Finite Group Databasebeta

← C2×#10612760 #12977802

Subgroups — C2×#10612760 #12977802

Order 1280.

1854 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66066C211yesmaximal, characteristic
2C2×#65918C211yesmaximal, characteristic
3C2×C5×#475C211yesmaximal, characteristic, complemented
4C2×#65367C211yesmaximal, characteristic, complemented
5C2×C4×#689C211yesmaximal, characteristic, complemented
6#10612760C2116yesmaximal, direct product factor
7C2×#65149C212yesmaximal, complemented
8C2×#65156C212yesmaximal, complemented
9C2×#65140C211yesmaximal, characteristic, complemented
10C2×#65123C211yesmaximal, characteristic
11C2×#65104C212yesmaximal, complemented
12C2×#65102C212yesmaximal, complemented
13C2×C5×C4⋊Q8C2211yescharacteristic
14C2×C4×C5×Q8C2211yescharacteristic
15C2×#4093—21no—
16C2×#4092C2212yes—
17C2×Q8×Dic5—21no—
18C22×#689C2211yescharacteristic
19C22×C8×D5C2212yes—
20#66066—24no—
21#66066C2218yes—
22#65918—24no—
23#65918C2218yes—
24C2×#4044—22no—
25C2×#4044—22no—
26C2×#4043C2212yes—
27C2×#4038C2212yes—
28C2×#4038C2212yes—
29C2×#4021—21no—
30C2×C4×D20C2211yescharacteristic
31C2×#4015C2211yescharacteristic
32C2×C4×Dic10—21no—
33C2×C4×Dic10C2211yescharacteristic
34C5×#475C2218yescomplemented
35C2×C5×#106C2212yes—
36C2×C5×C4⋊C8C2211yescharacteristic
37C2×C5×#103C2212yes—
38C2×#3967C2211yescharacteristic
39C2×#3952C2212yes—
40C2×#3949C2212yes—
41C2×#3948C2212yes—
42#65367C2218yescomplemented
43C4×#689C2218yescomplemented
44C2×#3942C2211yescharacteristic
45C2×#3939C2212yes—
46C2×#3939C2212yes—
47C2×#3939C2211yescharacteristic
48C2×C4×#168C2211yescharacteristic
49#65156C22116yescomplemented
50#65149C22116yescomplemented
51#65140C2218yescomplemented
52#65123C2218yes—
53#65104C22116yescomplemented
54#65102C22116yescomplemented
55C2×#3342—51nomaximal, Sylow
56C22×C5×Q8—21no—
57C22×C5×Q8C2311yescharacteristic
58C22×D20D411yescharacteristic
59C22×C4×D5C2312yes—
60C22×Dic10—21no—
61C22×Dic10C2311yescharacteristic
62C22×C5×C8C2312yes—
63C5×C4⋊Q8—22no—
64C5×C4⋊Q8C2314yes—
65C4×C5×Q8—22no—
66C4×C5×Q8C2314yes—
67C2×C5×C4⋊C4—21no—
68C2×C5×C4⋊C4—21no—
69C2×C5×C4⋊C4C2312yes—
70C2×C5×C4⋊C4C2311yescharacteristic
71C2×C42×C5—21no—
72C2×C42×C5C2311yescharacteristic
73#4093—28no—
74#4092—24no—
75#4092C2318yes—
76Q8×Dic5—28no—
77C2×#689—24no—
78C2×#689C2314yes—
79C2×#685—22no—
80C2×#685—22no—
81C2×#685—22no—
82C2×#685—22no—
83C2×#685D412yes—
84C2×C4⋊Dic5—21no—
85C2×C4⋊Dic5—22no—
86C2×C4⋊Dic5D411yescharacteristic
87C2×C4⋊Dic5C2311yescharacteristic
88C2×C4⋊Dic5C2312yes—
89C2×#683—21no—
90C2×#683—22no—
91C2×C4×Dic5—21no—
92C2×C4×Dic5—22no—
93C2×C4×Dic5—22no—
94C22×#168C2311yescharacteristic
95C22×#168C2311yescharacteristic
96C2×C8×D5—28no—
97C2×C8×D5C2318yes—
98#4044—216no—
99#4044—216no—
100#4043—24no—
101#4043C2318yes—
102#4038—24no—
103#4038—24no—
104#4038C2318yes—
105#4038C2318yes—
106#4021—28no—
107C4×D20D414yes—
108C4×D20C2314yes—
109#4015—22no—
110#4015C2314yes—
111C4×Dic10—22no—
112C4×Dic10—28no—
113C4×Dic10C2314yes—
114C5×#106C2318yes—
115C5×C4⋊C8C2314yes—
116C5×#103C2318yes—
117#3967C2314yes—
118#3952C2318yes—
119#3949C2318yes—
120#3948C2318yes—
121#3942C2314yes—
122#3939C2314yes—
123#3939C2318yes—
124#3939C2318yes—
125C4×#168C2314yes—
126C2×#556—51no—
127C2×#556—51no—
128#3342—516no—
129C2×#500—51no—
130C2×#485—52no—
131C2×#485—52no—
132C2×#475—51no—
133C2×#475D511yescharacteristic, complemented
134C2×#467—52no—
135C2×C4×Q16—51no—
136C2×C8×D4—51no—
137C2×#427—52no—
138C23×D5C2×D412yes—
139C2×C5×Q8—21no—
140C2×C5×Q8—24no—
141C2×C5×Q8—22no—
142C2×C5×Q8—22no—
143C2×C5×Q8—44no—
144C2×C5×Q8C2412yes—
145C22×C4×C5—21no—
146C22×C4×C5—21no—
147C22×C4×C5—22no—
148C22×C4×C5C2×D411yescharacteristic
149C22×C4×C5C2×D411yescharacteristic
150C22×C4×C5C2411yescharacteristic
151C22×Dic5—22no—
152C22×Dic5—21no—
153C22×Dic5—21no—
154C22×Dic5C2×D412yes—
155C2×D20—24no—
156C2×D20C2×D414yes—
157C2×C4×D5—24no—
158C2×C4×D5—24no—
159C2×C4×D5C2×D414yes—
160C2×C4×D5C2414yes—
161C2×Dic10—22no—
162C2×Dic10—22no—
163C2×Dic10—24no—
164C2×Dic10—21no—
165C2×Dic10—44no—
166C2×Dic10C2412yes—
167C2×C5×C8—24no—
168C2×C5×C8C2414yes—
169C5×C4⋊C4—24no—
170C5×C4⋊C4—24no—
171C5×C4⋊C4—21no—
172C5×C4⋊C4—22no—
173C5×C4⋊C4C2412yes—
174C5×C4⋊C4C2414yes—
175C42×C5—24no—
176C42×C5C2×D412yes—
177C42×C5C2412yes—
178#689—28no—
179#685—28no—
180#685—28no—
181#685—28no—
182#685—28no—
183#685C2×D418yes—
184C4⋊Dic5—21no—
185C4⋊Dic5—22no—
186C4⋊Dic5—24no—
187C4⋊Dic5—28no—
188C4⋊Dic5C2×D414yes—
189C4⋊Dic5C2414yes—
190C4⋊Dic5C2412yes—
191#683—24no—
192#683—28no—
193C4×Dic5—24no—
194C4×Dic5—28no—
195C4×Dic5—28no—
196C2×#168—22no—
197C2×#168#4114yes—
198C2×#168C2412yes—
199C2×#168C2412yes—
200C8×D5—216no—
201C22×Q16—51no—
202C23×C8—52no—
203#556—58no—
204#556—58no—
205#556—104no—
206#556—104no—
207C2×C4⋊Q8—51no—
208C2×C4⋊Q8C2×D511yescharacteristic
209C2×#126—102no—
210C2×#126—102no—
211C2×#124—101no—
212C2×#124—101no—
213C2×C22⋊Q8—52no—
214C2×C22⋊Q8—52no—
215C2×C4×Q8—51no—
216C2×C4×Q8—101no—
217C2×C4×Q8—101no—
218C2×C4×Q8C2×D511yescharacteristic
219C2×C4×D4—51no—
220C2×#117—52no—
221C2×#117—52no—
222#500—58no—
223#485—516no—
224#485—516no—
225#475—58no—
226#475C2×D518yescomplemented
227#467—516no—
228C4×Q16—58no—
229C8×D4—58no—
230#427—516no—
231C2×#107—52no—
232C2×#107—52no—
233C2×#107—51no—
234C2×#106—52no—
235C2×#106C2×D512yes—
236C2×C4⋊C8C2×D511yescharacteristic
237C2×#103—51no—
238C2×#103—52no—
239C2×#103—51no—
240C2×#103C2×D512yes—
241C2×C22⋊C8—52no—
242C2×C4×C8—51no—
243C23×C5C22×D411yescharacteristic
244C22×D5—24no—
245C22×D5—24no—
246C22×D5—24no—
247C22×D5C22×D414yes—
248C5×Q8—22no—
249C5×Q8—22no—
250C5×Q8—42no—
251C5×Q8—48no—
252C2×C4×C5—22no—
253C2×C4×C5—22no—
254C2×C4×C5—24no—
255C2×C4×C5—22no—
256C2×C4×C5—42no—
257C2×C4×C5—44no—
258C2×C4×C5—42no—
259C2×C4×C5#13514yes—
260C2×C4×C5C22×D412yes—
261C2×C4×C5C22×D411yescharacteristic
262C2×C4×C5C22×D412yes—
263C2×C4×C5C2×#4112yes—
264C2×C4×C52−1+412yes—
265C2×C4×C5C2511yescharacteristic
266C2×Dic5—24no—
267C2×Dic5—22no—
268C2×Dic5—22no—
269C2×Dic5—24no—
270C2×Dic5—42no—
271C2×Dic5—44no—
272C2×Dic5—42no—
273C2×Dic5C22×D414yes—
274D20—28no—
275C4×D5—28no—
276C4×D5—24no—
277C4×D5—24no—
278Dic10—22no—
279Dic10—22no—
280Dic10—42no—
281Dic10—48no—
282C5×C8—24no—
283#168—22no—
284#168C2×#4114yes—
285C22×Q8—21no—
286C22×Q8—51no—
287C22×Q8—101no—
288C22×Q8C22×D511yescharacteristic
289C22×D4—51no—
290C23×C4—52no—
291C2×Q16—54no—
292C2×Q16—104no—
293C22×C8—51no—
294C22×C8—51no—
295C22×C8—58no—
296C22×C8—108no—
297C22×C8C22×D512yes—
298C4⋊Q8—22no—
299C4⋊Q8—54no—
300C4⋊Q8—102no—
301C4⋊Q8C22×D514yes—
302#126—1016no—
303#126—1016no—
304#124—108no—
305#124—108no—
306C22⋊Q8—58no—
307C22⋊Q8—58no—
308C22⋊Q8—104no—
309C22⋊Q8—104no—
310C4×Q8—22no—
311C4×Q8—54no—
312C4×Q8—108no—
313C4×Q8—108no—
314C4×Q8—102no—
315C4×Q8C22×D514yes—
316C4×D4—54no—
317C4×D4—54no—
318#117—58no—
319#117—58no—
320#117—104no—
321#117—104no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—51no—
325C2×C4⋊C4—51no—
326C2×C4⋊C4—52no—
327C2×C4⋊C4—102no—
328C2×C4⋊C4—102no—
329C2×C4⋊C4—101no—
330C2×C4⋊C4—101no—
331C2×C4⋊C4C22×D511yescharacteristic
332C2×C4⋊C4C22×D512yes—
333C2×C22⋊C4—52no—
334C2×C22⋊C4—102no—
335C2×C22⋊C4—102no—
336C2×C22⋊C4—102no—
337C2×C22⋊C4—102no—
338C2×C42—21no—
339C2×C42—102no—
340C2×C42—102no—
341C2×C42—101no—
342C2×C42C22×D511yescharacteristic
343#107—54no—
344#107—58no—
345#107—58no—
346#106—58no—
347#106C22×D518yes—
348C4⋊C8C22×D514yes—
349#103—54no—
350#103—58no—
351#103—54no—
352#103C22×D518yes—
353C22⋊C8—58no—
354C4×C8—54no—
355C22×C5#54712yes—
356C22×C5C2×#13512yes—
357C22×C5#57812yes—
358C22×C5C23×D411yescharacteristic
359C2×D5—24no—
360C2×D5—24no—
361C2×D5—24no—
362C2×D5—48no—
363C4×C5—22no—
364C4×C5—42no—
365C4×C5—44no—
366C4×C5—42no—
367C4×C5#54712yes—
368C4×C5C2×#13514yes—
369C4×C5C22×#4111yescharacteristic
370C4×C5C2×2−1+411yescharacteristic
371Dic5—24no—
372Dic5—42no—
373Dic5—44no—
374Dic5—42no—
375C24—52no—
376C2×Q8—21no—
377C2×Q8—24no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—44no—
381C2×Q8—52no—
382C2×Q8—101no—
383C2×Q8—104no—
384C2×Q8—102no—
385C2×Q8—102no—
386C2×Q8—204no—
387C2×Q8C23×D512yes—
388C2×D4—54no—
389C2×D4—104no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—52no—
394C22×C4—54no—
395C22×C4—54no—
396C22×C4—104no—
397C22×C4—104no—
398C22×C4—101no—
399C22×C4—102no—
400C22×C4—101no—
401C22×C4D4×D511yescharacteristic
402C22×C4D4×D511yescharacteristic
403C22×C4C23×D511yescharacteristic
404Q16—108no—
405C2×C8—24no—
406C2×C8—54no—
407C2×C8—52no—
408C2×C8—52no—
409C2×C8—1016no—
410C2×C8—102no—
411C2×C8C23×D514yes—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—21no—
415C4⋊C4—22no—
416C4⋊C4—54no—
417C4⋊C4—52no—
418C4⋊C4—54no—
419C4⋊C4—108no—
420C4⋊C4—108no—
421C4⋊C4—101no—
422C4⋊C4—102no—
423C4⋊C4—104no—
424C4⋊C4—104no—
425C4⋊C4C23×D512yes—
426C4⋊C4C23×D514yes—
427C22⋊C4—58no—
428C22⋊C4—108no—
429C22⋊C4—108no—
430C22⋊C4—108no—
431C22⋊C4—108no—
432C42—24no—
433C42—108no—
434C42—108no—
435C42—104no—
436C42D4×D512yes—
437C42C23×D512yes—
438C2×C5#334214yescomplemented
439C2×C5C2×#54711yescharacteristic
440C2×C5C22×#13511yescharacteristic
441C2×C5C2×#57811yescharacteristic
442D5—44no—
443C23—54no—
444C23—104no—
445C23—104no—
446C23—104no—
447C23C2×D4×D511yescharacteristic
448Q8—22no—
449Q8—22no—
450Q8—48no—
451Q8—42no—
452Q8—102no—
453Q8—102no—
454Q8—208no—
455Q8—202no—
456D4—108no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—54no—
465C2×C4—104no—
466C2×C4—108no—
467C2×C4—104no—
468C2×C4—104no—
469C2×C4—102no—
470C2×C4—104no—
471C2×C4—102no—
472C2×C4—204no—
473C2×C4—202no—
474C2×C4—202no—
475C2×C4#406214yes—
476C2×C4C2×D4×D512yes—
477C2×C4C2×D4×D512yes—
478C2×C4C2×D4×D511yescharacteristic
479C2×C4#414712yes—
480C2×C4D5×#4112yes—
481C2×C4C24×D511yescharacteristic
482C8—24no—
483C8—54no—
484C8—102no—
485C5C2×#334211yescharacteristic, complemented, Sylow
486C22—104no—
487C22—104no—
488C22—104no—
489C22—208no—
490C22#6596512yes—
491C22C2×#406212yes—
492C22#6611312yes—
493C22C22×D4×D511yescharacteristic
494C4—22no—
495C4—42no—
496C4—44no—
497C4—42no—
498C4—104no—
499C4—202no—
500C4—204no—
501C4—202no—
502C4#6596512yes—
503C4C2×#406214yes—
504C4C2×#414711yescharacteristic
505C4C2×D5×#4111yescharacteristic
506C2—204no—
507C2#1061276014yesdirect product factor
508C2C2×#6596511yescharacteristic
509C2C22×#406211yescharacteristic
510C2C2×#6611311yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#475C2
socleC23×C5C22×D4
prime radicalC22×D20D4
hypercenterC2×#475D5
hypocenterC5C2×#3342
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
3. C2×C5×#475C2semidirect
4. C2×#65367C2semidirect
5. C2×C4×#689C2semidirect
6. #10612760C2direct (N × H)
7. C2×#65149C2semidirect
8. C2×#65156C2semidirect
9. C2×#65140C2semidirect
11. C2×#65104C2semidirect
12. C2×#65102C2semidirect
34. C5×#475C22semidirect
42. #65367C22semidirect
43. C4×#689C22semidirect
49. #65156C22semidirect
50. #65149C22semidirect
51. #65140C22semidirect
53. #65104C22semidirect
54. #65102C22semidirect
133. C2×#475D5semidirect
226. #475C2×D5semidirect
438. C2×C5#3342semidirect
485. C5C2×#3342semidirect
507. C2#10612760direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#660662
C2×#659182
C2×C5×#4752
C2×#653672
C2×C4×#6892
#106127602
C2×#651492
C2×#651562
C2×#651402
C2×#651232
C2×#651042
C2×#651022
C2×#33425