Finite Group Databasebeta

← #13079448

Subgroups — #13079448

Order 1408.

414 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 121 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#10623322C211yesmaximal, characteristic
2C2×#66977C211yesmaximal, characteristic, complemented
3C11×#407C211yesmaximal, characteristic, complemented
4#10622750C212yesmaximal, complemented
5#10622747C212yesmaximal, complemented
6#10622352C211yesmaximal, characteristic, complemented
7#10622351C211yesmaximal, characteristic, complemented
8#10622348C211yesmaximal, characteristic, complemented
9#10622346C212yesmaximal, complemented
10#10622336C212yesmaximal, complemented
11#10622335C211yesmaximal, characteristic, complemented
12C23×C4×C11C2211yescharacteristic
13C2×#4339C2211yescharacteristic
14C22×Dic22C2211yescharacteristic
15C2×C11×M16C2212yes—
16#67006—21no—
17C2×C22⋊Dic11—21no—
18#66983C2211yescharacteristic
19#66983C2211yescharacteristic
20#66982—21no—
21C4×C22⋊D11—21no—
22C4×C22⋊D11—21no—
23#66979C2211yescharacteristic
24C2×C4⋊Dic11C2211yescharacteristic
25#66977—21no—
26#66977—21no—
27#66977C2212yes—
28#66977C2211yescharacteristic
29#66977C2212yes—
30#66977C2211yescharacteristic
31C2×Dic11⋊C4—21no—
32C2×Dic44C2212yescomplemented
33C2×#4314C2212yescomplemented
34C11×C22⋊C8C2211yescharacteristic
35C11×C22⋊C8C2212yes—
36C11×C22⋊C8C2211yescharacteristic
37#66885C2212yescomplemented
38#66882C2212yescomplemented
39#66881C2212yescomplemented
40#66880C2212yescomplemented
41#66880C2212yescomplemented
42#66880C2212yescomplemented
43C24×C11D411yescharacteristic
44C22×C4×C11—42no—
45C22×C4×C11D411yescharacteristic
46C22×C4×C11D411yescharacteristic
47C22×C4×C11D411yescharacteristic
48C22×C4×C11C2311yescharacteristic
49C22×C4×C11C2311yescharacteristic
50C22×C4×C11C2311yescharacteristic
51C2×C22⋊D11—21no—
52C2×C22⋊D11—21no—
53C2×C22⋊D11—21no—
54C22×Dic11—21no—
55C22×Dic11—21no—
56#4339—42no—
57C2×D44C2311yescharacteristic
58C2×C4×D11—21no—
59C2×Dic22—21no—
60C2×Dic22—42no—
61C2×Dic22C2312yes—
62C2×Dic22C2311yescharacteristic
63C11×M16—42no—
64C2×C8×C11C2312yes—
65C2×C8×C11C2312yes—
66C22⋊Dic11—22no—
67C22⋊Dic11—22no—
68C22⋊Dic11—21no—
69C22⋊Dic11—21no—
70C22⋊Dic11—21no—
71(C2×D11)⋊C4—21no—
72(C2×D11)⋊C4—21no—
73C4⋊Dic11—21no—
74C4⋊Dic11C2311yescharacteristic
75C4⋊Dic11C2311yescharacteristic
76C4⋊Dic11C2312yes—
77Dic11⋊C4—22no—
78Dic11⋊C4—22no—
79Dic11⋊C4—21no—
80Dic11⋊C4—21no—
81C4×Dic11—21no—
82Dic44—42no—
83#4314—42no—
84#3196—111nomaximal, Sylow
85C23×C11—22no—
86C23×C11—41no—
87C23×C11—41no—
88C23×C11C2×D411yescharacteristic
89C23×C11C2×D411yescharacteristic
90C23×C11C2×D411yescharacteristic
91C22×D11—21no—
92C2×C4×C11—22no—
93C2×C4×C11—22no—
94C2×C4×C11—22no—
95C2×C4×C11—41no—
96C2×C4×C11—41no—
97C2×C4×C11—41no—
98C2×C4×C11C2×D411yescharacteristic
99C2×C4×C11C2×D411yescharacteristic
100C2×C4×C11C2×D411yescharacteristic
101C2×C4×C11C2411yescharacteristic
102C22⋊D11—42no—
103C22⋊D11—42no—
104C22⋊D11—42no—
105C2×Dic11—21no—
106C2×Dic11—21no—
107C2×Dic11—22no—
108C2×Dic11—22no—
109C2×Dic11—21no—
110C2×Dic11—42no—
111C2×Dic11—42no—
112D44—22no—
113C4×D11—42no—
114Dic22—22no—
115Dic22—22no—
116Dic22—22no—
117Dic22—42no—
118C8×C11—22no—
119C8×C11—22no—
120#525—111no—
121C2×C22⋊Q8—111no—
122#485—112no—
123#484—111no—
124#481—111no—
125#470—112no—
126#468—112no—
127#452—112no—
128#451—111no—
129#448—111no—
130#407D1111yescharacteristic, complemented
131C22×C11—21no—
132C22×C11—21no—
133C22×C11—21no—
134C22×C11—21no—
135C22×C11—21no—
136C22×C11—21no—
137C22×C11—21no—
138C22×C11—21no—
139C22×C11—44no—
140C22×C11#13712yes—
141C22×C11C22×D411yescharacteristic
142C2×D11—41no—
143C2×D11—41no—
144C2×D11—41no—
145C4×C11—21no—
146C4×C11—21no—
147C4×C11—21no—
148C4×C11H(5,𝔽2)12yes—
149Dic11—42no—
150Dic11—41no—
151Dic11—41no—
152Dic11—42no—
153Dic11—41no—
154C2×#41—111no—
155C22×Q8—111no—
156C23×C4C2×D1111yescharacteristic
157C2×Q16—112no—
158C2×SD16—112no—
159C2×M16C2×D1112yes—
160#123—221no—
161C22⋊Q8—112no—
162C22⋊Q8—112no—
163C22⋊Q8—111no—
164C22⋊Q8—111no—
165C22⋊Q8—221no—
166C22⋊Q8—221no—
167#121—111no—
168#121—111no—
169#120—221no—
170C4×D4—221no—
171C4×D4—221no—
172#117—111no—
173C2×C4⋊C4—111no—
174C2×C4⋊C4—221no—
175C2×C22⋊C4—221no—
176#107—112no—
177#106—112no—
178#103—112no—
179#103—112no—
180#103—112no—
181#102—112no—
182C22⋊C8C2×D1111yescharacteristic
183C22⋊C8C2×D1111yescharacteristic
184C22⋊C8C2×D1112yes—
185C2×C11—22no—
186C2×C11—41no—
187C2×C11—41no—
188C2×C11#53411yescharacteristic
189C2×C11C2×#13711yescharacteristic
190C2×C11#57511yescharacteristic
191D11—81no—
192C24D4411yescharacteristic
193#41—442no—
194C2×Q8—112no—
195C2×Q8—111no—
196C2×Q8—221no—
197C2×Q8—442no—
198C2×D4—111no—
199C2×D4—221no—
200C2×D4—221no—
201C2×D4—221no—
202C22×C4—42no—
203C22×C4—221no—
204C22×C4—221no—
205C22×C4—221no—
206C22×C4D4411yescharacteristic
207C22×C4D4411yescharacteristic
208C22×C4D4411yescharacteristic
209C22×C4C22×D1111yescharacteristic
210C22×C4C22×D1111yescharacteristic
211C22×C4C22×D1111yescharacteristic
212Q16—442no—
213SD16—442no—
214M16—42no—
215C2×C8C22×D1112yes—
216C2×C8C22×D1112yes—
217C4⋊C4—112no—
218C4⋊C4—111no—
219C4⋊C4—111no—
220C4⋊C4—221no—
221C4⋊C4—222no—
222C4⋊C4—222no—
223C4⋊C4—221no—
224C4⋊C4—221no—
225C22⋊C4—222no—
226C22⋊C4—222no—
227C22⋊C4—221no—
228C22⋊C4—221no—
229C22⋊C4—221no—
230C22⋊C4—221no—
231C22⋊C4—221no—
232C42—221no—
233C11#319611yescharacteristic, complemented, Sylow
234C23—22no—
235C23—41no—
236C23—41no—
237C23—221no—
238C23C2×D4411yescharacteristic
239C23C2×D4411yescharacteristic
240C23C2×D4411yescharacteristic
241Q8—222no—
242Q8—222no—
243Q8—222no—
244Q8—442no—
245D4—222no—
246D4—442no—
247D4—442no—
248D4—442no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—221no—
256C2×C4—221no—
257C2×C4—222no—
258C2×C4—221no—
259C2×C4—222no—
260C2×C4—442no—
261C2×C4—442no—
262C2×C4—442no—
263C2×C4C2×D4411yescharacteristic
264C2×C4C2×D4411yescharacteristic
265C2×C4C2×D4411yescharacteristic
266C2×C4C23×D1111yescharacteristic
267C8—22no—
268C8—22no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—44no—
278C22—441no—
279C22—441no—
280C22—441no—
281C22#6696212yes—
282C22C22×D4411yescharacteristic
283C4—21no—
284C4—21no—
285C4—21no—
286C4—441no—
287C4—442no—
288C4—441no—
289C4—441no—
290C4—442no—
291C4#6703712yes—
292C2—22no—
293C2—41no—
294C2—41no—
295C2—881no—
296C2#1062302411yescharacteristic
297C2C2×#6696211yescharacteristic
298C2#1062327911yescharacteristic
subgroupG/H
centerC22C22×D44
commutator, cosocleC2×C4×C11C24
FrattiniC2×C4C23×D11
FittingC11×#407C2
socleC22×C11C22×D4
prime radical#10623322C2
hypercenter#407D11
hypocenterC11#3196
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#66977C2semidirect
3. C11×#407C2semidirect
4. #10622750C2semidirect
5. #10622747C2semidirect
6. #10622352C2semidirect
7. #10622351C2semidirect
8. #10622348C2semidirect
9. #10622346C2semidirect
10. #10622336C2semidirect
11. #10622335C2semidirect
32. C2×Dic44C22semidirect
33. C2×#4314C22semidirect
37. #66885C22semidirect
38. #66882C22semidirect
39. #66881C22semidirect
40. #66880C22semidirect
41. #66880C22semidirect
42. #66880C22semidirect
130. #407D11semidirect
233. C11#3196semidirect

Maximal subgroups

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

subgroupindex
#106233222
C2×#669772
C11×#4072
#106227502
#106227472
#106223522
#106223512
#106223482
#106223462
#106223362
#106223352
#319611