Finite Group Databasebeta

← C2×#10612229 #12972154

Subgroups — C2×#10612229 #12972154

Order 1280.

1898 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66062C211yesmaximal, characteristic, complemented
2C2×#65914C211yesmaximal, characteristic
3C2×C5×#462C211yesmaximal, characteristic, complemented
4C2×#65478C212yesmaximal, complemented
5C2×#65474C212yesmaximal, complemented
6C2×#65372C211yesmaximal, characteristic, complemented
7C2×#65310C211yesmaximal, characteristic, complemented
8#10612229C2116yesmaximal, direct product factor
9C2×#65131C211yesmaximal, characteristic, complemented
10C2×#65126C211yesmaximal, characteristic, complemented
11C2×#65101C212yesmaximal
12C2×#65088C212yesmaximal, complemented
13C22×C5×Q16C2212yes—
14C2×C5×C4⋊Q8C2211yescharacteristic
15C2×C4×C5×Q8C2211yescharacteristic
16C2×#4093C2211yescharacteristic
17C2×#4092C2211yescharacteristic
18C2×#4092C2212yes—
19C2×Q8×Dic5C2212yes—
20C22×#688C2212yes—
21C22×#676C2212yes—
22#66062—24no—
23#66062C2218yescomplemented
24#65914—24no—
25#65914C2218yes—
26C2×#4044—21no—
27C2×#4043—21no—
28C2×#4041C2211yescharacteristic
29C2×#4040—21no—
30C2×#4038—21no—
31C2×#4036—21no—
32C2×#4036C2211yescharacteristic
33C2×#4033—21no—
34C2×C4×D20—21no—
35C2×C4×D20C2211yescharacteristic
36C2×#4018—21no—
37C2×C42×D5C2211yescharacteristic
38C2×#4015C2211yescharacteristic
39C5×#462C2218yescomplemented
40C2×C5×C4⋊C8C2211yescharacteristic
41C2×C5×#103C2212yes—
42C2×#3967C2212yes—
43C2×#3967C2212yes—
44#65478C22116yescomplemented
45#65474C22116yescomplemented
46C2×#3953C2212yes—
47C2×#3949C2212yes—
48C2×#3946C2212yes—
49#65372C2218yescomplemented
50#65310C2218yescomplemented
51C2×#3939C2212yes—
52C2×C4⋊#168C2211yescharacteristic
53#65131C2218yescomplemented
54#65126C2218yescomplemented
55#65101C22116yes—
56#65088C22116yescomplemented
57C2×#3252—51nomaximal, Sylow
58C22×C5×Q8C2312yes—
59C22×C5×Q8C2311yescharacteristic
60C22×D20—21no—
61C22×D20C2311yescharacteristic
62C22×C4×D5—21no—
63C22×C4×D5D411yescharacteristic
64C22×C4×D5D411yescharacteristic
65C22×C4×D5C2311yescharacteristic
66C22×Dic10—21no—
67C2×C5×Q16—44no—
68C2×C5×Q16C2318yes—
69C22×C5×C8C2312yes—
70C5×C4⋊Q8—22no—
71C5×C4⋊Q8C2314yes—
72C4×C5×Q8—22no—
73C4×C5×Q8C2314yes—
74C2×C5×C4⋊C4—21no—
75C2×C5×C4⋊C4—21no—
76C2×C5×C4⋊C4—21no—
77C2×C5×C4⋊C4C2311yescharacteristic
78C2×C42×C5—21no—
79C2×C42×C5C2311yescharacteristic
80#4093—22no—
81#4093C2314yes—
82#4092—24no—
83#4092—22no—
84#4092C2318yes—
85#4092C2314yes—
86Q8×Dic5—24no—
87Q8×Dic5C2318yes—
88C2×#688—44no—
89C2×#688C2318yes—
90C2×#685—21no—
91C2×#685—22no—
92C2×#685—21no—
93C2×#685—21no—
94C2×#685—21no—
95C2×#685—21no—
96C2×C4⋊Dic5—21no—
97C2×C4⋊Dic5—21no—
98C2×C4⋊Dic5—22no—
99C2×C4⋊Dic5C2312yes—
100C2×C4⋊Dic5C2311yescharacteristic
101C2×#683—21no—
102C2×#683—21no—
103C2×#683—22no—
104C2×C4×Dic5—22no—
105C2×C4×Dic5D411yescharacteristic
106C2×C4×Dic5D411yescharacteristic
107C2×C4×Dic5C2311yescharacteristic
108C22×#168C2312yes—
109C2×#676—44no—
110C2×#676C2318yes—
111#4044—28no—
112#4043—28no—
113#4041—22no—
114#4041C2314yes—
115#4040—28no—
116#4038—28no—
117#4036—28no—
118#4036—22no—
119#4036C2314yes—
120#4033—28no—
121C4×D20—28no—
122C4×D20—22no—
123C4×D20C2314yes—
124#4018—28no—
125C42×D5D414yes—
126C42×D5C2314yes—
127#4015—22no—
128#4015C2314yes—
129C5×C4⋊C8C2314yes—
130C5×#103C2318yes—
131#3967C2318yes—
132#3967C2318yes—
133#3953C2318yes—
134#3949C2318yes—
135#3946C2318yes—
136#3939C2318yes—
137C4⋊#168C2314yes—
138C2×#533—51no—
139C2×#529—51no—
140#3252—516no—
141C2×#479—52no—
142C2×#477—51no—
143C2×#476—51no—
144C2×#470—52no—
145C2×#468—52no—
146C2×#464—52no—
147C2×#462D511yescharacteristic, complemented
148C2×#461—51no—
149C2×#423—51no—
150C23×D5—21no—
151C23×D5C2×D411yescharacteristic
152C2×C5×Q8—21no—
153C2×C5×Q8—22no—
154C2×C5×Q8—22no—
155C2×C5×Q8—24no—
156C2×C5×Q8—24no—
157C2×C5×Q8—22no—
158C2×C5×Q8C2412yes—
159C2×C5×Q8C2414yes—
160C22×C4×C5—21no—
161C22×C4×C5—21no—
162C22×C4×C5—22no—
163C22×C4×C5C2×D411yescharacteristic
164C22×C4×C5C2×D411yescharacteristic
165C22×C4×C5C2411yescharacteristic
166C22×Dic5—22no—
167C22×Dic5—21no—
168C22×Dic5C2×D411yescharacteristic
169C22×Dic5C2×D411yescharacteristic
170C22×Dic5C2×D411yescharacteristic
171C2×D20—22no—
172C2×D20—22no—
173C2×D20—24no—
174C2×D20—21no—
175C2×D20—44no—
176C2×D20C2412yes—
177C2×C4×D5—24no—
178C2×C4×D5—22no—
179C2×C4×D5—22no—
180C2×C4×D5—24no—
181C2×C4×D5—42no—
182C2×C4×D5—44no—
183C2×C4×D5C2×D414yes—
184C2×C4×D5C2×D412yes—
185C2×C4×D5C2×D414yes—
186C2×C4×D5C2412yes—
187C2×Dic10—24no—
188C2×Dic10—44no—
189C5×Q16—48no—
190C2×C5×C8—24no—
191C2×C5×C8C2414yes—
192C5×C4⋊C4—24no—
193C5×C4⋊C4—24no—
194C5×C4⋊C4—24no—
195C5×C4⋊C4—21no—
196C5×C4⋊C4C2412yes—
197C42×C5—24no—
198C42×C5C2×D412yes—
199C42×C5C2412yes—
200#688—48no—
201#685—24no—
202#685—28no—
203#685—24no—
204#685—24no—
205#685—24no—
206#685—24no—
207C4⋊Dic5—22no—
208C4⋊Dic5—24no—
209C4⋊Dic5—21no—
210C4⋊Dic5—24no—
211C4⋊Dic5—28no—
212C4⋊Dic5C2414yes—
213C4⋊Dic5C2412yes—
214#683—24no—
215#683—24no—
216#683—28no—
217C4×Dic5—28no—
218C4×Dic5C2×D412yes—
219C4×Dic5C2×D414yes—
220C4×Dic5C2×D414yes—
221C4×Dic5C2412yes—
222C2×#168—24no—
223C2×#168C2414yes—
224#676—48no—
225C22×Q16C2×D512yes—
226C22×SD16—52no—
227C22×M16—52no—
228#533—58no—
229#533—104no—
230C2×C4⋊Q8—51no—
231C2×C4⋊Q8C2×D511yescharacteristic
232#529—58no—
233#529—104no—
234C2×#126—101no—
235C2×#125—51no—
236C2×#125—101no—
237C2×#124—51no—
238C2×#123—101no—
239C2×C22⋊Q8—51no—
240C2×C22⋊Q8—52no—
241C2×C22⋊Q8—101no—
242C2×#121—51no—
243C2×C4×Q8—52no—
244C2×C4×Q8—101no—
245C2×C4×Q8C2×D511yescharacteristic
246C2×C4×D4—51no—
247C2×C4×D4—101no—
248C2×#117—101no—
249C2×#117—101no—
250C22×C42—51no—
251#479—516no—
252#477—58no—
253#476—58no—
254#470—516no—
255#468—516no—
256#464—516no—
257#462C2×D518yescomplemented
258#461—58no—
259C2×#107—52no—
260C2×#106—52no—
261#423—58no—
262C2×C4⋊C8—52no—
263C2×C4⋊C8—51no—
264C2×C4⋊C8C2×D511yescharacteristic
265C2×#103—52no—
266C2×#103—52no—
267C2×#103C2×D512yes—
268C2×#102—52no—
269C23×C5C22×D411yescharacteristic
270C22×D5—22no—
271C22×D5—22no—
272C22×D5—22no—
273C22×D5—22no—
274C22×D5—42no—
275C22×D5—42no—
276C22×D5—42no—
277C22×D5C22×D412yes—
278C5×Q8—24no—
279C5×Q8—22no—
280C5×Q8—22no—
281C5×Q8—24no—
282C5×Q8—42no—
283C5×Q8—44no—
284C2×C4×C5—22no—
285C2×C4×C5—24no—
286C2×C4×C5—22no—
287C2×C4×C5—22no—
288C2×C4×C5—42no—
289C2×C4×C5—44no—
290C2×C4×C5—42no—
291C2×C4×C5#13714yes—
292C2×C4×C5C22×D412yes—
293C2×C4×C5C22×D411yescharacteristic
294C2×C4×C5C22×D412yes—
295C2×C4×C5H(5,𝔽2)12yes—
296C2×C4×C52−1+412yes—
297C2×C4×C5C2511yescharacteristic
298C2×Dic5—22no—
299C2×Dic5—22no—
300C2×Dic5—22no—
301C2×Dic5—24no—
302C2×Dic5—22no—
303C2×Dic5—42no—
304C2×Dic5—44no—
305C2×Dic5C22×D412yes—
306C2×Dic5C22×D412yes—
307C2×Dic5C22×D412yes—
308D20—22no—
309D20—22no—
310D20—48no—
311D20—42no—
312C4×D5—24no—
313C4×D5—28no—
314C4×D5—22no—
315C4×D5—22no—
316C4×D5—44no—
317C4×D5—48no—
318Dic10—48no—
319C5×C8—24no—
320#168—24no—
321C22×Q8—101no—
322C22×Q8C22×D512yes—
323C22×Q8C22×D511yescharacteristic
324C22×D4—51no—
325C22×D4—101no—
326C23×C4—51no—
327C23×C4—51no—
328C23×C4—51no—
329C23×C4—101no—
330C2×Q16—44no—
331C2×Q16C22×D518yes—
332C2×SD16—58no—
333C2×SD16—204no—
334C2×M16—58no—
335C2×M16—204no—
336C22×C8—52no—
337C22×C8C22×D512yes—
338C4⋊Q8—22no—
339C4⋊Q8—54no—
340C4⋊Q8—102no—
341C4⋊Q8C22×D514yes—
342#126—108no—
343#125—54no—
344#125—102no—
345#125—108no—
346#124—54no—
347#124—102no—
348#123—108no—
349C22⋊Q8—54no—
350C22⋊Q8—58no—
351C22⋊Q8—108no—
352C22⋊Q8—102no—
353C22⋊Q8—104no—
354#121—54no—
355#121—102no—
356C4×Q8—22no—
357C4×Q8—58no—
358C4×Q8—108no—
359C4×Q8—104no—
360C4×Q8C22×D514yes—
361C4×D4—54no—
362C4×D4—108no—
363C4×D4—102no—
364#117—108no—
365#117—108no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—51no—
370C2×C4⋊C4—52no—
371C2×C4⋊C4—101no—
372C2×C4⋊C4—101no—
373C2×C4⋊C4—101no—
374C2×C4⋊C4—102no—
375C2×C4⋊C4—102no—
376C2×C4⋊C4—101no—
377C2×C4⋊C4C22×D511yescharacteristic
378C2×C22⋊C4—101no—
379C2×C22⋊C4—101no—
380C2×C22⋊C4—101no—
381C2×C22⋊C4—102no—
382C2×C22⋊C4—101no—
383C2×C22⋊C4—101no—
384C2×C42—21no—
385C2×C42—51no—
386C2×C42—51no—
387C2×C42—51no—
388C2×C42—54no—
389C2×C42—54no—
390C2×C42—102no—
391C2×C42C22×D511yescharacteristic
392#107—58no—
393#106—58no—
394C4⋊C8—58no—
395C4⋊C8—54no—
396C4⋊C8C22×D514yes—
397#103—58no—
398#103—58no—
399#103C22×D518yes—
400#102—58no—
401C22×C5#53712yes—
402C22×C5C2×#13712yes—
403C22×C5#57512yes—
404C22×C5C23×D411yescharacteristic
405C2×D5—22no—
406C2×D5—22no—
407C2×D5—22no—
408C2×D5—44no—
409C2×D5—42no—
410C2×D5—42no—
411C2×D5—42no—
412C2×D5—84no—
413C4×C5—22no—
414C4×C5—44no—
415C4×C5—42no—
416C4×C5—42no—
417C4×C5#53712yes—
418C4×C5C2×#13714yes—
419C4×C5C2×H(5,𝔽2)11yescharacteristic
420C4×C5C2×2−1+411yescharacteristic
421Dic5—22no—
422Dic5—22no—
423Dic5—22no—
424Dic5—42no—
425Dic5—44no—
426C24—51no—
427C24—101no—
428C2×Q8—21no—
429C2×Q8—22no—
430C2×Q8—24no—
431C2×Q8—22no—
432C2×Q8—24no—
433C2×Q8—22no—
434C2×Q8—104no—
435C2×Q8—204no—
436C2×Q8C23×D512yes—
437C2×Q8C23×D514yes—
438C2×D4—52no—
439C2×D4—104no—
440C2×D4—101no—
441C2×D4—102no—
442C2×D4—102no—
443C2×D4—204no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—51no—
448C22×C4—51no—
449C22×C4—51no—
450C22×C4—54no—
451C22×C4—52no—
452C22×C4—52no—
453C22×C4—54no—
454C22×C4—102no—
455C22×C4—102no—
456C22×C4—104no—
457C22×C4—102no—
458C22×C4—101no—
459C22×C4—104no—
460C22×C4—202no—
461C22×C4—204no—
462C22×C4D4×D511yescharacteristic
463C22×C4D4×D511yescharacteristic
464C22×C4C23×D511yescharacteristic
465Q16—48no—
466SD16—208no—
467M16—208no—
468C2×C8—24no—
469C2×C8—54no—
470C2×C8—104no—
471C2×C8C23×D514yes—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—21no—
476C4⋊C4—54no—
477C4⋊C4—52no—
478C4⋊C4—104no—
479C4⋊C4—104no—
480C4⋊C4—104no—
481C4⋊C4—108no—
482C4⋊C4—108no—
483C4⋊C4—101no—
484C4⋊C4—102no—
485C4⋊C4—104no—
486C4⋊C4C23×D512yes—
487C22⋊C4—104no—
488C22⋊C4—104no—
489C22⋊C4—104no—
490C22⋊C4—108no—
491C22⋊C4—104no—
492C22⋊C4—104no—
493C42—24no—
494C42—54no—
495C42—52no—
496C42—54no—
497C42—52no—
498C42—108no—
499C42D4×D512yes—
500C42C23×D512yes—
501C2×C5#325214yescomplemented
502C2×C5C2×#53711yescharacteristic
503C2×C5C22×#13711yescharacteristic
504C2×C5C2×#57511yescharacteristic
505D5—42no—
506D5—82no—
507C23—52no—
508C23—102no—
509C23—102no—
510C23—102no—
511C23—102no—
512C23—202no—
513C23—202no—
514C23—202no—
515C23C2×D4×D511yescharacteristic
516Q8—24no—
517Q8—22no—
518Q8—22no—
519Q8—24no—
520Q8—42no—
521Q8—44no—
522Q8—208no—
523D4—102no—
524D4—102no—
525D4—208no—
526D4—202no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—52no—
535C2×C4—52no—
536C2×C4—52no—
537C2×C4—102no—
538C2×C4—102no—
539C2×C4—102no—
540C2×C4—108no—
541C2×C4—104no—
542C2×C4—102no—
543C2×C4—102no—
544C2×C4—104no—
545C2×C4—102no—
546C2×C4—204no—
547C2×C4—202no—
548C2×C4—204no—
549C2×C4—208no—
550C2×C4#406414yes—
551C2×C4C2×D4×D512yes—
552C2×C4C2×D4×D512yes—
553C2×C4C2×D4×D511yescharacteristic
554C2×C4#414412yes—
555C2×C4#415012yes—
556C2×C4C24×D511yescharacteristic
557C8—24no—
558C8—104no—
559C5C2×#325211yescharacteristic, complemented, Sylow
560C22—102no—
561C22—102no—
562C22—102no—
563C22—204no—
564C22—202no—
565C22—202no—
566C22—202no—
567C22—404no—
568C22#6594212yes—
569C22C2×#406412yes—
570C22#6611112yes—
571C22C22×D4×D511yescharacteristic
572C4—22no—
573C4—42no—
574C4—42no—
575C4—44no—
576C4—102no—
577C4—102no—
578C4—102no—
579C4—202no—
580C4—204no—
581C4#6594212yes—
582C4C2×#406414yes—
583C4C2×#414411yescharacteristic
584C4C2×#415011yescharacteristic
585C2—202no—
586C2—402no—
587C2#1061222914yesdirect product factor
588C2C2×#6594211yescharacteristic
589C2C22×#406411yescharacteristic
590C2C2×#6611111yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#462C2
socleC23×C5C22×D4
prime radicalC2×#4041C22
hypercenterC2×#462D5
hypocenterC5C2×#3252
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
4. C2×#65478C2semidirect
5. C2×#65474C2semidirect
6. C2×#65372C2semidirect
7. C2×#65310C2semidirect
8. #10612229C2direct (N × H)
9. C2×#65131C2semidirect
10. C2×#65126C2semidirect
12. C2×#65088C2semidirect
23. #66062C22semidirect
39. C5×#462C22semidirect
44. #65478C22semidirect
45. #65474C22semidirect
49. #65372C22semidirect
50. #65310C22semidirect
53. #65131C22semidirect
54. #65126C22semidirect
56. #65088C22semidirect
147. C2×#462D5semidirect
257. #462C2×D5semidirect
501. C2×C5#3252semidirect
559. C5C2×#3252semidirect
587. C2#10612229direct (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×#659142
C2×C5×#4622
C2×#654782
C2×#654742
C2×#653722
C2×#653102
#106122292
C2×#651312
C2×#651262
C2×#651012
C2×#650882
C2×#32525