Finite Group Databasebeta

← C2×#10600359 #12066564

Subgroups — C2×#10600359 #12066564

Order 1280.

1022 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65804C211yesmaximal, characteristic
2C2×C5×#383C211yesmaximal, characteristic
3C2×#65215C211yesmaximal, characteristic
4#10600359C218yesmaximal, direct product factor
5C2×#64672C214yesmaximal
6C22×C42×C5C2211yescharacteristic
7C2×#4072—21no—
8C2×#4070—21no—
9C2×#4070C2212yes—
10#65804—22no—
11#65804—22no—
12#65804C414yescomplemented
13#65804C2214yes—
14C2×#4016—21no—
15C2×#4015C411yescharacteristic, complemented
16C2×#4015C411yescharacteristic, complemented
17C2×C4×Dic10—21no—
18C2×C4×Dic10C412yescomplemented
19C5×#383—22no—
20C5×#383C2214yes—
21C2×C5×#95—21no—
22C2×C5×#95—21no—
23C2×C5×#95C2212yes—
24C2×C22⋊#168C2212yes—
25#65215—22no—
26#65215C2214yes—
27C2×C4⋊#168—21no—
28C2×#3935—21no—
29#64672C22116yes—
30C2×#1515—51nomaximal, Sylow
31C23×C4×C5—22no—
32C23×C4×C5D411yescharacteristic
33C23×C4×C5D411yescharacteristic
34C23×C4×C5C2311yescharacteristic
35C22×Dic10—21no—
36C22×Dic10C2×C412yes—
37C2×C42×C5C2×C411yescharacteristic
38C2×C42×C5C2×C411yescharacteristic
39C2×C42×C5C2×C412yes—
40C2×C42×C5D411yescharacteristic
41C2×C42×C5D411yescharacteristic
42C2×C42×C5D412yes—
43C2×C42×C5D412yes—
44C2×C42×C5C2312yes—
45C2×C22⋊Dic5—22no—
46#4072—24no—
47#4072—24no—
48C2×C4⋊Dic5—21no—
49C2×C4⋊Dic5—21no—
50C2×C4⋊Dic5—21no—
51C2×C4⋊Dic5C2×C412yes—
52#4070—24no—
53#4070—22no—
54#4070—22no—
55#4070—24no—
56#4070C2×C414yes—
57#4070C2314yes—
58C2×#683—22no—
59C2×#683—22no—
60C2×C4×Dic5—22no—
61C22×#168—22no—
62#4016—24no—
63#4016—24no—
64#4015—22no—
65#4015—22no—
66#4015C2×C414yescomplemented
67#4015C2×C414yescomplemented
68C4×Dic10—24no—
69C4×Dic10—24no—
70C4×Dic10—24no—
71C4×Dic10C2×C418yescomplemented
72C5×#95—24no—
73C5×#95—24no—
74C5×#95—22no—
75C5×#95C2314yes—
76C22⋊#168—22no—
77C22⋊#168C2314yes—
78C4⋊#168—24no—
79#3935—24no—
80C2×#533—51no—
81C2×#432—51no—
82C2×#383D511yescharacteristic
83#1515—58no—
84C2×#328—54no—
85C24×C5C2×D411yescharacteristic
86C22×C4×C5—24no—
87C22×C4×C5—24no—
88C22×C4×C5—42no—
89C22×C4×C5—42no—
90C22×C4×C5—44no—
91C22×C4×C5—44no—
92C22×C4×C5—44no—
93C22×C4×C5C22⋊C411yescharacteristic
94C22×C4×C5C22⋊C411yescharacteristic
95C22×C4×C5C22⋊C411yescharacteristic
96C22×C4×C5C22⋊C411yescharacteristic
97C22×C4×C5C22⋊C412yes—
98C22×C4×C5C22⋊C412yes—
99C22×C4×C5C22×C411yescharacteristic
100C22×C4×C5C22×C411yescharacteristic
101C22×C4×C5C2×D411yescharacteristic
102C22×C4×C5C2×D411yescharacteristic
103C22×C4×C5C2×D412yes—
104C22×C4×C5C2×D411yescharacteristic
105C22×C4×C5C2×D412yes—
106C22×C4×C5C2411yescharacteristic
107C22×Dic5—22no—
108C22×Dic5—22no—
109C2×Dic10—24no—
110C2×Dic10—22no—
111C2×Dic10—44no—
112C2×Dic10—44no—
113C2×Dic10C22×C414yes—
114C42×C5C22⋊C412yes—
115C42×C5C22⋊C412yes—
116C42×C5C22⋊C412yes—
117C42×C5C22⋊C412yes—
118C42×C5C22×C412yes—
119C42×C5C22×C412yes—
120C42×C5C2×D412yes—
121C42×C5C2×D412yes—
122C22⋊Dic5—24no—
123C22⋊Dic5—24no—
124C4⋊Dic5—22no—
125C4⋊Dic5—24no—
126C4⋊Dic5—24no—
127C4⋊Dic5—24no—
128C4⋊Dic5C22×C414yes—
129#683—24no—
130#683—28no—
131#683—24no—
132C4×Dic5—28no—
133C2×#168—24no—
134C2×#168—44no—
135#533—54no—
136#533—54no—
137#533—102no—
138#533—102no—
139C2×C4⋊Q8—51no—
140C2×C4⋊Q8—51no—
141C2×#125—101no—
142C2×C22⋊Q8—52no—
143C2×C22⋊Q8—101no—
144C2×C4×Q8—52no—
145C2×C4×Q8—101no—
146C2×#117—101no—
147C22×C42C2×D511yescharacteristic
148#432—54no—
149#432—102no—
150C2×C4⋊C8—101no—
151C2×C22⋊C8—52no—
152C2×#97—101no—
153#383—22no—
154#383C2×D514yes—
155C2×#95—21no—
156C2×#95—21no—
157C2×#95C2×D512yes—
158#328—516no—
159C23×C5—22no—
160C23×C5—22no—
161C23×C5C2×C22⋊C411yescharacteristic
162C23×C5C2×C22⋊C411yescharacteristic
163C23×C5#13714yes—
164C23×C5C22×D411yescharacteristic
165C2×C4×C5—24no—
166C2×C4×C5—22no—
167C2×C4×C5—22no—
168C2×C4×C5—24no—
169C2×C4×C5—22no—
170C2×C4×C5—22no—
171C2×C4×C5—22no—
172C2×C4×C5—22no—
173C2×C4×C5—42no—
174C2×C4×C5—41no—
175C2×C4×C5—41no—
176C2×C4×C5—41no—
177C2×C4×C5—42no—
178C2×C4×C5—41no—
179C2×C4×C5—44no—
180C2×C4×C5—44no—
181C2×C4×C5—44no—
182C2×C4×C5—88no—
183C2×C4×C5C2×C22⋊C412yes—
184C2×C4×C5C2×C22⋊C411yescharacteristic
185C2×C4×C5C2×C22⋊C412yes—
186C2×C4×C5C2×C22⋊C412yes—
187C2×C4×C5C2×C22⋊C411yescharacteristic
188C2×C4×C5C2×C22⋊C412yes—
189C2×C4×C5C23×C411yescharacteristic
190C2×C4×C5C22×D411yescharacteristic
191C2×Dic5—24no—
192C2×Dic5—24no—
193C2×Dic5—44no—
194C2×Dic5—44no—
195Dic10—44no—
196Dic10—48no—
197Dic10—44no—
198#168—44no—
199C22×Q8—52no—
200C22×Q8—101no—
201C23×C4—22no—
202C23×C4C22⋊D511yescharacteristic
203C23×C4C22⋊D511yescharacteristic
204C23×C4C22×D511yescharacteristic
205C22×C8—102no—
206C4⋊Q8—54no—
207C4⋊Q8—54no—
208C4⋊Q8—102no—
209C4⋊Q8—102no—
210#125—104no—
211#125—104no—
212C22⋊Q8—54no—
213C22⋊Q8—54no—
214C22⋊Q8—104no—
215C22⋊Q8—102no—
216C22⋊Q8—102no—
217C22⋊Q8—104no—
218C4×Q8—58no—
219C4×Q8—104no—
220C4×Q8—104no—
221C4×Q8—104no—
222#117—104no—
223#117—104no—
224C2×C4⋊C4—52no—
225C2×C4⋊C4—101no—
226C2×C4⋊C4—101no—
227C2×C4⋊C4—102no—
228C2×C4⋊C4—101no—
229C2×C4⋊C4—102no—
230C2×C22⋊C4—102no—
231C2×C42—102no—
232C2×C42C4×D511yescharacteristic
233C2×C42C4×D511yescharacteristic
234C2×C42C4×D512yes—
235C2×C42D2011yescharacteristic
236C2×C42D2011yescharacteristic
237C2×C42D2012yes—
238C2×C42C22⋊D512yes—
239C2×C42C22×D512yes—
240C4⋊C8—104no—
241C22⋊C8—54no—
242C22⋊C8—102no—
243#97—104no—
244#95—24no—
245#95—24no—
246#95—22no—
247#95C22×D514yes—
248C22×C5—21no—
249C22×C5—21no—
250C22×C5—21no—
251C22×C5—21no—
252C22×C5—28no—
253C22×C5#41012yes—
254C22×C5#41912yes—
255C22×C5#41912yes—
256C22×C5#42112yes—
257C22×C5C22×C22⋊C411yescharacteristic
258C22×C5C2×#13712yes—
259C4×C5—21no—
260C4×C5—21no—
261C4×C5—21no—
262C4×C5—21no—
263C4×C5—22no—
264C4×C5—22no—
265C4×C5—22no—
266C4×C5—22no—
267C4×C5—84no—
268Dic5—44no—
269Dic5—44no—
270C24C2×C22⋊D511yescharacteristic
271C2×Q8—54no—
272C2×Q8—104no—
273C2×Q8—102no—
274C2×Q8—204no—
275C2×Q8—204no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—102no—
284C22×C4—102no—
285C22×C4#68512yes—
286C22×C4#68511yescharacteristic
287C22×C4#68511yescharacteristic
288C22×C4#68511yescharacteristic
289C22×C4#68511yescharacteristic
290C22×C4#68512yes—
291C22×C4C2×C4×D511yescharacteristic
292C22×C4C2×C4×D511yescharacteristic
293C22×C4C2×D2011yescharacteristic
294C22×C4C2×D2011yescharacteristic
295C22×C4C2×C22⋊D512yes—
296C22×C4C2×C22⋊D511yescharacteristic
297C22×C4C2×C22⋊D512yes—
298C22×C4C23×D511yescharacteristic
299C2×C8—104no—
300C2×C8—204no—
301C4⋊C4—54no—
302C4⋊C4—104no—
303C4⋊C4—102no—
304C4⋊C4—104no—
305C4⋊C4—104no—
306C4⋊C4—108no—
307C4⋊C4—104no—
308C4⋊C4—104no—
309C22⋊C4—104no—
310C22⋊C4—104no—
311C42—108no—
312C42#68512yes—
313C42#68512yes—
314C42#68512yes—
315C42#68512yes—
316C42C2×C4×D512yes—
317C42C2×C4×D512yes—
318C42C2×D2012yes—
319C42C2×D2012yes—
320C2×C5—22no—
321C2×C5—22no—
322C2×C5#151514yescomplemented
323C2×C5C2×#41011yescharacteristic
324C2×C5C2×#41911yescharacteristic
325C2×C5C2×#42111yescharacteristic
326C23—22no—
327C23—22no—
328C23C2×#68511yescharacteristic
329C23C2×#68511yescharacteristic
330C23#409714yes—
331C23C22×C22⋊D511yescharacteristic
332Q8—204no—
333Q8—208no—
334Q8—204no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—88no—
353C2×C4—104no—
354C2×C4—104no—
355C2×C4—204no—
356C2×C4—204no—
357C2×C4C2×#68512yes—
358C2×C4C2×#68511yescharacteristic
359C2×C4C2×#68511yescharacteristic
360C2×C4C2×#68512yes—
361C2×C4C2×#68512yes—
362C2×C4C2×#68512yes—
363C2×C4C22×C4×D511yescharacteristic
364C2×C4C22×D2011yescharacteristic
365C8—204no—
366C5C2×#151511yescharacteristic, complemented, Sylow
367C22—28no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22#6529312yes—
373C22#6529312yes—
374C22#6529412yes—
375C22#6529512yes—
376C22C22×#68511yescharacteristic
377C22C2×#409712yes—
378C4—22no—
379C4—22no—
380C4—22no—
381C4—22no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—84no—
387C4—204no—
388C4—204no—
389C2—22no—
390C2—22no—
391C2#1060035914yesdirect product factor
392C2C2×#6529311yescharacteristic
393C2C2×#6529411yescharacteristic
394C2C2×#6529511yescharacteristic
subgroupG/H
centerC23C2×#685
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC2×C5×#383C2
socleC23×C5C2×C22⋊C4
prime radicalC24×C5C2×D4
cosocleC22×C4×C5C24
hypercenterC2×#383D5
hypocenterC5C2×#1515
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#383

Complements

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

NH ≅ G/Ntype
4. #10600359C2direct (N × H)
12. #65804C4semidirect
15. C2×#4015C4semidirect
16. C2×#4015C4semidirect
18. C2×C4×Dic10C4semidirect
66. #4015C2×C4semidirect
67. #4015C2×C4semidirect
71. C4×Dic10C2×C4semidirect
322. C2×C5#1515semidirect
366. C5C2×#1515semidirect
391. C2#10600359direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#658042
C2×C5×#3832
C2×#652152
#106003592
C2×#646722
C2×#15155