Finite Group Databasebeta

← #12001547

Subgroups — #12001547

Order 1280.

542 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 105 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×#3162C211yesmaximal, characteristic, complemented
2#10615086C212yesmaximal
3#10602631C212yesmaximal, complemented
4#10600616C211yesmaximal, characteristic
5#10598152C211yesmaximal, characteristic, complemented
6C5×#533C2212yes—
7C2×#4020C2211yescharacteristic
8C5×#501—22no—
9C5×#501—22no—
10C5×#427—22no—
11C2×C5×#97C2211yescharacteristic
12C2×#3953—22no—
13#65377—22no—
14#65374—22no—
15#65335—24no—
16#65316—24no—
17#65225C2211yescharacteristic
18#65214C2212yes—
19#64758—24no—
20#10220—51nomaximal, Sylow
21C22×D20—21no—
22C22×D20—22no—
23C22×C5×C8D412yes—
24C5×C4⋊Q8D412yes—
25C5×C4⋊Q8D412yes—
26C5×#125—22no—
27C5×C22⋊Q8—22no—
28C5×C22⋊Q8—22no—
29C4×C5×Q8—22no—
30C4×C5×Q8—22no—
31C5×#117—22no—
32C2×C42×C5C2311yescharacteristic
33C2×#688—28no—
34C2×C4⋊Dic5—21no—
35#4020—22no—
36#4020—21no—
37#4020D412yes—
38C5×#107—24no—
39C5×#106—22no—
40C5×#106—22no—
41C5×#97—22no—
42#3963—22no—
43C22⋊#168D414yes—
44#3953—44no—
45#3941—28no—
46C4⋊#168—22no—
47C4×#168—22no—
48#3208—52no—
49#3162D511yescharacteristic, complemented
50#1961—51no—
51#1518—52no—
52#1281—51no—
53C23×D5—41no—
54C2×C5×Q8—22no—
55C2×C5×Q8—22no—
56C22×C4×C5C2×D412yes—
57C22×C4×C5C2×D411yescharacteristic
58C22×Dic5—41no—
59C2×D20—24no—
60C2×D20—22no—
61C2×D20—42no—
62C2×D20—42no—
63C2×D20—41no—
64C2×D20—42no—
65C2×D20—84no—
66C2×C5×C8—22no—
67C2×C5×C8—22no—
68C2×C5×C8SD1614yescomplemented
69C5×C4⋊C4—22no—
70C5×C4⋊C4—22no—
71C5×C4⋊C4—22no—
72C5×C4⋊C4—22no—
73C5×C4⋊C4—42no—
74C5×C4⋊C4—42no—
75C5×C22⋊C4—42no—
76C42×C5—42no—
77C42×C5C2×D411yescharacteristic
78C42×C5C2×D412yes—
79C42×C5#4111yescharacteristic
80#688—88no—
81C4⋊Dic5—42no—
82C2×#168—24no—
83#533C2×D512yes—
84C2×#127—51no—
85#501—22no—
86#501—22no—
87#492—102no—
88#486—102no—
89#460—104no—
90#450—104no—
91#431—52no—
92#427—22no—
93C2×#102—102no—
94C2×#97C2×D511yescharacteristic
95#382—51no—
96#328—104no—
97C23×C5#12711yescharacteristic
98C22×D5—42no—
99C22×D5—84no—
100C22×D5—81no—
101C22×D5—81no—
102C5×Q8—44no—
103C5×Q8—82no—
104C2×C4×C5—42no—
105C2×C4×C5—42no—
106C2×C4×C5—42no—
107C2×C4×C5#12011yescharacteristic
108C2×C4×C5#12012yes—
109C2×C4×C5#12111yescharacteristic
110C2×C4×C5#12112yes—
111C2×C4×C5C2×SD1612yes—
112C2×C4×C5#13712yes—
113C2×Dic5—82no—
114C2×Dic5—81no—
115D20—48no—
116D20—44no—
117D20—84no—
118D20—82no—
119C5×C8—24no—
120#168—44no—
121C22×D4—102no—
122C22×D4—101no—
123C2×SD16—108no—
124C22×C8C22⋊D512yes—
125C4⋊Q8C22⋊D512yes—
126C4⋊Q8C22⋊D512yes—
127#127—52no—
128#127—101no—
129#127—102no—
130#125—22no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C4×Q8—22no—
134C4×Q8—22no—
135#117—22no—
136C2×C4⋊C4—101no—
137C2×C42C22×D511yescharacteristic
138#107—24no—
139#106—22no—
140#106—22no—
141C4⋊C8—102no—
142#102—108no—
143#102—204no—
144C22⋊C8—54no—
145#97—22no—
146C4×C8—102no—
147#95—102no—
148C22×C5—22no—
149C22×C5#39211yescharacteristic
150C22×C5#49211yescharacteristic
151C22×C5#49711yescharacteristic
152C2×D5—84no—
153C2×D5—82no—
154C2×D5—162no—
155C2×D5—162no—
156C4×C5—22no—
157C4×C5—22no—
158C4×C5—82no—
159C4×C5—82no—
160C4×C5#44812yes—
161C4×C5#46112yes—
162Dic5—161no—
163C24—201no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×D4—104no—
167C2×D4—102no—
168C2×D4—201no—
169C2×D4—202no—
170C2×D4—202no—
171C2×D4—202no—
172C2×D4—404no—
173C22×C4—201no—
174C22×C4D4×D512yes—
175C22×C4C2×C22⋊D511yescharacteristic
176SD16—408no—
177C2×C8—22no—
178C2×C8—22no—
179C2×C8—104no—
180C2×C8#68814yescomplemented
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—202no—
188C22⋊C4—42no—
189C42—42no—
190C42D4×D511yescharacteristic
191C42#71111yescharacteristic
192C42C2×C22⋊D512yes—
193C2×C5—21no—
194C2×C5—21no—
195C2×C5#200812yes—
196C2×C5#201311yescharacteristic
197D5—162no—
198C23—202no—
199C23—404no—
200C23—401no—
201C23—401no—
202C23#408611yescharacteristic
203Q8—44no—
204Q8—82no—
205D4—208no—
206D4—204no—
207D4—404no—
208D4—402no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—402no—
213C2×C4—401no—
214C2×C4#408312yes—
215C2×C4#408411yescharacteristic
216C2×C4#408512yes—
217C2×C4C2×#68812yes—
218C2×C4#408812yes—
219C2×C4#409911yescharacteristic
220C8—24no—
221C8—204no—
222C5#1022011yescharacteristic, complemented, Sylow
223C22—22no—
224C22—402no—
225C22—404no—
226C22—802no—
227C22—802no—
228C22#6537411yescharacteristic
229C22#6537511yescharacteristic
230C22#6551211yescharacteristic
231C4—22no—
232C4—22no—
233C4—82no—
234C4—82no—
235C4—801no—
236C4#6537212yes—
237C4#6551612yes—
238C2—21no—
239C2—21no—
240C2—802no—
241C2#1060258411yescharacteristic
242C2#1060291212yes—
subgroupG/H
centerC22#65512
commutator, cosocleC2×C42×C5C23
FrattiniC2×C42C22×D5
FittingC5×#3162C2
socleC22×C5#392
prime radicalC2×#4020C22
hypercenter#3162D5
hypocenterC5#10220
LCSG ▷ C2×C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3162

Complements

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

NH ≅ G/Ntype
1. C5×#3162C2semidirect
3. #10602631C2semidirect
5. #10598152C2semidirect
49. #3162D5semidirect
68. C2×C5×C8SD16semidirect
180. C2×C8#688semidirect
222. C5#10220semidirect

Maximal subgroups

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

subgroupindex
C5×#31622
#106150862
#106026312
#106006162
#105981522
#102205