Finite Group Databasebeta

← #27484916

Subgroups — #27484916

Order 13568.

926 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277221C211yesmaximal, characteristic
2C2×#16810522C211yesmaximal, characteristic, complemented
3#20276207C212yesmaximal, complemented
4#20276098C212yesmaximal, complemented
5#20275288C212yesmaximal, complemented
6#20275032C211yesmaximal, characteristic
7#20275030C211yesmaximal, characteristic
8#20274963C212yesmaximal, complemented
9#20274902C211yesmaximal, characteristic, complemented
10#20274693C211yesmaximal, characteristic, complemented
11C53×#1852C211yesmaximal, characteristic, complemented
12C2×#13133191C2211yescharacteristic
13C2×#13133179C2211yescharacteristic
14C2×#13133177—21no—
15C22×C4⋊Dic53—21no—
16C22×C4⋊Dic53C2211yescharacteristic
17C22×C4×Dic53C2211yescharacteristic
18C2×#13133150C2212yes—
19C2×#13133146C2212yes—
20C2×#13133139C2212yes—
21C2×#13133137C2212yes—
22C2×C22⋊#88322C2211yescharacteristic
23#16810524C2211yescharacteristic
24#16810522—21no—
25#16810522—21no—
26#16810522C2212yes—
27#16810522C2212yes—
28#16810519C2211yescharacteristic
29#16810512—21no—
30#16810500—21no—
31C4⋊C4×Dic53C2211yescharacteristic
32#16810470—21no—
33C2×#13133124—21no—
34#16810386C2212yes—
35#16810385C2212yes—
36#16810385C2212yes—
37#16810377C2212yes—
38C22×C4⋊C4×C53C2211yescharacteristic
39C2×#107×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—42no—
53#13133177—44no—
54C2×C4⋊Dic53—21no—
55C2×C4⋊Dic53—22no—
56C2×C4⋊Dic53—22no—
57C2×C4⋊Dic53—42no—
58C2×C4⋊Dic53—44no—
59C2×C4⋊Dic53C2312yes—
60C2×C4⋊Dic53C2311yescharacteristic
61C2×#11719919—21no—
62C2×#11719921—21no—
63C2×#11719921—21no—
64C2×#11719921—21no—
65C2×#11719921—21no—
66C2×C4×Dic53—21no—
67C2×C4×Dic53—42no—
68C2×C4×Dic53D411yescharacteristic
69C2×C4×Dic53D411yescharacteristic
70C2×C4×Dic53C2311yescharacteristic
71C2×C4×Dic53C2311yescharacteristic
72#13133150—42no—
73#13133146—42no—
74#13133139—42no—
75#13133137—24no—
76C22⋊#88322—22no—
77#13133124—21no—
78#13133124—21no—
79#13133124—22no—
80#13133124—22no—
81C23×C4×C53—21no—
82C23×C4×C53C2311yescharacteristic
83C22×C8×C53C2312yes—
84C2×C4⋊C4×C53—24no—
85C2×C4⋊C4×C53—21no—
86C2×C4⋊C4×C53C2311yescharacteristic
87C2×C4⋊C4×C53C2311yescharacteristic
88C2×C4⋊C4×C53C2312yes—
89C2×C22⋊C4×C53—21no—
90#107×C53—24no—
91C22⋊C8×C53—22no—
92#95×C53—21no—
93C24×C53C2×D411yescharacteristic
94C23×D53—21no—
95C22×C4×C53—22no—
96C22×C4×C53—22no—
97C22×C4×C53—24no—
98C22×C4×C53—22no—
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53—42no—
102C22×C4×C53—42no—
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53C2×Q812yes—
105C22×C4×C53#4112yes—
106C22×C4×C53C2411yescharacteristic
107C2×C22⋊D53—24no—
108C2×C22⋊D53—42no—
109C2×C22⋊D53—42no—
110C22×Dic53—22no—
111C22×Dic53—21no—
112C22×Dic53—22no—
113C22×Dic53—41no—
114C22×Dic53—41no—
115C22×Dic53—41no—
116C22×Dic53—41no—
117C22×Dic53—42no—
118C22×Dic53—42no—
119C22×Dic53—42no—
120C22×Dic53D814yes—
121C22×Dic53C2×D411yescharacteristic
122C22×Dic53C2×D411yescharacteristic
123C22×Dic53C2×D411yescharacteristic
124C22×Dic53C2×D411yescharacteristic
125C2×D212—22no—
126C2×D212—22no—
127C2×D212—21no—
128C2×D212C2×Q812yes—
129C2×C4×D53—42no—
130C2×C4×D53—42no—
131C2×C4×D53—42no—
132C2×C8×C53—22no—
133C2×C8×C53—22no—
134C2×C8×C53—22no—
135C4⋊C4×C53—24no—
136C4⋊C4×C53—22no—
137C4⋊C4×C53—22no—
138C4⋊C4×C53—22no—
139C4⋊C4×C53#4114yes—
140C22⋊C4×C53—42no—
141#11719921—24no—
142#11719921—42no—
143#11719921—42no—
144#11719921—42no—
145C4⋊Dic53—21no—
146C4⋊Dic53—22no—
147C4⋊Dic53—22no—
148C4⋊Dic53—42no—
149C4⋊Dic53—44no—
150C4⋊Dic53—44no—
151C4⋊Dic53C2×Q812yes—
152#11719919—42no—
153C4×Dic53—22no—
154C4×Dic53—22no—
155C4×Dic53—41no—
156C4×Dic53—41no—
157C4×Dic53—42no—
158C2×#88322—22no—
159C2×#88322—22no—
160C2×#88322—22no—
161C23×C53—21no—
162C23×C53—21no—
163C23×C53—21no—
164C23×C53—21no—
165C23×C53C22⋊Q812yes—
166C23×C53C2×D812yes—
167C23×C53#13712yes—
168C23×C53C22×D411yescharacteristic
169C22×D53—22no—
170C22×D53—41no—
171C22×D53—41no—
172C22×D53—41no—
173C22×D53—41no—
174C22×D53—41no—
175C22×D53—41no—
176C2×C4×C53—21no—
177C2×C4×C53—21no—
178C2×C4×C53—24no—
179C2×C4×C53—24no—
180C2×C4×C53—24no—
181C2×C4×C53—48no—
182C2×C4×C53—42no—
183C2×C4×C53—42no—
184C2×C4×C53—41no—
185C2×C4×C53—41no—
186C2×C4×C53—41no—
187C2×C4×C53—41no—
188C2×C4×C53—41no—
189C2×C4×C53—41no—
190C2×C4×C53C22⋊Q812yes—
191C2×C4×C53C22×Q811yescharacteristic
192C2×C4×C53C2×#4112yes—
193C2×C4×C53C2×#4111yescharacteristic
194C2×C4×C53H(5,𝔽2)12yes—
195C22⋊D53—48no—
196C2×Dic53—21no—
197C2×Dic53—21no—
198C2×Dic53—21no—
199C2×Dic53—21no—
200C2×Dic53—22no—
201C2×Dic53—21no—
202C2×Dic53—21no—
203C2×Dic53—21no—
204C2×Dic53—21no—
205C2×Dic53—21no—
206C2×Dic53—21no—
207C2×Dic53—44no—
208C2×Dic53—44no—
209C2×Dic53—41no—
210C2×Dic53—41no—
211C2×Dic53—42no—
212C2×Dic53—42no—
213C2×Dic53—42no—
214C2×Dic53—84no—
215C2×Dic53C2×D812yes—
216C2×Dic53C2×D812yes—
217D212—24no—
218D212—42no—
219C4×D53—84no—
220C8×C53—42no—
221#88322—42no—
222#29234—531nomaximal, Sylow
223C22×C53—22no—
224C22×C53—22no—
225C22×C53—22no—
226C22×C53—22no—
227C22×C53—22no—
228C22×C53—22no—
229C22×C53#47414yes—
230C22×C53C2×C22⋊Q811yescharacteristic
231C22×C53#52511yescharacteristic
232C22×C53#53411yescharacteristic
233C22×C53C22×D811yescharacteristic
234C22×C53C2×#13711yescharacteristic
235C22×C53#57511yescharacteristic
236C22×C53#57511yescharacteristic
237C2×D53—42no—
238C2×D53—42no—
239C2×D53—42no—
240C2×D53—81no—
241C2×D53—81no—
242C2×D53—81no—
243C2×D53—81no—
244C4×C53—22no—
245C4×C53—44no—
246C4×C53—81no—
247C4×C53—81no—
248C4×C53#55312yes—
249C4×C53#55412yes—
250Dic53—22no—
251Dic53—22no—
252Dic53—41no—
253Dic53—41no—
254Dic53—42no—
255Dic53—82no—
256C4×#121—531no—
257C2×#389—531no—
258#2086—532no—
259#2067—532no—
260#1918—532no—
261#1874—531no—
262#1874—531no—
263#1862—532no—
264#1852—531no—
265#1852D5311yescharacteristic, complemented
266#1797—531no—
267C2×C53—24no—
268C2×C53#257911yescharacteristic
269C2×C53C2×#47411yescharacteristic
270C2×C53#308011yescharacteristic
271C2×C53#310011yescharacteristic
272C2×C53#311711yescharacteristic
273C2×C53#318611yescharacteristic
274C2×C53#319611yescharacteristic
275D53—82no—
276C2×#121—531no—
277C2×C4×D4—531no—
278C2×C4×D4—1061no—
279C22×C4⋊C4—531no—
280C22×C4⋊C4—1061no—
281C22×C4⋊C4C2×D5311yescharacteristic
282C22×C42—531no—
283C2×#107—532no—
284C2×#107C2×D5312yes—
285C2×C4⋊C8—532no—
286C2×#102—532no—
287C2×#102—532no—
288C2×C22⋊C8—531no—
289C2×C22⋊C8C2×D5311yescharacteristic
290#390—531no—
291#389—532no—
292#389—531no—
293#389—532no—
294#389—1061no—
295#389—1061no—
296#388—1061no—
297#386—1061no—
298#380C2×D5311yescharacteristic
299C4×C4⋊C4—531no—
300C4×C22⋊C4—1061no—
301C2×#95—1061no—
302#340—532no—
303#340—532no—
304#340—532no—
305#340C2×D5312yes—
306#336—532no—
307C53#2923411yescharacteristic, complemented, Sylow
308C22×D4—531no—
309C22×D4—1061no—
310C23×C4—21no—
311C23×C4—531no—
312C23×C4—531no—
313C23×C4—1061no—
314C23×C4—1061no—
315C23×C4C22×D5311yescharacteristic
316C22×C8—532no—
317C22×C8C22×D5312yes—
318#121—534no—
319#121—1062no—
320C4×D4—2122no—
321C4×D4—2124no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—531no—
325C2×C4⋊C4—532no—
326C2×C4⋊C4—1062no—
327C2×C4⋊C4—1062no—
328C2×C4⋊C4—1061no—
329C2×C4⋊C4—1061no—
330C2×C4⋊C4—2122no—
331C2×C4⋊C4—2124no—
332C2×C4⋊C4C22×D5311yescharacteristic
333C2×C4⋊C4C22×D5311yescharacteristic
334C2×C4⋊C4C22×D5312yes—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—1061no—
337C2×C22⋊C4—1061no—
338C2×C22⋊C4—1061no—
339C2×C22⋊C4—1061no—
340C2×C42—531no—
341C2×C42—531no—
342C2×C42—531no—
343C2×C42—531no—
344C2×C42—1061no—
345C2×C42—2122no—
346#107—24no—
347#107—2122no—
348C4⋊C8—2122no—
349#102—1064no—
350#102—2122no—
351C22⋊C8—22no—
352C22⋊C8—1062no—
353#95—21no—
354#95—1062no—
355#95—1062no—
356#95—1061no—
357#95—1061no—
358C24—1061no—
359C24D4×D5311yescharacteristic
360C2×D4—532no—
361C2×D4—1064no—
362C2×D4—1062no—
363C2×D4—1061no—
364C2×D4—1062no—
365C2×D4—2122no—
366C2×D4—2122no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—531no—
376C22×C4—531no—
377C22×C4—534no—
378C22×C4—531no—
379C22×C4—531no—
380C22×C4—1062no—
381C22×C4—1061no—
382C22×C4—1062no—
383C22×C4—2122no—
384C22×C4—2122no—
385C22×C4—2122no—
386C22×C4—2121no—
387C22×C4—2121no—
388C22×C4—2121no—
389C22×C4—2121no—
390C22×C4—2122no—
391C22×C4—2122no—
392C22×C4—2122no—
393C22×C4#1171994512yes—
394C22×C4D4×D5311yescharacteristic
395C22×C4Q8×D5312yes—
396C22×C4C23×D5311yescharacteristic
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—1062no—
401C2×C8—1062no—
402C2×C8—1062no—
403C4⋊C4—24no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—532no—
408C4⋊C4—1061no—
409C4⋊C4—1062no—
410C4⋊C4—1062no—
411C4⋊C4—2122no—
412C4⋊C4—2124no—
413C4⋊C4—2124no—
414C4⋊C4—2122no—
415C4⋊C4#1171994714yes—
416C22⋊C4—42no—
417C22⋊C4—1064no—
418C22⋊C4—2122no—
419C22⋊C4—2122no—
420C22⋊C4—2122no—
421C42—1062no—
422C42—1062no—
423C42—2121no—
424C42—2121no—
425C42—2122no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—1062no—
431C23—2121no—
432C23—2121no—
433C23—2121no—
434C23—2121no—
435C23—2121no—
436C23—2121no—
437C23#1313319512yes—
438C23D8×D5312yes—
439C23#1313324912yes—
440C23C2×D4×D5311yescharacteristic
441D4—1064no—
442D4—2128no—
443D4—2122no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—48no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—532no—
459C2×C4—532no—
460C2×C4—1061no—
461C2×C4—1061no—
462C2×C4—1061no—
463C2×C4—1061no—
464C2×C4—1062no—
465C2×C4—1061no—
466C2×C4—1061no—
467C2×C4—1061no—
468C2×C4—1061no—
469C2×C4—1061no—
470C2×C4—1061no—
471C2×C4—4244no—
472C2×C4—4244no—
473C2×C4—2124no—
474C2×C4—2122no—
475C2×C4—2122no—
476C2×C4—2122no—
477C2×C4—2124no—
478C2×C4—2121no—
479C2×C4—2121no—
480C2×C4#1313319512yes—
481C2×C4C2×Q8×D5311yescharacteristic
482C2×C4C2×#1171994712yes—
483C2×C4C2×#1171994511yescharacteristic
484C2×C4#1313326312yes—
485C8—42no—
486C8—2122no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—4241no—
494C22—4241no—
495C22—4241no—
496C22—4241no—
497C22—2122no—
498C22—2122no—
499C22—2122no—
500C22#1681090214yes—
501C22C2×#1313319511yescharacteristic
502C22#1681112911yescharacteristic
503C22#1681118411yescharacteristic
504C22C2×D8×D5311yescharacteristic
505C22C2×#1313324911yescharacteristic
506C22#1681141911yescharacteristic
507C22#1681142211yescharacteristic
508C4—22no—
509C4—44no—
510C4—81no—
511C4—81no—
512C4—1062no—
513C4—1062no—
514C4—4242no—
515C4—2122no—
516C4—2121no—
517C4—2121no—
518C4#1681131912yes—
519C4#1681133112yes—
520C2—24no—
521C2—4242no—
522C2#2028006511yescharacteristic
523C2C2×#1681090211yescharacteristic
524C2#2028453611yescharacteristic
525C2#2028471011yescharacteristic
526C2#2028488711yescharacteristic
527C2#2028546511yescharacteristic
528C2#2028555011yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1852C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#1852D53
hypocenterC53#29234
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
2. C2×#16810522C2semidirect
3. #20276207C2semidirect
4. #20276098C2semidirect
5. #20275288C2semidirect
8. #20274963C2semidirect
9. #20274902C2semidirect
10. #20274693C2semidirect
11. C53×#1852C2semidirect
265. #1852D53semidirect
307. C53#29234semidirect

Maximal subgroups

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

subgroupindex
#202772212
C2×#168105222
#202762072
#202760982
#202752882
#202750322
#202750302
#202749632
#202749022
#202746932
C53×#18522
#2923453