Finite Group Databasebeta

← #27441688

Subgroups — #27441688

Order 13568.

802 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#20278413C211yesmaximal, characteristic
2#20277583C211yesmaximal, characteristic
3#20276276C211yesmaximal, characteristic, complemented
4#20276206C211yesmaximal, characteristic, complemented
5#20276205C211yesmaximal, characteristic, complemented
6#20275668C211yesmaximal, characteristic, complemented
7#20275470C211yesmaximal, characteristic, complemented
8#20275288C211yesmaximal, characteristic, complemented
9#20275287C211yesmaximal, characteristic, complemented
10#20274901C211yesmaximal, characteristic
11#20274894C211yesmaximal, characteristic
12#20274800C211yesmaximal, characteristic
13#20274741C211yesmaximal, characteristic
14#340×D53C211yesmaximal, characteristic
15C53×#2057C211yesmaximal, characteristic, complemented
16C22×C8×D53C2211yescharacteristic
17C2×#13133193—21no—
18C2×#13133175—21no—
19C2×C4⋊C4×D53C2211yescharacteristic
20C2×C4⋊C4×D53C2211yescharacteristic
21C2×#13133151C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×#13133146C2211yescharacteristic
25C2×#13133129C2211yescharacteristic
26#16810556—21no—
27#16810542—21no—
28#16810522C2211yescharacteristic
29#16810522C2211yescharacteristic
30#16810521C2211yescharacteristic
31#16810510—21no—
32#16810504—21no—
33#16810494—21no—
34#16810492C2211yescharacteristic
35#16810485—21no—
36#16810480C2211yescharacteristic
37#16810480C2211yescharacteristic
38C4⋊C4×Dic53C2211yescharacteristic
39C4×C4⋊Dic53C2211yescharacteristic
40C4×#11719921C2211yescharacteristic
41#16810386C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×C4⋊Q8×C53C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic
54C2×#107×C53C2211yescharacteristic
55C53×#389C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×C4×D53—21no—
60C22×C4×D53—21no—
61C22×C4×D53C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C2×C8×D53—22no—
65C2×C8×D53—22no—
66C2×C8×D53—41no—
67#13133193—44no—
68#13133175—44no—
69C2×C4⋊Dic53—21no—
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—
79C2×#11719919—21no—
80C4⋊C4×D53—22no—
81C4⋊C4×D53—22no—
82C4⋊C4×D53—41no—
83C4⋊C4×D53Q814yescomplemented
84C2×#11719921—21no—
85C2×#11719921—21no—
86C2×#11719921D411yescharacteristic
87C2×#11719921D411yescharacteristic
88C2×C4×Dic53—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53C2311yescharacteristic
91#13133151—22no—
92#13133150—41no—
93C4⋊#88322—41no—
94#13133146—22no—
95#13133129—22no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124D411yescharacteristic
104#13133124D411yescharacteristic
105C22×Q8×C53—21no—
106C22×C8×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108C4⋊Q8×C53—22no—
109C4⋊Q8×C53—41no—
110C2×C4⋊C4×C53—21no—
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C42×C53C2311yescharacteristic
117#107×C53—22no—
118#107×C53—41no—
119#95×C53—21no—
120C23×D53C2×D411yescharacteristic
121C2×Q8×C53—42no—
122C2×Q8×C53—42no—
123C2×Q8×C53—81no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2411yescharacteristic
131C22×Dic53—21no—
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53C2×D411yescharacteristic
136C22×Dic53C2×D411yescharacteristic
137C22×Dic53C2×D411yescharacteristic
138C2×C4×D53—21no—
139C2×C4×D53—21no—
140C2×C4×D53—21no—
141C2×C4×D53—21no—
142C2×C4×D53—24no—
143C2×C4×D53—42no—
144C2×C4×D53—42no—
145C2×C4×D53—42no—
146C2×C4×D53—42no—
147C2×C4×D53—81no—
148C2×C4×D53C2×Q812yes—
149C2×C4×D53#4112yes—
150C2×C8×C53—21no—
151C2×C8×C53—21no—
152C2×C8×C53—21no—
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53#4112yes—
156C4⋊C4×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—42no—
163C4⋊C4×C53—42no—
164C4⋊C4×C53C2×Q812yes—
165C4⋊C4×C53#4112yes—
166C42×C53—21no—
167C42×C53—21no—
168#11719921—22no—
169#11719921—42no—
170#11719921—42no—
171#11719921D814yes—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—42no—
180C4⋊Dic53—42no—
181C4⋊Dic53C2×Q812yes—
182#11719919—24no—
183#11719919—42no—
184#11719919—42no—
185#11719919—42no—
186#11719919—81no—
187C4×Dic53—21no—
188C4×Dic53—21no—
189C4×Dic53—42no—
190C4×Dic53—42no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C8×D53—44no—
198C23×C53C22×D411yescharacteristic
199C22×D53—21no—
200C22×D53—21no—
201C22×D53—21no—
202C22×D53—21no—
203C22×D53C22⋊Q812yes—
204C22×D53C2×D812yes—
205C22×D53#13712yes—
206Q8×C53—84no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—22no—
210C2×C4×C53—22no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
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×C53—41no—
224C2×C4×C53C2×D812yes—
225C2×C4×C53C22×Q811yescharacteristic
226C2×C4×C53C2×#4111yescharacteristic
227C2×C4×C53C2×#4111yescharacteristic
228C2×C4×C53C2×#4111yescharacteristic
229C2×C4×C53H(5,𝔽2)11yescharacteristic
230C2×C4×C532−1+411yescharacteristic
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—22no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
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×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53C22⋊Q812yes—
251C2×Dic53C2×D812yes—
252C4×D53—22no—
253C4×D53—22no—
254C4×D53—22no—
255C4×D53—22no—
256C4×D53—48no—
257C4×D53—84no—
258C8×C53—22no—
259C8×C53—41no—
260#88322—41no—
261#88322—41no—
262#25988—531nomaximal, Sylow
263C22×C53C2×C22⋊Q811yescharacteristic
264C22×C53#54611yescharacteristic
265C22×C53#54711yescharacteristic
266C22×C53C22×D811yescharacteristic
267C22×C53C2×#13711yescharacteristic
268C22×C53#57711yescharacteristic
269C22×C53#57811yescharacteristic
270C2×D53—22no—
271C2×D53—22no—
272C2×D53—22no—
273C2×D53—22no—
274C2×D53—22no—
275C2×D53—22no—
276C2×D53#47414yes—
277C4×C53—22no—
278C4×C53—42no—
279C4×C53—41no—
280C4×C53—42no—
281C4×C53—81no—
282C4×C53—81no—
283C4×C53#53311yescharacteristic
284C4×C53#54411yescharacteristic
285C4×C53#55111yescharacteristic
286C4×C53#55211yescharacteristic
287Dic53—22no—
288Dic53—22no—
289Dic53—41no—
290Dic53—42no—
291Dic53—81no—
292Dic53—81no—
293Dic53—81no—
294#2440—531no—
295#2344—531no—
296#2099—531no—
297#2086—531no—
298#2086—531no—
299#2057D5311yescharacteristic, complemented
300#1988—531no—
301#1947—531no—
302#1918—531no—
303#1918—531no—
304#1852—531no—
305#1850—531no—
306#1823—531no—
307#1808—531no—
308C2×#340—531no—
309C2×C53#243811yescharacteristic
310C2×C53C2×#47411yescharacteristic
311C2×C53#308211yescharacteristic
312C2×C53#329411yescharacteristic
313C2×C53#331211yescharacteristic
314C2×C53#333411yescharacteristic
315C2×C53#334411yescharacteristic
316D53—24no—
317C23×C8—531no—
318C2×C4⋊Q8C2×D5311yescharacteristic
319C2×C22⋊Q8—1061no—
320C2×#117—1061no—
321C22×C4⋊C4—531no—
322C22×C4⋊C4—531no—
323C2×#107—531no—
324C2×#107—531no—
325C2×#107C2×D5311yescharacteristic
326C2×#107C2×D5311yescharacteristic
327C2×C4⋊C8—531no—
328C2×C4⋊C8—531no—
329C2×C22⋊C8—531no—
330#398—1061no—
331#395—1061no—
332#389—531no—
333#389—531no—
334#389—531no—
335#389C2×D5311yescharacteristic
336#387—1061no—
337#387—1061no—
338#385—1061no—
339#384—531no—
340#381—1061no—
341#380—531no—
342#380—531no—
343C4×C4⋊C4—531no—
344C4×C4⋊C4—531no—
345C4×C22⋊C4—531no—
346#340—531no—
347#340—531no—
348#340—531no—
349#340—531no—
350#340—531no—
351#340—531no—
352#340—531no—
353#340—531no—
354#340—531no—
355#340—531no—
356#340C2×D5311yescharacteristic
357#340C2×D5311yescharacteristic
358#336—531no—
359#336C2×D5311yescharacteristic
360C53#2598811yescharacteristic, complemented, Sylow
361C22×Q8—21no—
362C23×C4—531no—
363C23×C4—1061no—
364C23×C4—1061no—
365C22×C8—531no—
366C22×C8—531no—
367C22×C8—1062no—
368C22×C8—1062no—
369C22×C8—2121no—
370C22×C8C22×D5311yescharacteristic
371C22×C8C22×D5311yescharacteristic
372C4⋊Q8—22no—
373C4⋊Q8—41no—
374C22⋊Q8—2124no—
375#117—2124no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—534no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—531no—
383C2×C4⋊C4—1062no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4—1062no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4—1061no—
389C2×C4⋊C4—1061no—
390C2×C4⋊C4—1061no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4—2121no—
393C2×C4⋊C4C22×D5311yescharacteristic
394C2×C4⋊C4C22×D5311yescharacteristic
395C2×C4⋊C4C22×D5311yescharacteristic
396C2×C4⋊C4C22×D5311yescharacteristic
397C2×C22⋊C4—531no—
398C2×C22⋊C4—531no—
399C2×C22⋊C4—1061no—
400C2×C22⋊C4—1061no—
401C2×C42—531no—
402C2×C42—1061no—
403C2×C42—1061no—
404C2×C42C22×D5311yescharacteristic
405#107—22no—
406#107—41no—
407#107—1062no—
408#107—2121no—
409C4⋊C8—1062no—
410C4⋊C8—2121no—
411C22⋊C8—1062no—
412#95—21no—
413#95—531no—
414#95—531no—
415#95—1061no—
416#95—1061no—
417#95—1061no—
418#95—1061no—
419#95—1061no—
420#95—1061no—
421#95—1061no—
422C24—531no—
423C2×Q8—42no—
424C2×Q8—42no—
425C2×Q8—81no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—531no—
431C22×C4—531no—
432C22×C4—532no—
433C22×C4—532no—
434C22×C4—531no—
435C22×C4—1064no—
436C22×C4—1061no—
437C22×C4—1061no—
438C22×C4—1061no—
439C22×C4—1061no—
440C22×C4—1061no—
441C22×C4—1061no—
442C22×C4—1061no—
443C22×C4—1061no—
444C22×C4—4241no—
445C22×C4—2122no—
446C22×C4—2122no—
447C22×C4—2122no—
448C22×C4—2122no—
449C22×C4D4×D5311yescharacteristic
450C22×C4D4×D5311yescharacteristic
451C22×C4C23×D5311yescharacteristic
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8—1061no—
462C2×C8—1061no—
463C2×C8—2124no—
464C2×C8#1171994912yes—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—532no—
474C4⋊C4—1064no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4—1061no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—4241no—
483C4⋊C4—2122no—
484C4⋊C4—2122no—
485C4⋊C4—2122no—
486C4⋊C4—2122no—
487C4⋊C4—2122no—
488C4⋊C4#1171994712yes—
489C4⋊C4Q8×D5312yes—
490C22⋊C4—534no—
491C22⋊C4—1062no—
492C22⋊C4—2122no—
493C22⋊C4—2122no—
494C42—21no—
495C42—21no—
496C42—1061no—
497C42—1061no—
498C42—2122no—
499C42—2122no—
500C23—532no—
501C23—532no—
502C23—532no—
503C23—1061no—
504C23—1061no—
505C23—1061no—
506C23—1061no—
507C23C2×D4×D5311yescharacteristic
508Q8—84no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
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—41no—
526C2×C4—532no—
527C2×C4—532no—
528C2×C4—1061no—
529C2×C4—1061no—
530C2×C4—1061no—
531C2×C4—1061no—
532C2×C4—1062no—
533C2×C4—1062no—
534C2×C4—1062no—
535C2×C4—1062no—
536C2×C4—1061no—
537C2×C4—1062no—
538C2×C4—1061no—
539C2×C4—1061no—
540C2×C4—4244no—
541C2×C4—2128no—
542C2×C4—2121no—
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×C4D8×D5312yes—
554C2×C4C2×Q8×D5311yescharacteristic
555C2×C4#41×D5311yescharacteristic
556C2×C4C2×#1171994711yescharacteristic
557C2×C4C2×#1171994911yescharacteristic
558C2×C4#1313326211yescharacteristic
559C2×C4#1313326511yescharacteristic
560C8—22no—
561C8—41no—
562C8—2121no—
563C8—2121no—
564C22—534no—
565C22—1062no—
566C22—1062no—
567C22—1062no—
568C22—1062no—
569C22—1062no—
570C22—1062no—
571C22C22⋊Q8×D5311yescharacteristic
572C22#1681126011yescharacteristic
573C22#1681127611yescharacteristic
574C22C2×D8×D5311yescharacteristic
575C22#137×D5311yescharacteristic
576C22#1681143311yescharacteristic
577C22#1681143611yescharacteristic
578C4—22no—
579C4—41no—
580C4—42no—
581C4—42no—
582C4—81no—
583C4—81no—
584C4—1062no—
585C4—1062no—
586C4—4241no—
587C4—4241no—
588C4—4241no—
589C4—2122no—
590C4—2121no—
591C4#1681117811yescharacteristic
592C4#1681125011yescharacteristic
593C4#1681130011yescharacteristic
594C4#1681130711yescharacteristic
595C2—1064no—
596C2#2027838411yescharacteristic
597C2#474×D5311yescharacteristic
598C2#2028455211yescharacteristic
599C2#2028652111yescharacteristic
600C2#2028675911yescharacteristic
601C2#2028706211yescharacteristic
602C2#2028716911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2057C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#2057D53
hypocenterC53#25988
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2057

Complements

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

NH ≅ G/Ntype
3. #20276276C2semidirect
4. #20276206C2semidirect
5. #20276205C2semidirect
6. #20275668C2semidirect
7. #20275470C2semidirect
8. #20275288C2semidirect
9. #20275287C2semidirect
15. C53×#2057C2semidirect
83. C4⋊C4×D53Q8semidirect
299. #2057D53semidirect
360. C53#25988semidirect

Maximal subgroups

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

subgroupindex
#202784132
#202775832
#202762762
#202762062
#202762052
#202756682
#202754702
#202752882
#202752872
#202749012
#202748942
#202748002
#202747412
#340×D532
C53×#20572
#2598853