Finite Group Databasebeta

← C2×#10613709 #12971406

Subgroups — C2×#10613709 #12971406

Order 1280.

2050 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66059C211yesmaximal, characteristic
2C2×D5×#125C211yesmaximal, characteristic, complemented
3C2×C5×#501C211yesmaximal, characteristic, complemented
4C2×#65369C211yesmaximal, characteristic, complemented
5C2×#65356C211yesmaximal, characteristic, complemented
6#10613709C2116yesmaximal, direct product factor
7C2×#65180C212yesmaximal, complemented
8C2×#65171C212yesmaximal, complemented
9C2×#65160C212yesmaximal, complemented
10C2×#65152C212yesmaximal, complemented
11C2×#65002C211yesmaximal, characteristic, complemented
12C2×#64994C211yesmaximal, characteristic, complemented
13C22×#713—21no—
14C2×C5×C4⋊Q8C2211yescharacteristic
15C2×C5×#125C2211yescharacteristic
16C2×#4093—21no—
17#66059—24no—
18#66059C2218yes—
19D5×#125—24no—
20D5×#125C2218yescomplemented
21C2×#4041—21no—
22C2×#4041C2212yes—
23C2×#4039—21no—
24C2×#4039C2212yes—
25C2×D5×C4⋊C4—21no—
26C2×D5×C4⋊C4—21no—
27C2×D5×C4⋊C4C2211yescharacteristic
28C2×D5×C4⋊C4C2211yescharacteristic
29C2×#4036C2211yescharacteristic
30C2×#4036C2211yescharacteristic
31C2×#4035—21no—
32C2×#4035—21no—
33C2×#4020C2211yescharacteristic
34C2×C42×D5C2211yescharacteristic
35C2×#4016C2211yescharacteristic
36C5×#501C2218yescomplemented
37C2×C5×#107C2212yes—
38C2×C5×#106C2212yes—
39C2×C5×#97C2211yescharacteristic
40C2×#3953C2212yes—
41C2×#3953C2212yes—
42C2×#3952C2212yes—
43C2×#3946C2212yes—
44#65369C2218yescomplemented
45#65356C2218yescomplemented
46C2×#3941C2212yes—
47C2×#3941C2212yes—
48C2×#3940C2212yes—
49C2×#3939C2212yes—
50C2×#3935C2211yescharacteristic
51#65180C22116yescomplemented
52#65171C22116yescomplemented
53#65160C22116yescomplemented
54#65152C22116yescomplemented
55#65002C2218yescomplemented
56#64994C2218yescomplemented
57C2×#3234—51nomaximal, Sylow
58C22×C5×Q8—21no—
59C2×#713—28no—
60C2×#713—48no—
61C22×D20—22no—
62C22×D20—21no—
63C22×D20—21no—
64C22×D20C2312yes—
65C22×C4×D5—21no—
66C22×C4×D5—21no—
67C22×C4×D5—22no—
68C22×C4×D5D411yescharacteristic
69C22×C4×D5D411yescharacteristic
70C22×C4×D5C2311yescharacteristic
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⋊C4C2311yescharacteristic
81C2×C5×C4⋊C4C2312yes—
82C2×C42×C5C2311yescharacteristic
83#4093—28no—
84C2×#685—22no—
85C2×#685—22no—
86C2×C4⋊Dic5—21no—
87C2×C4⋊Dic5—21no—
88C2×C4⋊Dic5C2311yescharacteristic
89C2×C4⋊Dic5C2311yescharacteristic
90C2×#683—21no—
91C2×#683—21no—
92C2×#683—21no—
93C2×#683—21no—
94C2×#683—21no—
95C2×#683—21no—
96C2×C4×Dic5D411yescharacteristic
97C2×C4×Dic5D411yescharacteristic
98C2×C4×Dic5C2311yescharacteristic
99C22×#168C2312yes—
100#4041—24no—
101#4041—28no—
102#4041C2318yes—
103#4039—24no—
104#4039—28no—
105#4039C2318yes—
106D5×C4⋊C4—28no—
107D5×C4⋊C4—28no—
108D5×C4⋊C4—22no—
109D5×C4⋊C4—22no—
110D5×C4⋊C4C2314yes—
111D5×C4⋊C4C2314yes—
112#4036—22no—
113#4036—22no—
114#4036C2314yes—
115#4036C2314yes—
116#4035—28no—
117#4035—28no—
118#4020—22no—
119#4020C2314yes—
120C42×D5D414yes—
121C42×D5C2314yes—
122#4016—22no—
123#4016C2314yes—
124C5×#107C2318yes—
125C5×#106C2318yes—
126C5×#97C2314yes—
127#3953C2318yes—
128#3953C2318yes—
129#3952C2318yes—
130#3946C2318yes—
131#3941C2318yes—
132#3941C2318yes—
133#3940C2318yes—
134#3939C2318yes—
135#3935C2314yes—
136C2×#532—51no—
137C22×#125—51no—
138#3234—516no—
139C2×#501—51no—
140C2×#501D511yescharacteristic, complemented
141C2×#490—51no—
142C2×#490—51no—
143C2×#481—52no—
144C2×#480—52no—
145C2×#478—52no—
146C2×#478—52no—
147C2×#432—51no—
148C23×D5—22no—
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—21no—
159C22×Dic5—21no—
160C22×Dic5C2×D411yescharacteristic
161C22×Dic5C2×D411yescharacteristic
162C22×Dic5C2×D411yescharacteristic
163#713—432no—
164C2×D20—28no—
165C2×D20—28no—
166C2×D20—24no—
167C2×D20—24no—
168C2×D20—22no—
169C2×D20—44no—
170C2×D20—44no—
171C2×D20—48no—
172C2×D20C2414yes—
173C2×C4×D5—24no—
174C2×C4×D5—24no—
175C2×C4×D5—24no—
176C2×C4×D5—24no—
177C2×C4×D5—28no—
178C2×C4×D5—42no—
179C2×C4×D5—48no—
180C2×C4×D5—44no—
181C2×C4×D5—44no—
182C2×C4×D5C2×D414yes—
183C2×C4×D5C2×D412yes—
184C2×C4×D5C2×D414yes—
185C2×C4×D5C2412yes—
186C2×C5×C8—24no—
187C2×C5×C8C2414yes—
188C5×C4⋊C4—24no—
189C5×C4⋊C4—24no—
190C5×C4⋊C4—24no—
191C5×C4⋊C4—21no—
192C5×C4⋊C4—21no—
193C5×C4⋊C4—22no—
194C5×C4⋊C4C2412yes—
195C5×C4⋊C4C2412yes—
196C5×C4⋊C4C2414yes—
197C42×C5C2×D412yes—
198C42×C5C2412yes—
199#685—28no—
200#685—28no—
201C4⋊Dic5—21no—
202C4⋊Dic5—21no—
203C4⋊Dic5—24no—
204C4⋊Dic5—24no—
205C4⋊Dic5C2412yes—
206C4⋊Dic5C2412yes—
207#683—24no—
208#683—24no—
209#683—24no—
210#683—24no—
211#683—24no—
212#683—24no—
213C4×Dic5C2×D412yes—
214C4×Dic5C2×D414yes—
215C4×Dic5C2×D414yes—
216C4×Dic5C2412yes—
217C2×#168—24no—
218C2×#168C2414yes—
219C22×#41—101no—
220#532—58no—
221#532—104no—
222C2×C4⋊Q8C2×D511yescharacteristic
223C2×#127—51no—
224C2×#125—58no—
225C2×#125—51no—
226C2×#125—51no—
227C2×#125—51no—
228C2×#125—101no—
229C2×#125—101no—
230C2×#125—104no—
231C2×#125C2×D511yescharacteristic
232C2×#124—101no—
233C2×#121—52no—
234C2×#121—101no—
235C2×C4×D4—52no—
236C2×C4×D4—101no—
237C22×C4⋊C4—51no—
238C22×C4⋊C4—51no—
239C22×C4⋊C4—101no—
240C22×C4⋊C4—101no—
241C22×C42—51no—
242#501—58no—
243#501C2×D518yescomplemented
244#490—58no—
245#490—58no—
246#481—516no—
247#480—516no—
248#478—516no—
249#478—516no—
250#432—58no—
251C2×#107—52no—
252C2×#107C2×D512yes—
253C2×#106—52no—
254C2×#106C2×D512yes—
255C2×C4⋊C8—52no—
256C2×#102—52no—
257C2×#102—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—24no—
267C22×D5—44no—
268C22×D5—44no—
269C22×D5—44no—
270C22×D5AGL(1,ℤ/8)14yes—
271C22×D5C22×D412yes—
272C5×Q8—48no—
273C2×C4×C5—22no—
274C2×C4×C5—22no—
275C2×C4×C5—22no—
276C2×C4×C5—22no—
277C2×C4×C5—24no—
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—22no—
290C2×Dic5—22no—
291C2×Dic5—42no—
292C2×Dic5—42no—
293C2×Dic5C22×D412yes—
294C2×Dic5C22×D412yes—
295C2×Dic5C22×D412yes—
296D20—28no—
297D20—48no—
298D20—48no—
299D20—416no—
300D20—44no—
301C4×D5—28no—
302C4×D5—24no—
303C4×D5—24no—
304C4×D5—44no—
305C4×D5—416no—
306C4×D5—48no—
307C4×D5—48no—
308C5×C8—24no—
309#168—24no—
310C2×#41—108no—
311C2×#41—208no—
312C22×Q8—21no—
313C22×D4—52no—
314C22×D4—102no—
315C22×D4—101no—
316C22×D4—101no—
317C23×C4—51no—
318C23×C4—51no—
319C23×C4—51no—
320C23×C4—102no—
321C23×C4—101no—
322C23×C4—101no—
323C22×C8—52no—
324C22×C8C22×D512yes—
325C4⋊Q8—22no—
326C4⋊Q8C22×D514yes—
327#127—54no—
328#127—102no—
329#125—22no—
330#125—54no—
331#125—54no—
332#125—54no—
333#125—108no—
334#125—108no—
335#125—102no—
336#125—102no—
337#125—102no—
338#125C22×D514yes—
339#124—108no—
340#121—58no—
341#121—104no—
342#121—108no—
343C4×D4—58no—
344C4×D4—104no—
345C4×D4—108no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—51no—
350C2×C4⋊C4—51no—
351C2×C4⋊C4—54no—
352C2×C4⋊C4—54no—
353C2×C4⋊C4—102no—
354C2×C4⋊C4—102no—
355C2×C4⋊C4—108no—
356C2×C4⋊C4—108no—
357C2×C4⋊C4—101no—
358C2×C4⋊C4—101no—
359C2×C4⋊C4—101no—
360C2×C4⋊C4—101no—
361C2×C4⋊C4—101no—
362C2×C4⋊C4—101no—
363C2×C4⋊C4—101no—
364C2×C4⋊C4—101no—
365C2×C4⋊C4C22×D511yescharacteristic
366C2×C4⋊C4C22×D511yescharacteristic
367C2×C4⋊C4C22×D512yes—
368C2×C22⋊C4—102no—
369C2×C22⋊C4—102no—
370C2×C42—51no—
371C2×C42—51no—
372C2×C42—51no—
373C2×C42—54no—
374C2×C42—54no—
375C2×C42C22×D511yescharacteristic
376#107—58no—
377#107C22×D518yes—
378#106—58no—
379#106C22×D518yes—
380C4⋊C8—58no—
381#102—58no—
382#102—58no—
383#102—58no—
384#102—58no—
385C22⋊C8—58no—
386#97—54no—
387#97C22×D514yes—
388C22×C5#53612yes—
389C22×C5C2×AGL(1,ℤ/8)12yes—
390C22×C5#57512yes—
391C22×C5C23×D411yescharacteristic
392C2×D5—22no—
393C2×D5—28no—
394C2×D5—22no—
395C2×D5—44no—
396C2×D5—44no—
397C2×D5—44no—
398C2×D5—88no—
399C2×D5C2×AGL(1,ℤ/8)14yes—
400C4×C5—22no—
401C4×C5—22no—
402C4×C5—44no—
403C4×C5—42no—
404C4×C5—42no—
405C4×C5#53612yes—
406C4×C5C2×2−1+412yes—
407Dic5—22no—
408Dic5—22no—
409Dic5—22no—
410Dic5—42no—
411Dic5—42no—
412C24—51no—
413C24—102no—
414#41—2032no—
415C2×Q8—24no—
416C2×Q8—44no—
417C2×D4—54no—
418C2×D4—108no—
419C2×D4—104no—
420C2×D4—102no—
421C2×D4—104no—
422C2×D4—108no—
423C2×D4—204no—
424C2×D4—204no—
425C2×D4—208no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—51no—
430C22×C4—51no—
431C22×C4—51no—
432C22×C4—52no—
433C22×C4—54no—
434C22×C4—52no—
435C22×C4—54no—
436C22×C4—108no—
437C22×C4—104no—
438C22×C4—104no—
439C22×C4—104no—
440C22×C4—104no—
441C22×C4—101no—
442C22×C4—101no—
443C22×C4—202no—
444C22×C4—208no—
445C22×C4—204no—
446C22×C4—204no—
447C22×C4D4×D511yescharacteristic
448C22×C4D4×D511yescharacteristic
449C22×C4C23×D511yescharacteristic
450C2×C8—24no—
451C2×C8—54no—
452C2×C8—104no—
453C2×C8C23×D514yes—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—52no—
461C4⋊C4—52no—
462C4⋊C4—104no—
463C4⋊C4—104no—
464C4⋊C4—104no—
465C4⋊C4—104no—
466C4⋊C4—104no—
467C4⋊C4—104no—
468C4⋊C4—101no—
469C4⋊C4—101no—
470C4⋊C4—104no—
471C4⋊C4—104no—
472C4⋊C4C23×D512yes—
473C4⋊C4C23×D512yes—
474C4⋊C4C23×D514yes—
475C22⋊C4—108no—
476C22⋊C4—108no—
477C42—54no—
478C42—52no—
479C42—54no—
480C42—52no—
481C42D4×D512yes—
482C42C23×D512yes—
483C2×C5#323414yescomplemented
484C2×C5C2×#53611yescharacteristic
485C2×C5C22×AGL(1,ℤ/8)11yescharacteristic
486C2×C5C2×#57511yescharacteristic
487D5—24no—
488D5—84no—
489C23—54no—
490C23—52no—
491C23—102no—
492C23—102no—
493C23—104no—
494C23—204no—
495C23—204no—
496C23—204no—
497C23C2×D4×D511yescharacteristic
498Q8—48no—
499D4—108no—
500D4—208no—
501D4—208no—
502D4—2016no—
503D4—204no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—52no—
513C2×C4—52no—
514C2×C4—52no—
515C2×C4—108no—
516C2×C4—102no—
517C2×C4—102no—
518C2×C4—102no—
519C2×C4—104no—
520C2×C4—104no—
521C2×C4—102no—
522C2×C4—102no—
523C2×C4—204no—
524C2×C4—2016no—
525C2×C4—208no—
526C2×C4—208no—
527C2×C4—202no—
528C2×C4—202no—
529C2×C4C2×D4×D512yes—
530C2×C4C2×D4×D512yes—
531C2×C4C2×D4×D511yescharacteristic
532C2×C4#414714yes—
533C2×C4C24×D511yescharacteristic
534C8—24no—
535C8—104no—
536C5C2×#323411yescharacteristic, complemented, Sylow
537C22—54no—
538C22—108no—
539C22—102no—
540C22—102no—
541C22—204no—
542C22—204no—
543C22—204no—
544C22—408no—
545C22#6605812yes—
546C22D5×AGL(1,ℤ/8)12yes—
547C22#6611112yes—
548C22C22×D4×D511yescharacteristic
549C4—22no—
550C4—22no—
551C4—44no—
552C4—42no—
553C4—42no—
554C4—102no—
555C4—102no—
556C4—102no—
557C4—202no—
558C4—202no—
559C4#6605812yes—
560C4C2×#414712yes—
561C2—104no—
562C2—404no—
563C2#1061370914yesdirect product factor
564C2C2×#6605811yescharacteristic
565C2C2×D5×AGL(1,ℤ/8)11yescharacteristic
566C2C2×#6611111yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#501C2
socleC23×C5C22×D4
prime radicalC2×#66059C2
hypercenterC2×#501D5
hypocenterC5C2×#3234
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
2. C2×D5×#125C2semidirect
3. C2×C5×#501C2semidirect
4. C2×#65369C2semidirect
5. C2×#65356C2semidirect
6. #10613709C2direct (N × H)
7. C2×#65180C2semidirect
8. C2×#65171C2semidirect
9. C2×#65160C2semidirect
10. C2×#65152C2semidirect
11. C2×#65002C2semidirect
12. C2×#64994C2semidirect
20. D5×#125C22semidirect
36. C5×#501C22semidirect
44. #65369C22semidirect
45. #65356C22semidirect
51. #65180C22semidirect
52. #65171C22semidirect
53. #65160C22semidirect
54. #65152C22semidirect
55. #65002C22semidirect
56. #64994C22semidirect
140. C2×#501D5semidirect
243. #501C2×D5semidirect
483. C2×C5#3234semidirect
536. C5C2×#3234semidirect
563. C2#10613709direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#660592
C2×D5×#1252
C2×C5×#5012
C2×#653692
C2×#653562
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