Finite Group Databasebeta

← #12008861

Subgroups — #12008861

Order 1280.

408 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 87 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×#2500C211yesmaximal, characteristic
2#10602365C211yesmaximal, characteristic, complemented
3#10602364C211yesmaximal, characteristic, complemented
4#10598131C212yesmaximal, complemented
5#10598123C212yesmaximal, complemented
6C2×C5×#121C411yescharacteristic
7C2×C5×#117C2211yescharacteristic
8C5×#394—21no—
9C5×#394—21no—
10C5×#390C411yescharacteristic
11C5×#390C2211yescharacteristic
12C5×#388—21no—
13C5×#388—21no—
14C5×#384C411yescharacteristic
15C5×#382C2211yescharacteristic
16C4×C5×C22⋊C4C411yescharacteristic
17#65215—21no—
18#65214—21no—
19#64748C2212yes—
20#64748C2212yes—
21#64745—21no—
22#64745—21no—
23#11550—51nomaximal, Sylow
24C22×C5×D4—21no—
25C22×C5×D4C2×C411yescharacteristic
26C23×C4×C5C2×C411yescharacteristic
27C5×#121—41no—
28C5×#121—41no—
29C5×#117—41no—
30C5×#117—41no—
31C2×C5×C4⋊C4—21no—
32C2×C5×C4⋊C4—21no—
33C2×C5×C4⋊C4C2×C411yescharacteristic
34C2×C5×C4⋊C4C2×C411yescharacteristic
35C2×C5×C4⋊C4C2311yescharacteristic
36C2×C5×C22⋊C4—21no—
37C2×C5×C22⋊C4—21no—
38C2×C5×C22⋊C4—21no—
39C2×C5×C22⋊C4—21no—
40C2×C5×C22⋊C4—21no—
41C2×C5×C22⋊C4—21no—
42C2×C5×C22⋊C4D411yescharacteristic
43C2×C5×C22⋊C4D411yescharacteristic
44C2×C42×C5C2×C411yescharacteristic
45C2×C42×C5C2×C411yescharacteristic
46C2×C42×C5D411yescharacteristic
47C2×C42×C5D411yescharacteristic
48C5×#95—21no—
49C5×#95—21no—
50C5×#95—21no—
51C22⋊#168—22no—
52C22⋊#168—22no—
53C4⋊#168—41no—
54C4⋊#168—41no—
55#3935—41no—
56C4×#168—41no—
57#2500D511yescharacteristic
58#1464—51no—
59#1458—51no—
60#1324—52no—
61#1318—52no—
62C24×C5—21no—
63C24×C5C22⋊C411yescharacteristic
64C2×C5×D4—22no—
65C2×C5×D4—41no—
66C2×C5×D4—41no—
67C2×C5×D4—42no—
68C2×C5×D4—81no—
69C2×C5×D4—81no—
70C22×C4×C5—21no—
71C22×C4×C5—21no—
72C22×C4×C5—21no—
73C22×C4×C5—22no—
74C22×C4×C5—21no—
75C22×C4×C5—21no—
76C22×C4×C5—41no—
77C22×C4×C5C22⋊C411yescharacteristic
78C22×C4×C5C22⋊C411yescharacteristic
79C22×C4×C5C22⋊C411yescharacteristic
80C22×C4×C5C22×C411yescharacteristic
81C22×C4×C5C2×D411yescharacteristic
82C22×C4×C5C2×D411yescharacteristic
83C5×C4⋊C4—41no—
84C5×C4⋊C4—41no—
85C5×C4⋊C4—41no—
86C5×C4⋊C4—81no—
87C5×C4⋊C4—81no—
88C5×C22⋊C4—41no—
89C5×C22⋊C4—41no—
90C5×C22⋊C4—42no—
91C5×C22⋊C4—42no—
92C5×C22⋊C4—42no—
93C5×C22⋊C4—41no—
94C5×C22⋊C4—41no—
95C5×C22⋊C4—81no—
96C5×C22⋊C4—81no—
97C42×C5—21no—
98C42×C5—21no—
99C42×C5—21no—
100C42×C5—21no—
101C42×C5—22no—
102C42×C5—41no—
103C2×#168—42no—
104C2×#168—42no—
105C2×#121Dic511yescharacteristic
106C2×#117C2×D511yescharacteristic
107#432—101no—
108#431—101no—
109#394—21no—
110#394—21no—
111#390Dic511yescharacteristic
112#390C2×D511yescharacteristic
113#388—21no—
114#388—21no—
115#384Dic511yescharacteristic
116#382C2×D511yescharacteristic
117C4×C22⋊C4Dic511yescharacteristic
118#324—52no—
119#324—52no—
120#323—101no—
121#323—101no—
122C23×C5—21no—
123C23×C5—21no—
124C23×C5—21no—
125C23×C5—41no—
126C23×C5—41no—
127C23×C5—41no—
128C23×C5—41no—
129C23×C5—41no—
130C23×C5—41no—
131C23×C5—41no—
132C23×C5—41no—
133C23×C5—41no—
134C23×C5C2×C22⋊C411yescharacteristic
135C5×D4—82no—
136C5×D4—84no—
137C2×C4×C5—21no—
138C2×C4×C5—21no—
139C2×C4×C5—21no—
140C2×C4×C5—21no—
141C2×C4×C5—22no—
142C2×C4×C5—21no—
143C2×C4×C5—21no—
144C2×C4×C5—21no—
145C2×C4×C5—21no—
146C2×C4×C5—42no—
147C2×C4×C5—42no—
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#9911yescharacteristic
161C2×C4×C5#9911yescharacteristic
162C2×C4×C5#13112yes—
163#168—82no—
164#168—82no—
165C22×D4—21no—
166C22×D4C2×Dic511yescharacteristic
167C23×C4C2×Dic511yescharacteristic
168#121—41no—
169#121—41no—
170#117—41no—
171#117—41no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4C2×Dic511yescharacteristic
175C2×C4⋊C4C2×Dic511yescharacteristic
176C2×C4⋊C4C22×D511yescharacteristic
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4C22⋊D511yescharacteristic
184C2×C22⋊C4C22⋊D511yescharacteristic
185C2×C42C2×Dic511yescharacteristic
186C2×C42C2×Dic511yescharacteristic
187C2×C42C22⋊D511yescharacteristic
188C2×C42C22⋊D511yescharacteristic
189C4⋊C8—201no—
190C4⋊C8—201no—
191C22⋊C8—102no—
192C22⋊C8—102no—
193#97—201no—
194C4×C8—201no—
195#95—21no—
196#95—21no—
197#95—21no—
198C22×C5—21no—
199C22×C5—21no—
200C22×C5—41no—
201C22×C5—41no—
202C22×C5—41no—
203C22×C5—81no—
204C22×C5—81no—
205C22×C5—81no—
206C22×C5—81no—
207C22×C5—81no—
208C22×C5—81no—
209C22×C5—81no—
210C22×C5—81no—
211C22×C5—81no—
212C22×C5#40811yescharacteristic
213C22×C5C2×#9911yescharacteristic
214C22×C5#41311yescharacteristic
215C4×C5—21no—
216C4×C5—21no—
217C4×C5—22no—
218C4×C5—42no—
219C4×C5—41no—
220C4×C5—41no—
221C4×C5—41no—
222C4×C5—41no—
223C4×C5—81no—
224C4×C5—81no—
225C24—21no—
226C24C22⋊Dic511yescharacteristic
227C2×D4—22no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—42no—
231C2×D4—81no—
232C2×D4—81no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—41no—
240C22×C4C22⋊Dic511yescharacteristic
241C22×C4C22⋊Dic511yescharacteristic
242C22×C4C22⋊Dic511yescharacteristic
243C22×C4C22×Dic511yescharacteristic
244C22×C4C2×C22⋊D511yescharacteristic
245C22×C4C2×C22⋊D511yescharacteristic
246C2×C8—202no—
247C2×C8—202no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—81no—
252C4⋊C4—81no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—81no—
261C22⋊C4—81no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—22no—
267C42—41no—
268C2×C5—21no—
269C2×C5—21no—
270C2×C5—81no—
271C2×C5—161no—
272C2×C5#146811yescharacteristic
273C2×C5#212611yescharacteristic
274C2×C5#213411yescharacteristic
275C23—21no—
276C23—21no—
277C23—21no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23C2×C22⋊Dic511yescharacteristic
288D4—82no—
289D4—84no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
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#396611yescharacteristic
314C2×C4#396611yescharacteristic
315C2×C4#409412yes—
316C8—402no—
317C8—402no—
318C5#1155011yescharacteristic, complemented, Sylow
319C22—21no—
320C22—21no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22C2×#396611yescharacteristic
334C22#6552311yescharacteristic
335C22#6552611yescharacteristic
336C4—21no—
337C4—21no—
338C4—22no—
339C4—42no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—81no—
345C4—81no—
346C2—21no—
347C2—21no—
348C2—81no—
349C2—161no—
350C2#1060249511yescharacteristic
351C2#1060261211yescharacteristic
352C2#1060265511yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C4⋊C4C22×D5
FittingC5×#2500C2
socleC22×C5C2×#99
prime radicalC2×C5×#121C4
cosocleC2×C5×C4⋊C4C23
hypercenter#2500D5
hypocenterC5#11550
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2500

Complements

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

NH ≅ G/Ntype
2. #10602365C2semidirect
3. #10602364C2semidirect
4. #10598131C2semidirect
5. #10598123C2semidirect
318. C5#11550semidirect

Maximal subgroups

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

subgroupindex
C5×#25002
#106023652
#106023642
#105981312
#105981232
#115505