Finite Group Databasebeta

← C2×#10605249 #12268593

Subgroups — C2×#10605249 #12268593

Order 1280.

2918 conjugacy classes of subgroups.

The subgroup lattice has 336 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×#65232C214yesmaximal
5C2×#65218C212yesmaximal, complemented
6C2×#65225C212yesmaximal, complemented
7#10605249C2116yesmaximal, direct product factor
8C2×#64948C214yesmaximal, 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—
20C2×#4070—28no—
21C2×#4070C2218yes—
22C22×#683C2212yes—
23C4×C5×C22⋊C4C2218yes—
24C2×C5×#95C2212yes—
25C2×#3963C2218yes—
26C2×#3963C2218yes—
27#65232C22132yes—
28#65225C22116yescomplemented
29#65218C22116yescomplemented
30#64948C22132yescomplemented
31C2×#2503—51nomaximal, Sylow
32C25×C5C2311yescharacteristic
33C23×C4×C5—28no—
34C23×C4×C5D418yes—
35C23×C4×C5C2314yes—
36C23×C4×C5C2311yescharacteristic
37C23×C4×C5C2314yes—
38C23×C4×C5C2311yescharacteristic
39C23×Dic5—28no—
40C23×Dic5—216no—
41C23×Dic5C2312yes—
42C23×Dic5C2312yes—
43C23×Dic5C2318yes—
44C23×Dic5C2314yes—
45C22×Dic10—24no—
46C22×Dic10—28no—
47C22×Dic10C2314yes—
48C2×C5×C22⋊C4—28no—
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⋊Dic5C2314yes—
59C2×C4⋊Dic5C2318yes—
60C2×C4⋊Dic5C2318yes—
61#4070—232no—
62C2×#683—28no—
63C2×#683C2318yes—
64C5×#95C2318yes—
65#3963C23132yes—
66#3963C23132yes—
67C22×C22⋊Q8—51no—
68C23×C4⋊C4—51no—
69#2503—516no—
70C2×#400—54no—
71C2×#391—52no—
72C2×#382—52no—
73C2×#381—54no—
74C2×C4×C22⋊C4D511yescharacteristic
75C24×C5—24no—
76C24×C5—24no—
77C24×C5C2×D414yes—
78C24×C5#4118yes—
79C24×C5C2412yes—
80C24×C5C2411yescharacteristic
81C22×C4×C5—24no—
82C22×C4×C5—232no—
83C22×C4×C5—216no—
84C22×C4×C5—28no—
85C22×C4×C5—28no—
86C22×C4×C5—28no—
87C22×C4×C5C2×D414yes—
88C22×C4×C5C2×D4116yes—
89C22×C4×C5C2×D414yes—
90C22×C4×C5#4118yes—
91C22×C4×C5C2418yes—
92C22×C4×C5C2412yes—
93C22×C4×C5C2412yes—
94C22×Dic5—24no—
95C22×Dic5—24no—
96C22×Dic5—216no—
97C22×Dic5—28no—
98C22×Dic5—28no—
99C22×Dic5—232no—
100C22×Dic5—216no—
101C22×Dic5—28no—
102C22×Dic5—432no—
103C22×Dic5C2414yes—
104C22×Dic5C2414yes—
105C22×Dic5C2418yes—
106C2×Dic10—28no—
107C2×Dic10—216no—
108C2×Dic10—416no—
109C5×C22⋊C4—216no—
110C42×C5—216no—
111C22⋊Dic5—216no—
112C4⋊Dic5—28no—
113C4⋊Dic5—216no—
114C4⋊Dic5—216no—
115#683—216no—
116C23×Q8—51no—
117C24×C4—52no—
118C24×C4C2×D511yescharacteristic
119C2×C22⋊Q8—58no—
120C2×C22⋊Q8—108no—
121C22×C4⋊C4—52no—
122C22×C4⋊C4—52no—
123C22×C4⋊C4—58no—
124C22×C4⋊C4—52no—
125C22×C4⋊C4—51no—
126C22×C4⋊C4—108no—
127C22×C22⋊C4—52no—
128C22×C22⋊C4C2×D512yes—
129C22×C42C2×D512yes—
130#400—532no—
131#391—516no—
132#382—516no—
133#381—532no—
134C4×C22⋊C4C2×D518yes—
135C2×#95—58no—
136C2×#95—58no—
137C2×#95C2×D512yes—
138C23×C5—28no—
139C23×C5—28no—
140C23×C5—232no—
141C23×C5—24no—
142C23×C5—24no—
143C23×C5#124116yes—
144C23×C5C22×D414yes—
145C23×C5C22×D412yes—
146C23×C5C2×#4118yes—
147C23×C5C2×#4114yes—
148C23×C5C2×#4114yes—
149C23×C52−1+414yes—
150C23×C5C2511yescharacteristic
151C2×C4×C5—28no—
152C2×C4×C5—24no—
153C2×C4×C5—232no—
154C2×C4×C5—28no—
155C2×C4×C5—28no—
156C2×C4×C5—28no—
157C2×C4×C5—416no—
158C2×C4×C5#123116yes—
159C2×C4×C5C22×D414yes—
160C2×C4×C5C22×D414yes—
161C2×C4×C5C2×#4118yes—
162C2×Dic5—24no—
163C2×Dic5—24no—
164C2×Dic5—28no—
165C2×Dic5—28no—
166C2×Dic5—216no—
167C2×Dic5—28no—
168C2×Dic5—48no—
169C2×Dic5—432no—
170C2×Dic5—416no—
171C2×Dic5—48no—
172Dic10—416no—
173C25C22×D511yescharacteristic
174C22×Q8—54no—
175C22×Q8—104no—
176C22×Q8—108no—
177C23×C4—28no—
178C23×C4—58no—
179C23×C4—52no—
180C23×C4—52no—
181C23×C4—54no—
182C23×C4—108no—
183C23×C4—1016no—
184C23×C4D2018yes—
185C23×C4C22×D514yes—
186C23×C4C22×D514yes—
187C23×C4C22×D511yescharacteristic
188C23×C4C22×D511yescharacteristic
189C22⋊Q8—1032no—
190C2×C4⋊C4—58no—
191C2×C4⋊C4—58no—
192C2×C4⋊C4—54no—
193C2×C4⋊C4—58no—
194C2×C4⋊C4—108no—
195C2×C4⋊C4—1032no—
196C2×C4⋊C4—108no—
197C2×C4⋊C4—108no—
198C2×C4⋊C4—104no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—58no—
201C2×C22⋊C4—108no—
202C2×C22⋊C4C22×D518yes—
203C2×C42—28no—
204C2×C42C22×D518yes—
205#95—532no—
206#95—532no—
207#95C22×D518yes—
208C22×C5—24no—
209C22×C5—24no—
210C22×C5—216no—
211C22×C5—232no—
212C22×C5C2×#12314yes—
213C22×C5C2×#12418yes—
214C22×C5C2×#12414yes—
215C22×C5#53614yes—
216C22×C5#552116yes—
217C22×C5C23×D411yescharacteristic
218C22×C5C22×#4112yes—
219C22×C5C22×#4112yes—
220C22×C5C2×2−1+412yes—
221C4×C5—24no—
222C4×C5—48no—
223C4×C5C2×#12318yes—
224Dic5—44no—
225Dic5—48no—
226Dic5—44no—
227C24—24no—
228C24—24no—
229C24C2×D2014yes—
230C24#70918yes—
231C24C23×D512yes—
232C24C23×D511yescharacteristic
233C2×Q8—108no—
234C2×Q8—1016no—
235C2×Q8—2016no—
236C22×C4—24no—
237C22×C4—216no—
238C22×C4—232no—
239C22×C4—28no—
240C22×C4—28no—
241C22×C4—28no—
242C22×C4—54no—
243C22×C4—54no—
244C22×C4—58no—
245C22×C4—104no—
246C22×C4—104no—
247C22×C4—1016no—
248C22×C4—108no—
249C22×C4—1032no—
250C22×C4—1016no—
251C22×C4—108no—
252C22×C4—108no—
253C22×C4—2032no—
254C22×C4C2×D2014yes—
255C22×C4C2×D20116yes—
256C22×C4C2×D2014yes—
257C22×C4#71118yes—
258C22×C4C23×D518yes—
259C22×C4C23×D512yes—
260C22×C4C23×D512yes—
261C4⋊C4—1016no—
262C4⋊C4—1016no—
263C4⋊C4—1016no—
264C4⋊C4—108no—
265C22⋊C4—216no—
266C22⋊C4—1016no—
267C42—216no—
268C2×C5—28no—
269C2×C5#250318yescomplemented
270C2×C5C22×#12311yescharacteristic
271C2×C5C22×#12411yescharacteristic
272C2×C5C2×#53611yescharacteristic
273C2×C5C2×#55214yes—
274C23—28no—
275C23—28no—
276C23—232no—
277C23—24no—
278C23—24no—
279C23#4021116yes—
280C23C22×D2014yes—
281C23C22×D2012yes—
282C23C2×#70918yes—
283C23C2×#70914yes—
284C23C2×#71114yes—
285C23#415014yes—
286C23C24×D511yescharacteristic
287Q8—2016no—
288C2×C4—28no—
289C2×C4—24no—
290C2×C4—232no—
291C2×C4—28no—
292C2×C4—28no—
293C2×C4—28no—
294C2×C4—416no—
295C2×C4—104no—
296C2×C4—104no—
297C2×C4—108no—
298C2×C4—1016no—
299C2×C4—108no—
300C2×C4—108no—
301C2×C4—2032no—
302C2×C4—2016no—
303C2×C4—208no—
304C2×C4—208no—
305C2×C4#4032116yes—
306C2×C4C22×D2014yes—
307C2×C4C22×D2014yes—
308C2×C4C2×#71118yes—
309C5C2×#250311yescharacteristic, complemented, Sylow
310C22—24no—
311C22—216no—
312C22—232no—
313C22—24no—
314C22C2×#402118yes—
315C22C2×#402114yes—
316C22C2×#403214yes—
317C22#6586014yes—
318C22#65875116yes—
319C22C23×D2011yescharacteristic
320C22C22×#70912yes—
321C22C22×#71112yes—
322C22C2×#415012yes—
323C4—24no—
324C4—48no—
325C4—204no—
326C4—208no—
327C4—204no—
328C4C2×#403218yes—
329C2—28no—
330C2#1060524918yesdirect product factor
331C2C22×#402111yescharacteristic
332C2C22×#403211yescharacteristic
333C2C2×#6586011yescharacteristic
334C2C2×#6587514yes—
subgroupG/H
centerC24C23×D5
commutator, cosocleC23×C5C25
FrattiniC23C24×D5
FittingC2×C4×C5×C22⋊C4C2
socleC24×C5C24
prime radicalC25×C5C23
hypercenterC2×C4×C22⋊C4D5
hypocenterC5C2×#2503
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×#65225C2semidirect
7. #10605249C2direct (N × H)
8. C2×#64948C2semidirect
28. #65225C22semidirect
29. #65218C22semidirect
30. #64948C22semidirect
269. C2×C5#2503semidirect
309. C5C2×#2503semidirect
330. C2#10605249direct (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×#652322
C2×#652182
C2×#652252
#106052492
C2×#649482
C2×#25035