Finite Group Databasebeta

← C2×#10600750 #12100415

Subgroups — C2×#10600750 #12100415

Order 1280.

2438 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C22⋊D5C211yesmaximal, characteristic
2C23×C8×D5C211yesmaximal, characteristic, complemented
3C22×C5×C22⋊C8C211yesmaximal, characteristic, complemented
4C22×C22⋊#168C211yesmaximal, characteristic, complemented
5C22×#3952C211yesmaximal, characteristic, complemented
6C2×#64770C211yesmaximal, characteristic, complemented
7C2×#64770C211yesmaximal, characteristic, complemented
8#10600750C218yesmaximal, direct product factor
9C24×C4×C5C2211yescharacteristic
10C23×C22⋊D5C411yescharacteristic
11C23×C4×D5C2211yescharacteristic
12C23×C5×C8C2211yescharacteristic
13C23×C5×C8C2211yescharacteristic
14C22×C22⋊Dic5C411yescharacteristic
15C2×C4×C22⋊D5—24no—
16C2×C4×C22⋊D5—24no—
17C2×C4×C22⋊D5C414yes—
18C2×C4×C22⋊D5C2214yes—
19C22×#685C411yescharacteristic
20C22×#683C411yescharacteristic
21C22×C4×Dic5C2211yescharacteristic
22C23×#168C2211yescharacteristic
23C23×#168C2211yescharacteristic
24C22×C8×D5—24no—
25C22×C8×D5—24no—
26C22×C8×D5—24no—
27C22×C8×D5C2214yescomplemented
28C2×C5×C22⋊C8—24no—
29C2×C5×C22⋊C8C2214yescomplemented
30C2×C22⋊#168—24no—
31C2×C22⋊#168C2214yescomplemented
32C2×#3952—24no—
33C2×#3952C2214yescomplemented
34#64770C2214yescomplemented
35#64770C2214yescomplemented
36C2×#1869—51nomaximal, Sylow
37C25×C5C2×C411yescharacteristic
38C24×D5C2×C411yescharacteristic
39C23×C4×C5—22no—
40C23×C4×C5—22no—
41C23×C4×C5—22no—
42C23×C4×C5—22no—
43C23×C4×C5—22no—
44C23×C4×C5—22no—
45C23×C4×C5C2×C411yescharacteristic
46C23×C4×C5C2×C412yes—
47C23×C4×C5C2312yes—
48C23×C4×C5C2311yescharacteristic
49C22×C22⋊D5—24no—
50C22×C22⋊D5—24no—
51C22×C22⋊D5C818yescomplemented
52C22×C22⋊D5C2×C414yes—
53C23×Dic5C2×C411yescharacteristic
54C23×Dic5C2×C411yescharacteristic
55C23×Dic5C2×C411yescharacteristic
56C22×C4×D5—22no—
57C22×C4×D5—22no—
58C22×C4×D5—22no—
59C22×C4×D5—22no—
60C22×C4×D5C2×C412yes—
61C22×C4×D5D414yes—
62C22×C4×D5D414yes—
63C22×C4×D5C2312yes—
64C22×C5×C8—22no—
65C22×C5×C8—22no—
66C22×C5×C8—22no—
67C22×C5×C8—22no—
68C22×C5×C8—22no—
69C22×C5×C8D414yescomplemented
70C22×C5×C8C2312yes—
71C22×C5×C8C2312yes—
72C2×C22⋊Dic5—24no—
73C2×C22⋊Dic5C2×C414yes—
74C4×C22⋊D5—216no—
75C4×C22⋊D5—28no—
76C4×C22⋊D5—28no—
77C2×#685—24no—
78C2×#685C2×C414yes—
79C2×#683—24no—
80C2×#683C2×C414yes—
81C2×C4×Dic5—22no—
82C2×C4×Dic5—22no—
83C2×C4×Dic5C2×C412yes—
84C2×C4×Dic5C2312yes—
85C22×#168—22no—
86C22×#168—22no—
87C22×#168—22no—
88C22×#168—22no—
89C22×#168—22no—
90C22×#168D414yescomplemented
91C22×#168C2312yes—
92C22×#168C2312yes—
93C2×C8×D5—232no—
94C2×C8×D5—28no—
95C2×C8×D5—28no—
96C2×C8×D5—28no—
97C5×C22⋊C8—28no—
98C22⋊#168—28no—
99#3952—28no—
100C24×C8—51no—
101C22×C4×D4—51no—
102C22×C22⋊C8—51no—
103C22×C22⋊C8—51no—
104C22×C22⋊C8D511yescharacteristic, complemented
105#1869—58no—
106C2×#336—51no—
107C2×#336—51no—
108C24×C5—22no—
109C24×C5—22no—
110C24×C5—22no—
111C24×C5—22no—
112C24×C5—22no—
113C24×C5—22no—
114C24×C5C2×C814yescomplemented
115C24×C5C22×C412yes—
116C24×C5C22×C411yescharacteristic
117C23×D5—22no—
118C23×D5—22no—
119C23×D5—22no—
120C23×D5—22no—
121C23×D5C22⋊C414yes—
122C23×D5C2×C814yes—
123C23×D5M1614yes—
124C23×D5C22×C412yes—
125C22×C4×C5—22no—
126C22×C4×C5—22no—
127C22×C4×C5—22no—
128C22×C4×C5—24no—
129C22×C4×C5—24no—
130C22×C4×C5—24no—
131C22×C4×C5—22no—
132C22×C4×C5—22no—
133C22×C4×C5—22no—
134C22×C4×C5—22no—
135C22×C4×C5—22no—
136C22×C4×C5—22no—
137C22×C4×C5—44no—
138C22×C4×C5—44no—
139C22×C4×C5—44no—
140C22×C4×C5—44no—
141C22×C4×C5C22×C412yes—
142C22×C4×C5C22×C411yescharacteristic
143C22×C4×C5C2×D412yes—
144C22×C4×C5C2×D412yes—
145C22×C4×C5C2×D412yes—
146C22×C4×C5C2×D412yes—
147C22×C4×C5#4112yes—
148C22×C4×C5#4112yes—
149C22×C4×C5C2411yescharacteristic
150C2×C22⋊D5—232no—
151C2×C22⋊D5—28no—
152C2×C22⋊D5—28no—
153C2×C22⋊D5C2×C8116yescomplemented
154C22×Dic5—22no—
155C22×Dic5—22no—
156C22×Dic5—22no—
157C22×Dic5—22no—
158C22×Dic5—22no—
159C22×Dic5—22no—
160C22×Dic5—22no—
161C22×Dic5C22⋊C414yes—
162C22×Dic5C2×C814yescomplemented
163C22×Dic5C22×C412yes—
164C22×Dic5C22×C412yes—
165C22×Dic5C22×C412yes—
166C2×C4×D5—24no—
167C2×C4×D5—24no—
168C2×C4×D5—28no—
169C2×C4×D5—28no—
170C2×C4×D5—28no—
171C2×C4×D5—28no—
172C2×C4×D5—22no—
173C2×C4×D5—22no—
174C2×C4×D5—22no—
175C2×C4×D5—22no—
176C2×C4×D5C22⋊C418yes—
177C2×C4×D5C2×D414yes—
178C2×C4×D5C2×D414yes—
179C2×C5×C8—28no—
180C2×C5×C8—22no—
181C2×C5×C8—22no—
182C2×C5×C8—22no—
183C2×C5×C8—22no—
184C2×C5×C8—22no—
185C2×C5×C8—44no—
186C2×C5×C8C2×D414yescomplemented
187C22⋊Dic5—28no—
188#685—28no—
189#683—28no—
190C4×Dic5—24no—
191C4×Dic5—22no—
192C4×Dic5—22no—
193C2×#168—28no—
194C2×#168—22no—
195C2×#168—22no—
196C2×#168—22no—
197C2×#168—22no—
198C2×#168—22no—
199C2×#168—44no—
200C2×#168C2×D414yescomplemented
201C8×D5—232no—
202C23×D4—51no—
203C24×C4—51no—
204C24×C4C2×D511yescharacteristic
205C23×C8—51no—
206C23×C8—51no—
207C23×C8—54no—
208C23×C8—104no—
209C23×C8—104no—
210C23×C8—104no—
211C23×C8C2×D511yescharacteristic
212C23×C8C2×D511yescharacteristic
213C2×C4×D4—54no—
214C2×C4×D4—54no—
215C2×C4×D4—104no—
216C2×C4×D4—104no—
217C22×C4⋊C4—51no—
218C22×C22⋊C4—51no—
219C22×C22⋊C4—51no—
220C22×C42—51no—
221C2×C22⋊C8—24no—
222C2×C22⋊C8—54no—
223C2×C22⋊C8—54no—
224C2×C22⋊C8—104no—
225C2×C22⋊C8—104no—
226C2×C22⋊C8C2×D514yescomplemented
227#336—54no—
228#336—54no—
229C23×C5—22no—
230C23×C5—22no—
231C23×C5—22no—
232C23×C5—22no—
233C23×C5—22no—
234C23×C5—22no—
235C23×C5—28no—
236C23×C5—28no—
237C23×C5—28no—
238C23×C5—44no—
239C23×C5—44no—
240C23×C5—44no—
241C23×C5—44no—
242C23×C5C2×C22⋊C412yes—
243C23×C5C4×D412yes—
244C23×C5C4×D412yes—
245C23×C5C22×C812yes—
246C23×C5C22×C814yescomplemented
247C23×C5C2×M1612yes—
248C23×C5#13112yes—
249C23×C5#13112yes—
250C23×C5C23×C411yescharacteristic
251C22×D5—22no—
252C22×D5—22no—
253C22×D5—22no—
254C22×D5—22no—
255C22×D5—28no—
256C22×D5—28no—
257C22×D5—28no—
258C22×D5—28no—
259C22×D5—28no—
260C22×D5—28no—
261C22×D5C22⋊C8116yescomplemented
262C22×D5C2×C22⋊C414yes—
263C22×D5C22×C814yes—
264C22×D5C2×M1614yes—
265C2×C4×C5—22no—
266C2×C4×C5—22no—
267C2×C4×C5—22no—
268C2×C4×C5—44no—
269C2×C4×C5—42no—
270C2×C4×C5—42no—
271C2×C4×C5—42no—
272C2×C4×C5—42no—
273C2×C4×C5—48no—
274C2×C4×C5C2×C22⋊C412yes—
275C2×C4×C5C4×D412yes—
276C2×C4×C5C4×D412yes—
277C2×C4×C5#12014yes—
278C2×C4×C5#12114yes—
279C2×C4×C5#12114yes—
280C2×C4×C5#12314yes—
281C2×C4×C5C22×D411yescharacteristic
282C2×C4×C5C22×D411yescharacteristic
283C2×C4×C5C22×D411yescharacteristic
284C2×C4×C5C22×D411yescharacteristic
285C2×C4×C5C2×#4111yescharacteristic
286C2×C4×C5C2×#4111yescharacteristic
287C22⋊D5—232no—
288C2×Dic5—22no—
289C2×Dic5—22no—
290C2×Dic5—22no—
291C2×Dic5—22no—
292C2×Dic5—28no—
293C2×Dic5—28no—
294C2×Dic5—22no—
295C2×Dic5—22no—
296C2×Dic5—22no—
297C2×Dic5—44no—
298C2×Dic5C2×C22⋊C414yes—
299C2×Dic5C22×C814yescomplemented
300C4×D5—24no—
301C4×D5—24no—
302C4×D5—24no—
303C4×D5—24no—
304C4×D5—216no—
305C5×C8—24no—
306C5×C8—42no—
307#168—24no—
308#168—42no—
309C25—51no—
310C25C4×D511yescharacteristic
311C22×D4—58no—
312C22×D4—54no—
313C22×D4—104no—
314C22×D4—104no—
315C23×C4—22no—
316C23×C4—22no—
317C23×C4—22no—
318C23×C4—22no—
319C23×C4—22no—
320C23×C4—22no—
321C23×C4—51no—
322C23×C4—51no—
323C23×C4—52no—
324C23×C4—54no—
325C23×C4—54no—
326C23×C4—52no—
327C23×C4—51no—
328C23×C4—102no—
329C23×C4—102no—
330C23×C4—102no—
331C23×C4—102no—
332C23×C4C4×D511yescharacteristic
333C23×C4C4×D512yes—
334C23×C4C22×D512yes—
335C23×C4C22×D511yescharacteristic
336C22×C8—22no—
337C22×C8—22no—
338C22×C8—22no—
339C22×C8—22no—
340C22×C8—22no—
341C22×C8—54no—
342C22×C8—52no—
343C22×C8—52no—
344C22×C8—108no—
345C22×C8—108no—
346C22×C8—1032no—
347C22×C8—102no—
348C22×C8—102no—
349C22×C8—108no—
350C22×C8—102no—
351C22×C8—102no—
352C22×C8—102no—
353C22×C8C22⋊D514yescomplemented
354C22×C8C22×D512yes—
355C22×C8C22×D512yes—
356C4×D4—108no—
357C4×D4—108no—
358C4×D4—1016no—
359C2×C4⋊C4—54no—
360C2×C4⋊C4—104no—
361C2×C22⋊C4—54no—
362C2×C22⋊C4—54no—
363C2×C22⋊C4—104no—
364C2×C22⋊C4—104no—
365C2×C42—52no—
366C2×C42—52no—
367C2×C42—102no—
368C2×C42—102no—
369C22⋊C8—28no—
370C22⋊C8—108no—
371C22⋊C8—108no—
372C22×C5—24no—
373C22×C5—24no—
374C22×C5—24no—
375C22×C5—42no—
376C22×C5—416no—
377C22×C5—42no—
378C22×C5—42no—
379C22×C5—42no—
380C22×C5#38614yes—
381C22×C5C2×C22⋊C814yes—
382C22×C5#40814yes—
383C22×C5C8×D414yes—
384C22×C5C8×D414yes—
385C22×C5#43514yes—
386C22×C5#43514yes—
387C22×C5C22×C22⋊C411yescharacteristic
388C22×C5C2×C4×D411yescharacteristic
389C22×C5C2×C4×D411yescharacteristic
390C22×C5C23×C811yescharacteristic
391C22×C5C22×M1611yescharacteristic
392C22×C5C2×#13111yescharacteristic
393C22×C5C2×#13111yescharacteristic
394C2×D5—24no—
395C2×D5—232no—
396C2×D5—24no—
397C2×D5—24no—
398C2×D5—24no—
399C2×D5—24no—
400C2×D5—24no—
401C2×D5C2×C22⋊C818yescomplemented
402C4×C5—42no—
403C4×C5#38614yes—
404C4×C5C2×#12011yescharacteristic
405C4×C5C2×#12111yescharacteristic
406C4×C5C2×#12111yescharacteristic
407C4×C5C2×#12311yescharacteristic
408Dic5—24no—
409Dic5—24no—
410Dic5—42no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—22no—
416C24—22no—
417C24—52no—
418C24—54no—
419C24—54no—
420C24—54no—
421C24—102no—
422C24—102no—
423C24—102no—
424C24—102no—
425C24C8×D514yescomplemented
426C24C2×C4×D512yes—
427C24C2×C4×D511yescharacteristic
428C2×D4—516no—
429C2×D4—108no—
430C2×D4—108no—
431C2×D4—1032no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—52no—
449C22×C4—52no—
450C22×C4—54no—
451C22×C4—54no—
452C22×C4—54no—
453C22×C4—54no—
454C22×C4—58no—
455C22×C4—52no—
456C22×C4—102no—
457C22×C4—102no—
458C22×C4—102no—
459C22×C4—102no—
460C22×C4—102no—
461C22×C4—102no—
462C22×C4—102no—
463C22×C4—102no—
464C22×C4—104no—
465C22×C4—108no—
466C22×C4—108no—
467C22×C4—104no—
468C22×C4—108no—
469C22×C4—108no—
470C22×C4—102no—
471C22×C4—102no—
472C22×C4—102no—
473C22×C4C2×C4×D512yes—
474C22×C4C2×C4×D511yescharacteristic
475C22×C4#70912yes—
476C22×C4D4×D512yes—
477C22×C4D4×D512yes—
478C22×C4D4×D512yes—
479C22×C4#71112yes—
480C22×C4C2×C22⋊D512yes—
481C22×C4C23×D511yescharacteristic
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—28no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—44no—
489C2×C8—54no—
490C2×C8—1032no—
491C2×C8—102no—
492C2×C8—102no—
493C2×C8—108no—
494C2×C8—102no—
495C2×C8—102no—
496C2×C8—102no—
497C2×C8—204no—
498C2×C8C2×C22⋊D514yescomplemented
499C4⋊C4—108no—
500C22⋊C4—108no—
501C22⋊C4—108no—
502C42—102no—
503C42—102no—
504C42—104no—
505C2×C5—44no—
506C2×C5#186918yescomplemented
507C2×C5C2×#38611yescharacteristic
508C2×C5C22×C22⋊C811yescharacteristic
509C2×C5C2×#40811yescharacteristic
510C2×C5C2×C8×D411yescharacteristic
511C2×C5C2×C8×D411yescharacteristic
512C2×C5C2×#43511yescharacteristic
513C2×C5C2×#43511yescharacteristic
514D5—28no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—54no—
529C23—54no—
530C23—54no—
531C23—516no—
532C23—102no—
533C23—102no—
534C23—102no—
535C23—102no—
536C23—108no—
537C23—108no—
538C23—108no—
539C23—108no—
540C23—108no—
541C23—108no—
542C23D5×C22⋊C412yes—
543C23#402712yes—
544C23C2×C8×D512yes—
545C23C2×C8×D514yescomplemented
546C23#404712yes—
547C23D5×M1612yes—
548C23#405312yes—
549C23C4×C22⋊D512yes—
550C23C22×C4×D511yescharacteristic
551D4—1032no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—48no—
561C2×C4—54no—
562C2×C4—54no—
563C2×C4—102no—
564C2×C4—102no—
565C2×C4—102no—
566C2×C4—102no—
567C2×C4—108no—
568C2×C4—108no—
569C2×C4—104no—
570C2×C4—104no—
571C2×C4—104no—
572C2×C4—104no—
573C2×C4—1016no—
574C2×C4—102no—
575C2×C4—102no—
576C2×C4—102no—
577C2×C4—204no—
578C2×C4D5×C22⋊C412yes—
579C2×C4#402712yes—
580C2×C4#402914yes—
581C2×C4#403014yes—
582C2×C4C4×C22⋊D512yes—
583C2×C4#408314yes—
584C2×C4#408414yes—
585C2×C4C2×#70911yescharacteristic
586C2×C4C2×D4×D511yescharacteristic
587C2×C4C2×D4×D511yescharacteristic
588C2×C4C2×D4×D511yescharacteristic
589C2×C4C2×#71111yescharacteristic
590C2×C4C22×C22⋊D511yescharacteristic
591C8—24no—
592C8—42no—
593C8—104no—
594C8—202no—
595C5C2×#186911yescharacteristic, complemented, Sylow
596C22—24no—
597C22—24no—
598C22—24no—
599C22—416no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—58no—
605C22—1032no—
606C22—104no—
607C22—104no—
608C22—104no—
609C22—104no—
610C22—104no—
611C22—104no—
612C22D5×C22⋊C814yes—
613C22#6501414yes—
614C22#6501614yes—
615C22#6501714yes—
616C22#6524514yes—
617C22C8×C22⋊D514yes—
618C22#6541614yes—
619C22C2×D5×C22⋊C411yescharacteristic
620C22C2×#402711yescharacteristic
621C22C22×C8×D511yescharacteristic
622C22C2×#404711yescharacteristic
623C22C2×D5×M1611yescharacteristic
624C22C2×#405311yescharacteristic
625C22C2×C4×C22⋊D511yescharacteristic
626C4—42no—
627C4—104no—
628C4—104no—
629C4—202no—
630C4#6524514yes—
631C4C2×#402911yescharacteristic
632C4C2×#403011yescharacteristic
633C4C2×#408311yescharacteristic
634C4C2×#408411yescharacteristic
635C2—44no—
636C2—108no—
637C2#1060075018yesdirect product factor
638C2C2×#6501411yescharacteristic
639C2C2×#6501711yescharacteristic
640C2C2×D5×C22⋊C811yescharacteristic
641C2C2×#6501611yescharacteristic
642C2C2×#6524511yescharacteristic
643C2C2×C8×C22⋊D511yescharacteristic
644C2C2×#6541611yescharacteristic
subgroupG/H
centerC23×C4C22×D5
commutatorC22×C5C23×C8
FrattiniC22×C4C23×D5
FittingC22×C5×C22⋊C8C2
socleC24×C5C22×C4
prime radicalC23×C22⋊D5C4
cosocleC22×C4×C5C24
hypercenterC22×C22⋊C8D5
hypocenterC5C2×#1869
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23×C4 ◁ C22×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C23×C8×D5C2semidirect
3. C22×C5×C22⋊C8C2semidirect
4. C22×C22⋊#168C2semidirect
5. C22×#3952C2semidirect
6. C2×#64770C2semidirect
7. C2×#64770C2semidirect
8. #10600750C2direct (N × H)
27. C22×C8×D5C22semidirect
29. C2×C5×C22⋊C8C22semidirect
31. C2×C22⋊#168C22semidirect
33. C2×#3952C22semidirect
34. #64770C22semidirect
35. #64770C22semidirect
51. C22×C22⋊D5C8semidirect
69. C22×C5×C8D4semidirect
90. C22×#168D4semidirect
104. C22×C22⋊C8D5semidirect
114. C24×C5C2×C8semidirect
153. C2×C22⋊D5C2×C8semidirect
162. C22×Dic5C2×C8semidirect
186. C2×C5×C8C2×D4semidirect
200. C2×#168C2×D4semidirect
226. C2×C22⋊C8C2×D5semidirect
246. C23×C5C22×C8semidirect
261. C22×D5C22⋊C8semidirect
299. C2×Dic5C22×C8semidirect
353. C22×C8C22⋊D5semidirect
401. C2×D5C2×C22⋊C8semidirect
425. C24C8×D5semidirect
498. C2×C8C2×C22⋊D5semidirect
506. C2×C5#1869semidirect
545. C23C2×C8×D5semidirect
595. C5C2×#1869semidirect
637. C2#10600750direct (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
C23×C8×D52
C22×C5×C22⋊C82
C22×C22⋊#1682
C22×#39522
C2×#647702
#106007502
C2×#18695