Finite Group Databasebeta

← #27533937

Subgroups — #27533937

Order 13568.

1292 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810958C211yesmaximal, characteristic, complemented
2C2×#16810785C211yesmaximal, characteristic
3#20280512C211yesmaximal, characteristic, complemented
4#20275056C212yesmaximal, complemented
5#20275049C212yesmaximal, complemented
6#20274969C212yesmaximal, complemented
7#20274962C212yesmaximal, complemented
8#20274801C211yesmaximal, characteristic, complemented
9#20274786C211yesmaximal, characteristic, complemented
10#20274655C211yesmaximal, characteristic
11C2×C53×#520C211yesmaximal, characteristic
12C22×#11719925C2211yescharacteristic
13C22×#11719923C2211yescharacteristic
14C2×#13133196C2211yescharacteristic
15#16810958—24no—
16#16810785—24no—
17C2×#13133150C2212yes—
18C2×#13133141C2211yescharacteristic
19C2×#13133141C2212yes—
20C2×#13133140C2212yes—
21C2×#13133138C2211yescharacteristic
22C2×#13133138C2212yes—
23C2×C22⋊#88322C2211yescharacteristic
24C2×C22⋊#88322C2211yescharacteristic
25#16810561—21no—
26#16810554—21no—
27#16810551—22no—
28#16810529C2212yes—
29#16810521C2212yes—
30#16810483C2211yescharacteristic
31#16810385C2212yes—
32#16810385C2212yes—
33#16810378C2212yes—
34#16810378C2212yes—
35C22×#41×C53C2211yescharacteristic
36C53×#520—24no—
37C53×#520—24no—
38C2×C4×Q8×C53C2212yes—
39C2×C4×D4×C53—22no—
40C2×C4×D4×C53C2212yes—
41C2×#117×C53—21no—
42C2×#117×C53C2211yescharacteristic
43C22×C4⋊C4×C53—21no—
44C22×C4⋊C4×C53C2211yescharacteristic
45C22×Dic106C2311yescharacteristic
46C2×#11719925—22no—
47C2×#11719925—24no—
48C2×#11719923—22no—
49C2×#11719923—24no—
50C22×#88322C2312yes—
51C22×#88322C2311yescharacteristic
52C22×#88322C2311yescharacteristic
53#13133196—22no—
54#13133196—22no—
55C2×C4⋊Dic53C2311yescharacteristic
56C2×C4⋊Dic53C2312yes—
57C2×#11719919—21no—
58C2×#11719919—22no—
59C2×C22⋊Dic53—21no—
60C2×C22⋊Dic53—21no—
61#13133150—24no—
62#13133141—22no—
63#13133141—24no—
64#13133140—24no—
65#13133138—22no—
66#13133138—24no—
67C22⋊#88322—22no—
68C22⋊#88322—22no—
69#13133124—22no—
70#13133124—22no—
71#13133124—22no—
72#13133124—21no—
73C2×#41×C53—24no—
74C2×#41×C53—24no—
75C2×#41×C53—22no—
76C2×#41×C53D414yes—
77C22×Q8×C53C2311yescharacteristic
78C22×D4×C53—21no—
79C22×D4×C53C2311yescharacteristic
80C23×C4×C53—21no—
81C23×C4×C53—21no—
82C23×C4×C53—22no—
83C23×C4×C53C2311yescharacteristic
84C4×Q8×C53—24no—
85C4×Q8×C53—24no—
86C4×D4×C53—216no—
87C4×D4×C53—24no—
88C4×D4×C53—24no—
89#117×C53—28no—
90#117×C53—22no—
91#117×C53—22no—
92C2×C4⋊C4×C53—28no—
93C2×C4⋊C4×C53—21no—
94C2×C4⋊C4×C53—22no—
95C2×C4⋊C4×C53—22no—
96C2×C4⋊C4×C53D414yes—
97C2×C4⋊C4×C53D411yescharacteristic
98C2×C4⋊C4×C53D411yescharacteristic
99C2×C4⋊C4×C53C2312yes—
100C2×C22⋊C4×C53—22no—
101C2×C22⋊C4×C53D412yes—
102C2×C42×C53—22no—
103C2×C42×C53C2312yes—
104C24×C53—21no—
105C24×C53C2×D411yescharacteristic
106#41×C53—216no—
107#41×C53—28no—
108#41×C53—28no—
109C2×Q8×C53—21no—
110C2×Q8×C53—22no—
111C2×Q8×C53C2×D412yes—
112C2×Q8×C53#4114yes—
113C2×D4×C53—28no—
114C2×D4×C53—24no—
115C2×D4×C53—21no—
116C2×D4×C53—22no—
117C2×D4×C53C2×D412yes—
118C2×D4×C53#4114yes—
119C22×C4×C53—28no—
120C22×C4×C53—21no—
121C22×C4×C53—24no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—28no—
126C22×C4×C53—24no—
127C22×C4×C53—82no—
128C22×C4×C53—82no—
129C22×C4×C53—81no—
130C22×C4×C53—81no—
131C22×C4×C53C2×D412yes—
132C22×C4×C53C2×D412yes—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D412yes—
135C22×C4×C53C2×D412yes—
136C22×C4×C53C2411yescharacteristic
137C22×Dic53—22no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C2×Dic106—21no—
141C2×Dic106—21no—
142C2×Dic106—22no—
143C2×Dic106—22no—
144C4⋊C4×C53—28no—
145C4⋊C4×C53—24no—
146C4⋊C4×C53—22no—
147C4⋊C4×C53—22no—
148C4⋊C4×C53—22no—
149C4⋊C4×C53C2×D414yes—
150C22⋊C4×C53—28no—
151C22⋊C4×C53—24no—
152C42×C53—28no—
153C42×C53—22no—
154C42×C53—22no—
155C22⋊Dic53—42no—
156C22⋊Dic53—81no—
157#11719925—44no—
158#11719923—44no—
159C4⋊Dic53—21no—
160C4⋊Dic53—21no—
161C4⋊Dic53—22no—
162C4⋊Dic53—22no—
163#11719919—42no—
164#11719919—82no—
165C2×#88322—21no—
166C2×#88322—22no—
167C2×#88322—21no—
168C2×#88322—22no—
169C2×#88322—22no—
170C2×#88322—22no—
171C2×#88322—22no—
172C23×C53—22no—
173C23×C53—22no—
174C23×C53—24no—
175C23×C53—21no—
176C23×C53—21no—
177C23×C53—22no—
178C23×C53—41no—
179C23×C53—41no—
180C23×C53—42no—
181C23×C53#12012yes—
182C23×C53C22×D411yescharacteristic
183Q8×C53—22no—
184Q8×C53—24no—
185Q8×C53#12314yes—
186D4×C53—216no—
187D4×C53—22no—
188D4×C53—24no—
189D4×C53#12314yes—
190C2×C4×C53—216no—
191C2×C4×C53—22no—
192C2×C4×C53—22no—
193C2×C4×C53—24no—
194C2×C4×C53—22no—
195C2×C4×C53—22no—
196C2×C4×C53—24no—
197C2×C4×C53—24no—
198C2×C4×C53—22no—
199C2×C4×C53—22no—
200C2×C4×C53—22no—
201C2×C4×C53—22no—
202C2×C4×C53—24no—
203C2×C4×C53—42no—
204C2×C4×C53—42no—
205C2×C4×C53—84no—
206C2×C4×C53—88no—
207C2×C4×C53#12012yes—
208C2×C4×C53C22×D411yescharacteristic
209C2×C4×C53C22×D411yescharacteristic
210C2×C4×C53C2×#4112yes—
211C2×C4×C53C2×#4112yes—
212C2×Dic53—41no—
213C2×Dic53—41no—
214C2×Dic53—42no—
215C2×Dic53—42no—
216C2×Dic53—41no—
217C2×Dic53—42no—
218C2×Dic53—41no—
219C2×Dic53—42no—
220Dic106—41no—
221Dic106—41no—
222Dic106—42no—
223#88322—41no—
224#88322—41no—
225#88322—42no—
226#33017—531nomaximal, Sylow
227C22×C53—22no—
228C22×C53—24no—
229C22×C53—22no—
230C22×C53—22no—
231C22×C53—42no—
232C22×C53—44no—
233C22×C53—42no—
234C22×C53—44no—
235C22×C53—42no—
236C22×C53—42no—
237C22×C53C2×#12011yescharacteristic
238C22×C53#52512yes—
239C22×C53#57611yescharacteristic
240C22×C53#57711yescharacteristic
241C22×C53#57812yes—
242C4×C53—22no—
243C4×C53—24no—
244C4×C53—42no—
245C4×C53—42no—
246C4×C53—44no—
247C4×C53C2×#12311yescharacteristic
248C4×C53C2×#12311yescharacteristic
249C4×C53#52512yes—
250Dic53—81no—
251Dic53—82no—
252Dic53—81no—
253C2×#520D5311yescharacteristic
254C2×#484—531no—
255C2×#452—531no—
256#2625—531no—
257#1878—532no—
258#1877—532no—
259#1863—532no—
260#1862—532no—
261#1823—531no—
262#1818—531no—
263#1788—531no—
264C2×C53—42no—
265C2×C53—44no—
266C2×C53#284911yescharacteristic
267C2×C53#301111yescharacteristic
268C2×C53#301211yescharacteristic
269C2×C53#312012yes—
270C2×C53#312412yes—
271C22×#41C2×D5311yescharacteristic
272C22×Q16—531no—
273C22×SD16—531no—
274C2×C22⋊Q8—531no—
275#520—24no—
276#520—24no—
277C2×C4×Q8C2×D5312yes—
278C2×C4×D4—22no—
279C2×C4×D4C2×D5312yes—
280C2×#117—21no—
281C2×#117C2×D5311yescharacteristic
282C22×C4⋊C4—21no—
283C22×C4⋊C4C2×D5311yescharacteristic
284#484—1064no—
285#452—1064no—
286C2×#107—532no—
287C2×#103—531no—
288C2×#103—532no—
289C2×#103—532no—
290C2×#102—531no—
291C2×#102—532no—
292C2×C22⋊C8—531no—
293C2×C22⋊C8—531no—
294#400—1061no—
295#398—1061no—
296#397—1062no—
297#391—532no—
298#389—532no—
299#380—531no—
300#340—532no—
301#340—532no—
302#336—532no—
303#336—532no—
304C53#3301711yescharacteristic, complemented, Sylow
305C2×#41—24no—
306C2×#41—24no—
307C2×#41—22no—
308C2×#41C22⋊D5314yes—
309C22×Q8—531no—
310C22×Q8C22×D5311yescharacteristic
311C22×D4—21no—
312C22×D4C22×D5311yescharacteristic
313C23×C4—21no—
314C23×C4—21no—
315C23×C4—22no—
316C23×C4C22×D5311yescharacteristic
317C2×Q16—1064no—
318C2×Q16—1062no—
319C2×SD16—1064no—
320C2×SD16—1062no—
321C22×C8—531no—
322C22×C8—531no—
323C22×C8—532no—
324C22⋊Q8—1062no—
325C22⋊Q8—1062no—
326C4×Q8—24no—
327C4×Q8—24no—
328C4×D4—216no—
329C4×D4—24no—
330C4×D4—24no—
331#117—28no—
332#117—22no—
333#117—22no—
334C2×C4⋊C4—28no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—531no—
339C2×C4⋊C4—532no—
340C2×C4⋊C4—1062no—
341C2×C4⋊C4—1061no—
342C2×C4⋊C4C22⋊D5311yescharacteristic
343C2×C4⋊C4C22⋊D5311yescharacteristic
344C2×C4⋊C4C22⋊D5314yes—
345C2×C4⋊C4C22×D5312yes—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—1061no—
348C2×C22⋊C4—1061no—
349C2×C22⋊C4C22⋊D5312yes—
350C2×C42—22no—
351C2×C42C22×D5312yes—
352#107—1064no—
353#103—1064no—
354#103—1062no—
355#103—1064no—
356#102—1064no—
357#102—1062no—
358C22⋊C8—1062no—
359C22⋊C8—1062no—
360#95—1061no—
361#95—1062no—
362#95—1062no—
363#95—1062no—
364C24—21no—
365C24C2×C22⋊D5311yescharacteristic
366#41—216no—
367#41—28no—
368#41—28no—
369C2×Q8—21no—
370C2×Q8—22no—
371C2×Q8—1061no—
372C2×Q8—1061no—
373C2×Q8—1062no—
374C2×Q8—1062no—
375C2×Q8#1171994514yes—
376C2×Q8C2×C22⋊D5312yes—
377C2×D4—28no—
378C2×D4—24no—
379C2×D4—21no—
380C2×D4—22no—
381C2×D4#1171994514yes—
382C2×D4C2×C22⋊D5312yes—
383C22×C4—28no—
384C22×C4—21no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—28no—
390C22×C4—24no—
391C22×C4—82no—
392C22×C4—82no—
393C22×C4—81no—
394C22×C4—81no—
395C22×C4—1062no—
396C22×C4—1061no—
397C22×C4—1061no—
398C22×C4C2×C22⋊D5312yes—
399C22×C4C2×C22⋊D5312yes—
400C22×C4C2×C22⋊D5311yescharacteristic
401C22×C4C2×C22⋊D5312yes—
402C22×C4C2×C22⋊D5312yes—
403C22×C4C23×D5311yescharacteristic
404Q16—2124no—
405SD16—2124no—
406C2×C8—1061no—
407C2×C8—1062no—
408C2×C8—1061no—
409C2×C8—1062no—
410C2×C8—1062no—
411C2×C8—1062no—
412C2×C8—1062no—
413C4⋊C4—28no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—1061no—
419C4⋊C4—1061no—
420C4⋊C4—1062no—
421C4⋊C4—1062no—
422C4⋊C4—4242no—
423C4⋊C4—2122no—
424C4⋊C4C2×C22⋊D5314yes—
425C22⋊C4—28no—
426C22⋊C4—24no—
427C22⋊C4—4241no—
428C22⋊C4—2122no—
429C42—28no—
430C42—22no—
431C42—22no—
432C23—22no—
433C23—22no—
434C23—24no—
435C23—21no—
436C23—21no—
437C23—22no—
438C23—41no—
439C23—41no—
440C23—42no—
441C23#1313318612yes—
442C23C22×C22⋊D5311yescharacteristic
443Q8—22no—
444Q8—24no—
445Q8—2121no—
446Q8—2121no—
447Q8—2122no—
448Q8#1313320014yes—
449D4—216no—
450D4—22no—
451D4—24no—
452D4#1313320014yes—
453C2×C4—216no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—84no—
469C2×C4—88no—
470C2×C4—2121no—
471C2×C4—2121no—
472C2×C4—2122no—
473C2×C4—2122no—
474C2×C4—2122no—
475C2×C4—2121no—
476C2×C4—2121no—
477C2×C4—2122no—
478C2×C4#1313318612yes—
479C2×C4C22×C22⋊D5311yescharacteristic
480C2×C4C22×C22⋊D5311yescharacteristic
481C2×C4C2×#1171994512yes—
482C2×C4C2×#1171994512yes—
483C8—2121no—
484C8—2121no—
485C8—2122no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—24no—
490C22—42no—
491C22—44no—
492C22—42no—
493C22—44no—
494C22—42no—
495C22—42no—
496C22C2×#1313318611yescharacteristic
497C22#1681113412yes—
498C22#1681142611yescharacteristic
499C22#1681143411yescharacteristic
500C22#1681143812yes—
501C4—22no—
502C4—24no—
503C4—42no—
504C4—42no—
505C4—44no—
506C4—4242no—
507C4—4241no—
508C4—4241no—
509C4C2×#1313320011yescharacteristic
510C4C2×#1313320011yescharacteristic
511C4#1681113412yes—
512C2—42no—
513C2—44no—
514C2#2028300711yescharacteristic
515C2#2028401011yescharacteristic
516C2#2028401911yescharacteristic
517C2#2028492912yes—
518C2#2028497712yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#520C2
socleC23×C53C22×D4
prime radicalC22×#41×C53C22
hypercenterC2×#520D53
hypocenterC53#33017
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
1. C2×#16810958C2semidirect
3. #20280512C2semidirect
4. #20275056C2semidirect
5. #20275049C2semidirect
6. #20274969C2semidirect
7. #20274962C2semidirect
8. #20274801C2semidirect
9. #20274786C2semidirect
304. C53#33017semidirect

Maximal subgroups

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

subgroupindex
C2×#168109582
C2×#168107852
#202805122
#202750562
#202750492
#202749692
#202749622
#202748012
#202747862
#202746552
C2×C53×#5202
#3301753