Finite Group Databasebeta

← C2×#10610901 #12968903

Subgroups — C2×#10610901 #12968903

Order 1280.

1942 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65851C212yesmaximal
2C2×C5×#427C211yesmaximal, characteristic
3C2×#65395C211yesmaximal, characteristic
4C22×C8×Dic5C211yesmaximal, characteristic
5C2×#65287C212yesmaximal
6#10610901C2116yesmaximal, direct product factor
7C2×#65167C212yesmaximal, complemented
8C2×#65163C212yesmaximal, complemented
9C2×#65147C212yesmaximal, complemented
10C2×#65144C212yesmaximal, complemented
11C23×C5×C8C2211yescharacteristic
12C2×C5×#117C2212yes—
13C2×#4072C2212yes—
14C22×C4×Dic5C2211yescharacteristic
15C23×#168C2211yescharacteristic
16#65851—28no—
17#65851C22116yes—
18C2×#4036C2214yes—
19C2×#4034C2214yes—
20C2×#4024—24no—
21C2×C4×Dic10—24no—
22C5×#427C2218yes—
23C2×C5×#107C2212yes—
24C2×C5×#106C2212yes—
25#65395C2218yes—
26C2×#3950C2212yes—
27C2×#3949C2212yes—
28C2×C8×Dic5C2211yescharacteristic
29C2×C8×Dic5C2211yescharacteristic
30C2×C8×Dic5C2211yescharacteristic
31C2×C8×Dic5C2211yescharacteristic
32C2×C8×Dic5C2218yes—
33#65287C22116yes—
34C2×#3940C2214yes—
35C2×#3939C2214yes—
36#65167C22116yescomplemented
37#65163C22116yescomplemented
38#65147C22116yescomplemented
39#65144C22116yescomplemented
40C2×#3160—51nomaximal, Sylow
41C23×C4×C5C2311yescharacteristic
42C23×Dic5D412yes—
43C22×Dic10—24no—
44C22×C5×C8Q818yescomplemented
45C22×C5×C8C2311yescharacteristic
46C22×C5×C8C2311yescharacteristic
47C22×C5×C8C2314yes—
48C5×#117—24no—
49C5×#117C2318yes—
50C2×C5×C4⋊C4C2314yes—
51C2×C5×C22⋊C4—22no—
52C2×C42×C5—22no—
53C2×C22⋊Dic5—22no—
54#4072—24no—
55#4072C2318yes—
56C2×C4⋊Dic5—24no—
57C2×C4⋊Dic5C2314yes—
58C2×#683—24no—
59C2×#683—24no—
60C2×C4×Dic5—22no—
61C2×C4×Dic5D412yes—
62C2×C4×Dic5D414yes—
63C2×C4×Dic5C2311yescharacteristic
64C2×C4×Dic5C2311yescharacteristic
65C2×C4×Dic5C2314yes—
66C22×#168Q818yescomplemented
67C22×#168C2311yescharacteristic
68C22×#168C2311yescharacteristic
69C22×#168C2314yes—
70#4036—28no—
71#4036C23116yes—
72#4034—28no—
73#4034C23116yes—
74#4024—232no—
75C4×Dic10—232no—
76C5×#107C2318yes—
77C5×#106C2318yes—
78#3950C2318yes—
79#3949C2318yes—
80C8×Dic5C2314yes—
81C8×Dic5C2314yes—
82C8×Dic5C2314yes—
83C8×Dic5C2314yes—
84#3940C23116yes—
85#3939C23116yes—
86C2×#533—52no—
87#3160—516no—
88C2×#500—52no—
89C2×#499—52no—
90C2×#499—52no—
91C2×#498—52no—
92C2×#427—51no—
93C2×#427—52no—
94C2×#427D511yescharacteristic
95C22×C4×C8—51no—
96C24×C5C2×D411yescharacteristic
97C22×C4×C5—24no—
98C22×C4×C5C2×D412yes—
99C22×C4×C5C2×D411yescharacteristic
100C22×C4×C5C2×Q814yes—
101C22×C4×C5C2×Q814yes—
102C22×C4×C5C2411yescharacteristic
103C22×C4×C5C2412yes—
104C22×Dic5—28no—
105C22×Dic5—24no—
106C22×Dic5C2×D418yes—
107C22×Dic5C2×D412yes—
108C22×Dic5C2×D412yes—
109C2×Dic10—216no—
110C2×Dic10—416no—
111C2×C5×C8C2×Q814yes—
112C2×C5×C8C2×Q814yes—
113C2×C5×C8C2×Q8116yescomplemented
114C2×C5×C8C2412yes—
115C2×C5×C8C2412yes—
116C5×C4⋊C4—24no—
117C5×C4⋊C4C2418yes—
118C5×C22⋊C4—28no—
119C42×C5—28no—
120C22⋊Dic5—28no—
121C4⋊Dic5—24no—
122C4⋊Dic5—216no—
123C4⋊Dic5C2418yes—
124#683—216no—
125#683—216no—
126C4×Dic5—28no—
127C4×Dic5C2×D418yes—
128C4×Dic5C2×D412yes—
129C4×Dic5C2×D412yes—
130C4×Dic5C2412yes—
131C4×Dic5C2412yes—
132C2×#168C2×Q814yes—
133C2×#168C2×Q814yes—
134C2×#168C2×Q8116yescomplemented
135C2×#168C2412yes—
136C2×#168C2412yes—
137C23×C8—51no—
138C23×C8C2×D511yescharacteristic
139#533—516no—
140#533—108no—
141C2×C4⋊Q8—54no—
142C2×#125—54no—
143C2×C22⋊Q8—104no—
144C2×C4×Q8—104no—
145C2×#117—52no—
146C2×#117C2×D512yes—
147C22×C42—51no—
148#500—516no—
149#499—516no—
150#499—516no—
151#498—516no—
152#427—58no—
153#427—516no—
154#427C2×D518yes—
155C2×#107—52no—
156C2×#107—54no—
157C2×#107C2×D512yes—
158C2×#106—52no—
159C2×#106—54no—
160C2×#106C2×D512yes—
161C2×C4×C8—51no—
162C2×C4×C8—51no—
163C2×C4×C8—51no—
164C2×C4×C8—51no—
165C2×C4×C8—58no—
166C23×C5—24no—
167C23×C5C4⋊Q814yes—
168C23×C5C22×D412yes—
169C23×C5C22×D411yescharacteristic
170C2×C4×C5—28no—
171C2×C4×C5—48no—
172C2×C4×C5C4⋊Q818yes—
173C2×C4×C5C4⋊Q814yes—
174C2×C4×C5C22×D412yes—
175C2×C4×C5C22×D411yescharacteristic
176C2×C4×C5C22×Q812yes—
177C2×C4×C5C22×Q814yes—
178C2×C4×C5C22×Q812yes—
179C2×C4×C5C22×Q814yes—
180C2×C4×C5C2511yescharacteristic
181C2×Dic5—24no—
182C2×Dic5—216no—
183C2×Dic5—28no—
184C2×Dic5—48no—
185C2×Dic5#13518yes—
186C2×Dic5C22×D414yes—
187C2×Dic5C22×D414yes—
188Dic10—432no—
189C5×C8C22×Q814yes—
190C5×C8C22×Q814yes—
191#168C22×Q814yes—
192#168C22×Q814yes—
193C22×Q8—104no—
194C23×C4—52no—
195C23×C4C22×D511yescharacteristic
196C22×C8—58no—
197C22×C8—51no—
198C22×C8—51no—
199C22×C8—54no—
200C22×C8Dic1018yescomplemented
201C22×C8C22×D511yescharacteristic
202C22×C8C22×D511yescharacteristic
203C22×C8C22×D514yes—
204C4⋊Q8—516no—
205C4⋊Q8—108no—
206#125—516no—
207#125—108no—
208C22⋊Q8—1032no—
209C4×Q8—1032no—
210#117—24no—
211#117—58no—
212#117—104no—
213#117C22×D518yes—
214C2×C4⋊C4—54no—
215C2×C4⋊C4—104no—
216C2×C4⋊C4—104no—
217C2×C4⋊C4—104no—
218C2×C4⋊C4C22×D514yes—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—102no—
221C2×C42—22no—
222C2×C42—54no—
223C2×C42—52no—
224C2×C42—51no—
225C2×C42—51no—
226C2×C42—54no—
227C2×C42—102no—
228#107—58no—
229#107—516no—
230#107C22×D518yes—
231#106—58no—
232#106—516no—
233#106C22×D518yes—
234C4×C8—54no—
235C4×C8—54no—
236C4×C8—54no—
237C4×C8—54no—
238C22×C5—28no—
239C22×C5—24no—
240C22×C5C2×C4⋊Q814yes—
241C22×C5C2×C4⋊Q812yes—
242C22×C5C2×#13514yes—
243C22×C5C23×D411yescharacteristic
244C4×C5—48no—
245C4×C5C2×C4⋊Q814yes—
246C4×C5C2×C4⋊Q812yes—
247C4×C5C23×Q811yescharacteristic
248C4×C5C23×Q811yescharacteristic
249Dic5—24no—
250Dic5—48no—
251Dic5C2×#13518yes—
252C24D4×D511yescharacteristic
253C2×Q8—1016no—
254C2×Q8—2016no—
255C22×C4—24no—
256C22×C4—52no—
257C22×C4—52no—
258C22×C4—58no—
259C22×C4—108no—
260C22×C4—104no—
261C22×C4C2×Dic1014yes—
262C22×C4D4×D512yes—
263C22×C4D4×D511yescharacteristic
264C22×C4Q8×D514yes—
265C22×C4C23×D511yescharacteristic
266C22×C4C23×D512yes—
267C2×C8—54no—
268C2×C8—54no—
269C2×C8—516no—
270C2×C8—52no—
271C2×C8—52no—
272C2×C8C2×Dic1014yes—
273C2×C8C2×Dic1014yes—
274C2×C8C2×Dic10116yescomplemented
275C2×C8C23×D512yes—
276C2×C8C23×D512yes—
277C4⋊C4—24no—
278C4⋊C4—58no—
279C4⋊C4—1016no—
280C4⋊C4—1016no—
281C4⋊C4—1016no—
282C4⋊C4—104no—
283C4⋊C4C23×D518yes—
284C22⋊C4—28no—
285C22⋊C4—108no—
286C42—28no—
287C42—58no—
288C42—52no—
289C42—52no—
290C42—52no—
291C42—52no—
292C42—108no—
293C2×C5—24no—
294C2×C5#316014yescomplemented
295C2×C5C22×C4⋊Q811yescharacteristic
296C2×C5C22×#13512yes—
297C23—24no—
298C23#403414yes—
299C23C2×D4×D512yes—
300C23C2×D4×D511yescharacteristic
301Q8—2032no—
302C2×C4—28no—
303C2×C4—48no—
304C2×C4—58no—
305C2×C4—54no—
306C2×C4—54no—
307C2×C4—1016no—
308C2×C4—104no—
309C2×C4—108no—
310C2×C4—208no—
311C2×C4#403418yes—
312C2×C4#403414yes—
313C2×C4C22×Dic1012yes—
314C2×C4C22×Dic1014yes—
315C2×C4C2×D4×D512yes—
316C2×C4C2×D4×D511yescharacteristic
317C2×C4C2×Q8×D514yes—
318C2×C4C2×Q8×D512yes—
319C2×C4C24×D511yescharacteristic
320C8—54no—
321C8—54no—
322C8C22×Dic1014yes—
323C8C22×Dic1014yes—
324C5C2×#316011yescharacteristic, complemented, Sylow
325C22—24no—
326C22—28no—
327C22C2×#403414yes—
328C22C2×#403412yes—
329C22D5×#13514yes—
330C22C22×D4×D511yescharacteristic
331C4—48no—
332C4—58no—
333C4—104no—
334C4—208no—
335C4C2×#403412yes—
336C4C2×#403414yes—
337C4C23×Dic1011yescharacteristic
338C4C22×Q8×D511yescharacteristic
339C2—24no—
340C2#1061090114yesdirect product factor
341C2C22×#403411yescharacteristic
342C2C2×D5×#13512yes—
subgroupG/H
centerC22×C4D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#427C2
socleC23×C5C22×D4
prime radicalC24×C5C2×D4
hypercenterC2×#427D5
hypocenterC5C2×#3160
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×#427

Complements

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

NH ≅ G/Ntype
6. #10610901C2direct (N × H)
7. C2×#65167C2semidirect
8. C2×#65163C2semidirect
9. C2×#65147C2semidirect
10. C2×#65144C2semidirect
36. #65167C22semidirect
37. #65163C22semidirect
38. #65147C22semidirect
39. #65144C22semidirect
44. C22×C5×C8Q8semidirect
66. C22×#168Q8semidirect
113. C2×C5×C8C2×Q8semidirect
134. C2×#168C2×Q8semidirect
200. C22×C8Dic10semidirect
274. C2×C8C2×Dic10semidirect
294. C2×C5#3160semidirect
324. C5C2×#3160semidirect
340. C2#10610901direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#658512
C2×C5×#4272
C2×#653952
C22×C8×Dic52
C2×#652872
#106109012
C2×#651672
C2×#651632
C2×#651472
C2×#651442
C2×#31605