Finite Group Databasebeta

← #13021674

Subgroups — #13021674

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×#4038C211yesmaximal, characteristic
2C2×#65764C212yesmaximal
3C2×#64917C214yesmaximal
4C23×C4×D5C2211yescharacteristic
5C22×C22⋊AGL(1,𝔽5)C2214yes—
6C22×C5×C4⋊C4C411yescharacteristic, complemented
7C22×#685C2212yes—
8C22×C4⋊Dic5C411yescharacteristic, complemented
9C22×C4×Dic5C412yescomplemented
10C2×#4038—28no—
11C2×#4038—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×C5×C4⋊C4—24no—
34C2×C5×C4⋊C4—22no—
35C2×#685—24no—
36C2×#685—24no—
37C2×#685D418yes—
38C2×C4⋊Dic5—22no—
39C2×C4⋊Dic5—24no—
40C2×C4×Dic5—24no—
41C2×C4×Dic5—24no—
42C2×C4×Dic5—24no—
43#4038—232no—
44(C2×AGL(1,𝔽5))⋊C4—216no—
45C22×#117—51no—
46C2×#381—52no—
47C2×#342—54no—
48C24×C5C22×C411yescharacteristic
49C23×D5—42no—
50C23×D5—42no—
51C23×D5C22×C412yes—
52C23×D5C2×D412yes—
53C23×D5C2×D411yescharacteristic
54C23×D5C2×D414yes—
55C23×D5C2×Q811yescharacteristic
56C23×D5#4114yes—
57C22×AGL(1,𝔽5)—28no—
58C22×AGL(1,𝔽5)—48no—
59C22×AGL(1,𝔽5)—48no—
60C22×AGL(1,𝔽5)—44no—
61C22×AGL(1,𝔽5)—44no—
62C22×C4×C5—22no—
63C22×C4×C5—22no—
64C22×C4×C5—22no—
65C22×C4×C5—22no—
66C22×C4×C5—44no—
67C22×C4×C5C4212yes—
68C22×C4×C5C22⋊C412yes—
69C22×C4×C5C22⋊C414yes—
70C22×C4×C5C4⋊C412yes—
71C22×Dic5—22no—
72C22×Dic5—22no—
73C22×Dic5—22no—
74C22×Dic5—22no—
75C22×Dic5—44no—
76C22×Dic5C4212yes—
77C22×Dic5C22⋊C412yes—
78C22×Dic5C22⋊C414yes—
79C22×Dic5C4⋊C412yes—
80C2×C4×D5—216no—
81C2×C4×D5—28no—
82C2×C4×D5—28no—
83C2×C4×D5—28no—
84C2×C4×D5—28no—
85C2×C4×D5C22⋊C418yes—
86C2×C4×D5C4⋊C418yes—
87C22⋊AGL(1,𝔽5)—216no—
88C22⋊AGL(1,𝔽5)—48no—
89C5×C4⋊C4—28no—
90#685—216no—
91C4⋊Dic5—28no—
92C4×Dic5—216no—
93C24×C4—51no—
94C2×#117—108no—
95C2×#117—104no—
96C22×C4⋊C4—51no—
97C22×C4⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
98C22×C22⋊C4—52no—
99C22×C22⋊C4—54no—
100C22×C42—52no—
101#381—58no—
102#381—104no—
103C2×#95—104no—
104#342—108no—
105C23×C5—22no—
106C23×C5—22no—
107C23×C5C2×C4211yescharacteristic
108C23×C5C2×C22⋊C412yes—
109C23×C5C2×C22⋊C411yescharacteristic
110C23×C5C2×C4⋊C411yescharacteristic
111C23×C5#11712yes—
112C22×D5—24no—
113C22×D5—24no—
114C22×D5—44no—
115C22×D5—42no—
116C22×D5—44no—
117C22×D5—44no—
118C22×D5—42no—
119C22×D5—48no—
120C22×D5C2×C22⋊C412yes—
121C22×D5C2×C4⋊C412yes—
122C22×D5C4×D418yes—
123C22×D5#12014yes—
124C22×D5#12114yes—
125C22×D5C22⋊Q814yes—
126C22×D5#12314yes—
127C2×AGL(1,𝔽5)—416no—
128C2×AGL(1,𝔽5)—48no—
129C2×AGL(1,𝔽5)—88no—
130C2×AGL(1,𝔽5)—84no—
131C2×AGL(1,𝔽5)—84no—
132C2×C4×C5—24no—
133C2×C4×C5—22no—
134C2×C4×C5—24no—
135C2×C4×C5—22no—
136C2×C4×C5—24no—
137C2×C4×C5—44no—
138C2×C4×C5—44no—
139C2×C4×C5—44no—
140C2×C4×C5#9514yes—
141C2×Dic5—24no—
142C2×Dic5—22no—
143C2×Dic5—24no—
144C2×Dic5—22no—
145C2×Dic5—24no—
146C2×Dic5—44no—
147C2×Dic5—44no—
148C2×Dic5—44no—
149C2×Dic5#9514yes—
150C4×D5—216no—
151C4×D5—216no—
152C25—51no—
153C23×C4—51no—
154C23×C4—54no—
155C23×C4—52no—
156C23×C4—52no—
157C23×C4—54no—
158C23×C4—52no—
159C23×C4—104no—
160C23×C4—104no—
161C23×C4—104no—
162C23×C4C2×AGL(1,𝔽5)12yes—
163C23×C4C2×AGL(1,𝔽5)11yescharacteristic
164#117—1032no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—104no—
168C2×C4⋊C4—102no—
169C2×C22⋊C4—58no—
170C2×C22⋊C4—58no—
171C2×C22⋊C4—104no—
172C2×C22⋊C4—104no—
173C2×C22⋊C4—108no—
174C2×C22⋊C4—108no—
175C2×C22⋊C4—104no—
176C2×C42—104no—
177C2×C42—104no—
178C2×C42—104no—
179#95—1016no—
180C22×C5—24no—
181C22×C5—22no—
182C22×C5—22no—
183C22×C5—22no—
184C22×C5—22no—
185C22×C5C2×#9511yescharacteristic
186C22×C5C4×C22⋊C412yes—
187C22×C5#37911yescharacteristic
188C22×C5#38012yes—
189C22×C5#38111yescharacteristic
190C22×C5#41012yes—
191C22×C5#41012yes—
192C2×D5—28no—
193C2×D5—24no—
194C2×D5—24no—
195C2×D5—24no—
196C2×D5—24no—
197C2×D5—48no—
198C2×D5—44no—
199C2×D5—44no—
200C2×D5#38514yes—
201C2×D5#38614yes—
202AGL(1,𝔽5)—88no—
203C4×C5—24no—
204C4×C5—44no—
205Dic5—24no—
206Dic5—44no—
207C24—52no—
208C24—52no—
209C24—51no—
210C24—51no—
211C24—54no—
212C24—54no—
213C24—202no—
214C24—202no—
215C24C22×AGL(1,𝔽5)11yescharacteristic
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—44no—
221C22×C4—52no—
222C22×C4—52no—
223C22×C4—52no—
224C22×C4—58no—
225C22×C4—58no—
226C22×C4—54no—
227C22×C4—108no—
228C22×C4—1016no—
229C22×C4—108no—
230C22×C4—108no—
231C22×C4—108no—
232C22×C4—102no—
233C22×C4—102no—
234C22×C4—102no—
235C22×C4—102no—
236C22×C4—108no—
237C22×C4—204no—
238C22×C4—204no—
239C22×C4—204no—
240C22×C4—208no—
241C22×C4—208no—
242C22×C4C4×AGL(1,𝔽5)12yes—
243C22×C4C4⋊AGL(1,𝔽5)12yes—
244C22×C4C22⋊AGL(1,𝔽5)12yes—
245C22×C4C22⋊AGL(1,𝔽5)14yes—
246C4⋊C4—28no—
247C4⋊C4—108no—
248C22⋊C4—1016no—
249C22⋊C4—1016no—
250C22⋊C4—208no—
251C42—1016no—
252C2×C5—22no—
253C2×C5—22no—
254C2×C5C22⋊#9511yescharacteristic
255C2×C5#174012yes—
256C2×C5#178211yescharacteristic
257C2×C5#178211yescharacteristic
258C2×C5#182012yes—
259D5—24no—
260D5—24no—
261C23—22no—
262C23—22no—
263C23—54no—
264C23—52no—
265C23—52no—
266C23—54no—
267C23—58no—
268C23—54no—
269C23—54no—
270C23—104no—
271C23—104no—
272C23—204no—
273C23—202no—
274C23—202no—
275C23—208no—
276C23—204no—
277C23—204no—
278C23C2×C4×AGL(1,𝔽5)11yescharacteristic
279C23C2×C4⋊AGL(1,𝔽5)11yescharacteristic
280C23#413012yes—
281C23C2×C22⋊AGL(1,𝔽5)12yes—
282C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—22no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—54no—
292C2×C4—1016no—
293C2×C4—1016no—
294C2×C4—102no—
295C2×C4—104no—
296C2×C4—102no—
297C2×C4—104no—
298C2×C4—104no—
299C2×C4—204no—
300C2×C4—204no—
301C2×C4—204no—
302C2×C4—208no—
303C2×C4—2016no—
304C2×C4—408no—
305C2×C4—404no—
306C2×C4—404no—
307C2×C4(C2×AGL(1,𝔽5))⋊C414yes—
308C5#1072611yescharacteristic, complemented, Sylow
309C22—22no—
310C22—22no—
311C22—24no—
312C22—22no—
313C22—22no—
314C22—54no—
315C22—54no—
316C22—104no—
317C22—104no—
318C22—108no—
319C22—104no—
320C22—104no—
321C22—208no—
322C22—204no—
323C22—204no—
324C22C2×(C2×AGL(1,𝔽5))⋊C411yescharacteristic
325C22C4×C22⋊AGL(1,𝔽5)12yes—
326C22#6576411yescharacteristic
327C22#6576512yes—
328C22#6577112yes—
329C22#6578512yes—
330C22#6580011yescharacteristic
331C4—24no—
332C4—44no—
333C4—104no—
334C4—204no—
335C4—408no—
336C2—22no—
337C2—22no—
338C2—104no—
339C2—104no—
340C2#1060430412yes—
341C2#1060430712yes—
342C2#1060490911yescharacteristic
343C2#1060494211yescharacteristic
344C2#1060498111yescharacteristic
subgroupG/H
centerC23C2×C22⋊AGL(1,𝔽5)
commutatorC23×C5C2×C42
FrattiniC24C22×AGL(1,𝔽5)
FittingC22×C5×C4⋊C4C4
socleC23×C5C2×C22⋊C4
prime radical, cosocleC24×D5C23
hypercenterC22×C4⋊C4AGL(1,𝔽5)
hypocenterC5#10726
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
6. C22×C5×C4⋊C4C4semidirect
8. C22×C4⋊Dic5C4semidirect
9. C22×C4×Dic5C4semidirect
13. #65764C4semidirect
97. C22×C4⋊C4AGL(1,𝔽5)semidirect
308. C5#10726semidirect

Maximal subgroups

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

subgroupindex
C22×#40382
C2×#657642
C2×#649172
#107265