Finite Group Databasebeta

← C2×#10613472 #12970575

Subgroups — C2×#10613472 #12970575

Order 1280.

2146 conjugacy classes of subgroups.

The subgroup lattice has 604 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×#491C211yesmaximal, characteristic, complemented
4C2×#65375C211yesmaximal, characteristic, complemented
5C2×#65358C211yesmaximal, characteristic, complemented
6#10613472C2116yesmaximal, direct product factor
7C2×#65082C212yesmaximal, complemented
8C2×#65107C212yesmaximal, complemented
9C2×#65097C212yesmaximal, complemented
10C2×#65095C212yesmaximal
11C2×#65004C211yesmaximal, characteristic, complemented
12C2×#64994C211yesmaximal, characteristic, complemented
13C22×Q8×D5C2212yes—
14C2×C5×C4⋊Q8C2211yescharacteristic
15C2×C5×#125C2211yescharacteristic
16C2×#4090C2212yes—
17C22×#689C2212yes—
18C22×#688C2212yes—
19C22×Dic20C2212yes—
20C22×#677C2212yes—
21D5×C4⋊Q8—24no—
22D5×C4⋊Q8C2218yescomplemented
23#66025—24no—
24#66025C2218yes—
25C2×#4044—21no—
26C2×#4043C2211yescharacteristic
27C2×#4041C2211yescharacteristic
28C2×#4040—21no—
29C2×#4039—21no—
30C2×#4039C2211yescharacteristic
31C2×#4038—21no—
32C2×D5×C4⋊C4—21no—
33C2×D5×C4⋊C4—21no—
34C2×#4034—21no—
35C2×#4034—21no—
36C2×#4033C2211yescharacteristic
37C2×#4021C2211yescharacteristic
38C2×C42×D5C2211yescharacteristic
39C2×#4015C2211yescharacteristic
40C5×#491C2218yescomplemented
41C2×C5×#103C2212yes—
42C2×C5×#103C2212yes—
43C2×C5×#97C2211yescharacteristic
44C2×#3952C2212yes—
45C2×#3946C2212yes—
46#65375C2218yescomplemented
47#65358C2218yescomplemented
48C2×#3942C2212yes—
49C2×#3942C2212yes—
50C2×#3935C2211yescharacteristic
51#65107C22116yescomplemented
52#65097C22116yescomplemented
53#65095C22116yes—
54#65082C22116yescomplemented
55#65004C2218yescomplemented
56#64994C2218yescomplemented
57C2×#3218—51nomaximal, Sylow
58C22×C5×Q8C2312yes—
59C2×Q8×D5—24no—
60C2×Q8×D5—44no—
61C2×Q8×D5—44no—
62C2×Q8×D5C2318yes—
63C22×D20—21no—
64C22×D20C2311yescharacteristic
65C22×C4×D5—22no—
66C22×C4×D5—21no—
67C22×C4×D5D411yescharacteristic
68C22×C4×D5D411yescharacteristic
69C22×C4×D5C2311yescharacteristic
70C22×Dic10—22no—
71C22×Dic10C2312yes—
72C22×Dic10C2311yescharacteristic
73C22×C5×C8C2312yes—
74C5×C4⋊Q8—22no—
75C5×C4⋊Q8C2314yes—
76C5×#125—22no—
77C5×#125C2314yes—
78C2×C5×C4⋊C4—21no—
79C2×C5×C4⋊C4—21no—
80C2×C5×C4⋊C4—21no—
81C2×C5×C4⋊C4—21no—
82C2×C5×C4⋊C4C2311yescharacteristic
83C2×C5×C4⋊C4C2311yescharacteristic
84C2×C42×C5C2311yescharacteristic
85#4090—24no—
86#4090C2318yes—
87C2×#689—44no—
88C2×#689C2318yes—
89C2×#688—44no—
90C2×#688C2318yes—
91C2×#685—21no—
92C2×#685—21no—
93C2×#685—21no—
94C2×#685—21no—
95C2×#685—21no—
96C2×C4⋊Dic5—21no—
97C2×C4⋊Dic5—21no—
98C2×C4⋊Dic5—21no—
99C2×#683—21no—
100C2×#683—22no—
101C2×#683—22no—
102C2×C4×Dic5—21no—
103C2×C4×Dic5D411yescharacteristic
104C2×C4×Dic5D411yescharacteristic
105C2×C4×Dic5C2311yescharacteristic
106C22×#168C2312yes—
107C2×Dic20—44no—
108C2×Dic20C2318yes—
109C2×#677—44no—
110C2×#677C2318yes—
111#4044—28no—
112#4043—22no—
113#4043C2314yes—
114#4041—22no—
115#4041C2314yes—
116#4040—28no—
117#4039—28no—
118#4039—22no—
119#4039C2314yes—
120#4038—28no—
121D5×C4⋊C4—28no—
122D5×C4⋊C4—28no—
123#4034—28no—
124#4034—28no—
125#4033—22no—
126#4033C2314yes—
127#4021—22no—
128#4021C2314yes—
129C42×D5D414yes—
130C42×D5C2314yes—
131#4015—22no—
132#4015C2314yes—
133C5×#103C2318yes—
134C5×#103C2318yes—
135C5×#97C2314yes—
136#3952C2318yes—
137#3946C2318yes—
138#3942C2318yes—
139#3942C2318yes—
140#3935C2314yes—
141C22×C4⋊Q8—51no—
142C2×#529—51no—
143#3218—516no—
144C2×#497—51no—
145C2×#497—51no—
146C2×#491—51no—
147C2×#491D511yescharacteristic, complemented
148C2×#462—52no—
149C2×#461—52no—
150C2×#451—52no—
151C2×#448—52no—
152C2×#432—51no—
153C23×D5—21no—
154C23×D5C2×D411yescharacteristic
155C2×C5×Q8—22no—
156C2×C5×Q8—24no—
157C2×C5×Q8—24no—
158C2×C5×Q8C2414yes—
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—22no—
166C22×Dic5—21no—
167C22×Dic5C2×D411yescharacteristic
168C22×Dic5C2×D411yescharacteristic
169C22×Dic5C2×D411yescharacteristic
170Q8×D5—48no—
171Q8×D5—48no—
172Q8×D5—416no—
173C2×D20—24no—
174C2×D20—24no—
175C2×D20—21no—
176C2×D20—44no—
177C2×D20C2412yes—
178C2×C4×D5—24no—
179C2×C4×D5—24no—
180C2×C4×D5—24no—
181C2×C4×D5—28no—
182C2×C4×D5—24no—
183C2×C4×D5—48no—
184C2×C4×D5—44no—
185C2×C4×D5C2×D414yes—
186C2×C4×D5C2×D414yes—
187C2×C4×D5C2×D412yes—
188C2×C4×D5C2412yes—
189C2×Dic10—24no—
190C2×Dic10—24no—
191C2×Dic10—24no—
192C2×Dic10—22no—
193C2×Dic10—21no—
194C2×Dic10—28no—
195C2×Dic10—48no—
196C2×Dic10C2414yes—
197C2×Dic10C2412yes—
198C2×C5×C8—24no—
199C2×C5×C8C2414yes—
200C5×C4⋊C4—24no—
201C5×C4⋊C4—24no—
202C5×C4⋊C4—24no—
203C5×C4⋊C4—24no—
204C5×C4⋊C4—21no—
205C5×C4⋊C4—21no—
206C5×C4⋊C4C2412yes—
207C5×C4⋊C4C2412yes—
208C42×C5C2×D412yes—
209C42×C5C2412yes—
210#689—48no—
211#688—48no—
212#685—24no—
213#685—24no—
214#685—24no—
215#685—24no—
216#685—24no—
217C4⋊Dic5—24no—
218C4⋊Dic5—24no—
219C4⋊Dic5—24no—
220#683—24no—
221#683—28no—
222#683—28no—
223C4×Dic5—24no—
224C4×Dic5C2×D412yes—
225C4×Dic5C2×D414yes—
226C4×Dic5C2×D414yes—
227C4×Dic5C2412yes—
228C2×#168—24no—
229C2×#168C2414yes—
230Dic20—48no—
231#677—48no—
232C23×Q8—52no—
233C22×Q16—52no—
234C22×Q16—52no—
235C22×SD16—52no—
236C22×SD16—52no—
237C2×C4⋊Q8—52no—
238C2×C4⋊Q8—58no—
239C2×C4⋊Q8—51no—
240C2×C4⋊Q8—101no—
241C2×C4⋊Q8—101no—
242C2×C4⋊Q8—104no—
243C2×C4⋊Q8C2×D511yescharacteristic
244#529—58no—
245#529—104no—
246C2×#126—101no—
247C2×#125C2×D511yescharacteristic
248C2×#124—51no—
249C2×#123—101no—
250C2×C22⋊Q8—51no—
251C2×#121—51no—
252C2×C4×Q8—51no—
253C2×C4×D4—51no—
254C2×C4×D4—101no—
255C2×#117—101no—
256C22×C4⋊C4—101no—
257C22×C4⋊C4—101no—
258C22×C42—51no—
259#497—58no—
260#497—58no—
261#491—58no—
262#491C2×D518yescomplemented
263#462—516no—
264#461—516no—
265#451—516no—
266#448—516no—
267#432—58no—
268C2×C4⋊C8—52no—
269C2×#103—52no—
270C2×#103—52no—
271C2×#103C2×D512yes—
272C2×#103C2×D512yes—
273C2×C22⋊C8—52no—
274C2×#97—51no—
275C2×#97C2×D511yescharacteristic
276C23×C5C22×D411yescharacteristic
277C22×D5—22no—
278C22×D5—22no—
279C22×D5—22no—
280C22×D5—42no—
281C22×D5—42no—
282C22×D5—42no—
283C22×D5#13714yes—
284C22×D5C22×D412yes—
285C5×Q8—24no—
286C5×Q8—24no—
287C5×Q8—44no—
288C2×C4×C5—22no—
289C2×C4×C5—22no—
290C2×C4×C5—22no—
291C2×C4×C5—24no—
292C2×C4×C5—22no—
293C2×C4×C5—44no—
294C2×C4×C5—42no—
295C2×C4×C5—42no—
296C2×C4×C5C22×D412yes—
297C2×C4×C5C22×D412yes—
298C2×C4×C5C22×D411yescharacteristic
299C2×C4×C5H(5,𝔽2)14yes—
300C2×C4×C5C2511yescharacteristic
301C2×Dic5—22no—
302C2×Dic5—22no—
303C2×Dic5—22no—
304C2×Dic5—24no—
305C2×Dic5—44no—
306C2×Dic5—42no—
307C2×Dic5#13714yes—
308C2×Dic5C22×D412yes—
309C2×Dic5C22×D412yes—
310C2×Dic5C22×D412yes—
311D20—24no—
312D20—48no—
313D20—42no—
314C4×D5—28no—
315C4×D5—24no—
316C4×D5—24no—
317C4×D5—28no—
318C4×D5—416no—
319C4×D5—48no—
320Dic10—24no—
321Dic10—24no—
322Dic10—24no—
323Dic10—416no—
324Dic10—42no—
325Dic10—44no—
326C5×C8—24no—
327#168—24no—
328C22×Q8—58no—
329C22×Q8—51no—
330C22×Q8—52no—
331C22×Q8—102no—
332C22×Q8—104no—
333C22×Q8—204no—
334C22×Q8—204no—
335C22×Q8C22×D512yes—
336C22×D4—51no—
337C22×D4—101no—
338C23×C4—51no—
339C23×C4—51no—
340C23×C4—51no—
341C23×C4—101no—
342C23×C4—102no—
343C2×Q16—58no—
344C2×Q16—58no—
345C2×Q16—204no—
346C2×Q16—204no—
347C2×SD16—58no—
348C2×SD16—58no—
349C2×SD16—204no—
350C2×SD16—204no—
351C22×C8—52no—
352C22×C8C22×D512yes—
353C4⋊Q8—22no—
354C4⋊Q8—58no—
355C4⋊Q8—54no—
356C4⋊Q8—108no—
357C4⋊Q8—108no—
358C4⋊Q8—104no—
359C4⋊Q8—102no—
360C4⋊Q8C22×D514yes—
361#126—108no—
362#125—22no—
363#125C22×D514yes—
364#124—54no—
365#124—102no—
366#123—108no—
367C22⋊Q8—54no—
368C22⋊Q8—102no—
369#121—54no—
370#121—102no—
371C4×Q8—54no—
372C4×Q8—102no—
373C4×D4—54no—
374C4×D4—102no—
375C4×D4—108no—
376#117—108no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—108no—
382C2×C4⋊C4—108no—
383C2×C4⋊C4—102no—
384C2×C4⋊C4—102no—
385C2×C4⋊C4—101no—
386C2×C4⋊C4—101no—
387C2×C4⋊C4—101no—
388C2×C4⋊C4—101no—
389C2×C4⋊C4C22×D511yescharacteristic
390C2×C4⋊C4C22×D511yescharacteristic
391C2×C22⋊C4—101no—
392C2×C22⋊C4—101no—
393C2×C22⋊C4—101no—
394C2×C22⋊C4—101no—
395C2×C22⋊C4—101no—
396C2×C42—51no—
397C2×C42—51no—
398C2×C42—51no—
399C2×C42—54no—
400C2×C42—54no—
401C2×C42—101no—
402C2×C42C22×D511yescharacteristic
403C4⋊C8—58no—
404#103—58no—
405#103—58no—
406#103C22×D518yes—
407#103C22×D518yes—
408C22⋊C8—58no—
409#97—54no—
410#97C22×D514yes—
411C22×C5#53512yes—
412C22×C5C2×#13712yes—
413C22×C5C2×#13712yes—
414C22×C5C23×D411yescharacteristic
415C2×D5—22no—
416C2×D5—28no—
417C2×D5—22no—
418C2×D5—42no—
419C2×D5—42no—
420C2×D5—42no—
421C2×D5—84no—
422C2×D5C2×#13714yes—
423C4×C5—22no—
424C4×C5—22no—
425C4×C5—44no—
426C4×C5—42no—
427C4×C5—42no—
428C4×C5#53512yes—
429C4×C5C2×H(5,𝔽2)12yes—
430Dic5—22no—
431Dic5—22no—
432Dic5—44no—
433Dic5—42no—
434Dic5C2×#13714yes—
435C24—51no—
436C24—101no—
437C2×Q8—22no—
438C2×Q8—24no—
439C2×Q8—24no—
440C2×Q8—54no—
441C2×Q8—52no—
442C2×Q8—108no—
443C2×Q8—101no—
444C2×Q8—102no—
445C2×Q8—104no—
446C2×Q8—104no—
447C2×Q8—104no—
448C2×Q8—208no—
449C2×Q8—208no—
450C2×Q8—208no—
451C2×Q8—2016no—
452C2×Q8C23×D514yes—
453C2×D4—52no—
454C2×D4—104no—
455C2×D4—101no—
456C2×D4—104no—
457C2×D4—204no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—51no—
462C22×C4—51no—
463C22×C4—51no—
464C22×C4—54no—
465C22×C4—54no—
466C22×C4—52no—
467C22×C4—52no—
468C22×C4—104no—
469C22×C4—104no—
470C22×C4—104no—
471C22×C4—108no—
472C22×C4—104no—
473C22×C4—101no—
474C22×C4—102no—
475C22×C4—204no—
476C22×C4—208no—
477C22×C4D4×D511yescharacteristic
478C22×C4D4×D511yescharacteristic
479C22×C4C23×D511yescharacteristic
480Q16—208no—
481Q16—208no—
482SD16—208no—
483SD16—208no—
484C2×C8—24no—
485C2×C8—54no—
486C2×C8—104no—
487C2×C8C23×D514yes—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—108no—
495C4⋊C4—108no—
496C4⋊C4—104no—
497C4⋊C4—104no—
498C4⋊C4—104no—
499C4⋊C4—104no—
500C4⋊C4C23×D512yes—
501C4⋊C4C23×D512yes—
502C22⋊C4—104no—
503C22⋊C4—104no—
504C22⋊C4—104no—
505C22⋊C4—104no—
506C22⋊C4—104no—
507C42—54no—
508C42—52no—
509C42—54no—
510C42—52no—
511C42—104no—
512C42D4×D512yes—
513C42C23×D512yes—
514C2×C5#321814yescomplemented
515C2×C5C2×#53511yescharacteristic
516C2×C5C22×#13711yescharacteristic
517C2×C5C22×#13711yescharacteristic
518D5—24no—
519D5—82no—
520C23—54no—
521C23—52no—
522C23—102no—
523C23—102no—
524C23—102no—
525C23—202no—
526C23—202no—
527C23—202no—
528C23C2×D4×D511yescharacteristic
529Q8—24no—
530Q8—24no—
531Q8—44no—
532Q8—104no—
533Q8—104no—
534Q8—104no—
535Q8—2016no—
536Q8—204no—
537Q8—202no—
538D4—104no—
539D4—208no—
540D4—202no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—54no—
550C2×C4—52no—
551C2×C4—52no—
552C2×C4—52no—
553C2×C4—108no—
554C2×C4—108no—
555C2×C4—102no—
556C2×C4—102no—
557C2×C4—104no—
558C2×C4—104no—
559C2×C4—102no—
560C2×C4—104no—
561C2×C4—208no—
562C2×C4—2016no—
563C2×C4—204no—
564C2×C4—202no—
565C2×C4C2×D4×D512yes—
566C2×C4C2×D4×D512yes—
567C2×C4C2×D4×D511yescharacteristic
568C2×C4#414914yes—
569C2×C4C24×D511yescharacteristic
570C8—24no—
571C8—104no—
572C5C2×#321811yescharacteristic, complemented, Sylow
573C22—54no—
574C22—102no—
575C22—102no—
576C22—108no—
577C22—202no—
578C22—202no—
579C22—202no—
580C22—404no—
581C22#6600812yes—
582C22D5×#13712yes—
583C22D5×#13712yes—
584C22C22×D4×D511yescharacteristic
585C4—22no—
586C4—22no—
587C4—44no—
588C4—42no—
589C4—42no—
590C4—54no—
591C4—102no—
592C4—102no—
593C4—204no—
594C4—202no—
595C4#6600812yes—
596C4C2×#414912yes—
597C2—104no—
598C2—402no—
599C2#1061347214yesdirect product factor
600C2C2×#6600811yescharacteristic
601C2C2×D5×#13711yescharacteristic
602C2C2×D5×#13711yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#491C2
socleC23×C5C22×D4
prime radicalC2×#4041C22
hypercenterC2×#491D5
hypocenterC5C2×#3218
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#491

Complements

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

NH ≅ G/Ntype
1. C2×D5×C4⋊Q8C2semidirect
3. C2×C5×#491C2semidirect
4. C2×#65375C2semidirect
5. C2×#65358C2semidirect
6. #10613472C2direct (N × H)
7. C2×#65082C2semidirect
8. C2×#65107C2semidirect
9. C2×#65097C2semidirect
11. C2×#65004C2semidirect
12. C2×#64994C2semidirect
22. D5×C4⋊Q8C22semidirect
40. C5×#491C22semidirect
46. #65375C22semidirect
47. #65358C22semidirect
51. #65107C22semidirect
52. #65097C22semidirect
54. #65082C22semidirect
55. #65004C22semidirect
56. #64994C22semidirect
147. C2×#491D5semidirect
262. #491C2×D5semidirect
514. C2×C5#3218semidirect
572. C5C2×#3218semidirect
599. C2#10613472direct (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×#4912
C2×#653752
C2×#653582
#106134722
C2×#650822
C2×#651072
C2×#650972
C2×#650952
C2×#650042
C2×#649942
C2×#32185