Finite Group Databasebeta

← #27275143

Subgroups — #27275143

Order 13568.

436 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274401C211yesmaximal, characteristic, complemented
2#20273303C211yesmaximal, characteristic
3#20273201C211yesmaximal, characteristic
4#20273004C211yesmaximal, characteristic, complemented
5#20272613C211yesmaximal, characteristic, complemented
6#20272607C211yesmaximal, characteristic, complemented
7C53×#2966C211yesmaximal, characteristic
8#16810676C2211yescharacteristic
9C4×#11719917C2211yescharacteristic
10C2×#13133128C2211yescharacteristic
11C2×#13133127—21no—
12C2×C8×Dic53—21no—
13#16810469C2211yescharacteristic
14#16810365C412yescomplemented
15#16810363C412yescomplemented
16#16810360C412yes—
17#16810357C412yes—
18#16810345—21no—
19#16810344—21no—
20C53×#529C2211yescharacteristic
21C53×#446C411yescharacteristic, complemented
22C8×Q8×C53C411yescharacteristic, complemented
23C53×#436C411yescharacteristic, complemented
24C53×#435—21no—
25C8×D4×C53—21no—
26C8×D4×C53C411yescharacteristic, complemented
27C53×#433—21no—
28C53×#432C2211yescharacteristic
29C53×#405—21no—
30C2×C4×C8×C53C2211yescharacteristic
31C2×#11719917—21no—
32C22×#88322—21no—
33C2×C4×Dic53—21no—
34C4⋊#88322—21no—
35C22⋊#88322C2×C412yes—
36#13133128—21no—
37#13133128C2×C412yes—
38#13133128C2×C412yes—
39#13133127—24no—
40C4×#88322—21no—
41C4×#88322—21no—
42C8×Dic53—24no—
43#13133124—21no—
44#13133124C2×C412yes—
45C2×M16×C53—21no—
46C22×C8×C53—21no—
47C22×C8×C53D411yescharacteristic
48C22×C8×C53D411yescharacteristic
49#126×C53—21no—
50#125×C53C2×C411yescharacteristic
51#124×C53C2×C411yescharacteristic
52#123×C53—21no—
53C22⋊Q8×C53C2×C411yescharacteristic
54#121×C53C2×C411yescharacteristic
55C4×Q8×C53C2×C411yescharacteristic
56C4×D4×C53—21no—
57C4×D4×C53C2×C411yescharacteristic
58#117×C53—21no—
59C2×C42×C53C2311yescharacteristic
60C4⋊C8×C53—21no—
61C4⋊C8×C53—21no—
62C4⋊C8×C53C2×C411yescharacteristic
63C4⋊C8×C53C2×C411yescharacteristic
64C22⋊C8×C53—21no—
65C22⋊C8×C53—21no—
66C22⋊C8×C53—21no—
67#97×C53—21no—
68#97×C53—21no—
69C4×C8×C53—21no—
70C4×C8×C53C2×C411yescharacteristic
71C4×C8×C53C2×C411yescharacteristic
72C4×C8×C53D411yescharacteristic
73C4×C8×C53D411yescharacteristic
74C2×Q8×C53C4211yescharacteristic
75C2×D4×C53—21no—
76C2×D4×C53C4211yescharacteristic
77C22×C4×C53—21no—
78C22×C4×C53C22×C411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53#4111yescharacteristic
81C22×Dic53—22no—
82M16×C53—42no—
83C2×C8×C53—22no—
84C2×C8×C53—22no—
85C2×C8×C53—21no—
86C2×C8×C53—21no—
87C2×C8×C53—42no—
88C2×C8×C53C22⋊C411yescharacteristic
89C2×C8×C53C22⋊C411yescharacteristic
90C2×C8×C53C22⋊C411yescharacteristic
91C2×C8×C53C22⋊C411yescharacteristic
92C4⋊C4×C53—21no—
93C4⋊C4×C53—21no—
94C4⋊C4×C53—21no—
95C4⋊C4×C53—21no—
96C4⋊C4×C53C4211yescharacteristic
97C4⋊C4×C53C4211yescharacteristic
98C22⋊C4×C53—21no—
99C22⋊C4×C53—21no—
100C22⋊C4×C53—21no—
101C22⋊C4×C53—21no—
102C22⋊C4×C53—21no—
103C42×C53—21no—
104C42×C53C22×C411yescharacteristic
105C42×C53C22×C411yescharacteristic
106C42×C53C2×D411yescharacteristic
107C42×C53#4111yescharacteristic
108C4×Dic53—22no—
109C4×Dic53—22no—
110#11719917—42no—
111C2×#88322—22no—
112C2×#88322—22no—
113C2×#88322—22no—
114C2×#88322—22no—
115C2×#88322—22no—
116C23×C53—21no—
117C23×C53C4×D411yescharacteristic
118Q8×C53—41no—
119D4×C53—41no—
120D4×C53—42no—
121C2×C4×C53—22no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53C2×C4211yescharacteristic
131C2×C4×C53C2×C22⋊C411yescharacteristic
132C2×C4×C53#11711yescharacteristic
133C2×C4×C53C4×D411yescharacteristic
134C2×C4×C53C4×D411yescharacteristic
135C2×C4×C53C4×D411yescharacteristic
136C2×Dic53—24no—
137C2×Dic53—42no—
138C2×Dic53—42no—
139C8×C53—21no—
140C8×C53—21no—
141C8×C53—41no—
142C8×C53—41no—
143C8×C53#10412yes—
144C8×C53#10412yes—
145#88322—24no—
146#88322—42no—
147#88322—42no—
148#8249—531nomaximal, Sylow
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53C4×C22⋊C411yescharacteristic
155C22×C53#44412yes—
156C4×C53—21no—
157C4×C53—21no—
158C4×C53—21no—
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—41no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53C2×#10411yescharacteristic
165C4×C53C2×#10411yescharacteristic
166Dic53—44no—
167#2966D5311yescharacteristic
168#1726—531no—
169#1524—531no—
170#1500—531no—
171#1451—531no—
172#1302—531no—
173#1300—531no—
174C2×C53—22no—
175C2×C53—81no—
176C2×C53#175711yescharacteristic
177C2×C53C4×#10411yescharacteristic
178C2×C53#176511yescharacteristic
179#529C2×D5311yescharacteristic
180#446Dic5311yescharacteristic, complemented
181C8×Q8Dic5311yescharacteristic, complemented
182#436Dic5311yescharacteristic, complemented
183#435—21no—
184C8×D4—21no—
185C8×D4Dic5311yescharacteristic, complemented
186#433—21no—
187#432C2×D5311yescharacteristic
188#431—531no—
189#405—21no—
190C4×M16—531no—
191C2×#97—531no—
192C2×#97—1061no—
193C2×C4×C8—1061no—
194C2×C4×C8C2×D5311yescharacteristic
195#376—531no—
196#330—532no—
197#329—532no—
198#328—532no—
199#327—532no—
200#322—1061no—
201#322—1061no—
202C53#824911yescharacteristic, complemented, Sylow
203C2×M16—21no—
204C2×M16—1061no—
205C22×C8—21no—
206C22×C8—1061no—
207C22×C8C22⋊D5311yescharacteristic
208C22×C8C22⋊D5311yescharacteristic
209#126—21no—
210#125C4×D5311yescharacteristic
211#124C4×D5311yescharacteristic
212#123—21no—
213C22⋊Q8C4×D5311yescharacteristic
214#121C4×D5311yescharacteristic
215C4×Q8C2×Dic5311yescharacteristic
216C4×D4—21no—
217C4×D4C2×Dic5311yescharacteristic
218#117—21no—
219C2×C42—1061no—
220C2×C42C22×D5311yescharacteristic
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—1061no—
224C4⋊C8C2×Dic5311yescharacteristic
225C4⋊C8C2×Dic5311yescharacteristic
226C22⋊C8—21no—
227C22⋊C8—21no—
228C22⋊C8—21no—
229C22⋊C8—532no—
230#97—21no—
231#97—21no—
232#97—532no—
233#97—532no—
234#97—1064no—
235#97—1061no—
236C4×C8—21no—
237C4×C8—1064no—
238C4×C8—1061no—
239C4×C8—1061no—
240C4×C8C2×Dic5311yescharacteristic
241C4×C8C2×Dic5311yescharacteristic
242C4×C8C22⋊D5311yescharacteristic
243C4×C8C22⋊D5311yescharacteristic
244#95—532no—
245#95—1061no—
246C2×Q8C4×Dic5311yescharacteristic
247C2×D4—21no—
248C2×D4C4×Dic5311yescharacteristic
249C22×C4—21no—
250C22×C4—1062no—
251C22×C4C2×C4×D5311yescharacteristic
252C22×C4#1171994511yescharacteristic
253C22×C4C2×C22⋊D5311yescharacteristic
254M16—42no—
255M16—2122no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—42no—
261C2×C8—1062no—
262C2×C8—1062no—
263C2×C8—1062no—
264C2×C8—1062no—
265C2×C8—1062no—
266C2×C8C22⋊Dic5311yescharacteristic
267C2×C8C22⋊Dic5311yescharacteristic
268C2×C8C22⋊Dic5311yescharacteristic
269C2×C8C22⋊Dic5311yescharacteristic
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4C4×Dic5311yescharacteristic
275C4⋊C4C4×Dic5311yescharacteristic
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—21no—
281C42—21no—
282C42—1062no—
283C42—1062no—
284C42C2×C4×D5311yescharacteristic
285C42#1171994511yescharacteristic
286C42C22×Dic5311yescharacteristic
287C42C2×C22⋊D5311yescharacteristic
288C23—21no—
289C23C4×C22⋊D5311yescharacteristic
290Q8—41no—
291D4—41no—
292D4—42no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—1064no—
303C2×C4—2122no—
304C2×C4—2122no—
305C2×C4C2×C4×Dic5311yescharacteristic
306C2×C4C2×C22⋊Dic5311yescharacteristic
307C2×C4#1313317611yescharacteristic
308C2×C4C4×C22⋊D5311yescharacteristic
309C2×C4C4×C22⋊D5311yescharacteristic
310C2×C4C4×C22⋊D5311yescharacteristic
311C8—21no—
312C8—21no—
313C8—41no—
314C8—41no—
315C8—1064no—
316C8—2122no—
317C8—2122no—
318C8#1313314412yes—
319C8#1313314412yes—
320C22—21no—
321C22—21no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22C4×C22⋊Dic5311yescharacteristic
326C22#1681074312yes—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—2124no—
336C4C2×#1313314411yescharacteristic
337C4C2×#1313314411yescharacteristic
338C2—22no—
339C2—81no—
340C2#2027451511yescharacteristic
341C2C4×#1313314411yescharacteristic
342C2#2027455711yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊Dic53
commutatorC2×C4×C53C2×C42
FrattiniC2×C42C22×D53
FittingC53×#2966C2
socleC22×C53C4×C22⋊C4
prime radical#121×C53C2×C4
cosocleC2×C42×C53C23
hypercenter#2966D53
hypocenterC53#8249
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ #2966

Complements

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

NH ≅ G/Ntype
1. #20274401C2semidirect
4. #20273004C2semidirect
5. #20272613C2semidirect
6. #20272607C2semidirect
14. #16810365C4semidirect
15. #16810363C4semidirect
21. C53×#446C4semidirect
22. C8×Q8×C53C4semidirect
23. C53×#436C4semidirect
26. C8×D4×C53C4semidirect
180. #446Dic53semidirect
181. C8×Q8Dic53semidirect
182. #436Dic53semidirect
185. C8×D4Dic53semidirect
202. C53#8249semidirect

Maximal subgroups

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

subgroupindex
#202744012
#202733032
#202732012
#202730042
#202726132
#202726072
C53×#29662
#824953