Finite Group Databasebeta

← C2×#10612701 #12971376

Subgroups — C2×#10612701 #12971376

Order 1280.

1810 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66062C211yesmaximal, characteristic, complemented
2C2×#65891C211yesmaximal, characteristic
3C2×C5×#474C211yesmaximal, characteristic, complemented
4C2×#65383C211yesmaximal, characteristic, complemented
5C2×#65297C211yesmaximal, characteristic, complemented
6#10612701C2116yesmaximal, direct product factor
7C2×#65177C212yesmaximal, complemented
8C2×#65167C212yesmaximal, complemented
9C2×#65139C211yesmaximal, characteristic, complemented
10C2×#65122C211yesmaximal, characteristic, complemented
11C2×#65073C212yesmaximal
12C2×#65057C212yesmaximal, complemented
13C2×C5×C4⋊Q8C2211yescharacteristic
14C2×C4×C5×D4C2211yescharacteristic
15C2×#4092—21no—
16C2×Q8×Dic5—21no—
17C2×#4084C2211yescharacteristic
18C2×#4082C2211yescharacteristic
19C2×C4×C22⋊D5—21no—
20C2×#4072—21no—
21#66062—24no—
22#66062C2218yescomplemented
23#65891—24no—
24#65891C2218yes—
25C2×#4043—21no—
26C2×#4043C2211yescharacteristic
27C2×#4038C2212yes—
28C2×#4036C2212yes—
29C2×#4036C2211yescharacteristic
30C2×#4033—21no—
31C2×#4029—21no—
32C2×#4025—21no—
33C2×C42×D5C2211yescharacteristic
34C2×#4015C2211yescharacteristic
35C2×C4×Dic10C2211yescharacteristic
36C5×#474C2218yescomplemented
37C2×C5×#107C2212yes—
38C2×C5×C4⋊C8C2211yescharacteristic
39C2×C5×#102C2212yes—
40C2×#3964C2212yes—
41C2×#3952C2212yes—
42C2×#3950C2212yes—
43C2×#3948C2212yes—
44C2×C8×Dic5C2212yes—
45#65383C2218yescomplemented
46#65297C2218yescomplemented
47C2×#3942C2212yes—
48C2×#3940C2212yes—
49C2×#3940C2212yes—
50C2×C4⋊#168C2211yescharacteristic
51#65177C22116yescomplemented
52#65167C22116yescomplemented
53#65139C2218yescomplemented
54#65122C2218yescomplemented
55#65073C22116yes—
56#65057C22116yescomplemented
57C2×#3233—51nomaximal, Sylow
58C22×C5×Q8—21no—
59C22×C5×D4C2311yescharacteristic
60C23×C4×C5—21no—
61C22×C22⋊D5—21no—
62C22×C4×D5D411yescharacteristic
63C22×C4×D5D411yescharacteristic
64C22×C4×D5C2311yescharacteristic
65C22×Dic10—21no—
66C22×Dic10C2311yescharacteristic
67C22×C5×C8C2312yes—
68C5×C4⋊Q8—22no—
69C5×C4⋊Q8C2314yes—
70C4×C5×D4—22no—
71C4×C5×D4C2314yes—
72C2×C5×C4⋊C4—21no—
73C2×C5×C4⋊C4C2312yes—
74C2×C5×C4⋊C4C2311yescharacteristic
75C2×C5×C22⋊C4—21no—
76C2×C42×C5C2311yescharacteristic
77C2×C22⋊Dic5—21no—
78C2×C22⋊Dic5—21no—
79C2×C22⋊Dic5—21no—
80#4092—28no—
81Q8×Dic5—28no—
82#4084—22no—
83#4084C2314yes—
84#4082—22no—
85#4082C2314yes—
86C4×C22⋊D5—28no—
87C2×#685—21no—
88C2×#685—22no—
89#4072—28no—
90C2×C4⋊Dic5—21no—
91C2×C4⋊Dic5—21no—
92C2×C4⋊Dic5—21no—
93C2×C4⋊Dic5—22no—
94C2×C4⋊Dic5C2311yescharacteristic
95C2×C4⋊Dic5C2312yes—
96C2×#683—21no—
97C2×#683—21no—
98C2×#683—22no—
99C2×C4×Dic5—21no—
100C2×C4×Dic5—22no—
101C2×C4×Dic5D411yescharacteristic
102C2×C4×Dic5D411yescharacteristic
103C2×C4×Dic5C2311yescharacteristic
104C22×#168C2312yes—
105#4043—28no—
106#4043—22no—
107#4043C2314yes—
108#4038—24no—
109#4038C2318yes—
110#4036—24no—
111#4036—22no—
112#4036C2318yes—
113#4036C2314yes—
114#4033—28no—
115#4029—28no—
116#4025—28no—
117C42×D5D414yes—
118C42×D5C2314yes—
119#4015—22no—
120#4015C2314yes—
121C4×Dic10—22no—
122C4×Dic10C2314yes—
123C5×#107C2318yes—
124C5×C4⋊C8C2314yes—
125C5×#102C2318yes—
126#3964C2318yes—
127#3952C2318yes—
128#3950C2318yes—
129#3948C2318yes—
130C8×Dic5C2318yes—
131#3942C2318yes—
132#3940C2318yes—
133#3940C2318yes—
134C4⋊#168C2314yes—
135C2×#533—51no—
136C2×#529—51no—
137#3233—516no—
138C2×#499—52no—
139C2×#488—52no—
140C2×#485—52no—
141C2×#482—52no—
142C2×#477—51no—
143C2×#475—51no—
144C2×#476—51no—
145C2×#474D511yescharacteristic, complemented
146C2×#431—51no—
147C24×C5—21no—
148C23×D5C2×D411yescharacteristic
149C2×C5×Q8—24no—
150C2×C5×Q8—44no—
151C2×C5×D4—21no—
152C2×C5×D4—22no—
153C2×C5×D4—22no—
154C2×C5×D4C2412yes—
155C22×C4×C5—21no—
156C22×C4×C5—22no—
157C22×C4×C5—24no—
158C22×C4×C5—44no—
159C22×C4×C5C2×D411yescharacteristic
160C22×C4×C5C2×D411yescharacteristic
161C22×C4×C5C2411yescharacteristic
162C2×C22⋊D5—24no—
163C2×C22⋊D5—44no—
164C22×Dic5—21no—
165C22×Dic5—21no—
166C22×Dic5—22no—
167C22×Dic5C2×D411yescharacteristic
168C22×Dic5C2×D411yescharacteristic
169C22×Dic5C2×D411yescharacteristic
170C2×C4×D5—24no—
171C2×C4×D5—22no—
172C2×C4×D5—22no—
173C2×C4×D5—42no—
174C2×C4×D5C2×D414yes—
175C2×C4×D5C2×D412yes—
176C2×C4×D5C2×D414yes—
177C2×C4×D5C2412yes—
178C2×Dic10—22no—
179C2×Dic10—22no—
180C2×Dic10—24no—
181C2×Dic10—21no—
182C2×Dic10—44no—
183C2×Dic10C2412yes—
184C2×C5×C8—24no—
185C2×C5×C8C2414yes—
186C5×C4⋊C4—24no—
187C5×C4⋊C4—21no—
188C5×C4⋊C4—22no—
189C5×C4⋊C4C2412yes—
190C5×C4⋊C4C2414yes—
191C5×C22⋊C4—24no—
192C42×C5C2×D412yes—
193C42×C5C2412yes—
194C22⋊Dic5—24no—
195C22⋊Dic5—24no—
196C22⋊Dic5—24no—
197#685—24no—
198#685—28no—
199C4⋊Dic5—21no—
200C4⋊Dic5—22no—
201C4⋊Dic5—24no—
202C4⋊Dic5—24no—
203C4⋊Dic5—24no—
204C4⋊Dic5—28no—
205C4⋊Dic5C2412yes—
206C4⋊Dic5C2414yes—
207#683—24no—
208#683—24no—
209#683—28no—
210C4×Dic5—24no—
211C4×Dic5—28no—
212C4×Dic5C2×D412yes—
213C4×Dic5C2×D414yes—
214C4×Dic5C2×D414yes—
215C4×Dic5C2412yes—
216C2×#168—24no—
217C2×#168C2414yes—
218#533—58no—
219#533—104no—
220C2×C4⋊Q8—51no—
221C2×C4⋊Q8C2×D511yescharacteristic
222#529—58no—
223#529—104no—
224C2×#126—101no—
225C2×#125—52no—
226C2×#125—51no—
227C2×#124—51no—
228C2×#123—101no—
229C2×C22⋊Q8—51no—
230C2×C22⋊Q8—101no—
231C2×C22⋊Q8—101no—
232C2×#121—51no—
233C2×C4×Q8—51no—
234C2×C4×Q8—101no—
235C2×C4×Q8—101no—
236C2×C4×D4—101no—
237C2×C4×D4C2×D511yescharacteristic
238C2×#117—52no—
239C2×#117—101no—
240C22×C42—51no—
241#499—516no—
242#488—516no—
243#485—516no—
244#482—516no—
245#477—58no—
246#476—58no—
247#475—58no—
248#474C2×D518yescomplemented
249#431—58no—
250C2×#107—52no—
251C2×#107C2×D512yes—
252C2×#106—52no—
253C2×#106—52no—
254C2×C4⋊C8—51no—
255C2×C4⋊C8C2×D511yescharacteristic
256C2×#103—52no—
257C2×#103—52no—
258C2×#102—52no—
259C2×#102C2×D512yes—
260C2×C22⋊C8—52no—
261C2×C4×C8—52no—
262C23×C5—22no—
263C23×C5—42no—
264C23×C5—42no—
265C23×C5—42no—
266C23×C5C22×D411yescharacteristic
267C22×D5—22no—
268C22×D5—22no—
269C22×D5—22no—
270C22×D5C22×D412yes—
271C5×Q8—48no—
272C5×D4—22no—
273C5×D4—22no—
274C5×D4—42no—
275C2×C4×C5—22no—
276C2×C4×C5—24no—
277C2×C4×C5—22no—
278C2×C4×C5—42no—
279C2×C4×C5—44no—
280C2×C4×C5—48no—
281C2×C4×C5#13514yes—
282C2×C4×C5C22×D412yes—
283C2×C4×C5C22×D411yescharacteristic
284C2×C4×C5C22×D412yes—
285C2×C4×C52−1+412yes—
286C2×C4×C52−1+412yes—
287C2×C4×C5C2511yescharacteristic
288C22⋊D5—48no—
289C2×Dic5—22no—
290C2×Dic5—22no—
291C2×Dic5—22no—
292C2×Dic5—22no—
293C2×Dic5—24no—
294C2×Dic5—22no—
295C2×Dic5—42no—
296C2×Dic5—44no—
297C2×Dic5—42no—
298C2×Dic5C22×D412yes—
299C2×Dic5C22×D412yes—
300C2×Dic5C22×D412yes—
301C4×D5—24no—
302C4×D5—28no—
303C4×D5—22no—
304C4×D5—22no—
305C4×D5—44no—
306Dic10—22no—
307Dic10—22no—
308Dic10—48no—
309Dic10—42no—
310C5×C8—24no—
311#168—24no—
312C22×Q8—21no—
313C22×Q8—51no—
314C22×Q8—101no—
315C22×D4—101no—
316C22×D4C22×D511yescharacteristic
317C23×C4—21no—
318C23×C4—51no—
319C23×C4—51no—
320C23×C4—51no—
321C22×C8—52no—
322C22×C8C22×D512yes—
323C4⋊Q8—22no—
324C4⋊Q8—54no—
325C4⋊Q8—102no—
326C4⋊Q8C22×D514yes—
327#126—108no—
328#125—54no—
329#125—58no—
330#125—102no—
331#125—104no—
332#124—54no—
333#124—102no—
334#123—108no—
335C22⋊Q8—54no—
336C22⋊Q8—102no—
337C22⋊Q8—108no—
338C22⋊Q8—108no—
339#121—54no—
340#121—102no—
341C4×Q8—54no—
342C4×Q8—108no—
343C4×Q8—108no—
344C4×Q8—102no—
345C4×D4—22no—
346C4×D4—108no—
347C4×D4C22×D514yes—
348#117—58no—
349#117—108no—
350#117—104no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—52no—
353C2×C4⋊C4—51no—
354C2×C4⋊C4—101no—
355C2×C4⋊C4—101no—
356C2×C4⋊C4—101no—
357C2×C4⋊C4—102no—
358C2×C4⋊C4—102no—
359C2×C4⋊C4—101no—
360C2×C4⋊C4—101no—
361C2×C4⋊C4C22×D512yes—
362C2×C4⋊C4C22×D511yescharacteristic
363C2×C22⋊C4—21no—
364C2×C22⋊C4—102no—
365C2×C22⋊C4—101no—
366C2×C22⋊C4—101no—
367C2×C22⋊C4—101no—
368C2×C22⋊C4—101no—
369C2×C42—51no—
370C2×C42—51no—
371C2×C42—51no—
372C2×C42—54no—
373C2×C42—54no—
374C2×C42—102no—
375C2×C42—101no—
376C2×C42C22×D511yescharacteristic
377#107—58no—
378#107C22×D518yes—
379#106—58no—
380#106—58no—
381C4⋊C8—54no—
382C4⋊C8C22×D514yes—
383#103—58no—
384#103—58no—
385#102—58no—
386#102C22×D518yes—
387C22⋊C8—58no—
388C4×C8—58no—
389C22×C5—42no—
390C22×C5—42no—
391C22×C5—42no—
392C22×C5—84no—
393C22×C5#53612yes—
394C22×C5C2×#13512yes—
395C22×C5#57512yes—
396C22×C5C23×D411yescharacteristic
397C2×D5—22no—
398C2×D5—22no—
399C2×D5—22no—
400C2×D5—44no—
401C4×C5—22no—
402C4×C5—42no—
403C4×C5—44no—
404C4×C5#53612yes—
405C4×C5C2×#13514yes—
406C4×C5C2×2−1+411yescharacteristic
407C4×C5C2×2−1+411yescharacteristic
408Dic5—22no—
409Dic5—22no—
410Dic5—22no—
411Dic5—42no—
412Dic5—44no—
413Dic5—42no—
414C24—21no—
415C24—51no—
416C2×Q8—24no—
417C2×Q8—44no—
418C2×Q8—52no—
419C2×Q8—101no—
420C2×Q8—104no—
421C2×Q8—102no—
422C2×Q8—102no—
423C2×Q8—204no—
424C2×D4—21no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—104no—
428C2×D4—204no—
429C2×D4C23×D512yes—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—44no—
434C22×C4—51no—
435C22×C4—51no—
436C22×C4—51no—
437C22×C4—52no—
438C22×C4—54no—
439C22×C4—52no—
440C22×C4—54no—
441C22×C4—102no—
442C22×C4—102no—
443C22×C4—104no—
444C22×C4—101no—
445C22×C4—101no—
446C22×C4—102no—
447C22×C4—202no—
448C22×C4D4×D511yescharacteristic
449C22×C4D4×D511yescharacteristic
450C22×C4C23×D511yescharacteristic
451C2×C8—24no—
452C2×C8—54no—
453C2×C8—104no—
454C2×C8C23×D514yes—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—54no—
459C4⋊C4—52no—
460C4⋊C4—104no—
461C4⋊C4—104no—
462C4⋊C4—104no—
463C4⋊C4—108no—
464C4⋊C4—108no—
465C4⋊C4—102no—
466C4⋊C4—101no—
467C4⋊C4—104no—
468C4⋊C4—104no—
469C4⋊C4C23×D514yes—
470C4⋊C4C23×D512yes—
471C22⋊C4—24no—
472C22⋊C4—108no—
473C22⋊C4—104no—
474C22⋊C4—104no—
475C22⋊C4—104no—
476C22⋊C4—104no—
477C42—54no—
478C42—54no—
479C42—52no—
480C42—52no—
481C42—108no—
482C42—104no—
483C42D4×D512yes—
484C42C23×D512yes—
485C2×C5—82no—
486C2×C5#323314yescomplemented
487C2×C5C2×#53611yescharacteristic
488C2×C5C22×#13511yescharacteristic
489C2×C5C2×#57511yescharacteristic
490D5—42no—
491C23—22no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—52no—
496C23—102no—
497C23—102no—
498C23—102no—
499C23C2×D4×D511yescharacteristic
500Q8—48no—
501Q8—102no—
502Q8—102no—
503Q8—208no—
504Q8—202no—
505D4—22no—
506D4—22no—
507D4—42no—
508D4—208no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—48no—
515C2×C4—52no—
516C2×C4—52no—
517C2×C4—52no—
518C2×C4—102no—
519C2×C4—102no—
520C2×C4—102no—
521C2×C4—108no—
522C2×C4—104no—
523C2×C4—102no—
524C2×C4—102no—
525C2×C4—102no—
526C2×C4—102no—
527C2×C4—104no—
528C2×C4—204no—
529C2×C4—202no—
530C2×C4—204no—
531C2×C4—202no—
532C2×C4#405814yes—
533C2×C4C2×D4×D512yes—
534C2×C4C2×D4×D512yes—
535C2×C4C2×D4×D511yescharacteristic
536C2×C4#414712yes—
537C2×C4#415012yes—
538C2×C4C24×D511yescharacteristic
539C8—24no—
540C8—104no—
541C5C2×#323311yescharacteristic, complemented, Sylow
542C22—42no—
543C22—42no—
544C22—42no—
545C22—84no—
546C22—102no—
547C22—102no—
548C22—102no—
549C22—204no—
550C22#6596212yes—
551C22C2×#405812yes—
552C22#6611112yes—
553C22C22×D4×D511yescharacteristic
554C4—22no—
555C4—42no—
556C4—44no—
557C4—102no—
558C4—102no—
559C4—102no—
560C4—202no—
561C4—202no—
562C4—204no—
563C4#6596212yes—
564C4C2×#405814yes—
565C4C2×#414711yescharacteristic
566C4C2×#415011yescharacteristic
567C2—82no—
568C2—202no—
569C2#1061270114yesdirect product factor
570C2C2×#6596211yescharacteristic
571C2C22×#405811yescharacteristic
572C2C2×#6611111yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#474C2
socleC23×C5C22×D4
prime radicalC2×#4084C22
hypercenterC2×#474D5
hypocenterC5C2×#3233
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. C2×#66062C2semidirect
3. C2×C5×#474C2semidirect
4. C2×#65383C2semidirect
5. C2×#65297C2semidirect
6. #10612701C2direct (N × H)
7. C2×#65177C2semidirect
8. C2×#65167C2semidirect
9. C2×#65139C2semidirect
10. C2×#65122C2semidirect
12. C2×#65057C2semidirect
22. #66062C22semidirect
36. C5×#474C22semidirect
45. #65383C22semidirect
46. #65297C22semidirect
51. #65177C22semidirect
52. #65167C22semidirect
53. #65139C22semidirect
54. #65122C22semidirect
56. #65057C22semidirect
145. C2×#474D5semidirect
248. #474C2×D5semidirect
486. C2×C5#3233semidirect
541. C5C2×#3233semidirect
569. C2#10612701direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#660622
C2×#658912
C2×C5×#4742
C2×#653832
C2×#652972
#106127012
C2×#651772
C2×#651672
C2×#651392
C2×#651222
C2×#650732
C2×#650572
C2×#32335