Finite Group Databasebeta

← #27279679

Subgroups — #27279679

Order 13568.

674 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810677C211yesmaximal, characteristic, complemented
2#20283041C211yesmaximal, characteristic
3C2×#16810357C211yesmaximal, characteristic
4#20272939C211yesmaximal, characteristic, complemented
5#20272595C211yesmaximal, characteristic, complemented
6#20272594C211yesmaximal, characteristic, complemented
7C2×C53×#327C211yesmaximal, characteristic
8C2×#13133198—21no—
9C22×C4×Dic53C2211yescharacteristic
10#16810677—22no—
11#16810677—22no—
12C2×#13133146—21no—
13C2×C22⋊#88322C2211yescharacteristic
14C2×#13133127—21no—
15#16810526C411yescharacteristic, complemented
16#16810526C411yescharacteristic, complemented
17#16810519C411yescharacteristic, complemented
18#16810519C411yescharacteristic, complemented
19#16810511—21no—
20#16810511—21no—
21#16810503—21no—
22#16810483C2211yescharacteristic
23C2×#13133124—21no—
24C2×#13133124—21no—
25C2×#13133124C2211yescharacteristic
26#16810357—22no—
27C2×#121×C53C2211yescharacteristic
28C2×C22⋊C8×C53C2211yescharacteristic
29C2×#95×C53C2211yescharacteristic
30C53×#327—22no—
31C23×Dic53—21no—
32C23×Dic53D411yescharacteristic
33C23×Dic53D411yescharacteristic
34C22×#88322—21no—
35#13133198—42no—
36#13133198—42no—
37C2×C4⋊Dic53C2×C411yescharacteristic
38C2×C4⋊Dic53C2×C411yescharacteristic
39C2×#11719919—21no—
40C2×#11719919—21no—
41C2×C22⋊Dic53—21no—
42C2×C22⋊Dic53—21no—
43C2×C22⋊Dic53—21no—
44C2×C22⋊Dic53—21no—
45C2×C22⋊Dic53—21no—
46C2×C4×Dic53—21no—
47C2×C4×Dic53—21no—
48C2×C4×Dic53—21no—
49C2×C4×Dic53—21no—
50C2×C4×Dic53C2×C411yescharacteristic
51C2×C4×Dic53C2×C411yescharacteristic
52C2×C4×Dic53D411yescharacteristic
53C2×C4×Dic53D411yescharacteristic
54#13133146—24no—
55C22⋊#88322—21no—
56C22⋊#88322D412yes—
57#13133127—24no—
58#13133124—24no—
59#13133124—21no—
60#13133124—21no—
61#13133124—21no—
62#13133124—21no—
63#13133124—21no—
64#13133124—42no—
65C22×D4×C53—21no—
66C22×D4×C53C2×C411yescharacteristic
67C23×C4×C53—21no—
68C23×C4×C53C2311yescharacteristic
69C22×C8×C53—21no—
70#121×C53—21no—
71#121×C53—21no—
72#121×C53—21no—
73#121×C53—21no—
74C2×C4⋊C4×C53C2×C411yescharacteristic
75C2×C22⋊C4×C53—21no—
76C22⋊C8×C53—21no—
77C22⋊C8×C53D412yes—
78#95×C53—21no—
79#95×C53—21no—
80C24×C53—21no—
81C24×C53C2×D411yescharacteristic
82C2×D4×C53—21no—
83C2×D4×C53—21no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C2×D4×C53—41no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53—41no—
90C2×D4×C53—41no—
91C2×D4×C53—41no—
92C22×C4×C53—22no—
93C22×C4×C53—21no—
94C22×C4×C53—41no—
95C22×C4×C53—41no—
96C22×C4×C53—41no—
97C22×C4×C53—41no—
98C22×C4×C53—41no—
99C22×C4×C53—41no—
100C22×C4×C53—41no—
101C22×C4×C53—41no—
102C22×C4×C53C22×C411yescharacteristic
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53#4111yescharacteristic
107C22×C4×C53#4111yescharacteristic
108C22×Dic53—21no—
109C22×Dic53—21no—
110C22×Dic53—21no—
111C22×Dic53—21no—
112C22×Dic53—21no—
113C22×Dic53—21no—
114C22×Dic53—21no—
115C22×Dic53—21no—
116C22×Dic53—22no—
117C22×Dic53—22no—
118C22×Dic53—21no—
119C22×Dic53—21no—
120C22×Dic53—41no—
121C22×Dic53—41no—
122C22×Dic53—41no—
123C22×Dic53—41no—
124C22×Dic53—41no—
125C22×Dic53—41no—
126C22×Dic53—41no—
127C22×Dic53C22⋊C411yescharacteristic
128C22×Dic53C22⋊C411yescharacteristic
129C22×Dic53C22⋊C411yescharacteristic
130C22×Dic53C22⋊C411yescharacteristic
131C2×C8×C53—22no—
132C2×C8×C53—41no—
133C2×C8×C53—41no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—21no—
136C22⋊C4×C53—41no—
137C22⋊C4×C53—41no—
138C22⋊Dic53—42no—
139C22⋊Dic53—42no—
140C22⋊Dic53—42no—
141C22⋊Dic53—41no—
142C22⋊Dic53—41no—
143C22⋊Dic53—42no—
144C4⋊Dic53—41no—
145C4⋊Dic53—41no—
146#11719919—42no—
147#11719919—42no—
148C4×Dic53—21no—
149C4×Dic53—21no—
150C4×Dic53—21no—
151C4×Dic53—21no—
152C4×Dic53—21no—
153C4×Dic53—21no—
154C4×Dic53—21no—
155C4×Dic53—21no—
156C2×#88322—22no—
157C2×#88322—41no—
158C2×#88322—41no—
159C23×C53—21no—
160C23×C53—21no—
161C23×C53—21no—
162C23×C53—21no—
163C23×C53—41no—
164C23×C53—41no—
165C23×C53—41no—
166C23×C53—41no—
167C23×C53—41no—
168C23×C53—41no—
169C23×C53—41no—
170C23×C53C2×C22⋊C411yescharacteristic
171C23×C53#12111yescharacteristic
172C23×C53#12111yescharacteristic
173C23×C53AGL(1,ℤ/8)11yescharacteristic
174C23×C53AGL(1,ℤ/8)11yescharacteristic
175C23×C53AGL(1,ℤ/8)11yescharacteristic
176C23×C53#13711yescharacteristic
177D4×C53—81no—
178D4×C53—81no—
179D4×C53—81no—
180D4×C53—81no—
181D4×C53—81no—
182D4×C53—81no—
183C2×C4×C53—21no—
184C2×C4×C53—21no—
185C2×C4×C53—21no—
186C2×C4×C53—21no—
187C2×C4×C53—41no—
188C2×C4×C53—41no—
189C2×C4×C53—41no—
190C2×C4×C53—41no—
191C2×C4×C53—42no—
192C2×C4×C53—42no—
193C2×C4×C53—42no—
194C2×C4×C53—41no—
195C2×C4×C53—41no—
196C2×C4×C53—41no—
197C2×C4×C53—81no—
198C2×C4×C53—81no—
199C2×C4×C53—81no—
200C2×C4×C53—81no—
201C2×C4×C53C4×D411yescharacteristic
202C2×C4×C53C4×D411yescharacteristic
203C2×C4×C53#12011yescharacteristic
204C2×C4×C53#12311yescharacteristic
205C2×Dic53—21no—
206C2×Dic53—21no—
207C2×Dic53—21no—
208C2×Dic53—21no—
209C2×Dic53—21no—
210C2×Dic53—21no—
211C2×Dic53—21no—
212C2×Dic53—21no—
213C2×Dic53—21no—
214C2×Dic53—21no—
215C2×Dic53—21no—
216C2×Dic53—42no—
217C2×Dic53—42no—
218C2×Dic53—41no—
219C2×Dic53—41no—
220C2×Dic53—41no—
221C2×Dic53—41no—
222C2×Dic53—42no—
223C2×Dic53—42no—
224C2×Dic53—42no—
225C2×Dic53—41no—
226C2×Dic53—41no—
227C2×Dic53—41no—
228C2×Dic53—41no—
229C2×Dic53—41no—
230C2×Dic53—41no—
231C2×Dic53—81no—
232C2×Dic53—81no—
233C2×Dic53—81no—
234C2×Dic53—81no—
235C2×Dic53#10412yes—
236C8×C53—42no—
237#88322—42no—
238#9118—531nomaximal, Sylow
239C22×C53—21no—
240C22×C53—21no—
241C22×C53—21no—
242C22×C53—21no—
243C22×C53—21no—
244C22×C53—21no—
245C22×C53—21no—
246C22×C53—21no—
247C22×C53—21no—
248C22×C53—21no—
249C22×C53—21no—
250C22×C53—21no—
251C22×C53—81no—
252C22×C53—81no—
253C22×C53—81no—
254C22×C53—81no—
255C22×C53—81no—
256C22×C53—81no—
257C22×C53—81no—
258C22×C53#38611yescharacteristic
259C22×C53#41011yescharacteristic
260C22×C53#41011yescharacteristic
261C22×C53#41711yescharacteristic
262C22×C53#41811yescharacteristic
263C22×C53C2×#10411yescharacteristic
264C22×C53#42111yescharacteristic
265C22×C53#46811yescharacteristic
266C22×C53#46811yescharacteristic
267C22×C53#46911yescharacteristic
268C22×C53#46911yescharacteristic
269C4×C53—21no—
270C4×C53—21no—
271C4×C53—21no—
272C4×C53—21no—
273C4×C53—82no—
274C4×C53—81no—
275Dic53—22no—
276Dic53—41no—
277Dic53—41no—
278Dic53—41no—
279Dic53—82no—
280Dic53—81no—
281Dic53—81no—
282C2×#432—531no—
283#2854—531no—
284C2×#327—531no—
285C2×#327D5311yescharacteristic
286#1434—531no—
287#1294—531no—
288#1294—531no—
289C2×C53—21no—
290C2×C53—21no—
291C2×C53—21no—
292C2×C53—21no—
293C2×C53—161no—
294C2×C53#150511yescharacteristic
295C2×C53#150911yescharacteristic
296C2×C53#151211yescharacteristic
297C2×C53#151411yescharacteristic
298C2×C53#187111yescharacteristic
299C2×C53#189511yescharacteristic
300C2×C53#189911yescharacteristic
301C2×#123—1061no—
302C2×#121C2×D5311yescharacteristic
303C22×C42—531no—
304#432—1062no—
305#432—1062no—
306C2×C4⋊C8—1061no—
307C2×C22⋊C8—531no—
308C2×C22⋊C8C2×D5311yescharacteristic
309C2×#97—1061no—
310#390—531no—
311#390—531no—
312#389—531no—
313#389—531no—
314#388—1061no—
315#388—1061no—
316#386—1061no—
317#380—531no—
318C2×#95—531no—
319C2×#95—1061no—
320C2×#95—1061no—
321C2×#95C2×D5311yescharacteristic
322#327—22no—
323#327—1062no—
324C53#911811yescharacteristic, complemented, Sylow
325C22×D4—21no—
326C22×D4C4×D5311yescharacteristic
327C23×C4—21no—
328C23×C4—531no—
329C23×C4—531no—
330C23×C4—1061no—
331C23×C4C22×D5311yescharacteristic
332C22×C8—21no—
333C22×C8—1061no—
334#123—2122no—
335#123—2122no—
336#121—21no—
337#121—21no—
338#121—21no—
339#121—21no—
340C2×C4⋊C4—531no—
341C2×C4⋊C4—531no—
342C2×C4⋊C4—1061no—
343C2×C4⋊C4—1061no—
344C2×C4⋊C4C4×D5311yescharacteristic
345C2×C22⋊C4—21no—
346C2×C22⋊C4—1061no—
347C2×C22⋊C4—1061no—
348C2×C22⋊C4—1061no—
349C2×C22⋊C4—1061no—
350C2×C22⋊C4—1061no—
351C2×C42—531no—
352C2×C42—531no—
353C2×C42—531no—
354C2×C42—531no—
355C2×C42—1061no—
356C2×C42—1061no—
357C2×C42—1061no—
358C2×C42—1061no—
359C4⋊C8—1064no—
360C22⋊C8—21no—
361C22⋊C8—532no—
362C22⋊C8—1061no—
363C22⋊C8C22⋊D5312yes—
364#97—1064no—
365#95—21no—
366#95—21no—
367#95—1061no—
368#95—1061no—
369#95—1064no—
370#95—1061no—
371#95—1061no—
372#95—1061no—
373#95—2122no—
374C24—21no—
375C24D4×D5311yescharacteristic
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4—41no—
385C2×D4—41no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—531no—
397C22×C4—531no—
398C22×C4—531no—
399C22×C4—531no—
400C22×C4—1062no—
401C22×C4—1061no—
402C22×C4—1061no—
403C22×C4—1061no—
404C22×C4—1061no—
405C22×C4—1061no—
406C22×C4—1061no—
407C22×C4—1061no—
408C22×C4—1061no—
409C22×C4—1062no—
410C22×C4—1061no—
411C22×C4—1061no—
412C22×C4—2121no—
413C22×C4—2121no—
414C22×C4—2121no—
415C22×C4—2121no—
416C22×C4—2121no—
417C22×C4—2121no—
418C22×C4—2121no—
419C22×C4C2×C4×D5311yescharacteristic
420C22×C4#1171994511yescharacteristic
421C22×C4D4×D5311yescharacteristic
422C22×C4D4×D5311yescharacteristic
423C22×C4#1171994711yescharacteristic
424C22×C4C2×C22⋊D5311yescharacteristic
425C2×C8—22no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—1062no—
429C2×C8—2121no—
430C2×C8—2121no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—2121no—
434C4⋊C4—2121no—
435C4⋊C4—2122no—
436C4⋊C4—2122no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—2122no—
440C22⋊C4—2122no—
441C22⋊C4—2122no—
442C22⋊C4—2122no—
443C22⋊C4—2121no—
444C22⋊C4—2121no—
445C42—1061no—
446C42—1061no—
447C42—1061no—
448C42—1061no—
449C42—1061no—
450C42—1061no—
451C42—1061no—
452C42—1061no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23C22⋊C4×D5311yescharacteristic
465C23#1313318811yescharacteristic
466C23#1313319011yescharacteristic
467C23#1313324111yescharacteristic
468C23#1313324111yescharacteristic
469C23#1313324411yescharacteristic
470C23#1313324611yescharacteristic
471D4—81no—
472D4—81no—
473D4—81no—
474D4—81no—
475D4—81no—
476D4—81no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—532no—
496C2×C4—1061no—
497C2×C4—1061no—
498C2×C4—1061no—
499C2×C4—1061no—
500C2×C4—1061no—
501C2×C4—1061no—
502C2×C4—1061no—
503C2×C4—1061no—
504C2×C4—1061no—
505C2×C4—1061no—
506C2×C4—1061no—
507C2×C4—4241no—
508C2×C4—4241no—
509C2×C4—4241no—
510C2×C4—4241no—
511C2×C4—2122no—
512C2×C4—2122no—
513C2×C4—2122no—
514C2×C4—2122no—
515C2×C4—2122no—
516C2×C4—2121no—
517C2×C4—2121no—
518C2×C4—2121no—
519C2×C4—2121no—
520C2×C4—2121no—
521C2×C4—2121no—
522C2×C4—2121no—
523C2×C4—2121no—
524C2×C4—2121no—
525C2×C4—2121no—
526C2×C4#1313317711yescharacteristic
527C2×C4C4×C22⋊D5311yescharacteristic
528C2×C4#1313318511yescharacteristic
529C2×C4#1313319911yescharacteristic
530C8—42no—
531C8—2122no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22—81no—
551C22#1681049911yescharacteristic
552C22#1681059511yescharacteristic
553C22#1681059511yescharacteristic
554C22#1681062111yescharacteristic
555C22#1681062611yescharacteristic
556C22#104×D5311yescharacteristic
557C22#1681063811yescharacteristic
558C22#1681086611yescharacteristic
559C22#1681086811yescharacteristic
560C22#1681087311yescharacteristic
561C22#1681087511yescharacteristic
562C4—21no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—82no—
567C4—81no—
568C4—1062no—
569C4—4242no—
570C4—4241no—
571C4—4241no—
572C4—2121no—
573C4—2121no—
574C4—2121no—
575C2—21no—
576C2—21no—
577C2—21no—
578C2—21no—
579C2—161no—
580C2#2027322111yescharacteristic
581C2#2027323711yescharacteristic
582C2#2027325011yescharacteristic
583C2#2027326011yescharacteristic
584C2#2027501411yescharacteristic
585C2#2027515011yescharacteristic
586C2#2027517011yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#327C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#121×C53C22
cosocleC23×C4×C53C23
hypercenterC2×#327D53
hypocenterC53#9118
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#327

Complements

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

NH ≅ G/Ntype
1. C2×#16810677C2semidirect
4. #20272939C2semidirect
5. #20272595C2semidirect
6. #20272594C2semidirect
15. #16810526C4semidirect
16. #16810526C4semidirect
17. #16810519C4semidirect
18. #16810519C4semidirect
324. C53#9118semidirect

Maximal subgroups

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

subgroupindex
C2×#168106772
#202830412
C2×#168103572
#202729392
#202725952
#202725942
C2×C53×#3272
#911853