Finite Group Databasebeta

← #12009887

Subgroups — #12009887

Order 1280.

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

normal subgroups only

This group has 464 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×#2486C211yesmaximal, characteristic
2C4×#3965C211yesmaximal, characteristic
3#10602395C211yesmaximal, characteristic
4#10598141C214yesmaximal, complemented
5C2×C5×#123—21no—
6C2×C4×C5×D4C2211yescharacteristic
7C22×C5×C4⋊C4C411yescharacteristic
8C5×#398C2212yes—
9C5×#397C412yes—
10C5×#389—21no—
11C5×#381—21no—
12C5×#380—21no—
13C5×#380C411yescharacteristic
14C2×#3965—21no—
15#65496—21no—
16C4×#681—21no—
17#64750C2212yes—
18#64748C2212yes—
19#64745—21no—
20#11780—51nomaximal, Sylow
21C22×C5×D4D411yescharacteristic
22C23×C4×C5—21no—
23C23×C4×C5C2×C411yescharacteristic
24C23×C4×C5D411yescharacteristic
25C5×#123—44no—
26C4×C5×D4—41no—
27C4×C5×D4—41no—
28C2×C5×C4⋊C4—21no—
29C2×C5×C4⋊C4—21no—
30C2×C5×C4⋊C4—22no—
31C2×C5×C4⋊C4—41no—
32C2×C5×C4⋊C4—41no—
33C2×C5×C4⋊C4C2×C412yes—
34C2×C5×C4⋊C4C2311yescharacteristic
35C2×C5×C22⋊C4—22no—
36C2×C5×C22⋊C4—21no—
37C2×C5×C22⋊C4C2×C411yescharacteristic
38C2×C5×C22⋊C4D411yescharacteristic
39C2×C42×C5D411yescharacteristic
40C2×#681—22no—
41C2×#681—41no—
42C5×#95—22no—
43C5×#95—22no—
44C5×#95C2×C412yes—
45#3965—24no—
46C22⋊#168—22no—
47C22⋊#168—41no—
48#3935—41no—
49C4×#168—41no—
50#2486D511yescharacteristic
51C4×#100—51no—
52#1465—51no—
53#1346—54no—
54C24×C5C22⋊C411yescharacteristic
55C24×C5C22⋊C411yescharacteristic
56C2×C5×D4—22no—
57C2×C5×D4—22no—
58C2×C5×D4—41no—
59C22×C4×C5—21no—
60C22×C4×C5—21no—
61C22×C4×C5—21no—
62C22×C4×C5—21no—
63C22×C4×C5—22no—
64C22×C4×C5—22no—
65C22×C4×C5—42no—
66C22×C4×C5—42no—
67C22×C4×C5—42no—
68C22×C4×C5—42no—
69C22×C4×C5—42no—
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—42no—
77C5×C4⋊C4—42no—
78C5×C4⋊C4—41no—
79C5×C4⋊C4—41no—
80C5×C4⋊C4—44no—
81C5×C22⋊C4—41no—
82C5×C22⋊C4—41no—
83C5×C22⋊C4—42no—
84C5×C22⋊C4—44no—
85C42×C5—21no—
86C42×C5—21no—
87#681—24no—
88#681—42no—
89#681—82no—
90C2×#168—42no—
91C2×#168—42no—
92C2×#123—21no—
93C2×C4×D4C2×D511yescharacteristic
94C22×C4⋊C4Dic511yescharacteristic
95C2×#100—101no—
96#407—101no—
97C4×M16—101no—
98#398C2×D512yes—
99#397Dic512yes—
100#389—21no—
101#381—21no—
102#380—21no—
103#380Dic511yescharacteristic
104#343—52no—
105#324—52no—
106#323—101no—
107C23×C5—21no—
108C23×C5—21no—
109C23×C5—21no—
110C23×C5—21no—
111C23×C5—21no—
112C23×C5—41no—
113C23×C5—41no—
114C23×C5—41no—
115C23×C5—41no—
116C23×C5#10412yes—
117C23×C5C2×C22⋊C411yescharacteristic
118C5×D4—44no—
119C2×C4×C5—21no—
120C2×C4×C5—21no—
121C2×C4×C5—22no—
122C2×C4×C5—21no—
123C2×C4×C5—21no—
124C2×C4×C5—21no—
125C2×C4×C5—42no—
126C2×C4×C5—42no—
127C2×C4×C5—42no—
128C2×C4×C5—41no—
129C2×C4×C5—41no—
130C2×C4×C5—41no—
131C2×C4×C5—42no—
132C2×C4×C5—42no—
133C2×C4×C5—42no—
134C2×C4×C5—42no—
135C2×C4×C5—42no—
136C2×C4×C5—42no—
137C2×C4×C5—81no—
138C2×C4×C5—84no—
139C2×C4×C5#9911yescharacteristic
140C2×C4×C5#9911yescharacteristic
141C2×C4×C5AGL(1,ℤ/8)12yes—
142#168—44no—
143#168—82no—
144C22×D4C22⋊D511yescharacteristic
145C23×C4—21no—
146C23×C4C2×Dic511yescharacteristic
147C23×C4C22⋊D511yescharacteristic
148C2×M16—102no—
149C2×M16—201no—
150#123—44no—
151C4×D4—41no—
152C4×D4—41no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4C2×Dic512yes—
159C2×C4⋊C4C22×D511yescharacteristic
160C2×C22⋊C4—22no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4C2×Dic511yescharacteristic
163C2×C22⋊C4C22⋊D511yescharacteristic
164C2×C42C22⋊D511yescharacteristic
165#100—104no—
166C22⋊C8—102no—
167C22⋊C8—201no—
168#97—201no—
169C4×C8—201no—
170#95—22no—
171#95—22no—
172#95C2×Dic512yes—
173C22×C5—21no—
174C22×C5—21no—
175C22×C5—22no—
176C22×C5—42no—
177C22×C5—42no—
178C22×C5—42no—
179C22×C5—41no—
180C22×C5—41no—
181C22×C5—41no—
182C22×C5—41no—
183C22×C5—41no—
184C22×C5—81no—
185C22×C5C2×#9911yescharacteristic
186C22×C5#41811yescharacteristic
187C22×C5C2×#10411yescharacteristic
188C4×C5—21no—
189C4×C5—22no—
190C4×C5—41no—
191C4×C5—41no—
192C4×C5—41no—
193C4×C5—82no—
194C4×C5—82no—
195C4×C5#35212yes—
196C24C22⋊Dic511yescharacteristic
197C24C22⋊Dic511yescharacteristic
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—41no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4C22⋊Dic511yescharacteristic
213C22×C4C22⋊Dic511yescharacteristic
214C22×C4C22×Dic511yescharacteristic
215C22×C4C2×C22⋊D511yescharacteristic
216C22×C4C2×C22⋊D511yescharacteristic
217M16—104no—
218M16—202no—
219M16—402no—
220C2×C8—202no—
221C2×C8—202no—
222C4⋊C4—41no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—44no—
228C22⋊C4—41no—
229C22⋊C4—41no—
230C22⋊C4—42no—
231C22⋊C4—44no—
232C42—21no—
233C42—21no—
234C2×C5—21no—
235C2×C5—21no—
236C2×C5—42no—
237C2×C5—81no—
238C2×C5#150711yescharacteristic
239C2×C5C2×#35211yescharacteristic
240C2×C5#213511yescharacteristic
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23#396912yes—
251C23C2×C22⋊Dic511yescharacteristic
252D4—44no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—81no—
272C2×C4—84no—
273C2×C4#396611yescharacteristic
274C2×C4#396611yescharacteristic
275C2×C4#407812yes—
276C8—204no—
277C8—402no—
278C5#1178011yescharacteristic, complemented, Sylow
279C22—21no—
280C22—21no—
281C22—22no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—81no—
291C22#6550411yescharacteristic
292C22C2×#396611yescharacteristic
293C22C2×#396911yescharacteristic
294C4—21no—
295C4—22no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—82no—
300C4—82no—
301C4#6475912yes—
302C2—21no—
303C2—21no—
304C2—42no—
305C2—81no—
306C2#1060246611yescharacteristic
307C2C2×#6475911yescharacteristic
308C2#1060266611yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C4⋊C4C22×D5
FittingC5×#2486C2
socleC22×C5C2×#99
prime radicalC22×C5×D4D4
cosocleC2×C5×C4⋊C4C23
hypercenter#2486D5
hypocenterC5#11780
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2486

Complements

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

NH ≅ G/Ntype
4. #10598141C2semidirect
278. C5#11780semidirect

Maximal subgroups

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

subgroupindex
C5×#24862
C4×#39652
#106023952
#105981412
#117805