Finite Group Databasebeta

← #12097456

Subgroups — #12097456

Order 1280.

930 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#433C211yesmaximal, characteristic
2Dic5×#117C211yesmaximal, characteristic
3C2×#65289C211yesmaximal, characteristic
4#10601235C212yesmaximal, complemented
5#10601234C212yesmaximal, complemented
6#10600700C212yesmaximal
7#10600699C212yesmaximal
8#10600599C211yesmaximal, characteristic, complemented
9Dic5×#97C211yesmaximal, characteristic, complemented
10#10600544C211yesmaximal, characteristic, complemented
11C4×#3947C211yesmaximal, characteristic, complemented
12C2×C5×#117C2211yescharacteristic
13C2×#4072C2211yescharacteristic
14C22×C4×Dic5C2211yescharacteristic
15C5×#433C418yescomplemented
16C2×C5×C4⋊C8C2212yes—
17C2×C5×C22⋊C8C2212yes—
18C2×C5×#97C2211yescharacteristic
19C2×C4×C5×C8C2211yescharacteristic
20C2×C22⋊#168C2212yes—
21C2×#3947C2212yes—
22C2×#3947C2212yes—
23C2×#3947C2212yes—
24C2×C8×Dic5C2212yes—
25#65289C418yescomplemented
26C4⋊C4×Dic5C414yes—
27C22⋊C4×Dic5C414yes—
28#65221C2211yescharacteristic
29#65221C2211yescharacteristic
30C42×Dic5C2211yescharacteristic
31C42×Dic5C2211yescharacteristic
32C2×C4⋊#168C2212yes—
33C2×#3935C2211yescharacteristic
34C2×C4×#168C2211yescharacteristic
35#64770C2212yes—
36#64770C2212yes—
37#64770C2212yes—
38#64770C2212yes—
39#20465—51nomaximal, Sylow
40C23×C4×C5C2311yescharacteristic
41C23×Dic5C2×C412yes—
42C22×C5×C8C2312yes—
43C22×C5×C8C2312yes—
44C5×#117C2×C414yes—
45C5×#117C2×C414yes—
46C2×C5×C4⋊C4C2×C412yes—
47C2×C5×C22⋊C4C2×C412yes—
48C2×C42×C5C2311yescharacteristic
49C2×C42×C5C2311yescharacteristic
50C2×C22⋊Dic5C2×C412yes—
51#4072—24no—
52C2×C4⋊Dic5C2×C412yes—
53C2×C4×Dic5—24no—
54C2×C4×Dic5C2×C412yes—
55C2×C4×Dic5C2×C414yes—
56C2×C4×Dic5C2×C414yes—
57C2×C4×Dic5C2311yescharacteristic
58C2×C4×Dic5C2311yescharacteristic
59C2×C4×Dic5C2311yescharacteristic
60C2×C4×Dic5C2311yescharacteristic
61C22×#168C2312yes—
62C22×#168C2312yes—
63C5×C4⋊C8C2×C418yescomplemented
64C5×C22⋊C8C2×C418yes—
65C5×#97C2×C414yescomplemented
66C4×C5×C8C2×C414yescomplemented
67#3963C2×C414yes—
68#3963C2×C414yes—
69C22⋊#168C2×C418yes—
70#3947—24no—
71#3947—24no—
72#3947—24no—
73C8×Dic5—24no—
74C4⋊#168C2×C418yescomplemented
75#3935C2×C414yescomplemented
76C4×#168C2×C414yescomplemented
77C2×#433—51no—
78C2×#433D511yescharacteristic
79C4×#117—51no—
80#1768—52no—
81#1767—52no—
82#1750—52no—
83#1749—52no—
84#1746—51no—
85#1746—51no—
86C4×#97—51no—
87C4×#97—51no—
88C24×C5C22×C411yescharacteristic
89C22×C4×C5—22no—
90C22×C4×C5—22no—
91C22×C4×C5C22×C412yes—
92C22×C4×C5C22×C412yes—
93C22×C4×C5C22×C411yescharacteristic
94C22×C4×C5C22×C412yes—
95C22×C4×C5C22×C412yes—
96C22×C4×C5C2411yescharacteristic
97C22×Dic5—24no—
98C22×Dic5—44no—
99C22×Dic5C22×C412yes—
100C22×Dic5C22×C412yes—
101C22×Dic5C22×C412yes—
102C22×Dic5C22×C412yes—
103C2×C5×C8—22no—
104C2×C5×C8—22no—
105C2×C5×C8—22no—
106C2×C5×C8—22no—
107C2×C5×C8C22×C414yes—
108C2×C5×C8C22×C414yes—
109C5×C4⋊C4C4218yes—
110C5×C22⋊C4C4218yes—
111C42×C5C22×C412yes—
112C42×C5C22×C412yes—
113C42×C5C22×C412yes—
114C42×C5C22×C412yes—
115C22⋊Dic5—24no—
116C4⋊Dic5—24no—
117C4×Dic5—22no—
118C4×Dic5—22no—
119C4×Dic5—24no—
120C4×Dic5—44no—
121C4×Dic5—44no—
122C4×Dic5#4114yes—
123C4×Dic5#4114yes—
124C2×#168—22no—
125C2×#168—22no—
126C2×#168—22no—
127C2×#168—22no—
128C2×#168C22×C414yes—
129C2×#168C22×C414yes—
130C2×#117—51no—
131C2×#117C2×D511yescharacteristic
132C22×C42—51no—
133#433—58no—
134#433Dic518yescomplemented
135C2×C4⋊C8—52no—
136C2×C4⋊C8C2×D512yes—
137C2×C22⋊C8—52no—
138C2×C22⋊C8C2×D512yes—
139C2×#97—51no—
140C2×#97—52no—
141C2×#97—52no—
142C2×#97—52no—
143C2×#97C2×D511yescharacteristic
144C2×C4×C8—51no—
145C2×C4×C8—52no—
146C2×C4×C8C2×D511yescharacteristic
147C4×C4⋊C4—54no—
148C4×C22⋊C4—54no—
149#376—51no—
150#376—51no—
151C43—51no—
152C43—51no—
153#336—52no—
154#336—52no—
155#336—52no—
156#336—52no—
157C23×C5—21no—
158C23×C5—21no—
159C23×C5—24no—
160C23×C5C2×C4212yes—
161C23×C5C23×C411yescharacteristic
162C2×C4×C5—21no—
163C2×C4×C5—21no—
164C2×C4×C5—24no—
165C2×C4×C5—24no—
166C2×C4×C5—24no—
167C2×C4×C5—24no—
168C2×C4×C5C2×C4214yes—
169C2×C4×C5C2×C4214yes—
170C2×C4×C5C2×C4212yes—
171C2×C4×C5C23×C411yescharacteristic
172C2×C4×C5C23×C411yescharacteristic
173C2×C4×C5C2×#4112yes—
174C2×C4×C5C2×#4112yes—
175C2×Dic5—24no—
176C2×Dic5—24no—
177C2×Dic5—22no—
178C2×Dic5—22no—
179C2×Dic5—24no—
180C2×Dic5—24no—
181C2×Dic5—48no—
182C2×Dic5#11714yes—
183C2×Dic5#11714yes—
184C5×C8—24no—
185C5×C8—24no—
186#168—24no—
187#168—24no—
188C23×C4—52no—
189C23×C4C22×D511yescharacteristic
190C22×C8—52no—
191C22×C8—52no—
192C22×C8C22×D512yes—
193C22×C8C22×D512yes—
194#117—104no—
195#117C4×D514yes—
196#117C2×Dic514yes—
197C2×C4⋊C4—52no—
198C2×C4⋊C4C4×D512yes—
199C2×C22⋊C4—52no—
200C2×C22⋊C4C4×D512yes—
201C2×C42—51no—
202C2×C42—51no—
203C2×C42—52no—
204C2×C42—51no—
205C2×C42—51no—
206C2×C42—54no—
207C2×C42—54no—
208C2×C42—104no—
209C2×C42C22×D511yescharacteristic
210C2×C42C22×D511yescharacteristic
211C4⋊C8—58no—
212C4⋊C8C2×Dic518yescomplemented
213C22⋊C8—58no—
214C22⋊C8C2×Dic518yes—
215#97—54no—
216#97—104no—
217#97—104no—
218#97—104no—
219#97C2×Dic514yescomplemented
220C4×C8—54no—
221C4×C8—104no—
222C4×C8C2×Dic514yescomplemented
223#95—54no—
224#95—54no—
225C22×C5—24no—
226C22×C5—22no—
227C22×C5—42no—
228C22×C5—42no—
229C22×C5C22×C4211yescharacteristic
230C22×C5C2×#11711yescharacteristic
231C22×C5C2×#11711yescharacteristic
232C22×C5#56814yes—
233C4×C5—24no—
234C4×C5—24no—
235C4×C5—22no—
236C4×C5C4×#4112yes—
237C4×C5C4×#4112yes—
238Dic5—24no—
239Dic5—24no—
240Dic5—42no—
241Dic5—42no—
242C24C2×C4×D511yescharacteristic
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—52no—
246C22×C4—52no—
247C22×C4—52no—
248C22×C4—52no—
249C22×C4—104no—
250C22×C4—204no—
251C22×C4C2×C4×D512yes—
252C22×C4C2×C4×D512yes—
253C22×C4C2×C4×D512yes—
254C22×C4C2×C4×D511yescharacteristic
255C22×C4C22×Dic512yes—
256C22×C4C23×D511yescharacteristic
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—54no—
262C2×C8—54no—
263C2×C8—102no—
264C2×C8—102no—
265C2×C8—102no—
266C2×C8—102no—
267C2×C8C22×Dic514yes—
268C2×C8C22×Dic514yes—
269C4⋊C4—104no—
270C4⋊C4C4×Dic518yes—
271C22⋊C4—104no—
272C22⋊C4C4×Dic518yes—
273C42—54no—
274C42—54no—
275C42—104no—
276C42—102no—
277C42—102no—
278C42—204no—
279C42—204no—
280C42C2×C4×D512yes—
281C42C2×C4×D512yes—
282C42C22×Dic512yes—
283C42C22×Dic512yes—
284C2×C5—42no—
285C2×C5C4×#11711yescharacteristic
286C2×C5#287512yes—
287C2×C5#292112yes—
288C2×C5#292312yes—
289C23—24no—
290C23—21no—
291C23—21no—
292C23C2×C4×Dic512yes—
293C23C22×C4×D511yescharacteristic
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—54no—
301C2×C4—54no—
302C2×C4—104no—
303C2×C4—104no—
304C2×C4—104no—
305C2×C4—104no—
306C2×C4—102no—
307C2×C4—102no—
308C2×C4—208no—
309C2×C4C2×C4×Dic514yes—
310C2×C4C2×C4×Dic514yes—
311C2×C4C2×C4×Dic512yes—
312C2×C4C22×C4×D511yescharacteristic
313C2×C4D5×#4112yes—
314C2×C4D5×#4112yes—
315C2×C4C23×Dic511yescharacteristic
316C8—24no—
317C8—24no—
318C8—104no—
319C8—104no—
320C5#2046511yescharacteristic, complemented, Sylow
321C22—22no—
322C22—24no—
323C22—42no—
324C22—42no—
325C22D5×#11711yescharacteristic
326C22D5×#11711yescharacteristic
327C22#6608414yes—
328C22C22×C4×Dic511yescharacteristic
329C4—22no—
330C4—24no—
331C4—24no—
332C4—104no—
333C4—104no—
334C4—202no—
335C4—202no—
336C4#6585612yes—
337C4#41×Dic512yes—
338C2—42no—
339C2#1061110212yes—
340C2#1061110412yes—
341C2Dic5×#11711yescharacteristic
342C2#1061551312yes—
subgroupG/H
center, FrattiniC22×C4C23×D5
commutatorC22×C5C22×C42
FittingC2×C5×#433C2
socleC23×C5C23×C4
prime radicalC24×C5C22×C4
cosocleC22×C4×C5C24
hypercenterC2×#433D5
hypocenterC5#20465
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C4 ◁ C2×#433

Complements

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

NH ≅ G/Ntype
4. #10601235C2semidirect
5. #10601234C2semidirect
8. #10600599C2semidirect
9. Dic5×#97C2semidirect
10. #10600544C2semidirect
11. C4×#3947C2semidirect
15. C5×#433C4semidirect
25. #65289C4semidirect
63. C5×C4⋊C8C2×C4semidirect
65. C5×#97C2×C4semidirect
66. C4×C5×C8C2×C4semidirect
74. C4⋊#168C2×C4semidirect
75. #3935C2×C4semidirect
76. C4×#168C2×C4semidirect
134. #433Dic5semidirect
212. C4⋊C8C2×Dic5semidirect
219. #97C2×Dic5semidirect
222. C4×C8C2×Dic5semidirect
320. C5#20465semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#4332
Dic5×#1172
C2×#652892
#106012352
#106012342
#106007002
#106006992
#106005992
Dic5×#972
#106005442
C4×#39472
#204655