Finite Group Databasebeta

← #13076915

Subgroups — #13076915

Order 1408.

696 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 177 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×C11×#121C211yesmaximal, characteristic
2C2×#66989C211yesmaximal, characteristic, complemented
3C2×#66977C211yesmaximal, characteristic, complemented
4C22×C4×Dic11C211yesmaximal, characteristic, complemented
5#10622834C211yesmaximal, characteristic, complemented
6#10622833C214yesmaximal, complemented
7#10622590C211yesmaximal, characteristic, complemented
8#10622587C211yesmaximal, characteristic, complemented
9#10622558C214yesmaximal, complemented
10C22×D4×C11C2211yescharacteristic
11C22×D4×C11C2212yes—
12C23×C4×C11C2211yescharacteristic
13C23×Dic11C2212yes—
14C22×Dic22C2211yescharacteristic
15C11×#121—24no—
16C2×C11×C4⋊C4C2211yescharacteristic
17C2×C11×C22⋊C4C2212yes—
18C2×C22⋊Dic11C2212yes—
19C2×C22⋊Dic11C2214yes—
20C2×C22⋊Dic11C2214yes—
21#66989—24no—
22C2×C4⋊Dic11C2211yescharacteristic
23#66977—24no—
24C2×Dic11⋊C4C2212yes—
25C2×C4×Dic11—24no—
26C2×C4×Dic11C2212yes—
27C2×C4×Dic11C2211yescharacteristic
28C2×C4×Dic11C2211yescharacteristic
29C11⋊#95C2214yes—
30C11⋊#95C2214yes—
31C24×C11C2312yes—
32C24×C11C2311yescharacteristic
33C2×D4×C11—24no—
34C2×D4×C11—28no—
35C2×D4×C11—22no—
36C2×D4×C11—22no—
37C2×D4×C11—22no—
38C22×C4×C11—22no—
39C22×C4×C11—22no—
40C22×C4×C11D414yes—
41C22×C4×C11C2312yes—
42C22×C4×C11C2311yescharacteristic
43C22×C4×C11C2311yescharacteristic
44C22×Dic11—24no—
45C22×Dic11—24no—
46C22×Dic11—24no—
47C22×Dic11C2312yes—
48C22×Dic11C2312yes—
49C22×Dic11C2312yes—
50C22×Dic11C2312yes—
51C2×Dic22—22no—
52C2×Dic22—22no—
53C2×Dic22—22no—
54C11×C4⋊C4—22no—
55C11×C22⋊C4—24no—
56C22⋊Dic11—28no—
57C22⋊Dic11—28no—
58C22⋊Dic11—24no—
59C4⋊Dic11—22no—
60Dic11⋊C4—24no—
61C4×Dic11—24no—
62C4×Dic11—22no—
63C4×Dic11D414yescomplemented
64#2850—111nomaximal, Sylow
65C23×C11—21no—
66C23×C11—22no—
67C23×C11—21no—
68C23×C11—22no—
69C23×C11—22no—
70C23×C11—24no—
71C23×C11—22no—
72C23×C11—24no—
73C23×C11C2×D412yes—
74C23×C11#4114yes—
75C23×C11C2411yescharacteristic
76D4×C11—48no—
77D4×C11—44no—
78C2×C4×C11—21no—
79C2×C4×C11—21no—
80C2×C4×C11—28no—
81C2×C4×C11—22no—
82C2×C4×C11—24no—
83C2×C4×C11C2×D412yes—
84C2×C4×C11C2×D412yes—
85C2×C4×C11C2×D412yes—
86C2×C4×C11#4112yes—
87C2×Dic11—216no—
88C2×Dic11—22no—
89C2×Dic11—22no—
90C2×Dic11—22no—
91C2×Dic11—22no—
92C2×Dic11—22no—
93C2×Dic11—22no—
94C2×Dic11—22no—
95C2×Dic11—24no—
96C2×Dic11—22no—
97C2×Dic11C2×D414yes—
98C2×Dic11#4114yes—
99Dic22—44no—
100C2×#124—111no—
101C2×C22⋊Q8—111no—
102C2×#121D1111yescharacteristic
103C22×C42—111no—
104#391—111no—
105#390—111no—
106#388—114no—
107#386—114no—
108#382—111no—
109C22×C11—22no—
110C22×C11—24no—
111C22×C11—24no—
112C22×C11—22no—
113C22×C11—42no—
114C22×C11—44no—
115C22×C11—42no—
116C22×C11—44no—
117C22×C11—42no—
118C22×C11#12414yescomplemented
119C22×C11C22×D411yescharacteristic
120C22×C11C22×D411yescharacteristic
121C22×C11C2×#4111yescharacteristic
122C22×C11C2×#4112yes—
123C22×C11C2×#4112yes—
124C4×C11—22no—
125C4×C11—42no—
126C4×C11#12114yes—
127Dic11—24no—
128Dic11—24no—
129Dic11—42no—
130Dic11—42no—
131C22×Q8—111no—
132C22×D4C2×D1111yescharacteristic
133C22×D4C2×D1112yes—
134C23×C4—112no—
135C23×C4C2×D1111yescharacteristic
136#124—224no—
137C22⋊Q8—224no—
138#121—24no—
139C2×C4⋊C4—111no—
140C2×C4⋊C4—112no—
141C2×C4⋊C4C2×D1111yescharacteristic
142C2×C22⋊C4—112no—
143C2×C22⋊C4—114no—
144C2×C22⋊C4—114no—
145C2×C22⋊C4C2×D1112yes—
146C2×C42—111no—
147C2×C42—112no—
148C2×C42—111no—
149C2×C42—224no—
150#95—114no—
151#95—114no—
152C2×C11—24no—
153C2×C11—82no—
154C2×C11C2×#12111yescharacteristic
155C2×C11#52512yes—
156C2×C11C2×#12411yescharacteristic
157C2×C11#52912yes—
158C2×C11#53211yescharacteristic
159C24C22×D1112yes—
160C24C22×D1111yescharacteristic
161C2×Q8—222no—
162C2×Q8—222no—
163C2×Q8—222no—
164C2×D4—28no—
165C2×D4—24no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—112no—
172C22×C4—112no—
173C22×C4—112no—
174C22×C4—112no—
175C22×C4—224no—
176C22×C4—224no—
177C22×C4—224no—
178C22×C4C22⋊D1114yes—
179C22×C4C22×D1112yes—
180C22×C4C22×D1111yescharacteristic
181C22×C4C22×D1111yescharacteristic
182C4⋊C4—22no—
183C4⋊C4—224no—
184C4⋊C4—222no—
185C22⋊C4—24no—
186C22⋊C4—228no—
187C22⋊C4—228no—
188C22⋊C4—224no—
189C42—114no—
190C42—224no—
191C42—222no—
192C11#285011yescharacteristic, complemented, Sylow
193C23—21no—
194C23—21no—
195C23—22no—
196C23—22no—
197C23—24no—
198C23—22no—
199C23—24no—
200C23—22no—
201C23#434114yes—
202C23C2×C22⋊D1112yes—
203C23C23×D1111yescharacteristic
204Q8—444no—
205D4—48no—
206D4—44no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—28no—
210C2×C4—24no—
211C2×C4—22no—
212C2×C4—114no—
213C2×C4—114no—
214C2×C4—2216no—
215C2×C4—222no—
216C2×C4—222no—
217C2×C4—222no—
218C2×C4—222no—
219C2×C4—222no—
220C2×C4—222no—
221C2×C4—224no—
222C2×C4—222no—
223C2×C4—222no—
224C2×C4D4×D1112yes—
225C2×C4#434112yes—
226C2×C4C2×C22⋊D1112yes—
227C2×C4C2×C22⋊D1112yes—
228C22—22no—
229C22—22no—
230C22—24no—
231C22—24no—
232C22—42no—
233C22—42no—
234C22—42no—
235C22—44no—
236C22—44no—
237C22#6698914yescomplemented
238C22C2×D4×D1111yescharacteristic
239C22C2×#434112yes—
240C22C2×#434111yescharacteristic
241C22#41×D1112yes—
242C22C22×C22⋊D1111yescharacteristic
243C4—22no—
244C4—42no—
245C4—224no—
246C4—224no—
247C4—442no—
248C4—442no—
249C4#6699214yes—
250C2—24no—
251C2—82no—
252C2#1062313112yes—
253C2#1062313712yes—
254C2C2×#6698911yescharacteristic
255C2C2×#6699211yescharacteristic
256C2#1062336311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
Fitting, prime radicalC2×C11×#121C2
hypercenterC2×#121D11
hypocenterC11#2850
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
2. C2×#66989C2semidirect
3. C2×#66977C2semidirect
4. C22×C4×Dic11C2semidirect
5. #10622834C2semidirect
6. #10622833C2semidirect
7. #10622590C2semidirect
8. #10622587C2semidirect
9. #10622558C2semidirect
63. C4×Dic11D4semidirect
118. C22×C11#124semidirect
192. C11#2850semidirect
237. C22#66989semidirect

Maximal subgroups

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

subgroupindex
C2×C11×#1212
C2×#669892
C2×#669772
C22×C4×Dic112
#106228342
#106228332
#106225902
#106225872
#106225582
#285011