Finite Group Databasebeta

← #11997045

Subgroups — #11997045

Order 1280.

610 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 107 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×#2501C211yesmaximal, characteristic
2#10613886C211yesmaximal, characteristic, complemented
3#10602467C212yesmaximal
4#10602355C211yesmaximal, characteristic, complemented
5#10598107C212yesmaximal, complemented
6C2×C5×#123—21no—
7C2×C5×#123—21no—
8C2×C5×#121C2211yescharacteristic
9C22×C5×C4⋊C4C2211yescharacteristic
10C5×#399—22no—
11C5×#390C412yescomplemented
12C5×#384C412yescomplemented
13C4×C5×C22⋊C4C2211yescharacteristic
14#65501—21no—
15#65500C2212yes—
16C4×C22⋊Dic5—21no—
17#65496C2212yes—
18#65270—24no—
19#64757—24no—
20#64750—22no—
21#9464—51nomaximal, Sylow
22C22×C5×D4—21no—
23C22×C5×D4C2×C411yescharacteristic
24C23×C4×C5D411yescharacteristic
25C23×C4×C5D411yescharacteristic
26C23×C4×C5C2311yescharacteristic
27C5×#123—44no—
28C5×#123—44no—
29C5×#121—22no—
30C5×#121—22no—
31C2×C5×C4⋊C4—22no—
32C2×C5×C4⋊C4—22no—
33C2×C5×C4⋊C4—22no—
34C2×C5×C4⋊C4—22no—
35C2×C5×C4⋊C4C2×C411yescharacteristic
36C2×C5×C4⋊C4C2×C412yes—
37C2×C5×C4⋊C4D412yes—
38C2×C5×C22⋊C4—21no—
39C2×C5×C22⋊C4—22no—
40C2×C5×C22⋊C4—22no—
41C2×C5×C22⋊C4—21no—
42C2×C42×C5C2×C412yes—
43C2×C22⋊Dic5—22no—
44C2×C4⋊Dic5—22no—
45C2×C4×Dic5—22no—
46C2×#681—22no—
47C5×#95—21no—
48#3963—22no—
49#3963—22no—
50#3963—22no—
51#3963D414yes—
52C22⋊#168—22no—
53C22⋊#168—22no—
54#2501D511yescharacteristic
55#2313—51no—
56#1821—51no—
57#1510—52no—
58#1284—52no—
59C24×C5—21no—
60C24×C5C2×D411yescharacteristic
61C2×C5×D4—22no—
62C2×C5×D4—42no—
63C2×C5×D4—42no—
64C2×C5×D4—42no—
65C2×C5×D4—41no—
66C2×C5×D4—41no—
67C22×C4×C5—22no—
68C22×C4×C5—21no—
69C22×C4×C5—21no—
70C22×C4×C5—21no—
71C22×C4×C5—21no—
72C22×C4×C5—21no—
73C22×C4×C5—42no—
74C22×C4×C5—42no—
75C22×C4×C5—42no—
76C22×C4×C5C22⋊C412yes—
77C22×C4×C5C22⋊C412yes—
78C22×C4×C5C22×C411yescharacteristic
79C22×C4×C5C2×D412yes—
80C22×C4×C5C2×D411yescharacteristic
81C22×C4×C5#4112yes—
82C22×Dic5—24no—
83C5×C4⋊C4—24no—
84C5×C4⋊C4—24no—
85C5×C4⋊C4—22no—
86C5×C4⋊C4—44no—
87C5×C4⋊C4—44no—
88C5×C22⋊C4—24no—
89C5×C22⋊C4—44no—
90C5×C22⋊C4—44no—
91C5×C22⋊C4—42no—
92C42×C5—42no—
93C22⋊Dic5—44no—
94C4⋊Dic5—82no—
95C4×Dic5—82no—
96#681—44no—
97C2×#168—42no—
98C2×#123—21no—
99C2×#123—21no—
100C2×#121C2×D511yescharacteristic
101C22×C4⋊C4C2×D511yescharacteristic
102#407—52no—
103#399—22no—
104#390Dic512yescomplemented
105#387—104no—
106#384Dic512yescomplemented
107#381—52no—
108#380—101no—
109C4×C22⋊C4—101no—
110C4×C22⋊C4C2×D511yescharacteristic
111#343—102no—
112#327—104no—
113C23×C5—22no—
114C23×C5—22no—
115C23×C5—41no—
116C23×C5—41no—
117C23×C5—41no—
118C23×C5—42no—
119C23×C5—41no—
120C23×C5—42no—
121C23×C5C2×C22⋊C411yescharacteristic
122C23×C5#12011yescharacteristic
123C23×C5#12311yescharacteristic
124C5×D4—82no—
125C5×D4—82no—
126C5×D4—82no—
127C2×C4×C5—22no—
128C2×C4×C5—22no—
129C2×C4×C5—24no—
130C2×C4×C5—22no—
131C2×C4×C5—22no—
132C2×C4×C5—42no—
133C2×C4×C5—42no—
134C2×C4×C5—42no—
135C2×C4×C5—42no—
136C2×C4×C5—42no—
137C2×C4×C5—42no—
138C2×C4×C5—42no—
139C2×C4×C5—42no—
140C2×C4×C5—41no—
141C2×C4×C5—41no—
142C2×C4×C5—41no—
143C2×C4×C5—81no—
144C2×C4×C5—81no—
145C2×C4×C5C4×D412yes—
146C2×C4×C5#12112yes—
147C2×Dic5—44no—
148C2×Dic5—44no—
149C2×Dic5—44no—
150#168—82no—
151C22×D4—21no—
152C22×D4C2×Dic511yescharacteristic
153C23×C4C22⋊D511yescharacteristic
154C23×C4C22⋊D511yescharacteristic
155C23×C4C22×D511yescharacteristic
156C2×M16—102no—
157#123—44no—
158#123—44no—
159#121—22no—
160#121—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—102no—
166C2×C4⋊C4C2×Dic511yescharacteristic
167C2×C4⋊C4C2×Dic512yes—
168C2×C4⋊C4C22⋊D512yes—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—102no—
174C2×C42—102no—
175C2×C42C2×Dic512yes—
176C22⋊C8—102no—
177C22⋊C8—102no—
178#95—21no—
179#95—54no—
180#95—102no—
181#95—102no—
182#95—102no—
183C22×C5—21no—
184C22×C5—21no—
185C22×C5—21no—
186C22×C5—21no—
187C22×C5—21no—
188C22×C5—21no—
189C22×C5—21no—
190C22×C5—44no—
191C22×C5—82no—
192C22×C5—82no—
193C22×C5—81no—
194C22×C5—81no—
195C22×C5—81no—
196C22×C5#38611yescharacteristic
197C22×C5#41012yes—
198C22×C5#45312yes—
199C4×C5—22no—
200C4×C5—22no—
201C4×C5—44no—
202C4×C5—42no—
203C4×C5—42no—
204C4×C5—81no—
205Dic5—84no—
206C24—21no—
207C24C2×C22⋊D511yescharacteristic
208C2×D4—22no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—41no—
213C2×D4—41no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—104no—
224C22×C4C22⋊Dic512yes—
225C22×C4C22⋊Dic512yes—
226C22×C4D4×D512yes—
227C22×C4#71112yes—
228C22×C4C22×Dic511yescharacteristic
229C22×C4C2×C22⋊D511yescharacteristic
230M16—204no—
231C2×C8—202no—
232C4⋊C4—24no—
233C4⋊C4—24no—
234C4⋊C4—22no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—402no—
238C22⋊C4—24no—
239C22⋊C4—44no—
240C22⋊C4—44no—
241C22⋊C4—42no—
242C22⋊C4—204no—
243C42—42no—
244C42—402no—
245C2×C5—22no—
246C2×C5—22no—
247C2×C5—41no—
248C2×C5—161no—
249C2×C5#188711yescharacteristic
250C2×C5#189911yescharacteristic
251C2×C5#191211yescharacteristic
252C23—22no—
253C23—22no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—42no—
258C23—41no—
259C23—42no—
260C23#408111yescharacteristic
261C23#408311yescharacteristic
262C23C2×C22⋊Dic511yescharacteristic
263D4—82no—
264D4—82no—
265D4—82no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—24no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—204no—
285C2×C4—204no—
286C2×C4—204no—
287C2×C4D4×Dic512yes—
288C2×C4#408512yes—
289C8—402no—
290C5#946411yescharacteristic, complemented, Sylow
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—44no—
299C22—82no—
300C22—82no—
301C22—81no—
302C22—81no—
303C22—81no—
304C22#6536612yes—
305C22#6550911yescharacteristic
306C22#6552912yes—
307C4—22no—
308C4—22no—
309C4—44no—
310C4—42no—
311C4—42no—
312C4—81no—
313C4—404no—
314C2—22no—
315C2—22no—
316C2—41no—
317C2—161no—
318C2#1060249811yescharacteristic
319C2#1060255611yescharacteristic
320C2#1060265711yescharacteristic
subgroupG/H
centerC22#65509
commutatorC22×C4×C5C22×C4
FrattiniC23×C4C22×D5
FittingC5×#2501C2
socleC22×C5#386
prime radicalC2×C5×#121C22
cosocleC23×C4×C5C23
hypercenter#2501D5
hypocenterC5#9464
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2501

Complements

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

NH ≅ G/Ntype
2. #10613886C2semidirect
4. #10602355C2semidirect
5. #10598107C2semidirect
11. C5×#390C4semidirect
12. C5×#384C4semidirect
104. #390Dic5semidirect
106. #384Dic5semidirect
290. C5#9464semidirect

Maximal subgroups

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

subgroupindex
C5×#25012
#106138862
#106024672
#106023552
#105981072
#94645