Finite Group Databasebeta

← #28301312

Subgroups — #28301312

Order 13568.

994 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 148 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291159C212yesmaximal
2#20289614C214yesmaximal
3#20277095C211yesmaximal, characteristic
4C2×#13133296—22no—
5#16811566—22no—
6#16811565—22no—
7#16811565C414yescomplemented
8#16811564C2214yes—
9C4×C22⋊#6157—22no—
10C4×C22⋊#6157C414yescomplemented
11C23×C4×D53C2211yescharacteristic
12C2×#13133173C412yescomplemented
13#16810486C411yescharacteristic, complemented
14#16810482—22no—
15#16810478C2212yes—
16#16810470—22no—
17C53×#381C411yescharacteristic, complemented
18#13133296—48no—
19C2×C4⋊#6157—24no—
20C2×C22⋊#6157—24no—
21C2×C22⋊#6157—24no—
22C2×C22⋊#6157C2×C418yes—
23C2×C4×#6157—24no—
24(C2×#6157)⋊C4—24no—
25(C2×#6157)⋊C4—24no—
26C24×D53C2311yescharacteristic
27C23×Dic53C2×C411yescharacteristic
28C22×C4×D53—44no—
29C22×C4×D53C2×C412yes—
30C22×C4×D53C2×C412yes—
31C22×C4×D53D412yes—
32C22×C4×D53D411yescharacteristic
33C22×C4×D53D411yescharacteristic
34C22×C4×D53D412yes—
35C22×C4×D53Q811yescharacteristic
36C22×C4×D53Q811yescharacteristic
37#13133173—44no—
38C2×#11719919—22no—
39C2×C22⋊Dic53C2×C412yes—
40C2×#11719921—24no—
41C2×C4×Dic53—22no—
42#13133124—22no—
43C23×C4×C53C2×C411yescharacteristic
44C2×C22⋊C4×C53C2×C412yes—
45#95×C53—22no—
46C24×C53C22×C411yescharacteristic
47C23×D53—44no—
48C23×D53—42no—
49C23×D53C22×C412yes—
50C23×D53C2×D411yescharacteristic
51C23×D53C2×D412yes—
52C23×D53C2×Q811yescharacteristic
53C22×#6157—28no—
54C22×C4×C53—22no—
55C22×C4×C53—42no—
56C22×C4×C53C4212yes—
57C22×C4×C53C22⋊C412yes—
58C22×C4×C53C4⋊C411yescharacteristic
59C22×C4×C53C4⋊C411yescharacteristic
60C22×Dic53—24no—
61C22×Dic53—22no—
62C22×Dic53C4211yescharacteristic
63C22×Dic53C4211yescharacteristic
64C22×Dic53C22⋊C411yescharacteristic
65C22×Dic53C22⋊C411yescharacteristic
66C22×Dic53C4⋊C411yescharacteristic
67C22×Dic53C4⋊C411yescharacteristic
68C2×C4×D53—28no—
69C2×C4×D53—44no—
70C2×C4×D53—44no—
71C2×C4×D53—42no—
72C2×C4×D53—42no—
73C2×C4×D53—42no—
74C2×C4×D53—42no—
75C2×C4×D53—44no—
76C2×C4×D53C22⋊C412yes—
77C2×C4×D53C22⋊C412yes—
78C2×C4×D53C22⋊C412yes—
79C2×C4×D53C22⋊C412yes—
80C2×C4×D53C4⋊C412yes—
81C2×C4×D53C4⋊C412yes—
82C2×C4×D53C4⋊C412yes—
83C2×C4×D53C4⋊C412yes—
84C22⋊#6157—48no—
85C22⋊#6157—48no—
86C22⋊#6157—48no—
87C4⋊#6157—48no—
88C4×#6157—48no—
89C22⋊C4×C53—42no—
90C22⋊Dic53—42no—
91#11719921—48no—
92#11719919—44no—
93C4×Dic53—44no—
94C23×C53—22no—
95C23×C53—42no—
96C23×C53C2×C4211yescharacteristic
97C23×C53C2×C22⋊C411yescharacteristic
98C23×C53C2×C4⋊C411yescharacteristic
99C22×D53—24no—
100C22×D53—24no—
101C22×D53—24no—
102C22×D53—24no—
103C22×D53—42no—
104C22×D53—41no—
105C22×D53—41no—
106C22×D53—42no—
107C22×D53—42no—
108C22×D53—42no—
109C22×D53—416no—
110C22×D53C2×C22⋊C412yes—
111C22×D53C2×C4⋊C412yes—
112C2×#6157—48no—
113C2×#6157—48no—
114C2×#6157—48no—
115C2×C4×C53—24no—
116C2×C4×C53—42no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—44no—
120C2×C4×C53—42no—
121C2×C4×C53#9512yes—
122C2×C4×C53#9512yes—
123C2×Dic53—22no—
124C2×Dic53—22no—
125C2×Dic53—22no—
126C2×Dic53—22no—
127C2×Dic53—22no—
128C2×Dic53—44no—
129C2×Dic53—42no—
130C2×Dic53#9511yescharacteristic
131C2×Dic53#9511yescharacteristic
132C2×Dic53#9511yescharacteristic
133C2×Dic53#9511yescharacteristic
134C2×Dic53#9914yes—
135C4×D53—24no—
136C4×D53—24no—
137C4×D53—24no—
138C4×D53—24no—
139C4×D53—42no—
140C4×D53—42no—
141C4×D53—42no—
142C4×D53—42no—
143#8100—531nomaximal, Sylow
144C22×C53—22no—
145C22×C53—21no—
146C22×C53—21no—
147C22×C53—22no—
148C22×C53—22no—
149C22×C53—44no—
150C22×C53C2×#9511yescharacteristic
151C22×C53C2×#9912yes—
152C2×D53—22no—
153C2×D53—22no—
154C2×D53—22no—
155C2×D53—22no—
156C2×D53—22no—
157C2×D53—44no—
158C2×D53—48no—
159C2×D53—48no—
160C2×D53—44no—
161C2×D53#41014yes—
162#6157—88no—
163C4×C53—22no—
164C4×C53—22no—
165C4×C53—82no—
166Dic53—21no—
167Dic53—21no—
168Dic53—82no—
169Dic53#34212yes—
170Dic53#34212yes—
171#2280—531no—
172#2280—532no—
173#1304—534no—
174C2×C53—22no—
175C2×C53—41no—
176C2×C53—41no—
177C2×C53C2×#34211yescharacteristic
178C2×C53#174012yes—
179D53—24no—
180D53—42no—
181C24×C4—531no—
182C2×#117—532no—
183C2×#117—1062no—
184#381—531no—
185#381—1062no—
186#381#615711yescharacteristic, complemented
187#380—534no—
188#380—1062no—
189#380—1062no—
190#379—534no—
191#379—532no—
192C4×C22⋊C4—534no—
193C4×C22⋊C4—1062no—
194C4×C22⋊C4—1062no—
195C53#810011yescharacteristic, complemented, Sylow
196C25—531no—
197C23×C4—531no—
198C23×C4—532no—
199C23×C4—532no—
200C23×C4—532no—
201C23×C4—531no—
202C23×C4—531no—
203C23×C4—531no—
204C23×C4—531no—
205C23×C4—532no—
206C23×C4—2124no—
207C23×C4C2×#615711yescharacteristic
208#117—2128no—
209#117—2124no—
210C2×C4⋊C4—1062no—
211C2×C4⋊C4—1064no—
212C2×C22⋊C4—538no—
213C2×C22⋊C4—532no—
214C2×C22⋊C4—1064no—
215C2×C22⋊C4—1064no—
216C2×C22⋊C4—1064no—
217C2×C22⋊C4C2×#615712yes—
218C2×C42—1062no—
219C2×C42—1064no—
220#95—22no—
221#95—1064no—
222#95—1064no—
223#95—1062no—
224C24—532no—
225C24—531no—
226C24—532no—
227C24—531no—
228C24—2122no—
229C24—2124no—
230C24C22×#615711yescharacteristic
231C22×C4—22no—
232C22×C4—42no—
233C22×C4—531no—
234C22×C4—531no—
235C22×C4—531no—
236C22×C4—531no—
237C22×C4—531no—
238C22×C4—531no—
239C22×C4—532no—
240C22×C4—532no—
241C22×C4—532no—
242C22×C4—532no—
243C22×C4—532no—
244C22×C4—532no—
245C22×C4—532no—
246C22×C4—532no—
247C22×C4—1068no—
248C22×C4—1064no—
249C22×C4—1068no—
250C22×C4—1062no—
251C22×C4—2124no—
252C22×C4—2124no—
253C22×C4—2124no—
254C22×C4—2122no—
255C22×C4—2122no—
256C22×C4—2122no—
257C22×C4—2122no—
258C22×C4C4×#615712yes—
259C22×C4C4⋊#615711yescharacteristic
260C22×C4C4⋊#615711yescharacteristic
261C22×C4C22⋊#615712yes—
262C4⋊C4—2124no—
263C4⋊C4—2128no—
264C22⋊C4—42no—
265C22⋊C4—2128no—
266C22⋊C4—2128no—
267C22⋊C4—2128no—
268C22⋊C4—2122no—
269C22⋊C4—2128no—
270C42—2124no—
271C42—2128no—
272C23—22no—
273C23—42no—
274C23—532no—
275C23—532no—
276C23—1064no—
277C23—1064no—
278C23—1064no—
279C23—1064no—
280C23—2122no—
281C23—2121no—
282C23—2121no—
283C23—2122no—
284C23—2122no—
285C23—2122no—
286C23—21216no—
287C23C2×C4×#615711yescharacteristic
288C23C2×C22⋊#615711yescharacteristic
289C23C2×C4⋊#615711yescharacteristic
290C2×C4—24no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—531no—
297C2×C4—531no—
298C2×C4—534no—
299C2×C4—531no—
300C2×C4—531no—
301C2×C4—1064no—
302C2×C4—1064no—
303C2×C4—1064no—
304C2×C4—1064no—
305C2×C4—1062no—
306C2×C4—1062no—
307C2×C4—1062no—
308C2×C4—1062no—
309C2×C4—1062no—
310C2×C4—2122no—
311C2×C4—2122no—
312C2×C4—2122no—
313C2×C4—2122no—
314C2×C4—2124no—
315C2×C4—2128no—
316C2×C4—2128no—
317C2×C4—2128no—
318C2×C4—2122no—
319C2×C4(C2×#6157)⋊C412yes—
320C2×C4(C2×#6157)⋊C412yes—
321C22—22no—
322C22—21no—
323C22—21no—
324C22—22no—
325C22—22no—
326C22—44no—
327C22—534no—
328C22—1062no—
329C22—1062no—
330C22—1062no—
331C22—1062no—
332C22—1062no—
333C22—2128no—
334C22—2124no—
335C22—2128no—
336C22—2124no—
337C22C2×(C2×#6157)⋊C411yescharacteristic
338C22D53⋊#9912yes—
339C4—22no—
340C4—22no—
341C4—82no—
342C4—532no—
343C4—532no—
344C4—1061no—
345C4—1061no—
346C4—4242no—
347C4—4248no—
348C2—22no—
349C2—41no—
350C2—41no—
351C2—1064no—
352C2—2122no—
353C2#2029041211yescharacteristic
354C2#2029041312yes—
subgroupG/H
centerC22C2×(C2×#6157)⋊C4
commutatorC23×C53C2×C42
FrattiniC24C22×#6157
FittingC53×#381C4
socleC22×C53C2×#95
prime radical, cosocleC24×D53C23
hypercenter#381#6157
hypocenterC53#8100
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #381

Complements

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

NH ≅ G/Ntype
7. #16811565C4semidirect
10. C4×C22⋊#6157C4semidirect
12. C2×#13133173C4semidirect
13. #16810486C4semidirect
17. C53×#381C4semidirect
186. #381#6157semidirect
195. C53#8100semidirect

Maximal subgroups

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

subgroupindex
#202911592
#202896142
#202770952
#810053