Finite Group Databasebeta

← #13073063

Subgroups — #13073063

Order 1408.

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

normal subgroups only

This group has 546 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×#66945C211yesmaximal, characteristic, complemented
2C2×#66936C211yesmaximal, characteristic
3C11×#397C211yesmaximal, characteristic, complemented
4#10622593C211yesmaximal, characteristic, complemented
5#10622569C212yesmaximal
6#10622562C211yesmaximal, characteristic, complemented
7#10622558C212yesmaximal, complemented
8#10622280C212yesmaximal, complemented
9D11×#95C211yesmaximal, characteristic, complemented
10#10622272C211yesmaximal, characteristic, complemented
11#10622268C212yesmaximal, complemented
12C22×C22⋊D11C2211yescharacteristic
13C22×C4×D11C2212yes—
14C22×C4×D11C2211yescharacteristic
15C2×C11×C4⋊C4C2211yescharacteristic
16C2×C11×C22⋊C4C2211yescharacteristic
17C2×C11×C22⋊C4C2212yes—
18C2×C22⋊Dic11C2211yescharacteristic
19C2×C22⋊Dic11C2212yes—
20C2×(C2×D11)⋊C4C2211yescharacteristic
21C2×(C2×D11)⋊C4C2212yes—
22C2×C4⋊Dic11C2211yescharacteristic
23C2×C4⋊Dic11C2211yescharacteristic
24C2×Dic11⋊C4C2211yescharacteristic
25C2×Dic11⋊C4C2212yes—
26C2×Dic11⋊C4C2212yes—
27C2×C4×Dic11C2212yes—
28#66945—24no—
29#66936—24no—
30C11×#95C2212yes—
31C11×#95C2211yescharacteristic
32C11⋊#95C2212yes—
33C11⋊#95C2212yes—
34C11⋊#95C2212yes—
35C11⋊#95C2212yes—
36C11⋊#95C2211yescharacteristic
37C24×C11C2311yescharacteristic
38C23×D11C2311yescharacteristic
39C22×C4×C11C2312yes—
40C22×C4×C11C2312yes—
41C22×C4×C11C2311yescharacteristic
42C22×C4×C11C2311yescharacteristic
43C2×C22⋊D11—22no—
44C2×C22⋊D11—24no—
45C22×Dic11C2311yescharacteristic
46C22×Dic11C2312yes—
47C22×Dic11C2311yescharacteristic
48C22×Dic11C2311yescharacteristic
49C22×Dic11C2312yes—
50C2×C4×D11—24no—
51C2×C4×D11—24no—
52C2×C4×D11—24no—
53C2×C4×D11—42no—
54C2×C4×D11D414yes—
55C11×C4⋊C4—22no—
56C11×C22⋊C4—22no—
57C11×C22⋊C4—42no—
58C22⋊Dic11—22no—
59C22⋊Dic11—24no—
60(C2×D11)⋊C4—22no—
61(C2×D11)⋊C4—24no—
62C4⋊Dic11—22no—
63C4⋊Dic11—22no—
64Dic11⋊C4—24no—
65Dic11⋊C4—22no—
66Dic11⋊C4—42no—
67C4×Dic11—24no—
68#2498—111nomaximal, Sylow
69C23×C11—21no—
70C23×C11—22no—
71C23×C11—22no—
72C23×C11—21no—
73C23×C11—21no—
74C23×C11C2411yescharacteristic
75C22×D11—22no—
76C22×D11—21no—
77C22×D11—21no—
78C22×D11C2×D412yes—
79C22×D11#4114yes—
80C2×C4×C11—22no—
81C2×C4×C11—22no—
82C2×C4×C11—21no—
83C2×C4×C11—22no—
84C2×C4×C11—22no—
85C2×C4×C11—22no—
86C2×C4×C11—22no—
87C2×C4×C11—22no—
88C2×C4×C11—22no—
89C2×C4×C11C2×D412yes—
90C22⋊D11—44no—
91C2×Dic11—22no—
92C2×Dic11—22no—
93C2×Dic11—22no—
94C2×Dic11—22no—
95C2×Dic11—21no—
96C2×Dic11—21no—
97C2×Dic11—22no—
98C2×Dic11—22no—
99C2×Dic11—22no—
100C2×Dic11—21no—
101C2×Dic11—21no—
102C2×Dic11C2×D412yes—
103C2×Dic11#4114yes—
104C4×D11—28no—
105C4×D11—48no—
106C2×#123—111no—
107C2×#117—111no—
108#400—111no—
109#399—111no—
110#398—111no—
111#397D1111yescharacteristic, complemented
112#388—112no—
113#387—112no—
114#386—112no—
115#380—112no—
116C2×#95—111no—
117C22×C11—41no—
118C22×C11—41no—
119C22×C11—41no—
120C22×C11—42no—
121C22×C11—42no—
122C22×C11C22×D411yescharacteristic
123C22×C11C2×#4112yes—
124C22×C11C2×#4112yes—
125C22×C11H(5,𝔽2)11yescharacteristic
126C22×C112−1+411yescharacteristic
127C2×D11—22no—
128C2×D11—22no—
129C2×D11—24no—
130C2×D11—24no—
131C2×D11#12314yes—
132C4×C11—22no—
133C4×C11—41no—
134C4×C11—42no—
135C4×C11—42no—
136Dic11—22no—
137Dic11—24no—
138Dic11—41no—
139Dic11—42no—
140Dic11—41no—
141C22×D4—111no—
142C23×C4—111no—
143C23×C4—112no—
144#123—224no—
145#117—224no—
146C2×C4⋊C4—111no—
147C2×C4⋊C4—111no—
148C2×C4⋊C4—111no—
149C2×C4⋊C4—112no—
150C2×C4⋊C4—112no—
151C2×C4⋊C4C2×D1111yescharacteristic
152C2×C22⋊C4—111no—
153C2×C22⋊C4—112no—
154C2×C22⋊C4—112no—
155C2×C22⋊C4—111no—
156C2×C22⋊C4C2×D1111yescharacteristic
157C2×C22⋊C4C2×D1112yes—
158C2×C42—112no—
159#95—112no—
160#95—111no—
161#95—112no—
162#95—112no—
163#95—112no—
164#95C2×D1111yescharacteristic
165#95C2×D1112yes—
166C2×C11—81no—
167C2×C11C2×#12311yescharacteristic
168C2×C11#53211yescharacteristic
169C2×C11#53711yescharacteristic
170C2×C11#55112yes—
171C2×C11#55212yes—
172D11—24no—
173C24—111no—
174C24C22×D1111yescharacteristic
175C2×D4—222no—
176C2×D4—224no—
177C22×C4—112no—
178C22×C4—111no—
179C22×C4—114no—
180C22×C4—111no—
181C22×C4—112no—
182C22×C4—111no—
183C22×C4—224no—
184C22×C4—224no—
185C22×C4—224no—
186C22×C4—442no—
187C22×C4C22×D1112yes—
188C22×C4C22×D1112yes—
189C22×C4C22×D1111yescharacteristic
190C22×C4C22×D1111yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—222no—
193C4⋊C4—224no—
194C4⋊C4—222no—
195C4⋊C4—222no—
196C4⋊C4—442no—
197C22⋊C4—22no—
198C22⋊C4—42no—
199C22⋊C4—224no—
200C22⋊C4—222no—
201C22⋊C4—222no—
202C22⋊C4—224no—
203C42—224no—
204C11#249811yescharacteristic, complemented, Sylow
205C23—21no—
206C23—21no—
207C23—22no—
208C23—22no—
209C23—21no—
210C23—112no—
211C23—114no—
212C23—221no—
213C23—222no—
214C23—221no—
215C23C23×D1111yescharacteristic
216D4—444no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—112no—
227C2×C4—114no—
228C2×C4—228no—
229C2×C4—222no—
230C2×C4—222no—
231C2×C4—222no—
232C2×C4—222no—
233C2×C4—221no—
234C2×C4—221no—
235C2×C4—221no—
236C2×C4—222no—
237C2×C4—222no—
238C2×C4—222no—
239C2×C4—221no—
240C2×C4—448no—
241C2×C4D4×D1112yes—
242C22—42no—
243C22—41no—
244C22—41no—
245C22—42no—
246C22—41no—
247C22—114no—
248C22—224no—
249C22—222no—
250C22—224no—
251C22—222no—
252C22C2×D4×D1111yescharacteristic
253C22#6703711yescharacteristic
254C22#41×D1112yes—
255C22#41×D1112yes—
256C22#6704311yescharacteristic
257C4—22no—
258C4—42no—
259C4—42no—
260C4—41no—
261C4—222no—
262C4—224no—
263C4—441no—
264C4—441no—
265C4—442no—
266C2—81no—
267C2—224no—
268C2#1062313911yescharacteristic
269C2D11×#12311yescharacteristic
270C2#1062319012yes—
271C2#1062319911yescharacteristic
272C2#1062323912yes—
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC11×#397C2
prime radicalC22×C22⋊D11C22
hypercenter#397D11
hypocenterC11#2498
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#66945C2semidirect
3. C11×#397C2semidirect
4. #10622593C2semidirect
6. #10622562C2semidirect
7. #10622558C2semidirect
8. #10622280C2semidirect
9. D11×#95C2semidirect
10. #10622272C2semidirect
11. #10622268C2semidirect
111. #397D11semidirect
204. C11#2498semidirect

Maximal subgroups

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

subgroupindex
C2×#669452
C2×#669362
C11×#3972
#106225932
#106225692
#106225622
#106225582
#106222802
D11×#952
#106222722
#106222682
#249811