Finite Group Databasebeta

← #28303406

Subgroups — #28303406

Order 13568.

418 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 72 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#20290616C211yesmaximal, characteristic
2#20290612C211yesmaximal, characteristic
3#20289563C212yesmaximal
4#20289563C212yesmaximal
5#20277208C211yesmaximal, characteristic
6#16811595C2212yes—
7#16811566C2212yes—
8#16811503—21no—
9#16811503—21no—
10#16811501—21no—
11#16811501—21no—
12C2×#13133174C2211yescharacteristic
13#16810515—21no—
14#16810513—21no—
15#16810510—21no—
16#16810504—21no—
17C4×C4⋊Dic53C411yescharacteristic, complemented
18C4×#11719921C2211yescharacteristic
19C4×#11719921C2211yescharacteristic
20#16810469C411yescharacteristic, complemented
21C42×Dic53C411yescharacteristic, complemented
22C4×C4⋊C4×C53C411yescharacteristic, complemented
23C2×#11719940—22no—
24C2×C22⋊#6157—22no—
25#13133274—24no—
26(C2×#6157)⋊C4—24no—
27C22×C4×D53C2311yescharacteristic
28#13133174—24no—
29C2×C4⋊Dic53—21no—
30C2×#11719919D411yescharacteristic
31C2×#11719919D411yescharacteristic
32C2×#11719921—21no—
33C2×#11719921—21no—
34C2×#11719921D411yescharacteristic
35C2×#11719921D411yescharacteristic
36C2×C4×Dic53—21no—
37C2×C4×Dic53—21no—
38C2×C4×Dic53C2×C411yescharacteristic
39C2×C4×Dic53C2×C411yescharacteristic
40C2×C4×Dic53C2×C411yescharacteristic
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46C2×C4⋊C4×C53—21no—
47C2×C42×C53C2×C411yescharacteristic
48C2×C42×C53C2×C411yescharacteristic
49C2×C42×C53C2×C411yescharacteristic
50#95×C53—21no—
51C23×D53C2×D411yescharacteristic
52C22×#6157—42no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22×C411yescharacteristic
58C22×Dic53—21no—
59C22×Dic53—21no—
60C22×Dic53C22⋊C411yescharacteristic
61C22×Dic53C22⋊C411yescharacteristic
62C22×Dic53C2×D411yescharacteristic
63C2×C4×D53—41no—
64C2×C4×D53—41no—
65C2×C4×D53#4114yes—
66C22⋊#6157—42no—
67C22⋊#6157—42no—
68#11719940—42no—
69#11719940—42no—
70C2×#88324—42no—
71C4⋊C4×C53—42no—
72C42×C53—22no—
73C42×C53—41no—
74C42×C53—41no—
75#11719921—22no—
76#11719921—22no—
77#11719921—42no—
78#11719921—42no—
79C4⋊Dic53—42no—
80#11719919—22no—
81#11719919—22no—
82C4×Dic53—22no—
83C4×Dic53—42no—
84C4×Dic53—42no—
85C4×Dic53—41no—
86C4×Dic53—41no—
87C23×C53C2×C22⋊C411yescharacteristic
88C22×D53—21no—
89C22×D53—21no—
90C22×D53—41no—
91C22×D53—41no—
92C22×D53#12312yes—
93C2×#6157—82no—
94C2×#6157—82no—
95C2×#6157—82no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—22no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53#11712yes—
109C2×Dic53—21no—
110C2×Dic53—21no—
111C2×Dic53—22no—
112C2×Dic53—21no—
113C2×Dic53—21no—
114C2×Dic53—41no—
115C2×Dic53—41no—
116C2×Dic53—41no—
117C2×Dic53—41no—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53—41no—
121C2×Dic53#12312yes—
122C4×D53—44no—
123#88324—82no—
124#8740—531nomaximal, Sylow
125C22×C53—21no—
126C22×C53—21no—
127C22×C53#38111yescharacteristic
128C22×C53#41011yescharacteristic
129C22×C53#41011yescharacteristic
130C2×D53—22no—
131C2×D53—22no—
132C2×D53—22no—
133C2×D53—81no—
134C2×D53—81no—
135#6157—162no—
136C4×C53—22no—
137C4×C53—42no—
138C4×C53—41no—
139C4×C53—41no—
140C4×C53—81no—
141Dic53—22no—
142Dic53—42no—
143Dic53—41no—
144Dic53—41no—
145Dic53—81no—
146#2298—531no—
147#1822—531no—
148#1820—531no—
149#1284—532no—
150#1284—532no—
151C2×C53—21no—
152C2×C53—21no—
153C2×C53#182011yescharacteristic
154C2×C53#182711yescharacteristic
155C2×C53#182711yescharacteristic
156D53—42no—
157C2×#117—531no—
158#407—532no—
159#388—1061no—
160#388—1061no—
161#387—1061no—
162#387—1061no—
163#381—532no—
164C4×C4⋊C4—531no—
165C4×C4⋊C4#615711yescharacteristic, complemented
166C4×C22⋊C4—531no—
167C4×C22⋊C4—531no—
168#376—531no—
169C43—531no—
170#343—1061no—
171#343—1061no—
172#342—1061no—
173#342—1061no—
174C53#874011yescharacteristic, complemented, Sylow
175C23×C4—531no—
176C2×M16—1062no—
177#117—1064no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—531no—
180C2×C4⋊C4—531no—
181C2×C4⋊C4—1061no—
182C2×C22⋊C4—531no—
183C2×C22⋊C4—531no—
184C2×C22⋊C4—1061no—
185C2×C22⋊C4—1061no—
186C2×C22⋊C4—1062no—
187C2×C42—531no—
188C2×C42—531no—
189C2×C42—531no—
190C2×C42—1061no—
191C2×C42—1061no—
192C2×C42C2×#615711yescharacteristic
193C2×C42C2×#615711yescharacteristic
194C2×C42C2×#615711yescharacteristic
195C22⋊C8—1064no—
196#95—21no—
197#95—1061no—
198#95—1061no—
199#95—1061no—
200#95—1061no—
201#95—1064no—
202#95—1061no—
203C24—531no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—531no—
207C22×C4—534no—
208C22×C4—531no—
209C22×C4—531no—
210C22×C4—1061no—
211C22×C4—1061no—
212C22×C4—2121no—
213C22×C4—2121no—
214C22×C4—2122no—
215C22×C4C22⋊#615711yescharacteristic
216C22×C4C22⋊#615711yescharacteristic
217C22×C4C22×#615711yescharacteristic
218M16—2122no—
219M16—2122no—
220C2×C8—2122no—
221C4⋊C4—42no—
222C4⋊C4—1062no—
223C4⋊C4—1062no—
224C4⋊C4—2122no—
225C22⋊C4—1062no—
226C22⋊C4—1062no—
227C22⋊C4—2122no—
228C22⋊C4—2122no—
229C22⋊C4—2122no—
230C22⋊C4—2122no—
231C42—22no—
232C42—41no—
233C42—41no—
234C42—1062no—
235C42—2121no—
236C42—2121no—
237C42—2122no—
238C42—2122no—
239C23—532no—
240C23—1061no—
241C23—1061no—
242C23—2121no—
243C23—2121no—
244C23C2×C22⋊#615711yescharacteristic
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—532no—
258C2×C4—1061no—
259C2×C4—1061no—
260C2×C4—1062no—
261C2×C4—1061no—
262C2×C4—1061no—
263C2×C4—4242no—
264C2×C4—4242no—
265C2×C4—4242no—
266C2×C4—2124no—
267C2×C4—2121no—
268C2×C4—2121no—
269C2×C4—2121no—
270C2×C4—2121no—
271C2×C4—2121no—
272C2×C4—2121no—
273C2×C4—2121no—
274C2×C4#1313329612yes—
275C8—4242no—
276C22—21no—
277C22—21no—
278C22—1062no—
279C22—1062no—
280C22—1062no—
281C22—4241no—
282C22—4241no—
283C22#1681156611yescharacteristic
284C22#1681160311yescharacteristic
285C22#1681160411yescharacteristic
286C4—22no—
287C4—42no—
288C4—41no—
289C4—41no—
290C4—81no—
291C4—8482no—
292C4—1062no—
293C4—4241no—
294C4—2122no—
295C4—2121no—
296C4—2121no—
297C2—21no—
298C2—21no—
299C2—2122no—
300C2#2029061311yescharacteristic
301C2#2029062411yescharacteristic
302C2#2029062511yescharacteristic
subgroupG/H
centerC22#16811566
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC4×C4⋊C4×C53C4
socleC22×C53#381
prime radicalC23×D53C2×D4
cosocleC22×C4×D53C23
hypercenterC4×C4⋊C4#6157
hypocenterC53#8740
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
17. C4×C4⋊Dic53C4semidirect
20. #16810469C4semidirect
21. C42×Dic53C4semidirect
22. C4×C4⋊C4×C53C4semidirect
165. C4×C4⋊C4#6157semidirect
174. C53#8740semidirect

Maximal subgroups

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

subgroupindex
#202906162
#202906122
#202895632
#202772082
#874053