Finite Group Databasebeta

← C2×#10605248 #12268482

Subgroups — C2×#10605248 #12268482

Order 1280.

3094 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#4070C211yesmaximal, characteristic
2C23×C4⋊Dic5C211yesmaximal, characteristic
3C2×C4×C5×C22⋊C4C211yesmaximal, characteristic
4C2×#65239C214yesmaximal
5C2×#65218C212yesmaximal, complemented
6C2×#65224C212yesmaximal, complemented
7#10605248C2116yesmaximal, direct product factor
8C2×#64950C214yesmaximal, complemented
9C24×C4×C5C2211yescharacteristic
10C24×Dic5C2212yes—
11C23×Dic10C2211yescharacteristic
12C22×C5×C22⋊C4C2212yes—
13C22×C42×C5C2212yes—
14C22×C22⋊Dic5C2212yes—
15C22×C4⋊Dic5—28no—
16C22×C4⋊Dic5C2212yes—
17C22×C4⋊Dic5C2211yescharacteristic
18C22×C4⋊Dic5C2212yes—
19C22×C4⋊Dic5C2218yes—
20C22×C4⋊Dic5C2218yes—
21C2×#4070—28no—
22C2×#4070C2218yes—
23C22×#683C2212yes—
24C4×C5×C22⋊C4C2218yes—
25C2×C5×#95C2212yes—
26C2×#3963C2218yes—
27#65239C22132yes—
28#65224C22116yescomplemented
29#65218C22116yescomplemented
30#64950C22132yescomplemented
31C2×#2502—51nomaximal, Sylow
32C25×C5C2311yescharacteristic
33C23×C4×C5—28no—
34C23×C4×C5D418yes—
35C23×C4×C5C2311yescharacteristic
36C23×C4×C5C2314yes—
37C23×C4×C5C2314yes—
38C23×C4×C5C2311yescharacteristic
39C23×Dic5—28no—
40C23×Dic5—48no—
41C23×Dic5C2312yes—
42C23×Dic5C2312yes—
43C23×Dic5C2314yes—
44C23×Dic5C2318yes—
45C22×Dic10—24no—
46C22×Dic10—28no—
47C22×Dic10C2314yes—
48C2×C5×C22⋊C4Q8116yescomplemented
49C2×C5×C22⋊C4C2318yes—
50C2×C42×C5—28no—
51C2×C42×C5C2318yes—
52C2×C22⋊Dic5—28no—
53C2×C22⋊Dic5C2318yes—
54C2×C4⋊Dic5—24no—
55C2×C4⋊Dic5—28no—
56C2×C4⋊Dic5—232no—
57C2×C4⋊Dic5—28no—
58C2×C4⋊Dic5—416no—
59C2×C4⋊Dic5C2314yes—
60C2×C4⋊Dic5C2318yes—
61C2×C4⋊Dic5C2318yes—
62C2×C4⋊Dic5C23132yes—
63#4070—232no—
64C2×#683—28no—
65C2×#683C2318yes—
66C5×#95C2318yes—
67#3963C23132yes—
68C22×C22⋊Q8—51no—
69C23×C4⋊C4—51no—
70#2502—516no—
71C2×#398—54no—
72C2×#391—52no—
73C2×#385—54no—
74C2×#384—52no—
75C2×C4×C22⋊C4D511yescharacteristic
76C24×C5—24no—
77C24×C5—24no—
78C24×C5C2×D414yes—
79C24×C5C2×Q818yes—
80C24×C5C2412yes—
81C24×C5C2411yescharacteristic
82C22×C4×C5—28no—
83C22×C4×C5—24no—
84C22×C4×C5—28no—
85C22×C4×C5—232no—
86C22×C4×C5—216no—
87C22×C4×C5C2×D414yes—
88C22×C4×C5C2×D4116yes—
89C22×C4×C5C2×D414yes—
90C22×C4×C5C2×Q8116yes—
91C22×C4×C5#4118yes—
92C22×C4×C5C2412yes—
93C22×C4×C5C2418yes—
94C22×C4×C5C2412yes—
95C22×Dic5—24no—
96C22×Dic5—24no—
97C22×Dic5—216no—
98C22×Dic5—28no—
99C22×Dic5—28no—
100C22×Dic5—216no—
101C22×Dic5—28no—
102C22×Dic5—416no—
103C22×Dic5—432no—
104C22×Dic5C2414yes—
105C22×Dic5C2414yes—
106C22×Dic5C2418yes—
107C2×Dic10—28no—
108C2×Dic10—216no—
109C2×Dic10—416no—
110C5×C22⋊C4C2×Q8132yescomplemented
111C42×C5—216no—
112C22⋊Dic5—216no—
113C4⋊Dic5—28no—
114C4⋊Dic5—216no—
115C4⋊Dic5—216no—
116C4⋊Dic5—432no—
117#683—216no—
118C23×Q8—51no—
119C24×C4—52no—
120C24×C4C2×D511yescharacteristic
121C2×C22⋊Q8—58no—
122C2×C22⋊Q8—108no—
123C22×C4⋊C4—52no—
124C22×C4⋊C4—52no—
125C22×C4⋊C4—58no—
126C22×C4⋊C4—52no—
127C22×C4⋊C4—51no—
128C22×C4⋊C4—58no—
129C22×C4⋊C4—108no—
130C22×C22⋊C4—52no—
131C22×C22⋊C4C2×D512yes—
132C22×C42C2×D512yes—
133#398—532no—
134#391—516no—
135#385—532no—
136#384—516no—
137C4×C22⋊C4C2×D518yes—
138C2×#95—58no—
139C2×#95C2×D512yes—
140C23×C5—28no—
141C23×C5—28no—
142C23×C5—232no—
143C23×C5—24no—
144C23×C5—24no—
145C23×C5C4⋊Q8116yes—
146C23×C5C22×D414yes—
147C23×C5C22×D412yes—
148C23×C5C22×Q818yes—
149C23×C5C22×Q814yes—
150C23×C5C2×#4114yes—
151C23×C5H(5,𝔽2)14yes—
152C23×C5C2511yescharacteristic
153C2×C4×C5—28no—
154C2×C4×C5—232no—
155C2×C4×C5—28no—
156C2×C4×C5—24no—
157C2×C4×C5—28no—
158C2×C4×C5—232no—
159C2×C4×C5#123116yes—
160C2×C4×C5C22×D414yes—
161C2×C4×C5C22×D414yes—
162C2×C4×C5C22×Q8116yes—
163C2×C4×C5C2×#4118yes—
164C2×Dic5—24no—
165C2×Dic5—24no—
166C2×Dic5—28no—
167C2×Dic5—28no—
168C2×Dic5—28no—
169C2×Dic5—216no—
170C2×Dic5—432no—
171C2×Dic5—48no—
172C2×Dic5—48no—
173C2×Dic5—416no—
174Dic10—416no—
175C25C22×D511yescharacteristic
176C22×Q8—54no—
177C22×Q8—104no—
178C22×Q8—108no—
179C23×C4—28no—
180C23×C4—52no—
181C23×C4—52no—
182C23×C4—54no—
183C23×C4—58no—
184C23×C4—108no—
185C23×C4—208no—
186C23×C4D2018yes—
187C23×C4C22×D514yes—
188C23×C4C22×D514yes—
189C23×C4C22×D511yescharacteristic
190C23×C4C22×D511yescharacteristic
191C22⋊Q8—1032no—
192C2×C4⋊C4—58no—
193C2×C4⋊C4—58no—
194C2×C4⋊C4—54no—
195C2×C4⋊C4—58no—
196C2×C4⋊C4—532no—
197C2×C4⋊C4—108no—
198C2×C4⋊C4—1032no—
199C2×C4⋊C4—108no—
200C2×C4⋊C4—108no—
201C2×C4⋊C4—104no—
202C2×C4⋊C4—2016no—
203C2×C22⋊C4—58no—
204C2×C22⋊C4—108no—
205C2×C22⋊C4Dic10116yescomplemented
206C2×C22⋊C4C22×D518yes—
207C2×C42—28no—
208C2×C42C22×D518yes—
209#95—532no—
210#95C22×D518yes—
211C22×C5—24no—
212C22×C5—24no—
213C22×C5—216no—
214C22×C5—232no—
215C22×C5C2×#12314yes—
216C22×C5C2×C4⋊Q818yes—
217C22×C5C2×C4⋊Q814yes—
218C22×C5#53514yes—
219C22×C5#554116yes—
220C22×C5C23×D411yescharacteristic
221C22×C5C23×Q812yes—
222C22×C5C22×#4112yes—
223C22×C5C2×H(5,𝔽2)12yes—
224C4×C5—24no—
225C4×C5—216no—
226C4×C5C2×#12318yes—
227Dic5—44no—
228Dic5—44no—
229Dic5—48no—
230C24—24no—
231C24—24no—
232C24C2×Dic1018yes—
233C24C2×D2014yes—
234C24C23×D512yes—
235C24C23×D511yescharacteristic
236C2×Q8—108no—
237C2×Q8—1016no—
238C2×Q8—2016no—
239C22×C4—28no—
240C22×C4—24no—
241C22×C4—28no—
242C22×C4—216no—
243C22×C4—232no—
244C22×C4—54no—
245C22×C4—58no—
246C22×C4—54no—
247C22×C4—104no—
248C22×C4—104no—
249C22×C4—1016no—
250C22×C4—108no—
251C22×C4—108no—
252C22×C4—108no—
253C22×C4—1016no—
254C22×C4—2016no—
255C22×C4—2032no—
256C22×C4C2×Dic10116yes—
257C22×C4C2×D2014yes—
258C22×C4C2×D20116yes—
259C22×C4C2×D2014yes—
260C22×C4#71118yes—
261C22×C4C23×D518yes—
262C22×C4C23×D512yes—
263C22×C4C23×D512yes—
264C4⋊C4—1016no—
265C4⋊C4—1016no—
266C4⋊C4—1016no—
267C4⋊C4—108no—
268C4⋊C4—2032no—
269C22⋊C4—1016no—
270C22⋊C4C2×Dic10132yescomplemented
271C42—216no—
272C2×C5—28no—
273C2×C5#250218yescomplemented
274C2×C5C22×#12311yescharacteristic
275C2×C5C22×C4⋊Q811yescharacteristic
276C2×C5C2×#53511yescharacteristic
277C2×C5C2×#55414yes—
278C23—28no—
279C23—28no—
280C23—232no—
281C23—24no—
282C23—24no—
283C23#4015116yes—
284C23C22×Dic1018yes—
285C23C22×Dic1014yes—
286C23C22×D2014yes—
287C23C22×D2012yes—
288C23C2×#71114yes—
289C23#414914yes—
290C23C24×D511yescharacteristic
291Q8—2016no—
292C2×C4—28no—
293C2×C4—232no—
294C2×C4—28no—
295C2×C4—24no—
296C2×C4—28no—
297C2×C4—232no—
298C2×C4—104no—
299C2×C4—104no—
300C2×C4—108no—
301C2×C4—108no—
302C2×C4—108no—
303C2×C4—1016no—
304C2×C4—2032no—
305C2×C4—208no—
306C2×C4—2016no—
307C2×C4—208no—
308C2×C4#4032116yes—
309C2×C4C22×Dic10116yes—
310C2×C4C22×D2014yes—
311C2×C4C22×D2014yes—
312C2×C4C2×#71118yes—
313C5C2×#250211yescharacteristic, complemented, Sylow
314C22—24no—
315C22—216no—
316C22—232no—
317C22—24no—
318C22C2×#401518yes—
319C22C2×#401514yes—
320C22C2×#403214yes—
321C22#6585914yes—
322C22#65877116yes—
323C22C23×Dic1012yes—
324C22C23×D2011yescharacteristic
325C22C22×#71112yes—
326C22C2×#414912yes—
327C4—216no—
328C4—24no—
329C4—208no—
330C4—204no—
331C4—204no—
332C4C2×#403218yes—
333C2—28no—
334C2#1060524818yesdirect product factor
335C2C22×#401511yescharacteristic
336C2C22×#403211yescharacteristic
337C2C2×#6585911yescharacteristic
338C2C2×#6587714yes—
subgroupG/H
centerC24C23×D5
commutator, cosocleC23×C5C25
FrattiniC23C24×D5
FittingC2×C4×C5×C22⋊C4C2
socleC24×C5C24
prime radicalC25×C5C23
hypercenterC2×C4×C22⋊C4D5
hypocenterC5C2×#2502
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
5. C2×#65218C2semidirect
6. C2×#65224C2semidirect
7. #10605248C2direct (N × H)
8. C2×#64950C2semidirect
28. #65224C22semidirect
29. #65218C22semidirect
30. #64950C22semidirect
48. C2×C5×C22⋊C4Q8semidirect
110. C5×C22⋊C4C2×Q8semidirect
205. C2×C22⋊C4Dic10semidirect
270. C22⋊C4C2×Dic10semidirect
273. C2×C5#2502semidirect
313. C5C2×#2502semidirect
334. C2#10605248direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#40702
C23×C4⋊Dic52
C2×C4×C5×C22⋊C42
C2×#652392
C2×#652182
C2×#652242
#106052482
C2×#649502
C2×#25025