Finite Group Databasebeta

← #27279929

Subgroups — #27279929

Order 13568.

486 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284814C211yesmaximal, characteristic
2#20274540C211yesmaximal, characteristic
3#20273331C211yesmaximal, characteristic, complemented
4#20273215C211yesmaximal, characteristic, complemented
5#20272615C211yesmaximal, characteristic, complemented
6#20272570C211yesmaximal, characteristic, complemented
7C53×#2974C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9#16810958—21no—
10#16810903C411yescharacteristic, complemented
11#16810903C411yescharacteristic, complemented
12#16810828C411yescharacteristic, complemented
13#16810828C411yescharacteristic, complemented
14#16810774—21no—
15#16810723—21no—
16#16810723—21no—
17#16810680C2211yescharacteristic
18C2×#13133138—21no—
19C2×#13133128C2211yescharacteristic
20C4×C4⋊Dic53C2211yescharacteristic
21#16810385—21no—
22#16810364—22no—
23#16810361—21no—
24#16810361—21no—
25C2×C4×D4×C53C2211yescharacteristic
26C53×#436—21no—
27C53×#435—22no—
28C8×D4×C53—21no—
29C53×#432C2211yescharacteristic
30C53×#431C2211yescharacteristic
31C53×#407—21no—
32C2×C22⋊C8×C53—21no—
33C2×#11719923—21no—
34C2×#11719923—21no—
35C22×#88322—21no—
36#13133217C2×C411yescharacteristic
37#13133217C2×C411yescharacteristic
38#13133209—21no—
39#13133196—21no—
40#13133196—21no—
41C4×Dic106—21no—
42C4×Dic106—21no—
43#13133176—21no—
44C2×C4⋊Dic53—21no—
45C2×C4×Dic53—21no—
46#13133150—21no—
47#13133150—21no—
48C4⋊#88322C2×C411yescharacteristic
49C4⋊#88322C2×C411yescharacteristic
50#13133140—21no—
51#13133140—21no—
52#13133138—24no—
53#13133138—21no—
54#13133138—21no—
55C22⋊#88322—21no—
56#13133128—21no—
57#13133128—21no—
58#13133128—21no—
59#13133124—21no—
60C22×D4×C53D411yescharacteristic
61C23×C4×C53—21no—
62C2×M16×C53—22no—
63C22×C8×C53—22no—
64C4×D4×C53—21no—
65C4×D4×C53—21no—
66C4×D4×C53C2×C411yescharacteristic
67C4×D4×C53C2×C411yescharacteristic
68C4×D4×C53D411yescharacteristic
69C4×D4×C53D411yescharacteristic
70C2×C4⋊C4×C53D411yescharacteristic
71C2×C22⋊C4×C53—21no—
72C2×C42×C53C2311yescharacteristic
73C4⋊C8×C53—21no—
74C4⋊C8×C53—21no—
75C22⋊C8×C53—22no—
76C22⋊C8×C53—22no—
77C22⋊C8×C53—21no—
78C22⋊C8×C53D411yescharacteristic
79C22⋊C8×C53D411yescharacteristic
80#97×C53D412yes—
81C4×C8×C53—21no—
82C24×C53—21no—
83C2×D4×C53—22no—
84C2×D4×C53—21no—
85C2×D4×C53—41no—
86C2×D4×C53C22⋊C411yescharacteristic
87C2×D4×C53C22⋊C411yescharacteristic
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53—21no—
92C22×C4×C53—21no—
93C22×C4×C53—82no—
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53#4111yescharacteristic
97C22×Dic53—41no—
98C2×Dic106—21no—
99C2×Dic106—21no—
100M16×C53—82no—
101C2×C8×C53—21no—
102C2×C8×C53—21no—
103C2×C8×C53—22no—
104C2×C8×C53—82no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53C22⋊C411yescharacteristic
107C4⋊C4×C53C22⋊C411yescharacteristic
108C22⋊C4×C53—21no—
109C22⋊C4×C53—21no—
110C22⋊C4×C53—21no—
111C22⋊C4×C53—21no—
112C42×C53C22×C411yescharacteristic
113C42×C53C2×D411yescharacteristic
114C42×C53C2×D411yescharacteristic
115C42×C53#4111yescharacteristic
116C22⋊Dic53—41no—
117#11719923—42no—
118#11719923—42no—
119C4⋊Dic53—22no—
120C4⋊Dic53—21no—
121C4⋊Dic53—21no—
122C4⋊Dic53—21no—
123C4⋊Dic53—21no—
124C4⋊Dic53—41no—
125#11719919—41no—
126#11719919—41no—
127C4×Dic53—41no—
128C4×Dic53—81no—
129C2×#88322—22no—
130C2×#88322—21no—
131C2×#88322—21no—
132C2×#88322—41no—
133C2×#88322—41no—
134C23×C53—21no—
135C23×C53—21no—
136C23×C53—42no—
137C23×C53—41no—
138C23×C53—41no—
139C23×C53—81no—
140C23×C53#12111yescharacteristic
141D4×C53—21no—
142D4×C53—21no—
143D4×C53—21no—
144D4×C53—42no—
145D4×C53#10412yes—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—41no—
153C2×C4×C53—42no—
154C2×C4×C53—42no—
155C2×C4×C53—81no—
156C2×C4×C53—81no—
157C2×C4×C53—81no—
158C2×C4×C53C2×C22⋊C411yescharacteristic
159C2×C4×C53C4×D411yescharacteristic
160C2×C4×C53C4×D411yescharacteristic
161C2×C4×C53#12011yescharacteristic
162C2×C4×C53#12111yescharacteristic
163C2×C4×C53#12311yescharacteristic
164C2×Dic53—42no—
165C2×Dic53—41no—
166C2×Dic53—41no—
167C2×Dic53—81no—
168C2×Dic53—81no—
169Dic106—41no—
170Dic106—41no—
171Dic106—81no—
172C8×C53—41no—
173C8×C53—42no—
174C8×C53—41no—
175#88322—42no—
176#88322—41no—
177#88322—41no—
178#9165—531nomaximal, Sylow
179C22×C53—21no—
180C22×C53—21no—
181C22×C53—22no—
182C22×C53—41no—
183C22×C53—41no—
184C22×C53—41no—
185C22×C53—41no—
186C22×C53—41no—
187C22×C53—82no—
188C22×C53—82no—
189C22×C53#38611yescharacteristic
190C22×C53#47111yescharacteristic
191C22×C53#47311yescharacteristic
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—41no—
198C4×C53—41no—
199C4×C53C2×#10411yescharacteristic
200C4×C53#42111yescharacteristic
201Dic53—82no—
202Dic53—81no—
203Dic53—81no—
204#3110—531no—
205#2974D5311yescharacteristic
206#1762—531no—
207#1531—531no—
208#1503—531no—
209#1302—531no—
210#1285—531no—
211C2×C53—21no—
212C2×C53—21no—
213C2×C53—42no—
214C2×C53—81no—
215C2×C53#187111yescharacteristic
216C2×C53#190411yescharacteristic
217C2×C53#190811yescharacteristic
218#533—531no—
219C2×C4×D4C2×D5311yescharacteristic
220#484—1061no—
221#474—531no—
222#474—531no—
223#461—531no—
224#461—531no—
225#450—1061no—
226#440—1061no—
227#440—1061no—
228#436—21no—
229#435—22no—
230C8×D4—21no—
231#432—531no—
232#432C2×D5311yescharacteristic
233#431C2×D5311yescharacteristic
234C2×#102—1061no—
235#407—21no—
236C2×C22⋊C8—21no—
237C2×#97—531no—
238C4×C4⋊C4—531no—
239#340—1061no—
240#330—1062no—
241#328—1061no—
242#328—1061no—
243C53#916511yescharacteristic, complemented, Sylow
244C22×D4D21211yescharacteristic
245C23×C4—21no—
246C2×SD16—1061no—
247C2×SD16—1061no—
248C2×M16—22no—
249C22×C8—22no—
250C22×C8—1061no—
251C4⋊Q8—531no—
252C4⋊Q8—531no—
253#125—1061no—
254C22⋊Q8—1061no—
255C22⋊Q8—1061no—
256C4×Q8—1061no—
257C4×Q8—1061no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4C4×D5311yescharacteristic
261C4×D4C4×D5311yescharacteristic
262C4×D4C22⋊D5311yescharacteristic
263C4×D4C22⋊D5311yescharacteristic
264#117—1061no—
265C2×C4⋊C4—1061no—
266C2×C4⋊C4D21211yescharacteristic
267C2×C22⋊C4—21no—
268C2×C42—1061no—
269C2×C42C22×D5311yescharacteristic
270#107—1061no—
271#107—1061no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—531no—
275C4⋊C8—531no—
276#103—1061no—
277#103—1061no—
278#102—1064no—
279#102—1061no—
280#102—1061no—
281C22⋊C8—22no—
282C22⋊C8—22no—
283C22⋊C8—21no—
284C22⋊C8—1061no—
285C22⋊C8C22⋊D5311yescharacteristic
286C22⋊C8C22⋊D5311yescharacteristic
287#97—1061no—
288#97—1061no—
289#97—1061no—
290#97C22⋊D5312yes—
291C4×C8—21no—
292#95—1061no—
293C24—21no—
294C2×Q8—1061no—
295C2×Q8—1061no—
296C2×D4—22no—
297C2×D4—21no—
298C2×D4—41no—
299C2×D4#1171992111yescharacteristic
300C2×D4#1171992111yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—82no—
307C22×C4—2121no—
308C22×C4C2×D21211yescharacteristic
309C22×C4#1171994511yescharacteristic
310C22×C4C2×C22⋊D5311yescharacteristic
311SD16—2122no—
312SD16—2122no—
313M16—82no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—22no—
317C2×C8—82no—
318C2×C8—1062no—
319C2×C8—1061no—
320C2×C8—1061no—
321C2×C8—2121no—
322C2×C8—2121no—
323C4⋊C4—21no—
324C4⋊C4—1062no—
325C4⋊C4—1061no—
326C4⋊C4—1061no—
327C4⋊C4—1061no—
328C4⋊C4—1061no—
329C4⋊C4—2121no—
330C4⋊C4—2121no—
331C4⋊C4—2121no—
332C4⋊C4#1171992111yescharacteristic
333C4⋊C4#1171992111yescharacteristic
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—21no—
337C22⋊C4—21no—
338C22⋊C4—2121no—
339C42—4241no—
340C42—2121no—
341C42C2×C4×D5311yescharacteristic
342C42#1171994511yescharacteristic
343C42C2×C22⋊D5311yescharacteristic
344C42C2×C22⋊D5311yescharacteristic
345C23—21no—
346C23—21no—
347C23—42no—
348C23—41no—
349C23—41no—
350C23—81no—
351C23#1313319111yescharacteristic
352Q8—4241no—
353Q8—2121no—
354Q8—2121no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—42no—
359D4#1313314312yes—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—4241no—
373C2×C4—4241no—
374C2×C4—2122no—
375C2×C4—2121no—
376C2×C4—2121no—
377C2×C4C2×#1171992111yescharacteristic
378C2×C4C4×C22⋊D5311yescharacteristic
379C2×C4C4×C22⋊D5311yescharacteristic
380C2×C4#1313318611yescharacteristic
381C2×C4#1313319111yescharacteristic
382C2×C4#1313320011yescharacteristic
383C8—41no—
384C8—42no—
385C8—41no—
386C8—2122no—
387C8—2121no—
388C8—2121no—
389C22—21no—
390C22—21no—
391C22—22no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—82no—
398C22—82no—
399C22#1681050211yescharacteristic
400C22#1681088611yescharacteristic
401C22#1681089811yescharacteristic
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—41no—
408C4—41no—
409C4—4242no—
410C4—4241no—
411C4—4241no—
412C4C2×#1313314311yescharacteristic
413C4#1681063911yescharacteristic
414C2—21no—
415C2—21no—
416C2—42no—
417C2—81no—
418C2#2027501711yescharacteristic
419C2#2027520211yescharacteristic
420C2#2027523011yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2974C2
socleC22×C53#386
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenter#2974D53
hypocenterC53#9165
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2974

Complements

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

NH ≅ G/Ntype
3. #20273331C2semidirect
4. #20273215C2semidirect
5. #20272615C2semidirect
6. #20272570C2semidirect
10. #16810903C4semidirect
11. #16810903C4semidirect
12. #16810828C4semidirect
13. #16810828C4semidirect
243. C53#9165semidirect

Maximal subgroups

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

subgroupindex
#202848142
#202745402
#202733312
#202732152
#202726152
#202725702
C53×#29742
#916553