Finite Group Databasebeta

← #27530920

Subgroups — #27530920

Order 13568.

528 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20276293C211yesmaximal, characteristic, complemented
2#20276183C211yesmaximal, characteristic, complemented
3#20276179C211yesmaximal, characteristic, complemented
4#20276116C211yesmaximal, characteristic, complemented
5#20275992C211yesmaximal, characteristic
6#20275560C211yesmaximal, characteristic, complemented
7#20275381C211yesmaximal, characteristic
8#20275268C211yesmaximal, characteristic
9#20275253C211yesmaximal, characteristic
10#20275190C211yesmaximal, characteristic
11#20275109C211yesmaximal, characteristic, complemented
12#20274957C211yesmaximal, characteristic
13#20274861C211yesmaximal, characteristic, complemented
14#20274827C211yesmaximal, characteristic, complemented
15C53×#2005C211yesmaximal, characteristic
16#16811390C2211yescharacteristic
17#16811101C2211yescharacteristic
18#16810672C2211yescharacteristic
19#16810671C2211yescharacteristic
20#16810671C2211yescharacteristic
21#16810647C2211yescharacteristic
22#16810640C2211yescharacteristic
23#16810640C2211yescharacteristic
24#16810632C2211yescharacteristic
25#16810632C2211yescharacteristic
26#16810607C2211yescharacteristic
27C2×#13133136C2211yescharacteristic
28C2×#13133136C2211yescharacteristic
29C2×#13133136C2211yescharacteristic
30C2×#13133136C2211yescharacteristic
31C2×#13133136C2211yescharacteristic
32C4×#11719917C2211yescharacteristic
33#16810395C2211yescharacteristic
34#16810395C2211yescharacteristic
35#16810394C2211yescharacteristic
36#16810394C2211yescharacteristic
37#16810393C2211yescharacteristic
38#16810393C2211yescharacteristic
39#16810383C2211yescharacteristic
40#16810383C2211yescharacteristic
41#16810381C2211yescharacteristic
42#16810379C2211yescharacteristic
43#16810379C2211yescharacteristic
44C2×#137×C53C2211yescharacteristic
45C53×#536C2211yescharacteristic
46C2×C4⋊Q8×C53C2211yescharacteristic
47C2×#104×C53C2211yescharacteristic
48C53×#419C2211yescharacteristic
49C2×#101×C53C2211yescharacteristic
50C53×#339C2211yescharacteristic
51#13133224—41no—
52#13133224—41no—
53C2×#11719917—41no—
54C2×#11719917C2311yescharacteristic
55C2×#11719917C2311yescharacteristic
56C2×#11719917C2311yescharacteristic
57C2×#11719917C2311yescharacteristic
58C2×#11719917C2311yescharacteristic
59C2×#11719917C2311yescharacteristic
60Q8×Dic53—41no—
61Q8×Dic53—41no—
62#13133176—22no—
63#13133176C2311yescharacteristic
64#13133176C2311yescharacteristic
65#13133153—41no—
66#13133152—41no—
67#13133152—41no—
68C4⋊#88322—41no—
69#13133144—41no—
70#13133144—41no—
71#13133141—41no—
72#13133141—41no—
73#13133136—21no—
74#13133136—41no—
75#13133136—41no—
76#13133136—41no—
77#13133136—41no—
78#13133136—41no—
79#13133134—21no—
80#13133128—21no—
81C4×#88322—21no—
82C2×#41×C53C2311yescharacteristic
83C22×Q8×C53C2311yescharacteristic
84C22×Q8×C53C2311yescharacteristic
85#137×C53—41no—
86#137×C53—41no—
87C2×Q16×C53—21no—
88C2×SD16×C53—21no—
89C2×M16×C53C2311yescharacteristic
90C2×M16×C53C2311yescharacteristic
91C4⋊Q8×C53—21no—
92C4⋊Q8×C53—21no—
93C4⋊Q8×C53—21no—
94C4⋊Q8×C53—41no—
95#124×C53—21no—
96#123×C53—22no—
97C22⋊Q8×C53—41no—
98#117×C53C2311yescharacteristic
99C2×C4⋊C4×C53—22no—
100C2×C42×C53C2311yescharacteristic
101#104×C53—21no—
102#104×C53—21no—
103#103×C53—41no—
104#101×C53—41no—
105#41×C53—21no—
106#41×C53—21no—
107#41×C53—21no—
108#41×C53—21no—
109C2×Q8×C53—21no—
110C2×Q8×C53—21no—
111C2×Q8×C53—21no—
112C2×Q8×C53—21no—
113C2×Q8×C53—21no—
114C2×Q8×C53—21no—
115C2×Q8×C53—21no—
116C2×Q8×C53—21no—
117C2×Q8×C53—21no—
118C2×Q8×C53—21no—
119C2×Q8×C53—21no—
120C2×Q8×C53—21no—
121C2×Q8×C53#4111yescharacteristic
122C2×D4×C53—21no—
123C2×D4×C53#4111yescharacteristic
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53#4112yes—
128C22×C4×C53C2411yescharacteristic
129Q16×C53—42no—
130SD16×C53—42no—
131M16×C53—21no—
132M16×C53—21no—
133M16×C53—21no—
134M16×C53—21no—
135C2×C8×C53—21no—
136C2×C8×C53—21no—
137C4⋊C4×C53—21no—
138C4⋊C4×C53—42no—
139C4⋊C4×C53—42no—
140C4⋊C4×C53—42no—
141C22⋊C4×C53—21no—
142C22⋊C4×C53—42no—
143C42×C53—21no—
144C42×C53—21no—
145C42×C53#4111yescharacteristic
146C42×C53#4111yescharacteristic
147C22⋊Dic53—21no—
148C22⋊Dic53—21no—
149C4⋊Dic53—21no—
150C4⋊Dic53—21no—
151C4⋊Dic53—42no—
152C4×Dic53—21no—
153C4×Dic53—21no—
154C4×Dic53—42no—
155#11719917—21no—
156#11719917—21no—
157#11719917—21no—
158#11719917—21no—
159#11719917—21no—
160#11719917—21no—
161#11719917—21no—
162#11719917—21no—
163#11719917—21no—
164#11719917—21no—
165#11719917—21no—
166#11719917—21no—
167#11719917—42no—
168C2×#88322—21no—
169C2×#88322—21no—
170C2×#88322—21no—
171C2×#88322—21no—
172C2×#88322—21no—
173C2×#88322—21no—
174C2×#88322—42no—
175C23×C53—21no—
176C23×C53C2×#4111yescharacteristic
177Q8×C53—21no—
178Q8×C53—21no—
179Q8×C53—41no—
180Q8×C53—41no—
181Q8×C53—41no—
182Q8×C53—41no—
183Q8×C53—41no—
184Q8×C53—41no—
185Q8×C53—41no—
186Q8×C53—41no—
187D4×C53—21no—
188D4×C53—21no—
189D4×C53—41no—
190D4×C53—41no—
191C2×C4×C53—21no—
192C2×C4×C53—22no—
193C2×C4×C53—21no—
194C2×C4×C53—21no—
195C2×C4×C53—22no—
196C2×C4×C53—22no—
197C2×C4×C53—42no—
198C2×C4×C53—41no—
199C2×C4×C53—41no—
200C2×C4×C53—41no—
201C2×C4×C53—41no—
202C2×C4×C53—41no—
203C2×C4×C53—41no—
204C2×C4×C53C2×#4111yescharacteristic
205C2×C4×C53C2×#4111yescharacteristic
206C2×C4×C53H(5,𝔽2)11yescharacteristic
207C2×C4×C53H(5,𝔽2)11yescharacteristic
208C2×C4×C53H(5,𝔽2)11yescharacteristic
209C2×C4×C53H(5,𝔽2)11yescharacteristic
210C2×Dic53—41no—
211C2×Dic53—41no—
212C8×C53—41no—
213C8×C53—41no—
214#88322—41no—
215#88322—41no—
216#88322—41no—
217#88322—41no—
218#88322—41no—
219#88322—41no—
220#32697—531nomaximal, Sylow
221C22×C53—41no—
222C22×C53—41no—
223C22×C53—81no—
224C22×C53#53811yescharacteristic
225C22×C53#55111yescharacteristic
226C22×C53#55111yescharacteristic
227C4×C53—22no—
228C4×C53—41no—
229C4×C53—42no—
230C4×C53—41no—
231C4×C53—81no—
232C4×C53—81no—
233C4×C53#53811yescharacteristic
234C4×C53#54011yescharacteristic
235C4×C53#55311yescharacteristic
236C4×C53#56011yescharacteristic
237Dic53—81no—
238Dic53—81no—
239#2102—531no—
240#2082—531no—
241#2082—531no—
242#2070—531no—
243#2049—531no—
244#2005D5311yescharacteristic
245#1966—531no—
246#1932—531no—
247#1915—531no—
248#1912—531no—
249#1902—531no—
250#1886—531no—
251#1861—531no—
252#1841—531no—
253#1830—531no—
254C2×C53—21no—
255C2×C53—21no—
256C2×C53—21no—
257C2×C53—81no—
258C2×C53#274911yescharacteristic
259C2×#137C2×D5311yescharacteristic
260#568—531no—
261#536C2×D5311yescharacteristic
262C2×C4⋊Q8C2×D5311yescharacteristic
263#519—531no—
264#430—531no—
265#430—531no—
266#430—531no—
267#423—531no—
268#421—531no—
269#421—531no—
270C2×#104C2×D5311yescharacteristic
271#419—531no—
272#419—531no—
273#419C2×D5311yescharacteristic
274#413—531no—
275C2×#101—531no—
276C2×#101—531no—
277C2×#101—531no—
278C2×#101—531no—
279C2×#101—531no—
280C2×#101C2×D5311yescharacteristic
281C4×M16—531no—
282#344—531no—
283#344—531no—
284#344—531no—
285#344—531no—
286#344—531no—
287#344—531no—
288#339—531no—
289#339—531no—
290#339C2×D5311yescharacteristic
291#338—531no—
292H(3,ℤ/4)—531no—
293H(3,ℤ/4)—531no—
294C53#3269711yescharacteristic, complemented, Sylow
295C2×#41C22×D5311yescharacteristic
296C22×Q8C22×D5311yescharacteristic
297C22×Q8C22×D5311yescharacteristic
298#137—41no—
299#137—41no—
300C2×Q16—21no—
301C2×SD16—21no—
302#131—2121no—
303#131—2121no—
304C2×M16—531no—
305C2×M16—531no—
306C2×M16—531no—
307C2×M16—531no—
308C2×M16—531no—
309C2×M16—531no—
310C2×M16—2121no—
311C2×M16C22×D5311yescharacteristic
312C2×M16C22×D5311yescharacteristic
313C4⋊Q8—21no—
314C4⋊Q8—21no—
315C4⋊Q8—21no—
316C4⋊Q8—41no—
317#124—21no—
318#123—22no—
319C22⋊Q8—41no—
320C4×Q8—2121no—
321C4×Q8—2121no—
322#117—531no—
323#117—531no—
324#117—1062no—
325#117C22×D5311yescharacteristic
326C2×C4⋊C4—22no—
327C2×C42C22×D5311yescharacteristic
328#108—2121no—
329#108—2121no—
330#108—2121no—
331C4⋊C8—2121no—
332#104—21no—
333#104—21no—
334#104—2121no—
335#104—2121no—
336#103—41no—
337#103—2121no—
338#103—2121no—
339#101—41no—
340#101—1061no—
341#101—2121no—
342#101—2121no—
343#101—2121no—
344#101—2121no—
345#101—2121no—
346#100—1061no—
347#97—1061no—
348C4×C8—1061no—
349#41—21no—
350#41—21no—
351#41—21no—
352#41—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8#1171994711yescharacteristic
366C2×D4—21no—
367C2×D4#1171994711yescharacteristic
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4#1171994712yes—
372C22×C4C23×D5311yescharacteristic
373Q16—42no—
374SD16—42no—
375M16—21no—
376M16—21no—
377M16—21no—
378M16—21no—
379M16—1061no—
380M16—1061no—
381M16—1061no—
382M16—1061no—
383M16—1061no—
384M16—1061no—
385M16—1061no—
386M16—1061no—
387M16—1061no—
388M16—1061no—
389M16—1061no—
390M16—1061no—
391M16—2122no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—1061no—
395C2×C8—1061no—
396C2×C8—1061no—
397C2×C8—1061no—
398C2×C8—1061no—
399C2×C8—1061no—
400C2×C8—2122no—
401C4⋊C4—21no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—1061no—
406C4⋊C4—1061no—
407C4⋊C4—2122no—
408C22⋊C4—21no—
409C22⋊C4—42no—
410C22⋊C4—1061no—
411C22⋊C4—1061no—
412C42—21no—
413C42—21no—
414C42—1061no—
415C42—1061no—
416C42—2122no—
417C42#1171994711yescharacteristic
418C42#1171994711yescharacteristic
419C23—21no—
420C23C2×#1171994711yescharacteristic
421Q8—21no—
422Q8—21no—
423Q8—41no—
424Q8—41no—
425Q8—41no—
426Q8—41no—
427Q8—41no—
428Q8—41no—
429Q8—41no—
430Q8—41no—
431D4—21no—
432D4—21no—
433D4—41no—
434D4—41no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—2121no—
449C2×C4—2121no—
450C2×C4C2×#1171994711yescharacteristic
451C2×C4C2×#1171994711yescharacteristic
452C2×C4#1313326211yescharacteristic
453C2×C4#1313326211yescharacteristic
454C2×C4#1313326211yescharacteristic
455C2×C4#1313326311yescharacteristic
456C8—41no—
457C8—41no—
458C8—2121no—
459C8—2121no—
460C8—2121no—
461C8—2121no—
462C8—2121no—
463C8—2121no—
464C22—41no—
465C22—41no—
466C22—81no—
467C22#1681121011yescharacteristic
468C22#1681130411yescharacteristic
469C22#1681130411yescharacteristic
470C4—22no—
471C4—41no—
472C4—42no—
473C4—41no—
474C4—81no—
475C4—81no—
476C4—4241no—
477C4—4241no—
478C4#1681120811yescharacteristic
479C4#1681122711yescharacteristic
480C4#1681132511yescharacteristic
481C4#1681136111yescharacteristic
482C2—21no—
483C2—21no—
484C2—21no—
485C2—81no—
486C2#2028194011yescharacteristic
subgroupG/H
centerC2#20281940
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2005C2
socleC2×C53#2749
prime radicalC2×#41×C53C23
hypercenter#2005D53
hypocenterC53#32697
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ #2005

Complements

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

NH ≅ G/Ntype
1. #20276293C2semidirect
2. #20276183C2semidirect
3. #20276179C2semidirect
4. #20276116C2semidirect
6. #20275560C2semidirect
11. #20275109C2semidirect
13. #20274861C2semidirect
14. #20274827C2semidirect
294. C53#32697semidirect

Maximal subgroups

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

subgroupindex
#202762932
#202761832
#202761792
#202761162
#202759922
#202755602
#202753812
#202752682
#202752532
#202751902
#202751092
#202749572
#202748612
#202748272
C53×#20052
#3269753