Finite Group Databasebeta

← #13021668

Subgroups — #13021668

Order 1280.

1294 conjugacy classes of subgroups; only the 228 normal subgroups are shown.

normal subgroups only

This group has 1294 conjugacy classes of subgroups — too many to draw. Only its 228 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
2C2×#65756C212yesmaximal
3C2×#64896C214yesmaximal, complemented
4C23×C4×D5C2211yescharacteristic
5C22×#700C2214yes—
6C22×C5×M16C411yescharacteristic
7C22×#681C411yescharacteristic
8C2×D5×M16—28no—
9C2×D5×M16—24no—
10C22×#676C412yes—
11C22×C8×D5C2212yes—
12C2×C22⋊#170—22no—
13#65756—24no—
14#65756C418yes—
15C2×#4003—22no—
16#64896—28no—
17#10724—51nomaximal, Sylow
18C24×D5Q811yescharacteristic
19C23×C4×C5C2×C411yescharacteristic
20C23×Dic5D411yescharacteristic
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
29C22×#170—24no—
30C2×#700—28no—
31C2×#700—28no—
32C2×#700—24no—
33C2×#700C2×C418yes—
34C2×C5×M16—22no—
35C2×C5×M16—24no—
36C22×C5×C8C2×C412yes—
37C2×#681—22no—
38C2×#681—24no—
39C22×#168C2×C412yes—
40D5×M16—232no—
41C2×#676—24no—
42C2×#676—24no—
43C2×#676—24no—
44C2×C8×D5—24no—
45C2×C8×D5—24no—
46C2×C8×D5D418yes—
47C22⋊#170—28no—
48#4003—28no—
49C23×M16—51no—
50C2×#407—52no—
51C2×#341—54no—
52C24×C5C4⋊C411yescharacteristic
53C23×D5—24no—
54C23×D5—44no—
55C23×D5C4⋊C411yescharacteristic
56C23×D5C4⋊C414yes—
57C23×D5C2×Q811yescharacteristic
58C22×C4×C5—22no—
59C22×C4×C5—22no—
60C22×C4×C5C4212yes—
61C22×C4×C5C22⋊C412yes—
62C22×C4×C5C4⋊C411yescharacteristic
63C22×C4×C5C22×C411yescharacteristic
64C22×Dic5—24no—
65C22×Dic5C22⋊C414yes—
66C22×Dic5C4⋊C411yescharacteristic
67C22×Dic5C2×D411yescharacteristic
68C2×C4×D5—24no—
69C2×C4×D5—22no—
70C2×C4×D5—22no—
71C2×C4×D5—24no—
72C2×C4×D5—216no—
73C2×C4×D5—28no—
74C2×C4×D5—28no—
75C2×C4×D5C4212yes—
76C2×C4×D5C22⋊C412yes—
77C2×C4×D5C22⋊C414yes—
78C2×C4×D5C4⋊C414yes—
79C2×C4×D5C22×C412yes—
80C2×C4×D5C2×D414yes—
81C2×C4×D5C2×D412yes—
82C2×C4×D5#4114yes—
83C2×#170—28no—
84C2×#170—48no—
85#700—216no—
86#700—44no—
87#700—44no—
88C5×M16—28no—
89C2×C5×C8—22no—
90C2×C5×C8—22no—
91C2×C5×C8C22⋊C414yes—
92#681—28no—
93C2×#168—22no—
94C2×#168—22no—
95C2×#168C22⋊C414yes—
96#676—216no—
97C8×D5—216no—
98C24×C4—51no—
99C22×M16—51no—
100C22×M16—52no—
101C22×M16—54no—
102C22×M16—108no—
103C22×M16—104no—
104C22×M16AGL(1,𝔽5)11yescharacteristic
105C23×C8—52no—
106#407—58no—
107#407—104no—
108C2×C22⋊C8—102no—
109C2×C22⋊C8—102no—
110#341—108no—
111C23×C5—22no—
112C23×C5—24no—
113C23×C5#9512yes—
114C23×C5C2×C4⋊C411yescharacteristic
115C22×D5—24no—
116C22×D5—22no—
117C22×D5—24no—
118C22×D5—28no—
119C22×D5—48no—
120C22×D5—416no—
121C22×D5#9512yes—
122C22×D5C2×C4⋊C412yes—
123C22×D5C22⋊Q814yes—
124C2×C4×C5—22no—
125C2×C4×C5—24no—
126C2×C4×C5—22no—
127C2×C4×C5—22no—
128C2×C4×C5#9512yes—
129C2×C4×C5C2×C4211yescharacteristic
130C2×C4×C5C2×C22⋊C412yes—
131C2×C4×C5C2×C22⋊C411yescharacteristic
132C2×C4×C5#11712yes—
133C2×Dic5—22no—
134C2×Dic5—28no—
135C2×Dic5#9512yes—
136C2×Dic5C2×C22⋊C412yes—
137C2×Dic5#12114yes—
138C4×D5—28no—
139C4×D5—24no—
140C4×D5—22no—
141C4×D5—22no—
142C4×D5—28no—
143C4×D5C4×D418yes—
144C4×D5#12014yes—
145C4×D5#12314yes—
146#170—48no—
147C5×C8—24no—
148#168—24no—
149C25—51no—
150C23×C4—51no—
151C23×C4—51no—
152C23×C4—54no—
153C23×C4—51no—
154C23×C4—51no—
155C23×C4—54no—
156C23×C4—51no—
157C23×C4—104no—
158C23×C4—104no—
159C23×C4C2×AGL(1,𝔽5)11yescharacteristic
160C2×M16—22no—
161C2×M16—24no—
162C2×M16—58no—
163C2×M16—104no—
164C2×M16—104no—
165C2×M16—104no—
166C2×M16—1032no—
167C2×M16—102no—
168C2×M16—104no—
169C2×M16—104no—
170C2×M16—108no—
171C2×M16—108no—
172C22×C8—52no—
173C22×C8—58no—
174C22×C8—104no—
175C22×C8—104no—
176C22×C8—104no—
177C22×C8C2×AGL(1,𝔽5)12yes—
178C22⋊C8—108no—
179C22⋊C8—108no—
180C22×C5—22no—
181C22×C5—24no—
182C22×C5—44no—
183C22×C5C2×#9511yescharacteristic
184C22×C5#38012yes—
185C22×C5#43014yes—
186C2×D5—28no—
187C2×D5—24no—
188C2×D5—22no—
189C2×D5—24no—
190C2×D5—416no—
191C2×D5—44no—
192C2×D5#38514yes—
193C4×C5—22no—
194C4×C5C4×C22⋊C412yes—
195C4×C5#37911yescharacteristic
196C4×C5#38111yescharacteristic
197Dic5—22no—
198Dic5#38614yes—
199C24—51no—
200C24—51no—
201C24—54no—
202C24—104no—
203C24—204no—
204C24C4⋊AGL(1,𝔽5)11yescharacteristic
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—51no—
208C22×C4—52no—
209C22×C4—54no—
210C22×C4—54no—
211C22×C4—52no—
212C22×C4—51no—
213C22×C4—54no—
214C22×C4—52no—
215C22×C4—52no—
216C22×C4—54no—
217C22×C4—54no—
218C22×C4—108no—
219C22×C4—104no—
220C22×C4—1016no—
221C22×C4—108no—
222C22×C4—104no—
223C22×C4—104no—
224C22×C4—102no—
225C22×C4—102no—
226C22×C4C4×AGL(1,𝔽5)12yes—
227C22×C4C4⋊AGL(1,𝔽5)11yescharacteristic
228C22×C4C22⋊AGL(1,𝔽5)12yes—
229C22×C4C22×AGL(1,𝔽5)11yescharacteristic
230M16—28no—
231M16—1016no—
232M16—108no—
233M16—1016no—
234M16—204no—
235M16—204no—
236C2×C8—22no—
237C2×C8—22no—
238C2×C8—54no—
239C2×C8—1016no—
240C2×C8—102no—
241C2×C8—102no—
242C2×C8—108no—
243C2×C8—208no—
244C2×C8C22⋊AGL(1,𝔽5)14yes—
245C2×C5—42no—
246C2×C5C22⋊#9511yescharacteristic
247C2×C5#173812yes—
248C2×C5#181114yes—
249D5—24no—
250D5—42no—
251C23—24no—
252C23—22no—
253C23—52no—
254C23—54no—
255C23—52no—
256C23—108no—
257C23—104no—
258C23—104no—
259C23—102no—
260C23—208no—
261C23—2016no—
262C23(C2×AGL(1,𝔽5))⋊C412yes—
263C23C2×C4⋊AGL(1,𝔽5)11yescharacteristic
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—24no—
267C2×C4—22no—
268C2×C4—52no—
269C2×C4—54no—
270C2×C4—54no—
271C2×C4—58no—
272C2×C4—54no—
273C2×C4—52no—
274C2×C4—108no—
275C2×C4—108no—
276C2×C4—108no—
277C2×C4—102no—
278C2×C4—102no—
279C2×C4—104no—
280C2×C4—102no—
281C2×C4(C2×AGL(1,𝔽5))⋊C412yes—
282C2×C4C2×C4×AGL(1,𝔽5)11yescharacteristic
283C2×C4#413012yes—
284C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
285C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
286C8—24no—
287C8—104no—
288C8—208no—
289C5#1072411yescharacteristic, complemented, Sylow
290C22—22no—
291C22—24no—
292C22—44no—
293C22—54no—
294C22—104no—
295C22—108no—
296C22—102no—
297C22—104no—
298C22—2016no—
299C22—204no—
300C22#6572914yes—
301C22C2×(C2×AGL(1,𝔽5))⋊C411yescharacteristic
302C22#6576512yes—
303C4—22no—
304C4—54no—
305C4—102no—
306C4C4×C22⋊AGL(1,𝔽5)12yes—
307C4#6576411yescharacteristic
308C4#6580011yescharacteristic
309C2—42no—
310C2—104no—
311C2—202no—
312C2#1060476912yes—
313C2#1060477214yes—
314C2#1060490911yescharacteristic
subgroupG/H
centerC23C2×C4⋊AGL(1,𝔽5)
commutatorC2×C4×C5C2×C42
FrattiniC22×C4C22×AGL(1,𝔽5)
FittingC22×C5×M16C4
socleC23×C5C2×C4⋊C4
prime radicalC24×D5Q8
cosocleC22×C4×D5C23
hypercenterC22×M16AGL(1,𝔽5)
hypocenterC5#10724
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
3. C2×#64896C2semidirect
289. C5#10724semidirect

Maximal subgroups

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

subgroupindex
C22×D5×M162
C2×#657562
C2×#648962
#107245