Finite Group Databasebeta

← C23×#87831 #72223470

Subgroups — C23×#87831 #72223470

Order 3200.

10626 conjugacy classes of subgroups; only the 1291 normal subgroups are shown.

normal subgroups only

This group has 10626 conjugacy classes of subgroups — too many to draw. Only its 1291 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
1C24×D52C211yesmaximal, characteristic
2C24×#1044C211yesmaximal, characteristic, complemented
3C24×#1043C211yesmaximal, characteristic, complemented
4C22×#87831C2128yesmaximal, direct product factor
5C24×C5⋊D5C2211yescharacteristic
6C24×C5×D5C412yescomplemented
7C23×D52—28no—
8C23×D52—28no—
9C23×D52C4114yescomplemented
10C23×D52C22114yes—
11C23×#1044—28no—
12C23×#1044C22114yescomplemented
13C23×#1043—28no—
14C23×#1043C22114yescomplemented
15C2×#87831C221112yesdirect product factor
16C23×C22⋊AGL(1,𝔽5)—52nomaximal
17C24×C52C2×C411yescharacteristic
18C23×C5⋊D5C2×C417yes—
19C23×C5⋊D5D418yes—
20C23×C5⋊D5D418yes—
21C23×C5⋊D5C2317yes—
22C23×C5×D5—216no—
23C23×C5×D5C2×C4128yescomplemented
24C22×D52—2112no—
25C22×D52—256no—
26C22×D52—256no—
27C22×D52C2×C4184yescomplemented
28C22×D52C23128yes—
29C22×#1044—256no—
30C22×#1044C23128yescomplemented
31C22×#1043—256no—
32C22×#1043C23128yescomplemented
33#87831C23164yesdirect product factor
34C25×D5—52no—
35C24×AGL(1,𝔽5)—52no—
36C24×AGL(1,𝔽5)—52no—
37C24×AGL(1,𝔽5)—102no—
38C22×C22⋊AGL(1,𝔽5)—556no—
39C23×C52C22⋊C418yescomplemented
40C23×C52C22×C417yes—
41C22×C5⋊D5C22⋊C4156yescomplemented
42C22×C5⋊D5C22×C4121yes—
43C22×C5⋊D5C2×D4128yes—
44C22×C5⋊D5C2×D4128yes—
45C22×C5⋊D5C2417yes—
46C22×C5×D5—2112no—
47C22×C5×D5C22×C4156yescomplemented
48C2×D52—2336no—
49C2×D52—256no—
50C2×D52—256no—
51C2×D52C22×C4156yescomplemented
52C2×D52C2418yes—
53C2×#1044—256no—
54C2×#1044C2418yescomplemented
55C2×#1043—256no—
56C2×#1043C2418yescomplemented
57C25×C5—52no—
58C24×D5—52no—
59C24×D5—528no—
60C24×D5—528no—
61C24×D5—102no—
62C24×D5—1016no—
63C24×D5—1016no—
64C24×D5AGL(1,𝔽5)12yescomplemented
65C23×AGL(1,𝔽5)—528no—
66C23×AGL(1,𝔽5)—528no—
67C23×AGL(1,𝔽5)—1028no—
68C23×AGL(1,𝔽5)—1016no—
69C23×AGL(1,𝔽5)—1016no—
70C23×AGL(1,𝔽5)—1032no—
71C2×C22⋊AGL(1,𝔽5)—5224no—
72C23×C22⋊C4—251noSylow
73C22×C52C2×C22⋊C4128yescomplemented
74C22×C52C23×C417yes—
75C2×C5⋊D5C2×C22⋊C4184yescomplemented
76C2×C5⋊D5C23×C417yes—
77C2×C5⋊D5C22×D4114yes—
78C2×C5⋊D5C22×D4114yes—
79C2×C5⋊D5C2511yescharacteristic
80C2×C5×D5—2112no—
81C2×C5×D5C23×C4116yescomplemented
82D52—2112no—
83D52—28no—
84D52—28no—
85#1044—28no—
86#1043—28no—
87C24×C5—22no—
88C24×C5—528no—
89C24×C5—1016no—
90C24×C5C2×AGL(1,𝔽5)12yes—
91C23×D5—216no—
92C23×D5—516no—
93C23×D5—516no—
94C23×D5—514no—
95C23×D5—514no—
96C23×D5—556no—
97C23×D5—5168no—
98C23×D5—1016no—
99C23×D5—1014no—
100C23×D5—1014no—
101C23×D5—1016no—
102C23×D5—10224no—
103C23×D5—10112no—
104C23×D5—10112no—
105C23×D5C2×AGL(1,𝔽5)128yescomplemented
106C22×AGL(1,𝔽5)—556no—
107C22×AGL(1,𝔽5)—556no—
108C22×AGL(1,𝔽5)—1056no—
109C22×AGL(1,𝔽5)—10112no—
110C22×AGL(1,𝔽5)—10112no—
111C22×AGL(1,𝔽5)—10224no—
112C22⋊AGL(1,𝔽5)—5128no—
113C26—251no—
114C24×C4—251no—
115C24×C4—251no—
116C22×C22⋊C4—2528no—
117C2×C52C22×C22⋊C4114yescomplemented
118C2×C52C24×C411yescharacteristic
119C5⋊D5C22×C22⋊C4114yescomplemented
120C5⋊D5C23×D411yescharacteristic
121C5⋊D5C23×D411yescharacteristic
122C5×D5—216no—
123C23×C5—216no—
124C23×C5—214no—
125C23×C5—556no—
126C23×C5—10112no—
127C23×C5C22⋊AGL(1,𝔽5)116yescomplemented
128C23×C5C22×AGL(1,𝔽5)114yes—
129C22×D5—2112no—
130C22×D5—5112no—
131C22×D5—556no—
132C22×D5—556no—
133C22×D5—542no—
134C22×D5—514no—
135C22×D5—5112no—
136C22×D5—516no—
137C22×D5—10112no—
138C22×D5—1042no—
139C22×D5—1056no—
140C22×D5—1056no—
141C22×D5—1014no—
142C22×D5—10672no—
143C22×D5—10112no—
144C22×D5—10112no—
145C22×D5C22×AGL(1,𝔽5)156yescomplemented
146C2×AGL(1,𝔽5)—516no—
147C2×AGL(1,𝔽5)—516no—
148C2×AGL(1,𝔽5)—1016no—
149C2×AGL(1,𝔽5)—10112no—
150C2×AGL(1,𝔽5)—10112no—
151C2×AGL(1,𝔽5)—10224no—
152C25—52no—
153C25—2514no—
154C25—2514no—
155C25—251no—
156C25—508no—
157C25—508no—
158C23×C4—2514no—
159C23×C4—2514no—
160C23×C4—508no—
161C23×C4—508no—
162C2×C22⋊C4—25112no—
163C52C23×C22⋊C411yescharacteristic, complemented, Sylow
164C22×C5—214no—
165C22×C5—256no—
166C22×C5—516no—
167C22×C5—10112no—
168C22×C5C2×C22⋊AGL(1,𝔽5)156yescomplemented
169C22×C5C23×AGL(1,𝔽5)114yes—
170C2×D5—2112no—
171C2×D5—5168no—
172C2×D5—528no—
173C2×D5—528no—
174C2×D5—514no—
175C2×D5—52no—
176C2×D5—10168no—
177C2×D5—1014no—
178C2×D5—1028no—
179C2×D5—1028no—
180C2×D5—102no—
181C2×D5—10224no—
182C2×D5—1016no—
183C2×D5—1016no—
184C2×D5C23×AGL(1,𝔽5)116yescomplemented
185AGL(1,𝔽5)—1016no—
186AGL(1,𝔽5)—1016no—
187AGL(1,𝔽5)—1032no—
188C24—528no—
189C24—1016no—
190C24—2528no—
191C24—2584no—
192C24—258no—
193C24—258no—
194C24—257no—
195C24—257no—
196C24—5056no—
197C24—5056no—
198C24—50112no—
199C24D5⋊AGL(1,𝔽5)11yescharacteristic
200C22×C4—2528no—
201C22×C4—2528no—
202C22×C4—5056no—
203C22×C4—5056no—
204C22⋊C4—2564no—
205C2×C5—228no—
206C2×C5—22no—
207C2×C5—1016no—
208C2×C5C22×C22⋊AGL(1,𝔽5)128yescomplemented
209C2×C5C24×AGL(1,𝔽5)12yes—
210D5—216no—
211D5—528no—
212D5—52no—
213D5—52no—
214D5—1028no—
215D5—102no—
216D5—102no—
217C23—556no—
218C23—10112no—
219C23—258no—
220C23—2556no—
221C23—257no—
222C23—2521no—
223C23—2528no—
224C23—2528no—
225C23—2556no—
226C23—5056no—
227C23—5056no—
228C23—50336no—
229C23#8783118yesdirect product factor
230C23C2×D5⋊AGL(1,𝔽5)17yes—
231C2×C4—258no—
232C2×C4—258no—
233C2×C4—5056no—
234C2×C4—5056no—
235C5—22no—
236C5C23×C22⋊AGL(1,𝔽5)12yescomplemented
237C22—516no—
238C22—10112no—
239C22—2514no—
240C22—2514no—
241C22—2584no—
242C22—251no—
243C22—257no—
244C22—508no—
245C22—508no—
246C22—50112no—
247C22C2×#87831128yesdirect product factor
248C22C22×D5⋊AGL(1,𝔽5)17yes—
249C4—508no—
250C4—508no—
251C2—1016no—
252C2—251no—
253C2—251no—
254C2—2514no—
255C2C22×#87831114yesdirect product factor
256C2C23×D5⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, hypercenterC24D5⋊AGL(1,𝔽5)
commutatorC2×C52C24×C4
FrattiniC2C23×D5⋊AGL(1,𝔽5)
Fitting, socleC24×C52C2×C4
prime radicalC24×D52C2
cosocleC2×C5⋊D5C25
hypocenterC52C23×C22⋊C4
LCSG ▷ C2×C52 ▷ C52
UCSC1 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C24×#1044C2semidirect
3. C24×#1043C2semidirect
4. C22×#87831C2direct (N × H)
6. C24×C5×D5C4semidirect
9. C23×D52C4semidirect
12. C23×#1044C22semidirect
14. C23×#1043C22semidirect
15. C2×#87831C22direct (N × H)
23. C23×C5×D5C2×C4semidirect
27. C22×D52C2×C4semidirect
30. C22×#1044C23semidirect
32. C22×#1043C23semidirect
33. #87831C23direct (N × H)
39. C23×C52C22⋊C4semidirect
41. C22×C5⋊D5C22⋊C4semidirect
47. C22×C5×D5C22×C4semidirect
51. C2×D52C22×C4semidirect
54. C2×#1044C24semidirect
56. C2×#1043C24semidirect
64. C24×D5AGL(1,𝔽5)semidirect
73. C22×C52C2×C22⋊C4semidirect
75. C2×C5⋊D5C2×C22⋊C4semidirect
81. C2×C5×D5C23×C4semidirect
105. C23×D5C2×AGL(1,𝔽5)semidirect
117. C2×C52C22×C22⋊C4semidirect
119. C5⋊D5C22×C22⋊C4semidirect
127. C23×C5C22⋊AGL(1,𝔽5)semidirect
145. C22×D5C22×AGL(1,𝔽5)semidirect
163. C52C23×C22⋊C4semidirect
168. C22×C5C2×C22⋊AGL(1,𝔽5)semidirect
184. C2×D5C23×AGL(1,𝔽5)semidirect
208. C2×C5C22×C22⋊AGL(1,𝔽5)semidirect
229. C23#87831direct (N × H)
236. C5C23×C22⋊AGL(1,𝔽5)semidirect
247. C22C2×#87831direct (N × H)
255. C2C22×#87831direct (N × H)

Maximal subgroups

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

subgroupindex
C24×D522
C24×#10442
C24×#10432
C22×#878312
C23×C22⋊AGL(1,𝔽5)5