Finite Group Databasebeta

← C2×#10610972 #12966838

Subgroups — C2×#10610972 #12966838

Order 1280.

2142 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65856C211yesmaximal, characteristic
2C22×#4036C211yesmaximal, characteristic, complemented
3C2×C5×#428C211yesmaximal, characteristic, complemented
4C2×#65422C211yesmaximal, characteristic
5C2×#65407C211yesmaximal, characteristic, complemented
6C2×#65284C211yesmaximal, characteristic, complemented
7C2×#65255C211yesmaximal, characteristic
8#10610972C2116yesmaximal, direct product factor
9C2×#65166C212yesmaximal, complemented
10C2×#65180C212yesmaximal, complemented
11C2×#65146C212yesmaximal, complemented
12C2×#65160C212yesmaximal, complemented
13C22×#709C2211yescharacteristic
14C22×C5×M16C2211yescharacteristic
15C2×C5×#117C2211yescharacteristic
16C22×C5×C4⋊C4C2211yescharacteristic
17C22×C4⋊Dic5—21no—
18C22×C4⋊Dic5C2211yescharacteristic
19C22×#683—21no—
20C22×C4×Dic5C2211yescharacteristic
21C22×#681C2211yescharacteristic
22#65856—24no—
23#65856C2218yes—
24C2×#4039C2211yescharacteristic
25C2×#4039C2211yescharacteristic
26C2×#4036—24no—
27C2×#4036—21no—
28C2×#4036—21no—
29C2×#4036C2212yes—
30C2×#4036C2212yes—
31C2×#4036C2218yescomplemented
32C2×#4033C2211yescharacteristic
33C2×#4033C2211yescharacteristic
34C2×#4027—21no—
35C2×#4027—21no—
36C2×#4018—21no—
37C2×C42×D5—21no—
38C5×#428C2218yescomplemented
39C2×C5×#107C2212yes—
40C2×C5×#106C2212yes—
41#65422C2218yes—
42#65407C2218yescomplemented
43C2×#3953C2212yes—
44C2×#3948C2212yes—
45C2×#3946C2212yes—
46C2×#3946C2212yes—
47#65284C2218yescomplemented
48C2×#3942C2212yes—
49#65255C2218yes—
50C2×#3941C2212yes—
51C2×#3940C2212yes—
52C2×#3939C2212yes—
53#65180C22116yescomplemented
54#65166C22116yescomplemented
55#65160C22116yescomplemented
56#65146C22116yescomplemented
57C2×#3080—51nomaximal, Sylow
58C23×C4×C5—21no—
59C23×C4×C5C2311yescharacteristic
60C22×C22⋊D5—21no—
61C23×Dic5—21no—
62C23×Dic5D411yescharacteristic
63C23×Dic5D411yescharacteristic
64C2×#709—22no—
65C2×#709—24no—
66C2×#709Q818yes—
67C2×#709C2314yes—
68C22×D20C2311yescharacteristic
69C22×C4×D5—21no—
70C22×C4×D5—21no—
71C22×C4×D5—21no—
72C22×Dic10C2311yescharacteristic
73C2×C5×M16—24no—
74C2×C5×M16C2314yes—
75C22×C5×C8C2312yes—
76C5×#117—22no—
77C5×#117C2314yes—
78C2×C5×C4⋊C4—21no—
79C2×C5×C4⋊C4—22no—
80C2×C5×C4⋊C4C2311yescharacteristic
81C2×C5×C4⋊C4C2311yescharacteristic
82C2×C5×C4⋊C4C2312yes—
83C2×C5×C4⋊C4C2314yes—
84C2×C5×C22⋊C4—21no—
85C2×C42×C5—21no—
86C2×#685—21no—
87C2×#685—21no—
88C2×C4⋊Dic5—21no—
89C2×C4⋊Dic5—22no—
90C2×C4⋊Dic5—22no—
91C2×C4⋊Dic5—22no—
92C2×C4⋊Dic5—28no—
93C2×C4⋊Dic5C2314yes—
94C2×C4⋊Dic5C2312yes—
95C2×#683—21no—
96C2×#683—21no—
97C2×#683—22no—
98C2×#683—22no—
99C2×#683—28no—
100C2×C4×Dic5—21no—
101C2×C4×Dic5—21no—
102C2×C4×Dic5D411yescharacteristic
103C2×C4×Dic5D411yescharacteristic
104C2×C4×Dic5D414yes—
105C2×C4×Dic5C2311yescharacteristic
106C2×C4×Dic5C2311yescharacteristic
107C2×C4×Dic5C2314yes—
108C2×#681—24no—
109C2×#681C2314yes—
110C22×#168C2312yes—
111#4039—22no—
112#4039—22no—
113#4039C2314yes—
114#4039C2314yes—
115#4036—24no—
116#4036—24no—
117#4036—28no—
118#4036—28no—
119#4036C2318yes—
120#4036C2318yes—
121#4033—22no—
122#4033—22no—
123#4033C2314yes—
124#4033C2314yes—
125#4027—28no—
126#4027—28no—
127#4018—28no—
128C42×D5—28no—
129C5×#107C2318yes—
130C5×#106C2318yes—
131#3953C2318yes—
132#3948C2318yes—
133#3946C2318yes—
134#3946C2318yes—
135#3942C2318yes—
136#3941C2318yes—
137#3940C2318yes—
138#3939C2318yes—
139C22×#125—51no—
140C2×C4×#41—51no—
141#3080—516no—
142C2×#479—52no—
143C2×#479—52no—
144C2×#478—52no—
145C2×#478—52no—
146C2×#428—51no—
147C2×#428D511yescharacteristic, complemented
148C2×#423—51no—
149C2×#417—51no—
150C2×#417—51no—
151C24×C5C2×D411yescharacteristic
152C23×D5—21no—
153C22×C4×C5—21no—
154C22×C4×C5—21no—
155C22×C4×C5—24no—
156C22×C4×C5—22no—
157C22×C4×C5—44no—
158C22×C4×C5C2×D411yescharacteristic
159C22×C4×C5C2×D412yes—
160C22×C4×C5C2×Q814yes—
161C22×C4×C5#4114yes—
162C22×C4×C5C2411yescharacteristic
163C22×C4×C5C2412yes—
164C2×C22⋊D5—28no—
165C2×C22⋊D5—24no—
166C22×Dic5—21no—
167C22×Dic5—24no—
168C22×Dic5—22no—
169C22×Dic5—42no—
170C22×Dic5—42no—
171C22×Dic5—44no—
172C22×Dic5C2×D411yescharacteristic
173C22×Dic5C2×D411yescharacteristic
174C22×Dic5C2×D411yescharacteristic
175C22×Dic5C2×D411yescharacteristic
176C22×Dic5C2×D414yes—
177C22×Dic5C2×D414yes—
178#709—216no—
179#709—28no—
180#709C2×Q8116yes—
181C2×D20—21no—
182C2×D20—22no—
183C2×D20C2×Q814yes—
184C2×D20C2412yes—
185C2×C4×D5—28no—
186C2×C4×D5—24no—
187C2×C4×D5—24no—
188C2×C4×D5—24no—
189C2×C4×D5—44no—
190C2×C4×D5—44no—
191C2×Dic10—21no—
192C2×Dic10—22no—
193C2×Dic10C2×Q814yes—
194C2×Dic10C2412yes—
195C5×M16—28no—
196C2×C5×C8—24no—
197C2×C5×C8C2414yes—
198C5×C4⋊C4—24no—
199C5×C4⋊C4—22no—
200C5×C4⋊C4—21no—
201C5×C4⋊C4—21no—
202C5×C4⋊C4C2414yes—
203C5×C4⋊C4C2412yes—
204C5×C4⋊C4C2412yes—
205C5×C22⋊C4—24no—
206C42×C5—24no—
207#685—24no—
208#685—24no—
209C4⋊Dic5—24no—
210C4⋊Dic5—22no—
211C4⋊Dic5—28no—
212C4⋊Dic5—28no—
213C4⋊Dic5C2414yes—
214#683—24no—
215#683—24no—
216#683—28no—
217#683—28no—
218C4×Dic5—24no—
219C4×Dic5—24no—
220C4×Dic5C2×D414yes—
221C4×Dic5C2×D414yes—
222C4×Dic5C2×D412yes—
223C4×Dic5C2×D412yes—
224C4×Dic5C2412yes—
225C4×Dic5C2412yes—
226#681—28no—
227C2×#168—24no—
228C2×#168C2414yes—
229C22×#41—51no—
230C22×M16—51no—
231C22×M16C2×D511yescharacteristic
232C2×#125—52no—
233C2×#125—52no—
234C2×#125—58no—
235C2×#125—101no—
236C2×#125—104no—
237C2×#125—101no—
238C4×#41—58no—
239C4×#41—104no—
240C2×C4×Q8—51no—
241C2×C4×Q8—51no—
242C2×C4×D4—51no—
243C2×C4×D4—51no—
244C2×C4×D4—101no—
245C2×C4×D4—101no—
246C2×#117—101no—
247C2×#117C2×D511yescharacteristic
248C22×C4⋊C4—51no—
249C22×C4⋊C4—101no—
250C22×C4⋊C4—101no—
251C22×C4⋊C4C2×D511yescharacteristic
252C22×C42—51no—
253C22×C42—101no—
254#479—516no—
255#479—516no—
256#478—516no—
257#478—516no—
258#428—58no—
259#428C2×D518yescomplemented
260C2×#107—52no—
261C2×#107C2×D512yes—
262C2×#106—52no—
263C2×#106C2×D512yes—
264#423—58no—
265C2×C4⋊C8—52no—
266C2×C4⋊C8—52no—
267#417—58no—
268#417—58no—
269C2×#103—52no—
270C2×#103—52no—
271C2×#102—52no—
272C2×#102—52no—
273C23×C5—22no—
274C23×C5—22no—
275C23×C5C22⋊Q814yes—
276C23×C5C22×D412yes—
277C23×C5C22×D411yescharacteristic
278C22×D5—24no—
279C22×D5—22no—
280C22×D5—42no—
281C22×D5—42no—
282C2×C4×C5—22no—
283C2×C4×C5—22no—
284C2×C4×C5—24no—
285C2×C4×C5—48no—
286C2×C4×C5—42no—
287C2×C4×C5—42no—
288C2×C4×C5—44no—
289C2×C4×C5C22⋊Q814yes—
290C2×C4×C5C22⋊Q818yes—
291C2×C4×C5C22×D412yes—
292C2×C4×C5C22×D411yescharacteristic
293C2×C4×C5C22×Q812yes—
294C2×C4×C5C22×Q814yes—
295C2×C4×C5C2×#4112yes—
296C2×C4×C5C2×#4114yes—
297C2×C4×C5C2511yescharacteristic
298C22⋊D5—216no—
299C2×Dic5—24no—
300C2×Dic5—22no—
301C2×Dic5—22no—
302C2×Dic5—22no—
303C2×Dic5—22no—
304C2×Dic5—22no—
305C2×Dic5—24no—
306C2×Dic5—44no—
307C2×Dic5—44no—
308C2×Dic5—48no—
309C2×Dic5—44no—
310C2×Dic5C22×D412yes—
311C2×Dic5C22×D412yes—
312C2×Dic5C22×D412yes—
313C2×Dic5C22×D412yes—
314D20—24no—
315D20—22no—
316D20C22×Q814yes—
317C4×D5—216no—
318C4×D5—48no—
319C4×D5—48no—
320Dic10—24no—
321Dic10—22no—
322Dic10C22×Q814yes—
323C5×C8—24no—
324#168—24no—
325C2×#41—58no—
326C2×#41—54no—
327C2×#41—104no—
328C2×#41—102no—
329C22×Q8—51no—
330C22×D4—51no—
331C22×D4—101no—
332C23×C4—21no—
333C23×C4—51no—
334C23×C4—51no—
335C23×C4—101no—
336C23×C4—101no—
337C23×C4—101no—
338C23×C4—101no—
339C23×C4C22×D511yescharacteristic
340C2×M16—24no—
341C2×M16—54no—
342C2×M16—104no—
343C2×M16C22×D514yes—
344C22×C8—52no—
345C22×C8C22×D512yes—
346#125—58no—
347#125—58no—
348#125—108no—
349#125—108no—
350#125—104no—
351#125—104no—
352C4×Q8—54no—
353C4×Q8—54no—
354C4×Q8—102no—
355C4×Q8—102no—
356C4×D4—54no—
357C4×D4—54no—
358C4×D4—102no—
359C4×D4—102no—
360C4×D4—108no—
361C4×D4—108no—
362#117—22no—
363#117—108no—
364#117C22×D514yes—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—52no—
368C2×C4⋊C4—54no—
369C2×C4⋊C4—108no—
370C2×C4⋊C4—102no—
371C2×C4⋊C4—102no—
372C2×C4⋊C4—108no—
373C2×C4⋊C4—102no—
374C2×C4⋊C4—102no—
375C2×C4⋊C4—101no—
376C2×C4⋊C4—101no—
377C2×C4⋊C4—102no—
378C2×C4⋊C4—101no—
379C2×C4⋊C4C22×D512yes—
380C2×C4⋊C4C22×D514yes—
381C2×C4⋊C4C22×D511yescharacteristic
382C2×C4⋊C4C22×D511yescharacteristic
383C2×C22⋊C4—21no—
384C2×C22⋊C4—101no—
385C2×C22⋊C4—101no—
386C2×C42—21no—
387C2×C42—51no—
388C2×C42—54no—
389C2×C42—51no—
390C2×C42—51no—
391C2×C42—51no—
392C2×C42—54no—
393C2×C42—108no—
394C2×C42—101no—
395C2×C42—101no—
396#107—58no—
397#107C22×D518yes—
398#106—58no—
399#106C22×D518yes—
400C4⋊C8—58no—
401C4⋊C8—58no—
402#103—58no—
403#103—58no—
404#102—58no—
405#102—58no—
406C22×C5—28no—
407C22×C5—22no—
408C22×C5—22no—
409C22×C5C2×C22⋊Q814yes—
410C22×C5C2×C22⋊Q812yes—
411C22×C5#57512yes—
412C22×C5#57512yes—
413C22×C5C23×D411yescharacteristic
414C2×D5—24no—
415C2×D5—48no—
416C2×D5—42no—
417C2×D5—42no—
418C4×C5—42no—
419C4×C5—42no—
420C4×C5—44no—
421C4×C5C2×C22⋊Q812yes—
422C4×C5C2×C22⋊Q814yes—
423C4×C5C23×Q811yescharacteristic
424C4×C5C22×#4111yescharacteristic
425Dic5—24no—
426Dic5—22no—
427Dic5—22no—
428Dic5—22no—
429Dic5—22no—
430Dic5—44no—
431C24—101no—
432C24D4×D511yescharacteristic
433#41—516no—
434#41—108no—
435#41—1016no—
436C2×Q8—54no—
437C2×Q8—52no—
438C2×Q8—102no—
439C2×Q8—101no—
440C2×D4—54no—
441C2×D4—52no—
442C2×D4—104no—
443C2×D4—108no—
444C2×D4—102no—
445C2×D4—101no—
446C22×C4—24no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—44no—
451C22×C4—51no—
452C22×C4—51no—
453C22×C4—51no—
454C22×C4—51no—
455C22×C4—54no—
456C22×C4—54no—
457C22×C4—104no—
458C22×C4—104no—
459C22×C4—101no—
460C22×C4—104no—
461C22×C4—108no—
462C22×C4—104no—
463C22×C4—102no—
464C22×C4—204no—
465C22×C4—204no—
466C22×C4—202no—
467C22×C4—202no—
468C22×C4—204no—
469C22×C4#70914yes—
470C22×C4D4×D512yes—
471C22×C4D4×D511yescharacteristic
472C22×C4Q8×D514yes—
473C22×C4C23×D512yes—
474C22×C4C23×D511yescharacteristic
475M16—28no—
476M16—108no—
477C2×C8—24no—
478C2×C8—54no—
479C2×C8—104no—
480C2×C8C23×D514yes—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—54no—
486C4⋊C4—108no—
487C4⋊C4—108no—
488C4⋊C4—108no—
489C4⋊C4—108no—
490C4⋊C4—104no—
491C4⋊C4—104no—
492C4⋊C4—104no—
493C4⋊C4—102no—
494C4⋊C4C23×D514yes—
495C4⋊C4C23×D512yes—
496C4⋊C4C23×D512yes—
497C22⋊C4—24no—
498C22⋊C4—104no—
499C22⋊C4—104no—
500C42—24no—
501C42—54no—
502C42—54no—
503C42—52no—
504C42—52no—
505C42—52no—
506C42—52no—
507C42—104no—
508C42—104no—
509C2×C5—24no—
510C2×C5#308014yescomplemented
511C2×C5C22×C22⋊Q811yescharacteristic
512C2×C5C2×#57511yescharacteristic
513C2×C5C2×#57511yescharacteristic
514D5—44no—
515C23—22no—
516C23—22no—
517C23—104no—
518C23—102no—
519C23—202no—
520C23—202no—
521C23#404214yes—
522C23C2×D4×D512yes—
523C23C2×D4×D511yescharacteristic
524Q8—54no—
525Q8—104no—
526Q8—102no—
527D4—54no—
528D4—1016no—
529D4—104no—
530D4—102no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—48no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—52no—
539C2×C4—52no—
540C2×C4—52no—
541C2×C4—52no—
542C2×C4—104no—
543C2×C4—102no—
544C2×C4—102no—
545C2×C4—102no—
546C2×C4—102no—
547C2×C4—102no—
548C2×C4—1016no—
549C2×C4—104no—
550C2×C4—208no—
551C2×C4—208no—
552C2×C4—204no—
553C2×C4—204no—
554C2×C4—208no—
555C2×C4—204no—
556C2×C4#404214yes—
557C2×C4#404218yes—
558C2×C4C2×#70912yes—
559C2×C4C2×#70914yes—
560C2×C4C2×D4×D512yes—
561C2×C4C2×D4×D511yescharacteristic
562C2×C4C2×Q8×D512yes—
563C2×C4C2×Q8×D514yes—
564C2×C4C24×D511yescharacteristic
565C8—24no—
566C8—104no—
567C5C2×#308011yescharacteristic, complemented, Sylow
568C22—28no—
569C22—22no—
570C22—22no—
571C22—104no—
572C22—208no—
573C22—202no—
574C22—202no—
575C22C2×#404214yes—
576C22C2×#404212yes—
577C22#6610712yes—
578C22#6611112yes—
579C22C22×D4×D511yescharacteristic
580C4—44no—
581C4—42no—
582C4—42no—
583C4—104no—
584C4—102no—
585C4—102no—
586C4—102no—
587C4—102no—
588C4—204no—
589C4C2×#404214yes—
590C4C2×#404212yes—
591C4C22×#70911yescharacteristic
592C4C22×Q8×D511yescharacteristic
593C2—24no—
594C2—204no—
595C2#1061097214yesdirect product factor
596C2C22×#404211yescharacteristic
597C2C2×#6610711yescharacteristic
598C2C2×#6611111yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#428C2
socleC23×C5C22×D4
prime radicalC22×#709C22
hypercenterC2×#428D5
hypocenterC5C2×#3080
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
2. C22×#4036C2semidirect
3. C2×C5×#428C2semidirect
5. C2×#65407C2semidirect
6. C2×#65284C2semidirect
8. #10610972C2direct (N × H)
9. C2×#65166C2semidirect
10. C2×#65180C2semidirect
11. C2×#65146C2semidirect
12. C2×#65160C2semidirect
31. C2×#4036C22semidirect
38. C5×#428C22semidirect
42. #65407C22semidirect
47. #65284C22semidirect
53. #65180C22semidirect
54. #65166C22semidirect
55. #65160C22semidirect
56. #65146C22semidirect
147. C2×#428D5semidirect
259. #428C2×D5semidirect
510. C2×C5#3080semidirect
567. C5C2×#3080semidirect
595. C2#10610972direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#658562
C22×#40362
C2×C5×#4282
C2×#654222
C2×#654072
C2×#652842
C2×#652552
#106109722
C2×#651662
C2×#651802
C2×#651462
C2×#651602
C2×#30805