Finite Group Databasebeta

← #27355242

Subgroups — #27355242

Order 13568.

650 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284762C211yesmaximal, characteristic
2#20283871C211yesmaximal, characteristic, complemented
3#20283750C211yesmaximal, characteristic, complemented
4#20273608C211yesmaximal, characteristic, complemented
5#20273597C211yesmaximal, characteristic, complemented
6#20273596C211yesmaximal, characteristic, complemented
7#20273582C211yesmaximal, characteristic, complemented
8#20273576C211yesmaximal, characteristic, complemented
9#20273569C211yesmaximal, characteristic, complemented
10#20273518C211yesmaximal, characteristic, complemented
11#20273507C211yesmaximal, characteristic, complemented
12#20273498C211yesmaximal, characteristic, complemented
13#20273176C211yesmaximal, characteristic, complemented
14#20272998C211yesmaximal, characteristic, complemented
15C2×C53×D4⋊C8C211yesmaximal, characteristic, complemented
16#16811157C2211yescharacteristic
17#16810958C411yescharacteristic
18#16810941C411yescharacteristic
19#16810923C411yescharacteristic
20#16810923C411yescharacteristic
21#16810838C411yescharacteristic
22#16810838C411yescharacteristic
23#16810774C411yescharacteristic
24#16810757C411yescharacteristic
25#16810750C2211yescharacteristic, complemented
26#16810747C2211yescharacteristic, complemented
27C4×#11719923C2211yescharacteristic
28C4×#11719923C2211yescharacteristic
29C4×#11719922C2211yescharacteristic
30C4×#11719922C2211yescharacteristic
31#16810701C2211yescharacteristic
32#16810701C2211yescharacteristic
33#16810700C2211yescharacteristic, complemented
34#16810695—21no—
35#16810695—21no—
36#16810694—21no—
37D4×#88322C2211yescharacteristic
38D4×#88322C2211yescharacteristic
39#16810688C2211yescharacteristic, complemented
40C8×C22⋊D53—21no—
41#16810683—21no—
42#16810680C2211yescharacteristic
43#16810676C2211yescharacteristic
44#16810676C2211yescharacteristic
45#16810648—21no—
46#16810587—21no—
47C2×C22⋊#88322—21no—
48#16810375C2211yescharacteristic, complemented
49#16810375C2211yescharacteristic, complemented
50#16810373C2211yescharacteristic, complemented
51#16810373C2211yescharacteristic, complemented
52#16810355C2211yescharacteristic, complemented
53#16810355C2211yescharacteristic, complemented
54#16810352C2211yescharacteristic, complemented
55#16810352C2211yescharacteristic, complemented
56#16810344C2211yescharacteristic, complemented
57#16810344C2211yescharacteristic, complemented
58C8×#88322C2211yescharacteristic, complemented
59C8×#88322C2211yescharacteristic, complemented
60C2×C4×D4×C53C2211yescharacteristic
61C2×C4⋊C8×C53C2211yescharacteristic
62C2×C4×C8×C53C2211yescharacteristic
63C53×D4⋊C8C2211yescharacteristic
64C53×D4⋊C8C2211yescharacteristic
65C53×D4⋊C8C2211yescharacteristic
66C53×D4⋊C8C2211yescharacteristic
67C2×#11719923C2×C411yescharacteristic
68C2×#11719923C2×C411yescharacteristic
69C2×#11719922C2×C411yescharacteristic
70C2×#11719922C2×C411yescharacteristic
71C2×#11719917—21no—
72C2×#11719917—21no—
73C2×#11719912—21no—
74C22×#88322—21no—
75C22×#88322—21no—
76C2×C8×D53—21no—
77#13133211—21no—
78#13133209C2×C411yescharacteristic
79#13133205C2×C411yescharacteristic
80#13133200—21no—
81#13133196C2×C411yescharacteristic
82#13133191C2×C411yescharacteristic
83C4×Dic106C2311yescharacteristic
84C4×D212C2311yescharacteristic
85C4×C22⋊D53—21no—
86#13133174—21no—
87#13133150C2×C411yescharacteristic
88#13133150C2×C411yescharacteristic
89#13133148C2×C411yescharacteristic
90#13133148C2×C411yescharacteristic
91C4⋊#88322C2×C411yescharacteristic
92C4⋊#88322C2×C411yescharacteristic
93C4⋊#88322D411yescharacteristic
94C4⋊#88322D411yescharacteristic
95C4⋊#88322C2311yescharacteristic
96C4⋊#88322C2311yescharacteristic
97#13133146—21no—
98#13133146—21no—
99#13133140C2×C411yescharacteristic
100#13133140C2×C411yescharacteristic
101#13133138C2×C411yescharacteristic
102#13133138C2×C411yescharacteristic
103#13133138C2×C411yescharacteristic
104#13133138C2×C411yescharacteristic
105#13133137C2×C411yescharacteristic
106#13133137C2×C411yescharacteristic
107C22⋊#88322—21no—
108C22⋊#88322—21no—
109C22⋊#88322—21no—
110C22⋊#88322—21no—
111C22⋊#88322—21no—
112C22⋊#88322C2×C411yescharacteristic
113C22⋊#88322C2×C411yescharacteristic
114C22⋊#88322D411yescharacteristic
115C22⋊#88322D411yescharacteristic
116#13133129—21no—
117#13133129—21no—
118#13133128—21no—
119#13133127—21no—
120C4×#88322C2311yescharacteristic
121C4×#88322C2311yescharacteristic
122C4×#88322C2311yescharacteristic
123C4×#88322C2311yescharacteristic
124C8×Dic53—21no—
125C22×D4×C53C2×C411yescharacteristic
126C23×C4×C53—21no—
127C22×C8×C53—21no—
128C22×C8×C53—21no—
129C4×D4×C53—21no—
130C4×D4×C53—21no—
131C4×D4×C53C2×C411yescharacteristic
132C4×D4×C53C2×C411yescharacteristic
133C4×D4×C53C2311yescharacteristic
134C4×D4×C53C2311yescharacteristic
135C2×C4⋊C4×C53C2×C411yescharacteristic
136C2×C22⋊C4×C53—21no—
137C2×C42×C53C2311yescharacteristic
138C4⋊C8×C53—21no—
139C4⋊C8×C53C2311yescharacteristic
140C4⋊C8×C53C2311yescharacteristic
141C4×C8×C53—21no—
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C24×C53—21no—
145C2×D4×C53—22no—
146C2×D4×C53—21no—
147C2×D4×C53—41no—
148C2×D4×C53C22×C411yescharacteristic
149C2×D4×C53C22×C411yescharacteristic
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53—21no—
155C22×C4×C53—81no—
156C22×C4×C53—81no—
157C22×C4×C53C22×C411yescharacteristic
158C22×C4×C53C2×D411yescharacteristic
159C22×C4×C53#4111yescharacteristic
160C2×C22⋊D53—21no—
161C2×D212C22×C411yescharacteristic
162C2×C4×D53—21no—
163C2×Dic106C22×C411yescharacteristic
164C2×C8×C53—24no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53—81no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53C22×C411yescharacteristic
172C4⋊C4×C53C22×C411yescharacteristic
173C22⋊C4×C53—21no—
174C22⋊C4×C53—21no—
175C22⋊C4×C53—21no—
176C22⋊C4×C53—21no—
177C42×C53C22×C411yescharacteristic
178C42×C53C2×D411yescharacteristic
179C42×C53#4111yescharacteristic
180C42×C53C2411yescharacteristic
181C22⋊Dic53—21no—
182#11719923—22no—
183#11719923—41no—
184#11719922—22no—
185#11719922—41no—
186#11719921—21no—
187#11719921—21no—
188#11719921—21no—
189#11719921—21no—
190C4⋊Dic53C22×C411yescharacteristic
191C4⋊Dic53C22×C411yescharacteristic
192#11719919—21no—
193#11719919—21no—
194#11719919—21no—
195#11719919—21no—
196C4×Dic53—21no—
197#11719917—81no—
198#11719917—81no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—81no—
202C2×#88322—81no—
203C2×#88322C22×C411yescharacteristic
204C2×#88322C22×C411yescharacteristic
205C2×#88322C22×C411yescharacteristic
206C2×#88322C22×C411yescharacteristic
207C2×#88322C2×D411yescharacteristic
208C2×#88322C2×D411yescharacteristic
209C2×#88322C2×D411yescharacteristic
210C2×#88322C2×D411yescharacteristic
211#11719912—81no—
212C8×D53—81no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—42no—
216C23×C53—42no—
217C23×C53—81no—
218C23×C53C4×D411yescharacteristic
219C22×D53—21no—
220D4×C53—21no—
221D4×C53—21no—
222D4×C53—21no—
223D4×C53—42no—
224D4×C53#13112yes—
225C2×C4×C53—22no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—81no—
237C2×C4×C53—81no—
238C2×C4×C53—81no—
239C2×C4×C53C4×D411yescharacteristic
240C2×C4×C53C4×D411yescharacteristic
241C2×C4×C53C4×D411yescharacteristic
242C2×C4×C53C23×C411yescharacteristic
243C2×C4×C53C22×D411yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C22⋊D53—42no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249D212—21no—
250D212—21no—
251C4×D53—41no—
252C4×D53—41no—
253Dic106—21no—
254Dic106—21no—
255C8×C53—22no—
256C8×C53—22no—
257C8×C53—41no—
258C8×C53—41no—
259#88322—21no—
260#88322—21no—
261#88322—21no—
262#88322—21no—
263#88322—21no—
264#88322—21no—
265#88322—21no—
266#88322—21no—
267#88322—41no—
268#88322—41no—
269#18014—531nomaximal, Sylow
270C22×C53—21no—
271C22×C53—21no—
272C22×C53—22no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53—41no—
276C22×C53—41no—
277C22×C53—41no—
278C22×C53—84no—
279C22×C53#44312yes—
280C22×C53C2×C4×D411yescharacteristic
281C2×D53—41no—
282C2×D53—41no—
283C2×D53—41no—
284C4×C53—21no—
285C4×C53—21no—
286C4×C53—21no—
287C4×C53—21no—
288C4×C53—41no—
289C4×C53—41no—
290C4×C53C2×#13111yescharacteristic
291C4×C53#56811yescharacteristic
292C4×C53#57611yescharacteristic
293C4×C53#57711yescharacteristic
294Dic53—41no—
295Dic53—41no—
296Dic53—41no—
297#3105—531no—
298#2994—531no—
299#2974—531no—
300#1590—531no—
301#1589—531no—
302#1588—531no—
303#1586—531no—
304#1586—531no—
305#1585—531no—
306C8×SD16—531no—
307C8×D8—531no—
308#1573—531no—
309#1494—531no—
310C2×D4⋊C8D5311yescharacteristic, complemented
311#1450—531no—
312C2×C53—22no—
313C2×C53—42no—
314C2×C53—81no—
315C2×C53#293311yescharacteristic
316C2×C53#294611yescharacteristic
317C2×C53C2×#44311yescharacteristic
318D53—81no—
319#529—531no—
320C2×C4×D4C2×D5311yescharacteristic
321#484—531no—
322#481—531no—
323#478—531no—
324#478—531no—
325#463—531no—
326#463—531no—
327#450—531no—
328#447—531no—
329#446—531no—
330C8×Q8—531no—
331C4×SD16—531no—
332C4×SD16—531no—
333C4×D8—531no—
334C4×D8—531no—
335#436—531no—
336#436—531no—
337#436—531no—
338#435—1061no—
339#435—1061no—
340#435—1061no—
341C8×D4—531no—
342C8×D4—531no—
343C8×D4—531no—
344C8×D4—1061no—
345#433—1061no—
346#432—531no—
347#431—531no—
348#431—531no—
349#424—1061no—
350C2×C4⋊C8C2×D5311yescharacteristic
351#407—1061no—
352C2×C22⋊C8—1061no—
353C2×C4×C8C2×D5311yescharacteristic
354#335—531no—
355#335—531no—
356#334—531no—
357#334—531no—
358Q8⋊C8—531no—
359Q8⋊C8—531no—
360D4⋊C8—531no—
361D4⋊C8—531no—
362D4⋊C8C2×D5311yescharacteristic
363D4⋊C8C2×D5311yescharacteristic
364D4⋊C8C2×D5311yescharacteristic
365D4⋊C8C2×D5311yescharacteristic
366#322—531no—
367#322—531no—
368C82—531no—
369C82—531no—
370C53#1801411yescharacteristic, complemented, Sylow
371C22×D4C4×D5311yescharacteristic
372C23×C4—21no—
373C2×SD16—531no—
374C2×SD16—531no—
375C2×D8—531no—
376C2×D8—531no—
377C2×M16—1061no—
378C2×M16—1061no—
379C2×M16—1061no—
380C22×C8—21no—
381C22×C8—21no—
382C22×C8—1061no—
383C22×C8—1061no—
384C22×C8—1061no—
385#126—1061no—
386#125—531no—
387#124—531no—
388#123—1061no—
389C22⋊Q8—531no—
390#121—531no—
391C4×Q8—531no—
392C4×D4—21no—
393C4×D4—21no—
394C4×D4—531no—
395C4×D4—1061no—
396C4×D4C4×D5311yescharacteristic
397C4×D4C4×D5311yescharacteristic
398C4×D4C22×D5311yescharacteristic
399C4×D4C22×D5311yescharacteristic
400#117—1061no—
401C2×C4⋊C4C4×D5311yescharacteristic
402C2×C22⋊C4—21no—
403C2×C42C22×D5311yescharacteristic
404#107—531no—
405#107—531no—
406#106—531no—
407#106—531no—
408C4⋊C8—21no—
409C4⋊C8—531no—
410C4⋊C8—531no—
411C4⋊C8—531no—
412C4⋊C8—531no—
413C4⋊C8—531no—
414C4⋊C8—531no—
415C4⋊C8—1061no—
416C4⋊C8—1061no—
417C4⋊C8C22×D5311yescharacteristic
418C4⋊C8C22×D5311yescharacteristic
419#103—531no—
420#103—531no—
421#102—531no—
422#102—531no—
423#102—531no—
424#102—531no—
425#102—531no—
426#102—531no—
427C22⋊C8—531no—
428C22⋊C8—531no—
429C22⋊C8—531no—
430C22⋊C8—531no—
431C22⋊C8—1061no—
432C22⋊C8—1061no—
433C22⋊C8—1061no—
434C22⋊C8—1061no—
435C22⋊C8—1061no—
436C22⋊C8—1061no—
437C22⋊C8—1061no—
438#97—1061no—
439#97—1061no—
440C4×C8—21no—
441C4×C8—531no—
442C4×C8—531no—
443C4×C8—531no—
444C4×C8—531no—
445C4×C8—1061no—
446C4×C8C22×D5311yescharacteristic
447C4×C8C22×D5311yescharacteristic
448C24—21no—
449C2×Q8—531no—
450C2×D4—22no—
451C2×D4—21no—
452C2×D4—41no—
453C2×D4—531no—
454C2×D4—1061no—
455C2×D4C2×C4×D5311yescharacteristic
456C2×D4C2×C4×D5311yescharacteristic
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—81no—
463C22×C4—81no—
464C22×C4—1061no—
465C22×C4C2×C4×D5311yescharacteristic
466C22×C4D4×D5311yescharacteristic
467C22×C4#1171994711yescharacteristic
468SD16—1062no—
469SD16—2121no—
470D8—1062no—
471D8—2121no—
472M16—4241no—
473M16—4241no—
474M16—4241no—
475C2×C8—24no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—81no—
481C2×C8—531no—
482C2×C8—531no—
483C2×C8—531no—
484C2×C8—531no—
485C2×C8—531no—
486C2×C8—531no—
487C2×C8—531no—
488C2×C8—531no—
489C2×C8—1061no—
490C2×C8—1061no—
491C2×C8—4241no—
492C2×C8—4241no—
493C2×C8—4241no—
494C4⋊C4—21no—
495C4⋊C4—531no—
496C4⋊C4—531no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4C2×C4×D5311yescharacteristic
502C4⋊C4C2×C4×D5311yescharacteristic
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—1061no—
508C22⋊C4—1061no—
509C22⋊C4—1061no—
510C22⋊C4—1061no—
511C22⋊C4—1061no—
512C42—1061no—
513C42C2×C4×D5311yescharacteristic
514C42D4×D5311yescharacteristic
515C42#1171994711yescharacteristic
516C42C23×D5311yescharacteristic
517C23—21no—
518C23—21no—
519C23—42no—
520C23—42no—
521C23—81no—
522C23—1061no—
523C23#1313317711yescharacteristic
524Q8—1061no—
525Q8—1061no—
526D4—21no—
527D4—21no—
528D4—21no—
529D4—42no—
530D4—1061no—
531D4—1061no—
532D4—2122no—
533D4#1313322312yes—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—1061no—
549C2×C4—1061no—
550C2×C4—1061no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4#1313317711yescharacteristic
554C2×C4#1313317711yescharacteristic
555C2×C4#1313317711yescharacteristic
556C2×C4C22×C4×D5311yescharacteristic
557C2×C4C2×D4×D5311yescharacteristic
558C2×C4C2×#1171994711yescharacteristic
559C8—22no—
560C8—22no—
561C8—41no—
562C8—41no—
563C8—1061no—
564C8—1061no—
565C8—1061no—
566C8—1061no—
567C8—1061no—
568C8—1061no—
569C8—1061no—
570C8—1061no—
571C8—2121no—
572C8—2121no—
573C22—21no—
574C22—21no—
575C22—22no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—84no—
582C22—2121no—
583C22—2121no—
584C22—2121no—
585C22#1681073712yes—
586C22C2×#1313317711yescharacteristic
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—41no—
592C4—41no—
593C4—2121no—
594C4—2121no—
595C4—2121no—
596C4C2×#1313322311yescharacteristic
597C4#1681138911yescharacteristic
598C4#1681142411yescharacteristic
599C4#1681142911yescharacteristic
600C2—22no—
601C2—42no—
602C2—81no—
603C2—4241no—
604C2#2028343611yescharacteristic
605C2#2028354111yescharacteristic
606C2C2×#1681073711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC2×C53×D4⋊C8C2
socleC22×C53C2×C4×D4
prime radical#16810757C4
cosocleC42×C53C24
hypercenterC2×D4⋊C8D53
hypocenterC53#18014
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #20283871C2semidirect
3. #20283750C2semidirect
4. #20273608C2semidirect
5. #20273597C2semidirect
6. #20273596C2semidirect
7. #20273582C2semidirect
8. #20273576C2semidirect
9. #20273569C2semidirect
10. #20273518C2semidirect
11. #20273507C2semidirect
12. #20273498C2semidirect
13. #20273176C2semidirect
14. #20272998C2semidirect
15. C2×C53×D4⋊C8C2semidirect
25. #16810750C22semidirect
26. #16810747C22semidirect
33. #16810700C22semidirect
39. #16810688C22semidirect
48. #16810375C22semidirect
49. #16810375C22semidirect
50. #16810373C22semidirect
51. #16810373C22semidirect
52. #16810355C22semidirect
53. #16810355C22semidirect
54. #16810352C22semidirect
55. #16810352C22semidirect
56. #16810344C22semidirect
57. #16810344C22semidirect
58. C8×#88322C22semidirect
59. C8×#88322C22semidirect
310. C2×D4⋊C8D53semidirect
370. C53#18014semidirect

Maximal subgroups

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

subgroupindex
#202847622
#202838712
#202837502
#202736082
#202735972
#202735962
#202735822
#202735762
#202735692
#202735182
#202735072
#202734982
#202731762
#202729982
C2×C53×D4⋊C82
#1801453