Finite Group Databasebeta

← #13021980

Subgroups — #13021980

Order 1280.

674 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#676C211yesmaximal, characteristic
2C2×#64897C211yesmaximal, characteristic
3C2×#64894C212yesmaximal
4C2×#64895C211yesmaximal, characteristic
5C2×#64862C212yesmaximal
6C2×#4130C2212yes—
7C2×D5⋊M16C2212yes—
8C22×#676C2211yescharacteristic
9C22×#676C2211yescharacteristic
10C2×C42×D5C2211yescharacteristic
11C2×C4×C5×C8C411yescharacteristic, complemented
12C2×#3947C411yescharacteristic, complemented
13C2×#3947C411yescharacteristic, complemented
14C2×C4×#168C411yescharacteristic, complemented
15C4×#676—24no—
16#64897—22no—
17#64895—22no—
18#64894—24no—
19#64862C418yescomplemented
20#10825—51nomaximal, Sylow
21C22×C4×D5—21no—
22C22×C4×D5C2311yescharacteristic
23C2×C22⋊AGL(1,𝔽5)—22no—
24C2×#704—22no—
25#4130—22no—
26#4130—22no—
27#4130—22no—
28#4130C2×C414yes—
29C2×C4⋊AGL(1,𝔽5)—22no—
30C2×C4×AGL(1,𝔽5)—22no—
31D5⋊M16—22no—
32D5⋊M16—22no—
33D5⋊M16—22no—
34D5⋊M16C2×C414yes—
35C2×#700—22no—
36C2×D5⋊C8—22no—
37C22×C5×C8C2×C411yescharacteristic
38C22×C5×C8C2×C411yescharacteristic
39C2×C42×C5C2×C411yescharacteristic
40C2×C4×Dic5—21no—
41C2×C4×Dic5C2×C411yescharacteristic
42C22×#168C2×C411yescharacteristic
43C22×#168C2×C411yescharacteristic
44C2×#676—22no—
45C2×#676—22no—
46C2×#676—21no—
47C2×#676—21no—
48C2×#676—22no—
49C2×#676D414yes—
50C2×#676D412yes—
51C2×#676Q812yes—
52C42×D5—22no—
53C42×D5C2×C414yes—
54C4×C5×C8—22no—
55#3947—22no—
56#3947—22no—
57C4×#168—22no—
58C2×C4×M16—51no—
59C2×#344—52no—
60C2×#339—52no—
61C2×#338—51no—
62C2×H(3,ℤ/4)—51no—
63C23×D5#4111yescharacteristic
64C22×AGL(1,𝔽5)—42no—
65C22×C4×C5—21no—
66C22×C4×C5C22×C411yescharacteristic
67C22×Dic5—21no—
68C22×Dic5#4111yescharacteristic
69C2×C4×D5—24no—
70C2×C4×D5—44no—
71C2×C4×D5C22×C412yes—
72C2×C4×D5C2×D411yescharacteristic
73C2×C4×D5C2×D412yes—
74C2×C4×D5C2×Q811yescharacteristic
75C2×C4×D5#4112yes—
76C2×C4×D5#4111yescharacteristic
77C2×C4×D5#4111yescharacteristic
78C2×C4×D5#4112yes—
79C22⋊AGL(1,𝔽5)—24no—
80C22⋊AGL(1,𝔽5)—42no—
81#704—24no—
82#704—42no—
83C2×#170—42no—
84C4⋊AGL(1,𝔽5)—24no—
85C4⋊AGL(1,𝔽5)—42no—
86C4×AGL(1,𝔽5)—24no—
87C4×AGL(1,𝔽5)—42no—
88#700—24no—
89#700—42no—
90D5⋊C8—24no—
91D5⋊C8—42no—
92C2×C5×C8—21no—
93C2×C5×C8—21no—
94C2×C5×C8—21no—
95C2×C5×C8—21no—
96C2×C5×C8C22⋊C412yes—
97C2×C5×C8C4⋊C412yes—
98C42×C5—21no—
99C42×C5C4212yes—
100C4×Dic5—21no—
101C4×Dic5—24no—
102C4×Dic5C4212yes—
103C2×#168—21no—
104C2×#168—21no—
105C2×#168—21no—
106C2×#168—21no—
107C2×#168C22⋊C412yes—
108C2×#168C4⋊C412yes—
109#676—24no—
110#676—24no—
111#676—24no—
112#676—24no—
113C22×M16—51no—
114C22×M16—51no—
115C22×M16—52no—
116C2×#117—52no—
117C22×C42—51no—
118C4×M16—104no—
119C2×#97—51no—
120C2×#97—51no—
121C2×C4×C8—51no—
122C2×C4×C8AGL(1,𝔽5)11yescharacteristic, complemented
123#344—104no—
124#339—58no—
125#338—102no—
126H(3,ℤ/4)—102no—
127C23×C5#11711yescharacteristic
128C22×D5—42no—
129C22×D5C4×D412yes—
130C22×D5#12411yescharacteristic
131C22×D5#12511yescharacteristic
132C22×D5#12612yes—
133C2×AGL(1,𝔽5)—44no—
134C2×AGL(1,𝔽5)—84no—
135C2×C4×C5—24no—
136C2×C4×C5—22no—
137C2×C4×C5C2×C4211yescharacteristic
138C2×C4×C5C2×C22⋊C411yescharacteristic
139C2×C4×C5C2×C4⋊C411yescharacteristic
140C2×C4×C5#11711yescharacteristic
141C2×C4×C5#11711yescharacteristic
142C2×C4×C5#11711yescharacteristic
143C2×Dic5—22no—
144C2×Dic5—42no—
145C2×Dic5C4×D412yes—
146C2×Dic5#12411yescharacteristic
147C2×Dic5#12511yescharacteristic
148C2×Dic5#12612yes—
149C4×D5—48no—
150C4×D5#11712yes—
151C4×D5#11712yes—
152C4×D5C4×D412yes—
153C4×D5C4×Q812yes—
154C4×D5#12112yes—
155C4×D5C22⋊Q812yes—
156C4×D5#12312yes—
157C4×D5#12312yes—
158#170—44no—
159C5×C8—22no—
160C5×C8—22no—
161#168—22no—
162#168—22no—
163C23×C4—51no—
164C23×C4—101no—
165C2×M16—52no—
166C2×M16—52no—
167C2×M16—54no—
168C2×M16—54no—
169C2×M16—102no—
170C2×M16—101no—
171C2×M16—101no—
172C2×M16—102no—
173C2×M16—102no—
174C2×M16—102no—
175C2×M16—102no—
176C2×M16—102no—
177C2×M16—102no—
178C2×M16—102no—
179C22×C8—51no—
180C22×C8—51no—
181C22×C8—102no—
182C22×C8C2×AGL(1,𝔽5)11yescharacteristic
183C22×C8C2×AGL(1,𝔽5)11yescharacteristic
184#117—54no—
185#117—102no—
186#117—102no—
187#117—102no—
188C2×C4⋊C4—102no—
189C2×C22⋊C4—102no—
190C2×C42—51no—
191C2×C42—54no—
192C2×C42—102no—
193C2×C42—101no—
194C2×C42—102no—
195C2×C42C2×AGL(1,𝔽5)11yescharacteristic
196#97—102no—
197#97—102no—
198C4×C8—22no—
199C4×C8—102no—
200C22×C5—22no—
201C22×C5C4×C22⋊C411yescharacteristic
202C22×C5#38211yescharacteristic
203C22×C5#38311yescharacteristic
204C2×D5—24no—
205C2×D5—42no—
206C2×D5—41no—
207C2×D5—41no—
208C2×D5#38712yes—
209C2×D5#38812yes—
210AGL(1,𝔽5)—84no—
211C4×C5—24no—
212C4×C5#37611yescharacteristic
213C4×C5C4×C4⋊C411yescharacteristic
214C4×C5#38011yescharacteristic
215C4×C5#38111yescharacteristic
216Dic5—42no—
217Dic5#38712yes—
218Dic5#38812yes—
219C24—51no—
220C22×C4—21no—
221C22×C4—51no—
222C22×C4—52no—
223C22×C4—51no—
224C22×C4—51no—
225C22×C4—52no—
226C22×C4—51no—
227C22×C4—51no—
228C22×C4—52no—
229C22×C4—52no—
230C22×C4—104no—
231C22×C4—101no—
232C22×C4—204no—
233C22×C4—202no—
234C22×C4C22×AGL(1,𝔽5)11yescharacteristic
235M16—104no—
236M16—104no—
237M16—104no—
238M16—104no—
239M16—104no—
240M16—104no—
241M16—202no—
242M16—202no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—52no—
248C2×C8—52no—
249C2×C8—101no—
250C2×C8—101no—
251C2×C8—101no—
252C2×C8—101no—
253C2×C8—104no—
254C2×C8—202no—
255C2×C8—202no—
256C2×C8C4⋊AGL(1,𝔽5)12yes—
257C2×C8C22⋊AGL(1,𝔽5)12yes—
258C4⋊C4—104no—
259C4⋊C4—202no—
260C22⋊C4—104no—
261C22⋊C4—202no—
262C42—21no—
263C42—52no—
264C42—104no—
265C42—104no—
266C42—101no—
267C42—202no—
268C42C4×AGL(1,𝔽5)12yes—
269C2×C5—21no—
270C2×C5—21no—
271C2×C5#143411yescharacteristic
272C2×C5#184112yes—
273D5—22no—
274D5—22no—
275C23—52no—
276C23—51no—
277C23—51no—
278C23—52no—
279C23—202no—
280C23#413011yescharacteristic
281C2×C4—24no—
282C2×C4—22no—
283C2×C4—51no—
284C2×C4—51no—
285C2×C4—52no—
286C2×C4—52no—
287C2×C4—52no—
288C2×C4—52no—
289C2×C4—52no—
290C2×C4—52no—
291C2×C4—52no—
292C2×C4—52no—
293C2×C4—52no—
294C2×C4—52no—
295C2×C4—102no—
296C2×C4—208no—
297C2×C4—202no—
298C2×C4—204no—
299C2×C4—404no—
300C2×C4C2×C4×AGL(1,𝔽5)11yescharacteristic
301C2×C4C2×C4⋊AGL(1,𝔽5)11yescharacteristic
302C2×C4#413011yescharacteristic
303C2×C4#413011yescharacteristic
304C2×C4#413011yescharacteristic
305C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
306C8—22no—
307C8—22no—
308C8—102no—
309C8—102no—
310C8—204no—
311C5#1082511yescharacteristic, complemented, Sylow
312C22—22no—
313C22—52no—
314C22—52no—
315C22—104no—
316C22—202no—
317C22—201no—
318C22—201no—
319C22#6568911yescharacteristic
320C22#6569011yescharacteristic
321C22C4×C22⋊AGL(1,𝔽5)11yescharacteristic
322C4—24no—
323C4—52no—
324C4—52no—
325C4—202no—
326C4—404no—
327C4#6568611yescharacteristic
328C4C4×C4⋊AGL(1,𝔽5)11yescharacteristic
329C4#6576411yescharacteristic
330C4#6576511yescharacteristic
331C2—21no—
332C2—21no—
333C2—102no—
334C2—102no—
335C2#1060385312yes—
336C2#1060419711yescharacteristic
subgroupG/H
centerC2×C4#4130
commutatorC2×C4×C5C2×C42
FrattiniC22×C4C22×AGL(1,𝔽5)
FittingC2×C4×C5×C8C4
socleC22×C5#383
prime radicalC23×D5#41
cosocleC22×C4×D5C23
hypercenterC2×C4×C8AGL(1,𝔽5)
hypocenterC5#10825
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8

Complements

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

NH ≅ G/Ntype
11. C2×C4×C5×C8C4semidirect
12. C2×#3947C4semidirect
13. C2×#3947C4semidirect
14. C2×C4×#168C4semidirect
19. #64862C4semidirect
122. C2×C4×C8AGL(1,𝔽5)semidirect
311. C5#10825semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#6762
C2×#648972
C2×#648942
C2×#648952
C2×#648622
#108255