Finite Group Databasebeta

← #11996899

Subgroups — #11996899

Order 1280.

504 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 107 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×#2979C211yesmaximal, characteristic
2#10615019C211yesmaximal, characteristic
3#10602603C211yesmaximal, characteristic
4#10601573C212yesmaximal, complemented
5#10598234C212yesmaximal, complemented
6C5×#532C2211yescharacteristic
7C2×#4097—22no—
8C2×#4015C2211yescharacteristic
9C5×#436—22no—
10C5×C8×D4—22no—
11C5×#431C2212yes—
12C5×#408—22no—
13#65381—21no—
14#65365—21no—
15#65349C414yes—
16#65213C2212yes—
17C4×#681C2211yescharacteristic
18#64766—21no—
19#64764—21no—
20#64702—22no—
21#64702—22no—
22#9435—51nomaximal, Sylow
23C2×C5×#41D412yes—
24C22×Dic10—21no—
25C2×C5×M16—22no—
26C22×C5×C8—22no—
27C5×C4⋊Q8D411yescharacteristic
28C5×#127D411yescharacteristic
29C5×#124C2×C412yes—
30C5×#121—22no—
31C4×C5×D4—22no—
32C2×C42×C5C2311yescharacteristic
33#4097—44no—
34#4097—44no—
35C2×#689—24no—
36C2×#687—24no—
37C2×C4⋊Dic5—22no—
38C2×#681—21no—
39C2×#681—22no—
40#4015—22no—
41#4015—21no—
42#4015C2×C412yes—
43C5×C4⋊C8D412yes—
44C5×C4⋊C8D412yes—
45C5×C22⋊C8—24no—
46C5×C22⋊C8—22no—
47C4×C5×C8—22no—
48C4⋊#168—22no—
49C4⋊#168—22no—
50#3935C2×C412yes—
51C4×#168—21no—
52#3152—51no—
53#2979D511yescharacteristic
54#1560—51no—
55#1549—52no—
56#1299—52no—
57C5×#41—42no—
58C5×#41—42no—
59C2×C5×Q8C22⋊C412yes—
60C2×C5×D4—22no—
61C2×C5×D4—22no—
62C2×C5×D4C22⋊C412yes—
63C22×C4×C5—22no—
64C22×C4×C5C2×D411yescharacteristic
65C22×C4×C5#4112yes—
66C22×Dic5—41no—
67C2×Dic10—22no—
68C2×Dic10—42no—
69C2×Dic10—42no—
70C2×Dic10—41no—
71C5×M16—82no—
72C2×C5×C8—22no—
73C2×C5×C8—22no—
74C2×C5×C8—82no—
75C5×C4⋊C4—22no—
76C5×C22⋊C4—24no—
77C42×C5C22×C411yescharacteristic
78C42×C5C2×D411yescharacteristic
79C42×C5C2×D412yes—
80#689—48no—
81#687—48no—
82C4⋊Dic5—24no—
83C4⋊Dic5—42no—
84#681—22no—
85#681—22no—
86#681—82no—
87C2×#168—22no—
88C2×#168—42no—
89C2×#137—102no—
90#532C2×D511yescharacteristic
91C2×C4⋊Q8—51no—
92#497—54no—
93#494—101no—
94#492—101no—
95#436—22no—
96C8×D4—22no—
97#431C2×D512yes—
98#423—52no—
99#408—22no—
100C4×M16—51no—
101#333—101no—
102#332—102no—
103#332—102no—
104#331—101no—
105C23×C5—22no—
106C23×C5#12311yescharacteristic
107C5×Q8—42no—
108C5×D4—42no—
109C5×D4—42no—
110C5×D4—42no—
111C5×D4—82no—
112C2×C4×C5—21no—
113C2×C4×C5—21no—
114C2×C4×C5—22no—
115C2×C4×C5—22no—
116C2×C4×C5—22no—
117C2×C4×C5—42no—
118C2×C4×C5—42no—
119C2×C4×C5C2×C22⋊C411yescharacteristic
120C2×C4×C5C4×D412yes—
121C2×C4×C5#12011yescharacteristic
122C2×C4×C5#12112yes—
123C2×Dic5—42no—
124C2×Dic5—82no—
125Dic10—44no—
126Dic10—81no—
127Dic10—81no—
128C5×C8—42no—
129C5×C8—42no—
130#168—24no—
131#168—82no—
132C2×#41C22⋊D512yes—
133C22×Q8—101no—
134#137—204no—
135#137—204no—
136C2×Q16—104no—
137C2×SD16—104no—
138C2×M16—22no—
139C2×M16—101no—
140C2×M16—102no—
141C22×C8—22no—
142C4⋊Q8—52no—
143C4⋊Q8—101no—
144C4⋊Q8—102no—
145C4⋊Q8D2011yescharacteristic
146#127D2011yescharacteristic
147#124C4×D512yes—
148#121—22no—
149C4×D4—22no—
150C2×C4⋊C4—102no—
151C2×C42C22×D511yescharacteristic
152C4⋊C8—102no—
153C4⋊C8—102no—
154C4⋊C8C22⋊D512yes—
155C4⋊C8C22⋊D512yes—
156C22⋊C8—24no—
157C22⋊C8—22no—
158#97—52no—
159C4×C8—22no—
160C4×C8—101no—
161C22×C5—21no—
162C22×C5—22no—
163C22×C5—42no—
164C22×C5—44no—
165C22×C5#38611yescharacteristic
166C4×C5—21no—
167C4×C5—22no—
168C4×C5—42no—
169C4×C5#45311yescharacteristic
170C4×C5#45311yescharacteristic
171C4×C5#47112yes—
172C4×C5#47312yes—
173Dic5—82no—
174#41—42no—
175#41—42no—
176C2×Q8—102no—
177C2×Q8—202no—
178C2×Q8—202no—
179C2×Q8—201no—
180C2×Q8#68512yes—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4#68512yes—
184C22×C4—22no—
185C22×C4—201no—
186C22×C4#70912yes—
187C22×C4C2×C22⋊D511yescharacteristic
188Q16—208no—
189SD16—208no—
190M16—82no—
191M16—102no—
192M16—102no—
193M16—402no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—82no—
197C2×C8—102no—
198C2×C8—202no—
199C4⋊C4—22no—
200C4⋊C4—104no—
201C4⋊C4—202no—
202C22⋊C4—24no—
203C42C2×C4×D511yescharacteristic
204C42C2×D2011yescharacteristic
205C42C2×C22⋊D512yes—
206C2×C5—41no—
207C2×C5—82no—
208C2×C5#188711yescharacteristic
209C2×C5#190812yes—
210C23—22no—
211C23#407511yescharacteristic
212Q8—42no—
213Q8—204no—
214Q8—401no—
215Q8—401no—
216D4—42no—
217D4—42no—
218D4—42no—
219D4—82no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—202no—
228C2×C4—402no—
229C2×C4C2×#68511yescharacteristic
230C2×C4C4×C22⋊D512yes—
231C2×C4#407612yes—
232C2×C4#409911yescharacteristic
233C8—42no—
234C8—42no—
235C8—104no—
236C8—402no—
237C5#943511yescharacteristic, complemented, Sylow
238C22—22no—
239C22—21no—
240C22—44no—
241C22—42no—
242C22#6550211yescharacteristic
243C4—22no—
244C4—21no—
245C4—42no—
246C4—402no—
247C4#6543012yes—
248C4#6543212yes—
249C4#6553011yescharacteristic
250C4#6553011yescharacteristic
251C2—41no—
252C2—82no—
253C2#1060244311yescharacteristic
254C2#1060281212yes—
subgroupG/H
centerC22#65502
commutatorC42×C5C22×C4
FrattiniC2×C42C22×D5
FittingC5×#2979C2
socleC22×C5#386
prime radicalC5×#532C22
cosocleC2×C42×C5C23
hypercenter#2979D5
hypocenterC5#9435
LCSG ▷ C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2979

Complements

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

NH ≅ G/Ntype
4. #10601573C2semidirect
5. #10598234C2semidirect
237. C5#9435semidirect

Maximal subgroups

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

subgroupindex
C5×#29792
#106150192
#106026032
#106015732
#105982342
#94355