Finite Group Databasebeta

← #27275136

Subgroups — #27275136

Order 13568.

392 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274466C211yesmaximal, 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
11#16810469C2211yescharacteristic
12#16810377—21no—
13#16810377—21no—
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—
39C4×#88322—21no—
40C4×#88322—21no—
41#13133124—21no—
42#13133124C2×C412yes—
43C2×M16×C53—21no—
44C22×C8×C53—21no—
45C22×C8×C53D411yescharacteristic
46C22×C8×C53D411yescharacteristic
47#126×C53—21no—
48#125×C53C2×C411yescharacteristic
49#124×C53C2×C411yescharacteristic
50#123×C53—21no—
51C22⋊Q8×C53C2×C411yescharacteristic
52#121×C53C2×C411yescharacteristic
53C4×Q8×C53C2×C411yescharacteristic
54C4×D4×C53—21no—
55C4×D4×C53C2×C411yescharacteristic
56#117×C53—21no—
57C2×C42×C53C2311yescharacteristic
58C4⋊C8×C53—21no—
59C4⋊C8×C53—21no—
60C4⋊C8×C53C2×C411yescharacteristic
61C4⋊C8×C53C2×C411yescharacteristic
62C22⋊C8×C53—21no—
63C22⋊C8×C53—21no—
64C22⋊C8×C53—21no—
65#97×C53—21no—
66#97×C53—21no—
67C4×C8×C53—21no—
68C4×C8×C53C2×C411yescharacteristic
69C4×C8×C53C2×C411yescharacteristic
70C4×C8×C53D411yescharacteristic
71C4×C8×C53D411yescharacteristic
72C2×Q8×C53C4211yescharacteristic
73C2×D4×C53—21no—
74C2×D4×C53C4211yescharacteristic
75C22×C4×C53—21no—
76C22×C4×C53C22×C411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53#4111yescharacteristic
79C22×Dic53—22no—
80M16×C53—42no—
81C2×C8×C53—21no—
82C2×C8×C53—21no—
83C2×C8×C53—41no—
84C2×C8×C53—41no—
85C2×C8×C53—42no—
86C2×C8×C53C22⋊C411yescharacteristic
87C2×C8×C53C22⋊C411yescharacteristic
88C2×C8×C53C22⋊C411yescharacteristic
89C2×C8×C53C22⋊C411yescharacteristic
90C4⋊C4×C53—21no—
91C4⋊C4×C53—21no—
92C4⋊C4×C53—21no—
93C4⋊C4×C53—21no—
94C4⋊C4×C53C4211yescharacteristic
95C4⋊C4×C53C4211yescharacteristic
96C22⋊C4×C53—21no—
97C22⋊C4×C53—21no—
98C22⋊C4×C53—21no—
99C22⋊C4×C53—21no—
100C22⋊C4×C53—21no—
101C42×C53—21no—
102C42×C53C22×C411yescharacteristic
103C42×C53C22×C411yescharacteristic
104C42×C53C2×D411yescharacteristic
105C42×C53#4111yescharacteristic
106C4×Dic53—41no—
107C4×Dic53—41no—
108#11719917—42no—
109C2×#88322—22no—
110C2×#88322—22no—
111C2×#88322—22no—
112C2×#88322—41no—
113C2×#88322—41no—
114C23×C53—21no—
115C23×C53C4×D411yescharacteristic
116Q8×C53—41no—
117D4×C53—41no—
118D4×C53—42no—
119C2×C4×C53—22no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53C2×C4211yescharacteristic
129C2×C4×C53C2×C22⋊C411yescharacteristic
130C2×C4×C53#11711yescharacteristic
131C2×C4×C53C4×D411yescharacteristic
132C2×C4×C53C4×D411yescharacteristic
133C2×C4×C53C4×D411yescharacteristic
134C2×Dic53—42no—
135C2×Dic53—42no—
136C2×Dic53—42no—
137C8×C53—21no—
138C8×C53—21no—
139C8×C53—21no—
140C8×C53—21no—
141C8×C53—41no—
142C8×C53—41no—
143#88322—42no—
144#88322—42no—
145#88322—42no—
146#8248—531nomaximal, Sylow
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53C4×C22⋊C411yescharacteristic
153C22×C53#44412yes—
154C4×C53—21no—
155C4×C53—21no—
156C4×C53—21no—
157C4×C53—21no—
158C4×C53—21no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53#42111yescharacteristic
163C4×C53#42111yescharacteristic
164Dic53—82no—
165#2966D5311yescharacteristic
166#1746—531no—
167#1524—531no—
168#1500—531no—
169#1451—531no—
170#1302—531no—
171#1300—531no—
172C2×C53—22no—
173C2×C53—81no—
174C2×C53#175711yescharacteristic
175C2×C53#175911yescharacteristic
176C2×C53#176511yescharacteristic
177#529C2×D5311yescharacteristic
178#446Dic5311yescharacteristic, complemented
179C8×Q8Dic5311yescharacteristic, complemented
180#436Dic5311yescharacteristic, complemented
181#435—21no—
182C8×D4—21no—
183C8×D4Dic5311yescharacteristic, complemented
184#433—21no—
185#432C2×D5311yescharacteristic
186#431—531no—
187#405—21no—
188C4×M16—531no—
189C2×#97—531no—
190C2×C4×C8C2×D5311yescharacteristic
191#376—531no—
192#336—1061no—
193#336—1061no—
194#330—532no—
195#329—532no—
196#328—532no—
197#327—532no—
198#322—1061no—
199#322—1061no—
200C53#824811yescharacteristic, complemented, Sylow
201C2×M16—21no—
202C2×M16—1061no—
203C22×C8—21no—
204C22×C8—1061no—
205C22×C8C22⋊D5311yescharacteristic
206C22×C8C22⋊D5311yescharacteristic
207#126—21no—
208#125C4×D5311yescharacteristic
209#124C4×D5311yescharacteristic
210#123—21no—
211C22⋊Q8C4×D5311yescharacteristic
212#121C4×D5311yescharacteristic
213C4×Q8C2×Dic5311yescharacteristic
214C4×D4—21no—
215C4×D4C2×Dic5311yescharacteristic
216#117—21no—
217C2×C42—1061no—
218C2×C42C22×D5311yescharacteristic
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—1061no—
222C4⋊C8C2×Dic5311yescharacteristic
223C4⋊C8C2×Dic5311yescharacteristic
224C22⋊C8—21no—
225C22⋊C8—21no—
226C22⋊C8—21no—
227C22⋊C8—532no—
228#97—21no—
229#97—21no—
230#97—532no—
231#97—532no—
232#97—1061no—
233C4×C8—21no—
234C4×C8—1061no—
235C4×C8—1061no—
236C4×C8C2×Dic5311yescharacteristic
237C4×C8C2×Dic5311yescharacteristic
238C4×C8C22⋊D5311yescharacteristic
239C4×C8C22⋊D5311yescharacteristic
240#95—532no—
241#95—1061no—
242C2×Q8C4×Dic5311yescharacteristic
243C2×D4—21no—
244C2×D4C4×Dic5311yescharacteristic
245C22×C4—21no—
246C22×C4—1062no—
247C22×C4C2×C4×D5311yescharacteristic
248C22×C4#1171994511yescharacteristic
249C22×C4C2×C22⋊D5311yescharacteristic
250M16—42no—
251M16—2122no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—42no—
257C2×C8—1062no—
258C2×C8—1062no—
259C2×C8—1062no—
260C2×C8—2121no—
261C2×C8—2121no—
262C2×C8C22⋊Dic5311yescharacteristic
263C2×C8C22⋊Dic5311yescharacteristic
264C2×C8C22⋊Dic5311yescharacteristic
265C2×C8C22⋊Dic5311yescharacteristic
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4C4×Dic5311yescharacteristic
271C4⋊C4C4×Dic5311yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C42—21no—
278C42—2121no—
279C42—2121no—
280C42C2×C4×D5311yescharacteristic
281C42#1171994511yescharacteristic
282C42C22×Dic5311yescharacteristic
283C42C2×C22⋊D5311yescharacteristic
284C23—21no—
285C23C4×C22⋊D5311yescharacteristic
286Q8—41no—
287D4—41no—
288D4—42no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—2122no—
299C2×C4—2122no—
300C2×C4—2122no—
301C2×C4C2×C4×Dic5311yescharacteristic
302C2×C4C2×C22⋊Dic5311yescharacteristic
303C2×C4#1313317611yescharacteristic
304C2×C4C4×C22⋊D5311yescharacteristic
305C2×C4C4×C22⋊D5311yescharacteristic
306C2×C4C4×C22⋊D5311yescharacteristic
307C8—21no—
308C8—21no—
309C8—21no—
310C8—21no—
311C8—41no—
312C8—41no—
313C8—2122no—
314C8—2122no—
315C8—2122no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22C4×C22⋊Dic5311yescharacteristic
322C22#1681074312yes—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—4242no—
332C4#1681064011yescharacteristic
333C4#1681064011yescharacteristic
334C2—22no—
335C2—81no—
336C2#2027451511yescharacteristic
337C2#2027452511yescharacteristic
338C2#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#8248
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. #20274466C2semidirect
4. #20273004C2semidirect
5. #20272613C2semidirect
6. #20272607C2semidirect
14. #16810365C4semidirect
15. #16810363C4semidirect
21. C53×#446C4semidirect
22. C8×Q8×C53C4semidirect
23. C53×#436C4semidirect
26. C8×D4×C53C4semidirect
178. #446Dic53semidirect
179. C8×Q8Dic53semidirect
180. #436Dic53semidirect
183. C8×D4Dic53semidirect
200. C53#8248semidirect

Maximal subgroups

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

subgroupindex
#202744662
#202733032
#202732012
#202730042
#202726132
#202726072
C53×#29662
#824853