Finite Group Databasebeta

← #27280196

Subgroups — #27280196

Order 13568.

510 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284170C211yesmaximal, characteristic
2#20274735C211yesmaximal, characteristic, complemented
3#20273211C211yesmaximal, characteristic, complemented
4#20273167C211yesmaximal, characteristic, complemented
5#20272611C211yesmaximal, characteristic, complemented
6#20272555C211yesmaximal, characteristic, complemented
7C53×#2965C211yesmaximal, characteristic, complemented
8C2×#13133237—21no—
9#16811173C2211yescharacteristic
10#16810845—21no—
11#16810838—21no—
12#16810833C411yescharacteristic, complemented
13#16810828C411yescharacteristic, complemented
14#16810823C411yescharacteristic, complemented
15#16810818C411yescharacteristic, complemented
16#16810647C2211yescharacteristic
17C2×C4⋊#88322C2211yescharacteristic
18C2×#13133144—21no—
19#16810617—21no—
20C4×C4⋊Dic53C2211yescharacteristic
21#16810382—21no—
22#16810358—22no—
23#16810352—21no—
24#16810352—21no—
25C4×#41×C53C2211yescharacteristic
26C53×#446—21no—
27C53×#436—21no—
28C53×#435—22no—
29C53×#431—21no—
30C53×#431C2211yescharacteristic
31C4×M16×C53—21no—
32C2×C4×C8×C53C2211yescharacteristic
33C2×#11719945—21no—
34#13133237—42no—
35#13133237—42no—
36C2×#11719925—21no—
37C2×#11719923—21no—
38C2×#11719924—21no—
39C2×#11719922—21no—
40C2×#11719917—21no—
41C22×#88322—21no—
42#13133217C2×C411yescharacteristic
43#13133214C2×C411yescharacteristic
44#13133205—21no—
45#13133191—21no—
46#13133191—21no—
47C4×D212—21no—
48C4×D212—21no—
49C2×C4⋊Dic53—21no—
50C2×C4×Dic53—21no—
51C4⋊#88322—21no—
52C4⋊#88322—21no—
53C4⋊#88322—21no—
54C4⋊#88322C2×C411yescharacteristic
55C4⋊#88322C2×C411yescharacteristic
56#13133144—42no—
57#13133140—21no—
58#13133140—21no—
59#13133137—21no—
60#13133137—21no—
61#13133124—21no—
62C2×#41×C53D411yescharacteristic
63C2×M16×C53—22no—
64C22×C8×C53—21no—
65C4×Q8×C53C2×C411yescharacteristic
66C4×Q8×C53D411yescharacteristic
67C4×D4×C53—21no—
68C4×D4×C53—21no—
69C4×D4×C53C2×C411yescharacteristic
70C4×D4×C53D411yescharacteristic
71#117×C53—21no—
72#117×C53D411yescharacteristic
73C2×C42×C53—21no—
74C2×C42×C53C2311yescharacteristic
75C4⋊C8×C53—22no—
76C4⋊C8×C53—21no—
77C22⋊C8×C53—22no—
78C22⋊C8×C53D412yes—
79#97×C53—22no—
80C4×C8×C53—21no—
81C4×C8×C53—21no—
82C4×C8×C53D411yescharacteristic
83C4×C8×C53D411yescharacteristic
84#41×C53—21no—
85#41×C53—21no—
86#41×C53—21no—
87#41×C53—21no—
88C2×Q8×C53C22⋊C411yescharacteristic
89C2×D4×C53—21no—
90C2×D4×C53C22⋊C411yescharacteristic
91C22×C4×C53—21no—
92C22×C4×C53—21no—
93C22×C4×C53—21no—
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53#4111yescharacteristic
97C2×C22⋊D53—41no—
98C22×Dic53—41no—
99#11719945—41no—
100#11719945—41no—
101#11719945—41no—
102#11719945—41no—
103C2×D212—21no—
104C2×D212—21no—
105C2×D212—21no—
106C2×C4×D53—41no—
107C2×Dic106—21no—
108M16×C53—44no—
109C2×C8×C53—21no—
110C2×C8×C53—21no—
111C2×C8×C53—22no—
112C2×C8×C53—42no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53C22⋊C411yescharacteristic
116C4⋊C4×C53C22⋊C411yescharacteristic
117C22⋊C4×C53—21no—
118C22⋊C4×C53—21no—
119C22⋊C4×C53—21no—
120C42×C53—21no—
121C42×C53—21no—
122C42×C53—21no—
123C42×C53C22×C411yescharacteristic
124C42×C53C2×D411yescharacteristic
125C42×C53C2×D411yescharacteristic
126C42×C53#4111yescharacteristic
127#11719925—42no—
128#11719924—42no—
129#11719923—42no—
130#11719922—42no—
131#11719921—41no—
132#11719921—41no—
133C4⋊Dic53—22no—
134C4⋊Dic53—21no—
135C4⋊Dic53—21no—
136C4⋊Dic53—41no—
137C4×Dic53—41no—
138C4×Dic53—41no—
139#11719917—41no—
140#11719917—41no—
141C2×#88322—21no—
142C2×#88322—21no—
143C2×#88322—41no—
144C2×#88322—41no—
145C2×#88322—41no—
146C23×C53—21no—
147C23×C53#12111yescharacteristic
148C22×D53—41no—
149Q8×C53—21no—
150Q8×C53—21no—
151D4×C53—21no—
152D4×C53—21no—
153D4×C53—41no—
154D4×C53—41no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—41no—
161C2×C4×C53—42no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53C2×C22⋊C411yescharacteristic
167C2×C4×C53C4×D411yescharacteristic
168C2×C4×C53C4×D411yescharacteristic
169C2×C4×C53#12011yescharacteristic
170C2×C4×C53#12111yescharacteristic
171C2×C4×C53#12311yescharacteristic
172C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
173C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
174C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
175C2×C4×C53#13711yescharacteristic
176C22⋊D53—81no—
177C22⋊D53—81no—
178C2×Dic53—42no—
179C2×Dic53—41no—
180C2×Dic53—81no—
181C2×Dic53—81no—
182D212—41no—
183D212—41no—
184D212—41no—
185D212—41no—
186D212—81no—
187C4×D53—81no—
188C4×D53—81no—
189Dic106—41no—
190Dic106—41no—
191C8×C53—22no—
192C8×C53—41no—
193C8×C53—42no—
194#88322—41no—
195#88322—41no—
196#88322—81no—
197#9210—531nomaximal, Sylow
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—41no—
201C22×C53—41no—
202C22×C53—41no—
203C22×C53#38611yescharacteristic
204C22×C53#46811yescharacteristic
205C22×C53#46911yescharacteristic
206C2×D53—81no—
207C2×D53—81no—
208C2×D53—81no—
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—21no—
212C4×C53—41no—
213C4×C53—41no—
214C4×C53—41no—
215C4×C53#41711yescharacteristic
216C4×C53#41811yescharacteristic
217C4×C53#44311yescharacteristic
218C4×C53#44411yescharacteristic
219C4×C53#46811yescharacteristic
220C4×C53#46911yescharacteristic
221Dic53—82no—
222Dic53—81no—
223#3035—531no—
224#2965D5311yescharacteristic, complemented
225#1806—531no—
226#1502—531no—
227#1492—531no—
228#1301—531no—
229#1279—531no—
230C2×C53—21no—
231C2×C53—21no—
232C2×C53—81no—
233C2×C53#187611yescharacteristic
234C2×C53#189511yescharacteristic
235C2×C53#190611yescharacteristic
236D53—161no—
237C2×#135—1061no—
238#532—531no—
239C4×#41C2×D5311yescharacteristic
240#464—1061no—
241#463—1061no—
242#462—531no—
243#461—531no—
244#460—531no—
245#459—531no—
246#446—21no—
247#436—21no—
248#435—22no—
249#431—21no—
250#431C2×D5311yescharacteristic
251#423—531no—
252C2×C4⋊C8—531no—
253C2×#104—1061no—
254#416—1061no—
255C4×M16—21no—
256C2×C4×C8C2×D5311yescharacteristic
257C4×C4⋊C4—531no—
258#339—1061no—
259#327—1062no—
260D4⋊C8—1061no—
261D4⋊C8—1061no—
262C53#921011yescharacteristic, complemented, Sylow
263C2×#41—1061no—
264C2×#41D21211yescharacteristic
265#135—2122no—
266#135—2122no—
267C2×Q16—1061no—
268C2×SD16—1061no—
269C2×SD16—1061no—
270C2×D8—1061no—
271C2×M16—22no—
272C2×M16—1061no—
273C22×C8—21no—
274C22×C8—1061no—
275C4⋊Q8—531no—
276#127—531no—
277#124—1061no—
278#121—1061no—
279#121—1061no—
280C4×Q8C4×D5311yescharacteristic
281C4×Q8C22⋊D5311yescharacteristic
282C4×D4—21no—
283C4×D4—21no—
284C4×D4—1061no—
285C4×D4—1061no—
286C4×D4C4×D5311yescharacteristic
287C4×D4C22⋊D5311yescharacteristic
288#117—21no—
289#117D21211yescharacteristic
290C2×C4⋊C4—1061no—
291C2×C42—21no—
292C2×C42—1061no—
293C2×C42C22×D5311yescharacteristic
294C4⋊C8—22no—
295C4⋊C8—21no—
296C4⋊C8—531no—
297C4⋊C8—531no—
298C4⋊C8—1061no—
299C4⋊C8—1061no—
300C4⋊C8—1061no—
301#104—2122no—
302#103—1061no—
303#103—1061no—
304#102—1061no—
305#102—1061no—
306C22⋊C8—22no—
307C22⋊C8C22⋊D5312yes—
308#97—22no—
309C4×C8—21no—
310C4×C8—21no—
311C4×C8C22⋊D5311yescharacteristic
312C4×C8C22⋊D5311yescharacteristic
313#95—1061no—
314#41—21no—
315#41—21no—
316#41—21no—
317#41—21no—
318#41—2121no—
319#41—2121no—
320#41—2121no—
321#41—2121no—
322C2×Q8—1061no—
323C2×Q8#1171992111yescharacteristic
324C2×D4—21no—
325C2×D4—1061no—
326C2×D4—1061no—
327C2×D4—1061no—
328C2×D4—2121no—
329C2×D4#1171992111yescharacteristic
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—2121no—
334C22×C4—2121no—
335C22×C4C2×D21211yescharacteristic
336C22×C4#1171994511yescharacteristic
337C22×C4C2×C22⋊D5311yescharacteristic
338Q16—2122no—
339SD16—2122no—
340SD16—2122no—
341D8—2122no—
342M16—44no—
343M16—2121no—
344M16—2121no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—22no—
348C2×C8—42no—
349C2×C8—1061no—
350C2×C8—1061no—
351C2×C8—2121no—
352C2×C8—2121no—
353C2×C8—2121no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—1062no—
357C4⋊C4—1061no—
358C4⋊C4—1061no—
359C4⋊C4—2121no—
360C4⋊C4#1171992111yescharacteristic
361C4⋊C4#1171992111yescharacteristic
362C22⋊C4—21no—
363C22⋊C4—21no—
364C22⋊C4—21no—
365C22⋊C4—2121no—
366C22⋊C4—2121no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—2121no—
371C42—2121no—
372C42C2×C4×D5311yescharacteristic
373C42#1171994511yescharacteristic
374C42C2×C22⋊D5311yescharacteristic
375C42C2×C22⋊D5311yescharacteristic
376C23—21no—
377C23—2121no—
378C23#1313319111yescharacteristic
379Q8—21no—
380Q8—21no—
381Q8—2121no—
382Q8—2121no—
383D4—21no—
384D4—21no—
385D4—41no—
386D4—41no—
387D4—4241no—
388D4—4241no—
389D4—4241no—
390D4—2121no—
391D4—2121no—
392D4—2121no—
393D4—2121no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—4241no—
406C2×C4—4241no—
407C2×C4—4241no—
408C2×C4—4241no—
409C2×C4—2122no—
410C2×C4—2121no—
411C2×C4C2×#1171992111yescharacteristic
412C2×C4C4×C22⋊D5311yescharacteristic
413C2×C4C4×C22⋊D5311yescharacteristic
414C2×C4#1313318611yescharacteristic
415C2×C4#1313319111yescharacteristic
416C2×C4#1313320011yescharacteristic
417C2×C4#1313324511yescharacteristic
418C2×C4#1313324511yescharacteristic
419C2×C4#1313324511yescharacteristic
420C2×C4#1313325011yescharacteristic
421C8—22no—
422C8—41no—
423C8—42no—
424C8—4241no—
425C8—2121no—
426C8—2121no—
427C22—21no—
428C22—21no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—4241no—
433C22—4241no—
434C22—4241no—
435C22#1681050211yescharacteristic
436C22#1681086911yescharacteristic
437C22#1681087611yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—41no—
442C4—41no—
443C4—41no—
444C4—4242no—
445C4—4241no—
446C4#1681062511yescharacteristic
447C4#1681062911yescharacteristic
448C4#1681073811yescharacteristic
449C4#1681074311yescharacteristic
450C4#1681086911yescharacteristic
451C4#1681087611yescharacteristic
452C2—21no—
453C2—21no—
454C2—81no—
455C2—8481no—
456C2#2027504211yescharacteristic
457C2#2027515311yescharacteristic
458C2#2027521611yescharacteristic
subgroupG/H
centerC2×C4#13133191
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2965C2
socleC22×C53#386
prime radical#20284170C2
cosocleC2×C42×C53C23
hypercenter#2965D53
hypocenterC53#9210
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2965

Complements

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

NH ≅ G/Ntype
2. #20274735C2semidirect
3. #20273211C2semidirect
4. #20273167C2semidirect
5. #20272611C2semidirect
6. #20272555C2semidirect
7. C53×#2965C2semidirect
12. #16810833C4semidirect
13. #16810828C4semidirect
14. #16810823C4semidirect
15. #16810818C4semidirect
224. #2965D53semidirect
262. C53#9210semidirect

Maximal subgroups

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

subgroupindex
#202841702
#202747352
#202732112
#202731672
#202726112
#202725552
C53×#29652
#921053