Finite Group Databasebeta

← #27466168

Subgroups — #27466168

Order 13568.

882 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277221C211yesmaximal, characteristic
2C2×#16810488C211yesmaximal, characteristic, complemented
3#20276285C212yesmaximal, complemented
4#20276237C212yesmaximal, complemented
5#20275484C212yesmaximal, complemented
6#20275032C211yesmaximal, characteristic
7#20275030C211yesmaximal, characteristic
8#20274902C211yesmaximal, characteristic, complemented
9#20274724C212yesmaximal, complemented
10#20274693C211yesmaximal, characteristic, complemented
11C53×#1853C211yesmaximal, characteristic, complemented
12C2×#13133191C2211yescharacteristic
13C2×#13133179C2211yescharacteristic
14C2×#13133177—21no—
15C22×C4⋊Dic53C2211yescharacteristic
16C22×C4×Dic53C2211yescharacteristic
17C2×#13133150C2212yes—
18C2×#13133146C2212yes—
19C2×#13133139C2212yes—
20C2×#13133137C2212yes—
21C2×C22⋊#88322C2211yescharacteristic
22#16810524C2211yescharacteristic
23#16810519C2211yescharacteristic
24#16810512—21no—
25#16810500—21no—
26#16810488—21no—
27#16810488—21no—
28#16810488C2212yes—
29#16810488C2212yes—
30C4⋊C4×Dic53C2211yescharacteristic
31#16810470—21no—
32C2×#13133124—21no—
33C2×#13133124—21no—
34#16810386C2212yes—
35#16810385C2212yes—
36#16810385C2212yes—
37#16810377C2212yes—
38C22×C4⋊C4×C53C2211yescharacteristic
39C2×#106×C53C2212yes—
40C2×C22⋊C8×C53C2211yescharacteristic
41C53×#380C2211yescharacteristic
42C53×#340C2212yes—
43C22×D212C2311yescharacteristic
44C22×C22⋊D53—21no—
45C23×Dic53—21no—
46C23×Dic53D411yescharacteristic
47C23×Dic53D411yescharacteristic
48C22×C4×D53—21no—
49C22×#88322C2312yes—
50#13133191—22no—
51#13133191Q814yes—
52#13133179—24no—
53#13133177—44no—
54C2×C4⋊Dic53—21no—
55C2×C4⋊Dic53—42no—
56C2×C4⋊Dic53C2312yes—
57C2×C4⋊Dic53C2311yescharacteristic
58C2×#11719919—21no—
59C2×#11719921—21no—
60C2×#11719921—21no—
61C2×#11719921—21no—
62C2×#11719921—21no—
63C2×C4×Dic53—24no—
64C2×C4×Dic53—21no—
65C2×C4×Dic53D411yescharacteristic
66C2×C4×Dic53D411yescharacteristic
67C2×C4×Dic53C2311yescharacteristic
68C2×C4×Dic53C2311yescharacteristic
69#13133150—42no—
70#13133146—24no—
71#13133139—42no—
72#13133137—42no—
73C22⋊#88322—22no—
74#13133124—21no—
75#13133124—21no—
76#13133124—22no—
77#13133124—22no—
78#13133124—22no—
79#13133124—22no—
80C23×C4×C53—21no—
81C23×C4×C53C2311yescharacteristic
82C22×C8×C53C2312yes—
83C2×C4⋊C4×C53—21no—
84C2×C4⋊C4×C53—42no—
85C2×C4⋊C4×C53C2311yescharacteristic
86C2×C4⋊C4×C53C2312yes—
87C2×C4⋊C4×C53C2311yescharacteristic
88C2×C22⋊C4×C53—21no—
89#106×C53—42no—
90C22⋊C8×C53—22no—
91#95×C53—21no—
92C24×C53C2×D411yescharacteristic
93C23×D53—21no—
94C22×C4×C53—22no—
95C22×C4×C53—22no—
96C22×C4×C53—21no—
97C22×C4×C53—21no—
98C22×C4×C53—22no—
99C22×C4×C53—42no—
100C22×C4×C53—42no—
101C22×C4×C53—42no—
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×Q812yes—
104C22×C4×C53#4112yes—
105C22×C4×C53C2411yescharacteristic
106C2×C22⋊D53—24no—
107C2×C22⋊D53—42no—
108C2×C22⋊D53—42no—
109C22×Dic53—22no—
110C22×Dic53—22no—
111C22×Dic53—21no—
112C22×Dic53—22no—
113C22×Dic53—22no—
114C22×Dic53—22no—
115C22×Dic53—41no—
116C22×Dic53—41no—
117C22×Dic53—42no—
118C22×Dic53—42no—
119C22×Dic53—42no—
120C22×Dic53C2×D411yescharacteristic
121C22×Dic53C2×D411yescharacteristic
122C22×Dic53C2×D411yescharacteristic
123C22×Dic53C2×D411yescharacteristic
124C2×D212—22no—
125C2×D212—22no—
126C2×D212—21no—
127C2×D212C2×Q812yes—
128C2×C4×D53—42no—
129C2×C4×D53—42no—
130C2×C4×D53—42no—
131C2×C8×C53—22no—
132C2×C8×C53—22no—
133C2×C8×C53—22no—
134C4⋊C4×C53—22no—
135C4⋊C4×C53—22no—
136C4⋊C4×C53—22no—
137C4⋊C4×C53—22no—
138C4⋊C4×C53—42no—
139C22⋊C4×C53—42no—
140#11719921—24no—
141#11719921—42no—
142#11719921—42no—
143#11719921—42no—
144C4⋊Dic53—21no—
145C4⋊Dic53—22no—
146C4⋊Dic53—22no—
147C4⋊Dic53—42no—
148C4⋊Dic53C2×Q812yes—
149#11719919—42no—
150C4×Dic53—22no—
151C4×Dic53—22no—
152C4×Dic53—22no—
153C4×Dic53—42no—
154C4×Dic53#4114yes—
155C2×#88322—22no—
156C2×#88322—22no—
157C2×#88322—22no—
158C23×C53—21no—
159C23×C53—21no—
160C23×C53—21no—
161C23×C53—21no—
162C23×C53C22⋊Q812yes—
163C23×C53#13512yes—
164C23×C53AGL(1,ℤ/8)12yes—
165C23×C53C22×D411yescharacteristic
166C22×D53—22no—
167C22×D53—41no—
168C22×D53—41no—
169C22×D53—41no—
170C22×D53—41no—
171C22×D53—41no—
172C22×D53—41no—
173C2×C4×C53—21no—
174C2×C4×C53—21no—
175C2×C4×C53—24no—
176C2×C4×C53—24no—
177C2×C4×C53—41no—
178C2×C4×C53—42no—
179C2×C4×C53—41no—
180C2×C4×C53—42no—
181C2×C4×C53—41no—
182C2×C4×C53—41no—
183C2×C4×C53—41no—
184C2×C4×C53—42no—
185C2×C4×C53—41no—
186C2×C4×C53—84no—
187C2×C4×C53C22⋊Q812yes—
188C2×C4×C53C22×Q811yescharacteristic
189C2×C4×C53C2×#4112yes—
190C2×C4×C53C2×#4111yescharacteristic
191C2×C4×C53H(5,𝔽2)12yes—
192C22⋊D53—48no—
193C2×Dic53—21no—
194C2×Dic53—21no—
195C2×Dic53—21no—
196C2×Dic53—21no—
197C2×Dic53—22no—
198C2×Dic53—21no—
199C2×Dic53—21no—
200C2×Dic53—21no—
201C2×Dic53—21no—
202C2×Dic53—21no—
203C2×Dic53—21no—
204C2×Dic53—44no—
205C2×Dic53—44no—
206C2×Dic53—41no—
207C2×Dic53—41no—
208C2×Dic53—42no—
209C2×Dic53—42no—
210C2×Dic53—42no—
211C2×Dic53—84no—
212C2×Dic53#12312yes—
213C2×Dic53#12312yes—
214D212—24no—
215D212—42no—
216C4×D53—84no—
217C8×C53—42no—
218#88322—42no—
219#27684—531nomaximal, Sylow
220C22×C53—22no—
221C22×C53—22no—
222C22×C53—22no—
223C22×C53—22no—
224C22×C53—22no—
225C22×C53—22no—
226C22×C53#47814yes—
227C22×C53C2×C22⋊Q811yescharacteristic
228C22×C53C2×#12311yescharacteristic
229C22×C53#53511yescharacteristic
230C22×C53C2×#13511yescharacteristic
231C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
232C22×C53#57511yescharacteristic
233C22×C53#57511yescharacteristic
234C2×D53—42no—
235C2×D53—42no—
236C2×D53—42no—
237C2×D53—81no—
238C2×D53—81no—
239C2×D53—81no—
240C2×D53—81no—
241C4×C53—22no—
242C4×C53—81no—
243C4×C53—82no—
244C4×C53—81no—
245C4×C53#55312yes—
246C4×C53#55412yes—
247Dic53—22no—
248Dic53—22no—
249Dic53—42no—
250Dic53—41no—
251Dic53—41no—
252Dic53—82no—
253C4×#121—531no—
254C2×#382—531no—
255#2101—532no—
256#2092—532no—
257#1949—532no—
258#1874—531no—
259#1874—531no—
260#1853D5311yescharacteristic, complemented
261#1852—531no—
262#1804—532no—
263#1797—531no—
264C2×C53—24no—
265C2×C53#248711yescharacteristic
266C2×C53C2×#47811yescharacteristic
267C2×C53#308011yescharacteristic
268C2×C53#309111yescharacteristic
269C2×C53#309411yescharacteristic
270C2×C53#321111yescharacteristic
271C2×C53#321311yescharacteristic
272D53—82no—
273C2×#121—531no—
274C2×C4×D4—531no—
275C2×C4×D4—1061no—
276C22×C4⋊C4—531no—
277C22×C4⋊C4C2×D5311yescharacteristic
278C22×C42—531no—
279C2×#107—532no—
280C2×#106C2×D5312yes—
281C2×C4⋊C8—532no—
282C2×#102—532no—
283C2×#102—532no—
284C2×C22⋊C8—531no—
285C2×C22⋊C8C2×D5311yescharacteristic
286#390—531no—
287#389—531no—
288#388—1061no—
289#386—1061no—
290#382—532no—
291#382—532no—
292#382—1061no—
293#382—1061no—
294#380C2×D5311yescharacteristic
295C4×C4⋊C4—531no—
296C4×C22⋊C4—1061no—
297C2×#95—1061no—
298C2×#95—1061no—
299#340—532no—
300#340—532no—
301#340—532no—
302#340C2×D5312yes—
303#336—532no—
304C53#2768411yescharacteristic, complemented, Sylow
305C22×D4—531no—
306C22×D4—1061no—
307C23×C4—21no—
308C23×C4—531no—
309C23×C4—531no—
310C23×C4—1061no—
311C23×C4—1061no—
312C23×C4C22×D5311yescharacteristic
313C22×C8—532no—
314C22×C8C22×D5312yes—
315#121—534no—
316#121—1062no—
317C4×D4—1064no—
318C4×D4—2124no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—42no—
321C2×C4⋊C4—531no—
322C2×C4⋊C4—532no—
323C2×C4⋊C4—1061no—
324C2×C4⋊C4—1061no—
325C2×C4⋊C4—2122no—
326C2×C4⋊C4C22×D5312yes—
327C2×C4⋊C4C22×D5311yescharacteristic
328C2×C4⋊C4C22×D5311yescharacteristic
329C2×C22⋊C4—21no—
330C2×C22⋊C4—1061no—
331C2×C22⋊C4—1061no—
332C2×C22⋊C4—1061no—
333C2×C22⋊C4—1061no—
334C2×C42—531no—
335C2×C42—531no—
336C2×C42—531no—
337C2×C42—531no—
338C2×C42—1064no—
339C2×C42—1061no—
340#107—2122no—
341#106—42no—
342C4⋊C8—1064no—
343#102—2122no—
344#102—2122no—
345C22⋊C8—22no—
346C22⋊C8—1062no—
347#95—21no—
348#95—1062no—
349#95—1062no—
350#95—1062no—
351#95—1062no—
352#95—1061no—
353#95—1061no—
354C24—1061no—
355C24D4×D5311yescharacteristic
356C2×D4—532no—
357C2×D4—1064no—
358C2×D4—1062no—
359C2×D4—1061no—
360C2×D4—1062no—
361C2×D4—2122no—
362C2×D4—2122no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—531no—
372C22×C4—531no—
373C22×C4—531no—
374C22×C4—531no—
375C22×C4—1062no—
376C22×C4—1062no—
377C22×C4—1062no—
378C22×C4—1062no—
379C22×C4—1061no—
380C22×C4—1062no—
381C22×C4—2122no—
382C22×C4—2122no—
383C22×C4—2122no—
384C22×C4—2121no—
385C22×C4—2121no—
386C22×C4—2122no—
387C22×C4—2122no—
388C22×C4—2122no—
389C22×C4#1171994512yes—
390C22×C4D4×D5311yescharacteristic
391C22×C4Q8×D5312yes—
392C22×C4C23×D5311yescharacteristic
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—1062no—
397C2×C8—1062no—
398C2×C8—1062no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—42no—
404C4⋊C4—532no—
405C4⋊C4—1061no—
406C4⋊C4—1062no—
407C4⋊C4—1062no—
408C4⋊C4—2122no—
409C4⋊C4—2122no—
410C22⋊C4—42no—
411C22⋊C4—1064no—
412C22⋊C4—2122no—
413C22⋊C4—2122no—
414C22⋊C4—2122no—
415C42—534no—
416C42—1062no—
417C42—1062no—
418C42—1062no—
419C42—2122no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—1062no—
425C23—2121no—
426C23—2121no—
427C23—2121no—
428C23—2121no—
429C23—2121no—
430C23—2121no—
431C23#1313319512yes—
432C23#1313323812yes—
433C23#1313324412yes—
434C23C2×D4×D5311yescharacteristic
435D4—1064no—
436D4—2128no—
437D4—2122no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—84no—
452C2×C4—532no—
453C2×C4—532no—
454C2×C4—1061no—
455C2×C4—1061no—
456C2×C4—1061no—
457C2×C4—1061no—
458C2×C4—1062no—
459C2×C4—1061no—
460C2×C4—1061no—
461C2×C4—1061no—
462C2×C4—1061no—
463C2×C4—1061no—
464C2×C4—1061no—
465C2×C4—4244no—
466C2×C4—4244no—
467C2×C4—2124no—
468C2×C4—2124no—
469C2×C4—2122no—
470C2×C4—2122no—
471C2×C4—2122no—
472C2×C4—2121no—
473C2×C4—2121no—
474C2×C4#1313319512yes—
475C2×C4C2×Q8×D5311yescharacteristic
476C2×C4#41×D5312yes—
477C2×C4C2×#1171994511yescharacteristic
478C2×C4#1313326212yes—
479C8—42no—
480C8—2122no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—4241no—
488C22—4241no—
489C22—4241no—
490C22—4241no—
491C22—2122no—
492C22—2122no—
493C22—2122no—
494C22#1681092214yes—
495C22C2×#1313319511yescharacteristic
496C22#123×D5311yescharacteristic
497C22#1681118811yescharacteristic
498C22C2×#1313323811yescharacteristic
499C22C2×#1313324411yescharacteristic
500C22#1681141911yescharacteristic
501C22#1681142211yescharacteristic
502C4—22no—
503C4—81no—
504C4—81no—
505C4—82no—
506C4—1062no—
507C4—1062no—
508C4—4242no—
509C4—2122no—
510C4—2121no—
511C4—2121no—
512C4#1681132212yes—
513C4#1681132712yes—
514C2—24no—
515C2—4242no—
516C2#2027903111yescharacteristic
517C2C2×#1681092211yescharacteristic
518C2#2028453611yescharacteristic
519C2#2028462911yescharacteristic
520C2#2028465211yescharacteristic
521C2#2028568311yescharacteristic
522C2#2028570111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1853C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#1853D53
hypocenterC53#27684
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
2. C2×#16810488C2semidirect
3. #20276285C2semidirect
4. #20276237C2semidirect
5. #20275484C2semidirect
8. #20274902C2semidirect
9. #20274724C2semidirect
10. #20274693C2semidirect
11. C53×#1853C2semidirect
260. #1853D53semidirect
304. C53#27684semidirect

Maximal subgroups

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

subgroupindex
#202772212
C2×#168104882
#202762852
#202762372
#202754842
#202750322
#202750302
#202749022
#202747242
#202746932
C53×#18532
#2768453