Finite Group Databasebeta

← #13074783

Subgroups — #13074783

Order 1408.

434 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 129 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×#400C211yesmaximal, characteristic, complemented
4#10622605C211yesmaximal, characteristic
5#10622596C211yesmaximal, characteristic, complemented
6#10622289C212yesmaximal
7#10622287C211yesmaximal, characteristic
8#10622283C212yesmaximal
9#10622270C212yesmaximal, complemented
10#10622262C211yesmaximal, characteristic, complemented
11#10622260C212yesmaximal, complemented
12C2×C11×C4⋊C4C2211yescharacteristic
13C2×C11×C4⋊C4C2211yescharacteristic
14C2×(C2×D11)⋊C4C2211yescharacteristic
15C2×(C2×D11)⋊C4C2211yescharacteristic
16C2×(C2×D11)⋊C4C2211yescharacteristic
17C2×(C2×D11)⋊C4C2212yes—
18C2×(C2×D11)⋊C4C2212yes—
19C2×C4⋊Dic11C2212yes—
20C2×C4⋊Dic11C2211yescharacteristic
21C2×C4⋊Dic11C2211yescharacteristic
22C2×Dic11⋊C4C2211yescharacteristic
23C2×Dic11⋊C4C2212yes—
24C2×Dic11⋊C4C2212yes—
25C2×Dic11⋊C4C2212yes—
26C2×Dic11⋊C4C2212yes—
27C2×C4×Dic11C2212yes—
28C2×C4×Dic11C2211yescharacteristic
29#66951—24no—
30#66943—24no—
31C11×#95C2212yes—
32C11×#95C2211yescharacteristic
33C11×#95C2212yes—
34C11⋊#95C2212yes—
35C11⋊#95C2211yescharacteristic
36C11⋊#95C2212yes—
37C23×D11C2311yescharacteristic
38C22×C4×C11C2311yescharacteristic
39C22×C4×C11C2312yes—
40C22×C4×C11C2311yescharacteristic
41C22×C4×C11C2312yes—
42C22×C4×C11C2311yescharacteristic
43C22×Dic11C2311yescharacteristic
44C22×Dic11C2312yes—
45C22×Dic11C2311yescharacteristic
46C22×Dic11C2311yescharacteristic
47C22×Dic11C2312yes—
48C11×C4⋊C4—22no—
49C11×C4⋊C4—22no—
50(C2×D11)⋊C4—22no—
51(C2×D11)⋊C4—22no—
52(C2×D11)⋊C4—24no—
53(C2×D11)⋊C4—22no—
54(C2×D11)⋊C4—42no—
55C4⋊Dic11—24no—
56C4⋊Dic11—22no—
57C4⋊Dic11—22no—
58Dic11⋊C4—24no—
59Dic11⋊C4—22no—
60Dic11⋊C4—24no—
61Dic11⋊C4—24no—
62Dic11⋊C4—42no—
63C4×Dic11—42no—
64C4×Dic11D414yes—
65#2664—111nomaximal, Sylow
66C23×C11C2411yescharacteristic
67C22×D11—21no—
68C22×D11—22no—
69C22×D11—21no—
70C22×D11—21no—
71C22×D11—22no—
72C2×C4×C11—22no—
73C2×C4×C11—22no—
74C2×C4×C11—21no—
75C2×C4×C11—21no—
76C2×C4×C11—22no—
77C2×C4×C11—22no—
78C2×C4×C11—22no—
79C2×C4×C11—22no—
80C2×C4×C11—21no—
81C2×C4×C11—21no—
82C2×C4×C11—22no—
83C2×C4×C11—22no—
84C2×C4×C11C2×D412yes—
85C2×Dic11—22no—
86C2×Dic11—21no—
87C2×Dic11—22no—
88C2×Dic11—21no—
89C2×Dic11—22no—
90C2×Dic11—22no—
91C2×Dic11—21no—
92C2×Dic11—22no—
93C2×Dic11—22no—
94C2×Dic11—22no—
95C2×Dic11C2×D412yes—
96C2×Dic11C2×D412yes—
97C2×Dic11#4114yes—
98C2×#126—111no—
99C2×#125—111no—
100#400D1111yescharacteristic, complemented
101#399—111no—
102#398—111no—
103#398—112no—
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+411yescharacteristic
113C22×C112−1+412yes—
114C2×D11—41no—
115C2×D11—41no—
116C2×D11—41no—
117C2×D11—42no—
118C2×D11—42no—
119C4×C11—22no—
120C4×C11—42no—
121C4×C11—41no—
122C4×C11—41no—
123C4×C11—42no—
124Dic11—22no—
125Dic11—22no—
126Dic11—24no—
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⋊C4—112no—
139C2×C4⋊C4C2×D1111yescharacteristic
140C2×C4⋊C4C2×D1111yescharacteristic
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#95C2×D1111yescharacteristic
152#95C2×D1112yes—
153#95C2×D1112yes—
154C2×C11#53211yescharacteristic
155C2×C11#53611yescharacteristic
156C2×C11#53711yescharacteristic
157C2×C11#53912yes—
158C2×C11#54112yes—
159D11—81no—
160C24—111no—
161C22×C4—112no—
162C22×C4—111no—
163C22×C4—111no—
164C22×C4—111no—
165C22×C4—112no—
166C22×C4C22×D1111yescharacteristic
167C22×C4C22×D1112yes—
168C22×C4C22×D1111yescharacteristic
169C22×C4C22×D1112yes—
170C22×C4C22×D1111yescharacteristic
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—224no—
174C4⋊C4—224no—
175C4⋊C4—222no—
176C4⋊C4—222no—
177C4⋊C4—224no—
178C4⋊C4—224no—
179C4⋊C4—222no—
180C4⋊C4—442no—
181C22⋊C4—222no—
182C22⋊C4—222no—
183C22⋊C4—224no—
184C22⋊C4—222no—
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—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—112no—
208C2×C4—112no—
209C2×C4—114no—
210C2×C4—221no—
211C2×C4—222no—
212C2×C4—221no—
213C2×C4—222no—
214C2×C4—222no—
215C2×C4—221no—
216C2×C4—222no—
217C2×C4—222no—
218C2×C4—222no—
219C2×C4—222no—
220C2×C4D4×D1112yes—
221C22—442no—
222C22—441no—
223C22—441no—
224C22—441no—
225C22—442no—
226C22C2×D4×D1111yescharacteristic
227C22#6704011yescharacteristic
228C22#41×D1112yes—
229C22#6704211yescharacteristic
230C22#6704312yes—
231C4—22no—
232C4—42no—
233C4—41no—
234C4—41no—
235C4—42no—
236C4—224no—
237C4—222no—
238C4—222no—
239C4—442no—
240C4—441no—
241C2—881no—
242C2#1062316011yescharacteristic
243C2#1062317911yescharacteristic
244C2#1062319911yescharacteristic
245C2#1062322712yes—
246C2#1062323812yes—
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC11×#400C2
prime radicalC23×D11C23
hypercenter#400D11
hypocenterC11#2664
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. C2×#66943C2semidirect
3. C11×#400C2semidirect
5. #10622596C2semidirect
9. #10622270C2semidirect
10. #10622262C2semidirect
11. #10622260C2semidirect
100. #400D11semidirect
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×#4002
#106226052
#106225962
#106222892
#106222872
#106222832
#106222702
#106222622
#106222602
#266411