Finite Group Databasebeta

← #12016129

Subgroups — #12016129

Order 1280.

424 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 99 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×#2381C211yesmaximal, characteristic
2C2×#64768C211yesmaximal, characteristic
3#10602577C211yesmaximal, characteristic
4#10600465C212yesmaximal
5#10598135C212yesmaximal
6C2×C5×C4⋊Q8C2211yescharacteristic
7#66146C2211yescharacteristic
8C5×#391—21no—
9C5×#391C2212yes—
10C5×#387—22no—
11C5×#387—22no—
12C5×#383—21no—
13C2×#3968C2211yescharacteristic
14C2×#3968C2212yes—
15#65514—21no—
16#65253—22no—
17#64768C414yescomplemented
18#64743—22no—
19#64743—21no—
20#12971—51nomaximal, Sylow
21C22×C5×Q8D411yescharacteristic
22C22×C5×Q8C2311yescharacteristic
23C5×C4⋊Q8—22no—
24C5×C4⋊Q8C2×C412yes—
25C5×C4⋊Q8Q812yes—
26C2×C5×C4⋊C4—22no—
27C2×C5×C4⋊C4—21no—
28C2×C5×C4⋊C4D412yes—
29C2×C42×C5—21no—
30C2×C42×C5D411yescharacteristic
31C2×C42×C5D412yes—
32Q8×Dic5—42no—
33#4072—22no—
34#4072—21no—
35C2×#681—22no—
36C2×#681—21no—
37C5×#95—21no—
38C5×#95—22no—
39C5×#95—21no—
40C5×#95—22no—
41C5×#95—22no—
42C5×#95—22no—
43#3968—22no—
44#3968—22no—
45#3968C2×C414yes—
46#3967—42no—
47#3940—42no—
48#3939—42no—
49#2381D511yescharacteristic
50C2×#356—51no—
51#1886—51no—
52#1857—52no—
53#1394—52no—
54C2×C5×Q8—21no—
55C2×C5×Q8—41no—
56C2×C5×Q8—41no—
57C2×C5×Q8—41no—
58C2×C5×Q8—41no—
59C2×C5×Q8C4⋊C412yes—
60C2×C5×Q8C22×C411yescharacteristic
61C2×C5×Q8C2×Q811yescharacteristic
62C2×C5×Q8#4112yes—
63C22×C4×C5—21no—
64C22×C4×C5—21no—
65C22×C4×C5—22no—
66C22×C4×C5—22no—
67C22×C4×C5C2×D411yescharacteristic
68C22×C4×C5C2×D412yes—
69C5×C4⋊C4—24no—
70C5×C4⋊C4—24no—
71C5×C4⋊C4—42no—
72C5×C4⋊C4—81no—
73C42×C5—21no—
74C42×C5—42no—
75C42×C5—81no—
76C42×C5C4⋊C412yes—
77C22⋊Dic5—41no—
78C4⋊Dic5—22no—
79C4⋊Dic5—22no—
80C4⋊Dic5—22no—
81C4×Dic5—42no—
82C4×Dic5—41no—
83#681—24no—
84#681—42no—
85#681—41no—
86#681—41no—
87C2×#168—42no—
88C2×#168—41no—
89C2×C4⋊Q8C2×D511yescharacteristic
90#519—51no—
91#428—102no—
92#419—101no—
93C2×#101—52no—
94C2×#101—51no—
95#391—21no—
96#391C2×D512yes—
97#387—22no—
98#387—22no—
99#383—21no—
100#356—54no—
101#339—102no—
102#339—101no—
103C23×C5#12011yescharacteristic
104C5×Q8—41no—
105C5×Q8—41no—
106C5×Q8—42no—
107C5×Q8—42no—
108C5×Q8—81no—
109C2×C4×C5—22no—
110C2×C4×C5—21no—
111C2×C4×C5—22no—
112C2×C4×C5—21no—
113C2×C4×C5—22no—
114C2×C4×C5—22no—
115C2×C4×C5—42no—
116C2×C4×C5—41no—
117C2×C4×C5—41no—
118C2×C4×C5—44no—
119C2×C4×C5—44no—
120C2×C4×C5—42no—
121C2×C4×C5—42no—
122C2×C4×C5C2×C4⋊C411yescharacteristic
123C2×C4×C5C4×D412yes—
124C2×C4×C5C22⋊Q812yes—
125C2×C4×C5#12311yescharacteristic
126C2×Dic5—42no—
127#168—44no—
128#168—81no—
129C22×Q8C22⋊D511yescharacteristic
130C22×Q8C22×D511yescharacteristic
131C2×M16—102no—
132C2×M16—101no—
133C4⋊Q8—22no—
134C4⋊Q8Dic1012yes—
135C4⋊Q8C4×D512yes—
136C4×Q8—202no—
137#117—101no—
138#117—102no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4C22⋊D512yes—
142C2×C42—21no—
143C2×C42C22⋊D511yescharacteristic
144C2×C42C22⋊D512yes—
145#107—202no—
146#106—202no—
147#103—202no—
148#101—54no—
149#101—102no—
150#101—102no—
151#95—21no—
152#95—22no—
153#95—21no—
154#95—22no—
155#95—22no—
156#95—22no—
157C22×C5—22no—
158C22×C5#38511yescharacteristic
159C22×C5B(4,𝔽2)11yescharacteristic
160C22×C5#45811yescharacteristic
161C4×C5—22no—
162C4×C5—21no—
163C4×C5—21no—
164C4×C5—22no—
165C4×C5—42no—
166C4×C5—42no—
167C4×C5—82no—
168C4×C5—82no—
169Dic5—82no—
170C2×Q8—21no—
171C2×Q8—41no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8#68312yes—
176C2×Q8C2×Dic1011yescharacteristic
177C2×Q8C2×C4×D511yescharacteristic
178C2×Q8#70912yes—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4C2×C22⋊D511yescharacteristic
184C22×C4C2×C22⋊D512yes—
185M16—104no—
186M16—202no—
187M16—201no—
188M16—201no—
189C2×C8—202no—
190C2×C8—201no—
191C4⋊C4—24no—
192C4⋊C4—24no—
193C4⋊C4—42no—
194C4⋊C4—81no—
195C4⋊C4—102no—
196C4⋊C4—102no—
197C4⋊C4—102no—
198C22⋊C4—201no—
199C42—21no—
200C42—42no—
201C42—81no—
202C42—202no—
203C42—201no—
204C42#68312yes—
205C2×C5—21no—
206C2×C5—21no—
207C2×C5#186711yescharacteristic
208C2×C5#220312yes—
209C23#409911yescharacteristic
210Q8—41no—
211Q8—41no—
212Q8—42no—
213Q8—42no—
214Q8—81no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—202no—
229C2×C4C2×#68311yescharacteristic
230C2×C4#407012yes—
231C2×C4C4×C22⋊D512yes—
232C2×C4#407511yescharacteristic
233C8—204no—
234C8—401no—
235C5#1297111yescharacteristic, complemented, Sylow
236C22—22no—
237C22#6549911yescharacteristic
238C22#6553211yescharacteristic
239C22#6553311yescharacteristic
240C4—22no—
241C4—21no—
242C4—21no—
243C4—22no—
244C4—42no—
245C4—42no—
246C4—82no—
247C4—82no—
248C4—402no—
249C2—21no—
250C2—21no—
251C2#1060245011yescharacteristic
252C2#1060275212yes—
subgroupG/H
centerC22#65533
commutatorC2×C5×Q8C22×C4
FrattiniC22×Q8C22×D5
FittingC5×#2381C2
socleC22×C5B(4,𝔽2)
prime radicalC23×C5#120
cosocleC22×C5×Q8C23
hypercenter#2381D5
hypocenterC5#12971
LCSG ▷ C2×C5×Q8 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #2381

Complements

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

NH ≅ G/Ntype
17. #64768C4semidirect
235. C5#12971semidirect

Maximal subgroups

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

subgroupindex
C5×#23812
C2×#647682
#106025772
#106004652
#105981352
#129715