Finite Group Databasebeta

← #27393782

Subgroups — #27393782

Order 13568.

1166 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811152C211yesmaximal, characteristic
2C2×#16810674C211yesmaximal, characteristic, complemented
3#382×D53C211yesmaximal, characteristic, complemented
4#20275680C211yesmaximal, characteristic, complemented
5#20275679C211yesmaximal, characteristic, complemented
6#20275527C211yesmaximal, characteristic, complemented
7#20275029C212yesmaximal
8#20275019C212yesmaximal, complemented
9#20274728C212yesmaximal, complemented
10#20274723C212yesmaximal, complemented
11C53×#1958C211yesmaximal, characteristic, complemented
12#16811152—24no—
13#16811152C418yescomplemented
14C2×#13133210—21no—
15C2×#13133209C2211yescharacteristic
16C2×#13133199—21no—
17C2×#13133193C2211yescharacteristic
18C2×#13133188C2211yescharacteristic
19C2×Q8×Dic53C2211yescharacteristic
20C2×D4×Dic53C2211yescharacteristic
21C2×#13133177—21no—
22C2×#13133173—21no—
23C2×C4⋊C4×D53C2212yes—
24C2×C42×D53C2211yescharacteristic
25#16810674—24no—
26C2×#13133146C2212yes—
27C2×#13133141C2212yes—
28C2×#13133138C2212yes—
29C2×#13133129C2212yes—
30C2×C4×#88322C2211yescharacteristic
31#16810489C2211yescharacteristic
32#16810488C2212yes—
33#16810487—21no—
34#16810487—21no—
35#95×D53—21no—
36#95×D53—21no—
37#16810386C2212yes—
38#16810386C2212yes—
39#16810385C2212yes—
40#16810385C2212yes—
41C2×#124×C53C2211yescharacteristic
42C2×#103×C53C2212yes—
43C2×#102×C53C2212yes—
44C2×C4×C8×C53C2211yescharacteristic
45C53×#382C2211yescharacteristic
46C22×C22⋊D53—21no—
47C23×Dic53—21no—
48C22×C4×D53—22no—
49C22×C4×D53D411yescharacteristic
50C22×C4×D53D411yescharacteristic
51C22×C4×D53C2311yescharacteristic
52C22×#88322C2312yes—
53#13133210—28no—
54#13133209—22no—
55#13133209C2×C414yescomplemented
56#13133199—28no—
57#13133193—22no—
58#13133193C2×C414yes—
59#13133188—22no—
60#13133188C2×C414yes—
61Q8×Dic53—22no—
62Q8×Dic53C2×C414yescomplemented
63D4×Dic53—22no—
64D4×Dic53C2×C414yescomplemented
65#13133177—28no—
66#13133173—28no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53C2312yes—
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×#11719919—21no—
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—22no—
75C4⋊C4×D53—42no—
76C4⋊C4×D53—42no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×#11719921—21no—
80C2×C4×Dic53—21no—
81C2×C4×Dic53D411yescharacteristic
82C2×C4×Dic53D411yescharacteristic
83C2×C4×Dic53C2311yescharacteristic
84C42×D53C2×C414yes—
85C42×D53D414yes—
86#13133146—24no—
87#13133141—42no—
88#13133138—42no—
89#13133129—24no—
90C4×#88322—22no—
91#13133124—21no—
92#13133124—21no—
93#13133124—22no—
94#13133124—22no—
95C22×Q8×C53C2311yescharacteristic
96C22×D4×C53C2311yescharacteristic
97C22×C8×C53C2312yes—
98#124×C53—22no—
99#124×C53C2×C414yescomplemented
100C2×C4⋊C4×C53C2312yes—
101C2×C22⋊C4×C53—21no—
102C2×C22⋊C4×C53—21no—
103C2×C42×C53C2311yescharacteristic
104#103×C53—42no—
105#102×C53—42no—
106C4×C8×C53—22no—
107#95×C53—21no—
108#95×C53—21no—
109C24×C53—21no—
110C23×D53C2×D411yescharacteristic
111C2×Q8×C53—21no—
112C2×Q8×C53—22no—
113C2×Q8×C53—22no—
114C2×Q8×C53C22×C412yes—
115C2×D4×C53—21no—
116C2×D4×C53—22no—
117C2×D4×C53—22no—
118C2×D4×C53C22×C412yes—
119C22×C4×C53—21no—
120C22×C4×C53—22no—
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2411yescharacteristic
124C2×C22⋊D53—24no—
125C2×C22⋊D53—44no—
126C22×Dic53—21no—
127C22×Dic53—21no—
128C22×Dic53—24no—
129C22×Dic53—22no—
130C22×Dic53—44no—
131C22×Dic53C2×D411yescharacteristic
132C22×Dic53C2×D411yescharacteristic
133C22×Dic53C2×D411yescharacteristic
134C2×C4×D53—24no—
135C2×C4×D53—22no—
136C2×C4×D53—22no—
137C2×C4×D53—42no—
138C2×C4×D53—44no—
139C2×C4×D53—44no—
140C2×C4×D53—44no—
141C2×C4×D53C22⋊C414yes—
142C2×C4×D53C22⋊C414yes—
143C2×C4×D53C22×C412yes—
144C2×C4×D53C2×D412yes—
145C2×C8×C53—22no—
146C2×C8×C53—22no—
147C2×C8×C53—22no—
148C4⋊C4×C53—22no—
149C4⋊C4×C53—22no—
150C22⋊C4×C53—24no—
151C22⋊C4×C53—24no—
152C42×C53C22×C412yes—
153C42×C53C2×D412yes—
154C22⋊Dic53—24no—
155C22⋊Dic53—24no—
156#11719921—24no—
157C4⋊Dic53—21no—
158C4⋊Dic53—21no—
159C4⋊Dic53—24no—
160C4⋊Dic53—22no—
161C4⋊Dic53—22no—
162C4⋊Dic53C22×C412yes—
163C4⋊Dic53C22×C412yes—
164#11719919—24no—
165#11719919—24no—
166#11719919—24no—
167#11719919—44no—
168C4×Dic53—24no—
169C4×Dic53C22⋊C414yes—
170C4×Dic53C22⋊C414yes—
171C4×Dic53C22×C412yes—
172C4×Dic53C2×D412yes—
173C2×#88322—22no—
174C2×#88322—22no—
175C2×#88322—22no—
176C23×C53—22no—
177C23×C53—41no—
178C23×C53—41no—
179C23×C53—42no—
180C23×C53—42no—
181C23×C53C22×D411yescharacteristic
182C22×D53—21no—
183C22×D53—21no—
184C22×D53—22no—
185C22×D53C2×C22⋊C412yes—
186C22×D53#13514yes—
187Q8×C53—24no—
188Q8×C53—42no—
189D4×C53—24no—
190D4×C53—42no—
191C2×C4×C53—22no—
192C2×C4×C53—22no—
193C2×C4×C53—22no—
194C2×C4×C53—41no—
195C2×C4×C53—41no—
196C2×C4×C53—42no—
197C2×C4×C53—42no—
198C2×C4×C53—42no—
199C2×C4×C53C2×C22⋊C412yes—
200C2×C4×C53C2×C22⋊C412yes—
201C2×C4×C53C23×C411yescharacteristic
202C2×C4×C53C22×D411yescharacteristic
203C2×C4×C53H(5,𝔽2)12yes—
204C2×C4×C532−1+412yes—
205C22⋊D53—48no—
206C2×Dic53—21no—
207C2×Dic53—21no—
208C2×Dic53—22no—
209C2×Dic53—22no—
210C2×Dic53—22no—
211C2×Dic53—22no—
212C2×Dic53—48no—
213C2×Dic53—42no—
214C2×Dic53—42no—
215C2×Dic53—42no—
216C2×Dic53—42no—
217C2×Dic53—42no—
218C2×Dic53C2×C22⋊C412yes—
219C2×Dic53C2×C22⋊C412yes—
220C2×Dic53C2×C22⋊C412yes—
221C4×D53—28no—
222C4×D53—24no—
223C4×D53—24no—
224C4×D53—44no—
225C4×D53—88no—
226C8×C53—42no—
227#88322—42no—
228#21861—531nomaximal, Sylow
229C22×C53—42no—
230C22×C53—42no—
231C22×C53—42no—
232C22×C53—82no—
233C22×C53—82no—
234C22×C53C22×C22⋊C411yescharacteristic
235C22×C53#53511yescharacteristic
236C22×C53#53611yescharacteristic
237C22×C53C2×#13512yes—
238C22×C53#57512yes—
239C2×D53—22no—
240C2×D53—22no—
241C2×D53—24no—
242C2×D53—24no—
243C2×D53#41614yes—
244C4×C53—22no—
245C4×C53—22no—
246C4×C53—42no—
247C4×C53—82no—
248C4×C53#52012yes—
249C4×C53#53712yes—
250Dic53—22no—
251Dic53—22no—
252Dic53—22no—
253Dic53—42no—
254Dic53—42no—
255Dic53—82no—
256C2×#529—531no—
257C2×#431—531no—
258C2×#382—531no—
259#1991—531no—
260#1991—531no—
261#1958—531no—
262#1958D5311yescharacteristic, complemented
263#1874—532no—
264#1872—532no—
265#1805—532no—
266#1804—532no—
267C2×C53—82no—
268C2×C53#231111yescharacteristic
269C2×C53C2×#41611yescharacteristic
270C2×C53#289611yescharacteristic
271C2×C53#321112yes—
272C2×C53#323212yes—
273D53—24no—
274#529—538no—
275#529—1064no—
276C2×#126—1061no—
277C2×#125—531no—
278C2×#124C2×D5311yescharacteristic
279C2×#123—1061no—
280C2×C22⋊Q8—531no—
281C2×#121—531no—
282C2×C4×Q8—531no—
283C2×C4×D4—531no—
284C2×C4×D4—1061no—
285C2×#117—1061no—
286C22×C4⋊C4—532no—
287C22×C42—531no—
288#431—1064no—
289C2×C4⋊C8—532no—
290C2×#103—532no—
291C2×#103C2×D5312yes—
292C2×#102—532no—
293C2×#102C2×D5312yes—
294C2×C22⋊C8—532no—
295C2×C4×C8—531no—
296C2×C4×C8C2×D5311yescharacteristic
297#382—532no—
298#382—531no—
299#382—1061no—
300#382—1061no—
301#382C2×D5311yescharacteristic
302C2×#95—1061no—
303C2×#95—1061no—
304#340—532no—
305#340—532no—
306#340—532no—
307#340—532no—
308C53#2186111yescharacteristic, complemented, Sylow
309C22×Q8C22×D5311yescharacteristic
310C22×D4—1061no—
311C22×D4C22×D5311yescharacteristic
312C23×C4—531no—
313C23×C4—531no—
314C23×C4—531no—
315C23×C4—1062no—
316C23×C4—1061no—
317C22×C8—532no—
318C22×C8C22×D5312yes—
319#126—1068no—
320#125—534no—
321#125—1062no—
322#124—22no—
323#124C4×D5314yescomplemented
324#123—1068no—
325C22⋊Q8—534no—
326C22⋊Q8—1062no—
327#121—534no—
328#121—1062no—
329C4×Q8—534no—
330C4×Q8—1062no—
331C4×D4—534no—
332C4×D4—1062no—
333C4×D4—1068no—
334#117—1068no—
335C2×C4⋊C4—531no—
336C2×C4⋊C4—531no—
337C2×C4⋊C4—532no—
338C2×C4⋊C4—1061no—
339C2×C4⋊C4—1061no—
340C2×C4⋊C4—1061no—
341C2×C4⋊C4—1061no—
342C2×C4⋊C4—1062no—
343C2×C4⋊C4—2122no—
344C2×C4⋊C4—2122no—
345C2×C4⋊C4C22×D5312yes—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—1061no—
349C2×C22⋊C4—1061no—
350C2×C22⋊C4—1061no—
351C2×C42—531no—
352C2×C42—531no—
353C2×C42—531no—
354C2×C42—534no—
355C2×C42—534no—
356C2×C42—1061no—
357C2×C42C22×D5311yescharacteristic
358C4⋊C8—1064no—
359#103—42no—
360#103—2122no—
361#102—42no—
362#102—2122no—
363C22⋊C8—1064no—
364C4×C8—22no—
365C4×C8—1062no—
366#95—21no—
367#95—21no—
368#95—1062no—
369#95—1062no—
370#95—1061no—
371#95—1061no—
372C24—21no—
373C24—531no—
374C2×Q8—21no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8C2×C4×D5312yes—
378C2×D4—21no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—1064no—
382C2×D4—2124no—
383C2×D4C2×C4×D5312yes—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—531no—
387C22×C4—531no—
388C22×C4—531no—
389C22×C4—534no—
390C22×C4—532no—
391C22×C4—532no—
392C22×C4—534no—
393C22×C4—1064no—
394C22×C4—1064no—
395C22×C4—1062no—
396C22×C4—1062no—
397C22×C4—1061no—
398C22×C4—1061no—
399C22×C4—1062no—
400C22×C4—2122no—
401C22×C4—2124no—
402C22×C4—2124no—
403C22×C4—2124no—
404C22×C4—2124no—
405C22×C4D4×D5311yescharacteristic
406C22×C4D4×D5311yescharacteristic
407C22×C4C23×D5311yescharacteristic
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—1062no—
412C2×C8—1062no—
413C2×C8—1062no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—532no—
417C4⋊C4—532no—
418C4⋊C4—1064no—
419C4⋊C4—1064no—
420C4⋊C4—1064no—
421C4⋊C4—1064no—
422C4⋊C4—1061no—
423C4⋊C4—1061no—
424C4⋊C4—1062no—
425C4⋊C4—1062no—
426C4⋊C4—2124no—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—1064no—
430C22⋊C4—1064no—
431C22⋊C4—1064no—
432C42—534no—
433C42—534no—
434C42—532no—
435C42—532no—
436C42—1064no—
437C42C2×C4×D5312yes—
438C42D4×D5312yes—
439C23—22no—
440C23—42no—
441C23—42no—
442C23—41no—
443C23—41no—
444C23—532no—
445C23—534no—
446C23—1062no—
447C23—1061no—
448C23—1061no—
449C23C2×D4×D5311yescharacteristic
450Q8—24no—
451Q8—42no—
452D4—24no—
453D4—42no—
454D4—2128no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—532no—
464C2×C4—532no—
465C2×C4—532no—
466C2×C4—1068no—
467C2×C4—1062no—
468C2×C4—1062no—
469C2×C4—1061no—
470C2×C4—1061no—
471C2×C4—1064no—
472C2×C4—1064no—
473C2×C4—1062no—
474C2×C4—1062no—
475C2×C4—4248no—
476C2×C4—2124no—
477C2×C4—2128no—
478C2×C4—2122no—
479C2×C4—2122no—
480C2×C4—2122no—
481C2×C4—2122no—
482C2×C4—2122no—
483C2×C4C22⋊C4×D5312yes—
484C2×C4C22⋊C4×D5312yes—
485C2×C4C22×C4×D5311yescharacteristic
486C2×C4C2×D4×D5311yescharacteristic
487C2×C4#1313326212yes—
488C2×C4#1313326412yes—
489C8—42no—
490C8—2122no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—82no—
495C22—82no—
496C22—534no—
497C22—1062no—
498C22—1062no—
499C22—1064no—
500C22—1064no—
501C22C2×C22⋊C4×D5311yescharacteristic
502C22#1681118611yescharacteristic
503C22#1681119311yescharacteristic
504C22#135×D5312yes—
505C22#1681142012yes—
506C4—22no—
507C4—22no—
508C4—42no—
509C4—82no—
510C4—1062no—
511C4—1062no—
512C4—1062no—
513C4—4242no—
514C4—2122no—
515C4—2122no—
516C4#1681110212yes—
517C4#1681120212yes—
518C2—82no—
519C2—1064no—
520C2#2027730511yescharacteristic
521C2#416×D5311yescharacteristic
522C2#2028323011yescharacteristic
523C2#2028567512yes—
524C2#2028587812yes—
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1958C2
socleC23×C53C22×D4
prime radicalC2×#13133188C22
cosocleC22×C4×C53C24
hypercenter#1958D53
hypocenterC53#21861
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1958

Complements

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

NH ≅ G/Ntype
2. C2×#16810674C2semidirect
3. #382×D53C2semidirect
4. #20275680C2semidirect
5. #20275679C2semidirect
6. #20275527C2semidirect
8. #20275019C2semidirect
9. #20274728C2semidirect
10. #20274723C2semidirect
11. C53×#1958C2semidirect
13. #16811152C4semidirect
55. #13133209C2×C4semidirect
62. Q8×Dic53C2×C4semidirect
64. D4×Dic53C2×C4semidirect
99. #124×C53C2×C4semidirect
262. #1958D53semidirect
308. C53#21861semidirect
323. #124C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168111522
C2×#168106742
#382×D532
#202756802
#202756792
#202755272
#202750292
#202750192
#202747282
#202747232
C53×#19582
#2186153