Finite Group Databasebeta

← #13021651

Subgroups — #13021651

Order 1280.

1374 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 1374 conjugacy classes of subgroups — too many to draw. Only its 216 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×D5×M16C211yesmaximal, characteristic
2#10617480C212yesmaximal
3C2×#64895C214yesmaximal, complemented
4C23×C4×D5C2211yescharacteristic
5C2×#4130C2214yes—
6C22×C5×M16C411yescharacteristic, complemented
7C22×#681C411yescharacteristic, complemented
8C2×D5×M16—28no—
9C2×D5×M16—24no—
10C22×#676C2212yes—
11C22×C8×D5C412yescomplemented
12#65800—22no—
13#65765—24no—
14#65764—22no—
15C4×C22⋊AGL(1,𝔽5)C418yescomplemented
16#64895—28no—
17#10717—51nomaximal, Sylow
18C24×D5D411yescharacteristic
19C23×C4×C5C2×C411yescharacteristic
20C23×Dic5Q811yescharacteristic
21C22×C4×D5—24no—
22C22×C4×D5—24no—
23C22×C4×D5C2×C411yescharacteristic
24C22×C4×D5C2×C414yes—
25C22×C4×D5D411yescharacteristic
26C22×C4×D5D414yes—
27C22×C4×D5Q811yescharacteristic
28C22×C4×D5C2311yescharacteristic
29C2×C22⋊AGL(1,𝔽5)—28no—
30C2×C22⋊AGL(1,𝔽5)—24no—
31#4130—28no—
32#4130—28no—
33C2×C4⋊AGL(1,𝔽5)—24no—
34C2×C4×AGL(1,𝔽5)C2×C418yes—
35C2×C5×M16—22no—
36C2×C5×M16—24no—
37C22×C5×C8C2×C412yes—
38C2×#681—22no—
39C2×#681—24no—
40C22×#168C2×C412yes—
41D5×M16—232no—
42C2×#676—24no—
43C2×#676—24no—
44C2×#676D418yes—
45C2×C8×D5—24no—
46C2×C8×D5—24no—
47C2×C8×D5—24no—
48(C2×AGL(1,𝔽5))⋊C4—28no—
49C23×M16—51no—
50#2280—52no—
51C2×H(3,ℤ/4)—54no—
52C24×C5C4⋊C411yescharacteristic
53C23×D5—24no—
54C23×D5—44no—
55C23×D5C22⋊C414yes—
56C23×D5C4⋊C411yescharacteristic
57C23×D5C2×D411yescharacteristic
58C22×AGL(1,𝔽5)—28no—
59C22×C4×C5—22no—
60C22×C4×C5—22no—
61C22×C4×C5C4212yes—
62C22×C4×C5C22⋊C412yes—
63C22×C4×C5C4⋊C411yescharacteristic
64C22×C4×C5C22×C411yescharacteristic
65C22×Dic5—24no—
66C22×Dic5C4⋊C411yescharacteristic
67C22×Dic5C4⋊C414yes—
68C22×Dic5C2×Q811yescharacteristic
69C2×C4×D5—24no—
70C2×C4×D5—22no—
71C2×C4×D5—22no—
72C2×C4×D5—24no—
73C2×C4×D5—216no—
74C2×C4×D5—28no—
75C2×C4×D5—28no—
76C2×C4×D5C4212yes—
77C2×C4×D5C22⋊C414yes—
78C2×C4×D5C22⋊C412yes—
79C2×C4×D5C4⋊C414yes—
80C2×C4×D5C22×C412yes—
81C2×C4×D5C2×D414yes—
82C2×C4×D5C2×D412yes—
83C2×C4×D5#4114yes—
84C22⋊AGL(1,𝔽5)—416no—
85C22⋊AGL(1,𝔽5)—48no—
86C4⋊AGL(1,𝔽5)—44no—
87C4⋊AGL(1,𝔽5)—44no—
88C4×AGL(1,𝔽5)—216no—
89C5×M16—28no—
90C2×C5×C8—22no—
91C2×C5×C8—22no—
92C2×C5×C8C22⋊C414yescomplemented
93#681—28no—
94C2×#168—22no—
95C2×#168—22no—
96C2×#168C22⋊C414yescomplemented
97#676—216no—
98C8×D5—216no—
99C24×C4—51no—
100C22×M16—51no—
101C22×M16—52no—
102C22×M16—108no—
103C22×M16—104no—
104C22×M16AGL(1,𝔽5)11yescharacteristic, complemented
105C23×C8—52no—
106C2×#117—54no—
107#381—102no—
108#380—104no—
109#379—102no—
110C4×C22⋊C4—58no—
111H(3,ℤ/4)—108no—
112C23×C5—22no—
113C23×C5—42no—
114C23×C5#9512yes—
115C23×C5C2×C4⋊C411yescharacteristic
116C22×D5—22no—
117C22×D5—24no—
118C22×D5—28no—
119C22×D5—28no—
120C22×D5—44no—
121C22×D5—42no—
122C22×D5—416no—
123C22×D5#9512yes—
124C22×D5C2×C22⋊C412yes—
125C22×D5#12114yes—
126C2×AGL(1,𝔽5)—416no—
127C2×AGL(1,𝔽5)—48no—
128C2×C4×C5—22no—
129C2×C4×C5—24no—
130C2×C4×C5—22no—
131C2×C4×C5—22no—
132C2×C4×C5#9512yes—
133C2×C4×C5C2×C4211yescharacteristic
134C2×C4×C5C2×C22⋊C412yes—
135C2×C4×C5C2×C22⋊C411yescharacteristic
136C2×C4×C5#11712yes—
137C2×Dic5—22no—
138C2×Dic5—28no—
139C2×Dic5#9512yes—
140C2×Dic5C2×C4⋊C412yes—
141C2×Dic5C22⋊Q814yes—
142C4×D5—28no—
143C4×D5—24no—
144C4×D5—22no—
145C4×D5—22no—
146C4×D5—28no—
147C4×D5C4×D418yes—
148C4×D5#12014yes—
149C4×D5#12314yes—
150C5×C8—24no—
151#168—24no—
152C25—51no—
153C23×C4—51no—
154C23×C4—51no—
155C23×C4—54no—
156C23×C4—51no—
157C23×C4—51no—
158C23×C4—54no—
159C23×C4—51no—
160C23×C4—104no—
161C23×C4—104no—
162C23×C4C2×AGL(1,𝔽5)11yescharacteristic
163C2×M16—22no—
164C2×M16—24no—
165C2×M16—58no—
166C2×M16—104no—
167C2×M16—104no—
168C2×M16—104no—
169C2×M16—1032no—
170C2×M16—102no—
171C22×C8—52no—
172C22×C8—104no—
173C22×C8—104no—
174C22×C8—104no—
175C22×C8C2×AGL(1,𝔽5)12yes—
176#117—108no—
177#117—108no—
178C2×C4⋊C4—104no—
179C2×C22⋊C4—104no—
180C2×C22⋊C4—108no—
181C2×C42—58no—
182#95—108no—
183C22×C5—22no—
184C22×C5—22no—
185C22×C5—24no—
186C22×C5—44no—
187C22×C5C2×#9511yescharacteristic
188C22×C5#38012yes—
189C2×D5—28no—
190C2×D5—22no—
191C2×D5—24no—
192C2×D5—22no—
193C2×D5—44no—
194C2×D5—416no—
195C2×D5H(3,ℤ/4)14yescomplemented
196C2×D5#38614yes—
197AGL(1,𝔽5)—88no—
198C4×C5—22no—
199C4×C5C4×C22⋊C412yes—
200C4×C5#37911yescharacteristic
201C4×C5#38111yescharacteristic
202Dic5—22no—
203Dic5#38514yes—
204C24—51no—
205C24—51no—
206C24—54no—
207C24—104no—
208C24—204no—
209C24C4⋊AGL(1,𝔽5)11yescharacteristic
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—51no—
213C22×C4—52no—
214C22×C4—54no—
215C22×C4—52no—
216C22×C4—54no—
217C22×C4—51no—
218C22×C4—54no—
219C22×C4—52no—
220C22×C4—52no—
221C22×C4—54no—
222C22×C4—54no—
223C22×C4—108no—
224C22×C4—104no—
225C22×C4—1016no—
226C22×C4—108no—
227C22×C4—104no—
228C22×C4—104no—
229C22×C4—102no—
230C22×C4—102no—
231C22×C4—108no—
232C22×C4C4×AGL(1,𝔽5)12yes—
233C22×C4C4⋊AGL(1,𝔽5)11yescharacteristic
234C22×C4C22⋊AGL(1,𝔽5)12yes—
235C22×C4C22×AGL(1,𝔽5)11yescharacteristic
236M16—28no—
237M16—1016no—
238M16—108no—
239C2×C8—22no—
240C2×C8—22no—
241C2×C8—54no—
242C2×C8—1016no—
243C2×C8—102no—
244C2×C8—102no—
245C2×C8C22⋊AGL(1,𝔽5)14yescomplemented
246C4⋊C4—204no—
247C4⋊C4—204no—
248C22⋊C4—208no—
249C22⋊C4—2016no—
250C42—1016no—
251C2×C5—22no—
252C2×C5—42no—
253C2×C5C22⋊#9511yescharacteristic
254C2×C5C2×H(3,ℤ/4)12yes—
255D5—24no—
256D5—24no—
257C23—22no—
258C23—42no—
259C23—52no—
260C23—52no—
261C23—54no—
262C23—104no—
263C23—108no—
264C23—108no—
265C23—102no—
266C23—202no—
267C23—204no—
268C23—2016no—
269C23(C2×AGL(1,𝔽5))⋊C412yes—
270C23C2×C4⋊AGL(1,𝔽5)11yescharacteristic
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—52no—
276C2×C4—54no—
277C2×C4—54no—
278C2×C4—58no—
279C2×C4—54no—
280C2×C4—52no—
281C2×C4—108no—
282C2×C4—108no—
283C2×C4—108no—
284C2×C4—102no—
285C2×C4—102no—
286C2×C4—104no—
287C2×C4—102no—
288C2×C4—208no—
289C2×C4—2016no—
290C2×C4(C2×AGL(1,𝔽5))⋊C412yes—
291C2×C4C2×C4×AGL(1,𝔽5)11yescharacteristic
292C2×C4#413012yes—
293C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
294C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
295C8—24no—
296C8—104no—
297C5#1071711yescharacteristic, complemented, Sylow
298C22—22no—
299C22—22no—
300C22—24no—
301C22—44no—
302C22—54no—
303C22—54no—
304C22—108no—
305C22—102no—
306C22—104no—
307C22—102no—
308C22—204no—
309C22—2016no—
310C22C2×(C2×AGL(1,𝔽5))⋊C411yescharacteristic
311C22#6576512yes—
312C4—22no—
313C4—54no—
314C4—102no—
315C4—408no—
316C4C4×C22⋊AGL(1,𝔽5)12yes—
317C4#6576411yescharacteristic
318C4#6580011yescharacteristic
319C2—22no—
320C2—42no—
321C2—104no—
322C2—104no—
323C2#1060476212yes—
324C2#1060490911yescharacteristic
subgroupG/H
centerC2×C4C2×C4×AGL(1,𝔽5)
commutatorC2×C4×C5C2×C42
FrattiniC22×C4C22×AGL(1,𝔽5)
FittingC22×C5×M16C4
socleC22×C5C2×#95
prime radicalC24×D5D4
cosocleC22×C4×D5C23
hypercenterC22×M16AGL(1,𝔽5)
hypocenterC5#10717
LCSG ▷ C2×C4×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
3. C2×#64895C2semidirect
6. C22×C5×M16C4semidirect
7. C22×#681C4semidirect
11. C22×C8×D5C4semidirect
15. C4×C22⋊AGL(1,𝔽5)C4semidirect
92. C2×C5×C8C22⋊C4semidirect
96. C2×#168C22⋊C4semidirect
104. C22×M16AGL(1,𝔽5)semidirect
195. C2×D5H(3,ℤ/4)semidirect
245. C2×C8C22⋊AGL(1,𝔽5)semidirect
297. C5#10717semidirect

Maximal subgroups

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

subgroupindex
C22×D5×M162
#106174802
C2×#648952
#107175