Finite Group Databasebeta

← #13036978

Subgroups — #13036978

Order 1280.

1294 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 1294 conjugacy classes of subgroups — too many to draw. Only its 216 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
1C2×#66058C211yesmaximal, characteristic
2#10617527C212yesmaximal, complemented
3#10604981C212yesmaximal
4#10604308C218yesmaximal, complemented
5#10604168C212yesmaximal, complemented
6C22×#713C2211yescharacteristic
7C22×Q8×D5C2211yescharacteristic
8C2×#4130C2214yes—
9C2×C5×C4⋊Q8C411yescharacteristic, complemented
10C2×#4093C411yescharacteristic, complemented
11C2×#4090C411yescharacteristic, complemented
12#66058—28no—
13C2×#4042C414yescomplemented
14C2×#4040C2214yes—
15C2×#4021C411yescharacteristic, complemented
16C2×#4018C2211yescharacteristic
17(C2×AGL(1,𝔽5))⋊Q8—28no—
18#65764C2214yes—
19#65764C2214yes—
20C2×#3999C2218yes—
21#64917C2218yes—
22#21751—51nomaximal, Sylow
23C22×C5×Q8C2×C411yescharacteristic
24C22×C5×Q8C2×C411yescharacteristic
25C2×#713—24no—
26C2×#713—44no—
27C2×Q8×D5—44no—
28C2×Q8×D5D418yes—
29C22×D20C2×C411yescharacteristic
30C22×D20C2312yes—
31C22×C4×D5C2×C412yes—
32C22×C4×D5C2311yescharacteristic
33C22×C4×D5C2312yes—
34C22×Dic10C2×C411yescharacteristic
35C22×Dic10C2×C412yes—
36C2×C22⋊AGL(1,𝔽5)C2318yes—
37#4130—48no—
38C2×C4⋊AGL(1,𝔽5)—24no—
39C2×C4×AGL(1,𝔽5)—24no—
40C5×C4⋊Q8—24no—
41C2×C5×C4⋊C4C2×C414yes—
42C2×C42×C5C2×C411yescharacteristic
43#4093—24no—
44#4090—24no—
45C2×#685C2×C414yes—
46C2×#685C2×C414yes—
47C2×#685C2312yes—
48C2×#683C2×C412yes—
49C2×#683C2×C414yes—
50C2×C4×Dic5C2×C411yescharacteristic
51#4042—216no—
52#4040—28no—
53#4040—28no—
54#4021—24no—
55#4018—24no—
56(C2×AGL(1,𝔽5))⋊C4—28no—
57(C2×AGL(1,𝔽5))⋊C4—28no—
58#3999—48no—
59C2×#536—51no—
60#2310—52no—
61#1899—58no—
62#1820—52no—
63#1782—52no—
64C23×D5C22×C412yes—
65C23×D5C2411yescharacteristic
66C22×AGL(1,𝔽5)—28no—
67C2×C5×Q8—22no—
68C2×C5×Q8—42no—
69C2×C5×Q8—42no—
70C2×C5×Q8C22⋊C414yes—
71C22×C4×C5C22×C412yes—
72C22×C4×C5C22×C411yescharacteristic
73C22×C4×C5C22×C412yes—
74C22×C4×C5C22×C412yes—
75C22×Dic5C22×C411yescharacteristic
76C22×Dic5C22×C412yes—
77C22×Dic5C22×C412yes—
78#713—416no—
79Q8×D5—48no—
80Q8×D5—48no—
81C2×D20—24no—
82C2×D20—28no—
83C2×D20—22no—
84C2×D20—42no—
85C2×C4×D5—28no—
86C2×C4×D5—24no—
87C2×C4×D5—24no—
88C2×C4×D5—44no—
89C2×C4×D5C2×D414yes—
90C2×C4×D5C2×D414yes—
91C2×C4×D5C2×D414yes—
92C2×Dic10—44no—
93C2×Dic10—42no—
94C2×Dic10C22⋊C414yes—
95C2×Dic10C22⋊C418yes—
96C22⋊AGL(1,𝔽5)—216no—
97C4⋊AGL(1,𝔽5)—48no—
98C4×AGL(1,𝔽5)—48no—
99C5×C4⋊C4—28no—
100C42×C5—22no—
101#685—24no—
102#685—28no—
103#685—28no—
104#683—24no—
105#683—28no—
106C4×Dic5—22no—
107C22×#41—51no—
108C23×Q8—51no—
109#536—108no—
110C2×C4⋊Q8—51no—
111C2×C4⋊Q8AGL(1,𝔽5)11yescharacteristic, complemented
112C2×#124—51no—
113C2×#124—51no—
114C2×#123—54no—
115C2×C22⋊Q8—54no—
116C2×#117—51no—
117C2×#117—54no—
118C2×#99—58no—
119#391—108no—
120#381—54no—
121#381—54no—
122#342—58no—
123C23×C5C23×C411yescharacteristic
124C22×D5—22no—
125C22×D5—24no—
126C22×D5—42no—
127C22×D5—44no—
128C22×D5C22×D412yes—
129C22×D5H(5,𝔽2)14yes—
130C2×AGL(1,𝔽5)—416no—
131C2×AGL(1,𝔽5)—48no—
132C5×Q8—44no—
133C5×Q8—44no—
134C2×C4×C5—22no—
135C2×C4×C5—24no—
136C2×C4×C5—22no—
137C2×C4×C5—22no—
138C2×C4×C5—24no—
139C2×C4×C5—42no—
140C2×C4×C5C2×C22⋊C412yes—
141C2×C4×C5C2×C22⋊C414yes—
142C2×Dic5—22no—
143C2×Dic5—24no—
144C2×Dic5—22no—
145C2×Dic5—42no—
146C2×Dic5C2×C22⋊C414yes—
147C2×Dic5C2×C22⋊C412yes—
148D20—44no—
149D20—48no—
150C4×D5—24no—
151C4×D5—24no—
152C4×D5—28no—
153C4×D5—28no—
154C4×D5—48no—
155Dic10—48no—
156Dic10—44no—
157C2×#41—104no—
158C2×#41—204no—
159C22×Q8—58no—
160C22×Q8—52no—
161C22×Q8—51no—
162C22×Q8—204no—
163C22×Q8C2×AGL(1,𝔽5)11yescharacteristic
164C22×Q8C2×AGL(1,𝔽5)11yescharacteristic
165C22×D4—52no—
166C22×D4—51no—
167C23×C4—52no—
168C23×C4—51no—
169C23×C4—52no—
170C4⋊Q8—24no—
171C4⋊Q8—104no—
172#124—104no—
173#124—104no—
174#123—108no—
175#123—108no—
176C22⋊Q8—1016no—
177#117—104no—
178#117—208no—
179C2×C4⋊C4—52no—
180C2×C4⋊C4—54no—
181C2×C4⋊C4—104no—
182C2×C4⋊C4C2×AGL(1,𝔽5)14yes—
183C2×C22⋊C4—52no—
184C2×C22⋊C4—54no—
185C2×C22⋊C4—54no—
186C2×C22⋊C4—58no—
187C2×C42—51no—
188C2×C42—104no—
189C2×C42C2×AGL(1,𝔽5)11yescharacteristic
190#99—208no—
191#95—108no—
192#95—108no—
193C22×C5—22no—
194C22×C5C22×C22⋊C411yescharacteristic
195C22×C5#51812yes—
196C2×D5—24no—
197C2×D5—44no—
198C2×D5—42no—
199C2×D5—44no—
200C2×D5—84no—
201C2×D5#53414yes—
202AGL(1,𝔽5)—88no—
203C4×C5—22no—
204C4×C5—24no—
205C4×C5—42no—
206C4×C5—42no—
207Dic5—22no—
208Dic5—24no—
209Dic5—42no—
210C24—51no—
211C24—52no—
212#41—2016no—
213C2×Q8—22no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—58no—
217C2×Q8—54no—
218C2×Q8—202no—
219C2×Q8—204no—
220C2×Q8—208no—
221C2×Q8—208no—
222C2×Q8C22⋊AGL(1,𝔽5)14yes—
223C2×D4—108no—
224C2×D4—104no—
225C2×D4—102no—
226C2×D4—202no—
227C22×C4—52no—
228C22×C4—51no—
229C22×C4—54no—
230C22×C4—54no—
231C22×C4—54no—
232C22×C4—52no—
233C22×C4—104no—
234C22×C4—108no—
235C22×C4—104no—
236C22×C4—108no—
237C22×C4—204no—
238C22×C4C22×AGL(1,𝔽5)12yes—
239C22×C4C22×AGL(1,𝔽5)11yescharacteristic
240C22×C4C22×AGL(1,𝔽5)12yes—
241C22×C4C22×AGL(1,𝔽5)12yes—
242C4⋊C4—28no—
243C4⋊C4—104no—
244C4⋊C4—108no—
245C4⋊C4—208no—
246C22⋊C4—108no—
247C22⋊C4—104no—
248C22⋊C4—108no—
249C22⋊C4—1016no—
250C42—22no—
251C42—102no—
252C42—208no—
253C2×C5—22no—
254C2×C5#230811yescharacteristic
255C2×C5#288412yes—
256D5—24no—
257D5—82no—
258C23—52no—
259C23—54no—
260C23—102no—
261C23—104no—
262C23—202no—
263C23—204no—
264C23C23×AGL(1,𝔽5)11yescharacteristic
265Q8—44no—
266Q8—44no—
267Q8—204no—
268Q8—208no—
269D4—204no—
270D4—208no—
271C2×C4—22no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—54no—
278C2×C4—52no—
279C2×C4—104no—
280C2×C4—104no—
281C2×C4—108no—
282C2×C4—108no—
283C2×C4—102no—
284C2×C4—104no—
285C2×C4—102no—
286C2×C4—208no—
287C2×C4—202no—
288C2×C4—208no—
289C2×C4—2016no—
290C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
291C2×C4C2×C22⋊AGL(1,𝔽5)14yes—
292C5#2175111yescharacteristic, complemented, Sylow
293C22—22no—
294C22—54no—
295C22—104no—
296C22—204no—
297C22—202no—
298C22—204no—
299C22—404no—
300C22#6625812yes—
301C22C22×C22⋊AGL(1,𝔽5)11yescharacteristic
302C4—22no—
303C4—24no—
304C4—42no—
305C4—42no—
306C4—102no—
307C4—104no—
308C4—202no—
309C4—408no—
310C2—22no—
311C2—104no—
312C2—402no—
313C2#1061750611yescharacteristic
314C2#1061751412yes—
subgroupG/H
centerC22C22×C22⋊AGL(1,𝔽5)
commutatorC23×C5C23×C4
FrattiniC23C23×AGL(1,𝔽5)
FittingC2×C5×C4⋊Q8C4
socleC22×C5C22×C22⋊C4
prime radicalC22×#713C22
cosocleC23×D5C24
hypercenterC2×C4⋊Q8AGL(1,𝔽5)
hypocenterC5#21751
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #10617527C2semidirect
4. #10604308C2semidirect
5. #10604168C2semidirect
9. C2×C5×C4⋊Q8C4semidirect
10. C2×#4093C4semidirect
11. C2×#4090C4semidirect
13. C2×#4042C4semidirect
15. C2×#4021C4semidirect
111. C2×C4⋊Q8AGL(1,𝔽5)semidirect
292. C5#21751semidirect

Maximal subgroups

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

subgroupindex
C2×#660582
#106175272
#106049812
#106043082
#106041682
#217515