Finite Group Databasebeta

← #27279922

Subgroups — #27279922

Order 13568.

458 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284837C211yesmaximal, characteristic
2#20274545C211yesmaximal, characteristic, complemented
3#20273327C211yesmaximal, characteristic
4#20273211C211yesmaximal, characteristic
5#20272615C211yesmaximal, characteristic, complemented
6#20272570C211yesmaximal, characteristic, complemented
7C53×#2984C211yesmaximal, characteristic, complemented
8#16811173C2211yescharacteristic
9#16810936—21no—
10#16810918C411yescharacteristic, complemented
11#16810918C411yescharacteristic, complemented
12#16810823C411yescharacteristic, complemented
13#16810823C411yescharacteristic, complemented
14#16810762—21no—
15#16810721—21no—
16#16810721—21no—
17#16810680C2211yescharacteristic
18C2×#13133141—21no—
19C2×#13133128C2211yescharacteristic
20C4×C4⋊Dic53C2211yescharacteristic
21#16810385—21no—
22#16810362—22no—
23#16810358—21no—
24#16810358—21no—
25C2×C4×Q8×C53C2211yescharacteristic
26C53×#446—21no—
27C53×#446—22no—
28C8×Q8×C53—21no—
29C53×#432—21no—
30C53×#432C2211yescharacteristic
31C53×#431—21no—
32C53×#431C2211yescharacteristic
33C2×#11719945—21no—
34C2×#11719924—21no—
35C2×#11719924—21no—
36C22×#88322—21no—
37#13133217C2×C411yescharacteristic
38#13133214C2×C411yescharacteristic
39#13133205—21no—
40#13133191—21no—
41#13133191—21no—
42C4×D212—21no—
43C4×D212—21no—
44C2×C4⋊Dic53—21no—
45C2×C4×Dic53—21no—
46#13133150—21no—
47#13133150—21no—
48C4⋊#88322C2×C411yescharacteristic
49C4⋊#88322C2×C411yescharacteristic
50#13133141—24no—
51#13133141—21no—
52#13133141—21no—
53#13133137—21no—
54#13133137—21no—
55C22⋊#88322—21no—
56#13133128—21no—
57#13133128—21no—
58#13133128—21no—
59#13133124—21no—
60C22×Q8×C53D411yescharacteristic
61C4×Q8×C53—21no—
62C4×Q8×C53—21no—
63C4×Q8×C53C2×C411yescharacteristic
64C4×Q8×C53C2×C411yescharacteristic
65C4×Q8×C53D411yescharacteristic
66C4×Q8×C53D411yescharacteristic
67C2×C4⋊C4×C53—21no—
68C2×C4⋊C4×C53D411yescharacteristic
69C2×C42×C53—21no—
70C2×C42×C53C2311yescharacteristic
71C4⋊C8×C53—22no—
72C4⋊C8×C53—22no—
73C4⋊C8×C53—21no—
74C4⋊C8×C53—21no—
75C22⋊C8×C53—21no—
76C22⋊C8×C53D411yescharacteristic
77C22⋊C8×C53D411yescharacteristic
78#97×C53—22no—
79#97×C53D412yes—
80C4×C8×C53—22no—
81C4×C8×C53—21no—
82C2×Q8×C53—21no—
83C2×Q8×C53—22no—
84C2×Q8×C53—41no—
85C2×Q8×C53C22⋊C411yescharacteristic
86C2×Q8×C53C22⋊C411yescharacteristic
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53#4111yescharacteristic
92C2×C22⋊D53—41no—
93C22×Dic53—41no—
94#11719945—81no—
95#11719945—81no—
96C2×D212—21no—
97C2×D212—21no—
98C2×D212—21no—
99C2×C4×D53—41no—
100C2×Dic106—21no—
101C2×C8×C53—21no—
102C2×C8×C53—21no—
103C2×C8×C53—22no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53C22⋊C411yescharacteristic
110C4⋊C4×C53C22⋊C411yescharacteristic
111C42×C53—21no—
112C42×C53—21no—
113C42×C53—21no—
114C42×C53—21no—
115C42×C53C22×C411yescharacteristic
116C42×C53C2×D411yescharacteristic
117C42×C53C2×D411yescharacteristic
118C42×C53#4111yescharacteristic
119#11719924—42no—
120#11719924—42no—
121#11719921—41no—
122#11719921—41no—
123C4⋊Dic53—22no—
124C4⋊Dic53—21no—
125C4⋊Dic53—21no—
126C4⋊Dic53—41no—
127C4×Dic53—81no—
128C2×#88322—22no—
129C2×#88322—21no—
130C2×#88322—21no—
131C2×#88322—41no—
132C2×#88322—41no—
133C23×C53#12111yescharacteristic
134C22×D53—41no—
135Q8×C53—21no—
136Q8×C53—21no—
137Q8×C53—21no—
138Q8×C53—42no—
139Q8×C53#10412yes—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—41no—
149C2×C4×C53—42no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53C2×C22⋊C411yescharacteristic
153C2×C4×C53C4×D411yescharacteristic
154C2×C4×C53C4×D411yescharacteristic
155C2×C4×C53#12011yescharacteristic
156C2×C4×C53#12111yescharacteristic
157C2×C4×C53#12311yescharacteristic
158C22⋊D53—81no—
159C22⋊D53—81no—
160C2×Dic53—42no—
161C2×Dic53—41no—
162C2×Dic53—81no—
163C2×Dic53—81no—
164D212—41no—
165D212—41no—
166D212—81no—
167D212—81no—
168C4×D53—82no—
169Dic106—81no—
170C8×C53—41no—
171C8×C53—41no—
172C8×C53—42no—
173#88322—42no—
174#88322—41no—
175#88322—41no—
176#9164—531nomaximal, Sylow
177C22×C53—21no—
178C22×C53—21no—
179C22×C53#38611yescharacteristic
180C22×C53#47111yescharacteristic
181C22×C53#47211yescharacteristic
182C2×D53—81no—
183C2×D53—81no—
184C2×D53—81no—
185C4×C53—21no—
186C4×C53—22no—
187C4×C53—21no—
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—41no—
192C4×C53—41no—
193C4×C53—41no—
194C4×C53C2×#10411yescharacteristic
195C4×C53#42111yescharacteristic
196Dic53—82no—
197Dic53—81no—
198#3112—531no—
199#2984D5311yescharacteristic, complemented
200#1763—531no—
201#1530—531no—
202#1502—531no—
203#1302—531no—
204#1285—531no—
205C2×C53—21no—
206C2×C53—21no—
207C2×C53#187111yescharacteristic
208C2×C53#190511yescharacteristic
209C2×C53#190711yescharacteristic
210D53—161no—
211#532—531no—
212C2×C4×Q8C2×D5311yescharacteristic
213#480—1061no—
214#477—531no—
215#477—531no—
216#460—531no—
217#460—531no—
218#448—1061no—
219#446—21no—
220#446—22no—
221C8×Q8—21no—
222#440—1061no—
223#440—1061no—
224#432—21no—
225#432—531no—
226#432C2×D5311yescharacteristic
227#431—21no—
228#431C2×D5311yescharacteristic
229C2×#103—1061no—
230C2×#97—531no—
231C4×C4⋊C4—531no—
232#340—1061no—
233#329—1062no—
234#327—1061no—
235#327—1061no—
236C53#916411yescharacteristic, complemented, Sylow
237C2×#41—1061no—
238C22×Q8D21211yescharacteristic
239C2×SD16—1061no—
240C2×SD16—1061no—
241C22×C8—1061no—
242C4⋊Q8—531no—
243#127—531no—
244#124—1061no—
245#121—1061no—
246#121—1061no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8C4×D5311yescharacteristic
250C4×Q8C4×D5311yescharacteristic
251C4×Q8C22⋊D5311yescharacteristic
252C4×Q8C22⋊D5311yescharacteristic
253C4×D4—1061no—
254C4×D4—1061no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—1061no—
257C2×C4⋊C4D21211yescharacteristic
258C2×C42—21no—
259C2×C42—1061no—
260C2×C42C22×D5311yescharacteristic
261#107—1061no—
262#107—1061no—
263C4⋊C8—22no—
264C4⋊C8—22no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—531no—
268C4⋊C8—531no—
269#103—1064no—
270#103—1061no—
271#103—1061no—
272#102—1061no—
273#102—1061no—
274C22⋊C8—21no—
275C22⋊C8—1061no—
276C22⋊C8C22⋊D5311yescharacteristic
277C22⋊C8C22⋊D5311yescharacteristic
278#97—22no—
279#97—1061no—
280#97—1061no—
281#97—1061no—
282#97C22⋊D5312yes—
283C4×C8—22no—
284C4×C8—21no—
285#95—1061no—
286#41—4241no—
287#41—4241no—
288C2×Q8—21no—
289C2×Q8—22no—
290C2×Q8—41no—
291C2×Q8—1061no—
292C2×Q8#1171992111yescharacteristic
293C2×Q8#1171992111yescharacteristic
294C2×D4—1061no—
295C2×D4—1061no—
296C2×D4—1061no—
297C2×D4—2121no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—2121no—
301C22×C4—2121no—
302C22×C4C2×D21211yescharacteristic
303C22×C4#1171994511yescharacteristic
304C22×C4C2×C22⋊D5311yescharacteristic
305SD16—2122no—
306SD16—2122no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—22no—
310C2×C8—1062no—
311C2×C8—1061no—
312C2×C8—1061no—
313C2×C8—2121no—
314C2×C8—2121no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—1062no—
321C4⋊C4—1061no—
322C4⋊C4—1061no—
323C4⋊C4—2121no—
324C4⋊C4#1171992111yescharacteristic
325C4⋊C4#1171992111yescharacteristic
326C22⋊C4—2121no—
327C22⋊C4—2121no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—4241no—
333C42C2×C4×D5311yescharacteristic
334C42#1171994511yescharacteristic
335C42C2×C22⋊D5311yescharacteristic
336C42C2×C22⋊D5311yescharacteristic
337C23—2121no—
338C23#1313319111yescharacteristic
339Q8—21no—
340Q8—21no—
341Q8—21no—
342Q8—42no—
343Q8—4241no—
344Q8#1313314312yes—
345D4—4241no—
346D4—4241no—
347D4—4241no—
348D4—4241no—
349D4—2121no—
350D4—2121no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—4242no—
364C2×C4—4241no—
365C2×C4—4241no—
366C2×C4—2122no—
367C2×C4—2121no—
368C2×C4C2×#1171992111yescharacteristic
369C2×C4C4×C22⋊D5311yescharacteristic
370C2×C4C4×C22⋊D5311yescharacteristic
371C2×C4#1313318611yescharacteristic
372C2×C4#1313319111yescharacteristic
373C2×C4#1313320011yescharacteristic
374C8—41no—
375C8—41no—
376C8—42no—
377C8—2122no—
378C8—2121no—
379C8—2121no—
380C22—21no—
381C22—21no—
382C22—4241no—
383C22—4241no—
384C22—4241no—
385C22#1681050211yescharacteristic
386C22#1681088611yescharacteristic
387C22#1681089311yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—22no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—4242no—
398C4—4241no—
399C4C2×#1313314311yescharacteristic
400C4#1681063911yescharacteristic
401C2—21no—
402C2—21no—
403C2—8481no—
404C2#2027501711yescharacteristic
405C2#2027520911yescharacteristic
406C2#2027522311yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2984C2
socleC22×C53#386
prime radical#16811173C22
cosocleC2×C42×C53C23
hypercenter#2984D53
hypocenterC53#9164
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2984

Complements

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

NH ≅ G/Ntype
2. #20274545C2semidirect
5. #20272615C2semidirect
6. #20272570C2semidirect
7. C53×#2984C2semidirect
10. #16810918C4semidirect
11. #16810918C4semidirect
12. #16810823C4semidirect
13. #16810823C4semidirect
199. #2984D53semidirect
236. C53#9164semidirect

Maximal subgroups

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

subgroupindex
#202848372
#202745452
#202733272
#202732112
#202726152
#202725702
C53×#29842
#916453