Finite Group Databasebeta

← #11997090

Subgroups — #11997090

Order 1280.

556 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 123 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
2#10614033C211yesmaximal, characteristic
3#10602412C211yesmaximal, characteristic
4#10602347C212yesmaximal, complemented
5#10598116C212yesmaximal, complemented
6C2×C4×C5×D4C2211yescharacteristic
7C22×C5×C4⋊C4C2211yescharacteristic
8C5×#400C412yescomplemented
9C5×#399C412yescomplemented
10C5×#389—21no—
11C5×#386—21no—
12C5×#385—21no—
13C5×#380—21no—
14C5×#380C2211yescharacteristic
15C2×#3966—21no—
16#65501C2212yes—
17#65496—21no—
18C2×C22⋊#168C2212yes—
19#65300—24no—
20#65215—21no—
21#64749—24no—
22#64746—21no—
23#9473—51nomaximal, Sylow
24C22×C5×D4D411yescharacteristic
25C23×C4×C5—21no—
26C23×C4×C5D411yescharacteristic
27C23×C4×C5C2311yescharacteristic
28C4×C5×D4—22no—
29C4×C5×D4—22no—
30C2×C5×C4⋊C4—22no—
31C2×C5×C4⋊C4—22no—
32C2×C5×C4⋊C4—21no—
33C2×C5×C4⋊C4—21no—
34C2×C5×C4⋊C4—22no—
35C2×C5×C4⋊C4C2×C412yes—
36C2×C5×C4⋊C4C2×C411yescharacteristic
37C2×C5×C22⋊C4—21no—
38C2×C5×C22⋊C4—22no—
39C2×C5×C22⋊C4C2×C411yescharacteristic
40C2×C5×C22⋊C4D411yescharacteristic
41C2×C42×C5D411yescharacteristic
42C2×C22⋊Dic5—22no—
43C2×C4⋊Dic5—22no—
44C2×#681—22no—
45C22×#168D414yes—
46C5×#95—22no—
47C5×#95—22no—
48C5×#95C2×C412yes—
49#3966—42no—
50#3963—22no—
51C22⋊#168—22no—
52C22⋊#168—22no—
53C22⋊#168—22no—
54C4⋊#168—22no—
55#3935—22no—
56#2973—51no—
57#2487D511yescharacteristic
58#1824—52no—
59#1799—51no—
60#1289—52no—
61C24×C5C22⋊C411yescharacteristic
62C24×C5C2×D411yescharacteristic
63C2×C5×D4—22no—
64C2×C5×D4—21no—
65C2×C5×D4—21no—
66C2×C5×D4—22no—
67C22×C4×C5—21no—
68C22×C4×C5—21no—
69C22×C4×C5—22no—
70C22×C4×C5—22no—
71C22×C4×C5—42no—
72C22×C4×C5—42no—
73C22×C4×C5—42no—
74C22×C4×C5—82no—
75C22×C4×C5C22⋊C411yescharacteristic
76C22×C4×C5C22⋊C411yescharacteristic
77C22×C4×C5C22⋊C411yescharacteristic
78C22×C4×C5C22×C411yescharacteristic
79C22×C4×C5C2×D412yes—
80C22×C4×C5C2×D411yescharacteristic
81C22×C4×C5#4112yes—
82C22×Dic5—42no—
83C5×C4⋊C4—22no—
84C5×C4⋊C4—24no—
85C5×C4⋊C4—42no—
86C5×C4⋊C4—41no—
87C5×C4⋊C4—41no—
88C5×C4⋊C4—82no—
89C5×C22⋊C4—22no—
90C5×C22⋊C4—22no—
91C5×C22⋊C4—41no—
92C5×C22⋊C4—41no—
93C5×C22⋊C4—82no—
94C42×C5—21no—
95C42×C5—21no—
96C22⋊Dic5—42no—
97C22⋊Dic5—42no—
98C4⋊Dic5—44no—
99#681—82no—
100C2×#168—24no—
101C2×#168—44no—
102C2×C4×D4C2×D511yescharacteristic
103C22×C4⋊C4C2×D511yescharacteristic
104#435—104no—
105#432—101no—
106C2×#99—101no—
107#407—101no—
108C2×C22⋊C8—52no—
109#400Dic512yescomplemented
110#399Dic512yescomplemented
111#389—21no—
112#386—21no—
113#385—21no—
114#380—21no—
115#380—52no—
116#380C2×D511yescharacteristic
117#342—101no—
118#340—104no—
119C23×C5—21no—
120C23×C5—21no—
121C23×C5—21no—
122C23×C5—41no—
123C23×C5—41no—
124C23×C5—41no—
125C23×C5—41no—
126C23×C5C2×C22⋊C411yescharacteristic
127C23×C5#12112yes—
128C23×C5AGL(1,ℤ/8)11yescharacteristic
129C23×C5#13711yescharacteristic
130C23×C5#13712yes—
131C5×D4—42no—
132C5×D4—42no—
133C2×C4×C5—24no—
134C2×C4×C5—22no—
135C2×C4×C5—21no—
136C2×C4×C5—21no—
137C2×C4×C5—21no—
138C2×C4×C5—21no—
139C2×C4×C5—21no—
140C2×C4×C5—41no—
141C2×C4×C5—42no—
142C2×C4×C5—42no—
143C2×C4×C5—41no—
144C2×C4×C5—41no—
145C2×C4×C5—42no—
146C2×C4×C5—42no—
147C2×C4×C5—42no—
148C2×C4×C5—42no—
149C2×C4×C5—42no—
150C2×C4×C5—42no—
151C2×C4×C5—81no—
152C2×C4×C5—84no—
153C2×C4×C5C4×D412yes—
154C2×C4×C5#12011yescharacteristic
155C2×C4×C5#12311yescharacteristic
156C2×Dic5—82no—
157C2×Dic5—82no—
158C2×Dic5—82no—
159#168—44no—
160C22×D4C22⋊D511yescharacteristic
161C23×C4—21no—
162C23×C4C22⋊D511yescharacteristic
163C23×C4C22×D511yescharacteristic
164C2×M16—102no—
165C22×C8—54no—
166C4×D4—22no—
167C4×D4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—102no—
174C2×C4⋊C4C2×Dic512yes—
175C2×C4⋊C4C2×Dic511yescharacteristic
176C2×C22⋊C4—21no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—102no—
179C2×C22⋊C4C2×Dic511yescharacteristic
180C2×C22⋊C4C22⋊D511yescharacteristic
181C2×C42C22⋊D511yescharacteristic
182C4⋊C8—102no—
183#99—202no—
184C22⋊C8—102no—
185C22⋊C8—102no—
186C22⋊C8—102no—
187#97—102no—
188#95—22no—
189#95—22no—
190#95—102no—
191#95C2×Dic512yes—
192C22×C5—22no—
193C22×C5—21no—
194C22×C5—22no—
195C22×C5—21no—
196C22×C5—41no—
197C22×C5—41no—
198C22×C5—42no—
199C22×C5—42no—
200C22×C5—41no—
201C22×C5—41no—
202C22×C5—41no—
203C22×C5—81no—
204C22×C5#38611yescharacteristic
205C22×C5#41711yescharacteristic
206C22×C5#41911yescharacteristic
207C22×C5#46812yes—
208C22×C5#47012yes—
209C4×C5—21no—
210C4×C5—22no—
211C4×C5—41no—
212C4×C5—41no—
213C4×C5—41no—
214C4×C5—82no—
215C4×C5—82no—
216C4×C5#45812yes—
217Dic5—162no—
218C24C22⋊Dic511yescharacteristic
219C24C2×C22⋊D511yescharacteristic
220C2×D4—21no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—82no—
232C22×C4—202no—
233C22×C4C22⋊Dic511yescharacteristic
234C22×C4C22⋊Dic511yescharacteristic
235C22×C4C22⋊Dic511yescharacteristic
236C22×C4D4×D512yes—
237C22×C4#71112yes—
238C22×C4C22×Dic511yescharacteristic
239C22×C4C2×C22⋊D511yescharacteristic
240M16—402no—
241C2×C8—104no—
242C2×C8—204no—
243C4⋊C4—22no—
244C4⋊C4—24no—
245C4⋊C4—42no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—82no—
249C4⋊C4—204no—
250C22⋊C4—22no—
251C22⋊C4—22no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—82no—
255C22⋊C4—202no—
256C22⋊C4—202no—
257C42—21no—
258C42—21no—
259C2×C5—22no—
260C2×C5—21no—
261C2×C5—21no—
262C2×C5—22no—
263C2×C5—81no—
264C2×C5#189011yescharacteristic
265C2×C5#189411yescharacteristic
266C2×C5#191011yescharacteristic
267C23—21no—
268C23—21no—
269C23—21no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23#408512yes—
275C23#408812yes—
276C23#409511yescharacteristic
277C23#409711yescharacteristic
278C23C2×C22⋊Dic511yescharacteristic
279D4—42no—
280D4—42no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—81no—
300C2×C4—84no—
301C2×C4—402no—
302C2×C4—402no—
303C2×C4—402no—
304C2×C4D4×Dic512yes—
305C2×C4#408111yescharacteristic
306C2×C4#408311yescharacteristic
307C8—204no—
308C5#947311yescharacteristic, complemented, Sylow
309C22—21no—
310C22—22no—
311C22—21no—
312C22—22no—
313C22—41no—
314C22—42no—
315C22—41no—
316C22—42no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22#6533812yes—
322C22#6534112yes—
323C22#6550911yescharacteristic
324C22#6551411yescharacteristic
325C22#6552111yescharacteristic
326C4—21no—
327C4—22no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—82no—
332C4—82no—
333C4—802no—
334C4#6534212yes—
335C2—21no—
336C2—22no—
337C2—21no—
338C2—22no—
339C2—81no—
340C2#1060249011yescharacteristic
341C2#1060249711yescharacteristic
342C2#1060264711yescharacteristic
subgroupG/H
centerC22#65509
commutatorC22×C4×C5C22×C4
FrattiniC23×C4C22×D5
FittingC5×#2487C2
socleC22×C5#386
prime radicalC22×C5×D4D4
cosocleC23×C4×C5C23
hypercenter#2487D5
hypocenterC5#9473
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2487

Complements

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

NH ≅ G/Ntype
4. #10602347C2semidirect
5. #10598116C2semidirect
8. C5×#400C4semidirect
9. C5×#399C4semidirect
109. #400Dic5semidirect
110. #399Dic5semidirect
308. C5#9473semidirect

Maximal subgroups

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

subgroupindex
C5×#24872
#106140332
#106024122
#106023472
#105981162
#94735