Finite Group Databasebeta

← #13074396

Subgroups — #13074396

Order 1408.

526 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×C11×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#66982C211yesmaximal, characteristic
3#10622826C212yesmaximal
4#10622824C211yesmaximal, characteristic, complemented
5#10622823C212yesmaximal, complemented
6#10622606C212yesmaximal, complemented
7#10622605C212yesmaximal, complemented
8#10622595C212yesmaximal, complemented
9#10622593C212yesmaximal, complemented
10C23×C4×C11C2211yescharacteristic
11C23×C4×C11C2212yes—
12C22×C22⋊D11C2211yescharacteristic
13C2×C11×C4⋊C4—24no—
14C2×C11×C4⋊C4C2212yes—
15C2×C11×C4⋊C4C2212yes—
16C2×C22⋊Dic11C2211yescharacteristic
17C2×C22⋊Dic11C2212yes—
18#66982—24no—
19C2×(C2×D11)⋊C4C2211yescharacteristic
20C2×(C2×D11)⋊C4C2211yescharacteristic
21C2×(C2×D11)⋊C4C2214yes—
22C2×Dic11⋊C4C2211yescharacteristic
23C2×Dic11⋊C4C2211yescharacteristic
24C2×Dic11⋊C4C2214yes—
25C11⋊#95C2214yes—
26C11⋊#95C2212yes—
27C11⋊#95C2212yes—
28C11⋊#95C2214yes—
29C24×C11C2311yescharacteristic
30C23×D11C2311yescharacteristic
31C22×C4×C11—24no—
32C22×C4×C11—24no—
33C22×C4×C11—44no—
34C22×C4×C11D414yes—
35C22×C4×C11C2314yes—
36C22×C4×C11C2311yescharacteristic
37C22×C4×C11C2311yescharacteristic
38C2×C22⋊D11—22no—
39C2×C22⋊D11—24no—
40C22×Dic11C2311yescharacteristic
41C22×Dic11C2311yescharacteristic
42C22×Dic11C2314yes—
43C22×Dic11C2311yescharacteristic
44C11×C4⋊C4—24no—
45C11×C4⋊C4—24no—
46C22⋊Dic11—22no—
47C22⋊Dic11—24no—
48(C2×D11)⋊C4—22no—
49(C2×D11)⋊C4—22no—
50(C2×D11)⋊C4—44no—
51Dic11⋊C4—22no—
52Dic11⋊C4—22no—
53Dic11⋊C4—44no—
54#2623—111nomaximal, Sylow
55C23×C11—22no—
56C23×C11—22no—
57C23×C11C2×D412yes—
58C23×C11#4114yes—
59C23×C11C2411yescharacteristic
60C22×D11—21no—
61C22×D11—22no—
62C22×D11—22no—
63C22×D11—22no—
64C2×C4×C11—28no—
65C2×C4×C11—22no—
66C2×C4×C11—22no—
67C2×C4×C11—24no—
68C2×C4×C11—24no—
69C2×C4×C11—24no—
70C2×C4×C11—48no—
71C2×C4×C11C2×D412yes—
72C2×C4×C11C2×D412yes—
73C22⋊D11—44no—
74C2×Dic11—21no—
75C2×Dic11—21no—
76C2×Dic11—21no—
77C2×Dic11—24no—
78C2×Dic11—22no—
79C2×Dic11—22no—
80C2×Dic11—24no—
81C2×Dic11—22no—
82C2×Dic11—24no—
83C2×#123—111no—
84C22×C4⋊C4D1111yescharacteristic, complemented
85#400—112no—
86#399—112no—
87#398—112no—
88#397—112no—
89#386—112no—
90#385—112no—
91#381—111no—
92C22×C11—22no—
93C22×C11—22no—
94C22×C11—28no—
95C22×C11#12314yes—
96C22×C11C22×D411yescharacteristic
97C22×C11C2×#4112yes—
98C22×C11H(5,𝔽2)12yes—
99C22×C112−1+412yes—
100C2×D11—41no—
101C2×D11—41no—
102C2×D11—41no—
103C2×D11—44no—
104C4×C11—22no—
105C4×C11—22no—
106C4×C11—44no—
107Dic11—41no—
108Dic11—41no—
109Dic11—41no—
110Dic11—44no—
111C22×D4—111no—
112C23×C4C2×D1111yescharacteristic
113C23×C4C2×D1112yes—
114#123—224no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—111no—
117C2×C4⋊C4—111no—
118C2×C4⋊C4—114no—
119C2×C4⋊C4C2×D1112yes—
120C2×C4⋊C4C2×D1112yes—
121C2×C22⋊C4—111no—
122C2×C22⋊C4—112no—
123C2×C22⋊C4—111no—
124C2×C22⋊C4—111no—
125C2×C22⋊C4—114no—
126#95—112no—
127#95—112no—
128#95—114no—
129#95—114no—
130C2×C11—24no—
131C2×C11C2×#12311yescharacteristic
132C2×C11#53511yescharacteristic
133C2×C11#53611yescharacteristic
134C2×C11#53914yes—
135D11—81no—
136C24—111no—
137C24C22×D1111yescharacteristic
138C2×D4—224no—
139C2×D4—222no—
140C22×C4—24no—
141C22×C4—24no—
142C22×C4—44no—
143C22×C4—114no—
144C22×C4—111no—
145C22×C4—111no—
146C22×C4—111no—
147C22×C4C22⋊D1114yes—
148C22×C4C22×D1114yes—
149C22×C4C22×D1111yescharacteristic
150C22×C4C22×D1111yescharacteristic
151C4⋊C4—24no—
152C4⋊C4—24no—
153C4⋊C4—222no—
154C4⋊C4—222no—
155C4⋊C4—444no—
156C22⋊C4—224no—
157C22⋊C4—222no—
158C22⋊C4—222no—
159C22⋊C4—222no—
160C22⋊C4—444no—
161C11#262311yescharacteristic, complemented, Sylow
162C23—22no—
163C23—22no—
164C23—222no—
165C23—222no—
166C23—222no—
167C23—221no—
168C23#433914yes—
169C23C2×C22⋊D1112yes—
170C23C23×D1111yescharacteristic
171D4—444no—
172C2×C4—28no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—24no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—48no—
179C2×C4—224no—
180C2×C4—222no—
181C2×C4—224no—
182C2×C4—222no—
183C2×C4—224no—
184C2×C4—222no—
185C2×C4—221no—
186C2×C4—221no—
187C2×C4—221no—
188C2×C4C2×C22⋊D1112yes—
189C2×C4C2×C22⋊D1112yes—
190C22—28no—
191C22—22no—
192C22—22no—
193C22—444no—
194C22—441no—
195C22—441no—
196C22—441no—
197C22#6698214yes—
198C22C2×#433912yes—
199C22#6703712yes—
200C22#6704012yes—
201C22C22×C22⋊D1111yescharacteristic
202C4—22no—
203C4—22no—
204C4—44no—
205C4—444no—
206C4—441no—
207C4—441no—
208C4—441no—
209C2—24no—
210C2—881no—
211C2#1062304614yes—
212C2C2×#6698211yescharacteristic
213C2#1062333611yescharacteristic
214C2#1062334711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC22×C11×C4⋊C4C2
prime radicalC22×C22⋊D11C22
hypercenterC22×C4⋊C4D11
hypocenterC11#2623
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C11×C4⋊C4C2semidirect
4. #10622824C2semidirect
5. #10622823C2semidirect
6. #10622606C2semidirect
7. #10622605C2semidirect
8. #10622595C2semidirect
9. #10622593C2semidirect
84. C22×C4⋊C4D11semidirect
161. C11#2623semidirect

Maximal subgroups

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

subgroupindex
C22×C11×C4⋊C42
C2×#669822
#106228262
#106228242
#106228232
#106226062
#106226052
#106225952
#106225932
#262311