Finite Group Databasebeta

← #27280665

Subgroups — #27280665

Order 13568.

648 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285463C211yesmaximal, characteristic
2#20275551C211yesmaximal, characteristic, complemented
3#20273283C211yesmaximal, characteristic, complemented
4C2×#16810356C211yesmaximal, characteristic, complemented
5#20272594C211yesmaximal, characteristic, complemented
6#20272563C211yesmaximal, characteristic, complemented
7C53×#2364C211yesmaximal, characteristic, complemented
8C2×#13133191C2211yescharacteristic
9#16810936—21no—
10#16810934C412yescomplemented
11#16810857—22no—
12#16810757—21no—
13#16810756C412yescomplemented
14C2×#13133131—21no—
15C2×#13133131—21no—
16C2×C22⋊#88322C2211yescharacteristic
17C2×C22⋊#88322C2211yescharacteristic
18#16810483C2211yescharacteristic
19#16810357—22no—
20#16810356—21no—
21#16810356—21no—
22C2×#121×C53C2211yescharacteristic
23C2×C4×D4×C53—21no—
24C53×#388—21no—
25C53×#387—21no—
26C53×#386—21no—
27C53×#386—21no—
28C4×C22⋊C4×C53—21no—
29C4×C22⋊C4×C53C2211yescharacteristic
30C2×#95×C53C2211yescharacteristic
31C22×D212C2×C411yescharacteristic, complemented
32C22×C22⋊D53—21no—
33C2×#11719922—22no—
34C22×#88322—21no—
35C22×#88322—21no—
36#13133191—21no—
37#13133191—21no—
38#13133191—21no—
39#13133191—21no—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53C2×C411yescharacteristic, complemented
42C2×C22⋊Dic53—21no—
43C2×#11719921—21no—
44#13133150—22no—
45#13133138—22no—
46#13133137—22no—
47#13133131—42no—
48#13133131—42no—
49C22⋊#88322—21no—
50C22⋊#88322—21no—
51C22⋊#88322—21no—
52C22⋊#88322C2×C412yescomplemented
53#13133124—21no—
54C22×D4×C53—21no—
55C22×D4×C53D411yescharacteristic
56C23×C4×C53—21no—
57C23×C4×C53—21no—
58C23×C4×C53C2311yescharacteristic
59#121×C53—21no—
60#121×C53—21no—
61#121×C53C2×C412yes—
62#121×C53D412yes—
63C4×D4×C53—42no—
64C4×D4×C53—42no—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53D411yescharacteristic
67C2×C22⋊C4×C53—21no—
68C2×C22⋊C4×C53—21no—
69C2×C22⋊C4×C53—21no—
70C2×C22⋊C4×C53—21no—
71C2×C22⋊C4×C53D411yescharacteristic
72C2×C22⋊C4×C53D411yescharacteristic
73C2×C42×C53—21no—
74C2×C42×C53—21no—
75#95×C53—21no—
76#95×C53—21no—
77#95×C53—21no—
78#95×C53—21no—
79#95×C53D411yescharacteristic
80#95×C53D411yescharacteristic
81C24×C53—21no—
82C24×C53C2×D411yescharacteristic
83C23×D53—21no—
84C2×D4×C53—22no—
85C2×D4×C53—22no—
86C2×D4×C53—21no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53—41no—
90C2×D4×C53—41no—
91C2×D4×C53—81no—
92C2×D4×C53C22⋊C412yes—
93C22×C4×C53—21no—
94C22×C4×C53—21no—
95C22×C4×C53—21no—
96C22×C4×C53—21no—
97C22×C4×C53—21no—
98C22×C4×C53—42no—
99C22×C4×C53—42no—
100C22×C4×C53—42no—
101C22×C4×C53—41no—
102C22×C4×C53—41no—
103C22×C4×C53—81no—
104C22×C4×C53—81no—
105C22×C4×C53—81no—
106C22×C4×C53C22×C411yescharacteristic
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53#4111yescharacteristic
111C22×C4×C53#4111yescharacteristic
112C2×C22⋊D53—41no—
113C2×C22⋊D53—41no—
114C2×C22⋊D53—41no—
115C2×C22⋊D53—41no—
116C2×C22⋊D53—41no—
117C2×C22⋊D53—41no—
118C22×Dic53—21no—
119C22×Dic53—41no—
120C2×D212—21no—
121C2×D212—21no—
122C2×D212—81no—
123C4⋊C4×C53—21no—
124C4⋊C4×C53—42no—
125C4⋊C4×C53C22⋊C412yes—
126C22⋊C4×C53—21no—
127C22⋊C4×C53—22no—
128C22⋊C4×C53—22no—
129C22⋊C4×C53—21no—
130C22⋊C4×C53—21no—
131C22⋊C4×C53—42no—
132C22⋊C4×C53—42no—
133C22⋊C4×C53—81no—
134C22⋊C4×C53D812yescomplemented
135C42×C53—42no—
136C42×C53—81no—
137C22⋊Dic53—41no—
138C22⋊Dic53—41no—
139#11719922—82no—
140#11719921—41no—
141#11719921—41no—
142C4⋊Dic53—21no—
143C4⋊Dic53—21no—
144C4⋊Dic53—81no—
145C2×#88322—22no—
146C2×#88322—41no—
147C2×#88322—81no—
148C2×#88322—81no—
149C23×C53—22no—
150C23×C53—41no—
151C23×C53—41no—
152C23×C53—41no—
153C23×C53—41no—
154C23×C53—41no—
155C23×C53—41no—
156C23×C53—41no—
157C23×C53—41no—
158C23×C53C4×D411yescharacteristic
159C23×C53#12011yescharacteristic
160C23×C53#12111yescharacteristic
161C23×C53C2×D811yescharacteristic
162C23×C53#13511yescharacteristic
163C23×C53AGL(1,ℤ/8)11yescharacteristic
164C23×C53AGL(1,ℤ/8)11yescharacteristic
165C22×D53—41no—
166C22×D53—41no—
167C22×D53—41no—
168C22×D53—81no—
169C22×D53—81no—
170D4×C53—42no—
171D4×C53—42no—
172D4×C53—42no—
173D4×C53—42no—
174D4×C53—42no—
175D4×C53—81no—
176C2×C4×C53—22no—
177C2×C4×C53—22no—
178C2×C4×C53—22no—
179C2×C4×C53—41no—
180C2×C4×C53—41no—
181C2×C4×C53—41no—
182C2×C4×C53—41no—
183C2×C4×C53—41no—
184C2×C4×C53—41no—
185C2×C4×C53—41no—
186C2×C4×C53—41no—
187C2×C4×C53—41no—
188C2×C4×C53—41no—
189C2×C4×C53—41no—
190C2×C4×C53—41no—
191C2×C4×C53—41no—
192C2×C4×C53—41no—
193C2×C4×C53—41no—
194C2×C4×C53—41no—
195C2×C4×C53—41no—
196C2×C4×C53—41no—
197C2×C4×C53—84no—
198C2×C4×C53—82no—
199C2×C4×C53—82no—
200C2×C4×C53C2×C22⋊C411yescharacteristic
201C2×C4×C53C4×D411yescharacteristic
202C2×C4×C53#12111yescharacteristic
203C2×C4×C53#12311yescharacteristic
204C22⋊D53—81no—
205C22⋊D53—81no—
206C22⋊D53—81no—
207C22⋊D53—81no—
208C2×Dic53—41no—
209C2×Dic53—41no—
210C2×Dic53—41no—
211C2×Dic53—81no—
212C2×Dic53—81no—
213C2×Dic53—81no—
214D212—42no—
215D212—81no—
216#88322—42no—
217#88322—81no—
218#9277—531nomaximal, Sylow
219C22×C53—21no—
220C22×C53—21no—
221C22×C53—21no—
222C22×C53—21no—
223C22×C53—21no—
224C22×C53—21no—
225C22×C53—41no—
226C22×C53—41no—
227C22×C53—41no—
228C22×C53—41no—
229C22×C53—42no—
230C22×C53—42no—
231C22×C53—42no—
232C22×C53—81no—
233C22×C53—81no—
234C22×C53—81no—
235C22×C53—81no—
236C22×C53#38611yescharacteristic
237C22×C53C4×D811yescharacteristic
238C22×C53#44211yescharacteristic
239C22×C53#44711yescharacteristic
240C22×C53#44911yescharacteristic
241C22×C53#45911yescharacteristic
242C22×C53#46311yescharacteristic
243C2×D53—81no—
244C2×D53—81no—
245C2×D53—81no—
246C2×D53—161no—
247C2×D53—161no—
248C4×C53—21no—
249C4×C53—21no—
250C4×C53—21no—
251C4×C53—21no—
252C4×C53—42no—
253C4×C53—42no—
254C4×C53—42no—
255C4×C53—81no—
256C4×C53—81no—
257Dic53—81no—
258Dic53—161no—
259#3186—531no—
260#2364D5311yescharacteristic, complemented
261#1964—531no—
262#1519—531no—
263C2×#327—531no—
264#1294—531no—
265#1282—531no—
266C2×C53—21no—
267C2×C53—21no—
268C2×C53—21no—
269C2×C53—21no—
270C2×C53—21no—
271C2×C53—21no—
272C2×C53—82no—
273C2×C53#187911yescharacteristic
274C2×C53#190611yescharacteristic
275C2×C53#191311yescharacteristic
276D53—161no—
277C2×#121—531no—
278C2×#121C2×D5311yescharacteristic
279C2×C4×D4—21no—
280#480—532no—
281#480—1061no—
282#466—1062no—
283#447—532no—
284#447—1061no—
285C2×#99—1061no—
286C2×#99—1061no—
287C2×C22⋊C8—531no—
288C2×C22⋊C8—531no—
289#388—21no—
290#387—21no—
291#386—21no—
292#386—21no—
293#380—531no—
294C4×C22⋊C4—21no—
295C4×C22⋊C4C2×D5311yescharacteristic
296C2×#95C2×D5311yescharacteristic
297#327—1062no—
298#327—1061no—
299#327—1061no—
300C53#927711yescharacteristic, complemented, Sylow
301C22×D4—21no—
302C22×D4—531no—
303C22×D4—1061no—
304C22×D4C22⋊D5311yescharacteristic
305C23×C4—21no—
306C23×C4—21no—
307C23×C4C22×D5311yescharacteristic
308C2×D8—1062no—
309C22×C8—1061no—
310C22×C8—1061no—
311#121—21no—
312#121—21no—
313#121—1061no—
314#121—1061no—
315#121—1061no—
316#121—1061no—
317#121C4×D5312yes—
318#121D21212yes—
319C4×D4—42no—
320C4×D4—42no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—531no—
323C2×C4⋊C4—1061no—
324C2×C4⋊C4C22⋊D5311yescharacteristic
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—1061no—
330C2×C22⋊C4—1061no—
331C2×C22⋊C4C22⋊D5311yescharacteristic
332C2×C22⋊C4C22⋊D5311yescharacteristic
333C2×C42—21no—
334C2×C42—21no—
335#107—1062no—
336#102—1062no—
337#102—1062no—
338#99—2122no—
339#99—2122no—
340C22⋊C8—532no—
341C22⋊C8—1061no—
342C22⋊C8—1061no—
343C22⋊C8—1061no—
344#95—21no—
345#95—21no—
346#95—21no—
347#95—21no—
348#95—1061no—
349#95C22⋊D5311yescharacteristic
350#95C22⋊D5311yescharacteristic
351C24—21no—
352C24—1061no—
353C24C2×C22⋊D5311yescharacteristic
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—21no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—81no—
362C2×D4—1061no—
363C2×D4—1061no—
364C2×D4—4241no—
365C2×D4—2121no—
366C2×D4—2121no—
367C2×D4—2121no—
368C2×D4—2121no—
369C2×D4—2121no—
370C2×D4—2121no—
371C2×D4#1171992112yes—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—81no—
383C22×C4—81no—
384C22×C4—81no—
385C22×C4—1061no—
386C22×C4—2121no—
387C22×C4C2×C4×D5311yescharacteristic
388C22×C4C2×D21211yescharacteristic
389C22×C4#1171994511yescharacteristic
390C22×C4#1171994511yescharacteristic
391C22×C4C2×C22⋊D5311yescharacteristic
392C22×C4C2×C22⋊D5311yescharacteristic
393D8—4242no—
394C2×C8—1062no—
395C2×C8—4241no—
396C2×C8—4241no—
397C2×C8—2121no—
398C4⋊C4—21no—
399C4⋊C4—42no—
400C4⋊C4—1061no—
401C4⋊C4—1061no—
402C4⋊C4—4241no—
403C4⋊C4#1171992112yes—
404C22⋊C4—21no—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—21no—
408C22⋊C4—21no—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C22⋊C4—81no—
412C22⋊C4—2121no—
413C22⋊C4—2121no—
414C22⋊C4—2121no—
415C22⋊C4—2121no—
416C22⋊C4#1171992212yescomplemented
417C42—42no—
418C42—81no—
419C23—22no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—4241no—
429C23—4241no—
430C23—2121no—
431C23—2121no—
432C23—2121no—
433C23C4×C22⋊D5311yescharacteristic
434C23#1313318611yescharacteristic
435C23#1313319111yescharacteristic
436C23C2×#1171992211yescharacteristic
437C23#1313323711yescharacteristic
438C23#1313324211yescharacteristic
439C23#1313324311yescharacteristic
440D4—42no—
441D4—42no—
442D4—42no—
443D4—42no—
444D4—42no—
445D4—81no—
446D4—4241no—
447D4—4241no—
448D4—4241no—
449D4—4241no—
450D4—4241no—
451D4—2122no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—84no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—4241no—
477C2×C4—4241no—
478C2×C4—4241no—
479C2×C4—2121no—
480C2×C4—2121no—
481C2×C4—2121no—
482C2×C4C2×#1171992111yescharacteristic
483C2×C4C4×C22⋊D5311yescharacteristic
484C2×C4#1313319111yescharacteristic
485C2×C4#1313320011yescharacteristic
486C8—4241no—
487C8—2122no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22—8481no—
506C22—8481no—
507C22—4241no—
508C22—4241no—
509C22—4241no—
510C22#1681050211yescharacteristic
511C22C4×#1171992211yescharacteristic
512C22#1681073311yescharacteristic
513C22#1681075711yescharacteristic
514C22#1681076811yescharacteristic
515C22#1681081811yescharacteristic
516C22#1681083811yescharacteristic
517C4—21no—
518C4—21no—
519C4—21no—
520C4—21no—
521C4—42no—
522C4—42no—
523C4—42no—
524C4—81no—
525C4—81no—
526C4—8481no—
527C4—4241no—
528C2—21no—
529C2—21no—
530C2—21no—
531C2—21no—
532C2—21no—
533C2—21no—
534C2—82no—
535C2—8481no—
536C2#2027506311yescharacteristic
537C2#2027521611yescharacteristic
538C2#2027525711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2364C2
socleC22×C53#386
prime radical#20285463C2
cosocleC23×C4×C53C23
hypercenter#2364D53
hypocenterC53#9277
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2364

Complements

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

NH ≅ G/Ntype
2. #20275551C2semidirect
3. #20273283C2semidirect
4. C2×#16810356C2semidirect
5. #20272594C2semidirect
6. #20272563C2semidirect
7. C53×#2364C2semidirect
10. #16810934C4semidirect
13. #16810756C4semidirect
31. C22×D212C2×C4semidirect
41. C2×C4⋊Dic53C2×C4semidirect
52. C22⋊#88322C2×C4semidirect
134. C22⋊C4×C53D8semidirect
260. #2364D53semidirect
300. C53#9277semidirect
416. C22⋊C4#11719922semidirect

Maximal subgroups

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

subgroupindex
#202854632
#202755512
#202732832
C2×#168103562
#202725942
#202725632
C53×#23642
#927753