Finite Group Databasebeta

← #13074782

Subgroups — #13074782

Order 1408.

434 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 133 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×#66951C211yesmaximal, characteristic
2C2×#66943C211yesmaximal, characteristic, complemented
3C11×#398C211yesmaximal, characteristic, complemented
4#10622607C212yesmaximal
5#10622606C211yesmaximal, characteristic
6#10622599C211yesmaximal, characteristic, complemented
7#10622586C212yesmaximal, complemented
8#10622290C211yesmaximal, characteristic
9#10622282C212yesmaximal
10#10622274C212yesmaximal, complemented
11#10622262C211yesmaximal, characteristic, complemented
12C2×C11×C4⋊C4C2211yescharacteristic
13C2×C11×C4⋊C4C2211yescharacteristic
14C2×C11×C4⋊C4C2212yes—
15C2×(C2×D11)⋊C4C2211yescharacteristic
16C2×(C2×D11)⋊C4C2211yescharacteristic
17C2×(C2×D11)⋊C4C2211yescharacteristic
18C2×(C2×D11)⋊C4C2212yes—
19C2×(C2×D11)⋊C4C2212yes—
20C2×C4⋊Dic11C2212yes—
21C2×C4⋊Dic11C2211yescharacteristic
22C2×C4⋊Dic11C2211yescharacteristic
23C2×C4⋊Dic11C2212yes—
24C2×Dic11⋊C4C2212yes—
25C2×Dic11⋊C4C2212yes—
26C2×Dic11⋊C4C2211yescharacteristic
27C2×C4×Dic11C2212yes—
28C2×C4×Dic11C2211yescharacteristic
29#66951—24no—
30#66943—24no—
31C11×#95C2211yescharacteristic
32C11×#95C2212yes—
33C11⋊#95C2211yescharacteristic
34C11⋊#95C2212yes—
35C11⋊#95C2212yes—
36C11⋊#95C2212yes—
37C23×D11C2311yescharacteristic
38C22×C4×C11C2311yescharacteristic
39C22×C4×C11C2311yescharacteristic
40C22×C4×C11C2312yes—
41C22×C4×C11C2311yescharacteristic
42C22×C4×C11C2312yes—
43C22×Dic11C2312yes—
44C22×Dic11C2312yes—
45C22×Dic11C2311yescharacteristic
46C22×Dic11C2311yescharacteristic
47C22×Dic11C2311yescharacteristic
48C11×C4⋊C4—22no—
49C11×C4⋊C4—22no—
50C11×C4⋊C4—42no—
51(C2×D11)⋊C4—22no—
52(C2×D11)⋊C4—22no—
53(C2×D11)⋊C4—22no—
54(C2×D11)⋊C4—24no—
55(C2×D11)⋊C4—42no—
56C4⋊Dic11—24no—
57C4⋊Dic11—22no—
58C4⋊Dic11—22no—
59C4⋊Dic11—24no—
60Dic11⋊C4—24no—
61Dic11⋊C4—22no—
62Dic11⋊C4—24no—
63C4×Dic11—42no—
64C4×Dic11D414yes—
65#2664—111nomaximal, Sylow
66C23×C11C2411yescharacteristic
67C22×D11—21no—
68C22×D11—22no—
69C22×D11—21no—
70C22×D11—22no—
71C22×D11—21no—
72C2×C4×C11—22no—
73C2×C4×C11—22no—
74C2×C4×C11—22no—
75C2×C4×C11—22no—
76C2×C4×C11—21no—
77C2×C4×C11—21no—
78C2×C4×C11—22no—
79C2×C4×C11—21no—
80C2×C4×C11—22no—
81C2×C4×C11—21no—
82C2×C4×C11—22no—
83C2×C4×C11C2×D412yes—
84C2×C4×C11#4114yes—
85C2×Dic11—22no—
86C2×Dic11—21no—
87C2×Dic11—21no—
88C2×Dic11—22no—
89C2×Dic11—21no—
90C2×Dic11—22no—
91C2×Dic11—22no—
92C2×Dic11—22no—
93C2×Dic11—22no—
94C2×Dic11—22no—
95C2×Dic11—22no—
96C2×Dic11C2×D412yes—
97C2×Dic11C2×D412yes—
98C2×#126—111no—
99C2×#125—111no—
100#400—111no—
101#399—111no—
102#398—112no—
103#398D1111yescharacteristic, complemented
104#397—111no—
105#396—112no—
106#389—112no—
107#388—112no—
108C4×C4⋊C4—111no—
109C22×C11C22×D411yescharacteristic
110C22×C11C2×#4112yes—
111C22×C11H(5,𝔽2)11yescharacteristic
112C22×C112−1+412yes—
113C22×C112−1+411yescharacteristic
114C2×D11—41no—
115C2×D11—41no—
116C2×D11—41no—
117C2×D11—42no—
118C2×D11—42no—
119C4×C11—22no—
120C4×C11—24no—
121C4×C11—41no—
122C4×C11—42no—
123C4×C11—41no—
124Dic11—22no—
125Dic11—22no—
126Dic11—42no—
127Dic11—41no—
128Dic11—42no—
129#126—224no—
130#125—224no—
131C2×C4⋊C4—112no—
132C2×C4⋊C4—112no—
133C2×C4⋊C4—111no—
134C2×C4⋊C4—111no—
135C2×C4⋊C4—111no—
136C2×C4⋊C4—112no—
137C2×C4⋊C4—112no—
138C2×C4⋊C4C2×D1111yescharacteristic
139C2×C4⋊C4C2×D1111yescharacteristic
140C2×C4⋊C4C2×D1112yes—
141C2×C22⋊C4—111no—
142C2×C22⋊C4—111no—
143C2×C22⋊C4—111no—
144C2×C22⋊C4—112no—
145C2×C22⋊C4—112no—
146C2×C42—111no—
147C2×C42—112no—
148#95—111no—
149#95—112no—
150#95—112no—
151#95—112no—
152#95C2×D1111yescharacteristic
153#95C2×D1112yes—
154C2×C11#53211yescharacteristic
155C2×C11#53611yescharacteristic
156C2×C11#53711yescharacteristic
157C2×C11#53912yes—
158C2×C11#54112yes—
159D11—81no—
160C24—111no—
161C22×C4—111no—
162C22×C4—111no—
163C22×C4—112no—
164C22×C4—111no—
165C22×C4—112no—
166C22×C4C22×D1111yescharacteristic
167C22×C4C22×D1111yescharacteristic
168C22×C4C22×D1112yes—
169C22×C4C22×D1111yescharacteristic
170C22×C4C22×D1112yes—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—42no—
174C4⋊C4—224no—
175C4⋊C4—224no—
176C4⋊C4—222no—
177C4⋊C4—222no—
178C4⋊C4—222no—
179C4⋊C4—224no—
180C4⋊C4—224no—
181C22⋊C4—222no—
182C22⋊C4—222no—
183C22⋊C4—222no—
184C22⋊C4—224no—
185C22⋊C4—442no—
186C42—114no—
187C42—442no—
188C11#266411yescharacteristic, complemented, Sylow
189C23—221no—
190C23—221no—
191C23—222no—
192C23—221no—
193C23—222no—
194C23C23×D1111yescharacteristic
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—112no—
207C2×C4—112no—
208C2×C4—221no—
209C2×C4—222no—
210C2×C4—221no—
211C2×C4—221no—
212C2×C4—222no—
213C2×C4—222no—
214C2×C4—222no—
215C2×C4—222no—
216C2×C4—222no—
217C2×C4—222no—
218C2×C4—222no—
219C2×C4D4×D1112yes—
220C2×C4#434314yes—
221C22—441no—
222C22—441no—
223C22—442no—
224C22—441no—
225C22—442no—
226C22C2×D4×D1111yescharacteristic
227C22#6703711yescharacteristic
228C22C2×#434312yes—
229C22#6704311yescharacteristic
230C22#6704312yes—
231C4—24no—
232C4—22no—
233C4—41no—
234C4—42no—
235C4—41no—
236C4—222no—
237C4—222no—
238C4—441no—
239C4—442no—
240C4—442no—
241C2—881no—
242C2#1062313911yescharacteristic
243C2#1062317612yes—
244C2#1062317911yescharacteristic
245C2#1062322812yes—
246C2#1062325611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC11×#398C2
prime radicalC23×D11C23
hypercenter#398D11
hypocenterC11#2664
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
2. C2×#66943C2semidirect
3. C11×#398C2semidirect
6. #10622599C2semidirect
7. #10622586C2semidirect
10. #10622274C2semidirect
11. #10622262C2semidirect
103. #398D11semidirect
188. C11#2664semidirect

Maximal subgroups

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

subgroupindex
C2×#669512
C2×#669432
C11×#3982
#106226072
#106226062
#106225992
#106225862
#106222902
#106222822
#106222742
#106222622
#266411