Finite Group Databasebeta

← #13017538

Subgroups — #13017538

Order 1280.

1030 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 1030 conjugacy classes of subgroups — too many to draw. Only its 160 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
1C22×#4090C211yesmaximal, characteristic
2C2×#64931C214yesmaximal, complemented
3#10604900C212yesmaximal
4C23×C5×Q8C411yescharacteristic
5C23×Dic10C411yescharacteristic
6C2×#4090—28no—
7C2×#4090—22no—
8C2×#4090C412yes—
9C2×#4090C2212yes—
10C22×#683—22no—
11C22×C4×Dic5C2211yescharacteristic
12C2×C22⋊#170—22no—
13C2×#4004C2214yes—
14#64931—28no—
15#64918—22no—
16#8342—51nomaximal, Sylow
17C22×C5×Q8—24no—
18C22×C5×Q8—21no—
19C22×C5×Q8—42no—
20C22×C5×Q8—42no—
21C22×C5×Q8C2×C412yes—
22C23×C4×C5—21no—
23C23×C4×C5C2×C411yescharacteristic
24C23×Dic5—21no—
25C23×Dic5D412yes—
26C22×Dic10—21no—
27C22×Dic10—24no—
28C22×Dic10—42no—
29C22×Dic10—42no—
30C22×Dic10C2×C412yes—
31C22×#170—24no—
32#4090—216no—
33#4090—24no—
34#4090—24no—
35#4090—24no—
36#4090D418yes—
37C2×#683—216no—
38C2×#683—24no—
39C2×#683—24no—
40C2×C4×Dic5—24no—
41C2×C4×Dic5C2×C411yescharacteristic
42C2×C4×Dic5D412yes—
43C2×C4×Dic5C2311yescharacteristic
44C22⋊#170—44no—
45#4004—28no—
46C22×C4⋊Q8—51no—
47#1797—52no—
48C2×#333—54no—
49C24×C5C22⋊C411yescharacteristic
50C2×C5×Q8—24no—
51C2×C5×Q8—22no—
52C2×C5×Q8—416no—
53C2×C5×Q8—42no—
54C2×C5×Q8—42no—
55C2×C5×Q8—42no—
56C2×C5×Q8—42no—
57C2×C5×Q8C22⋊C414yes—
58C22×C4×C5—22no—
59C22×C4×C5—22no—
60C22×C4×C5—22no—
61C22×C4×C5—24no—
62C22×C4×C5—22no—
63C22×C4×C5—42no—
64C22×C4×C5—42no—
65C22×C4×C5C22⋊C411yescharacteristic
66C22×C4×C5C22×C411yescharacteristic
67C22×Dic5—24no—
68C22×Dic5—24no—
69C22×Dic5—24no—
70C22×Dic5—22no—
71C22×Dic5—42no—
72C22×Dic5—42no—
73C22×Dic5C22⋊C412yes—
74C22×Dic5SD1614yes—
75C22×Dic5Q1614yes—
76C22×Dic5C2×D412yes—
77C2×Dic10—24no—
78C2×Dic10—22no—
79C2×Dic10—42no—
80C2×Dic10—416no—
81C2×Dic10—42no—
82C2×Dic10—42no—
83C2×Dic10—42no—
84C2×Dic10C22⋊C414yes—
85C2×#170—44no—
86C2×#170—44no—
87C2×#170—44no—
88#683—216no—
89#683—216no—
90C4×Dic5—22no—
91C4×Dic5—24no—
92C4×Dic5C2×D414yes—
93C23×Q8—51no—
94C23×Q8AGL(1,𝔽5)11yescharacteristic
95C2×C4⋊Q8—52no—
96C2×C4⋊Q8—52no—
97C2×C4⋊Q8—102no—
98C2×C4⋊Q8—108no—
99C22×C4⋊C4—102no—
100C22×C42—51no—
101C2×C4⋊C8—54no—
102C2×C22⋊C8—102no—
103#336—102no—
104#333—108no—
105C23×C5—21no—
106C23×C5—21no—
107C23×C5—22no—
108C23×C5#10012yes—
109C23×C5#10314yes—
110C23×C5C2×C22⋊C411yescharacteristic
111C5×Q8—44no—
112C5×Q8—48no—
113C5×Q8—44no—
114C2×C4×C5—28no—
115C2×C4×C5—21no—
116C2×C4×C5—21no—
117C2×C4×C5—21no—
118C2×C4×C5—21no—
119C2×C4×C5—24no—
120C2×C4×C5—48no—
121C2×C4×C5—42no—
122C2×C4×C5—42no—
123C2×C4×C5#10012yes—
124C2×C4×C5C2×C22⋊C412yes—
125C2×Dic5—22no—
126C2×Dic5—22no—
127C2×Dic5—28no—
128C2×Dic5—28no—
129C2×Dic5—24no—
130C2×Dic5—48no—
131C2×Dic5—42no—
132C2×Dic5—42no—
133C2×Dic5#10314yes—
134C2×Dic5#12014yes—
135C2×Dic5C2×SD1612yes—
136C2×Dic5C2×Q1612yes—
137C2×Dic5#13714yes—
138Dic10—48no—
139Dic10—44no—
140Dic10—44no—
141#170—84no—
142C22×Q8—24no—
143C22×Q8—21no—
144C22×Q8—42no—
145C22×Q8—42no—
146C22×Q8—52no—
147C22×Q8—104no—
148C22×Q8—101no—
149C22×Q8—202no—
150C22×Q8—202no—
151C22×Q8C2×AGL(1,𝔽5)12yes—
152C23×C4—21no—
153C23×C4—52no—
154C23×C4—101no—
155C23×C4C2×AGL(1,𝔽5)11yescharacteristic
156C22×C8—104no—
157C4⋊Q8—58no—
158C4⋊Q8—104no—
159C4⋊Q8—104no—
160C4⋊Q8—1016no—
161C4⋊Q8—104no—
162C2×C4⋊C4—104no—
163C2×C4⋊C4—104no—
164C2×C4⋊C4—1016no—
165C2×C42—51no—
166C2×C42—52no—
167C2×C42—51no—
168C2×C42—104no—
169C4⋊C8—108no—
170C22⋊C8—204no—
171C22×C5—24no—
172C22×C5—22no—
173C22×C5—22no—
174C22×C5—24no—
175C22×C5#33314yescomplemented
176C22×C5#37911yescharacteristic
177C22×C5C2×#10011yescharacteristic
178C22×C5C2×#10011yescharacteristic
179C22×C5C2×#10312yes—
180C22×C5#41912yes—
181C4×C5—22no—
182C4×C5—22no—
183C4×C5—44no—
184Dic5—24no—
185Dic5—44no—
186Dic5#44814yes—
187Dic5#45114yes—
188C24C22⋊AGL(1,𝔽5)11yescharacteristic
189C2×Q8—24no—
190C2×Q8—22no—
191C2×Q8—416no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8—54no—
197C2×Q8—104no—
198C2×Q8—102no—
199C2×Q8—2016no—
200C2×Q8—202no—
201C2×Q8—202no—
202C2×Q8—202no—
203C2×Q8—202no—
204C2×Q8C22⋊AGL(1,𝔽5)14yes—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—52no—
213C22×C4—52no—
214C22×C4—54no—
215C22×C4—54no—
216C22×C4—104no—
217C22×C4—104no—
218C22×C4—104no—
219C22×C4—102no—
220C22×C4—202no—
221C22×C4—202no—
222C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
223C22×C4C22×AGL(1,𝔽5)11yescharacteristic
224C2×C8—204no—
225C2×C8—204no—
226C2×C8—204no—
227C4⋊C4—1016no—
228C4⋊C4—1016no—
229C42—54no—
230C42—104no—
231C42—102no—
232C2×C5—24no—
233C2×C5C2×#33311yescharacteristic
234C2×C5#154812yes—
235C2×C5#155811yescharacteristic
236C2×C5#178311yescharacteristic
237C2×C5#178712yes—
238C23—21no—
239C23—21no—
240C23—22no—
241C23#400914yes—
242C23#401312yes—
243C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
244Q8—44no—
245Q8—44no—
246Q8—48no—
247Q8—204no—
248Q8—204no—
249Q8—208no—
250C2×C4—28no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—48no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—54no—
260C2×C4—52no—
261C2×C4—52no—
262C2×C4—54no—
263C2×C4—54no—
264C2×C4—108no—
265C2×C4—108no—
266C2×C4—102no—
267C2×C4—102no—
268C2×C4—104no—
269C2×C4—208no—
270C2×C4—202no—
271C2×C4—202no—
272C2×C4#401312yes—
273C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
274C8—404no—
275C5#834211yescharacteristic, complemented, Sylow
276C22—24no—
277C22—22no—
278C22—24no—
279C22—22no—
280C22#6493114yescomplemented
281C22C2×#400912yes—
282C22#6578212yes—
283C22C2×#401311yescharacteristic
284C22C2×#401311yescharacteristic
285C22#6580011yescharacteristic
286C4—22no—
287C4—22no—
288C4—44no—
289C4—54no—
290C4—54no—
291C4—104no—
292C4—204no—
293C2—24no—
294C2C2×#6493111yescharacteristic
295C2#1060497112yes—
296C2#1060497211yescharacteristic
297C2#1060497312yes—
298C2#1060502611yescharacteristic
subgroupG/H
centerC23C2×C22⋊AGL(1,𝔽5)
commutatorC22×C4×C5C22×C4
FrattiniC22×C4C22×AGL(1,𝔽5)
FittingC23×C5×Q8C4
socleC23×C5C2×C22⋊C4
prime radicalC24×C5C22⋊C4
cosocleC2×C4×Dic5C23
hypercenterC23×Q8AGL(1,𝔽5)
hypocenterC5#8342
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#64931C2semidirect
175. C22×C5#333semidirect
275. C5#8342semidirect
280. C22#64931semidirect

Maximal subgroups

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

subgroupindex
C22×#40902
C2×#649312
#106049002
#83425