Finite Group Databasebeta

← #13028392

Subgroups — #13028392

Order 1280.

1646 conjugacy classes of subgroups; only the 288 normal subgroups are shown.

normal subgroups only

This group has 1646 conjugacy classes of subgroups — too many to draw. Only its 288 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×C4×C22⋊AGL(1,𝔽5)C212yesmaximal
2D5×#381C211yesmaximal, characteristic
3#10604254C214yesmaximal
4#10603801C218yesmaximal, complemented
5C23×C4×D5C2211yescharacteristic
6C22×C22⋊AGL(1,𝔽5)C2214yes—
7C22×C4⋊AGL(1,𝔽5)C2214yes—
8C22×C4×AGL(1,𝔽5)C2214yes—
9C2×D5×C22⋊C4C2212yes—
10C4×C22⋊AGL(1,𝔽5)—28no—
11C2×(C2×AGL(1,𝔽5))⋊C4C2214yes—
12C2×(C2×AGL(1,𝔽5))⋊C4C2218yes—
13C2×(C2×AGL(1,𝔽5))⋊C4C2214yes—
14C5×#381C411yescharacteristic, complemented
15#65500C411yescharacteristic, complemented
16#65232C412yescomplemented
17#64953C414yescomplemented
18D5×#95C2214yes—
19#14165—51nomaximal, Sylow
20C24×D5C2311yescharacteristic
21C23×AGL(1,𝔽5)C2318yes—
22C23×C4×C5C2×C411yescharacteristic
23C23×Dic5C2×C411yescharacteristic
24C22×C4×D5—42no—
25C22×C4×D5—42no—
26C22×C4×D5C2×C412yes—
27C22×C4×D5D414yes—
28C22×C4×D5Q814yes—
29C22×C4×D5C2314yes—
30C22×C4×D5C2312yes—
31C2×C22⋊AGL(1,𝔽5)—28no—
32C2×C22⋊AGL(1,𝔽5)—216no—
33C2×C4⋊AGL(1,𝔽5)—216no—
34C2×C4⋊AGL(1,𝔽5)—44no—
35C2×C4×AGL(1,𝔽5)—28no—
36C2×C4×AGL(1,𝔽5)—44no—
37C2×C4×AGL(1,𝔽5)—44no—
38C2×C5×C22⋊C4C2×C412yes—
39C2×C22⋊Dic5C2×C412yes—
40C2×#685C2×C414yes—
41D5×C22⋊C4—42no—
42D5×C22⋊C4—42no—
43(C2×AGL(1,𝔽5))⋊C4—216no—
44(C2×AGL(1,𝔽5))⋊C4—28no—
45(C2×AGL(1,𝔽5))⋊C4—44no—
46C5×#95C2×C414yes—
47#3963C2×C418yes—
48#3963C2×C414yes—
49C2×#381—51no—
50C2×C4×C22⋊C4—52no—
51#1439—54no—
52#1436—58no—
53C24×C5C22×C411yescharacteristic
54C23×D5—28no—
55C23×D5—22no—
56C23×D5—22no—
57C23×D5C22×C411yescharacteristic
58C23×D5C2×D412yes—
59C23×D5C2×Q812yes—
60C23×D5C2411yescharacteristic
61C22×AGL(1,𝔽5)—28no—
62C22×AGL(1,𝔽5)—28no—
63C22×AGL(1,𝔽5)—28no—
64C22×AGL(1,𝔽5)—28no—
65C22×AGL(1,𝔽5)—28no—
66C22×AGL(1,𝔽5)—28no—
67C22×AGL(1,𝔽5)#41116yes—
68C22×C4×C5—42no—
69C22×C4×C5C4⋊C414yescomplemented
70C22×C4×C5C22×C414yes—
71C22×C4×C5C22×C412yes—
72C22×Dic5—42no—
73C22×Dic5C4⋊C414yescomplemented
74C22×Dic5C22×C414yes—
75C22×Dic5C22×C412yes—
76C2×C4×D5—24no—
77C2×C4×D5—24no—
78C2×C4×D5—28no—
79C2×C4×D5—28no—
80C2×C4×D5—24no—
81C2×C4×D5—24no—
82C2×C4×D5—48no—
83C2×C4×D5—44no—
84C2×C4×D5—44no—
85C2×C4×D5—44no—
86C2×C4×D5C4⋊C418yescomplemented
87C2×C4×D5C2×D414yes—
88C2×C4×D5C2×Q814yes—
89C22⋊AGL(1,𝔽5)—232no—
90C4⋊AGL(1,𝔽5)—416no—
91C4×AGL(1,𝔽5)—416no—
92C5×C22⋊C4—42no—
93C22⋊Dic5—42no—
94#685—44no—
95C24×C4—51no—
96C22×C4⋊C4—54no—
97C22×C22⋊C4—52no—
98C22×C22⋊C4—54no—
99C22×C42—54no—
100#381—51no—
101#381—52no—
102#381—54no—
103#381AGL(1,𝔽5)11yescharacteristic, complemented
104C4×C22⋊C4—108no—
105C2×#95—54no—
106C2×#95—58no—
107C2×#95—54no—
108C2×#95—54no—
109C23×C5—24no—
110C23×C5—22no—
111C23×C5C2×C4⋊C412yes—
112C23×C5C23×C411yescharacteristic
113C22×D5—24no—
114C22×D5—22no—
115C22×D5—28no—
116C22×D5—28no—
117C22×D5—24no—
118C22×D5—22no—
119C22×D5—22no—
120C22×D5—44no—
121C22×D5—48no—
122C22×D5—44no—
123C22×D5C2×C4⋊C412yes—
124C22×D5C22×D411yescharacteristic
125C22×D5C22×Q811yescharacteristic
126C22×D5C2×#4118yes—
127C22×D5H(5,𝔽2)12yes—
128C22×D5H(5,𝔽2)12yes—
129C2×AGL(1,𝔽5)—216no—
130C2×AGL(1,𝔽5)—216no—
131C2×AGL(1,𝔽5)—216no—
132C2×AGL(1,𝔽5)—48no—
133C2×AGL(1,𝔽5)—48no—
134C2×AGL(1,𝔽5)—48no—
135C2×AGL(1,𝔽5)—48no—
136C2×C4×C5—24no—
137C2×C4×C5—28no—
138C2×C4×C5—24no—
139C2×C4×C5—44no—
140C2×C4×C5C2×C4⋊C414yes—
141C2×Dic5—24no—
142C2×Dic5—28no—
143C2×Dic5—24no—
144C2×Dic5—44no—
145C2×Dic5C2×C4⋊C414yes—
146C4×D5—28no—
147C4×D5—24no—
148C4×D5—24no—
149C4×D5—48no—
150C4×D5—44no—
151C4×D5—48no—
152C4×D5—44no—
153C25—51no—
154C23×C4—51no—
155C23×C4—52no—
156C23×C4—54no—
157C23×C4—54no—
158C23×C4—52no—
159C23×C4—58no—
160C23×C4—54no—
161C23×C4—202no—
162C23×C4—202no—
163C23×C4C2×AGL(1,𝔽5)11yescharacteristic
164C2×C4⋊C4—1016no—
165C2×C4⋊C4—204no—
166C2×C22⋊C4—52no—
167C2×C22⋊C4—54no—
168C2×C22⋊C4—1016no—
169C2×C22⋊C4—108no—
170C2×C22⋊C4—202no—
171C2×C22⋊C4—202no—
172C2×C22⋊C4C2×AGL(1,𝔽5)12yes—
173C2×C42—108no—
174C2×C42—204no—
175C2×C42—204no—
176#95—58no—
177#95—54no—
178#95—108no—
179#95—1016no—
180#95—204no—
181#95C2×AGL(1,𝔽5)14yes—
182C22×C5—22no—
183C22×C5—24no—
184C22×C5—44no—
185C22×C5C22×C4⋊C411yescharacteristic
186C22×C5C4×#4114yes—
187C22×C5#51812yes—
188C2×D5—22no—
189C2×D5—24no—
190C2×D5—44no—
191C2×D5—48no—
192C2×D5—44no—
193C2×D5—44no—
194C2×D5#52514yes—
195C2×D5#53314yes—
196C2×D5#53412yes—
197C2×D5#54311yescharacteristic
198C2×D5#54311yescharacteristic
199C2×D5#55114yes—
200C2×D5#55318yes—
201C2×D5#55914yes—
202AGL(1,𝔽5)—416no—
203C4×C5—24no—
204C4×C5—44no—
205Dic5—24no—
206Dic5—44no—
207C24—51no—
208C24—52no—
209C24—52no—
210C24—51no—
211C24—108no—
212C24—102no—
213C24—102no—
214C24C22×AGL(1,𝔽5)11yescharacteristic
215C22×C4—42no—
216C22×C4—52no—
217C22×C4—54no—
218C22×C4—54no—
219C22×C4—54no—
220C22×C4—58no—
221C22×C4—516no—
222C22×C4—54no—
223C22×C4—104no—
224C22×C4—104no—
225C22×C4—108no—
226C22×C4—104no—
227C22×C4—104no—
228C22×C4—108no—
229C22×C4—108no—
230C22×C4—108no—
231C22×C4—108no—
232C22×C4—108no—
233C22×C4—108no—
234C22×C4—108no—
235C22×C4—204no—
236C22×C4—208no—
237C22×C4—204no—
238C22×C4—204no—
239C22×C4—202no—
240C22×C4C4⋊AGL(1,𝔽5)14yescomplemented
241C22×C4C22×AGL(1,𝔽5)14yes—
242C22×C4C22×AGL(1,𝔽5)12yes—
243C4⋊C4—2016no—
244C22⋊C4—42no—
245C22⋊C4—1032no—
246C22⋊C4—204no—
247C22⋊C4—202no—
248C42—2016no—
249C2×C5—42no—
250C2×C5#229612yes—
251C2×C5#231511yescharacteristic
252C2×C5#237414yes—
253D5—42no—
254D5#260014yes—
255D5#260712yes—
256D5#260812yes—
257C23—22no—
258C23—24no—
259C23—51no—
260C23—51no—
261C23—52no—
262C23—52no—
263C23—58no—
264C23—52no—
265C23—108no—
266C23—104no—
267C23—102no—
268C23—102no—
269C23—108no—
270C23—102no—
271C23—104no—
272C23—204no—
273C23—208no—
274C23—204no—
275C23C2×C4⋊AGL(1,𝔽5)12yes—
276C23C23×AGL(1,𝔽5)11yescharacteristic
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—28no—
280C2×C4—44no—
281C2×C4—54no—
282C2×C4—108no—
283C2×C4—104no—
284C2×C4—104no—
285C2×C4—104no—
286C2×C4—1016no—
287C2×C4—1016no—
288C2×C4—1016no—
289C2×C4—104no—
290C2×C4—108no—
291C2×C4—208no—
292C2×C4—204no—
293C2×C4—208no—
294C2×C4—204no—
295C2×C4—204no—
296C2×C4—208no—
297C2×C4—208no—
298C2×C4—208no—
299C2×C4—208no—
300C2×C4C2×C4⋊AGL(1,𝔽5)14yes—
301C5#1416511yescharacteristic, complemented, Sylow
302C22—24no—
303C22—22no—
304C22—44no—
305C22—54no—
306C22—54no—
307C22—54no—
308C22—58no—
309C22—54no—
310C22—52no—
311C22—51no—
312C22—51no—
313C22—104no—
314C22—102no—
315C22—208no—
316C22—204no—
317C22—204no—
318C22—204no—
319C22C22×C4⋊AGL(1,𝔽5)11yescharacteristic
320C22#6625812yes—
321C22#41×AGL(1,𝔽5)14yes—
322C4—24no—
323C4—44no—
324C4—104no—
325C4—2016no—
326C4—204no—
327C2—42no—
328C2—54no—
329C2—52no—
330C2—52no—
331C2—202no—
332C2#1061707111yescharacteristic
333C2#1061712812yes—
334C2#1061730714yes—
subgroupG/H
center, FrattiniC23C23×AGL(1,𝔽5)
commutator, socleC23×C5C23×C4
FittingC5×#381C4
prime radicalC24×D5C23
cosocleC23×D5C24
hypercenter#381AGL(1,𝔽5)
hypocenterC5#14165
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
4. #10603801C2semidirect
14. C5×#381C4semidirect
15. #65500C4semidirect
16. #65232C4semidirect
17. #64953C4semidirect
69. C22×C4×C5C4⋊C4semidirect
73. C22×Dic5C4⋊C4semidirect
86. C2×C4×D5C4⋊C4semidirect
103. #381AGL(1,𝔽5)semidirect
240. C22×C4C4⋊AGL(1,𝔽5)semidirect
301. C5#14165semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊AGL(1,𝔽5)2
D5×#3812
#106042542
#106038012
#141655