Finite Group Databasebeta

← #13076652

Subgroups — #13076652

Order 1408.

444 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 125 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
2C2×#66949C211yesmaximal, characteristic
3C11×#400C211yesmaximal, characteristic, complemented
4#10622595C211yesmaximal, characteristic, complemented
5#10622592C211yesmaximal, characteristic, complemented
6#10622288C212yesmaximal
7#10622287C212yesmaximal
8#10622277C211yesmaximal, characteristic
9#10622273C212yesmaximal, complemented
10#10622269C212yesmaximal, complemented
11#10622269C211yesmaximal, characteristic, complemented
12C22×C4×D11C2211yescharacteristic
13C22×Dic22C2211yescharacteristic
14C22×Dic22C2211yescharacteristic
15C2×C11×C4⋊C4C2211yescharacteristic
16C2×C11×C4⋊C4C2211yescharacteristic
17C2×(C2×D11)⋊C4C2211yescharacteristic
18C2×(C2×D11)⋊C4C2211yescharacteristic
19C2×(C2×D11)⋊C4C2212yes—
20C2×(C2×D11)⋊C4C2212yes—
21C2×C4⋊Dic11C2212yes—
22C2×C4⋊Dic11C2212yes—
23C2×Dic11⋊C4C2211yescharacteristic
24C2×Dic11⋊C4C2211yescharacteristic
25C2×Dic11⋊C4C2212yes—
26#66950—24no—
27#66949—24no—
28C11×#95C2211yescharacteristic
29C11×#95C2212yes—
30C11×#95C2212yes—
31C11⋊#95C2212yes—
32C11⋊#95C2212yes—
33C11⋊#95C2212yes—
34C11⋊#95C2212yes—
35C11⋊#95C2212yes—
36C11⋊#95C2211yescharacteristic
37C23×D11C2311yescharacteristic
38C22×C4×C11C2312yes—
39C22×C4×C11C2311yescharacteristic
40C22×C4×C11C2312yes—
41C22×C4×C11C2311yescharacteristic
42C22×C4×C11C2311yescharacteristic
43C22×Dic11C2312yes—
44C22×Dic11C2311yescharacteristic
45C22×Dic11C2311yescharacteristic
46C22×Dic11C2312yes—
47C22×Dic11C2311yescharacteristic
48C2×C4×D11—42no—
49C2×C4×D11D414yes—
50C2×Dic22—22no—
51C2×Dic22—22no—
52C2×Dic22—42no—
53C2×Dic22—42no—
54C11×C4⋊C4—22no—
55C11×C4⋊C4—22no—
56(C2×D11)⋊C4—22no—
57(C2×D11)⋊C4—22no—
58(C2×D11)⋊C4—24no—
59(C2×D11)⋊C4—42no—
60C4⋊Dic11—24no—
61C4⋊Dic11—42no—
62Dic11⋊C4—22no—
63Dic11⋊C4—22no—
64Dic11⋊C4—42no—
65#2817—111nomaximal, Sylow
66C23×C11C2411yescharacteristic
67C22×D11—22no—
68C22×D11—21no—
69C22×D11—21no—
70C22×D11—22no—
71C22×D11C2×D412yes—
72C2×C4×C11—22no—
73C2×C4×C11—21no—
74C2×C4×C11—22no—
75C2×C4×C11—21no—
76C2×C4×C11—22no—
77C2×C4×C11—22no—
78C2×C4×C11—21no—
79C2×C4×C11—21no—
80C2×C4×C11—24no—
81C2×C4×C11—22no—
82C2×C4×C11—22no—
83C2×C4×C11C2×D412yes—
84C2×Dic11—22no—
85C2×Dic11—21no—
86C2×Dic11—21no—
87C2×Dic11—22no—
88C2×Dic11—22no—
89C2×Dic11—22no—
90C2×Dic11—22no—
91C2×Dic11—22no—
92C2×Dic11—21no—
93C2×Dic11—22no—
94C2×Dic11—21no—
95C2×Dic11—22no—
96C2×Dic11C2×D412yes—
97C4×D11—44no—
98Dic22—44no—
99Dic22—44no—
100C2×C22⋊Q8—111no—
101C2×C22⋊Q8—111no—
102#400—111no—
103#400—112no—
104#400D1111yescharacteristic, complemented
105#397—112no—
106#397—112no—
107#395—111no—
108#395—112no—
109#395—111no—
110#381—111no—
111C22×C11C22×D411yescharacteristic
112C22×C11H(5,𝔽2)12yes—
113C22×C112−1+411yescharacteristic
114C22×C112−1+411yescharacteristic
115C22×C112−1+412yes—
116C2×D11—22no—
117C2×D11—22no—
118C2×D11—22no—
119C2×D11—42no—
120C2×D11—42no—
121C4×C11—22no—
122C4×C11—41no—
123C4×C11—41no—
124C4×C11—42no—
125C4×C11—42no—
126Dic11—22no—
127Dic11—41no—
128Dic11—42no—
129Dic11—41no—
130Dic11—42no—
131C22×Q8—111no—
132C22×Q8—111no—
133C23×C4—111no—
134C22⋊Q8—224no—
135C22⋊Q8—224no—
136C2×C4⋊C4—111no—
137C2×C4⋊C4—112no—
138C2×C4⋊C4—111no—
139C2×C4⋊C4—112no—
140C2×C4⋊C4—112no—
141C2×C4⋊C4C2×D1111yescharacteristic
142C2×C4⋊C4C2×D1111yescharacteristic
143C2×C22⋊C4—111no—
144C2×C22⋊C4—111no—
145C2×C22⋊C4—112no—
146C2×C22⋊C4—112no—
147#95—112no—
148#95—111no—
149#95—112no—
150#95—112no—
151#95—112no—
152#95—112no—
153#95C2×D1111yescharacteristic
154#95C2×D1112yes—
155#95C2×D1112yes—
156C2×C11#53411yescharacteristic
157C2×C11#53611yescharacteristic
158C2×C11#53611yescharacteristic
159C2×C11#56312yes—
160C2×C11#56312yes—
161D11—42no—
162C24—111no—
163C2×Q8—222no—
164C2×Q8—222no—
165C2×Q8—442no—
166C2×Q8—442no—
167C22×C4—111no—
168C22×C4—114no—
169C22×C4—112no—
170C22×C4—111no—
171C22×C4—111no—
172C22×C4—112no—
173C22×C4—442no—
174C22×C4C22×D1112yes—
175C22×C4C22×D1111yescharacteristic
176C22×C4C22×D1112yes—
177C22×C4C22×D1111yescharacteristic
178C22×C4C22×D1111yescharacteristic
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—222no—
182C4⋊C4—224no—
183C4⋊C4—222no—
184C4⋊C4—442no—
185C4⋊C4—442no—
186C22⋊C4—222no—
187C22⋊C4—224no—
188C22⋊C4—222no—
189C22⋊C4—442no—
190C11#281711yescharacteristic, complemented, Sylow
191C23—112no—
192C23—221no—
193C23—222no—
194C23—221no—
195C23—222no—
196C23C23×D1111yescharacteristic
197Q8—444no—
198Q8—444no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—24no—
210C2×C4—112no—
211C2×C4—222no—
212C2×C4—221no—
213C2×C4—222no—
214C2×C4—221no—
215C2×C4—221no—
216C2×C4—222no—
217C2×C4—222no—
218C2×C4—222no—
219C2×C4—222no—
220C2×C4—222no—
221C2×C4—221no—
222C2×C4—222no—
223C2×C4—444no—
224C2×C4D4×D1112yes—
225C22—222no—
226C22—222no—
227C22—222no—
228C22—442no—
229C22—442no—
230C22C2×D4×D1111yescharacteristic
231C22#6704011yescharacteristic
232C22#6704212yes—
233C22#6704311yescharacteristic
234C22#6704312yes—
235C4—22no—
236C4—42no—
237C4—41no—
238C4—41no—
239C4—42no—
240C4—222no—
241C4—441no—
242C4—442no—
243C4—441no—
244C4—442no—
245C2—442no—
246C2#1062315911yescharacteristic
247C2#1062318411yescharacteristic
248C2#1062320611yescharacteristic
249C2#1062323012yes—
250C2#1062324312yes—
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC11×#400C2
prime radicalC23×D11C23
hypercenter#400D11
hypocenterC11#2817
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
3. C11×#400C2semidirect
4. #10622595C2semidirect
5. #10622592C2semidirect
9. #10622273C2semidirect
10. #10622269C2semidirect
11. #10622269C2semidirect
104. #400D11semidirect
190. C11#2817semidirect

Maximal subgroups

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

subgroupindex
C2×#669502
C2×#669492
C11×#4002
#106225952
#106225922
#106222882
#106222872
#106222772
#106222732
#106222692
#281711