Finite Group Databasebeta

← #12009891

Subgroups — #12009891

Order 1280.

482 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C5×#2548C211yesmaximal, characteristic
2C4×#3968C211yesmaximal, characteristic, complemented
3#10602418C211yesmaximal, characteristic, complemented
4#10598138C214yesmaximal, complemented
5C2×C5×#127C411yescharacteristic
6C2×C5×#124—21no—
7C2×C4×C5×Q8C2211yescharacteristic
8C5×#394C412yes—
9C5×#394C2212yes—
10C5×#390—21no—
11C5×#390—21no—
12C5×#390—21no—
13C4×C5×C4⋊C4C411yescharacteristic
14C2×#3968—21no—
15#65215—21no—
16C4×#681—21no—
17#64750C2212yes—
18#64748—21no—
19#64748C2212yes—
20#11781—51nomaximal, Sylow
21C22×C5×Q8D411yescharacteristic
22C22×C5×D4—21no—
23C22×C5×D4—21no—
24C22×C5×D4C2×C412yes—
25C5×#127—41no—
26C5×#127—41no—
27C5×#124—44no—
28C4×C5×Q8—41no—
29C4×C5×Q8—41no—
30C2×C5×C4⋊C4—21no—
31C2×C5×C4⋊C4C2×C411yescharacteristic
32C2×C5×C4⋊C4D411yescharacteristic
33C2×C5×C4⋊C4C2311yescharacteristic
34C2×C5×C22⋊C4—22no—
35C2×C5×C22⋊C4—22no—
36C2×C5×C22⋊C4—22no—
37C2×C5×C22⋊C4—22no—
38C2×C42×C5—21no—
39C2×C42×C5C2×C411yescharacteristic
40C2×C42×C5D411yescharacteristic
41C2×C42×C5D411yescharacteristic
42C2×#681—22no—
43C5×#95C2×C412yes—
44#3968—24no—
45C22⋊#168—22no—
46C4⋊#168—41no—
47#3935—41no—
48#3935—41no—
49C4×#168—41no—
50#2548D511yescharacteristic
51C4×#101—51no—
52#1467—51no—
53#1345—54no—
54C24×C5—22no—
55C2×C5×Q8—22no—
56C2×C5×Q8—22no—
57C2×C5×Q8—41no—
58C2×C5×D4—42no—
59C2×C5×D4—42no—
60C2×C5×D4—42no—
61C2×C5×D4—42no—
62C2×C5×D4—42no—
63C2×C5×D4—41no—
64C2×C5×D4—41no—
65C2×C5×D4—42no—
66C2×C5×D4—82no—
67C22×C4×C5—22no—
68C22×C4×C5C22⋊C411yescharacteristic
69C22×C4×C5C22⋊C411yescharacteristic
70C22×C4×C5C22⋊C411yescharacteristic
71C22×C4×C5C22⋊C411yescharacteristic
72C22×C4×C5C22×C411yescharacteristic
73C22×C4×C5C2×D411yescharacteristic
74C22×C4×C5C2×D411yescharacteristic
75C5×C4⋊C4—41no—
76C5×C4⋊C4—41no—
77C5×C4⋊C4—41no—
78C5×C4⋊C4—42no—
79C5×C22⋊C4—44no—
80C5×C22⋊C4—44no—
81C5×C22⋊C4—82no—
82C5×C22⋊C4—82no—
83C42×C5—21no—
84C42×C5—21no—
85C42×C5—21no—
86C42×C5—21no—
87C42×C5—21no—
88C42×C5—21no—
89C42×C5—42no—
90#681—24no—
91#681—42no—
92C2×#168—42no—
93C2×#168—42no—
94C2×#127Dic511yescharacteristic
95C2×#124—21no—
96C2×C4×Q8C2×D511yescharacteristic
97#432—101no—
98C2×#101—101no—
99C4×M16—101no—
100#394Dic512yes—
101#394C2×D512yes—
102#390—21no—
103#390—21no—
104#390—21no—
105C4×C4⋊C4Dic511yescharacteristic
106#343—52no—
107#324—52no—
108#324—101no—
109C23×C5—42no—
110C23×C5—42no—
111C23×C5—42no—
112C23×C5—42no—
113C23×C5—42no—
114C23×C5—42no—
115C23×C5—42no—
116C23×C5C2×C22⋊C411yescharacteristic
117C5×Q8—44no—
118C5×D4—82no—
119C5×D4—84no—
120C5×D4—84no—
121C5×D4—82no—
122C2×C4×C5—21no—
123C2×C4×C5—21no—
124C2×C4×C5—21no—
125C2×C4×C5—21no—
126C2×C4×C5—22no—
127C2×C4×C5—21no—
128C2×C4×C5—21no—
129C2×C4×C5—21no—
130C2×C4×C5—21no—
131C2×C4×C5—21no—
132C2×C4×C5—21no—
133C2×C4×C5—41no—
134C2×C4×C5—41no—
135C2×C4×C5—41no—
136C2×C4×C5—42no—
137C2×C4×C5—42no—
138C2×C4×C5—42no—
139C2×C4×C5#9911yescharacteristic
140C2×C4×C5#9911yescharacteristic
141C2×C4×C5#10412yes—
142C2×C4×C5AGL(1,ℤ/8)12yes—
143#168—44no—
144#168—82no—
145C22×Q8C22⋊D511yescharacteristic
146C22×D4—21no—
147C22×D4—21no—
148C22×D4C2×Dic512yes—
149C2×M16—102no—
150#127—41no—
151#127—41no—
152#124—44no—
153C4×Q8—41no—
154C4×Q8—41no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4C2×Dic511yescharacteristic
157C2×C4⋊C4C22⋊D511yescharacteristic
158C2×C4⋊C4C22×D511yescharacteristic
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C42—21no—
164C2×C42C2×Dic511yescharacteristic
165C2×C42C22⋊D511yescharacteristic
166C2×C42C22⋊D511yescharacteristic
167C4⋊C8—201no—
168#101—104no—
169C22⋊C8—102no—
170#97—201no—
171#97—201no—
172C4×C8—201no—
173#95C2×Dic512yes—
174C22×C5—21no—
175C22×C5—21no—
176C22×C5—82no—
177C22×C5—82no—
178C22×C5—82no—
179C22×C5—82no—
180C22×C5—82no—
181C22×C5—82no—
182C22×C5—82no—
183C22×C5C2×#9911yescharacteristic
184C22×C5#41811yescharacteristic
185C22×C5C2×#10411yescharacteristic
186C4×C5—21no—
187C4×C5—22no—
188C4×C5—22no—
189C4×C5—41no—
190C4×C5—41no—
191C4×C5—41no—
192C4×C5—41no—
193C4×C5—82no—
194C4×C5#35512yes—
195C24—22no—
196C2×Q8—22no—
197C2×Q8—22no—
198C2×Q8—41no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—42no—
207C2×D4—82no—
208C22×C4—22no—
209C22×C4C22⋊Dic511yescharacteristic
210C22×C4C22⋊Dic511yescharacteristic
211C22×C4C22⋊Dic511yescharacteristic
212C22×C4C22⋊Dic511yescharacteristic
213C22×C4C22×Dic511yescharacteristic
214C22×C4C2×C22⋊D511yescharacteristic
215C22×C4C2×C22⋊D511yescharacteristic
216M16—104no—
217M16—202no—
218C2×C8—202no—
219C2×C8—202no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—42no—
224C22⋊C4—44no—
225C22⋊C4—44no—
226C22⋊C4—82no—
227C22⋊C4—82no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—42no—
235C2×C5—21no—
236C2×C5—21no—
237C2×C5—162no—
238C2×C5#150711yescharacteristic
239C2×C5#212211yescharacteristic
240C2×C5C2×#35511yescharacteristic
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23C2×C22⋊Dic511yescharacteristic
249Q8—44no—
250D4—82no—
251D4—84no—
252D4—84no—
253D4—82no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4#396611yescharacteristic
272C2×C4#396611yescharacteristic
273C2×C4#396912yes—
274C2×C4#407812yes—
275C8—204no—
276C8—402no—
277C5#1178111yescharacteristic, complemented, Sylow
278C22—21no—
279C22—21no—
280C22—82no—
281C22—82no—
282C22—82no—
283C22—82no—
284C22—82no—
285C22—82no—
286C22—82no—
287C22#6550411yescharacteristic
288C22C2×#396611yescharacteristic
289C22C2×#396911yescharacteristic
290C4—21no—
291C4—22no—
292C4—22no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—82no—
298C4#6476212yes—
299C2—21no—
300C2—21no—
301C2—162no—
302C2#1060246611yescharacteristic
303C2C2×#6476211yescharacteristic
304C2#1060265211yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C4⋊C4C22×D5
FittingC5×#2548C2
socleC22×C5C2×#99
prime radicalC2×C5×#127C4
cosocleC2×C5×C4⋊C4C23
hypercenter#2548D5
hypocenterC5#11781
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2548

Complements

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

NH ≅ G/Ntype
2. C4×#3968C2semidirect
3. #10602418C2semidirect
4. #10598138C2semidirect
277. C5#11781semidirect

Maximal subgroups

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

subgroupindex
C5×#25482
C4×#39682
#106024182
#105981382
#117815