Finite Group Databasebeta

← C23×C22⋊#1044 #72223516

Subgroups — C23×C22⋊#1044 #72223516

Order 3200.

14914 conjugacy classes of subgroups; only the 1342 normal subgroups are shown.

normal subgroups only

This group has 14914 conjugacy classes of subgroups — too many to draw. Only its 1342 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
1C25×C5⋊D5C211yesmaximal, characteristic
2C24×#1044C212yesmaximal, complemented
3C22×C22⋊#1044C2128yesmaximal, direct product factor
4C25×C52C411yescharacteristic, complemented
5C24×C5⋊D5—216no—
6C24×C5⋊D5C4114yescomplemented
7C24×C5⋊D5C411yescharacteristic, complemented
8C24×C5⋊D5C22114yes—
9C24×C5⋊D5C2211yescharacteristic
10C23×#1044—216no—
11C23×#1044C22128yescomplemented
12C2×C22⋊#1044C221112yesdirect product factor
13C23×C22⋊AGL(1,𝔽5)—52nomaximal
14C24×C52—28no—
15C24×C52C2×C411yescharacteristic
16C24×C52C2×C4114yescomplemented
17C23×C5⋊D5—28no—
18C23×C5⋊D5—2112no—
19C23×C5⋊D5—2112no—
20C23×C5⋊D5C2×C417yes—
21C23×C5⋊D5C2×C4184yescomplemented
22C23×C5⋊D5C2×C4114yescomplemented
23C23×C5⋊D5D4116yes—
24C23×C5⋊D5C23128yes—
25C23×C5⋊D5C2317yes—
26C22×#1044—2112no—
27C22×#1044C23156yescomplemented
28C22⋊#1044C23164yesdirect product factor
29C25×D5—52no—
30C25×D5—102no—
31C24×AGL(1,𝔽5)—54no—
32C22×C22⋊AGL(1,𝔽5)—556no—
33C23×C52—256no—
34C23×C52C22⋊C418yescomplemented
35C23×C52C22×C417yes—
36C23×C52C22×C4128yescomplemented
37C22×C5⋊D5—256no—
38C22×C5⋊D5—2336no—
39C22×C5⋊D5—2112no—
40C22×C5⋊D5C22⋊C4156yescomplemented
41C22×C5⋊D5C22×C4121yes—
42C22×C5⋊D5C22×C4156yescomplemented
43C22×C5⋊D5C22×C4128yescomplemented
44C22×C5⋊D5C2×D4156yes—
45C22×C5⋊D5C2418yes—
46C22×C5⋊D5C2417yes—
47C2×#1044—2112no—
48C2×#1044C24116yescomplemented
49C25×C5—22no—
50C25×C5AGL(1,𝔽5)12yescomplemented
51C24×D5—52no—
52C24×D5—52no—
53C24×D5—528no—
54C24×D5—528no—
55C24×D5—102no—
56C24×D5—1028no—
57C24×D5—102no—
58C24×D5—1028no—
59C24×D5—1032no—
60C24×D5—1064no—
61C23×AGL(1,𝔽5)—556no—
62C23×AGL(1,𝔽5)—1032no—
63C2×C22⋊AGL(1,𝔽5)—5224no—
64C23×C22⋊C4—251noSylow
65C22×C52—256no—
66C22×C52C2×C22⋊C4128yescomplemented
67C22×C52C23×C417yes—
68C22×C52C23×C418yescomplemented
69C2×C5⋊D5—256no—
70C2×C5⋊D5—2112no—
71C2×C5⋊D5—216no—
72C2×C5⋊D5C2×C22⋊C4184yescomplemented
73C2×C5⋊D5C23×C417yes—
74C2×C5⋊D5C23×C418yescomplemented
75C2×C5⋊D5C22×D4128yes—
76C2×C5⋊D5C2511yescharacteristic
77#1044—216no—
78C24×C5—232no—
79C24×C5—216no—
80C24×C5—22no—
81C24×C5—228no—
82C24×C5C2×AGL(1,𝔽5)12yes—
83C24×C5C2×AGL(1,𝔽5)128yescomplemented
84C23×D5—532no—
85C23×D5—514no—
86C23×D5—514no—
87C23×D5—528no—
88C23×D5—556no—
89C23×D5—5168no—
90C23×D5—1014no—
91C23×D5—1028no—
92C23×D5—10168no—
93C23×D5—1032no—
94C23×D5—1014no—
95C23×D5—1056no—
96C23×D5—10224no—
97C23×D5—10224no—
98C23×D5—10448no—
99C23×D5—10448no—
100C23×D5—1032no—
101C23×D5—1016no—
102C22×AGL(1,𝔽5)—5112no—
103C22×AGL(1,𝔽5)—10224no—
104C22⋊AGL(1,𝔽5)—5128no—
105C26—251no—
106C24×C4—252no—
107C22×C22⋊C4—2528no—
108C2×C52—28no—
109C2×C52C22×C22⋊C4114yescomplemented
110C2×C52C24×C411yescharacteristic
111C5⋊D5—28no—
112C5⋊D5C22×C22⋊C4114yescomplemented
113C5⋊D5C23×D412yes—
114C23×C5—2224no—
115C23×C5—2112no—
116C23×C5—216no—
117C23×C5—214no—
118C23×C5—256no—
119C23×C5C22⋊AGL(1,𝔽5)116yescomplemented
120C23×C5C22×AGL(1,𝔽5)114yes—
121C23×C5C22×AGL(1,𝔽5)156yescomplemented
122C22×D5—5112no—
123C22×D5—5112no—
124C22×D5—542no—
125C22×D5—514no—
126C22×D5—556no—
127C22×D5—5112no—
128C22×D5—516no—
129C22×D5—10112no—
130C22×D5—1042no—
131C22×D5—1056no—
132C22×D5—10112no—
133C22×D5—10112no—
134C22×D5—1014no—
135C22×D5—1016no—
136C22×D5—10672no—
137C22×D5—10224no—
138C22×D5—101344no—
139C22×D5—10448no—
140C22×D5—10224no—
141C22×D5—10112no—
142C2×AGL(1,𝔽5)—532no—
143C2×AGL(1,𝔽5)—10224no—
144C25—251no—
145C25—251no—
146C25—2514no—
147C25—2514no—
148C25—5016no—
149C25#104411yescharacteristic, complemented
150C23×C4—2528no—
151C23×C4—5016no—
152C2×C22⋊C4—25112no—
153C52C23×C22⋊C411yescharacteristic, complemented, Sylow
154C22×C5—2224no—
155C22×C5—2112no—
156C22×C5—256no—
157C22×C5—214no—
158C22×C5—216no—
159C22×C5C2×C22⋊AGL(1,𝔽5)156yescomplemented
160C22×C5C23×AGL(1,𝔽5)114yes—
161C22×C5C23×AGL(1,𝔽5)116yescomplemented
162C2×D5—5168no—
163C2×D5—556no—
164C2×D5—514no—
165C2×D5—52no—
166C2×D5—516no—
167C2×D5—10168no—
168C2×D5—1014no—
169C2×D5—1016no—
170C2×D5—1056no—
171C2×D5—102no—
172C2×D5—10224no—
173C2×D5—1032no—
174C2×D5—10448no—
175C2×D5—1064no—
176C2×D5—10224no—
177C2×D5—10112no—
178AGL(1,𝔽5)—1032no—
179C24—28no—
180C24—2516no—
181C24—257no—
182C24—257no—
183C24—2514no—
184C24—2528no—
185C24—2584no—
186C24—50112no—
187C24—50112no—
188C24—508no—
189C24C2×#104411yescharacteristic
190C24C2×#1044114yescomplemented
191C22×C4—2556no—
192C22×C4—50112no—
193C22⋊C4—2564no—
194C2×C5—232no—
195C2×C5—216no—
196C2×C5—228no—
197C2×C5—22no—
198C2×C5C22×C22⋊AGL(1,𝔽5)128yescomplemented
199C2×C5C24×AGL(1,𝔽5)12yes—
200D5—528no—
201D5—54no—
202D5—1028no—
203D5—104no—
204D5—1032no—
205D5—1016no—
206C23—256no—
207C23—2556no—
208C23—2556no—
209C23—257no—
210C23—2521no—
211C23—2528no—
212C23—258no—
213C23—2556no—
214C23—50112no—
215C23—50336no—
216C23—5056no—
217C23C22⋊#104418yesdirect product factor
218C23C22×#104417yes—
219C23C22×#1044128yescomplemented
220C2×C4—2516no—
221C2×C4—50112no—
222C5—22no—
223C5C23×C22⋊AGL(1,𝔽5)12yescomplemented
224C22—256no—
225C22—2528no—
226C22—2584no—
227C22—251no—
228C22—257no—
229C22—258no—
230C22—5016no—
231C22—50112no—
232C22—5056no—
233C22C2×C22⋊#1044128yesdirect product factor
234C22C23×#104417yes—
235C22C23×#104418yescomplemented
236C4—5016no—
237C2—28no—
238C2—252no—
239C2—2514no—
240C2—508no—
241C2C22×C22⋊#1044114yesdirect product factor
242C2C24×#104411yescharacteristic
subgroupG/H
centerC24C2×#1044
commutatorC2×C52C24×C4
FrattiniC2C24×#1044
FittingC25×C52C4
socleC24×C52C2×C4
prime radicalC25×C5⋊D5C2
cosocleC2×C5⋊D5C25
hypercenterC25#1044
hypocenterC52C23×C22⋊C4
LCSG ▷ C2×C52 ▷ C52
UCSC1 ◁ C24 ◁ C25

Complements

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

NH ≅ G/Ntype
2. C24×#1044C2semidirect
3. C22×C22⋊#1044C2direct (N × H)
4. C25×C52C4semidirect
6. C24×C5⋊D5C4semidirect
7. C24×C5⋊D5C4semidirect
11. C23×#1044C22semidirect
12. C2×C22⋊#1044C22direct (N × H)
16. C24×C52C2×C4semidirect
21. C23×C5⋊D5C2×C4semidirect
22. C23×C5⋊D5C2×C4semidirect
27. C22×#1044C23semidirect
28. C22⋊#1044C23direct (N × H)
34. C23×C52C22⋊C4semidirect
36. C23×C52C22×C4semidirect
40. C22×C5⋊D5C22⋊C4semidirect
42. C22×C5⋊D5C22×C4semidirect
43. C22×C5⋊D5C22×C4semidirect
48. C2×#1044C24semidirect
50. C25×C5AGL(1,𝔽5)semidirect
66. C22×C52C2×C22⋊C4semidirect
68. C22×C52C23×C4semidirect
72. C2×C5⋊D5C2×C22⋊C4semidirect
74. C2×C5⋊D5C23×C4semidirect
83. C24×C5C2×AGL(1,𝔽5)semidirect
109. C2×C52C22×C22⋊C4semidirect
112. C5⋊D5C22×C22⋊C4semidirect
119. C23×C5C22⋊AGL(1,𝔽5)semidirect
121. C23×C5C22×AGL(1,𝔽5)semidirect
149. C25#1044semidirect
153. C52C23×C22⋊C4semidirect
159. C22×C5C2×C22⋊AGL(1,𝔽5)semidirect
161. C22×C5C23×AGL(1,𝔽5)semidirect
190. C24C2×#1044semidirect
198. C2×C5C22×C22⋊AGL(1,𝔽5)semidirect
217. C23C22⋊#1044direct (N × H)
219. C23C22×#1044semidirect
223. C5C23×C22⋊AGL(1,𝔽5)semidirect
233. C22C2×C22⋊#1044direct (N × H)
235. C22C23×#1044semidirect
241. C2C22×C22⋊#1044direct (N × H)

Maximal subgroups

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

subgroupindex
C25×C5⋊D52
C24×#10442
C22×C22⋊#10442
C23×C22⋊AGL(1,𝔽5)5