Finite Group Databasebeta

← C2×#20282712 #27590210

Subgroups — C2×#20282712 #27590210

Order 13568.

4438 conjugacy classes of subgroups; only the 667 normal subgroups are shown.

normal subgroups only

This group has 4438 conjugacy classes of subgroups — too many to draw. Only its 667 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×#13133186C212yesmaximal, complemented
2#20282712C2116yesmaximal, direct product factor
3C2×#16810548C213yesmaximal, complemented
4C2×#16810539C216yesmaximal, complemented
5C2×#16810532C213yesmaximal, complemented
6C2×C53×#379C211yesmaximal, characteristic, complemented
7C23×D212C2211yescharacteristic, complemented
8C23×C22⋊D53C2216yes—
9C2×#13133186—216no—
10C2×#13133186C22116yescomplemented
11C22×C4⋊Dic53C2213yescomplemented
12C22×C22⋊Dic53C2216yes—
13C22×#11719921C2216yescomplemented
14#16810548C22124yescomplemented
15#16810539C22148yescomplemented
16#16810532C22124yescomplemented
17C2×#13133124C2213yescomplemented
18C2×#13133124C2213yescomplemented
19C26×C53C2211yescharacteristic
20C22×C22⋊C4×C53C2213yes—
21C22×C22⋊C4×C53C2213yes—
22C53×#379C2218yescomplemented
23C24×D53C2312yes—
24C22×D212—46no—
25C22×D212C2314yescomplemented
26C22×C22⋊D53—224no—
27C22×C22⋊D53—412no—
28C22×C22⋊D53—412no—
29C22×C22⋊D53C23124yes—
30C23×Dic53C2316yes—
31#13133186—264no—
32C2×C4⋊Dic53—46no—
33C2×C4⋊Dic53C23112yescomplemented
34C2×C22⋊Dic53—224no—
35C2×C22⋊Dic53C23124yes—
36C2×#11719921—412no—
37C2×#11719921C23124yescomplemented
38#13133124C23112yescomplemented
39#13133124C23112yescomplemented
40C25×C53—46no—
41C25×C53—46no—
42C25×C53D418yescomplemented
43C25×C53C2313yes—
44C25×C53C2314yes—
45C23×C4×C53C2313yes—
46C23×C4×C53C2311yescharacteristic
47C2×C22⋊C4×C53—46no—
48C2×C22⋊C4×C53—46no—
49C2×C22⋊C4×C53C23112yes—
50C2×C22⋊C4×C53C23112yes—
51C24×C53—26no—
52C24×C53—26no—
53C24×C53—212no—
54C24×C53—212no—
55C24×C53—448no—
56C24×C53—412no—
57C24×C53—412no—
58C24×C53—448no—
59C24×C53—416no—
60C24×C53C2×D4112yes—
61C24×C53C2×D4116yescomplemented
62C24×C53C2416yes—
63C24×C53C2411yescharacteristic
64C23×D53—24no—
65C23×D53—212no—
66C23×D53—212no—
67C23×D53C2414yes—
68C22×C4×C53—212no—
69C22×C4×C53—26no—
70C22×C4×C53—26no—
71C22×C4×C53C2412yes—
72C22×C4×C53C2416yes—
73C2×C22⋊D53—248no—
74C2×C22⋊D53—424no—
75C2×C22⋊D53—424no—
76C2×C22⋊D53—496no—
77C22×Dic53—212no—
78C22×Dic53—212no—
79C22×Dic53—212no—
80C22×Dic53C24112yes—
81C2×D212—412no—
82C2×D212—416no—
83C22⋊C4×C53—412no—
84C22⋊C4×C53—412no—
85C22⋊Dic53—248no—
86#11719921—424no—
87C4⋊Dic53—412no—
88C23×C53—248no—
89C23×C53—224no—
90C23×C53—212no—
91C23×C53—212no—
92C23×C53—26no—
93C23×C53—26no—
94C23×C53—448no—
95C23×C53—448no—
96C23×C53—416no—
97C23×C53—4128no—
98C23×C53—424no—
99C23×C53—412no—
100C23×C53—412no—
101C23×C53C22×D4112yes—
102C23×C53C22×D412yes—
103C23×C53H(5,𝔽2)16yes—
104C23×C53H(5,𝔽2)16yes—
105C23×C53C2511yescharacteristic
106C22×D53—24no—
107C22×D53—212no—
108C22×D53—212no—
109C22×D53—424no—
110C22×D53—424no—
111C22×D53—48no—
112C2×C4×C53—26no—
113C2×C4×C53—26no—
114C2×C4×C53—212no—
115C2×C4×C53—412no—
116C2×C4×C53—44no—
117C22⋊D53—496no—
118C2×Dic53—212no—
119C2×Dic53—212no—
120C2×Dic53—212no—
121C2×Dic53—424no—
122D212—416no—
123C2×#2812—531nomaximal, Sylow
124C22×C53—224no—
125C22×C53—212no—
126C22×C53—464no—
127C22×C53—46no—
128C22×C53—412no—
129C22×C53—46no—
130C22×C53—448no—
131C22×C53#534112yes—
132C22×C53#560112yes—
133C22×C53H(3,𝔽4)14yes—
134C22×C53C23×D411yescharacteristic
135C22×C53C2×H(5,𝔽2)13yes—
136C22×C53C2×H(5,𝔽2)13yes—
137C2×D53—412no—
138C2×D53—412no—
139C2×D53—44no—
140C2×D53—816no—
141C4×C53—46no—
142C4×C53—42no—
143Dic53—412no—
144C22×#120—532no—
145#2812—5316no—
146C2×#397—533no—
147C2×#394—536no—
148C2×#392—533no—
149C2×#379D5311yescharacteristic, complemented
150C2×C53—412no—
151C2×C53#281218yescomplemented
152C2×C53C2×#53413yes—
153C2×C53C2×#56013yes—
154C2×C53C2×H(3,𝔽4)11yescharacteristic
155D53—84no—
156C26C2×D5311yescharacteristic
157C23×D4—536no—
158C23×D4—531no—
159C2×#120—5316no—
160C2×#120—10616no—
161C22×C4⋊C4—533no—
162C22×C22⋊C4—536no—
163C22×C22⋊C4—536no—
164C22×C22⋊C4C2×D5313yes—
165C22×C22⋊C4C2×D5313yes—
166#397—5324no—
167#394—5348no—
168#392—5324no—
169#379C2×D5318yescomplemented
170C2×#95—533no—
171C2×#95—533no—
172C53C2×#281211yescharacteristic, complemented, Sylow
173C25—46no—
174C25—46no—
175C25—532no—
176C25D21218yescomplemented
177C25C22×D5313yes—
178C25C22×D5314yes—
179C22×D4—5324no—
180C22×D4—534no—
181C22×D4—10624no—
182C22×D4—2126no—
183C22×D4—21212no—
184C22×D4—21212no—
185C23×C4—536no—
186C23×C4C22×D5313yes—
187C23×C4C22×D5311yescharacteristic
188#120—10664no—
189C2×C4⋊C4—5312no—
190C2×C4⋊C4—2126no—
191C2×C22⋊C4—46no—
192C2×C22⋊C4—46no—
193C2×C22⋊C4—5324no—
194C2×C22⋊C4—5324no—
195C2×C22⋊C4—10624no—
196C2×C22⋊C4—21212no—
197C2×C22⋊C4C22×D53112yes—
198C2×C22⋊C4C22×D53112yes—
199#95—5312no—
200#95—5312no—
201C24—26no—
202C24—26no—
203C24—212no—
204C24—212no—
205C24—448no—
206C24—412no—
207C24—412no—
208C24—448no—
209C24—416no—
210C24—534no—
211C24—1064no—
212C24—10612no—
213C24—10612no—
214C24C2×D212112yes—
215C24C2×D212116yescomplemented
216C24C23×D5316yes—
217C24C23×D5311yescharacteristic
218C2×D4—10648no—
219C2×D4—21212no—
220C2×D4—21216no—
221C2×D4—21224no—
222C2×D4—21224no—
223C2×D4—21296no—
224C22×C4—212no—
225C22×C4—26no—
226C22×C4—26no—
227C22×C4—5312no—
228C22×C4—10612no—
229C22×C4—10612no—
230C22×C4—10612no—
231C22×C4C23×D5312yes—
232C22×C4C23×D5316yes—
233C4⋊C4—21212no—
234C22⋊C4—412no—
235C22⋊C4—412no—
236C22⋊C4—10648no—
237C22⋊C4—21224no—
238C23—248no—
239C23—224no—
240C23—26no—
241C23—26no—
242C23—212no—
243C23—212no—
244C23—448no—
245C23—448no—
246C23—416no—
247C23—4128no—
248C23—424no—
249C23—412no—
250C23—412no—
251C23—1064no—
252C23—10612no—
253C23—10612no—
254C23—21224no—
255C23—2128no—
256C23—21224no—
257C23C22×D212112yes—
258C23C22×D21212yes—
259C23#1313326216yes—
260C23#1313326216yes—
261C23C24×D5311yescharacteristic
262D4—21216no—
263D4—21296no—
264C2×C4—212no—
265C2×C4—26no—
266C2×C4—26no—
267C2×C4—412no—
268C2×C4—44no—
269C2×C4—10612no—
270C2×C4—10612no—
271C2×C4—10612no—
272C2×C4—21224no—
273C22—224no—
274C22—212no—
275C22—464no—
276C22—412no—
277C22—46no—
278C22—46no—
279C22—448no—
280C22—42416no—
281C22—21212no—
282C22—21212no—
283C22—2124no—
284C22#16811185112yes—
285C22#16811362112yes—
286C22#1681136614yes—
287C22C23×D21211yescharacteristic
288C22C2×#1313326213yes—
289C22C2×#1313326213yes—
290C4—46no—
291C4—42no—
292C4—21212no—
293C2—412no—
294C2—4244no—
295C2#2028271218yesdirect product factor
296C2C2×#1681118513yes—
297C2C2×#1681136213yes—
298C2C2×#1681136611yescharacteristic
subgroupG/H
centerC24C23×D53
commutator, cosocleC23×C53C25
FrattiniC23C24×D53
FittingC2×C53×#379C2
socleC24×C53C24
hypercenterC2×#379D53
hypocenterC53C2×#2812
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#379

Complements

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

NH ≅ G/Ntype
1. C22×#13133186C2semidirect
2. #20282712C2direct (N × H)
3. C2×#16810548C2semidirect
4. C2×#16810539C2semidirect
5. C2×#16810532C2semidirect
6. C2×C53×#379C2semidirect
7. C23×D212C22semidirect
10. C2×#13133186C22semidirect
11. C22×C4⋊Dic53C22semidirect
13. C22×#11719921C22semidirect
14. #16810548C22semidirect
15. #16810539C22semidirect
16. #16810532C22semidirect
17. C2×#13133124C22semidirect
18. C2×#13133124C22semidirect
22. C53×#379C22semidirect
25. C22×D212C23semidirect
33. C2×C4⋊Dic53C23semidirect
37. C2×#11719921C23semidirect
38. #13133124C23semidirect
39. #13133124C23semidirect
42. C25×C53D4semidirect
61. C24×C53C2×D4semidirect
149. C2×#379D53semidirect
151. C2×C53#2812semidirect
169. #379C2×D53semidirect
172. C53C2×#2812semidirect
176. C25D212semidirect
215. C24C2×D212semidirect
295. C2#20282712direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131331862
#202827122
C2×#168105482
C2×#168105392
C2×#168105322
C2×C53×#3792
C2×#281253