Finite Group Databasebeta

← #27425397

Subgroups — #27425397

Order 13568.

912 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810900C211yesmaximal, characteristic, complemented
2C2×#16810680C211yesmaximal, characteristic, complemented
3C4×#13133191C211yesmaximal, characteristic
4#20276146C212yesmaximal, complemented
5#20275904C212yesmaximal
6#20275668C211yesmaximal, characteristic, complemented
7#20275476C211yesmaximal, characteristic, complemented
8#20275455C212yesmaximal, complemented
9#20275411C212yesmaximal, complemented
10#20274712C211yesmaximal, characteristic, complemented
11C53×#2857C211yesmaximal, characteristic, complemented
12C2×#13133191C2211yescharacteristic
13C2×C4×D212C2211yescharacteristic
14C2×C4×C22⋊D53—21no—
15#16810900—24no—
16#16810680—24no—
17C2×#13133150C2212yes—
18C2×#13133148C2212yes—
19C2×C4⋊#88322C2211yescharacteristic
20C2×C4⋊#88322C2211yescharacteristic
21C2×#13133137C2212yes—
22C2×#13133137C2212yes—
23C2×C22⋊#88322C2212yes—
24C2×#13133128C2212yes—
25#16810525C2211yescharacteristic
26#16810521C2211yescharacteristic
27#16810514—21no—
28#16810502—21no—
29#16810483C2211yescharacteristic
30C4×C4⋊Dic53C2211yescharacteristic
31C4×#11719921—21no—
32#16810385C2212yes—
33#16810385C2212yes—
34#16810385C2212yes—
35#16810385C2212yes—
36C2×C4⋊Q8×C53C2211yescharacteristic
37C2×C22⋊Q8×C53—21no—
38C22×C42×C53C2211yescharacteristic
39C53×#391—21no—
40C53×#389—21no—
41C53×#389C2212yes—
42C53×#384C2211yescharacteristic
43C53×#380—21no—
44C53×#380C2212yes—
45C22×D212C2311yescharacteristic
46C22×C22⋊D53—21no—
47C22×C4×D53—21no—
48C22×#88322C2312yes—
49C22×#88322C2312yes—
50#13133191—22no—
51#13133191—22no—
52C4×D212—22no—
53C4×D212—22no—
54C4×C22⋊D53—44no—
55C2×C4⋊Dic53C2311yescharacteristic
56C2×C4⋊Dic53C2311yescharacteristic
57C2×C4⋊Dic53C2311yescharacteristic
58C2×#11719919—21no—
59C2×C22⋊Dic53—21no—
60C2×#11719921—21no—
61C2×#11719921—21no—
62C2×#11719921—21no—
63C2×#11719921—21no—
64C2×C4×Dic53—21no—
65#13133150—24no—
66#13133148—24no—
67C4⋊#88322—22no—
68C4⋊#88322D414yescomplemented
69#13133137—24no—
70#13133137—24no—
71C22⋊#88322—24no—
72#13133128—24no—
73#13133124—21no—
74#13133124—21no—
75#13133124—21no—
76C22×Q8×C53—21no—
77C23×C4×C53D411yescharacteristic
78C23×C4×C53D411yescharacteristic
79C23×C4×C53C2311yescharacteristic
80C4⋊Q8×C53—22no—
81C4⋊Q8×C53—22no—
82C22⋊Q8×C53—44no—
83C2×C4⋊C4×C53—21no—
84C2×C4⋊C4×C53—22no—
85C2×C4⋊C4×C53—21no—
86C2×C4⋊C4×C53—21no—
87C2×C4⋊C4×C53C2312yes—
88C2×C4⋊C4×C53C2312yes—
89C2×C22⋊C4×C53—22no—
90C2×C42×C53—22no—
91C2×C42×C53—22no—
92C2×C42×C53D411yescharacteristic
93C2×C42×C53D411yescharacteristic
94C2×C42×C53C2311yescharacteristic
95C2×C42×C53C2311yescharacteristic
96#95×C53—22no—
97#95×C53—22no—
98C24×C53C2×D411yescharacteristic
99C23×D53—21no—
100C2×Q8×C53—42no—
101C2×Q8×C53—42no—
102C2×Q8×C53—42no—
103C22×C4×C53—22no—
104C22×C4×C53—22no—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—21no—
109C22×C4×C53—21no—
110C22×C4×C53—21no—
111C22×C4×C53—21no—
112C22×C4×C53—21no—
113C22×C4×C53—21no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—41no—
117C22×C4×C53C2×D411yescharacteristic
118C22×C4×C53C2×D411yescharacteristic
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2411yescharacteristic
123C2×C22⋊D53—24no—
124C2×C22⋊D53—42no—
125C2×C22⋊D53—42no—
126C22×Dic53—21no—
127C22×Dic53—21no—
128C22×Dic53—21no—
129C2×D212—21no—
130C2×D212—21no—
131C2×D212—21no—
132C2×D212—21no—
133C2×D212—21no—
134C2×D212#4112yes—
135C2×C4×D53—42no—
136C2×C4×D53—42no—
137C2×C4×D53—42no—
138C4⋊C4×C53—22no—
139C4⋊C4×C53—22no—
140C4⋊C4×C53—22no—
141C4⋊C4×C53—22no—
142C4⋊C4×C53—42no—
143C4⋊C4×C53—44no—
144C4⋊C4×C53—42no—
145C4⋊C4×C53—42no—
146C22⋊C4×C53—44no—
147C42×C53—22no—
148C42×C53—22no—
149C42×C53—21no—
150C42×C53—21no—
151C42×C53C2×D412yes—
152C42×C53#4112yes—
153C22⋊Dic53—42no—
154#11719921—24no—
155#11719921—42no—
156#11719921—42no—
157#11719921—42no—
158C4⋊Dic53—21no—
159C4⋊Dic53—21no—
160C4⋊Dic53—21no—
161C4⋊Dic53—21no—
162C4⋊Dic53—21no—
163C4⋊Dic53#4112yes—
164#11719919—42no—
165C4×Dic53—42no—
166C2×#88322—22no—
167C2×#88322—22no—
168C2×#88322—22no—
169C2×#88322—22no—
170C2×#88322—22no—
171C2×#88322C2×D414yes—
172C23×C53—21no—
173C23×C53—21no—
174C23×C53—21no—
175C23×C53—21no—
176C23×C53—21no—
177C23×C53—21no—
178C23×C53#13712yes—
179C23×C53C22×D411yescharacteristic
180C22×D53—22no—
181C22×D53—41no—
182C22×D53—41no—
183C22×D53—41no—
184C22×D53—41no—
185C22×D53—41no—
186C22×D53—41no—
187Q8×C53—84no—
188C2×C4×C53—21no—
189C2×C4×C53—21no—
190C2×C4×C53—21no—
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—21no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—44no—
200C2×C4×C53—41no—
201C2×C4×C53—41no—
202C2×C4×C53—41no—
203C2×C4×C53—41no—
204C2×C4×C53—44no—
205C2×C4×C53—42no—
206C2×C4×C53—42no—
207C2×C4×C53—42no—
208C2×C4×C53—42no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53#12112yes—
212C2×C4×C53#12112yes—
213C2×C4×C53AGL(1,ℤ/8)12yes—
214C2×C4×C53AGL(1,ℤ/8)12yes—
215C2×C4×C53C22×D411yescharacteristic
216C2×C4×C53C2×#4111yescharacteristic
217C2×C4×C53C2×#4111yescharacteristic
218C2×C4×C53C2×#4111yescharacteristic
219C2×C4×C532−1+411yescharacteristic
220C2×C4×C532−1+411yescharacteristic
221C22⋊D53—48no—
222C2×Dic53—22no—
223C2×Dic53—41no—
224C2×Dic53—41no—
225C2×Dic53—41no—
226C2×Dic53—41no—
227C2×Dic53—41no—
228C2×Dic53—41no—
229C2×Dic53—41no—
230C2×Dic53—41no—
231D212—22no—
232D212—22no—
233D212—41no—
234D212—41no—
235C4×D53—84no—
236#88322—24no—
237#88322—42no—
238#24709—531nomaximal, Sylow
239C22×C53—22no—
240C22×C53—22no—
241C22×C53—22no—
242C22×C53—41no—
243C22×C53—41no—
244C22×C53—41no—
245C22×C53—41no—
246C22×C53C2×#12111yescharacteristic
247C22×C53#52511yescharacteristic
248C22×C53#53611yescharacteristic
249C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
250C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
251C22×C53C2×#13711yescharacteristic
252C22×C53#57511yescharacteristic
253C2×D53—42no—
254C2×D53—42no—
255C2×D53—42no—
256C2×D53—81no—
257C2×D53—81no—
258C2×D53—81no—
259C2×D53—81no—
260C4×C53—22no—
261C4×C53—41no—
262C4×C53—41no—
263C4×C53—41no—
264C4×C53—82no—
265C4×C53—82no—
266C4×C53#46914yes—
267C4×C53#54711yescharacteristic
268C4×C53#54811yescharacteristic
269C4×C53#55211yescharacteristic
270C4×C53#55611yescharacteristic
271Dic53—42no—
272Dic53—81no—
273Dic53—81no—
274C2×#474—531no—
275C2×#432—531no—
276#2857D5311yescharacteristic, complemented
277C4×#121—531no—
278#2076—532no—
279#2034—532no—
280#1988—531no—
281#1948—531no—
282#1944—532no—
283#1936—532no—
284#1802—531no—
285C2×C53—42no—
286C2×C53#241511yescharacteristic
287C2×C53C2×#46911yescharacteristic
288C2×C53#305611yescharacteristic
289C2×C53#311011yescharacteristic
290C2×C53#311511yescharacteristic
291C2×C53#323511yescharacteristic
292C2×C53#323911yescharacteristic
293D53—82no—
294C2×C4⋊Q8C2×D5311yescharacteristic
295C2×C22⋊Q8—21no—
296C2×#121—531no—
297C2×C4×D4—531no—
298C2×C4×D4—1061no—
299C22×C42C2×D5311yescharacteristic
300#474—1064no—
301#432—1064no—
302C2×#107—532no—
303C2×#106—532no—
304C2×C4⋊C8—531no—
305C2×C4⋊C8—531no—
306C2×#102—532no—
307C2×#102—532no—
308C2×C22⋊C8—532no—
309C2×#97—532no—
310#391—21no—
311#390—531no—
312#389—21no—
313#389—531no—
314#389C2×D5312yes—
315#388—1061no—
316#386—1061no—
317#384C2×D5311yescharacteristic
318#380—21no—
319#380—531no—
320#380C2×D5312yes—
321C4×C4⋊C4—531no—
322C4×C22⋊C4—1061no—
323#340—532no—
324#340—532no—
325#340—532no—
326#340—532no—
327C53#2470911yescharacteristic, complemented, Sylow
328C22×Q8—21no—
329C22×D4—531no—
330C22×D4—1061no—
331C23×C4—1061no—
332C23×C4C22⋊D5311yescharacteristic
333C23×C4C22⋊D5311yescharacteristic
334C23×C4C22×D5311yescharacteristic
335C22×C8—532no—
336C22×C8—532no—
337C4⋊Q8—22no—
338C4⋊Q8—22no—
339C22⋊Q8—44no—
340#121—1062no—
341#121—1062no—
342C4×D4—1062no—
343C4×D4—1062no—
344C4×D4—2124no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—531no—
350C2×C4⋊C4—531no—
351C2×C4⋊C4—531no—
352C2×C4⋊C4—1061no—
353C2×C4⋊C4C22×D5312yes—
354C2×C4⋊C4C22×D5312yes—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—1061no—
357C2×C22⋊C4—1061no—
358C2×C22⋊C4—1061no—
359C2×C22⋊C4—1061no—
360C2×C22⋊C4—1061no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—1061no—
364C2×C42C22⋊D5311yescharacteristic
365C2×C42C22⋊D5311yescharacteristic
366C2×C42C22×D5311yescharacteristic
367C2×C42C22×D5311yescharacteristic
368#107—1064no—
369#106—1064no—
370C4⋊C8—534no—
371C4⋊C8—1062no—
372#102—1064no—
373#102—1064no—
374C22⋊C8—1064no—
375#97—1064no—
376#95—22no—
377#95—22no—
378#95—1061no—
379#95—1061no—
380#95—1061no—
381C24—1061no—
382C24C2×C22⋊D5311yescharacteristic
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×D4—532no—
387C2×D4—1064no—
388C2×D4—1061no—
389C2×D4—1061no—
390C2×D4—1061no—
391C2×D4—1061no—
392C2×D4—1061no—
393C2×D4—2122no—
394C2×D4—2122no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—41no—
409C22×C4—1061no—
410C22×C4—1061no—
411C22×C4—1061no—
412C22×C4—2122no—
413C22×C4—2122no—
414C22×C4—2122no—
415C22×C4C2×C22⋊D5311yescharacteristic
416C22×C4C2×C22⋊D5311yescharacteristic
417C22×C4C2×C22⋊D5311yescharacteristic
418C22×C4C2×C22⋊D5311yescharacteristic
419C22×C4C2×C22⋊D5311yescharacteristic
420C22×C4C23×D5311yescharacteristic
421C2×C8—534no—
422C2×C8—1062no—
423C2×C8—1062no—
424C2×C8—1062no—
425C2×C8—1062no—
426C2×C8—1062no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—44no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—532no—
436C4⋊C4—1061no—
437C4⋊C4—1061no—
438C4⋊C4—1061no—
439C4⋊C4—1061no—
440C4⋊C4—1061no—
441C4⋊C4—2122no—
442C22⋊C4—44no—
443C22⋊C4—1064no—
444C22⋊C4—2122no—
445C22⋊C4—2122no—
446C22⋊C4—2122no—
447C22⋊C4—2122no—
448C42—22no—
449C42—22no—
450C42—21no—
451C42—21no—
452C42—2122no—
453C42D4×D5312yes—
454C42#1171994712yes—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—1062no—
462C23—2121no—
463C23—2121no—
464C23—2121no—
465C23—2121no—
466C23—2121no—
467C23—2121no—
468C23#1313324812yes—
469C23C22×C22⋊D5311yescharacteristic
470Q8—84no—
471D4—1062no—
472D4—1062no—
473D4—2128no—
474D4—2121no—
475D4—2121no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—44no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—1062no—
500C2×C4—4244no—
501C2×C4—2121no—
502C2×C4—2121no—
503C2×C4—2121no—
504C2×C4—2121no—
505C2×C4—2121no—
506C2×C4—2121no—
507C2×C4—2121no—
508C2×C4—2121no—
509C2×C4#1313318912yes—
510C2×C4#1313318912yes—
511C2×C4#1313324212yes—
512C2×C4#1313324312yes—
513C2×C4C2×D4×D5311yescharacteristic
514C2×C4#41×D5311yescharacteristic
515C2×C4#41×D5311yescharacteristic
516C2×C4C2×#1171994711yescharacteristic
517C2×C4#1313326511yescharacteristic
518C2×C4#1313326511yescharacteristic
519C8—1064no—
520C8—2122no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—4241no—
529C22—4241no—
530C22—4241no—
531C22—4241no—
532C22—2122no—
533C22—2122no—
534C22—2122no—
535C22C2×#1313318911yescharacteristic
536C22#1681113011yescharacteristic
537C22#1681119711yescharacteristic
538C22C2×#1313324211yescharacteristic
539C22C2×#1313324311yescharacteristic
540C22C2×#1313324811yescharacteristic
541C22#1681142111yescharacteristic
542C4—22no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—82no—
547C4—82no—
548C4—4241no—
549C4—4241no—
550C4—2122no—
551C4#1681087414yes—
552C4#1681127511yescharacteristic
553C4#1681128311yescharacteristic
554C4#1681130711yescharacteristic
555C4#1681134411yescharacteristic
556C2—42no—
557C2—4242no—
558C2#2027814511yescharacteristic
559C2C2×#1681087411yescharacteristic
560C2#2028434111yescharacteristic
561C2#2028481011yescharacteristic
562C2#2028486011yescharacteristic
563C2#2028591311yescharacteristic
564C2#2028594511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2857C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#2857D53
hypocenterC53#24709
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2857

Complements

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

NH ≅ G/Ntype
1. C2×#16810900C2semidirect
2. C2×#16810680C2semidirect
4. #20276146C2semidirect
6. #20275668C2semidirect
7. #20275476C2semidirect
8. #20275455C2semidirect
9. #20275411C2semidirect
10. #20274712C2semidirect
11. C53×#2857C2semidirect
68. C4⋊#88322D4semidirect
276. #2857D53semidirect
327. C53#24709semidirect

Maximal subgroups

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

subgroupindex
C2×#168109002
C2×#168106802
C4×#131331912
#202761462
#202759042
#202756682
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C53×#28572
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