Finite Group Databasebeta

← #12016139

Subgroups — #12016139

Order 1280.

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

normal subgroups only

This group has 460 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×#2369C211yesmaximal, characteristic
2C2×#64768C211yesmaximal, characteristic
3#10602577C211yesmaximal, characteristic
4#10600465C212yesmaximal
5#10598135C212yesmaximal
6C2×C5×C4⋊Q8C2211yescharacteristic
7#66146C2211yescharacteristic
8C5×#391—21no—
9C5×#391C2212yes—
10C5×#389—22no—
11C4×C5×C4⋊C4—21no—
12C4×C5×C4⋊C4—22no—
13C2×#3968C2211yescharacteristic
14C2×#3968C2212yes—
15#65514—21no—
16#65253—22no—
17#64768C414yescomplemented
18#64743—22no—
19#64743—21no—
20#12973—51nomaximal, Sylow
21C22×C5×Q8D411yescharacteristic
22C22×C5×Q8C2311yescharacteristic
23C5×C4⋊Q8—22no—
24C5×C4⋊Q8C2×C412yes—
25C5×C4⋊Q8Q812yes—
26C2×C5×C4⋊C4—21no—
27C2×C5×C4⋊C4—22no—
28C2×C5×C4⋊C4—21no—
29C2×C5×C4⋊C4—21no—
30C2×C5×C4⋊C4D412yes—
31C2×C42×C5—22no—
32C2×C42×C5—21no—
33C2×C42×C5D411yescharacteristic
34C2×C42×C5D412yes—
35Q8×Dic5—41no—
36Q8×Dic5—41no—
37#4072—22no—
38#4072—21no—
39C2×#681—22no—
40C2×#681—21no—
41C5×#95—22no—
42C5×#95—22no—
43C5×#95—22no—
44#3968—22no—
45#3968—22no—
46#3968C2×C414yes—
47#3967—41no—
48#3967—41no—
49#3940—42no—
50#3939—42no—
51#2369D511yescharacteristic
52C2×#356—51no—
53#1886—51no—
54#1857—52no—
55#1394—52no—
56C2×C5×Q8—21no—
57C2×C5×Q8—41no—
58C2×C5×Q8—41no—
59C2×C5×Q8—41no—
60C2×C5×Q8—41no—
61C2×C5×Q8C4⋊C412yes—
62C2×C5×Q8C22×C411yescharacteristic
63C2×C5×Q8C2×Q811yescharacteristic
64C2×C5×Q8#4112yes—
65C22×C4×C5—21no—
66C22×C4×C5—22no—
67C22×C4×C5—21no—
68C22×C4×C5—22no—
69C22×C4×C5C2×D411yescharacteristic
70C22×C4×C5C2×D412yes—
71C5×C4⋊C4—24no—
72C5×C4⋊C4—24no—
73C5×C4⋊C4—24no—
74C5×C4⋊C4—42no—
75C5×C4⋊C4—42no—
76C5×C4⋊C4—81no—
77C42×C5—21no—
78C42×C5—42no—
79C42×C5—44no—
80C42×C5—81no—
81C42×C5C4⋊C412yes—
82C22⋊Dic5—41no—
83C4⋊Dic5—22no—
84C4⋊Dic5—22no—
85C4⋊Dic5—21no—
86C4⋊Dic5—21no—
87C4×Dic5—42no—
88C4×Dic5—41no—
89#681—24no—
90#681—42no—
91#681—41no—
92#681—41no—
93C2×#168—42no—
94C2×#168—41no—
95C2×C4⋊Q8C2×D511yescharacteristic
96#519—51no—
97#428—102no—
98#419—101no—
99C2×#101—52no—
100C2×#101—51no—
101#391—21no—
102#391C2×D512yes—
103#389—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—21no—
106#356—54no—
107#339—102no—
108#339—101no—
109C23×C5#12011yescharacteristic
110C5×Q8—41no—
111C5×Q8—41no—
112C5×Q8—42no—
113C5×Q8—42no—
114C5×Q8—81no—
115C2×C4×C5—22no—
116C2×C4×C5—22no—
117C2×C4×C5—22no—
118C2×C4×C5—22no—
119C2×C4×C5—21no—
120C2×C4×C5—22no—
121C2×C4×C5—21no—
122C2×C4×C5—24no—
123C2×C4×C5—22no—
124C2×C4×C5—42no—
125C2×C4×C5—41no—
126C2×C4×C5—42no—
127C2×C4×C5—41no—
128C2×C4×C5—42no—
129C2×C4×C5—42no—
130C2×C4×C5—42no—
131C2×C4×C5C2×C4⋊C411yescharacteristic
132C2×C4×C5C4×D412yes—
133C2×C4×C5C22⋊Q812yes—
134C2×C4×C5#12311yescharacteristic
135C2×Dic5—41no—
136C2×Dic5—41no—
137#168—44no—
138#168—81no—
139C22×Q8C22⋊D511yescharacteristic
140C22×Q8C22×D511yescharacteristic
141C2×M16—102no—
142C2×M16—101no—
143C4⋊Q8—22no—
144C4⋊Q8Dic1012yes—
145C4⋊Q8C4×D512yes—
146C4×Q8—201no—
147C4×Q8—201no—
148#117—101no—
149#117—102no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4C22⋊D512yes—
155C2×C42—22no—
156C2×C42—21no—
157C2×C42C22⋊D511yescharacteristic
158C2×C42C22⋊D512yes—
159#107—202no—
160#106—202no—
161#103—201no—
162#103—201no—
163#101—54no—
164#101—102no—
165#101—102no—
166#95—22no—
167#95—22no—
168#95—22no—
169C22×C5—21no—
170C22×C5—21no—
171C22×C5#38511yescharacteristic
172C22×C5B(4,𝔽2)11yescharacteristic
173C22×C5#45811yescharacteristic
174C4×C5—24no—
175C4×C5—22no—
176C4×C5—21no—
177C4×C5—21no—
178C4×C5—22no—
179C4×C5—44no—
180C4×C5—42no—
181C4×C5—82no—
182Dic5—81no—
183Dic5—81no—
184C2×Q8—21no—
185C2×Q8—41no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8#68312yes—
190C2×Q8C2×Dic1011yescharacteristic
191C2×Q8C2×C4×D511yescharacteristic
192C2×Q8#70912yes—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4C2×C22⋊D511yescharacteristic
198C22×C4C2×C22⋊D512yes—
199M16—104no—
200M16—202no—
201M16—201no—
202M16—201no—
203C2×C8—202no—
204C2×C8—201no—
205C4⋊C4—24no—
206C4⋊C4—24no—
207C4⋊C4—24no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—81no—
211C4⋊C4—101no—
212C4⋊C4—101no—
213C4⋊C4—102no—
214C4⋊C4—102no—
215C22⋊C4—201no—
216C42—21no—
217C42—42no—
218C42—44no—
219C42—81no—
220C42—202no—
221C42—201no—
222C42#68312yes—
223C2×C5—21no—
224C2×C5—21no—
225C2×C5#186711yescharacteristic
226C2×C5#220411yescharacteristic
227C2×C5#220511yescharacteristic
228C23#409911yescharacteristic
229Q8—41no—
230Q8—41no—
231Q8—42no—
232Q8—42no—
233Q8—81no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—201no—
251C2×C4—201no—
252C2×C4C2×#68311yescharacteristic
253C2×C4#407012yes—
254C2×C4C4×C22⋊D512yes—
255C2×C4#407511yescharacteristic
256C8—204no—
257C8—401no—
258C5#1297311yescharacteristic, complemented, Sylow
259C22—21no—
260C22—21no—
261C22#6549911yescharacteristic
262C22#6553211yescharacteristic
263C22#6553311yescharacteristic
264C4—24no—
265C4—22no—
266C4—21no—
267C4—21no—
268C4—22no—
269C4—44no—
270C4—42no—
271C4—82no—
272C4—401no—
273C4—401no—
274C2—21no—
275C2—21no—
276C2#1060245011yescharacteristic
277C2#1060272311yescharacteristic
278C2#1060273511yescharacteristic
subgroupG/H
centerC22#65533
commutatorC2×C5×Q8C22×C4
FrattiniC22×Q8C22×D5
FittingC5×#2369C2
socleC22×C5B(4,𝔽2)
prime radicalC23×C5#120
cosocleC22×C5×Q8C23
hypercenter#2369D5
hypocenterC5#12973
LCSG ▷ C2×C5×Q8 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #2369

Complements

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

NH ≅ G/Ntype
17. #64768C4semidirect
258. C5#12973semidirect

Maximal subgroups

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

subgroupindex
C5×#23692
C2×#647682
#106025772
#106004652
#105981352
#129735