Finite Group Databasebeta

← #12008657

Subgroups — #12008657

Order 1280.

1866 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 1866 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
1C2×C5×#379C211yesmaximal, characteristic
2#10602332C212yesmaximal, complemented
3#10598113C214yesmaximal, complemented
4C26×C5C411yescharacteristic
5C22×C5×C22⋊C4—22no—
6C22×C5×C22⋊C4C411yescharacteristic
7C22×C5×C22⋊C4C2211yescharacteristic
8C5×#379—22no—
9C5×#379C412yes—
10C5×#379C2212yes—
11#65496—22no—
12#64745—22no—
13#64745C2214yescomplemented
14#11499—51nomaximal, Sylow
15C25×C5—21no—
16C25×C5—21no—
17C25×C5—24no—
18C25×C5—48no—
19C25×C5—42no—
20C25×C5—42no—
21C25×C5C2×C411yescharacteristic
22C25×C5C2×C412yes—
23C23×C4×C5—21no—
24C23×C4×C5D412yes—
25C2×C5×C22⋊C4—21no—
26C2×C5×C22⋊C4—24no—
27C2×C5×C22⋊C4—24no—
28C2×C5×C22⋊C4—42no—
29C2×C5×C22⋊C4—48no—
30C2×C5×C22⋊C4—42no—
31C2×C5×C22⋊C4C2×C412yes—
32C2×C5×C22⋊C4C2×C411yescharacteristic
33C2×C5×C22⋊C4D412yes—
34C2×C5×C22⋊C4C2311yescharacteristic
35C2×#681—42no—
36C22⋊#168—24no—
37C22⋊#168—42no—
38C2×#379D511yescharacteristic
39#1462—52no—
40#1316—54no—
41C24×C5—28no—
42C24×C5—24no—
43C24×C5—22no—
44C24×C5—22no—
45C24×C5—22no—
46C24×C5—22no—
47C24×C5—24no—
48C24×C5—22no—
49C24×C5—44no—
50C24×C5—48no—
51C24×C5—42no—
52C24×C5—42no—
53C24×C5—44no—
54C24×C5—42no—
55C24×C5—48no—
56C24×C5—432no—
57C24×C5—416no—
58C24×C5—42no—
59C24×C5—48no—
60C24×C5—416no—
61C24×C5—432no—
62C24×C5—48no—
63C24×C5—42no—
64C24×C5—42no—
65C24×C5C22⋊C411yescharacteristic
66C24×C5C22⋊C411yescharacteristic
67C24×C5M1614yescomplemented
68C24×C5C22×C411yescharacteristic
69C22×C4×C5—22no—
70C22×C4×C5—22no—
71C22×C4×C5—22no—
72C22×C4×C5—44no—
73C22×C4×C5—44no—
74C22×C4×C5—42no—
75C22×C4×C5C22⋊C412yes—
76C22×C4×C5C2×D412yes—
77C5×C22⋊C4—41no—
78C5×C22⋊C4—42no—
79C5×C22⋊C4—41no—
80C5×C22⋊C4—48no—
81C5×C22⋊C4—48no—
82C5×C22⋊C4—42no—
83C5×C22⋊C4—42no—
84#681—84no—
85C2×#168—44no—
86C26Dic511yescharacteristic
87C22×C22⋊C4—22no—
88C22×C22⋊C4Dic511yescharacteristic
89C22×C22⋊C4C2×D511yescharacteristic
90#407—102no—
91#379—22no—
92#379Dic512yes—
93#379C2×D512yes—
94#323—54no—
95#323—102no—
96C23×C5—24no—
97C23×C5—21no—
98C23×C5—21no—
99C23×C5—28no—
100C23×C5—216no—
101C23×C5—28no—
102C23×C5—21no—
103C23×C5—24no—
104C23×C5—21no—
105C23×C5—21no—
106C23×C5—21no—
107C23×C5—42no—
108C23×C5—48no—
109C23×C5—44no—
110C23×C5—42no—
111C23×C5—48no—
112C23×C5—416no—
113C23×C5—48no—
114C23×C5—48no—
115C23×C5—44no—
116C23×C5—48no—
117C23×C5—44no—
118C23×C5—48no—
119C23×C5—48no—
120C23×C5—416no—
121C23×C5—48no—
122C23×C5—44no—
123C23×C5—48no—
124C23×C5—416no—
125C23×C5—416no—
126C23×C5—464no—
127C23×C5—464no—
128C23×C5—48no—
129C23×C5—416no—
130C23×C5—48no—
131C23×C5—44no—
132C23×C5—42no—
133C23×C5—42no—
134C23×C5#9912yes—
135C23×C5#10012yes—
136C23×C5C2×C22⋊C411yescharacteristic
137C23×C5C2×M1612yes—
138C2×C4×C5—22no—
139C2×C4×C5—22no—
140C2×C4×C5—44no—
141C2×C4×C5—42no—
142C2×C4×C5—42no—
143C2×C4×C5—44no—
144C2×C4×C5—42no—
145C2×C4×C5—82no—
146C2×C4×C5—84no—
147#168—84no—
148C25—21no—
149C25—24no—
150C25—21no—
151C25—48no—
152C25—42no—
153C25—42no—
154C25C2×Dic511yescharacteristic
155C25C2×Dic512yes—
156C23×C4—21no—
157C23×C4C22⋊D512yes—
158C2×M16—202no—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—42no—
163C2×C22⋊C4—48no—
164C2×C22⋊C4—42no—
165C2×C22⋊C4C2×Dic512yes—
166C2×C22⋊C4C2×Dic511yescharacteristic
167C2×C22⋊C4C22⋊D512yes—
168C2×C22⋊C4C22×D511yescharacteristic
169C22⋊C8—104no—
170C22⋊C8—202no—
171C22×C5—24no—
172C22×C5—24no—
173C22×C5—22no—
174C22×C5—22no—
175C22×C5—22no—
176C22×C5—28no—
177C22×C5—22no—
178C22×C5—22no—
179C22×C5—42no—
180C22×C5—42no—
181C22×C5—48no—
182C22×C5—42no—
183C22×C5—42no—
184C22×C5—44no—
185C22×C5—432no—
186C22×C5—416no—
187C22×C5—48no—
188C22×C5—48no—
189C22×C5—416no—
190C22×C5—432no—
191C22×C5—48no—
192C22×C5—42no—
193C22×C5—42no—
194C22×C5—44no—
195C22×C5AΣL(1,𝔽16)14yes—
196C22×C5#40711yescharacteristic
197C22×C5C2×#9911yescharacteristic
198C22×C5C2×#10011yescharacteristic
199C4×C5—42no—
200C4×C5—42no—
201C4×C5—82no—
202C24—28no—
203C24—24no—
204C24—22no—
205C24—22no—
206C24—22no—
207C24—22no—
208C24—24no—
209C24—22no—
210C24—44no—
211C24—48no—
212C24—42no—
213C24—42no—
214C24—42no—
215C24—44no—
216C24—48no—
217C24—432no—
218C24—416no—
219C24—42no—
220C24—48no—
221C24—416no—
222C24—432no—
223C24—48no—
224C24—42no—
225C24—42no—
226C24#68114yescomplemented
227C24C22⋊Dic511yescharacteristic
228C24C22⋊Dic511yescharacteristic
229C24C22×Dic511yescharacteristic
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—42no—
236C22×C4C22⋊Dic512yes—
237C22×C4C2×C22⋊D512yes—
238M16—404no—
239C2×C8—204no—
240C22⋊C4—41no—
241C22⋊C4—42no—
242C22⋊C4—41no—
243C22⋊C4—48no—
244C22⋊C4—48no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C2×C5—21no—
248C2×C5—21no—
249C2×C5—24no—
250C2×C5—48no—
251C2×C5—42no—
252C2×C5—42no—
253C2×C5#146211yescharacteristic
254C2×C5C2×AΣL(1,𝔽16)12yes—
255C23—21no—
256C23—21no—
257C23—24no—
258C23—28no—
259C23—216no—
260C23—28no—
261C23—24no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—48no—
267C23—44no—
268C23—42no—
269C23—42no—
270C23—48no—
271C23—416no—
272C23—48no—
273C23—48no—
274C23—48no—
275C23—44no—
276C23—48no—
277C23—44no—
278C23—48no—
279C23—416no—
280C23—48no—
281C23—44no—
282C23—48no—
283C23—416no—
284C23—416no—
285C23—464no—
286C23—464no—
287C23—48no—
288C23—416no—
289C23—42no—
290C23—42no—
291C23—48no—
292C23—44no—
293C23#396512yes—
294C23#396612yes—
295C23C2×#68112yes—
296C23C2×C22⋊Dic511yescharacteristic
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4—82no—
305C2×C4—84no—
306C8—404no—
307C5#1149911yescharacteristic, complemented, Sylow
308C22—24no—
309C22—24no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—28no—
314C22—22no—
315C22—22no—
316C22—42no—
317C22—42no—
318C22—48no—
319C22—42no—
320C22—44no—
321C22—42no—
322C22—432no—
323C22—416no—
324C22—48no—
325C22—48no—
326C22—416no—
327C22—432no—
328C22—44no—
329C22—48no—
330C22—42no—
331C22—42no—
332C22#6475614yes—
333C22#6549611yescharacteristic
334C22C2×#396511yescharacteristic
335C22C2×#396611yescharacteristic
336C4—42no—
337C4—42no—
338C4—82no—
339C2—21no—
340C2—21no—
341C2—24no—
342C2—48no—
343C2—42no—
344C2—42no—
345C2#1060233211yescharacteristic
346C2C2×#6475612yes—
subgroupG/H
centerC22C2×#3966
commutatorC24×C5C22×C4
FrattiniC2×C22⋊C4C22×D5
FittingC2×C5×#379C2
socleC22×C5C2×#99
prime radicalC26×C5C4
cosocleC2×C5×C22⋊C4C23
hypercenterC2×#379D5
hypocenterC5#11499
LCSG ▷ C24×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#379

Complements

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

NH ≅ G/Ntype
2. #10602332C2semidirect
3. #10598113C2semidirect
13. #64745C22semidirect
67. C24×C5M16semidirect
226. C24#681semidirect
307. C5#11499semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#3792
#106023322
#105981132
#114995