Finite Group Databasebeta

← #27355400

Subgroups — #27355400

Order 13568.

656 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284716C211yesmaximal, characteristic
2#20283781C211yesmaximal, characteristic, complemented
3#20283749C211yesmaximal, characteristic, complemented
4#20273630C211yesmaximal, characteristic, complemented
5#20273601C211yesmaximal, characteristic, complemented
6#20273581C211yesmaximal, characteristic, complemented
7#20273581C211yesmaximal, characteristic, complemented
8#20273546C211yesmaximal, characteristic, complemented
9#20273546C211yesmaximal, characteristic, complemented
10#20273540C211yesmaximal, characteristic, complemented
11C8×D424C211yesmaximal, characteristic, complemented
12#20273495C211yesmaximal, characteristic, complemented
13#20273185C211yesmaximal, characteristic, complemented
14#20273107C211yesmaximal, characteristic, complemented
15C53×#1454C211yesmaximal, characteristic, complemented
16#16811173C2211yescharacteristic
17C2×C4×D212C2211yescharacteristic
18#16810937C411yescharacteristic
19#16810937C411yescharacteristic
20#16810904C411yescharacteristic
21#16810904C411yescharacteristic
22#16810819C411yescharacteristic
23#16810819C411yescharacteristic
24#16810759C411yescharacteristic
25#16810759C411yescharacteristic
26C4×D424C2211yescharacteristic
27C4×D424C2211yescharacteristic
28C4×D424C2211yescharacteristic
29C4×D424C2211yescharacteristic
30#16810702C2211yescharacteristic
31#16810702C2211yescharacteristic
32#16810702C2211yescharacteristic, complemented
33#16810702C2211yescharacteristic, complemented
34#16810700—21no—
35#16810693—21no—
36#16810693—21no—
37C8×D212C2211yescharacteristic
38C8×D212C2211yescharacteristic
39C8×D212C2211yescharacteristic, complemented
40C8×D212C2211yescharacteristic, complemented
41#16810688—21no—
42#16810680C2211yescharacteristic
43#16810676C2211yescharacteristic
44#16810588—21no—
45#16810588—21no—
46#16810585—21no—
47C22⋊C8×D53—21no—
48#16810376C2211yescharacteristic, complemented
49#16810376C2211yescharacteristic, complemented
50#16810374C2211yescharacteristic, complemented
51#16810374C2211yescharacteristic, complemented
52#16810352C2211yescharacteristic
53#16810352C2211yescharacteristic
54#16810352C2211yescharacteristic
55#16810352C2211yescharacteristic
56#16810352C2211yescharacteristic, complemented
57#16810352C2211yescharacteristic, complemented
58#16810352C2211yescharacteristic, complemented
59#16810352C2211yescharacteristic, complemented
60C53×#431C2211yescharacteristic
61C53×#431C2211yescharacteristic
62C53×#431C2211yescharacteristic
63C53×#322C2211yescharacteristic, complemented
64C53×#322C2211yescharacteristic, complemented
65C53×#322C2211yescharacteristic, complemented
66C82×C53C2211yescharacteristic, complemented
67C2×#11719945—21no—
68C22×D212C2×C411yescharacteristic
69C22×C4×D53—21no—
70C2×D424C2×C411yescharacteristic
71C2×D424C2×C411yescharacteristic
72C2×D424C2×C411yescharacteristic
73C2×D424C2×C411yescharacteristic
74C2×#11719912—21no—
75C2×#11719912—21no—
76C2×#11719912—21no—
77C2×#11719912—21no—
78C2×C8×D53—21no—
79C2×C8×D53—21no—
80C2×C8×D53—21no—
81C2×C8×D53—21no—
82#13133217C2×C411yescharacteristic
83#13133214C2×C411yescharacteristic
84#13133205—21no—
85#13133191—21no—
86#13133191C2×C411yescharacteristic
87#13133191C2×C411yescharacteristic
88C4×D212—21no—
89C4×D212—21no—
90C4×D212—21no—
91C4×D212C2×C411yescharacteristic
92C4×D212C2×C411yescharacteristic
93C4×D212C2311yescharacteristic
94C4×D212C2311yescharacteristic
95C4×D212C2311yescharacteristic
96C4×D212C2311yescharacteristic
97C2×C4⋊Dic53C2×C411yescharacteristic
98C2×#11719921—21no—
99#13133151C2×C411yescharacteristic
100#13133151C2×C411yescharacteristic
101#13133151C2×C411yescharacteristic
102#13133151C2×C411yescharacteristic
103C4⋊#88322C2311yescharacteristic
104C4⋊#88322C2311yescharacteristic
105C4⋊#88322C2311yescharacteristic
106C4⋊#88322C2311yescharacteristic
107#13133139C2×C411yescharacteristic
108#13133139C2×C411yescharacteristic
109#13133139C2×C411yescharacteristic
110#13133139C2×C411yescharacteristic
111#13133139C2×C411yescharacteristic
112#13133139C2×C411yescharacteristic
113#13133139C2×C411yescharacteristic
114#13133139C2×C411yescharacteristic
115C22⋊#88322—21no—
116C22⋊#88322—21no—
117#13133129—21no—
118#13133129—21no—
119#13133129—21no—
120#13133129—21no—
121#13133129—21no—
122#13133129—21no—
123#13133129—21no—
124#13133129—21no—
125#13133128—21no—
126C4×#88322—21no—
127C2×C42×C53C2311yescharacteristic
128C4⋊C8×C53—21no—
129C4⋊C8×C53C2×C411yescharacteristic
130C4⋊C8×C53C2×C411yescharacteristic
131C4⋊C8×C53D411yescharacteristic
132C4⋊C8×C53D411yescharacteristic
133C22⋊C8×C53—21no—
134C22⋊C8×C53C2×C411yescharacteristic
135C22⋊C8×C53C2×C411yescharacteristic
136C22⋊C8×C53D411yescharacteristic
137C22⋊C8×C53D411yescharacteristic
138C4×C8×C53C2311yescharacteristic
139C4×C8×C53C2311yescharacteristic
140C4×C8×C53C2311yescharacteristic
141C4×C8×C53C2311yescharacteristic
142C4×C8×C53C2311yescharacteristic
143C4×C8×C53C2311yescharacteristic
144C23×D53—21no—
145C22×C4×C53C22×C411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53#4111yescharacteristic
148C2×C22⋊D53—21no—
149C2×C22⋊D53—21no—
150C22×Dic53—21no—
151#11719945—81no—
152#11719945—81no—
153C2×D212—22no—
154C2×D212—21no—
155C2×D212—21no—
156C2×D212—21no—
157C2×D212—41no—
158C2×D212C22×C411yescharacteristic
159C2×D212C22×C411yescharacteristic
160C2×D212C22×C411yescharacteristic
161C2×D212C22×C411yescharacteristic
162C2×C4×D53—21no—
163C2×C4×D53—21no—
164C2×C4×D53—21no—
165C2×C4×D53—21no—
166C2×C4×D53—21no—
167C2×C4×D53—21no—
168C2×C4×D53—81no—
169C2×C4×D53—81no—
170C2×Dic106—21no—
171C2×C8×C53—21no—
172C2×C8×C53—21no—
173C2×C8×C53C22×C411yescharacteristic
174C2×C8×C53C22×C411yescharacteristic
175C2×C8×C53C22×C411yescharacteristic
176C2×C8×C53C22×C411yescharacteristic
177C2×C8×C53C2×D411yescharacteristic
178C2×C8×C53C2×D411yescharacteristic
179C2×C8×C53C2×D411yescharacteristic
180C2×C8×C53C2×D411yescharacteristic
181C42×C53C22×C411yescharacteristic
182C42×C53C2×D411yescharacteristic
183C42×C53#4111yescharacteristic
184C42×C53C2411yescharacteristic
185#11719921—21no—
186#11719921—21no—
187#11719921—21no—
188#11719921—21no—
189#11719921—21no—
190#11719921—21no—
191#11719921—21no—
192#11719921—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53C22×C411yescharacteristic
196C4⋊Dic53C22×C411yescharacteristic
197C4⋊Dic53C22×C411yescharacteristic
198C4⋊Dic53C22×C411yescharacteristic
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203D424—22no—
204D424—22no—
205D424—41no—
206D424—41no—
207#11719912—81no—
208#11719912—81no—
209#11719912—81no—
210#11719912—81no—
211C8×D53—81no—
212C8×D53—81no—
213C8×D53—81no—
214C8×D53—81no—
215C23×C53C4×D411yescharacteristic
216C22×D53—21no—
217C22×D53—21no—
218C22×D53—21no—
219C22×D53—21no—
220C22×D53—42no—
221C22×D53—42no—
222C22×D53—81no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—41no—
228C2×C4×C53C4×D411yescharacteristic
229C2×C4×C53C4×D411yescharacteristic
230C2×C4×C53C4×D411yescharacteristic
231C2×C4×C53C23×C411yescharacteristic
232C2×C4×C53C22×D411yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C22⋊D53—81no—
235C22⋊D53—81no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—81no—
241D212—21no—
242D212—21no—
243D212—21no—
244D212—21no—
245D212—21no—
246D212—21no—
247D212—21no—
248D212—42no—
249D212—42no—
250D212—81no—
251D212#13112yes—
252C4×D53—41no—
253C4×D53—41no—
254C4×D53—41no—
255C4×D53—41no—
256C4×D53—41no—
257C4×D53—41no—
258C4×D53—41no—
259C4×D53—41no—
260C4×D53—81no—
261C4×D53—81no—
262Dic106—81no—
263C8×C53—21no—
264C8×C53—21no—
265C8×C53—21no—
266C8×C53—21no—
267C8×C53—21no—
268C8×C53—21no—
269C8×C53—21no—
270C8×C53—21no—
271C8×C53—41no—
272C8×C53—41no—
273#88322—41no—
274#88322—41no—
275#88322—41no—
276#88322—41no—
277#18026—531nomaximal, Sylow
278C22×C53—21no—
279C22×C53—21no—
280C22×C53—21no—
281C22×C53C2×C4×D411yescharacteristic
282C2×D53—22no—
283C2×D53—41no—
284C2×D53—41no—
285C2×D53—41no—
286C2×D53—41no—
287C2×D53—41no—
288C2×D53—41no—
289C2×D53—41no—
290C2×D53—41no—
291C2×D53—41no—
292C2×D53—41no—
293C2×D53—41no—
294C2×D53—84no—
295C4×C53—21no—
296C4×C53—21no—
297C4×C53—21no—
298C4×C53—21no—
299C4×C53C2×#13111yescharacteristic
300C4×C53#56811yescharacteristic
301C4×C53#57611yescharacteristic
302C4×C53#57611yescharacteristic
303Dic53—41no—
304Dic53—41no—
305Dic53—41no—
306Dic53—41no—
307#3100—531no—
308#2979—531no—
309#2974—531no—
310#1594—531no—
311#1589—531no—
312#1586—531no—
313#1586—531no—
314#1581—531no—
315#1581—531no—
316#1580—531no—
317C8×D8—531no—
318#1572—531no—
319#1496—531no—
320#1478—531no—
321#1454D5311yescharacteristic, complemented
322C2×C53—41no—
323C2×C53#293311yescharacteristic
324C2×C53#294811yescharacteristic
325C2×C53#295511yescharacteristic
326D53—42no—
327D53—81no—
328D53—81no—
329D53—81no—
330#532—531no—
331C2×C4×D4—531no—
332#480—531no—
333#480—531no—
334#474—531no—
335#474—531no—
336#459—531no—
337#459—531no—
338#447—531no—
339#447—531no—
340C4×D8—531no—
341C4×D8—531no—
342C4×D8—531no—
343C4×D8—531no—
344#436—531no—
345#436—531no—
346#436—531no—
347#436—531no—
348#436—1061no—
349#435—1061no—
350#435—1061no—
351C8×D4—531no—
352C8×D4—531no—
353C8×D4—531no—
354C8×D4—531no—
355C8×D4—1061no—
356#432—531no—
357#431—531no—
358#431C2×D5311yescharacteristic
359#431C2×D5311yescharacteristic
360#431C2×D5311yescharacteristic
361#408—1061no—
362#408—1061no—
363#407—1061no—
364C2×C22⋊C8—1061no—
365#335—531no—
366#335—531no—
367#334—531no—
368#334—531no—
369D4⋊C8—531no—
370D4⋊C8—531no—
371D4⋊C8—531no—
372D4⋊C8—531no—
373D4⋊C8—531no—
374D4⋊C8—531no—
375D4⋊C8—531no—
376D4⋊C8—531no—
377#322C2×D5311yescharacteristic, complemented
378#322C2×D5311yescharacteristic, complemented
379#322C2×D5311yescharacteristic, complemented
380C82C2×D5311yescharacteristic, complemented
381C53#1802611yescharacteristic, complemented, Sylow
382C2×#41—1061no—
383C22×D4—531no—
384C23×C4—1061no—
385C2×D8—531no—
386C2×D8—531no—
387C2×D8—531no—
388C2×D8—531no—
389C2×M16—1061no—
390C2×M16—1061no—
391C2×M16—1061no—
392C2×M16—1061no—
393C22×C8—1061no—
394C22×C8—1061no—
395C22×C8—1061no—
396C22×C8—1061no—
397C4⋊Q8—531no—
398#127—531no—
399#124—1061no—
400#121—531no—
401#121—531no—
402#121—1061no—
403C4×D4—531no—
404C4×D4—531no—
405C4×D4—531no—
406C4×D4—531no—
407C4×D4—531no—
408C4×D4—531no—
409C4×D4—1061no—
410C4×D4—1061no—
411C4×D4—1061no—
412C2×C4⋊C4—531no—
413C2×C22⋊C4—1061no—
414C2×C42C22×D5311yescharacteristic
415#107—531no—
416#107—531no—
417#107—531no—
418#107—531no—
419C4⋊C8—21no—
420C4⋊C8—531no—
421C4⋊C8—531no—
422C4⋊C8—531no—
423C4⋊C8—531no—
424C4⋊C8C4×D5311yescharacteristic
425C4⋊C8C4×D5311yescharacteristic
426C4⋊C8D21211yescharacteristic
427C4⋊C8D21211yescharacteristic
428#102—531no—
429#102—531no—
430#102—531no—
431#102—531no—
432#102—531no—
433#102—531no—
434#102—531no—
435#102—531no—
436C22⋊C8—21no—
437C22⋊C8—1061no—
438C22⋊C8—1061no—
439C22⋊C8—1061no—
440C22⋊C8—1061no—
441C22⋊C8—1061no—
442C22⋊C8—1061no—
443C22⋊C8—1061no—
444C22⋊C8—1061no—
445C22⋊C8—1061no—
446C22⋊C8—1061no—
447C22⋊C8C4×D5311yescharacteristic
448C22⋊C8C4×D5311yescharacteristic
449C22⋊C8D21211yescharacteristic
450C22⋊C8D21211yescharacteristic
451#97—1061no—
452C4×C8—1061no—
453C4×C8C22×D5311yescharacteristic
454C4×C8C22×D5311yescharacteristic
455C4×C8C22×D5311yescharacteristic
456C4×C8C22×D5311yescharacteristic
457C4×C8C22×D5311yescharacteristic
458C4×C8C22×D5311yescharacteristic
459C24—1061no—
460#41—4241no—
461#41—4241no—
462C2×Q8—1061no—
463C2×D4—531no—
464C2×D4—531no—
465C2×D4—531no—
466C2×D4—531no—
467C2×D4—1062no—
468C2×D4—1061no—
469C2×D4—1061no—
470C2×D4—1061no—
471C2×D4—1061no—
472C2×D4—1061no—
473C2×D4—2121no—
474C22×C4—1061no—
475C22×C4—1061no—
476C22×C4—1061no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4—1061no—
480C22×C4—1061no—
481C22×C4—4241no—
482C22×C4—4241no—
483C22×C4C2×C4×D5311yescharacteristic
484C22×C4C2×D21211yescharacteristic
485C22×C4#1171994511yescharacteristic
486D8—1062no—
487D8—1062no—
488D8—2121no—
489D8—2121no—
490M16—4241no—
491M16—4241no—
492M16—4241no—
493M16—4241no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—1061no—
497C2×C8—1061no—
498C2×C8—1061no—
499C2×C8—1061no—
500C2×C8—4241no—
501C2×C8—4241no—
502C2×C8—4241no—
503C2×C8—4241no—
504C2×C8C2×C4×D5311yescharacteristic
505C2×C8C2×C4×D5311yescharacteristic
506C2×C8C2×C4×D5311yescharacteristic
507C2×C8C2×C4×D5311yescharacteristic
508C2×C8C2×D21211yescharacteristic
509C2×C8C2×D21211yescharacteristic
510C2×C8C2×D21211yescharacteristic
511C2×C8C2×D21211yescharacteristic
512C4⋊C4—531no—
513C4⋊C4—531no—
514C4⋊C4—531no—
515C4⋊C4—531no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C22⋊C4—1061no—
519C22⋊C4—1061no—
520C22⋊C4—1061no—
521C22⋊C4—1061no—
522C22⋊C4—1061no—
523C22⋊C4—1061no—
524C22⋊C4—1061no—
525C22⋊C4—1061no—
526C42C2×C4×D5311yescharacteristic
527C42C2×D21211yescharacteristic
528C42#1171994511yescharacteristic
529C42C23×D5311yescharacteristic
530C23—1061no—
531C23—1061no—
532C23—1061no—
533C23—1061no—
534C23—4241no—
535C23—2122no—
536C23—2122no—
537C23C4×D21211yescharacteristic
538Q8—4241no—
539D4—532no—
540D4—1061no—
541D4—1061no—
542D4—1061no—
543D4—1061no—
544D4—1061no—
545D4—1061no—
546D4—1061no—
547D4—4241no—
548D4—4241no—
549D4—4241no—
550D4—2122no—
551D4—2122no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—41no—
557C2×C4—1061no—
558C2×C4—1061no—
559C2×C4—1061no—
560C2×C4—1061no—
561C2×C4—4241no—
562C2×C4—4241no—
563C2×C4—4241no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4C4×D21211yescharacteristic
573C2×C4C4×D21211yescharacteristic
574C2×C4C4×D21211yescharacteristic
575C2×C4C22×C4×D5311yescharacteristic
576C2×C4C22×D21211yescharacteristic
577C2×C4C2×#1171994511yescharacteristic
578C8—21no—
579C8—21no—
580C8—21no—
581C8—21no—
582C8—21no—
583C8—21no—
584C8—21no—
585C8—21no—
586C8—41no—
587C8—41no—
588C8—2121no—
589C8—2121no—
590C8—2121no—
591C8—2121no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—1062no—
596C22—4244no—
597C22—2121no—
598C22—2121no—
599C22—2121no—
600C22—2121no—
601C22—2121no—
602C22—2121no—
603C22—2121no—
604C22—2121no—
605C22—2121no—
606C22—2121no—
607C22—2121no—
608C22C2×C4×D21211yescharacteristic
609C4—21no—
610C4—21no—
611C4—21no—
612C4—21no—
613C4—2121no—
614C4—2121no—
615C4—2121no—
616C4—2121no—
617C4#131×D5311yescharacteristic
618C4#1681138811yescharacteristic
619C4#1681142811yescharacteristic
620C4#1681142811yescharacteristic
621C2—41no—
622C2—4241no—
623C2—4241no—
624C2—4241no—
625C2—2122no—
626C2#2028344311yescharacteristic
627C2#2028356811yescharacteristic
628C2#2028361911yescharacteristic
subgroupG/H
centerC2×C4C22×D212
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1454C2
socleC22×C53C2×C4×D4
prime radical#20284716C2
cosocleC42×C53C24
hypercenter#1454D53
hypocenterC53#18026
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1454

Complements

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

NH ≅ G/Ntype
2. #20283781C2semidirect
3. #20283749C2semidirect
4. #20273630C2semidirect
5. #20273601C2semidirect
6. #20273581C2semidirect
7. #20273581C2semidirect
8. #20273546C2semidirect
9. #20273546C2semidirect
10. #20273540C2semidirect
11. C8×D424C2semidirect
12. #20273495C2semidirect
13. #20273185C2semidirect
14. #20273107C2semidirect
15. C53×#1454C2semidirect
32. #16810702C22semidirect
33. #16810702C22semidirect
39. C8×D212C22semidirect
40. C8×D212C22semidirect
48. #16810376C22semidirect
49. #16810376C22semidirect
50. #16810374C22semidirect
51. #16810374C22semidirect
56. #16810352C22semidirect
57. #16810352C22semidirect
58. #16810352C22semidirect
59. #16810352C22semidirect
63. C53×#322C22semidirect
64. C53×#322C22semidirect
65. C53×#322C22semidirect
66. C82×C53C22semidirect
321. #1454D53semidirect
377. #322C2×D53semidirect
378. #322C2×D53semidirect
379. #322C2×D53semidirect
380. C82C2×D53semidirect
381. C53#18026semidirect

Maximal subgroups

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

subgroupindex
#202847162
#202837812
#202837492
#202736302
#202736012
#202735812
#202735462
#202735402
C8×D4242
#202734952
#202731852
#202731072
C53×#14542
#1802653