Finite Group Databasebeta

← #27393791

Subgroups — #27393791

Order 13568.

1138 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
1C2×#16811152C211yesmaximal, characteristic
2C2×#16810674C211yesmaximal, characteristic, complemented
3#20277121C211yesmaximal, characteristic, complemented
4#20275680C211yesmaximal, characteristic, complemented
5#20275679C211yesmaximal, characteristic, complemented
6#20275527C211yesmaximal, characteristic, complemented
7#20275034C212yesmaximal
8#20275024C212yesmaximal, complemented
9#20274718C212yesmaximal, complemented
10#20274713C212yesmaximal, 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×#13133175C2212yes—
23C2×#13133173—21no—
24C2×C42×D53C2211yescharacteristic
25#16810674—24no—
26C2×#13133146C2212yes—
27C2×#13133141C2212yes—
28C2×#13133138C2212yes—
29C2×#13133129C2212yes—
30C2×C4×#88322C2211yescharacteristic
31#16810492C2212yes—
32#16810489C2211yescharacteristic
33#16810482—21no—
34#16810482—21no—
35#16810474—21no—
36#16810474—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×D53D411yescharacteristic
49C22×C4×D53D411yescharacteristic
50C22×C4×D53C2311yescharacteristic
51C22×#88322C2312yes—
52#13133210—28no—
53#13133209—22no—
54#13133209C2×C414yes—
55#13133199—28no—
56#13133193—22no—
57#13133193C2×C414yes—
58#13133188—22no—
59#13133188C2×C414yes—
60Q8×Dic53—22no—
61Q8×Dic53C2×C414yes—
62D4×Dic53—22no—
63D4×Dic53C2×C414yes—
64#13133177—28no—
65#13133175—42no—
66#13133175—42no—
67#13133173—28no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53C2312yes—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—22no—
76C2×#11719919—22no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×#11719921—21no—
80C2×#11719921—22no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53—22no—
83C2×C4×Dic53D411yescharacteristic
84C2×C4×Dic53D411yescharacteristic
85C2×C4×Dic53C2311yescharacteristic
86C42×D53C2×C414yes—
87C42×D53D414yes—
88#13133146—24no—
89#13133141—42no—
90#13133138—42no—
91#13133129—24no—
92C4×#88322—22no—
93#13133124—21no—
94#13133124—21no—
95C22×Q8×C53C2311yescharacteristic
96C22×D4×C53C2311yescharacteristic
97C22×C8×C53C2312yes—
98#124×C53—22no—
99#124×C53C2×C414yes—
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—22no—
135C2×C4×D53—22no—
136C2×C4×D53—24no—
137C2×C4×D53—42no—
138C2×C4×D53C22⋊C414yes—
139C2×C4×D53C22⋊C414yes—
140C2×C4×D53C22×C412yes—
141C2×C4×D53C2×D412yes—
142C2×C8×C53—22no—
143C2×C8×C53—22no—
144C2×C8×C53—22no—
145C4⋊C4×C53—22no—
146C4⋊C4×C53—22no—
147C22⋊C4×C53—24no—
148C22⋊C4×C53—24no—
149C42×C53C22×C412yes—
150C42×C53C2×D412yes—
151C22⋊Dic53—24no—
152C22⋊Dic53—24no—
153#11719921—24no—
154#11719921—44no—
155C4⋊Dic53—21no—
156C4⋊Dic53—21no—
157C4⋊Dic53—24no—
158C4⋊Dic53—22no—
159C4⋊Dic53—22no—
160C4⋊Dic53C22×C412yes—
161C4⋊Dic53C22×C412yes—
162#11719919—24no—
163#11719919—24no—
164#11719919—24no—
165#11719919—44no—
166#11719919—44no—
167C4×Dic53—24no—
168C4×Dic53—44no—
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×D53—22no—
186C22×D53C2×C22⋊C412yes—
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—42no—
212C2×Dic53—42no—
213C2×Dic53—48no—
214C2×Dic53—42no—
215C2×Dic53—42no—
216C2×Dic53—42no—
217C2×Dic53C2×C22⋊C412yes—
218C2×Dic53C2×C22⋊C412yes—
219C2×Dic53C2×C22⋊C412yes—
220C2×Dic53#13514yes—
221C4×D53—24no—
222C4×D53—24no—
223C4×D53—28no—
224C4×D53—44no—
225C8×C53—42no—
226#88322—42no—
227#21862—531nomaximal, Sylow
228C22×C53—42no—
229C22×C53—42no—
230C22×C53—42no—
231C22×C53—82no—
232C22×C53—82no—
233C22×C53C22×C22⋊C411yescharacteristic
234C22×C53#53511yescharacteristic
235C22×C53#53611yescharacteristic
236C22×C53C2×#13512yes—
237C22×C53#57512yes—
238C2×D53—22no—
239C2×D53—22no—
240C2×D53—22no—
241C2×D53—42no—
242C2×D53—42no—
243C4×C53—22no—
244C4×C53—22no—
245C4×C53—42no—
246C4×C53—82no—
247C4×C53#52012yes—
248C4×C53#53712yes—
249Dic53—22no—
250Dic53—22no—
251Dic53—42no—
252Dic53—42no—
253Dic53—82no—
254Dic53#41614yes—
255C2×#529—531no—
256C2×#431—531no—
257#2285—531no—
258#1991—531no—
259#1991—531no—
260#1958—531no—
261#1958D5311yescharacteristic, complemented
262#1875—532no—
263#1873—532no—
264#1803—532no—
265#1802—532no—
266C2×C53—82no—
267C2×C53#231111yescharacteristic
268C2×C53C2×#41611yescharacteristic
269C2×C53#289611yescharacteristic
270C2×C53#321212yes—
271C2×C53#323312yes—
272D53—42no—
273#529—538no—
274#529—1064no—
275C2×#126—1061no—
276C2×#125—531no—
277C2×#124C2×D5311yescharacteristic
278C2×#123—1061no—
279C2×C22⋊Q8—531no—
280C2×#121—531no—
281C2×C4×Q8—531no—
282C2×C4×D4—531no—
283C2×C4×D4—1061no—
284C2×#117—532no—
285C2×#117—1061no—
286C22×C42—531no—
287#431—1064no—
288C2×C4⋊C8—532no—
289C2×#103—532no—
290C2×#103C2×D5312yes—
291C2×#102—532no—
292C2×#102C2×D5312yes—
293C2×C22⋊C8—532no—
294C2×C4×C8—531no—
295C2×C4×C8C2×D5311yescharacteristic
296#384—532no—
297#382—531no—
298#382C2×D5311yescharacteristic
299#380—1061no—
300#380—1061no—
301C4×C4⋊C4—1061no—
302C4×C4⋊C4—1061no—
303#340—532no—
304#340—532no—
305#340—532no—
306#340—532no—
307C53#2186211yescharacteristic, complemented, Sylow
308C22×Q8C22×D5311yescharacteristic
309C22×D4—1061no—
310C22×D4C22×D5311yescharacteristic
311C23×C4—531no—
312C23×C4—531no—
313C23×C4—531no—
314C23×C4—1061no—
315C22×C8—532no—
316C22×C8C22×D5312yes—
317#126—1068no—
318#125—534no—
319#125—1062no—
320#124—22no—
321#124C4×D5314yes—
322#123—1068no—
323C22⋊Q8—534no—
324C22⋊Q8—1062no—
325#121—534no—
326#121—1062no—
327C4×Q8—534no—
328C4×Q8—1062no—
329C4×D4—534no—
330C4×D4—1062no—
331C4×D4—1068no—
332#117—1068no—
333#117—2122no—
334#117—2122no—
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—1062no—
344C2×C4⋊C4C22×D5312yes—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—1061no—
348C2×C22⋊C4—1061no—
349C2×C22⋊C4—1061no—
350C2×C22⋊C4—1062no—
351C2×C42—531no—
352C2×C42—531no—
353C2×C42—531no—
354C2×C42—534no—
355C2×C42—534no—
356C2×C42—1061no—
357C2×C42—1062no—
358C2×C42C22×D5311yescharacteristic
359C4⋊C8—1064no—
360#103—42no—
361#103—2122no—
362#102—42no—
363#102—2122no—
364C22⋊C8—1064no—
365C4×C8—22no—
366C4×C8—1062no—
367#95—21no—
368#95—21no—
369#95—1061no—
370#95—1061no—
371C24—21no—
372C24—531no—
373C2×Q8—21no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8C2×C4×D5312yes—
377C2×D4—21no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—1064no—
381C2×D4—2124no—
382C2×D4C2×C4×D5312yes—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—531no—
386C22×C4—531no—
387C22×C4—531no—
388C22×C4—534no—
389C22×C4—532no—
390C22×C4—532no—
391C22×C4—534no—
392C22×C4—1064no—
393C22×C4—1064no—
394C22×C4—1062no—
395C22×C4—1062no—
396C22×C4—1061no—
397C22×C4—1061no—
398C22×C4—1062no—
399C22×C4—2122no—
400C22×C4—2124no—
401C22×C4D4×D5311yescharacteristic
402C22×C4D4×D5311yescharacteristic
403C22×C4C23×D5311yescharacteristic
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—1062no—
408C2×C8—1062no—
409C2×C8—1062no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—532no—
413C4⋊C4—532no—
414C4⋊C4—1064no—
415C4⋊C4—1064no—
416C4⋊C4—1064no—
417C4⋊C4—1064no—
418C4⋊C4—1061no—
419C4⋊C4—1061no—
420C4⋊C4—1062no—
421C4⋊C4—1062no—
422C4⋊C4—2124no—
423C4⋊C4—2124no—
424C22⋊C4—24no—
425C22⋊C4—24no—
426C22⋊C4—1064no—
427C22⋊C4—1064no—
428C22⋊C4—1064no—
429C22⋊C4—2124no—
430C42—534no—
431C42—534no—
432C42—532no—
433C42—532no—
434C42—1064no—
435C42—2124no—
436C42C2×C4×D5312yes—
437C42D4×D5312yes—
438C23—22no—
439C23—42no—
440C23—42no—
441C23—41no—
442C23—41no—
443C23—532no—
444C23—1062no—
445C23—1062no—
446C23—1061no—
447C23—1061no—
448C23C2×D4×D5311yescharacteristic
449Q8—24no—
450Q8—42no—
451D4—24no—
452D4—42no—
453D4—2128no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—532no—
463C2×C4—534no—
464C2×C4—532no—
465C2×C4—532no—
466C2×C4—1068no—
467C2×C4—1062no—
468C2×C4—1061no—
469C2×C4—1061no—
470C2×C4—1064no—
471C2×C4—1064no—
472C2×C4—1062no—
473C2×C4—1062no—
474C2×C4—2124no—
475C2×C4—2128no—
476C2×C4—2122no—
477C2×C4—2122no—
478C2×C4—2122no—
479C2×C4—2122no—
480C2×C4—2122no—
481C2×C4C22⋊C4×D5312yes—
482C2×C4C22⋊C4×D5312yes—
483C2×C4C22×C4×D5311yescharacteristic
484C2×C4C2×D4×D5311yescharacteristic
485C2×C4#1313326212yes—
486C2×C4#1313326412yes—
487C8—42no—
488C8—2122no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—82no—
493C22—82no—
494C22—1062no—
495C22—1062no—
496C22—1062no—
497C22—2122no—
498C22—2122no—
499C22C2×C22⋊C4×D5311yescharacteristic
500C22#1681118611yescharacteristic
501C22#1681119311yescharacteristic
502C22#135×D5312yes—
503C22#1681142012yes—
504C4—22no—
505C4—22no—
506C4—42no—
507C4—82no—
508C4—534no—
509C4—1062no—
510C4—1062no—
511C4—4242no—
512C4—2122no—
513C4—2122no—
514C4#1681110212yes—
515C4#1681120212yes—
516C2—82no—
517C2—2122no—
518C2#2027730511yescharacteristic
519C2#416×D5311yescharacteristic
520C2#2028323011yescharacteristic
521C2#2028568612yes—
522C2#2028588912yes—
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#21862
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. #20277121C2semidirect
4. #20275680C2semidirect
5. #20275679C2semidirect
6. #20275527C2semidirect
8. #20275024C2semidirect
9. #20274718C2semidirect
10. #20274713C2semidirect
11. C53×#1958C2semidirect
13. #16811152C4semidirect
261. #1958D53semidirect
307. C53#21862semidirect

Maximal subgroups

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

subgroupindex
C2×#168111522
C2×#168106742
#202771212
#202756802
#202756792
#202755272
#202750342
#202750242
#202747182
#202747132
C53×#19582
#2186253