Finite Group Databasebeta

← #12104178

Subgroups — #12104178

Order 1280.

938 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×C5×#432C211yesmaximal, characteristic
2#10613791C211yesmaximal, characteristic
3C2×C4×C4⋊Dic5C211yesmaximal, characteristic
4#10601248C214yesmaximal, complemented
5#10600721C212yesmaximal
6#10600720C212yesmaximal
7#10600607C214yesmaximal, complemented
8C22×C42×C5C2211yescharacteristic
9C22×C4⋊Dic5C2212yes—
10C22×C4×Dic5—21no—
11C5×#432C418yescomplemented
12C2×C5×C4⋊C8C2212yes—
13C2×C5×C22⋊C8C2212yes—
14C2×C5×#97C2212yes—
15#65501C2211yescharacteristic
16#65501C2211yescharacteristic
17#65500—21no—
18C4×C22⋊Dic5—21no—
19C2×#3963—21no—
20C2×#3950C2214yes—
21C2×#3949C2214yes—
22#65226—21no—
23#65225C2212yes—
24C4×C4⋊Dic5—21no—
25C4×C4⋊Dic5—21no—
26C4×C4⋊Dic5C2212yes—
27C4×C4⋊Dic5C2212yes—
28C4×C4⋊Dic5C2212yes—
29#65221—21no—
30#64771C2214yes—
31#64771C2214yes—
32#21363—51nomaximal, Sylow
33C23×C4×C5D411yescharacteristic
34C23×C4×C5D411yescharacteristic
35C23×C4×C5C2311yescharacteristic
36C23×Dic5—22no—
37C22×C5×C8C2314yes—
38C2×C42×C5C2×C414yes—
39C2×C42×C5D411yescharacteristic
40C2×C42×C5D411yescharacteristic
41C2×C42×C5Q814yes—
42C2×C42×C5C2311yescharacteristic
43C2×C42×C5C2311yescharacteristic
44C2×C22⋊Dic5—21no—
45C2×C22⋊Dic5—21no—
46C2×C4⋊Dic5—22no—
47C2×C4⋊Dic5—28no—
48C2×C4⋊Dic5C2312yes—
49C2×C4⋊Dic5C2312yes—
50C2×C4⋊Dic5C2314yes—
51C2×C4×Dic5—22no—
52C2×C4×Dic5—22no—
53C2×C4×Dic5—22no—
54C2×C4×Dic5—44no—
55C5×C4⋊C8C2×C418yescomplemented
56C5×C22⋊C8C2×C418yes—
57C5×#97C2×C418yescomplemented
58#3963—22no—
59#3963—21no—
60#3963—21no—
61#3963—22no—
62#3963—22no—
63#3963—22no—
64#3963—22no—
65#3950—28no—
66#3949—28no—
67C2×#432D511yescharacteristic
68#2283—51no—
69C2×C4×C4⋊C4—51no—
70#1949—54no—
71#1853—52no—
72#1852—52no—
73#1776—54no—
74C24×C5C2×D411yescharacteristic
75C22×C4×C5—22no—
76C22×C4×C5—22no—
77C22×C4×C5—24no—
78C22×C4×C5—42no—
79C22×C4×C5C4⋊C414yes—
80C22×C4×C5C4⋊C414yes—
81C22×C4×C5C22×C412yes—
82C22×C4×C5C2×D411yescharacteristic
83C22×C4×C5C2×D411yescharacteristic
84C22×C4×C5C2×D411yescharacteristic
85C22×C4×C5C2×D411yescharacteristic
86C22×C4×C5C2×D411yescharacteristic
87C22×C4×C5C2×Q812yes—
88C22×C4×C5C2411yescharacteristic
89C22×Dic5—28no—
90C22×Dic5—24no—
91C22×Dic5—22no—
92C22×Dic5—22no—
93C22×Dic5—44no—
94C22×Dic5—44no—
95C2×C5×C8—24no—
96C2×C5×C8—24no—
97C2×C5×C8C22×C418yes—
98C42×C5C4⋊C414yes—
99C42×C5C4⋊C414yes—
100C42×C5C22×C412yes—
101C42×C5C22×C412yes—
102C42×C5C2×Q812yes—
103C42×C5C2×Q812yes—
104C22⋊Dic5—42no—
105C22⋊Dic5—42no—
106C4⋊Dic5—28no—
107C4⋊Dic5—24no—
108C4⋊Dic5—24no—
109C4⋊Dic5—24no—
110C4⋊Dic5#4118yes—
111C4×Dic5—44no—
112C4×Dic5—44no—
113C4×Dic5—44no—
114C22×C4⋊C4—52no—
115C22×C42—101no—
116C22×C42C2×D511yescharacteristic
117#432Dic518yescomplemented
118C2×#107—54no—
119C2×#106—54no—
120C2×C4⋊C8C2×D512yes—
121C2×C22⋊C8C2×D512yes—
122C2×#97C2×D512yes—
123#383—101no—
124#382—52no—
125#381—101no—
126#380—51no—
127#380—51no—
128C4×C4⋊C4—52no—
129C4×C4⋊C4—52no—
130C4×C4⋊C4—52no—
131C4×C4⋊C4—101no—
132C4×C4⋊C4—101no—
133C4×C22⋊C4—101no—
134#376—101no—
135C2×#95—101no—
136#340—54no—
137#340—54no—
138C23×C5—22no—
139C23×C5—22no—
140C23×C5C2×C4⋊C412yes—
141C23×C5AGL(1,ℤ/8)12yes—
142C23×C5#13712yes—
143C23×C5C22×D411yescharacteristic
144C2×C4×C5—22no—
145C2×C4×C5—21no—
146C2×C4×C5—21no—
147C2×C4×C5—21no—
148C2×C4×C5—21no—
149C2×C4×C5—22no—
150C2×C4×C5—22no—
151C2×C4×C5—28no—
152C2×C4×C5—24no—
153C2×C4×C5—24no—
154C2×C4×C5—44no—
155C2×C4×C5C2×C4⋊C412yes—
156C2×C4×C5C2×C4⋊C412yes—
157C2×C4×C5C2×C4⋊C412yes—
158C2×C4×C5C2×C4⋊C412yes—
159C2×C4×C5C2×C4⋊C412yes—
160C2×C4×C5C23×C411yescharacteristic
161C2×C4×C5C22×Q811yescharacteristic
162C2×C4×C5C2×#4114yes—
163C2×Dic5—28no—
164C2×Dic5—416no—
165C2×Dic5—44no—
166C2×Dic5—44no—
167C2×Dic5—42no—
168C2×Dic5—42no—
169C2×Dic5—42no—
170C2×Dic5—42no—
171C2×Dic5—42no—
172C2×Dic5—42no—
173C5×C8—28no—
174C23×C4—102no—
175C23×C4D2011yescharacteristic
176C23×C4D2011yescharacteristic
177C23×C4C22×D511yescharacteristic
178C22×C8C22×D514yes—
179C2×C4⋊C4—54no—
180C2×C4⋊C4—52no—
181C2×C4⋊C4—52no—
182C2×C4⋊C4—108no—
183C2×C4⋊C4—102no—
184C2×C22⋊C4—101no—
185C2×C22⋊C4—101no—
186C2×C42—102no—
187C2×C42—102no—
188C2×C42—102no—
189C2×C42—204no—
190C2×C42Dic1014yes—
191C2×C42D2011yescharacteristic
192C2×C42D2011yescharacteristic
193C2×C42C2×Dic514yes—
194C2×C42C22×D511yescharacteristic
195C2×C42C22×D511yescharacteristic
196#107—108no—
197#106—108no—
198C4⋊C8C2×Dic518yescomplemented
199C22⋊C8C2×Dic518yes—
200#97C2×Dic518yescomplemented
201#95—102no—
202#95—102no—
203#95—101no—
204#95—101no—
205#95—102no—
206#95—102no—
207#95—102no—
208C22×C5—24no—
209C22×C5—24no—
210C22×C5—22no—
211C22×C5—22no—
212C22×C5#42814yes—
213C22×C5C22×C4⋊C411yescharacteristic
214C22×C5#52512yes—
215C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
216C22×C5C2×#13711yescharacteristic
217C22×C5#57512yes—
218C4×C5—22no—
219C4×C5—22no—
220C4×C5—22no—
221C4×C5—22no—
222C4×C5—22no—
223C4×C5C4×#4112yes—
224C4×C5#53312yes—
225Dic5—48no—
226Dic5—82no—
227Dic5—82no—
228C24C2×D2011yescharacteristic
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—42no—
233C22×C4—108no—
234C22×C4—104no—
235C22×C4—102no—
236C22×C4—102no—
237C22×C4—204no—
238C22×C4—204no—
239C22×C4C4⋊Dic514yes—
240C22×C4C4⋊Dic514yes—
241C22×C4C2×Dic1012yes—
242C22×C4C2×D2011yescharacteristic
243C22×C4C2×D2011yescharacteristic
244C22×C4C2×D2011yescharacteristic
245C22×C4C2×D2011yescharacteristic
246C22×C4C2×D2011yescharacteristic
247C22×C4C22×Dic512yes—
248C22×C4C23×D511yescharacteristic
249C2×C8—24no—
250C2×C8—24no—
251C2×C8C22×Dic518yes—
252C4⋊C4—58no—
253C4⋊C4—108no—
254C4⋊C4—104no—
255C4⋊C4—104no—
256C4⋊C4—104no—
257C22⋊C4—202no—
258C22⋊C4—202no—
259C42—204no—
260C42—204no—
261C42—204no—
262C42C4⋊Dic514yes—
263C42C4⋊Dic514yes—
264C42C2×Dic1012yes—
265C42C2×Dic1012yes—
266C42C22×Dic512yes—
267C42C22×Dic512yes—
268C2×C5—24no—
269C2×C5#229611yescharacteristic
270C2×C5C2×#42811yescharacteristic
271C2×C5#291111yescharacteristic
272C2×C5#311412yes—
273C2×C5#311712yes—
274C23—22no—
275C23—22no—
276C23#405412yes—
277C23#405512yes—
278C23C2×C4⋊Dic512yes—
279C23C22×D2011yescharacteristic
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—28no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—44no—
291C2×C4—108no—
292C2×C4—2016no—
293C2×C4—204no—
294C2×C4—204no—
295C2×C4—202no—
296C2×C4—202no—
297C2×C4—202no—
298C2×C4—202no—
299C2×C4—202no—
300C2×C4—202no—
301C2×C4C2×C4⋊Dic512yes—
302C2×C4C2×C4⋊Dic512yes—
303C2×C4C2×C4⋊Dic512yes—
304C2×C4C2×C4⋊Dic512yes—
305C2×C4C2×C4⋊Dic512yes—
306C2×C4C22×Dic1011yescharacteristic
307C2×C4D5×#4114yes—
308C2×C4C23×Dic511yescharacteristic
309C8—28no—
310C5#2136311yescharacteristic, complemented, Sylow
311C22—24no—
312C22—24no—
313C22—22no—
314C22—22no—
315C22#6541014yes—
316C22#6585812yes—
317C22C2×#405411yescharacteristic
318C22C2×#405511yescharacteristic
319C22#6608212yes—
320C22C22×C4⋊Dic511yescharacteristic
321C4—22no—
322C4—22no—
323C4—22no—
324C4—22no—
325C4—22no—
326C4—208no—
327C4—402no—
328C4—402no—
329C4#6585112yes—
330C4#41×Dic512yes—
331C2—24no—
332C2#1061107712yes—
333C2#1061108812yes—
334C2#1061395811yescharacteristic
335C2C2×#6541011yescharacteristic
336C2#1061548311yescharacteristic
subgroupG/H
centerC23C22×D20
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC2×C5×#432C2
socleC23×C5C22×D4
prime radicalC24×C5C2×D4
cosocleC22×C4×C5C24
hypercenterC2×#432D5
hypocenterC5#21363
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#432

Complements

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

NH ≅ G/Ntype
4. #10601248C2semidirect
7. #10600607C2semidirect
11. C5×#432C4semidirect
55. C5×C4⋊C8C2×C4semidirect
57. C5×#97C2×C4semidirect
117. #432Dic5semidirect
198. C4⋊C8C2×Dic5semidirect
200. #97C2×Dic5semidirect
310. C5#21363semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#4322
#106137912
C2×C4×C4⋊Dic52
#106012482
#106007212
#106007202
#106006072
#213635