Finite Group Databasebeta

← #12009913

Subgroups — #12009913

Order 1280.

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

normal subgroups only

This group has 462 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×#2487C211yesmaximal, characteristic
2C4×#3966C211yesmaximal, characteristic
3#10602396C211yesmaximal, characteristic
4#10598142C214yesmaximal, complemented
5C2×C4×C5×D4C2211yescharacteristic
6C22×C5×C4⋊C4C411yescharacteristic, complemented
7C5×#400C412yescomplemented
8C5×#399C2212yes—
9C5×#389—21no—
10C5×#386—21no—
11C5×#385—21no—
12C5×#380—21no—
13C5×#380C411yescharacteristic, complemented
14C2×#3966—21no—
15C4×C22⋊Dic5—21no—
16#65215—21no—
17#64746C2212yes—
18#64745—21no—
19#64745C2212yes—
20#11786—51nomaximal, Sylow
21C22×C5×D4D411yescharacteristic
22C23×C4×C5—21no—
23C23×C4×C5C2×C411yescharacteristic
24C23×C4×C5D411yescharacteristic
25C4×C5×D4—41no—
26C4×C5×D4—41no—
27C2×C5×C4⋊C4—21no—
28C2×C5×C4⋊C4—21no—
29C2×C5×C4⋊C4—22no—
30C2×C5×C4⋊C4—41no—
31C2×C5×C4⋊C4—41no—
32C2×C5×C4⋊C4C2×C412yes—
33C2×C5×C4⋊C4C2×C411yescharacteristic
34C2×C5×C22⋊C4—21no—
35C2×C5×C22⋊C4—22no—
36C2×C5×C22⋊C4D411yescharacteristic
37C2×C5×C22⋊C4C2311yescharacteristic
38C2×C42×C5D411yescharacteristic
39C2×C22⋊Dic5—22no—
40C2×C4×Dic5—41no—
41C5×#95—22no—
42C5×#95—22no—
43C5×#95C2×C412yes—
44#3966—24no—
45#3963—41no—
46C22⋊#168—22no—
47C4⋊#168—41no—
48#3935—41no—
49#2487D511yescharacteristic
50C4×#99—51no—
51#1466—51no—
52#1352—54no—
53C24×C5C22⋊C411yescharacteristic
54C24×C5C2×D411yescharacteristic
55C2×C5×D4—22no—
56C2×C5×D4—22no—
57C2×C5×D4—41no—
58C22×C4×C5—21no—
59C22×C4×C5—21no—
60C22×C4×C5—21no—
61C22×C4×C5—21no—
62C22×C4×C5—22no—
63C22×C4×C5—22no—
64C22×C4×C5—42no—
65C22×C4×C5—42no—
66C22×C4×C5—42no—
67C22×C4×C5—82no—
68C22×C4×C5C22⋊C411yescharacteristic
69C22×C4×C5C22⋊C411yescharacteristic
70C22×C4×C5C22⋊C411yescharacteristic
71C22×C4×C5C22×C411yescharacteristic
72C22×C4×C5C2×D411yescharacteristic
73C22×Dic5—42no—
74C5×C4⋊C4—41no—
75C5×C4⋊C4—42no—
76C5×C4⋊C4—42no—
77C5×C4⋊C4—41no—
78C5×C4⋊C4—41no—
79C5×C4⋊C4—82no—
80C5×C22⋊C4—41no—
81C5×C22⋊C4—41no—
82C5×C22⋊C4—42no—
83C5×C22⋊C4—82no—
84C42×C5—21no—
85C42×C5—21no—
86C22⋊Dic5—24no—
87C22⋊Dic5—42no—
88C4×Dic5—82no—
89C2×#168—42no—
90C2×C4×D4C2×D511yescharacteristic
91C22×C4⋊C4Dic511yescharacteristic, complemented
92#432—101no—
93C2×#99—101no—
94#400Dic512yescomplemented
95#399C2×D512yes—
96#389—21no—
97#386—21no—
98#385—21no—
99#380—21no—
100#380Dic511yescharacteristic, complemented
101C4×C22⋊C4—101no—
102#342—52no—
103#323—52no—
104#323—101no—
105C23×C5—21no—
106C23×C5—21no—
107C23×C5—21no—
108C23×C5—21no—
109C23×C5—41no—
110C23×C5—41no—
111C23×C5—41no—
112C23×C5—41no—
113C23×C5#10412yes—
114C23×C5C2×C22⋊C411yescharacteristic
115C23×C5#13712yes—
116C5×D4—44no—
117C2×C4×C5—22no—
118C2×C4×C5—21no—
119C2×C4×C5—21no—
120C2×C4×C5—21no—
121C2×C4×C5—21no—
122C2×C4×C5—21no—
123C2×C4×C5—21no—
124C2×C4×C5—42no—
125C2×C4×C5—42no—
126C2×C4×C5—42no—
127C2×C4×C5—41no—
128C2×C4×C5—41no—
129C2×C4×C5—41no—
130C2×C4×C5—42no—
131C2×C4×C5—42no—
132C2×C4×C5—42no—
133C2×C4×C5—42no—
134C2×C4×C5—42no—
135C2×C4×C5—42no—
136C2×C4×C5—81no—
137C2×C4×C5—84no—
138C2×C4×C5#9911yescharacteristic
139C2×C4×C5#9911yescharacteristic
140C2×Dic5—44no—
141C2×Dic5—82no—
142C2×Dic5—82no—
143#168—82no—
144C22×D4C22⋊D511yescharacteristic
145C23×C4—21no—
146C23×C4C2×Dic511yescharacteristic
147C23×C4C22⋊D511yescharacteristic
148C4×D4—41no—
149C4×D4—41no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4C2×Dic512yes—
156C2×C4⋊C4C2×Dic511yescharacteristic
157C2×C22⋊C4—21no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—102no—
160C2×C22⋊C4C22⋊D511yescharacteristic
161C2×C22⋊C4C22×D511yescharacteristic
162C2×C42—201no—
163C2×C42C22⋊D511yescharacteristic
164C4⋊C8—201no—
165#99—104no—
166C22⋊C8—102no—
167#97—201no—
168#95—22no—
169#95—22no—
170#95—201no—
171#95C2×Dic512yes—
172C22×C5—21no—
173C22×C5—21no—
174C22×C5—22no—
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—81no—
184C22×C5C2×#9911yescharacteristic
185C22×C5#41911yescharacteristic
186C22×C5C2×#10411yescharacteristic
187C4×C5—21no—
188C4×C5—41no—
189C4×C5—41no—
190C4×C5—41no—
191C4×C5—41no—
192C4×C5—82no—
193C4×C5—82no—
194C4×C5#35212yes—
195Dic5—84no—
196C24C22⋊Dic511yescharacteristic
197C24C2×C22⋊D511yescharacteristic
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—82no—
211C22×C4—202no—
212C22×C4C22⋊Dic511yescharacteristic
213C22×C4C22⋊Dic511yescharacteristic
214C22×C4C22⋊Dic511yescharacteristic
215C22×C4C22×Dic511yescharacteristic
216C22×C4C2×C22⋊D511yescharacteristic
217C2×C8—202no—
218C4⋊C4—41no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—82no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4—42no—
227C22⋊C4—82no—
228C22⋊C4—104no—
229C22⋊C4—202no—
230C42—21no—
231C42—21no—
232C42—402no—
233C2×C5—21no—
234C2×C5—21no—
235C2×C5—42no—
236C2×C5—42no—
237C2×C5#150811yescharacteristic
238C2×C5C2×#35211yescharacteristic
239C2×C5#212811yescharacteristic
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23#396912yes—
249C23#408812yes—
250C23C2×C22⋊Dic511yescharacteristic
251D4—44no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
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—204no—
274C2×C4—402no—
275C2×C4—402no—
276C2×C4#396611yescharacteristic
277C2×C4#396611yescharacteristic
278C8—402no—
279C5#1178611yescharacteristic, complemented, Sylow
280C22—21no—
281C22—21no—
282C22—22no—
283C22—22no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—81no—
292C22C2×#396611yescharacteristic
293C22#6551411yescharacteristic
294C22C2×#396911yescharacteristic
295C4—21no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—82no—
301C4—82no—
302C4—404no—
303C4#6475912yes—
304C2—21no—
305C2—21no—
306C2—42no—
307C2—42no—
308C2#1060248111yescharacteristic
309C2C2×#6475911yescharacteristic
310C2#1060265411yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C22⋊C4C22×D5
FittingC5×#2487C2
socleC22×C5C2×#99
prime radicalC22×C5×D4D4
cosocleC2×C5×C22⋊C4C23
hypercenter#2487D5
hypocenterC5#11786
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2487

Complements

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

NH ≅ G/Ntype
4. #10598142C2semidirect
6. C22×C5×C4⋊C4C4semidirect
7. C5×#400C4semidirect
13. C5×#380C4semidirect
91. C22×C4⋊C4Dic5semidirect
94. #400Dic5semidirect
100. #380Dic5semidirect
279. C5#11786semidirect

Maximal subgroups

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

subgroupindex
C5×#24872
C4×#39662
#106023962
#105981422
#117865