Finite Group Databasebeta

← C2×#10605672 #12270585

Subgroups — C2×#10605672 #12270585

Order 1280.

3006 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C22⋊D5C211yesmaximal, characteristic
2C22×#4036C211yesmaximal, characteristic, complemented
3C2×C5×#380C211yesmaximal, characteristic, complemented
4C2×#65501C211yesmaximal, characteristic, complemented
5C2×#65276C211yesmaximal, characteristic, complemented
6C2×#65276C211yesmaximal, characteristic, complemented
7C2×C4⋊C4×Dic5C211yesmaximal, characteristic, complemented
8C2×#65250C212yesmaximal
9C2×#65233C212yesmaximal, complemented
10#10605672C2116yesmaximal, direct product factor
11C2×#64949C212yesmaximal, complemented
12C2×#64958C212yesmaximal, complemented
13C24×C4×C5C2211yescharacteristic
14C23×C22⋊D5C2211yescharacteristic
15C23×C4×D5C2211yescharacteristic
16C22×C5×C4⋊C4C2211yescharacteristic
17C22×C5×C4⋊C4C2211yescharacteristic
18C22×C5×C22⋊C4C2212yes—
19C22×C22⋊Dic5C2211yescharacteristic
20C22×C22⋊Dic5C2212yes—
21C2×C4×C22⋊D5—28no—
22C2×C4×C22⋊D5C2218yes—
23C22×#685C2211yescharacteristic
24C22×#685C2212yes—
25C22×#685C2212yes—
26C22×C4⋊Dic5C2212yes—
27C22×C4⋊Dic5C2211yescharacteristic
28C22×C4⋊Dic5C2211yescharacteristic
29C22×#683C2212yes—
30C22×#683C2211yescharacteristic
31C22×C4×Dic5C2212yes—
32C22×C4×Dic5C2211yescharacteristic
33C2×#4036—28no—
34C2×#4036C2218yescomplemented
35C5×#380C2218yescomplemented
36C2×C5×#95C2212yes—
37#65501C2218yescomplemented
38C2×#3963C2212yes—
39C2×#3963C2212yes—
40C2×#3963C2212yes—
41C2×#3963C2212yes—
42#65276C2218yescomplemented
43#65276C2218yescomplemented
44C4⋊C4×Dic5C2218yescomplemented
45#65250C22116yes—
46#65233C22116yescomplemented
47#64958C22116yescomplemented
48#64949C22116yescomplemented
49C2×#2562—51nomaximal, Sylow
50C25×C5C2311yescharacteristic
51C24×D5C2311yescharacteristic
52C23×C4×C5—24no—
53C23×C4×C5—24no—
54C23×C4×C5—24no—
55C23×C4×C5C2311yescharacteristic
56C23×C4×C5C2312yes—
57C23×C4×C5C2312yes—
58C23×C4×C5C2314yes—
59C23×C4×C5C2311yescharacteristic
60C22×C22⋊D5—24no—
61C22×C22⋊D5—24no—
62C22×C22⋊D5—24no—
63C22×C22⋊D5C2314yes—
64C23×Dic5C2312yes—
65C23×Dic5C2312yes—
66C23×Dic5C2311yescharacteristic
67C23×Dic5C2311yescharacteristic
68C23×Dic5C2311yescharacteristic
69C22×C4×D5—24no—
70C22×C4×D5—24no—
71C22×C4×D5—24no—
72C22×C4×D5C2314yes—
73C2×C5×C4⋊C4—24no—
74C2×C5×C4⋊C4—24no—
75C2×C5×C4⋊C4C2314yes—
76C2×C5×C4⋊C4C2314yes—
77C2×C5×C22⋊C4—44no—
78C2×C5×C22⋊C4C2318yes—
79C2×C22⋊Dic5—28no—
80C2×C22⋊Dic5—24no—
81C2×C22⋊Dic5C2314yes—
82C2×C22⋊Dic5C2318yes—
83C4×C22⋊D5—232no—
84C2×#685—28no—
85C2×#685—24no—
86C2×#685—44no—
87C2×#685C2314yes—
88C2×#685C2318yes—
89C2×#685C2318yes—
90C2×C4⋊Dic5—24no—
91C2×C4⋊Dic5—24no—
92C2×C4⋊Dic5—28no—
93C2×C4⋊Dic5C2318yes—
94C2×C4⋊Dic5C2314yes—
95C2×C4⋊Dic5C2314yes—
96C2×#683—28no—
97C2×#683—24no—
98C2×#683C2318yes—
99C2×#683C2314yes—
100C2×C4×Dic5—44no—
101C2×C4×Dic5D418yes—
102C2×C4×Dic5C2318yes—
103C2×C4×Dic5C2314yes—
104#4036—232no—
105C5×#95C2318yes—
106#3963C2318yes—
107#3963C2318yes—
108#3963C2318yes—
109#3963C2318yes—
110C22×#125—51no—
111C22×C4×D4—51no—
112#2562—516no—
113C2×#400—52no—
114C2×#394—52no—
115C2×#388—52no—
116C2×#388—52no—
117C2×#380—51no—
118C2×#380—51no—
119C2×#380—51no—
120C2×#380D511yescharacteristic, complemented
121C2×C4×C4⋊C4—51no—
122C24×C5—22no—
123C24×C5—22no—
124C24×C5—22no—
125C24×C5—22no—
126C24×C5—22no—
127C24×C5—22no—
128C24×C5#4114yes—
129C24×C5C2412yes—
130C24×C5C2411yescharacteristic
131C23×D5—22no—
132C23×D5—22no—
133C23×D5—22no—
134C23×D5—22no—
135C23×D5—22no—
136C23×D5—22no—
137C23×D5#4114yes—
138C23×D5C2412yes—
139C22×C4×C5—24no—
140C22×C4×C5—22no—
141C22×C4×C5—24no—
142C22×C4×C5—22no—
143C22×C4×C5—22no—
144C22×C4×C5—24no—
145C22×C4×C5—24no—
146C22×C4×C5—24no—
147C22×C4×C5—24no—
148C22×C4×C5—28no—
149C22×C4×C5—28no—
150C22×C4×C5—28no—
151C22×C4×C5—416no—
152C22×C4×C5C2×D414yes—
153C22×C4×C5#4114yes—
154C22×C4×C5#4114yes—
155C22×C4×C5C2414yes—
156C22×C4×C5C2414yes—
157C22×C4×C5C2412yes—
158C22×C4×C5C2412yes—
159C2×C22⋊D5—232no—
160C2×C22⋊D5—28no—
161C2×C22⋊D5—28no—
162C2×C22⋊D5—28no—
163C22×Dic5—22no—
164C22×Dic5—22no—
165C22×Dic5—22no—
166C22×Dic5—22no—
167C22×Dic5—22no—
168C22×Dic5—24no—
169C22×Dic5—24no—
170C22×Dic5—24no—
171C22×Dic5—22no—
172C22×Dic5—22no—
173C22×Dic5—24no—
174C22×Dic5—24no—
175C22×Dic5—24no—
176C22×Dic5C2×D414yes—
177C22×Dic5C2×D414yes—
178C22×Dic5C2412yes—
179C22×Dic5C2414yes—
180C22×Dic5C2414yes—
181C22×Dic5C2412yes—
182C22×Dic5C2412yes—
183C2×C4×D5—28no—
184C2×C4×D5—28no—
185C2×C4×D5—28no—
186C2×C4×D5—416no—
187C5×C4⋊C4—28no—
188C5×C4⋊C4—28no—
189C5×C22⋊C4—48no—
190C22⋊Dic5—216no—
191C22⋊Dic5—28no—
192#685—216no—
193#685—28no—
194#685—48no—
195C4⋊Dic5—28no—
196C4⋊Dic5—28no—
197C4⋊Dic5—216no—
198#683—216no—
199#683—28no—
200C4×Dic5—48no—
201C4×Dic5C2×D4116yes—
202C23×D4—51no—
203C24×C4—51no—
204C24×C4C2×D511yescharacteristic
205C2×#125—58no—
206C2×#125—108no—
207C2×C4×D4—58no—
208C2×C4×D4—108no—
209C22×C4⋊C4—51no—
210C22×C4⋊C4—52no—
211C22×C4⋊C4—52no—
212C22×C4⋊C4—51no—
213C22×C4⋊C4—51no—
214C22×C4⋊C4C2×D511yescharacteristic
215C22×C4⋊C4C2×D511yescharacteristic
216C22×C22⋊C4—52no—
217C22×C22⋊C4—51no—
218C22×C22⋊C4—52no—
219C22×C22⋊C4—51no—
220C22×C22⋊C4—52no—
221C22×C22⋊C4C2×D512yes—
222C22×C42—51no—
223C22×C42—52no—
224#400—516no—
225#394—516no—
226#388—516no—
227#388—516no—
228#380—58no—
229#380—58no—
230#380—58no—
231#380C2×D518yescomplemented
232C4×C4⋊C4—58no—
233C2×#95—52no—
234C2×#95—52no—
235C2×#95—52no—
236C2×#95—52no—
237C2×#95C2×D512yes—
238C23×C5—28no—
239C23×C5—28no—
240C23×C5—28no—
241C23×C5—22no—
242C23×C5—22no—
243C23×C5—22no—
244C23×C5—22no—
245C23×C5—22no—
246C23×C5—22no—
247C23×C5—44no—
248C23×C5—44no—
249C23×C5—44no—
250C23×C5—44no—
251C23×C5C22×D412yes—
252C23×C5C2×#4112yes—
253C23×C5C2×#4112yes—
254C23×C5C2×#4114yes—
255C23×C5C2×#4112yes—
256C23×C5C2×#4112yes—
257C23×C5H(5,𝔽2)12yes—
258C23×C52−1+412yes—
259C23×C5C2511yescharacteristic
260C22×D5—28no—
261C22×D5—28no—
262C22×D5—28no—
263C22×D5—22no—
264C22×D5—22no—
265C22×D5—22no—
266C22×D5—22no—
267C22×D5—22no—
268C22×D5—22no—
269C22×D5—44no—
270C22×D5—44no—
271C22×D5—44no—
272C22×D5—44no—
273C22×D5C2×#4114yes—
274C2×C4×C5—24no—
275C2×C4×C5—22no—
276C2×C4×C5—24no—
277C2×C4×C5—24no—
278C2×C4×C5—22no—
279C2×C4×C5—22no—
280C2×C4×C5—24no—
281C2×C4×C5—24no—
282C2×C4×C5—24no—
283C2×C4×C5—44no—
284C2×C4×C5—48no—
285C2×C4×C5—48no—
286C2×C4×C5—416no—
287C2×C4×C5#124116yes—
288C2×C4×C5C22×D414yes—
289C2×C4×C5C2×#4114yes—
290C2×C4×C5C2×#4114yes—
291C22⋊D5—232no—
292C2×Dic5—28no—
293C2×Dic5—28no—
294C2×Dic5—22no—
295C2×Dic5—22no—
296C2×Dic5—22no—
297C2×Dic5—22no—
298C2×Dic5—24no—
299C2×Dic5—22no—
300C2×Dic5—24no—
301C2×Dic5—24no—
302C2×Dic5—24no—
303C2×Dic5—24no—
304C2×Dic5—22no—
305C2×Dic5—22no—
306C2×Dic5—24no—
307C2×Dic5—48no—
308C2×Dic5—44no—
309C2×Dic5—48no—
310C2×Dic5C22×D414yes—
311C2×Dic5C22×D414yes—
312C4×D5—416no—
313C25—51no—
314C25C22×D511yescharacteristic
315C22×D4—54no—
316C22×D4—104no—
317C22×D4—104no—
318C22×D4—104no—
319C23×C4—24no—
320C23×C4—24no—
321C23×C4—24no—
322C23×C4—51no—
323C23×C4—51no—
324C23×C4—52no—
325C23×C4—52no—
326C23×C4—51no—
327C23×C4—54no—
328C23×C4—104no—
329C23×C4—104no—
330C23×C4—104no—
331C23×C4C22×D514yes—
332C23×C4C22×D512yes—
333C23×C4C22×D512yes—
334C23×C4C22×D511yescharacteristic
335C23×C4C22×D511yescharacteristic
336#125—1032no—
337C4×D4—1032no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—54no—
341C2×C4⋊C4—54no—
342C2×C4⋊C4—58no—
343C2×C4⋊C4—58no—
344C2×C4⋊C4—54no—
345C2×C4⋊C4—104no—
346C2×C4⋊C4—108no—
347C2×C4⋊C4—108no—
348C2×C4⋊C4—104no—
349C2×C4⋊C4—104no—
350C2×C4⋊C4C22×D514yes—
351C2×C4⋊C4C22×D514yes—
352C2×C22⋊C4—44no—
353C2×C22⋊C4—54no—
354C2×C22⋊C4—58no—
355C2×C22⋊C4—54no—
356C2×C22⋊C4—58no—
357C2×C22⋊C4—58no—
358C2×C22⋊C4—104no—
359C2×C22⋊C4—108no—
360C2×C22⋊C4—104no—
361C2×C22⋊C4—108no—
362C2×C22⋊C4—204no—
363C2×C22⋊C4C22×D518yes—
364C2×C42—58no—
365C2×C42—54no—
366C2×C42—58no—
367C2×C42—204no—
368#95—58no—
369#95—58no—
370#95—58no—
371#95—58no—
372#95C22×D518yes—
373C22×C5—24no—
374C22×C5—24no—
375C22×C5—24no—
376C22×C5—416no—
377C22×C5—42no—
378C22×C5—42no—
379C22×C5—42no—
380C22×C5—42no—
381C22×C5#52514yes—
382C22×C5C2×#12414yes—
383C22×C5#53714yes—
384C22×C5#55214yes—
385C22×C5#55214yes—
386C22×C5#55314yes—
387C22×C5#55314yes—
388C22×C5C23×D411yescharacteristic
389C22×C5C22×#4111yescharacteristic
390C22×C5C22×#4111yescharacteristic
391C22×C5C22×#4111yescharacteristic
392C22×C5C22×#4111yescharacteristic
393C22×C5C2×H(5,𝔽2)11yescharacteristic
394C22×C5C2×2−1+411yescharacteristic
395C2×D5—24no—
396C2×D5—24no—
397C2×D5—24no—
398C2×D5—416no—
399C2×D5—42no—
400C2×D5—42no—
401C2×D5—42no—
402C2×D5—42no—
403C4×C5—42no—
404C4×C5—44no—
405C4×C5—44no—
406C4×C5C2×#12418yes—
407Dic5—24no—
408Dic5—24no—
409Dic5—44no—
410Dic5—42no—
411Dic5—44no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24—22no—
417C24—22no—
418C24—54no—
419C24—52no—
420C24—102no—
421C24—102no—
422C24—102no—
423C24—102no—
424C24—102no—
425C24—102no—
426C24#71314yes—
427C24C23×D512yes—
428C24C23×D511yescharacteristic
429C2×D4—108no—
430C2×D4—108no—
431C2×D4—1032no—
432C2×D4—108no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—28no—
443C22×C4—28no—
444C22×C4—28no—
445C22×C4—416no—
446C22×C4—54no—
447C22×C4—54no—
448C22×C4—52no—
449C22×C4—52no—
450C22×C4—52no—
451C22×C4—54no—
452C22×C4—54no—
453C22×C4—102no—
454C22×C4—102no—
455C22×C4—102no—
456C22×C4—102no—
457C22×C4—104no—
458C22×C4—102no—
459C22×C4—102no—
460C22×C4—104no—
461C22×C4—102no—
462C22×C4—104no—
463C22×C4—104no—
464C22×C4—104no—
465C22×C4—104no—
466C22×C4—108no—
467C22×C4—108no—
468C22×C4—108no—
469C22×C4—2016no—
470C22×C4#70914yes—
471C22×C4D4×D514yes—
472C22×C4#71114yes—
473C22×C4C23×D512yes—
474C22×C4C23×D514yes—
475C22×C4C23×D514yes—
476C22×C4C23×D512yes—
477C4⋊C4—28no—
478C4⋊C4—28no—
479C4⋊C4—108no—
480C4⋊C4—1016no—
481C4⋊C4—1016no—
482C4⋊C4—108no—
483C4⋊C4—108no—
484C22⋊C4—48no—
485C22⋊C4—108no—
486C22⋊C4—1016no—
487C22⋊C4—108no—
488C22⋊C4—1016no—
489C22⋊C4—208no—
490C42—516no—
491C42—208no—
492C2×C5—44no—
493C2×C5#256218yescomplemented
494C2×C5C2×#52511yescharacteristic
495C2×C5C22×#12411yescharacteristic
496C2×C5C2×#53711yescharacteristic
497C2×C5C2×#55211yescharacteristic
498C2×C5C2×#55211yescharacteristic
499C2×C5C2×#55311yescharacteristic
500C2×C5C2×#55311yescharacteristic
501D5—44no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—54no—
516C23—108no—
517C23—108no—
518C23—108no—
519C23—102no—
520C23—102no—
521C23—102no—
522C23—102no—
523C23—102no—
524C23—102no—
525C23—204no—
526C23—204no—
527C23—204no—
528C23—204no—
529C23C2×#70912yes—
530C23C2×D4×D512yes—
531C23C2×#71112yes—
532C23#414412yes—
533C23C2×#71314yes—
534C23C2×#71312yes—
535C23D5×#4112yes—
536C23#415012yes—
537C23C24×D511yescharacteristic
538D4—1032no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—44no—
549C2×C4—48no—
550C2×C4—48no—
551C2×C4—416no—
552C2×C4—54no—
553C2×C4—54no—
554C2×C4—108no—
555C2×C4—108no—
556C2×C4—102no—
557C2×C4—102no—
558C2×C4—102no—
559C2×C4—102no—
560C2×C4—104no—
561C2×C4—102no—
562C2×C4—102no—
563C2×C4—104no—
564C2×C4—102no—
565C2×C4—104no—
566C2×C4—104no—
567C2×C4—104no—
568C2×C4—104no—
569C2×C4—204no—
570C2×C4—208no—
571C2×C4—208no—
572C2×C4—2016no—
573C2×C4#4031116yes—
574C2×C4C2×#70914yes—
575C2×C4C2×D4×D514yes—
576C2×C4C2×#71114yes—
577C5C2×#256211yescharacteristic, complemented, Sylow
578C22—24no—
579C22—24no—
580C22—24no—
581C22—42no—
582C22—416no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—104no—
587C22—104no—
588C22—104no—
589C22—202no—
590C22—202no—
591C22—2016no—
592C22—202no—
593C22—202no—
594C22C2×#403114yes—
595C22#6584814yes—
596C22#6586314yes—
597C22#6594214yes—
598C22#6594814yes—
599C22#6596114yes—
600C22#6597614yes—
601C22C22×#70911yescharacteristic
602C22C22×D4×D511yescharacteristic
603C22C22×#71111yescharacteristic
604C22C2×#414411yescharacteristic
605C22C22×#71311yescharacteristic
606C22C2×D5×#4111yescharacteristic
607C22C2×#415011yescharacteristic
608C4—42no—
609C4—44no—
610C4—44no—
611C4—104no—
612C4—104no—
613C4—204no—
614C4—204no—
615C4—202no—
616C4C2×#403118yes—
617C2—44no—
618C2—204no—
619C2#1060567218yesdirect product factor
620C2C22×#403111yescharacteristic
621C2C2×#6584811yescharacteristic
622C2C2×#6586311yescharacteristic
623C2C2×#6594211yescharacteristic
624C2C2×#6594811yescharacteristic
625C2C2×#6596111yescharacteristic
626C2C2×#6597611yescharacteristic
subgroupG/H
centerC24C23×D5
commutator, cosocleC23×C5C25
FrattiniC23C24×D5
FittingC2×C5×#380C2
socleC24×C5C24
prime radicalC23×C22⋊D5C22
hypercenterC2×#380D5
hypocenterC5C2×#2562
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C22×#4036C2semidirect
3. C2×C5×#380C2semidirect
4. C2×#65501C2semidirect
5. C2×#65276C2semidirect
6. C2×#65276C2semidirect
7. C2×C4⋊C4×Dic5C2semidirect
9. C2×#65233C2semidirect
10. #10605672C2direct (N × H)
11. C2×#64949C2semidirect
12. C2×#64958C2semidirect
34. C2×#4036C22semidirect
35. C5×#380C22semidirect
37. #65501C22semidirect
42. #65276C22semidirect
43. #65276C22semidirect
44. C4⋊C4×Dic5C22semidirect
46. #65233C22semidirect
47. #64958C22semidirect
48. #64949C22semidirect
120. C2×#380D5semidirect
231. #380C2×D5semidirect
493. C2×C5#2562semidirect
577. C5C2×#2562semidirect
619. C2#10605672direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C22⋊D52
C22×#40362
C2×C5×#3802
C2×#655012
C2×#652762
C2×C4⋊C4×Dic52
C2×#652502
C2×#652332
#106056722
C2×#649492
C2×#649582
C2×#25625