Finite Group Databasebeta

← C2×#10612838 #12966804

Subgroups — C2×#10612838 #12966804

Order 1280.

2162 conjugacy classes of subgroups.

The subgroup lattice has 596 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×#477C211yesmaximal, characteristic, complemented
4C2×#65373C211yesmaximal, characteristic, complemented
5C2×#65309C211yesmaximal, characteristic, complemented
6#10612838C2116yesmaximal, direct product factor
7C2×#65180C212yesmaximal, complemented
8C2×#65169C212yesmaximal, complemented
9C2×#65135C211yesmaximal, characteristic, complemented
10C2×D5×C4⋊C8C211yesmaximal, characteristic, complemented
11C2×#65093C212yesmaximal
12C2×#65085C212yesmaximal, complemented
13C22×#713C2211yescharacteristic
14C2×C5×C4⋊Q8C2211yescharacteristic
15C2×C4×C5×Q8C2211yescharacteristic
16C2×#4092—21no—
17C2×Q8×Dic5—21no—
18C2×Q8×Dic5C2211yescharacteristic
19C22×C8×D5C2212yes—
20#66062—24no—
21#66062C2218yescomplemented
22C4×#713—24no—
23C4×#713C2218yes—
24C2×#4043—21no—
25C2×#4039—21no—
26C2×#4039C2211yescharacteristic
27C2×#4038—21no—
28C2×#4038C2212yes—
29C2×#4038C2211yescharacteristic
30C2×#4036C2212yes—
31C2×#4033—21no—
32C2×C4×D20—21no—
33C2×C4×D20C2211yescharacteristic
34C2×C42×D5—21no—
35C2×C42×D5C2211yescharacteristic
36C2×#4015C2211yescharacteristic
37C5×#477C2218yescomplemented
38C2×C5×#107C2212yes—
39C2×C5×C4⋊C8C2211yescharacteristic
40C2×C5×#103C2212yes—
41C2×#3967C2212yes—
42C2×#3953C2212yes—
43C2×#3950C2212yes—
44C2×#3946C2212yes—
45#65373C2218yescomplemented
46#65309C2218yescomplemented
47C2×#3941C2212yes—
48C2×#3940C2212yes—
49C2×#3940C2212yes—
50C2×C4⋊#168C2211yescharacteristic
51#65180C22116yescomplemented
52#65169C22116yescomplemented
53#65135C2218yescomplemented
54D5×C4⋊C8C2218yescomplemented
55#65093C22116yes—
56#65085C22116yescomplemented
57C2×#3076—51nomaximal, Sylow
58C22×C5×Q8—21no—
59C22×C5×Q8C2311yescharacteristic
60C2×#713—22no—
61C2×#713—24no—
62C2×#713Q818yescomplemented
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—
73C22×C5×C8C2312yes—
74C5×C4⋊Q8—22no—
75C5×C4⋊Q8C2314yes—
76C4×C5×Q8—22no—
77C4×C5×Q8C2314yes—
78C2×C5×C4⋊C4—21no—
79C2×C5×C4⋊C4—21no—
80C2×C5×C4⋊C4C2312yes—
81C2×C5×C4⋊C4C2311yescharacteristic
82C2×C42×C5—21no—
83C2×C42×C5C2311yescharacteristic
84#4092—28no—
85Q8×Dic5—28no—
86Q8×Dic5—22no—
87Q8×Dic5C2314yes—
88C2×#685—21no—
89C2×#685—22no—
90C2×#685—21no—
91C2×#685—21no—
92C2×C4⋊Dic5—21no—
93C2×C4⋊Dic5—21no—
94C2×C4⋊Dic5—22no—
95C2×C4⋊Dic5C2312yes—
96C2×C4⋊Dic5C2311yescharacteristic
97C2×#683—22no—
98C2×C4×Dic5—21no—
99C2×C4×Dic5—21no—
100C2×C4×Dic5—22no—
101C2×C4×Dic5D411yescharacteristic
102C2×C4×Dic5D411yescharacteristic
103C2×C4×Dic5C2311yescharacteristic
104C22×#168C2312yes—
105C2×C8×D5—28no—
106C2×C8×D5C2318yes—
107#4043—28no—
108#4039—28no—
109#4039—22no—
110#4039C2314yes—
111#4038—24no—
112#4038—22no—
113#4038—28no—
114#4038C2318yes—
115#4038C2314yes—
116#4036—24no—
117#4036C2318yes—
118#4033—28no—
119C4×D20—28no—
120C4×D20—22no—
121C4×D20C2314yes—
122C42×D5—28no—
123C42×D5D414yes—
124C42×D5C2314yes—
125#4015—22no—
126#4015C2314yes—
127C5×#107C2318yes—
128C5×C4⋊C8C2314yes—
129C5×#103C2318yes—
130#3967C2318yes—
131#3953C2318yes—
132#3950C2318yes—
133#3946C2318yes—
134#3941C2318yes—
135#3940C2318yes—
136#3940C2318yes—
137C4⋊#168C2314yes—
138C2×#533—51no—
139C2×C4×#41—51no—
140#3076—516no—
141C2×#479—52no—
142C2×#477D511yescharacteristic, complemented
143C2×#475—51no—
144C2×#478—52no—
145C2×#476—51no—
146C2×#474—51no—
147C2×#427—52no—
148C22×C4⋊C8—51no—
149C2×#416—52no—
150C23×D5—21no—
151C23×D5C2×D411yescharacteristic
152C2×C5×Q8—21no—
153C2×C5×Q8—24no—
154C2×C5×Q8—22no—
155C2×C5×Q8—44no—
156C2×C5×Q8C2×Q814yes—
157C2×C5×Q8C2412yes—
158C22×C4×C5—21no—
159C22×C4×C5—21no—
160C22×C4×C5—22no—
161C22×C4×C5C2×D411yescharacteristic
162C22×C4×C5C2×D411yescharacteristic
163C22×C4×C5C2411yescharacteristic
164C22×Dic5—22no—
165C22×Dic5—21no—
166C22×Dic5C2×D411yescharacteristic
167C22×Dic5C2×D411yescharacteristic
168C22×Dic5C2×D411yescharacteristic
169#713—216no—
170#713—28no—
171#713C2×Q8116yescomplemented
172C2×D20—28no—
173C2×D20—24no—
174C2×D20—21no—
175C2×D20—22no—
176C2×D20C2×Q814yes—
177C2×D20C2412yes—
178C2×C4×D5—24no—
179C2×C4×D5—28no—
180C2×C4×D5—24no—
181C2×C4×D5—24no—
182C2×C4×D5—24no—
183C2×C4×D5—42no—
184C2×C4×D5—44no—
185C2×C4×D5—44no—
186C2×C4×D5C2×D414yes—
187C2×C4×D5C2×D414yes—
188C2×C4×D5C2×D412yes—
189C2×C4×D5C2×Q814yes—
190C2×C4×D5#4114yes—
191C2×C4×D5C2412yes—
192C2×Dic10—24no—
193C2×Dic10—44no—
194C2×C5×C8—24no—
195C2×C5×C8C2414yes—
196C5×C4⋊C4—24no—
197C5×C4⋊C4—24no—
198C5×C4⋊C4—21no—
199C5×C4⋊C4—22no—
200C5×C4⋊C4C2412yes—
201C5×C4⋊C4C2414yes—
202C42×C5—24no—
203C42×C5C2×D412yes—
204C42×C5C2412yes—
205#685—24no—
206#685—28no—
207#685—24no—
208#685—24no—
209C4⋊Dic5—22no—
210C4⋊Dic5—24no—
211C4⋊Dic5—21no—
212C4⋊Dic5—24no—
213C4⋊Dic5—28no—
214C4⋊Dic5C2412yes—
215C4⋊Dic5C2414yes—
216#683—28no—
217C4×Dic5—24no—
218C4×Dic5—24no—
219C4×Dic5—28no—
220C4×Dic5C2×D414yes—
221C4×Dic5C2×D412yes—
222C4×Dic5C2×D414yes—
223C4×Dic5C2412yes—
224C2×#168—24no—
225C2×#168C2414yes—
226C8×D5—216no—
227C22×#41—51no—
228C23×C8—52no—
229#533—58no—
230#533—104no—
231C2×C4⋊Q8—51no—
232C2×C4⋊Q8C2×D511yescharacteristic
233C2×#125—52no—
234C2×C22⋊Q8—101no—
235C2×C22⋊Q8—101no—
236C4×#41—58no—
237C4×#41—104no—
238C2×C4×Q8—51no—
239C2×C4×Q8—101no—
240C2×C4×Q8—101no—
241C2×C4×Q8C2×D511yescharacteristic
242C2×C4×D4—51no—
243C2×C4×D4—51no—
244C2×C4×D4—101no—
245C2×C4×D4—101no—
246C2×#117—52no—
247C2×#117—51no—
248C2×#117—101no—
249C22×C42—51no—
250C22×C42—101no—
251#479—516no—
252#478—516no—
253#477C2×D518yescomplemented
254#476—58no—
255#475—58no—
256#474—58no—
257#427—516no—
258C2×#107—52no—
259C2×#107C2×D512yes—
260C2×#106—52no—
261C2×#106—52no—
262C2×C4⋊C8—52no—
263C2×C4⋊C8—51no—
264C2×C4⋊C8—58no—
265C2×C4⋊C8C2×D511yescharacteristic
266#416—516no—
267C2×#103—52no—
268C2×#103C2×D512yes—
269C2×#102—52no—
270C2×#102—52no—
271C23×C5C22×D411yescharacteristic
272C22×D5—22no—
273C22×D5—22no—
274C22×D5—24no—
275C22×D5—22no—
276C22×D5—42no—
277C22×D5—42no—
278C22×D5C22⋊Q814yes—
279C22×D5C22×D412yes—
280C5×Q8—24no—
281C5×Q8—22no—
282C5×Q8—48no—
283C5×Q8C22×Q814yes—
284C2×C4×C5—24no—
285C2×C4×C5—22no—
286C2×C4×C5—22no—
287C2×C4×C5—24no—
288C2×C4×C5—22no—
289C2×C4×C5—42no—
290C2×C4×C5—44no—
291C2×C4×C5#13514yes—
292C2×C4×C5C22×D412yes—
293C2×C4×C5C22×D411yescharacteristic
294C2×C4×C5C22×D412yes—
295C2×C4×C5C22×Q812yes—
296C2×C4×C5C2×#4112yes—
297C2×C4×C5C2511yescharacteristic
298C2×Dic5—22no—
299C2×Dic5—22no—
300C2×Dic5—22no—
301C2×Dic5—24no—
302C2×Dic5—44no—
303C2×Dic5—42no—
304C2×Dic5C22⋊Q814yes—
305C2×Dic5C22×D412yes—
306C2×Dic5C22×D412yes—
307C2×Dic5C22×D412yes—
308D20—216no—
309D20—24no—
310D20—22no—
311D20C22×Q814yes—
312C4×D5—28no—
313C4×D5—216no—
314C4×D5—44no—
315C4×D5—48no—
316C4×D5—48no—
317C4×D5C22⋊Q818yes—
318C4×D5C22×Q814yes—
319C4×D5C2×#4114yes—
320Dic10—48no—
321C5×C8—24no—
322#168—24no—
323C2×#41—58no—
324C2×#41—54no—
325C2×#41—102no—
326C2×#41—104no—
327C22×Q8—21no—
328C22×Q8—101no—
329C22×Q8C22×D511yescharacteristic
330C22×D4—51no—
331C22×D4—101no—
332C23×C4—51no—
333C23×C4—51no—
334C23×C4—51no—
335C23×C4—101no—
336C23×C4—101no—
337C23×C4—101no—
338C22×C8—52no—
339C22×C8—58no—
340C22×C8—108no—
341C22×C8C22×D512yes—
342C4⋊Q8—22no—
343C4⋊Q8—54no—
344C4⋊Q8—102no—
345C4⋊Q8C22×D514yes—
346#125—58no—
347#125—104no—
348C22⋊Q8—108no—
349C22⋊Q8—108no—
350C4×Q8—22no—
351C4×Q8—54no—
352C4×Q8—108no—
353C4×Q8—102no—
354C4×Q8—108no—
355C4×Q8C22×D514yes—
356C4×D4—54no—
357C4×D4—54no—
358C4×D4—108no—
359C4×D4—102no—
360C4×D4—108no—
361C4×D4—102no—
362#117—54no—
363#117—58no—
364#117—108no—
365#117—102no—
366#117—104no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—51no—
370C2×C4⋊C4—52no—
371C2×C4⋊C4—101no—
372C2×C4⋊C4—102no—
373C2×C4⋊C4—102no—
374C2×C4⋊C4—101no—
375C2×C4⋊C4C22×D511yescharacteristic
376C2×C4⋊C4C22×D512yes—
377C2×C22⋊C4—101no—
378C2×C22⋊C4—101no—
379C2×C22⋊C4—102no—
380C2×C22⋊C4—101no—
381C2×C42—21no—
382C2×C42—51no—
383C2×C42—51no—
384C2×C42—51no—
385C2×C42—54no—
386C2×C42—54no—
387C2×C42—108no—
388C2×C42—101no—
389C2×C42—102no—
390C2×C42—101no—
391C2×C42C22×D511yescharacteristic
392#107—58no—
393#107C22×D518yes—
394#106—58no—
395#106—58no—
396C4⋊C8—58no—
397C4⋊C8—54no—
398C4⋊C8C22×D514yes—
399#103—58no—
400#103C22×D518yes—
401#102—58no—
402#102—58no—
403C22×C5C2×C22⋊Q812yes—
404C22×C5C2×#13512yes—
405C22×C5#57512yes—
406C22×C5C23×D411yescharacteristic
407C2×D5—28no—
408C2×D5—22no—
409C2×D5—22no—
410C2×D5—24no—
411C2×D5—48no—
412C2×D5—42no—
413C2×D5—42no—
414C2×D5C2×C22⋊Q814yes—
415C4×C5—24no—
416C4×C5—22no—
417C4×C5—42no—
418C4×C5—44no—
419C4×C5C2×C22⋊Q812yes—
420C4×C5C2×#13514yes—
421C4×C5C23×Q811yescharacteristic
422C4×C5C22×#4111yescharacteristic
423Dic5—22no—
424Dic5—22no—
425Dic5—44no—
426Dic5—42no—
427Dic5C2×C22⋊Q814yes—
428C24—51no—
429C24—101no—
430#41—516no—
431#41—1016no—
432#41—108no—
433C2×Q8—21no—
434C2×Q8—24no—
435C2×Q8—22no—
436C2×Q8—44no—
437C2×Q8—104no—
438C2×Q8—204no—
439C2×Q8Q8×D514yes—
440C2×Q8C23×D512yes—
441C2×D4—54no—
442C2×D4—52no—
443C2×D4—104no—
444C2×D4—108no—
445C2×D4—101no—
446C2×D4—102no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—51no—
451C22×C4—52no—
452C22×C4—54no—
453C22×C4—54no—
454C22×C4—52no—
455C22×C4—51no—
456C22×C4—51no—
457C22×C4—54no—
458C22×C4—54no—
459C22×C4—104no—
460C22×C4—108no—
461C22×C4—104no—
462C22×C4—104no—
463C22×C4—101no—
464C22×C4—102no—
465C22×C4—104no—
466C22×C4—202no—
467C22×C4—204no—
468C22×C4—204no—
469C22×C4D4×D511yescharacteristic
470C22×C4D4×D511yescharacteristic
471C22×C4C23×D511yescharacteristic
472C2×C8—24no—
473C2×C8—54no—
474C2×C8—1016no—
475C2×C8—104no—
476C2×C8C23×D514yes—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—54no—
482C4⋊C4—52no—
483C4⋊C4—104no—
484C4⋊C4—108no—
485C4⋊C4—108no—
486C4⋊C4—101no—
487C4⋊C4—102no—
488C4⋊C4—104no—
489C4⋊C4C23×D514yes—
490C4⋊C4C23×D512yes—
491C22⋊C4—104no—
492C22⋊C4—104no—
493C22⋊C4—108no—
494C22⋊C4—104no—
495C42—24no—
496C42—54no—
497C42—54no—
498C42—52no—
499C42—52no—
500C42—104no—
501C42—104no—
502C42—108no—
503C42D4×D512yes—
504C42C23×D512yes—
505C2×C5#307614yescomplemented
506C2×C5C22×C22⋊Q811yescharacteristic
507C2×C5C22×#13511yescharacteristic
508C2×C5C2×#57511yescharacteristic
509D5—24no—
510D5—44no—
511C23—54no—
512C23—52no—
513C23—102no—
514C23—102no—
515C23—104no—
516C23—102no—
517C23—202no—
518C23—202no—
519C23C2×D4×D511yescharacteristic
520Q8—24no—
521Q8—22no—
522Q8—48no—
523Q8—208no—
524Q8C2×Q8×D514yes—
525D4—54no—
526D4—1016no—
527D4—104no—
528D4—102no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—54no—
537C2×C4—52no—
538C2×C4—58no—
539C2×C4—54no—
540C2×C4—54no—
541C2×C4—52no—
542C2×C4—52no—
543C2×C4—1016no—
544C2×C4—102no—
545C2×C4—102no—
546C2×C4—108no—
547C2×C4—102no—
548C2×C4—104no—
549C2×C4—204no—
550C2×C4—208no—
551C2×C4—202no—
552C2×C4—204no—
553C2×C4—208no—
554C2×C4#406514yes—
555C2×C4C2×D4×D512yes—
556C2×C4C2×D4×D512yes—
557C2×C4C2×D4×D511yescharacteristic
558C2×C4C2×Q8×D512yes—
559C2×C4D5×#4112yes—
560C2×C4C24×D511yescharacteristic
561C8—24no—
562C8—104no—
563C5C2×#307611yescharacteristic, complemented, Sylow
564C22—54no—
565C22—108no—
566C22—102no—
567C22—102no—
568C22—104no—
569C22—208no—
570C22—202no—
571C22—202no—
572C22D5×C22⋊Q812yes—
573C22C2×#406512yes—
574C22#6610712yes—
575C22C22×D4×D511yescharacteristic
576C4—24no—
577C4—22no—
578C4—42no—
579C4—44no—
580C4—54no—
581C4—102no—
582C4—102no—
583C4—202no—
584C4—204no—
585C4D5×C22⋊Q812yes—
586C4C2×#406514yes—
587C4C22×Q8×D511yescharacteristic
588C4C2×D5×#4111yescharacteristic
589C2—104no—
590C2—204no—
591C2#1061283814yesdirect product factor
592C2C2×D5×C22⋊Q811yescharacteristic
593C2C22×#406511yescharacteristic
594C2C2×#6610711yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#477C2
socleC23×C5C22×D4
prime radicalC22×#713C22
hypercenterC2×#477D5
hypocenterC5C2×#3076
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
1. C2×#66062C2semidirect
3. C2×C5×#477C2semidirect
4. C2×#65373C2semidirect
5. C2×#65309C2semidirect
6. #10612838C2direct (N × H)
7. C2×#65180C2semidirect
8. C2×#65169C2semidirect
9. C2×#65135C2semidirect
10. C2×D5×C4⋊C8C2semidirect
12. C2×#65085C2semidirect
21. #66062C22semidirect
37. C5×#477C22semidirect
45. #65373C22semidirect
46. #65309C22semidirect
51. #65180C22semidirect
52. #65169C22semidirect
53. #65135C22semidirect
54. D5×C4⋊C8C22semidirect
56. #65085C22semidirect
62. C2×#713Q8semidirect
142. C2×#477D5semidirect
171. #713C2×Q8semidirect
253. #477C2×D5semidirect
505. C2×C5#3076semidirect
563. C5C2×#3076semidirect
591. C2#10612838direct (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×#4772
C2×#653732
C2×#653092
#106128382
C2×#651802
C2×#651692
C2×#651352
C2×D5×C4⋊C82
C2×#650932
C2×#650852
C2×#30765