Finite Group Databasebeta

← #13088732

Subgroups — #13088732

Order 1440.

430 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C2×ΓL(1,𝔽16)⋊D3C212yesmaximal, complemented
2D5×C4⋊(C3⋊D3)C211yesmaximal, characteristic
3C3×C4⋊ΓL(1,𝔽16)C211yesmaximal, characteristic, complemented
4C3×C4×ΓL(1,𝔽16)C211yesmaximal, characteristic, complemented
5#10623978C212yesmaximal, complemented
6#91921—31nomaximal
7AGL(1,𝔽5)×D12—31nomaximal
8C2×D5×C3⋊D3C2212yes—
9C2×C3×ΓL(1,𝔽16)C2212yes—
10C2×C3×ΓL(1,𝔽16)C2211yescharacteristic
11C2×C3×ΓL(1,𝔽16)C2211yescharacteristic
12ΓL(1,𝔽16)⋊D3—24no—
13C4⋊(C3⋊D15)C411yescharacteristic, complemented
14C5×C4⋊(C3⋊D3)C411yescharacteristic, complemented
15C32×C4×D5C2211yescharacteristic
16#67140C412yescomplemented
17#63894—51nomaximal, Hall
18C2×D3×AGL(1,𝔽5)—32no—
19C2×D3×AGL(1,𝔽5)—32no—
20D5×D12—31no—
21D5×D12—31no—
22D5×D12—61no—
23C4⋊ΓL(1,𝔽16)D311yescharacteristic, complemented
24C4×ΓL(1,𝔽16)D311yescharacteristic, complemented
25C3×C4⋊AGL(1,𝔽5)—31no—
26C3×C4×AGL(1,𝔽5)—31no—
27(C2×D3)⋊AGL(1,𝔽5)—32no—
28(C2×D3)⋊AGL(1,𝔽5)—32no—
29C2×C3⋊D15C2×C412yescomplemented
30C2×C5×C3⋊D3C2×C412yescomplemented
31C2×C32×D5C2311yescharacteristic
32D5×C3⋊D3—22no—
33D5×C3⋊D3—22no—
34C3×ΓL(1,𝔽16)—22no—
35C3×ΓL(1,𝔽16)—21no—
36C3×ΓL(1,𝔽16)D412yescomplemented
37C32×C4×C5C2×C411yescharacteristic, complemented
38C32×Dic5C2×C411yescharacteristic, complemented
39D4×AGL(1,𝔽5)—91noHall
40C2×C4⋊(C3⋊D3)—51no—
41C2×#624—52no—
42C3×C4⋊Dic3—51no—
43C3×C4×Dic3—51no—
44#3722—52no—
45C2×D3×D5—32no—
46C2×D3×D5—32no—
47C2×D3×D5—62no—
48C2×ΓL(1,𝔽16)C2×D311yescharacteristic
49C2×ΓL(1,𝔽16)C2×D312yes—
50C2×ΓL(1,𝔽16)C2×D311yescharacteristic
51C2×C3×AGL(1,𝔽5)—31no—
52C2×C3×AGL(1,𝔽5)—31no—
53C2×C3×AGL(1,𝔽5)—32no—
54D3×AGL(1,𝔽5)—64no—
55D3×AGL(1,𝔽5)—64no—
56D60—31no—
57D60—31no—
58D60—61no—
59C5×D12—31no—
60C5×D12—31no—
61C5×D12—61no—
62C3×C4×D5—21no—
63C3×C4×D5C2×D311yescharacteristic
64C3×C4×D5C2×D311yescharacteristic
65#1203—32no—
66#1203—32no—
67#1203—62no—
68#893—151no—
69C4×D12—151no—
70C2×C32×C5C22×C411yescharacteristic
71C3⋊D15—22no—
72C5×C3⋊D3—22no—
73C32×D5C2×D411yescharacteristic
74C32×D5#4111yescharacteristic
75C22×AGL(1,𝔽5)—92no—
76D4×D5—91no—
77C22⋊AGL(1,𝔽5)—92no—
78C4⋊AGL(1,𝔽5)—31no—
79C4×AGL(1,𝔽5)—31no—
80C22×C3⋊D3—52no—
81C2×C32×C4—51no—
82C4⋊(C3⋊D3)—52no—
83C4⋊(C3⋊D3)—51no—
84C4⋊(C3⋊D3)AGL(1,𝔽5)11yescharacteristic, complemented
85C2×C3×Dic3—51no—
86C2×C3×Dic3—51no—
87C2×C3×Dic3—52no—
88#624—104no—
89C2×D15—32no—
90C2×D15—32no—
91C2×D15—62no—
92C2×C5×D3—32no—
93C2×C5×D3—32no—
94C2×C5×D3—62no—
95C2×C3×D5—21no—
96C2×C3×D5C22×D311yescharacteristic
97C2×C3×D5C22×D311yescharacteristic
98D3×D5—62no—
99D3×D5—62no—
100D3×D5—62no—
101D3×D5—62no—
102D3×D5—122no—
103D3×D5—122no—
104ΓL(1,𝔽16)—21no—
105ΓL(1,𝔽16)—22no—
106ΓL(1,𝔽16)D1212yescomplemented
107C3×AGL(1,𝔽5)—32no—
108C3×AGL(1,𝔽5)—62no—
109C3×AGL(1,𝔽5)—61no—
110C3×C4×C5—21no—
111C3×C4×C5C4×D311yescharacteristic, complemented
112C3×C4×C5C4×D311yescharacteristic, complemented
113C3×Dic5—21no—
114C3×Dic5C4×D311yescharacteristic, complemented
115C3×Dic5C4×D311yescharacteristic, complemented
116C2×D12—151no—
117C2×D12—151no—
118C2×D12—301no—
119C2×C4×D3—152no—
120C2×C4×D3—152no—
121C3×C4⋊C4—151no—
122C3×C42—151no—
123(C2×D3)⋊C4—152no—
124(C2×D3)⋊C4—152no—
125C4⋊Dic3—51no—
126C4×Dic3—51no—
127C32×C5C4×D411yescharacteristic, complemented, Hall
128C22×D5—92no—
129C2×AGL(1,𝔽5)—31no—
130C2×AGL(1,𝔽5)—31no—
131C2×AGL(1,𝔽5)—32no—
132C2×AGL(1,𝔽5)—184no—
133C5×D4—91no—
134C22⋊D5—92no—
135D20—91no—
136C4×D5D3211yescharacteristic
137C22×C32—51no—
138C2×C3⋊D3—52no—
139C2×C3⋊D3—102no—
140C2×C3⋊D3—102no—
141C2×C3⋊D3C2×AGL(1,𝔽5)12yescomplemented
142C32×C4—51no—
143C32×C4C2×AGL(1,𝔽5)11yescharacteristic, complemented
144C3×Dic3—52no—
145C3×Dic3—102no—
146C3×Dic3—101no—
147C4×D4—451noSylow
148C2×C3×C5—21no—
149C2×C3×C5C2×C4×D311yescharacteristic
150C2×C3×C5C2×C4×D311yescharacteristic
151D15—62no—
152D15—62no—
153D15—122no—
154C3×D5—21no—
155C3×D5—21no—
156C3×D5C2×D1211yescharacteristic
157C3×D5#23511yescharacteristic
158C3×D5D3×D411yescharacteristic
159C3×D5#23911yescharacteristic
160C5×D3—62no—
161C5×D3—62no—
162C5×D3—122no—
163C22×D3—152no—
164C22×D3—152no—
165C22×D3—302no—
166C2×C3×C4—51no—
167C2×C3×C4—51no—
168C2×C3×C4—101no—
169C2×C3×C4—151no—
170C2×C3×C4—151no—
171C2×C3×C4—152no—
172C2×Dic3—51no—
173C2×Dic3—51no—
174C2×Dic3—52no—
175D12—31no—
176D12—31no—
177D12—61no—
178D12—152no—
179D12—152no—
180D12—151no—
181D12—151no—
182D12—301no—
183D12—302no—
184C4×D3—304no—
185C4×D3—304no—
186C22×C5—92no—
187C2×D5—92no—
188C2×D5—182no—
189C2×D5—182no—
190C2×D5C2×D3211yescharacteristic
191AGL(1,𝔽5)—32no—
192AGL(1,𝔽5)—61no—
193AGL(1,𝔽5)—62no—
194C4×C5#62411yescharacteristic, complemented
195Dic5#62411yescharacteristic, complemented
196C2×C32—51no—
197C2×C32—51no—
198C2×C32C22×AGL(1,𝔽5)11yescharacteristic
199C3⋊D3—22no—
200C3⋊D3—102no—
201C2×D4—451no—
202C22×C4—452no—
203C4⋊C4—151no—
204C22⋊C4—452no—
205C42—151no—
206C3×C5—21no—
207C3×C5C4×D1211yescharacteristic, complemented
208C3×C5#89311yescharacteristic, complemented
209C22×C3—51no—
210C22×C3—51no—
211C22×C3—101no—
212C2×D3—32no—
213C2×D3—32no—
214C2×D3—62no—
215C2×D3—152no—
216C2×D3—152no—
217C2×D3—302no—
218C2×D3—302no—
219C2×D3—302no—
220C2×D3—302no—
221C2×D3—302no—
222C2×D3—602no—
223C2×D3—602no—
224C3×C4—21no—
225C3×C4—51no—
226C3×C4—51no—
227C3×C4—101no—
228C3×C4—152no—
229C3×C4—302no—
230C3×C4—301no—
231C3×C4D3×AGL(1,𝔽5)11yescharacteristic, complemented
232C3×C4D3×AGL(1,𝔽5)11yescharacteristic, complemented
233Dic3—52no—
234Dic3—102no—
235Dic3—101no—
236C2×C5—182no—
237C2×C5C2×#62411yescharacteristic
238D5—182no—
239D5#379911yescharacteristic
240D5D3×D1211yescharacteristic
241C32D4×AGL(1,𝔽5)11yescharacteristic, complemented, Sylow
242C23—452no—
243D4—91no—
244D4—452no—
245D4—451no—
246C2×C4—51no—
247C2×C4—151no—
248C2×C4—151no—
249C2×C4—152no—
250C2×C4—904no—
251C2×C3—21no—
252C2×C3—51no—
253C2×C3—51no—
254C2×C3—51no—
255C2×C3—51no—
256C2×C3—101no—
257C2×C3—101no—
258C2×C3C2×D3×AGL(1,𝔽5)11yescharacteristic
259C2×C3C2×D3×AGL(1,𝔽5)11yescharacteristic
260D3—62no—
261D3—62no—
262D3—122no—
263D3—302no—
264D3—302no—
265D3—602no—
266C5#6389411yescharacteristic, complemented, Sylow
267C22—51no—
268C22—92no—
269C22—452no—
270C22—902no—
271C22—902no—
272C4—51no—
273C4—152no—
274C4—302no—
275C4—301no—
276C4ΓL(1,𝔽16)⋊D311yescharacteristic, complemented
277C3—21no—
278C3AGL(1,𝔽5)×D1211yescharacteristic, complemented
279C3#9192111yescharacteristic, complemented
280C2—51no—
281C2—51no—
282C2—182no—
283C2—902no—
284C2C2×ΓL(1,𝔽16)⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C2×ΓL(1,𝔽16)⋊D3
commutator, socleC2×C32×C5C22×C4
FittingC32×C4×C5C2×C4
prime radicalD5×C4⋊(C3⋊D3)C2
cosocleC2×C32×D5C23
hypercenterC4ΓL(1,𝔽16)⋊D3
hypocenterC32×C5C4×D4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C2×ΓL(1,𝔽16)⋊D3C2semidirect
3. C3×C4⋊ΓL(1,𝔽16)C2semidirect
4. C3×C4×ΓL(1,𝔽16)C2semidirect
5. #10623978C2semidirect
13. C4⋊(C3⋊D15)C4semidirect
14. C5×C4⋊(C3⋊D3)C4semidirect
16. #67140C4semidirect
23. C4⋊ΓL(1,𝔽16)D3semidirect
24. C4×ΓL(1,𝔽16)D3semidirect
29. C2×C3⋊D15C2×C4semidirect
30. C2×C5×C3⋊D3C2×C4semidirect
36. C3×ΓL(1,𝔽16)D4semidirect
37. C32×C4×C5C2×C4semidirect
38. C32×Dic5C2×C4semidirect
84. C4⋊(C3⋊D3)AGL(1,𝔽5)semidirect
106. ΓL(1,𝔽16)D12semidirect
111. C3×C4×C5C4×D3semidirect
112. C3×C4×C5C4×D3semidirect
114. C3×Dic5C4×D3semidirect
115. C3×Dic5C4×D3semidirect
127. C32×C5C4×D4semidirect
141. C2×C3⋊D3C2×AGL(1,𝔽5)semidirect
143. C32×C4C2×AGL(1,𝔽5)semidirect
194. C4×C5#624semidirect
195. Dic5#624semidirect
207. C3×C5C4×D12semidirect
208. C3×C5#893semidirect
231. C3×C4D3×AGL(1,𝔽5)semidirect
232. C3×C4D3×AGL(1,𝔽5)semidirect
241. C32D4×AGL(1,𝔽5)semidirect
266. C5#63894semidirect
276. C4ΓL(1,𝔽16)⋊D3semidirect
278. C3AGL(1,𝔽5)×D12semidirect
279. C3#91921semidirect

Maximal subgroups

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

subgroupindex
C2×ΓL(1,𝔽16)⋊D32
D5×C4⋊(C3⋊D3)2
C3×C4⋊ΓL(1,𝔽16)2
C3×C4×ΓL(1,𝔽16)2
#106239782
#919213
AGL(1,𝔽5)×D123
#638945