Finite Group Databasebeta

← #13074376

Subgroups — #13074376

Order 1408.

612 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C2×#66950C211yesmaximal, characteristic, complemented
2C2×#66948C211yesmaximal, characteristic
3C2×#66928C211yesmaximal, characteristic, complemented
4C2×#66925C211yesmaximal, characteristic, complemented
5C11×#384C211yesmaximal, characteristic, complemented
6#10622607C214yesmaximal, complemented
7#10622604C214yesmaximal, complemented
8#10622601C211yesmaximal, characteristic, complemented
9#10622587C211yesmaximal, characteristic, complemented
10C22×D44C2211yescharacteristic
11C22×D44C2212yes—
12C22×C4×D11C2211yescharacteristic
13C22×Dic22C2211yescharacteristic
14C2×C11×C4⋊C4C2211yescharacteristic
15C2×C11×C4⋊C4C2211yescharacteristic
16C2×C11×C4⋊C4C2214yes—
17C2×C42×C11C2211yescharacteristic
18C2×(C2×D11)⋊C4C2212yes—
19C2×(C2×D11)⋊C4C2214yes—
20C2×(C2×D11)⋊C4C2212yes—
21C2×(C2×D11)⋊C4C2212yes—
22C2×(C2×D11)⋊C4C2214yes—
23C2×C4⋊Dic11C2212yes—
24C2×Dic11⋊C4C2212yes—
25C2×C4×Dic11C2211yescharacteristic
26#66950—24no—
27#66948—24no—
28#66928—24no—
29#66925—24no—
30C11⋊#95C2214yes—
31C23×D11C2312yes—
32C23×D11C2311yescharacteristic
33C22×C4×C11C2312yes—
34C22×C4×C11C2312yes—
35C22×C4×C11C2312yes—
36C22×C4×C11C2311yescharacteristic
37C22×Dic11C2312yes—
38C22×Dic11C2312yes—
39C22×Dic11C2311yescharacteristic
40C2×D44—24no—
41C2×D44—22no—
42C2×D44—22no—
43C2×D44—22no—
44C2×D44—44no—
45C2×C4×D11—22no—
46C2×C4×D11—22no—
47C2×C4×D11D414yes—
48C2×Dic22—22no—
49C2×Dic22—22no—
50C2×Dic22—22no—
51C11×C4⋊C4—22no—
52C11×C4⋊C4—22no—
53C11×C4⋊C4—44no—
54C42×C11D414yescomplemented
55(C2×D11)⋊C4—24no—
56(C2×D11)⋊C4—24no—
57(C2×D11)⋊C4—24no—
58(C2×D11)⋊C4—28no—
59(C2×D11)⋊C4—44no—
60C4⋊Dic11—24no—
61Dic11⋊C4—24no—
62C4×Dic11—22no—
63#2620—111nomaximal, Sylow
64C23×C11C2411yescharacteristic
65C22×D11—21no—
66C22×D11—21no—
67C22×D11—22no—
68C22×D11—22no—
69C22×D11—22no—
70C22×D11—22no—
71C22×D11—22no—
72C22×D11—24no—
73C22×D11—24no—
74C22×D11C2×D412yes—
75C2×C4×C11—22no—
76C2×C4×C11—22no—
77C2×C4×C11—22no—
78C2×C4×C11—22no—
79C2×C4×C11—24no—
80C2×C4×C11—24no—
81C2×C4×C11C2×D414yes—
82C2×C4×C11C2×D412yes—
83C2×C4×C11C2×D412yes—
84C2×C4×C11#4112yes—
85C2×Dic11—21no—
86C2×Dic11—21no—
87C2×Dic11—22no—
88C2×Dic11—22no—
89C2×Dic11—22no—
90C2×Dic11—22no—
91C2×Dic11—22no—
92C2×Dic11—22no—
93C2×Dic11C2×D412yes—
94D44—48no—
95D44—44no—
96C4×D11—28no—
97Dic22—44no—
98C2×#124—111no—
99C2×C22⋊Q8—111no—
100C2×#121—111no—
101C2×#117—111no—
102#396—114no—
103#394—114no—
104#391—111no—
105#390—111no—
106#384D1111yescharacteristic, complemented
107C22×C11C22×D411yescharacteristic
108C22×C11C22×D411yescharacteristic
109C22×C11C2×#4111yescharacteristic
110C22×C11H(5,𝔽2)12yes—
111C22×C112−1+412yes—
112C2×D11—22no—
113C2×D11—22no—
114C2×D11—22no—
115C2×D11—42no—
116C2×D11—42no—
117C2×D11—42no—
118C2×D11—42no—
119C2×D11—42no—
120C2×D11—44no—
121C2×D11—44no—
122C4×C11—24no—
123C4×C11—42no—
124C4×C11—42no—
125C4×C11#12114yes—
126Dic11—22no—
127Dic11—42no—
128Dic11—42no—
129C22×Q8—111no—
130C22×D4—111no—
131C22×D4—112no—
132C23×C4—111no—
133#124—224no—
134C22⋊Q8—224no—
135#121—224no—
136#117—224no—
137C2×C4⋊C4—112no—
138C2×C4⋊C4—112no—
139C2×C4⋊C4C2×D1111yescharacteristic
140C2×C4⋊C4C2×D1111yescharacteristic
141C2×C4⋊C4C2×D1114yes—
142C2×C22⋊C4—112no—
143C2×C22⋊C4—112no—
144C2×C22⋊C4—112no—
145C2×C22⋊C4—114no—
146C2×C22⋊C4—114no—
147C2×C42—111no—
148C2×C42C2×D1111yescharacteristic
149#95—114no—
150C2×C11C2×#12111yescharacteristic
151C2×C11#53511yescharacteristic
152C2×C11#53611yescharacteristic
153C2×C11#53712yes—
154C2×C11#54212yes—
155D11—42no—
156D11—82no—
157C24—111no—
158C24—112no—
159C2×Q8—222no—
160C2×Q8—222no—
161C2×Q8—222no—
162C2×D4—224no—
163C2×D4—222no—
164C2×D4—222no—
165C2×D4—222no—
166C2×D4—444no—
167C22×C4—111no—
168C22×C4—114no—
169C22×C4—112no—
170C22×C4—112no—
171C22×C4—222no—
172C22×C4—222no—
173C22×C4C22×D1112yes—
174C22×C4C22×D1112yes—
175C22×C4C22×D1112yes—
176C22×C4C22×D1111yescharacteristic
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—44no—
180C4⋊C4—224no—
181C4⋊C4—224no—
182C22⋊C4—224no—
183C22⋊C4—224no—
184C22⋊C4—224no—
185C22⋊C4—228no—
186C22⋊C4—444no—
187C42—222no—
188C42D4414yescomplemented
189C11#262011yescharacteristic, complemented, Sylow
190C23—112no—
191C23—221no—
192C23—222no—
193C23—221no—
194C23—222no—
195C23—222no—
196C23—222no—
197C23—224no—
198C23—222no—
199C23—224no—
200C23C23×D1111yescharacteristic
201Q8—444no—
202D4—448no—
203D4—444no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—24no—
209C2×C4—24no—
210C2×C4—112no—
211C2×C4—221no—
212C2×C4—221no—
213C2×C4—228no—
214C2×C4—222no—
215C2×C4—222no—
216C2×C4—222no—
217C2×C4—222no—
218C2×C4—222no—
219C2×C4—222no—
220C2×C4C2×D4414yes—
221C2×C4C2×D4412yes—
222C2×C4D4×D1112yes—
223C2×C4#434312yes—
224C22—222no—
225C22—222no—
226C22—222no—
227C22—442no—
228C22—442no—
229C22—442no—
230C22—442no—
231C22—444no—
232C22—442no—
233C22—444no—
234C22C22×D4411yescharacteristic
235C22C2×D4×D1111yescharacteristic
236C22#6703712yes—
237C22C2×#434311yescharacteristic
238C22#6704012yes—
239C4—24no—
240C4—42no—
241C4—42no—
242C4—222no—
243C4—442no—
244C4—442no—
245C4#6694814yes—
246C2—442no—
247C2—882no—
248C2C2×#6694811yescharacteristic
249C2#1062304112yes—
250C2#1062324611yescharacteristic
251C2#1062325511yescharacteristic
252C2#1062326412yes—
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC11×#384C2
prime radicalC2×#66948C2
hypercenter#384D11
hypocenterC11#2620
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #384

Complements

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

NH ≅ G/Ntype
1. C2×#66950C2semidirect
3. C2×#66928C2semidirect
4. C2×#66925C2semidirect
5. C11×#384C2semidirect
6. #10622607C2semidirect
7. #10622604C2semidirect
8. #10622601C2semidirect
9. #10622587C2semidirect
54. C42×C11D4semidirect
106. #384D11semidirect
188. C42D44semidirect
189. C11#2620semidirect

Maximal subgroups

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

subgroupindex
C2×#669502
C2×#669482
C2×#669282
C2×#669252
C11×#3842
#106226072
#106226042
#106226012
#106225872
#262011