Finite Group Databasebeta

← #13075454

Subgroups — #13075454

Order 1408.

512 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 141 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×#66989C211yesmaximal, characteristic
2C2×#66988C211yesmaximal, characteristic
3C11×#386C211yesmaximal, characteristic
4#10622825C212yesmaximal, complemented
5#10622566C212yesmaximal, complemented
6#10622566C212yesmaximal, complemented
7#10622563C212yesmaximal, complemented
8#10622269C212yesmaximal, complemented
9#10622266C212yesmaximal, complemented
10C22×D4×C11C2211yescharacteristic
11C23×C4×C11C2211yescharacteristic
12C23×Dic11C2211yescharacteristic
13C22×Dic22C2211yescharacteristic
14C2×C11×C22⋊C4C2212yes—
15C2×C11×C22⋊C4C2211yescharacteristic
16C2×C22⋊Dic11C2211yescharacteristic
17C2×C22⋊Dic11C2212yes—
18C2×C22⋊Dic11C2212yes—
19C2×C22⋊Dic11C2212yes—
20#66989—24no—
21#66988—24no—
22C2×C4⋊Dic11C2212yes—
23C2×C4⋊Dic11C2212yes—
24C2×C4⋊Dic11C2212yes—
25C2×Dic11⋊C4C2212yes—
26C2×C4×Dic11C2211yescharacteristic
27C11×#95C2212yes—
28C11⋊#95C2212yes—
29C11⋊#95C2212yes—
30C11⋊#95C2212yes—
31C11⋊#95C2212yes—
32C11⋊#95C2212yes—
33C24×C11C2311yescharacteristic
34C24×C11C2311yescharacteristic
35C2×D4×C11—41no—
36C2×D4×C11—41no—
37C2×D4×C11D414yes—
38C22×C4×C11—24no—
39C22×C4×C11—41no—
40C22×C4×C11C2312yes—
41C22×C4×C11C2311yescharacteristic
42C22×C4×C11C2312yes—
43C22×Dic11—22no—
44C22×Dic11—41no—
45C22×Dic11—41no—
46C22×Dic11C2312yes—
47C22×Dic11C2312yes—
48C22×Dic11C2312yes—
49C22×Dic11C2312yes—
50C2×Dic22—22no—
51C2×Dic22—41no—
52C2×Dic22—41no—
53C11×C22⋊C4—24no—
54C11×C22⋊C4—41no—
55C22⋊Dic11—24no—
56C22⋊Dic11—22no—
57C22⋊Dic11—24no—
58C22⋊Dic11—24no—
59C4⋊Dic11—24no—
60C4⋊Dic11—42no—
61C4⋊Dic11—42no—
62Dic11⋊C4—24no—
63C4×Dic11—22no—
64#2717—111nomaximal, Sylow
65C23×C11—22no—
66C23×C11—22no—
67C23×C11—21no—
68C23×C11—21no—
69C23×C11—21no—
70C23×C11—21no—
71C23×C11—22no—
72C23×C11—22no—
73C23×C11C2×D412yes—
74C23×C11C2×D412yes—
75C23×C11C2411yescharacteristic
76D4×C11—44no—
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×C11—44no—
85C2×C4×C11C2×D412yes—
86C2×C4×C11#4114yes—
87C2×Dic11—22no—
88C2×Dic11—22no—
89C2×Dic11—22no—
90C2×Dic11—22no—
91C2×Dic11—22no—
92C2×Dic11—22no—
93C2×Dic11—22no—
94C2×Dic11—22no—
95C2×Dic11—22no—
96C2×Dic11—22no—
97C2×Dic11—22no—
98C2×Dic11—22no—
99C2×Dic11—44no—
100Dic22—44no—
101C2×#124—111no—
102C2×#123—111no—
103#397—112no—
104#397—112no—
105#395—112no—
106#387—112no—
107#386D1111yescharacteristic
108#385—112no—
109#380—112no—
110C22×C11—22no—
111C22×C11—22no—
112C22×C11—24no—
113C22×C11—24no—
114C22×C11—42no—
115C22×C11—42no—
116C22×C11—41no—
117C22×C11—41no—
118C22×C11—41no—
119C22×C11—41no—
120C22×C11C22×D411yescharacteristic
121C22×C11C2×#4112yes—
122C22×C11H(5,𝔽2)11yescharacteristic
123C22×C11H(5,𝔽2)11yescharacteristic
124C22×C112−1+412yes—
125C4×C11—22no—
126C4×C11—24no—
127C4×C11—42no—
128Dic11—42no—
129Dic11—42no—
130Dic11—42no—
131Dic11—42no—
132C22×Q8—111no—
133C22×D4C2×D1111yescharacteristic
134C23×C4—111no—
135C23×C4C2×D1111yescharacteristic
136#124—224no—
137#123—224no—
138C2×C4⋊C4—112no—
139C2×C4⋊C4—112no—
140C2×C4⋊C4—112no—
141C2×C4⋊C4—112no—
142C2×C22⋊C4—112no—
143C2×C22⋊C4—111no—
144C2×C22⋊C4—112no—
145C2×C22⋊C4—112no—
146C2×C22⋊C4C2×D1112yes—
147C2×C22⋊C4C2×D1111yescharacteristic
148C2×C42—111no—
149#95—112no—
150#95—112no—
151#95—112no—
152#95—112no—
153#95—112no—
154#95C2×D1112yes—
155C2×C11—42no—
156C2×C11—42no—
157C2×C11#53411yescharacteristic
158C2×C11#54212yes—
159C2×C11#54712yes—
160C2×C11#55111yescharacteristic
161C2×C11#56311yescharacteristic
162C24C22×D1111yescharacteristic
163C24C22×D1111yescharacteristic
164C2×Q8—222no—
165C2×Q8—441no—
166C2×Q8—441no—
167C2×D4—41no—
168C2×D4—41no—
169C2×D4D4414yes—
170C22×C4—24no—
171C22×C4—41no—
172C22×C4—112no—
173C22×C4—112no—
174C22×C4—112no—
175C22×C4—112no—
176C22×C4—222no—
177C22×C4—441no—
178C22×C4—441no—
179C22×C4C22×D1112yes—
180C22×C4C22×D1111yescharacteristic
181C22×C4C22×D1112yes—
182C4⋊C4—224no—
183C4⋊C4—224no—
184C4⋊C4—442no—
185C4⋊C4—442no—
186C22⋊C4—24no—
187C22⋊C4—41no—
188C22⋊C4—222no—
189C22⋊C4—224no—
190C22⋊C4—224no—
191C22⋊C4—224no—
192C42—222no—
193C11#271711yescharacteristic, complemented, Sylow
194C23—22no—
195C23—22no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—22no—
201C23—22no—
202C23C2×D4412yes—
203C23C2×D4412yes—
204C23C23×D1111yescharacteristic
205Q8—444no—
206D4—44no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—44no—
215C2×C4—222no—
216C2×C4—222no—
217C2×C4—222no—
218C2×C4—222no—
219C2×C4—222no—
220C2×C4—222no—
221C2×C4—222no—
222C2×C4—222no—
223C2×C4—222no—
224C2×C4—222no—
225C2×C4—222no—
226C2×C4—222no—
227C2×C4—444no—
228C2×C4C2×D4412yes—
229C2×C4#434114yes—
230C22—22no—
231C22—22no—
232C22—24no—
233C22—24no—
234C22—41no—
235C22—41no—
236C22—42no—
237C22—42no—
238C22—41no—
239C22—41no—
240C22C22×D4411yescharacteristic
241C22C2×#434112yes—
242C22#6703711yescharacteristic
243C22#6703711yescharacteristic
244C22#6704312yes—
245C4—22no—
246C4—24no—
247C4—42no—
248C4—442no—
249C4—442no—
250C4—442no—
251C4—442no—
252C2—42no—
253C2—42no—
254C2#1062302411yescharacteristic
255C2#1062308612yes—
256C2#1062311811yescharacteristic
257C2#1062313412yes—
258C2#1062318211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC11×#386C2
prime radicalC22×D4×C11C22
hypercenter#386D11
hypocenterC11#2717
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
4. #10622825C2semidirect
5. #10622566C2semidirect
6. #10622566C2semidirect
7. #10622563C2semidirect
8. #10622269C2semidirect
9. #10622266C2semidirect
193. C11#2717semidirect

Maximal subgroups

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

subgroupindex
C2×#669892
C2×#669882
C11×#3862
#106228252
#106225662
#106225632
#106222692
#106222662
#271711