Finite Group Databasebeta

← #27414019

Subgroups — #27414019

Order 13568.

814 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277196C211yesmaximal, characteristic
2#382×D53C211yesmaximal, characteristic
3#20276288C211yesmaximal, characteristic, complemented
4#20276287C211yesmaximal, characteristic, complemented
5#20276286C211yesmaximal, characteristic, complemented
6#20276285C211yesmaximal, characteristic, complemented
7#20276284C211yesmaximal, characteristic, complemented
8#20276283C211yesmaximal, characteristic, complemented
9#20276282C211yesmaximal, characteristic, complemented
10#20274433C211yesmaximal, characteristic
11#20274433C211yesmaximal, characteristic
12#20274433C211yesmaximal, characteristic
13#20274433C211yesmaximal, characteristic
14#20274407C211yesmaximal, characteristic
15C53×#2101C211yesmaximal, characteristic, complemented
16C22×#11719912C2211yescharacteristic
17C22×#11719912C2211yescharacteristic
18C2×#13133175C2211yescharacteristic
19C2×#13133175C2211yescharacteristic
20C2×C4⋊C4×D53C2211yescharacteristic
21C2×C4⋊C4×D53C2211yescharacteristic
22C2×C42×D53C2211yescharacteristic
23#16810522C2211yescharacteristic
24#16810521C2211yescharacteristic
25#16810519C2211yescharacteristic
26#16810497—21no—
27#16810494—21no—
28#16810489C2211yescharacteristic
29#16810488C2211yescharacteristic
30#16810488C2211yescharacteristic
31#16810487—21no—
32#16810487—21no—
33C4⋊C4×Dic53—21no—
34#16810474—21no—
35#95×D53—21no—
36#95×D53—21no—
37#16810386C2211yescharacteristic
38#16810386C2211yescharacteristic
39#16810386C2211yescharacteristic
40#16810386C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810378C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C53×#389C2211yescharacteristic
53C53×#382C2211yescharacteristic
54C53×#340C2211yescharacteristic
55C53×#340C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×C4×D53—21no—
60C22×C4×D53—21no—
61C22×C4×D53D411yescharacteristic
62C22×C4×D53D411yescharacteristic
63C22×C4×D53C2311yescharacteristic
64C2×#11719912—22no—
65C2×#11719912—22no—
66C2×#11719912—22no—
67C2×#11719912—22no—
68C2×#11719912—22no—
69C2×#11719912—22no—
70C22×#88322C2311yescharacteristic
71C22×#88322C2311yescharacteristic
72#13133175—22no—
73#13133175—22no—
74#13133175—22no—
75#13133175Q814yes—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C4⋊C4×D53—22no—
86C4⋊C4×D53—22no—
87C4⋊C4×D53—22no—
88C4⋊C4×D53—22no—
89C2×#11719921—21no—
90C2×#11719921—21no—
91C2×C4×Dic53—21no—
92C2×C4×Dic53—21no—
93C2×C4×Dic53D411yescharacteristic
94C2×C4×Dic53D411yescharacteristic
95C2×C4×Dic53C2311yescharacteristic
96C42×D53—22no—
97C42×D53—22no—
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—
107C22×C8×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109C2×C4⋊C4×C53—21no—
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C42×C53C2311yescharacteristic
115#95×C53—21no—
116#95×C53—21no—
117#95×C53—21no—
118C23×D53C2×D411yescharacteristic
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2411yescharacteristic
126C22×Dic53—21no—
127C22×Dic53—21no—
128C22×Dic53—21no—
129C22×Dic53—21no—
130C22×Dic53C2×D411yescharacteristic
131C22×Dic53C2×D411yescharacteristic
132C22×Dic53C2×D411yescharacteristic
133C2×C4×D53—22no—
134C2×C4×D53—22no—
135C2×C4×D53—41no—
136C2×C4×D53—41no—
137C2×C4×D53—41no—
138C2×C4×D53—41no—
139C2×C4×D53—42no—
140C2×C4×D53—42no—
141C2×C4×D53—42no—
142C2×C4×D53—42no—
143C2×C4×D53—42no—
144C2×C4×D53—42no—
145C2×C4×D53C2×Q812yes—
146C2×C4×D53#4112yes—
147C2×C4×D53#4112yes—
148C2×C4×D53#4112yes—
149C2×C4×D53#4112yes—
150C2×C4×D53#4112yes—
151C2×C8×C53—21no—
152C2×C8×C53—21no—
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—42no—
165C4⋊C4×C53C2×Q812yes—
166C42×C53—21no—
167C42×C53—21no—
168#11719921—24no—
169#11719921—42no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—42no—
178C4⋊Dic53C2×Q812yes—
179#11719919—42no—
180#11719919—42no—
181#11719919—42no—
182#11719919—42no—
183C4×Dic53—21no—
184C4×Dic53—21no—
185C4×Dic53—24no—
186C4×Dic53—22no—
187C4×Dic53—22no—
188C4×Dic53—42no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195#11719912—44no—
196#11719912—44no—
197C23×C53C22×D411yescharacteristic
198C22×D53—21no—
199C22×D53—21no—
200C22×D53—21no—
201C22×D53—21no—
202C22×D53C22⋊Q812yes—
203C22×D53#12312yes—
204C22×D53#12412yes—
205C2×C4×C53—22no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
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—41no—
223C2×C4×C53C22×Q811yescharacteristic
224C2×C4×C53C2×#4111yescharacteristic
225C2×C4×C53C2×#4111yescharacteristic
226C2×C4×C53C2×#4111yescharacteristic
227C2×C4×C53C2×#4111yescharacteristic
228C2×C4×C53C2×#4111yescharacteristic
229C2×Dic53—22no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—41no—
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×Dic53C22⋊Q812yes—
248C2×Dic53#12312yes—
249C2×Dic53#12412yes—
250C4×D53—44no—
251C4×D53—44no—
252C4×D53—84no—
253C4×D53—84no—
254C4×D53#12514yes—
255C4×D53#12514yes—
256C4×D53#12614yes—
257C4×D53#12614yes—
258C8×C53—41no—
259C8×C53—41no—
260#88322—41no—
261#88322—41no—
262#23746—531nomaximal, Sylow
263C22×C53C2×C22⋊Q811yescharacteristic
264C22×C53C2×#12311yescharacteristic
265C22×C53C2×#12411yescharacteristic
266C22×C53#57511yescharacteristic
267C22×C53#57511yescharacteristic
268C22×C53#57511yescharacteristic
269C22×C53#57511yescharacteristic
270C2×D53—22no—
271C2×D53—22no—
272C2×D53—22no—
273C2×D53—22no—
274C2×D53—22no—
275C2×D53—22no—
276C2×D53#40014yes—
277C4×C53—42no—
278C4×C53—41no—
279C4×C53—41no—
280C4×C53—81no—
281C4×C53—81no—
282C4×C53—81no—
283C4×C53C2×#12511yescharacteristic
284C4×C53C2×#12511yescharacteristic
285C4×C53C2×#12611yescharacteristic
286C4×C53C2×#12611yescharacteristic
287Dic53—42no—
288Dic53—41no—
289Dic53—41no—
290Dic53—81no—
291Dic53—81no—
292Dic53—81no—
293Dic53#40014yes—
294#2296—531no—
295C2×#382—531no—
296#2101—531no—
297#2101—531no—
298#2101—531no—
299#2101—531no—
300#2101—531no—
301#2101—531no—
302#2101—531no—
303#2101D5311yescharacteristic, complemented
304#1736—531no—
305#1736—531no—
306#1736—531no—
307#1736—531no—
308#1728—531no—
309C2×C53C2×#40011yescharacteristic
310C2×C53#308011yescharacteristic
311C2×C53#308011yescharacteristic
312C2×C53#309311yescharacteristic
313C2×C53#309311yescharacteristic
314C2×C53#314011yescharacteristic
315C2×C53#314011yescharacteristic
316D53—24no—
317C22×M16—531no—
318C22×M16—531no—
319C2×#117—531no—
320C2×#117—531no—
321C22×C4⋊C4—531no—
322C22×C4⋊C4—531no—
323C22×C42—531no—
324#389—531no—
325#389—531no—
326#389—531no—
327#389C2×D5311yescharacteristic
328#385—1061no—
329#385—1061no—
330#382—531no—
331#382—531no—
332#382—531no—
333#382—1061no—
334#382—1061no—
335#382C2×D5311yescharacteristic
336C4×C4⋊C4—1061no—
337C4×C4⋊C4—1061no—
338C2×#95—1061no—
339C2×#95—1061no—
340#340—531no—
341#340—531no—
342#340—531no—
343#340—531no—
344#340—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#340C2×D5311yescharacteristic
356#336—531no—
357#336—531no—
358#336—531no—
359#336C2×D5311yescharacteristic
360C53#2374611yescharacteristic, complemented, Sylow
361C23×C4—531no—
362C23×C4—531no—
363C23×C4—531no—
364C23×C4—1061no—
365C23×C4—1061no—
366C2×M16—1062no—
367C2×M16—1062no—
368C2×M16—1062no—
369C2×M16—1062no—
370C2×M16—1062no—
371C2×M16—1062no—
372C22×C8—531no—
373C22×C8—531no—
374C22×C8C22×D5311yescharacteristic
375C22×C8C22×D5311yescharacteristic
376#117—534no—
377#117—1062no—
378#117—1062no—
379#117—1062no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—531no—
383C2×C4⋊C4—531no—
384C2×C4⋊C4—531no—
385C2×C4⋊C4—1062no—
386C2×C4⋊C4—1062no—
387C2×C4⋊C4—1062no—
388C2×C4⋊C4—1062no—
389C2×C4⋊C4—1061no—
390C2×C4⋊C4—1061no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4C22×D5311yescharacteristic
395C2×C4⋊C4C22×D5311yescharacteristic
396C2×C4⋊C4C22×D5311yescharacteristic
397C2×C4⋊C4C22×D5311yescharacteristic
398C2×C22⋊C4—1061no—
399C2×C22⋊C4—1061no—
400C2×C42—531no—
401C2×C42—531no—
402C2×C42—531no—
403C2×C42—1062no—
404C2×C42—1062no—
405C2×C42—1061no—
406C2×C42—1061no—
407C2×C42C22×D5311yescharacteristic
408#95—21no—
409#95—21no—
410#95—21no—
411#95—1061no—
412#95—1061no—
413#95—1061no—
414#95—1061no—
415#95—1061no—
416#95—1061no—
417#95—1061no—
418#95—1061no—
419#95—1061no—
420C24—531no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—531no—
426C22×C4—532no—
427C22×C4—532no—
428C22×C4—532no—
429C22×C4—532no—
430C22×C4—532no—
431C22×C4—532no—
432C22×C4—531no—
433C22×C4—531no—
434C22×C4—1062no—
435C22×C4—1062no—
436C22×C4—1061no—
437C22×C4—1061no—
438C22×C4—1061no—
439C22×C4—1061no—
440C22×C4—2121no—
441C22×C4—2121no—
442C22×C4—2121no—
443C22×C4—2121no—
444C22×C4—2122no—
445C22×C4—2122no—
446C22×C4—2122no—
447C22×C4—2122no—
448C22×C4—2122no—
449C22×C4—2122no—
450C22×C4D4×D5311yescharacteristic
451C22×C4D4×D5311yescharacteristic
452C22×C4C23×D5311yescharacteristic
453M16—2124no—
454M16—2124no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—1061no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C2×C8—1061no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—42no—
475C4⋊C4—532no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4—2122no—
484C4⋊C4—2122no—
485C4⋊C4—2122no—
486C4⋊C4—2122no—
487C4⋊C4—2122no—
488C4⋊C4Q8×D5312yes—
489C22⋊C4—1064no—
490C22⋊C4—2122no—
491C42—21no—
492C42—21no—
493C42—1062no—
494C42—1062no—
495C42—1064no—
496C42—1061no—
497C42—1061no—
498C42—2122no—
499C23—532no—
500C23—532no—
501C23—532no—
502C23—1061no—
503C23—1061no—
504C23—1061no—
505C23—1061no—
506C23C2×D4×D5311yescharacteristic
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—532no—
526C2×C4—532no—
527C2×C4—532no—
528C2×C4—534no—
529C2×C4—534no—
530C2×C4—534no—
531C2×C4—534no—
532C2×C4—1062no—
533C2×C4—1061no—
534C2×C4—1061no—
535C2×C4—1061no—
536C2×C4—1061no—
537C2×C4—1061no—
538C2×C4—1061no—
539C2×C4—4244no—
540C2×C4—4244no—
541C2×C4—2124no—
542C2×C4—2124no—
543C2×C4—2121no—
544C2×C4—2121no—
545C2×C4—2121no—
546C2×C4—2121no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4—2121no—
554C2×C4C2×Q8×D5311yescharacteristic
555C2×C4#41×D5311yescharacteristic
556C2×C4#41×D5311yescharacteristic
557C2×C4#41×D5311yescharacteristic
558C2×C4#41×D5311yescharacteristic
559C2×C4#41×D5311yescharacteristic
560C8—41no—
561C8—41no—
562C8—2121no—
563C8—2121no—
564C22—534no—
565C22—1062no—
566C22—1062no—
567C22—1062no—
568C22—1062no—
569C22—1062no—
570C22—1062no—
571C22C22⋊Q8×D5311yescharacteristic
572C22#123×D5311yescharacteristic
573C22#124×D5311yescharacteristic
574C22#1681141911yescharacteristic
575C22#1681142011yescharacteristic
576C22#1681142011yescharacteristic
577C22#1681142211yescharacteristic
578C4—42no—
579C4—41no—
580C4—41no—
581C4—81no—
582C4—81no—
583C4—81no—
584C4—534no—
585C4—4241no—
586C4—4241no—
587C4—4241no—
588C4—2122no—
589C4—2121no—
590C4—2121no—
591C4#125×D5311yescharacteristic
592C4#125×D5311yescharacteristic
593C4#126×D5311yescharacteristic
594C4#126×D5311yescharacteristic
595C2—1064no—
596C2#400×D5311yescharacteristic
597C2#2028453211yescharacteristic
598C2#2028453411yescharacteristic
599C2#2028464311yescharacteristic
600C2#2028464711yescharacteristic
601C2#2028509811yescharacteristic
602C2#2028510011yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2101C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#2101D53
hypocenterC53#23746
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2101

Complements

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

NH ≅ G/Ntype
3. #20276288C2semidirect
4. #20276287C2semidirect
5. #20276286C2semidirect
6. #20276285C2semidirect
7. #20276284C2semidirect
8. #20276283C2semidirect
9. #20276282C2semidirect
15. C53×#2101C2semidirect
303. #2101D53semidirect
360. C53#23746semidirect

Maximal subgroups

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

subgroupindex
#202771962
#382×D532
#202762882
#202762872
#202762862
#202762852
#202762842
#202762832
#202762822
#202744332
#202744072
C53×#21012
#2374653