Finite Group Databasebeta

← C2×#10602161 #12087097

Subgroups — C2×#10602161 #12087097

Order 1280.

798 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#500C211yesmaximal, characteristic
2C2×#65367C211yesmaximal, characteristic
3C2×#65368C211yesmaximal, characteristic
4C2×#65381C211yesmaximal, characteristic
5C2×#65370C211yesmaximal, characteristic
6C2×#64970C211yesmaximal, characteristic
7#10602161C218yesmaximal, direct product factor
8C2×#64674C211yesmaximal, characteristic
9C2×C5×C4⋊Q8C2211yescharacteristic
10C2×C5×C4⋊Q8C2211yescharacteristic
11C22×#689—22no—
12C2×#4015C2211yescharacteristic
13C2×#4015C2211yescharacteristic
14C5×#500—22no—
15C5×#500C2214yes—
16C2×C5×#107—22no—
17C2×C4×C5×C8C2211yescharacteristic
18C2×#3967—22no—
19C2×#3948—22no—
20#65381—22no—
21#65381C2214yes—
22#65370—22no—
23#65370C2214yes—
24#65368—22no—
25#65368C2214yes—
26#65367—22no—
27#65367C2214yes—
28C2×#3940—22no—
29C2×#3939—22no—
30C2×C4×#168C2211yescharacteristic
31C2×C4×#168C2211yescharacteristic
32#64970—22no—
33#64970C2214yes—
34#64674C2214yes—
35C2×#1723—51nomaximal, Sylow
36C22×C5×Q8—21no—
37C22×C5×Q8—21no—
38C22×Dic10—21no—
39C22×Dic10—21no—
40C22×C5×C8D412yes—
41C5×C4⋊Q8—21no—
42C5×C4⋊Q8—21no—
43C5×C4⋊Q8—21no—
44C5×C4⋊Q8—21no—
45C5×C4⋊Q8—21no—
46C5×C4⋊Q8—21no—
47C5×C4⋊Q8C2312yes—
48C5×C4⋊Q8C2312yes—
49C2×C5×C4⋊C4—21no—
50C2×C5×C4⋊C4—21no—
51C2×C5×C4⋊C4—21no—
52C2×C5×C4⋊C4—21no—
53C2×C42×C5C2311yescharacteristic
54C2×#689—28no—
55C2×#689—48no—
56C2×C4⋊Dic5—21no—
57C2×C4⋊Dic5—21no—
58C2×C4⋊Dic5—21no—
59C2×C4⋊Dic5—21no—
60C22×#168D412yes—
61C22×#168D412yes—
62#4015—21no—
63#4015—21no—
64#4015—21no—
65#4015—21no—
66#4015—21no—
67#4015—21no—
68#4015C2312yes—
69#4015C2312yes—
70C5×#107—28no—
71C4×C5×C8D412yes—
72C4×C5×C8C2312yes—
73#3967—28no—
74#3948—28no—
75#3940—28no—
76#3939—28no—
77C4×#168D412yes—
78C4×#168D412yes—
79C4×#168C2312yes—
80C4×#168C2312yes—
81C2×#500—51no—
82C2×#500D511yescharacteristic
83C2×#498—51no—
84C2×#494—51no—
85C2×#487—51no—
86C2×#487—51no—
87#1723—58no—
88C2×#322—51no—
89C2×C5×Q8—22no—
90C2×C5×Q8—22no—
91C2×C5×Q8—41no—
92C2×C5×Q8—41no—
93C2×C5×Q8—44no—
94C2×C5×Q8—44no—
95C22×C4×C5—41no—
96C22×C4×C5—41no—
97C22×C4×C5C2×D411yescharacteristic
98C22×C4×C5C2×D411yescharacteristic
99C22×C4×C5C2×D411yescharacteristic
100C22×Dic5—41no—
101C22×Dic5—41no—
102C2×Dic10—22no—
103C2×Dic10—22no—
104C2×Dic10—44no—
105C2×Dic10—44no—
106C2×Dic10—41no—
107C2×Dic10—41no—
108C2×C5×C8—24no—
109C2×C5×C8C2×D414yes—
110C5×C4⋊C4—22no—
111C5×C4⋊C4—22no—
112C5×C4⋊C4—22no—
113C5×C4⋊C4—22no—
114C5×C4⋊C4—41no—
115C5×C4⋊C4—41no—
116C5×C4⋊C4—41no—
117C5×C4⋊C4—41no—
118C42×C5C2×D411yescharacteristic
119C42×C5C2×D411yescharacteristic
120C42×C5C2×D411yescharacteristic
121C42×C5C2411yescharacteristic
122#689—416no—
123C4⋊Dic5—22no—
124C4⋊Dic5—22no—
125C4⋊Dic5—22no—
126C4⋊Dic5—22no—
127C4⋊Dic5—41no—
128C4⋊Dic5—41no—
129C4⋊Dic5—41no—
130C4⋊Dic5—41no—
131C2×#168—24no—
132C2×#168Q1618yescomplemented
133C2×#168C2×D414yes—
134C2×#168C2×D414yes—
135C22×Q16—102no—
136C2×C4⋊Q8—51no—
137C2×C4⋊Q8—51no—
138C2×C4⋊Q8C2×D511yescharacteristic
139C2×C4⋊Q8C2×D511yescharacteristic
140#500—22no—
141#500—54no—
142#500—102no—
143#500C2×D514yes—
144#498—54no—
145#498—102no—
146#494—54no—
147#494—102no—
148#487—54no—
149#487—54no—
150#487—102no—
151#487—102no—
152C2×#107—22no—
153C2×#107—102no—
154C2×#106—102no—
155C2×#103—102no—
156C2×#103—102no—
157C2×C4×C8—51no—
158C2×C4×C8—51no—
159C2×C4×C8C2×D511yescharacteristic
160#322—54no—
161C23×C5#12711yescharacteristic
162C5×Q8—44no—
163C5×Q8—44no—
164C5×Q8—82no—
165C5×Q8—82no—
166C2×C4×C5—42no—
167C2×C4×C5—42no—
168C2×C4×C5—82no—
169C2×C4×C5—82no—
170C2×C4×C5#12711yescharacteristic
171C2×C4×C5#12711yescharacteristic
172C2×C4×C5#12711yescharacteristic
173C2×C4×C5C2×Q1614yes—
174C2×C4×C5AGL(1,ℤ/8)14yes—
175C2×C4×C5#13714yes—
176C2×C4×C5C22×D411yescharacteristic
177C2×C4×C5C22×D411yescharacteristic
178C2×C4×C5C22×D411yescharacteristic
179C2×Dic5—42no—
180C2×Dic5—42no—
181C2×Dic5—82no—
182C2×Dic5—82no—
183Dic10—44no—
184Dic10—44no—
185Dic10—82no—
186Dic10—82no—
187C5×C8—24no—
188#168—24no—
189#168C2×Q1618yescomplemented
190C22×Q8—21no—
191C22×Q8—21no—
192C22×Q8—101no—
193C22×Q8—101no—
194C2×Q16—108no—
195C2×Q16—208no—
196C22×C8—52no—
197C22×C8—52no—
198C22×C8C22⋊D512yes—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8—52no—
206C4⋊Q8—52no—
207C4⋊Q8—101no—
208C4⋊Q8—101no—
209C4⋊Q8—101no—
210C4⋊Q8—101no—
211C4⋊Q8—101no—
212C4⋊Q8—101no—
213C4⋊Q8C22×D512yes—
214C4⋊Q8C22×D512yes—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—101no—
220C2×C4⋊C4—101no—
221C2×C4⋊C4—101no—
222C2×C4⋊C4—101no—
223C2×C42C22×D511yescharacteristic
224#107—28no—
225#107—108no—
226#106—108no—
227#103—108no—
228#103—108no—
229C4×C8—52no—
230C4×C8—52no—
231C4×C8—52no—
232C4×C8—52no—
233C4×C8C22⋊D512yes—
234C4×C8C22×D512yes—
235C22×C5#49412yes—
236C22×C5#49612yes—
237C22×C5#49712yes—
238C22×C5C2×#12711yescharacteristic
239C4×C5—82no—
240C4×C5—82no—
241C4×C5#49412yes—
242C4×C5#49612yes—
243C4×C5#49712yes—
244C4×C5C22×Q1612yes—
245C4×C5C2×AGL(1,ℤ/8)12yes—
246C4×C5C2×#13712yes—
247Dic5—82no—
248Dic5—82no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—44no—
254C2×Q8—44no—
255C2×Q8—102no—
256C2×Q8—102no—
257C2×Q8—201no—
258C2×Q8—201no—
259C2×Q8—204no—
260C2×Q8—204no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—201no—
264C22×C4—201no—
265C22×C4D4×D511yescharacteristic
266C22×C4D4×D511yescharacteristic
267C22×C4C2×C22⋊D511yescharacteristic
268Q16—2016no—
269C2×C8—24no—
270C2×C8—54no—
271C2×C8—58no—
272C2×C8—54no—
273C2×C8—104no—
274C2×C8C2×C22⋊D514yes—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—102no—
284C4⋊C4—102no—
285C4⋊C4—102no—
286C4⋊C4—102no—
287C4⋊C4—201no—
288C4⋊C4—201no—
289C4⋊C4—201no—
290C4⋊C4—201no—
291C42D4×D511yescharacteristic
292C42D4×D511yescharacteristic
293C42C2×C22⋊D511yescharacteristic
294C42C23×D511yescharacteristic
295C2×C5#172314yescomplemented
296C2×C5C2×#49411yescharacteristic
297C2×C5C2×#49611yescharacteristic
298C2×C5C2×#49711yescharacteristic
299C23#408611yescharacteristic
300Q8—44no—
301Q8—44no—
302Q8—82no—
303Q8—82no—
304Q8—204no—
305Q8—204no—
306Q8—402no—
307Q8—402no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—202no—
313C2×C4—202no—
314C2×C4—402no—
315C2×C4—402no—
316C2×C4#405714yes—
317C2×C4D5×Q1614yes—
318C2×C4#408611yescharacteristic
319C2×C4#408611yescharacteristic
320C2×C4#408611yescharacteristic
321C2×C4#408814yes—
322C2×C4C2×D4×D511yescharacteristic
323C2×C4C2×D4×D511yescharacteristic
324C2×C4C22×C22⋊D511yescharacteristic
325C8—24no—
326C8—58no—
327C8—104no—
328C5C2×#172311yescharacteristic, complemented, Sylow
329C22#6537512yes—
330C22#6544312yes—
331C22#6547012yes—
332C22C2×#408611yescharacteristic
333C4—82no—
334C4—82no—
335C4—402no—
336C4—402no—
337C4#6537512yes—
338C4#6544312yes—
339C4#6547012yes—
340C4C2×#405712yes—
341C4C2×D5×Q1612yes—
342C4C2×#408812yes—
343C2#1060216114yesdirect product factor
344C2C2×#6537511yescharacteristic
345C2C2×#6544311yescharacteristic
346C2C2×#6547011yescharacteristic
subgroupG/H
centerC23#4086
commutator, cosocleC42×C5C24
FrattiniC42C23×D5
FittingC2×C5×#500C2
socle, prime radicalC23×C5#127
hypercenterC2×#500D5
hypocenterC5C2×#1723
LCSG ▷ C42×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#500

Complements

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

NH ≅ G/Ntype
7. #10602161C2direct (N × H)
132. C2×#168Q16semidirect
189. #168C2×Q16semidirect
295. C2×C5#1723semidirect
328. C5C2×#1723semidirect
343. C2#10602161direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5002
C2×#653672
C2×#653682
C2×#653812
C2×#653702
C2×#649702
#106021612
C2×#646742
C2×#17235