Finite Group Databasebeta

← #12009905

Subgroups — #12009905

Order 1280.

460 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 91 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×#2580C211yesmaximal, characteristic
2C4×#3966C211yesmaximal, characteristic
3#10602395C211yesmaximal, characteristic
4#10598142C212yesmaximal, complemented
5#10598137C212yesmaximal, complemented
6C2×C4×C5×Q8C411yescharacteristic, complemented
7C2×C4×C5×D4C2211yescharacteristic
8C5×#399C411yescharacteristic, complemented
9C5×#399C2211yescharacteristic
10C5×#393C411yescharacteristic, complemented
11C5×#393C2211yescharacteristic
12C5×#390—21no—
13C5×#389—21no—
14C5×#385—21no—
15C5×#385—21no—
16C5×#382C411yescharacteristic, complemented
17C2×#3966—21no—
18C4×C22⋊Dic5—21no—
19#65496—21no—
20#64748—21no—
21#64746C2212yes—
22#64745C2212yes—
23#11784—51nomaximal, Sylow
24C22×C5×Q8C2×C411yescharacteristic
25C22×C5×D4D411yescharacteristic
26C23×C4×C5D411yescharacteristic
27C23×C4×C5D411yescharacteristic
28C4×C5×Q8—41no—
29C4×C5×Q8—41no—
30C4×C5×D4—41no—
31C4×C5×D4—41no—
32C2×C5×C4⋊C4—21no—
33C2×C5×C4⋊C4—21no—
34C2×C5×C4⋊C4—21no—
35C2×C5×C4⋊C4—21no—
36C2×C5×C4⋊C4C2×C411yescharacteristic
37C2×C5×C4⋊C4D411yescharacteristic
38C2×C5×C22⋊C4—21no—
39C2×C5×C22⋊C4—21no—
40C2×C5×C22⋊C4—21no—
41C2×C5×C22⋊C4—21no—
42C2×C5×C22⋊C4C2×C411yescharacteristic
43C2×C5×C22⋊C4C2311yescharacteristic
44C2×C42×C5—21no—
45C2×C42×C5C2×C411yescharacteristic
46C2×C22⋊Dic5—22no—
47C2×C4×Dic5—41no—
48C2×#681—41no—
49C5×#95—21no—
50C5×#95—21no—
51C5×#95—21no—
52C5×#95C2×C411yescharacteristic
53C5×#95C2×C411yescharacteristic
54#3966—24no—
55#3963—41no—
56C22⋊#168—22no—
57C22⋊#168—41no—
58#2580D511yescharacteristic
59C4×#99—51no—
60#1465—51no—
61#1352—52no—
62#1351—52no—
63C24×C5C22⋊C411yescharacteristic
64C24×C5C2×D411yescharacteristic
65C2×C5×Q8—42no—
66C2×C5×Q8—41no—
67C2×C5×D4—22no—
68C2×C5×D4—22no—
69C2×C5×D4—41no—
70C22×C4×C5—21no—
71C22×C4×C5—21no—
72C22×C4×C5—21no—
73C22×C4×C5—21no—
74C22×C4×C5—21no—
75C22×C4×C5—21no—
76C22×C4×C5—21no—
77C22×C4×C5—21no—
78C22×C4×C5—81no—
79C22×C4×C5—81no—
80C22×C4×C5C22⋊C411yescharacteristic
81C22×C4×C5C22⋊C411yescharacteristic
82C22×C4×C5C22⋊C411yescharacteristic
83C22×C4×C5C22×C411yescharacteristic
84C22×C4×C5C2×D411yescharacteristic
85C22×Dic5—42no—
86C5×C4⋊C4—41no—
87C5×C4⋊C4—41no—
88C5×C4⋊C4—41no—
89C5×C4⋊C4—41no—
90C5×C4⋊C4—42no—
91C5×C4⋊C4—81no—
92C5×C4⋊C4—81no—
93C5×C22⋊C4—41no—
94C5×C22⋊C4—41no—
95C5×C22⋊C4—42no—
96C5×C22⋊C4—42no—
97C5×C22⋊C4—81no—
98C5×C22⋊C4—81no—
99C42×C5—21no—
100C42×C5—21no—
101C42×C5—42no—
102C22⋊Dic5—24no—
103C22⋊Dic5—42no—
104C4×Dic5—82no—
105#681—82no—
106C2×#168—42no—
107C2×C4×Q8Dic511yescharacteristic, complemented
108C2×C4×D4C2×D511yescharacteristic
109C2×#99—101no—
110#407—101no—
111#399Dic511yescharacteristic, complemented
112#399C2×D511yescharacteristic
113#393Dic511yescharacteristic, complemented
114#393C2×D511yescharacteristic
115#390—21no—
116#389—21no—
117#385—21no—
118#385—21no—
119#382Dic511yescharacteristic, complemented
120C4×C22⋊C4—101no—
121#342—52no—
122#324—101no—
123#323—52no—
124C23×C5—21no—
125C23×C5—21no—
126C23×C5—21no—
127C23×C5—21no—
128C23×C5—21no—
129C23×C5—41no—
130C23×C5—41no—
131C23×C5—41no—
132C23×C5—41no—
133C23×C5C2×C22⋊C411yescharacteristic
134C23×C5#13712yes—
135C5×Q8—42no—
136C5×Q8—42no—
137C5×D4—44no—
138C2×C4×C5—21no—
139C2×C4×C5—21no—
140C2×C4×C5—22no—
141C2×C4×C5—21no—
142C2×C4×C5—21no—
143C2×C4×C5—21no—
144C2×C4×C5—41no—
145C2×C4×C5—42no—
146C2×C4×C5—42no—
147C2×C4×C5—41no—
148C2×C4×C5—41no—
149C2×C4×C5—41no—
150C2×C4×C5—41no—
151C2×C4×C5—41no—
152C2×C4×C5—41no—
153C2×C4×C5—41no—
154C2×C4×C5—41no—
155C2×C4×C5—41no—
156C2×C4×C5—41no—
157C2×C4×C5—41no—
158C2×C4×C5—41no—
159C2×C4×C5—41no—
160C2×C4×C5—41no—
161C2×C4×C5—81no—
162C2×C4×C5—81no—
163C2×C4×C5#9911yescharacteristic
164C2×C4×C5#9911yescharacteristic
165C2×C4×C5#10412yes—
166C2×Dic5—44no—
167C2×Dic5—82no—
168C2×Dic5—82no—
169#168—82no—
170C22×Q8C2×Dic511yescharacteristic
171C22×D4C22⋊D511yescharacteristic
172C23×C4C22⋊D511yescharacteristic
173C23×C4C22⋊D511yescharacteristic
174C2×M16—201no—
175C4×Q8—41no—
176C4×Q8—41no—
177C4×D4—41no—
178C4×D4—41no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4C2×Dic511yescharacteristic
184C2×C4⋊C4C22⋊D511yescharacteristic
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—102no—
190C2×C22⋊C4C2×Dic511yescharacteristic
191C2×C22⋊C4C22×D511yescharacteristic
192C2×C42—21no—
193C2×C42—201no—
194C2×C42C2×Dic511yescharacteristic
195#99—104no—
196C22⋊C8—102no—
197C22⋊C8—201no—
198#95—21no—
199#95—21no—
200#95—21no—
201#95—201no—
202#95C2×Dic511yescharacteristic
203#95C2×Dic511yescharacteristic
204C22×C5—21no—
205C22×C5—21no—
206C22×C5—22no—
207C22×C5—42no—
208C22×C5—42no—
209C22×C5—41no—
210C22×C5—41no—
211C22×C5—41no—
212C22×C5—41no—
213C22×C5—41no—
214C22×C5—81no—
215C22×C5—81no—
216C22×C5C2×#9911yescharacteristic
217C22×C5#41911yescharacteristic
218C22×C5C2×#10411yescharacteristic
219C4×C5—21no—
220C4×C5—21no—
221C4×C5—22no—
222C4×C5—42no—
223C4×C5—41no—
224C4×C5—41no—
225C4×C5—41no—
226C4×C5—81no—
227C4×C5—81no—
228C4×C5—81no—
229Dic5—84no—
230C24C22⋊Dic511yescharacteristic
231C24C2×C22⋊D511yescharacteristic
232C2×Q8—42no—
233C2×Q8—41no—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—41no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4—202no—
248C22×C4C22⋊Dic511yescharacteristic
249C22×C4C22⋊Dic511yescharacteristic
250C22×C4C22⋊Dic511yescharacteristic
251C22×C4C22×Dic511yescharacteristic
252C22×C4C2×C22⋊D511yescharacteristic
253M16—402no—
254C2×C8—202no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C4⋊C4—81no—
261C4⋊C4—81no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—81no—
267C22⋊C4—81no—
268C22⋊C4—104no—
269C22⋊C4—202no—
270C42—21no—
271C42—21no—
272C42—42no—
273C42—402no—
274C2×C5—21no—
275C2×C5—21no—
276C2×C5—42no—
277C2×C5—81no—
278C2×C5#150811yescharacteristic
279C2×C5#212011yescharacteristic
280C2×C5#212711yescharacteristic
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23#408812yes—
291C23C2×C22⋊Dic511yescharacteristic
292Q8—42no—
293Q8—42no—
294D4—44no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—204no—
321C2×C4—402no—
322C2×C4—402no—
323C2×C4#396611yescharacteristic
324C2×C4#396611yescharacteristic
325C2×C4#396912yes—
326C8—402no—
327C5#1178411yescharacteristic, complemented, Sylow
328C22—21no—
329C22—21no—
330C22—22no—
331C22—42no—
332C22—42no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—81no—
339C22—81no—
340C22C2×#396611yescharacteristic
341C22#6551411yescharacteristic
342C22C2×#396911yescharacteristic
343C4—21no—
344C4—21no—
345C4—22no—
346C4—42no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—81no—
351C4—81no—
352C4—81no—
353C4—404no—
354C2—21no—
355C2—21no—
356C2—42no—
357C2—81no—
358C2#1060248111yescharacteristic
359C2#1060250911yescharacteristic
360C2#1060265311yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C22⋊C4C22×D5
FittingC5×#2580C2
socleC22×C5C2×#99
prime radicalC22×C5×D4D4
cosocleC2×C5×C22⋊C4C23
hypercenter#2580D5
hypocenterC5#11784
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2580

Complements

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

NH ≅ G/Ntype
4. #10598142C2semidirect
5. #10598137C2semidirect
6. C2×C4×C5×Q8C4semidirect
8. C5×#399C4semidirect
10. C5×#393C4semidirect
16. C5×#382C4semidirect
107. C2×C4×Q8Dic5semidirect
111. #399Dic5semidirect
113. #393Dic5semidirect
119. #382Dic5semidirect
327. C5#11784semidirect

Maximal subgroups

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

subgroupindex
C5×#25802
C4×#39662
#106023952
#105981422
#105981372
#117845