2×D4×D5311yescharacteristic
565C2×C4C2×Q8×D5311yescharacteristic
566C2×C4#41×D5311yescharacteristic
567C2×C4C2×#1171994711yescharacteristic
568C2×C4#1313326411yescharacteristic
569C2×C4#1313326411yescharacteristic
570C8—22no—
571C8—41no—
572C8—1062no—
573C8—1062no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22#121×D5311yescharacteristic
582C22#1681127911yescharacteristic
583C22#1681128611yescharacteristic
584C22C2×#1313324111yescharacteristic
585C22#1681142011yescharacteristic
586C22#1681143211yescharacteristic
587C22#1681143311yescharacteristic
588C4—22no—
589C4—42no—
590C4—41no—
591C4—81no—
592C4—81no—
593C4—1062no—
594C4—1062no—
595C4—4241no—
596C4—4241no—
597C4—4241no—
598C4—2122no—
599C4—2121no—
600C4—2121no—
601C4#1681117511yescharacteristic
602C4#1681123011yescharacteristic
603C4#1681127311yescharacteristic
604C4#1681128611yescharacteristic
605C2—82no—
606C2#2027830711yescharacteristic
607C2#2028433111yescharacteristic
608C2#2028435211yescharacteristic
609C2#2028748711yescharacteristic
610C2#2028751111yescharacteristic
611C2#2028761811yescharacteristic
612C2#2028766411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2067C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2067D53
hypocenterC53#25485
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2067

Complements

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

NH ≅ G/Ntype
2. C2×#16810750C2semidirect
4. #20279702C2semidirect
5. #20276219C2semidirect
6. #20276213C2semidirect
8. #20275686C2semidirect
9. #20275480C2semidirect
10. #20275454C2semidirect
11. #20275440C2semidirect
98. #13133138Q8semidirect
367. C53#25485semidirect

Maximal subgroups

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

subgroupindex
C2×#168109112
C2×#168107502
#202800712
#202797022
#202762192
#202762132
#202760852
#202756862
#202754802
#202754542
#202754402
#202754092
#202754012
#202745422
C53×#20672
#2548553