Finite Group Databasebeta

← #13075625

Subgroups — #13075625

Order 1408.

492 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 137 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×#67000C211yesmaximal, characteristic
2C2×#66997C211yesmaximal, characteristic, complemented
3C11×#391C211yesmaximal, characteristic, complemented
4#10622607C212yesmaximal, complemented
5#10622553C212yesmaximal, complemented
6#10622286C212yesmaximal, complemented
7#10622285C212yesmaximal
8#10622283C212yesmaximal, complemented
9#10622266C212yesmaximal, complemented
10C22×Q8×C11C2211yescharacteristic
11C22×D44C2211yescharacteristic
12C22×Dic22C2211yescharacteristic
13C2×C11×C4⋊C4C2211yescharacteristic
14C2×C42×C11C2211yescharacteristic
15#67000—24no—
16#66997—24no—
17C2×(C2×D11)⋊C4C2212yes—
18C2×(C2×D11)⋊C4C2212yes—
19C2×(C2×D11)⋊C4C2212yes—
20C2×(C2×D11)⋊C4C2212yes—
21C2×(C2×D11)⋊C4C2212yes—
22C2×(C2×D11)⋊C4C2212yes—
23C2×C4⋊Dic11C2212yes—
24C2×Dic11⋊C4C2212yes—
25C2×Dic11⋊C4C2212yes—
26C2×C4×Dic11C2211yescharacteristic
27C2×C4×Dic11C2211yescharacteristic
28C11×#95C2212yes—
29C11×#95C2212yes—
30C11⋊#95C2212yes—
31C11⋊#95C2212yes—
32C11⋊#95C2212yes—
33C23×D11C2312yes—
34C2×Q8×C11—41no—
35C2×Q8×C11—41no—
36C2×Q8×C11D414yes—
37C22×C4×C11C2312yes—
38C22×C4×C11C2312yes—
39C22×C4×C11C2311yescharacteristic
40C22×C4×C11C2311yescharacteristic
41C22×C4×C11C2311yescharacteristic
42C22×Dic11C2312yes—
43C22×Dic11C2312yes—
44C22×Dic11C2312yes—
45C2×D44—22no—
46C2×D44—41no—
47C2×D44—41no—
48C2×Dic22—22no—
49C2×Dic22—41no—
50C2×Dic22—41no—
51C11×C4⋊C4—41no—
52C42×C11—41no—
53(C2×D11)⋊C4—24no—
54(C2×D11)⋊C4—24no—
55(C2×D11)⋊C4—24no—
56(C2×D11)⋊C4—24no—
57(C2×D11)⋊C4—42no—
58(C2×D11)⋊C4—42no—
59C4⋊Dic11—24no—
60Dic11⋊C4—24no—
61Dic11⋊C4—24no—
62C4×Dic11—22no—
63C4×Dic11—22no—
64#2731—111nomaximal, Sylow
65C23×C11C2411yescharacteristic
66C22×D11—22no—
67C22×D11—22no—
68C22×D11—22no—
69C22×D11—22no—
70C22×D11—22no—
71C22×D11—22no—
72C22×D11—22no—
73Q8×C11—44no—
74C2×C4×C11—21no—
75C2×C4×C11—21no—
76C2×C4×C11—21no—
77C2×C4×C11—21no—
78C2×C4×C11—21no—
79C2×C4×C11—21no—
80C2×C4×C11—22no—
81C2×C4×C11—22no—
82C2×C4×C11—22no—
83C2×C4×C11—22no—
84C2×C4×C11—22no—
85C2×C4×C11—22no—
86C2×C4×C11C2×D412yes—
87C2×C4×C11C2×D412yes—
88C2×C4×C11C2×D412yes—
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#4114yes—
98D44—44no—
99Dic22—44no—
100C2×C4⋊Q8—111no—
101C2×#124—111no—
102#396—112no—
103#394—112no—
104#393—112no—
105#391D1111yescharacteristic, complemented
106#388—112no—
107#388—112no—
108#387—112no—
109C22×C11C22×D411yescharacteristic
110C22×C11C2×#4112yes—
111C22×C11H(5,𝔽2)11yescharacteristic
112C22×C11H(5,𝔽2)11yescharacteristic
113C22×C112−1+412yes—
114C2×D11—42no—
115C2×D11—42no—
116C2×D11—42no—
117C2×D11—42no—
118C2×D11—42no—
119C2×D11—42no—
120C2×D11—42no—
121C4×C11—22no—
122C4×C11—22no—
123C4×C11—22no—
124C4×C11—42no—
125C4×C11—42no—
126Dic11—24no—
127Dic11—42no—
128Dic11—42no—
129C22×Q8—111no—
130C22×Q8C2×D1111yescharacteristic
131C22×D4—111no—
132C4⋊Q8—224no—
133#124—224no—
134C2×C4⋊C4—112no—
135C2×C4⋊C4—112no—
136C2×C4⋊C4—112no—
137C2×C4⋊C4C2×D1111yescharacteristic
138C2×C22⋊C4—112no—
139C2×C22⋊C4—112no—
140C2×C22⋊C4—112no—
141C2×C22⋊C4—112no—
142C2×C22⋊C4—112no—
143C2×C22⋊C4—112no—
144C2×C42—111no—
145C2×C42—111no—
146C2×C42C2×D1111yescharacteristic
147#95—112no—
148#95—112no—
149#95—112no—
150#95C2×D1112yes—
151#95C2×D1112yes—
152C2×C11#53411yescharacteristic
153C2×C11#54212yes—
154C2×C11#54812yes—
155C2×C11#55511yescharacteristic
156C2×C11#56311yescharacteristic
157D11—82no—
158C24—112no—
159C2×Q8—41no—
160C2×Q8—41no—
161C2×Q8—222no—
162C2×Q8—441no—
163C2×Q8—441no—
164C2×Q8D4414yes—
165C2×D4—222no—
166C2×D4—441no—
167C2×D4—441no—
168C22×C4—112no—
169C22×C4—112no—
170C22×C4—112no—
171C22×C4C22×D1112yes—
172C22×C4C22×D1112yes—
173C22×C4C22×D1111yescharacteristic
174C22×C4C22×D1111yescharacteristic
175C22×C4C22×D1111yescharacteristic
176C4⋊C4—41no—
177C4⋊C4—224no—
178C4⋊C4—224no—
179C4⋊C4—224no—
180C22⋊C4—224no—
181C22⋊C4—224no—
182C22⋊C4—224no—
183C22⋊C4—224no—
184C22⋊C4—442no—
185C22⋊C4—442no—
186C42—41no—
187C42—222no—
188C42—222no—
189C11#273111yescharacteristic, complemented, Sylow
190C23—222no—
191C23—222no—
192C23—222no—
193C23—222no—
194C23—222no—
195C23—222no—
196C23—222no—
197C23C23×D1111yescharacteristic
198Q8—44no—
199Q8—444no—
200D4—444no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—114no—
214C2×C4—222no—
215C2×C4—222no—
216C2×C4—222no—
217C2×C4—222no—
218C2×C4—222no—
219C2×C4—222no—
220C2×C4—222no—
221C2×C4—222no—
222C2×C4C2×D4412yes—
223C2×C4C2×D4412yes—
224C2×C4C2×D4412yes—
225C22—442no—
226C22—442no—
227C22—442no—
228C22—442no—
229C22—442no—
230C22—442no—
231C22—442no—
232C22C22×D4411yescharacteristic
233C22#6703711yescharacteristic
234C22#6703711yescharacteristic
235C22#6704012yes—
236C22#41×D1112yes—
237C4—22no—
238C4—22no—
239C4—22no—
240C4—42no—
241C4—42no—
242C4—224no—
243C4—442no—
244C4—442no—
245C2—882no—
246C2#1062302411yescharacteristic
247C2#1062310712yes—
248C2#1062317112yes—
249C2#1062320111yescharacteristic
250C2#1062326611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC11×#391C2
prime radicalC22×D44C22
hypercenter#391D11
hypocenterC11#2731
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×#66997C2semidirect
3. C11×#391C2semidirect
4. #10622607C2semidirect
5. #10622553C2semidirect
6. #10622286C2semidirect
8. #10622283C2semidirect
9. #10622266C2semidirect
105. #391D11semidirect
189. C11#2731semidirect

Maximal subgroups

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

subgroupindex
C2×#670002
C2×#669972
C11×#3912
#106226072
#106225532
#106222862
#106222852
#106222832
#106222662
#273111