Finite Group Databasebeta

← #12097221

Subgroups — #12097221

Order 1280.

1146 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C5×#117C211yesmaximal, characteristic
2C2×#65286C212yesmaximal
3C2×#65213C211yesmaximal, characteristic
4C22×C4⋊#168C211yesmaximal, characteristic
5#10602372C212yesmaximal
6#10600229C214yesmaximal, complemented
7#10600184C212yesmaximal, complemented
8C4×C4⋊#168C212yesmaximal, complemented
9C2×C5×#117C2211yescharacteristic
10C2×C5×#117C2211yescharacteristic
11C22×C42×C5C2211yescharacteristic
12C22×#681C2212yes—
13C23×#168C2212yes—
14C4×C5×C4⋊C4C412yes—
15C4×C5×C4⋊C4C412yes—
16C4×C5×C22⋊C4C412yes—
17C4×C5×C22⋊C4C412yes—
18C5×#376C2212yes—
19C43×C5C2212yes—
20#65286—28no—
21#65213C418yescomplemented
22C2×C4⋊#168C418yescomplemented
23C2×C4⋊#168C2212yes—
24C2×C4⋊#168C2211yescharacteristic
25C2×C4⋊#168C2211yescharacteristic
26C2×C4⋊#168C2212yes—
27C2×C4⋊#168C2212yes—
28C2×#3935C2214yes—
29C2×C4×#168C2214yes—
30#64744C2214yes—
31#64744C2214yes—
32#20431—51nomaximal, Sylow
33C23×C4×C5C2×C411yescharacteristic
34C23×C4×C5C2×C411yescharacteristic
35C23×C4×C5C2311yescharacteristic
36C5×#117—22no—
37C5×#117—22no—
38C5×#117D414yes—
39C5×#117Q814yes—
40C2×C5×C4⋊C4C2×C412yes—
41C2×C5×C4⋊C4C2×C412yes—
42C2×C5×C22⋊C4C2×C412yes—
43C2×C5×C22⋊C4C2×C412yes—
44C2×C42×C5C2×C414yes—
45C2×C42×C5C2×C414yes—
46C2×C42×C5C2×C414yes—
47C2×C42×C5C2×C414yes—
48C2×C42×C5C2×C411yescharacteristic
49C2×C42×C5C2×C411yescharacteristic
50C2×C42×C5C2312yes—
51C2×C42×C5C2311yescharacteristic
52C2×C42×C5C2311yescharacteristic
53C2×C42×C5C2312yes—
54C2×#681—24no—
55C2×#681—24no—
56C2×#681C2×C418yes—
57C22×#168—24no—
58C22×#168—24no—
59C22×#168C2×C418yes—
60C22×#168C2312yes—
61C22×#168C2312yes—
62C22×#168C2314yes—
63C5×#95C2×C414yes—
64C5×#95C2×C414yes—
65C4⋊#168C2×C418yes—
66C4⋊#168C2×C414yes—
67C4⋊#168C2×C414yes—
68C4⋊#168C2×C418yes—
69C4⋊#168C2×C418yes—
70#3935—28no—
71C4×#168—28no—
72C2×#423—51no—
73C22×C4⋊C8—51no—
74C2×#405—52no—
75C4×#117D511yescharacteristic
76#1768—54no—
77#1767—52no—
78C4×C4⋊C8—52no—
79#1729—52no—
80C24×C5C22×C411yescharacteristic
81C22×C4×C5—24no—
82C22×C4×C5—42no—
83C22×C4×C5C4214yes—
84C22×C4×C5C4214yes—
85C22×C4×C5C22×C412yes—
86C22×C4×C5C22×C412yes—
87C22×C4×C5C22×C414yes—
88C22×C4×C5C22×C411yescharacteristic
89C22×C4×C5C22×C411yescharacteristic
90C22×C4×C5C22×C411yescharacteristic
91C22×C4×C5C22×C414yes—
92C22×C4×C5C22×C411yescharacteristic
93C22×C4×C5C22×C411yescharacteristic
94C22×C4×C5C2×D412yes—
95C22×C4×C5C2×Q812yes—
96C22×C4×C5#4112yes—
97C22×C4×C5#4112yes—
98C22×C4×C5C2411yescharacteristic
99C5×C4⋊C4—24no—
100C5×C4⋊C4C4⋊C418yes—
101C5×C22⋊C4—24no—
102C5×C22⋊C4C4⋊C418yes—
103C42×C5—28no—
104C42×C5—22no—
105C42×C5—22no—
106C42×C5—44no—
107C42×C5C4214yes—
108C42×C5C4214yes—
109C42×C5C22×C412yes—
110C42×C5C22×C412yes—
111C42×C5C22×C412yes—
112C42×C5C22×C412yes—
113C42×C5C2×D414yes—
114C42×C5C2×Q814yes—
115#681—216no—
116C2×#168—216no—
117C2×#168—22no—
118C2×#168—24no—
119C2×#168—22no—
120C2×#168—22no—
121C2×#168—22no—
122C2×#168—24no—
123C2×#168C22×C414yes—
124C2×#168C22×C414yes—
125C2×#168C22×C418yes—
126C22×M16—52no—
127C23×C8—52no—
128C2×#117C2×D511yescharacteristic
129C2×#117C2×D511yescharacteristic
130C22×C42C2×D511yescharacteristic
131#423—58no—
132C2×C4⋊C8—58no—
133C2×C4⋊C8—51no—
134C2×C4⋊C8—51no—
135C2×C4⋊C8—52no—
136C2×C4⋊C8—52no—
137C2×C4⋊C8—52no—
138#405—108no—
139C2×#97—54no—
140C2×C4×C8—54no—
141C4×C4⋊C4Dic512yes—
142C4×C4⋊C4Dic512yes—
143C4×C22⋊C4Dic512yes—
144C4×C22⋊C4Dic512yes—
145#376C2×D512yes—
146C43C2×D512yes—
147#336—54no—
148#336—54no—
149C23×C5—22no—
150C23×C5—22no—
151C23×C5C2×C4212yes—
152C23×C5C2×C4⋊C412yes—
153C23×C5#11712yes—
154C23×C5C23×C411yescharacteristic
155C2×C4×C5—24no—
156C2×C4×C5—22no—
157C2×C4×C5—22no—
158C2×C4×C5—24no—
159C2×C4×C5—22no—
160C2×C4×C5—28no—
161C2×C4×C5—24no—
162C2×C4×C5—24no—
163C2×C4×C5—24no—
164C2×C4×C5—44no—
165C2×C4×C5C2×C4212yes—
166C2×C4×C5C2×C4212yes—
167C2×C4×C5C2×C4212yes—
168C2×C4×C5C2×C4212yes—
169C2×C4×C5C2×C4212yes—
170C2×C4×C5C2×C4⋊C418yes—
171C2×C4×C5C2×C4⋊C412yes—
172C2×C4×C5#11712yes—
173C2×C4×C5C4×D414yes—
174C2×C4×C5C4×D414yes—
175C2×C4×C5C4×Q814yes—
176C2×C4×C5C4×Q814yes—
177C2×C4×C5#13114yes—
178C2×C4×C5C23×C411yescharacteristic
179C2×C4×C5C23×C411yescharacteristic
180C2×C4×C5C22×D411yescharacteristic
181C2×C4×C5C22×Q811yescharacteristic
182C2×C4×C5C2×#4111yescharacteristic
183C2×C4×C5C2×#4111yescharacteristic
184#168—24no—
185#168—24no—
186#168—28no—
187C23×C4C2×Dic511yescharacteristic
188C23×C4C2×Dic511yescharacteristic
189C23×C4C22×D511yescharacteristic
190C2×M16—58no—
191C2×M16—104no—
192C2×M16—104no—
193C22×C8—58no—
194C22×C8—52no—
195C22×C8—52no—
196C22×C8—54no—
197C22×C8—104no—
198C22×C8—104no—
199#117—22no—
200#117—22no—
201#117Dic1014yes—
202#117D2014yes—
203C2×C4⋊C4C2×Dic512yes—
204C2×C4⋊C4C2×Dic512yes—
205C2×C22⋊C4C2×Dic512yes—
206C2×C22⋊C4C2×Dic512yes—
207C2×C42C4×D514yes—
208C2×C42C4×D514yes—
209C2×C42C2×Dic514yes—
210C2×C42C2×Dic514yes—
211C2×C42C2×Dic511yescharacteristic
212C2×C42C2×Dic511yescharacteristic
213C2×C42C22×D512yes—
214C2×C42C22×D512yes—
215C2×C42C22×D511yescharacteristic
216C2×C42C22×D511yescharacteristic
217C4⋊C8—54no—
218C4⋊C8—54no—
219C4⋊C8—58no—
220C4⋊C8—58no—
221C4⋊C8—58no—
222#97—108no—
223C4×C8—108no—
224#95C2×Dic514yes—
225#95C2×Dic514yes—
226C22×C5—22no—
227C22×C5—22no—
228C22×C5—24no—
229C22×C5—24no—
230C22×C5C4×C4⋊C414yes—
231C22×C5C22×C4211yescharacteristic
232C22×C5C22×C4⋊C411yescharacteristic
233C22×C5C2×#11711yescharacteristic
234C22×C5C2×#13112yes—
235C22×C5#56812yes—
236C4×C5—22no—
237C4×C5—22no—
238C4×C5—22no—
239C4×C5—28no—
240C4×C5—44no—
241C4×C5C4×C4⋊C414yes—
242C4×C5#40514yescomplemented
243C4×C5C2×C4×D411yescharacteristic
244C4×C5C2×C4×D411yescharacteristic
245C4×C5C2×C4×Q811yescharacteristic
246C4×C5C2×C4×Q811yescharacteristic
247C24C22×Dic511yescharacteristic
248C22×C4—24no—
249C22×C4—42no—
250C22×C4C4×Dic514yes—
251C22×C4C4×Dic514yes—
252C22×C4C2×Dic1012yes—
253C22×C4C2×C4×D512yes—
254C22×C4C2×C4×D512yes—
255C22×C4C2×D2012yes—
256C22×C4#70912yes—
257C22×C4#70912yes—
258C22×C4C22×Dic514yes—
259C22×C4C22×Dic511yescharacteristic
260C22×C4C22×Dic511yescharacteristic
261C22×C4C22×Dic511yescharacteristic
262C22×C4C22×Dic514yes—
263C22×C4C22×Dic511yescharacteristic
264C22×C4C22×Dic511yescharacteristic
265C22×C4C23×D511yescharacteristic
266M16—1016no—
267C2×C8—54no—
268C2×C8—54no—
269C2×C8—58no—
270C2×C8—102no—
271C2×C8—104no—
272C2×C8—102no—
273C2×C8—102no—
274C2×C8—102no—
275C2×C8—104no—
276C2×C8—1016no—
277C4⋊C4—24no—
278C4⋊C4C4⋊Dic518yes—
279C22⋊C4—24no—
280C22⋊C4C4⋊Dic518yes—
281C42—22no—
282C42—22no—
283C42—28no—
284C42—44no—
285C42C4×Dic514yes—
286C42C4×Dic514yes—
287C42C2×Dic1014yes—
288C42C2×C4×D512yes—
289C42C2×C4×D512yes—
290C42C2×C4×D512yes—
291C42C2×C4×D512yes—
292C42C2×D2014yes—
293C2×C5—24no—
294C2×C5C2×C4×C4⋊C411yescharacteristic
295C2×C5C2×#40511yescharacteristic
296C2×C5#287311yescharacteristic
297C2×C5#290612yes—
298C2×C5#292512yes—
299C23—22no—
300C23—22no—
301C23C2×C4×Dic512yes—
302C23C2×C4⋊Dic512yes—
303C23#407212yes—
304C23C23×Dic511yescharacteristic
305C2×C4—24no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—28no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—44no—
315C2×C4C4×Dic1014yes—
316C2×C4C4×Dic1014yes—
317C2×C4C4×D2014yes—
318C2×C4C4×D2014yes—
319C2×C4C2×C4×Dic512yes—
320C2×C4C2×C4×Dic512yes—
321C2×C4C2×C4×Dic512yes—
322C2×C4C2×C4×Dic512yes—
323C2×C4C2×C4×Dic512yes—
324C2×C4C2×C4⋊Dic518yes—
325C2×C4C2×C4⋊Dic512yes—
326C2×C4#407212yes—
327C2×C4#409414yes—
328C2×C4C22×Dic1011yescharacteristic
329C2×C4C22×C4×D511yescharacteristic
330C2×C4C22×C4×D511yescharacteristic
331C2×C4C22×D2011yescharacteristic
332C2×C4C2×#70911yescharacteristic
333C2×C4C2×#70911yescharacteristic
334C8—104no—
335C8—104no—
336C8—108no—
337C5#2043111yescharacteristic, complemented, Sylow
338C22—22no—
339C22—22no—
340C22—24no—
341C22—24no—
342C22C4×C4⋊Dic514yes—
343C22C22×C4×Dic511yescharacteristic
344C22C22×C4⋊Dic511yescharacteristic
345C22C2×#407211yescharacteristic
346C22C2×#409412yes—
347C22#6615312yes—
348C4—22no—
349C4—22no—
350C4—22no—
351C4—28no—
352C4—44no—
353C4C4×C4⋊Dic514yes—
354C4#6528614yescomplemented
355C4C2×C4×Dic1011yescharacteristic
356C4C2×C4×Dic1011yescharacteristic
357C4C2×C4×D2011yescharacteristic
358C4C2×C4×D2011yescharacteristic
359C2—24no—
360C2C2×C4×C4⋊Dic511yescharacteristic
361C2C2×#6528611yescharacteristic
362C2#1061393311yescharacteristic
363C2#1061424312yes—
364C2#1061424612yes—
subgroupG/H
center, FrattiniC22×C4C23×D5
commutatorC22×C5C22×C42
FittingC4×C5×#117C2
socleC23×C5C23×C4
prime radicalC24×C5C22×C4
cosocleC22×C4×C5C24
hypercenterC4×#117D5
hypocenterC5#20431
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C4 ◁ C4×#117

Complements

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

NH ≅ G/Ntype
6. #10600229C2semidirect
7. #10600184C2semidirect
8. C4×C4⋊#168C2semidirect
21. #65213C4semidirect
22. C2×C4⋊#168C4semidirect
242. C4×C5#405semidirect
337. C5#20431semidirect
354. C4#65286semidirect

Maximal subgroups

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

subgroupindex
C4×C5×#1172
C2×#652862
C2×#652132
C22×C4⋊#1682
#106023722
#106002292
#106001842
C4×C4⋊#1682
#204315