Finite Group Databasebeta

← #27280978

Subgroups — #27280978

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#20283832C211yesmaximal, characteristic, complemented
3#20275642C211yesmaximal, characteristic, complemented
4#20273384C211yesmaximal, characteristic, complemented
5#20272618C211yesmaximal, characteristic, complemented
6#20272589C211yesmaximal, characteristic, complemented
7C53×#1806C211yesmaximal, characteristic, complemented
8C2×#13133242—21no—
9C2×#13133214C2211yescharacteristic
10#16810823—21no—
11#16810823C412yescomplemented
12#16810818—21no—
13#16810818C412yescomplemented
14#16810754—21no—
15#16810701—21no—
16#16810695—22no—
17#16810680—21no—
18#16810676C2211yescharacteristic
19#16810647—21no—
20#16810647C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22#16810627—21no—
23C2×#13133137—22no—
24#16810366—21no—
25#16810366—21no—
26C4×#41×C53C2211yescharacteristic
27C53×#423C2211yescharacteristic
28C2×#104×C53—21no—
29C4×C4⋊C4×C53C2211yescharacteristic
30C53×#339—21no—
31C22×D212—21no—
32C22×D212—21no—
33C22×D212—21no—
34#13133242—42no—
35#13133242—42no—
36C2×#11719924—22no—
37C2×#11719922—22no—
38C2×#11719917—22no—
39C2×#11719917—21no—
40C22×#88322D412yes—
41#13133214—21no—
42#13133214—21no—
43#13133214—21no—
44#13133214C2×C412yes—
45C4⋊#88322—22no—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48C4⋊#88322—21no—
49C4⋊#88322C2×C412yes—
50C4⋊#88322D411yescharacteristic
51C4⋊#88322D411yescharacteristic
52#13133137—22no—
53#13133137—22no—
54#13133137—44no—
55C22⋊#88322—22no—
56C22⋊#88322—21no—
57#13133128—22no—
58C4×#88322—21no—
59C2×#41×C53D411yescharacteristic
60C2×M16×C53—21no—
61C4×Q8×C53C2×C411yescharacteristic
62C4×Q8×C53D411yescharacteristic
63C4×D4×C53—21no—
64C4×D4×C53—21no—
65C4×D4×C53C2×C411yescharacteristic
66C4×D4×C53D411yescharacteristic
67#117×C53—21no—
68#117×C53D411yescharacteristic
69C2×C4⋊C4×C53—21no—
70C2×C42×C53—21no—
71C2×C42×C53—21no—
72C2×C42×C53C2311yescharacteristic
73C4⋊C8×C53—21no—
74C4⋊C8×C53—21no—
75#104×C53—42no—
76#95×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×C53—41no—
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53#4111yescharacteristic
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—41no—
106M16×C53—41no—
107C2×C8×C53—41no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—22no—
111C4⋊C4×C53—41no—
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×C53—41no—
121C42×C53—41no—
122C42×C53C22×C411yescharacteristic
123C42×C53C2×D411yescharacteristic
124C42×C53C2×D411yescharacteristic
125C42×C53#4111yescharacteristic
126#11719924—44no—
127#11719922—44no—
128#11719917—41no—
129#11719917—41no—
130#11719917—82no—
131C2×#88322—22no—
132C2×#88322—21no—
133C2×#88322—21no—
134C2×#88322—22no—
135C2×#88322—42no—
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—42no—
160C2×C4×C53—81no—
161C2×C4×C53—81no—
162C2×C4×C53—81no—
163C2×C4×C53C2×C22⋊C411yescharacteristic
164C2×C4×C53C4×D411yescharacteristic
165C2×C4×C53C4×D411yescharacteristic
166C2×C4×C53#12011yescharacteristic
167C2×C4×C53#12111yescharacteristic
168C2×C4×C53#12311yescharacteristic
169C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
170C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
171C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
172C2×C4×C53#13711yescharacteristic
173D212—44no—
174D212—42no—
175D212—42no—
176D212—82no—
177D212—82no—
178D212—81no—
179D212—81no—
180D212—81no—
181D212—81no—
182C8×C53—81no—
183#88322—42no—
184#88322—42no—
185#88322—41no—
186#88322—41no—
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—82no—
208C4×C53#41711yescharacteristic
209C4×C53#41811yescharacteristic
210C4×C53#45311yescharacteristic
211C4×C53#45411yescharacteristic
212C4×C53#46811yescharacteristic
213C4×C53#46911yescharacteristic
214#3038—531no—
215#2989—531no—
216#1982—531no—
217#1806D5311yescharacteristic, complemented
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—1061no—
236#436—1061no—
237#435—1062no—
238#432—1061no—
239#431—531no—
240#423—531no—
241#423—1061no—
242#423C2×D5311yescharacteristic
243C2×C4⋊C8—531no—
244C2×#104—21no—
245#418—1061no—
246C2×#102—1062no—
247C4×C4⋊C4C2×D5311yescharacteristic
248#339—21no—
249#331—1061no—
250#331—1061no—
251C53#932211yescharacteristic, complemented, Sylow
252C2×#41C22⋊D5311yescharacteristic
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—21no—
261C2×M16—1062no—
262C2×M16—1061no—
263C22×C8—532no—
264#127—532no—
265#127—1061no—
266#127—1061no—
267#127—1061no—
268C4×Q8C4×D5311yescharacteristic
269C4×Q8D21211yescharacteristic
270C4×D4—21no—
271C4×D4—21no—
272C4×D4C4×D5311yescharacteristic
273C4×D4D21211yescharacteristic
274#117—21no—
275#117C22⋊D5311yescharacteristic
276C2×C4⋊C4—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42C22×D5311yescharacteristic
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—532no—
283C4⋊C8—531no—
284C4⋊C8—531no—
285C4⋊C8—1062no—
286C4⋊C8—1061no—
287C4⋊C8—1061no—
288C4⋊C8—1061no—
289#104—42no—
290#102—1062no—
291#102—1062no—
292#102—2124no—
293C22⋊C8—1062no—
294C22⋊C8—1061no—
295#97—1062no—
296C4×C8—1061no—
297#95—21no—
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—41no—
323C22×C4D4×D5311yescharacteristic
324C22×C4#1171994711yescharacteristic
325C22×C4C2×C22⋊D5311yescharacteristic
326SD16—2124no—
327D8—2124no—
328M16—41no—
329M16—41no—
330M16—4242no—
331M16—2121no—
332M16—2121no—
333C2×C8—41no—
334C2×C8—1062no—
335C2×C8—1061no—
336C2×C8—1061no—
337C2×C8—1062no—
338C2×C8—2122no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—22no—
342C4⋊C4—41no—
343C4⋊C4#1171992111yescharacteristic
344C4⋊C4#1171992111yescharacteristic
345C22⋊C4—21no—
346C22⋊C4—21no—
347C22⋊C4—21no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—41no—
352C42—41no—
353C42C2×C4×D5311yescharacteristic
354C42C2×D21211yescharacteristic
355C42D4×D5311yescharacteristic
356C42#1171994711yescharacteristic
357C23—21no—
358C23—4242no—
359C23—4242no—
360C23—4241no—
361C23—4241no—
362C23—2122no—
363C23#1313318911yescharacteristic
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—42no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4C2×#1171992111yescharacteristic
394C2×C4#1313317711yescharacteristic
395C2×C4#1313317711yescharacteristic
396C2×C4#1313318711yescharacteristic
397C2×C4#1313318911yescharacteristic
398C2×C4#1313320111yescharacteristic
399C2×C4#1313324211yescharacteristic
400C2×C4#1313324211yescharacteristic
401C2×C4#1313324311yescharacteristic
402C2×C4#1313324811yescharacteristic
403C8—81no—
404C8—2122no—
405C8—2121no—
406C8—2121no—
407C8—2122no—
408C22—21no—
409C22—21no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—8482no—
414C22—8482no—
415C22—4242no—
416C22—4242no—
417C22—4242no—
418C22#1681050011yescharacteristic
419C22#1681087111yescharacteristic
420C22#1681087411yescharacteristic
421C4—21no—
422C4—21no—
423C4—21no—
424C4—41no—
425C4—41no—
426C4—41no—
427C4—82no—
428C4#1681062211yescharacteristic
429C4#1681062711yescharacteristic
430C4#1681079211yescharacteristic
431C4#1681079711yescharacteristic
432C4#1681087111yescharacteristic
433C4#1681087411yescharacteristic
434C2—21no—
435C2—21no—
436C2—81no—
437C2—8482no—
438C2#2027508911yescharacteristic
439C2#2027515111yescharacteristic
440C2#2027519311yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1806C2
socleC22×C53#386
prime radical#20284195C2
cosocleC2×C42×C53C23
hypercenter#1806D53
hypocenterC53#9322
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
2. #20283832C2semidirect
3. #20275642C2semidirect
4. #20273384C2semidirect
5. #20272618C2semidirect
6. #20272589C2semidirect
7. C53×#1806C2semidirect
11. #16810823C4semidirect
13. #16810818C4semidirect
217. #1806D53semidirect
251. C53#9322semidirect

Maximal subgroups

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

subgroupindex
#202841952
#202838322
#202756422
#202733842
#202726182
#202725892
C53×#18062
#932253