Finite Group Databasebeta

← #11997035

Subgroups — #11997035

Order 1280.

700 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 115 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×#2434C211yesmaximal, characteristic
2#10615793C211yesmaximal, characteristic
3#10602926C211yesmaximal, characteristic
4#10602482C212yesmaximal
5#10598107C212yesmaximal
6C23×C5×Q8C2211yescharacteristic
7C2×C5×C4⋊Q8C412yescomplemented
8C2×C5×C22⋊Q8C2211yescharacteristic
9C2×#4072—21no—
10C5×#395—22no—
11C5×#391C412yescomplemented
12C5×#385—21no—
13C5×#385—21no—
14C4×C5×C22⋊C4C2211yescharacteristic
15#65519—24no—
16C2×#3968—22no—
17#65500—21no—
18#65500C2212yes—
19#65496C2212yes—
20#64758—24no—
21#9462—51nomaximal, Sylow
22C22×C5×Q8—22no—
23C22×C5×Q8—22no—
24C22×C5×Q8—44no—
25C22×C5×Q8C2×C411yescharacteristic
26C22×C5×Q8C2×C412yes—
27C22×C5×Q8D412yes—
28C23×C4×C5D411yescharacteristic
29C23×C4×C5D411yescharacteristic
30C23×C4×C5C2311yescharacteristic
31C5×C4⋊Q8—44no—
32C5×C22⋊Q8—22no—
33C5×C22⋊Q8—22no—
34C2×C5×C4⋊C4—22no—
35C2×C5×C4⋊C4—22no—
36C2×C5×C4⋊C4—21no—
37C2×C5×C4⋊C4C2×C411yescharacteristic
38C2×C5×C22⋊C4—21no—
39C2×C5×C22⋊C4—21no—
40C2×C42×C5C2×C412yes—
41C2×C22⋊Dic5—22no—
42#4072—44no—
43C2×C4⋊Dic5—22no—
44C2×C4×Dic5—22no—
45C2×#681—22no—
46C5×#95—22no—
47C5×#95—22no—
48C5×#95—21no—
49#3968—44no—
50#3963—24no—
51#3963—22no—
52#3963D414yes—
53C22⋊#168—22no—
54C22⋊#168—22no—
55#2434D511yescharacteristic
56#2310—51no—
57#1784—51no—
58#1511—52no—
59#1284—52no—
60C24×C5C2×D411yescharacteristic
61C2×C5×Q8—24no—
62C2×C5×Q8—24no—
63C2×C5×Q8—24no—
64C2×C5×Q8—24no—
65C2×C5×Q8—22no—
66C2×C5×Q8—42no—
67C2×C5×Q8—42no—
68C2×C5×Q8—44no—
69C2×C5×Q8—44no—
70C2×C5×Q8—48no—
71C2×C5×Q8—42no—
72C22×C4×C5—21no—
73C22×C4×C5—21no—
74C22×C4×C5—24no—
75C22×C4×C5—22no—
76C22×C4×C5—21no—
77C22×C4×C5—21no—
78C22×C4×C5—21no—
79C22×C4×C5—21no—
80C22×C4×C5—42no—
81C22×C4×C5—42no—
82C22×C4×C5C22⋊C412yes—
83C22×C4×C5C22⋊C412yes—
84C22×C4×C5C22×C411yescharacteristic
85C22×C4×C5C2×D412yes—
86C22×C4×C5C2×D411yescharacteristic
87C22×C4×C5#4112yes—
88C22×Dic5—24no—
89C5×C4⋊C4—22no—
90C5×C4⋊C4—44no—
91C5×C4⋊C4—44no—
92C5×C4⋊C4—42no—
93C5×C22⋊C4—24no—
94C5×C22⋊C4—41no—
95C5×C22⋊C4—41no—
96C42×C5—42no—
97C22⋊Dic5—44no—
98C4⋊Dic5—44no—
99C4×Dic5—44no—
100#681—44no—
101C2×#168—42no—
102C23×Q8C2×D511yescharacteristic
103C2×C4⋊Q8Dic512yescomplemented
104C2×C22⋊Q8C2×D511yescharacteristic
105C2×#117—101no—
106C2×#101—102no—
107#407—52no—
108#395—22no—
109#391—104no—
110#391Dic512yescomplemented
111#385—21no—
112#385—21no—
113#381—52no—
114#381—101no—
115C4×C22⋊C4C2×D511yescharacteristic
116#328—104no—
117C23×C5—22no—
118C23×C5—22no—
119C23×C5—41no—
120C23×C5C2×C22⋊C411yescharacteristic
121C23×C5#12011yescharacteristic
122C23×C5#12311yescharacteristic
123C5×Q8—48no—
124C5×Q8—48no—
125C5×Q8—82no—
126C2×C4×C5—22no—
127C2×C4×C5—22no—
128C2×C4×C5—24no—
129C2×C4×C5—22no—
130C2×C4×C5—22no—
131C2×C4×C5—22no—
132C2×C4×C5—22no—
133C2×C4×C5—24no—
134C2×C4×C5—22no—
135C2×C4×C5—22no—
136C2×C4×C5—22no—
137C2×C4×C5—42no—
138C2×C4×C5—42no—
139C2×C4×C5—42no—
140C2×C4×C5—42no—
141C2×C4×C5—42no—
142C2×C4×C5—42no—
143C2×C4×C5—42no—
144C2×C4×C5—41no—
145C2×C4×C5—41no—
146C2×C4×C5—42no—
147C2×C4×C5#9914yes—
148C2×C4×C5C4×D412yes—
149C2×C4×C5#12112yes—
150C2×Dic5—44no—
151C2×Dic5—44no—
152C2×Dic5—44no—
153#168—82no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—44no—
157C22×Q8C2×Dic511yescharacteristic
158C22×Q8C2×Dic512yes—
159C22×Q8C22⋊D512yes—
160C23×C4C22⋊D511yescharacteristic
161C23×C4C22⋊D511yescharacteristic
162C23×C4C22×D511yescharacteristic
163C2×M16—102no—
164C4⋊Q8—44no—
165C22⋊Q8—22no—
166C22⋊Q8—22no—
167#117—204no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—102no—
172C2×C4⋊C4C2×Dic511yescharacteristic
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—102no—
176C2×C42—102no—
177C2×C42C2×Dic512yes—
178#101—204no—
179C22⋊C8—102no—
180C22⋊C8—102no—
181#95—22no—
182#95—22no—
183#95—21no—
184#95—54no—
185#95—104no—
186#95—102no—
187C22×C5—21no—
188C22×C5—21no—
189C22×C5—21no—
190C22×C5—21no—
191C22×C5—21no—
192C22×C5—21no—
193C22×C5—21no—
194C22×C5—44no—
195C22×C5#38611yescharacteristic
196C22×C5C2×#9912yes—
197C22×C5#45612yes—
198C4×C5—22no—
199C4×C5—24no—
200C4×C5—22no—
201C4×C5—44no—
202C4×C5—42no—
203C4×C5—81no—
204C4×C5—81no—
205Dic5—84no—
206C24C2×C22⋊D511yescharacteristic
207C2×Q8—24no—
208C2×Q8—24no—
209C2×Q8—24no—
210C2×Q8—24no—
211C2×Q8—22no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—44no—
215C2×Q8—44no—
216C2×Q8—48no—
217C2×Q8—42no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—104no—
229C22×C4C22⋊Dic512yes—
230C22×C4C22⋊Dic512yes—
231C22×C4D4×D512yes—
232C22×C4#71112yes—
233C22×C4C22×Dic511yescharacteristic
234C22×C4C2×C22⋊D511yescharacteristic
235M16—204no—
236C2×C8—202no—
237C4⋊C4—22no—
238C4⋊C4—44no—
239C4⋊C4—44no—
240C4⋊C4—42no—
241C4⋊C4—204no—
242C22⋊C4—24no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C22⋊C4—204no—
246C42—42no—
247C42—204no—
248C2×C5—22no—
249C2×C5—22no—
250C2×C5—41no—
251C2×C5#188611yescharacteristic
252C2×C5#189811yescharacteristic
253C2×C5#191211yescharacteristic
254C23—22no—
255C23—22no—
256C23—41no—
257C23#408111yescharacteristic
258C23#408311yescharacteristic
259C23C2×C22⋊Dic511yescharacteristic
260Q8—48no—
261Q8—48no—
262Q8—82no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—24no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—204no—
285C2×C4—204no—
286C2×C4—204no—
287C2×C4#396614yes—
288C2×C4D4×Dic512yes—
289C2×C4#408512yes—
290C8—402no—
291C5#946211yescharacteristic, complemented, Sylow
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—44no—
300C22#6538412yes—
301C22C2×#396612yes—
302C22#6550911yescharacteristic
303C4—22no—
304C4—24no—
305C4—22no—
306C4—44no—
307C4—42no—
308C4—81no—
309C4—81no—
310C4—404no—
311C2—22no—
312C2—22no—
313C2—41no—
314C2#1060246311yescharacteristic
315C2#1060257711yescharacteristic
316C2#1060265711yescharacteristic
subgroupG/H
centerC22#65509
commutatorC22×C4×C5C22×C4
FrattiniC23×C4C22×D5
FittingC5×#2434C2
socleC22×C5#386
prime radicalC24×C5C2×D4
cosocleC23×C4×C5C23
hypercenter#2434D5
hypocenterC5#9462
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2434

Complements

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

NH ≅ G/Ntype
7. C2×C5×C4⋊Q8C4semidirect
11. C5×#391C4semidirect
103. C2×C4⋊Q8Dic5semidirect
110. #391Dic5semidirect
291. C5#9462semidirect

Maximal subgroups

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

subgroupindex
C5×#24342
#106157932
#106029262
#106024822
#105981072
#94625