Finite Group Databasebeta

← #12009806

Subgroups — #12009806

Order 1280.

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

normal subgroups only

This group has 428 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×#2599C211yesmaximal, characteristic
2#10602380C211yesmaximal, characteristic
3#10602365C211yesmaximal, characteristic
4#10598141C212yesmaximal, complemented
5#10598139C212yesmaximal, complemented
6C2×C4×C5×Q8C411yescharacteristic
7C2×C5×#117C2211yescharacteristic
8C22×#681—21no—
9C5×#398C2211yescharacteristic
10C5×#397C411yescharacteristic
11C5×#395C2211yescharacteristic
12C5×#393C411yescharacteristic
13C5×#388—21no—
14C5×#387—21no—
15C5×#387—21no—
16C5×#385—21no—
17C5×#383C411yescharacteristic
18#65215—21no—
19#64750—21no—
20#64750C2212yes—
21#64748C2212yes—
22#64745—21no—
23#11763—51nomaximal, Sylow
24C22×C5×Q8C2×C411yescharacteristic
25C23×C4×C5D411yescharacteristic
26C4×C5×Q8—41no—
27C4×C5×Q8—41no—
28C5×#117—41no—
29C5×#117—41no—
30C2×C5×C4⋊C4—21no—
31C2×C5×C4⋊C4—21no—
32C2×C5×C4⋊C4—21no—
33C2×C5×C4⋊C4C2×C411yescharacteristic
34C2×C5×C4⋊C4D411yescharacteristic
35C2×C5×C4⋊C4C2311yescharacteristic
36C2×C5×C22⋊C4—21no—
37C2×C5×C22⋊C4—21no—
38C2×C5×C22⋊C4C2×C411yescharacteristic
39C2×C5×C22⋊C4D411yescharacteristic
40C2×C42×C5—21no—
41C2×C42×C5C2×C411yescharacteristic
42C2×C42×C5D411yescharacteristic
43C2×#681—24no—
44C2×#681—22no—
45C2×#681—41no—
46C2×#681—42no—
47C22×#168—41no—
48C5×#95—21no—
49C5×#95—21no—
50C5×#95—21no—
51C5×#95—21no—
52C5×#95—21no—
53C5×#95—21no—
54C5×#95C2×C411yescharacteristic
55C5×#95C2×C411yescharacteristic
56C22⋊#168—22no—
57C4⋊#168—41no—
58#3935—41no—
59#2599D511yescharacteristic
60#1742—51no—
61#1464—51no—
62#1354—52no—
63#1346—52no—
64C24×C5C22⋊C411yescharacteristic
65C2×C5×Q8—42no—
66C2×C5×Q8—41no—
67C22×C4×C5—22no—
68C22×C4×C5—21no—
69C22×C4×C5—21no—
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×C5C22⋊C411yescharacteristic
77C22×C4×C5C22⋊C411yescharacteristic
78C22×C4×C5C22⋊C411yescharacteristic
79C22×C4×C5C22×C411yescharacteristic
80C22×C4×C5C2×D411yescharacteristic
81C22×C4×C5C2×D411yescharacteristic
82C5×C4⋊C4—41no—
83C5×C4⋊C4—41no—
84C5×C4⋊C4—41no—
85C5×C4⋊C4—42no—
86C5×C4⋊C4—41no—
87C5×C4⋊C4—41no—
88C5×C4⋊C4—41no—
89C5×C4⋊C4—41no—
90C5×C22⋊C4—41no—
91C5×C22⋊C4—41no—
92C5×C22⋊C4—42no—
93C5×C22⋊C4—42no—
94C42×C5—21no—
95C42×C5—21no—
96C42×C5—21no—
97C42×C5—21no—
98C42×C5—42no—
99#681—24no—
100#681—44no—
101#681—42no—
102C2×#168—44no—
103C2×#168—42no—
104C2×#168—42no—
105C22×M16—101no—
106C2×C4×Q8Dic511yescharacteristic
107C2×#117C2×D511yescharacteristic
108#432—101no—
109#398C2×D511yescharacteristic
110#397Dic511yescharacteristic
111#395C2×D511yescharacteristic
112#393Dic511yescharacteristic
113#388—21no—
114#387—21no—
115#387—21no—
116#385—21no—
117#383Dic511yescharacteristic
118#343—52no—
119#343—101no—
120#324—52no—
121#323—101no—
122C23×C5—21no—
123C23×C5—21no—
124C23×C5—41no—
125C23×C5—41no—
126C23×C5#10412yes—
127C23×C5C2×C22⋊C411yescharacteristic
128C5×Q8—42no—
129C5×Q8—42no—
130C2×C4×C5—22no—
131C2×C4×C5—22no—
132C2×C4×C5—22no—
133C2×C4×C5—22no—
134C2×C4×C5—21no—
135C2×C4×C5—21no—
136C2×C4×C5—21no—
137C2×C4×C5—21no—
138C2×C4×C5—22no—
139C2×C4×C5—21no—
140C2×C4×C5—21no—
141C2×C4×C5—21no—
142C2×C4×C5—21no—
143C2×C4×C5—41no—
144C2×C4×C5—41no—
145C2×C4×C5—41no—
146C2×C4×C5—41no—
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#9911yescharacteristic
159C2×C4×C5#9911yescharacteristic
160C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C5#13711yescharacteristic
162#168—44no—
163#168—82no—
164C22×Q8C2×Dic511yescharacteristic
165C23×C4C22⋊D511yescharacteristic
166C2×M16—104no—
167C2×M16—102no—
168C2×M16—201no—
169C2×M16—202no—
170C22×C8—201no—
171C4×Q8—41no—
172C4×Q8—41no—
173#117—41no—
174#117—41no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4C2×Dic511yescharacteristic
179C2×C4⋊C4C22⋊D511yescharacteristic
180C2×C4⋊C4C22×D511yescharacteristic
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4C2×Dic511yescharacteristic
184C2×C22⋊C4C22⋊D511yescharacteristic
185C2×C42—21no—
186C2×C42C2×Dic511yescharacteristic
187C2×C42C22⋊D511yescharacteristic
188C4⋊C8—201no—
189C22⋊C8—102no—
190#97—201no—
191#95—21no—
192#95—21no—
193#95—21no—
194#95—21no—
195#95—21no—
196#95—21no—
197#95C2×Dic511yescharacteristic
198#95C2×Dic511yescharacteristic
199C22×C5—21no—
200C22×C5—21no—
201C22×C5—22no—
202C22×C5—42no—
203C22×C5—41no—
204C22×C5—41no—
205C22×C5—81no—
206C22×C5C2×#9911yescharacteristic
207C22×C5#41711yescharacteristic
208C22×C5C2×#10411yescharacteristic
209C4×C5—21no—
210C4×C5—21no—
211C4×C5—21no—
212C4×C5—21no—
213C4×C5—41no—
214C4×C5—42no—
215C4×C5—41no—
216C4×C5—41no—
217C4×C5—41no—
218C4×C5—81no—
219C4×C5—81no—
220C4×C5—81no—
221C24C22⋊Dic511yescharacteristic
222C2×Q8—42no—
223C2×Q8—41no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4C22⋊Dic511yescharacteristic
234C22×C4C22⋊Dic511yescharacteristic
235C22×C4C22⋊Dic511yescharacteristic
236C22×C4C22×Dic511yescharacteristic
237C22×C4C2×C22⋊D511yescharacteristic
238C22×C4C2×C22⋊D511yescharacteristic
239M16—104no—
240M16—204no—
241M16—202no—
242C2×C8—204no—
243C2×C8—202no—
244C2×C8—202no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—42no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C42—21no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—42no—
262C2×C5—21no—
263C2×C5—21no—
264C2×C5—42no—
265C2×C5#150411yescharacteristic
266C2×C5#212011yescharacteristic
267C2×C5#213511yescharacteristic
268C23—21no—
269C23—21no—
270C23—41no—
271C23—41no—
272C23#396912yes—
273C23C2×C22⋊Dic511yescharacteristic
274Q8—42no—
275Q8—42no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4#396611yescharacteristic
305C2×C4#396611yescharacteristic
306C2×C4#409511yescharacteristic
307C2×C4#409711yescharacteristic
308C8—204no—
309C8—402no—
310C5#1176311yescharacteristic, complemented, Sylow
311C22—21no—
312C22—21no—
313C22—22no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—81no—
318C22C2×#396611yescharacteristic
319C22#6552111yescharacteristic
320C22C2×#396911yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—42no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—81no—
331C4—81no—
332C4—81no—
333C2—21no—
334C2—21no—
335C2—42no—
336C2#1060250111yescharacteristic
337C2#1060250911yescharacteristic
338C2#1060266611yescharacteristic
subgroupG/H
centerC22C2×#3966
commutatorC22×C4×C5C22×C4
FrattiniC2×C4⋊C4C22×D5
FittingC5×#2599C2
socleC22×C5C2×#99
prime radicalC24×C5C22⋊C4
cosocleC2×C5×C4⋊C4C23
hypercenter#2599D5
hypocenterC5#11763
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2599

Complements

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

NH ≅ G/Ntype
4. #10598141C2semidirect
5. #10598139C2semidirect
310. C5#11763semidirect

Maximal subgroups

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

subgroupindex
C5×#25992
#106023802
#106023652
#105981412
#105981392
#117635