Finite Group Databasebeta

← C2×#10613714 #12971433

Subgroups — C2×#10613714 #12971433

Order 1280.

2018 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D5×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#66025C211yesmaximal, characteristic
3C2×C5×#501C211yesmaximal, characteristic, complemented
4C2×#65378C211yesmaximal, characteristic, complemented
5C2×#65354C211yesmaximal, characteristic, complemented
6#10613714C2116yesmaximal, direct product factor
7C2×#65179C212yesmaximal, complemented
8C2×#65173C212yesmaximal, complemented
9C2×#65143C212yesmaximal, complemented
10C2×#65152C212yesmaximal
11C2×#65003C211yesmaximal, characteristic, complemented
12C2×#64994C211yesmaximal, characteristic, complemented
13C22×Q8×D5—21no—
14C2×C5×C4⋊Q8C2211yescharacteristic
15C2×C5×#125C2211yescharacteristic
16C2×#4090—21no—
17D5×C4⋊Q8—24no—
18D5×C4⋊Q8C2218yescomplemented
19#66025—24no—
20#66025C2218yes—
21C2×#4044—21no—
22C2×#4043C2211yescharacteristic
23C2×#4041C2211yescharacteristic
24C2×#4040—21no—
25C2×#4039—21no—
26C2×#4039C2211yescharacteristic
27C2×#4038—21no—
28C2×D5×C4⋊C4—21no—
29C2×D5×C4⋊C4C2212yes—
30C2×#4034—21no—
31C2×#4034C2212yes—
32C2×#4033C2211yescharacteristic
33C2×#4021C2211yescharacteristic
34C2×C42×D5C2211yescharacteristic
35C2×#4015C2211yescharacteristic
36C5×#501C2218yescomplemented
37C2×C5×#107C2212yes—
38C2×C5×#106C2212yes—
39C2×C5×#97C2211yescharacteristic
40C2×#3953C2212yes—
41C2×#3952C2212yes—
42C2×#3948C2212yes—
43C2×#3946C2212yes—
44#65378C2218yescomplemented
45#65354C2218yescomplemented
46C2×#3942C2212yes—
47C2×#3941C2212yes—
48C2×#3940C2212yes—
49C2×#3939C2212yes—
50C2×#3935C2211yescharacteristic
51#65179C22116yescomplemented
52#65173C22116yescomplemented
53#65152C22116yes—
54#65143C22116yescomplemented
55#65003C2218yescomplemented
56#64994C2218yescomplemented
57C2×#3239—51nomaximal, Sylow
58C22×C5×Q8—21no—
59C2×Q8×D5—28no—
60C2×Q8×D5—48no—
61C22×D20—21no—
62C22×D20C2311yescharacteristic
63C22×C4×D5—22no—
64C22×C4×D5—21no—
65C22×C4×D5D411yescharacteristic
66C22×C4×D5D411yescharacteristic
67C22×C4×D5C2311yescharacteristic
68C22×Dic10—21no—
69C22×Dic10—22no—
70C22×Dic10C2311yescharacteristic
71C22×C5×C8C2312yes—
72C5×C4⋊Q8—22no—
73C5×C4⋊Q8C2314yes—
74C5×#125—22no—
75C5×#125C2314yes—
76C2×C5×C4⋊C4—21no—
77C2×C5×C4⋊C4—21no—
78C2×C5×C4⋊C4—21no—
79C2×C5×C4⋊C4C2311yescharacteristic
80C2×C5×C4⋊C4C2312yes—
81C2×C5×C4⋊C4C2311yescharacteristic
82C2×C42×C5C2311yescharacteristic
83#4090—28no—
84C2×#685—21no—
85C2×#685—21no—
86C2×#685—21no—
87C2×#685—21no—
88C2×#685—21no—
89C2×C4⋊Dic5—21no—
90C2×C4⋊Dic5—21no—
91C2×C4⋊Dic5C2312yes—
92C2×#683—21no—
93C2×#683—22no—
94C2×#683—22no—
95C2×C4×Dic5—21no—
96C2×C4×Dic5D411yescharacteristic
97C2×C4×Dic5D411yescharacteristic
98C2×C4×Dic5C2311yescharacteristic
99C22×#168C2312yes—
100#4044—28no—
101#4043—22no—
102#4043C2314yes—
103#4041—22no—
104#4041C2314yes—
105#4040—28no—
106#4039—28no—
107#4039—22no—
108#4039C2314yes—
109#4038—28no—
110D5×C4⋊C4—24no—
111D5×C4⋊C4—28no—
112D5×C4⋊C4C2318yes—
113#4034—24no—
114#4034—28no—
115#4034C2318yes—
116#4033—22no—
117#4033C2314yes—
118#4021—22no—
119#4021C2314yes—
120C42×D5D414yes—
121C42×D5C2314yes—
122#4015—22no—
123#4015C2314yes—
124C5×#107C2318yes—
125C5×#106C2318yes—
126C5×#97C2314yes—
127#3953C2318yes—
128#3952C2318yes—
129#3948C2318yes—
130#3946C2318yes—
131#3942C2318yes—
132#3941C2318yes—
133#3940C2318yes—
134#3939C2318yes—
135#3935C2314yes—
136C22×C4⋊Q8—51no—
137C2×#529—51no—
138#3239—516no—
139C2×#501—51no—
140C2×#501D511yescharacteristic, complemented
141C2×#489—51no—
142C2×#489—51no—
143C2×#484—52no—
144C2×#481—52no—
145C2×#475—52no—
146C2×#474—52no—
147C2×#432—51no—
148C23×D5—21no—
149C23×D5C2×D411yescharacteristic
150C2×C5×Q8—24no—
151C2×C5×Q8—44no—
152C22×C4×C5—21no—
153C22×C4×C5—21no—
154C22×C4×C5—22no—
155C22×C4×C5C2×D411yescharacteristic
156C22×C4×C5C2×D411yescharacteristic
157C22×C4×C5C2411yescharacteristic
158C22×Dic5—22no—
159C22×Dic5—21no—
160C22×Dic5C2×D411yescharacteristic
161C22×Dic5C2×D411yescharacteristic
162C22×Dic5C2×D411yescharacteristic
163Q8×D5—432no—
164C2×D20—24no—
165C2×D20—24no—
166C2×D20—21no—
167C2×D20—44no—
168C2×D20C2412yes—
169C2×C4×D5—24no—
170C2×C4×D5—24no—
171C2×C4×D5—24no—
172C2×C4×D5—28no—
173C2×C4×D5—24no—
174C2×C4×D5—48no—
175C2×C4×D5—44no—
176C2×C4×D5C2×D414yes—
177C2×C4×D5C2×D414yes—
178C2×C4×D5C2×D412yes—
179C2×C4×D5C2412yes—
180C2×Dic10—24no—
181C2×Dic10—24no—
182C2×Dic10—21no—
183C2×Dic10—28no—
184C2×Dic10—44no—
185C2×Dic10—48no—
186C2×Dic10C2412yes—
187C2×C5×C8—24no—
188C2×C5×C8C2414yes—
189C5×C4⋊C4—24no—
190C5×C4⋊C4—24no—
191C5×C4⋊C4—24no—
192C5×C4⋊C4—21no—
193C5×C4⋊C4—21no—
194C5×C4⋊C4—22no—
195C5×C4⋊C4C2412yes—
196C5×C4⋊C4C2412yes—
197C5×C4⋊C4C2414yes—
198C42×C5C2×D412yes—
199C42×C5C2412yes—
200#685—24no—
201#685—24no—
202#685—24no—
203#685—24no—
204#685—24no—
205C4⋊Dic5—22no—
206C4⋊Dic5—24no—
207C4⋊Dic5—24no—
208C4⋊Dic5C2414yes—
209#683—24no—
210#683—28no—
211#683—28no—
212C4×Dic5—24no—
213C4×Dic5C2×D412yes—
214C4×Dic5C2×D414yes—
215C4×Dic5C2×D414yes—
216C4×Dic5C2412yes—
217C2×#168—24no—
218C2×#168C2414yes—
219C23×Q8—101no—
220C2×C4⋊Q8—58no—
221C2×C4⋊Q8—52no—
222C2×C4⋊Q8—51no—
223C2×C4⋊Q8—101no—
224C2×C4⋊Q8—104no—
225C2×C4⋊Q8—101no—
226C2×C4⋊Q8C2×D511yescharacteristic
227#529—58no—
228#529—104no—
229C2×#126—101no—
230C2×#125C2×D511yescharacteristic
231C2×#124—51no—
232C2×#123—101no—
233C2×C22⋊Q8—51no—
234C2×#121—51no—
235C2×C4×Q8—51no—
236C2×C4×D4—51no—
237C2×C4×D4—101no—
238C2×#117—101no—
239C22×C4⋊C4—52no—
240C22×C4⋊C4—101no—
241C22×C42—51no—
242#501—58no—
243#501C2×D518yescomplemented
244#489—58no—
245#489—58no—
246#484—516no—
247#481—516no—
248#475—516no—
249#474—516no—
250#432—58no—
251C2×#107—52no—
252C2×#107C2×D512yes—
253C2×#106—52no—
254C2×#106C2×D512yes—
255C2×C4⋊C8—52no—
256C2×#103—52no—
257C2×#103—52no—
258C2×#102—52no—
259C2×#102—52no—
260C2×C22⋊C8—52no—
261C2×#97—51no—
262C2×#97C2×D511yescharacteristic
263C23×C5C22×D411yescharacteristic
264C22×D5—22no—
265C22×D5—22no—
266C22×D5—22no—
267C22×D5—42no—
268C22×D5—42no—
269C22×D5—42no—
270C22×D5#13714yes—
271C22×D5C22×D412yes—
272C5×Q8—48no—
273C2×C4×C5—22no—
274C2×C4×C5—22no—
275C2×C4×C5—22no—
276C2×C4×C5—24no—
277C2×C4×C5—22no—
278C2×C4×C5—44no—
279C2×C4×C5—42no—
280C2×C4×C5—42no—
281C2×C4×C5C22×D412yes—
282C2×C4×C5C22×D412yes—
283C2×C4×C5C22×D411yescharacteristic
284C2×C4×C52−1+414yes—
285C2×C4×C5C2511yescharacteristic
286C2×Dic5—22no—
287C2×Dic5—22no—
288C2×Dic5—22no—
289C2×Dic5—24no—
290C2×Dic5—42no—
291C2×Dic5—44no—
292C2×Dic5AGL(1,ℤ/8)14yes—
293C2×Dic5C22×D412yes—
294C2×Dic5C22×D412yes—
295C2×Dic5C22×D412yes—
296D20—24no—
297D20—48no—
298D20—42no—
299C4×D5—28no—
300C4×D5—24no—
301C4×D5—24no—
302C4×D5—28no—
303C4×D5—416no—
304C4×D5—48no—
305Dic10—24no—
306Dic10—48no—
307Dic10—42no—
308Dic10—416no—
309C5×C8—24no—
310#168—24no—
311C22×Q8—21no—
312C22×Q8—51no—
313C22×Q8—102no—
314C22×Q8—108no—
315C22×Q8—101no—
316C22×Q8—208no—
317C22×D4—51no—
318C22×D4—101no—
319C23×C4—51no—
320C23×C4—51no—
321C23×C4—51no—
322C23×C4—101no—
323C23×C4—102no—
324C22×C8—52no—
325C22×C8C22×D512yes—
326C4⋊Q8—22no—
327C4⋊Q8—58no—
328C4⋊Q8—54no—
329C4⋊Q8—108no—
330C4⋊Q8—104no—
331C4⋊Q8—108no—
332C4⋊Q8—102no—
333C4⋊Q8C22×D514yes—
334#126—108no—
335#125—22no—
336#125C22×D514yes—
337#124—54no—
338#124—102no—
339#123—108no—
340C22⋊Q8—54no—
341C22⋊Q8—102no—
342#121—54no—
343#121—102no—
344C4×Q8—54no—
345C4×Q8—102no—
346C4×D4—54no—
347C4×D4—102no—
348C4×D4—108no—
349#117—108no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—52no—
354C2×C4⋊C4—58no—
355C2×C4⋊C4—104no—
356C2×C4⋊C4—108no—
357C2×C4⋊C4—102no—
358C2×C4⋊C4—102no—
359C2×C4⋊C4—101no—
360C2×C4⋊C4—101no—
361C2×C4⋊C4—101no—
362C2×C4⋊C4C22×D512yes—
363C2×C4⋊C4C22×D511yescharacteristic
364C2×C4⋊C4C22×D511yescharacteristic
365C2×C22⋊C4—101no—
366C2×C22⋊C4—101no—
367C2×C22⋊C4—101no—
368C2×C22⋊C4—101no—
369C2×C22⋊C4—101no—
370C2×C42—51no—
371C2×C42—51no—
372C2×C42—51no—
373C2×C42—54no—
374C2×C42—54no—
375C2×C42—101no—
376C2×C42C22×D511yescharacteristic
377#107—58no—
378#107C22×D518yes—
379#106—58no—
380#106C22×D518yes—
381C4⋊C8—58no—
382#103—58no—
383#103—58no—
384#102—58no—
385#102—58no—
386C22⋊C8—58no—
387#97—54no—
388#97C22×D514yes—
389C22×C5#53612yes—
390C22×C5C2×AGL(1,ℤ/8)12yes—
391C22×C5C2×#13712yes—
392C22×C5C23×D411yescharacteristic
393C2×D5—22no—
394C2×D5—28no—
395C2×D5—22no—
396C2×D5—42no—
397C2×D5—42no—
398C2×D5—42no—
399C2×D5—84no—
400C2×D5C2×#13714yes—
401C4×C5—22no—
402C4×C5—22no—
403C4×C5—44no—
404C4×C5—42no—
405C4×C5—42no—
406C4×C5#53612yes—
407C4×C5C2×2−1+412yes—
408Dic5—22no—
409Dic5—22no—
410Dic5—42no—
411Dic5—44no—
412Dic5C2×AGL(1,ℤ/8)14yes—
413C24—51no—
414C24—101no—
415C2×Q8—24no—
416C2×Q8—44no—
417C2×Q8—52no—
418C2×Q8—108no—
419C2×Q8—101no—
420C2×Q8—104no—
421C2×Q8—104no—
422C2×Q8—204no—
423C2×Q8—208no—
424C2×Q8—2032no—
425C2×D4—52no—
426C2×D4—104no—
427C2×D4—101no—
428C2×D4—104no—
429C2×D4—204no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—51no—
434C22×C4—51no—
435C22×C4—51no—
436C22×C4—52no—
437C22×C4—54no—
438C22×C4—54no—
439C22×C4—52no—
440C22×C4—104no—
441C22×C4—104no—
442C22×C4—104no—
443C22×C4—108no—
444C22×C4—104no—
445C22×C4—101no—
446C22×C4—102no—
447C22×C4—204no—
448C22×C4—208no—
449C22×C4D4×D511yescharacteristic
450C22×C4D4×D511yescharacteristic
451C22×C4C23×D511yescharacteristic
452C2×C8—24no—
453C2×C8—54no—
454C2×C8—104no—
455C2×C8C23×D514yes—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—54no—
463C4⋊C4—108no—
464C4⋊C4—108no—
465C4⋊C4—104no—
466C4⋊C4—104no—
467C4⋊C4—102no—
468C4⋊C4—104no—
469C4⋊C4C23×D514yes—
470C4⋊C4C23×D512yes—
471C4⋊C4C23×D512yes—
472C22⋊C4—104no—
473C22⋊C4—104no—
474C22⋊C4—104no—
475C22⋊C4—104no—
476C22⋊C4—104no—
477C42—54no—
478C42—52no—
479C42—54no—
480C42—52no—
481C42—104no—
482C42D4×D512yes—
483C42C23×D512yes—
484C2×C5#323914yescomplemented
485C2×C5C2×#53611yescharacteristic
486C2×C5C22×AGL(1,ℤ/8)11yescharacteristic
487C2×C5C22×#13711yescharacteristic
488D5—24no—
489D5—82no—
490C23—54no—
491C23—52no—
492C23—102no—
493C23—102no—
494C23—102no—
495C23—202no—
496C23—202no—
497C23—202no—
498C23C2×D4×D511yescharacteristic
499Q8—48no—
500Q8—104no—
501Q8—208no—
502Q8—2016no—
503Q8—202no—
504D4—104no—
505D4—208no—
506D4—202no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—54no—
516C2×C4—52no—
517C2×C4—52no—
518C2×C4—52no—
519C2×C4—108no—
520C2×C4—108no—
521C2×C4—102no—
522C2×C4—102no—
523C2×C4—104no—
524C2×C4—104no—
525C2×C4—102no—
526C2×C4—104no—
527C2×C4—208no—
528C2×C4—2016no—
529C2×C4—202no—
530C2×C4—204no—
531C2×C4C2×D4×D512yes—
532C2×C4C2×D4×D512yes—
533C2×C4C2×D4×D511yescharacteristic
534C2×C4#414714yes—
535C2×C4C24×D511yescharacteristic
536C8—24no—
537C8—104no—
538C5C2×#323911yescharacteristic, complemented, Sylow
539C22—54no—
540C22—102no—
541C22—108no—
542C22—102no—
543C22—202no—
544C22—202no—
545C22—202no—
546C22—404no—
547C22#6605812yes—
548C22D5×AGL(1,ℤ/8)12yes—
549C22D5×#13712yes—
550C22C22×D4×D511yescharacteristic
551C4—22no—
552C4—22no—
553C4—44no—
554C4—42no—
555C4—42no—
556C4—54no—
557C4—102no—
558C4—102no—
559C4—204no—
560C4—202no—
561C4#6605812yes—
562C4C2×#414712yes—
563C2—104no—
564C2—402no—
565C2#1061371414yesdirect product factor
566C2C2×#6605811yescharacteristic
567C2C2×D5×AGL(1,ℤ/8)11yescharacteristic
568C2C2×D5×#13711yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#501C2
socleC23×C5C22×D4
prime radicalC2×#4041C22
hypercenterC2×#501D5
hypocenterC5C2×#3239
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
1. C2×D5×C4⋊Q8C2semidirect
3. C2×C5×#501C2semidirect
4. C2×#65378C2semidirect
5. C2×#65354C2semidirect
6. #10613714C2direct (N × H)
7. C2×#65179C2semidirect
8. C2×#65173C2semidirect
9. C2×#65143C2semidirect
11. C2×#65003C2semidirect
12. C2×#64994C2semidirect
18. D5×C4⋊Q8C22semidirect
36. C5×#501C22semidirect
44. #65378C22semidirect
45. #65354C22semidirect
51. #65179C22semidirect
52. #65173C22semidirect
54. #65143C22semidirect
55. #65003C22semidirect
56. #64994C22semidirect
140. C2×#501D5semidirect
243. #501C2×D5semidirect
484. C2×C5#3239semidirect
538. C5C2×#3239semidirect
565. C2#10613714direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D5×C4⋊Q82
C2×#660252
C2×C5×#5012
C2×#653782
C2×#653542
#106137142
C2×#651792
C2×#651732
C2×#651432
C2×#651522
C2×#650032
C2×#649942
C2×#32395