Finite Group Databasebeta

← #12305726

Subgroups — #12305726

Order 1280.

1778 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2280C211yesmaximal, characteristic
2#10615758C212yesmaximal
3#10613976C214yesmaximal, complemented
4#10605846C214yesmaximal, complemented
5#10605728C218yesmaximal, complemented
6#10605576C212yesmaximal
7#10605576C211yesmaximal, characteristic
8#10605575C211yesmaximal, characteristic
9#10605211C218yesmaximal, complemented
10C24×C4×C5C2211yescharacteristic
11C2×C5×#117C2212yes—
12C2×#4072C2214yes—
13C2×C4×Dic10C2214yescomplemented
14C5×#381C2212yes—
15C5×#380C2214yes—
16C5×#379C2211yescharacteristic
17C5×#379C2211yescharacteristic
18C4×C5×C22⋊C4C2214yes—
19#65536C2212yes—
20#65536C2212yes—
21#65501C2214yes—
22#65501C2214yes—
23#65500C2214yes—
24C4×C22⋊Dic5C2214yes—
25C4×C22⋊Dic5C2214yes—
26#65275C2214yescomplemented
27#65273C2218yescomplemented
28#65268C2214yescomplemented
29#65238C2214yes—
30#65238C2218yes—
31#65237C2218yes—
32#65237C2218yes—
33#65237C2218yes—
34#65236C2214yes—
35#65236C2218yes—
36C4×C4⋊Dic5C2214yescomplemented
37C4×#683C2218yescomplemented
38#64950C2218yescomplemented
39#64949C2218yescomplemented
40#64946C2218yescomplemented
41#64945C2218yescomplemented
42#36674—51nomaximal, Sylow
43C25×C5C2311yescharacteristic
44C23×C4×C5—44no—
45C23×C4×C5Q818yescomplemented
46C23×C4×C5C2312yes—
47C23×C4×C5C2311yescharacteristic
48C23×C4×C5C2312yes—
49C23×C4×C5C2311yescharacteristic
50C22×Dic10C2314yes—
51C5×#117—44no—
52C2×C5×C4⋊C4C2314yes—
53C2×C5×C22⋊C4C2314yes—
54C2×C5×C22⋊C4C2314yes—
55C2×C5×C22⋊C4C2312yes—
56C2×C5×C22⋊C4C2312yes—
57C2×C42×C5C2314yes—
58C2×C22⋊Dic5C2314yes—
59C2×C22⋊Dic5C2314yes—
60C2×C22⋊Dic5C2314yes—
61C2×C22⋊Dic5C2314yes—
62C2×C22⋊Dic5C2314yes—
63C2×C22⋊Dic5C2314yes—
64#4072—48no—
65C2×C4⋊Dic5C2314yes—
66C2×C4⋊Dic5C2318yes—
67C2×C4⋊Dic5C2318yes—
68C2×#683C2318yes—
69C2×#683C2318yes—
70C2×#683C2318yes—
71C2×C4×Dic5C2318yes—
72C4×Dic10—48no—
73C5×#95C2318yes—
74#3963C2318yes—
75#3963C2318yes—
76#3963C2314yes—
77#3963C2314yes—
78#3963C2318yes—
79#3963C2318yes—
80#2842—52no—
81#2842—51no—
82#2841—54no—
83#2841—58no—
84#2840—51no—
85#2396—58no—
86#2396—54no—
87#2280—52no—
88#2280D511yescharacteristic
89C24×C5—42no—
90C24×C5—44no—
91C24×C5C2×Q814yes—
92C24×C5C2412yes—
93C24×C5C2411yescharacteristic
94C22×C4×C5—24no—
95C22×C4×C5—28no—
96C22×C4×C5—416no—
97C22×C4×C5—42no—
98C22×C4×C5—44no—
99C22×C4×C5C2×Q818yes—
100C22×C4×C5C2×Q814yes—
101C22×C4×C5C2×Q818yes—
102C22×C4×C5C2×Q814yes—
103C22×C4×C5C2414yes—
104C22×C4×C5C2414yes—
105C22×C4×C5C2411yescharacteristic
106C22×C4×C5C2412yes—
107C22×C4×C5C2411yescharacteristic
108C22×Dic5C2418yes—
109C22×Dic5C2418yes—
110C2×Dic10—48no—
111C2×Dic10—44no—
112C5×C4⋊C4—44no—
113C5×C22⋊C4—44no—
114C5×C22⋊C4—44no—
115C5×C22⋊C4—42no—
116C5×C22⋊C4—42no—
117C42×C5—28no—
118C22⋊Dic5—44no—
119C22⋊Dic5—44no—
120C22⋊Dic5—44no—
121C22⋊Dic5—44no—
122C22⋊Dic5—44no—
123C22⋊Dic5—44no—
124C4⋊Dic5—44no—
125C4⋊Dic5—48no—
126C4⋊Dic5—48no—
127#683—48no—
128#683—48no—
129#683—48no—
130C4×Dic5—216no—
131C24×C4C2×D511yescharacteristic
132C2×C4×Q8—54no—
133C2×#117—54no—
134C2×#117C2×D512yes—
135#400—54no—
136#400—58no—
137#399—58no—
138#399—54no—
139#398—58no—
140#398—58no—
141#398—58no—
142#396—58no—
143#396—58no—
144#396—58no—
145#395—54no—
146#395—58no—
147#393—58no—
148#393—54no—
149#381—54no—
150#381C2×D512yes—
151#380—54no—
152#380—54no—
153#380C2×D514yes—
154#379—52no—
155#379—52no—
156#379C2×D511yescharacteristic
157#379C2×D511yescharacteristic
158C4×C4⋊C4—58no—
159C4×C4⋊C4—54no—
160C4×C22⋊C4—54no—
161C4×C22⋊C4—54no—
162C4×C22⋊C4C2×D514yes—
163C23×C5—24no—
164C23×C5—22no—
165C23×C5—24no—
166C23×C5—24no—
167C23×C5—24no—
168C23×C5—44no—
169C23×C5—48no—
170C23×C5—416no—
171C23×C5C22×Q814yes—
172C23×C5C22×Q812yes—
173C23×C5C2511yescharacteristic
174C2×C4×C5—24no—
175C2×C4×C5—22no—
176C2×C4×C5—28no—
177C2×C4×C5—216no—
178C2×C4×C5—24no—
179C2×C4×C5—24no—
180C2×C4×C5—28no—
181C2×C4×C5—28no—
182C2×C4×C5—44no—
183C2×C4×C5—48no—
184C2×C4×C5C22×Q812yes—
185C2×C4×C5C22×Q814yes—
186C2×C4×C5C22×Q812yes—
187C2×C4×C5H(5,𝔽2)14yes—
188C2×Dic5—28no—
189C2×Dic5—28no—
190C2×Dic5—28no—
191C2×Dic5—28no—
192C2×Dic5—28no—
193C2×Dic5—28no—
194Dic10—416no—
195C25C22×D511yescharacteristic
196C22×Q8—54no—
197C23×C4—44no—
198C23×C4Dic1018yescomplemented
199C23×C4C22×D512yes—
200C23×C4C22×D511yescharacteristic
201C23×C4C22×D511yescharacteristic
202C23×C4C22×D512yes—
203C4×Q8—208no—
204#117—44no—
205#117—208no—
206C2×C4⋊C4—58no—
207C2×C4⋊C4—58no—
208C2×C4⋊C4—54no—
209C2×C4⋊C4—58no—
210C2×C4⋊C4—58no—
211C2×C4⋊C4—58no—
212C2×C4⋊C4C22×D514yes—
213C2×C22⋊C4—54no—
214C2×C22⋊C4—54no—
215C2×C22⋊C4—54no—
216C2×C22⋊C4—54no—
217C2×C22⋊C4—54no—
218C2×C22⋊C4—54no—
219C2×C22⋊C4C22×D512yes—
220C2×C22⋊C4C22×D512yes—
221C2×C22⋊C4C22×D514yes—
222C2×C22⋊C4C22×D514yes—
223C2×C42—58no—
224C2×C42C22×D514yes—
225#95—58no—
226#95—58no—
227#95—58no—
228#95—54no—
229#95—58no—
230#95—54no—
231#95C22×D518yes—
232C22×C5—24no—
233C22×C5—22no—
234C22×C5—24no—
235C22×C5—24no—
236C22×C5—24no—
237C22×C5—416no—
238C22×C5—48no—
239C22×C5—44no—
240C22×C5C23×Q811yescharacteristic
241C22×C5C2×H(5,𝔽2)12yes—
242C22×C5#58514yes—
243C4×C5—24no—
244C4×C5—22no—
245C4×C5—44no—
246C4×C5—44no—
247C4×C5#54314yes—
248Dic5—48no—
249Dic5—48no—
250C24—42no—
251C24—44no—
252C24C2×Dic1014yes—
253C24C23×D511yescharacteristic
254C24C23×D512yes—
255C2×Q8—204no—
256C2×Q8—208no—
257C22×C4—24no—
258C22×C4—28no—
259C22×C4—416no—
260C22×C4—44no—
261C22×C4—42no—
262C22×C4—58no—
263C22×C4—58no—
264C22×C4C2×Dic1018yes—
265C22×C4C2×Dic1014yes—
266C22×C4C2×Dic1014yes—
267C22×C4C2×Dic1018yes—
268C22×C4C23×D514yes—
269C22×C4C23×D514yes—
270C22×C4C23×D511yescharacteristic
271C22×C4C23×D512yes—
272C22×C4C23×D511yescharacteristic
273C4⋊C4—44no—
274C4⋊C4—208no—
275C4⋊C4—208no—
276C4⋊C4—204no—
277C4⋊C4—208no—
278C4⋊C4—208no—
279C4⋊C4—208no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—44no—
283C22⋊C4—44no—
284C22⋊C4—204no—
285C22⋊C4—204no—
286C22⋊C4—204no—
287C22⋊C4—204no—
288C22⋊C4—204no—
289C22⋊C4—204no—
290C42—28no—
291C42—1016no—
292C2×C5—44no—
293C2×C5—42no—
294C2×C5C2×#54311yescharacteristic
295C2×C5#345812yes—
296C2×C5#352914yes—
297C23—22no—
298C23—24no—
299C23—24no—
300C23—24no—
301C23—24no—
302C23—44no—
303C23—48no—
304C23—416no—
305C23C22×Dic1014yes—
306C23C22×Dic1012yes—
307C23C24×D511yescharacteristic
308Q8—2016no—
309C2×C4—22no—
310C2×C4—24no—
311C2×C4—28no—
312C2×C4—216no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—28no—
316C2×C4—28no—
317C2×C4—48no—
318C2×C4—44no—
319C2×C4—108no—
320C2×C4—108no—
321C2×C4—108no—
322C2×C4—108no—
323C2×C4—108no—
324C2×C4—108no—
325C2×C4C22×Dic1012yes—
326C2×C4C22×Dic1014yes—
327C2×C4C22×Dic1012yes—
328C2×C4#414414yes—
329C5#3667411yescharacteristic, complemented, Sylow
330C22—24no—
331C22—24no—
332C22—24no—
333C22—22no—
334C22—24no—
335C22—416no—
336C22—44no—
337C22—48no—
338C22C23×Dic1011yescharacteristic
339C22C2×#414412yes—
340C22#6628314yes—
341C4—22no—
342C4—24no—
343C4—44no—
344C4—44no—
345C4—208no—
346C4—208no—
347C4#6581714yes—
348C2—44no—
349C2—42no—
350C2C2×#6581711yescharacteristic
351C2#1061762612yes—
352C2#1061794914yes—
subgroupG/H
centerC22×C4C23×D5
commutator, socle, cosocleC23×C5C25
FrattiniC23C24×D5
FittingC5×#2280C2
prime radicalC25×C5C23
hypercenter#2280D5
hypocenterC5#36674
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C22×C4 ◁ #2280

Complements

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

NH ≅ G/Ntype
3. #10613976C2semidirect
4. #10605846C2semidirect
5. #10605728C2semidirect
9. #10605211C2semidirect
13. C2×C4×Dic10C22semidirect
26. #65275C22semidirect
27. #65273C22semidirect
28. #65268C22semidirect
36. C4×C4⋊Dic5C22semidirect
37. C4×#683C22semidirect
38. #64950C22semidirect
39. #64949C22semidirect
40. #64946C22semidirect
41. #64945C22semidirect
45. C23×C4×C5Q8semidirect
198. C23×C4Dic10semidirect
329. C5#36674semidirect

Maximal subgroups

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

subgroupindex
C5×#22802
#106157582
#106139762
#106058462
#106057282
#106055762
#106055752
#106052112
#366745