Finite Group Databasebeta

← #27280976

Subgroups — #27280976

Order 13568.

474 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284524C211yesmaximal, characteristic
2#20283833C211yesmaximal, characteristic
3#20274736C211yesmaximal, characteristic
4#20273400C211yesmaximal, characteristic, complemented
5#20272618C211yesmaximal, characteristic, complemented
6#20272590C211yesmaximal, characteristic, complemented
7C53×#1981C211yesmaximal, characteristic, complemented
8C4×#11719945C2211yescharacteristic
9#16810915C412yescomplemented
10#16810910C412yescomplemented
11#16810905—21no—
12#16810900—21no—
13#16810750—21no—
14#16810700—21no—
15#16810694—22no—
16#16810679—21no—
17#16810676C2211yescharacteristic
18#16810663—21no—
19#16810647C2211yescharacteristic
20#16810647C2211yescharacteristic
21#16810645—21no—
22C2×#13133145—21no—
23#16810622—21no—
24#16810383—21no—
25#16810369—21no—
26#16810369—21no—
27C2×C4⋊Q8×C53C2211yescharacteristic
28C2×C4⋊C8×C53C2211yescharacteristic
29C2×#103×C53—22no—
30C4×C4⋊C4×C53C2211yescharacteristic
31C2×#11719945D411yescharacteristic
32C2×#11719917—21no—
33C2×#11719917—21no—
34C2×#11719912—22no—
35C4×Dic106C2×C411yescharacteristic
36C4×Dic106D411yescharacteristic
37C4×D212C2×C411yescharacteristic
38C4×D212D411yescharacteristic
39C4×C22⋊D53—21no—
40C4×C22⋊D53—21no—
41#13133176D411yescharacteristic
42#13133174—21no—
43C42×D53—21no—
44#13133150—22no—
45#13133148—22no—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48C4⋊#88322—21no—
49C4⋊#88322C2×C412yes—
50C4⋊#88322D411yescharacteristic
51C4⋊#88322D411yescharacteristic
52#13133146—22no—
53#13133145—42no—
54#13133140—22no—
55#13133137—22no—
56C22⋊#88322—21no—
57#13133129—22no—
58#13133127—22no—
59C4×#88322—21no—
60C22×Q8×C53—21no—
61C22×C8×C53D412yes—
62C4⋊Q8×C53—21no—
63C4⋊Q8×C53—21no—
64C4⋊Q8×C53—21no—
65C4⋊Q8×C53C2×C412yes—
66C2×C4⋊C4×C53—21no—
67C2×C4⋊C4×C53—21no—
68C2×C4⋊C4×C53—21no—
69C2×C42×C53—21no—
70C2×C42×C53C2311yescharacteristic
71C4⋊C8×C53—21no—
72C4⋊C8×C53—21no—
73#103×C53—44no—
74#95×C53—21no—
75C2×Q8×C53—22no—
76C2×Q8×C53—42no—
77C2×Q8×C53—42no—
78C2×Q8×C53—41no—
79C22×C4×C53—41no—
80C22×C4×C53—41no—
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53#4111yescharacteristic
84C2×C22⋊D53—21no—
85#11719945—21no—
86#11719945—21no—
87#11719945—21no—
88#11719945—21no—
89C2×D212C22⋊C411yescharacteristic
90C2×C4×D53—21no—
91C2×C4×D53—21no—
92C2×C4×D53—21no—
93C2×Dic106C22⋊C411yescharacteristic
94C2×C8×C53—22no—
95C2×C8×C53—42no—
96C4⋊C4×C53—22no—
97C4⋊C4×C53—22no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—41no—
100C4⋊C4×C53—41no—
101C4⋊C4×C53—41no—
102C42×C53—41no—
103C42×C53—41no—
104C42×C53C22×C411yescharacteristic
105C42×C53C2×D411yescharacteristic
106C42×C53C2×D411yescharacteristic
107C42×C53#4111yescharacteristic
108C22⋊Dic53—21no—
109#11719921—21no—
110#11719921—21no—
111C4⋊Dic53C22⋊C411yescharacteristic
112C4⋊Dic53C22⋊C411yescharacteristic
113#11719919—21no—
114#11719919—21no—
115C4×Dic53—21no—
116C4×Dic53—21no—
117C4×Dic53—21no—
118#11719917—41no—
119#11719917—41no—
120#11719917—41no—
121#11719917—41no—
122C2×#88322—22no—
123C2×#88322—21no—
124C2×#88322—21no—
125C2×#88322—41no—
126#11719912—82no—
127C23×C53#12111yescharacteristic
128C22×D53—21no—
129Q8×C53—44no—
130Q8×C53—81no—
131Q8×C53—81no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—41no—
135C2×C4×C53—42no—
136C2×C4×C53—42no—
137C2×C4×C53—81no—
138C2×C4×C53—82no—
139C2×C4×C53—81no—
140C2×C4×C53C2×C22⋊C411yescharacteristic
141C2×C4×C53C4×D411yescharacteristic
142C2×C4×C53C4×D411yescharacteristic
143C2×C4×C53#12011yescharacteristic
144C2×C4×C53#12111yescharacteristic
145C2×C4×C53#12311yescharacteristic
146C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C53#13711yescharacteristic
148C2×C4×C53#13711yescharacteristic
149C2×C4×C53#13711yescharacteristic
150C22⋊D53—41no—
151C22⋊D53—41no—
152C2×Dic53—21no—
153C2×Dic53—21no—
154C2×Dic53—21no—
155D212—21no—
156D212—21no—
157C4×D53—41no—
158C4×D53—41no—
159C4×D53—41no—
160C4×D53—41no—
161C4×D53—81no—
162Dic106—21no—
163Dic106—21no—
164C8×C53—42no—
165#88322—42no—
166#88322—41no—
167#88322—41no—
168#88322—81no—
169#9321—531nomaximal, Sylow
170C22×C53—21no—
171C22×C53—21no—
172C22×C53#38611yescharacteristic
173C22×C53#46811yescharacteristic
174C22×C53#47011yescharacteristic
175C2×D53—41no—
176C2×D53—41no—
177C2×D53—41no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—21no—
181C4×C53—82no—
182C4×C53—82no—
183C4×C53#41711yescharacteristic
184C4×C53#41911yescharacteristic
185C4×C53#45311yescharacteristic
186C4×C53#45411yescharacteristic
187C4×C53#46811yescharacteristic
188C4×C53#47011yescharacteristic
189Dic53—41no—
190Dic53—41no—
191Dic53—41no—
192#3079—531no—
193#2989—531no—
194#1981D5311yescharacteristic, complemented
195#1806—531no—
196#1549—531no—
197#1303—531no—
198#1292—531no—
199C2×C53—21no—
200C2×C53—21no—
201C2×C53#188311yescharacteristic
202C2×C53#189411yescharacteristic
203C2×C53#190211yescharacteristic
204D53—81no—
205C2×C4⋊Q8C2×D5311yescharacteristic
206C4×#41—531no—
207#477—532no—
208#476—532no—
209#475—1061no—
210#474—1061no—
211#446—1061no—
212#436—1061no—
213#435—1062no—
214#432—1061no—
215#431—531no—
216#428—1061no—
217#423—531no—
218#423—531no—
219#423—1061no—
220C2×C4⋊C8C2×D5311yescharacteristic
221C2×#104—1061no—
222#417—1061no—
223C2×#103—22no—
224C4×C4⋊C4C2×D5311yescharacteristic
225#339—1061no—
226#332—1061no—
227#332—1061no—
228C53#932111yescharacteristic, complemented, Sylow
229C2×#41—531no—
230C22×Q8—21no—
231C2×M16—1062no—
232C2×M16—1061no—
233C2×M16—1061no—
234C22×C8C22⋊D5312yes—
235C4⋊Q8—21no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8C4×D5312yes—
239C4×Q8—531no—
240C4×Q8—531no—
241C4×D4—531no—
242C4×D4—531no—
243C4×D4—1061no—
244C4×D4—1061no—
245#117—531no—
246#117—1061no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C42—21no—
251C2×C42—1061no—
252C2×C42C22×D5311yescharacteristic
253#107—1062no—
254#106—1062no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—532no—
258C4⋊C8—531no—
259C4⋊C8—531no—
260C4⋊C8—1062no—
261C4⋊C8—1061no—
262C4⋊C8—1061no—
263C4⋊C8—1061no—
264#104—2122no—
265#103—44no—
266#103—1062no—
267#102—1062no—
268C22⋊C8—1062no—
269C22⋊C8—1061no—
270#97—1062no—
271C4×C8—1061no—
272#95—21no—
273#41—1061no—
274#41—1061no—
275#41—1061no—
276#41—1061no—
277C2×Q8—22no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—41no—
281C2×Q8—531no—
282C2×D4—531no—
283C2×D4—1061no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—1061no—
287C22×C4—1061no—
288C22×C4—1061no—
289C22×C4D4×D5311yescharacteristic
290C22×C4#1171994711yescharacteristic
291C22×C4C2×C22⋊D5311yescharacteristic
292M16—4242no—
293M16—2121no—
294M16—2121no—
295M16—2121no—
296M16—2121no—
297C2×C8—22no—
298C2×C8—42no—
299C2×C8—1062no—
300C2×C8—1061no—
301C2×C8—1061no—
302C2×C8—2121no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—531no—
310C4⋊C4—531no—
311C4⋊C4—1061no—
312C4⋊C4—1061no—
313C22⋊C4—1061no—
314C22⋊C4—1061no—
315C22⋊C4—1061no—
316C42—41no—
317C42—41no—
318C42—1061no—
319C42—1061no—
320C42—1061no—
321C42C2×C4×D5311yescharacteristic
322C42#1171994511yescharacteristic
323C42D4×D5311yescharacteristic
324C42D4×D5311yescharacteristic
325C23—1061no—
326C23#1313318811yescharacteristic
327Q8—44no—
328Q8—81no—
329Q8—81no—
330Q8—1061no—
331Q8—1061no—
332D4—1061no—
333D4—1061no—
334D4—2121no—
335D4—2121no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—81no—
342C2×C4—82no—
343C2×C4—81no—
344C2×C4—1061no—
345C2×C4—1061no—
346C2×C4—1061no—
347C2×C4—4241no—
348C2×C4—2121no—
349C2×C4—2121no—
350C2×C4—2121no—
351C2×C4—2121no—
352C2×C4C22⋊C4×D5311yescharacteristic
353C2×C4#1313317711yescharacteristic
354C2×C4C4×C22⋊D5311yescharacteristic
355C2×C4#1313318511yescharacteristic
356C2×C4#1313319011yescharacteristic
357C2×C4#1313319911yescharacteristic
358C2×C4#1313324411yescharacteristic
359C2×C4#1313324611yescharacteristic
360C2×C4#1313324911yescharacteristic
361C2×C4#1313324911yescharacteristic
362C8—42no—
363C8—4241no—
364C8—2121no—
365C8—2121no—
366C8—2122no—
367C22—21no—
368C22—21no—
369C22—2121no—
370C22—2121no—
371C22—2121no—
372C22#1681049911yescharacteristic
373C22#1681086611yescharacteristic
374C22#1681087811yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—82no—
379C4—82no—
380C4—2121no—
381C4—2121no—
382C4—2121no—
383C4#1681062311yescharacteristic
384C4#1681063011yescharacteristic
385C4#1681079011yescharacteristic
386C4#1681079511yescharacteristic
387C4#1681087211yescharacteristic
388C4#1681088011yescharacteristic
389C2—21no—
390C2—21no—
391C2—4241no—
392C2#2027509411yescharacteristic
393C2#2027514511yescharacteristic
394C2#2027518511yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1981C2
socleC22×C53#386
prime radicalC2×#11719945D4
cosocleC2×C42×C53C23
hypercenter#1981D53
hypocenterC53#9321
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1981

Complements

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

NH ≅ G/Ntype
4. #20273400C2semidirect
5. #20272618C2semidirect
6. #20272590C2semidirect
7. C53×#1981C2semidirect
9. #16810915C4semidirect
10. #16810910C4semidirect
194. #1981D53semidirect
228. C53#9321semidirect

Maximal subgroups

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

subgroupindex
#202845242
#202838332
#202747362
#202734002
#202726182
#202725902
C53×#19812
#932153