Finite Group Databasebeta

← #27438458

Subgroups — #27438458

Order 13568.

1032 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811026C211yesmaximal, characteristic, complemented
2C2×#16810982C211yesmaximal, characteristic, complemented
3#20280536C211yesmaximal, characteristic, complemented
4#20275915C212yesmaximal, complemented
5#20275894C212yesmaximal
6#20275689C211yesmaximal, characteristic, complemented
7#20275527C211yesmaximal, characteristic, complemented
8#20274969C212yesmaximal, complemented
9#20274962C212yesmaximal, complemented
10#20274481C211yesmaximal, characteristic, complemented
11C2×C53×#542C211yesmaximal, characteristic
12C2×#13133217C2211yescharacteristic
13#16811026—24no—
14#16810982—24no—
15C2×#13133150C2212yes—
16C2×#13133150C2212yes—
17C2×#13133141C2212yes—
18C2×#13133141C2212yes—
19C2×#13133138C2212yes—
20C2×#13133138C2212yes—
21C2×C22⋊#88322C2212yes—
22C2×#13133128C2211yescharacteristic
23C2×C4×#88322C2211yescharacteristic
24#16810562—21no—
25#16810559—21no—
26#16810543—21no—
27#16810535—21no—
28#16810521C2212yes—
29#16810489C2211yescharacteristic
30#16810483C2212yes—
31#16810385C2212yes—
32#16810385C2212yes—
33#16810378C2212yes—
34C53×#542—24no—
35C53×#542—24no—
36C2×C4⋊Q8×C53C2211yescharacteristic
37C2×#126×C53—21no—
38C2×#124×C53—21no—
39C2×#124×C53C2211yescharacteristic
40C2×#123×C53—21no—
41C2×C22⋊Q8×C53—21no—
42C2×C22⋊Q8×C53C2211yescharacteristic
43C2×#121×C53C2211yescharacteristic
44C2×C4×Q8×C53C2211yescharacteristic
45C2×C4×D4×C53C2211yescharacteristic
46C2×#117×C53C2211yescharacteristic
47C22×Dic106—21no—
48C22×#88322C2312yes—
49C22×#88322C2312yes—
50#13133217—22no—
51#13133217—22no—
52C2×C4⋊Dic53—21no—
53C2×C4⋊Dic53C2312yes—
54C2×C4⋊Dic53C2312yes—
55C2×#11719919—22no—
56C2×C22⋊Dic53—22no—
57#13133150—24no—
58#13133150—24no—
59#13133141—24no—
60#13133141—24no—
61#13133138—24no—
62#13133138—24no—
63C22⋊#88322—24no—
64#13133128—22no—
65C4×#88322—22no—
66#13133124—22no—
67#13133124—21no—
68#13133124—22no—
69#13133124—21no—
70C22×Q8×C53—21no—
71C22×Q8×C53C2311yescharacteristic
72C22×D4×C53—21no—
73C22×D4×C53C2311yescharacteristic
74C23×C4×C53—21no—
75C23×C4×C53C2311yescharacteristic
76C4⋊Q8×C53—22no—
77C4⋊Q8×C53D414yes—
78#126×C53—28no—
79#124×C53—28no—
80#124×C53—22no—
81#124×C53D414yes—
82#123×C53—28no—
83C22⋊Q8×C53—28no—
84C22⋊Q8×C53—22no—
85C22⋊Q8×C53—22no—
86#121×C53—22no—
87#121×C53—22no—
88C4×Q8×C53—22no—
89C4×Q8×C53—22no—
90C4×D4×C53—22no—
91C4×D4×C53—22no—
92#117×C53—22no—
93#117×C53—22no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53D411yescharacteristic
98C2×C4⋊C4×C53D411yescharacteristic
99C2×C4⋊C4×C53C2311yescharacteristic
100C2×C4⋊C4×C53C2311yescharacteristic
101C2×C22⋊C4×C53—21no—
102C2×C22⋊C4×C53—21no—
103C2×C22⋊C4×C53—21no—
104C2×C22⋊C4×C53—21no—
105C2×C22⋊C4×C53—21no—
106C2×C22⋊C4×C53D411yescharacteristic
107C2×C22⋊C4×C53D411yescharacteristic
108C2×C42×C53—21no—
109C2×C42×C53C2311yescharacteristic
110C2×C42×C53C2311yescharacteristic
111C24×C53—21no—
112C24×C53C2×D411yescharacteristic
113C2×Q8×C53—24no—
114C2×Q8×C53—21no—
115C2×Q8×C53—22no—
116C2×Q8×C53—22no—
117C2×Q8×C53—82no—
118C2×Q8×C53C2×D412yes—
119C2×D4×C53—24no—
120C2×D4×C53—21no—
121C2×D4×C53—22no—
122C2×D4×C53—22no—
123C2×D4×C53—82no—
124C2×D4×C53C2×D412yes—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—24no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—82no—
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2411yescharacteristic
140C22×Dic53—22no—
141C22×Dic53—22no—
142C2×Dic106—42no—
143C2×Dic106—82no—
144C4⋊C4×C53—24no—
145C4⋊C4×C53—24no—
146C4⋊C4×C53—24no—
147C4⋊C4×C53—22no—
148C4⋊C4×C53—22no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53C2×D412yes—
152C4⋊C4×C53C2×D412yes—
153C22⋊C4×C53—24no—
154C22⋊C4×C53—24no—
155C22⋊C4×C53—24no—
156C22⋊C4×C53—24no—
157C22⋊C4×C53—24no—
158C22⋊C4×C53—22no—
159C22⋊C4×C53—22no—
160C42×C53—24no—
161C42×C53—21no—
162C42×C53—21no—
163C42×C53C2×D412yes—
164C42×C53C2×D412yes—
165C22⋊Dic53—82no—
166C4⋊Dic53—22no—
167C4⋊Dic53—22no—
168C4⋊Dic53—22no—
169C4⋊Dic53—22no—
170C4⋊Dic53—42no—
171#11719919—82no—
172C2×#88322—22no—
173C2×#88322—22no—
174C2×#88322—22no—
175C2×#88322—22no—
176C2×#88322—22no—
177C2×#88322#4114yes—
178C23×C53—21no—
179C23×C53—22no—
180C23×C53—21no—
181C23×C53—21no—
182C23×C53—22no—
183C23×C53—22no—
184C23×C53—41no—
185C23×C53—41no—
186C23×C53—42no—
187C23×C53—42no—
188C23×C53C22×D411yescharacteristic
189Q8×C53—24no—
190Q8×C53—42no—
191Q8×C53—84no—
192D4×C53—24no—
193D4×C53—42no—
194D4×C53—84no—
195C2×C4×C53—22no—
196C2×C4×C53—22no—
197C2×C4×C53—21no—
198C2×C4×C53—22no—
199C2×C4×C53—21no—
200C2×C4×C53—22no—
201C2×C4×C53—22no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—22no—
206C2×C4×C53—22no—
207C2×C4×C53—41no—
208C2×C4×C53—41no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—84no—
214C2×C4×C53#12012yes—
215C2×C4×C53#12012yes—
216C2×C4×C53C22×D411yescharacteristic
217C2×C4×C53C22×D411yescharacteristic
218C2×C4×C53C2×#4112yes—
219C2×C4×C532−1+412yes—
220C2×Dic53—42no—
221C2×Dic53—42no—
222C2×Dic53—42no—
223C2×Dic53—42no—
224C2×Dic53—42no—
225C2×Dic53—42no—
226Dic106—84no—
227#88322—24no—
228#88322—42no—
229#25681—531nomaximal, Sylow
230C22×C53—42no—
231C22×C53—41no—
232C22×C53—42no—
233C22×C53—42no—
234C22×C53—42no—
235C22×C53—41no—
236C22×C53—41no—
237C22×C53—42no—
238C22×C53—82no—
239C22×C53—82no—
240C22×C53C2×#12011yescharacteristic
241C22×C53#53211yescharacteristic
242C22×C53#53611yescharacteristic
243C22×C53#57612yes—
244C22×C53#57812yes—
245C4×C53—22no—
246C4×C53—22no—
247C4×C53—41no—
248C4×C53—42no—
249C4×C53—41no—
250C4×C53—42no—
251C4×C53—42no—
252C4×C53—41no—
253C4×C53#54712yes—
254C4×C53#54812yes—
255Dic53—82no—
256Dic53—82no—
257C2×#542D5311yescharacteristic
258C2×#500—531no—
259C2×#489—531no—
260#2628—531no—
261#2036—532no—
262#2033—532no—
263#1993—531no—
264#1958—531no—
265#1863—532no—
266#1862—532no—
267#1750—531no—
268C2×C53—81no—
269C2×C53—82no—
270C2×C53#243211yescharacteristic
271C2×C53#300811yescharacteristic
272C2×C53#301111yescharacteristic
273C2×C53#317512yes—
274C2×C53#324312yes—
275#542—24no—
276#542—24no—
277C2×C4⋊Q8—531no—
278C2×C4⋊Q8C2×D5311yescharacteristic
279C2×#126—21no—
280C2×#124—21no—
281C2×#124C2×D5311yescharacteristic
282C2×#123—21no—
283C2×C22⋊Q8—21no—
284C2×C22⋊Q8C2×D5311yescharacteristic
285C2×#121C2×D5311yescharacteristic
286C2×C4×Q8C2×D5311yescharacteristic
287C2×C4×D4C2×D5311yescharacteristic
288C2×#117C2×D5311yescharacteristic
289#500—1064no—
290#489—1064no—
291C2×#107—532no—
292C2×#107—532no—
293C2×#103—532no—
294C2×#103—532no—
295C2×#102—532no—
296C2×#102—532no—
297C2×C22⋊C8—532no—
298C2×#97—531no—
299C2×C4×C8—531no—
300#400—1061no—
301#399—1061no—
302#395—1061no—
303#393—1061no—
304#389—532no—
305#382—531no—
306#380—532no—
307#340—532no—
308#340—532no—
309#336—532no—
310C53#2568111yescharacteristic, complemented, Sylow
311C22×Q8—21no—
312C22×Q8—1061no—
313C22×Q8C22×D5311yescharacteristic
314C22×D4—21no—
315C22×D4C22×D5311yescharacteristic
316C23×C4—21no—
317C23×C4C22×D5311yescharacteristic
318C22×C8—532no—
319C22×C8—532no—
320C4⋊Q8—22no—
321C4⋊Q8—1062no—
322C4⋊Q8—1062no—
323C4⋊Q8C22⋊D5314yes—
324#126—28no—
325#124—28no—
326#124—22no—
327#124C22⋊D5314yes—
328#123—28no—
329C22⋊Q8—28no—
330C22⋊Q8—22no—
331C22⋊Q8—22no—
332#121—22no—
333#121—22no—
334C4×Q8—22no—
335C4×Q8—22no—
336C4×D4—22no—
337C4×D4—22no—
338#117—22no—
339#117—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—532no—
344C2×C4⋊C4—532no—
345C2×C4⋊C4—1062no—
346C2×C4⋊C4—1061no—
347C2×C4⋊C4C22⋊D5311yescharacteristic
348C2×C4⋊C4C22⋊D5311yescharacteristic
349C2×C4⋊C4C22×D5311yescharacteristic
350C2×C4⋊C4C22×D5311yescharacteristic
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—1062no—
357C2×C22⋊C4C22⋊D5311yescharacteristic
358C2×C22⋊C4C22⋊D5311yescharacteristic
359C2×C42—21no—
360C2×C42C22×D5311yescharacteristic
361C2×C42C22×D5311yescharacteristic
362#107—1064no—
363#107—1064no—
364#103—1064no—
365#103—1064no—
366#102—1064no—
367#102—1064no—
368C22⋊C8—1064no—
369#97—1062no—
370C4×C8—1062no—
371#95—1062no—
372#95—1061no—
373#95—1061no—
374#95—1062no—
375C24—21no—
376C24C2×C22⋊D5311yescharacteristic
377C2×Q8—24no—
378C2×Q8—21no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—82no—
382C2×Q8—4242no—
383C2×Q8—2122no—
384C2×Q8C2×C22⋊D5312yes—
385C2×D4—24no—
386C2×D4—21no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—82no—
390C2×D4C2×C22⋊D5312yes—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—82no—
400C22×C4—1062no—
401C22×C4—1062no—
402C22×C4C2×C22⋊D5311yescharacteristic
403C22×C4C2×C22⋊D5311yescharacteristic
404C22×C4C2×C22⋊D5311yescharacteristic
405C22×C4C2×C22⋊D5311yescharacteristic
406C22×C4C2×C22⋊D5311yescharacteristic
407C22×C4C23×D5311yescharacteristic
408C2×C8—534no—
409C2×C8—1062no—
410C2×C8—1062no—
411C2×C8—1062no—
412C2×C8—1062no—
413C2×C8—1062no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—1062no—
422C4⋊C4—1062no—
423C4⋊C4—1062no—
424C4⋊C4—1062no—
425C4⋊C4—4242no—
426C4⋊C4—2122no—
427C4⋊C4C2×C22⋊D5312yes—
428C4⋊C4C2×C22⋊D5312yes—
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—24no—
433C22⋊C4—24no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—4242no—
437C42—24no—
438C42—21no—
439C42—21no—
440C42C2×C22⋊D5312yes—
441C42C2×C22⋊D5312yes—
442C23—21no—
443C23—22no—
444C23—21no—
445C23—22no—
446C23—22no—
447C23—21no—
448C23—41no—
449C23—42no—
450C23—42no—
451C23—41no—
452C23C22×C22⋊D5311yescharacteristic
453Q8—24no—
454Q8—42no—
455Q8—84no—
456Q8—4244no—
457D4—24no—
458D4—42no—
459D4—84no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—84no—
479C2×C4—2122no—
480C2×C4—2122no—
481C2×C4—2122no—
482C2×C4—2122no—
483C2×C4—2122no—
484C2×C4—2122no—
485C2×C4#1313318612yes—
486C2×C4#1313318612yes—
487C2×C4C22×C22⋊D5311yescharacteristic
488C2×C4C22×C22⋊D5311yescharacteristic
489C2×C4#41×D5312yes—
490C2×C4#1313326412yes—
491C8—1064no—
492C8—2122no—
493C22—42no—
494C22—41no—
495C22—42no—
496C22—42no—
497C22—41no—
498C22—41no—
499C22—42no—
500C22—42no—
501C22—82no—
502C22—82no—
503C22C2×#1313318611yescharacteristic
504C22#1681116911yescharacteristic
505C22#1681119411yescharacteristic
506C22#1681142612yes—
507C22#1681143812yes—
508C4—22no—
509C4—22no—
510C4—41no—
511C4—42no—
512C4—41no—
513C4—42no—
514C4—42no—
515C4—41no—
516C4—4242no—
517C4—4242no—
518C4#1681127212yes—
519C4#1681128012yes—
520C2—81no—
521C2—82no—
522C2#2027833911yescharacteristic
523C2#2028398311yescharacteristic
524C2#2028401011yescharacteristic
525C2#2028534812yes—
526C2#2028599212yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#542C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenterC2×#542D53
hypocenterC53#25681
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#542

Complements

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

NH ≅ G/Ntype
1. C2×#16811026C2semidirect
2. C2×#16810982C2semidirect
3. #20280536C2semidirect
4. #20275915C2semidirect
6. #20275689C2semidirect
7. #20275527C2semidirect
8. #20274969C2semidirect
9. #20274962C2semidirect
10. #20274481C2semidirect
310. C53#25681semidirect

Maximal subgroups

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

subgroupindex
C2×#168110262
C2×#168109822
#202805362
#202759152
#202758942
#202756892
#202755272
#202749692
#202749622
#202744812
C2×C53×#5422
#2568153