Finite Group Databasebeta

← #13076681

Subgroups — #13076681

Order 1408.

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

normal subgroups only

This group has 430 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×#66951C211yesmaximal, characteristic
2C2×#66941C211yesmaximal, characteristic, complemented
3C11×#400C211yesmaximal, characteristic, complemented
4#10622605C211yesmaximal, characteristic
5#10622599C211yesmaximal, characteristic, complemented
6#10622290C211yesmaximal, characteristic
7#10622288C212yesmaximal
8#10622286C212yesmaximal
9#10622270C212yesmaximal, complemented
10#10622269C212yesmaximal, complemented
11#10622264C211yesmaximal, characteristic, complemented
12C22×Dic22C2212yes—
13C2×C11×C4⋊C4C2211yescharacteristic
14C2×C11×C4⋊C4C2211yescharacteristic
15C2×(C2×D11)⋊C4C2211yescharacteristic
16C2×(C2×D11)⋊C4C2211yescharacteristic
17C2×(C2×D11)⋊C4C2211yescharacteristic
18C2×(C2×D11)⋊C4C2212yes—
19C2×(C2×D11)⋊C4C2212yes—
20C2×C4⋊Dic11C2211yescharacteristic
21C2×C4⋊Dic11C2211yescharacteristic
22C2×C4⋊Dic11C2212yes—
23C2×Dic11⋊C4C2212yes—
24C2×Dic11⋊C4C2211yescharacteristic
25C2×Dic11⋊C4C2212yes—
26C2×C4×Dic11C2211yescharacteristic
27#66951—24no—
28#66941—24no—
29C11×#95C2211yescharacteristic
30C11×#95C2212yes—
31C11×#95C2212yes—
32C11⋊#95C2212yes—
33C11⋊#95C2212yes—
34C11⋊#95C2211yescharacteristic
35C11⋊#95C2212yes—
36C11⋊#95C2212yes—
37C23×D11C2311yescharacteristic
38C22×C4×C11C2312yes—
39C22×C4×C11C2311yescharacteristic
40C22×C4×C11C2311yescharacteristic
41C22×C4×C11C2312yes—
42C22×C4×C11C2311yescharacteristic
43C22×Dic11C2312yes—
44C22×Dic11C2312yes—
45C22×Dic11C2311yescharacteristic
46C22×Dic11C2311yescharacteristic
47C22×Dic11C2311yescharacteristic
48C2×Dic22—24no—
49C2×Dic22—42no—
50C2×Dic22—42no—
51C11×C4⋊C4—22no—
52C11×C4⋊C4—22no—
53(C2×D11)⋊C4—22no—
54(C2×D11)⋊C4—22no—
55(C2×D11)⋊C4—22no—
56(C2×D11)⋊C4—42no—
57(C2×D11)⋊C4—42no—
58C4⋊Dic11—22no—
59C4⋊Dic11—22no—
60C4⋊Dic11—42no—
61Dic11⋊C4—24no—
62Dic11⋊C4—22no—
63Dic11⋊C4—42no—
64C4×Dic11D414yes—
65#2821—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—21no—
74C2×C4×C11—22no—
75C2×C4×C11—21no—
76C2×C4×C11—22no—
77C2×C4×C11—22no—
78C2×C4×C11—22no—
79C2×C4×C11—21no—
80C2×C4×C11—21no—
81C2×C4×C11—22no—
82C2×C4×C11—22no—
83C2×C4×C11—22no—
84C2×C4×C11C2×D412yes—
85C2×Dic11—22no—
86C2×Dic11—21no—
87C2×Dic11—21no—
88C2×Dic11—21no—
89C2×Dic11—22no—
90C2×Dic11—22no—
91C2×Dic11—22no—
92C2×Dic11—22no—
93C2×Dic11—22no—
94C2×Dic11—22no—
95C2×Dic11—22no—
96C2×Dic11C2×D412yes—
97C2×Dic11C2×D412yes—
98Dic22—48no—
99C2×C4⋊Q8—111no—
100C2×#126—111no—
101#400—111no—
102#400D1111yescharacteristic, complemented
103#399—111no—
104#399—111no—
105#398—112no—
106#397—112no—
107#395—112no—
108#393—112no—
109#383—111no—
110C22×C11C22×D411yescharacteristic
111C22×C11H(5,𝔽2)12yes—
112C22×C112−1+411yescharacteristic
113C22×C112−1+411yescharacteristic
114C22×C112−1+412yes—
115C2×D11—41no—
116C2×D11—41no—
117C2×D11—41no—
118C2×D11—42no—
119C2×D11—42no—
120C4×C11—22no—
121C4×C11—42no—
122C4×C11—41no—
123C4×C11—41no—
124C4×C11—42no—
125Dic11—22no—
126Dic11—22no—
127Dic11—42no—
128Dic11—41no—
129Dic11—42no—
130C22×Q8—112no—
131C4⋊Q8—224no—
132#126—224no—
133C2×C4⋊C4—111no—
134C2×C4⋊C4—111no—
135C2×C4⋊C4—112no—
136C2×C4⋊C4—111no—
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—
147#95—112no—
148#95—112no—
149#95—111no—
150#95—112no—
151#95—112no—
152#95C2×D1111yescharacteristic
153#95C2×D1112yes—
154#95C2×D1112yes—
155C2×C11#53511yescharacteristic
156C2×C11#53611yescharacteristic
157C2×C11#53611yescharacteristic
158C2×C11#56312yes—
159C2×C11#56312yes—
160D11—81no—
161C24—111no—
162C2×Q8—224no—
163C2×Q8—442no—
164C2×Q8—442no—
165C22×C4—111no—
166C22×C4—111no—
167C22×C4—112no—
168C22×C4—112no—
169C22×C4—111no—
170C22×C4C22×D1112yes—
171C22×C4C22×D1111yescharacteristic
172C22×C4C22×D1111yescharacteristic
173C22×C4C22×D1112yes—
174C22×C4C22×D1111yescharacteristic
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—222no—
178C4⋊C4—222no—
179C4⋊C4—224no—
180C4⋊C4—222no—
181C4⋊C4—442no—
182C4⋊C4—442no—
183C22⋊C4—222no—
184C22⋊C4—222no—
185C22⋊C4—222no—
186C22⋊C4—442no—
187C22⋊C4—442no—
188C42—114no—
189C11#282111yescharacteristic, complemented, Sylow
190C23—221no—
191C23—221no—
192C23—222no—
193C23—221no—
194C23—222no—
195C23C23×D1111yescharacteristic
196Q8—448no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—112no—
210C2×C4—112no—
211C2×C4—221no—
212C2×C4—222no—
213C2×C4—221no—
214C2×C4—221no—
215C2×C4—222no—
216C2×C4—222no—
217C2×C4—222no—
218C2×C4—222no—
219C2×C4—222no—
220C2×C4—222no—
221C2×C4—222no—
222C2×C4D4×D1112yes—
223C22—442no—
224C22—441no—
225C22—441no—
226C22—441no—
227C22—442no—
228C22C2×D4×D1111yescharacteristic
229C22#6704011yescharacteristic
230C22#6704212yes—
231C22#6704311yescharacteristic
232C22#6704312yes—
233C4—22no—
234C4—42no—
235C4—41no—
236C4—41no—
237C4—42no—
238C4—222no—
239C4—222no—
240C4—442no—
241C4—441no—
242C4—442no—
243C2—881no—
244C2#1062315911yescharacteristic
245C2#1062317911yescharacteristic
246C2#1062320511yescharacteristic
247C2#1062323012yes—
248C2#1062324312yes—
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC11×#400C2
prime radicalC23×D11C23
hypercenter#400D11
hypocenterC11#2821
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. C2×#66941C2semidirect
3. C11×#400C2semidirect
5. #10622599C2semidirect
9. #10622270C2semidirect
10. #10622269C2semidirect
11. #10622264C2semidirect
102. #400D11semidirect
189. C11#2821semidirect

Maximal subgroups

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

subgroupindex
C2×#669512
C2×#669412
C11×#4002
#106226052
#106225992
#106222902
#106222882
#106222862
#106222702
#106222692
#106222642
#282111