Finite Group Databasebeta

← #12065226

Subgroups — #12065226

Order 1280.

1206 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65908C211yesmaximal, characteristic
2#10611019C212yesmaximal, complemented
3C5×#1461C211yesmaximal, characteristic, complemented
4#10602417C211yesmaximal, characteristic, complemented
5#10601575C212yesmaximal, complemented
6#10598938C214yesmaximal, complemented
7#10598937C214yesmaximal, complemented
8C22×#713C411yescharacteristic
9C2×C4×C5×Q8C2211yescharacteristic
10C2×Q8×Dic5C411yescharacteristic
11#65908—24no—
12#65908—24no—
13C2×#4039C411yescharacteristic
14C2×#4039C2212yes—
15C2×D5×C4⋊C4C411yescharacteristic
16C2×D5×C4⋊C4C2212yes—
17C2×C4×D20C412yes—
18C2×C4×D20C2211yescharacteristic
19C2×#4018C412yes—
20C2×#4018C2211yescharacteristic
21C5×#431C2212yes—
22#65214C2212yes—
23#65127—24no—
24#65015C2218yes—
25#64977—24no—
26C5×#324C2214yes—
27#64748C2214yes—
28#64689C2214yes—
29#64688C2214yes—
30#17370—51nomaximal, Sylow
31C22×C5×Q8C2×C411yescharacteristic
32C2×#713—24no—
33C2×#713—24no—
34C2×#713—24no—
35C22×D20C2×C412yes—
36C22×D20C2×C411yescharacteristic
37C22×C4×D5C2×C412yes—
38C22×C4×D5C2×C411yescharacteristic
39C22×C4×D5C2×C412yes—
40C22×C4×D5C2×C412yes—
41C22×C4×D5C2312yes—
42C4×C5×Q8—22no—
43C4×C5×Q8—22no—
44C2×C5×C4⋊C4C2×C411yescharacteristic
45C2×C5×C4⋊C4C2312yes—
46C2×C42×C5C2×C412yes—
47C2×C42×C5C2311yescharacteristic
48Q8×Dic5—24no—
49C2×#685C2×C412yes—
50C2×#685C2×C411yescharacteristic
51C2×#685C2×C412yes—
52C2×#685C2311yescharacteristic
53C2×C4⋊Dic5C2×C412yes—
54C2×C4⋊Dic5C2×C411yescharacteristic
55C2×#683C2×C411yescharacteristic
56C2×#683C2×C412yes—
57C2×#683C2311yescharacteristic
58C2×C4×Dic5C2×C412yes—
59C2×C4×Dic5C2×C411yescharacteristic
60C2×#676—28no—
61#4039—24no—
62#4039—28no—
63D5×C4⋊C4—28no—
64D5×C4⋊C4—24no—
65C4×D20—24no—
66C4×D20—24no—
67C4×D20D414yes—
68C4×D20D414yes—
69#4018—22no—
70#4018—22no—
71#4018—24no—
72#4018—24no—
73C5×C4⋊C8—22no—
74C5×C22⋊C8C2314yes—
75C4×C5×C8—22no—
76C22⋊#168C2314yes—
77#3952—28no—
78C4⋊#168—22no—
79C4×#168—22no—
80C2×#520—51no—
81#2975—52no—
82#1465—54no—
83#1463—54no—
84#1461—51no—
85#1461D511yescharacteristic, complemented
86#1458—52no—
87C23×D5C22×C412yes—
88C23×D5C22×C411yescharacteristic
89C2×C5×Q8—22no—
90C2×C5×Q8—22no—
91C2×C5×Q8M1614yescomplemented
92C22×C4×C5C22×C412yes—
93C22×C4×C5C22×C412yes—
94C22×C4×C5C22×C411yescharacteristic
95C22×C4×C5C22×C411yescharacteristic
96C22×C4×C5C2411yescharacteristic
97C22×Dic5C22×C411yescharacteristic
98C22×Dic5C22×C411yescharacteristic
99C22×Dic5C22×C411yescharacteristic
100C22×Dic5C22×C412yes—
101#713—232no—
102C2×D20—24no—
103C2×D20—24no—
104C2×D20—24no—
105C2×D20—22no—
106C2×D20—22no—
107C2×D20C22⋊C414yes—
108C2×C4×D5—24no—
109C2×C4×D5—22no—
110C2×C4×D5—24no—
111C2×C4×D5—22no—
112C2×C4×D5—24no—
113C2×C4×D5—24no—
114C2×C4×D5—24no—
115C2×C4×D5—82no—
116C2×C4×D5—82no—
117C2×C4×D5—82no—
118C2×C4×D5M1614yescomplemented
119C2×C4×D5C2×D418yes—
120C2×C5×C8—24no—
121C5×C4⋊C4—22no—
122C5×C4⋊C4—24no—
123C42×C5—22no—
124C42×C5—22no—
125C42×C5C2×D412yes—
126C42×C5C2×D412yes—
127#685—24no—
128#685—21no—
129#685—21no—
130#685—22no—
131#685C22⋊C414yes—
132#685C22⋊C414yes—
133C4⋊Dic5—24no—
134C4⋊Dic5C22⋊C414yes—
135#683—21no—
136#683—21no—
137#683—22no—
138#683—24no—
139C4×Dic5—22no—
140C4×Dic5—24no—
141C2×#168—24no—
142#676—88no—
143C22×#41—51no—
144#520—104no—
145#520—104no—
146C2×C4×Q8—51no—
147C2×C4×Q8C2×D511yescharacteristic
148C2×C4×D4—51no—
149C2×C4×D4—52no—
150C2×C4×D4—52no—
151C2×C4×D4—51no—
152C2×#117—52no—
153C2×#117—51no—
154C22×C4⋊C4—51no—
155C22×C4⋊C4—52no—
156#436—104no—
157#436—104no—
158#431—52no—
159#431C2×D512yes—
160#407—58no—
161#324—54no—
162#324—54no—
163#324C2×D514yes—
164#323—54no—
165C23×C5C23×C411yescharacteristic
166C22×D5—21no—
167C22×D5—22no—
168C22×D5—21no—
169C22×D5—22no—
170C22×D5—42no—
171C22×D5—42no—
172C22×D5—41no—
173C22×D5—41no—
174C22×D5C2×C22⋊C414yes—
175C22×D5C2×M1612yes—
176C5×Q8—28no—
177C2×C4×C5—22no—
178C2×C4×C5—22no—
179C2×C4×C5—22no—
180C2×C4×C5—41no—
181C2×C4×C5—41no—
182C2×C4×C5—42no—
183C2×C4×C5C2×C22⋊C411yescharacteristic
184C2×C4×C5C2×C22⋊C411yescharacteristic
185C2×C4×C5C2×C22⋊C412yes—
186C2×C4×C5C2×M1614yes—
187C2×C4×C5C2×M1612yes—
188C2×C4×C5C2×M1612yes—
189C2×C4×C5C22×D412yes—
190C2×Dic5—21no—
191C2×Dic5—21no—
192C2×Dic5—21no—
193C2×Dic5—21no—
194C2×Dic5—42no—
195C2×Dic5—41no—
196C2×Dic5—41no—
197C2×Dic5C2×C22⋊C414yes—
198C2×Dic5C2×M1612yes—
199D20—216no—
200D20—28no—
201C4×D5—216no—
202C4×D5—28no—
203C4×D5—48no—
204C4×D5—84no—
205C4×D5—84no—
206C5×C8—44no—
207#168—44no—
208C2×#41—104no—
209C2×#41—104no—
210C2×#41—104no—
211C22×Q8C4×D511yescharacteristic
212C22×D4—51no—
213C22×D4—52no—
214C23×C4—51no—
215C23×C4—52no—
216C23×C4—52no—
217C23×C4—52no—
218C23×C4—52no—
219C2×M16—108no—
220C4×Q8—22no—
221C4×Q8—22no—
222C4×Q8—104no—
223C4×D4—54no—
224C4×D4—54no—
225C4×D4—104no—
226C4×D4—104no—
227C4×D4—108no—
228C4×D4—104no—
229#117—104no—
230#117—104no—
231#117—102no—
232#117—102no—
233C2×C4⋊C4—51no—
234C2×C4⋊C4—52no—
235C2×C4⋊C4—52no—
236C2×C4⋊C4—51no—
237C2×C4⋊C4—51no—
238C2×C4⋊C4—104no—
239C2×C4⋊C4—108no—
240C2×C4⋊C4C4×D511yescharacteristic
241C2×C4⋊C4C22×D512yes—
242C2×C22⋊C4—52no—
243C2×C22⋊C4—52no—
244C2×C22⋊C4—51no—
245C2×C22⋊C4—51no—
246C2×C42—52no—
247C2×C42—51no—
248C2×C42C4×D512yes—
249C2×C42C22×D511yescharacteristic
250C4⋊C8—22no—
251C4⋊C8—102no—
252C22⋊C8—54no—
253C22⋊C8—108no—
254C22⋊C8C22×D514yes—
255C4×C8—22no—
256C4×C8—102no—
257C22×C5—21no—
258C22×C5—21no—
259C22×C5C22×C22⋊C411yescharacteristic
260C22×C5C22×M1611yescharacteristic
261C22×C5C22×M1611yescharacteristic
262C2×D5—24no—
263C2×D5—22no—
264C2×D5—24no—
265C2×D5—24no—
266C2×D5—44no—
267C2×D5—44no—
268C2×D5—42no—
269C2×D5—41no—
270C2×D5—41no—
271C2×D5—42no—
272C4×C5—24no—
273C4×C5—22no—
274C4×C5—21no—
275C4×C5—21no—
276C4×C5—42no—
277C4×C5#40714yes—
278Dic5—22no—
279Dic5—24no—
280Dic5—41no—
281Dic5—41no—
282C24—52no—
283C24—51no—
284#41—1032no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8#67614yescomplemented
288C2×D4—54no—
289C2×D4—104no—
290C2×D4—104no—
291C2×D4—104no—
292C2×D4—102no—
293C2×D4—102no—
294C22×C4—51no—
295C22×C4—52no—
296C22×C4—54no—
297C22×C4—58no—
298C22×C4—51no—
299C22×C4—51no—
300C22×C4—104no—
301C22×C4—104no—
302C22×C4—104no—
303C22×C4—104no—
304C22×C4—102no—
305C22×C4—104no—
306C22×C4—102no—
307C22×C4—402no—
308C22×C4—402no—
309C22×C4—402no—
310C22×C4C2×C4×D512yes—
311C22×C4C2×C4×D512yes—
312C22×C4C2×C4×D511yescharacteristic
313C22×C4C2×C4×D511yescharacteristic
314C22×C4C23×D511yescharacteristic
315M16—408no—
316C2×C8—24no—
317C2×C8—104no—
318C4⋊C4—24no—
319C4⋊C4—22no—
320C4⋊C4—54no—
321C4⋊C4—104no—
322C4⋊C4—104no—
323C4⋊C4—102no—
324C4⋊C4—101no—
325C4⋊C4—101no—
326C22⋊C4—54no—
327C22⋊C4—54no—
328C22⋊C4—104no—
329C22⋊C4—102no—
330C22⋊C4—101no—
331C22⋊C4—101no—
332C42—22no—
333C42—22no—
334C42—104no—
335C42—102no—
336C42D4×D512yes—
337C42D4×D512yes—
338C2×C5—21no—
339C2×C5—21no—
340C2×C5C2×#40711yescharacteristic
341C2×C5#288311yescharacteristic
342C2×C5#288911yescharacteristic
343D5—44no—
344D5—42no—
345C23—54no—
346C23—52no—
347C23—102no—
348C23—101no—
349C23—101no—
350C23—102no—
351C23—202no—
352C23—202no—
353C23—201no—
354C23—201no—
355C23C22×C4×D511yescharacteristic
356Q8—28no—
357D4—1016no—
358D4—108no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—54no—
366C2×C4—52no—
367C2×C4—1016no—
368C2×C4—101no—
369C2×C4—101no—
370C2×C4—108no—
371C2×C4—101no—
372C2×C4—101no—
373C2×C4—208no—
374C2×C4—202no—
375C2×C4—201no—
376C2×C4—201no—
377C2×C4—404no—
378C2×C4—404no—
379C2×C4D5×C22⋊C411yescharacteristic
380C2×C4D5×C22⋊C411yescharacteristic
381C2×C4D5×C22⋊C412yes—
382C2×C4C2×#67614yes—
383C2×C4C2×#67612yes—
384C2×C4D5×M1612yes—
385C2×C4C2×D4×D512yes—
386C8—44no—
387C8—204no—
388C5#1737011yescharacteristic, complemented, Sylow
389C22—21no—
390C22—21no—
391C22—104no—
392C22—104no—
393C22—102no—
394C22—104no—
395C22—202no—
396C22—204no—
397C22—202no—
398C22—204no—
399C22—201no—
400C22—201no—
401C22C2×D5×C22⋊C411yescharacteristic
402C22C22×#67611yescharacteristic
403C22C2×D5×M1611yescharacteristic
404C4—21no—
405C4—21no—
406C4—24no—
407C4—22no—
408C4—42no—
409C4—102no—
410C4—104no—
411C4—201no—
412C4—201no—
413C4#6501514yes—
414C2—21no—
415C2—21no—
416C2—204no—
417C2—202no—
418C2C2×#6501511yescharacteristic
419C2#1061042511yescharacteristic
420C2#1061043711yescharacteristic
subgroupG/H
centerC22C2×D5×C22⋊C4
commutatorC23×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1461C2
socleC22×C5C22×C22⋊C4
prime radicalC22×#713C4
cosocleC22×C4×C5C24
hypercenter#1461D5
hypocenterC5#17370
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
2. #10611019C2semidirect
3. C5×#1461C2semidirect
4. #10602417C2semidirect
5. #10601575C2semidirect
6. #10598938C2semidirect
7. #10598937C2semidirect
85. #1461D5semidirect
91. C2×C5×Q8M16semidirect
118. C2×C4×D5M16semidirect
287. C2×Q8#676semidirect
388. C5#17370semidirect

Maximal subgroups

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

subgroupindex
C2×#659082
#106110192
C5×#14612
#106024172
#106015752
#105989382
#105989372
#173705