Finite Group Databasebeta

← C2×#10599551 #12100104

Subgroups — C2×#10599551 #12100104

Order 1280.

1706 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65855C211yesmaximal, characteristic
2C22×#3955C211yesmaximal, characteristic
3C2×#65410C211yesmaximal, characteristic, complemented
4C2×#65284C211yesmaximal, characteristic, complemented
5C2×#65228C211yesmaximal, characteristic
6C2×C5×#339C211yesmaximal, characteristic, complemented
7C2×#64743C211yesmaximal, characteristic, complemented
8#10599551C218yesmaximal, direct product factor
9C23×D20C2211yescharacteristic
10C22×C5×M16C2211yescharacteristic
11C2×C5×#117C2211yescharacteristic
12C22×C42×C5C2211yescharacteristic
13C22×#685—22no—
14C2×#4072C2211yescharacteristic
15C22×C4⋊Dic5C2211yescharacteristic
16C22×#681C2211yescharacteristic
17#65855—22no—
18#65855—22no—
19#65855—22no—
20#65855C2214yes—
21C2×#4039—21no—
22C2×#4039—21no—
23C2×D5×C22⋊C4—22no—
24C2×C4×D20—21no—
25C2×C4×D20—21no—
26C2×#3955C418yescomplemented
27C2×#3955C2214yes—
28#65410—22no—
29#65410C2214yescomplemented
30C2×#3950—21no—
31C2×#3949—21no—
32#65284—22no—
33#65284C2214yescomplemented
34C2×#3940—21no—
35C2×#3939—21no—
36#65228—22no—
37#65228C2214yes—
38C5×#339C2214yescomplemented
39#64743C2214yescomplemented
40C2×#1856—51nomaximal, Sylow
41C24×D5D412yes—
42C23×C4×C5—21no—
43C23×C4×C5C2311yescharacteristic
44C23×Dic5—21no—
45C22×D20—21no—
46C22×D20—21no—
47C22×D20—21no—
48C22×D20—21no—
49C22×D20—22no—
50C22×D20—22no—
51C22×D20C2×C414yes—
52C22×D20D412yes—
53C22×D20Q814yes—
54C22×D20C2312yes—
55C22×C4×D5—22no—
56C22×C4×D5—22no—
57C2×C5×M16—21no—
58C2×C5×M16—22no—
59C2×C5×M16C2×C414yescomplemented
60C2×C5×M16C2312yes—
61C22×C5×C8—21no—
62C5×#117—21no—
63C5×#117—21no—
64C5×#117D412yes—
65C5×#117C2312yes—
66C2×C5×C4⋊C4D411yescharacteristic
67C2×C5×C4⋊C4D411yescharacteristic
68C2×C5×C22⋊C4—21no—
69C2×C42×C5—21no—
70C2×C42×C5—22no—
71C2×C42×C5—22no—
72C2×C42×C5—21no—
73C2×C42×C5C2312yes—
74C2×C22⋊Dic5—21no—
75C2×#685—22no—
76C2×#685—22no—
77C2×#685—28no—
78C2×#685—48no—
79#4072—21no—
80#4072—21no—
81#4072D412yes—
82#4072C2312yes—
83C2×C4⋊Dic5—21no—
84C2×C4⋊Dic5—21no—
85C2×C4⋊Dic5—21no—
86C2×C4⋊Dic5—21no—
87C2×C4⋊Dic5—21no—
88C2×C4⋊Dic5D411yescharacteristic
89C2×C4⋊Dic5D411yescharacteristic
90C2×C4⋊Dic5C2312yes—
91C2×C4×Dic5—21no—
92C2×#681—21no—
93C2×#681—22no—
94C2×#681C2×C414yescomplemented
95C2×#681C2312yes—
96C22×#168—21no—
97#4039—24no—
98#4039—24no—
99#4039—24no—
100#4039—24no—
101D5×C22⋊C4—28no—
102D5×C22⋊C4—28no—
103C4×D20—24no—
104C4×D20—24no—
105C4×D20—24no—
106C4×D20—24no—
107#3955C2×C4116yescomplemented
108#3950—24no—
109#3949—24no—
110#3940—24no—
111#3939—24no—
112C2×#518—51no—
113C2×#428—51no—
114C2×#428—51no—
115C22×#100—51no—
116C2×#390—51no—
117#1856—58no—
118C2×#339—51no—
119C2×#339D511yescharacteristic, complemented
120C24×C5C2×D411yescharacteristic
121C23×D5—22no—
122C23×D5—22no—
123C23×D5—24no—
124C23×D5—44no—
125C23×D5—44no—
126C23×D5C4⋊C418yes—
127C23×D5C2×D414yes—
128C22×C4×C5—24no—
129C22×C4×C5—21no—
130C22×C4×C5—21no—
131C22×C4×C5—22no—
132C22×C4×C5—42no—
133C22×C4×C5—42no—
134C22×C4×C5C22×C412yes—
135C22×C4×C5C2×D411yescharacteristic
136C22×C4×C5C2×D411yescharacteristic
137C22×C4×C5C2×D411yescharacteristic
138C22×C4×C5C2×D411yescharacteristic
139C22×C4×C5C2×D411yescharacteristic
140C22×C4×C5C2×Q812yes—
141C22×C4×C5#4112yes—
142C22×C4×C5#4112yes—
143C22×C4×C5C2411yescharacteristic
144C22×Dic5—21no—
145C22×Dic5—21no—
146C22×Dic5—22no—
147C22×Dic5—44no—
148C22×Dic5—41no—
149C22×Dic5—41no—
150C2×D20—24no—
151C2×D20—24no—
152C2×D20—24no—
153C2×D20—24no—
154C2×D20—22no—
155C2×D20—22no—
156C2×D20—24no—
157C2×D20—24no—
158C2×D20—24no—
159C2×D20—24no—
160C2×D20—24no—
161C2×D20—22no—
162C2×D20—22no—
163C2×D20—22no—
164C2×D20—22no—
165C2×D20C4⋊C418yes—
166C2×D20C22×C414yes—
167C2×D20C2×Q814yes—
168C2×C4×D5—28no—
169C2×C4×D5—28no—
170C2×C4×D5—84no—
171C2×C4×D5—84no—
172C5×M16—24no—
173C5×M16—22no—
174C5×M16C22×C414yescomplemented
175C2×C5×C8—24no—
176C2×C5×C8—22no—
177C5×C4⋊C4—21no—
178C5×C4⋊C4—21no—
179C5×C4⋊C4C2×D412yes—
180C5×C4⋊C4C2×D412yes—
181C5×C22⋊C4—22no—
182C5×C22⋊C4—22no—
183C42×C5—22no—
184C42×C5—22no—
185C42×C5—22no—
186C42×C5—22no—
187C42×C5—42no—
188C22⋊Dic5—22no—
189C22⋊Dic5—22no—
190#685—28no—
191#685—28no—
192#685—416no—
193C4⋊Dic5—21no—
194C4⋊Dic5—21no—
195C4⋊Dic5—22no—
196C4⋊Dic5—22no—
197C4⋊Dic5—41no—
198C4⋊Dic5—41no—
199C4⋊Dic5C2×D412yes—
200C4⋊Dic5C2×D412yes—
201C4×Dic5—22no—
202C4×Dic5—22no—
203#681—24no—
204#681—22no—
205#681C22×C414yescomplemented
206C2×#168—24no—
207C2×#168—22no—
208C23×D4—51no—
209C22×M16—51no—
210C22×M16C2×D511yescharacteristic
211#518—54no—
212#518—102no—
213#518—102no—
214#518—102no—
215C2×C4×D4—101no—
216C2×C4×D4—101no—
217C2×C4×D4—101no—
218C2×C4×D4—101no—
219C2×#117—51no—
220C2×#117C2×D511yescharacteristic
221C22×C4⋊C4—51no—
222C22×C22⋊C4—102no—
223C22×C22⋊C4—102no—
224C22×C42C2×D511yescharacteristic
225#428—54no—
226#428—54no—
227#428—102no—
228#428—102no—
229C2×#107—101no—
230C2×#107—101no—
231C2×#106—101no—
232C2×#106—101no—
233C2×#100—58no—
234C2×#100—54no—
235#390—54no—
236#390—102no—
237#339—54no—
238#339C2×D514yescomplemented
239C23×C5—22no—
240C23×C5—22no—
241C23×C5C2×C4⋊C412yes—
242C23×C5#12011yescharacteristic
243C23×C5#12011yescharacteristic
244C23×C5#12312yes—
245C23×C5C22×D411yescharacteristic
246C22×D5—24no—
247C22×D5—28no—
248C22×D5—24no—
249C22×D5—28no—
250C22×D5—44no—
251C22×D5—44no—
252C22×D5—44no—
253C22×D5—416no—
254C22×D5—44no—
255C22×D5—88no—
256C22×D5—82no—
257C22×D5—82no—
258C22×D5C2×C4⋊C418yes—
259C2×C4×C5—24no—
260C2×C4×C5—22no—
261C2×C4×C5—22no—
262C2×C4×C5—48no—
263C2×C4×C5—42no—
264C2×C4×C5—42no—
265C2×C4×C5—42no—
266C2×C4×C5—42no—
267C2×C4×C5C2×C4⋊C412yes—
268C2×C4×C5C4×D412yes—
269C2×C4×C5C4×D412yes—
270C2×C4×C5C4×D412yes—
271C2×C4×C5C4×D412yes—
272C2×C4×C5#12011yescharacteristic
273C2×C4×C5#12011yescharacteristic
274C2×C4×C5C22⋊Q812yes—
275C2×C4×C5C22⋊Q812yes—
276C2×C4×C5C22⋊Q812yes—
277C2×C4×C5C22⋊Q812yes—
278C2×C4×C5#12312yes—
279C2×C4×C5C23×C411yescharacteristic
280C2×C4×C5C22×D411yescharacteristic
281C2×C4×C5C22×D411yescharacteristic
282C2×C4×C5C22×Q811yescharacteristic
283C2×C4×C5C2×#4111yescharacteristic
284C2×C4×C5C2×#4111yescharacteristic
285C2×Dic5—22no—
286C2×Dic5—22no—
287C2×Dic5—42no—
288C2×Dic5—42no—
289C2×Dic5—44no—
290C2×Dic5—42no—
291C2×Dic5—82no—
292C2×Dic5—82no—
293D20—24no—
294D20—24no—
295D20—24no—
296D20—24no—
297D20—28no—
298D20—28no—
299C4×D5—88no—
300C4×D5—88no—
301C5×C8—24no—
302#168—24no—
303C25—52no—
304C22×D4—54no—
305C22×D4—52no—
306C22×D4—54no—
307C22×D4—52no—
308C22×D4—101no—
309C22×D4—101no—
310C22×D4—102no—
311C22×D4—101no—
312C22×D4—101no—
313C22×D4—102no—
314C23×C4—21no—
315C23×C4—102no—
316C23×C4—102no—
317C23×C4—101no—
318C23×C4C22×D511yescharacteristic
319C2×M16—21no—
320C2×M16—22no—
321C2×M16—54no—
322C2×M16—52no—
323C2×M16—101no—
324C2×M16—102no—
325C2×M16C4×D514yescomplemented
326C2×M16C22×D512yes—
327C22×C8—21no—
328C22×C8—101no—
329C4×D4—104no—
330C4×D4—104no—
331C4×D4—104no—
332C4×D4—104no—
333C4×D4—104no—
334C4×D4—104no—
335C4×D4—104no—
336C4×D4—104no—
337#117—21no—
338#117—21no—
339#117—52no—
340#117—52no—
341#117—101no—
342#117—101no—
343#117D2012yes—
344#117C22×D512yes—
345C2×C4⋊C4—51no—
346C2×C4⋊C4—51no—
347C2×C4⋊C4—52no—
348C2×C4⋊C4—101no—
349C2×C4⋊C4—101no—
350C2×C4⋊C4—101no—
351C2×C4⋊C4—101no—
352C2×C4⋊C4—101no—
353C2×C4⋊C4D2011yescharacteristic
354C2×C4⋊C4D2011yescharacteristic
355C2×C22⋊C4—21no—
356C2×C22⋊C4—108no—
357C2×C22⋊C4—108no—
358C2×C22⋊C4—102no—
359C2×C22⋊C4—102no—
360C2×C22⋊C4—101no—
361C2×C22⋊C4—108no—
362C2×C22⋊C4—208no—
363C2×C42—21no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—21no—
367C2×C42—101no—
368C2×C42C22×D512yes—
369#107—104no—
370#107—104no—
371#106—104no—
372#106—104no—
373#100—516no—
374C22×C5—24no—
375C22×C5—24no—
376C22×C5—21no—
377C22×C5—21no—
378C22×C5—21no—
379C22×C5—21no—
380C22×C5#38512yes—
381C22×C5#38512yes—
382C22×C5#45312yes—
383C22×C5#45512yes—
384C22×C5C22×C4⋊C411yescharacteristic
385C22×C5C2×#12011yescharacteristic
386C22×C5C2×#12311yescharacteristic
387C2×D5—28no—
388C2×D5—416no—
389C2×D5—44no—
390C2×D5—44no—
391C2×D5—48no—
392C2×D5—84no—
393C2×D5—88no—
394C4×C5—44no—
395C4×C5—42no—
396C4×C5—42no—
397C4×C5#38512yes—
398C4×C5#38512yes—
399C4×C5C2×C4×D411yescharacteristic
400C4×C5C2×C4×D411yescharacteristic
401C4×C5C2×C22⋊Q811yescharacteristic
402C4×C5C2×C22⋊Q811yescharacteristic
403Dic5—42no—
404Dic5—42no—
405Dic5—82no—
406C24—54no—
407C24—58no—
408C24—102no—
409C24—102no—
410C24—104no—
411C24—204no—
412C24—204no—
413C24D4×D511yescharacteristic
414C2×D4—54no—
415C2×D4—54no—
416C2×D4—58no—
417C2×D4—102no—
418C2×D4—102no—
419C2×D4—102no—
420C2×D4—102no—
421C2×D4—104no—
422C2×D4—104no—
423C2×D4—104no—
424C2×D4—104no—
425C2×D4—102no—
426C2×D4—102no—
427C2×D4—104no—
428C2×D4—104no—
429C2×D4—104no—
430C2×D4—104no—
431C2×D4—104no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—108no—
439C22×C4—108no—
440C22×C4—101no—
441C22×C4—101no—
442C22×C4—102no—
443C22×C4—204no—
444C22×C4—201no—
445C22×C4—201no—
446C22×C4—404no—
447C22×C4—404no—
448C22×C4C2×C4×D512yes—
449C22×C4C2×D2011yescharacteristic
450C22×C4C2×D2011yescharacteristic
451C22×C4#70912yes—
452C22×C4D4×D511yescharacteristic
453C22×C4D4×D511yescharacteristic
454C22×C4D4×D511yescharacteristic
455C22×C4#71112yes—
456C22×C4Q8×D512yes—
457C22×C4C23×D511yescharacteristic
458M16—22no—
459M16—24no—
460M16—54no—
461M16—102no—
462M16—104no—
463M16C2×C4×D514yescomplemented
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—102no—
467C2×C8—104no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—52no—
471C4⋊C4—52no—
472C4⋊C4—101no—
473C4⋊C4—101no—
474C4⋊C4—102no—
475C4⋊C4—102no—
476C4⋊C4—201no—
477C4⋊C4—201no—
478C4⋊C4C2×D2012yes—
479C4⋊C4C2×D2012yes—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—108no—
483C22⋊C4—108no—
484C22⋊C4—102no—
485C22⋊C4—102no—
486C22⋊C4—2016no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—42no—
492C42—102no—
493C42—102no—
494C2×C5—21no—
495C2×C5—21no—
496C2×C5—22no—
497C2×C5#185614yescomplemented
498C2×C5C2×#38511yescharacteristic
499C2×C5C2×#45311yescharacteristic
500C2×C5C2×#45511yescharacteristic
501D5—48no—
502C23—22no—
503C23—22no—
504C23—58no—
505C23—104no—
506C23—108no—
507C23—104no—
508C23—108no—
509C23—204no—
510C23—204no—
511C23—2016no—
512C23—204no—
513C23—204no—
514C23—402no—
515C23—402no—
516C23—408no—
517C23#402811yescharacteristic
518C23#402811yescharacteristic
519C23#402912yes—
520C23D5×C4⋊C412yes—
521C23C2×D4×D511yescharacteristic
522D4—104no—
523D4—104no—
524D4—104no—
525D4—104no—
526D4—108no—
527D4—108no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—42no—
532C2×C4—48no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—102no—
537C2×C4—102no—
538C2×C4—202no—
539C2×C4—202no—
540C2×C4—204no—
541C2×C4—202no—
542C2×C4—408no—
543C2×C4—408no—
544C2×C4—402no—
545C2×C4—402no—
546C2×C4C4×D2012yes—
547C2×C4C4×D2012yes—
548C2×C4#402712yes—
549C2×C4#402712yes—
550C2×C4#402811yescharacteristic
551C2×C4#402811yescharacteristic
552C2×C4#402912yes—
553C2×C4D5×C4⋊C412yes—
554C2×C4#404212yes—
555C2×C4#404212yes—
556C2×C4#404312yes—
557C2×C4#404312yes—
558C2×C4C22×C4×D511yescharacteristic
559C2×C4C22×D2011yescharacteristic
560C2×C4C2×#70911yescharacteristic
561C2×C4C2×D4×D511yescharacteristic
562C2×C4C2×#71111yescharacteristic
563C2×C4C2×Q8×D511yescharacteristic
564C8—24no—
565C8—104no—
566C5C2×#185611yescharacteristic, complemented, Sylow
567C22—21no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—24no—
572C22—24no—
573C22—108no—
574C22—208no—
575C22—204no—
576C22—204no—
577C22—2016no—
578C22—404no—
579C22—408no—
580C22#6495712yes—
581C22#6495712yes—
582C22#6511112yes—
583C22#6511412yes—
584C22C2×#402811yescharacteristic
585C22C2×#402911yescharacteristic
586C22C2×D5×C4⋊C411yescharacteristic
587C4—44no—
588C4—42no—
589C4—42no—
590C4—202no—
591C4—202no—
592C4—402no—
593C4#6495712yes—
594C4#6495712yes—
595C4C2×C4×D2011yescharacteristic
596C4C2×#402711yescharacteristic
597C4C2×#404211yescharacteristic
598C4C2×#404311yescharacteristic
599C2—21no—
600C2—21no—
601C2—22no—
602C2—208no—
603C2#1059955114yesdirect product factor
604C2C2×#6495711yescharacteristic
605C2C2×#6511111yescharacteristic
606C2C2×#6511411yescharacteristic
subgroupG/H
centerC23#4028
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC2×C5×#339C2
socleC23×C5#120
prime radicalC23×D20C22
cosocleC22×C4×C5C24
hypercenterC2×#339D5
hypocenterC5C2×#1856
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#339

Complements

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

NH ≅ G/Ntype
3. C2×#65410C2semidirect
4. C2×#65284C2semidirect
6. C2×C5×#339C2semidirect
7. C2×#64743C2semidirect
8. #10599551C2direct (N × H)
26. C2×#3955C4semidirect
29. #65410C22semidirect
33. #65284C22semidirect
38. C5×#339C22semidirect
39. #64743C22semidirect
59. C2×C5×M16C2×C4semidirect
94. C2×#681C2×C4semidirect
107. #3955C2×C4semidirect
119. C2×#339D5semidirect
174. C5×M16C22×C4semidirect
205. #681C22×C4semidirect
238. #339C2×D5semidirect
325. C2×M16C4×D5semidirect
463. M16C2×C4×D5semidirect
497. C2×C5#1856semidirect
566. C5C2×#1856semidirect
603. C2#10599551direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#658552
C22×#39552
C2×#654102
C2×#652842
C2×#652282
C2×C5×#3392
C2×#647432
#105995512
C2×#18565