Finite Group Databasebeta

← #11997129

Subgroups — #11997129

Order 1280.

614 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 103 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×#2312C211yesmaximal, characteristic
2#10613844C211yesmaximal, characteristic
3#10602332C211yesmaximal, characteristic
4#10600235C212yesmaximal, complemented
5#10598107C212yesmaximal, complemented
6C22×C5×C22⋊C4C2211yescharacteristic
7C2×#4072C2211yescharacteristic
8C5×#388—24no—
9C5×#380—21no—
10C5×#379—21no—
11C4×C5×C22⋊C4C2212yes—
12#65536—21no—
13#65500C2211yescharacteristic
14#65499—21no—
15#65496C2212yes—
16#65242C412yescomplemented
17#65238C412yescomplemented
18#65233—22no—
19#64745—22no—
20#64743—24no—
21#9481—51nomaximal, Sylow
22C25×C5D411yescharacteristic
23C23×C4×C5D411yescharacteristic
24C23×C4×C5C2311yescharacteristic
25C2×C5×C4⋊C4—22no—
26C2×C5×C22⋊C4—22no—
27C2×C5×C22⋊C4—22no—
28C2×C5×C22⋊C4—22no—
29C2×C5×C22⋊C4—22no—
30C2×C5×C22⋊C4—22no—
31C2×C5×C22⋊C4C2×C412yes—
32C2×C5×C22⋊C4D412yes—
33C2×C42×C5D414yes—
34C2×C22⋊Dic5—22no—
35C2×C22⋊Dic5—21no—
36C2×C22⋊Dic5—21no—
37C2×C22⋊Dic5—22no—
38#4072—22no—
39#4072—22no—
40C2×C4⋊Dic5C2×C411yescharacteristic
41C2×C4⋊Dic5C2×C411yescharacteristic
42C2×#683—22no—
43C2×C4×Dic5—21no—
44C2×#681—22no—
45C5×#95—22no—
46C5×#95—22no—
47#3963—22no—
48#3963—21no—
49#3963C2×C412yes—
50C22⋊#168—22no—
51C22⋊#168—22no—
52#2482—51no—
53#2312D511yescharacteristic
54#1826—52no—
55#1462—51no—
56#1284—52no—
57C24×C5—21no—
58C24×C5—21no—
59C24×C5—42no—
60C24×C5—42no—
61C24×C5—81no—
62C24×C5C22⋊C412yes—
63C24×C5C2×D411yescharacteristic
64C22×C4×C5—21no—
65C22×C4×C5—21no—
66C22×C4×C5—24no—
67C22×C4×C5—42no—
68C22×C4×C5—81no—
69C22×C4×C5C22⋊C412yes—
70C22×C4×C5C22×C411yescharacteristic
71C22×C4×C5C2×D411yescharacteristic
72C22×C4×C5C2×D412yes—
73C22×C4×C5#4112yes—
74C22×Dic5—22no—
75C22×Dic5—21no—
76C22×Dic5—21no—
77C5×C4⋊C4—82no—
78C5×C22⋊C4—24no—
79C5×C22⋊C4—24no—
80C5×C22⋊C4—44no—
81C5×C22⋊C4—82no—
82C5×C22⋊C4—82no—
83C42×C5—44no—
84C22⋊Dic5—42no—
85C22⋊Dic5—44no—
86C22⋊Dic5—41no—
87C22⋊Dic5—41no—
88C22⋊Dic5—82no—
89C4⋊Dic5—22no—
90C4⋊Dic5—22no—
91#683—82no—
92C4×Dic5—42no—
93#681—44no—
94C2×#168—42no—
95C2×#117—51no—
96C22×C22⋊C4C2×D511yescharacteristic
97#407—52no—
98#399—52no—
99#397—52no—
100#388—24no—
101#388—102no—
102#385—101no—
103#381—51no—
104#380—21no—
105#379—21no—
106#379—101no—
107C4×C22⋊C4C2×D512yes—
108#339—104no—
109#323—102no—
110C23×C5—22no—
111C23×C5—22no—
112C23×C5—22no—
113C23×C5—22no—
114C23×C5—22no—
115C23×C5—41no—
116C23×C5—42no—
117C23×C5—42no—
118C23×C5—42no—
119C23×C5—42no—
120C23×C5—42no—
121C23×C5—42no—
122C23×C5—81no—
123C23×C5—81no—
124C23×C5—88no—
125C23×C5C2×C22⋊C411yescharacteristic
126C23×C5#12011yescharacteristic
127C23×C5#12311yescharacteristic
128C2×C4×C5—24no—
129C2×C4×C5—22no—
130C2×C4×C5—41no—
131C2×C4×C5—41no—
132C2×C4×C5—42no—
133C2×C4×C5—42no—
134C2×C4×C5—42no—
135C2×C4×C5—44no—
136C2×C4×C5—44no—
137C2×C4×C5—44no—
138C2×C4×C5—81no—
139C2×C4×C5C4×D412yes—
140C2×C4×C5#12112yes—
141C2×Dic5—42no—
142C2×Dic5—42no—
143C2×Dic5—42no—
144C2×Dic5—41no—
145C2×Dic5—41no—
146C2×Dic5—42no—
147C2×Dic5—42no—
148#168—82no—
149C25C22⋊D511yescharacteristic
150C23×C4C22⋊D511yescharacteristic
151C23×C4C22×D511yescharacteristic
152C2×M16—102no—
153#117—102no—
154#117—102no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—51no—
157C2×C4⋊C4—51no—
158C2×C4⋊C4—102no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—102no—
165C2×C22⋊C4—101no—
166C2×C22⋊C4—101no—
167C2×C22⋊C4—102no—
168C2×C22⋊C4C4×D512yes—
169C2×C22⋊C4D2012yes—
170C2×C42—101no—
171C2×C42C22⋊D514yes—
172C22⋊C8—102no—
173C22⋊C8—102no—
174#95—22no—
175#95—22no—
176#95—52no—
177#95—102no—
178#95—101no—
179C22×C5—21no—
180C22×C5—21no—
181C22×C5—21no—
182C22×C5—21no—
183C22×C5—21no—
184C22×C5—21no—
185C22×C5—21no—
186C22×C5—44no—
187C22×C5—44no—
188C22×C5—42no—
189C22×C5—42no—
190C22×C5—42no—
191C22×C5—82no—
192C22×C5—84no—
193C22×C5—84no—
194C22×C5#38611yescharacteristic
195C22×C5#42112yes—
196C22×C5#45812yes—
197C4×C5—22no—
198C4×C5—22no—
199C4×C5—42no—
200C4×C5—84no—
201Dic5—82no—
202Dic5—81no—
203Dic5—81no—
204C24—21no—
205C24—21no—
206C24—42no—
207C24—42no—
208C24—81no—
209C24#68512yes—
210C24C2×C22⋊D511yescharacteristic
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—24no—
214C22×C4—42no—
215C22×C4—81no—
216C22×C4—102no—
217C22×C4—101no—
218C22×C4—101no—
219C22×C4#68512yes—
220C22×C4C2×C4×D511yescharacteristic
221C22×C4C2×D2011yescharacteristic
222C22×C4#70912yes—
223C22×C4C2×C22⋊D512yes—
224M16—204no—
225C2×C8—202no—
226C4⋊C4—82no—
227C4⋊C4—102no—
228C4⋊C4—102no—
229C4⋊C4—402no—
230C22⋊C4—24no—
231C22⋊C4—24no—
232C22⋊C4—44no—
233C22⋊C4—82no—
234C22⋊C4—82no—
235C22⋊C4—202no—
236C22⋊C4—204no—
237C22⋊C4—201no—
238C22⋊C4—201no—
239C22⋊C4—402no—
240C42—44no—
241C42—202no—
242C2×C5—22no—
243C2×C5—22no—
244C2×C5—41no—
245C2×C5—82no—
246C2×C5#189011yescharacteristic
247C2×C5#189711yescharacteristic
248C2×C5#191111yescharacteristic
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—41no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—81no—
262C23—81no—
263C23—88no—
264C23C2×#68511yescharacteristic
265C23#407511yescharacteristic
266C23#409911yescharacteristic
267C2×C4—24no—
268C2×C4—22no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—81no—
278C2×C4—202no—
279C2×C4—202no—
280C2×C4—202no—
281C2×C4—202no—
282C2×C4—202no—
283C2×C4—201no—
284C2×C4—201no—
285C2×C4C4×C22⋊D512yes—
286C2×C4#407612yes—
287C8—402no—
288C5#948111yescharacteristic, complemented, Sylow
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—44no—
297C22—44no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—82no—
302C22—84no—
303C22—84no—
304C22#6529412yes—
305C22#6550211yescharacteristic
306C22#6553212yes—
307C4—22no—
308C4—22no—
309C4—42no—
310C4—84no—
311C4—402no—
312C4—401no—
313C4—401no—
314C2—22no—
315C2—22no—
316C2—41no—
317C2—82no—
318C2#1060238411yescharacteristic
319C2#1060244511yescharacteristic
320C2#1060244811yescharacteristic
subgroupG/H
centerC22#65502
commutatorC22×C4×C5C22×C4
FrattiniC23×C4C22×D5
FittingC5×#2312C2
socleC22×C5#386
prime radicalC25×C5D4
cosocleC23×C4×C5C23
hypercenter#2312D5
hypocenterC5#9481
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
4. #10600235C2semidirect
5. #10598107C2semidirect
16. #65242C4semidirect
17. #65238C4semidirect
288. C5#9481semidirect

Maximal subgroups

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

subgroupindex
C5×#23122
#106138442
#106023322
#106002352
#105981072
#94815