Finite Group Databasebeta

← C2×#10612829 #12972171

Subgroups — C2×#10612829 #12972171

Order 1280.

2146 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66059C211yesmaximal, characteristic
2C2×#65912C211yesmaximal, characteristic, complemented
3C2×C5×#477C211yesmaximal, characteristic, complemented
4C2×#65380C211yesmaximal, characteristic, complemented
5C2×#65316C211yesmaximal, characteristic, complemented
6#10612829C2116yesmaximal, direct product factor
7C2×#65179C212yesmaximal, complemented
8C2×#65175C212yesmaximal, complemented
9C2×#65137C211yesmaximal, characteristic, complemented
10C2×#65126C211yesmaximal, characteristic, complemented
11C2×#65107C212yesmaximal, complemented
12C2×#65101C212yesmaximal, complemented
13C22×#713—21no—
14C2×C5×C4⋊Q8C2211yescharacteristic
15C2×C4×C5×Q8C2211yescharacteristic
16C2×#4093—21no—
17C2×#4092C2211yescharacteristic
18C2×#4090C2211yescharacteristic
19C22×#688C2212yes—
20C22×#677C2212yes—
21C22×#676C2212yes—
22#66059—24no—
23#66059C2218yes—
24#65912—24no—
25#65912C2218yescomplemented
26C2×#4043C2211yescharacteristic
27C2×#4042—21no—
28C2×#4041—21no—
29C2×#4041C2212yes—
30C2×#4039—21no—
31C2×#4039C2212yes—
32C2×#4035—21no—
33C2×#4034C2211yescharacteristic
34C2×#4020C2211yescharacteristic
35C2×#4018—21no—
36C2×C42×D5C2211yescharacteristic
37C2×C4×Dic10—21no—
38C2×C4×Dic10C2211yescharacteristic
39C5×#477C2218yescomplemented
40C2×C5×#107C2212yes—
41C2×C5×C4⋊C8C2211yescharacteristic
42C2×C5×#103C2212yes—
43C2×#3953C2212yes—
44C2×#3946C2212yes—
45#65380C2218yescomplemented
46#65316C2218yescomplemented
47C2×#3942C2212yes—
48C2×#3941C2212yes—
49C2×#3939C2212yes—
50C2×C4⋊#168C2211yescharacteristic
51#65179C22116yescomplemented
52#65175C22116yescomplemented
53#65137C2218yescomplemented
54#65126C2218yescomplemented
55#65107C22116yescomplemented
56#65101C22116yescomplemented
57C2×#3254—51nomaximal, Sylow
58C22×C5×Q8—21no—
59C22×C5×Q8C2311yescharacteristic
60C2×#713—28no—
61C2×#713—48no—
62C22×D20—22no—
63C22×D20—21no—
64C22×D20—21no—
65C22×D20C2312yes—
66C22×C4×D5—22no—
67C22×C4×D5D411yescharacteristic
68C22×C4×D5D411yescharacteristic
69C22×C4×D5C2311yescharacteristic
70C22×Dic10—21no—
71C22×Dic10C2311yescharacteristic
72C22×C5×C8C2312yes—
73C5×C4⋊Q8—22no—
74C5×C4⋊Q8C2314yes—
75C4×C5×Q8—22no—
76C4×C5×Q8C2314yes—
77C2×C5×C4⋊C4—21no—
78C2×C5×C4⋊C4—21no—
79C2×C5×C4⋊C4C2311yescharacteristic
80C2×C5×C4⋊C4C2312yes—
81C2×C42×C5—21no—
82C2×C42×C5C2311yescharacteristic
83#4093—28no—
84#4092—22no—
85#4092C2314yes—
86#4090—22no—
87#4090C2314yes—
88C2×#688—44no—
89C2×#688C2318yes—
90C2×#685—21no—
91C2×#685—21no—
92C2×#685—22no—
93C2×#685—22no—
94C2×C4⋊Dic5—21no—
95C2×C4⋊Dic5C2311yescharacteristic
96C2×#683—21no—
97C2×#683—21no—
98C2×#683—21no—
99C2×#683—21no—
100C2×#683—21no—
101C2×C4×Dic5—21no—
102C2×C4×Dic5D411yescharacteristic
103C2×C4×Dic5D411yescharacteristic
104C2×C4×Dic5C2311yescharacteristic
105C22×#168C2312yes—
106C2×#677—44no—
107C2×#677C2318yes—
108C2×#676—44no—
109C2×#676C2318yes—
110#4043—22no—
111#4043C2314yes—
112#4042—28no—
113#4041—24no—
114#4041—28no—
115#4041C2318yes—
116#4039—24no—
117#4039—28no—
118#4039C2318yes—
119#4035—28no—
120#4034—22no—
121#4034C2314yes—
122#4020—22no—
123#4020C2314yes—
124#4018—28no—
125C42×D5D414yes—
126C42×D5C2314yes—
127C4×Dic10—22no—
128C4×Dic10—28no—
129C4×Dic10C2314yes—
130C5×#107C2318yes—
131C5×C4⋊C8C2314yes—
132C5×#103C2318yes—
133#3953C2318yes—
134#3946C2318yes—
135#3942C2318yes—
136#3941C2318yes—
137#3939C2318yes—
138C4⋊#168C2314yes—
139C2×#533—51no—
140C2×#532—51no—
141#3254—516no—
142C2×#477—51no—
143C2×#477D511yescharacteristic, complemented
144C2×#474—52no—
145C2×#468—52no—
146C2×#468—52no—
147C2×#460—51no—
148C2×#460—51no—
149C2×#461—52no—
150C2×#423—51no—
151C23×D5—22no—
152C23×D5C2×D411yescharacteristic
153C2×C5×Q8—21no—
154C2×C5×Q8—24no—
155C2×C5×Q8—22no—
156C2×C5×Q8—22no—
157C2×C5×Q8—44no—
158C2×C5×Q8C2412yes—
159C22×C4×C5—21no—
160C22×C4×C5—21no—
161C22×C4×C5—22no—
162C22×C4×C5C2×D411yescharacteristic
163C22×C4×C5C2×D411yescharacteristic
164C22×C4×C5C2411yescharacteristic
165C22×Dic5—21no—
166C22×Dic5—21no—
167C22×Dic5C2×D411yescharacteristic
168C22×Dic5C2×D411yescharacteristic
169C22×Dic5C2×D411yescharacteristic
170#713—432no—
171C2×D20—24no—
172C2×D20—24no—
173C2×D20—28no—
174C2×D20—24no—
175C2×D20—24no—
176C2×D20—22no—
177C2×D20—44no—
178C2×D20—48no—
179C2×D20—44no—
180C2×D20C2414yes—
181C2×C4×D5—24no—
182C2×C4×D5—22no—
183C2×C4×D5—22no—
184C2×C4×D5—24no—
185C2×C4×D5—28no—
186C2×C4×D5—48no—
187C2×C4×D5C2×D414yes—
188C2×C4×D5C2×D414yes—
189C2×C4×D5C2×D412yes—
190C2×C4×D5C2412yes—
191C2×Dic10—22no—
192C2×Dic10—22no—
193C2×Dic10—21no—
194C2×Dic10—24no—
195C2×Dic10—44no—
196C2×Dic10C2412yes—
197C2×C5×C8—24no—
198C2×C5×C8C2414yes—
199C5×C4⋊C4—24no—
200C5×C4⋊C4—24no—
201C5×C4⋊C4—21no—
202C5×C4⋊C4—22no—
203C5×C4⋊C4C2412yes—
204C5×C4⋊C4C2414yes—
205C42×C5—24no—
206C42×C5C2×D412yes—
207C42×C5C2412yes—
208#688—48no—
209#685—24no—
210#685—24no—
211#685—28no—
212#685—28no—
213C4⋊Dic5—21no—
214C4⋊Dic5—24no—
215C4⋊Dic5C2412yes—
216#683—24no—
217#683—24no—
218#683—24no—
219#683—24no—
220#683—24no—
221C4×Dic5—24no—
222C4×Dic5C2×D414yes—
223C4×Dic5C2×D412yes—
224C4×Dic5C2×D414yes—
225C4×Dic5C2412yes—
226C2×#168—24no—
227C2×#168C2414yes—
228#677—48no—
229#676—48no—
230C22×#41—101no—
231C22×SD16—52no—
232C22×SD16—52no—
233C22×M16—52no—
234#533—58no—
235#533—104no—
236#532—58no—
237#532—104no—
238C2×C4⋊Q8—51no—
239C2×C4⋊Q8—51no—
240C2×C4⋊Q8C2×D511yescharacteristic
241C2×#127—51no—
242C2×#125—101no—
243C2×#124—101no—
244C2×C22⋊Q8—51no—
245C2×C22⋊Q8—51no—
246C2×C22⋊Q8—101no—
247C2×#121—52no—
248C2×#121—101no—
249C2×C4×Q8—51no—
250C2×C4×Q8—101no—
251C2×C4×Q8C2×D511yescharacteristic
252C2×C4×D4—52no—
253C2×C4×D4—101no—
254C2×#117—101no—
255C22×C42—51no—
256#477—58no—
257#477C2×D518yescomplemented
258#474—516no—
259#468—516no—
260#468—516no—
261#461—516no—
262#460—58no—
263#460—58no—
264C2×#107—52no—
265C2×#107C2×D512yes—
266#423—58no—
267C2×C4⋊C8—52no—
268C2×C4⋊C8—51no—
269C2×C4⋊C8C2×D511yescharacteristic
270C2×#103—52no—
271C2×#103C2×D512yes—
272C2×#102—52no—
273C2×#102—52no—
274C23×C5C22×D411yescharacteristic
275C22×D5—22no—
276C22×D5—22no—
277C22×D5—22no—
278C22×D5—24no—
279C22×D5—44no—
280C22×D5—44no—
281C22×D5—44no—
282C22×D5C22×D412yes—
283C5×Q8—22no—
284C5×Q8—22no—
285C5×Q8—48no—
286C5×Q8—42no—
287C2×C4×C5—22no—
288C2×C4×C5—22no—
289C2×C4×C5—22no—
290C2×C4×C5—24no—
291C2×C4×C5—42no—
292C2×C4×C5—42no—
293C2×C4×C5—44no—
294C2×C4×C5#13714yes—
295C2×C4×C5C22×D412yes—
296C2×C4×C5C22×D411yescharacteristic
297C2×C4×C5C22×D412yes—
298C2×C4×C5H(5,𝔽2)12yes—
299C2×C4×C52−1+412yes—
300C2×C4×C5C2511yescharacteristic
301C2×Dic5—22no—
302C2×Dic5—22no—
303C2×Dic5—22no—
304C2×Dic5—22no—
305C2×Dic5—42no—
306C2×Dic5—42no—
307C2×Dic5AGL(1,ℤ/8)14yes—
308C2×Dic5C22×D412yes—
309C2×Dic5C22×D412yes—
310C2×Dic5C22×D412yes—
311D20—24no—
312D20—24no—
313D20—48no—
314D20—416no—
315D20—44no—
316D20—48no—
317C4×D5—24no—
318C4×D5—28no—
319C4×D5—22no—
320C4×D5—22no—
321C4×D5—28no—
322C4×D5—416no—
323Dic10—22no—
324Dic10—22no—
325Dic10—42no—
326Dic10—48no—
327C5×C8—24no—
328#168—24no—
329C2×#41—108no—
330C2×#41—208no—
331C22×Q8—21no—
332C22×Q8—51no—
333C22×Q8—101no—
334C22×Q8C22×D511yescharacteristic
335C22×D4—52no—
336C22×D4—102no—
337C22×D4—101no—
338C22×D4—101no—
339C23×C4—51no—
340C23×C4—51no—
341C23×C4—51no—
342C23×C4—102no—
343C2×SD16—58no—
344C2×SD16—58no—
345C2×SD16—204no—
346C2×SD16—204no—
347C2×M16—58no—
348C2×M16—204no—
349C22×C8—52no—
350C22×C8C22×D512yes—
351C4⋊Q8—22no—
352C4⋊Q8—54no—
353C4⋊Q8—54no—
354C4⋊Q8—102no—
355C4⋊Q8—102no—
356C4⋊Q8C22×D514yes—
357#127—54no—
358#127—102no—
359#125—108no—
360#124—108no—
361C22⋊Q8—54no—
362C22⋊Q8—54no—
363C22⋊Q8—102no—
364C22⋊Q8—108no—
365C22⋊Q8—102no—
366#121—58no—
367#121—104no—
368#121—108no—
369C4×Q8—22no—
370C4×Q8—54no—
371C4×Q8—108no—
372C4×Q8—102no—
373C4×Q8C22×D514yes—
374C4×D4—58no—
375C4×D4—104no—
376C4×D4—108no—
377#117—108no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—51no—
381C2×C4⋊C4—101no—
382C2×C4⋊C4—101no—
383C2×C4⋊C4—101no—
384C2×C4⋊C4—101no—
385C2×C4⋊C4—101no—
386C2×C4⋊C4—101no—
387C2×C4⋊C4C22×D512yes—
388C2×C4⋊C4C22×D511yescharacteristic
389C2×C22⋊C4—102no—
390C2×C22⋊C4—102no—
391C2×C22⋊C4—101no—
392C2×C22⋊C4—101no—
393C2×C42—21no—
394C2×C42—51no—
395C2×C42—51no—
396C2×C42—51no—
397C2×C42—54no—
398C2×C42—54no—
399C2×C42—101no—
400C2×C42C22×D511yescharacteristic
401#107—58no—
402#107C22×D518yes—
403C4⋊C8—58no—
404C4⋊C8—54no—
405C4⋊C8C22×D514yes—
406#103—58no—
407#103C22×D518yes—
408#102—58no—
409#102—58no—
410C22×C5#53712yes—
411C22×C5C2×AGL(1,ℤ/8)12yes—
412C22×C5C2×#13712yes—
413C22×C5C23×D411yescharacteristic
414C2×D5—22no—
415C2×D5—22no—
416C2×D5—22no—
417C2×D5—44no—
418C2×D5—44no—
419C2×D5—44no—
420C2×D5—44no—
421C2×D5—88no—
422C4×C5—22no—
423C4×C5—42no—
424C4×C5—42no—
425C4×C5—44no—
426C4×C5#53712yes—
427C4×C5C2×#13714yes—
428C4×C5C2×H(5,𝔽2)11yescharacteristic
429C4×C5C2×2−1+411yescharacteristic
430Dic5—22no—
431Dic5—22no—
432Dic5—42no—
433Dic5—42no—
434Dic5C2×AGL(1,ℤ/8)14yes—
435C24—51no—
436C24—102no—
437#41—2032no—
438C2×Q8—21no—
439C2×Q8—24no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—44no—
443C2×Q8—52no—
444C2×Q8—104no—
445C2×Q8—101no—
446C2×Q8—102no—
447C2×Q8—102no—
448C2×Q8—204no—
449C2×Q8C23×D512yes—
450C2×D4—54no—
451C2×D4—108no—
452C2×D4—104no—
453C2×D4—102no—
454C2×D4—104no—
455C2×D4—104no—
456C2×D4—104no—
457C2×D4—204no—
458C2×D4—208no—
459C2×D4—204no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—51no—
464C22×C4—51no—
465C22×C4—51no—
466C22×C4—54no—
467C22×C4—52no—
468C22×C4—54no—
469C22×C4—52no—
470C22×C4—104no—
471C22×C4—108no—
472C22×C4—102no—
473C22×C4—102no—
474C22×C4—104no—
475C22×C4—101no—
476C22×C4—101no—
477C22×C4—208no—
478C22×C4D4×D511yescharacteristic
479C22×C4D4×D511yescharacteristic
480C22×C4C23×D511yescharacteristic
481SD16—208no—
482SD16—208no—
483M16—208no—
484C2×C8—24no—
485C2×C8—54no—
486C2×C8—104no—
487C2×C8C23×D514yes—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—52no—
493C4⋊C4—104no—
494C4⋊C4—104no—
495C4⋊C4—104no—
496C4⋊C4—104no—
497C4⋊C4—104no—
498C4⋊C4—101no—
499C4⋊C4—104no—
500C4⋊C4C23×D512yes—
501C4⋊C4C23×D514yes—
502C22⋊C4—108no—
503C22⋊C4—108no—
504C22⋊C4—104no—
505C22⋊C4—104no—
506C42—24no—
507C42—54no—
508C42—52no—
509C42—54no—
510C42—52no—
511C42—104no—
512C42D4×D512yes—
513C42C23×D512yes—
514C2×C5#325414yescomplemented
515C2×C5C2×#53711yescharacteristic
516C2×C5C22×AGL(1,ℤ/8)11yescharacteristic
517C2×C5C22×#13711yescharacteristic
518D5—42no—
519D5—84no—
520C23—52no—
521C23—102no—
522C23—102no—
523C23—102no—
524C23—104no—
525C23—204no—
526C23—204no—
527C23—204no—
528C23C2×D4×D511yescharacteristic
529Q8—22no—
530Q8—22no—
531Q8—48no—
532Q8—42no—
533Q8—102no—
534Q8—102no—
535Q8—208no—
536Q8—202no—
537D4—104no—
538D4—104no—
539D4—208no—
540D4—2016no—
541D4—208no—
542D4—204no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—54no—
551C2×C4—52no—
552C2×C4—52no—
553C2×C4—52no—
554C2×C4—108no—
555C2×C4—102no—
556C2×C4—102no—
557C2×C4—108no—
558C2×C4—104no—
559C2×C4—102no—
560C2×C4—102no—
561C2×C4—102no—
562C2×C4—102no—
563C2×C4—2016no—
564C2×C4—202no—
565C2×C4—202no—
566C2×C4#406414yes—
567C2×C4C2×D4×D512yes—
568C2×C4C2×D4×D512yes—
569C2×C4C2×D4×D511yescharacteristic
570C2×C4#414712yes—
571C2×C4#414912yes—
572C2×C4C24×D511yescharacteristic
573C8—24no—
574C8—104no—
575C5C2×#325411yescharacteristic, complemented, Sylow
576C22—102no—
577C22—102no—
578C22—102no—
579C22—204no—
580C22—204no—
581C22—204no—
582C22—204no—
583C22—408no—
584C22#6596312yes—
585C22C2×#406412yes—
586C22D5×AGL(1,ℤ/8)12yes—
587C22C22×D4×D511yescharacteristic
588C4—22no—
589C4—42no—
590C4—44no—
591C4—42no—
592C4—54no—
593C4—102no—
594C4—102no—
595C4—202no—
596C4—202no—
597C4#6596312yes—
598C4C2×#406414yes—
599C4C2×#414711yescharacteristic
600C4C2×#414911yescharacteristic
601C2—202no—
602C2—404no—
603C2#1061282914yesdirect product factor
604C2C2×#6596311yescharacteristic
605C2C22×#406411yescharacteristic
606C2C2×D5×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#477C2
socleC23×C5C22×D4
prime radicalC2×#66059C2
hypercenterC2×#477D5
hypocenterC5C2×#3254
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
2. C2×#65912C2semidirect
3. C2×C5×#477C2semidirect
4. C2×#65380C2semidirect
5. C2×#65316C2semidirect
6. #10612829C2direct (N × H)
7. C2×#65179C2semidirect
8. C2×#65175C2semidirect
9. C2×#65137C2semidirect
10. C2×#65126C2semidirect
11. C2×#65107C2semidirect
12. C2×#65101C2semidirect
25. #65912C22semidirect
39. C5×#477C22semidirect
45. #65380C22semidirect
46. #65316C22semidirect
51. #65179C22semidirect
52. #65175C22semidirect
53. #65137C22semidirect
54. #65126C22semidirect
55. #65107C22semidirect
56. #65101C22semidirect
143. C2×#477D5semidirect
257. #477C2×D5semidirect
514. C2×C5#3254semidirect
575. C5C2×#3254semidirect
603. C2#10612829direct (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×#659122
C2×C5×#4772
C2×#653802
C2×#653162
#106128292
C2×#651792
C2×#651752
C2×#651372
C2×#651262
C2×#651072
C2×#651012
C2×#32545