Finite Group Databasebeta

← C2×#10612188 #12967372

Subgroups — C2×#10612188 #12967372

Order 1280.

2170 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66062C211yesmaximal, characteristic, complemented
2C2×C4×#713C211yesmaximal, characteristic
3C2×C5×#462C211yesmaximal, characteristic, complemented
4C2×#65476C212yesmaximal
5C2×Q16×Dic5C212yesmaximal, complemented
6C2×#65372C211yesmaximal, characteristic, complemented
7C2×#65309C211yesmaximal, characteristic, complemented
8#10612188C2116yesmaximal, direct product factor
9C2×#65135C211yesmaximal, characteristic, complemented
10C2×#65122C211yesmaximal, characteristic, complemented
11C2×#65103C212yesmaximal, complemented
12C2×#65077C212yesmaximal, complemented
13C22×#713C2211yescharacteristic
14C22×C5×Q16C2212yes—
15C2×C5×C4⋊Q8C2211yescharacteristic
16C2×C4×C5×Q8C2211yescharacteristic
17C2×#4092C2212yes—
18C2×Q8×Dic5C2211yescharacteristic
19C2×Q8×Dic5C2212yes—
20C22×#688C2212yes—
21#66062—24no—
22#66062C2218yescomplemented
23C4×#713—24no—
24C4×#713C2218yes—
25C2×#4043—21no—
26C2×#4039—21no—
27C2×#4039C2211yescharacteristic
28C2×#4038—21no—
29C2×#4038—21no—
30C2×#4038C2211yescharacteristic
31C2×#4036—21no—
32C2×#4033—21no—
33C2×C4×D20—21no—
34C2×C4×D20C2211yescharacteristic
35C2×C42×D5—21no—
36C2×C42×D5C2211yescharacteristic
37C2×#4015C2211yescharacteristic
38C5×#462C2218yescomplemented
39C2×C5×C4⋊C8C2211yescharacteristic
40C2×C5×#103C2212yes—
41C2×#3967C2212yes—
42C2×#3967C2212yes—
43#65476C22116yes—
44Q16×Dic5C22116yescomplemented
45C2×#3953C2212yes—
46C2×#3952C2212yes—
47C2×#3950C2212yes—
48C2×C8×Dic5C2212yes—
49#65372C2218yescomplemented
50#65309C2218yescomplemented
51C2×#3940C2212yes—
52C2×C4⋊#168C2211yescharacteristic
53#65135C2218yescomplemented
54#65122C2218yescomplemented
55#65103C22116yescomplemented
56#65077C22116yescomplemented
57C2×#3107—51nomaximal, Sylow
58C22×C5×Q8C2312yes—
59C22×C5×Q8C2311yescharacteristic
60C2×#713—24no—
61C2×#713—24no—
62C2×#713—22no—
63C2×#713C2314yes—
64C22×D20—21no—
65C22×D20C2311yescharacteristic
66C22×C4×D5—21no—
67C22×C4×D5—21no—
68C22×C4×D5—21no—
69C22×C4×D5D411yescharacteristic
70C22×C4×D5D411yescharacteristic
71C22×C4×D5C2311yescharacteristic
72C22×Dic10—21no—
73C2×C5×Q16—28no—
74C2×C5×Q16C2318yes—
75C22×C5×C8C2312yes—
76C5×C4⋊Q8—22no—
77C5×C4⋊Q8C2314yes—
78C4×C5×Q8—22no—
79C4×C5×Q8C2314yes—
80C2×C5×C4⋊C4—21no—
81C2×C5×C4⋊C4—21no—
82C2×C5×C4⋊C4—21no—
83C2×C5×C4⋊C4C2311yescharacteristic
84C2×C42×C5—21no—
85C2×C42×C5C2311yescharacteristic
86#4092—24no—
87#4092C2318yes—
88Q8×Dic5—24no—
89Q8×Dic5—22no—
90Q8×Dic5C2318yes—
91Q8×Dic5C2314yes—
92C2×#688—28no—
93C2×#688C2318yes—
94C2×#685—21no—
95C2×#685—22no—
96C2×#685—21no—
97C2×#685—21no—
98C2×C4⋊Dic5—21no—
99C2×C4⋊Dic5—21no—
100C2×C4⋊Dic5—22no—
101C2×C4⋊Dic5C2312yes—
102C2×C4⋊Dic5C2311yescharacteristic
103C2×#683—22no—
104C2×C4×Dic5—21no—
105C2×C4×Dic5—21no—
106C2×C4×Dic5—22no—
107C2×C4×Dic5D411yescharacteristic
108C2×C4×Dic5D411yescharacteristic
109C2×C4×Dic5C2311yescharacteristic
110C22×#168C2312yes—
111#4043—28no—
112#4039—28no—
113#4039—22no—
114#4039C2314yes—
115#4038—28no—
116#4038—22no—
117#4038—28no—
118#4038C2314yes—
119#4036—28no—
120#4033—28no—
121C4×D20—28no—
122C4×D20—22no—
123C4×D20C2314yes—
124C42×D5—28no—
125C42×D5D414yes—
126C42×D5C2314yes—
127#4015—22no—
128#4015C2314yes—
129C5×C4⋊C8C2314yes—
130C5×#103C2318yes—
131#3967C2318yes—
132#3967C2318yes—
133#3953C2318yes—
134#3952C2318yes—
135#3950C2318yes—
136C8×Dic5C2318yes—
137#3940C2318yes—
138C4⋊#168C2314yes—
139C2×#533—51no—
140C2×C4×#41—51no—
141#3107—516no—
142C2×#485—52no—
143C2×#475—51no—
144C2×#474—51no—
145C2×#462D511yescharacteristic, complemented
146C2×#461—51no—
147C2×#452—52no—
148C2×C4×Q16—52no—
149C2×C4×SD16—52no—
150C2×#431—51no—
151C23×D5—21no—
152C23×D5C2×D411yescharacteristic
153C2×C5×Q8—21no—
154C2×C5×Q8—22no—
155C2×C5×Q8—22no—
156C2×C5×Q8—24no—
157C2×C5×Q8—24no—
158C2×C5×Q8#4114yes—
159C2×C5×Q8C2414yes—
160C2×C5×Q8C2412yes—
161C22×C4×C5—21no—
162C22×C4×C5—21no—
163C22×C4×C5—22no—
164C22×C4×C5C2×D411yescharacteristic
165C22×C4×C5C2×D411yescharacteristic
166C22×C4×C5C2411yescharacteristic
167C22×Dic5—22no—
168C22×Dic5—21no—
169C22×Dic5C2×D411yescharacteristic
170C22×Dic5C2×D411yescharacteristic
171C22×Dic5C2×D411yescharacteristic
172#713—216no—
173#713—28no—
174#713—28no—
175C2×D20—28no—
176C2×D20—24no—
177C2×D20—21no—
178C2×D20—22no—
179C2×D20#4114yes—
180C2×D20C2412yes—
181C2×C4×D5—28no—
182C2×C4×D5—24no—
183C2×C4×D5—22no—
184C2×C4×D5—22no—
185C2×C4×D5—24no—
186C2×C4×D5—24no—
187C2×C4×D5—24no—
188C2×C4×D5—24no—
189C2×C4×D5—44no—
190C2×C4×D5—44no—
191C2×C4×D5C2×D414yes—
192C2×C4×D5C2×D414yes—
193C2×C4×D5C2×D412yes—
194C2×C4×D5C2412yes—
195C2×Dic10—24no—
196C2×Dic10—44no—
197C5×Q16—216no—
198C2×C5×C8—24no—
199C2×C5×C8C2414yes—
200C5×C4⋊C4—24no—
201C5×C4⋊C4—24no—
202C5×C4⋊C4—24no—
203C5×C4⋊C4—21no—
204C5×C4⋊C4C2412yes—
205C42×C5—24no—
206C42×C5C2×D412yes—
207C42×C5C2412yes—
208#688—216no—
209#685—24no—
210#685—28no—
211#685—24no—
212#685—24no—
213C4⋊Dic5—22no—
214C4⋊Dic5—24no—
215C4⋊Dic5—21no—
216C4⋊Dic5—24no—
217C4⋊Dic5—28no—
218C4⋊Dic5C2414yes—
219C4⋊Dic5C2412yes—
220#683—28no—
221C4×Dic5—24no—
222C4×Dic5—24no—
223C4×Dic5—28no—
224C4×Dic5C2×D414yes—
225C4×Dic5C2×D412yes—
226C4×Dic5C2×D414yes—
227C4×Dic5C2412yes—
228C2×#168—24no—
229C2×#168C2414yes—
230C22×#41—51no—
231C22×Q16C2×D512yes—
232C22×SD16—52no—
233#533—58no—
234#533—104no—
235C2×C4⋊Q8—51no—
236C2×C4⋊Q8C2×D511yescharacteristic
237C2×#125—101no—
238C2×C22⋊Q8—52no—
239C2×C22⋊Q8—101no—
240C4×#41—58no—
241C4×#41—104no—
242C2×C4×Q8—51no—
243C2×C4×Q8—52no—
244C2×C4×Q8—101no—
245C2×C4×Q8C2×D511yescharacteristic
246C2×C4×D4—51no—
247C2×C4×D4—51no—
248C2×C4×D4—101no—
249C2×C4×D4—101no—
250C2×#117—51no—
251C2×#117—101no—
252C2×#117—101no—
253C22×C42—51no—
254C22×C42—101no—
255#485—516no—
256#475—58no—
257#474—58no—
258#462C2×D518yescomplemented
259#461—58no—
260#452—516no—
261C4×Q16—516no—
262C4×SD16—516no—
263#431—58no—
264C2×#107—52no—
265C2×#106—52no—
266C2×C4⋊C8—51no—
267C2×C4⋊C8C2×D511yescharacteristic
268C2×#103—52no—
269C2×#103—52no—
270C2×#103C2×D512yes—
271C2×#102—52no—
272C2×C22⋊C8—52no—
273C2×C4×C8—52no—
274C23×C5C22×D411yescharacteristic
275C22×D5—22no—
276C22×D5—24no—
277C22×D5—22no—
278C22×D5—22no—
279C22×D5—22no—
280C22×D5—42no—
281C22×D5—42no—
282C22×D5C22×D412yes—
283C5×Q8—24no—
284C5×Q8—22no—
285C5×Q8—24no—
286C5×Q8—24no—
287C5×Q8—44no—
288C5×Q8C2×#4114yes—
289C2×C4×C5—24no—
290C2×C4×C5—22no—
291C2×C4×C5—22no—
292C2×C4×C5—24no—
293C2×C4×C5—22no—
294C2×C4×C5—44no—
295C2×C4×C5—42no—
296C2×C4×C5#13514yes—
297C2×C4×C5C22×D412yes—
298C2×C4×C5C22×D411yescharacteristic
299C2×C4×C5C22×D412yes—
300C2×C4×C5C2×#4112yes—
301C2×C4×C5C2×#4112yes—
302C2×C4×C5C2511yescharacteristic
303C2×Dic5—22no—
304C2×Dic5—22no—
305C2×Dic5—22no—
306C2×Dic5—24no—
307C2×Dic5—44no—
308C2×Dic5—42no—
309C2×Dic5#13714yes—
310C2×Dic5C22×D412yes—
311C2×Dic5C22×D412yes—
312C2×Dic5C22×D412yes—
313D20—216no—
314D20—24no—
315D20—22no—
316D20C2×#4114yes—
317C4×D5—216no—
318C4×D5—24no—
319C4×D5—28no—
320C4×D5—22no—
321C4×D5—22no—
322C4×D5—28no—
323C4×D5—48no—
324C4×D5—48no—
325Dic10—48no—
326C5×C8—24no—
327#168—24no—
328C2×#41—54no—
329C2×#41—102no—
330C2×#41—104no—
331C2×#41—104no—
332C22×Q8—101no—
333C22×Q8C22×D512yes—
334C22×Q8C22×D511yescharacteristic
335C22×D4—51no—
336C22×D4—101no—
337C23×C4—51no—
338C23×C4—51no—
339C23×C4—51no—
340C23×C4—101no—
341C23×C4—101no—
342C23×C4—101no—
343C2×Q16—28no—
344C2×Q16C22×D518yes—
345C2×SD16—58no—
346C2×SD16—108no—
347C22×C8—52no—
348C22×C8C22×D512yes—
349C4⋊Q8—22no—
350C4⋊Q8—54no—
351C4⋊Q8—102no—
352C4⋊Q8C22×D514yes—
353#125—108no—
354C22⋊Q8—58no—
355C22⋊Q8—108no—
356C22⋊Q8—104no—
357C4×Q8—22no—
358C4×Q8—54no—
359C4×Q8—58no—
360C4×Q8—108no—
361C4×Q8—102no—
362C4×Q8—104no—
363C4×Q8C22×D514yes—
364C4×D4—54no—
365C4×D4—54no—
366C4×D4—108no—
367C4×D4—102no—
368C4×D4—108no—
369C4×D4—102no—
370#117—54no—
371#117—108no—
372#117—102no—
373#117—108no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—51no—
378C2×C4⋊C4—52no—
379C2×C4⋊C4—101no—
380C2×C4⋊C4—102no—
381C2×C4⋊C4—102no—
382C2×C4⋊C4—101no—
383C2×C4⋊C4C22×D511yescharacteristic
384C2×C22⋊C4—101no—
385C2×C22⋊C4—101no—
386C2×C22⋊C4—102no—
387C2×C22⋊C4—101no—
388C2×C42—21no—
389C2×C42—51no—
390C2×C42—51no—
391C2×C42—51no—
392C2×C42—54no—
393C2×C42—54no—
394C2×C42—108no—
395C2×C42—101no—
396C2×C42—102no—
397C2×C42—101no—
398C2×C42C22×D511yescharacteristic
399#107—58no—
400#106—58no—
401C4⋊C8—54no—
402C4⋊C8C22×D514yes—
403#103—58no—
404#103—58no—
405#103C22×D518yes—
406#102—58no—
407C22⋊C8—58no—
408C4×C8—58no—
409C22×C5#52512yes—
410C22×C5C2×#13512yes—
411C22×C5C2×#13712yes—
412C22×C5C23×D411yescharacteristic
413C2×D5—22no—
414C2×D5—24no—
415C2×D5—22no—
416C2×D5—22no—
417C2×D5—44no—
418C2×D5—48no—
419C2×D5—42no—
420C2×D5—42no—
421C4×C5—24no—
422C4×C5—22no—
423C4×C5—44no—
424C4×C5—42no—
425C4×C5#52512yes—
426C4×C5C2×#13514yes—
427C4×C5C22×#4111yescharacteristic
428C4×C5C22×#4111yescharacteristic
429Dic5—22no—
430Dic5—22no—
431Dic5—44no—
432Dic5—42no—
433Dic5C2×#13714yes—
434C24—51no—
435C24—101no—
436#41—108no—
437#41—1016no—
438#41—108no—
439C2×Q8—21no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—24no—
443C2×Q8—24no—
444C2×Q8—104no—
445C2×Q8—204no—
446C2×Q8#71114yes—
447C2×Q8C23×D514yes—
448C2×Q8C23×D512yes—
449C2×D4—54no—
450C2×D4—52no—
451C2×D4—104no—
452C2×D4—108no—
453C2×D4—101no—
454C2×D4—102no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—51no—
459C22×C4—51no—
460C22×C4—51no—
461C22×C4—54no—
462C22×C4—52no—
463C22×C4—54no—
464C22×C4—52no—
465C22×C4—104no—
466C22×C4—104no—
467C22×C4—104no—
468C22×C4—108no—
469C22×C4—102no—
470C22×C4—102no—
471C22×C4—104no—
472C22×C4—101no—
473C22×C4—102no—
474C22×C4—104no—
475C22×C4—204no—
476C22×C4—204no—
477C22×C4D4×D511yescharacteristic
478C22×C4D4×D511yescharacteristic
479C22×C4C23×D511yescharacteristic
480Q16—216no—
481SD16—1016no—
482C2×C8—24no—
483C2×C8—54no—
484C2×C8—104no—
485C2×C8C23×D514yes—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—21no—
490C4⋊C4—54no—
491C4⋊C4—52no—
492C4⋊C4—104no—
493C4⋊C4—108no—
494C4⋊C4—108no—
495C4⋊C4—101no—
496C4⋊C4—102no—
497C4⋊C4—104no—
498C4⋊C4C23×D512yes—
499C22⋊C4—104no—
500C22⋊C4—104no—
501C22⋊C4—108no—
502C22⋊C4—104no—
503C42—24no—
504C42—54no—
505C42—54no—
506C42—52no—
507C42—52no—
508C42—104no—
509C42—104no—
510C42—108no—
511C42D4×D512yes—
512C42C23×D512yes—
513C2×C5#310714yescomplemented
514C2×C5C2×#52511yescharacteristic
515C2×C5C22×#13511yescharacteristic
516C2×C5C22×#13711yescharacteristic
517D5—42no—
518D5—44no—
519C23—52no—
520C23—102no—
521C23—102no—
522C23—104no—
523C23—102no—
524C23—102no—
525C23—202no—
526C23—202no—
527C23C2×D4×D511yescharacteristic
528Q8—24no—
529Q8—22no—
530Q8—24no—
531Q8—24no—
532Q8—44no—
533Q8—208no—
534Q8C2×#71114yes—
535D4—54no—
536D4—1016no—
537D4—104no—
538D4—102no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—54no—
547C2×C4—52no—
548C2×C4—52no—
549C2×C4—52no—
550C2×C4—108no—
551C2×C4—1016no—
552C2×C4—102no—
553C2×C4—102no—
554C2×C4—108no—
555C2×C4—104no—
556C2×C4—102no—
557C2×C4—102no—
558C2×C4—102no—
559C2×C4—104no—
560C2×C4—208no—
561C2×C4—202no—
562C2×C4—204no—
563C2×C4—208no—
564C2×C4#406514yes—
565C2×C4C2×D4×D512yes—
566C2×C4C2×D4×D512yes—
567C2×C4C2×D4×D511yescharacteristic
568C2×C4C2×#71112yes—
569C2×C4D5×#4112yes—
570C2×C4C24×D511yescharacteristic
571C8—24no—
572C8—104no—
573C5C2×#310711yescharacteristic, complemented, Sylow
574C22—102no—
575C22—102no—
576C22—104no—
577C22—102no—
578C22—202no—
579C22—202no—
580C22—204no—
581C22—208no—
582C22#6594012yes—
583C22C2×#406512yes—
584C22D5×#13712yes—
585C22C22×D4×D511yescharacteristic
586C4—22no—
587C4—24no—
588C4—42no—
589C4—44no—
590C4—54no—
591C4—102no—
592C4—102no—
593C4—202no—
594C4—204no—
595C4#6594012yes—
596C4C2×#406514yes—
597C4C22×#71111yescharacteristic
598C4C2×D5×#4111yescharacteristic
599C2—202no—
600C2—204no—
601C2#1061218814yesdirect product factor
602C2C2×#6594011yescharacteristic
603C2C22×#406511yescharacteristic
604C2C2×D5×#13711yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#462C2
socleC23×C5C22×D4
prime radicalC22×#713C22
hypercenterC2×#462D5
hypocenterC5C2×#3107
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
1. C2×#66062C2semidirect
3. C2×C5×#462C2semidirect
5. C2×Q16×Dic5C2semidirect
6. C2×#65372C2semidirect
7. C2×#65309C2semidirect
8. #10612188C2direct (N × H)
9. C2×#65135C2semidirect
10. C2×#65122C2semidirect
11. C2×#65103C2semidirect
12. C2×#65077C2semidirect
22. #66062C22semidirect
38. C5×#462C22semidirect
44. Q16×Dic5C22semidirect
49. #65372C22semidirect
50. #65309C22semidirect
53. #65135C22semidirect
54. #65122C22semidirect
55. #65103C22semidirect
56. #65077C22semidirect
145. C2×#462D5semidirect
258. #462C2×D5semidirect
513. C2×C5#3107semidirect
573. C5C2×#3107semidirect
601. C2#10612188direct (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×C4×#7132
C2×C5×#4622
C2×#654762
C2×Q16×Dic52
C2×#653722
C2×#653092
#106121882
C2×#651352
C2×#651222
C2×#651032
C2×#650772
C2×#31075