Finite Group Databasebeta

← #12097263

Subgroups — #12097263

Order 1280.

1110 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#424C211yesmaximal, characteristic
2C2×#65411C211yesmaximal, characteristic
3C22×#3947C211yesmaximal, characteristic
4#10613958C211yesmaximal, characteristic
5C2×#65286C211yesmaximal, characteristic
6#10602772C211yesmaximal, characteristic
7#10602758C211yesmaximal, characteristic
8#10601260C214yesmaximal, complemented
9#10601259C214yesmaximal, complemented
10C22×C5×M16C2211yescharacteristic
11C23×C5×C8C2211yescharacteristic
12C2×C5×#117C2211yescharacteristic
13C2×#4072C2211yescharacteristic
14C22×C4×Dic5C2211yescharacteristic
15C22×#681C2211yescharacteristic
16C23×#168C2211yescharacteristic
17C5×#424C418yescomplemented
18C2×C5×C4⋊C8C2214yes—
19#65411—24no—
20C2×#3947—24no—
21C2×#3947C2212yes—
22C2×#3947C2212yes—
23C2×#3947C2212yes—
24C2×C8×Dic5C2212yes—
25#65286C418yescomplemented
26#65274C414yes—
27#65239C414yes—
28C4×C4⋊Dic5C2214yes—
29C2×#3935C2212yes—
30C2×C4×#168C2212yes—
31#64770C2214yes—
32#64770C2214yes—
33#20436—51nomaximal, Sylow
34C23×C4×C5C2311yescharacteristic
35C23×Dic5C2×C412yes—
36C2×C5×M16—22no—
37C2×C5×M16—22no—
38C2×C5×M16C2×C414yes—
39C22×C5×C8—22no—
40C22×C5×C8—22no—
41C22×C5×C8C2×C414yes—
42C22×C5×C8C2312yes—
43C22×C5×C8C2312yes—
44C5×#117C2×C414yes—
45C5×#117C2×C414yes—
46C2×C5×C4⋊C4C2×C412yes—
47C2×C5×C22⋊C4C2×C412yes—
48C2×C42×C5C2312yes—
49C2×C22⋊Dic5C2×C412yes—
50#4072—22no—
51#4072—22no—
52C2×C4⋊Dic5C2×C412yes—
53C2×C4⋊Dic5C2×C418yes—
54C2×C4×Dic5—22no—
55C2×C4×Dic5—22no—
56C2×C4×Dic5C2×C412yes—
57C2×C4×Dic5C2312yes—
58C2×C4×Dic5C2312yes—
59C2×#681—22no—
60C2×#681—22no—
61C2×#681C2×C414yes—
62C22×#168C2×C414yes—
63C22×#168D414yes—
64C22×#168Q814yes—
65C22×#168C2312yes—
66C22×#168C2311yescharacteristic
67C22×#168C2311yescharacteristic
68C5×C4⋊C8C2×C4116yescomplemented
69#3963C2×C418yes—
70#3947—24no—
71#3947—24no—
72#3947—24no—
73C8×Dic5—24no—
74#3935C2×C418yescomplemented
75C4×#168C2×C418yescomplemented
76C2×#424D511yescharacteristic
77C2×#405—51no—
78C2×#405—51no—
79C22×#97—51no—
80#2296—51no—
81#1771—54no—
82#1770—54no—
83#1729—51no—
84#1728—51no—
85C24×C5C22×C411yescharacteristic
86C22×C4×C5C22×C414yes—
87C22×C4×C5C22×C411yescharacteristic
88C22×C4×C5C22×C412yes—
89C22×C4×C5C22×C412yes—
90C22×C4×C5C2×D412yes—
91C22×C4×C5C2×Q812yes—
92C22×C4×C5#4112yes—
93C22×C4×C5#4112yes—
94C22×C4×C5C2411yescharacteristic
95C22×Dic5—24no—
96C22×Dic5—44no—
97C22×Dic5C22×C414yes—
98C22×Dic5C22×C414yes—
99C5×M16—28no—
100C2×C5×C8—28no—
101C2×C5×C8—22no—
102C2×C5×C8—22no—
103C2×C5×C8—22no—
104C2×C5×C8—22no—
105C2×C5×C8C22×C414yes—
106C2×C5×C8C22×C414yes—
107C5×C4⋊C4C4218yes—
108C5×C22⋊C4C4218yes—
109C42×C5C22×C414yes—
110C42×C5C22×C414yes—
111C22⋊Dic5—24no—
112C4⋊Dic5—24no—
113C4⋊Dic5—48no—
114C4×Dic5—24no—
115C4×Dic5—22no—
116C4×Dic5—22no—
117C4×Dic5—22no—
118C4×Dic5—22no—
119#681—28no—
120C2×#168—22no—
121C2×#168—22no—
122C2×#168C4⋊C4116yescomplemented
123C2×#168C22×C414yes—
124C2×#168C22×C412yes—
125C2×#168C22×C412yes—
126C2×#168C2×D412yes—
127C2×#168C2×D412yes—
128C2×#168C2×Q812yes—
129C2×#168C2×Q812yes—
130C22×M16—51no—
131C22×M16C2×D511yescharacteristic
132C23×C8—51no—
133C23×C8C2×D511yescharacteristic
134C2×#117—51no—
135C2×#117C2×D511yescharacteristic
136C22×C42—51no—
137#424Dic518yescomplemented
138C2×C4⋊C8C2×D514yes—
139#405—58no—
140#405—104no—
141C2×#97—52no—
142C2×#97—52no—
143C2×#97—52no—
144C2×#97—52no—
145C2×#97—104no—
146C2×C4×C8—52no—
147C2×C4×C8—52no—
148#389—54no—
149#385—54no—
150C4×C4⋊C4—54no—
151#336—54no—
152#336—54no—
153C23×C5—24no—
154C23×C5C2×C4212yes—
155C23×C5C4×D412yes—
156C23×C5C4×Q812yes—
157C23×C5C23×C411yescharacteristic
158C2×C4×C5—28no—
159C2×C4×C5C2×C4218yes—
160C2×C4×C5C2×C4212yes—
161C2×C4×C5C2×C4⋊C414yes—
162C2×C4×C5#11714yes—
163C2×C4×C5C4×D412yes—
164C2×C4×C5C4×D414yes—
165C2×C4×C5C4×Q812yes—
166C2×C4×C5C4×Q814yes—
167C2×C4×C5C23×C411yescharacteristic
168C2×C4×C5C23×C411yescharacteristic
169C2×C4×C5C22×D411yescharacteristic
170C2×C4×C5C22×Q811yescharacteristic
171C2×C4×C5C2×#4111yescharacteristic
172C2×C4×C5C2×#4111yescharacteristic
173C2×Dic5—24no—
174C2×Dic5—24no—
175C2×Dic5—24no—
176C2×Dic5—28no—
177C2×Dic5—24no—
178C2×Dic5—48no—
179C5×C8—24no—
180C5×C8—24no—
181#168—24no—
182#168C2×C4⋊C414yes—
183#168C2×C4⋊C414yes—
184C23×C4—52no—
185C23×C4C22×D511yescharacteristic
186C2×M16—22no—
187C2×M16—22no—
188C2×M16—54no—
189C2×M16—102no—
190C2×M16—102no—
191C2×M16C2×Dic514yes—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—54no—
195C22×C8—54no—
196C22×C8—54no—
197C22×C8—51no—
198C22×C8—51no—
199C22×C8—52no—
200C22×C8C2×Dic514yes—
201C22×C8C22×D512yes—
202C22×C8C22×D512yes—
203#117—102no—
204#117—102no—
205#117C4×D514yes—
206#117C2×Dic514yes—
207C2×C4⋊C4—52no—
208C2×C4⋊C4—58no—
209C2×C4⋊C4C4×D512yes—
210C2×C22⋊C4—52no—
211C2×C22⋊C4C4×D512yes—
212C2×C42—52no—
213C2×C42—52no—
214C2×C42—52no—
215C2×C42—102no—
216C2×C42—102no—
217C2×C42C22×D512yes—
218C4⋊C8C2×Dic5116yescomplemented
219#97—58no—
220#97—104no—
221#97—104no—
222#97—104no—
223C4×C8—58no—
224C4×C8—104no—
225#95—58no—
226C22×C5—24no—
227C22×C5—24no—
228C22×C5—24no—
229C22×C5C4×C4⋊C414yes—
230C22×C5C22×C4211yescharacteristic
231C22×C5C2×C4×D411yescharacteristic
232C22×C5C2×C4×Q811yescharacteristic
233C22×C5#56814yes—
234C4×C5—28no—
235C4×C5C4×C4⋊C414yes—
236C4×C5C22×C4⋊C411yescharacteristic
237C4×C5C2×#11711yescharacteristic
238C4×C5C2×C4×D411yescharacteristic
239C4×C5C2×C4×Q811yescharacteristic
240Dic5—44no—
241Dic5—44no—
242C24C2×C4×D511yescharacteristic
243C22×C4—54no—
244C22×C4—54no—
245C22×C4—104no—
246C22×C4—204no—
247C22×C4C2×C4×D514yes—
248C22×C4C2×C4×D512yes—
249C22×C4C2×C4×D511yescharacteristic
250C22×C4D4×D512yes—
251C22×C4#71112yes—
252C22×C4Q8×D512yes—
253C22×C4#71312yes—
254C22×C4C22×Dic512yes—
255C22×C4C23×D511yescharacteristic
256M16—28no—
257M16—108no—
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—28no—
263C2×C8—52no—
264C2×C8—52no—
265C2×C8—52no—
266C2×C8—52no—
267C2×C8—52no—
268C2×C8—52no—
269C2×C8—516no—
270C2×C8—54no—
271C2×C8—102no—
272C2×C8—102no—
273C2×C8C22×Dic514yes—
274C2×C8C22×Dic514yes—
275C4⋊C4—104no—
276C4⋊C4—208no—
277C4⋊C4C4×Dic518yes—
278C22⋊C4—104no—
279C22⋊C4C4×Dic518yes—
280C42—102no—
281C42—102no—
282C42—102no—
283C42—102no—
284C42—104no—
285C42C2×C4×D514yes—
286C42C22×Dic514yes—
287C2×C5—24no—
288C2×C5C2×C4×C4⋊C411yescharacteristic
289C2×C5#287512yes—
290C2×C5#293012yes—
291C2×C5#296112yes—
292C23—24no—
293C23#403312yes—
294C23#403912yes—
295C23C2×C4×Dic512yes—
296C23C22×C4×D511yescharacteristic
297C2×C4—28no—
298C2×C4—104no—
299C2×C4—108no—
300C2×C4—104no—
301C2×C4—104no—
302C2×C4—104no—
303C2×C4—208no—
304C2×C4#403312yes—
305C2×C4D5×C4⋊C414yes—
306C2×C4#403814yes—
307C2×C4#403912yes—
308C2×C4C2×C4×Dic518yes—
309C2×C4C2×C4×Dic512yes—
310C2×C4D4×Dic514yes—
311C2×C4Q8×Dic514yes—
312C2×C4C22×C4×D511yescharacteristic
313C2×C4C2×D4×D511yescharacteristic
314C2×C4C2×#71111yescharacteristic
315C2×C4C2×Q8×D511yescharacteristic
316C2×C4C2×#71311yescharacteristic
317C2×C4C23×Dic511yescharacteristic
318C8—24no—
319C8—24no—
320C8—54no—
321C8—54no—
322C8—104no—
323C5#2043611yescharacteristic, complemented, Sylow
324C22—24no—
325C22—24no—
326C22—24no—
327C22C4⋊C4×Dic514yes—
328C22C2×#403311yescharacteristic
329C22C2×#403911yescharacteristic
330C22#6608414yes—
331C22C22×C4×Dic511yescharacteristic
332C4—28no—
333C4—204no—
334C4—204no—
335C4C4⋊C4×Dic514yes—
336C4C2×D5×C4⋊C411yescharacteristic
337C4C2×#403811yescharacteristic
338C4C2×D4×Dic511yescharacteristic
339C4C2×Q8×Dic511yescharacteristic
340C2—24no—
341C2#1061080512yes—
342C2#1061080812yes—
343C2C2×C4⋊C4×Dic511yescharacteristic
344C2#1061551312yes—
subgroupG/H
center, FrattiniC22×C4C23×D5
commutatorC22×C5C22×C42
FittingC2×C5×#424C2
socleC23×C5C23×C4
prime radicalC24×C5C22×C4
cosocleC22×C4×C5C24
hypercenterC2×#424D5
hypocenterC5#20436
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C4 ◁ C2×#424

Complements

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

NH ≅ G/Ntype
8. #10601260C2semidirect
9. #10601259C2semidirect
17. C5×#424C4semidirect
25. #65286C4semidirect
68. C5×C4⋊C8C2×C4semidirect
74. #3935C2×C4semidirect
75. C4×#168C2×C4semidirect
122. C2×#168C4⋊C4semidirect
137. #424Dic5semidirect
218. C4⋊C8C2×Dic5semidirect
323. C5#20436semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#4242
C2×#654112
C22×#39472
#106139582
C2×#652862
#106027722
#106027582
#106012602
#106012592
#204365