Finite Group Databasebeta

← C2×#10607943 #12267255

Subgroups — C2×#10607943 #12267255

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×#4043C211yesmaximal, characteristic, complemented
2C22×#4043C211yesmaximal, characteristic, complemented
3C22×D5×C4⋊C4C211yesmaximal, characteristic, complemented
4C22×C4×D20C211yesmaximal, characteristic
5C2×C5×#389C211yesmaximal, characteristic, complemented
6C2×#65283C212yesmaximal, complemented
7C2×#65276C211yesmaximal, characteristic, complemented
8C2×#65276C212yesmaximal, complemented
9C2×#65218C211yesmaximal, characteristic, complemented
10#10607943C2116yesmaximal, direct product factor
11C2×#64954C212yesmaximal
12C2×#64950C212yesmaximal, complemented
13C23×D20C2211yescharacteristic
14C23×C4×D5C2211yescharacteristic
15C23×C4×D5C2212yes—
16C23×C4×D5C2211yescharacteristic
17C23×Dic10C2211yescharacteristic
18C22×C5×C4⋊C4C2211yescharacteristic
19C22×C5×C4⋊C4C2212yes—
20C22×C5×C4⋊C4C2211yescharacteristic
21C22×C42×C5C2211yescharacteristic
22C22×#685C2212yes—
23C22×#685C2212yes—
24C22×#685C2211yescharacteristic
25C22×#685C2211yescharacteristic
26C22×#685C2212yes—
27C22×C4⋊Dic5C2211yescharacteristic
28C22×C4⋊Dic5C2212yes—
29C22×C4⋊Dic5C2212yes—
30C22×C4⋊Dic5C2211yescharacteristic
31C22×#683C2212yes—
32C2×#4043—28no—
33C2×#4043—28no—
34C2×#4043C2218yescomplemented
35C2×#4043C2218yescomplemented
36C2×D5×C4⋊C4—28no—
37C2×D5×C4⋊C4C2218yescomplemented
38C2×C4×D20—28no—
39C2×C4×D20C2218yes—
40C5×#389C2218yescomplemented
41C2×C5×#95C2212yes—
42C2×#3963C2212yes—
43C2×#3963C2212yes—
44C2×#3963C2212yes—
45#65283C22116yescomplemented
46#65276C2218yescomplemented
47#65276C22116yescomplemented
48#65218C2218yescomplemented
49#64954C22116yes—
50#64950C22116yescomplemented
51C2×#2417—51nomaximal, Sylow
52C24×D5C2311yescharacteristic
53C24×D5C2311yescharacteristic
54C23×C4×C5C2312yes—
55C23×C4×C5C2311yescharacteristic
56C23×C4×C5C2312yes—
57C23×C4×C5C2311yescharacteristic
58C23×C4×C5C2311yescharacteristic
59C23×Dic5C2312yes—
60C23×Dic5C2311yescharacteristic
61C23×Dic5C2312yes—
62C23×Dic5C2311yescharacteristic
63C22×D20—24no—
64C22×D20—24no—
65C22×D20Q818yescomplemented
66C22×D20C2314yes—
67C22×C4×D5—24no—
68C22×C4×D5—24no—
69C22×C4×D5—28no—
70C22×C4×D5—24no—
71C22×C4×D5—28no—
72C22×C4×D5—24no—
73C22×C4×D5—24no—
74C22×C4×D5—44no—
75C22×C4×D5D418yes—
76C22×C4×D5C2314yes—
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—44no—
85C2×C5×C4⋊C4D418yescomplemented
86C2×C5×C4⋊C4C2314yes—
87C2×C5×C4⋊C4C2318yes—
88C2×C5×C4⋊C4C2314yes—
89C2×C42×C5—24no—
90C2×C42×C5C2314yes—
91C2×#685—24no—
92C2×#685—28no—
93C2×#685—28no—
94C2×#685—24no—
95C2×#685—44no—
96C2×#685C2314yes—
97C2×#685C2318yes—
98C2×#685C2314yes—
99C2×#685C2318yes—
100C2×#685C2318yes—
101C2×C4⋊Dic5—24no—
102C2×C4⋊Dic5—28no—
103C2×C4⋊Dic5—24no—
104C2×C4⋊Dic5—28no—
105C2×C4⋊Dic5C2318yes—
106C2×C4⋊Dic5C2314yes—
107C2×C4⋊Dic5C2314yes—
108C2×C4⋊Dic5C2318yes—
109C2×#683—28no—
110C2×#683C2318yes—
111#4043—232no—
112#4043—232no—
113D5×C4⋊C4—232no—
114C4×D20—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×#389D511yescharacteristic, complemented
128C2×#386—52no—
129C2×#380—52no—
130C2×#380—51no—
131C24×C5C2411yescharacteristic
132C23×D5—22no—
133C23×D5—22no—
134C23×D5—22no—
135C23×D5—22no—
136C23×D5—22no—
137C23×D5—22no—
138C23×D5—22no—
139C23×D5—22no—
140C23×D5—22no—
141C23×D5—22no—
142C23×D5C2×D414yes—
143C23×D5C2×Q814yes—
144C23×D5C2×Q814yes—
145C23×D5#4114yes—
146C23×D5C2412yes—
147C23×D5C2412yes—
148C22×C4×C5—22no—
149C22×C4×C5—24no—
150C22×C4×C5—22no—
151C22×C4×C5—24no—
152C22×C4×C5—22no—
153C22×C4×C5—24no—
154C22×C4×C5—22no—
155C22×C4×C5—24no—
156C22×C4×C5—24no—
157C22×C4×C5—22no—
158C22×C4×C5C2×D418yes—
159C22×C4×C5C2×D414yes—
160C22×C4×C5C2×D414yes—
161C22×C4×C5C2×Q814yes—
162C22×C4×C5#4114yes—
163C22×C4×C5C2412yes—
164C22×C4×C5C2414yes—
165C22×C4×C5C2414yes—
166C22×C4×C5C2412yes—
167C22×C4×C5C2412yes—
168C22×Dic5—22no—
169C22×Dic5—22no—
170C22×Dic5—24no—
171C22×Dic5—24no—
172C22×Dic5—24no—
173C22×Dic5—22no—
174C22×Dic5—22no—
175C22×Dic5—24no—
176C22×Dic5—22no—
177C22×Dic5—24no—
178C22×Dic5—24no—
179C22×Dic5C2×D414yes—
180C22×Dic5C2414yes—
181C22×Dic5C2414yes—
182C22×Dic5C2412yes—
183C22×Dic5C2412yes—
184C2×D20—232no—
185C2×D20—28no—
186C2×D20—28no—
187C2×D20C2×Q8116yescomplemented
188C2×C4×D5—232no—
189C2×C4×D5—28no—
190C2×C4×D5—28no—
191C2×C4×D5—216no—
192C2×C4×D5—28no—
193C2×C4×D5—216no—
194C2×C4×D5—28no—
195C2×C4×D5—28no—
196C2×C4×D5—48no—
197C2×C4×D5—432no—
198C2×C4×D5—416no—
199C2×C4×D5C2×D4116yes—
200C2×Dic10—28no—
201C2×Dic10—28no—
202C2×Dic10—28no—
203C2×Dic10—416no—
204C5×C4⋊C4—28no—
205C5×C4⋊C4—48no—
206C5×C4⋊C4C2×D4116yescomplemented
207C42×C5—28no—
208#685—28no—
209#685—216no—
210#685—216no—
211#685—28no—
212#685—48no—
213C4⋊Dic5—28no—
214C4⋊Dic5—216no—
215C4⋊Dic5—28no—
216C4⋊Dic5—216no—
217#683—216no—
218C23×Q8—51no—
219C23×D4—51no—
220C24×C4—51no—
221C24×C4—51no—
222C24×C4—52no—
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—52no—
230C22×C4⋊C4—52no—
231C22×C4⋊C4—51no—
232C22×C4⋊C4—52no—
233C22×C4⋊C4—51no—
234C22×C4⋊C4—58no—
235C22×C4⋊C4—108no—
236C22×C4⋊C4C2×D511yescharacteristic
237C22×C4⋊C4C2×D512yes—
238C22×C4⋊C4C2×D511yescharacteristic
239C22×C22⋊C4—51no—
240C22×C22⋊C4—52no—
241C22×C22⋊C4—51no—
242C22×C22⋊C4—52no—
243C22×C22⋊C4—52no—
244C22×C42C2×D511yescharacteristic
245#398—516no—
246#396—516no—
247#391—58no—
248#389C2×D518yescomplemented
249#386—516no—
250#380—58no—
251#380—516no—
252C2×#95—52no—
253C2×#95—52no—
254C2×#95—52no—
255C2×#95C2×D512yes—
256C23×C5C22×D412yes—
257C23×C5C22×D412yes—
258C23×C5C22×Q812yes—
259C23×C5C2×#4112yes—
260C23×C5C2×#4112yes—
261C23×C5H(5,𝔽2)12yes—
262C23×C52−1+412yes—
263C23×C5C2511yescharacteristic
264C22×D5—28no—
265C22×D5—28no—
266C22×D5—28no—
267C22×D5—28no—
268C22×D5—28no—
269C22×D5—28no—
270C22×D5—22no—
271C22×D5—22no—
272C22×D5—22no—
273C22×D5—22no—
274C22×D5—28no—
275C22×D5—28no—
276C22×D5—28no—
277C22×D5—22no—
278C22×D5—22no—
279C22×D5—22no—
280C22×D5—22no—
281C22×D5—22no—
282C22×D5—22no—
283C22×D5—44no—
284C22×D5—44no—
285C22×D5—44no—
286C22×D5—44no—
287C22×D5C22⋊Q8116yescomplemented
288C22×D5C22×D414yes—
289C22×D5C22×Q814yes—
290C22×D5C22×Q814yes—
291C22×D5C2×#4114yes—
292C2×C4×C5—28no—
293C2×C4×C5—216no—
294C2×C4×C5—22no—
295C2×C4×C5—24no—
296C2×C4×C5—22no—
297C2×C4×C5—24no—
298C2×C4×C5—24no—
299C2×C4×C5—22no—
300C2×C4×C5—24no—
301C2×C4×C5—22no—
302C2×C4×C5—22no—
303C2×C4×C5—24no—
304C2×C4×C5—44no—
305C2×C4×C5—48no—
306C2×C4×C5#121116yes—
307C2×C4×C5C22×D418yes—
308C2×C4×C5C22×D414yes—
309C2×C4×C5C22×D414yes—
310C2×C4×C5C22×Q814yes—
311C2×C4×C5C2×#4114yes—
312C2×Dic5—28no—
313C2×Dic5—22no—
314C2×Dic5—22no—
315C2×Dic5—24no—
316C2×Dic5—22no—
317C2×Dic5—24no—
318C2×Dic5—24no—
319C2×Dic5—22no—
320C2×Dic5—24no—
321C2×Dic5—22no—
322C2×Dic5—24no—
323C2×Dic5—24no—
324C2×Dic5—48no—
325C2×Dic5—44no—
326C2×Dic5—48no—
327C2×Dic5C22×D414yes—
328D20—232no—
329C4×D5—232no—
330C4×D5—432no—
331C4×D5—416no—
332Dic10—416no—
333C25—51no—
334C25—51no—
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—51no—
344C23×C4—58no—
345C23×C4—54no—
346C23×C4—52no—
347C23×C4—51no—
348C23×C4—52no—
349C23×C4—54no—
350C23×C4—58no—
351C23×C4—108no—
352C23×C4—104no—
353C23×C4—104no—
354C23×C4—108no—
355C23×C4—104no—
356C23×C4—104no—
357C23×C4—104no—
358C23×C4—204no—
359C23×C4C22×D512yes—
360C23×C4C22×D511yescharacteristic
361C23×C4C22×D512yes—
362C23×C4C22×D511yescharacteristic
363C23×C4C22×D511yescharacteristic
364C22⋊Q8—1032no—
365C22⋊Q8—1032no—
366C4×D4—1032no—
367C2×C4⋊C4—24no—
368C2×C4⋊C4—44no—
369C2×C4⋊C4—54no—
370C2×C4⋊C4—58no—
371C2×C4⋊C4—54no—
372C2×C4⋊C4—58no—
373C2×C4⋊C4—58no—
374C2×C4⋊C4—104no—
375C2×C4⋊C4—108no—
376C2×C4⋊C4—108no—
377C2×C4⋊C4—104no—
378C2×C4⋊C4—108no—
379C2×C4⋊C4—1032no—
380C2×C4⋊C4D2018yescomplemented
381C2×C4⋊C4C22×D514yes—
382C2×C4⋊C4C22×D518yes—
383C2×C4⋊C4C22×D514yes—
384C2×C22⋊C4—54no—
385C2×C22⋊C4—54no—
386C2×C22⋊C4—58no—
387C2×C22⋊C4—58no—
388C2×C22⋊C4—58no—
389C2×C22⋊C4—104no—
390C2×C22⋊C4—104no—
391C2×C22⋊C4—108no—
392C2×C22⋊C4—108no—
393C2×C22⋊C4—204no—
394C2×C42—24no—
395C2×C42C22×D514yes—
396#95—58no—
397#95—58no—
398#95—58no—
399#95C22×D518yes—
400C22×C5C2×#12114yes—
401C22×C5C2×C22⋊Q814yes—
402C22×C5#53714yes—
403C22×C5#54614yes—
404C22×C5D4×Q814yes—
405C22×C5#55214yes—
406C22×C5#55414yes—
407C22×C5C23×D411yescharacteristic
408C22×C5C23×D411yescharacteristic
409C22×C5C23×Q811yescharacteristic
410C22×C5C22×#4111yescharacteristic
411C22×C5C22×#4111yescharacteristic
412C22×C5C2×H(5,𝔽2)11yescharacteristic
413C22×C5C2×2−1+411yescharacteristic
414C2×D5—24no—
415C2×D5—232no—
416C2×D5—24no—
417C2×D5—24no—
418C2×D5—24no—
419C2×D5—24no—
420C2×D5—24no—
421C2×D5—24no—
422C2×D5—24no—
423C2×D5—24no—
424C2×D5—42no—
425C2×D5—416no—
426C2×D5—42no—
427C2×D5—42no—
428C2×D5—42no—
429C2×D5C2×C22⋊Q818yescomplemented
430C4×C5—24no—
431C4×C5—28no—
432C4×C5—44no—
433C4×C5—42no—
434C4×C5C2×#12118yes—
435Dic5—24no—
436Dic5—44no—
437Dic5—42no—
438Dic5—44no—
439C24—54no—
440C24—54no—
441C24—54no—
442C24—52no—
443C24—54no—
444C24—52no—
445C24—102no—
446C24—102no—
447C24—102no—
448C24—102no—
449C24—102no—
450C24—102no—
451C24—102no—
452C24—102no—
453C24—102no—
454C24—102no—
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—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—54no—
475C22×C4—52no—
476C22×C4—516no—
477C22×C4—54no—
478C22×C4—54no—
479C22×C4—52no—
480C22×C4—1016no—
481C22×C4—102no—
482C22×C4—102no—
483C22×C4—108no—
484C22×C4—108no—
485C22×C4—1032no—
486C22×C4—104no—
487C22×C4—102no—
488C22×C4—104no—
489C22×C4—102no—
490C22×C4—104no—
491C22×C4—104no—
492C22×C4—102no—
493C22×C4—104no—
494C22×C4—104no—
495C22×C4—1016no—
496C22×C4—108no—
497C22×C4—108no—
498C22×C4—108no—
499C22×C4—208no—
500C22×C4—2032no—
501C22×C4—2016no—
502C22×C4C2×D2018yes—
503C22×C4C2×D2014yes—
504C22×C4D4×D514yes—
505C22×C4Q8×D514yes—
506C22×C4#71314yes—
507C22×C4C23×D512yes—
508C22×C4C23×D514yes—
509C22×C4C23×D514yes—
510C22×C4C23×D512yes—
511C22×C4C23×D512yes—
512C4⋊C4—28no—
513C4⋊C4—48no—
514C4⋊C4—108no—
515C4⋊C4—1016no—
516C4⋊C4—1016no—
517C4⋊C4—108no—
518C4⋊C4—1016no—
519C4⋊C4C2×D20116yescomplemented
520C22⋊C4—108no—
521C22⋊C4—108no—
522C22⋊C4—1016no—
523C22⋊C4—1016no—
524C22⋊C4—208no—
525C42—28no—
526C2×C5#241718yescomplemented
527C2×C5C22×#12111yescharacteristic
528C2×C5C22×C22⋊Q811yescharacteristic
529C2×C5C2×#53711yescharacteristic
530C2×C5C2×#54611yescharacteristic
531C2×C5C2×D4×Q811yescharacteristic
532C2×C5C2×#55211yescharacteristic
533C2×C5C2×#55411yescharacteristic
534D5—28no—
535D5—44no—
536C23—516no—
537C23—54no—
538C23—54no—
539C23—54no—
540C23—54no—
541C23—108no—
542C23—108no—
543C23—108no—
544C23—108no—
545C23—108no—
546C23—102no—
547C23—102no—
548C23—102no—
549C23—102no—
550C23—108no—
551C23—108no—
552C23—108no—
553C23—108no—
554C23—102no—
555C23—102no—
556C23—102no—
557C23—102no—
558C23—102no—
559C23—102no—
560C23—204no—
561C23—204no—
562C23—204no—
563C23—204no—
564C23C22×D2012yes—
565C23C2×D4×D512yes—
566C23#414412yes—
567C23C2×Q8×D512yes—
568C23C2×#71312yes—
569C23D5×#4112yes—
570C23#415012yes—
571C23C24×D511yescharacteristic
572Q8—2016no—
573D4—1032no—
574C2×C4—216no—
575C2×C4—28no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—44no—
587C2×C4—48no—
588C2×C4—54no—
589C2×C4—108no—
590C2×C4—102no—
591C2×C4—102no—
592C2×C4—1032no—
593C2×C4—104no—
594C2×C4—102no—
595C2×C4—104no—
596C2×C4—102no—
597C2×C4—104no—
598C2×C4—104no—
599C2×C4—102no—
600C2×C4—104no—
601C2×C4—104no—
602C2×C4—2032no—
603C2×C4—204no—
604C2×C4—208no—
605C2×C4—208no—
606C2×C4—2016no—
607C2×C4#4041116yes—
608C2×C4C22×D2018yes—
609C2×C4C22×D2014yes—
610C2×C4C2×D4×D514yes—
611C2×C4C2×Q8×D514yes—
612C2×C4C2×#71314yes—
613C5C2×#241711yescharacteristic, complemented, Sylow
614C22—58no—
615C22—104no—
616C22—104no—
617C22—1032no—
618C22—104no—
619C22—104no—
620C22—104no—
621C22—104no—
622C22—104no—
623C22—104no—
624C22—104no—
625C22—202no—
626C22—2016no—
627C22—202no—
628C22—202no—
629C22—202no—
630C22C2×#404114yes—
631C22#6588814yes—
632C22Q8×D2014yes—
633C22#6594214yes—
634C22D5×C22⋊Q814yes—
635C22#6602814yes—
636C22#6606414yes—
637C22C23×D2011yescharacteristic
638C22C22×D4×D511yescharacteristic
639C22C2×#414411yescharacteristic
640C22C22×Q8×D511yescharacteristic
641C22C22×#71311yescharacteristic
642C22C2×D5×#4111yescharacteristic
643C22C2×#415011yescharacteristic
644C4—24no—
645C4—28no—
646C4—42no—
647C4—44no—
648C4—104no—
649C4—202no—
650C4—204no—
651C4—204no—
652C4C2×#404118yes—
653C2—108no—
654C2—204no—
655C2#1060794318yesdirect product factor
656C2C22×#404111yescharacteristic
657C2C2×#6588811yescharacteristic
658C2C2×Q8×D2011yescharacteristic
659C2C2×#6594211yescharacteristic
660C2C2×D5×C22⋊Q811yescharacteristic
661C2C2×#6602811yescharacteristic
662C2C2×#6606411yescharacteristic
subgroupG/H
centerC24C23×D5
commutator, cosocleC23×C5C25
FrattiniC23C24×D5
FittingC2×C5×#389C2
socleC24×C5C24
prime radicalC23×D20C22
hypercenterC2×#389D5
hypocenterC5C2×#2417
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
1. C22×#4043C2semidirect
2. C22×#4043C2semidirect
3. C22×D5×C4⋊C4C2semidirect
5. C2×C5×#389C2semidirect
6. C2×#65283C2semidirect
7. C2×#65276C2semidirect
8. C2×#65276C2semidirect
9. C2×#65218C2semidirect
10. #10607943C2direct (N × H)
12. C2×#64950C2semidirect
34. C2×#4043C22semidirect
35. C2×#4043C22semidirect
37. C2×D5×C4⋊C4C22semidirect
40. C5×#389C22semidirect
45. #65283C22semidirect
46. #65276C22semidirect
47. #65276C22semidirect
48. #65218C22semidirect
50. #64950C22semidirect
65. C22×D20Q8semidirect
85. C2×C5×C4⋊C4D4semidirect
127. C2×#389D5semidirect
187. C2×D20C2×Q8semidirect
206. C5×C4⋊C4C2×D4semidirect
248. #389C2×D5semidirect
287. C22×D5C22⋊Q8semidirect
380. C2×C4⋊C4D20semidirect
429. C2×D5C2×C22⋊Q8semidirect
519. C4⋊C4C2×D20semidirect
526. C2×C5#2417semidirect
613. C5C2×#2417semidirect
655. C2#10607943direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#40432
C22×D5×C4⋊C42
C22×C4×D202
C2×C5×#3892
C2×#652832
C2×#652762
C2×#652182
#106079432
C2×#649542
C2×#649502
C2×#24175