Finite Group Databasebeta

← #72231626

Subgroups — #72231626

Order 2880.

1116 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 270 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×C4⋊(C3⋊ΓL(1,𝔽16))C211yesmaximal, characteristic
2C2×C32×C4⋊AGL(1,𝔽5)C211yesmaximal, characteristic
3C2×#10623965C214yesmaximal
4C4×D5×C3⋊Dic3C211yesmaximal, characteristic
5#11751860—34nomaximal
6C22×C3⋊ΓL(1,𝔽16)C2212yes—
7C22×C32×AGL(1,𝔽5)C2212yes—
8C2×C32×C4×D5C2211yescharacteristic
9C2×D5×C3⋊Dic3C2212yes—
10C4⋊(C3⋊ΓL(1,𝔽16))—22no—
11C32×C4⋊AGL(1,𝔽5)—22no—
12#10623965—28no—
13C4×C3⋊Dic15C411yescharacteristic, complemented
14C4×C5×C3⋊Dic3C411yescharacteristic, complemented
15Dic5×C3⋊Dic3C412yescomplemented
16#10592674—51nomaximal, Hall
17C2×C4⋊ΓL(1,𝔽16)—34no—
18C2×C3×C4⋊AGL(1,𝔽5)D314yes—
19C2×Dic3⋊AGL(1,𝔽5)—316no—
20C4×D5×Dic3—34no—
21C22×C32×D5C2311yescharacteristic
22C2×C3⋊ΓL(1,𝔽16)—22no—
23C2×C3⋊ΓL(1,𝔽16)—24no—
24C2×C32×AGL(1,𝔽5)—22no—
25C2×C32×AGL(1,𝔽5)—24no—
26C2×C32×C4×C5C2×C411yescharacteristic
27C2×C3⋊Dic15C2×C412yes—
28C2×C5×C3⋊Dic3C2×C412yes—
29C2×C32×Dic5C2×C411yescharacteristic
30C32×C4×D5D412yes—
31C32×C4×D5Q812yes—
32D5×C3⋊Dic3D414yes—
33D5×C3⋊Dic3Q814yes—
34#65688—91noHall
35C2×C32×C4⋊C4—51no—
36C2×C4⋊(C3⋊Dic3)—51no—
37C2×#3750—54no—
38C2×C4×C3⋊Dic3—51no—
39C22×ΓL(1,𝔽16)—38no—
40C22×C3×AGL(1,𝔽5)C2×D318yes—
41C2×C3×C4×D5C2×D314yes—
42C2×D5×Dic3—38no—
43C4⋊ΓL(1,𝔽16)—68no—
44C3×C4⋊AGL(1,𝔽5)—28no—
45Dic3⋊AGL(1,𝔽5)—632no—
46C4×Dic15—34no—
47C4×C5×Dic3—34no—
48Dic3×Dic5—38no—
49#4975—154no—
50C22×C32×C5C22×C411yescharacteristic
51C2×C32×D5C2×D412yes—
52C2×C32×D5C2×D411yescharacteristic
53C2×C32×D5C2×Q811yescharacteristic
54C2×C32×D5C2×Q812yes—
55C3⋊ΓL(1,𝔽16)—42no—
56C32×AGL(1,𝔽5)—42no—
57C32×C4×C5C4⋊C412yescomplemented
58C3⋊Dic15C4⋊C414yescomplemented
59C5×C3⋊Dic3C4⋊C414yescomplemented
60C32×Dic5C4⋊C412yescomplemented
61C2×C4⋊AGL(1,𝔽5)—94no—
62C2×C4⋊AGL(1,𝔽5)—91no—
63C2×C4⋊AGL(1,𝔽5)C3⋊D311yescharacteristic
64C42×D5—91no—
65C22×C32×C4—51no—
66C22×C32×C4—52no—
67C22×C3⋊Dic3—52no—
68C22×C3⋊Dic3—52no—
69C32×C4⋊C4—102no—
70C4⋊(C3⋊Dic3)—102no—
71#3750—108no—
72C4×C3⋊Dic3—51no—
73C4×C3⋊Dic3—52no—
74C4×C3⋊Dic3AGL(1,𝔽5)11yescharacteristic, complemented
75C22×C3×D5C22×D314yes—
76C2×ΓL(1,𝔽16)—616no—
77C2×ΓL(1,𝔽16)—68no—
78C2×C3×AGL(1,𝔽5)—216no—
79C2×C3×AGL(1,𝔽5)—28no—
80C2×C3×C4×C5C4×D314yes—
81C2×Dic15—38no—
82C2×C5×Dic3—38no—
83C2×C3×Dic5C4×D314yes—
84C3×C4×D5Dic618yes—
85C3×C4×D5#6718yes—
86D5×Dic3—316no—
87D5×Dic3—316no—
88C2×C3×C4⋊C4—54no—
89C2×C4⋊Dic3—154no—
90C2×Dic3⋊C4—1516no—
91C2×C4×Dic3—154no—
92C2×C32×C5C2×C4⋊C412yes—
93C2×C32×C5C2×C4⋊C411yescharacteristic
94C32×D5#12711yescharacteristic
95C32×D5C4⋊Q812yes—
96C32×D5C4⋊Q811yescharacteristic
97C22×AGL(1,𝔽5)—92no—
98C22×AGL(1,𝔽5)C2×C3⋊D312yes—
99C2×C4×D5—92no—
100C2×C4×D5C2×C3⋊D311yescharacteristic
101C4⋊AGL(1,𝔽5)—22no—
102C4⋊AGL(1,𝔽5)—188no—
103C4⋊AGL(1,𝔽5)—182no—
104C42×C5—91no—
105C4×Dic5—91no—
106C4×Dic5—92no—
107C23×C32—51no—
108C2×C32×C4—51no—
109C2×C32×C4—52no—
110C2×C32×C4—52no—
111C2×C32×C4—104no—
112C2×C32×C4—102no—
113C2×C32×C4C2×AGL(1,𝔽5)11yescharacteristic
114C2×C3⋊Dic3—52no—
115C2×C3⋊Dic3—54no—
116C2×C3⋊Dic3—54no—
117C2×C3⋊Dic3—104no—
118C2×C3⋊Dic3—102no—
119C2×C3⋊Dic3C2×AGL(1,𝔽5)12yes—
120#384—451noSylow
121C22×C3×C5C2×C4×D314yes—
122C2×C3×D5C2×Dic614yes—
123C2×C3×D5D3×D418yes—
124C2×C3×D5D3×Q818yes—
125C2×C3×D5C2×#6714yes—
126ΓL(1,𝔽16)—128no—
127C3×AGL(1,𝔽5)—48no—
128C3×C4×C5Dic3⋊C418yescomplemented
129Dic15—316no—
130C3×Dic5Dic3⋊C418yescomplemented
131C5×Dic3—316no—
132C22×C3×C4—54no—
133C22×C3×C4—58no—
134C22×Dic3—158no—
135C22×Dic3—158no—
136C3×C4⋊C4—108no—
137C4⋊Dic3—308no—
138Dic3⋊C4—3032no—
139C4×Dic3—34no—
140C4×Dic3—158no—
141C4×Dic3—154no—
142C32×C5#38411yescharacteristic, complemented, Hall
143C22×D5C22×C3⋊D311yescharacteristic
144C2×AGL(1,𝔽5)—24no—
145C2×AGL(1,𝔽5)—22no—
146C2×AGL(1,𝔽5)—184no—
147C2×AGL(1,𝔽5)—182no—
148C2×C4×C5—92no—
149C2×C4×C5C4×C3⋊D311yescharacteristic
150C2×Dic5—92no—
151C2×Dic5C4×C3⋊D311yescharacteristic
152C4×D5—94no—
153C4×D5—94no—
154C4×D5#63412yes—
155C4×D5#63812yes—
156C22×C32—52no—
157C22×C32—51no—
158C22×C32—52no—
159C22×C32—51no—
160C22×C32C22×AGL(1,𝔽5)11yescharacteristic
161C32×C4—52no—
162C32×C4—202no—
163C32×C4C4⋊AGL(1,𝔽5)12yescomplemented
164C3⋊Dic3—54no—
165C3⋊Dic3—202no—
166C3⋊Dic3C4⋊AGL(1,𝔽5)14yescomplemented
167C2×C4⋊C4—51no—
168C2×C4⋊C4—454no—
169C2×C4⋊C4—451no—
170C2×C42—451no—
171C2×C3×C5D3×C4⋊C418yes—
172C2×C3×C5C2×Dic3⋊C414yes—
173C3×D5#88818yes—
174C3×D5#94014yes—
175C3×D5#94414yes—
176C23×C3—54no—
177C2×C3×C4—54no—
178C2×C3×C4—58no—
179C2×C3×C4—58no—
180C2×C3×C4—1016no—
181C2×C3×C4—108no—
182C2×C3×C4D3×AGL(1,𝔽5)14yes—
183C2×Dic3—38no—
184C2×Dic3—1516no—
185C2×Dic3—1516no—
186C2×Dic3—158no—
187C2×Dic3—3016no—
188C2×Dic3—308no—
189C22×C5C2×C4×C3⋊D311yescharacteristic
190C2×D5C2×#63411yescharacteristic
191C2×D5D4×C3⋊D312yes—
192C2×D5Q8×C3⋊D312yes—
193C2×D5C2×#63811yescharacteristic
194AGL(1,𝔽5)—42no—
195AGL(1,𝔽5)—362no—
196C4×C5—94no—
197C4×C5#375012yescomplemented
198Dic5—94no—
199Dic5#375012yescomplemented
200C2×C32—51no—
201C2×C32—52no—
202C2×C32—51no—
203C2×C32C2×C4⋊AGL(1,𝔽5)12yes—
204C2×C32C2×C4⋊AGL(1,𝔽5)11yescharacteristic
205C22×C4—51no—
206C22×C4—52no—
207C22×C4—452no—
208C22×C4—452no—
209C4⋊C4—102no—
210C4⋊C4—902no—
211C4⋊C4—908no—
212C42—91no—
213C42—452no—
214C42—451no—
215C3×C5#497514yescomplemented
216C22×C3—58no—
217C22×C3—58no—
218C22×C3—54no—
219C22×C3—54no—
220C22×C3C2×D3×AGL(1,𝔽5)14yes—
221C3×C4—58no—
222C3×C4—208no—
223C3×C4Dic3⋊AGL(1,𝔽5)18yescomplemented
224Dic3—316no—
225Dic3—1516no—
226Dic3—608no—
227C2×C5C4⋊C4×C3⋊D312yes—
228C2×C5C2×#375011yescharacteristic
229D5#6409712yes—
230D5#6414911yescharacteristic
231D5#6415311yescharacteristic
232C32#6568811yescharacteristic, complemented, Sylow
233C23—51no—
234C2×C4—51no—
235C2×C4—52no—
236C2×C4—52no—
237C2×C4—92no—
238C2×C4—102no—
239C2×C4—104no—
240C2×C4—454no—
241C2×C4—454no—
242C2×C4—452no—
243C2×C4—904no—
244C2×C4—902no—
245C2×C4C3⋊D3×AGL(1,𝔽5)11yescharacteristic
246C2×C3—54no—
247C2×C3—58no—
248C2×C3—54no—
249C2×C3D3×C4⋊AGL(1,𝔽5)18yes—
250C2×C3C2×Dic3⋊AGL(1,𝔽5)14yes—
251C5#1059267411yescharacteristic, complemented, Sylow
252C22—52no—
253C22—52no—
254C22—51no—
255C22—51no—
256C22C2×C3⋊D3×AGL(1,𝔽5)11yescharacteristic
257C4—52no—
258C4—94no—
259C4—202no—
260C4—454no—
261C4—1802no—
262C4#1062396512yescomplemented
263C3#1175186014yescomplemented
264C2—51no—
265C2—52no—
266C2—51no—
267C2C3⋊D3×C4⋊AGL(1,𝔽5)12yes—
268C2C2×#1062396511yescharacteristic
subgroupG/H
center, FrattiniC22C2×C3⋊D3×AGL(1,𝔽5)
commutator, socleC22×C32×C5C22×C4
FittingC2×C32×C4×C5C2×C4
prime radical, cosocleC22×C32×D5C23
hypercenterC2×C4C3⋊D3×AGL(1,𝔽5)
hypocenterC32×C5#384
LCSG ▷ C22×C32×C5 ▷ C32×C5
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
13. C4×C3⋊Dic15C4semidirect
14. C4×C5×C3⋊Dic3C4semidirect
15. Dic5×C3⋊Dic3C4semidirect
57. C32×C4×C5C4⋊C4semidirect
58. C3⋊Dic15C4⋊C4semidirect
59. C5×C3⋊Dic3C4⋊C4semidirect
60. C32×Dic5C4⋊C4semidirect
74. C4×C3⋊Dic3AGL(1,𝔽5)semidirect
128. C3×C4×C5Dic3⋊C4semidirect
130. C3×Dic5Dic3⋊C4semidirect
142. C32×C5#384semidirect
163. C32×C4C4⋊AGL(1,𝔽5)semidirect
166. C3⋊Dic3C4⋊AGL(1,𝔽5)semidirect
197. C4×C5#3750semidirect
199. Dic5#3750semidirect
215. C3×C5#4975semidirect
223. C3×C4Dic3⋊AGL(1,𝔽5)semidirect
232. C32#65688semidirect
251. C5#10592674semidirect
262. C4#10623965semidirect
263. C3#11751860semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(C3⋊ΓL(1,𝔽16))2
C2×C32×C4⋊AGL(1,𝔽5)2
C2×#106239652
C4×D5×C3⋊Dic32
#117518603
#105926745