Finite Group Databasebeta

← #13034122

Subgroups — #13034122

Order 1280.

5990 conjugacy classes of subgroups; only the 456 normal subgroups are shown.

normal subgroups only

This group has 5990 conjugacy classes of subgroups — too many to draw. Only its 456 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
1C24×C4×D5C211yesmaximal, characteristic
2C2×#65798C212yesmaximal, complemented
3C2×#65755C214yesmaximal, complemented
4#10604898C218yesmaximal, complemented
5C25×D5C411yescharacteristic
6C24×C4×C5C411yescharacteristic
7C24×Dic5C2211yescharacteristic
8C23×C4×D5—28no—
9C23×C4×D5—216no—
10C23×C4×D5C412yes—
11C23×C4×D5C414yes—
12C23×C4×D5C2214yes—
13C23×C4×D5C2212yes—
14C22×#704C2214yescomplemented
15#65798—28no—
16C2×C22⋊#170C2218yescomplemented
17#65755—216no—
18C2×#4003C22116yescomplemented
19#20579—51nomaximal, Sylow
20C25×C5C2×C411yescharacteristic
21C24×D5—24no—
22C24×D5—28no—
23C24×D5—216no—
24C24×D5C2×C412yes—
25C24×D5C2×C414yes—
26C23×C4×C5—24no—
27C23×C4×C5—28no—
28C23×C4×C5C2×C414yes—
29C23×C4×C5C2×C412yes—
30C23×Dic5—28no—
31C23×Dic5C2×C411yescharacteristic
32C23×Dic5C2×C412yes—
33C23×Dic5D418yes—
34C23×Dic5C2312yes—
35C23×Dic5C2311yescharacteristic
36C22×C4×D5—2128no—
37C22×C4×D5—28no—
38C22×C4×D5—28no—
39C22×C4×D5—28no—
40C22×C4×D5—216no—
41C22×C4×D5—216no—
42C22×C4×D5—28no—
43C22×C4×D5—28no—
44C22×C4×D5—232no—
45C22×C4×D5—232no—
46C22×C4×D5C2×C418yes—
47C22×C4×D5C2×C414yes—
48C22×C4×D5D4116yes—
49C22×C4×D5C2314yes—
50C2×#704—28no—
51C2×#704—216no—
52C22×#170C2318yescomplemented
53C22⋊#170—216no—
54#4003—232no—
55C25×C4—51no—
56C2×#407—54no—
57C2×#407—52no—
58#1779—58no—
59C24×C5—24no—
60C24×C5—28no—
61C24×C5C22⋊C414yes—
62C24×C5C22×C411yescharacteristic
63C24×C5C22×C412yes—
64C23×D5—232no—
65C23×D5—264no—
66C23×D5—2128no—
67C23×D5—28no—
68C23×D5—24no—
69C23×D5—24no—
70C23×D5—28no—
71C23×D5—216no—
72C23×D5—216no—
73C23×D5—216no—
74C23×D5C22⋊C418yes—
75C23×D5M16116yescomplemented
76C23×D5C22×C414yes—
77C22×C4×C5—232no—
78C22×C4×C5—28no—
79C22×C4×C5—24no—
80C22×C4×C5—24no—
81C22×C4×C5—216no—
82C22×C4×C5C22⋊C418yes—
83C22×C4×C5C22×C414yes—
84C22×Dic5—24no—
85C22×Dic5—28no—
86C22×Dic5—232no—
87C22×Dic5—28no—
88C22×Dic5C22⋊C414yes—
89C22×Dic5C22⋊C418yes—
90C22×Dic5C22×C412yes—
91C22×Dic5C22×C411yescharacteristic
92C22×Dic5C2×D418yes—
93C22×Dic5C2×D414yes—
94C22×Dic5C2×D418yes—
95C22×Dic5C2411yescharacteristic
96C2×C4×D5—264no—
97C2×C4×D5—2128no—
98C2×C4×D5—2128no—
99C2×C4×D5—216no—
100C2×C4×D5—216no—
101C2×C4×D5—216no—
102C2×C4×D5—28no—
103C2×C4×D5—28no—
104C2×C4×D5—264no—
105C2×C4×D5—28no—
106C2×C4×D5—28no—
107C2×C4×D5C22⋊C4116yes—
108C2×C4×D5C2×D4116yes—
109#704—416no—
110C2×#170—216no—
111C2×#170—28no—
112C26—51no—
113C24×C4—51no—
114C24×C4—52no—
115C24×C4—54no—
116C24×C4—52no—
117C24×C4—54no—
118C24×C4—108no—
119C24×C4—1016no—
120C24×C4AGL(1,𝔽5)11yescharacteristic
121C22×M16—54no—
122#407—108no—
123#407—1016no—
124C2×C22⋊C8—58no—
125C2×C22⋊C8—516no—
126C23×C5—232no—
127C23×C5—24no—
128C23×C5—24no—
129C23×C5—24no—
130C23×C5—28no—
131C23×C5—216no—
132C23×C5C2×C22⋊C414yes—
133C23×C5C2×C22⋊C412yes—
134C23×C5C2×C22⋊C414yes—
135C23×C5C2×M1618yes—
136C23×C5C23×C411yescharacteristic
137C22×D5—232no—
138C22×D5—2256no—
139C22×D5—264no—
140C22×D5—232no—
141C22×D5—232no—
142C22×D5—232no—
143C22×D5—24no—
144C22×D5—24no—
145C22×D5—28no—
146C22×D5—216no—
147C22×D5—232no—
148C22×D5—232no—
149C22×D5—264no—
150C22×D5C2×C22⋊C418yes—
151C22×D5C2×M16116yes—
152C2×C4×C5—232no—
153C2×C4×C5—24no—
154C2×C4×C5—24no—
155C2×C4×C5—216no—
156C2×C4×C5C2×C22⋊C418yes—
157C2×Dic5—24no—
158C2×Dic5—28no—
159C2×Dic5—216no—
160C2×Dic5—216no—
161C2×Dic5C2×C22⋊C414yes—
162C2×Dic5C2×C22⋊C412yes—
163C2×Dic5C2×C22⋊C414yes—
164C2×Dic5#120116yes—
165C2×Dic5C22×D412yes—
166C2×Dic5C22×D414yes—
167C4×D5—264no—
168C4×D5—232no—
169C4×D5—232no—
170#170—48no—
171C25—54no—
172C25—52no—
173C25—108no—
174C25—1016no—
175C25—104no—
176C25C2×AGL(1,𝔽5)11yescharacteristic
177C23×C4—24no—
178C23×C4—28no—
179C23×C4—51no—
180C23×C4—58no—
181C23×C4—52no—
182C23×C4—51no—
183C23×C4—52no—
184C23×C4—516no—
185C23×C4—54no—
186C23×C4—54no—
187C23×C4—58no—
188C23×C4—108no—
189C23×C4—108no—
190C23×C4—1016no—
191C23×C4—1016no—
192C23×C4—108no—
193C23×C4—1032no—
194C23×C4—1032no—
195C23×C4—108no—
196C23×C4—10128no—
197C23×C4—108no—
198C23×C4—108no—
199C23×C4C2×AGL(1,𝔽5)14yes—
200C23×C4C2×AGL(1,𝔽5)12yes—
201C2×M16—108no—
202C2×M16—1016no—
203C22×C8—58no—
204C22⋊C8—1032no—
205C22⋊C8—1016no—
206C22×C5—232no—
207C22×C5—24no—
208C22×C5—24no—
209C22×C5—28no—
210C22×C5—216no—
211C22×C5—24no—
212C22×C5#37914yes—
213C22×C5#40718yescomplemented
214C22×C5C22×C22⋊C412yes—
215C22×C5C22×C22⋊C411yescharacteristic
216C22×C5C22×M1614yes—
217C2×D5—264no—
218C2×D5—2128no—
219C2×D5—28no—
220C2×D5—216no—
221C2×D5—216no—
222C2×D5—28no—
223C2×D5#407116yescomplemented
224C4×C5—28no—
225Dic5—24no—
226Dic5#37914yes—
227Dic5C2×#12014yes—
228C24—28no—
229C24—24no—
230C24—54no—
231C24—516no—
232C24—58no—
233C24—1016no—
234C24—1016no—
235C24—1016no—
236C24—1032no—
237C24—10128no—
238C24—1064no—
239C24—108no—
240C24—108no—
241C24—104no—
242C24—104no—
243C24C22⋊AGL(1,𝔽5)14yes—
244C24C22×AGL(1,𝔽5)12yes—
245C24C22×AGL(1,𝔽5)11yescharacteristic
246C22×C4—232no—
247C22×C4—28no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—216no—
251C22×C4—54no—
252C22×C4—58no—
253C22×C4—51no—
254C22×C4—51no—
255C22×C4—58no—
256C22×C4—58no—
257C22×C4—54no—
258C22×C4—52no—
259C22×C4—516no—
260C22×C4—516no—
261C22×C4—1064no—
262C22×C4—108no—
263C22×C4—108no—
264C22×C4—108no—
265C22×C4—108no—
266C22×C4—1016no—
267C22×C4—1016no—
268C22×C4—10128no—
269C22×C4—10128no—
270C22×C4—1064no—
271C22×C4—1032no—
272C22×C4—108no—
273C22×C4—108no—
274C22×C4—104no—
275C22×C4—1016no—
276C22×C4C22⋊AGL(1,𝔽5)18yes—
277C22×C4C22×AGL(1,𝔽5)14yes—
278M16—2016no—
279C2×C8—1016no—
280C2×C8—108no—
281C2×C5—24no—
282C2×C5—28no—
283C2×C5C2×#37911yescharacteristic
284C2×C5C2×#40712yes—
285C2×C5C2×#40714yes—
286D5—216no—
287C23—232no—
288C23—28no—
289C23—24no—
290C23—24no—
291C23—216no—
292C23—24no—
293C23—516no—
294C23—58no—
295C23—1064no—
296C23—1032no—
297C23—1032no—
298C23—10256no—
299C23—1032no—
300C23—1032no—
301C23—1064no—
302C23—1032no—
303C23—1016no—
304C23—108no—
305C23—1032no—
306C23—104no—
307C23—104no—
308C23D5⋊M1618yes—
309C23C2×C22⋊AGL(1,𝔽5)12yes—
310C23C2×C22⋊AGL(1,𝔽5)14yes—
311C23C2×C22⋊AGL(1,𝔽5)14yes—
312C23C23×AGL(1,𝔽5)11yescharacteristic
313C2×C4—232no—
314C2×C4—216no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—516no—
318C2×C4—52no—
319C2×C4—54no—
320C2×C4—52no—
321C2×C4—54no—
322C2×C4—54no—
323C2×C4—1032no—
324C2×C4—1032no—
325C2×C4—1064no—
326C2×C4—1016no—
327C2×C4—1016no—
328C2×C4—108no—
329C2×C4—104no—
330C2×C4C2×C22⋊AGL(1,𝔽5)18yes—
331C8—208no—
332C5#2057911yescharacteristic, complemented, Sylow
333C22—232no—
334C22—24no—
335C22—24no—
336C22—216no—
337C22—28no—
338C22—24no—
339C22—516no—
340C22—1064no—
341C22—10128no—
342C22—1016no—
343C22—1016no—
344C22—108no—
345C22—108no—
346C22#6575518yescomplemented
347C22#6580014yes—
348C22C2×D5⋊M1614yes—
349C22C22×C22⋊AGL(1,𝔽5)11yescharacteristic
350C22C22×C22⋊AGL(1,𝔽5)12yes—
351C4—28no—
352C4—54no—
353C4—54no—
354C4—104no—
355C2—24no—
356C2—28no—
357C2—1016no—
358C2C2×#6575512yes—
359C2D5⋊#40714yes—
360C2C2×#6580011yescharacteristic
subgroupG/H
center, FrattiniC23C23×AGL(1,𝔽5)
commutator, socleC23×C5C23×C4
FittingC24×C4×C5C4
prime radicalC25×D5C4
cosocleC22×Dic5C24
hypercenterC24×C4AGL(1,𝔽5)
hypocenterC5#20579
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C24×C4

Complements

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

NH ≅ G/Ntype
2. C2×#65798C2semidirect
3. C2×#65755C2semidirect
4. #10604898C2semidirect
14. C22×#704C22semidirect
16. C2×C22⋊#170C22semidirect
18. C2×#4003C22semidirect
52. C22×#170C23semidirect
75. C23×D5M16semidirect
213. C22×C5#407semidirect
223. C2×D5#407semidirect
332. C5#20579semidirect
346. C22#65755semidirect

Maximal subgroups

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

subgroupindex
C24×C4×D52
C2×#657982
C2×#657552
#106048982
#205795