Finite Group Databasebeta

← C2×#10613186 #12967451

Subgroups — C2×#10613186 #12967451

Order 1280.

2150 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65958C211yesmaximal, characteristic
2C22×#4033C211yesmaximal, characteristic, complemented
3C2×C5×#484C211yesmaximal, characteristic, complemented
4C2×#65339C211yesmaximal, characteristic, complemented
5C2×#65257C211yesmaximal, characteristic
6#10613186C2116yesmaximal, direct product factor
7C2×#65166C212yesmaximal, complemented
8C2×#65178C212yesmaximal, complemented
9C2×#65087C212yesmaximal, complemented
10C2×#65078C212yesmaximal, complemented
11C2×#65027C211yesmaximal, characteristic
12C2×#65008C211yesmaximal, characteristic, complemented
13C23×Dic10C2211yescharacteristic
14C2×C5×C22⋊Q8C2211yescharacteristic
15C22×C5×C4⋊C4C2211yescharacteristic
16C2×#4093C2211yescharacteristic
17C2×Q8×Dic5C2211yescharacteristic
18C22×#689C2212yes—
19C2×#4076C2211yescharacteristic
20C22×#683—21no—
21C22×C4×Dic5—21no—
22C22×C4×Dic5C2211yescharacteristic
23C22×#677C2212yes—
24#65958—24no—
25#65958C2218yes—
26C2×#4044—21no—
27C2×#4039C2211yescharacteristic
28C2×#4038—21no—
29C2×#4036C2211yescharacteristic
30C2×#4033—24no—
31C2×#4033—21no—
32C2×#4033—21no—
33C2×#4033C2212yes—
34C2×#4033C2212yes—
35C2×#4033C2218yescomplemented
36C2×#4032—21no—
37C2×#4027—21no—
38C5×#484C2218yescomplemented
39C2×C5×#107C2212yes—
40C2×C5×#103C2212yes—
41C2×C5×C22⋊C8C2211yescharacteristic
42C2×C22⋊#168C2211yescharacteristic
43C2×#3948C2212yes—
44C2×#3947C2212yes—
45C2×#3946C2212yes—
46#65339C2218yescomplemented
47C2×#3942C2212yes—
48#65257C2218yes—
49C2×#3941C2212yes—
50C2×#3940C2212yes—
51#65178C22116yescomplemented
52#65166C22116yescomplemented
53#65087C22116yescomplemented
54#65078C22116yescomplemented
55#65027C2218yes—
56#65008C2218yescomplemented
57C2×#3117—51nomaximal, Sylow
58C22×C5×Q8C2311yescharacteristic
59C23×C4×C5—21no—
60C23×C4×C5C2311yescharacteristic
61C22×C22⋊D5—21no—
62C23×Dic5—21no—
63C23×Dic5D411yescharacteristic
64C23×Dic5D411yescharacteristic
65C22×D20C2311yescharacteristic
66C22×C4×D5—21no—
67C22×Dic10—21no—
68C22×Dic10—22no—
69C22×Dic10—28no—
70C22×Dic10C2314yes—
71C22×Dic10C2312yes—
72C22×C5×C8C2312yes—
73C5×C22⋊Q8—22no—
74C5×C22⋊Q8C2314yes—
75C2×C5×C4⋊C4—21no—
76C2×C5×C4⋊C4—22no—
77C2×C5×C4⋊C4—21no—
78C2×C5×C4⋊C4C2312yes—
79C2×C5×C4⋊C4C2314yes—
80C2×C5×C4⋊C4C2311yescharacteristic
81C2×C5×C22⋊C4—21no—
82#4093—22no—
83#4093C2314yes—
84Q8×Dic5—22no—
85Q8×Dic5C2314yes—
86C2×#689—28no—
87C2×#689C2318yes—
88#4076—22no—
89#4076C2314yes—
90C2×#685—21no—
91C2×#685—21no—
92C2×#685—21no—
93C2×#685—21no—
94C2×C4⋊Dic5—21no—
95C2×C4⋊Dic5—21no—
96C2×C4⋊Dic5C2311yescharacteristic
97C2×#683—21no—
98C2×#683—22no—
99C2×#683—22no—
100C2×#683—28no—
101C2×C4×Dic5—21no—
102C2×C4×Dic5—22no—
103C2×C4×Dic5—22no—
104C2×C4×Dic5—28no—
105C2×C4×Dic5D411yescharacteristic
106C2×C4×Dic5D411yescharacteristic
107C2×C4×Dic5D414yes—
108C2×C4×Dic5C2311yescharacteristic
109C2×C4×Dic5C2311yescharacteristic
110C2×C4×Dic5C2314yes—
111C22×#168C2312yes—
112C2×#677—28no—
113C2×#677C2318yes—
114#4044—28no—
115#4039—22no—
116#4039C2314yes—
117#4038—28no—
118#4036—22no—
119#4036C2314yes—
120#4033—24no—
121#4033—24no—
122#4033—28no—
123#4033—28no—
124#4033C2318yes—
125#4033C2318yes—
126#4032—28no—
127#4027—28no—
128C5×#107C2318yes—
129C5×#103C2318yes—
130C5×C22⋊C8C2314yes—
131C22⋊#168C2314yes—
132#3948C2318yes—
133#3947C2318yes—
134#3946C2318yes—
135#3942C2318yes—
136#3941C2318yes—
137#3940C2318yes—
138C2×#529—51no—
139C22×C4×Q8—51no—
140#3117—516no—
141C2×#484D511yescharacteristic, complemented
142C2×#481—51no—
143C2×#479—52no—
144C2×#464—52no—
145C2×#451—51no—
146C2×#448—51no—
147C2×#441—52no—
148C2×#440—52no—
149C2×#432—51no—
150C24×C5C2×D411yescharacteristic
151C23×D5—21no—
152C2×C5×Q8—21no—
153C2×C5×Q8—24no—
154C2×C5×Q8C2412yes—
155C22×C4×C5—24no—
156C22×C4×C5—21no—
157C22×C4×C5—21no—
158C22×C4×C5—24no—
159C22×C4×C5—22no—
160C22×C4×C5—44no—
161C22×C4×C5C2×D411yescharacteristic
162C22×C4×C5C2×D412yes—
163C22×C4×C5C2412yes—
164C22×C4×C5C2411yescharacteristic
165C2×C22⋊D5—24no—
166C2×C22⋊D5—44no—
167C22×Dic5—28no—
168C22×Dic5—24no—
169C22×Dic5—24no—
170C22×Dic5—22no—
171C22×Dic5—21no—
172C22×Dic5—42no—
173C22×Dic5C2×D411yescharacteristic
174C22×Dic5C2×D414yes—
175C22×Dic5C2×D411yescharacteristic
176C22×Dic5C2×D411yescharacteristic
177C22×Dic5C2×D411yescharacteristic
178C22×Dic5C2×D414yes—
179C2×D20—24no—
180C2×D20—21no—
181C2×D20C2412yes—
182C2×C4×D5—24no—
183C2×C4×D5—44no—
184C2×Dic10—28no—
185C2×Dic10—216no—
186C2×Dic10—24no—
187C2×Dic10—22no—
188C2×Dic10—24no—
189C2×Dic10—216no—
190C2×Dic10#4118yes—
191C2×Dic10C2414yes—
192C2×C5×C8—24no—
193C2×C5×C8C2414yes—
194C5×C4⋊C4—24no—
195C5×C4⋊C4—22no—
196C5×C4⋊C4—24no—
197C5×C4⋊C4—21no—
198C5×C4⋊C4C2414yes—
199C5×C4⋊C4C2412yes—
200C5×C22⋊C4—24no—
201#689—216no—
202#685—24no—
203#685—24no—
204#685—24no—
205#685—24no—
206C4⋊Dic5—21no—
207C4⋊Dic5—24no—
208C4⋊Dic5—24no—
209C4⋊Dic5C2412yes—
210#683—24no—
211#683—28no—
212#683—28no—
213C4×Dic5—24no—
214C4×Dic5—28no—
215C4×Dic5—28no—
216C4×Dic5C2×D412yes—
217C4×Dic5C2×D414yes—
218C4×Dic5C2×D412yes—
219C4×Dic5C2×D414yes—
220C4×Dic5C2412yes—
221C4×Dic5C2412yes—
222C2×#168—24no—
223C2×#168C2414yes—
224#677—216no—
225C23×Q8—51no—
226C22×Q16—52no—
227C22×SD16—52no—
228#529—58no—
229#529—104no—
230C2×#126—101no—
231C2×#125—51no—
232C2×#124—51no—
233C2×#123—101no—
234C2×C22⋊Q8C2×D511yescharacteristic
235C2×#121—51no—
236C2×C4×Q8—51no—
237C2×C4×Q8—52no—
238C2×C4×Q8—52no—
239C2×C4×Q8—58no—
240C2×C4×Q8—101no—
241C2×C4×Q8—101no—
242C2×C4×Q8—104no—
243C2×C4×D4—51no—
244C2×C4×D4—101no—
245C2×#117—101no—
246C22×C4⋊C4—101no—
247C22×C4⋊C4C2×D511yescharacteristic
248C22×C42—51no—
249C22×C42—101no—
250#484C2×D518yescomplemented
251#481—58no—
252#479—516no—
253#464—516no—
254#451—58no—
255#448—58no—
256#441—516no—
257#440—516no—
258#432—58no—
259C2×#107C2×D512yes—
260C2×#106—52no—
261C2×C4⋊C8—52no—
262C2×#103—52no—
263C2×#103—52no—
264C2×#103C2×D512yes—
265C2×#102—52no—
266C2×C22⋊C8—51no—
267C2×C22⋊C8C2×D511yescharacteristic
268C2×#97—52no—
269C23×C5—22no—
270C23×C5—22no—
271C23×C5#13714yes—
272C23×C5C22×D412yes—
273C23×C5C22×D411yescharacteristic
274C22×D5—22no—
275C22×D5—42no—
276C22×D5—42no—
277C22×D5—42no—
278C5×Q8—24no—
279C5×Q8—42no—
280C2×C4×C5—22no—
281C2×C4×C5—24no—
282C2×C4×C5—24no—
283C2×C4×C5—24no—
284C2×C4×C5—22no—
285C2×C4×C5—22no—
286C2×C4×C5—48no—
287C2×C4×C5—42no—
288C2×C4×C5—42no—
289C2×C4×C5—44no—
290C2×C4×C5C22×D412yes—
291C2×C4×C5C22×D411yescharacteristic
292C2×C4×C5C2×#4114yes—
293C2×C4×C5C2511yescharacteristic
294C22⋊D5—48no—
295C2×Dic5—216no—
296C2×Dic5—28no—
297C2×Dic5—28no—
298C2×Dic5—22no—
299C2×Dic5—22no—
300C2×Dic5—22no—
301C2×Dic5—22no—
302C2×Dic5—24no—
303C2×Dic5—22no—
304C2×Dic5—44no—
305C2×Dic5—42no—
306C2×Dic5C22×D412yes—
307C2×Dic5C22×D412yes—
308C2×Dic5C22×D412yes—
309C2×Dic5C22×D412yes—
310D20—24no—
311D20—42no—
312C4×D5—48no—
313Dic10—216no—
314Dic10—28no—
315Dic10—24no—
316Dic10C2×#4118yes—
317C5×C8—24no—
318#168—24no—
319C22×Q8—52no—
320C22×Q8—54no—
321C22×Q8—108no—
322C22×Q8—102no—
323C22×Q8—101no—
324C22×Q8C22×D511yescharacteristic
325C22×D4—51no—
326C22×D4—101no—
327C23×C4—21no—
328C23×C4—51no—
329C23×C4—51no—
330C23×C4—101no—
331C23×C4—101no—
332C23×C4C22×D511yescharacteristic
333C2×Q16—58no—
334C2×Q16—108no—
335C2×SD16—58no—
336C2×SD16—108no—
337C22×C8—52no—
338C22×C8C22×D512yes—
339#126—108no—
340#125—54no—
341#125—102no—
342#124—54no—
343#124—102no—
344#123—108no—
345C22⋊Q8—22no—
346C22⋊Q8C22×D514yes—
347#121—54no—
348#121—102no—
349C4×Q8—54no—
350C4×Q8—58no—
351C4×Q8—58no—
352C4×Q8—108no—
353C4×Q8—104no—
354C4×Q8—104no—
355C4×Q8—108no—
356C4×Q8—102no—
357C4×D4—54no—
358C4×D4—102no—
359C4×D4—108no—
360#117—108no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—51no—
365C2×C4⋊C4—108no—
366C2×C4⋊C4—102no—
367C2×C4⋊C4—102no—
368C2×C4⋊C4—101no—
369C2×C4⋊C4—101no—
370C2×C4⋊C4—101no—
371C2×C4⋊C4C22×D512yes—
372C2×C4⋊C4C22×D514yes—
373C2×C4⋊C4C22×D511yescharacteristic
374C2×C22⋊C4—21no—
375C2×C22⋊C4—101no—
376C2×C22⋊C4—101no—
377C2×C22⋊C4—101no—
378C2×C22⋊C4—101no—
379C2×C42—51no—
380C2×C42—51no—
381C2×C42—54no—
382C2×C42—51no—
383C2×C42—51no—
384C2×C42—54no—
385C2×C42—101no—
386C2×C42—108no—
387C2×C42—102no—
388C2×C42—102no—
389#107C22×D518yes—
390#106—58no—
391C4⋊C8—58no—
392#103—58no—
393#103—58no—
394#103C22×D518yes—
395#102—58no—
396C22⋊C8—54no—
397C22⋊C8C22×D514yes—
398#97—58no—
399C22×C5—28no—
400C22×C5—22no—
401C22×C5—22no—
402C22×C5#52512yes—
403C22×C5C2×#13714yes—
404C22×C5C2×#13712yes—
405C22×C5#57512yes—
406C22×C5C23×D411yescharacteristic
407C2×D5—42no—
408C2×D5—42no—
409C2×D5—42no—
410C2×D5—84no—
411C4×C5—22no—
412C4×C5—42no—
413C4×C5—42no—
414C4×C5—44no—
415C4×C5#52512yes—
416C4×C5C22×#4112yes—
417Dic5—28no—
418Dic5—22no—
419Dic5—22no—
420Dic5—22no—
421Dic5—22no—
422Dic5—42no—
423C24—101no—
424C24D4×D511yescharacteristic
425C2×Q8—21no—
426C2×Q8—24no—
427C2×Q8—58no—
428C2×Q8—54no—
429C2×Q8—1016no—
430C2×Q8—108no—
431C2×Q8—1016no—
432C2×Q8—104no—
433C2×Q8—102no—
434C2×Q8—104no—
435C2×Q8C23×D512yes—
436C2×D4—52no—
437C2×D4—104no—
438C2×D4—101no—
439C2×D4—104no—
440C2×D4—204no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—44no—
447C22×C4—54no—
448C22×C4—51no—
449C22×C4—51no—
450C22×C4—51no—
451C22×C4—51no—
452C22×C4—54no—
453C22×C4—104no—
454C22×C4—108no—
455C22×C4—104no—
456C22×C4—102no—
457C22×C4—104no—
458C22×C4—101no—
459C22×C4—204no—
460C22×C4—202no—
461C22×C4D4×D512yes—
462C22×C4D4×D511yescharacteristic
463C22×C4C23×D512yes—
464C22×C4C23×D511yescharacteristic
465Q16—1016no—
466SD16—1016no—
467C2×C8—24no—
468C2×C8—54no—
469C2×C8—104no—
470C2×C8C23×D514yes—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—21no—
475C4⋊C4—52no—
476C4⋊C4—108no—
477C4⋊C4—108no—
478C4⋊C4—104no—
479C4⋊C4—104no—
480C4⋊C4—104no—
481C4⋊C4—101no—
482C4⋊C4C23×D514yes—
483C4⋊C4C23×D512yes—
484C22⋊C4—24no—
485C22⋊C4—104no—
486C22⋊C4—104no—
487C22⋊C4—104no—
488C22⋊C4—104no—
489C42—54no—
490C42—52no—
491C42—52no—
492C42—54no—
493C42—52no—
494C42—52no—
495C42—108no—
496C42—108no—
497C42—104no—
498C2×C5—24no—
499C2×C5#311714yescomplemented
500C2×C5C2×#52511yescharacteristic
501C2×C5C22×#13711yescharacteristic
502C2×C5C2×#57511yescharacteristic
503D5—82no—
504C23—22no—
505C23—22no—
506C23—102no—
507C23—202no—
508C23—202no—
509C23—202no—
510C23#406414yes—
511C23C2×D4×D512yes—
512C23C2×D4×D511yescharacteristic
513Q8—24no—
514Q8—42no—
515Q8—58no—
516Q8—1016no—
517Q8—108no—
518Q8—104no—
519D4—104no—
520D4—208no—
521D4—202no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—48no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—52no—
533C2×C4—52no—
534C2×C4—52no—
535C2×C4—52no—
536C2×C4—1016no—
537C2×C4—108no—
538C2×C4—102no—
539C2×C4—102no—
540C2×C4—102no—
541C2×C4—102no—
542C2×C4—108no—
543C2×C4—104no—
544C2×C4—102no—
545C2×C4—208no—
546C2×C4—204no—
547C2×C4—202no—
548C2×C4C2×D4×D512yes—
549C2×C4C2×D4×D511yescharacteristic
550C2×C4D5×#4114yes—
551C2×C4C24×D511yescharacteristic
552C8—24no—
553C8—104no—
554C5C2×#311711yescharacteristic, complemented, Sylow
555C22—22no—
556C22—28no—
557C22—22no—
558C22—202no—
559C22—202no—
560C22—202no—
561C22—404no—
562C22#6599012yes—
563C22C2×#406414yes—
564C22C2×#406412yes—
565C22#6610712yes—
566C22C22×D4×D511yescharacteristic
567C4—22no—
568C4—42no—
569C4—44no—
570C4—42no—
571C4—108no—
572C4—102no—
573C4—102no—
574C4—102no—
575C4—102no—
576C4—202no—
577C4#6599012yes—
578C4C2×D5×#4112yes—
579C2—24no—
580C2—402no—
581C2#1061318614yesdirect product factor
582C2C2×#6599011yescharacteristic
583C2C22×#406411yescharacteristic
584C2C2×#6610711yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#484C2
socleC23×C5C22×D4
prime radicalC2×#4076C22
hypercenterC2×#484D5
hypocenterC5C2×#3117
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#484

Complements

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

NH ≅ G/Ntype
2. C22×#4033C2semidirect
3. C2×C5×#484C2semidirect
4. C2×#65339C2semidirect
6. #10613186C2direct (N × H)
7. C2×#65166C2semidirect
8. C2×#65178C2semidirect
9. C2×#65087C2semidirect
10. C2×#65078C2semidirect
12. C2×#65008C2semidirect
35. C2×#4033C22semidirect
38. C5×#484C22semidirect
46. #65339C22semidirect
51. #65178C22semidirect
52. #65166C22semidirect
53. #65087C22semidirect
54. #65078C22semidirect
56. #65008C22semidirect
141. C2×#484D5semidirect
250. #484C2×D5semidirect
499. C2×C5#3117semidirect
554. C5C2×#3117semidirect
581. C2#10613186direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#659582
C22×#40332
C2×C5×#4842
C2×#653392
C2×#652572
#106131862
C2×#651662
C2×#651782
C2×#650872
C2×#650782
C2×#650272
C2×#650082
C2×#31175