Finite Group Databasebeta

← #13075381

Subgroups — #13075381

Order 1408.

556 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 133 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×#66989C212yesmaximal, complemented
2C11×#392C211yesmaximal, characteristic
3#10622835C212yesmaximal, complemented
4#10622833C212yesmaximal, complemented
5#10622566C211yesmaximal, characteristic, complemented
6#10622564C212yesmaximal, complemented
7#10622560C212yesmaximal, complemented
8#10622559C211yesmaximal, characteristic, complemented
9#10622259C212yesmaximal, complemented
10C22×D4×C11C2212yes—
11C22×D4×C11C2211yescharacteristic
12C23×Dic11C2211yescharacteristic
13C22×Dic22C2212yes—
14C2×C11×C22⋊C4C2212yes—
15C2×C11×C22⋊C4C2211yescharacteristic
16C2×C22⋊Dic11C2212yes—
17C2×C22⋊Dic11C2212yes—
18C2×C22⋊Dic11C2212yes—
19C2×C22⋊Dic11C2212yes—
20C2×C22⋊Dic11C2212yes—
21C2×C22⋊Dic11C2211yescharacteristic
22#66989—28no—
23C2×C4⋊Dic11C2211yescharacteristic
24C2×Dic11⋊C4C2212yes—
25C2×C4×Dic11C2212yes—
26C11×#95C2211yescharacteristic
27C11⋊#95C2212yes—
28C11⋊#95C2212yes—
29C11⋊#95C2212yes—
30C11⋊#95C2212yes—
31C11⋊#95C2211yescharacteristic
32C24×C11C2312yes—
33C24×C11C2311yescharacteristic
34C2×D4×C11—24no—
35C2×D4×C11—24no—
36C2×D4×C11—42no—
37C2×D4×C11—42no—
38C2×D4×C11—41no—
39C22×C4×C11C2311yescharacteristic
40C22×C4×C11C2311yescharacteristic
41C22×C4×C11C2312yes—
42C22×Dic11—41no—
43C22×Dic11—42no—
44C22×Dic11C2312yes—
45C22×Dic11C2312yes—
46C22×Dic11C2312yes—
47C22×Dic11C2312yes—
48C2×Dic22—24no—
49C2×Dic22—24no—
50C2×Dic22—42no—
51C11×C22⋊C4—24no—
52C11×C22⋊C4—41no—
53C22⋊Dic11—24no—
54C22⋊Dic11—24no—
55C22⋊Dic11—24no—
56C22⋊Dic11—42no—
57C22⋊Dic11—42no—
58C22⋊Dic11D414yescomplemented
59C4⋊Dic11—41no—
60Dic11⋊C4—42no—
61C4×Dic11—24no—
62#2712—111nomaximal, Sylow
63C23×C11—22no—
64C23×C11—21no—
65C23×C11—22no—
66C23×C11—21no—
67C23×C11—22no—
68C23×C11—22no—
69C23×C11—22no—
70C23×C11—22no—
71C23×C11—22no—
72C23×C11—22no—
73C23×C11—22no—
74C23×C11C2×D412yes—
75C23×C11C2411yescharacteristic
76D4×C11—48no—
77D4×C11—44no—
78C2×C4×C11—22no—
79C2×C4×C11—22no—
80C2×C4×C11—22no—
81C2×C4×C11—22no—
82C2×C4×C11—21no—
83C2×C4×C11—21no—
84C2×C4×C11#4114yes—
85C2×Dic11—22no—
86C2×Dic11—22no—
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—44no—
97C2×Dic11C2×D414yes—
98Dic22—48no—
99C2×#124—112no—
100#397—111no—
101#394—112no—
102#393—112no—
103#392D1111yescharacteristic
104#391—112no—
105#388—112no—
106#386—112no—
107#385—111no—
108C22×C11—22no—
109C22×C11—24no—
110C22×C11—42no—
111C22×C11—41no—
112C22×C11—41no—
113C22×C11—42no—
114C22×C11—42no—
115C22×C11—42no—
116C22×C11—42no—
117C22×C11—42no—
118C22×C11—42no—
119C22×C11—42no—
120C22×C11C22×D411yescharacteristic
121C22×C11C2×#4112yes—
122C22×C11H(5,𝔽2)11yescharacteristic
123C22×C11H(5,𝔽2)12yes—
124C22×C112−1+411yescharacteristic
125C4×C11—24no—
126C4×C11—41no—
127C4×C11—41no—
128Dic11—24no—
129Dic11—42no—
130Dic11—42no—
131Dic11—42no—
132C22×Q8—112no—
133C22×D4C2×D1112yes—
134C22×D4C2×D1111yescharacteristic
135C23×C4—111no—
136#124—228no—
137C2×C4⋊C4—112no—
138C2×C4⋊C4—111no—
139C2×C22⋊C4—111no—
140C2×C22⋊C4—112no—
141C2×C22⋊C4—112no—
142C2×C22⋊C4—112no—
143C2×C22⋊C4—112no—
144C2×C22⋊C4—112no—
145C2×C22⋊C4C2×D1112yes—
146C2×C22⋊C4C2×D1111yescharacteristic
147C2×C42—112no—
148#95—112no—
149#95—112no—
150#95—112no—
151#95—112no—
152#95—111no—
153#95C2×D1111yescharacteristic
154C2×C11—42no—
155C2×C11—82no—
156C2×C11#53711yescharacteristic
157C2×C11#54212yes—
158C2×C11#54612yes—
159C2×C11#55911yescharacteristic
160C2×C11H(3,𝔽4)11yescharacteristic
161C24C22×D1112yes—
162C24C22×D1111yescharacteristic
163C2×Q8—224no—
164C2×Q8—224no—
165C2×Q8—442no—
166C2×D4—24no—
167C2×D4—24no—
168C2×D4—42no—
169C2×D4—41no—
170C2×D4—42no—
171C22×C4—112no—
172C22×C4—112no—
173C22×C4—112no—
174C22×C4—112no—
175C22×C4—441no—
176C22×C4—442no—
177C22×C4C22×D1111yescharacteristic
178C22×C4C22×D1111yescharacteristic
179C22×C4C22×D1112yes—
180C4⋊C4—442no—
181C4⋊C4—441no—
182C22⋊C4—24no—
183C22⋊C4—41no—
184C22⋊C4—114no—
185C22⋊C4—224no—
186C22⋊C4—224no—
187C22⋊C4—224no—
188C22⋊C4—442no—
189C22⋊C4—442no—
190C42—224no—
191C11#271211yescharacteristic, complemented, Sylow
192C23—22no—
193C23—21no—
194C23—22no—
195C23—21no—
196C23—22no—
197C23—22no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—22no—
203C23D4×D1112yes—
204C23C23×D1111yescharacteristic
205Q8—448no—
206D4—48no—
207D4—44no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—114no—
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—444no—
227C2×C4#434114yes—
228C22—24no—
229C22—22no—
230C22—41no—
231C22—41no—
232C22—42no—
233C22—42no—
234C22—42no—
235C22—42no—
236C22—42no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22C2×D4×D1111yescharacteristic
241C22C2×#434112yes—
242C22#6703711yescharacteristic
243C22#6703712yes—
244C22#6704311yescharacteristic
245C4—24no—
246C4—41no—
247C4—41no—
248C4—224no—
249C4—442no—
250C4—442no—
251C4—442no—
252C2—42no—
253C2—82no—
254C2#1062312212yes—
255C2#1062312511yescharacteristic
256C2#1062312811yescharacteristic
257C2#1062313412yes—
258C2#1062324411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
Fitting, prime radicalC11×#392C2
hypercenter#392D11
hypocenterC11#2712
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#66989C2semidirect
3. #10622835C2semidirect
4. #10622833C2semidirect
5. #10622566C2semidirect
6. #10622564C2semidirect
7. #10622560C2semidirect
8. #10622559C2semidirect
9. #10622259C2semidirect
58. C22⋊Dic11D4semidirect
191. C11#2712semidirect

Maximal subgroups

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

subgroupindex
C2×#669892
C11×#3922
#106228352
#106228332
#106225662
#106225642
#106225602
#106225592
#106222592
#271211