Finite Group Databasebeta

← #12002816

Subgroups — #12002816

Order 1280.

630 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 127 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
1#10606099C211yesmaximal, characteristic
2C5×#1844C211yesmaximal, characteristic, complemented
3C2×#64679C212yesmaximal, complemented
4#10600377C212yesmaximal, complemented
5#10598125C211yesmaximal, characteristic, complemented
6C2×#4041—22no—
7C2×C4×D20C2212yes—
8C2×C5×#106—22no—
9C2×C4×C5×C8C2211yescharacteristic
10C5×#384C2211yescharacteristic
11C5×#384C2211yescharacteristic
12#65280—22no—
13#65276—22no—
14C2×#3941—22no—
15C2×C4⋊#168C2212yes—
16#64749—22no—
17#64679—24no—
18#10432—51nomaximal, Sylow
19C22×D20—22no—
20C22×D20D412yes—
21C22×C4×D5—22no—
22C22×C5×C8—21no—
23C2×C5×C4⋊C4—22no—
24C2×C5×C4⋊C4—21no—
25C2×C5×C4⋊C4—22no—
26C2×C5×C4⋊C4D412yes—
27C2×C42×C5C2311yescharacteristic
28C2×#685—24no—
29C2×#685—22no—
30C2×C4⋊Dic5D412yes—
31C22×#168—22no—
32#4041—44no—
33#4041—44no—
34C4×D20—22no—
35C4×D20—22no—
36C4×D20—22no—
37C4×D20—22no—
38C5×#106—44no—
39C4×C5×C8—21no—
40C4×C5×C8D412yescomplemented
41#3963—22no—
42#3941—44no—
43C4⋊#168—22no—
44C4⋊#168—22no—
45#2537—51no—
46#1844D511yescharacteristic, complemented
47#1802—52no—
48C2×D4⋊C8—52no—
49#1296—51no—
50C23×D5—22no—
51C22×C4×C5—22no—
52C22×C4×C5—41no—
53C22×C4×C5C2×D412yes—
54C22×C4×C5C2×D411yescharacteristic
55C22×Dic5—22no—
56C2×D20—22no—
57C2×D20—22no—
58C2×D20—22no—
59C2×D20—22no—
60C2×D20—22no—
61C2×D20—48no—
62C2×D20—82no—
63C2×D20SD1614yes—
64C2×C4×D5—42no—
65C2×C4×D5—42no—
66C2×C4×D5—42no—
67C2×C4×D5—42no—
68C2×C4×D5—42no—
69C2×C4×D5—42no—
70C2×C5×C8—22no—
71C2×C5×C8—22no—
72C2×C5×C8—22no—
73C5×C4⋊C4—42no—
74C5×C4⋊C4—42no—
75C5×C4⋊C4—42no—
76C5×C4⋊C4—42no—
77C5×C4⋊C4—42no—
78C5×C4⋊C4—42no—
79C42×C5C2×D411yescharacteristic
80C42×C5#4111yescharacteristic
81C42×C5#4112yes—
82#685—48no—
83#685—42no—
84#685—42no—
85C4⋊Dic5—22no—
86C4⋊Dic5—22no—
87C2×#168—42no—
88C2×#168—42no—
89C2×#168—42no—
90C2×#121—102no—
91C2×C4×D4—52no—
92C2×#106—22no—
93C2×C4⋊C8—52no—
94C2×#102—102no—
95C2×C4×C8C2×D511yescharacteristic
96#394—102no—
97#384C2×D511yescharacteristic
98#384C2×D511yescharacteristic
99#380—102no—
100#340—102no—
101D4⋊C8—104no—
102C23×C5#12011yescharacteristic
103C22×D5—24no—
104C22×D5—42no—
105C22×D5—42no—
106C22×D5—42no—
107C22×D5—42no—
108C22×D5—42no—
109C22×D5—42no—
110C2×C4×C5—21no—
111C2×C4×C5—21no—
112C2×C4×C5—44no—
113C2×C4×C5—42no—
114C2×C4×C5—82no—
115C2×C4×C5—81no—
116C2×C4×C5#12112yes—
117C2×C4×C5#12111yescharacteristic
118C2×C4×C5#12312yes—
119C2×C4×C5#12411yescharacteristic
120C2×C4×C5C2×SD1612yes—
121C2×C4×C5#13512yes—
122C2×C4×C5AGL(1,ℤ/8)12yes—
123C2×C4×C5AGL(1,ℤ/8)12yes—
124C2×Dic5—42no—
125C2×Dic5—42no—
126C2×Dic5—42no—
127D20—24no—
128D20—24no—
129D20—42no—
130D20—42no—
131D20—88no—
132C4×D5—82no—
133C4×D5—82no—
134C4×D5—82no—
135C4×D5—82no—
136C5×C8—24no—
137#168—82no—
138C22×D4—52no—
139C22×D4—102no—
140C23×C4—102no—
141C22×C8—21no—
142C22×C8—102no—
143#121—204no—
144#121—204no—
145C4×D4—102no—
146C4×D4—102no—
147C4×D4—102no—
148C4×D4—102no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—52no—
153C2×C4⋊C4C22⋊D512yes—
154C2×C22⋊C4—104no—
155C2×C22⋊C4—102no—
156C2×C42C22×D511yescharacteristic
157#106—44no—
158C4⋊C8—102no—
159C4⋊C8—102no—
160#102—204no—
161C4×C8—21no—
162C4×C8D2012yescomplemented
163#95—102no—
164C22×C5#39411yescharacteristic
165C22×C5#44912yes—
166C22×C5#45012yes—
167C22×C5#45311yescharacteristic
168C22×C5#45611yescharacteristic
169C2×D5—44no—
170C2×D5—44no—
171C2×D5—44no—
172C2×D5—82no—
173C2×D5—82no—
174C2×D5—82no—
175C2×D5—82no—
176C4×C5—21no—
177C4×C5—21no—
178C4×C5—82no—
179C4×C5—161no—
180C4×C5#46512yes—
181C4×C5#46912yes—
182C4×C5#48112yes—
183C4×C5#48212yes—
184Dic5—82no—
185C24—102no—
186C2×D4—54no—
187C2×D4—102no—
188C2×D4—102no—
189C2×D4—102no—
190C2×D4—102no—
191C2×D4—102no—
192C2×D4—208no—
193C2×D4—402no—
194C22×C4—22no—
195C22×C4—41no—
196C22×C4—102no—
197C22×C4—202no—
198C22×C4—202no—
199C22×C4—202no—
200C22×C4—202no—
201C22×C4—202no—
202C22×C4—202no—
203C22×C4D4×D512yes—
204C22×C4C2×C22⋊D511yescharacteristic
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—202no—
209C2×C8—202no—
210C2×C8—202no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—102no—
218C4⋊C4—102no—
219C22⋊C4—208no—
220C22⋊C4—202no—
221C22⋊C4—202no—
222C42C2×D2011yescharacteristic
223C42#70911yescharacteristic
224C42#71312yes—
225C2×C5#162411yescharacteristic
226C2×C5#162911yescharacteristic
227C2×C5#165412yes—
228C2×C5#203412yes—
229C2×C5#203911yescharacteristic
230D5—84no—
231C23—104no—
232C23—202no—
233C23—202no—
234C23—202no—
235C23—202no—
236C23—202no—
237C23—202no—
238C23#408311yescharacteristic
239D4—104no—
240D4—104no—
241D4—202no—
242D4—202no—
243D4—408no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—42no—
247C2×C4—44no—
248C2×C4—81no—
249C2×C4—82no—
250C2×C4—202no—
251C2×C4—202no—
252C2×C4—202no—
253C2×C4—402no—
254C2×C4—402no—
255C2×C4—402no—
256C2×C4—402no—
257C2×C4#404012yes—
258C2×C4#404112yes—
259C2×C4D5×SD1612yes—
260C2×C4#406012yes—
261C2×C4#406012yes—
262C2×C4#406212yes—
263C2×C4#407611yescharacteristic
264C2×C4#409311yescharacteristic
265C8—24no—
266C8—402no—
267C5#1043211yescharacteristic, complemented, Sylow
268C22—204no—
269C22—204no—
270C22—204no—
271C22—402no—
272C22—402no—
273C22—402no—
274C22—402no—
275C22#6528011yescharacteristic
276C22#6536611yescharacteristic
277C22#6538411yescharacteristic
278C22#6545812yes—
279C22#6546112yes—
280C4—21no—
281C4—21no—
282C4—82no—
283C4—161no—
284C4—402no—
285C4#6515212yes—
286C4#6515312yes—
287C4#6515412yes—
288C4#6515512yes—
289C2—404no—
290C2#1060038611yescharacteristic
291C2#1060137412yes—
292C2#1060205811yescharacteristic
293C2#1060205911yescharacteristic
294C2#1060213612yes—
subgroupG/H
centerC23#4083
commutator, cosocleC2×C42×C5C23
FrattiniC2×C42C22×D5
FittingC5×#1844C2
socleC23×C5#120
prime radical#10606099C2
hypercenter#1844D5
hypocenterC5#10432
LCSG ▷ C2×C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1844

Complements

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

NH ≅ G/Ntype
2. C5×#1844C2semidirect
3. C2×#64679C2semidirect
4. #10600377C2semidirect
5. #10598125C2semidirect
40. C4×C5×C8D4semidirect
46. #1844D5semidirect
162. C4×C8D20semidirect
267. C5#10432semidirect

Maximal subgroups

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

subgroupindex
#106060992
C5×#18442
C2×#646792
#106003772
#105981252
#104325