Finite Group Databasebeta

← #12096788

Subgroups — #12096788

Order 1280.

1022 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×M16×Dic5C211yesmaximal, characteristic
2C22×#3947C211yesmaximal, characteristic
3Dic5×#117C211yesmaximal, characteristic
4C5×#1729C211yesmaximal, characteristic
5#10602773C211yesmaximal, characteristic
6#10602758C211yesmaximal, characteristic
7#10602372C211yesmaximal, characteristic
8#10599391C214yesmaximal, complemented
9#10599390C214yesmaximal, complemented
10C22×C5×M16C2211yescharacteristic
11C23×C5×C8C2211yescharacteristic
12C2×C5×#117C2211yescharacteristic
13C2×#4072C2211yescharacteristic
14C22×C4×Dic5C2211yescharacteristic
15C22×#681C2211yescharacteristic
16C23×#168C2211yescharacteristic
17M16×Dic5C418yescomplemented
18C2×#3947C418yescomplemented
19C2×#3947C2212yes—
20C2×#3947C2212yes—
21C2×#3947C2212yes—
22C2×C8×Dic5C2212yes—
23C4⋊C4×Dic5C414yes—
24C22⋊C4×Dic5C414yes—
25#65221C2212yes—
26C42×Dic5C2212yes—
27C5×#336C2214yes—
28#64770C2214yes—
29#64770C2214yes—
30#64744C2214yes—
31#20367—51nomaximal, Sylow
32C23×C4×C5C2311yescharacteristic
33C23×Dic5C2×C412yes—
34C2×C5×M16—22no—
35C2×C5×M16—22no—
36C2×C5×M16C2×C414yes—
37C22×C5×C8—22no—
38C22×C5×C8—22no—
39C22×C5×C8C2×C414yes—
40C22×C5×C8C2312yes—
41C22×C5×C8C2312yes—
42C5×#117—22no—
43C5×#117—22no—
44C2×C5×C4⋊C4C2×C412yes—
45C2×C5×C22⋊C4C2×C412yes—
46C2×C42×C5C2312yes—
47C2×C22⋊Dic5C2×C412yes—
48#4072—22no—
49#4072—22no—
50C2×C4⋊Dic5C2×C412yes—
51C2×C4×Dic5C2×C418yes—
52C2×C4×Dic5C2×C412yes—
53C2×C4×Dic5C2×C414yes—
54C2×C4×Dic5C2×C414yes—
55C2×C4×Dic5C2312yes—
56C2×C4×Dic5C2311yescharacteristic
57C2×C4×Dic5C2311yescharacteristic
58C2×#681—22no—
59C2×#681—22no—
60C2×#681C2×C414yes—
61C22×#168—22no—
62C22×#168—22no—
63C22×#168C2×C414yes—
64C22×#168C2312yes—
65C22×#168C2312yes—
66#3963C2×C418yes—
67#3947C2×C418yes—
68#3947C2×C418yes—
69#3947C2×C418yes—
70C8×Dic5C2×C418yes—
71C2×C4×M16—51no—
72C22×#97—51no—
73C4×#117—51no—
74#1746—54no—
75#1745—54no—
76#1729—51no—
77#1729—51no—
78#1729—51no—
79#1729D511yescharacteristic
80C24×C5C22×C411yescharacteristic
81C22×C4×C5C22×C414yes—
82C22×C4×C5C22×C411yescharacteristic
83C22×C4×C5C22×C412yes—
84C22×C4×C5C22×C412yes—
85C22×C4×C5#4112yes—
86C22×C4×C5#4112yes—
87C22×C4×C5#4112yes—
88C22×C4×C5#4112yes—
89C22×C4×C5C2411yescharacteristic
90C22×Dic5—44no—
91C22×Dic5C4218yes—
92C22×Dic5C22×C414yes—
93C22×Dic5C22×C412yes—
94C22×Dic5C22×C412yes—
95C5×M16—28no—
96C2×C5×C8—28no—
97C2×C5×C8—22no—
98C2×C5×C8—22no—
99C2×C5×C8—22no—
100C2×C5×C8—22no—
101C2×C5×C8C22×C414yes—
102C2×C5×C8C22×C414yes—
103C5×C4⋊C4—24no—
104C5×C22⋊C4—24no—
105C42×C5—22no—
106C42×C5—22no—
107C22⋊Dic5—24no—
108C4⋊Dic5—24no—
109C4×Dic5—22no—
110C4×Dic5—22no—
111C4×Dic5—48no—
112C4×Dic5C4218yes—
113C4×Dic5C22×C412yes—
114C4×Dic5C22×C412yes—
115C4×Dic5C22×C412yes—
116C4×Dic5C22×C412yes—
117#681—28no—
118C2×#168—28no—
119C2×#168—22no—
120C2×#168—22no—
121C2×#168—22no—
122C2×#168—22no—
123C2×#168C22×C414yes—
124C2×#168C22×C414yes—
125C22×M16—51no—
126C22×M16C2×D511yescharacteristic
127C23×C8—51no—
128C23×C8C2×D511yescharacteristic
129C2×#117—51no—
130C2×#117C2×D511yescharacteristic
131C22×C42—51no—
132C4×M16—58no—
133C2×#97—58no—
134C2×#97—52no—
135C2×#97—52no—
136C2×#97—52no—
137C2×C4×C8—52no—
138C4×C4⋊C4—54no—
139C4×C22⋊C4—54no—
140#376—52no—
141C43—52no—
142#336—54no—
143#336—54no—
144#336—54no—
145#336C2×D514yes—
146C23×C5—24no—
147C23×C5C2×C4212yes—
148C23×C5#11712yes—
149C23×C5#11712yes—
150C23×C5C23×C411yescharacteristic
151C2×C4×C5—24no—
152C2×C4×C5—28no—
153C2×C4×C5C2×C4212yes—
154C2×C4×C5#11714yes—
155C2×C4×C5#11714yes—
156C2×C4×C5#11712yes—
157C2×C4×C5#11714yes—
158C2×C4×C5#11714yes—
159C2×C4×C5#11712yes—
160C2×C4×C5C23×C411yescharacteristic
161C2×C4×C5C23×C411yescharacteristic
162C2×C4×C5C2×#4111yescharacteristic
163C2×C4×C5C2×#4111yescharacteristic
164C2×C4×C5C2×#4111yescharacteristic
165C2×C4×C5C2×#4111yescharacteristic
166C2×Dic5—24no—
167C2×Dic5—24no—
168C2×Dic5—24no—
169C2×Dic5—28no—
170C2×Dic5—48no—
171C2×Dic5C2×C4214yes—
172C2×Dic5C2×C4214yes—
173C5×C8—24no—
174C5×C8—24no—
175#168—24no—
176#168—24no—
177C23×C4—52no—
178C23×C4C22×D511yescharacteristic
179C2×M16—22no—
180C2×M16—22no—
181C2×M16—54no—
182C2×M16—102no—
183C2×M16—102no—
184C2×M16C4×D514yes—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—54no—
188C22×C8—52no—
189C22×C8—52no—
190C22×C8—102no—
191C22×C8—102no—
192C22×C8C4×D514yes—
193C22×C8C22×D512yes—
194C22×C8C22×D512yes—
195#117—22no—
196#117—22no—
197#117—102no—
198#117—102no—
199C2×C4⋊C4—52no—
200C2×C4⋊C4C4×D512yes—
201C2×C22⋊C4—52no—
202C2×C22⋊C4C4×D512yes—
203C2×C42—52no—
204C2×C42—54no—
205C2×C42—54no—
206C2×C42—51no—
207C2×C42—51no—
208C2×C42—52no—
209C2×C42—58no—
210C2×C42C22×D512yes—
211#97—58no—
212#97—58no—
213#97—58no—
214C4×C8—58no—
215#95—58no—
216C22×C5—24no—
217C22×C5—24no—
218C22×C5—24no—
219C22×C5#37614yes—
220C22×C5C22×C4211yescharacteristic
221C22×C5C2×#11711yescharacteristic
222C22×C5C2×#11711yescharacteristic
223C22×C5#56814yes—
224C4×C5—44no—
225C4×C5#37614yes—
226C4×C5C2×#11711yescharacteristic
227C4×C5C2×#11711yescharacteristic
228C4×C5C2×#11711yescharacteristic
229C4×C5C2×#11711yescharacteristic
230Dic5—24no—
231Dic5—24no—
232Dic5—44no—
233C24C2×C4×D511yescharacteristic
234C22×C4—52no—
235C22×C4—52no—
236C22×C4—58no—
237C22×C4—54no—
238C22×C4—204no—
239C22×C4C2×C4×D514yes—
240C22×C4C2×C4×D512yes—
241C22×C4C2×C4×D512yes—
242C22×C4C2×C4×D511yescharacteristic
243C22×C4#71112yes—
244C22×C4#71112yes—
245C22×C4#71112yes—
246C22×C4#71312yes—
247C22×C4C23×D511yescharacteristic
248M16—28no—
249M16—108no—
250C2×C8—22no—
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—28no—
255C2×C8—54no—
256C2×C8—54no—
257C2×C8—102no—
258C2×C8—102no—
259C2×C8—102no—
260C2×C8—102no—
261C2×C8—108no—
262C2×C8C2×C4×D514yes—
263C2×C8C2×C4×D514yes—
264C4⋊C4—24no—
265C4⋊C4—104no—
266C22⋊C4—24no—
267C22⋊C4—104no—
268C42—22no—
269C42—22no—
270C42—52no—
271C42—52no—
272C42—52no—
273C42—52no—
274C42—58no—
275C42—102no—
276C42—102no—
277C42—208no—
278C2×C5—24no—
279C2×C5C2×#37611yescharacteristic
280C2×C5#287312yes—
281C2×C5#292312yes—
282C2×C5#292312yes—
283C23—24no—
284C23C42×D512yes—
285C23#402312yes—
286C23#403812yes—
287C23C22×C4×D511yescharacteristic
288C2×C4—24no—
289C2×C4—28no—
290C2×C4—54no—
291C2×C4—54no—
292C2×C4—104no—
293C2×C4—104no—
294C2×C4—108no—
295C2×C4—104no—
296C2×C4—208no—
297C2×C4C42×D512yes—
298C2×C4#402314yes—
299C2×C4#402314yes—
300C2×C4#402312yes—
301C2×C4#403814yes—
302C2×C4#403814yes—
303C2×C4#403812yes—
304C2×C4C22×C4×D511yescharacteristic
305C2×C4C22×C4×D511yescharacteristic
306C2×C4C2×#71111yescharacteristic
307C2×C4C2×#71111yescharacteristic
308C2×C4C2×#71111yescharacteristic
309C2×C4C2×#71311yescharacteristic
310C8—24no—
311C8—24no—
312C8—104no—
313C8—104no—
314C5#2036711yescharacteristic, complemented, Sylow
315C22—24no—
316C22—24no—
317C22—24no—
318C22#6493714yes—
319C22C2×C42×D511yescharacteristic
320C22C2×#402311yescharacteristic
321C22C2×#403811yescharacteristic
322C22#6608414yes—
323C4—44no—
324C4—104no—
325C4—104no—
326C4—204no—
327C4#6493714yes—
328C4C2×#402311yescharacteristic
329C4C2×#402311yescharacteristic
330C4C2×#403811yescharacteristic
331C4C2×#403811yescharacteristic
332C2—24no—
333C2C2×#6493711yescharacteristic
334C2#1061027912yes—
335C2#1061038012yes—
336C2#1061080212yes—
subgroupG/H
center, FrattiniC22×C4C23×D5
commutatorC22×C5C22×C42
FittingC5×#1729C2
socleC23×C5C23×C4
prime radicalC24×C5C22×C4
cosocleC22×C4×C5C24
hypercenter#1729D5
hypocenterC5#20367
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C4 ◁ #1729

Complements

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

NH ≅ G/Ntype
8. #10599391C2semidirect
9. #10599390C2semidirect
17. M16×Dic5C4semidirect
18. C2×#3947C4semidirect
314. C5#20367semidirect

Maximal subgroups

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

subgroupindex
C2×M16×Dic52
C22×#39472
Dic5×#1172
C5×#17292
#106027732
#106027582
#106023722
#105993912
#105993902
#203675