Finite Group Databasebeta

← #13021673

Subgroups — #13021673

Order 1280.

1510 conjugacy classes of subgroups; only the 228 normal subgroups are shown.

normal subgroups only

This group has 1510 conjugacy classes of subgroups — too many to draw. Only its 228 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×#4018C211yesmaximal, characteristic
2C2×#65764C212yesmaximal
3C2×#64917C214yesmaximal
4C23×C4×D5C2211yescharacteristic
5C22×C22⋊AGL(1,𝔽5)C2214yes—
6C22×C42×C5C411yescharacteristic, complemented
7C22×#685C2212yes—
8C22×#683C412yescomplemented
9C22×C4×Dic5C411yescharacteristic, complemented
10C2×#4018—28no—
11C2×#4018—24no—
12#65764—24no—
13#65764C418yescomplemented
14C2×(C2×AGL(1,𝔽5))⋊C4—24no—
15#64917—28no—
16#10726—51nomaximal, Sylow
17C24×D5C2311yescharacteristic
18C23×AGL(1,𝔽5)—24no—
19C23×C4×C5C2×C412yes—
20C23×C4×C5C2×C411yescharacteristic
21C23×Dic5C2×C411yescharacteristic
22C23×Dic5C2×C412yes—
23C22×C4×D5—24no—
24C22×C4×D5—24no—
25C22×C4×D5C2×C414yes—
26C22×C4×D5D412yes—
27C22×C4×D5D414yes—
28C22×C4×D5Q812yes—
29C2×C22⋊AGL(1,𝔽5)—28no—
30C2×C22⋊AGL(1,𝔽5)—28no—
31C2×C22⋊AGL(1,𝔽5)—24no—
32C2×C22⋊AGL(1,𝔽5)C2×C418yes—
33C2×C42×C5—24no—
34C2×C42×C5—22no—
35C2×#685—24no—
36C2×#685—24no—
37C2×#685D418yes—
38C2×#683—24no—
39C2×#683—24no—
40C2×#683—24no—
41C2×C4×Dic5—22no—
42C2×C4×Dic5—24no—
43#4018—232no—
44(C2×AGL(1,𝔽5))⋊C4—216no—
45C22×#117—51no—
46C2×#381—52no—
47C2×#342—54no—
48C24×C5C22×C411yescharacteristic
49C23×D5—44no—
50C23×D5C22×C412yes—
51C23×D5C2×D412yes—
52C23×D5C2×D411yescharacteristic
53C23×D5C2×D414yes—
54C23×D5C2×Q811yescharacteristic
55C23×D5#4114yes—
56C22×AGL(1,𝔽5)—28no—
57C22×AGL(1,𝔽5)—416no—
58C22×AGL(1,𝔽5)—44no—
59C22×AGL(1,𝔽5)—44no—
60C22×C4×C5—22no—
61C22×C4×C5—22no—
62C22×C4×C5—22no—
63C22×C4×C5—22no—
64C22×C4×C5—44no—
65C22×C4×C5C4212yes—
66C22×C4×C5C22⋊C412yes—
67C22×C4×C5C22⋊C414yes—
68C22×C4×C5C4⋊C412yes—
69C22×Dic5—22no—
70C22×Dic5—22no—
71C22×Dic5—22no—
72C22×Dic5—22no—
73C22×Dic5—44no—
74C22×Dic5C4212yes—
75C22×Dic5C22⋊C412yes—
76C22×Dic5C22⋊C414yes—
77C22×Dic5C4⋊C412yes—
78C2×C4×D5—216no—
79C2×C4×D5—28no—
80C2×C4×D5—28no—
81C2×C4×D5—28no—
82C2×C4×D5—28no—
83C2×C4×D5C22⋊C418yes—
84C2×C4×D5C4⋊C418yes—
85C22⋊AGL(1,𝔽5)—216no—
86C22⋊AGL(1,𝔽5)—48no—
87C42×C5—28no—
88#685—216no—
89#683—216no—
90C4×Dic5—28no—
91C24×C4—51no—
92C2×#117—108no—
93C2×#117—104no—
94C22×C4⋊C4—52no—
95C22×C22⋊C4—52no—
96C22×C22⋊C4—54no—
97C22×C42—51no—
98C22×C42AGL(1,𝔽5)11yescharacteristic, complemented
99#381—58no—
100#381—104no—
101C2×#95—104no—
102#342—108no—
103C23×C5—24no—
104C23×C5C2×C4211yescharacteristic
105C23×C5C2×C22⋊C412yes—
106C23×C5C2×C22⋊C411yescharacteristic
107C23×C5C2×C4⋊C411yescharacteristic
108C23×C5#11712yes—
109C22×D5—28no—
110C22×D5—44no—
111C22×D5—42no—
112C22×D5—44no—
113C22×D5—44no—
114C22×D5—42no—
115C22×D5—48no—
116C22×D5C2×C22⋊C412yes—
117C22×D5C2×C4⋊C412yes—
118C22×D5C4×D418yes—
119C22×D5#12014yes—
120C22×D5#12114yes—
121C22×D5C22⋊Q814yes—
122C22×D5#12314yes—
123C2×AGL(1,𝔽5)—416no—
124C2×AGL(1,𝔽5)—48no—
125C2×AGL(1,𝔽5)—88no—
126C2×AGL(1,𝔽5)—88no—
127C2×C4×C5—24no—
128C2×C4×C5—22no—
129C2×C4×C5—24no—
130C2×C4×C5—22no—
131C2×C4×C5—24no—
132C2×C4×C5—44no—
133C2×C4×C5—44no—
134C2×C4×C5—44no—
135C2×C4×C5#9514yes—
136C2×Dic5—24no—
137C2×Dic5—22no—
138C2×Dic5—24no—
139C2×Dic5—22no—
140C2×Dic5—24no—
141C2×Dic5—44no—
142C2×Dic5—44no—
143C2×Dic5—44no—
144C2×Dic5#9514yes—
145C4×D5—216no—
146C4×D5—216no—
147C25—51no—
148C23×C4—51no—
149C23×C4—54no—
150C23×C4—52no—
151C23×C4—52no—
152C23×C4—54no—
153C23×C4—52no—
154C23×C4—104no—
155C23×C4—104no—
156C23×C4—104no—
157C23×C4C2×AGL(1,𝔽5)12yes—
158C23×C4C2×AGL(1,𝔽5)11yescharacteristic
159#117—1032no—
160C2×C4⋊C4—104no—
161C2×C4⋊C4—104no—
162C2×C4⋊C4—104no—
163C2×C22⋊C4—58no—
164C2×C22⋊C4—58no—
165C2×C22⋊C4—104no—
166C2×C22⋊C4—104no—
167C2×C22⋊C4—108no—
168C2×C22⋊C4—108no—
169C2×C22⋊C4—104no—
170C2×C42—24no—
171C2×C42—22no—
172C2×C42—104no—
173C2×C42—102no—
174#95—1016no—
175C22×C5—24no—
176C22×C5—22no—
177C22×C5—22no—
178C22×C5—22no—
179C22×C5—22no—
180C22×C5C2×#9511yescharacteristic
181C22×C5C4×C22⋊C412yes—
182C22×C5#37911yescharacteristic
183C22×C5#38012yes—
184C22×C5#38111yescharacteristic
185C22×C5#41014yes—
186C2×D5—28no—
187C2×D5—24no—
188C2×D5—24no—
189C2×D5—28no—
190C2×D5—48no—
191C2×D5—44no—
192C2×D5—44no—
193C2×D5#38514yes—
194C2×D5#38614yes—
195AGL(1,𝔽5)—88no—
196C4×C5—24no—
197C4×C5—44no—
198Dic5—24no—
199Dic5—44no—
200C24—52no—
201C24—52no—
202C24—51no—
203C24—51no—
204C24—54no—
205C24—54no—
206C24—204no—
207C24C22×AGL(1,𝔽5)11yescharacteristic
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—44no—
213C22×C4—52no—
214C22×C4—52no—
215C22×C4—52no—
216C22×C4—58no—
217C22×C4—58no—
218C22×C4—54no—
219C22×C4—108no—
220C22×C4—1016no—
221C22×C4—108no—
222C22×C4—108no—
223C22×C4—108no—
224C22×C4—102no—
225C22×C4—102no—
226C22×C4—102no—
227C22×C4—102no—
228C22×C4—108no—
229C22×C4—204no—
230C22×C4—204no—
231C22×C4—204no—
232C22×C4—2016no—
233C22×C4C4×AGL(1,𝔽5)12yes—
234C22×C4C4⋊AGL(1,𝔽5)12yes—
235C22×C4C22⋊AGL(1,𝔽5)12yes—
236C22×C4C22⋊AGL(1,𝔽5)14yes—
237C4⋊C4—1016no—
238C22⋊C4—1016no—
239C22⋊C4—1016no—
240C22⋊C4—208no—
241C42—28no—
242C42—108no—
243C2×C5—22no—
244C2×C5—22no—
245C2×C5C22⋊#9511yescharacteristic
246C2×C5#174012yes—
247C2×C5#178212yes—
248C2×C5#182012yes—
249D5—24no—
250D5—24no—
251C23—24no—
252C23—54no—
253C23—52no—
254C23—52no—
255C23—54no—
256C23—58no—
257C23—54no—
258C23—54no—
259C23—108no—
260C23—204no—
261C23—204no—
262C23—204no—
263C23—202no—
264C23—202no—
265C23—208no—
266C23C2×C4×AGL(1,𝔽5)11yescharacteristic
267C23C2×C4⋊AGL(1,𝔽5)11yescharacteristic
268C23#413012yes—
269C23C2×C22⋊AGL(1,𝔽5)12yes—
270C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
271C2×C4—22no—
272C2×C4—24no—
273C2×C4—22no—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—54no—
280C2×C4—1016no—
281C2×C4—1016no—
282C2×C4—102no—
283C2×C4—104no—
284C2×C4—102no—
285C2×C4—104no—
286C2×C4—104no—
287C2×C4—204no—
288C2×C4—204no—
289C2×C4—204no—
290C2×C4—208no—
291C2×C4—2016no—
292C2×C4—408no—
293C2×C4—408no—
294C2×C4(C2×AGL(1,𝔽5))⋊C414yes—
295C5#1072611yescharacteristic, complemented, Sylow
296C22—22no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—24no—
301C22—54no—
302C22—54no—
303C22—108no—
304C22—104no—
305C22—104no—
306C22—108no—
307C22—208no—
308C22—204no—
309C22—204no—
310C22#6574514yes—
311C22C2×(C2×AGL(1,𝔽5))⋊C411yescharacteristic
312C22C4×C22⋊AGL(1,𝔽5)12yes—
313C22#6576411yescharacteristic
314C22#6576512yes—
315C22#6580011yescharacteristic
316C4—24no—
317C4—44no—
318C4—104no—
319C4—204no—
320C4—408no—
321C2—22no—
322C2—22no—
323C2—104no—
324C2—104no—
325C2#1060416512yes—
326C2#1060416812yes—
327C2#1060489112yes—
328C2#1060490911yescharacteristic
subgroupG/H
centerC23C2×C22⋊AGL(1,𝔽5)
commutatorC23×C5C2×C42
FrattiniC24C22×AGL(1,𝔽5)
FittingC22×C42×C5C4
socleC23×C5C2×C22⋊C4
prime radical, cosocleC24×D5C23
hypercenterC22×C42AGL(1,𝔽5)
hypocenterC5#10726
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
6. C22×C42×C5C4semidirect
8. C22×#683C4semidirect
9. C22×C4×Dic5C4semidirect
13. #65764C4semidirect
98. C22×C42AGL(1,𝔽5)semidirect
295. C5#10726semidirect

Maximal subgroups

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

subgroupindex
C22×#40182
C2×#657642
C2×#649172
#107265