Finite Group Databasebeta

← #27279366

Subgroups — #27279366

Order 13568.

822 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×#16810999C211yesmaximal, characteristic
2#20277779C211yesmaximal, characteristic, complemented
3#20275643C211yesmaximal, characteristic, complemented
4C2×#16810366C211yesmaximal, characteristic, complemented
5#20272556C211yesmaximal, characteristic, complemented
6#20272554C211yesmaximal, characteristic, complemented
7C53×#1982C211yesmaximal, characteristic, complemented
8C22×#11719922—22no—
9C2×#13133214C2211yescharacteristic
10C2×C4×D212—21no—
11C2×#13133179—21no—
12#16810999—22no—
13#16810999C414yescomplemented
14C2×C4⋊#88322C2211yescharacteristic
15C2×#13133138—21no—
16C2×#13133138—21no—
17C2×C4×#88322C2211yescharacteristic
18#16810525—21no—
19#16810524—21no—
20#16810501—22no—
21C4×C4⋊Dic53C2211yescharacteristic
22#16810366—22no—
23C2×#127×C53C2211yescharacteristic
24C2×C4⋊C8×C53C2211yescharacteristic
25C2×#102×C53—21no—
26C2×#102×C53—21no—
27C4×C4⋊C4×C53C2211yescharacteristic
28C22×D212—21no—
29C22×D212—21no—
30C22×D212D412yes—
31C22×C4×D53—22no—
32C2×#11719922—28no—
33C2×#11719922—44no—
34C2×#11719922—44no—
35C22×#88322—21no—
36C22×#88322—21no—
37#13133214—21no—
38#13133214—21no—
39#13133214C2×C412yes—
40#13133214D412yes—
41C4×D212—42no—
42C4×D212—42no—
43#13133179—42no—
44#13133179—42no—
45C2×C4⋊Dic53D411yescharacteristic
46C2×C4⋊Dic53D411yescharacteristic
47C2×#11719921—22no—
48C2×#11719921—22no—
49C2×#11719921—22no—
50C2×C4×Dic53—21no—
51C4⋊#88322—21no—
52C4⋊#88322—21no—
53#13133138—42no—
54#13133138—42no—
55C4×#88322—21no—
56C4×#88322C2×C412yescomplemented
57#13133124—21no—
58C22×D4×C53—21no—
59C22×D4×C53—21no—
60C22×D4×C53—21no—
61C22×C8×C53—21no—
62#127×C53—21no—
63#127×C53—21no—
64#127×C53—21no—
65#127×C53C2×C412yescomplemented
66C2×C4⋊C4×C53D411yescharacteristic
67C2×C4⋊C4×C53D411yescharacteristic
68C2×C42×C53—21no—
69C2×C42×C53C2311yescharacteristic
70C4⋊C8×C53—21no—
71C4⋊C8×C53—21no—
72#102×C53—42no—
73#102×C53—42no—
74#95×C53—21no—
75C24×C53—41no—
76C23×D53—22no—
77C2×D4×C53—22no—
78C2×D4×C53—22no—
79C2×D4×C53—22no—
80C2×D4×C53—42no—
81C2×D4×C53—42no—
82C2×D4×C53—41no—
83C2×D4×C53—41no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C2×D4×C53—41no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53—41no—
90C2×D4×C53—41no—
91C2×D4×C53—41no—
92C2×D4×C53—41no—
93C22×C4×C53—21no—
94C22×C4×C53—21no—
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×Dic53—21no—
99C22×Dic53—21no—
100C2×D212—22no—
101C2×D212—22no—
102C2×D212—22no—
103C2×D212—22no—
104C2×D212—22no—
105C2×D212—41no—
106C2×D212—41no—
107C2×D212—41no—
108C2×D212—41no—
109C2×D212—41no—
110C2×D212—41no—
111C2×D212—41no—
112C2×D212—41no—
113C2×D212—42no—
114C2×D212—42no—
115C2×D212C22⋊C414yes—
116C2×C4×D53—44no—
117C2×C4×D53—44no—
118C2×C4×D53—82no—
119C2×C8×C53—41no—
120C2×C8×C53—41no—
121C2×C8×C53—41no—
122C4⋊C4×C53—21no—
123C4⋊C4×C53—21no—
124C4⋊C4×C53—21no—
125C4⋊C4×C53D812yescomplemented
126C42×C53—41no—
127C42×C53—41no—
128C42×C53C22×C411yescharacteristic
129C42×C53C2×D411yescharacteristic
130C42×C53#4111yescharacteristic
131C42×C53#4111yescharacteristic
132#11719922—416no—
133#11719921—44no—
134#11719921—44no—
135#11719921—82no—
136C4⋊Dic53—21no—
137C4⋊Dic53—21no—
138C4⋊Dic53—21no—
139C4⋊Dic53D812yescomplemented
140C4×Dic53—41no—
141C4×Dic53—41no—
142C2×#88322—22no—
143C2×#88322—22no—
144C2×#88322—22no—
145C2×#88322—41no—
146C2×#88322—41no—
147C2×#88322—41no—
148C23×C53—42no—
149C23×C53—81no—
150C23×C53—81no—
151C23×C53—81no—
152C23×C53—81no—
153C23×C53—81no—
154C23×C53—81no—
155C23×C53#12011yescharacteristic
156C22×D53—24no—
157C22×D53—42no—
158C22×D53—42no—
159C22×D53—42no—
160C22×D53—42no—
161C22×D53—42no—
162C22×D53—42no—
163D4×C53—44no—
164D4×C53—42no—
165D4×C53—42no—
166D4×C53—42no—
167D4×C53—42no—
168D4×C53—82no—
169D4×C53—81no—
170D4×C53—81no—
171D4×C53—81no—
172D4×C53—81no—
173C2×C4×C53—21no—
174C2×C4×C53—21no—
175C2×C4×C53—22no—
176C2×C4×C53—41no—
177C2×C4×C53—41no—
178C2×C4×C53—41no—
179C2×C4×C53—41no—
180C2×C4×C53—41no—
181C2×C4×C53C2×C22⋊C411yescharacteristic
182C2×C4×C53C4×D411yescharacteristic
183C2×C4×C53C4×D411yescharacteristic
184C2×C4×C53#12111yescharacteristic
185C2×C4×C53#12111yescharacteristic
186C2×C4×C53#12311yescharacteristic
187C2×C4×C53C2×D811yescharacteristic
188C2×C4×C53C2×D811yescharacteristic
189C2×C4×C53#13511yescharacteristic
190C2×C4×C53#13511yescharacteristic
191C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
192C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
193C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
194C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
195C2×Dic53—22no—
196C2×Dic53—41no—
197C2×Dic53—41no—
198C2×Dic53—41no—
199C2×Dic53—41no—
200C2×Dic53—41no—
201D212—44no—
202D212—42no—
203D212—42no—
204D212—42no—
205D212—42no—
206D212—42no—
207D212—42no—
208D212—42no—
209D212—42no—
210D212—82no—
211C4×D53—88no—
212C8×C53—81no—
213#88322—24no—
214#88322—81no—
215#9061—531nomaximal, Sylow
216C22×C53—82no—
217C22×C53—82no—
218C22×C53—82no—
219C22×C53—161no—
220C22×C53—161no—
221C22×C53—161no—
222C22×C53—161no—
223C22×C53#38611yescharacteristic
224C22×C53#44711yescharacteristic
225C22×C53#44711yescharacteristic
226C22×C53#44911yescharacteristic
227C22×C53#44911yescharacteristic
228C22×C53#45311yescharacteristic
229C22×C53#45311yescharacteristic
230C2×D53—44no—
231C2×D53—44no—
232C2×D53—44no—
233C2×D53—82no—
234C2×D53—82no—
235C2×D53—82no—
236C2×D53—82no—
237C4×C53—21no—
238C4×C53—21no—
239C4×C53—42no—
240C4×C53—81no—
241C4×C53C4×D811yescharacteristic
242C4×C53C4×D811yescharacteristic
243C4×C53#44211yescharacteristic
244C4×C53#44211yescharacteristic
245C4×C53#45911yescharacteristic
246C4×C53#45911yescharacteristic
247C4×C53#46311yescharacteristic
248C4×C53#46311yescharacteristic
249Dic53—42no—
250Dic53—81no—
251C2×#493—531no—
252#2366—531no—
253#1982—531no—
254#1982D5311yescharacteristic, complemented
255C2×#331—531no—
256#1279—531no—
257#1279—531no—
258C2×C53—162no—
259C2×C53#161911yescharacteristic
260C2×C53#162011yescharacteristic
261C2×C53#162111yescharacteristic
262C2×C53#162111yescharacteristic
263C2×C53#187911yescharacteristic
264C2×C53#187911yescharacteristic
265C2×C53#188711yescharacteristic
266D53—84no—
267C22×D8—1062no—
268C2×#127—531no—
269C2×#127C2×D5311yescharacteristic
270C2×C4×D4—1061no—
271C2×C4×D4—1061no—
272#493—534no—
273#493—1062no—
274C2×C4⋊C8—531no—
275C2×C4⋊C8C2×D5311yescharacteristic
276C2×#102—21no—
277C2×#102—21no—
278C2×#102—1061no—
279C2×#102—1061no—
280C2×C4×C8—531no—
281#390—1061no—
282#390—1061no—
283#386—1062no—
284C4×C4⋊C4—531no—
285C4×C4⋊C4C2×D5311yescharacteristic
286#331—1062no—
287C53#906111yescharacteristic, complemented, Sylow
288C22×D4—21no—
289C22×D4—21no—
290C22×D4—21no—
291C22×D4—532no—
292C22×D4—1061no—
293C22×D4—1061no—
294C23×C4—1062no—
295C2×D8—1068no—
296C2×D8—2124no—
297C2×D8—2124no—
298C22×C8—21no—
299C22×C8—1061no—
300C22×C8—1061no—
301#127—21no—
302#127—21no—
303#127—21no—
304#127—532no—
305#127—532no—
306#127—1061no—
307#127—1061no—
308#127C4×D5312yescomplemented
309C4×D4—2122no—
310C4×D4—2122no—
311C4×D4—2122no—
312C4×D4—2122no—
313C2×C4⋊C4—531no—
314C2×C4⋊C4—531no—
315C2×C4⋊C4C22⋊D5311yescharacteristic
316C2×C4⋊C4C22⋊D5311yescharacteristic
317C2×C22⋊C4—1062no—
318C2×C22⋊C4—1062no—
319C2×C22⋊C4—1062no—
320C2×C42—21no—
321C2×C42—1061no—
322C2×C42C22×D5311yescharacteristic
323C4⋊C8—21no—
324C4⋊C8—21no—
325C4⋊C8—1061no—
326C4⋊C8—1061no—
327#102—42no—
328#102—42no—
329#102—2122no—
330#102—2122no—
331C4×C8—532no—
332C4×C8—1061no—
333#95—21no—
334#95—1061no—
335C24—41no—
336C24—1062no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—534no—
354C2×D4—1062no—
355C2×D4—1062no—
356C2×D4—1062no—
357C2×D4—1062no—
358C2×D4—1062no—
359C2×D4—2122no—
360C2×D4—2122no—
361C2×D4—2121no—
362C2×D4—2121no—
363C2×D4—2121no—
364C2×D4—2121no—
365C2×D4—2121no—
366C2×D4—2121no—
367C2×D4—2121no—
368C2×D4—2121no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—1061no—
372C22×C4—1061no—
373C22×C4—4242no—
374C22×C4—2124no—
375C22×C4—2124no—
376C22×C4D4×D5311yescharacteristic
377C22×C4D4×D5311yescharacteristic
378C22×C4C2×C22⋊D5311yescharacteristic
379D8—21216no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—1062no—
384C2×C8—1062no—
385C2×C8—1062no—
386C2×C8—2121no—
387C2×C8—2121no—
388C2×C8—2121no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—532no—
393C4⋊C4—1061no—
394C4⋊C4—1061no—
395C4⋊C4—1061no—
396C4⋊C4#1171992212yescomplemented
397C22⋊C4—4242no—
398C22⋊C4—2124no—
399C22⋊C4—2124no—
400C42—41no—
401C42—41no—
402C42—2121no—
403C42—2121no—
404C42C2×C4×D5311yescharacteristic
405C42#1171994511yescharacteristic
406C42D4×D5311yescharacteristic
407C42#1171994711yescharacteristic
408C23—42no—
409C23—81no—
410C23—81no—
411C23—81no—
412C23—81no—
413C23—81no—
414C23—81no—
415C23—1064no—
416C23—2122no—
417C23—2122no—
418C23—2122no—
419C23—2122no—
420C23—2122no—
421C23—2122no—
422C23#1313318511yescharacteristic
423D4—44no—
424D4—42no—
425D4—42no—
426D4—42no—
427D4—42no—
428D4—82no—
429D4—81no—
430D4—81no—
431D4—81no—
432D4—81no—
433D4—4242no—
434D4—2124no—
435D4—2122no—
436D4—2122no—
437D4—2122no—
438D4—2122no—
439D4—2122no—
440D4—2122no—
441D4—2122no—
442D4—2122no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—1062no—
452C2×C4—4248no—
453C2×C4—2121no—
454C2×C4—2121no—
455C2×C4—2121no—
456C2×C4—2121no—
457C2×C4—2121no—
458C2×C4C22⋊C4×D5311yescharacteristic
459C2×C4#1313317711yescharacteristic
460C2×C4C4×C22⋊D5311yescharacteristic
461C2×C4#1313318811yescharacteristic
462C2×C4#1313319011yescharacteristic
463C2×C4#1313319911yescharacteristic
464C2×C4D8×D5311yescharacteristic
465C2×C4C2×#1171992211yescharacteristic
466C2×C4#1313323711yescharacteristic
467C2×C4#1313323811yescharacteristic
468C2×C4#1313324111yescharacteristic
469C2×C4#1313324211yescharacteristic
470C2×C4#1313324311yescharacteristic
471C2×C4#1313324411yescharacteristic
472C8—81no—
473C8—1064no—
474C8—4241no—
475C22—82no—
476C22—82no—
477C22—82no—
478C22—161no—
479C22—161no—
480C22—161no—
481C22—161no—
482C22—4242no—
483C22—4242no—
484C22—4242no—
485C22—4242no—
486C22—2124no—
487C22—2124no—
488C22—2124no—
489C22#1681049911yescharacteristic
490C22#1681075511yescharacteristic
491C22#1681075611yescharacteristic
492C22#1681076611yescharacteristic
493C22#1681076711yescharacteristic
494C22#1681078911yescharacteristic
495C22#1681078911yescharacteristic
496C4—21no—
497C4—21no—
498C4—42no—
499C4—81no—
500C4—4241no—
501C4—2122no—
502C4#1681070311yescharacteristic
503C4C4×#1171992211yescharacteristic
504C4#1681073211yescharacteristic
505C4#1681073311yescharacteristic
506C4#1681081511yescharacteristic
507C4#1681081711yescharacteristic
508C4#1681083511yescharacteristic
509C4#1681083711yescharacteristic
510C2—162no—
511C2—4244no—
512C2#2027376811yescharacteristic
513C2#2027377311yescharacteristic
514C2#2027377811yescharacteristic
515C2#2027377911yescharacteristic
516C2#2027506011yescharacteristic
517C2#2027506611yescharacteristic
518C2#2027511011yescharacteristic
subgroupG/H
centerC23#13133185
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1982C2
socleC23×C53#120
prime radicalC2×#16810999C2
cosocleC2×C42×C53C23
hypercenter#1982D53
hypocenterC53#9061
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1982

Complements

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

NH ≅ G/Ntype
2. #20277779C2semidirect
3. #20275643C2semidirect
4. C2×#16810366C2semidirect
5. #20272556C2semidirect
6. #20272554C2semidirect
7. C53×#1982C2semidirect
13. #16810999C4semidirect
56. C4×#88322C2×C4semidirect
65. #127×C53C2×C4semidirect
125. C4⋊C4×C53D8semidirect
139. C4⋊Dic53D8semidirect
254. #1982D53semidirect
287. C53#9061semidirect
308. #127C4×D53semidirect
396. C4⋊C4#11719922semidirect

Maximal subgroups

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

subgroupindex
C2×#168109992
#202777792
#202756432
C2×#168103662
#202725562
#202725542
C53×#19822
#906153