Finite Group Databasebeta

← C2×A4×#685 #14514179

Subgroups — C2×A4×#685 #14514179

Order 1920.

1984 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1984 conjugacy classes of subgroups — too many to draw. Only its 171 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×D5×A4C211yesmaximal, characteristic
2C22×C4×C5×A4C211yesmaximal, characteristic, complemented
3C22×A4×Dic5C211yesmaximal, characteristic, complemented
4A4×#685C214yesmaximal, direct product factor
5C23×#685C311yesmaximal, characteristic, complemented, Hall
6C23×C5×A4C2211yescharacteristic
7C22×D5×A4—22no—
8C22×D5×A4—22no—
9C22×D5×A4C414yescomplemented
10C22×D5×A4C2212yes—
11C2×C4×C5×A4—22no—
12C2×C4×C5×A4C2212yescomplemented
13C2×A4×Dic5—22no—
14C2×A4×Dic5C2212yescomplemented
15C2×C3×#685—41nomaximal
16C2×A4×C22⋊C4—51nomaximal, Hall
17C25×D5C2×C311yescharacteristic
18C24×C4×C5C2×C311yescharacteristic, complemented
19C24×Dic5C2×C311yescharacteristic, complemented
20C22×#685—31no—
21C22×#685—31no—
22C22×#685—31no—
23C22×#685—34no—
24C22×#685—31no—
25C22×#685C2×C314yescomplemented
26C23×C3×D5—41no—
27C22×C5×A4C2×C412yes—
28C22×C5×A4D412yes—
29C22×C5×A4D412yes—
30C22×C5×A4C2311yescharacteristic
31C2×D5×A4—28no—
32C2×D5×A4—22no—
33C2×D5×A4—22no—
34C2×D5×A4C2×C414yescomplemented
35C22×C3×C4×C5—41no—
36C22×C3×Dic5—41no—
37C4×C5×A4—22no—
38A4×Dic5—22no—
39C3×#685—44no—
40C24×A4—51no—
41C22×C4×A4—51no—
42C22×C4×A4D511yescharacteristic, complemented
43A4×C22⋊C4—54no—
44C25×C5C22×C311yescharacteristic
45C24×D5—22no—
46C24×D5—22no—
47C24×D5—31no—
48C24×D5—31no—
49C24×D5—32no—
50C24×D5—34no—
51C24×D5—62no—
52C24×D5—62no—
53C24×D5C3×C414yescomplemented
54C24×D5C22×C312yes—
55C23×C4×C5—22no—
56C23×C4×C5—31no—
57C23×C4×C5—31no—
58C23×C4×C5—32no—
59C23×C4×C5—62no—
60C23×C4×C5C22×C312yescomplemented
61C23×Dic5—22no—
62C23×Dic5—31no—
63C23×Dic5—31no—
64C23×Dic5—32no—
65C23×Dic5—62no—
66C23×Dic5C22×C312yescomplemented
67C2×#685—32no—
68C2×#685—34no—
69C2×#685—34no—
70C2×#685—34no—
71C2×#685—31no—
72C2×#685—31no—
73C2×#685—31no—
74C2×#685—34no—
75C2×#685—34no—
76C2×#685—34no—
77C2×#685—34no—
78C2×#685—34no—
79C2×#685A411yescharacteristic, direct product factor
80C23×C22⋊C4—51noSylow
81C23×C3×C5—41no—
82C22×C3×D5—44no—
83C22×C3×D5—42no—
84C22×C3×D5—82no—
85C22×C3×D5—82no—
86C2×C5×A4C22⋊C414yescomplemented
87C2×C5×A4C22×C411yescharacteristic
88C2×C5×A4C2×D411yescharacteristic
89C2×C5×A4C2×D411yescharacteristic
90D5×A4—24no—
91C2×C3×C4×C5—42no—
92C2×C3×C4×C5—82no—
93C2×C3×Dic5—42no—
94C2×C3×Dic5—82no—
95C23×A4—54no—
96C23×A4—52no—
97C23×A4—102no—
98C23×A4—102no—
99C23×A4C2×D511yescharacteristic
100C2×C4×A4—22no—
101C2×C4×A4—52no—
102C2×C4×A4—102no—
103C2×C4×A4C2×D512yescomplemented
104C2×C3×C22⋊C4—201no—
105C24×C5—31no—
106C24×C5—32no—
107C24×C5—32no—
108C24×C5—32no—
109C24×C5—31no—
110C24×C5C2×C3×C412yes—
111C24×C5C3×D412yes—
112C24×C5C3×D412yes—
113C24×C5C23×C311yescharacteristic
114C23×D5—28no—
115C23×D5—22no—
116C23×D5—22no—
117C23×D5—34no—
118C23×D5—38no—
119C23×D5—32no—
120C23×D5—31no—
121C23×D5—34no—
122C23×D5—34no—
123C23×D5—32no—
124C23×D5—32no—
125C23×D5—34no—
126C23×D5—38no—
127C23×D5—34no—
128C23×D5—32no—
129C23×D5—62no—
130C23×D5—62no—
131C23×D5—62no—
132C23×D5—62no—
133C23×D5—64no—
134C23×D5—64no—
135C23×D5—616no—
136C23×D5—64no—
137C23×D5—64no—
138C23×D5—68no—
139C23×D5—68no—
140C23×D5—62no—
141C23×D5—62no—
142C23×D5—62no—
143C23×D5—62no—
144C23×D5—68no—
145C23×D5—62no—
146C23×D5—62no—
147C23×D5C2×C3×C414yescomplemented
148C23×D5C2×A411yescharacteristic
149C22×C4×C5—22no—
150C22×C4×C5—32no—
151C22×C4×C5—31no—
152C22×C4×C5—32no—
153C22×C4×C5—32no—
154C22×C4×C5—32no—
155C22×C4×C5—62no—
156C22×C4×C5—62no—
157C22×C4×C5—64no—
158C22×C4×C5—64no—
159C22×C4×C5—62no—
160C22×C4×C5—62no—
161C22×C4×C5—62no—
162C22×C4×C5C2×A411yescharacteristic, complemented
163C22×Dic5—22no—
164C22×Dic5—32no—
165C22×Dic5—31no—
166C22×Dic5—32no—
167C22×Dic5—32no—
168C22×Dic5—32no—
169C22×Dic5—62no—
170C22×Dic5—62no—
171C22×Dic5—64no—
172C22×Dic5—64no—
173C22×Dic5—62no—
174C22×Dic5—62no—
175C22×Dic5—62no—
176C22×Dic5C2×A411yescharacteristic, complemented
177#685—38no—
178#685—34no—
179#685—34no—
180#685—34no—
181#685C2×A414yesdirect product factor
182C26—51no—
183C24×C4—51no—
184C24×C4C3×D511yescharacteristic, complemented
185C22×C22⋊C4—54no—
186C22×C22⋊C4—151no—
187C22×C22⋊C4—154no—
188C22×C22⋊C4—151no—
189C22×C22⋊C4—151no—
190C22×C22⋊C4—151no—
191C22×C3×C5—42no—
192C22×C3×C5—42no—
193C22×C3×C5—42no—
194C22×C3×C5—41no—
195C2×C3×D5—44no—
196C2×C3×D5—82no—
197C2×C3×D5—82no—
198C2×C3×D5—88no—
199C5×A4C2×C22⋊C411yescharacteristic, complemented
200C3×C4×C5—82no—
201C3×Dic5—82no—
202C24×C3—201no—
203C22×A4—54no—
204C22×A4—108no—
205C22×A4—102no—
206C22×A4—102no—
207C22×A4C4×D512yes—
208C22×A4D2012yes—
209C22×A4C22⋊D512yes—
210C22×A4C22×D511yescharacteristic
211C22×C3×C4—41no—
212C22×C3×C4—201no—
213C4×A4—22no—
214C4×A4—102no—
215C3×C22⋊C4—204no—
216C23×C5—32no—
217C23×C5—31no—
218C23×C5—32no—
219C23×C5—31no—
220C23×C5—32no—
221C23×C5—31no—
222C23×C5—34no—
223C23×C5—34no—
224C23×C5—34no—
225C23×C5—32no—
226C23×C5—32no—
227C23×C5—32no—
228C23×C5—38no—
229C23×C5—32no—
230C23×C5—32no—
231C23×C5—32no—
232C23×C5—32no—
233C23×C5—32no—
234C23×C5—32no—
235C23×C5—31no—
236C23×C5—31no—
237C23×C5C3×C22⋊C414yescomplemented
238C23×C5C22×C3×C411yescharacteristic
239C23×C5C2×C3×D411yescharacteristic
240C23×C5C2×C3×D411yescharacteristic
241C23×C5C22×A411yescharacteristic
242C22×D5—22no—
243C22×D5—22no—
244C22×D5—24no—
245C22×D5—34no—
246C22×D5—38no—
247C22×D5—34no—
248C22×D5—32no—
249C22×D5—34no—
250C22×D5—38no—
251C22×D5—34no—
252C22×D5—34no—
253C22×D5—62no—
254C22×D5—62no—
255C22×D5—68no—
256C22×D5—62no—
257C22×D5—64no—
258C22×D5—62no—
259C22×D5—64no—
260C22×D5—68no—
261C22×D5—68no—
262C22×D5—68no—
263C22×D5—616no—
264C22×D5—616no—
265C22×D5—616no—
266C22×D5—68no—
267C22×D5—64no—
268C22×D5—64no—
269C22×D5—68no—
270C22×D5—68no—
271C22×D5—62no—
272C22×D5—62no—
273C22×D5—616no—
274C22×D5—64no—
275C22×D5—64no—
276C22×D5—62no—
277C22×D5—62no—
278C22×D5—68no—
279C22×D5—62no—
280C22×D5—62no—
281C22×D5—68no—
282C22×D5—64no—
283C22×D5C4×A414yescomplemented
284C22×D5C22×A412yes—
285C2×C4×C5—22no—
286C2×C4×C5—32no—
287C2×C4×C5—62no—
288C2×C4×C5—62no—
289C2×C4×C5—64no—
290C2×C4×C5—64no—
291C2×C4×C5—62no—
292C2×C4×C5—62no—
293C2×C4×C5—62no—
294C2×C4×C5C22×A412yescomplemented
295C2×Dic5—22no—
296C2×Dic5—32no—
297C2×Dic5—62no—
298C2×Dic5—62no—
299C2×Dic5—64no—
300C2×Dic5—64no—
301C2×Dic5—62no—
302C2×Dic5—62no—
303C2×Dic5—62no—
304C2×Dic5C22×A412yescomplemented
305C25—54no—
306C25—52no—
307C25—102no—
308C25—102no—
309C25—151no—
310C25—152no—
311C25—151no—
312C25—154no—
313C25—302no—
314C25—302no—
315C25C2×C3×D511yescharacteristic
316C23×C4—22no—
317C23×C4—31no—
318C23×C4—31no—
319C23×C4—32no—
320C23×C4—52no—
321C23×C4—62no—
322C23×C4—102no—
323C23×C4—151no—
324C23×C4—152no—
325C23×C4—151no—
326C23×C4—302no—
327C23×C4C2×C3×D512yescomplemented
328C2×C22⋊C4—51no—
329C2×C22⋊C4—154no—
330C2×C22⋊C4—154no—
331C2×C22⋊C4—151no—
332C2×C22⋊C4—151no—
333C2×C22⋊C4—154no—
334C2×C22⋊C4—154no—
335C2×C22⋊C4—152no—
336C2×C22⋊C4—154no—
337C2×C22⋊C4—151no—
338C2×C22⋊C4—154no—
339C2×C22⋊C4—154no—
340C2×C22⋊C4—154no—
341C2×C3×C5—41no—
342C2×C3×C5—41no—
343C2×C3×C5—41no—
344C2×C3×C5—44no—
345C3×D5—84no—
346C23×C3—41no—
347C23×C3—202no—
348C23×C3—204no—
349C23×C3—402no—
350C23×C3—402no—
351C2×A4—104no—
352C2×A4#68514yesdirect product factor
353C2×A4C2×C4×D511yescharacteristic
354C2×A4C2×D2011yescharacteristic
355C2×A4C2×C22⋊D511yescharacteristic
356C2×C3×C4—42no—
357C2×C3×C4—82no—
358C2×C3×C4—202no—
359C2×C3×C4—402no—
360C22×C5—34no—
361C22×C5—31no—
362C22×C5—31no—
363C22×C5—31no—
364C22×C5—34no—
365C22×C5—32no—
366C22×C5—34no—
367C22×C5—34no—
368C22×C5—32no—
369C22×C5—31no—
370C22×C5—31no—
371C22×C5—32no—
372C22×C5—31no—
373C22×C5—31no—
374C22×C5—32no—
375C22×C5—31no—
376C22×C5—31no—
377C22×C5—34no—
378C22×C5—34no—
379C22×C5—34no—
380C22×C5—34no—
381C22×C5C2×C3×C22⋊C411yescharacteristic, complemented
382C22×C5C2×C4×A412yes—
383C22×C5D4×A412yes—
384C22×C5D4×A412yes—
385C22×C5C23×A411yescharacteristic
386C2×D5—22no—
387C2×D5—22no—
388C2×D5—28no—
389C2×D5—34no—
390C2×D5—64no—
391C2×D5—616no—
392C2×D5—64no—
393C2×D5—64no—
394C2×D5—64no—
395C2×D5—616no—
396C2×D5—64no—
397C2×D5—64no—
398C2×D5—62no—
399C2×D5—62no—
400C2×D5—68no—
401C2×D5—64no—
402C2×D5—64no—
403C2×D5C2×C4×A414yescomplemented
404C4×C5—22no—
405C4×C5—62no—
406Dic5—22no—
407Dic5—62no—
408C24—31no—
409C24—32no—
410C24—32no—
411C24—32no—
412C24—31no—
413C24—51no—
414C24—54no—
415C24—108no—
416C24—102no—
417C24—102no—
418C24—152no—
419C24—152no—
420C24—154no—
421C24—154no—
422C24—152no—
423C24—158no—
424C24—154no—
425C24—152no—
426C24—158no—
427C24—154no—
428C24—154no—
429C24—151no—
430C24—302no—
431C24—302no—
432C24—302no—
433C24—302no—
434C24—302no—
435C24—302no—
436C24—308no—
437C24—304no—
438C24—304no—
439C24—308no—
440C24—308no—
441C24—302no—
442C24—302no—
443C24—302no—
444C24—302no—
445C24—304no—
446C24—304no—
447C24—3016no—
448C24C3×C4×D512yes—
449C24C3×D2012yes—
450C24C3×C22⋊D512yes—
451C24C22×C3×D511yescharacteristic
452C22×C4—22no—
453C22×C4—32no—
454C22×C4—31no—
455C22×C4—32no—
456C22×C4—32no—
457C22×C4—32no—
458C22×C4—51no—
459C22×C4—62no—
460C22×C4—62no—
461C22×C4—64no—
462C22×C4—64no—
463C22×C4—62no—
464C22×C4—62no—
465C22×C4—62no—
466C22×C4—102no—
467C22×C4—152no—
468C22×C4—152no—
469C22×C4—152no—
470C22×C4—151no—
471C22×C4—152no—
472C22×C4—302no—
473C22×C4—302no—
474C22×C4—304no—
475C22×C4—304no—
476C22×C4—302no—
477C22×C4—302no—
478C22×C4—302no—
479C22×C4D5×A411yescharacteristic, complemented
480C22⋊C4—54no—
481C22⋊C4—154no—
482C22⋊C4—154no—
483C22⋊C4—158no—
484C22⋊C4—154no—
485C3×C5—41noHall
486C22×C3—42no—
487C22×C3—42no—
488C22×C3—42no—
489C22×C3—41no—
490C22×C3—204no—
491C22×C3—402no—
492C22×C3—402no—
493C22×C3—408no—
494A4C2×#68511yescharacteristic, direct product factor
495C3×C4—82no—
496C3×C4—402no—
497C2×C5—34no—
498C2×C5—31no—
499C2×C5—31no—
500C2×C5—31no—
501C2×C5—31no—
502C2×C5A4×C22⋊C414yescomplemented
503C2×C5C22×C4×A411yescharacteristic
504C2×C5C2×D4×A411yescharacteristic
505C2×C5C2×D4×A411yescharacteristic
506D5—24no—
507D5—64no—
508C23—32no—
509C23—31no—
510C23—32no—
511C23—31no—
512C23—32no—
513C23—31no—
514C23—34no—
515C23—34no—
516C23—34no—
517C23—32no—
518C23—32no—
519C23—32no—
520C23—38no—
521C23—32no—
522C23—32no—
523C23—32no—
524C23—32no—
525C23—32no—
526C23—32no—
527C23—31no—
528C23—31no—
529C23—54no—
530C23—52no—
531C23—102no—
532C23—102no—
533C23—104no—
534C23—154no—
535C23—154no—
536C23—158no—
537C23—158no—
538C23—154no—
539C23—154no—
540C23—154no—
541C23—152no—
542C23—302no—
543C23—302no—
544C23—308no—
545C23—302no—
546C23—304no—
547C23—302no—
548C23—304no—
549C23—308no—
550C23—308no—
551C23—308no—
552C23—304no—
553C23—3016no—
554C23—304no—
555C23—304no—
556C23—302no—
557C23—302no—
558C23—308no—
559C23—302no—
560C23—302no—
561C23—308no—
562C23—308no—
563C23—3016no—
564C23—3016no—
565C23—3016no—
566C23—304no—
567C23—304no—
568C23—308no—
569C23—308no—
570C23—302no—
571C23—302no—
572C23C3×#68514yescomplemented
573C23C2×C3×C4×D511yescharacteristic
574C23C2×C3×D2011yescharacteristic
575C23C2×C3×C22⋊D511yescharacteristic
576C23C2×D5×A411yescharacteristic
577C2×C4—22no—
578C2×C4—32no—
579C2×C4—52no—
580C2×C4—62no—
581C2×C4—62no—
582C2×C4—64no—
583C2×C4—64no—
584C2×C4—62no—
585C2×C4—62no—
586C2×C4—62no—
587C2×C4—102no—
588C2×C4—152no—
589C2×C4—302no—
590C2×C4—302no—
591C2×C4—304no—
592C2×C4—304no—
593C2×C4—302no—
594C2×C4—302no—
595C2×C4—302no—
596C2×C4C2×D5×A412yescomplemented
597C2×C3—41no—
598C2×C3—41no—
599C2×C3—41no—
600C2×C3—44no—
601C2×C3—404no—
602C5C2×A4×C22⋊C411yescharacteristic, complemented, Sylow
603C22—34no—
604C22—31no—
605C22—31no—
606C22—31no—
607C22—34no—
608C22—32no—
609C22—34no—
610C22—34no—
611C22—31no—
612C22—32no—
613C22—31no—
614C22—31no—
615C22—32no—
616C22—31no—
617C22—31no—
618C22—32no—
619C22—31no—
620C22—34no—
621C22—34no—
622C22—34no—
623C22—34no—
624C22—54no—
625C22—102no—
626C22—102no—
627C22—108no—
628C22—154no—
629C22—304no—
630C22—304no—
631C22—3016no—
632C22—304no—
633C22—304no—
634C22—304no—
635C22—304no—
636C22—3016no—
637C22—304no—
638C22—304no—
639C22—302no—
640C22—302no—
641C22—308no—
642C22C2×C3×#68511yescharacteristic, complemented
643C22C4×D5×A412yes—
644C22A4×D2012yes—
645C22A4×C22⋊D512yes—
646C22C22×D5×A411yescharacteristic
647C4—22no—
648C4—62no—
649C4—102no—
650C4—302no—
651C3—41noSylow
652C2—31no—
653C2—34no—
654C2—31no—
655C2—31no—
656C2—31no—
657C2—104no—
658C2—304no—
659C2A4×#68514yesdirect product factor
660C2C2×C4×D5×A411yescharacteristic
661C2C2×A4×D2011yescharacteristic
662C2C2×A4×C22⋊D511yescharacteristic
subgroupG/H
centerC23C2×D5×A4
commutatorC23×C5C22×C3×C4
FrattiniC22C22×D5×A4
FittingC24×C4×C5C2×C3
socleC25×C5C22×C3
prime radicalC23×D5×A4C2
cosocleC24×C5C23×C3
hypercenterC22×C4D5×A4
hypocenterC22×C5C2×C3×C22⋊C4
LCSG ▷ C23×C5 ▷ C22×C5
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C22×C4×C5×A4C2semidirect
3. C22×A4×Dic5C2semidirect
4. A4×#685C2direct (N × H)
5. C23×#685C3semidirect
9. C22×D5×A4C4semidirect
12. C2×C4×C5×A4C22semidirect
14. C2×A4×Dic5C22semidirect
18. C24×C4×C5C2×C3semidirect
19. C24×Dic5C2×C3semidirect
25. C22×#685C2×C3semidirect
34. C2×D5×A4C2×C4semidirect
42. C22×C4×A4D5semidirect
53. C24×D5C3×C4semidirect
60. C23×C4×C5C22×C3semidirect
66. C23×Dic5C22×C3semidirect
79. C2×#685A4direct (N × H)
86. C2×C5×A4C22⋊C4semidirect
103. C2×C4×A4C2×D5semidirect
147. C23×D5C2×C3×C4semidirect
162. C22×C4×C5C2×A4semidirect
176. C22×Dic5C2×A4semidirect
181. #685C2×A4direct (N × H)
184. C24×C4C3×D5semidirect
199. C5×A4C2×C22⋊C4semidirect
237. C23×C5C3×C22⋊C4semidirect
283. C22×D5C4×A4semidirect
294. C2×C4×C5C22×A4semidirect
304. C2×Dic5C22×A4semidirect
327. C23×C4C2×C3×D5semidirect
352. C2×A4#685direct (N × H)
381. C22×C5C2×C3×C22⋊C4semidirect
403. C2×D5C2×C4×A4semidirect
479. C22×C4D5×A4semidirect
494. A4C2×#685direct (N × H)
502. C2×C5A4×C22⋊C4semidirect
572. C23C3×#685semidirect
596. C2×C4C2×D5×A4semidirect
602. C5C2×A4×C22⋊C4semidirect
642. C22C2×C3×#685semidirect
659. C2A4×#685direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D5×A42
C22×C4×C5×A42
C22×A4×Dic52
A4×#6852
C23×#6853
C2×C3×#6854
C2×A4×C22⋊C45