Finite Group Databasebeta

← C2×#10611910 #12970577

Subgroups — C2×#10611910 #12970577

Order 1280.

2146 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#4090C211yesmaximal, characteristic, complemented
2C2×#65958C211yesmaximal, characteristic
3C2×C5×#451C211yesmaximal, characteristic, complemented
4C2×#65516C211yesmaximal, characteristic
5C2×#65479C212yesmaximal, complemented
6C2×#65471C212yesmaximal, complemented
7C2×#65339C211yesmaximal, characteristic, complemented
8#10611910C2116yesmaximal, direct product factor
9C2×#65107C212yesmaximal, complemented
10C2×#65086C212yesmaximal, complemented
11C2×#65027C211yesmaximal, characteristic
12C2×#65008C211yesmaximal, characteristic, complemented
13C23×C5×Q8C2211yescharacteristic
14C23×Dic10C2211yescharacteristic
15C22×C5×Q16C2212yes—
16C2×C5×C22⋊Q8C2211yescharacteristic
17C2×#4093C2211yescharacteristic
18C2×Q8×Dic5C2211yescharacteristic
19C2×#4090—21no—
20C2×#4090—21no—
21C2×#4090—24no—
22C2×#4090C2212yes—
23C2×#4090C2212yes—
24C2×#4090C2218yescomplemented
25C22×#689C2212yes—
26C22×#688C2212yes—
27C2×#4076C2211yescharacteristic
28C22×#683—21no—
29C22×#683—21no—
30C22×C4×Dic5C2211yescharacteristic
31C22×#677C2212yes—
32#65958—24no—
33#65958C2218yes—
34C2×#4044—21no—
35C2×#4039C2211yescharacteristic
36C2×#4038—21no—
37C2×#4036C2211yescharacteristic
38C2×#4032—21no—
39C2×#4027—21no—
40C5×#451C2218yescomplemented
41C2×C5×#103C2212yes—
42C2×C5×C22⋊C8C2211yescharacteristic
43#65516C2218yes—
44C2×#3967C2212yes—
45C2×C22⋊#168C2211yescharacteristic
46#65479C22116yescomplemented
47#65471C22116yescomplemented
48C2×#3948C2212yes—
49C2×#3947C2212yes—
50C2×#3946C2212yes—
51#65339C2218yescomplemented
52C2×#3942C2212yes—
53#65107C22116yescomplemented
54#65086C22116yescomplemented
55#65027C2218yes—
56#65008C2218yescomplemented
57C2×#3218—51nomaximal, Sylow
58C22×C5×Q8—21no—
59C22×C5×Q8—22no—
60C22×C5×Q8—44no—
61C22×C5×Q8C2312yes—
62C22×C5×Q8C2311yescharacteristic
63C22×C5×Q8C2314yes—
64C23×C4×C5—21no—
65C23×C4×C5C2311yescharacteristic
66C22×C22⋊D5—21no—
67C23×Dic5—21no—
68C23×Dic5D411yescharacteristic
69C23×Dic5D411yescharacteristic
70C22×D20C2311yescharacteristic
71C22×C4×D5—21no—
72C22×Dic10—21no—
73C22×Dic10—22no—
74C22×Dic10—44no—
75C22×Dic10C2312yes—
76C22×Dic10C2314yes—
77C2×C5×Q16—44no—
78C2×C5×Q16C2318yes—
79C22×C5×C8C2312yes—
80C5×C22⋊Q8—22no—
81C5×C22⋊Q8C2314yes—
82C2×C5×C4⋊C4—21no—
83C2×C5×C4⋊C4C2311yescharacteristic
84C2×C5×C22⋊C4—21no—
85#4093—22no—
86#4093C2314yes—
87Q8×Dic5—22no—
88Q8×Dic5C2314yes—
89#4090—24no—
90#4090—24no—
91#4090—28no—
92#4090—28no—
93#4090C2318yes—
94#4090C2318yes—
95C2×#689—44no—
96C2×#689C2318yes—
97C2×#688—44no—
98C2×#688C2318yes—
99#4076—22no—
100#4076C2314yes—
101C2×#685—21no—
102C2×#685—21no—
103C2×#685—21no—
104C2×#685—21no—
105C2×C4⋊Dic5—21no—
106C2×C4⋊Dic5—21no—
107C2×C4⋊Dic5C2311yescharacteristic
108C2×#683—21no—
109C2×#683—22no—
110C2×#683—22no—
111C2×#683—22no—
112C2×#683—22no—
113C2×#683—28no—
114C2×#683—28no—
115C2×C4×Dic5—21no—
116C2×C4×Dic5D411yescharacteristic
117C2×C4×Dic5D411yescharacteristic
118C2×C4×Dic5D414yes—
119C2×C4×Dic5C2311yescharacteristic
120C2×C4×Dic5C2311yescharacteristic
121C2×C4×Dic5C2314yes—
122C22×#168C2312yes—
123C2×#677—44no—
124C2×#677C2318yes—
125#4044—28no—
126#4039—22no—
127#4039C2314yes—
128#4038—28no—
129#4036—22no—
130#4036C2314yes—
131#4032—28no—
132#4027—28no—
133C5×#103C2318yes—
134C5×C22⋊C8C2314yes—
135#3967C2318yes—
136C22⋊#168C2314yes—
137#3948C2318yes—
138#3947C2318yes—
139#3946C2318yes—
140#3942C2318yes—
141C22×C4⋊Q8—51no—
142C2×#529—51no—
143#3218—516no—
144C2×#497—52no—
145C2×#491—52no—
146C2×#462—52no—
147C2×#461—52no—
148C2×#451—51no—
149C2×#451D511yescharacteristic, complemented
150C2×#448—51no—
151C2×#448—51no—
152C2×#432—51no—
153C24×C5C2×D411yescharacteristic
154C23×D5—21no—
155C2×C5×Q8—21no—
156C2×C5×Q8—24no—
157C2×C5×Q8—22no—
158C2×C5×Q8—24no—
159C2×C5×Q8—24no—
160C2×C5×Q8—24no—
161C2×C5×Q8—48no—
162C2×C5×Q8—44no—
163C2×C5×Q8—48no—
164C2×C5×Q8C2412yes—
165C2×C5×Q8C2414yes—
166C22×C4×C5—24no—
167C22×C4×C5—21no—
168C22×C4×C5—21no—
169C22×C4×C5—24no—
170C22×C4×C5—22no—
171C22×C4×C5—44no—
172C22×C4×C5C2×D411yescharacteristic
173C22×C4×C5C2×D412yes—
174C22×C4×C5C2412yes—
175C22×C4×C5C2411yescharacteristic
176C2×C22⋊D5—24no—
177C2×C22⋊D5—44no—
178C22×Dic5—24no—
179C22×Dic5—24no—
180C22×Dic5—24no—
181C22×Dic5—22no—
182C22×Dic5—21no—
183C22×Dic5—44no—
184C22×Dic5C2×D414yes—
185C22×Dic5C2×D414yes—
186C22×Dic5C2×D411yescharacteristic
187C22×Dic5C2×D411yescharacteristic
188C22×Dic5C2×D411yescharacteristic
189C22×Dic5C2×D411yescharacteristic
190C2×D20—24no—
191C2×D20—21no—
192C2×D20C2412yes—
193C2×C4×D5—24no—
194C2×C4×D5—44no—
195C2×Dic10—24no—
196C2×Dic10—24no—
197C2×Dic10—24no—
198C2×Dic10—22no—
199C2×Dic10—48no—
200C2×Dic10—44no—
201C2×Dic10—48no—
202C2×Dic10C2414yes—
203C5×Q16—48no—
204C2×C5×C8—24no—
205C2×C5×C8C2414yes—
206C5×C4⋊C4—24no—
207C5×C4⋊C4—21no—
208C5×C4⋊C4C2412yes—
209C5×C22⋊C4—24no—
210#689—48no—
211#688—48no—
212#685—24no—
213#685—24no—
214#685—24no—
215#685—24no—
216C4⋊Dic5—21no—
217C4⋊Dic5—24no—
218C4⋊Dic5—24no—
219C4⋊Dic5C2412yes—
220#683—24no—
221#683—28no—
222#683—28no—
223#683—28no—
224#683—28no—
225C4×Dic5—24no—
226C4×Dic5C2×D412yes—
227C4×Dic5C2×D414yes—
228C4×Dic5C2×D412yes—
229C4×Dic5C2×D414yes—
230C4×Dic5C2412yes—
231C4×Dic5C2412yes—
232C2×#168—24no—
233C2×#168C2414yes—
234#677—48no—
235C23×Q8—51no—
236C23×Q8C2×D511yescharacteristic
237C22×Q16—52no—
238C22×Q16C2×D512yes—
239C22×SD16—52no—
240C22×SD16—52no—
241C2×C4⋊Q8—52no—
242C2×C4⋊Q8—52no—
243C2×C4⋊Q8—58no—
244C2×C4⋊Q8—101no—
245C2×C4⋊Q8—101no—
246C2×C4⋊Q8—104no—
247#529—58no—
248#529—104no—
249C2×#126—101no—
250C2×#125—51no—
251C2×#124—51no—
252C2×#123—101no—
253C2×C22⋊Q8C2×D511yescharacteristic
254C2×#121—51no—
255C2×C4×Q8—51no—
256C2×C4×D4—51no—
257C2×C4×D4—101no—
258C2×#117—101no—
259C22×C4⋊C4—101no—
260C22×C4⋊C4—101no—
261C22×C42—51no—
262#497—516no—
263#491—516no—
264#462—516no—
265#461—516no—
266#451—58no—
267#451C2×D518yescomplemented
268#448—58no—
269#448—58no—
270#432—58no—
271C2×C4⋊C8—52no—
272C2×#103—52no—
273C2×#103—52no—
274C2×#103—52no—
275C2×#103C2×D512yes—
276C2×C22⋊C8—51no—
277C2×C22⋊C8C2×D511yescharacteristic
278C2×#97—52no—
279C23×C5—22no—
280C23×C5—22no—
281C23×C5#13714yes—
282C23×C5C22×D412yes—
283C23×C5C22×D411yescharacteristic
284C22×D5—22no—
285C22×D5—42no—
286C22×D5—42no—
287C22×D5—42no—
288C5×Q8—24no—
289C5×Q8—24no—
290C5×Q8—24no—
291C5×Q8—48no—
292C5×Q8—42no—
293C5×Q8—44no—
294C2×C4×C5—22no—
295C2×C4×C5—24no—
296C2×C4×C5—24no—
297C2×C4×C5—22no—
298C2×C4×C5—24no—
299C2×C4×C5—22no—
300C2×C4×C5—48no—
301C2×C4×C5—44no—
302C2×C4×C5—42no—
303C2×C4×C5—42no—
304C2×C4×C5C22×D412yes—
305C2×C4×C5C22×D411yescharacteristic
306C2×C4×C5H(5,𝔽2)14yes—
307C2×C4×C5C2511yescharacteristic
308C22⋊D5—48no—
309C2×Dic5—28no—
310C2×Dic5—22no—
311C2×Dic5—22no—
312C2×Dic5—22no—
313C2×Dic5—28no—
314C2×Dic5—22no—
315C2×Dic5—24no—
316C2×Dic5—48no—
317C2×Dic5—44no—
318C2×Dic5—42no—
319C2×Dic5#13714yes—
320C2×Dic5C22×D412yes—
321C2×Dic5C22×D412yes—
322C2×Dic5C22×D412yes—
323C2×Dic5C22×D412yes—
324D20—24no—
325D20—42no—
326C4×D5—48no—
327Dic10—24no—
328Dic10—24no—
329Dic10—48no—
330Dic10—44no—
331C5×C8—24no—
332#168—24no—
333C22×Q8—22no—
334C22×Q8—21no—
335C22×Q8—44no—
336C22×Q8—52no—
337C22×Q8—54no—
338C22×Q8—102no—
339C22×Q8—101no—
340C22×Q8—204no—
341C22×Q8C22×D511yescharacteristic
342C22×Q8C22×D512yes—
343C22×Q8C22×D514yes—
344C22×D4—51no—
345C22×D4—101no—
346C23×C4—21no—
347C23×C4—51no—
348C23×C4—51no—
349C23×C4—101no—
350C23×C4—101no—
351C23×C4C22×D511yescharacteristic
352C2×Q16—44no—
353C2×Q16—58no—
354C2×Q16—204no—
355C2×Q16C22×D518yes—
356C2×SD16—58no—
357C2×SD16—58no—
358C2×SD16—204no—
359C2×SD16—204no—
360C22×C8—52no—
361C22×C8C22×D512yes—
362C4⋊Q8—58no—
363C4⋊Q8—58no—
364C4⋊Q8—108no—
365C4⋊Q8—104no—
366C4⋊Q8—104no—
367C4⋊Q8—108no—
368#126—108no—
369#125—54no—
370#125—102no—
371#124—54no—
372#124—102no—
373#123—108no—
374C22⋊Q8—22no—
375C22⋊Q8C22×D514yes—
376#121—54no—
377#121—102no—
378C4×Q8—54no—
379C4×Q8—102no—
380C4×D4—54no—
381C4×D4—102no—
382C4×D4—108no—
383#117—108no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—51no—
386C2×C4⋊C4—108no—
387C2×C4⋊C4—108no—
388C2×C4⋊C4—102no—
389C2×C4⋊C4—102no—
390C2×C4⋊C4—102no—
391C2×C4⋊C4—102no—
392C2×C4⋊C4—101no—
393C2×C4⋊C4—101no—
394C2×C4⋊C4—101no—
395C2×C4⋊C4C22×D511yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—101no—
398C2×C22⋊C4—101no—
399C2×C22⋊C4—101no—
400C2×C22⋊C4—101no—
401C2×C42—51no—
402C2×C42—51no—
403C2×C42—54no—
404C2×C42—51no—
405C2×C42—51no—
406C2×C42—54no—
407C2×C42—101no—
408C4⋊C8—58no—
409#103—58no—
410#103—58no—
411#103—58no—
412#103C22×D518yes—
413C22⋊C8—54no—
414C22⋊C8C22×D514yes—
415#97—58no—
416C22×C5—22no—
417C22×C5—28no—
418C22×C5—22no—
419C22×C5#53512yes—
420C22×C5C2×#13714yes—
421C22×C5C2×#13712yes—
422C22×C5C2×#13712yes—
423C22×C5C23×D411yescharacteristic
424C2×D5—42no—
425C2×D5—42no—
426C2×D5—42no—
427C2×D5—84no—
428C4×C5—22no—
429C4×C5—44no—
430C4×C5—42no—
431C4×C5—42no—
432C4×C5#53512yes—
433C4×C5C2×H(5,𝔽2)12yes—
434Dic5—22no—
435Dic5—22no—
436Dic5—22no—
437Dic5—44no—
438Dic5—42no—
439Dic5C2×#13714yes—
440C24—101no—
441C24D4×D511yescharacteristic
442C2×Q8—21no—
443C2×Q8—24no—
444C2×Q8—22no—
445C2×Q8—24no—
446C2×Q8—24no—
447C2×Q8—24no—
448C2×Q8—48no—
449C2×Q8—44no—
450C2×Q8—48no—
451C2×Q8—54no—
452C2×Q8—104no—
453C2×Q8—102no—
454C2×Q8—104no—
455C2×Q8—104no—
456C2×Q8—208no—
457C2×Q8—204no—
458C2×Q8—208no—
459C2×Q8C23×D514yes—
460C2×Q8C23×D512yes—
461C2×D4—52no—
462C2×D4—104no—
463C2×D4—101no—
464C2×D4—104no—
465C2×D4—204no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—44no—
472C22×C4—51no—
473C22×C4—54no—
474C22×C4—54no—
475C22×C4—51no—
476C22×C4—51no—
477C22×C4—51no—
478C22×C4—104no—
479C22×C4—104no—
480C22×C4—104no—
481C22×C4—104no—
482C22×C4—101no—
483C22×C4—102no—
484C22×C4—204no—
485C22×C4—204no—
486C22×C4D4×D512yes—
487C22×C4D4×D511yescharacteristic
488C22×C4C23×D512yes—
489C22×C4C23×D511yescharacteristic
490Q16—48no—
491Q16—208no—
492SD16—208no—
493SD16—208no—
494C2×C8—24no—
495C2×C8—54no—
496C2×C8—104no—
497C2×C8C23×D514yes—
498C4⋊C4—24no—
499C4⋊C4—21no—
500C4⋊C4—52no—
501C4⋊C4—108no—
502C4⋊C4—108no—
503C4⋊C4—108no—
504C4⋊C4—108no—
505C4⋊C4—104no—
506C4⋊C4—104no—
507C4⋊C4—104no—
508C4⋊C4—101no—
509C4⋊C4C23×D512yes—
510C22⋊C4—24no—
511C22⋊C4—104no—
512C22⋊C4—104no—
513C22⋊C4—104no—
514C22⋊C4—104no—
515C42—54no—
516C42—52no—
517C42—52no—
518C42—54no—
519C42—52no—
520C42—52no—
521C42—104no—
522C2×C5—24no—
523C2×C5#321814yescomplemented
524C2×C5C2×#53511yescharacteristic
525C2×C5C22×#13711yescharacteristic
526C2×C5C22×#13711yescharacteristic
527D5—82no—
528C23—22no—
529C23—22no—
530C23—102no—
531C23—202no—
532C23—202no—
533C23—202no—
534C23#406414yes—
535C23C2×D4×D512yes—
536C23C2×D4×D511yescharacteristic
537Q8—24no—
538Q8—24no—
539Q8—24no—
540Q8—48no—
541Q8—44no—
542Q8—42no—
543Q8—104no—
544Q8—104no—
545Q8—208no—
546Q8—204no—
547D4—104no—
548D4—208no—
549D4—202no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—48no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—54no—
561C2×C4—52no—
562C2×C4—52no—
563C2×C4—52no—
564C2×C4—52no—
565C2×C4—108no—
566C2×C4—108no—
567C2×C4—102no—
568C2×C4—102no—
569C2×C4—102no—
570C2×C4—102no—
571C2×C4—104no—
572C2×C4—208no—
573C2×C4—208no—
574C2×C4—202no—
575C2×C4—204no—
576C2×C4C2×D4×D512yes—
577C2×C4C2×D4×D511yescharacteristic
578C2×C4#414414yes—
579C2×C4C24×D511yescharacteristic
580C8—24no—
581C8—104no—
582C5C2×#321811yescharacteristic, complemented, Sylow
583C22—28no—
584C22—22no—
585C22—22no—
586C22—202no—
587C22—202no—
588C22—202no—
589C22—404no—
590C22#6592612yes—
591C22C2×#406414yes—
592C22C2×#406412yes—
593C22D5×#13712yes—
594C22C22×D4×D511yescharacteristic
595C4—22no—
596C4—44no—
597C4—42no—
598C4—42no—
599C4—54no—
600C4—102no—
601C4—102no—
602C4—102no—
603C4—202no—
604C4—204no—
605C4#6592612yes—
606C4C2×#414412yes—
607C2—24no—
608C2—402no—
609C2#1061191014yesdirect product factor
610C2C2×#6592611yescharacteristic
611C2C22×#406411yescharacteristic
612C2C2×D5×#13711yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC2×C5×#451C2
socleC23×C5C22×D4
prime radicalC2×#4076C22
hypercenterC2×#451D5
hypocenterC5C2×#3218
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#451

Complements

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

NH ≅ G/Ntype
1. C22×#4090C2semidirect
3. C2×C5×#451C2semidirect
5. C2×#65479C2semidirect
6. C2×#65471C2semidirect
7. C2×#65339C2semidirect
8. #10611910C2direct (N × H)
9. C2×#65107C2semidirect
10. C2×#65086C2semidirect
12. C2×#65008C2semidirect
24. C2×#4090C22semidirect
40. C5×#451C22semidirect
46. #65479C22semidirect
47. #65471C22semidirect
51. #65339C22semidirect
53. #65107C22semidirect
54. #65086C22semidirect
56. #65008C22semidirect
149. C2×#451D5semidirect
267. #451C2×D5semidirect
523. C2×C5#3218semidirect
582. C5C2×#3218semidirect
609. C2#10611910direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#40902
C2×#659582
C2×C5×#4512
C2×#655162
C2×#654792
C2×#654712
C2×#653392
#106119102
C2×#651072
C2×#650862
C2×#650272
C2×#650082
C2×#32185