Finite Group Databasebeta

← C2×#10605682 #12266371

Subgroups — C2×#10605682 #12266371

Order 1280.

2590 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#4070C211yesmaximal, characteristic
2C22×#4072C211yesmaximal, characteristic
3C2×C5×#380C211yesmaximal, characteristic
4C2×#65263C212yesmaximal, complemented
5C2×#65274C212yesmaximal, complemented
6C2×#65242C214yesmaximal
7#10605682C2116yesmaximal, direct product factor
8C2×#64948C212yesmaximal, complemented
9C2×#64950C212yesmaximal, complemented
10C24×C4×C5C2211yescharacteristic
11C23×Dic10C2211yescharacteristic
12C22×C5×C4⋊C4C2212yes—
13C22×C5×C22⋊C4C2212yes—
14C22×C22⋊Dic5C2212yes—
15C22×C22⋊Dic5C2212yes—
16C2×#4072—28no—
17C2×#4072C2218yes—
18C22×C4⋊Dic5C2211yescharacteristic
19C22×C4⋊Dic5C2211yescharacteristic
20C22×C4⋊Dic5C2211yescharacteristic
21C22×C4⋊Dic5C2214yes—
22C2×#4070—28no—
23C2×#4070C2218yes—
24C22×#683C2212yes—
25C22×C4×Dic5C2212yes—
26C5×#380C2218yes—
27C2×C5×#95C2212yes—
28C2×#3963C2214yes—
29C2×#3963C2214yes—
30C2×#3963C2214yes—
31#65274C22116yescomplemented
32#65263C22116yescomplemented
33#65242C22132yes—
34#64950C22116yescomplemented
35#64948C22116yescomplemented
36C2×#2625—51nomaximal, Sylow
37C25×C5C2311yescharacteristic
38C23×C4×C5—28no—
39C23×C4×C5D418yes—
40C23×C4×C5C2314yes—
41C23×C4×C5C2311yescharacteristic
42C23×C4×C5C2314yes—
43C23×C4×C5C2311yescharacteristic
44C23×Dic5C2312yes—
45C23×Dic5C2312yes—
46C23×Dic5C2312yes—
47C23×Dic5C2312yes—
48C22×Dic10—24no—
49C22×Dic10—28no—
50C22×Dic10C2314yes—
51C2×C5×C4⋊C4—28no—
52C2×C5×C4⋊C4C2318yes—
53C2×C5×C22⋊C4—44no—
54C2×C5×C22⋊C4C2318yes—
55C2×C22⋊Dic5—28no—
56C2×C22⋊Dic5—28no—
57C2×C22⋊Dic5C2318yes—
58C2×C22⋊Dic5C2318yes—
59#4072—232no—
60C2×C4⋊Dic5—24no—
61C2×C4⋊Dic5—24no—
62C2×C4⋊Dic5—24no—
63C2×C4⋊Dic5—48no—
64C2×C4⋊Dic5C2314yes—
65C2×C4⋊Dic5C2314yes—
66C2×C4⋊Dic5C2314yes—
67C2×C4⋊Dic5C23116yes—
68#4070—232no—
69C2×#683—28no—
70C2×#683C2318yes—
71C2×C4×Dic5—28no—
72C2×C4×Dic5C2318yes—
73C5×#95C2318yes—
74#3963C23116yes—
75#3963C23116yes—
76#3963C23116yes—
77C22×C22⋊Q8—51no—
78C22×#117—51no—
79#2625—516no—
80C2×#400—52no—
81C2×#398—52no—
82C2×#397—54no—
83C2×#391—52no—
84C2×#389—52no—
85C2×#380D511yescharacteristic
86C24×C5—24no—
87C24×C5—22no—
88C24×C5—22no—
89C24×C5—24no—
90C24×C5C2×D414yes—
91C24×C5C2412yes—
92C24×C5C2411yescharacteristic
93C22×C4×C5—24no—
94C22×C4×C5—232no—
95C22×C4×C5—216no—
96C22×C4×C5—28no—
97C22×C4×C5—28no—
98C22×C4×C5—28no—
99C22×C4×C5C2×D414yes—
100C22×C4×C5C2×D4116yes—
101C22×C4×C5C2×D414yes—
102C22×C4×C5#4118yes—
103C22×C4×C5C2418yes—
104C22×C4×C5C2412yes—
105C22×C4×C5C2412yes—
106C22×Dic5—24no—
107C22×Dic5—24no—
108C22×Dic5—24no—
109C22×Dic5—24no—
110C22×Dic5—28no—
111C22×Dic5—24no—
112C22×Dic5—24no—
113C22×Dic5—24no—
114C22×Dic5—24no—
115C22×Dic5—28no—
116C22×Dic5C2414yes—
117C22×Dic5C2414yes—
118C22×Dic5C2414yes—
119C22×Dic5C2414yes—
120C2×Dic10—28no—
121C2×Dic10—216no—
122C2×Dic10—416no—
123C5×C4⋊C4—216no—
124C5×C22⋊C4—48no—
125C22⋊Dic5—216no—
126C22⋊Dic5—216no—
127C4⋊Dic5—28no—
128C4⋊Dic5—28no—
129C4⋊Dic5—28no—
130C4⋊Dic5—416no—
131#683—216no—
132C4×Dic5—216no—
133C23×Q8—51no—
134C24×C4C2×D511yescharacteristic
135C2×C22⋊Q8—58no—
136C2×C22⋊Q8—108no—
137C2×#117—58no—
138C2×#117—108no—
139C22×C4⋊C4—52no—
140C22×C4⋊C4—51no—
141C22×C4⋊C4—51no—
142C22×C4⋊C4—51no—
143C22×C4⋊C4—54no—
144C22×C4⋊C4C2×D512yes—
145C22×C22⋊C4—52no—
146C22×C22⋊C4—52no—
147C22×C22⋊C4C2×D512yes—
148C22×C42—52no—
149#400—516no—
150#398—516no—
151#397—532no—
152#391—516no—
153#389—516no—
154#380C2×D518yes—
155C2×#95—54no—
156C2×#95—54no—
157C2×#95—54no—
158C2×#95C2×D512yes—
159C23×C5—28no—
160C23×C5—28no—
161C23×C5—28no—
162C23×C5—24no—
163C23×C5—22no—
164C23×C5—22no—
165C23×C5—24no—
166C23×C5—48no—
167C23×C5—48no—
168C23×C5C22×D414yes—
169C23×C5C22×D412yes—
170C23×C5C2×#4114yes—
171C23×C5H(5,𝔽2)12yes—
172C23×C52−1+412yes—
173C23×C52−1+414yes—
174C23×C5C2511yescharacteristic
175C2×C4×C5—28no—
176C2×C4×C5—24no—
177C2×C4×C5—232no—
178C2×C4×C5—28no—
179C2×C4×C5—28no—
180C2×C4×C5—28no—
181C2×C4×C5—416no—
182C2×C4×C5#123116yes—
183C2×C4×C5C22×D414yes—
184C2×C4×C5C22×D414yes—
185C2×C4×C5C2×#4118yes—
186C2×Dic5—24no—
187C2×Dic5—24no—
188C2×Dic5—24no—
189C2×Dic5—24no—
190C2×Dic5—28no—
191C2×Dic5—24no—
192C2×Dic5—24no—
193C2×Dic5—24no—
194C2×Dic5—24no—
195C2×Dic5—28no—
196C2×Dic5—48no—
197C2×Dic5—48no—
198C2×Dic5—48no—
199C2×Dic5—48no—
200Dic10—416no—
201C25C22×D511yescharacteristic
202C22×Q8—54no—
203C22×Q8—104no—
204C22×Q8—108no—
205C23×C4—28no—
206C23×C4—52no—
207C23×C4—52no—
208C23×C4—52no—
209C23×C4—52no—
210C23×C4D2018yes—
211C23×C4C22×D514yes—
212C23×C4C22×D514yes—
213C23×C4C22×D511yescharacteristic
214C23×C4C22×D511yescharacteristic
215C22⋊Q8—1032no—
216#117—1032no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—54no—
219C2×C4⋊C4—54no—
220C2×C4⋊C4—54no—
221C2×C4⋊C4—58no—
222C2×C4⋊C4—516no—
223C2×C4⋊C4—104no—
224C2×C4⋊C4—104no—
225C2×C4⋊C4—108no—
226C2×C4⋊C4—104no—
227C2×C4⋊C4—208no—
228C2×C4⋊C4C22×D518yes—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—58no—
231C2×C22⋊C4—58no—
232C2×C22⋊C4—108no—
233C2×C22⋊C4—108no—
234C2×C22⋊C4C22×D518yes—
235C2×C42—58no—
236C2×C42—108no—
237#95—516no—
238#95—516no—
239#95—516no—
240#95C22×D518yes—
241C22×C5—24no—
242C22×C5—24no—
243C22×C5—24no—
244C22×C5—416no—
245C22×C5—44no—
246C22×C5—44no—
247C22×C5C2×#12314yes—
248C22×C5#53614yes—
249C22×C5#53714yes—
250C22×C5#54118yes—
251C22×C5#54218yes—
252C22×C5C23×D411yescharacteristic
253C22×C5C22×#4112yes—
254C22×C5C2×H(5,𝔽2)11yescharacteristic
255C22×C5C2×2−1+411yescharacteristic
256C22×C5C2×2−1+412yes—
257C4×C5—24no—
258C4×C5—48no—
259C4×C5C2×#12318yes—
260Dic5—44no—
261Dic5—44no—
262Dic5—44no—
263Dic5—44no—
264C24—24no—
265C24—22no—
266C24—22no—
267C24—24no—
268C24C2×D2014yes—
269C24C23×D512yes—
270C24C23×D511yescharacteristic
271C2×Q8—108no—
272C2×Q8—1016no—
273C2×Q8—2016no—
274C22×C4—24no—
275C22×C4—216no—
276C22×C4—232no—
277C22×C4—28no—
278C22×C4—28no—
279C22×C4—28no—
280C22×C4—54no—
281C22×C4—54no—
282C22×C4—54no—
283C22×C4—54no—
284C22×C4—104no—
285C22×C4—104no—
286C22×C4—104no—
287C22×C4—104no—
288C22×C4—108no—
289C22×C4—104no—
290C22×C4—104no—
291C22×C4—104no—
292C22×C4—104no—
293C22×C4—108no—
294C22×C4C2×D2014yes—
295C22×C4C2×D20116yes—
296C22×C4C2×D2014yes—
297C22×C4#71118yes—
298C22×C4C23×D518yes—
299C22×C4C23×D512yes—
300C22×C4C23×D512yes—
301C4⋊C4—216no—
302C4⋊C4—108no—
303C4⋊C4—108no—
304C4⋊C4—1016no—
305C4⋊C4—108no—
306C4⋊C4—2016no—
307C22⋊C4—48no—
308C22⋊C4—1016no—
309C22⋊C4—1016no—
310C42—1016no—
311C2×C5—44no—
312C2×C5#262518yescomplemented
313C2×C5C22×#12311yescharacteristic
314C2×C5C2×#53611yescharacteristic
315C2×C5C2×#53711yescharacteristic
316C2×C5C2×#54112yes—
317C2×C5C2×#54212yes—
318C23—28no—
319C23—28no—
320C23—28no—
321C23—24no—
322C23—22no—
323C23—22no—
324C23—24no—
325C23—48no—
326C23—48no—
327C23C22×D2014yes—
328C23C22×D2012yes—
329C23C2×#71114yes—
330C23#414412yes—
331C23#414712yes—
332C23#415014yes—
333C23C24×D511yescharacteristic
334Q8—2016no—
335C2×C4—28no—
336C2×C4—24no—
337C2×C4—232no—
338C2×C4—28no—
339C2×C4—28no—
340C2×C4—28no—
341C2×C4—416no—
342C2×C4—104no—
343C2×C4—104no—
344C2×C4—104no—
345C2×C4—104no—
346C2×C4—108no—
347C2×C4—104no—
348C2×C4—104no—
349C2×C4—104no—
350C2×C4—104no—
351C2×C4—108no—
352C2×C4—208no—
353C2×C4—208no—
354C2×C4—208no—
355C2×C4—208no—
356C2×C4#4032116yes—
357C2×C4C22×D2014yes—
358C2×C4C22×D2014yes—
359C2×C4C2×#71118yes—
360C5C2×#262511yescharacteristic, complemented, Sylow
361C22—24no—
362C22—24no—
363C22—24no—
364C22—44no—
365C22—416no—
366C22—44no—
367C22C2×#403214yes—
368C22#6584414yes—
369C22#6586014yes—
370C22#6593718yes—
371C22#6595918yes—
372C22C23×D2011yescharacteristic
373C22C22×#71112yes—
374C22C2×#414411yescharacteristic
375C22C2×#414711yescharacteristic
376C22C2×#415012yes—
377C4—24no—
378C4—48no—
379C4—204no—
380C4—204no—
381C4—204no—
382C4—204no—
383C4C2×#403218yes—
384C2—44no—
385C2#1060568218yesdirect product factor
386C2C22×#403211yescharacteristic
387C2C2×#6584411yescharacteristic
388C2C2×#6586011yescharacteristic
389C2C2×#6593712yes—
390C2C2×#6595912yes—
subgroupG/H
centerC24C23×D5
commutator, cosocleC23×C5C25
FrattiniC23C24×D5
FittingC2×C5×#380C2
socleC24×C5C24
prime radicalC25×C5C23
hypercenterC2×#380D5
hypocenterC5C2×#2625
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
4. C2×#65263C2semidirect
5. C2×#65274C2semidirect
7. #10605682C2direct (N × H)
8. C2×#64948C2semidirect
9. C2×#64950C2semidirect
31. #65274C22semidirect
32. #65263C22semidirect
34. #64950C22semidirect
35. #64948C22semidirect
312. C2×C5#2625semidirect
360. C5C2×#2625semidirect
385. C2#10605682direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#40702
C22×#40722
C2×C5×#3802
C2×#652632
C2×#652742
C2×#652422
#106056822
C2×#649482
C2×#649502
C2×#26255