Finite Group Databasebeta

← #27355252

Subgroups — #27355252

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#20284764C211yesmaximal, characteristic
2#20283877C211yesmaximal, characteristic, complemented
3#20283749C211yesmaximal, characteristic, complemented
4#20273607C211yesmaximal, characteristic, complemented
5#20273601C211yesmaximal, characteristic, complemented
6#20273595C211yesmaximal, characteristic, complemented
7#20273581C211yesmaximal, characteristic, complemented
8#20273581C211yesmaximal, characteristic, complemented
9#20273574C211yesmaximal, characteristic, complemented
10C8×#11719913C211yesmaximal, characteristic, complemented
11C8×D424C211yesmaximal, characteristic, complemented
12#20273500C211yesmaximal, characteristic, complemented
13#20273176C211yesmaximal, characteristic, complemented
14C2×#16810352C211yesmaximal, characteristic, complemented
15C53×#1450C211yesmaximal, characteristic, complemented
16#16811157C2211yescharacteristic
17C2×C4×D212C2211yescharacteristic
18#16810959C411yescharacteristic
19#16810942C411yescharacteristic
20#16810924C411yescharacteristic
21#16810924C411yescharacteristic
22#16810839C411yescharacteristic
23#16810839C411yescharacteristic
24#16810776C411yescharacteristic
25#16810759C411yescharacteristic
26#16810753C2211yescharacteristic, complemented
27C8×Dic106C2211yescharacteristic, complemented
28C4×#11719913C2211yescharacteristic
29C4×#11719913C2211yescharacteristic
30C4×D424C2211yescharacteristic
31C4×D424C2211yescharacteristic
32#16810702C2211yescharacteristic
33#16810702C2211yescharacteristic
34#16810702C2211yescharacteristic, complemented
35#16810693—21no—
36#16810693—21no—
37#16810691—21no—
38C8×D212C2211yescharacteristic
39C8×D212C2211yescharacteristic
40C8×D212C2211yescharacteristic, complemented
41#16810686—21no—
42#16810683—21no—
43#16810680C2211yescharacteristic
44#16810648—21no—
45C2×C4⋊#88322C2211yescharacteristic
46#16810585—21no—
47C22⋊C8×D53—21no—
48#16810376C2211yescharacteristic, complemented
49#16810376C2211yescharacteristic, complemented
50#16810374C2211yescharacteristic, complemented
51#16810374C2211yescharacteristic, complemented
52#16810355C2211yescharacteristic, complemented
53#16810355C2211yescharacteristic, complemented
54#16810352C2211yescharacteristic
55#16810352C2211yescharacteristic
56#16810352C2211yescharacteristic
57#16810352C2211yescharacteristic
58#16810352C2211yescharacteristic, complemented
59#16810352C2211yescharacteristic, complemented
60C53×#431C2211yescharacteristic
61C53×#431C2211yescharacteristic
62C2×C4×C8×C53C2211yescharacteristic
63C53×#322C2211yescharacteristic, complemented
64C53×#322C2211yescharacteristic, complemented
65C82×C53C2211yescharacteristic, complemented
66C82×C53C2211yescharacteristic, complemented
67C22×D212C2×C411yescharacteristic
68C22×C4×D53—21no—
69C2×#11719913C2×C411yescharacteristic
70C2×#11719913C2×C411yescharacteristic
71C2×D424C2×C411yescharacteristic
72C2×D424C2×C411yescharacteristic
73C2×#11719912—21no—
74C2×#11719912—21no—
75C2×#11719912—21no—
76C22×#88322—21no—
77C2×C8×D53—21no—
78C2×C8×D53—21no—
79C2×C8×D53—21no—
80#13133211—21no—
81#13133209C2×C411yescharacteristic
82#13133205C2×C411yescharacteristic
83#13133200—21no—
84#13133196C2×C411yescharacteristic
85#13133191C2×C411yescharacteristic
86C4×Dic106C2311yescharacteristic
87C4×D212—21no—
88C4×D212—21no—
89C4×D212C2×C411yescharacteristic
90C4×D212C2×C411yescharacteristic
91C4×D212C2311yescharacteristic
92C4×D212C2311yescharacteristic
93C4×D212C2311yescharacteristic
94C4×C22⋊D53—21no—
95#13133174—21no—
96C2×C4⋊Dic53C2×C411yescharacteristic
97C2×#11719921—21no—
98#13133151C2×C411yescharacteristic
99#13133151C2×C411yescharacteristic
100#13133149C2×C411yescharacteristic
101#13133149C2×C411yescharacteristic
102C4⋊#88322—21no—
103C4⋊#88322C2311yescharacteristic
104C4⋊#88322C2311yescharacteristic
105C4⋊#88322C2311yescharacteristic
106C4⋊#88322C2311yescharacteristic
107#13133146—21no—
108#13133146—21no—
109#13133142C2×C411yescharacteristic
110#13133142C2×C411yescharacteristic
111#13133139C2×C411yescharacteristic
112#13133139C2×C411yescharacteristic
113#13133139C2×C411yescharacteristic
114#13133139C2×C411yescharacteristic
115#13133139C2×C411yescharacteristic
116#13133139C2×C411yescharacteristic
117C22⋊#88322—21no—
118#13133129—21no—
119#13133129—21no—
120#13133129—21no—
121#13133129—21no—
122#13133129—21no—
123#13133129—21no—
124#13133128—21no—
125#13133127—21no—
126C8×Dic53—21no—
127C22×C8×C53—21no—
128C2×C42×C53C2311yescharacteristic
129C4⋊C8×C53C2×C411yescharacteristic
130C4⋊C8×C53C2×C411yescharacteristic
131C4⋊C8×C53D411yescharacteristic
132C4⋊C8×C53D411yescharacteristic
133C22⋊C8×C53C2×C411yescharacteristic
134C22⋊C8×C53C2×C411yescharacteristic
135C22⋊C8×C53D411yescharacteristic
136C22⋊C8×C53D411yescharacteristic
137C4×C8×C53—21no—
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—
149C22×Dic53—21no—
150C2×D212—22no—
151C2×D212—21no—
152C2×D212—41no—
153C2×D212C22×C411yescharacteristic
154C2×D212C22×C411yescharacteristic
155C2×D212C22×C411yescharacteristic
156C2×C4×D53—21no—
157C2×C4×D53—21no—
158C2×C4×D53—21no—
159C2×C4×D53—21no—
160C2×C4×D53—21no—
161C2×C4×D53—81no—
162C2×C4×D53—81no—
163C2×Dic106C22×C411yescharacteristic
164C2×C8×C53—24no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53C22×C411yescharacteristic
168C2×C8×C53C22×C411yescharacteristic
169C2×C8×C53C22×C411yescharacteristic
170C2×C8×C53C22×C411yescharacteristic
171C2×C8×C53C2×D411yescharacteristic
172C2×C8×C53C2×D411yescharacteristic
173C2×C8×C53C2×D411yescharacteristic
174C2×C8×C53C2×D411yescharacteristic
175C42×C53C22×C411yescharacteristic
176C42×C53C2×D411yescharacteristic
177C42×C53#4111yescharacteristic
178C42×C53C2411yescharacteristic
179C22⋊Dic53—21no—
180#11719921—21no—
181#11719921—21no—
182#11719921—21no—
183#11719921—21no—
184#11719921—21no—
185#11719921—21no—
186#11719921—21no—
187#11719921—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53C22×C411yescharacteristic
190C4⋊Dic53C22×C411yescharacteristic
191C4⋊Dic53C22×C411yescharacteristic
192C4⋊Dic53C22×C411yescharacteristic
193#11719919—21no—
194#11719919—21no—
195#11719919—21no—
196#11719919—21no—
197C4×Dic53—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—81no—
203D424—22no—
204D424—41no—
205#11719913—22no—
206#11719913—41no—
207#11719912—81no—
208#11719912—81no—
209#11719912—81no—
210C8×D53—81no—
211C8×D53—81no—
212C8×D53—81no—
213C23×C53C4×D411yescharacteristic
214C22×D53—21no—
215C22×D53—21no—
216C22×D53—21no—
217C22×D53—42no—
218C22×D53—42no—
219C22×D53—81no—
220C2×C4×C53—22no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53C4×D411yescharacteristic
226C2×C4×C53C4×D411yescharacteristic
227C2×C4×C53C4×D411yescharacteristic
228C2×C4×C53C23×C411yescharacteristic
229C2×C4×C53C22×D411yescharacteristic
230C2×C4×C53C2×#4111yescharacteristic
231C22⋊D53—42no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—81no—
238D212—21no—
239D212—21no—
240D212—21no—
241D212—21no—
242D212—21no—
243D212—42no—
244D212#13112yes—
245C4×D53—41no—
246C4×D53—41no—
247C4×D53—41no—
248C4×D53—41no—
249C4×D53—41no—
250C4×D53—41no—
251C4×D53—81no—
252C4×D53—81no—
253Dic106—21no—
254Dic106—21no—
255C8×C53—22no—
256C8×C53—22no—
257C8×C53—21no—
258C8×C53—21no—
259C8×C53—21no—
260C8×C53—21no—
261C8×C53—21no—
262C8×C53—21no—
263C8×C53—21no—
264C8×C53—21no—
265#88322—41no—
266#88322—41no—
267#88322—41no—
268#88322—41no—
269#18014—531nomaximal, Sylow
270C22×C53—21no—
271C22×C53—21no—
272C22×C53#44312yes—
273C22×C53C2×C4×D411yescharacteristic
274C2×D53—22no—
275C2×D53—41no—
276C2×D53—41no—
277C2×D53—41no—
278C2×D53—41no—
279C2×D53—41no—
280C2×D53—41no—
281C2×D53—41no—
282C2×D53—41no—
283C2×D53—84no—
284C4×C53—21no—
285C4×C53—21no—
286C4×C53—21no—
287C4×C53—21no—
288C4×C53C2×#13111yescharacteristic
289C4×C53#56811yescharacteristic
290C4×C53#57611yescharacteristic
291C4×C53#57711yescharacteristic
292Dic53—41no—
293Dic53—41no—
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⋊C8—531no—
311#1450D5311yescharacteristic, complemented
312C2×C53—22no—
313C2×C53#293311yescharacteristic
314C2×C53#294611yescharacteristic
315C2×C53C2×#44311yescharacteristic
316D53—42no—
317D53—81no—
318D53—81no—
319#529—531no—
320C2×C4×D4—531no—
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#431C2×D5311yescharacteristic
348#431C2×D5311yescharacteristic
349#424—1061no—
350C2×C4⋊C8—531no—
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⋊C8—531no—
363D4⋊C8—531no—
364D4⋊C8—531no—
365D4⋊C8—531no—
366#322C2×D5311yescharacteristic, complemented
367#322C2×D5311yescharacteristic, complemented
368C82C2×D5311yescharacteristic, complemented
369C82C2×D5311yescharacteristic, complemented
370C53#1801411yescharacteristic, complemented, Sylow
371C22×D4—531no—
372C23×C4—1061no—
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—1061no—
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—531no—
393C4×D4—531no—
394C4×D4—531no—
395C4×D4—531no—
396C4×D4—531no—
397C4×D4—1061no—
398C4×D4—1061no—
399C4×D4—1061no—
400#117—1061no—
401C2×C4⋊C4—531no—
402C2×C22⋊C4—1061no—
403C2×C42C22×D5311yescharacteristic
404#107—531no—
405#107—531no—
406#106—531no—
407#106—531no—
408C4⋊C8—531no—
409C4⋊C8—531no—
410C4⋊C8—531no—
411C4⋊C8—531no—
412C4⋊C8—1061no—
413C4⋊C8—1061no—
414C4⋊C8—1061no—
415C4⋊C8C4×D5311yescharacteristic
416C4⋊C8C4×D5311yescharacteristic
417C4⋊C8D21211yescharacteristic
418C4⋊C8D21211yescharacteristic
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—1061no—
428C22⋊C8—1061no—
429C22⋊C8—1061no—
430C22⋊C8—1061no—
431C22⋊C8—1061no—
432C22⋊C8—1061no—
433C22⋊C8—1061no—
434C22⋊C8C4×D5311yescharacteristic
435C22⋊C8C4×D5311yescharacteristic
436C22⋊C8D21211yescharacteristic
437C22⋊C8D21211yescharacteristic
438#97—1061no—
439#97—1061no—
440C4×C8—21no—
441C4×C8—1061no—
442C4×C8C22×D5311yescharacteristic
443C4×C8C22×D5311yescharacteristic
444C4×C8C22×D5311yescharacteristic
445C4×C8C22×D5311yescharacteristic
446C4×C8C22×D5311yescharacteristic
447C4×C8C22×D5311yescharacteristic
448C24—1061no—
449C2×Q8—531no—
450C2×D4—531no—
451C2×D4—531no—
452C2×D4—531no—
453C2×D4—1062no—
454C2×D4—1061no—
455C2×D4—1061no—
456C2×D4—2121no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—4241no—
464C22×C4—4241no—
465C22×C4C2×C4×D5311yescharacteristic
466C22×C4C2×D21211yescharacteristic
467C22×C4#1171994511yescharacteristic
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—1061no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8—1061no—
482C2×C8—4241no—
483C2×C8—4241no—
484C2×C8—4241no—
485C2×C8—4241no—
486C2×C8C2×C4×D5311yescharacteristic
487C2×C8C2×C4×D5311yescharacteristic
488C2×C8C2×C4×D5311yescharacteristic
489C2×C8C2×C4×D5311yescharacteristic
490C2×C8C2×D21211yescharacteristic
491C2×C8C2×D21211yescharacteristic
492C2×C8C2×D21211yescharacteristic
493C2×C8C2×D21211yescharacteristic
494C4⋊C4—531no—
495C4⋊C4—531no—
496C4⋊C4—531no—
497C4⋊C4—531no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C22⋊C4—1061no—
504C22⋊C4—1061no—
505C22⋊C4—1061no—
506C22⋊C4—1061no—
507C22⋊C4—1061no—
508C22⋊C4—1061no—
509C22⋊C4—1061no—
510C22⋊C4—1061no—
511C22⋊C4—1061no—
512C42—1061no—
513C42C2×C4×D5311yescharacteristic
514C42C2×D21211yescharacteristic
515C42#1171994511yescharacteristic
516C42C23×D5311yescharacteristic
517C23—1061no—
518C23—1061no—
519C23—1061no—
520C23—4241no—
521C23—2122no—
522C23—2122no—
523C23C4×D21211yescharacteristic
524Q8—1061no—
525Q8—1061no—
526D4—532no—
527D4—1061no—
528D4—1061no—
529D4—1061no—
530D4—1061no—
531D4—1061no—
532D4—2122no—
533D4—2122no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—1061no—
540C2×C4—1061no—
541C2×C4—1061no—
542C2×C4—1061no—
543C2×C4—1061no—
544C2×C4—4241no—
545C2×C4—4241no—
546C2×C4—4241no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4—2121no—
553C2×C4C4×D21211yescharacteristic
554C2×C4C4×D21211yescharacteristic
555C2×C4C4×D21211yescharacteristic
556C2×C4C22×C4×D5311yescharacteristic
557C2×C4C22×D21211yescharacteristic
558C2×C4C2×#1171994511yescharacteristic
559C8—22no—
560C8—22no—
561C8—21no—
562C8—21no—
563C8—21no—
564C8—21no—
565C8—21no—
566C8—21no—
567C8—21no—
568C8—21no—
569C8—2121no—
570C8—2121no—
571C8—2121no—
572C8—2121no—
573C22—21no—
574C22—21no—
575C22—1062no—
576C22—4244no—
577C22—2121no—
578C22—2121no—
579C22—2121no—
580C22—2121no—
581C22—2121no—
582C22—2121no—
583C22—2121no—
584C22—2121no—
585C22#1681074112yes—
586C22C2×C4×D21211yescharacteristic
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—2121no—
592C4—2121no—
593C4—2121no—
594C4—2121no—
595C4—2121no—
596C4#131×D5311yescharacteristic
597C4#1681138811yescharacteristic
598C4#1681142811yescharacteristic
599C4#1681143511yescharacteristic
600C2—22no—
601C2—4241no—
602C2—4241no—
603C2—2122no—
604C2#2028344311yescharacteristic
605C2#2028354911yescharacteristic
606C2C2×#1681074111yescharacteristic
subgroupG/H
centerC2×C4C22×D212
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1450C2
socleC22×C53C2×C4×D4
prime radical#16810759C4
cosocleC42×C53C24
hypercenter#1450D53
hypocenterC53#18014
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1450

Complements

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

NH ≅ G/Ntype
2. #20283877C2semidirect
3. #20283749C2semidirect
4. #20273607C2semidirect
5. #20273601C2semidirect
6. #20273595C2semidirect
7. #20273581C2semidirect
8. #20273581C2semidirect
9. #20273574C2semidirect
10. C8×#11719913C2semidirect
11. C8×D424C2semidirect
12. #20273500C2semidirect
13. #20273176C2semidirect
14. C2×#16810352C2semidirect
15. C53×#1450C2semidirect
26. #16810753C22semidirect
27. C8×Dic106C22semidirect
34. #16810702C22semidirect
40. C8×D212C22semidirect
48. #16810376C22semidirect
49. #16810376C22semidirect
50. #16810374C22semidirect
51. #16810374C22semidirect
52. #16810355C22semidirect
53. #16810355C22semidirect
58. #16810352C22semidirect
59. #16810352C22semidirect
63. C53×#322C22semidirect
64. C53×#322C22semidirect
65. C82×C53C22semidirect
66. C82×C53C22semidirect
311. #1450D53semidirect
366. #322C2×D53semidirect
367. #322C2×D53semidirect
368. C82C2×D53semidirect
369. C82C2×D53semidirect
370. C53#18014semidirect

Maximal subgroups

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

subgroupindex
#202847642
#202838772
#202837492
#202736072
#202736012
#202735952
#202735812
#202735742
C8×#117199132
C8×D4242
#202735002
#202731762
C2×#168103522
C53×#14502
#1801453