Finite Group Databasebeta

← C22×#10622857 #15620486

Subgroups — C22×#10622857 #15620486

Order 2816.

7014 conjugacy classes of subgroups; only the 827 normal subgroups are shown.

normal subgroups only

This group has 7014 conjugacy classes of subgroups — too many to draw. Only its 827 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×#66990C213yesmaximal, complemented
2C2×#10622857C2124yesmaximal, direct product factor
3C22×#66898C213yesmaximal, complemented
4C22×C11×H(5,𝔽2)C211yesmaximal, characteristic, complemented
5C25×D11C2211yescharacteristic, complemented
6C2×C11×H(5,𝔽2)—218no—
7C2×C11×H(5,𝔽2)C22112yescomplemented
8C22×C11×#41—23no—
9C23×D4×C11—23no—
10C23×D4×C11C2213yes—
11C23×C22⋊D11—23no—
12C22×C22⋊Dic11C2213yescomplemented
13C2×#66990—218no—
14C2×#66990C22136yescomplemented
15C22×(C2×D11)⋊C4—23no—
16#10622857C22164yesdirect product factor
17C2×#66898C22136yescomplemented
18C25×C11—21no—
19C25×C11D413yescomplemented
20C25×C11C2311yescharacteristic
21C24×D11—23no—
22C24×D11—29no—
23C24×D11—48no—
24C24×D11C2316yescomplemented
25C11×H(5,𝔽2)—248no—
26C11×H(5,𝔽2)—272no—
27C11×H(5,𝔽2)C23116yescomplemented
28C2×C11×#41—236no—
29C2×C11×#41—236no—
30C22×Q8×C11—21no—
31C22×D4×C11—23no—
32C22×D4×C11—236no—
33C22×D4×C11—26no—
34C22×D4×C11—218no—
35C22×D4×C11—236no—
36C22×D4×C11D4118yescomplemented
37C22×D4×C11C23118yes—
38C23×C4×C11—23no—
39C23×C4×C11D413yescomplemented
40C22×C22⋊D11—236no—
41C22×C22⋊D11—424no—
42C23×Dic11—23no—
43C2×C22⋊Dic11—29no—
44C2×C22⋊Dic11C23118yescomplemented
45#66990—272no—
46#66990C23148yescomplemented
47C2×(C2×D11)⋊C4—236no—
48#66898C23148yescomplemented
49C22×B(4,𝔽2)—111nomaximal, Sylow
50C24×C11—212no—
51C24×C11—23no—
52C24×C11—29no—
53C24×C11—44no—
54C24×C11—412no—
55C24×C11—84no—
56C24×C11C2×D413yes—
57C24×C11C2×D419yes—
58C24×C11C2×D4118yescomplemented
59C24×C11C2413yes—
60C23×D11—224no—
61C23×D11—218no—
62C23×D11—218no—
63C23×D11—424no—
64C23×D11—472no—
65C23×D11—412no—
66C23×D11—49no—
67C23×D11—46no—
68C23×D11—41no—
69C23×D11C2414yescomplemented
70C11×#41—2144no—
71C11×#41—248no—
72C2×Q8×C11—212no—
73C2×D4×C11—236no—
74C2×D4×C11—2144no—
75C2×D4×C11—236no—
76C2×D4×C11—236no—
77C2×D4×C11—236no—
78C2×D4×C11—236no—
79C2×D4×C11—248no—
80C2×D4×C11C2×D4136yescomplemented
81C2×D4×C11C24112yes—
82C22×C4×C11—236no—
83C22×C4×C11—23no—
84C22×C4×C11—29no—
85C22×C4×C11C2×D4118yescomplemented
86C2×C22⋊D11—248no—
87C2×C22⋊D11—4144no—
88C22×Dic11—218no—
89C22×Dic11—412no—
90C22⋊Dic11—218no—
91C22⋊Dic11C24112yescomplemented
92(C2×D11)⋊C4—248no—
93C22×H(5,𝔽2)D1111yescharacteristic, complemented
94C22×#120—113no—
95C2×B(4,𝔽2)—1124no—
96C22×#99—113no—
97C23×C11—212no—
98C23×C11—216no—
99C23×C11—218no—
100C23×C11—218no—
101C23×C11—418no—
102C23×C11—46no—
103C23×C11—436no—
104C23×C11—412no—
105C23×C11—818no—
106C23×C11—824no—
107C23×C11#12019yes—
108C23×C11#12016yes—
109C23×C11#12011yescharacteristic
110C23×C11C22×D419yes—
111C23×C11C22×D419yes—
112C23×C11C22×D4112yescomplemented
113C23×C11C2511yescharacteristic
114C22×D11—272no—
115C22×D11—212no—
116C22×D11—48no—
117C22×D11—472no—
118C22×D11—436no—
119C22×D11—472no—
120C22×D11—448no—
121C22×D11—418no—
122C22×D11—46no—
123C22×D11—48no—
124Q8×C11—216no—
125D4×C11—248no—
126D4×C11—272no—
127D4×C11—224no—
128C2×C4×C11—248no—
129C2×C4×C11—218no—
130C2×C4×C11—218no—
131C2×C4×C11C22×D4112yescomplemented
132C22⋊D11—496no—
133C2×Dic11—212no—
134C2×Dic11—436no—
135C26—111no—
136C2×H(5,𝔽2)—218no—
137C2×H(5,𝔽2)C2×D11112yescomplemented
138C22×#41—23no—
139C23×D4—23no—
140C23×D4—223no—
141C23×D4C2×D1113yes—
142C2×#120—1136no—
143C2×#120—2218no—
144C22×C22⋊C4—113no—
145C22×C22⋊C4—223no—
146B(4,𝔽2)—1164no—
147C2×#99—1136no—
148C22×C11—218no—
149C22×C11—218no—
150C22×C11—212no—
151C22×C11—46no—
152C22×C11—49no—
153C22×C11—41no—
154C22×C11—412no—
155C22×C11—436no—
156C22×C11—818no—
157C22×C11—816no—
158C22×C11B(4,𝔽2)14yescomplemented
159C22×C11C2×#12019yes—
160C22×C11C2×#12013yes—
161C22×C11C23×D413yes—
162C2×D11—224no—
163C2×D11—412no—
164C2×D11—472no—
165C2×D11—44no—
166C2×D11—424no—
167C4×C11—212no—
168Dic11—412no—
169C25—21no—
170C25—116no—
171C25—223no—
172C25—229no—
173C25—448no—
174C25C22⋊D1113yescomplemented
175C25C22×D1111yescharacteristic
176H(5,𝔽2)—248no—
177H(5,𝔽2)—272no—
178H(5,𝔽2)C22×D11116yescomplemented
179C2×#41—236no—
180C2×#41—236no—
181C22×Q8—21no—
182C22×D4—23no—
183C22×D4—236no—
184C22×D4—26no—
185C22×D4—218no—
186C22×D4—236no—
187C22×D4—2236no—
188C22×D4—4424no—
189C22×D4C22⋊D11118yescomplemented
190C22×D4C22×D11118yes—
191C23×C4—23no—
192C23×C4—223no—
193C23×C4C22⋊D1113yescomplemented
194#120—1148no—
195#120—2272no—
196C2×C22⋊C4—1118no—
197C2×C22⋊C4—229no—
198C2×C22⋊C4—2236no—
199#99—1148no—
200C2×C11—23no—
201C2×C11—29no—
202C2×C11—412no—
203C2×C11C2×B(4,𝔽2)16yescomplemented
204C2×C11C22×#12011yescharacteristic
205D11—48no—
206C24—212no—
207C24—23no—
208C24—29no—
209C24—44no—
210C24—412no—
211C24—84no—
212C24—114no—
213C24—2224no—
214C24—2218no—
215C24—2218no—
216C24—4412no—
217C24—4424no—
218C24—4472no—
219C24—449no—
220C24—446no—
221C24—441no—
222C24C2×C22⋊D1113yes—
223C24C2×C22⋊D1119yes—
224C24C2×C22⋊D11118yescomplemented
225C24C23×D1113yes—
226#41—2144no—
227#41—248no—
228C2×Q8—212no—
229C2×D4—236no—
230C2×D4—2144no—
231C2×D4—236no—
232C2×D4—236no—
233C2×D4—236no—
234C2×D4—236no—
235C2×D4—248no—
236C2×D4—2248no—
237C2×D4—44144no—
238C2×D4C2×C22⋊D11136yescomplemented
239C2×D4C23×D11112yes—
240C22×C4—236no—
241C22×C4—29no—
242C22×C4—23no—
243C22×C4—2218no—
244C22×C4—4412no—
245C22×C4C2×C22⋊D11118yescomplemented
246C22⋊C4—1112no—
247C22⋊C4—2218no—
248C22⋊C4—2248no—
249C11C22×B(4,𝔽2)11yescharacteristic, complemented, Sylow
250C23—212no—
251C23—216no—
252C23—218no—
253C23—218no—
254C23—418no—
255C23—46no—
256C23—436no—
257C23—412no—
258C23—818no—
259C23—824no—
260C23—2272no—
261C23—2212no—
262C23—4436no—
263C23—4472no—
264C23—448no—
265C23—4472no—
266C23—4448no—
267C23—448no—
268C23—4418no—
269C23—446no—
270C23#6700619yes—
271C23#6700616yes—
272C23#6700611yescharacteristic
273C23C22×C22⋊D1119yes—
274C23C22×C22⋊D1119yes—
275C23C22×C22⋊D11112yescomplemented
276C23C24×D1111yescharacteristic
277Q8—216no—
278D4—248no—
279D4—272no—
280D4—224no—
281D4—4496no—
282C2×C4—248no—
283C2×C4—218no—
284C2×C4—218no—
285C2×C4—2212no—
286C2×C4—4436no—
287C2×C4C22×C22⋊D11112yescomplemented
288C22—218no—
289C22—218no—
290C22—212no—
291C22—46no—
292C22—49no—
293C22—41no—
294C22—412no—
295C22—436no—
296C22—818no—
297C22—816no—
298C22—2224no—
299C22—4412no—
300C22—4472no—
301C22—444no—
302C22—4424no—
303C22#1062285714yesdirect product factor
304C22C2×#6700619yes—
305C22C2×#6700613yes—
306C22C23×C22⋊D1113yes—
307C4—212no—
308C4—4412no—
309C2—23no—
310C2—29no—
311C2—412no—
312C2—448no—
313C2C2×#1062285716yesdirect product factor
314C2C22×#6700611yescharacteristic
subgroupG/H
centerC23#67006
commutator, cosocleC23×C11C25
FrattiniC23C24×D11
FittingC22×C11×H(5,𝔽2)C2
socleC23×C11#120
hypercenterC22×H(5,𝔽2)D11
hypocenterC11C22×B(4,𝔽2)
LCSG ▷ C23×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C25 ◁ C22×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C22×#66990C2semidirect
2. C2×#10622857C2direct (N × H)
3. C22×#66898C2semidirect
4. C22×C11×H(5,𝔽2)C2semidirect
5. C25×D11C22semidirect
7. C2×C11×H(5,𝔽2)C22semidirect
12. C22×C22⋊Dic11C22semidirect
14. C2×#66990C22semidirect
16. #10622857C22direct (N × H)
17. C2×#66898C22semidirect
19. C25×C11D4semidirect
24. C24×D11C23semidirect
27. C11×H(5,𝔽2)C23semidirect
36. C22×D4×C11D4semidirect
39. C23×C4×C11D4semidirect
44. C2×C22⋊Dic11C23semidirect
46. #66990C23semidirect
48. #66898C23semidirect
58. C24×C11C2×D4semidirect
69. C23×D11C24semidirect
80. C2×D4×C11C2×D4semidirect
85. C22×C4×C11C2×D4semidirect
91. C22⋊Dic11C24semidirect
93. C22×H(5,𝔽2)D11semidirect
112. C23×C11C22×D4semidirect
131. C2×C4×C11C22×D4semidirect
137. C2×H(5,𝔽2)C2×D11semidirect
158. C22×C11B(4,𝔽2)semidirect
174. C25C22⋊D11semidirect
178. H(5,𝔽2)C22×D11semidirect
189. C22×D4C22⋊D11semidirect
193. C23×C4C22⋊D11semidirect
203. C2×C11C2×B(4,𝔽2)semidirect
224. C24C2×C22⋊D11semidirect
238. C2×D4C2×C22⋊D11semidirect
245. C22×C4C2×C22⋊D11semidirect
249. C11C22×B(4,𝔽2)semidirect
275. C23C22×C22⋊D11semidirect
287. C2×C4C22×C22⋊D11semidirect
303. C22#10622857direct (N × H)
313. C2C2×#10622857direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#669902
C2×#106228572
C22×#668982
C22×C11×H(5,𝔽2)2
C22×B(4,𝔽2)11