Finite Group Databasebeta

← #27272201

Subgroups — #27272201

Order 13568.

504 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277253C211yesmaximal, characteristic
2#20274782C211yesmaximal, characteristic
3#20274694C211yesmaximal, characteristic
4#20272541C211yesmaximal, characteristic, complemented
5#20272539C211yesmaximal, characteristic, complemented
6#20272537C211yesmaximal, characteristic, complemented
7C53×#1272C211yesmaximal, characteristic, complemented
8C2×#13133193C411yescharacteristic
9C2×Q8×Dic53C411yescharacteristic
10C2×C42×D53C2211yescharacteristic
11C2×#13133146—21no—
12C2×#13133129—21no—
13C2×#13133129—21no—
14#16810521C2211yescharacteristic
15#16810510—21no—
16#16810497—21no—
17#16810480C411yescharacteristic
18C4⋊C4×Dic53C411yescharacteristic
19#16810474—21no—
20#16810470—21no—
21#16810378C2211yescharacteristic
22#16810378C2211yescharacteristic
23#16810377—21no—
24C53×#391C2211yescharacteristic
25C53×#336C2211yescharacteristic
26C53×#336C2211yescharacteristic
27C22×C4×D53C2×C411yescharacteristic
28C22×C4×D53D411yescharacteristic
29C22×C4×D53D411yescharacteristic
30C22×#88322—21no—
31C22×#88322—21no—
32#13133193—22no—
33#13133193C814yescomplemented
34Q8×Dic53—42no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×C4⋊Dic53C2×C411yescharacteristic
38C2×#11719919—21no—
39C2×#11719919—21no—
40C2×#11719921—21no—
41C2×#11719921—21no—
42C2×C4×Dic53—21no—
43C2×C4×Dic53—21no—
44C2×C4×Dic53C2×C411yescharacteristic
45C2×C4×Dic53D411yescharacteristic
46C2×C4×Dic53D411yescharacteristic
47C42×D53—42no—
48#13133146—42no—
49#13133129—42no—
50#13133129—42no—
51#13133124—21no—
52#13133124—21no—
53#13133124—21no—
54C22×Q8×C53C2×C411yescharacteristic
55C22×C8×C53—21no—
56C22×C8×C53—21no—
57C2×C4⋊C4×C53C2×C411yescharacteristic
58C2×C42×C53C2311yescharacteristic
59#95×C53—21no—
60#95×C53—21no—
61C23×D53C22⋊C411yescharacteristic
62C2×Q8×C53—21no—
63C2×Q8×C53—41no—
64C2×Q8×C53—41no—
65C2×Q8×C53C2×C812yes—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53C22×C411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53C22⋊C411yescharacteristic
74C22×Dic53C22⋊C411yescharacteristic
75C22×Dic53C22⋊C411yescharacteristic
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×C4×D53—21no—
79C2×C4×D53—21no—
80C2×C4×D53—22no—
81C2×C4×D53—41no—
82C2×C4×D53—41no—
83C2×C4×D53—41no—
84C2×C4×D53—41no—
85C2×C4×D53—41no—
86C2×C4×D53C2×C812yes—
87C2×C4×D53M1612yes—
88C2×C8×C53—41no—
89C2×C8×C53—41no—
90C2×C8×C53—41no—
91C2×C8×C53—41no—
92C2×C8×C53—41no—
93C2×C8×C53—41no—
94C4⋊C4×C53—41no—
95C42×C53—41no—
96#11719921—24no—
97#11719921—42no—
98C4⋊Dic53—21no—
99C4⋊Dic53—42no—
100C4⋊Dic53—41no—
101C4⋊Dic53C2×C812yes—
102#11719919—24no—
103#11719919—41no—
104#11719919—41no—
105C4×Dic53—22no—
106C4×Dic53—22no—
107C4×Dic53—41no—
108C4×Dic53—42no—
109C4×Dic53—42no—
110C2×#88322—41no—
111C2×#88322—41no—
112C2×#88322—41no—
113C2×#88322—41no—
114C2×#88322—41no—
115C2×#88322—41no—
116C23×C53C2×C22⋊C411yescharacteristic
117C22×D53—21no—
118C22×D53—21no—
119C22×D53—21no—
120C22×D53—21no—
121C22×D53—21no—
122C22×D53—21no—
123C22×D53C22⋊C812yes—
124Q8×C53—42no—
125Q8×C53—42no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—22no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53C22×C811yescharacteristic
139C2×C4×C53C2×M1611yescharacteristic
140C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C53#13711yescharacteristic
142C2×C4×C53#13711yescharacteristic
143C2×C4×C53#13711yescharacteristic
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—22no—
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—21no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53C22⋊C812yes—
158C2×Dic53#9912yes—
159C4×D53—42no—
160C4×D53—42no—
161C4×D53—42no—
162C4×D53—42no—
163C4×D53—42no—
164C4×D53—42no—
165C8×C53—81no—
166C8×C53—81no—
167#88322—81no—
168#88322—81no—
169#7600—531nomaximal, Sylow
170C22×C53C2×C22⋊C811yescharacteristic
171C22×C53C2×#9911yescharacteristic
172C22×C53#41311yescharacteristic
173C22×C53#41711yescharacteristic
174C22×C53#41911yescharacteristic
175C22×C53#42111yescharacteristic
176C22×C53#42111yescharacteristic
177C2×D53—22no—
178C2×D53—22no—
179C2×D53—22no—
180C2×D53—41no—
181C2×D53—41no—
182C2×D53—41no—
183C2×D53—41no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—21no—
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—42no—
191C4×C53—81no—
192Dic53—22no—
193Dic53—22no—
194Dic53—42no—
195Dic53—41no—
196Dic53—81no—
197C4×C22⋊Q8—531no—
198#1817—531no—
199#1797—531no—
200#1272—531no—
201#1272D5311yescharacteristic, complemented
202#1271—531no—
203#1271—531no—
204C2×C53#145911yescharacteristic
205C2×C53#147711yescharacteristic
206C2×C53#147911yescharacteristic
207C2×C53#151111yescharacteristic
208C2×C53#151311yescharacteristic
209C2×C53#153511yescharacteristic
210C2×C53#155911yescharacteristic
211D53—42no—
212C2×C22⋊Q8—531no—
213C2×C4×Q8—531no—
214C22×C42—531no—
215C2×C4⋊C8—1061no—
216C2×C22⋊C8—1061no—
217C2×C22⋊C8—1061no—
218#391C2×D5311yescharacteristic
219#389—531no—
220#387—1061no—
221#385—1061no—
222#380—531no—
223C4×C4⋊C4—531no—
224C4×C4⋊C4—1061no—
225C4×C22⋊C4—1061no—
226#336—531no—
227#336—531no—
228#336—1061no—
229#336C2×D5311yescharacteristic
230#336C2×D5311yescharacteristic
231C53#760011yescharacteristic, complemented, Sylow
232C22×Q8C4×D5311yescharacteristic
233C23×C4—531no—
234C23×C4—531no—
235C23×C4—531no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—1061no—
239C22×C8—1061no—
240C22⋊Q8—534no—
241C22⋊Q8—1062no—
242C4×Q8—2122no—
243C2×C4⋊C4—531no—
244C2×C4⋊C4—531no—
245C2×C4⋊C4—1061no—
246C2×C4⋊C4—1061no—
247C2×C4⋊C4—1061no—
248C2×C4⋊C4C4×D5311yescharacteristic
249C2×C22⋊C4—1061no—
250C2×C22⋊C4—1061no—
251C2×C42—531no—
252C2×C42—531no—
253C2×C42—531no—
254C2×C42—1061no—
255C2×C42—1061no—
256C2×C42—2122no—
257C2×C42C22×D5311yescharacteristic
258C4⋊C8—2122no—
259C22⋊C8—2122no—
260C22⋊C8—2122no—
261#95—21no—
262#95—21no—
263#95—1061no—
264#95—1061no—
265#95—1061no—
266C24—531no—
267C2×Q8—21no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8C8×D5312yes—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—531no—
274C22×C4—531no—
275C22×C4—532no—
276C22×C4—532no—
277C22×C4—531no—
278C22×C4—1062no—
279C22×C4—1061no—
280C22×C4—1061no—
281C22×C4—1061no—
282C22×C4—1061no—
283C22×C4—1061no—
284C22×C4—1061no—
285C22×C4—2121no—
286C22×C4—2121no—
287C22×C4—2121no—
288C22×C4—2121no—
289C22×C4—2121no—
290C22×C4C2×C4×D5311yescharacteristic
291C22×C4D4×D5311yescharacteristic
292C22×C4D4×D5311yescharacteristic
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—2121no—
300C2×C8—2121no—
301C2×C8—2121no—
302C2×C8—2121no—
303C2×C8—2121no—
304C2×C8—2121no—
305C4⋊C4—41no—
306C4⋊C4—532no—
307C4⋊C4—1064no—
308C4⋊C4—1061no—
309C4⋊C4—2121no—
310C4⋊C4—2122no—
311C4⋊C4—2121no—
312C4⋊C4—2121no—
313C22⋊C4—1064no—
314C22⋊C4—2122no—
315C42—41no—
316C42—1062no—
317C42—1062no—
318C42—2121no—
319C42—2122no—
320C42—2122no—
321C23—532no—
322C23—1061no—
323C23—1061no—
324C23—1061no—
325C23—1061no—
326C23—1061no—
327C23—1061no—
328C23C22⋊C4×D5311yescharacteristic
329Q8—42no—
330Q8—42no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—532no—
344C2×C4—532no—
345C2×C4—1061no—
346C2×C4—1061no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—1062no—
350C2×C4—1061no—
351C2×C4—1061no—
352C2×C4—1061no—
353C2×C4—2122no—
354C2×C4—2122no—
355C2×C4—2122no—
356C2×C4—2122no—
357C2×C4—2122no—
358C2×C4—2122no—
359C2×C4—2121no—
360C2×C4—2121no—
361C2×C4—2121no—
362C2×C4—2121no—
363C2×C4—2121no—
364C2×C4C2×C8×D5311yescharacteristic
365C2×C4M16×D5311yescharacteristic
366C2×C4#1313324411yescharacteristic
367C2×C4#1313324611yescharacteristic
368C2×C4#1313324911yescharacteristic
369C2×C4#1313324911yescharacteristic
370C8—81no—
371C8—81no—
372C8—4241no—
373C8—4241no—
374C22—1062no—
375C22—1062no—
376C22—1062no—
377C22—2121no—
378C22—2121no—
379C22—2121no—
380C22—2121no—
381C22C22⋊C8×D5311yescharacteristic
382C22#99×D5311yescharacteristic
383C22#1681060411yescharacteristic
384C22#1681062311yescharacteristic
385C22#1681063011yescharacteristic
386C22#1681063811yescharacteristic
387C22#1681063811yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—42no—
395C4—81no—
396C4—1062no—
397C4—1062no—
398C4—4241no—
399C4—2121no—
400C4—2122no—
401C2—2122no—
402C2#2027303011yescharacteristic
403C2#2027310111yescharacteristic
404C2#2027310811yescharacteristic
405C2#2027324611yescharacteristic
406C2#2027325711yescharacteristic
407C2#2027334311yescharacteristic
408C2#2027344211yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC2×C4×C53C22×C8
FrattiniC2×C42C22×D53
FittingC53×#1272C2
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1272D53
hypocenterC53#7600
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
4. #20272541C2semidirect
5. #20272539C2semidirect
6. #20272537C2semidirect
7. C53×#1272C2semidirect
33. #13133193C8semidirect
201. #1272D53semidirect
231. C53#7600semidirect

Maximal subgroups

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

subgroupindex
#202772532
#202747822
#202746942
#202725412
#202725392
#202725372
C53×#12722
#760053