Finite Group Databasebeta

← C2×#10605619 #12267257

Subgroups — C2×#10605619 #12267257

Order 1280.

3830 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#4070C211yesmaximal, characteristic, complemented
2C22×C4×C22⋊D5C211yesmaximal, characteristic
3C22×#4043C211yesmaximal, characteristic, complemented
4C22×D5×C4⋊C4C211yesmaximal, characteristic, complemented
5C2×C5×#380C211yesmaximal, characteristic, complemented
6C2×#65263C211yesmaximal, characteristic, complemented
7C2×#65262C211yesmaximal, characteristic, complemented
8C2×#65245C212yesmaximal
9C2×#65237C212yesmaximal, complemented
10#10605619C2116yesmaximal, direct product factor
11C2×#64956C212yesmaximal, complemented
12C2×#64946C212yesmaximal, complemented
13C24×C4×C5C2211yescharacteristic
14C23×C22⋊D5C2211yescharacteristic
15C23×C4×D5C2212yes—
16C23×C4×D5C2211yescharacteristic
17C23×Dic10C2211yescharacteristic
18C22×C5×C4⋊C4C2211yescharacteristic
19C22×C5×C4⋊C4C2211yescharacteristic
20C22×C5×C22⋊C4C2212yes—
21C22×C22⋊Dic5C2211yescharacteristic
22C22×C22⋊Dic5C2212yes—
23C2×C4×C22⋊D5—28no—
24C2×C4×C22⋊D5C2218yes—
25C22×#685C2212yes—
26C22×#685C2211yescharacteristic
27C22×C4⋊Dic5C2212yes—
28C22×C4⋊Dic5C2211yescharacteristic
29C2×#4070—28no—
30C2×#4070C2218yescomplemented
31C22×#683C2212yes—
32C22×#683C2211yescharacteristic
33C22×#683C2212yes—
34C22×#683C2212yes—
35C22×C4×Dic5C2211yescharacteristic
36C2×#4043—28no—
37C2×#4043C2218yescomplemented
38C2×D5×C4⋊C4—28no—
39C2×D5×C4⋊C4C2218yescomplemented
40C5×#380C2218yescomplemented
41C2×C5×#95C2212yes—
42C2×#3963C2212yes—
43C2×#3963C2212yes—
44C2×#3963C2212yes—
45#65263C2218yescomplemented
46#65262C2218yescomplemented
47#65245C22116yes—
48#65237C22116yescomplemented
49#64956C22116yescomplemented
50#64946C22116yescomplemented
51C2×#2417—51nomaximal, Sylow
52C25×C5C2311yescharacteristic
53C24×D5C2311yescharacteristic
54C23×C4×C5—24no—
55C23×C4×C5—24no—
56C23×C4×C5—24no—
57C23×C4×C5C2311yescharacteristic
58C23×C4×C5C2312yes—
59C23×C4×C5C2312yes—
60C23×C4×C5C2314yes—
61C23×C4×C5C2311yescharacteristic
62C22×C22⋊D5—24no—
63C22×C22⋊D5—24no—
64C22×C22⋊D5Q818yescomplemented
65C22×C22⋊D5C2314yes—
66C23×Dic5C2311yescharacteristic
67C23×Dic5C2312yes—
68C23×Dic5C2311yescharacteristic
69C23×Dic5C2312yes—
70C23×Dic5C2311yescharacteristic
71C22×C4×D5—24no—
72C22×C4×D5—24no—
73C22×C4×D5—28no—
74C22×C4×D5—28no—
75C22×C4×D5—44no—
76C22×C4×D5D418yes—
77C22×C4×D5C2318yes—
78C22×C4×D5C2314yes—
79C22×Dic10—24no—
80C22×Dic10—24no—
81C22×Dic10—24no—
82C22×Dic10C2314yes—
83C2×C5×C4⋊C4—24no—
84C2×C5×C4⋊C4—24no—
85C2×C5×C4⋊C4C2314yes—
86C2×C5×C4⋊C4C2314yes—
87C2×C5×C22⋊C4—44no—
88C2×C5×C22⋊C4C2318yes—
89C2×C22⋊Dic5—28no—
90C2×C22⋊Dic5—24no—
91C2×C22⋊Dic5C2318yes—
92C2×C22⋊Dic5C2314yes—
93C4×C22⋊D5—232no—
94C2×#685—28no—
95C2×#685—24no—
96C2×#685C2318yes—
97C2×#685C2314yes—
98C2×C4⋊Dic5—28no—
99C2×C4⋊Dic5D418yescomplemented
100C2×C4⋊Dic5C2318yes—
101C2×C4⋊Dic5C2314yes—
102#4070—232no—
103C2×#683—28no—
104C2×#683—28no—
105C2×#683—24no—
106C2×#683—44no—
107C2×#683C2318yes—
108C2×#683C2318yes—
109C2×#683C2314yes—
110C2×#683C2318yes—
111C2×C4×Dic5—24no—
112C2×C4×Dic5C2314yes—
113#4043—232no—
114D5×C4⋊C4—232no—
115C5×#95C2318yes—
116#3963C2318yes—
117#3963C2318yes—
118#3963C2318yes—
119C22×C22⋊Q8—51no—
120C22×C22⋊Q8—51no—
121C22×C4×D4—51no—
122C23×C4⋊C4—51no—
123#2417—516no—
124C2×#398—52no—
125C2×#396—52no—
126C2×#391—51no—
127C2×#389—51no—
128C2×#386—52no—
129C2×#380—52no—
130C2×#380D511yescharacteristic, complemented
131C24×C5—22no—
132C24×C5—22no—
133C24×C5—22no—
134C24×C5—22no—
135C24×C5—22no—
136C24×C5—22no—
137C24×C5C2×Q814yes—
138C24×C5C2412yes—
139C24×C5C2411yescharacteristic
140C23×D5—22no—
141C23×D5—22no—
142C23×D5—22no—
143C23×D5—22no—
144C23×D5C2×D414yes—
145C23×D5C2×Q814yes—
146C23×D5#4114yes—
147C23×D5C2412yes—
148C22×C4×C5—24no—
149C22×C4×C5—24no—
150C22×C4×C5—22no—
151C22×C4×C5—22no—
152C22×C4×C5—24no—
153C22×C4×C5—22no—
154C22×C4×C5—24no—
155C22×C4×C5—24no—
156C22×C4×C5—24no—
157C22×C4×C5—28no—
158C22×C4×C5—28no—
159C22×C4×C5—28no—
160C22×C4×C5—416no—
161C22×C4×C5C2×D414yes—
162C22×C4×C5C2×D414yes—
163C22×C4×C5#4114yes—
164C22×C4×C5C2414yes—
165C22×C4×C5C2414yes—
166C22×C4×C5C2412yes—
167C22×C4×C5C2412yes—
168C2×C22⋊D5—232no—
169C2×C22⋊D5—28no—
170C2×C22⋊D5—28no—
171C2×C22⋊D5C2×Q8116yescomplemented
172C22×Dic5—22no—
173C22×Dic5—22no—
174C22×Dic5—22no—
175C22×Dic5—24no—
176C22×Dic5—22no—
177C22×Dic5—24no—
178C22×Dic5—24no—
179C22×Dic5—22no—
180C22×Dic5—24no—
181C22×Dic5—24no—
182C22×Dic5—22no—
183C22×Dic5—22no—
184C22×Dic5C2×D418yes—
185C22×Dic5C2×D414yes—
186C22×Dic5C2×Q814yes—
187C22×Dic5C2414yes—
188C22×Dic5C2412yes—
189C22×Dic5C2414yes—
190C22×Dic5C2412yes—
191C22×Dic5C2412yes—
192C2×C4×D5—232no—
193C2×C4×D5—28no—
194C2×C4×D5—28no—
195C2×C4×D5—216no—
196C2×C4×D5—216no—
197C2×C4×D5—48no—
198C2×C4×D5—432no—
199C2×C4×D5C2×D4116yes—
200C2×Dic10—28no—
201C2×Dic10—28no—
202C2×Dic10—28no—
203C2×Dic10—416no—
204C5×C4⋊C4—28no—
205C5×C4⋊C4—28no—
206C5×C22⋊C4—48no—
207C22⋊Dic5—216no—
208C22⋊Dic5—28no—
209#685—216no—
210#685—28no—
211C4⋊Dic5—216no—
212C4⋊Dic5C2×D4116yescomplemented
213#683—216no—
214#683—216no—
215#683—28no—
216#683—48no—
217C4×Dic5—28no—
218C23×Q8—51no—
219C23×D4—51no—
220C24×C4—51no—
221C24×C4—52no—
222C24×C4C2×D511yescharacteristic
223C2×C22⋊Q8—58no—
224C2×C22⋊Q8—58no—
225C2×C22⋊Q8—108no—
226C2×C22⋊Q8—108no—
227C2×C4×D4—58no—
228C2×C4×D4—108no—
229C22×C4⋊C4—51no—
230C22×C4⋊C4—52no—
231C22×C4⋊C4—51no—
232C22×C4⋊C4—52no—
233C22×C4⋊C4—52no—
234C22×C4⋊C4—52no—
235C22×C4⋊C4—58no—
236C22×C4⋊C4—108no—
237C22×C4⋊C4C2×D511yescharacteristic
238C22×C4⋊C4C2×D511yescharacteristic
239C22×C22⋊C4—51no—
240C22×C22⋊C4—52no—
241C22×C22⋊C4—51no—
242C22×C22⋊C4—52no—
243C22×C22⋊C4C2×D512yes—
244C22×C42—51no—
245#398—516no—
246#396—516no—
247#391—58no—
248#389—58no—
249#386—516no—
250#380—516no—
251#380C2×D518yescomplemented
252C2×#95—52no—
253C2×#95—52no—
254C2×#95—52no—
255C2×#95C2×D512yes—
256C23×C5—28no—
257C23×C5—28no—
258C23×C5—28no—
259C23×C5—22no—
260C23×C5—22no—
261C23×C5—22no—
262C23×C5—22no—
263C23×C5—22no—
264C23×C5—22no—
265C23×C5—44no—
266C23×C5—44no—
267C23×C5—44no—
268C23×C5—44no—
269C23×C5C22×D412yes—
270C23×C5C22×D412yes—
271C23×C5C22×Q814yes—
272C23×C5C22×Q812yes—
273C23×C5C2×#4112yes—
274C23×C5C2×#4112yes—
275C23×C5H(5,𝔽2)12yes—
276C23×C52−1+412yes—
277C23×C5C2511yescharacteristic
278C22×D5—28no—
279C22×D5—28no—
280C22×D5—28no—
281C22×D5—28no—
282C22×D5—28no—
283C22×D5—28no—
284C22×D5—22no—
285C22×D5—22no—
286C22×D5—22no—
287C22×D5—22no—
288C22×D5C22⋊Q8116yescomplemented
289C22×D5C22×D414yes—
290C22×D5C22×Q814yes—
291C22×D5C2×#4114yes—
292C2×C4×C5—22no—
293C2×C4×C5—22no—
294C2×C4×C5—24no—
295C2×C4×C5—24no—
296C2×C4×C5—24no—
297C2×C4×C5—22no—
298C2×C4×C5—24no—
299C2×C4×C5—24no—
300C2×C4×C5—24no—
301C2×C4×C5—44no—
302C2×C4×C5—48no—
303C2×C4×C5—48no—
304C2×C4×C5—416no—
305C2×C4×C5#121116yes—
306C2×C4×C5C22×D414yes—
307C2×C4×C5C22×D414yes—
308C2×C4×C5C2×#4114yes—
309C22⋊D5—232no—
310C2×Dic5—28no—
311C2×Dic5—28no—
312C2×Dic5—216no—
313C2×Dic5—22no—
314C2×Dic5—22no—
315C2×Dic5—22no—
316C2×Dic5—24no—
317C2×Dic5—22no—
318C2×Dic5—24no—
319C2×Dic5—22no—
320C2×Dic5—24no—
321C2×Dic5—24no—
322C2×Dic5—24no—
323C2×Dic5—22no—
324C2×Dic5—22no—
325C2×Dic5—48no—
326C2×Dic5—44no—
327C2×Dic5C22×D418yes—
328C2×Dic5C22×D414yes—
329C2×Dic5C22×Q814yes—
330C4×D5—232no—
331C4×D5—432no—
332Dic10—416no—
333C25—51no—
334C25C22×D511yescharacteristic
335C22×Q8—54no—
336C22×Q8—104no—
337C22×Q8—104no—
338C22×Q8—104no—
339C22×D4—58no—
340C22×D4—54no—
341C22×D4—104no—
342C22×D4—104no—
343C23×C4—24no—
344C23×C4—24no—
345C23×C4—24no—
346C23×C4—51no—
347C23×C4—52no—
348C23×C4—51no—
349C23×C4—58no—
350C23×C4—54no—
351C23×C4—52no—
352C23×C4—51no—
353C23×C4—58no—
354C23×C4—108no—
355C23×C4—108no—
356C23×C4—104no—
357C23×C4—104no—
358C23×C4—204no—
359C23×C4C22×D514yes—
360C23×C4C22×D512yes—
361C23×C4C22×D511yescharacteristic
362C23×C4C22×D512yes—
363C23×C4C22×D511yescharacteristic
364C22⋊Q8—1032no—
365C22⋊Q8—1032no—
366C4×D4—1032no—
367C2×C4⋊C4—24no—
368C2×C4⋊C4—24no—
369C2×C4⋊C4—58no—
370C2×C4⋊C4—54no—
371C2×C4⋊C4—58no—
372C2×C4⋊C4—54no—
373C2×C4⋊C4—58no—
374C2×C4⋊C4—58no—
375C2×C4⋊C4—58no—
376C2×C4⋊C4—1032no—
377C2×C4⋊C4—108no—
378C2×C4⋊C4—104no—
379C2×C4⋊C4—108no—
380C2×C4⋊C4—108no—
381C2×C4⋊C4—204no—
382C2×C4⋊C4C22×D514yes—
383C2×C4⋊C4C22×D514yes—
384C2×C22⋊C4—44no—
385C2×C22⋊C4—54no—
386C2×C22⋊C4—58no—
387C2×C22⋊C4—54no—
388C2×C22⋊C4—58no—
389C2×C22⋊C4—104no—
390C2×C22⋊C4—108no—
391C2×C22⋊C4—104no—
392C2×C22⋊C4—108no—
393C2×C22⋊C4C22×D518yes—
394C2×C42—54no—
395C2×C42—104no—
396#95—58no—
397#95—58no—
398#95—58no—
399#95C22×D518yes—
400C22×C5—24no—
401C22×C5—24no—
402C22×C5—24no—
403C22×C5—42no—
404C22×C5—416no—
405C22×C5—42no—
406C22×C5—42no—
407C22×C5—42no—
408C22×C5C2×#12114yes—
409C22×C5C2×C22⋊Q814yes—
410C22×C5#53714yes—
411C22×C5#54614yes—
412C22×C5D4×Q814yes—
413C22×C5#55214yes—
414C22×C5#55414yes—
415C22×C5C23×D411yescharacteristic
416C22×C5C23×D411yescharacteristic
417C22×C5C23×Q811yescharacteristic
418C22×C5C22×#4111yescharacteristic
419C22×C5C22×#4111yescharacteristic
420C22×C5C2×H(5,𝔽2)11yescharacteristic
421C22×C5C2×2−1+411yescharacteristic
422C2×D5—24no—
423C2×D5—232no—
424C2×D5—24no—
425C2×D5—24no—
426C2×D5—24no—
427C2×D5—24no—
428C2×D5—24no—
429C2×D5C2×C22⋊Q818yescomplemented
430C4×C5—42no—
431C4×C5—44no—
432C4×C5—44no—
433C4×C5C2×#12118yes—
434Dic5—24no—
435Dic5—24no—
436Dic5—28no—
437Dic5—44no—
438Dic5—42no—
439C24—22no—
440C24—22no—
441C24—22no—
442C24—22no—
443C24—22no—
444C24—22no—
445C24—54no—
446C24—54no—
447C24—54no—
448C24—52no—
449C24—102no—
450C24—102no—
451C24—102no—
452C24—102no—
453C24Q8×D514yes—
454C24C23×D512yes—
455C24C23×D511yescharacteristic
456C2×Q8—108no—
457C2×Q8—108no—
458C2×Q8—108no—
459C2×Q8—2016no—
460C2×D4—516no—
461C2×D4—108no—
462C2×D4—1032no—
463C2×D4—108no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—28no—
474C22×C4—28no—
475C22×C4—28no—
476C22×C4—416no—
477C22×C4—58no—
478C22×C4—54no—
479C22×C4—54no—
480C22×C4—52no—
481C22×C4—54no—
482C22×C4—52no—
483C22×C4—516no—
484C22×C4—52no—
485C22×C4—54no—
486C22×C4—1016no—
487C22×C4—1016no—
488C22×C4—102no—
489C22×C4—102no—
490C22×C4—102no—
491C22×C4—104no—
492C22×C4—104no—
493C22×C4—102no—
494C22×C4—108no—
495C22×C4—108no—
496C22×C4—1032no—
497C22×C4—104no—
498C22×C4—102no—
499C22×C4—104no—
500C22×C4—104no—
501C22×C4—102no—
502C22×C4—102no—
503C22×C4—208no—
504C22×C4—2032no—
505C22×C4#70914yes—
506C22×C4D4×D514yes—
507C22×C4D4×D514yes—
508C22×C4C23×D514yes—
509C22×C4C23×D514yes—
510C22×C4C23×D512yes—
511C22×C4C23×D512yes—
512C4⋊C4—28no—
513C4⋊C4—28no—
514C4⋊C4—516no—
515C4⋊C4—1016no—
516C4⋊C4—108no—
517C4⋊C4—1016no—
518C4⋊C4—1016no—
519C4⋊C4—208no—
520C22⋊C4—48no—
521C22⋊C4—108no—
522C22⋊C4—1016no—
523C22⋊C4—108no—
524C22⋊C4—1016no—
525C42—108no—
526C2×C5—44no—
527C2×C5#241718yescomplemented
528C2×C5C22×#12111yescharacteristic
529C2×C5C22×C22⋊Q811yescharacteristic
530C2×C5C2×#53711yescharacteristic
531C2×C5C2×#54611yescharacteristic
532C2×C5C2×D4×Q811yescharacteristic
533C2×C5C2×#55211yescharacteristic
534C2×C5C2×#55411yescharacteristic
535D5—28no—
536C23—28no—
537C23—28no—
538C23—28no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—516no—
550C23—54no—
551C23—54no—
552C23—54no—
553C23—108no—
554C23—108no—
555C23—108no—
556C23—108no—
557C23—108no—
558C23—108no—
559C23—102no—
560C23—102no—
561C23—102no—
562C23—102no—
563C23C2×#70912yes—
564C23C2×D4×D512yes—
565C23C2×D4×D512yes—
566C23#414412yes—
567C23C2×Q8×D514yes—
568C23C2×Q8×D512yes—
569C23D5×#4112yes—
570C23#415012yes—
571C23C24×D511yescharacteristic
572Q8—2016no—
573D4—1032no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—44no—
584C2×C4—48no—
585C2×C4—48no—
586C2×C4—416no—
587C2×C4—58no—
588C2×C4—54no—
589C2×C4—54no—
590C2×C4—108no—
591C2×C4—1016no—
592C2×C4—108no—
593C2×C4—102no—
594C2×C4—102no—
595C2×C4—102no—
596C2×C4—104no—
597C2×C4—104no—
598C2×C4—102no—
599C2×C4—1032no—
600C2×C4—104no—
601C2×C4—102no—
602C2×C4—104no—
603C2×C4—104no—
604C2×C4—102no—
605C2×C4—102no—
606C2×C4—2032no—
607C2×C4—208no—
608C2×C4—204no—
609C2×C4#4030116yes—
610C2×C4C2×#70914yes—
611C2×C4C2×D4×D514yes—
612C2×C4C2×D4×D514yes—
613C5C2×#241711yescharacteristic, complemented, Sylow
614C22—24no—
615C22—24no—
616C22—24no—
617C22—42no—
618C22—416no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—58no—
623C22—104no—
624C22—104no—
625C22—104no—
626C22—1032no—
627C22—104no—
628C22—104no—
629C22—104no—
630C22C2×#403014yes—
631C22#6584514yes—
632C22#6586314yes—
633C22#6594214yes—
634C22#6594614yes—
635C22D5×C22⋊Q814yes—
636C22#6596614yes—
637C22C22×#70911yescharacteristic
638C22C22×D4×D511yescharacteristic
639C22C22×D4×D511yescharacteristic
640C22C2×#414411yescharacteristic
641C22C22×Q8×D511yescharacteristic
642C22C2×D5×#4111yescharacteristic
643C22C2×#415011yescharacteristic
644C4—44no—
645C4—44no—
646C4—42no—
647C4—104no—
648C4—108no—
649C4—104no—
650C4—202no—
651C4—204no—
652C4C2×#403018yes—
653C2—44no—
654C2—108no—
655C2#1060561918yesdirect product factor
656C2C22×#403011yescharacteristic
657C2C2×#6584511yescharacteristic
658C2C2×#6586311yescharacteristic
659C2C2×#6594211yescharacteristic
660C2C2×#6594611yescharacteristic
661C2C2×#6596611yescharacteristic
662C2C2×D5×C22⋊Q811yescharacteristic
subgroupG/H
centerC24C23×D5
commutator, cosocleC23×C5C25
FrattiniC23C24×D5
FittingC2×C5×#380C2
socleC24×C5C24
prime radicalC23×C22⋊D5C22
hypercenterC2×#380D5
hypocenterC5C2×#2417
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
1. C22×#4070C2semidirect
3. C22×#4043C2semidirect
4. C22×D5×C4⋊C4C2semidirect
5. C2×C5×#380C2semidirect
6. C2×#65263C2semidirect
7. C2×#65262C2semidirect
9. C2×#65237C2semidirect
10. #10605619C2direct (N × H)
11. C2×#64956C2semidirect
12. C2×#64946C2semidirect
30. C2×#4070C22semidirect
37. C2×#4043C22semidirect
39. C2×D5×C4⋊C4C22semidirect
40. C5×#380C22semidirect
45. #65263C22semidirect
46. #65262C22semidirect
48. #65237C22semidirect
49. #64956C22semidirect
50. #64946C22semidirect
64. C22×C22⋊D5Q8semidirect
99. C2×C4⋊Dic5D4semidirect
130. C2×#380D5semidirect
171. C2×C22⋊D5C2×Q8semidirect
212. C4⋊Dic5C2×D4semidirect
251. #380C2×D5semidirect
288. C22×D5C22⋊Q8semidirect
429. C2×D5C2×C22⋊Q8semidirect
527. C2×C5#2417semidirect
613. C5C2×#2417semidirect
655. C2#10605619direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#40702
C22×C4×C22⋊D52
C22×#40432
C22×D5×C4⋊C42
C2×C5×#3802
C2×#652632
C2×#652622
C2×#652452
C2×#652372
#106056192
C2×#649562
C2×#649462
C2×#24175