Finite Group Databasebeta

← #12099739

Subgroups — #12099739

Order 1280.

1070 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65170C212yesmaximal
2C2×#65149C211yesmaximal, characteristic
3C2×D5×#106C211yesmaximal, characteristic
4C2×#64978C211yesmaximal, characteristic
5#10605956C212yesmaximal
6C5×#1846C211yesmaximal, characteristic, complemented
7#10601401C212yesmaximal, complemented
8#10601348C211yesmaximal, characteristic, complemented
9#10601347C211yesmaximal, characteristic, complemented
10#10600557C211yesmaximal, characteristic, complemented
11#10600343C212yesmaximal, complemented
12C22×#676C2211yescharacteristic
13C22×C8×D5C2211yescharacteristic
14C2×#4038C2212yes—
15C2×D5×C4⋊C4C2212yes—
16C2×#4018C2211yescharacteristic
17C2×C5×#107C2212yes—
18C2×C5×#106C2211yescharacteristic
19C2×C5×#106C2211yescharacteristic
20C2×C4×C5×C8C2211yescharacteristic
21C5×#382C2212yes—
22C2×#3949C2211yescharacteristic
23C2×#3949C2211yescharacteristic
24C2×#3949C2212yes—
25C2×#3947C2211yescharacteristic
26C2×C8×Dic5C2211yescharacteristic
27#65266C2212yes—
28#65265C2212yes—
29C2×#3940C2212yes—
30C2×#3940C2212yes—
31C2×#3939C2212yes—
32C2×#3939C2212yes—
33#65224C2212yes—
34C2×#3935C2211yescharacteristic
35#65170—28no—
36#65149—24no—
37D5×#106—24no—
38#64978C418yescomplemented
39#64957—24no—
40#64936—24no—
41#20802—51nomaximal, Sylow
42C22×C4×D5—22no—
43C22×C4×D5C2311yescharacteristic
44C22×C5×C8C2311yescharacteristic
45C22×C5×C8C2311yescharacteristic
46C2×C5×C4⋊C4C2312yes—
47C2×C5×C4⋊C4C2312yes—
48C2×C42×C5C2311yescharacteristic
49C2×#685—22no—
50C2×#685D412yes—
51C2×C4⋊Dic5—24no—
52C2×C4⋊Dic5C2312yes—
53C2×C4⋊Dic5C2312yes—
54C2×#683—22no—
55C2×#683—24no—
56C2×#683D412yes—
57C2×C4×Dic5—22no—
58C2×C4×Dic5C2311yescharacteristic
59C22×#168C2311yescharacteristic
60C22×#168C2311yescharacteristic
61C2×#676C2×C414yes—
62C2×#676D414yes—
63C2×#676Q814yes—
64C2×C8×D5C2×C414yes—
65C2×C8×D5D414yes—
66C2×C8×D5Q814yes—
67#4038—24no—
68#4038—24no—
69D5×C4⋊C4—24no—
70D5×C4⋊C4—24no—
71#4018C2×C414yes—
72#4018Q814yes—
73C5×#107—24no—
74C5×#106—22no—
75C5×#106—22no—
76C4×C5×C8C2×C414yescomplemented
77C5×#95—24no—
78#3963—24no—
79#3949—22no—
80#3949—22no—
81#3949—24no—
82#3947C2×C414yescomplemented
83C8×Dic5C2×C414yescomplemented
84#3940—24no—
85#3940—24no—
86#3939—24no—
87#3939—24no—
88#3935C2×C414yescomplemented
89C2×#428—52no—
90C2×#427—51no—
91C22×#106—51no—
92C2×#405—51no—
93#2317—52no—
94#1847—52no—
95#1847—52no—
96#1846—51no—
97#1846D511yescharacteristic, complemented
98#1844—51no—
99#1844—51no—
100C23×D5C2×D411yescharacteristic
101C22×C4×C5—22no—
102C22×C4×C5—22no—
103C22×C4×C5C2×D412yes—
104C22×C4×C5C2411yescharacteristic
105C22×Dic5—22no—
106C22×Dic5—22no—
107C22×Dic5C2×D412yes—
108C22×Dic5C2×D411yescharacteristic
109C2×C4×D5—44no—
110C2×C4×D5—84no—
111C2×C4×D5C22×C412yes—
112C2×C4×D5C2×D412yes—
113C2×C4×D5C2×D412yes—
114C2×C4×D5C2×Q812yes—
115C2×C4×D5C2×Q812yes—
116C2×C4×D5C2×Q812yes—
117C2×C5×C8C22×C412yes—
118C2×C5×C8C22×C412yes—
119C2×C5×C8C2×D412yes—
120C2×C5×C8C2×D412yes—
121C2×C5×C8C2×Q812yes—
122C2×C5×C8C2×Q812yes—
123C5×C4⋊C4—22no—
124C5×C4⋊C4—22no—
125C5×C4⋊C4—22no—
126C5×C4⋊C4—22no—
127C42×C5C22×C412yes—
128C42×C5C2×Q812yes—
129#685—44no—
130#685C4⋊C418yes—
131C4⋊Dic5—22no—
132C4⋊Dic5—22no—
133C4⋊Dic5—22no—
134C4⋊Dic5—22no—
135C4⋊Dic5—84no—
136#683—44no—
137#683—84no—
138#683C4⋊C418yes—
139C4×Dic5—44no—
140C4×Dic5C22×C412yes—
141C4×Dic5C2×Q812yes—
142C2×#168C22×C412yes—
143C2×#168C22×C412yes—
144C2×#168C2×D412yes—
145C2×#168C2×D412yes—
146C2×#168C2×Q812yes—
147C2×#168C2×Q812yes—
148#676C4⋊C4116yescomplemented
149C8×D5C4⋊C4116yescomplemented
150C22×M16—51no—
151C23×C8—51no—
152C2×#117—51no—
153C2×#117—52no—
154C22×C4⋊C4—52no—
155#428—108no—
156#427—104no—
157C2×#107—52no—
158C2×#107—52no—
159C2×#107C2×D512yes—
160C2×#106—51no—
161C2×#106—51no—
162C2×#106—52no—
163C2×#106—52no—
164C2×#106—52no—
165C2×#106—104no—
166C2×#106C2×D511yescharacteristic
167C2×#106C2×D511yescharacteristic
168#405—58no—
169C2×#97—51no—
170C2×#97—51no—
171C2×C4×C8—51no—
172C2×C4×C8C2×D511yescharacteristic
173#389—104no—
174#385—104no—
175#384—52no—
176#384—52no—
177#382—52no—
178#382C2×D512yes—
179C23×C5C22×D411yescharacteristic
180C22×D5—24no—
181C22×D5C2×C4⋊C412yes—
182C22×D5#12712yes—
183C22×D5C4⋊Q812yes—
184C2×C4×C5—24no—
185C2×C4×C5—42no—
186C2×C4×C5—42no—
187C2×C4×C5—44no—
188C2×C4×C5—44no—
189C2×C4×C5C2×C4⋊C414yes—
190C2×C4×C5C23×C411yescharacteristic
191C2×C4×C5C22×D411yescharacteristic
192C2×C4×C5C22×D411yescharacteristic
193C2×C4×C5C22×Q811yescharacteristic
194C2×C4×C5C22×Q811yescharacteristic
195C2×C4×C5C22×Q811yescharacteristic
196C2×Dic5—24no—
197C2×Dic5—42no—
198C2×Dic5—42no—
199C2×Dic5—44no—
200C2×Dic5—44no—
201C2×Dic5C2×C4⋊C414yes—
202C2×Dic5C2×C4⋊C412yes—
203C2×Dic5#12712yes—
204C2×Dic5C4⋊Q812yes—
205C4×D5—88no—
206C4×D5C2×C4⋊C414yes—
207C4×D5C2×C4⋊C414yes—
208C4×D5C4⋊Q814yes—
209C4×D5C4⋊Q814yes—
210C5×C8C2×C4⋊C414yes—
211C5×C8C2×C4⋊C414yes—
212#168C2×C4⋊C414yes—
213#168C2×C4⋊C414yes—
214C23×C4—51no—
215C23×C4—102no—
216C2×M16—54no—
217C2×M16—54no—
218C2×M16—54no—
219C22×C8—51no—
220C22×C8—51no—
221C22×C8—54no—
222C22×C8—54no—
223C22×C8—54no—
224C22×C8C22×D511yescharacteristic
225C22×C8C22×D511yescharacteristic
226#117—54no—
227#117—54no—
228#117—104no—
229#117—104no—
230C2×C4⋊C4—52no—
231C2×C4⋊C4—52no—
232C2×C4⋊C4—52no—
233C2×C4⋊C4—104no—
234C2×C4⋊C4—104no—
235C2×C4⋊C4—102no—
236C2×C4⋊C4—104no—
237C2×C4⋊C4—104no—
238C2×C4⋊C4C22×D512yes—
239C2×C4⋊C4C22×D512yes—
240C2×C22⋊C4—52no—
241C2×C22⋊C4—102no—
242C2×C42—51no—
243C2×C42—102no—
244C2×C42C22×D511yescharacteristic
245#107—24no—
246#107—104no—
247#107—104no—
248#106—22no—
249#106—22no—
250#106—104no—
251#106—104no—
252#106—102no—
253#106—102no—
254#106—104no—
255#97—54no—
256#97—54no—
257C4×C8—54no—
258C4×C8C4×D514yescomplemented
259#95—24no—
260#95—104no—
261C22×C5C22×C4⋊C411yescharacteristic
262C22×C5C2×#12711yescharacteristic
263C22×C5C2×C4⋊Q811yescharacteristic
264C22×C5#57714yes—
265C2×D5—24no—
266C2×D5—24no—
267C2×D5—24no—
268C2×D5#38414yes—
269C4×C5—24no—
270C4×C5—82no—
271C4×C5—82no—
272C4×C5C22×C4⋊C411yescharacteristic
273C4×C5C22×C4⋊C411yescharacteristic
274C4×C5C2×C4⋊Q811yescharacteristic
275C4×C5C2×C4⋊Q811yescharacteristic
276Dic5—24no—
277Dic5—82no—
278Dic5—82no—
279Dic5#38414yes—
280C24—51no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—51no—
284C22×C4—52no—
285C22×C4—52no—
286C22×C4—52no—
287C22×C4—52no—
288C22×C4—52no—
289C22×C4—52no—
290C22×C4—52no—
291C22×C4—102no—
292C22×C4—102no—
293C22×C4—204no—
294C22×C4—404no—
295C22×C4D4×D512yes—
296C22×C4C23×D511yescharacteristic
297M16—516no—
298C2×C8—52no—
299C2×C8—52no—
300C2×C8—52no—
301C2×C8—52no—
302C2×C8—52no—
303C2×C8—52no—
304C2×C8—516no—
305C2×C8C2×C4×D512yes—
306C2×C8C2×C4×D512yes—
307C2×C8D4×D512yes—
308C2×C8D4×D512yes—
309C2×C8Q8×D512yes—
310C2×C8Q8×D512yes—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—58no—
316C4⋊C4—102no—
317C4⋊C4—102no—
318C4⋊C4—102no—
319C4⋊C4—102no—
320C4⋊C4—204no—
321C4⋊C4—404no—
322C4⋊C4—404no—
323C22⋊C4—58no—
324C22⋊C4—204no—
325C42—52no—
326C42—52no—
327C42—204no—
328C42C2×C4×D512yes—
329C42Q8×D512yes—
330C2×C5C2×#38411yescharacteristic
331C2×C5#291212yes—
332C2×C5#315412yes—
333C2×C5#316312yes—
334D5—24no—
335C23—52no—
336C23—52no—
337C23—52no—
338C23—104no—
339C23C2×D4×D511yescharacteristic
340C2×C4—24no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—52no—
346C2×C4—52no—
347C2×C4—52no—
348C2×C4—54no—
349C2×C4—54no—
350C2×C4—54no—
351C2×C4—54no—
352C2×C4—54no—
353C2×C4—104no—
354C2×C4—202no—
355C2×C4—202no—
356C2×C4—204no—
357C2×C4—204no—
358C2×C4—408no—
359C2×C4D5×C4⋊C414yes—
360C2×C4C22×C4×D511yescharacteristic
361C2×C4C2×D4×D511yescharacteristic
362C2×C4C2×D4×D511yescharacteristic
363C2×C4C2×Q8×D511yescharacteristic
364C2×C4C2×Q8×D511yescharacteristic
365C2×C4C2×Q8×D511yescharacteristic
366C8—54no—
367C8—54no—
368C8D5×C4⋊C414yes—
369C8D5×C4⋊C414yes—
370C5#2080211yescharacteristic, complemented, Sylow
371C22—54no—
372C22—104no—
373C22—104no—
374C22—104no—
375C22C2×D5×C4⋊C411yescharacteristic
376C22D5×#12711yescharacteristic
377C22D5×C4⋊Q811yescharacteristic
378C22#6610414yes—
379C4—24no—
380C4—54no—
381C4—82no—
382C4—82no—
383C4—104no—
384C4—402no—
385C4—402no—
386C4C2×D5×C4⋊C411yescharacteristic
387C4C2×D5×C4⋊C411yescharacteristic
388C4D5×C4⋊Q811yescharacteristic
389C4D5×C4⋊Q811yescharacteristic
390C2—104no—
391C2D5×#38411yescharacteristic
392C2#1061090912yes—
393C2#1061355312yes—
394C2#1061364812yes—
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1846C2
socleC23×C5C22×D4
prime radicalC23×D5C2×D4
cosocleC22×C4×C5C24
hypercenter#1846D5
hypocenterC5#20802
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1846

Complements

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

NH ≅ G/Ntype
6. C5×#1846C2semidirect
7. #10601401C2semidirect
8. #10601348C2semidirect
9. #10601347C2semidirect
10. #10600557C2semidirect
11. #10600343C2semidirect
38. #64978C4semidirect
76. C4×C5×C8C2×C4semidirect
82. #3947C2×C4semidirect
83. C8×Dic5C2×C4semidirect
88. #3935C2×C4semidirect
97. #1846D5semidirect
148. #676C4⋊C4semidirect
149. C8×D5C4⋊C4semidirect
258. C4×C8C4×D5semidirect
370. C5#20802semidirect

Maximal subgroups

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

subgroupindex
C2×#651702
C2×#651492
C2×D5×#1062
C2×#649782
#106059562
C5×#18462
#106014012
#106013482
#106013472
#106005572
#106003432
#208025