Finite Group Databasebeta

← #12002959

Subgroups — #12002959

Order 1280.

408 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#65131C211yesmaximal, characteristic
2C5×#1571C211yesmaximal, characteristic, complemented
3#10601572C211yesmaximal, characteristic, complemented
4#10601549C211yesmaximal, characteristic
5#10601320C211yesmaximal, characteristic
6#10601301C211yesmaximal, characteristic, complemented
7#10598198C211yesmaximal, characteristic, complemented
8C2×C4×D20C2211yescharacteristic
9C2×#4015C2211yescharacteristic
10C5×#431C2211yescharacteristic
11C5×#423C2211yescharacteristic
12C2×C5×C4⋊C8C2211yescharacteristic
13C2×#3955—21no—
14C2×#3953—21no—
15C2×#3949—21no—
16#65214C2211yescharacteristic
17#65213C2211yescharacteristic
18#65131—22no—
19#65131—22no—
20C5×#335—22no—
21#64774—21no—
22#64706—22no—
23#64703—22no—
24#64701—22no—
25#10453—51nomaximal, Sylow
26C22×D20D411yescharacteristic
27C22×C4×D5—21no—
28C22×Dic10—21no—
29C2×C5×M16—21no—
30C22×C5×C8—21no—
31C2×C42×C5C2311yescharacteristic
32C2×#685—21no—
33C2×C4⋊Dic5—21no—
34C2×C4⋊Dic5—21no—
35C2×C4⋊Dic5D411yescharacteristic
36C2×#681—21no—
37C4×D20—21no—
38C4×D20—21no—
39C4×D20—21no—
40C4×D20—21no—
41#4015—21no—
42#4015—21no—
43#4015—21no—
44#4015D412yes—
45C5×C4⋊C8—21no—
46C5×C4⋊C8—21no—
47C5×C4⋊C8—21no—
48C5×C4⋊C8D412yes—
49C5×C4⋊C8D412yes—
50C5×C22⋊C8—21no—
51C4×C5×C8—21no—
52C22⋊#168—21no—
53#3955—42no—
54#3953—24no—
55#3949—24no—
56C4⋊#168—21no—
57C4⋊#168—21no—
58C4⋊#168—21no—
59C4×#168—21no—
60C2×#476—51no—
61#1800—51no—
62#1571D511yescharacteristic, complemented
63#1549—51no—
64#1548—51no—
65#1478—51no—
66#1301—51no—
67C23×D5—21no—
68C22×C4×C5C2×D411yescharacteristic
69C22×C4×C5#4111yescharacteristic
70C22×C4×C5#4111yescharacteristic
71C22×Dic5—21no—
72C22×Dic5—41no—
73C2×D20—21no—
74C2×D20—21no—
75C2×D20—21no—
76C2×D20—21no—
77C2×D20—21no—
78C2×D20Q1612yes—
79C2×C4×D5—41no—
80C2×C4×D5—41no—
81C2×C4×D5—82no—
82C2×Dic10—22no—
83C2×Dic10—41no—
84C2×Dic10—82no—
85C5×M16—41no—
86C5×M16—41no—
87C2×C5×C8—22no—
88C2×C5×C8—22no—
89C2×C5×C8—41no—
90C2×C5×C8—41no—
91C2×C5×C8—41no—
92C42×C5C2×D411yescharacteristic
93C42×C5C2×D411yescharacteristic
94C42×C5C2×D411yescharacteristic
95C42×C5#4111yescharacteristic
96#685—41no—
97#685—41no—
98C4⋊Dic5—21no—
99C4⋊Dic5—22no—
100C4⋊Dic5—21no—
101C4⋊Dic5—22no—
102C4⋊Dic5—41no—
103C4⋊Dic5—41no—
104#681—41no—
105#681—41no—
106C2×#168—41no—
107C2×#168—41no—
108C2×C4⋊Q8—51no—
109C2×C4×D4—51no—
110#476—102no—
111#476—102no—
112#431—51no—
113#431C2×D511yescharacteristic
114C2×#106—101no—
115#423—51no—
116#423C2×D511yescharacteristic
117C2×C4⋊C8C2×D511yescharacteristic
118C2×#102—101no—
119C2×#100—101no—
120#343—101no—
121#335—22no—
122#333—102no—
123#332—102no—
124D4⋊C8—102no—
125C23×C5#12311yescharacteristic
126C22×D5—22no—
127C22×D5—41no—
128C22×D5—41no—
129C22×D5—81no—
130C22×D5—81no—
131C2×C4×C5—22no—
132C2×C4×C5—41no—
133C2×C4×C5#12011yescharacteristic
134C2×C4×C5#12111yescharacteristic
135C2×C4×C5#12111yescharacteristic
136C2×C4×C5#12111yescharacteristic
137C2×C4×C5#12311yescharacteristic
138C2×C4×C5#12411yescharacteristic
139C2×C4×C5C2×Q1611yescharacteristic
140C2×C4×C5#13511yescharacteristic
141C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C5#13711yescharacteristic
143C2×Dic5—42no—
144C2×Dic5—41no—
145C2×Dic5—82no—
146C2×Dic5—81no—
147D20—22no—
148D20—22no—
149D20—41no—
150D20—41no—
151C4×D5—82no—
152C4×D5—82no—
153Dic10—82no—
154Dic10—82no—
155C5×C8—42no—
156C5×C8—42no—
157C5×C8—81no—
158#168—81no—
159#168—81no—
160C22×Q8—101no—
161C22×D4—51no—
162C23×C4—101no—
163C2×M16—21no—
164C2×M16—101no—
165C22×C8—21no—
166C4⋊Q8—52no—
167C4⋊Q8—101no—
168C4⋊Q8—101no—
169C4⋊Q8—101no—
170C4×D4—101no—
171C4×D4—101no—
172C4×D4—101no—
173C4×D4—101no—
174C2×C4⋊C4—51no—
175C2×C4⋊C4—101no—
176C2×C4⋊C4—101no—
177C2×C22⋊C4—101no—
178C2×C42C22×D511yescharacteristic
179#106—104no—
180C4⋊C8—21no—
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8—101no—
184C4⋊C8—101no—
185C4⋊C8—101no—
186C4⋊C8D2012yes—
187C4⋊C8C22⋊D512yes—
188#102—104no—
189#100—202no—
190C22⋊C8—21no—
191C22⋊C8—101no—
192C4×C8—21no—
193C4×C8—101no—
194C22×C5—21no—
195C22×C5—21no—
196C22×C5#39411yescharacteristic
197C22×C5#48411yescharacteristic
198C22×C5#48511yescharacteristic
199C2×D5—42no—
200C2×D5—82no—
201C2×D5—81no—
202C2×D5—81no—
203C2×D5—82no—
204C4×C5—21no—
205C4×C5—21no—
206C4×C5—21no—
207C4×C5#44911yescharacteristic
208C4×C5#45111yescharacteristic
209C4×C5#46711yescharacteristic
210C4×C5#46811yescharacteristic
211C4×C5#47112yes—
212Dic5—82no—
213Dic5—81no—
214C24—101no—
215C2×Q8—102no—
216C2×Q8—201no—
217C2×Q8—402no—
218C2×D4—52no—
219C2×D4—101no—
220C2×D4—101no—
221C2×D4—101no—
222C2×D4—101no—
223C2×D4—101no—
224C22×C4—101no—
225C22×C4—201no—
226C22×C4—201no—
227C22×C4—201no—
228C22×C4—402no—
229C22×C4#70911yescharacteristic
230C22×C4D4×D511yescharacteristic
231C22×C4#71311yescharacteristic
232M16—41no—
233M16—41no—
234M16—201no—
235M16—201no—
236C2×C8—22no—
237C2×C8—22no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—201no—
242C2×C8—201no—
243C4⋊C4—102no—
244C4⋊C4—101no—
245C4⋊C4—102no—
246C4⋊C4—101no—
247C4⋊C4—201no—
248C4⋊C4—201no—
249C22⋊C4—201no—
250C22⋊C4—201no—
251C42C2×D2011yescharacteristic
252C42D4×D511yescharacteristic
253C42#71311yescharacteristic
254C42C2×C22⋊D511yescharacteristic
255C2×C5—21no—
256C2×C5—21no—
257C2×C5#203611yescharacteristic
258C2×C5#204611yescharacteristic
259C2×C5#204811yescharacteristic
260D5—82no—
261C23—102no—
262C23—201no—
263C23—201no—
264C23—401no—
265C23—401no—
266C23#404011yescharacteristic
267Q8—402no—
268Q8—402no—
269D4—102no—
270D4—102no—
271D4—201no—
272D4—201no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—201no—
276C2×C4—202no—
277C2×C4—402no—
278C2×C4—402no—
279C2×C4—402no—
280C2×C4—401no—
281C2×C4#404011yescharacteristic
282C2×C4#404111yescharacteristic
283C2×C4#404111yescharacteristic
284C2×C4#405711yescharacteristic
285C2×C4#405811yescharacteristic
286C2×C4D5×Q1611yescharacteristic
287C2×C4#406411yescharacteristic
288C2×C4#407611yescharacteristic
289C2×C4#408311yescharacteristic
290C2×C4#409311yescharacteristic
291C8—42no—
292C8—42no—
293C8—81no—
294C8—401no—
295C8—401no—
296C5#1045311yescharacteristic, complemented, Sylow
297C22—21no—
298C22—21no—
299C22—202no—
300C22—402no—
301C22—402no—
302C22—401no—
303C22—401no—
304C22#6517311yescharacteristic
305C22#6517711yescharacteristic
306C22#6528011yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—402no—
311C4—401no—
312C4#6517511yescharacteristic
313C4#6517611yescharacteristic
314C4#6543012yes—
315C4#6544711yescharacteristic
316C4#6547511yescharacteristic
317C2—21no—
318C2—21no—
319C2—402no—
320C2#1060133411yescharacteristic
321C2#1060142811yescharacteristic
322C2#1060152911yescharacteristic
subgroupG/H
centerC22#65280
commutator, cosocleC2×C42×C5C23
FrattiniC2×C42C22×D5
FittingC5×#1571C2
socleC22×C5#394
prime radicalC22×D20D4
hypercenter#1571D5
hypocenterC5#10453
LCSG ▷ C2×C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1571

Complements

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

NH ≅ G/Ntype
2. C5×#1571C2semidirect
3. #10601572C2semidirect
6. #10601301C2semidirect
7. #10598198C2semidirect
62. #1571D5semidirect
296. C5#10453semidirect

Maximal subgroups

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

subgroupindex
C2×#651312
C5×#15712
#106015722
#106015492
#106013202
#106013012
#105981982
#104535