Finite Group Databasebeta

← #27496235

Subgroups — #27496235

Order 13568.

456 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20276178C211yesmaximal, characteristic, complemented
2#20276177C211yesmaximal, characteristic, complemented
3#20276176C211yesmaximal, characteristic, complemented
4#20276174C211yesmaximal, characteristic, complemented
5#20276172C211yesmaximal, characteristic, complemented
6#20276113C211yesmaximal, characteristic, complemented
7#20276112C211yesmaximal, characteristic, complemented
8#20275624C211yesmaximal, characteristic
9#20275624C211yesmaximal, characteristic
10#20275622C211yesmaximal, characteristic
11#20275504C211yesmaximal, characteristic, complemented
12#20275504C211yesmaximal, characteristic, complemented
13#20275502C211yesmaximal, characteristic, complemented
14#20274469C211yesmaximal, characteristic, complemented
15C53×#2081C211yesmaximal, characteristic, complemented
16C22×#11719922C2211yescharacteristic
17#16810672C2211yescharacteristic
18#16810672C2211yescharacteristic
19#16810671C2211yescharacteristic
20#16810671C2211yescharacteristic
21#16810671C2211yescharacteristic
22C2×#13133152C2211yescharacteristic
23C2×#13133152C2211yescharacteristic
24C2×#13133152C2211yescharacteristic
25#16810629C2211yescharacteristic
26#16810628C2211yescharacteristic
27#16810627C2211yescharacteristic
28#16810627C2211yescharacteristic
29C2×#13133134C2211yescharacteristic
30C2×#13133134C2211yescharacteristic
31C2×#13133133C2211yescharacteristic
32C2×#13133133C2211yescharacteristic
33C2×#13133133C2211yescharacteristic
34#16810581C2211yescharacteristic
35#16810579C2211yescharacteristic
36#16810579C2211yescharacteristic
37#16810395C2211yescharacteristic
38#16810394C2211yescharacteristic
39#16810393C2211yescharacteristic
40#16810393C2211yescharacteristic
41#16810381C2211yescharacteristic
42#16810379C2211yescharacteristic
43#16810379C2211yescharacteristic
44C53×#430C2211yescharacteristic
45C53×#418C2211yescharacteristic
46C53×#418C2211yescharacteristic
47C2×#100×C53C2211yescharacteristic
48C53×#344C2211yescharacteristic
49C53×#344C2211yescharacteristic
50C53×H(3,ℤ/4)C2211yescharacteristic
51C22×D212C2311yescharacteristic
52C2×#11719922—21no—
53C2×#11719922—21no—
54C2×#11719922—21no—
55C2×#11719922—21no—
56C2×#11719922—21no—
57C2×#11719922—21no—
58C2×#11719917C2311yescharacteristic
59C2×#11719917C2311yescharacteristic
60C2×#11719917C2311yescharacteristic
61C2×#11719917C2311yescharacteristic
62C2×#11719917C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64#13133176C2311yescharacteristic
65#13133153—41no—
66#13133153—41no—
67#13133152—21no—
68#13133152—21no—
69#13133152—21no—
70#13133152—21no—
71#13133152—21no—
72#13133152—21no—
73#13133152—41no—
74#13133152—41no—
75#13133152—41no—
76#13133139—41no—
77#13133138—41no—
78#13133137—41no—
79#13133137—41no—
80#13133134—41no—
81#13133134—41no—
82#13133133—41no—
83#13133133—41no—
84#13133133—41no—
85#13133128—21no—
86#13133128—21no—
87#13133127—21no—
88C4×#88322—21no—
89C4×#88322—21no—
90C8×Dic53—21no—
91C22×D4×C53C2311yescharacteristic
92C2×M16×C53C2311yescharacteristic
93C2×M16×C53C2311yescharacteristic
94C2×M16×C53C2311yescharacteristic
95C2×M16×C53C2311yescharacteristic
96#117×C53C2311yescharacteristic
97#117×C53C2311yescharacteristic
98#108×C53—41no—
99#102×C53—41no—
100#102×C53—41no—
101#100×C53—41no—
102C24×C53—21no—
103C23×D53—21no—
104C2×D4×C53—21no—
105C2×D4×C53—21no—
106C2×D4×C53—21no—
107C2×D4×C53—21no—
108C2×D4×C53—21no—
109C2×D4×C53—21no—
110C22×C4×C53C2411yescharacteristic
111C2×D212—21no—
112C2×D212—21no—
113C2×D212—21no—
114C2×D212—21no—
115C2×D212—21no—
116C2×D212—21no—
117M16×C53—21no—
118M16×C53—21no—
119M16×C53—21no—
120M16×C53—21no—
121M16×C53—21no—
122M16×C53—21no—
123M16×C53—21no—
124M16×C53—21no—
125C2×C8×C53—21no—
126C2×C8×C53—21no—
127C2×C8×C53—21no—
128C2×C8×C53—21no—
129C4⋊C4×C53—21no—
130C4⋊C4×C53—21no—
131C22⋊C4×C53—21no—
132C22⋊C4×C53—21no—
133C42×C53—21no—
134C42×C53—21no—
135C22⋊Dic53—21no—
136#11719922—41no—
137#11719922—41no—
138#11719922—41no—
139#11719922—41no—
140C4⋊Dic53—21no—
141C4×Dic53—21no—
142#11719917—21no—
143#11719917—21no—
144#11719917—21no—
145#11719917—21no—
146#11719917—21no—
147#11719917—21no—
148#11719917—21no—
149#11719917—21no—
150#11719917—21no—
151#11719917—21no—
152C2×#88322—21no—
153C2×#88322—21no—
154C2×#88322—21no—
155C2×#88322—21no—
156C2×#88322—21no—
157C2×#88322#4111yescharacteristic
158C2×#88322#4111yescharacteristic
159C2×#88322#4111yescharacteristic
160C2×#88322#4111yescharacteristic
161C2×#88322#4111yescharacteristic
162C2×#88322#4111yescharacteristic
163C23×C53—41no—
164C23×C53—41no—
165C23×C53—41no—
166C23×C53—161no—
167C23×C532−1+411yescharacteristic
168C22×D53—41no—
169C22×D53—41no—
170C22×D53—41no—
171C22×D53—161no—
172D4×C53—41no—
173D4×C53—41no—
174D4×C53—41no—
175D4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53—41no—
178C2×C4×C53C2×#4111yescharacteristic
179C2×C4×C53C2×#4111yescharacteristic
180C2×C4×C53C2×#4111yescharacteristic
181C2×C4×C53H(5,𝔽2)11yescharacteristic
182C2×C4×C53H(5,𝔽2)11yescharacteristic
183C2×C4×C53H(5,𝔽2)11yescharacteristic
184C2×Dic53—41no—
185D212—41no—
186D212—41no—
187D212—41no—
188D212—41no—
189C8×C53—41no—
190C8×C53—41no—
191C8×C53—41no—
192C8×C53—41no—
193#88322—41no—
194#88322—41no—
195#88322—41no—
196#88322—41no—
197#88322—41no—
198#88322#12611yescharacteristic
199#88322#12611yescharacteristic
200#88322#12611yescharacteristic
201#88322#12611yescharacteristic
202#30237—531nomaximal, Sylow
203C22×C53—41no—
204C22×C53—81no—
205C22×C53—161no—
206C22×C53—161no—
207C22×C53—161no—
208C22×C53#54211yescharacteristic
209C22×C53#54211yescharacteristic
210C22×C53#54211yescharacteristic
211C2×D53—81no—
212C2×D53—161no—
213C2×D53—161no—
214C2×D53—161no—
215C4×C53—81no—
216C4×C53—81no—
217C4×C53C2×#12611yescharacteristic
218C4×C53#54011yescharacteristic
219C4×C53#54011yescharacteristic
220C4×C53#54011yescharacteristic
221Dic53—81no—
222#2081—531no—
223#2081—531no—
224#2081—531no—
225#2081—531no—
226#2081—531no—
227#2081D5311yescharacteristic, complemented
228#2069—531no—
229#2069—531no—
230#1978—531no—
231#1978—531no—
232#1978—531no—
233#1953—531no—
234#1953—531no—
235#1953—531no—
236#1747—531no—
237C2×C53—21no—
238C2×C53—21no—
239C2×C53—21no—
240C2×C53—161no—
241C2×C53#264411yescharacteristic
242D53—161no—
243C22×D8—531no—
244#430—531no—
245#430—531no—
246#430—531no—
247#430—531no—
248#430—531no—
249#430C2×D5311yescharacteristic
250C2×#108—531no—
251C2×#108—531no—
252C2×#108—531no—
253#418—531no—
254#418—531no—
255#418—531no—
256#418—531no—
257#418C2×D5311yescharacteristic
258#418C2×D5311yescharacteristic
259C2×#100—531no—
260C2×#100—531no—
261C2×#100—531no—
262C2×#100—531no—
263C2×#100—531no—
264C2×#100C2×D5311yescharacteristic
265#405—531no—
266#405—531no—
267#405—531no—
268#344—531no—
269#344—531no—
270#344—531no—
271#344—531no—
272#344C2×D5311yescharacteristic
273#344C2×D5311yescharacteristic
274#338—531no—
275H(3,ℤ/4)—531no—
276H(3,ℤ/4)—531no—
277H(3,ℤ/4)C2×D5311yescharacteristic
278C53#3023711yescharacteristic, complemented, Sylow
279C22×D4—531no—
280C22×D4C22×D5311yescharacteristic
281C2×D8—1061no—
282C2×D8—1061no—
283C2×D8—1061no—
284C2×D8—1061no—
285C2×D8—1061no—
286C2×D8—1061no—
287C2×M16—531no—
288C2×M16—531no—
289C2×M16—531no—
290C2×M16—531no—
291C2×M16—531no—
292C2×M16C22×D5311yescharacteristic
293C2×M16C22×D5311yescharacteristic
294C2×M16C22×D5311yescharacteristic
295C2×M16C22×D5311yescharacteristic
296C22×C8—531no—
297#117—531no—
298#117C22×D5311yescharacteristic
299#117C22×D5311yescharacteristic
300#108—41no—
301#108—1061no—
302#108—1061no—
303#108—1061no—
304#108—1061no—
305#108—1061no—
306#108—1061no—
307#108—2121no—
308#108—2121no—
309#108—2121no—
310#108—2121no—
311#108—2121no—
312#102—41no—
313#102—41no—
314#102—2121no—
315#102—2121no—
316#102—2121no—
317#102—2121no—
318#100—41no—
319#100—2121no—
320#100—2121no—
321#100—2121no—
322#100—2121no—
323#100—2121no—
324#97—1061no—
325#97—1061no—
326#97—1061no—
327C4×C8—1061no—
328C4×C8—1061no—
329C4×C8—1061no—
330C24—21no—
331C24—1061no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—21no—
337C2×D4—21no—
338C2×D4—1061no—
339C2×D4—1061no—
340C2×D4—1061no—
341C2×D4—1061no—
342C2×D4—1061no—
343C2×D4—1061no—
344C22×C4C23×D5311yescharacteristic
345D8—2121no—
346D8—2121no—
347D8—2121no—
348D8—2121no—
349M16—21no—
350M16—21no—
351M16—21no—
352M16—21no—
353M16—21no—
354M16—21no—
355M16—21no—
356M16—21no—
357M16—1061no—
358M16—1061no—
359M16—1061no—
360M16—1061no—
361M16—1061no—
362M16—1061no—
363M16—1061no—
364M16—1061no—
365M16—1061no—
366M16—1061no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—531no—
372C2×C8—531no—
373C2×C8—531no—
374C2×C8—531no—
375C2×C8—531no—
376C2×C8—531no—
377C2×C8—1061no—
378C2×C8—1061no—
379C2×C8—1061no—
380C2×C8—1061no—
381C2×C8—1061no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—1061no—
385C22⋊C4—21no—
386C22⋊C4—21no—
387C22⋊C4—1061no—
388C42—21no—
389C42—21no—
390C42—1061no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—161no—
395C23—8481no—
396C23—2121no—
397C23—2121no—
398C23—2121no—
399C23#1313326411yescharacteristic
400D4—41no—
401D4—41no—
402D4—41no—
403D4—41no—
404D4—2121no—
405D4—2121no—
406D4—2121no—
407D4—2121no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—2121no—
411C2×C4#41×D5311yescharacteristic
412C2×C4#41×D5311yescharacteristic
413C2×C4#41×D5311yescharacteristic
414C2×C4#1313326211yescharacteristic
415C2×C4#1313326211yescharacteristic
416C2×C4#1313326311yescharacteristic
417C8—41no—
418C8—41no—
419C8—41no—
420C8—41no—
421C8—531no—
422C8—531no—
423C8—531no—
424C8—531no—
425C8—2121no—
426C8—2121no—
427C8—2121no—
428C8—2121no—
429C8—2121no—
430C22—41no—
431C22—81no—
432C22—161no—
433C22—161no—
434C22—161no—
435C22—8481no—
436C22—8481no—
437C22—8481no—
438C22—4241no—
439C22#1681123711yescharacteristic
440C22#1681123911yescharacteristic
441C22#1681124111yescharacteristic
442C4—81no—
443C4—81no—
444C4—4241no—
445C4#126×D5311yescharacteristic
446C4#1681122311yescharacteristic
447C4#1681122511yescharacteristic
448C4#1681122511yescharacteristic
449C2—21no—
450C2—21no—
451C2—21no—
452C2—161no—
453C2—8481no—
454C2#2028067411yescharacteristic
subgroupG/H
centerC2#20280674
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2081C2
socleC2×C53#2644
prime radicalC22×#11719922C22
hypercenter#2081D53
hypocenterC53#30237
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ #2081

Complements

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

NH ≅ G/Ntype
1. #20276178C2semidirect
2. #20276177C2semidirect
3. #20276176C2semidirect
4. #20276174C2semidirect
5. #20276172C2semidirect
6. #20276113C2semidirect
7. #20276112C2semidirect
11. #20275504C2semidirect
12. #20275504C2semidirect
13. #20275502C2semidirect
14. #20274469C2semidirect
15. C53×#2081C2semidirect
227. #2081D53semidirect
278. C53#30237semidirect

Maximal subgroups

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

subgroupindex
#202761782
#202761772
#202761762
#202761742
#202761722
#202761132
#202761122
#202756242
#202756222
#202755042
#202755022
#202744692
C53×#20812
#3023753