Finite Group Databasebeta

← #27280980

Subgroups — #27280980

Order 13568.

558 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284195C211yesmaximal, characteristic
2#20275641C211yesmaximal, characteristic, complemented
3#20274735C211yesmaximal, characteristic, complemented
4#20273382C211yesmaximal, characteristic, complemented
5#20272617C211yesmaximal, characteristic, complemented
6#20272590C211yesmaximal, characteristic, complemented
7C53×#2989C211yesmaximal, characteristic, complemented
8C2×#13133242—21no—
9C2×#13133214C2211yescharacteristic
10#16810823—21no—
11#16810823C412yescomplemented
12#16810818—21no—
13#16810818C412yescomplemented
14#16810647C2211yescharacteristic
15#16810647C2211yescharacteristic
16C2×#13133144—21no—
17#16810627—21no—
18C2×#13133139—22no—
19C4×C4⋊Dic53C2211yescharacteristic
20#16810382—21no—
21#16810366—21no—
22#16810366—21no—
23C4×#41×C53C2211yescharacteristic
24C53×#446—21no—
25C53×#436—21no—
26C53×#435—22no—
27C53×#432—21no—
28C53×#431C2211yescharacteristic
29C53×#423—21no—
30C2×C4⋊C8×C53C2211yescharacteristic
31C22×D212—21no—
32C22×D212—21no—
33C22×D212—21no—
34#13133242—42no—
35#13133242—42no—
36C2×#11719924—22no—
37C2×#11719922—22no—
38C2×#11719917—21no—
39C2×#11719917—21no—
40#13133214—21no—
41#13133214—21no—
42#13133214—21no—
43#13133214C2×C412yes—
44C2×C4⋊Dic53—21no—
45C2×C4×Dic53—21no—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48C4⋊#88322—21no—
49C4⋊#88322C2×C412yes—
50#13133144—42no—
51#13133139—44no—
52#13133137—22no—
53#13133137—22no—
54#13133124—21no—
55C2×#41×C53D411yescharacteristic
56C2×M16×C53—22no—
57C22×C8×C53D412yes—
58C4×Q8×C53C2×C411yescharacteristic
59C4×Q8×C53D411yescharacteristic
60C4×D4×C53—21no—
61C4×D4×C53—21no—
62C4×D4×C53C2×C411yescharacteristic
63C4×D4×C53D411yescharacteristic
64#117×C53—21no—
65#117×C53D411yescharacteristic
66C2×C42×C53—21no—
67C2×C42×C53C2311yescharacteristic
68C4⋊C8×C53—22no—
69C4⋊C8×C53—21no—
70C4⋊C8×C53—21no—
71C4⋊C8×C53D411yescharacteristic
72C4⋊C8×C53D411yescharacteristic
73C22⋊C8×C53—22no—
74C22⋊C8×C53—21no—
75#97×C53—22no—
76C4×C8×C53—21no—
77C23×D53—41no—
78#41×C53—21no—
79#41×C53—21no—
80#41×C53—21no—
81#41×C53—21no—
82C2×Q8×C53C22⋊C411yescharacteristic
83C2×D4×C53—21no—
84C2×D4×C53C22⋊C411yescharacteristic
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53#4111yescharacteristic
91C22×Dic53—41no—
92C2×D212—22no—
93C2×D212—22no—
94C2×D212—22no—
95C2×D212—42no—
96C2×D212—42no—
97C2×D212—41no—
98C2×D212—41no—
99C2×D212—41no—
100C2×D212—41no—
101C2×D212—41no—
102C2×D212—41no—
103C2×D212—41no—
104C2×D212—82no—
105M16×C53—82no—
106C2×C8×C53—21no—
107C2×C8×C53—21no—
108C2×C8×C53—22no—
109C2×C8×C53—42no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53C22⋊C411yescharacteristic
113C4⋊C4×C53C22⋊C411yescharacteristic
114C22⋊C4×C53—21no—
115C22⋊C4×C53—21no—
116C22⋊C4×C53—21no—
117C42×C53—21no—
118C42×C53—21no—
119C42×C53—21no—
120C42×C53C22×C411yescharacteristic
121C42×C53C2×D411yescharacteristic
122C42×C53C2×D411yescharacteristic
123C42×C53#4111yescharacteristic
124#11719924—44no—
125#11719922—44no—
126C4⋊Dic53—22no—
127C4⋊Dic53—41no—
128C4×Dic53—41no—
129C4×Dic53—41no—
130#11719917—41no—
131#11719917—41no—
132#11719917—41no—
133#11719917—41no—
134C2×#88322—22no—
135C2×#88322—41no—
136C23×C53—21no—
137C23×C53#12111yescharacteristic
138C22×D53—42no—
139C22×D53—82no—
140C22×D53—82no—
141C22×D53—81no—
142C22×D53—81no—
143Q8×C53—21no—
144Q8×C53—21no—
145D4×C53—21no—
146D4×C53—21no—
147D4×C53—41no—
148D4×C53—41no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—81no—
160C2×C4×C53C2×C22⋊C411yescharacteristic
161C2×C4×C53C4×D411yescharacteristic
162C2×C4×C53C4×D411yescharacteristic
163C2×C4×C53#12011yescharacteristic
164C2×C4×C53#12111yescharacteristic
165C2×C4×C53#12311yescharacteristic
166C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
167C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C53#13711yescharacteristic
170C2×Dic53—42no—
171C2×Dic53—81no—
172C2×Dic53—81no—
173D212—44no—
174D212—42no—
175D212—42no—
176D212—82no—
177D212—82no—
178D212—81no—
179D212—81no—
180D212—81no—
181D212—81no—
182C8×C53—42no—
183C8×C53—41no—
184C8×C53—41no—
185#88322—42no—
186#88322—81no—
187#9322—531nomaximal, Sylow
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—41no—
191C22×C53—41no—
192C22×C53—41no—
193C22×C53#38611yescharacteristic
194C22×C53#46811yescharacteristic
195C22×C53#46911yescharacteristic
196C2×D53—82no—
197C2×D53—82no—
198C2×D53—82no—
199C2×D53—162no—
200C2×D53—162no—
201C4×C53—21no—
202C4×C53—21no—
203C4×C53—21no—
204C4×C53—41no—
205C4×C53—41no—
206C4×C53—41no—
207C4×C53#41711yescharacteristic
208C4×C53#41811yescharacteristic
209C4×C53#45311yescharacteristic
210C4×C53#45411yescharacteristic
211C4×C53#46811yescharacteristic
212C4×C53#46911yescharacteristic
213Dic53—82no—
214#3038—531no—
215#2989D5311yescharacteristic, complemented
216#1982—531no—
217#1806—531no—
218#1545—531no—
219#1303—531no—
220#1292—531no—
221C2×C53—21no—
222C2×C53—21no—
223C2×C53—81no—
224C2×C53#188311yescharacteristic
225C2×C53#189511yescharacteristic
226C2×C53#190311yescharacteristic
227D53—162no—
228C2×AGL(1,ℤ/8)—1061no—
229C2×#127—531no—
230C4×#41C2×D5311yescharacteristic
231#460—532no—
232#460—1061no—
233#459—532no—
234#459—1061no—
235#446—21no—
236#436—21no—
237#435—22no—
238#432—21no—
239#431C2×D5311yescharacteristic
240#423—21no—
241#423—531no—
242#423—531no—
243C2×C4⋊C8C2×D5311yescharacteristic
244C2×#104—1061no—
245#418—1061no—
246C2×#102—1062no—
247C4×C4⋊C4—531no—
248#339—1061no—
249#331—1061no—
250#331—1061no—
251C53#932211yescharacteristic, complemented, Sylow
252C2×#41D21211yescharacteristic
253C22×D4—1061no—
254C22×D4—1061no—
255C22×D4—1061no—
256AGL(1,ℤ/8)—2122no—
257AGL(1,ℤ/8)—2122no—
258C2×SD16—1062no—
259C2×D8—1062no—
260C2×M16—22no—
261C2×M16—1061no—
262C2×M16—1061no—
263C22×C8C22⋊D5312yes—
264#127—532no—
265#127—1061no—
266#127—1061no—
267#127—1061no—
268C4×Q8C4×D5311yescharacteristic
269C4×Q8C22⋊D5311yescharacteristic
270C4×D4—21no—
271C4×D4—21no—
272C4×D4C4×D5311yescharacteristic
273C4×D4C22⋊D5311yescharacteristic
274#117—21no—
275#117D21211yescharacteristic
276C2×C4⋊C4—1061no—
277C2×C42—21no—
278C2×C42—1061no—
279C2×C42C22×D5311yescharacteristic
280C4⋊C8—22no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—532no—
284C4⋊C8—1061no—
285C4⋊C8—1061no—
286C4⋊C8—1061no—
287C4⋊C8C22⋊D5311yescharacteristic
288C4⋊C8C22⋊D5311yescharacteristic
289#104—2122no—
290#102—1062no—
291#102—1062no—
292#102—2124no—
293C22⋊C8—22no—
294C22⋊C8—21no—
295#97—22no—
296C4×C8—21no—
297#95—1061no—
298C24—2121no—
299#41—21no—
300#41—21no—
301#41—21no—
302#41—21no—
303C2×Q8#1171992111yescharacteristic
304C2×D4—21no—
305C2×D4—1062no—
306C2×D4—1062no—
307C2×D4—1062no—
308C2×D4—4242no—
309C2×D4—2122no—
310C2×D4—2122no—
311C2×D4—2121no—
312C2×D4—2121no—
313C2×D4—2121no—
314C2×D4—2121no—
315C2×D4—2121no—
316C2×D4—2121no—
317C2×D4—2121no—
318C2×D4#1171992111yescharacteristic
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—2121no—
323C22×C4C2×D21211yescharacteristic
324C22×C4#1171994511yescharacteristic
325C22×C4C2×C22⋊D5311yescharacteristic
326SD16—2124no—
327D8—2124no—
328M16—82no—
329M16—2121no—
330M16—2121no—
331M16—2121no—
332M16—2121no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—22no—
336C2×C8—42no—
337C2×C8—1062no—
338C2×C8—2121no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—1062no—
342C4⋊C4—2121no—
343C4⋊C4#1171992111yescharacteristic
344C4⋊C4#1171992111yescharacteristic
345C22⋊C4—21no—
346C22⋊C4—21no—
347C22⋊C4—21no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—2121no—
352C42—2121no—
353C42C2×C4×D5311yescharacteristic
354C42#1171994511yescharacteristic
355C42C2×C22⋊D5311yescharacteristic
356C42C2×C22⋊D5311yescharacteristic
357C23—21no—
358C23—4242no—
359C23—4242no—
360C23—4241no—
361C23—4241no—
362C23—2122no—
363C23#1313319111yescharacteristic
364Q8—21no—
365Q8—21no—
366D4—21no—
367D4—21no—
368D4—41no—
369D4—41no—
370D4—4242no—
371D4—4242no—
372D4—4241no—
373D4—4241no—
374D4—4241no—
375D4—4241no—
376D4—2124no—
377D4—2122no—
378D4—2122no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—81no—
390C2×C4—4241no—
391C2×C4—4241no—
392C2×C4—2122no—
393C2×C4C2×#1171992111yescharacteristic
394C2×C4C4×C22⋊D5311yescharacteristic
395C2×C4C4×C22⋊D5311yescharacteristic
396C2×C4#1313318611yescharacteristic
397C2×C4#1313319111yescharacteristic
398C2×C4#1313320011yescharacteristic
399C2×C4#1313324511yescharacteristic
400C2×C4#1313324511yescharacteristic
401C2×C4#1313324511yescharacteristic
402C2×C4#1313325011yescharacteristic
403C8—42no—
404C8—41no—
405C8—41no—
406C8—4241no—
407C8—2122no—
408C22—21no—
409C22—21no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—8482no—
414C22—8482no—
415C22—4242no—
416C22—4242no—
417C22—4242no—
418C22#1681050211yescharacteristic
419C22#1681086911yescharacteristic
420C22#1681087611yescharacteristic
421C4—21no—
422C4—21no—
423C4—21no—
424C4—41no—
425C4—41no—
426C4—41no—
427C4—4242no—
428C4#1681062511yescharacteristic
429C4#1681062911yescharacteristic
430C4#1681079111yescharacteristic
431C4#1681079611yescharacteristic
432C4#1681086911yescharacteristic
433C4#1681087611yescharacteristic
434C2—21no—
435C2—21no—
436C2—81no—
437C2—8482no—
438C2#2027509111yescharacteristic
439C2#2027515311yescharacteristic
440C2#2027519511yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2989C2
socleC22×C53#386
prime radical#20284195C2
cosocleC2×C42×C53C23
hypercenter#2989D53
hypocenterC53#9322
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2989

Complements

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

NH ≅ G/Ntype
2. #20275641C2semidirect
3. #20274735C2semidirect
4. #20273382C2semidirect
5. #20272617C2semidirect
6. #20272590C2semidirect
7. C53×#2989C2semidirect
11. #16810823C4semidirect
13. #16810818C4semidirect
215. #2989D53semidirect
251. C53#9322semidirect

Maximal subgroups

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

subgroupindex
#202841952
#202756412
#202747352
#202733822
#202726172
#202725902
C53×#29892
#932253