Finite Group Databasebeta

← C2×#10613901 #12268002

Subgroups — C2×#10613901 #12268002

Order 1280.

3570 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C5×C4⋊C4C211yesmaximal, characteristic, complemented
2C22×#4075C211yesmaximal, characteristic
3C22×#4075C211yesmaximal, characteristic
4C22×#4072C211yesmaximal, characteristic, complemented
5C2×#65502C211yesmaximal, characteristic
6C2×#65499C211yesmaximal, characteristic, complemented
7C2×#65500C211yesmaximal, characteristic, complemented
8C2×#65283C212yesmaximal, complemented
9#10613901C2116yesmaximal, direct product factor
10C2×#65273C212yesmaximal, complemented
11C2×#65261C212yesmaximal, complemented
12C2×#65279C212yesmaximal, complemented
13C24×C4×C5C2211yescharacteristic
14C24×C4×C5C2211yescharacteristic
15C24×C4×C5C2211yescharacteristic
16C23×C22⋊D5C2211yescharacteristic
17C22×C5×C4⋊C4—28no—
18C22×C5×C4⋊C4C2212yes—
19C22×C5×C4⋊C4C2212yes—
20C22×C5×C4⋊C4C2218yescomplemented
21C22×C22⋊Dic5C2211yescharacteristic
22C22×C22⋊Dic5C2211yescharacteristic
23C22×C22⋊Dic5C2211yescharacteristic
24C2×#4075—28no—
25C2×#4075—28no—
26C2×#4075C2218yes—
27C2×#4075C2218yes—
28C22×#685C2211yescharacteristic
29C22×#685C2212yes—
30C22×#685C2212yes—
31C22×#685C2211yescharacteristic
32C2×#4072—28no—
33C2×#4072C2218yescomplemented
34C22×C4⋊Dic5C2212yes—
35C22×#683C2211yescharacteristic
36C22×#683C2212yes—
37C22×#683C2212yes—
38C22×#683C2211yescharacteristic
39C22×C4×Dic5C2212yes—
40#65502C2218yes—
41#65500C2218yescomplemented
42#65499C2218yescomplemented
43C2×#3963C2212yes—
44C2×#3963C2212yes—
45C2×#3963C2212yes—
46C2×#3963C2212yes—
47#65283C22116yescomplemented
48#65279C22116yescomplemented
49#65273C22116yescomplemented
50#65261C22116yescomplemented
51C2×#2413—51nomaximal, Sylow
52C25×C5C2311yescharacteristic
53C24×D5C2311yescharacteristic
54C23×C4×C5—24no—
55C23×C4×C5—24no—
56C23×C4×C5—24no—
57C23×C4×C5—28no—
58C23×C4×C5—24no—
59C23×C4×C5—24no—
60C23×C4×C5—24no—
61C23×C4×C5D418yes—
62C23×C4×C5C2312yes—
63C23×C4×C5C2314yes—
64C23×C4×C5C2314yes—
65C23×C4×C5C2311yescharacteristic
66C23×C4×C5C2311yescharacteristic
67C23×C4×C5C2314yes—
68C23×C4×C5C2312yes—
69C22×C22⋊D5—24no—
70C22×C22⋊D5—24no—
71C22×C22⋊D5—24no—
72C22×C22⋊D5C2314yes—
73C23×Dic5C2312yes—
74C23×Dic5C2311yescharacteristic
75C23×Dic5C2311yescharacteristic
76C23×Dic5C2311yescharacteristic
77C23×Dic5C2312yes—
78C2×C5×C4⋊C4—28no—
79C2×C5×C4⋊C4—28no—
80C2×C5×C4⋊C4—232no—
81C2×C5×C4⋊C4C2318yes—
82C2×C5×C4⋊C4C2318yes—
83C2×C22⋊Dic5—24no—
84C2×C22⋊Dic5—24no—
85C2×C22⋊Dic5D418yes—
86C2×C22⋊Dic5C2314yes—
87C2×C22⋊Dic5C2314yes—
88C2×C22⋊Dic5C2314yes—
89#4075—232no—
90#4075—232no—
91C2×#685—24no—
92C2×#685—24no—
93C2×#685—28no—
94C2×#685—28no—
95C2×#685C2314yes—
96C2×#685C2318yes—
97C2×#685C2318yes—
98C2×#685C2314yes—
99#4072—232no—
100C2×C4⋊Dic5—28no—
101C2×C4⋊Dic5C2318yes—
102C2×#683—24no—
103C2×#683—24no—
104C2×#683—28no—
105C2×#683—28no—
106C2×#683C2318yes—
107C2×#683C2314yes—
108C2×#683C2318yes—
109C2×#683C2314yes—
110C2×C4×Dic5—28no—
111C2×C4×Dic5C2318yes—
112#3963C2318yes—
113#3963C2318yes—
114#3963C2318yes—
115#3963C2318yes—
116C22×#123—51no—
117C22×#123—51no—
118C22×#117—51no—
119C23×C4⋊C4D511yescharacteristic, complemented
120#2413—516no—
121C2×#398—52no—
122C2×#396—52no—
123C2×#389—52no—
124C2×#388—52no—
125C2×#386—51no—
126C2×#385—51no—
127C2×#381—51no—
128C24×C5—24no—
129C24×C5—24no—
130C24×C5C2×D414yes—
131C24×C5C2×D414yes—
132C24×C5#4114yes—
133C24×C5C2412yes—
134C24×C5C2411yescharacteristic
135C23×D5—22no—
136C23×D5—22no—
137C23×D5—22no—
138C23×D5—24no—
139C23×D5—24no—
140C23×D5C2412yes—
141C22×C4×C5—232no—
142C22×C4×C5—232no—
143C22×C4×C5—28no—
144C22×C4×C5—24no—
145C22×C4×C5—22no—
146C22×C4×C5—22no—
147C22×C4×C5—28no—
148C22×C4×C5—24no—
149C22×C4×C5—28no—
150C22×C4×C5—22no—
151C22×C4×C5—22no—
152C22×C4×C5—216no—
153C22×C4×C5—28no—
154C22×C4×C5—24no—
155C22×C4×C5—24no—
156C22×C4×C5—28no—
157C22×C4×C5—28no—
158C22×C4×C5—24no—
159C22×C4×C5—416no—
160C22×C4×C5C2×D414yes—
161C22×C4×C5C2×D414yes—
162C22×C4×C5C2×D4116yes—
163C22×C4×C5#4118yes—
164C22×C4×C5C2414yes—
165C22×C4×C5C2412yes—
166C22×C4×C5C2412yes—
167C22×C4×C5C2414yes—
168C2×C22⋊D5—28no—
169C2×C22⋊D5—28no—
170C2×C22⋊D5—28no—
171C2×C22⋊D5—416no—
172C22×Dic5—24no—
173C22×Dic5—24no—
174C22×Dic5—24no—
175C22×Dic5—22no—
176C22×Dic5—22no—
177C22×Dic5—22no—
178C22×Dic5—22no—
179C22×Dic5—22no—
180C22×Dic5—24no—
181C22×Dic5—24no—
182C22×Dic5—24no—
183C22×Dic5—24no—
184C22×Dic5C2×D418yes—
185C22×Dic5C2412yes—
186C22×Dic5C2414yes—
187C22×Dic5C2412yes—
188C22×Dic5C2412yes—
189C22×Dic5C2414yes—
190C5×C4⋊C4—216no—
191C5×C4⋊C4—216no—
192C22⋊Dic5—28no—
193C22⋊Dic5—28no—
194C22⋊Dic5C2×D4116yes—
195#685—28no—
196#685—28no—
197#685—216no—
198#685—216no—
199C4⋊Dic5—216no—
200#683—28no—
201#683—28no—
202#683—216no—
203#683—216no—
204C4×Dic5—216no—
205C23×D4—51no—
206C24×C4C2×D511yescharacteristic
207C24×C4C2×D511yescharacteristic
208C24×C4C2×D511yescharacteristic
209C2×#123—58no—
210C2×#123—58no—
211C2×#123—108no—
212C2×#123—108no—
213C2×#117—58no—
214C2×#117—108no—
215C22×C4⋊C4—28no—
216C22×C4⋊C4—52no—
217C22×C4⋊C4—51no—
218C22×C4⋊C4—52no—
219C22×C4⋊C4—51no—
220C22×C4⋊C4—52no—
221C22×C4⋊C4C2×D512yes—
222C22×C4⋊C4C2×D512yes—
223C22×C4⋊C4C2×D518yescomplemented
224C22×C22⋊C4—51no—
225C22×C22⋊C4—51no—
226C22×C22⋊C4—51no—
227C22×C22⋊C4—51no—
228C22×C22⋊C4—52no—
229C22×C22⋊C4—52no—
230C22×C22⋊C4—51no—
231C22×C42—52no—
232#398—516no—
233#396—516no—
234#389—516no—
235#388—516no—
236#386—58no—
237#385—58no—
238#381—58no—
239C2×#95—52no—
240C2×#95—52no—
241C2×#95—52no—
242C2×#95—52no—
243C23×C5—28no—
244C23×C5—28no—
245C23×C5—216no—
246C23×C5—216no—
247C23×C5—24no—
248C23×C5—24no—
249C23×C5#121116yes—
250C23×C5C22×D414yes—
251C23×C5C22×D412yes—
252C23×C5C22×D414yes—
253C23×C5C22×D412yes—
254C23×C5C2×#4114yes—
255C23×C5C2×#4114yes—
256C23×C5C2×#4112yes—
257C23×C52−1+414yes—
258C23×C5C2511yescharacteristic
259C22×D5—22no—
260C22×D5—22no—
261C22×D5—24no—
262C22×D5—22no—
263C22×D5—24no—
264C22×D5—44no—
265C22×D5—44no—
266C22×D5—44no—
267C22×D5—48no—
268C22×D5—48no—
269C2×C4×C5—232no—
270C2×C4×C5—232no—
271C2×C4×C5—28no—
272C2×C4×C5—28no—
273C2×C4×C5—216no—
274C2×C4×C5—22no—
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—416no—
284C2×C4×C5—48no—
285C2×C4×C5C22×D414yes—
286C2×C4×C5C22×D414yes—
287C2×C4×C5C2×#4118yes—
288C22⋊D5—416no—
289C2×Dic5—216no—
290C2×Dic5—24no—
291C2×Dic5—24no—
292C2×Dic5—24no—
293C2×Dic5—22no—
294C2×Dic5—22no—
295C2×Dic5—22no—
296C2×Dic5—22no—
297C2×Dic5—24no—
298C2×Dic5—24no—
299C2×Dic5—22no—
300C2×Dic5—24no—
301C2×Dic5—24no—
302C2×Dic5—44no—
303C2×Dic5—44no—
304C2×Dic5—44no—
305C2×Dic5—48no—
306C2×Dic5C22×D418yes—
307C25—51no—
308C25C22×D511yescharacteristic
309C22×D4—54no—
310C22×D4—104no—
311C22×D4—104no—
312C22×D4—104no—
313C23×C4—24no—
314C23×C4—24no—
315C23×C4—24no—
316C23×C4—28no—
317C23×C4—24no—
318C23×C4—24no—
319C23×C4—24no—
320C23×C4—52no—
321C23×C4—51no—
322C23×C4—51no—
323C23×C4—52no—
324C23×C4—51no—
325C23×C4C22⋊D518yes—
326C23×C4C22×D514yes—
327C23×C4C22×D514yes—
328C23×C4C22×D512yes—
329C23×C4C22×D512yes—
330C23×C4C22×D514yes—
331C23×C4C22×D511yescharacteristic
332C23×C4C22×D511yescharacteristic
333#123—1032no—
334#123—1032no—
335#117—1032no—
336C2×C4⋊C4—28no—
337C2×C4⋊C4—28no—
338C2×C4⋊C4—232no—
339C2×C4⋊C4—58no—
340C2×C4⋊C4—54no—
341C2×C4⋊C4—58no—
342C2×C4⋊C4—54no—
343C2×C4⋊C4—58no—
344C2×C4⋊C4—108no—
345C2×C4⋊C4—108no—
346C2×C4⋊C4—108no—
347C2×C4⋊C4—104no—
348C2×C4⋊C4—104no—
349C2×C4⋊C4C22×D518yes—
350C2×C4⋊C4C22×D518yes—
351C2×C22⋊C4—58no—
352C2×C22⋊C4—54no—
353C2×C22⋊C4—54no—
354C2×C22⋊C4—54no—
355C2×C22⋊C4—54no—
356C2×C22⋊C4—58no—
357C2×C22⋊C4—54no—
358C2×C22⋊C4—58no—
359C2×C22⋊C4—108no—
360C2×C22⋊C4—104no—
361C2×C22⋊C4—104no—
362C2×C22⋊C4—108no—
363C2×C22⋊C4—104no—
364C2×C22⋊C4—104no—
365C2×C42—58no—
366C2×C42—108no—
367#95—58no—
368#95—58no—
369#95—58no—
370#95—58no—
371C22×C5—24no—
372C22×C5—232no—
373C22×C5—24no—
374C22×C5—28no—
375C22×C5—28no—
376C22×C5C2×#12118yes—
377C22×C5C2×#12114yes—
378C22×C5#52514yes—
379C22×C5#53614yes—
380C22×C5#54718yes—
381C22×C5#55218yes—
382C22×C5C23×D411yescharacteristic
383C22×C5C23×D411yescharacteristic
384C22×C5C22×#4112yes—
385C22×C5C22×#4111yescharacteristic
386C22×C5C2×2−1+412yes—
387C2×D5—42no—
388C2×D5—42no—
389C2×D5—42no—
390C2×D5—44no—
391C2×D5—44no—
392C2×D5—88no—
393C4×C5—24no—
394C4×C5—24no—
395C4×C5—28no—
396C4×C5—44no—
397Dic5—28no—
398Dic5—42no—
399Dic5—42no—
400Dic5—42no—
401Dic5—44no—
402C24—24no—
403C24—24no—
404C24—52no—
405C24—104no—
406C24—102no—
407C24—102no—
408C24—104no—
409C24—102no—
410C24D4×D514yes—
411C24#71114yes—
412C24C2×C22⋊D514yes—
413C24C23×D512yes—
414C24C23×D511yescharacteristic
415C2×D4—108no—
416C2×D4—108no—
417C2×D4—108no—
418C2×D4—2016no—
419C22×C4—232no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—28no—
428C22×C4—24no—
429C22×C4—232no—
430C22×C4—216no—
431C22×C4—24no—
432C22×C4—28no—
433C22×C4—28no—
434C22×C4—24no—
435C22×C4—28no—
436C22×C4—24no—
437C22×C4—416no—
438C22×C4—58no—
439C22×C4—54no—
440C22×C4—52no—
441C22×C4—54no—
442C22×C4—52no—
443C22×C4—52no—
444C22×C4—104no—
445C22×C4—104no—
446C22×C4—104no—
447C22×C4—104no—
448C22×C4—104no—
449C22×C4—102no—
450C22×C4—102no—
451C22×C4—104no—
452C22×C4—104no—
453C22×C4—102no—
454C22×C4—102no—
455C22×C4—102no—
456C22×C4#70918yes—
457C22×C4C2×C22⋊D514yes—
458C22×C4C2×C22⋊D514yes—
459C22×C4C2×C22⋊D5116yes—
460C22×C4C23×D514yes—
461C22×C4C23×D514yes—
462C22×C4C23×D512yes—
463C22×C4C23×D512yes—
464C4⋊C4—216no—
465C4⋊C4—216no—
466C4⋊C4—1016no—
467C4⋊C4—1016no—
468C4⋊C4—1016no—
469C4⋊C4—108no—
470C4⋊C4—108no—
471C22⋊C4—516no—
472C22⋊C4—1016no—
473C22⋊C4—108no—
474C22⋊C4—108no—
475C22⋊C4—1016no—
476C22⋊C4—108no—
477C22⋊C4—108no—
478C42—1016no—
479C2×C5—28no—
480C2×C5#241318yescomplemented
481C2×C5C22×#12111yescharacteristic
482C2×C5C2×#52511yescharacteristic
483C2×C5C2×#53611yescharacteristic
484C2×C5C2×#54712yes—
485C2×C5C2×#55212yes—
486D5—82no—
487C23—216no—
488C23—216no—
489C23—28no—
490C23—28no—
491C23—24no—
492C23—24no—
493C23—104no—
494C23—102no—
495C23—102no—
496C23—104no—
497C23—102no—
498C23—208no—
499C23—208no—
500C23—204no—
501C23—204no—
502C23—204no—
503C23#4085116yes—
504C23C2×#70914yes—
505C23C2×D4×D514yes—
506C23C2×D4×D512yes—
507C23C2×#71114yes—
508C23C2×#71112yes—
509C23#414714yes—
510C23C22×C22⋊D514yes—
511C23C22×C22⋊D512yes—
512C23C24×D511yescharacteristic
513D4—2016no—
514C2×C4—232no—
515C2×C4—216no—
516C2×C4—232no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—416no—
529C2×C4—48no—
530C2×C4—58no—
531C2×C4—1016no—
532C2×C4—104no—
533C2×C4—104no—
534C2×C4—104no—
535C2×C4—104no—
536C2×C4—104no—
537C2×C4—102no—
538C2×C4—102no—
539C2×C4—104no—
540C2×C4—104no—
541C2×C4—102no—
542C2×C4—102no—
543C2×C4—102no—
544C2×C4—208no—
545C2×C4—204no—
546C2×C4—204no—
547C2×C4—204no—
548C2×C4C2×#70918yes—
549C2×C4C22×C22⋊D514yes—
550C2×C4C22×C22⋊D514yes—
551C5C2×#241311yescharacteristic, complemented, Sylow
552C22—28no—
553C22—24no—
554C22—28no—
555C22—232no—
556C22—24no—
557C22—204no—
558C22—202no—
559C22—204no—
560C22—202no—
561C22—202no—
562C22—408no—
563C22#6584118yes—
564C22#6584718yes—
565C22#6612514yes—
566C22C2×#408518yes—
567C22C2×#408514yes—
568C22#6615014yes—
569C22C22×#70912yes—
570C22C22×D4×D511yescharacteristic
571C22C22×#71111yescharacteristic
572C22C2×#414712yes—
573C22C23×C22⋊D511yescharacteristic
574C4—28no—
575C4—24no—
576C4—24no—
577C4—44no—
578C4—108no—
579C4—202no—
580C4—204no—
581C4—202no—
582C4—202no—
583C2—28no—
584C2—402no—
585C2#1061390118yesdirect product factor
586C2C2×#6584712yes—
587C2C2×#6584112yes—
588C2C2×#6612511yescharacteristic
589C2C22×#408511yescharacteristic
590C2C2×#6615011yescharacteristic
subgroupG/H
centerC24C23×D5
commutator, cosocleC23×C5C25
FrattiniC23C24×D5
FittingC23×C5×C4⋊C4C2
socleC24×C5C24
prime radicalC23×C22⋊D5C22
hypercenterC23×C4⋊C4D5
hypocenterC5C2×#2413
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C23×C5×C4⋊C4C2semidirect
4. C22×#4072C2semidirect
6. C2×#65499C2semidirect
7. C2×#65500C2semidirect
8. C2×#65283C2semidirect
9. #10613901C2direct (N × H)
10. C2×#65273C2semidirect
11. C2×#65261C2semidirect
12. C2×#65279C2semidirect
20. C22×C5×C4⋊C4C22semidirect
33. C2×#4072C22semidirect
41. #65500C22semidirect
42. #65499C22semidirect
47. #65283C22semidirect
48. #65279C22semidirect
49. #65273C22semidirect
50. #65261C22semidirect
119. C23×C4⋊C4D5semidirect
223. C22×C4⋊C4C2×D5semidirect
480. C2×C5#2413semidirect
551. C5C2×#2413semidirect
585. C2#10613901direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C5×C4⋊C42
C22×#40752
C22×#40722
C2×#655022
C2×#654992
C2×#655002
C2×#652832
#106139012
C2×#652732
C2×#652612
C2×#652792
C2×#24135