Finite Group Databasebeta

← #11997125

Subgroups — #11997125

Order 1280.

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

normal subgroups only

This group has 726 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×#2406C211yesmaximal, characteristic
2#10613886C211yesmaximal, characteristic, complemented
3#10602931C212yesmaximal
4#10602345C211yesmaximal, characteristic, complemented
5#10598119C212yesmaximal, complemented
6C22×C5×#41C2211yescharacteristic
7C2×C5×#123—21no—
8C22×C5×C4⋊C4C2211yescharacteristic
9C5×#395C412yescomplemented
10C5×#394C412yescomplemented
11C5×#386—21no—
12C5×#385—22no—
13C5×#381C2211yescharacteristic
14C2×#3969—24no—
15#65501—21no—
16#65500—21no—
17C4×C22⋊Dic5C2212yes—
18#65496C2212yes—
19#65270—24no—
20#64748—22no—
21#9480—51nomaximal, Sylow
22C2×C5×#41—22no—
23C2×C5×#41—22no—
24C2×C5×#41—44no—
25C22×C5×Q8C2×C411yescharacteristic
26C22×C5×D4—21no—
27C22×C5×D4C2×C411yescharacteristic
28C23×C4×C5—21no—
29C23×C4×C5D411yescharacteristic
30C23×C4×C5D411yescharacteristic
31C23×C4×C5C2311yescharacteristic
32C5×#123—44no—
33C2×C5×C4⋊C4—22no—
34C2×C5×C4⋊C4—22no—
35C2×C5×C4⋊C4—22no—
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×C22⋊Dic5—22no—
42C2×C4⋊Dic5—22no—
43C2×C4×Dic5D414yes—
44C2×#681—22no—
45C5×#95—22no—
46C5×#95—21no—
47C5×#95C2×C412yes—
48#3969—48no—
49#3963—24no—
50#3963—22no—
51#3963—22no—
52C22⋊#168—22no—
53C22⋊#168—22no—
54#2406D511yescharacteristic
55#2313—51no—
56#1789—52no—
57#1463—51no—
58#1284—52no—
59C24×C5—21no—
60C24×C5C2×D411yescharacteristic
61C5×#41—28no—
62C5×#41—44no—
63C5×#41—44no—
64C5×#41—44no—
65C2×C5×Q8—22no—
66C2×C5×Q8—42no—
67C2×C5×D4—24no—
68C2×C5×D4—22no—
69C2×C5×D4—42no—
70C2×C5×D4—42no—
71C2×C5×D4—41no—
72C2×C5×D4—41no—
73C22×C4×C5—21no—
74C22×C4×C5—21no—
75C22×C4×C5—21no—
76C22×C4×C5—21no—
77C22×C4×C5—21no—
78C22×C4×C5—22no—
79C22×C4×C5—42no—
80C22×C4×C5—42no—
81C22×C4×C5—42no—
82C22×C4×C5—42no—
83C22×C4×C5—42no—
84C22×C4×C5—42no—
85C22×C4×C5C22⋊C412yes—
86C22×C4×C5C22⋊C412yes—
87C22×C4×C5C22×C411yescharacteristic
88C22×C4×C5C2×D412yes—
89C22×C4×C5C2×D411yescharacteristic
90C22×C4×C5#4112yes—
91C22×Dic5—24no—
92C5×C4⋊C4—24no—
93C5×C4⋊C4—24no—
94C5×C4⋊C4—44no—
95C5×C22⋊C4—44no—
96C5×C22⋊C4—44no—
97C5×C22⋊C4—42no—
98C22⋊Dic5—44no—
99C4⋊Dic5—82no—
100C4×Dic5—44no—
101#681—44no—
102C2×#168—42no—
103C22×#41C2×D511yescharacteristic
104C2×#123—21no—
105C22×C4⋊C4C2×D511yescharacteristic
106C2×#104—104no—
107#407—52no—
108#395Dic512yescomplemented
109#394Dic512yescomplemented
110#387—104no—
111#386—21no—
112#385—22no—
113#381—101no—
114#381C2×D511yescharacteristic
115#380—101no—
116C4×C22⋊C4—52no—
117#324—102no—
118C23×C5—22no—
119C23×C5—22no—
120C23×C5—22no—
121C23×C5—41no—
122C23×C5—41no—
123C23×C5—42no—
124C23×C5—41no—
125C23×C5—42no—
126C23×C5C2×C22⋊C411yescharacteristic
127C23×C5#12011yescharacteristic
128C23×C5#12311yescharacteristic
129C5×Q8—24no—
130C5×Q8—42no—
131C5×D4—24no—
132C5×D4—42no—
133C5×D4—44no—
134C5×D4—44no—
135C2×C4×C5—24no—
136C2×C4×C5—22no—
137C2×C4×C5—22no—
138C2×C4×C5—22no—
139C2×C4×C5—22no—
140C2×C4×C5—22no—
141C2×C4×C5—41no—
142C2×C4×C5—41no—
143C2×C4×C5—42no—
144C2×C4×C5—48no—
145C2×C4×C5—42no—
146C2×C4×C5—42no—
147C2×C4×C5—41no—
148C2×C4×C5—41no—
149C2×C4×C5—42no—
150C2×C4×C5—42no—
151C2×C4×C5—42no—
152C2×C4×C5—42no—
153C2×C4×C5—42no—
154C2×C4×C5—81no—
155C2×C4×C5C4×D412yes—
156C2×C4×C5#12112yes—
157C2×Dic5—44no—
158C2×Dic5—44no—
159C2×Dic5—44no—
160#168—82no—
161C2×#41—22no—
162C2×#41—22no—
163C2×#41—44no—
164C22×Q8C2×Dic511yescharacteristic
165C22×D4—21no—
166C22×D4C2×Dic511yescharacteristic
167C23×C4—21no—
168C23×C4C22⋊D511yescharacteristic
169C23×C4C22⋊D511yescharacteristic
170C23×C4C22×D511yescharacteristic
171C2×M16—102no—
172#123—44no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—102no—
177C2×C4⋊C4C2×Dic512yes—
178C2×C4⋊C4C22⋊D512yes—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—102no—
183C2×C42—54no—
184#104—208no—
185C22⋊C8—102no—
186C22⋊C8—102no—
187#95—22no—
188#95—21no—
189#95—102no—
190#95—104no—
191#95—102no—
192#95C2×Dic512yes—
193C22×C5—21no—
194C22×C5—21no—
195C22×C5—21no—
196C22×C5—21no—
197C22×C5—21no—
198C22×C5—21no—
199C22×C5—21no—
200C22×C5—44no—
201C22×C5—44no—
202C22×C5—42no—
203C22×C5—82no—
204C22×C5—81no—
205C22×C5—81no—
206C22×C5#38611yescharacteristic
207C22×C5#42112yes—
208C22×C5B(4,𝔽2)12yes—
209C4×C5—22no—
210C4×C5—22no—
211C4×C5—42no—
212C4×C5—42no—
213C4×C5—42no—
214C4×C5—82no—
215Dic5—84no—
216C24—21no—
217C24C2×C22⋊D511yescharacteristic
218#41—28no—
219#41—44no—
220#41—44no—
221#41—44no—
222C2×Q8—22no—
223C2×Q8—42no—
224C2×D4—24no—
225C2×D4—22no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—41no—
229C2×D4—41no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—104no—
243C22×C4C22⋊Dic512yes—
244C22×C4C22⋊Dic512yes—
245C22×C4D4×D512yes—
246C22×C4#71112yes—
247C22×C4C22×Dic511yescharacteristic
248C22×C4C2×C22⋊D511yescharacteristic
249M16—204no—
250C2×C8—202no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—44no—
254C4⋊C4—402no—
255C22⋊C4—44no—
256C22⋊C4—44no—
257C22⋊C4—42no—
258C22⋊C4—204no—
259C42—204no—
260C2×C5—22no—
261C2×C5—22no—
262C2×C5—41no—
263C2×C5—82no—
264C2×C5#188911yescharacteristic
265C2×C5#189711yescharacteristic
266C2×C5#191211yescharacteristic
267C23—22no—
268C23—22no—
269C23—22no—
270C23—41no—
271C23—41no—
272C23—42no—
273C23—41no—
274C23—42no—
275C23#408111yescharacteristic
276C23#408311yescharacteristic
277C23C2×C22⋊Dic511yescharacteristic
278Q8—24no—
279Q8—42no—
280D4—24no—
281D4—42no—
282D4—44no—
283D4—44no—
284C2×C4—24no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—48no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—81no—
304C2×C4—204no—
305C2×C4—204no—
306C2×C4—204no—
307C2×C4D4×Dic512yes—
308C2×C4#408512yes—
309C8—402no—
310C5#948011yescharacteristic, complemented, Sylow
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—44no—
319C22—44no—
320C22—42no—
321C22—82no—
322C22—81no—
323C22—81no—
324C22#6532112yes—
325C22#6550911yescharacteristic
326C22#6552512yes—
327C4—22no—
328C4—22no—
329C4—42no—
330C4—42no—
331C4—42no—
332C4—82no—
333C4—404no—
334C2—22no—
335C2—22no—
336C2—41no—
337C2—82no—
338C2#1060246211yescharacteristic
339C2#1060250511yescharacteristic
340C2#1060265711yescharacteristic
subgroupG/H
centerC22#65509
commutatorC22×C4×C5C22×C4
FrattiniC23×C4C22×D5
FittingC5×#2406C2
socleC22×C5#386
prime radicalC22×C5×#41C22
cosocleC23×C4×C5C23
hypercenter#2406D5
hypocenterC5#9480
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2406

Complements

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

NH ≅ G/Ntype
2. #10613886C2semidirect
4. #10602345C2semidirect
5. #10598119C2semidirect
9. C5×#395C4semidirect
10. C5×#394C4semidirect
108. #395Dic5semidirect
109. #394Dic5semidirect
310. C5#9480semidirect

Maximal subgroups

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

subgroupindex
C5×#24062
#106138862
#106029312
#106023452
#105981192
#94805