Finite Group Databasebeta

← #13070615

Subgroups — #13070615

Order 1408.

270 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 41 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
1#10623125C211yesmaximal, characteristic, complemented
2C11×B(4,𝔽2)C211yesmaximal, characteristic, complemented
3#10622857C211yesmaximal, characteristic, complemented
4#10622855C211yesmaximal, characteristic, complemented
5#10622174C211yesmaximal, characteristic, complemented
6#10622111C211yesmaximal, characteristic, complemented
7#10622109C211yesmaximal, characteristic, complemented
8C11×H(5,𝔽2)C2211yescharacteristic
9C11×#120—21no—
10C11×#120C2211yescharacteristic
11#67006—21no—
12#67004—21no—
13#66990—21no—
14#66990—21no—
15#66990C2211yescharacteristic
16#66989—21no—
17#66989C2211yescharacteristic
18#66938—21no—
19#66938—21no—
20#66938—21no—
21C11×#99—21no—
22C11×#99C2211yescharacteristic
23#66901—21no—
24#66898—21no—
25#66898C2211yescharacteristic
26#66897C2211yescharacteristic
27C24×C11D411yescharacteristic, complemented
28C23×D11D411yescharacteristic, complemented
29C11×#41—21no—
30C11×#41—41no—
31C2×D4×C11—21no—
32C2×D4×C11—21no—
33C2×D4×C11—21no—
34C2×D4×C11—41no—
35C2×D4×C11—41no—
36C2×D4×C11C2311yescharacteristic
37C2×C22⋊D11—21no—
38C2×C22⋊D11—21no—
39C2×C22⋊D11—21no—
40C2×C22⋊D11—41no—
41C2×Dic22—21no—
42C2×Dic22D411yescharacteristic, complemented
43C11×C22⋊C4—21no—
44C11×C22⋊C4—21no—
45C11×C22⋊C4—41no—
46C11×C22⋊C4D411yescharacteristic, complemented
47C22⋊Dic11—21no—
48C22⋊Dic11—21no—
49C22⋊Dic11—21no—
50C22⋊Dic11—21no—
51C22⋊Dic11—21no—
52C22⋊Dic11D411yescharacteristic, complemented
53#4326—41no—
54#4324—41no—
55(C2×D11)⋊C4—21no—
56(C2×D11)⋊C4—21no—
57(C2×D11)⋊C4—21no—
58(C2×D11)⋊C4—41no—
59C4×Dic11—21no—
60C4×Dic11D411yescharacteristic, complemented
61#4318—21no—
62#2199—111nomaximal, Sylow
63C23×C11—21no—
64C23×C11—21no—
65C23×C11—21no—
66C23×C11—41no—
67C23×C11—81no—
68C23×C11C2×D411yescharacteristic
69C23×C11C2×D411yescharacteristic
70C22×D11—21no—
71C22×D11—41no—
72C22×D11—81no—
73Q8×C11—21no—
74D4×C11—21no—
75D4×C11—41no—
76D4×C11—41no—
77D4×C11—41no—
78D4×C11—41no—
79D4×C11—81no—
80D4×C11—82no—
81C2×C4×C11—21no—
82C2×C4×C11—21no—
83C2×C4×C11—21no—
84C2×C4×C11—41no—
85C2×C4×C11—41no—
86C2×C4×C11C2×D411yescharacteristic
87C22⋊D11—81no—
88C22⋊D11—81no—
89C22⋊D11—81no—
90C22⋊D11—82no—
91C2×Dic11—21no—
92C2×Dic11—21no—
93C2×Dic11—21no—
94C2×Dic11—41no—
95Dic22—41no—
96Dic22—81no—
97C11⋊C8—41no—
98H(3,𝔽4)—111no—
99B(4,𝔽2)—111no—
100B(4,𝔽2)D1111yescharacteristic, complemented
101#455—111no—
102#354—111no—
103AΣL(1,𝔽16)—111no—
104AΣL(1,𝔽16)—111no—
105C22×C11—21no—
106C22×C11—21no—
107C22×C11—41no—
108C22×C11—41no—
109C22×C11—41no—
110C22×C11—41no—
111C22×C11—42no—
112C22×C11—41no—
113C22×C11—41no—
114C22×C11—81no—
115C22×C11—81no—
116C22×C11—81no—
117C22×C11#12011yescharacteristic
118C2×D11—41no—
119C2×D11—41no—
120C2×D11—42no—
121C2×D11—41no—
122C2×D11—81no—
123C4×C11—21no—
124C4×C11—41no—
125C4×C11—41no—
126C4×C11—81no—
127Dic11—41no—
128Dic11—41no—
129Dic11—41no—
130Dic11—81no—
131H(5,𝔽2)C2×D1111yescharacteristic
132#137—221no—
133#124—111no—
134#124—221no—
135#124—221no—
136#124—221no—
137#124—221no—
138#120—21no—
139#120—111no—
140#120—221no—
141#120—221no—
142#120—221no—
143#120C2×D1111yescharacteristic
144#104—221no—
145#100—111no—
146#99—21no—
147#99—111no—
148#99—221no—
149#99C2×D1111yescharacteristic
150C2×C11—21no—
151C2×C11—41no—
152C2×C11—41no—
153C2×C11—81no—
154C2×C11—81no—
155C2×C11B(4,𝔽2)11yescharacteristic
156D11—81no—
157C24—111no—
158C24C22⋊D1111yescharacteristic, complemented
159#41—21no—
160#41—41no—
161C2×Q8—111no—
162C2×Q8—221no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—41no—
167C2×D4—41no—
168C2×D4—221no—
169C2×D4—221no—
170C2×D4—221no—
171C2×D4—441no—
172C2×D4C22×D1111yescharacteristic
173Q16—441no—
174SD16—441no—
175M16—221no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C22⋊C4—41no—
179C22⋊C4—111no—
180C22⋊C4—221no—
181C22⋊C4—221no—
182C22⋊C4—221no—
183C22⋊C4—221no—
184C22⋊C4—221no—
185C22⋊C4—221no—
186C22⋊C4—221no—
187C22⋊C4—221no—
188C22⋊C4—441no—
189C22⋊C4C22⋊D1111yescharacteristic, complemented
190C42—111no—
191C42—221no—
192C11#219911yescharacteristic, complemented, Sylow
193C23—21no—
194C23—21no—
195C23—21no—
196C23—41no—
197C23—81no—
198C23—221no—
199C23—441no—
200C23—881no—
201C23D4×D1111yescharacteristic
202C23C2×C22⋊D1111yescharacteristic
203Q8—21no—
204Q8—441no—
205Q8—881no—
206D4—21no—
207D4—41no—
208D4—41no—
209D4—41no—
210D4—41no—
211D4—81no—
212D4—82no—
213D4—881no—
214D4—881no—
215D4—881no—
216D4—882no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—221no—
223C2×C4—221no—
224C2×C4—221no—
225C2×C4—441no—
226C2×C4D4×D1111yescharacteristic
227C8—441no—
228C22—21no—
229C22—21no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—42no—
235C22—41no—
236C22—41no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22—441no—
241C22—441no—
242C22—442no—
243C22—441no—
244C22—881no—
245C22#6699011yescharacteristic
246C4—21no—
247C4—41no—
248C4—41no—
249C4—81no—
250C4—441no—
251C4—441no—
252C4—441no—
253C4—881no—
254C2—21no—
255C2—41no—
256C2—41no—
257C2—81no—
258C2—81no—
259C2—881no—
260C2#1062264511yescharacteristic
subgroupG/H
centerC2#10622645
commutator, cosocleC2×D4×C11C23
FrattiniC2×D4C22×D11
FittingC11×B(4,𝔽2)C2
socleC2×C11B(4,𝔽2)
hypercenterB(4,𝔽2)D11
hypocenterC11#2199
LCSG ▷ C2×D4×C11 ▷ C22×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ B(4,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #10623125C2semidirect
2. C11×B(4,𝔽2)C2semidirect
3. #10622857C2semidirect
4. #10622855C2semidirect
5. #10622174C2semidirect
6. #10622111C2semidirect
7. #10622109C2semidirect
27. C24×C11D4semidirect
28. C23×D11D4semidirect
42. C2×Dic22D4semidirect
46. C11×C22⋊C4D4semidirect
52. C22⋊Dic11D4semidirect
60. C4×Dic11D4semidirect
100. B(4,𝔽2)D11semidirect
158. C24C22⋊D11semidirect
189. C22⋊C4C22⋊D11semidirect
192. C11#2199semidirect

Maximal subgroups

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

subgroupindex
#106231252
C11×B(4,𝔽2)2
#106228572
#106228552
#106221742
#106221112
#106221092
#219911