Finite Group Databasebeta

← #13073590

Subgroups — #13073590

Order 1408.

704 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 153 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×#66993C211yesmaximal, characteristic
2C2×#66989C211yesmaximal, characteristic, complemented
3C2×#66946C211yesmaximal, characteristic, complemented
4C2×#66928C211yesmaximal, characteristic, complemented
5C11×#390C211yesmaximal, characteristic, complemented
6#10622587C211yesmaximal, characteristic, complemented
7#10622572C214yesmaximal, complemented
8#10622569C214yesmaximal, complemented
9C4×Dic11⋊C4C211yesmaximal, characteristic, complemented
10C22×D4×C11C2211yescharacteristic
11C22×C22⋊D11C2214yes—
12C22×D44C2211yescharacteristic
13C22×C4×D11C2212yes—
14C22×Dic22C2211yescharacteristic
15C2×C11×C4⋊C4C2211yescharacteristic
16C2×C11×C22⋊C4C2214yes—
17C2×C42×C11C2211yescharacteristic
18C2×C22⋊Dic11C2212yes—
19C2×C22⋊Dic11C2212yes—
20#66993—24no—
21#66989—24no—
22C2×(C2×D11)⋊C4C2214yes—
23C2×(C2×D11)⋊C4C2212yes—
24C2×Dic11⋊C4C2212yes—
25C2×C4×Dic11C2211yescharacteristic
26C2×C4×Dic11C2211yescharacteristic
27#66946—24no—
28#66928—24no—
29C11⋊#95C2212yes—
30C11⋊#95C2214yes—
31C24×C11C2312yes—
32C23×D11C2312yes—
33C2×D4×C11—22no—
34C2×D4×C11—22no—
35C2×D4×C11—22no—
36C22×C4×C11C2312yes—
37C22×C4×C11C2312yes—
38C22×C4×C11C2311yescharacteristic
39C2×C22⋊D11—28no—
40C2×C22⋊D11—28no—
41C2×C22⋊D11—44no—
42C22×Dic11C2312yes—
43C22×Dic11C2311yescharacteristic
44C22×Dic11C2311yescharacteristic
45C22×Dic11C2312yes—
46C2×D44—22no—
47C2×D44—22no—
48C2×D44D414yes—
49C2×C4×D11—24no—
50C2×C4×D11—44no—
51C2×Dic22—22no—
52C2×Dic22—22no—
53C2×Dic22—22no—
54C11×C4⋊C4—22no—
55C11×C22⋊C4—44no—
56C42×C11—22no—
57C22⋊Dic11—24no—
58C22⋊Dic11—24no—
59(C2×D11)⋊C4—24no—
60(C2×D11)⋊C4—28no—
61Dic11⋊C4—24no—
62C4×Dic11—22no—
63C4×Dic11D414yescomplemented
64#2544—111nomaximal, Sylow
65C23×C11—22no—
66C23×C11—22no—
67C23×C11—24no—
68C23×C11—22no—
69C23×C11—24no—
70C23×C11C2411yescharacteristic
71C22×D11—22no—
72C22×D11—22no—
73C22×D11—24no—
74C22×D11—24no—
75C22×D11C2×D414yes—
76D4×C11—44no—
77C2×C4×C11—22no—
78C2×C4×C11—22no—
79C2×C4×C11—22no—
80C2×C4×C11—22no—
81C2×C4×C11—24no—
82C2×C4×C11C2×D412yes—
83C2×C4×C11C2×D412yes—
84C2×C4×C11#4112yes—
85C22⋊D11—416no—
86C2×Dic11—22no—
87C2×Dic11—22no—
88C2×Dic11—22no—
89C2×Dic11—22no—
90C2×Dic11—22no—
91C2×Dic11—22no—
92C2×Dic11—22no—
93C2×Dic11C2×D412yes—
94C2×Dic11C2×D412yes—
95C2×Dic11#4114yes—
96D44—28no—
97C4×D11—48no—
98Dic22—44no—
99C2×#127—111no—
100C2×#124—111no—
101C2×#124—111no—
102C2×C4×D4—111no—
103#394—114no—
104#391—111no—
105#390D1111yescharacteristic, complemented
106#386—114no—
107C4×C4⋊C4—111no—
108C22×C11—42no—
109C22×C11—42no—
110C22×C11—42no—
111C22×C11—44no—
112C22×C11—44no—
113C22×C11C22×D411yescharacteristic
114C22×C11C22×D411yescharacteristic
115C22×C11C2×#4111yescharacteristic
116C22×C11C2×#4112yes—
117C22×C11H(5,𝔽2)12yes—
118C2×D11—24no—
119C2×D11—28no—
120C2×D11—42no—
121C2×D11—42no—
122C2×D11—44no—
123C4×C11—42no—
124C4×C11—42no—
125C4×C11#12114yes—
126Dic11—22no—
127Dic11—22no—
128Dic11—24no—
129Dic11—42no—
130C22×Q8—111no—
131C22×D4—114no—
132C22×D4—111no—
133C22×D4C2×D1111yescharacteristic
134C23×C4—112no—
135#127—224no—
136#124—224no—
137#124—224no—
138C4×D4—224no—
139C2×C4⋊C4—112no—
140C2×C4⋊C4C2×D1111yescharacteristic
141C2×C22⋊C4—112no—
142C2×C22⋊C4—112no—
143C2×C22⋊C4—112no—
144C2×C22⋊C4—114no—
145C2×C22⋊C4C2×D1114yes—
146C2×C42—111no—
147C2×C42—111no—
148C2×C42C2×D1111yescharacteristic
149#95—112no—
150#95—114no—
151C2×C11—82no—
152C2×C11C2×#12111yescharacteristic
153C2×C11#53211yescharacteristic
154C2×C11#53511yescharacteristic
155C2×C11#54612yes—
156C2×C11#55912yes—
157D11—44no—
158C24—112no—
159C24C22×D1112yes—
160C2×Q8—222no—
161C2×Q8—222no—
162C2×Q8—222no—
163C2×D4—22no—
164C2×D4—22no—
165C2×D4—22no—
166C2×D4—114no—
167C2×D4—228no—
168C2×D4—228no—
169C2×D4—222no—
170C2×D4—222no—
171C2×D4—444no—
172C22×C4—112no—
173C22×C4—111no—
174C22×C4—111no—
175C22×C4—112no—
176C22×C4—224no—
177C22×C4—444no—
178C22×C4C22×D1112yes—
179C22×C4C22×D1112yes—
180C22×C4C22×D1111yescharacteristic
181C4⋊C4—22no—
182C4⋊C4—224no—
183C22⋊C4—44no—
184C22⋊C4—224no—
185C22⋊C4—228no—
186C22⋊C4—224no—
187C22⋊C4—224no—
188C42—22no—
189C42—114no—
190C42—222no—
191C11#254411yescharacteristic, complemented, Sylow
192C23—22no—
193C23—22no—
194C23—22no—
195C23—24no—
196C23—24no—
197C23—114no—
198C23—222no—
199C23—222no—
200C23—224no—
201C23—224no—
202C23C23×D1111yescharacteristic
203Q8—444no—
204D4—44no—
205D4—228no—
206D4—4416no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—24no—
211C2×C4—22no—
212C2×C4—112no—
213C2×C4—112no—
214C2×C4—114no—
215C2×C4—222no—
216C2×C4—222no—
217C2×C4—222no—
218C2×C4—222no—
219C2×C4—222no—
220C2×C4—222no—
221C2×C4—222no—
222C2×C4—448no—
223C2×C4#433912yes—
224C2×C4D4×D1112yes—
225C2×C4D4×D1112yes—
226C22—42no—
227C22—42no—
228C22—44no—
229C22—44no—
230C22—42no—
231C22—224no—
232C22—228no—
233C22—442no—
234C22—444no—
235C22—442no—
236C22C2×#433911yescharacteristic
237C22C2×D4×D1111yescharacteristic
238C22C2×D4×D1111yescharacteristic
239C22#6703712yes—
240C22#41×D1112yes—
241C4—42no—
242C4—42no—
243C4—222no—
244C4—224no—
245C4—222no—
246C4—442no—
247C4#6693714yes—
248C2—82no—
249C2—444no—
250C2C2×#6693711yescharacteristic
251C2#1062309012yes—
252C2#1062314312yes—
253C2#1062319911yescharacteristic
254C2#1062324611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC11×#390C2
prime radicalC2×#66993C2
hypercenter#390D11
hypocenterC11#2544
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#66989C2semidirect
3. C2×#66946C2semidirect
4. C2×#66928C2semidirect
5. C11×#390C2semidirect
6. #10622587C2semidirect
7. #10622572C2semidirect
8. #10622569C2semidirect
9. C4×Dic11⋊C4C2semidirect
63. C4×Dic11D4semidirect
105. #390D11semidirect
191. C11#2544semidirect

Maximal subgroups

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

subgroupindex
C2×#669932
C2×#669892
C2×#669462
C2×#669282
C11×#3902
#106225872
#106225722
#106225692
C4×Dic11⋊C42
#254411