Finite Group Databasebeta

← C2×#10614509 #12271069

Subgroups — C2×#10614509 #12271069

Order 1280.

3614 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C5×C22⋊Q8C211yesmaximal, characteristic, complemented
2C22×#4092C211yesmaximal, characteristic, complemented
3C22×Q8×Dic5C211yesmaximal, characteristic, complemented
4C22×C4×C22⋊D5C211yesmaximal, characteristic
5C2×#65501C211yesmaximal, characteristic, complemented
6#10614509C2116yesmaximal, direct product factor
7C2×#65283C212yesmaximal, complemented
8C2×#65276C211yesmaximal, characteristic, complemented
9C2×#65261C212yesmaximal, complemented
10C2×#65265C211yesmaximal, characteristic, complemented
11C2×#65237C212yesmaximal, complemented
12C2×#65246C212yesmaximal
13C23×C5×Q8C2211yescharacteristic
14C24×C4×C5C2211yescharacteristic
15C23×C22⋊D5C2211yescharacteristic
16C23×C4×D5C2211yescharacteristic
17C2×C5×C22⋊Q8—28no—
18C2×C5×C22⋊Q8C2218yescomplemented
19C22×C5×C4⋊C4C2211yescharacteristic
20C22×C5×C4⋊C4C2212yes—
21C22×C5×C22⋊C4C2212yes—
22C22×C22⋊Dic5C2211yescharacteristic
23C22×C22⋊Dic5C2212yes—
24C2×#4092—28no—
25C2×#4092C2218yescomplemented
26C2×Q8×Dic5—28no—
27C2×Q8×Dic5C2218yescomplemented
28C2×C4×C22⋊D5—28no—
29C2×C4×C22⋊D5C2218yes—
30C22×#685C2212yes—
31C22×#685C2211yescharacteristic
32C22×#685C2212yes—
33C22×C4⋊Dic5C2212yes—
34C22×C4⋊Dic5C2211yescharacteristic
35C22×C4⋊Dic5C2211yescharacteristic
36C22×#683C2212yes—
37C22×#683C2212yes—
38C22×#683C2211yescharacteristic
39C22×#683C2212yes—
40C22×C4×Dic5C2212yes—
41C22×C4×Dic5C2211yescharacteristic
42#65501C2218yescomplemented
43C2×#3963C2212yes—
44C2×#3963C2212yes—
45#65283C22116yescomplemented
46#65276C2218yescomplemented
47#65265C2218yescomplemented
48#65261C22116yescomplemented
49#65246C22116yes—
50#65237C22116yescomplemented
51C2×#2538—51nomaximal, Sylow
52C25×C5C2311yescharacteristic
53C24×D5C2311yescharacteristic
54C22×C5×Q8—24no—
55C22×C5×Q8—24no—
56C22×C5×Q8D418yescomplemented
57C22×C5×Q8C2314yes—
58C23×C4×C5—24no—
59C23×C4×C5—24no—
60C23×C4×C5—24no—
61C23×C4×C5C2312yes—
62C23×C4×C5C2311yescharacteristic
63C23×C4×C5C2314yes—
64C23×C4×C5C2312yes—
65C23×C4×C5C2311yescharacteristic
66C22×C22⋊D5—24no—
67C22×C22⋊D5—24no—
68C22×C22⋊D5Q818yescomplemented
69C22×C22⋊D5C2314yes—
70C23×Dic5C2312yes—
71C23×Dic5C2312yes—
72C23×Dic5C2311yescharacteristic
73C23×Dic5C2311yescharacteristic
74C23×Dic5C2311yescharacteristic
75C22×C4×D5—24no—
76C22×C4×D5—24no—
77C22×C4×D5—24no—
78C22×C4×D5C2314yes—
79C5×C22⋊Q8—232no—
80C2×C5×C4⋊C4—24no—
81C2×C5×C4⋊C4—28no—
82C2×C5×C4⋊C4C2318yes—
83C2×C5×C4⋊C4C2314yes—
84C2×C5×C22⋊C4—28no—
85C2×C5×C22⋊C4C2318yes—
86C2×C22⋊Dic5—24no—
87C2×C22⋊Dic5—44no—
88C2×C22⋊Dic5C2314yes—
89C2×C22⋊Dic5C2318yes—
90#4092—232no—
91Q8×Dic5—232no—
92C4×C22⋊D5—232no—
93C2×#685—28no—
94C2×#685—24no—
95C2×#685—44no—
96C2×#685C2314yes—
97C2×#685C2318yes—
98C2×#685C2318yes—
99C2×C4⋊Dic5—24no—
100C2×C4⋊Dic5—24no—
101C2×C4⋊Dic5—28no—
102C2×C4⋊Dic5C2318yes—
103C2×C4⋊Dic5C2314yes—
104C2×C4⋊Dic5C2314yes—
105C2×#683—28no—
106C2×#683—24no—
107C2×#683—28no—
108C2×#683—44no—
109C2×#683C2314yes—
110C2×#683C2318yes—
111C2×#683C2318yes—
112C2×#683C2318yes—
113C2×C4×Dic5—28no—
114C2×C4×Dic5D418yes—
115C2×C4×Dic5C2318yes—
116C2×C4×Dic5C2314yes—
117#3963C2318yes—
118#3963C2318yes—
119C22×C22⋊Q8—51no—
120C22×C22⋊Q8D511yescharacteristic, complemented
121C22×C4×Q8—51no—
122C22×C4×D4—51no—
123#2538—516no—
124C2×#396—52no—
125C2×#396—52no—
126C2×#390—52no—
127C2×#389—52no—
128C2×#384—51no—
129C2×#380—51no—
130C2×#380—51no—
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×D5—22no—
145C23×D5—22no—
146C23×D5C2×Q814yes—
147C23×D5C2412yes—
148C2×C5×Q8—232no—
149C2×C5×Q8—28no—
150C2×C5×Q8—28no—
151C2×C5×Q8C2×D4116yescomplemented
152C22×C4×C5—24no—
153C22×C4×C5—24no—
154C22×C4×C5—24no—
155C22×C4×C5—22no—
156C22×C4×C5—22no—
157C22×C4×C5—22no—
158C22×C4×C5—24no—
159C22×C4×C5—24no—
160C22×C4×C5—28no—
161C22×C4×C5—28no—
162C22×C4×C5—28no—
163C22×C4×C5—416no—
164C22×C4×C5C2×D414yes—
165C22×C4×C5C2×D418yes—
166C22×C4×C5C2×D414yes—
167C22×C4×C5#4114yes—
168C22×C4×C5C2414yes—
169C22×C4×C5C2412yes—
170C22×C4×C5C2414yes—
171C22×C4×C5C2412yes—
172C2×C22⋊D5—232no—
173C2×C22⋊D5—28no—
174C2×C22⋊D5—28no—
175C2×C22⋊D5C2×Q8116yescomplemented
176C22×Dic5—22no—
177C22×Dic5—22no—
178C22×Dic5—22no—
179C22×Dic5—22no—
180C22×Dic5—24no—
181C22×Dic5—24no—
182C22×Dic5—22no—
183C22×Dic5—24no—
184C22×Dic5—24no—
185C22×Dic5—24no—
186C22×Dic5—24no—
187C22×Dic5C2×D414yes—
188C22×Dic5C2×D414yes—
189C22×Dic5C2×Q814yes—
190C22×Dic5#4114yes—
191C22×Dic5C2412yes—
192C22×Dic5C2414yes—
193C22×Dic5C2414yes—
194C22×Dic5C2412yes—
195C22×Dic5C2412yes—
196C2×C4×D5—28no—
197C2×C4×D5—28no—
198C2×C4×D5—28no—
199C2×C4×D5—416no—
200C5×C4⋊C4—28no—
201C5×C4⋊C4—216no—
202C5×C22⋊C4—216no—
203C22⋊Dic5—28no—
204C22⋊Dic5—48no—
205#685—216no—
206#685—28no—
207#685—48no—
208C4⋊Dic5—28no—
209C4⋊Dic5—28no—
210C4⋊Dic5—216no—
211#683—216no—
212#683—28no—
213#683—216no—
214#683—48no—
215C4×Dic5—216no—
216C4×Dic5C2×D4116yes—
217C23×Q8C2×D511yescharacteristic
218C23×D4—51no—
219C24×C4—51no—
220C24×C4C2×D511yescharacteristic
221C2×C22⋊Q8—28no—
222C2×C22⋊Q8—58no—
223C2×C22⋊Q8—108no—
224C2×C22⋊Q8C2×D518yescomplemented
225C2×C4×Q8—58no—
226C2×C4×Q8—108no—
227C2×C4×D4—58no—
228C2×C4×D4—108no—
229C22×C4⋊C4—52no—
230C22×C4⋊C4—51no—
231C22×C4⋊C4—52no—
232C22×C4⋊C4—52no—
233C22×C4⋊C4—51no—
234C22×C4⋊C4—51no—
235C22×C4⋊C4—52no—
236C22×C4⋊C4C2×D512yes—
237C22×C4⋊C4C2×D511yescharacteristic
238C22×C22⋊C4—51no—
239C22×C22⋊C4—51no—
240C22×C22⋊C4—52no—
241C22×C22⋊C4—52no—
242C22×C22⋊C4—52no—
243C22×C22⋊C4C2×D512yes—
244C22×C42—51no—
245C22×C42—52no—
246#396—516no—
247#396—516no—
248#390—516no—
249#389—516no—
250#384—58no—
251#380—58no—
252#380—58no—
253C2×#95—52no—
254C2×#95—52no—
255C23×C5—28no—
256C23×C5—28no—
257C23×C5—28no—
258C23×C5—22no—
259C23×C5—22no—
260C23×C5—22no—
261C23×C5—22no—
262C23×C5—22no—
263C23×C5—22no—
264C23×C5—44no—
265C23×C5—44no—
266C23×C5—44no—
267C23×C5—44no—
268C23×C5C22×D412yes—
269C23×C5C22×D412yes—
270C23×C5C22×Q814yes—
271C23×C5C22×Q812yes—
272C23×C5C2×#4112yes—
273C23×C5C2×#4112yes—
274C23×C5H(5,𝔽2)12yes—
275C23×C52−1+412yes—
276C23×C5C2511yescharacteristic
277C22×D5—28no—
278C22×D5—28no—
279C22×D5—28no—
280C22×D5—22no—
281C22×D5—22no—
282C22×D5—22no—
283C22×D5—22no—
284C22×D5—22no—
285C22×D5—22no—
286C22×D5—44no—
287C22×D5—44no—
288C22×D5—44no—
289C22×D5—44no—
290C22×D5C22×Q814yes—
291C5×Q8—232no—
292C2×C4×C5—216no—
293C2×C4×C5—24no—
294C2×C4×C5—24no—
295C2×C4×C5—22no—
296C2×C4×C5—24no—
297C2×C4×C5—22no—
298C2×C4×C5—24no—
299C2×C4×C5—22no—
300C2×C4×C5—24no—
301C2×C4×C5—44no—
302C2×C4×C5—48no—
303C2×C4×C5—416no—
304C2×C4×C5#121116yes—
305C2×C4×C5C22×D414yes—
306C2×C4×C5C22×D418yes—
307C2×C4×C5C22×D414yes—
308C2×C4×C5C2×#4114yes—
309C22⋊D5—232no—
310C2×Dic5—28no—
311C2×Dic5—22no—
312C2×Dic5—22no—
313C2×Dic5—24no—
314C2×Dic5—22no—
315C2×Dic5—24no—
316C2×Dic5—22no—
317C2×Dic5—24no—
318C2×Dic5—22no—
319C2×Dic5—24no—
320C2×Dic5—24no—
321C2×Dic5—24no—
322C2×Dic5—44no—
323C2×Dic5—48no—
324C2×Dic5—48no—
325C2×Dic5C22⋊Q8116yescomplemented
326C2×Dic5C22×D414yes—
327C2×Dic5C22×D414yes—
328C2×Dic5C22×Q814yes—
329C2×Dic5C2×#4114yes—
330C4×D5—416no—
331C25—51no—
332C25C22×D511yescharacteristic
333C22×Q8—24no—
334C22×Q8—24no—
335C22×Q8C22⋊D518yescomplemented
336C22×Q8C22×D514yes—
337C22×D4—58no—
338C22×D4—54no—
339C22×D4—104no—
340C22×D4—104no—
341C23×C4—24no—
342C23×C4—24no—
343C23×C4—24no—
344C23×C4—51no—
345C23×C4—51no—
346C23×C4—52no—
347C23×C4—52no—
348C23×C4—51no—
349C23×C4—54no—
350C23×C4—104no—
351C23×C4—104no—
352C23×C4—104no—
353C23×C4C22×D514yes—
354C23×C4C22×D512yes—
355C23×C4C22×D511yescharacteristic
356C23×C4C22×D512yes—
357C23×C4C22×D511yescharacteristic
358C22⋊Q8—232no—
359C22⋊Q8—1032no—
360C4×Q8—1032no—
361C4×D4—1032no—
362C2×C4⋊C4—24no—
363C2×C4⋊C4—28no—
364C2×C4⋊C4—54no—
365C2×C4⋊C4—54no—
366C2×C4⋊C4—58no—
367C2×C4⋊C4—54no—
368C2×C4⋊C4—58no—
369C2×C4⋊C4—58no—
370C2×C4⋊C4—58no—
371C2×C4⋊C4—104no—
372C2×C4⋊C4—108no—
373C2×C4⋊C4—108no—
374C2×C4⋊C4—104no—
375C2×C4⋊C4—108no—
376C2×C4⋊C4—104no—
377C2×C4⋊C4—204no—
378C2×C4⋊C4C22×D514yes—
379C2×C4⋊C4C22×D518yes—
380C2×C22⋊C4—28no—
381C2×C22⋊C4—54no—
382C2×C22⋊C4—54no—
383C2×C22⋊C4—58no—
384C2×C22⋊C4—58no—
385C2×C22⋊C4—58no—
386C2×C22⋊C4—104no—
387C2×C22⋊C4—104no—
388C2×C22⋊C4—108no—
389C2×C22⋊C4—204no—
390C2×C22⋊C4—204no—
391C2×C22⋊C4C22×D518yes—
392C2×C42—58no—
393C2×C42—54no—
394C2×C42—58no—
395C2×C42—108no—
396#95—58no—
397#95—58no—
398C22×C5—24no—
399C22×C5—24no—
400C22×C5—24no—
401C22×C5—42no—
402C22×C5—42no—
403C22×C5—42no—
404C22×C5—42no—
405C22×C5—416no—
406C22×C5C2×#12114yes—
407C22×C5C2×C22⋊Q814yes—
408C22×C5#53714yes—
409C22×C5D4×Q814yes—
410C22×C5#55014yes—
411C22×C5#55414yes—
412C22×C5#55614yes—
413C22×C5C23×D411yescharacteristic
414C22×C5C23×D411yescharacteristic
415C22×C5C23×Q811yescharacteristic
416C22×C5C22×#4111yescharacteristic
417C22×C5C22×#4111yescharacteristic
418C22×C5C2×H(5,𝔽2)11yescharacteristic
419C22×C5C2×2−1+411yescharacteristic
420C2×D5—24no—
421C2×D5—24no—
422C2×D5—24no—
423C2×D5—42no—
424C2×D5—42no—
425C2×D5—42no—
426C2×D5—42no—
427C2×D5—416no—
428C4×C5—28no—
429C4×C5—42no—
430C4×C5—44no—
431C4×C5C2×#12118yes—
432Dic5—24no—
433Dic5—42no—
434Dic5—44no—
435Dic5—44no—
436Dic5C2×C22⋊Q818yescomplemented
437C24—22no—
438C24—22no—
439C24—22no—
440C24—22no—
441C24—22no—
442C24—22no—
443C24—54no—
444C24—52no—
445C24—102no—
446C24—102no—
447C24—102no—
448C24—102no—
449C24—102no—
450C24—102no—
451C24Q8×D514yes—
452C24C23×D512yes—
453C24C23×D511yescharacteristic
454C2×Q8—28no—
455C2×Q8—28no—
456C2×Q8—232no—
457C2×Q8C2×C22⋊D5116yescomplemented
458C2×D4—516no—
459C2×D4—108no—
460C2×D4—1032no—
461C2×D4—108no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—28no—
471C22×C4—28no—
472C22×C4—28no—
473C22×C4—416no—
474C22×C4—54no—
475C22×C4—54no—
476C22×C4—54no—
477C22×C4—52no—
478C22×C4—54no—
479C22×C4—52no—
480C22×C4—54no—
481C22×C4—52no—
482C22×C4—54no—
483C22×C4—102no—
484C22×C4—102no—
485C22×C4—104no—
486C22×C4—102no—
487C22×C4—102no—
488C22×C4—104no—
489C22×C4—104no—
490C22×C4—104no—
491C22×C4—104no—
492C22×C4—104no—
493C22×C4—102no—
494C22×C4—108no—
495C22×C4—108no—
496C22×C4—108no—
497C22×C4—2016no—
498C22×C4D4×D514yes—
499C22×C4#71114yes—
500C22×C4C2×C22⋊D518yes—
501C22×C4C2×C22⋊D514yes—
502C22×C4C23×D512yes—
503C22×C4C23×D514yes—
504C22×C4C23×D514yes—
505C22×C4C23×D512yes—
506C4⋊C4—28no—
507C4⋊C4—216no—
508C4⋊C4—108no—
509C4⋊C4—1016no—
510C4⋊C4—1016no—
511C4⋊C4—108no—
512C4⋊C4—1016no—
513C4⋊C4—108no—
514C4⋊C4—208no—
515C22⋊C4—216no—
516C22⋊C4—108no—
517C22⋊C4—108no—
518C22⋊C4—1016no—
519C22⋊C4—208no—
520C22⋊C4—208no—
521C42—516no—
522C42—1016no—
523C2×C5—44no—
524C2×C5#253818yescomplemented
525C2×C5C22×#12111yescharacteristic
526C2×C5C22×C22⋊Q811yescharacteristic
527C2×C5C2×#53711yescharacteristic
528C2×C5C2×D4×Q811yescharacteristic
529C2×C5C2×#55011yescharacteristic
530C2×C5C2×#55411yescharacteristic
531C2×C5C2×#55611yescharacteristic
532D5—44no—
533C23—28no—
534C23—28no—
535C23—28no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—54no—
547C23—108no—
548C23—108no—
549C23—108no—
550C23—102no—
551C23—102no—
552C23—102no—
553C23—102no—
554C23—102no—
555C23—102no—
556C23—204no—
557C23—204no—
558C23—204no—
559C23—204no—
560C23C2×D4×D512yes—
561C23C2×#71112yes—
562C23C2×Q8×D514yes—
563C23C2×Q8×D512yes—
564C23#414712yes—
565C23D5×#4112yes—
566C23#414912yes—
567C23C22×C22⋊D512yes—
568C23C24×D511yescharacteristic
569Q8—232no—
570D4—1032no—
571C2×C4—216no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—44no—
581C2×C4—48no—
582C2×C4—416no—
583C2×C4—516no—
584C2×C4—54no—
585C2×C4—54no—
586C2×C4—54no—
587C2×C4—54no—
588C2×C4—108no—
589C2×C4—102no—
590C2×C4—102no—
591C2×C4—104no—
592C2×C4—102no—
593C2×C4—102no—
594C2×C4—104no—
595C2×C4—104no—
596C2×C4—104no—
597C2×C4—104no—
598C2×C4—104no—
599C2×C4—102no—
600C2×C4—204no—
601C2×C4—208no—
602C2×C4—208no—
603C2×C4—2016no—
604C2×C4#4085116yes—
605C2×C4C2×D4×D514yes—
606C2×C4C2×#71114yes—
607C2×C4C22×C22⋊D518yes—
608C2×C4C22×C22⋊D514yes—
609C5C2×#253811yescharacteristic, complemented, Sylow
610C22—24no—
611C22—24no—
612C22—24no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—416no—
617C22—42no—
618C22—104no—
619C22—104no—
620C22—104no—
621C22—202no—
622C22—202no—
623C22—202no—
624C22—2016no—
625C22—202no—
626C22D5×C22⋊Q814yes—
627C22#6596314yes—
628C22#6596914yes—
629C22#6597514yes—
630C22C2×#408514yes—
631C22Q8×C22⋊D514yes—
632C22#6616814yes—
633C22C22×D4×D511yescharacteristic
634C22C22×#71111yescharacteristic
635C22C22×Q8×D511yescharacteristic
636C22C2×#414711yescharacteristic
637C22C2×D5×#4111yescharacteristic
638C22C2×#414911yescharacteristic
639C22C23×C22⋊D511yescharacteristic
640C4—28no—
641C4—44no—
642C4—42no—
643C4—58no—
644C4—104no—
645C4—204no—
646C4—204no—
647C4—202no—
648C4C2×#408518yes—
649C2—44no—
650C2—204no—
651C2#1061450918yesdirect product factor
652C2C2×#6596911yescharacteristic
653C2C2×#6597511yescharacteristic
654C2C2×D5×C22⋊Q811yescharacteristic
655C2C2×#6596311yescharacteristic
656C2C22×#408511yescharacteristic
657C2C2×Q8×C22⋊D511yescharacteristic
658C2C2×#6616811yescharacteristic
subgroupG/H
centerC24C23×D5
commutator, cosocleC23×C5C25
FrattiniC23C24×D5
FittingC22×C5×C22⋊Q8C2
socleC24×C5C24
prime radicalC23×C22⋊D5C22
hypercenterC22×C22⋊Q8D5
hypocenterC5C2×#2538
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×C5×C22⋊Q8C2semidirect
2. C22×#4092C2semidirect
3. C22×Q8×Dic5C2semidirect
5. C2×#65501C2semidirect
6. #10614509C2direct (N × H)
7. C2×#65283C2semidirect
8. C2×#65276C2semidirect
9. C2×#65261C2semidirect
10. C2×#65265C2semidirect
11. C2×#65237C2semidirect
18. C2×C5×C22⋊Q8C22semidirect
25. C2×#4092C22semidirect
27. C2×Q8×Dic5C22semidirect
42. #65501C22semidirect
45. #65283C22semidirect
46. #65276C22semidirect
47. #65265C22semidirect
48. #65261C22semidirect
50. #65237C22semidirect
56. C22×C5×Q8D4semidirect
68. C22×C22⋊D5Q8semidirect
120. C22×C22⋊Q8D5semidirect
151. C2×C5×Q8C2×D4semidirect
175. C2×C22⋊D5C2×Q8semidirect
224. C2×C22⋊Q8C2×D5semidirect
325. C2×Dic5C22⋊Q8semidirect
335. C22×Q8C22⋊D5semidirect
436. Dic5C2×C22⋊Q8semidirect
457. C2×Q8C2×C22⋊D5semidirect
524. C2×C5#2538semidirect
609. C5C2×#2538semidirect
651. C2#10614509direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C5×C22⋊Q82
C22×#40922
C22×Q8×Dic52
C22×C4×C22⋊D52
C2×#655012
#106145092
C2×#652832
C2×#652762
C2×#652612
C2×#652652
C2×#652372
C2×#652462
C2×#25385