Finite Group Databasebeta

← #12097510

Subgroups — #12097510

Order 1280.

1050 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C5×M16C211yesmaximal, characteristic
2C4×#4072C211yesmaximal, characteristic
3C2×#65214C211yesmaximal, characteristic
4#10602776C214yesmaximal
5#10600603C214yesmaximal, complemented
6#10600543C211yesmaximal, characteristic, complemented
7#10600543C211yesmaximal, characteristic, complemented
8C4×#3947C211yesmaximal, characteristic, complemented
9C4×C8×Dic5C211yesmaximal, characteristic, complemented
10C22×C5×M16C2212yes—
11C22×C42×C5C2211yescharacteristic
12C2×#4072C412yes—
13C4×C5×M16C418yescomplemented
14C2×C5×#97C2212yes—
15C2×C4×C5×C8C2211yescharacteristic
16C2×C4×C5×C8C2211yescharacteristic
17C4×C22⋊Dic5C412yes—
18C4×C22⋊Dic5C2212yes—
19C2×C22⋊#168C2212yes—
20C2×#3947C2214yes—
21C2×C8×Dic5C2214yes—
22C4×C4⋊Dic5C412yes—
23C4×C4⋊Dic5C2212yes—
24#65221C412yes—
25C42×Dic5C2211yescharacteristic
26C42×Dic5C2211yescharacteristic
27#65214C418yescomplemented
28C2×C4⋊#168C2212yes—
29C2×C4×#168C2211yescharacteristic
30C2×C4×#168C2211yescharacteristic
31#64770C2214yes—
32#64770C2214yes—
33#20472—51nomaximal, Sylow
34C23×C4×C5C2×C411yescharacteristic
35C23×C4×C5C2×C411yescharacteristic
36C23×C4×C5C2311yescharacteristic
37C2×C5×M16—28no—
38C2×C5×M16C2×C418yes—
39C22×C5×C8C2312yes—
40C22×C5×C8C2312yes—
41C2×C42×C5C2×C411yescharacteristic
42C2×C42×C5C2×C411yescharacteristic
43C2×C42×C5C2×C414yes—
44C2×C42×C5C2×C414yes—
45C2×C42×C5C2311yescharacteristic
46C2×C42×C5C2311yescharacteristic
47C2×C22⋊Dic5C2×C414yes—
48#4072—28no—
49C2×C4⋊Dic5C2×C414yes—
50C2×C4×Dic5C2×C412yes—
51C2×C4×Dic5C2×C412yes—
52C2×C4×Dic5C2×C412yes—
53C2×C4×Dic5C2×C412yes—
54C2×C4×Dic5C2312yes—
55C2×C4×Dic5C2312yes—
56C22×#168C2312yes—
57C22×#168C2312yes—
58C5×#97C2×C418yes—
59C4×C5×C8C2×C414yes—
60C4×C5×C8C2×C414yes—
61#3963C2×C414yes—
62#3963C2×C414yes—
63C22⋊#168C2×C418yes—
64#3947—28no—
65C8×Dic5—28no—
66C4⋊#168C2×C418yes—
67C4×#168C2×C414yes—
68C4×#168C2×C414yes—
69C2×#431—51no—
70C2×C4×M16D511yescharacteristic
71C4×#117—51no—
72#1771—54no—
73#1753—54no—
74#1744—51no—
75#1744—51no—
76C4×#97—51no—
77C42×C8—51no—
78C24×C5C22×C411yescharacteristic
79C22×C4×C5—24no—
80C22×C4×C5—24no—
81C22×C4×C5—24no—
82C22×C4×C5C4214yes—
83C22×C4×C5C4214yes—
84C22×C4×C5C22×C411yescharacteristic
85C22×C4×C5C22×C411yescharacteristic
86C22×C4×C5C22×C411yescharacteristic
87C22×C4×C5C22×C412yes—
88C22×C4×C5C22×C411yescharacteristic
89C22×C4×C5C22×C411yescharacteristic
90C22×C4×C5C22×C412yes—
91C22×C4×C5C2411yescharacteristic
92C22×Dic5C22×C414yes—
93C22×Dic5C22×C414yes—
94C5×M16—216no—
95C2×C5×C8—24no—
96C2×C5×C8C22×C414yes—
97C2×C5×C8C22×C414yes—
98C2×C5×C8#4118yes—
99C42×C5C4214yes—
100C42×C5C4214yes—
101C42×C5C22×C412yes—
102C42×C5C22×C412yes—
103C42×C5C22×C412yes—
104C42×C5C22×C412yes—
105C22⋊Dic5—28no—
106C4⋊Dic5—28no—
107C4×Dic5—24no—
108C4×Dic5—24no—
109C4×Dic5—24no—
110C4×Dic5—24no—
111C4×Dic5—24no—
112C4×Dic5M1618yescomplemented
113C2×#168—24no—
114C2×#168—24no—
115C2×#168C22×C414yes—
116C2×#168C22×C414yes—
117C22×M16C2×D512yes—
118C2×#117—52no—
119C22×C42C2×D511yescharacteristic
120#431—58no—
121C2×C4⋊C8—52no—
122C2×C22⋊C8—52no—
123C4×M16Dic518yescomplemented
124C2×#97—54no—
125C2×#97C2×D512yes—
126C2×C4×C8—51no—
127C2×C4×C8—51no—
128C2×C4×C8—54no—
129C2×C4×C8C2×D511yescharacteristic
130C2×C4×C8C2×D511yescharacteristic
131C4×C4⋊C4—52no—
132C4×C4⋊C4—52no—
133C4×C22⋊C4—52no—
134C4×C22⋊C4—52no—
135#376—52no—
136C43—51no—
137C43—51no—
138#336—54no—
139#336—54no—
140C23×C5—22no—
141C23×C5—22no—
142C23×C5—22no—
143C23×C5C2×C4212yes—
144C23×C5C23×C411yescharacteristic
145C2×C4×C5—22no—
146C2×C4×C5—22no—
147C2×C4×C5—22no—
148C2×C4×C5—24no—
149C2×C4×C5—24no—
150C2×C4×C5—28no—
151C2×C4×C5—28no—
152C2×C4×C5C2×C4212yes—
153C2×C4×C5C2×C4212yes—
154C2×C4×C5C2×C4212yes—
155C2×C4×C5C2×C4212yes—
156C2×C4×C5C2×C4212yes—
157C2×C4×C5C2×M1614yes—
158C2×C4×C5#13114yes—
159C2×C4×C5C23×C411yescharacteristic
160C2×C4×C5C23×C411yescharacteristic
161C2×C4×C5C2×#4114yes—
162C2×Dic5—24no—
163C2×Dic5—24no—
164C2×Dic5—24no—
165C2×Dic5—24no—
166C2×Dic5—24no—
167C2×Dic5C2×M1618yes—
168C5×C8—24no—
169C5×C8#11718yescomplemented
170#168—24no—
171#168—24no—
172C23×C4C4×D511yescharacteristic
173C23×C4C4×D511yescharacteristic
174C23×C4C22×D511yescharacteristic
175C2×M16—28no—
176C2×M16C2×Dic518yes—
177C22×C8—52no—
178C22×C8—52no—
179C22×C8C22×D512yes—
180C22×C8C22×D512yes—
181#117—108no—
182C2×C4⋊C4—54no—
183C2×C22⋊C4—54no—
184C2×C42—52no—
185C2×C42—52no—
186C2×C42—52no—
187C2×C42—52no—
188C2×C42—52no—
189C2×C42—52no—
190C2×C42C4×D511yescharacteristic
191C2×C42C4×D511yescharacteristic
192C2×C42C4×D514yes—
193C2×C42C2×Dic514yes—
194C2×C42C22×D511yescharacteristic
195C2×C42C22×D511yescharacteristic
196C4⋊C8—58no—
197C22⋊C8—58no—
198#97—108no—
199#97C2×Dic518yes—
200C4×C8—54no—
201C4×C8—54no—
202C4×C8—108no—
203C4×C8C2×Dic514yes—
204C4×C8C2×Dic514yes—
205#95—54no—
206#95—54no—
207C22×C5—22no—
208C22×C5—22no—
209C22×C5—22no—
210C22×C5—44no—
211C22×C5C22×C4211yescharacteristic
212C22×C5C4×#4112yes—
213C22×C5C22×M1612yes—
214C22×C5C2×#13112yes—
215C4×C5—22no—
216C4×C5—22no—
217C4×C5—22no—
218C4×C5C4×M1614yes—
219C4×C5#40514yes—
220C4×C5C2×#11712yes—
221C4×C5C4×#4112yes—
222Dic5—28no—
223Dic5—44no—
224C24C2×C4×D511yescharacteristic
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—54no—
229C22×C4—54no—
230C22×C4C4×Dic514yes—
231C22×C4C4×Dic514yes—
232C22×C4C2×C4×D512yes—
233C22×C4C2×C4×D511yescharacteristic
234C22×C4C2×C4×D511yescharacteristic
235C22×C4C2×C4×D511yescharacteristic
236C22×C4C2×C4×D511yescharacteristic
237C22×C4C2×C4×D511yescharacteristic
238C22×C4C22×Dic512yes—
239C22×C4C23×D511yescharacteristic
240M16—216no—
241C2×C8—24no—
242C2×C8—54no—
243C2×C8—54no—
244C2×C8—104no—
245C2×C8—104no—
246C2×C8#70918yes—
247C2×C8C22×Dic514yes—
248C2×C8C22×Dic514yes—
249C4⋊C4—108no—
250C22⋊C4—108no—
251C42—58no—
252C42—104no—
253C42—104no—
254C42—104no—
255C42—104no—
256C42—104no—
257C42C4×Dic514yes—
258C42C4×Dic514yes—
259C42C2×C4×D512yes—
260C42C2×C4×D512yes—
261C42C22×Dic512yes—
262C42C22×Dic512yes—
263C2×C5—42no—
264C2×C5C4×#11711yescharacteristic
265C2×C5C2×C4×M1611yescharacteristic
266C2×C5C2×#40511yescharacteristic
267C2×C5#296514yes—
268C23—22no—
269C23—22no—
270C23—22no—
271C23C2×C4×Dic512yes—
272C23C22×C4×D511yescharacteristic
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—28no—
279C2×C4—28no—
280C2×C4—58no—
281C2×C4—104no—
282C2×C4—104no—
283C2×C4—104no—
284C2×C4—104no—
285C2×C4—104no—
286C2×C4D5×M1614yes—
287C2×C4#405314yes—
288C2×C4C2×C4×Dic512yes—
289C2×C4C2×C4×Dic512yes—
290C2×C4C2×C4×Dic512yes—
291C2×C4C2×C4×Dic512yes—
292C2×C4C2×C4×Dic512yes—
293C2×C4C22×C4×D511yescharacteristic
294C2×C4C2×#70914yes—
295C2×C4C23×Dic511yescharacteristic
296C8—24no—
297C8—104no—
298C8—104no—
299C8#407218yescomplemented
300C5#2047211yescharacteristic, complemented, Sylow
301C22—22no—
302C22—22no—
303C22—22no—
304C22—44no—
305C22C4×#70912yes—
306C22C2×D5×M1612yes—
307C22C2×#405312yes—
308C22C22×C4×Dic511yescharacteristic
309C4—22no—
310C4—22no—
311C4—22no—
312C4—108no—
313C4—204no—
314C4M16×Dic514yes—
315C4#6541114yes—
316C4C4×#70912yes—
317C4C2×#407212yes—
318C2—42no—
319C2#1061025414yes—
320C2C4×#407211yescharacteristic
321C2C2×M16×Dic511yescharacteristic
322C2C2×#6541111yescharacteristic
subgroupG/H
centerC2×C42C22×D5
commutatorC22×C5C22×C42
FrattiniC22×C4C23×D5
FittingC2×C4×C5×M16C2
socleC23×C5C23×C4
prime radicalC24×C5C22×C4
cosocleC22×C4×C5C24
hypercenterC2×C4×M16D5
hypocenterC5#20472
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C2×C42 ◁ C2×C4×M16

Complements

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

NH ≅ G/Ntype
5. #10600603C2semidirect
6. #10600543C2semidirect
7. #10600543C2semidirect
8. C4×#3947C2semidirect
9. C4×C8×Dic5C2semidirect
13. C4×C5×M16C4semidirect
27. #65214C4semidirect
112. C4×Dic5M16semidirect
123. C4×M16Dic5semidirect
169. C5×C8#117semidirect
299. C8#4072semidirect
300. C5#20472semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C5×M162
C4×#40722
C2×#652142
#106027762
#106006032
#106005432
C4×#39472
C4×C8×Dic52
#204725