Finite Group Databasebeta

← #27414031

Subgroups — #27414031

Order 13568.

858 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
1#389×D53C211yesmaximal, characteristic
2#20277121C211yesmaximal, characteristic
3#20276284C211yesmaximal, characteristic, complemented
4#20276283C211yesmaximal, characteristic, complemented
5#20276282C211yesmaximal, characteristic, complemented
6#20276281C211yesmaximal, characteristic, complemented
7#20276280C211yesmaximal, characteristic, complemented
8#20276276C211yesmaximal, characteristic, complemented
9#20276275C211yesmaximal, 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×D53—21no—
21C2×C4⋊C4×D53C2211yescharacteristic
22C2×C4⋊C4×D53C2211yescharacteristic
23C2×C42×D53C2211yescharacteristic
24#16810522C2211yescharacteristic
25#16810521C2211yescharacteristic
26#16810520—21no—
27#16810519C2211yescharacteristic
28#16810518—21no—
29#16810492C2211yescharacteristic
30#16810492C2211yescharacteristic
31#16810489C2211yescharacteristic
32#16810482—21no—
33#16810482—21no—
34#16810474—21no—
35#16810474—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#13133175—22no—
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—
85C2×#11719919—21no—
86C2×#11719919—21no—
87C2×#11719919—21no—
88C2×#11719919—21no—
89C4⋊C4×D53—22no—
90C4⋊C4×D53—22no—
91C4⋊C4×D53—22no—
92C4⋊C4×D53—44no—
93C4⋊C4×D53Q814yes—
94C2×#11719921—21no—
95C2×#11719921—21no—
96C2×C4×Dic53—21no—
97C2×C4×Dic53—21no—
98C2×C4×Dic53D411yescharacteristic
99C2×C4×Dic53D411yescharacteristic
100C2×C4×Dic53C2311yescharacteristic
101C42×D53—22no—
102C42×D53—22no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108C22×C8×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C2×C4⋊C4×C53—21no—
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C42×C53C2311yescharacteristic
116#95×C53—21no—
117#95×C53—21no—
118#95×C53—21no—
119C23×D53C2×D411yescharacteristic
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2411yescharacteristic
127C22×Dic53—21no—
128C22×Dic53—21no—
129C22×Dic53—21no—
130C22×Dic53—21no—
131C22×Dic53C2×D411yescharacteristic
132C22×Dic53C2×D411yescharacteristic
133C22×Dic53C2×D411yescharacteristic
134C2×C4×D53—24no—
135C2×C4×D53—22no—
136C2×C4×D53—22no—
137C2×C4×D53—22no—
138C2×C4×D53—22no—
139C2×C4×D53—41no—
140C2×C4×D53—41no—
141C2×C4×D53—42no—
142C2×C4×D53—42no—
143C2×C4×D53—42no—
144C2×C4×D53—42no—
145C2×C4×D53—42no—
146C2×C4×D53C2×Q812yes—
147C2×C4×D53#4112yes—
148C2×C4×D53#4112yes—
149C2×C4×D53#4112yes—
150C2×C4×D53#4112yes—
151C2×C4×D53#4112yes—
152C2×C8×C53—21no—
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×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—21no—
165C4⋊C4×C53—42no—
166C4⋊C4×C53C2×Q812yes—
167C42×C53—21no—
168C42×C53—21no—
169#11719921—42no—
170#11719921—42no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—42no—
179C4⋊Dic53C2×Q812yes—
180#11719919—24no—
181#11719919—42no—
182#11719919—42no—
183#11719919—42no—
184#11719919—42no—
185#11719919—42no—
186#11719919—42no—
187#11719919—42no—
188C4×Dic53—21no—
189C4×Dic53—21no—
190C4×Dic53—22no—
191C4×Dic53—22no—
192C4×Dic53—42no—
193C4×Dic53—42no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200#11719912—44no—
201#11719912—44no—
202C23×C53C22×D411yescharacteristic
203C22×D53—21no—
204C22×D53—21no—
205C22×D53—21no—
206C22×D53—21no—
207C22×D53C22⋊Q812yes—
208C22×D53#12312yes—
209C22×D53#12412yes—
210C2×C4×C53—22no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
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×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53C22×Q811yescharacteristic
229C2×C4×C53C2×#4111yescharacteristic
230C2×C4×C53C2×#4111yescharacteristic
231C2×C4×C53C2×#4111yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—22no—
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×Dic53#12312yes—
252C2×Dic53#12412yes—
253C2×Dic53AGL(1,ℤ/8)12yes—
254C2×Dic53#13712yes—
255C4×D53—44no—
256C4×D53—44no—
257C4×D53—48no—
258C4×D53—84no—
259C4×D53#12514yes—
260C4×D53#12514yes—
261C4×D53#12614yes—
262C4×D53#12614yes—
263C8×C53—41no—
264C8×C53—41no—
265#88322—41no—
266#88322—41no—
267#23747—531nomaximal, Sylow
268C22×C53C2×C22⋊Q811yescharacteristic
269C22×C53C2×#12311yescharacteristic
270C22×C53C2×#12411yescharacteristic
271C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
272C22×C53C2×#13711yescharacteristic
273C22×C53#57511yescharacteristic
274C22×C53#57511yescharacteristic
275C2×D53—22no—
276C2×D53—22no—
277C2×D53—22no—
278C2×D53—22no—
279C2×D53—22no—
280C2×D53—22no—
281C2×D53#40014yes—
282C4×C53—42no—
283C4×C53—41no—
284C4×C53—41no—
285C4×C53—81no—
286C4×C53—81no—
287C4×C53—81no—
288C4×C53C2×#12511yescharacteristic
289C4×C53C2×#12511yescharacteristic
290C4×C53C2×#12611yescharacteristic
291C4×C53C2×#12611yescharacteristic
292Dic53—22no—
293Dic53—22no—
294Dic53—42no—
295Dic53—81no—
296Dic53—81no—
297Dic53—81no—
298Dic53#40014yes—
299C2×#389—531no—
300#2285—531no—
301#2101—531no—
302#2101—531no—
303#2101—531no—
304#2101D5311yescharacteristic, complemented
305#2100—531no—
306#2100—531no—
307#2099—531no—
308#2099—531no—
309#1736—531no—
310#1736—531no—
311#1736—531no—
312#1736—531no—
313#1728—531no—
314C2×C53C2×#40011yescharacteristic
315C2×C53#307711yescharacteristic
316C2×C53#307811yescharacteristic
317C2×C53#309411yescharacteristic
318C2×C53#309511yescharacteristic
319C2×C53#314111yescharacteristic
320C2×C53#314211yescharacteristic
321D53—24no—
322C22×M16—531no—
323C22×M16—531no—
324C2×#117—531no—
325C2×#117—531no—
326C22×C4⋊C4—531no—
327C22×C4⋊C4—531no—
328C22×C4⋊C4—1061no—
329C22×C42—531no—
330#389—531no—
331#389—531no—
332#389—531no—
333#389—1061no—
334#389—1061no—
335#389C2×D5311yescharacteristic
336#384—531no—
337#384—531no—
338#382—531no—
339#382C2×D5311yescharacteristic
340#380—1061no—
341#380—1061no—
342C4×C4⋊C4—1061no—
343C4×C4⋊C4—1061no—
344C2×#95—1061no—
345#340—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#340—531no—
357#340C2×D5311yescharacteristic
358#340C2×D5311yescharacteristic
359#340C2×D5311yescharacteristic
360#340C2×D5311yescharacteristic
361#336—531no—
362#336—531no—
363#336—531no—
364#336C2×D5311yescharacteristic
365C53#2374711yescharacteristic, complemented, Sylow
366C23×C4—531no—
367C23×C4—531no—
368C23×C4—531no—
369C23×C4—1061no—
370C23×C4—1061no—
371C2×M16—1062no—
372C2×M16—1062no—
373C2×M16—1062no—
374C2×M16—1062no—
375C2×M16—1062no—
376C2×M16—1062no—
377C22×C8—531no—
378C22×C8—531no—
379C22×C8C22×D5311yescharacteristic
380C22×C8C22×D5311yescharacteristic
381#117—1062no—
382#117—1062no—
383#117—1062no—
384#117—1062no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—531no—
387C2×C4⋊C4—534no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—1062no—
392C2×C4⋊C4—1062no—
393C2×C4⋊C4—1062no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4—2124no—
404C2×C4⋊C4C22×D5311yescharacteristic
405C2×C4⋊C4C22×D5311yescharacteristic
406C2×C4⋊C4C22×D5311yescharacteristic
407C2×C4⋊C4C22×D5311yescharacteristic
408C2×C22⋊C4—1061no—
409C2×C22⋊C4—1061no—
410C2×C42—531no—
411C2×C42—531no—
412C2×C42—531no—
413C2×C42—1062no—
414C2×C42—1062no—
415C2×C42—1061no—
416C2×C42—1061no—
417C2×C42C22×D5311yescharacteristic
418#95—21no—
419#95—21no—
420#95—21no—
421#95—1061no—
422#95—1061no—
423#95—1061no—
424#95—1061no—
425#95—1061no—
426C24—531no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—531no—
432C22×C4—532no—
433C22×C4—532no—
434C22×C4—532no—
435C22×C4—532no—
436C22×C4—532no—
437C22×C4—532no—
438C22×C4—531no—
439C22×C4—531no—
440C22×C4—1062no—
441C22×C4—1062no—
442C22×C4—1062no—
443C22×C4—1062no—
444C22×C4—1064no—
445C22×C4—1061no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—2121no—
450C22×C4—2121no—
451C22×C4—2122no—
452C22×C4—2122no—
453C22×C4—2122no—
454C22×C4—2122no—
455C22×C4—2122no—
456C22×C4D4×D5311yescharacteristic
457C22×C4D4×D5311yescharacteristic
458C22×C4C23×D5311yescharacteristic
459M16—2124no—
460M16—2124no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8—1061no—
470C2×C8—1061no—
471C2×C8—1061no—
472C2×C8—1061no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—42no—
481C4⋊C4—532no—
482C4⋊C4—1064no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—2122no—
491C4⋊C4—2122no—
492C4⋊C4—2122no—
493C4⋊C4—2122no—
494C4⋊C4—2122no—
495C4⋊C4—2122no—
496C4⋊C4—2122no—
497C4⋊C4—2122no—
498C4⋊C4Q8×D5312yes—
499C22⋊C4—2122no—
500C22⋊C4—2122no—
501C42—21no—
502C42—21no—
503C42—1062no—
504C42—1062no—
505C42—1061no—
506C42—1061no—
507C42—2122no—
508C42—2122no—
509C23—532no—
510C23—532no—
511C23—532no—
512C23—1061no—
513C23—1061no—
514C23—1061no—
515C23—1061no—
516C23C2×D4×D5311yescharacteristic
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—532no—
536C2×C4—532no—
537C2×C4—532no—
538C2×C4—534no—
539C2×C4—534no—
540C2×C4—534no—
541C2×C4—534no—
542C2×C4—532no—
543C2×C4—532no—
544C2×C4—1061no—
545C2×C4—1061no—
546C2×C4—1061no—
547C2×C4—1061no—
548C2×C4—1062no—
549C2×C4—4244no—
550C2×C4—2124no—
551C2×C4—2124no—
552C2×C4—2128no—
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—2121no—
564C2×C4C2×Q8×D5311yescharacteristic
565C2×C4#41×D5311yescharacteristic
566C2×C4#41×D5311yescharacteristic
567C2×C4#41×D5311yescharacteristic
568C2×C4#41×D5311yescharacteristic
569C2×C4#41×D5311yescharacteristic
570C8—41no—
571C8—41no—
572C8—2121no—
573C8—2121no—
574C22—534no—
575C22—1062no—
576C22—1062no—
577C22—1062no—
578C22—1062no—
579C22—1062no—
580C22—1062no—
581C22C22⋊Q8×D5311yescharacteristic
582C22#123×D5311yescharacteristic
583C22#124×D5311yescharacteristic
584C22AGL(1,ℤ/8)×D5311yescharacteristic
585C22#137×D5311yescharacteristic
586C22#1681142011yescharacteristic
587C22#1681142011yescharacteristic
588C4—42no—
589C4—41no—
590C4—41no—
591C4—81no—
592C4—81no—
593C4—81no—
594C4—534no—
595C4—1062no—
596C4—1062no—
597C4—4241no—
598C4—4241no—
599C4—4241no—
600C4—2122no—
601C4#125×D5311yescharacteristic
602C4#125×D5311yescharacteristic
603C4#126×D5311yescharacteristic
604C4#126×D5311yescharacteristic
605C2—1064no—
606C2#400×D5311yescharacteristic
607C2#2028451011yescharacteristic
608C2#2028451811yescharacteristic
609C2#2028465411yescharacteristic
610C2#2028466211yescharacteristic
611C2#2028510711yescharacteristic
612C2#2028511211yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2101C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#2101D53
hypocenterC53#23747
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. #20276284C2semidirect
4. #20276283C2semidirect
5. #20276282C2semidirect
6. #20276281C2semidirect
7. #20276280C2semidirect
8. #20276276C2semidirect
9. #20276275C2semidirect
15. C53×#2101C2semidirect
304. #2101D53semidirect
365. C53#23747semidirect

Maximal subgroups

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

subgroupindex
#389×D532
#202771212
#202762842
#202762832
#202762822
#202762812
#202762802
#202762762
#202762752
#202744332
#202744072
C53×#21012
#2374753