Finite Group Databasebeta

← #27438228

Subgroups — #27438228

Order 13568.

1112 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810982C211yesmaximal, characteristic, complemented
2C2×#16810913C211yesmaximal, characteristic, complemented
3#20279700C211yesmaximal, characteristic, complemented
4#20275951C212yesmaximal, complemented
5#20275946C212yesmaximal
6#20275656C211yesmaximal, characteristic, complemented
7#20275648C211yesmaximal, characteristic, complemented
8#20274983C212yesmaximal, complemented
9#20274976C212yesmaximal, complemented
10#20274540C211yesmaximal, characteristic, complemented
11C2×C53×#559C211yesmaximal, characteristic
12C2×#13133217C2211yescharacteristic
13C2×#13133215—21no—
14#16810982—24no—
15#16810913—24no—
16C2×#13133148C2212yes—
17C2×C4⋊#88322C2211yescharacteristic
18C2×#13133141C2212yes—
19C2×#13133141C2212yes—
20C2×#13133138C2212yes—
21C2×#13133138C2212yes—
22C2×#13133138C2212yes—
23C2×#13133138C2212yes—
24C2×#13133128C2211yescharacteristic
25#16810553—21no—
26#16810553—21no—
27#16810528—21no—
28#16810521C2212yes—
29#16810521C2212yes—
30C4×C4⋊Dic53C2211yescharacteristic
31#16810385C2212yes—
32#16810378C2212yes—
33#16810378C2212yes—
34C53×#559—24no—
35C53×#559—24no—
36C2×#127×C53C2211yescharacteristic
37C2×#124×C53—21no—
38C2×#124×C53C2211yescharacteristic
39C2×#124×C53C2211yescharacteristic
40C2×#123×C53—21no—
41C2×#123×C53—21no—
42C2×C4×Q8×C53C2211yescharacteristic
43C2×C4×Q8×C53C2211yescharacteristic
44C2×C4×D4×C53—21no—
45C2×C4×D4×C53C2211yescharacteristic
46C2×C4×D4×C53C2211yescharacteristic
47C22×Dic106—21no—
48C22×#88322C2312yes—
49C22×#88322C2312yes—
50#13133217—22no—
51#13133217—22no—
52#13133215—44no—
53C2×C4⋊Dic53—21no—
54C2×C4⋊Dic53C2312yes—
55C2×C4⋊Dic53C2312yes—
56C2×#11719919—22no—
57C2×#11719919—22no—
58C2×C4×Dic53—21no—
59#13133148—24no—
60C4⋊#88322—22no—
61#13133141—24no—
62#13133141—24no—
63#13133138—24no—
64#13133138—24no—
65#13133138—24no—
66#13133138—24no—
67#13133128—22no—
68#13133124—22no—
69#13133124—22no—
70#13133124—21no—
71C22×Q8×C53D411yescharacteristic
72C22×Q8×C53C2311yescharacteristic
73C22×D4×C53—21no—
74C22×D4×C53—21no—
75C22×D4×C53C2311yescharacteristic
76C22×D4×C53C2311yescharacteristic
77C23×C4×C53—21no—
78C23×C4×C53—21no—
79#127×C53—22no—
80#127×C53—22no—
81#124×C53—28no—
82#124×C53—22no—
83#124×C53—22no—
84#124×C53—22no—
85#124×C53D414yes—
86#123×C53—28no—
87#123×C53—28no—
88C4×Q8×C53—22no—
89C4×Q8×C53—22no—
90C4×Q8×C53—22no—
91C4×Q8×C53—22no—
92C4×D4×C53—28no—
93C4×D4×C53—22no—
94C4×D4×C53—22no—
95C4×D4×C53—22no—
96C4×D4×C53D414yes—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53D411yescharacteristic
99C2×C4⋊C4×C53D411yescharacteristic
100C2×C4⋊C4×C53D411yescharacteristic
101C2×C4⋊C4×C53C2311yescharacteristic
102C2×C22⋊C4×C53—21no—
103C2×C22⋊C4×C53—21no—
104C2×C22⋊C4×C53—21no—
105C2×C22⋊C4×C53—21no—
106C2×C22⋊C4×C53—21no—
107C2×C22⋊C4×C53—21no—
108C2×C42×C53—21no—
109C2×C42×C53C2311yescharacteristic
110C2×C42×C53C2311yescharacteristic
111C2×C42×C53C2311yescharacteristic
112C24×C53—21no—
113C24×C53—21no—
114C2×Q8×C53—22no—
115C2×Q8×C53—22no—
116C2×Q8×C53—21no—
117C2×Q8×C53—22no—
118C2×Q8×C53—22no—
119C2×Q8×C53—41no—
120C2×Q8×C53C2×D412yes—
121C2×D4×C53—24no—
122C2×D4×C53—24no—
123C2×D4×C53—21no—
124C2×D4×C53—21no—
125C2×D4×C53—22no—
126C2×D4×C53—22no—
127C2×D4×C53—22no—
128C2×D4×C53—22no—
129C2×D4×C53—42no—
130C2×D4×C53—42no—
131C2×D4×C53—82no—
132C2×D4×C53C2×D412yes—
133C2×D4×C53C2×D412yes—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—24no—
137C22×C4×C53—24no—
138C22×C4×C53—82no—
139C22×C4×C53—82no—
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2411yescharacteristic
147C22×Dic53—22no—
148C22×Dic53—22no—
149C2×Dic106—42no—
150C2×Dic106—42no—
151C2×Dic106—42no—
152C4⋊C4×C53—24no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—22no—
155C4⋊C4×C53—22no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53C2×D412yes—
158C22⋊C4×C53—24no—
159C22⋊C4×C53—24no—
160C22⋊C4×C53—24no—
161C22⋊C4×C53—24no—
162C22⋊C4×C53—24no—
163C22⋊C4×C53—24no—
164C42×C53—24no—
165C42×C53—21no—
166C42×C53—21no—
167C42×C53—21no—
168C42×C53—21no—
169C42×C53C2×D412yes—
170C42×C53C2×D412yes—
171C4⋊Dic53—22no—
172C4⋊Dic53—22no—
173C4⋊Dic53—22no—
174C4⋊Dic53—42no—
175C4⋊Dic53#4114yes—
176#11719919—44no—
177#11719919—44no—
178C4×Dic53—42no—
179C2×#88322—22no—
180C2×#88322—22no—
181C2×#88322—22no—
182C2×#88322—22no—
183C2×#88322—22no—
184C2×#88322—22no—
185C23×C53—22no—
186C23×C53—22no—
187C23×C53—42no—
188C23×C53—42no—
189C23×C53—41no—
190C23×C53—41no—
191C23×C53—41no—
192C23×C53—41no—
193C23×C53—42no—
194C23×C53—42no—
195C23×C53C22×D411yescharacteristic
196Q8×C53—24no—
197Q8×C53—44no—
198Q8×C53—41no—
199Q8×C53—41no—
200D4×C53—22no—
201D4×C53—22no—
202D4×C53—24no—
203D4×C53—48no—
204D4×C53—41no—
205D4×C53—41no—
206D4×C53—42no—
207D4×C53—84no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—21no—
215C2×C4×C53—22no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—84no—
225C2×C4×C53—84no—
226C2×C4×C53#12012yes—
227C2×C4×C53#12012yes—
228C2×C4×C53AGL(1,ℤ/8)12yes—
229C2×C4×C53C22×D411yescharacteristic
230C2×C4×C53C22×D411yescharacteristic
231C2×C4×C53C2×#4112yes—
232C2×C4×C532−1+412yes—
233C2×Dic53—24no—
234C2×Dic53—42no—
235C2×Dic53—42no—
236C2×Dic53—42no—
237C2×Dic53—42no—
238C2×Dic53—42no—
239Dic106—84no—
240#88322—42no—
241#88322—42no—
242#25659—531nomaximal, Sylow
243C22×C53—42no—
244C22×C53—42no—
245C22×C53—42no—
246C22×C53—42no—
247C22×C53—42no—
248C22×C53—42no—
249C22×C53—82no—
250C22×C53—82no—
251C22×C53—82no—
252C22×C53—82no—
253C22×C53C2×#12011yescharacteristic
254C22×C53#54812yes—
255C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
256C22×C53#57511yescharacteristic
257C22×C53#57712yes—
258C4×C53—22no—
259C4×C53—22no—
260C4×C53—22no—
261C4×C53—42no—
262C4×C53—41no—
263C4×C53—41no—
264C4×C53—41no—
265C4×C53—42no—
266C4×C53#53211yescharacteristic
267C4×C53#53611yescharacteristic
268C4×C53#54712yes—
269Dic53—44no—
270Dic53—82no—
271C2×#559D5311yescharacteristic
272C2×#489—531no—
273C2×#476—531no—
274#2546—531no—
275#2042—532no—
276#2041—532no—
277#1985—531no—
278#1983—531no—
279#1865—532no—
280#1864—532no—
281#1762—531no—
282C2×C53—82no—
283C2×C53—82no—
284C2×C53#243211yescharacteristic
285C2×C53#300611yescharacteristic
286C2×C53#300911yescharacteristic
287C2×C53#337112yes—
288C2×C53#337312yes—
289#559—24no—
290#559—24no—
291C2×C4⋊Q8—531no—
292C2×C4⋊Q8—1061no—
293C2×#127C2×D5311yescharacteristic
294C2×#124—21no—
295C2×#124C2×D5311yescharacteristic
296C2×#124C2×D5311yescharacteristic
297C2×#123—21no—
298C2×#123—21no—
299C2×C4×Q8C2×D5311yescharacteristic
300C2×C4×Q8C2×D5311yescharacteristic
301C2×C4×D4—21no—
302C2×C4×D4C2×D5311yescharacteristic
303C2×C4×D4C2×D5311yescharacteristic
304#489—1064no—
305#476—1064no—
306C2×#106—532no—
307C2×C4⋊C8—531no—
308C2×#103—532no—
309C2×#103—532no—
310C2×#102—532no—
311C2×#102—532no—
312C2×#102—532no—
313C2×#102—532no—
314C2×#97—531no—
315#398—1061no—
316#398—1061no—
317#391—1061no—
318#389—532no—
319#389—532no—
320C4×C4⋊C4—531no—
321#340—532no—
322#336—532no—
323#336—532no—
324C53#2565911yescharacteristic, complemented, Sylow
325C22×Q8—1061no—
326C22×Q8C22⋊D5311yescharacteristic
327C22×Q8C22×D5311yescharacteristic
328C22×D4—21no—
329C22×D4—21no—
330C22×D4C22×D5311yescharacteristic
331C22×D4C22×D5311yescharacteristic
332C23×C4—21no—
333C23×C4—21no—
334C22×C8—532no—
335C22×C8—532no—
336C4⋊Q8—1062no—
337C4⋊Q8—1062no—
338C4⋊Q8—2124no—
339#127—22no—
340#127—22no—
341#124—28no—
342#124—22no—
343#124—22no—
344#124—22no—
345#124C22⋊D5314yes—
346#123—28no—
347#123—28no—
348C4×Q8—22no—
349C4×Q8—22no—
350C4×Q8—22no—
351C4×Q8—22no—
352C4×D4—28no—
353C4×D4—22no—
354C4×D4—22no—
355C4×D4—22no—
356C4×D4C22⋊D5314yes—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—532no—
359C2×C4⋊C4—532no—
360C2×C4⋊C4—1061no—
361C2×C4⋊C4—1062no—
362C2×C4⋊C4—1062no—
363C2×C4⋊C4C22⋊D5311yescharacteristic
364C2×C4⋊C4C22⋊D5311yescharacteristic
365C2×C4⋊C4C22⋊D5311yescharacteristic
366C2×C4⋊C4C22×D5311yescharacteristic
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C42—21no—
374C2×C42—1061no—
375C2×C42C22×D5311yescharacteristic
376C2×C42C22×D5311yescharacteristic
377C2×C42C22×D5311yescharacteristic
378#106—1064no—
379C4⋊C8—1062no—
380#103—1064no—
381#103—1064no—
382#102—1064no—
383#102—1064no—
384#102—1064no—
385#102—1064no—
386#97—1062no—
387#95—1061no—
388#95—1062no—
389#95—1062no—
390C24—21no—
391C24—21no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—41no—
398C2×Q8—2122no—
399C2×Q8—2122no—
400C2×Q8—2122no—
401C2×Q8C2×C22⋊D5312yes—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—82no—
413C2×D4C2×C22⋊D5312yes—
414C2×D4C2×C22⋊D5312yes—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—82no—
420C22×C4—82no—
421C22×C4—1062no—
422C22×C4—1062no—
423C22×C4C2×C22⋊D5311yescharacteristic
424C22×C4C2×C22⋊D5311yescharacteristic
425C22×C4C2×C22⋊D5311yescharacteristic
426C22×C4C2×C22⋊D5311yescharacteristic
427C22×C4C2×C22⋊D5311yescharacteristic
428C22×C4C2×C22⋊D5311yescharacteristic
429C22×C4C23×D5311yescharacteristic
430C2×C8—1062no—
431C2×C8—1062no—
432C2×C8—1062no—
433C2×C8—1062no—
434C2×C8—1062no—
435C2×C8—1062no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—534no—
442C4⋊C4—1062no—
443C4⋊C4—1062no—
444C4⋊C4—1062no—
445C4⋊C4—2124no—
446C4⋊C4—2122no—
447C4⋊C4—2124no—
448C4⋊C4C2×C22⋊D5312yes—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C42—24no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—2122no—
461C42C2×C22⋊D5312yes—
462C42C2×C22⋊D5312yes—
463C23—22no—
464C23—22no—
465C23—42no—
466C23—42no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—42no—
472C23—42no—
473C23C22×C22⋊D5311yescharacteristic
474Q8—24no—
475Q8—44no—
476Q8—41no—
477Q8—41no—
478Q8—4244no—
479D4—24no—
480D4—22no—
481D4—22no—
482D4—48no—
483D4—42no—
484D4—41no—
485D4—41no—
486D4—84no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—84no—
504C2×C4—84no—
505C2×C4—1064no—
506C2×C4—2122no—
507C2×C4—2122no—
508C2×C4—2122no—
509C2×C4—2122no—
510C2×C4—2122no—
511C2×C4#1313318612yes—
512C2×C4#1313318612yes—
513C2×C4#1313324212yes—
514C2×C4C22×C22⋊D5311yescharacteristic
515C2×C4C22×C22⋊D5311yescharacteristic
516C2×C4#41×D5312yes—
517C2×C4#1313326412yes—
518C8—2122no—
519C8—2122no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22C2×#1313318611yescharacteristic
531C22#1681128012yes—
532C22C2×#1313324211yescharacteristic
533C22#1681142111yescharacteristic
534C22#1681143412yes—
535C4—22no—
536C4—22no—
537C4—22no—
538C4—41no—
539C4—42no—
540C4—41no—
541C4—41no—
542C4—42no—
543C4—4242no—
544C4—2124no—
545C4#1681116911yescharacteristic
546C4#1681119411yescharacteristic
547C4#1681127212yes—
548C2—82no—
549C2—82no—
550C2#2027833911yescharacteristic
551C2#2028396711yescharacteristic
552C2#2028399211yescharacteristic
553C2#2028747712yes—
554C2#2028750312yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#559C2
socleC23×C53C22×D4
prime radicalC2×#127×C53C22
hypercenterC2×#559D53
hypocenterC53#25659
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
1. C2×#16810982C2semidirect
2. C2×#16810913C2semidirect
3. #20279700C2semidirect
4. #20275951C2semidirect
6. #20275656C2semidirect
7. #20275648C2semidirect
8. #20274983C2semidirect
9. #20274976C2semidirect
10. #20274540C2semidirect
324. C53#25659semidirect

Maximal subgroups

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

subgroupindex
C2×#168109822
C2×#168109132
#202797002
#202759512
#202759462
#202756562
#202756482
#202749832
#202749762
#202745402
C2×C53×#5592
#2565953