Finite Group Databasebeta

← #15620530

Subgroups — #15620530

Order 2816.

4732 conjugacy classes of subgroups; only the 807 normal subgroups are shown.

normal subgroups only

This group has 4732 conjugacy classes of subgroups — too many to draw. Only its 807 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#13082264C213yesmaximal, complemented
2#13078022C2112yesmaximal, complemented
3#13078008C2112yesmaximal, complemented
4#13077080C213yesmaximal, complemented
5C11×2−1+6C211yesmaximal, characteristic, complemented
6C11×#585—212no—
7C11×#585C22112yes—
8C2×C11×2−1+4—212no—
9C2×C11×2−1+4C2213yes—
10C22×Q8×D11C2211yescharacteristic, complemented
11#10623362—212no—
12#10623359C22118yescomplemented
13#10623356C22118yescomplemented
14Q8×C22⋊D11—212no—
15#10623343C2213yescomplemented
16#10622857C22116yescomplemented
17#10622856C22148yescomplemented
18#10622853C22136yescomplemented
19C11×2−1+4—248no—
20C11×2−1+4—236no—
21C11×2−1+4—224no—
22C11×H(5,𝔽2)—216no—
23C11×H(5,𝔽2)—236no—
24C11×H(5,𝔽2)C23116yes—
25C2×C11×#41—248no—
26C2×C11×#41—272no—
27C2×C11×#41D4112yescomplemented
28C2×C11×#41C23118yes—
29C22×Q8×C11—216no—
30C22×Q8×C11D4112yescomplemented
31C22×Q8×C11C2311yescharacteristic
32C2×Q8×D11—212no—
33C22×C4×D11C2313yes—
34C22×Dic22C2313yes—
35#66999—218no—
36Q8×Dic11—212no—
37#66997—218no—
38#66991—218no—
39#66990C23112yes—
40#66989—218no—
41#66988C23136yes—
42C4×C22⋊D11—236no—
43#66979C23118yes—
44#66977—236no—
45#66898C23148yes—
46#60400—111nomaximal, Sylow
47C23×D11C2411yescharacteristic
48C11×#41—2144no—
49C11×#41—248no—
50C11×#41—248no—
51C11×#41—248no—
52C11×#41—272no—
53C2×Q8×C11—272no—
54C2×Q8×C11—248no—
55C2×Q8×C11C2×D4118yes—
56C2×Q8×C11C2×D4112yes—
57C2×D4×C11—272no—
58C2×D4×C11—248no—
59C2×D4×C11C2×D4118yes—
60C2×D4×C11C24112yes—
61C22×C4×C11—248no—
62C22×C4×C11C2×D4136yes—
63C22×C4×C11C2413yes—
64C2×C22⋊D11—212no—
65C22×Dic11C2413yes—
66Q8×D11—416no—
67C2×C4×D11—218no—
68C2×Dic22—218no—
69C22⋊Dic11—236no—
70C22⋊Dic11C24112yes—
71(C2×D11)⋊C4—212no—
72C4⋊Dic11—218no—
73Dic11⋊C4—236no—
74C4×Dic11—218no—
752−1+6D1111yescharacteristic, complemented
76#3486—113no—
77#3028—1112no—
78#3025—1112no—
79#2884—113no—
80C23×C11—216no—
81C23×C11C22×D4112yes—
82C23×C11C2511yescharacteristic
83C22×D11—23no—
84C22×D11—42no—
85Q8×C11—216no—
86Q8×C11—236no—
87Q8×C11#120116yes—
88D4×C11—248no—
89D4×C11—224no—
90D4×C11—236no—
91C2×C4×C11—248no—
92C2×C4×C11—272no—
93C2×C4×C11C22×D4112yes—
94C2×C4×C11C22×D4118yes—
95C22⋊D11—424no—
96C2×Dic11—212no—
97C2×Dic11—29no—
98C4×D11—412no—
99Dic22—412no—
100#585—212no—
101#585C2×D11112yes—
102C2×2−1+4—212no—
103C2×2−1+4C2×D1113yes—
104C23×Q8—111no—
105D4×Q8—2212no—
106#548—2212no—
107#536—1118no—
108#525—1118no—
109#519—113no—
110#458—1148no—
111B(4,𝔽2)—1116no—
112#410—1136no—
113C22×C11—212no—
114C22×C11—41no—
115C22×C11—412no—
116C22×C11—816no—
117C22×C11C23×D413yes—
118C2×D11—26no—
119C2×D11—43no—
120C2×D11—41no—
121C4×C11—212no—
122C4×C11C2×#120112yes—
123Dic11—412no—
124Dic11—43no—
1252−1+4—248no—
1262−1+4—236no—
1272−1+4—224no—
128H(5,𝔽2)—216no—
129H(5,𝔽2)—236no—
130H(5,𝔽2)C22×D11116yes—
131C2×#41—248no—
132C2×#41—272no—
133C2×#41C22⋊D11112yescomplemented
134C2×#41C22×D11118yes—
135C22×Q8—216no—
136C22×Q8—113no—
137C22×Q8—2212no—
138C22×Q8C22⋊D11112yescomplemented
139C22×Q8C22×D1111yescharacteristic
140C23×C4—113no—
141C4⋊Q8—2218no—
142#124—2218no—
143#123—1136no—
144C22⋊Q8—2218no—
145C22⋊Q8—2236no—
146#121—2218no—
147#120—1112no—
148C4×Q8—2212no—
149C4×D4—2236no—
150#117—1118no—
151#99—1148no—
152C2×C11—23no—
153C2×C11—412no—
154C2×C11C22×#12011yescharacteristic
155D11—42no—
156C24—111no—
157#41—2144no—
158#41—248no—
159#41—248no—
160#41—248no—
161#41—272no—
162C2×Q8—272no—
163C2×Q8—248no—
164C2×Q8—2218no—
165C2×Q8—4416no—
166C2×Q8C2×C22⋊D11118yes—
167C2×Q8C2×C22⋊D11112yes—
168C2×D4—272no—
169C2×D4—248no—
170C2×D4—2212no—
171C2×D4C2×C22⋊D11118yes—
172C2×D4C23×D11112yes—
173C22×C4—248no—
174C22×C4—113no—
175C22×C4—2218no—
176C22×C4C2×C22⋊D11136yes—
177C22×C4C23×D1113yes—
178C4⋊C4—2218no—
179C4⋊C4—2236no—
180C22⋊C4—1112no—
181C22⋊C4—2236no—
182C22⋊C4—2212no—
183C42—2218no—
184C11#6040011yescharacteristic, complemented, Sylow
185C23—216no—
186C23—223no—
187C23—442no—
188C23C22×C22⋊D11112yes—
189C23C24×D1111yescharacteristic
190Q8—216no—
191Q8—236no—
192Q8—4412no—
193Q8#67006116yes—
194D4—248no—
195D4—224no—
196D4—236no—
197D4—4424no—
198C2×C4—248no—
199C2×C4—272no—
200C2×C4—2212no—
201C2×C4—229no—
202C2×C4—4412no—
203C2×C4C22×C22⋊D11112yes—
204C2×C4C22×C22⋊D11118yes—
205C22—212no—
206C22—41no—
207C22—412no—
208C22—816no—
209C22—226no—
210C22—443no—
211C22—441no—
212C22C23×C22⋊D1113yes—
213C4—212no—
214C4—4412no—
215C4—443no—
216C4C2×#67006112yes—
217C2—23no—
218C2—412no—
219C2—442no—
220C2C22×#6700611yescharacteristic
subgroupG/H
centerC2C22×#67006
commutator, cosocleC23×C11C25
FrattiniC23C24×D11
FittingC11×2−1+6C2
socleC2×C11C22×#120
hypercenter2−1+6D11
hypocenterC11#60400
LCSG ▷ C23×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C2 ◁ C22×Q8 ◁ 2−1+6

Complements

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

NH ≅ G/Ntype
1. #13082264C2semidirect
2. #13078022C2semidirect
3. #13078008C2semidirect
4. #13077080C2semidirect
5. C11×2−1+6C2semidirect
10. C22×Q8×D11C22semidirect
12. #10623359C22semidirect
13. #10623356C22semidirect
15. #10623343C22semidirect
16. #10622857C22semidirect
17. #10622856C22semidirect
18. #10622853C22semidirect
27. C2×C11×#41D4semidirect
30. C22×Q8×C11D4semidirect
75. 2−1+6D11semidirect
133. C2×#41C22⋊D11semidirect
138. C22×Q8C22⋊D11semidirect
184. C11#60400semidirect

Maximal subgroups

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

subgroupindex
#130822642
#130780222
#130780082
#130770802
C11×2−1+62
#6040011