Finite Group Databasebeta

← C22×#16810810 #28250805

Subgroups — C22×#16810810 #28250805

Order 13568.

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×#13133185C213yesmaximal, complemented
2C2×#16810810C2124yesmaximal, direct product factor
3C22×#13133132C213yesmaximal, complemented
4C22×H(5,𝔽2)×C53C211yesmaximal, characteristic, complemented
5C25×D53C2211yescharacteristic, complemented
6C23×C22⋊D53—23no—
7C2×#13133185—218no—
8C2×#13133185C22136yescomplemented
9C22×C22⋊Dic53C2213yescomplemented
10C22×#11719921—23no—
11#16810810C22164yesdirect product factor
12C2×#13133132C22136yescomplemented
13C2×H(5,𝔽2)×C53—218no—
14C2×H(5,𝔽2)×C53C22112yescomplemented
15C22×#41×C53—23no—
16C23×D4×C53—23no—
17C23×D4×C53C2213yes—
18C24×D53—23no—
19C24×D53—29no—
20C24×D53—48no—
21C24×D53C2316yescomplemented
22C22×C22⋊D53—236no—
23C22×C22⋊D53—424no—
24C23×Dic53—23no—
25#13133185—272no—
26#13133185C23148yescomplemented
27C2×C22⋊Dic53—29no—
28C2×C22⋊Dic53C23118yescomplemented
29C2×#11719921—236no—
30#13133132C23148yescomplemented
31C25×C53—21no—
32C25×C53D413yescomplemented
33C25×C53C2311yescharacteristic
34H(5,𝔽2)×C53—248no—
35H(5,𝔽2)×C53—272no—
36H(5,𝔽2)×C53C23116yescomplemented
37C2×#41×C53—236no—
38C2×#41×C53—236no—
39C22×Q8×C53—21no—
40C22×D4×C53—23no—
41C22×D4×C53—236no—
42C22×D4×C53—26no—
43C22×D4×C53—218no—
44C22×D4×C53—236no—
45C22×D4×C53D4118yescomplemented
46C22×D4×C53C23118yes—
47C23×C4×C53—23no—
48C23×C4×C53D413yescomplemented
49C24×C53—212no—
50C24×C53—23no—
51C24×C53—29no—
52C24×C53—44no—
53C24×C53—412no—
54C24×C53—84no—
55C24×C53C2×D413yes—
56C24×C53C2×D419yes—
57C24×C53C2×D4118yescomplemented
58C24×C53C2413yes—
59C23×D53—224no—
60C23×D53—218no—
61C23×D53—218no—
62C23×D53—424no—
63C23×D53—472no—
64C23×D53—412no—
65C23×D53—49no—
66C23×D53—46no—
67C23×D53—41no—
68C23×D53C2414yescomplemented
69#41×C53—2144no—
70#41×C53—248no—
71C2×Q8×C53—212no—
72C2×D4×C53—236no—
73C2×D4×C53—2144no—
74C2×D4×C53—236no—
75C2×D4×C53—236no—
76C2×D4×C53—236no—
77C2×D4×C53—236no—
78C2×D4×C53—248no—
79C2×D4×C53C2×D4136yescomplemented
80C2×D4×C53C24112yes—
81C22×C4×C53—236no—
82C22×C4×C53—23no—
83C22×C4×C53—29no—
84C22×C4×C53C2×D4118yescomplemented
85C2×C22⋊D53—248no—
86C2×C22⋊D53—4144no—
87C22×Dic53—218no—
88C22×Dic53—412no—
89C22⋊Dic53—218no—
90C22⋊Dic53C24112yescomplemented
91#11719921—248no—
92C23×C53—212no—
93C23×C53—216no—
94C23×C53—218no—
95C23×C53—218no—
96C23×C53—418no—
97C23×C53—46no—
98C23×C53—436no—
99C23×C53—412no—
100C23×C53—818no—
101C23×C53—824no—
102C23×C53#12019yes—
103C23×C53#12016yes—
104C23×C53#12011yescharacteristic
105C23×C53C22×D419yes—
106C23×C53C22×D419yes—
107C23×C53C22×D4112yescomplemented
108C23×C53C2511yescharacteristic
109C22×D53—272no—
110C22×D53—212no—
111C22×D53—48no—
112C22×D53—472no—
113C22×D53—436no—
114C22×D53—472no—
115C22×D53—448no—
116C22×D53—418no—
117C22×D53—46no—
118C22×D53—48no—
119Q8×C53—216no—
120D4×C53—248no—
121D4×C53—272no—
122D4×C53—224no—
123C2×C4×C53—248no—
124C2×C4×C53—218no—
125C2×C4×C53—218no—
126C2×C4×C53C22×D4112yescomplemented
127C22⋊D53—496no—
128C2×Dic53—212no—
129C2×Dic53—436no—
130C22×B(4,𝔽2)—531nomaximal, Sylow
131C22×C53—218no—
132C22×C53—218no—
133C22×C53—212no—
134C22×C53—46no—
135C22×C53—49no—
136C22×C53—41no—
137C22×C53—412no—
138C22×C53—436no—
139C22×C53—818no—
140C22×C53—816no—
141C22×C53B(4,𝔽2)14yescomplemented
142C22×C53C2×#12019yes—
143C22×C53C2×#12013yes—
144C22×C53C23×D413yes—
145C2×D53—224no—
146C2×D53—412no—
147C2×D53—472no—
148C2×D53—44no—
149C2×D53—424no—
150C4×C53—212no—
151Dic53—412no—
152C22×H(5,𝔽2)D5311yescharacteristic, complemented
153C22×#120—533no—
154C2×B(4,𝔽2)—5324no—
155C22×#99—533no—
156C2×C53—23no—
157C2×C53—29no—
158C2×C53—412no—
159C2×C53C2×B(4,𝔽2)16yescomplemented
160C2×C53C22×#12011yescharacteristic
161D53—48no—
162C26—531no—
163C2×H(5,𝔽2)—218no—
164C2×H(5,𝔽2)C2×D53112yescomplemented
165C22×#41—23no—
166C23×D4—23no—
167C23×D4—1063no—
168C23×D4C2×D5313yes—
169C2×#120—5336no—
170C2×#120—10618no—
171C22×C22⋊C4—533no—
172C22×C22⋊C4—1063no—
173B(4,𝔽2)—5364no—
174C2×#99—5336no—
175C53C22×B(4,𝔽2)11yescharacteristic, complemented, Sylow
176C25—21no—
177C25—536no—
178C25—1063no—
179C25—1069no—
180C25—2128no—
181C25C22⋊D5313yescomplemented
182C25C22×D5311yescharacteristic
183H(5,𝔽2)—248no—
184H(5,𝔽2)—272no—
185H(5,𝔽2)C22×D53116yescomplemented
186C2×#41—236no—
187C2×#41—236no—
188C22×Q8—21no—
189C22×D4—23no—
190C22×D4—236no—
191C22×D4—26no—
192C22×D4—218no—
193C22×D4—236no—
194C22×D4—10636no—
195C22×D4—21224no—
196C22×D4C22⋊D53118yescomplemented
197C22×D4C22×D53118yes—
198C23×C4—23no—
199C23×C4—1063no—
200C23×C4C22⋊D5313yescomplemented
201#120—5348no—
202#120—10672no—
203C2×C22⋊C4—5318no—
204C2×C22⋊C4—1069no—
205C2×C22⋊C4—10636no—
206#99—5348no—
207C24—212no—
208C24—23no—
209C24—29no—
210C24—44no—
211C24—412no—
212C24—84no—
213C24—534no—
214C24—10624no—
215C24—10618no—
216C24—10618no—
217C24—21212no—
218C24—21224no—
219C24—21272no—
220C24—2129no—
221C24—2126no—
222C24—2121no—
223C24C2×C22⋊D5313yes—
224C24C2×C22⋊D5319yes—
225C24C2×C22⋊D53118yescomplemented
226C24C23×D5313yes—
227#41—2144no—
228#41—248no—
229C2×Q8—212no—
230C2×D4—236no—
231C2×D4—2144no—
232C2×D4—236no—
233C2×D4—236no—
234C2×D4—236no—
235C2×D4—236no—
236C2×D4—248no—
237C2×D4—10648no—
238C2×D4—212144no—
239C2×D4C2×C22⋊D53136yescomplemented
240C2×D4C23×D53112yes—
241C22×C4—236no—
242C22×C4—29no—
243C22×C4—23no—
244C22×C4—10618no—
245C22×C4—21212no—
246C22×C4C2×C22⋊D53118yescomplemented
247C22⋊C4—5312no—
248C22⋊C4—10618no—
249C22⋊C4—10648no—
250C23—212no—
251C23—216no—
252C23—218no—
253C23—218no—
254C23—418no—
255C23—46no—
256C23—436no—
257C23—412no—
258C23—818no—
259C23—824no—
260C23—10672no—
261C23—10612no—
262C23—21236no—
263C23—21272no—
264C23—2128no—
265C23—21272no—
266C23—21248no—
267C23—2128no—
268C23—21218no—
269C23—2126no—
270C23#1313318619yes—
271C23#1313318616yes—
272C23#1313318611yescharacteristic
273C23C22×C22⋊D5319yes—
274C23C22×C22⋊D5319yes—
275C23C22×C22⋊D53112yescomplemented
276C23C24×D5311yescharacteristic
277Q8—216no—
278D4—248no—
279D4—272no—
280D4—224no—
281D4—21296no—
282C2×C4—248no—
283C2×C4—218no—
284C2×C4—218no—
285C2×C4—10612no—
286C2×C4—21236no—
287C2×C4C22×C22⋊D53112yescomplemented
288C22—218no—
289C22—218no—
290C22—212no—
291C22—46no—
292C22—49no—
293C22—41no—
294C22—412no—
295C22—436no—
296C22—818no—
297C22—816no—
298C22—10624no—
299C22—21212no—
300C22—21272no—
301C22—2124no—
302C22—21224no—
303C22#1681081014yesdirect product factor
304C22C2×#1313318619yes—
305C22C2×#1313318613yes—
306C22C23×C22⋊D5313yes—
307C4—212no—
308C4—21212no—
309C2—23no—
310C2—29no—
311C2—412no—
312C2—2128no—
313C2C2×#1681081016yesdirect product factor
314C2C22×#1313318611yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC23×C53C25
FrattiniC23C24×D53
FittingC22×H(5,𝔽2)×C53C2
socleC23×C53#120
hypercenterC22×H(5,𝔽2)D53
hypocenterC53C22×B(4,𝔽2)
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C25 ◁ C22×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C22×#13133185C2semidirect
2. C2×#16810810C2direct (N × H)
3. C22×#13133132C2semidirect
4. C22×H(5,𝔽2)×C53C2semidirect
5. C25×D53C22semidirect
8. C2×#13133185C22semidirect
9. C22×C22⋊Dic53C22semidirect
11. #16810810C22direct (N × H)
12. C2×#13133132C22semidirect
14. C2×H(5,𝔽2)×C53C22semidirect
21. C24×D53C23semidirect
26. #13133185C23semidirect
28. C2×C22⋊Dic53C23semidirect
30. #13133132C23semidirect
32. C25×C53D4semidirect
36. H(5,𝔽2)×C53C23semidirect
45. C22×D4×C53D4semidirect
48. C23×C4×C53D4semidirect
57. C24×C53C2×D4semidirect
68. C23×D53C24semidirect
79. C2×D4×C53C2×D4semidirect
84. C22×C4×C53C2×D4semidirect
90. C22⋊Dic53C24semidirect
107. C23×C53C22×D4semidirect
126. C2×C4×C53C22×D4semidirect
141. C22×C53B(4,𝔽2)semidirect
152. C22×H(5,𝔽2)D53semidirect
159. C2×C53C2×B(4,𝔽2)semidirect
164. C2×H(5,𝔽2)C2×D53semidirect
175. C53C22×B(4,𝔽2)semidirect
181. C25C22⋊D53semidirect
185. H(5,𝔽2)C22×D53semidirect
196. C22×D4C22⋊D53semidirect
200. C23×C4C22⋊D53semidirect
225. C24C2×C22⋊D53semidirect
239. C2×D4C2×C22⋊D53semidirect
246. C22×C4C2×C22⋊D53semidirect
275. C23C22×C22⋊D53semidirect
287. C2×C4C22×C22⋊D53semidirect
303. C22#16810810direct (N × H)
313. C2C2×#16810810direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131331852
C2×#168108102
C22×#131331322
C22×H(5,𝔽2)×C532
C22×B(4,𝔽2)53