Finite Group Databasebeta

← #27313050

Subgroups — #27313050

Order 13568.

1246 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810509C212yesmaximal
2#20272972C212yesmaximal
3#20272968C212yesmaximal
4#20272968C212yesmaximal
5#20272949C212yesmaximal
6#20272944C212yesmaximal
7#20272942C211yesmaximal, characteristic
8#20272942C211yesmaximal, characteristic
9C2×C53×#389C211yesmaximal, characteristic
10C22×C4⋊Dic53C2212yes—
11C22×C4⋊Dic53C2212yes—
12C22×C4×Dic53C2212yes—
13#16810509—28no—
14C2×#13133124C2212yes—
15C2×#13133124C2212yes—
16C2×#13133124C2212yes—
17C2×#13133124C2212yes—
18C2×#13133124C2212yes—
19C2×#13133124C2212yes—
20C2×#13133124C2212yes—
21C2×#13133124C2211yescharacteristic
22C2×#13133124C2211yescharacteristic
23C2×#13133124C2211yescharacteristic
24C2×#13133124C2211yescharacteristic
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C22×C4⋊C4×C53C2212yes—
28C22×C4⋊C4×C53C2211yescharacteristic
29C22×C4⋊C4×C53C2211yescharacteristic
30C22×C42×C53C2211yescharacteristic
31C53×#389C418yescomplemented
32C2×#95×C53C2212yes—
33C23×Dic53C2312yes—
34C23×Dic53C2312yes—
35C23×Dic53C2312yes—
36C23×Dic53C2312yes—
37C2×C4⋊Dic53—24no—
38C2×C4⋊Dic53—24no—
39C2×C4⋊Dic53—24no—
40C2×C4⋊Dic53—42no—
41C2×C4⋊Dic53—42no—
42C2×C4⋊Dic53—42no—
43C2×C4×Dic53—24no—
44C2×C4×Dic53—24no—
45C2×C4×Dic53—42no—
46#13133124—24no—
47#13133124—24no—
48#13133124—24no—
49#13133124—24no—
50#13133124—24no—
51#13133124—24no—
52#13133124—24no—
53#13133124—41no—
54#13133124—42no—
55#13133124—41no—
56#13133124—41no—
57#13133124—41no—
58#13133124—42no—
59C23×C4×C53C2311yescharacteristic
60C23×C4×C53C2311yescharacteristic
61C23×C4×C53C2312yes—
62C23×C4×C53C2311yescharacteristic
63C23×C4×C53C2312yes—
64C2×C4⋊C4×C53—41no—
65C2×C4⋊C4×C53—41no—
66C2×C4⋊C4×C53—42no—
67C2×C4⋊C4×C53—42no—
68C2×C4⋊C4×C53C2×C414yes—
69C2×C4⋊C4×C53C2×C418yes—
70C2×C4⋊C4×C53C2×C414yes—
71C2×C4⋊C4×C53D414yes—
72C2×C4⋊C4×C53Q814yes—
73C2×C42×C53—41no—
74C2×C42×C53—41no—
75C2×C42×C53C2×C414yes—
76#95×C53C2×C418yes—
77C24×C53C2411yescharacteristic
78C22×C4×C53—22no—
79C22×C4×C53—22no—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—22no—
83C22×C4×C53—22no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—22no—
87C22×C4×C53—22no—
88C22×C4×C53—22no—
89C22×C4×C53—22no—
90C22×C4×C53—21no—
91C22×C4×C53—21no—
92C22×C4×C53—21no—
93C22×C4×C53—21no—
94C22×C4×C53—22no—
95C22×C4×C53—22no—
96C22×C4×C53—22no—
97C22×C4×C53—22no—
98C22×C4×C53—22no—
99C22×C4×C53—22no—
100C22×C4×C53C22×C412yes—
101C22×C4×C53C22×C414yes—
102C22×C4×C53C22×C412yes—
103C22×C4×C53C22×C414yes—
104C22×C4×C53C22×C412yes—
105C22×C4×C53C2×D412yes—
106C22×C4×C53C2×D414yes—
107C22×C4×C53C2×Q812yes—
108C22×C4×C53C2×Q814yes—
109C22×Dic53—22no—
110C22×Dic53—22no—
111C22×Dic53—22no—
112C22×Dic53—22no—
113C22×Dic53—22no—
114C22×Dic53—22no—
115C22×Dic53—22no—
116C22×Dic53—22no—
117C22×Dic53—22no—
118C22×Dic53—22no—
119C22×Dic53—22no—
120C22×Dic53—22no—
121C22×Dic53—22no—
122C22×Dic53—22no—
123C22×Dic53—22no—
124C22×Dic53—22no—
125C22×Dic53—22no—
126C22×Dic53—22no—
127C22×Dic53—22no—
128C22×Dic53—22no—
129C22×Dic53—22no—
130C22×Dic53—22no—
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53#4114yes—
136C22×Dic53#4114yes—
137C4⋊C4×C53—44no—
138C4⋊C4×C53—48no—
139C4⋊C4×C53C4⋊C4116yescomplemented
140C42×C53—44no—
141C4⋊Dic53—48no—
142C4⋊Dic53—48no—
143C4×Dic53—48no—
144C23×C53C23×C411yescharacteristic
145C23×C53C22×D411yescharacteristic
146C23×C53C22×Q811yescharacteristic
147C23×C53C2×#4112yes—
148C23×C53C2×#4112yes—
149C23×C53H(5,𝔽2)11yescharacteristic
150C23×C53H(5,𝔽2)11yescharacteristic
151C23×C53H(5,𝔽2)12yes—
152C23×C532−1+411yescharacteristic
153C23×C532−1+411yescharacteristic
154C23×C532−1+412yes—
155C2×C4×C53—24no—
156C2×C4×C53—22no—
157C2×C4×C53—24no—
158C2×C4×C53—22no—
159C2×C4×C53—22no—
160C2×C4×C53—22no—
161C2×C4×C53—22no—
162C2×C4×C53—22no—
163C2×C4×C53—24no—
164C2×C4×C53—24no—
165C2×C4×C53—24no—
166C2×C4×C53—24no—
167C2×C4×C53—22no—
168C2×C4×C53—22no—
169C2×C4×C53—24no—
170C2×C4×C53—22no—
171C2×C4×C53—22no—
172C2×C4×C53—24no—
173C2×C4×C53—24no—
174C2×C4×C53—22no—
175C2×C4×C53—22no—
176C2×C4×C53—42no—
177C2×C4×C53—42no—
178C2×C4×C53—42no—
179C2×C4×C53—42no—
180C2×C4×C53—42no—
181C2×C4×C53—42no—
182C2×C4×C53—42no—
183C2×C4×C53—42no—
184C2×C4×C53C2×C4⋊C414yes—
185C2×C4×C53C2×C4⋊C418yes—
186C2×Dic53—24no—
187C2×Dic53—24no—
188C2×Dic53—24no—
189C2×Dic53—24no—
190C2×Dic53—24no—
191C2×Dic53—24no—
192C2×Dic53—42no—
193C2×Dic53—42no—
194C2×Dic53—42no—
195C2×Dic53—42no—
196C2×Dic53—42no—
197C2×Dic53—42no—
198C2×Dic53—42no—
199C2×Dic53—42no—
200C2×Dic53—42no—
201C2×Dic53—42no—
202C2×Dic53—42no—
203C2×Dic53—42no—
204C2×Dic53—42no—
205C2×Dic53—42no—
206C2×Dic53—42no—
207C2×Dic53—42no—
208C2×Dic53—42no—
209C2×Dic53—42no—
210C2×Dic53—42no—
211C2×Dic53—42no—
212C2×Dic53—42no—
213C2×Dic53—42no—
214#14487—531nomaximal, Sylow
215C22×C53C22×C4⋊C411yescharacteristic
216C22×C53C4×#4111yescharacteristic
217C22×C53C4×#4111yescharacteristic
218C22×C53#51811yescharacteristic
219C22×C53#51811yescharacteristic
220C22×C53#51911yescharacteristic
221C22×C53#51911yescharacteristic
222C22×C53#53711yescharacteristic
223C22×C53#53711yescharacteristic
224C22×C53#53812yes—
225C22×C53#53912yes—
226C22×C53#54012yes—
227C22×C53#54212yes—
228C22×C53#54411yescharacteristic
229C22×C53#54411yescharacteristic
230C22×C53#54612yes—
231C22×C53#54812yes—
232C22×C53#55212yes—
233C22×C53#55212yes—
234C22×C53#55412yes—
235C22×C53#55712yes—
236C22×C53#56312yes—
237C22×C53#56312yes—
238C4×C53—24no—
239C4×C53—28no—
240C4×C53—42no—
241C4×C53—42no—
242C4×C53—44no—
243Dic53—44no—
244Dic53—44no—
245Dic53—82no—
246Dic53—82no—
247C2×#389D5311yescharacteristic
248C2×#387—532no—
249#1442—532no—
250#1441—532no—
251#1441—532no—
252#1436—532no—
253C4⋊#95—532no—
254#1434—531no—
255#1434—531no—
256C2×C53#231611yescharacteristic
257C2×C53#233211yescharacteristic
258C2×C53#233311yescharacteristic
259C2×C53#234911yescharacteristic
260C2×C53#235111yescharacteristic
261C2×C53#237111yescharacteristic
262C2×C53#238111yescharacteristic
263C2×C53#270212yes—
264C2×C53#270712yes—
265C2×C53#277412yes—
266C2×C53#279112yes—
267C22×C4⋊C4—532no—
268C22×C4⋊C4—532no—
269C22×C4⋊C4C2×D5312yes—
270C22×C4⋊C4C2×D5311yescharacteristic
271C22×C4⋊C4C2×D5311yescharacteristic
272C22×C42—532no—
273C22×C42C2×D5311yescharacteristic
274#389Dic5318yescomplemented
275#387—1068no—
276C2×#95—532no—
277C2×#95—532no—
278C2×#95—532no—
279C2×#95—532no—
280C2×#95—532no—
281C2×#95—532no—
282C2×#95—532no—
283C2×#95—531no—
284C2×#95—531no—
285C2×#95—531no—
286C2×#95—531no—
287C2×#95—532no—
288C2×#95—532no—
289C2×#95C2×D5312yes—
290C53#1448711yescharacteristic, complemented, Sylow
291C23×C4—532no—
292C23×C4—532no—
293C23×C4—532no—
294C23×C4—532no—
295C23×C4C22×D5312yes—
296C23×C4C22×D5311yescharacteristic
297C23×C4C22×D5311yescharacteristic
298C23×C4C22×D5312yes—
299C23×C4C22×D5311yescharacteristic
300C2×C4⋊C4—41no—
301C2×C4⋊C4—41no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—1064no—
305C2×C4⋊C4—1064no—
306C2×C4⋊C4—1064no—
307C2×C4⋊C4—2122no—
308C2×C4⋊C4—2122no—
309C2×C4⋊C4—2122no—
310C2×C4⋊C4Dic10614yes—
311C2×C4⋊C4D21214yes—
312C2×C4⋊C4C2×Dic5314yes—
313C2×C4⋊C4C2×Dic5318yes—
314C2×C4⋊C4C2×Dic5314yes—
315C2×C42—41no—
316C2×C42—41no—
317C2×C42—1064no—
318C2×C42—1064no—
319C2×C42—2122no—
320C2×C42C2×Dic5314yes—
321#95—1064no—
322#95—1064no—
323#95—1064no—
324#95—1064no—
325#95—1064no—
326#95—1064no—
327#95—1064no—
328#95—2121no—
329#95—2121no—
330#95—2121no—
331#95—2121no—
332#95—2122no—
333#95—2122no—
334#95C2×Dic5318yes—
335C24C23×D5311yescharacteristic
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—534no—
359C22×C4—534no—
360C22×C4—1062no—
361C22×C4—1062no—
362C22×C4—1062no—
363C22×C4—1062no—
364C22×C4—1062no—
365C22×C4—1062no—
366C22×C4—1062no—
367C22×C4—1062no—
368C22×C4—1062no—
369C22×C4—1062no—
370C22×C4—1062no—
371C22×C4—1062no—
372C22×C4—1062no—
373C22×C4—1062no—
374C22×C4—1062no—
375C22×C4—1062no—
376C22×C4—1062no—
377C22×C4—1062no—
378C22×C4—1062no—
379C22×C4—1062no—
380C22×C4—1062no—
381C22×C4—1062no—
382C22×C4—1062no—
383C22×C4—1062no—
384C22×C4—1062no—
385C22×C4—1062no—
386C22×C4C2×Dic10614yes—
387C22×C4C2×Dic10612yes—
388C22×C4C2×D21212yes—
389C22×C4C2×D21214yes—
390C22×C4C22×Dic5312yes—
391C22×C4C22×Dic5314yes—
392C22×C4C22×Dic5312yes—
393C22×C4C22×Dic5314yes—
394C22×C4C22×Dic5312yes—
395C4⋊C4—44no—
396C4⋊C4—48no—
397C4⋊C4—2128no—
398C4⋊C4—2128no—
399C4⋊C4C4⋊Dic53116yescomplemented
400C42—44no—
401C42—2128no—
402C23C23×Dic5311yescharacteristic
403C23C22×D21211yescharacteristic
404C23C22×Dic10611yescharacteristic
405C23#41×D5312yes—
406C23#41×D5312yes—
407C23#1313326212yes—
408C23#1313326211yescharacteristic
409C23#1313326211yescharacteristic
410C23#1313326512yes—
411C23#1313326511yescharacteristic
412C23#1313326511yescharacteristic
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—1064no—
443C2×C4—1064no—
444C2×C4—1064no—
445C2×C4—1064no—
446C2×C4—1064no—
447C2×C4—1064no—
448C2×C4—2122no—
449C2×C4—2122no—
450C2×C4—2122no—
451C2×C4—2122no—
452C2×C4—2122no—
453C2×C4—2122no—
454C2×C4—2122no—
455C2×C4—2122no—
456C2×C4—2122no—
457C2×C4—2122no—
458C2×C4—2122no—
459C2×C4—2122no—
460C2×C4—2122no—
461C2×C4—2122no—
462C2×C4—2122no—
463C2×C4—2122no—
464C2×C4—2122no—
465C2×C4—2122no—
466C2×C4—2122no—
467C2×C4—2122no—
468C2×C4—2122no—
469C2×C4—2122no—
470C2×C4C2×C4⋊Dic5314yes—
471C2×C4C2×C4⋊Dic5318yes—
472C22C22×C4⋊Dic5311yescharacteristic
473C22#41×Dic5311yescharacteristic
474C22#41×Dic5311yescharacteristic
475C22#1681109811yescharacteristic
476C22#1681109811yescharacteristic
477C22#1681110111yescharacteristic
478C22#1681110111yescharacteristic
479C22#1681120411yescharacteristic
480C22#1681120411yescharacteristic
481C22#1681120512yes—
482C22#1681121912yes—
483C22#1681122312yes—
484C22#1681124312yes—
485C22#1681125411yescharacteristic
486C22#1681125411yescharacteristic
487C22#1681126812yes—
488C22#1681128512yes—
489C22#1681130512yes—
490C22#1681130512yes—
491C22#1681133212yes—
492C22#1681134912yes—
493C22#1681137212yes—
494C22#1681137212yes—
495C4—24no—
496C4—28no—
497C4—42no—
498C4—42no—
499C4—44no—
500C4—4242no—
501C4—4242no—
502C4—2124no—
503C4—2124no—
504C2#2027733311yescharacteristic
505C2#2027747311yescharacteristic
506C2#2027748211yescharacteristic
507C2#2027762511yescharacteristic
508C2#2027763611yescharacteristic
509C2#2027782111yescharacteristic
510C2#2027789611yescharacteristic
511C2#2028134112yes—
512C2#2028141612yes—
513C2#2028227412yes—
514C2#2028249512yes—
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC2×C53×#389C2
socle, prime radical, cosocleC24×C53C24
hypercenterC2×#389D53
hypocenterC53#14487
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
31. C53×#389C4semidirect
139. C4⋊C4×C53C4⋊C4semidirect
274. #389Dic53semidirect
290. C53#14487semidirect
399. C4⋊C4C4⋊Dic53semidirect

Maximal subgroups

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

subgroupindex
C2×#168105092
#202729722
#202729682
#202729492
#202729442
#202729422
C2×C53×#3892
#1448753