Finite Group Databasebeta

← #14480691

Subgroups — #14480691

Order 1920.

3112 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×#6779C212yesmaximal, complemented
2#11756252C211yesmaximal, characteristic, complemented
3C2×C5×#877C211yesmaximal, characteristic, complemented
4C3×#65827C211yesmaximal, characteristic, complemented
5#11755427C212yesmaximal, complemented
6#11754357C214yesmaximal, complemented
7#11754356C214yesmaximal, complemented
8#11752878C214yesmaximal, complemented
9#11752877C214yesmaximal, complemented
10#11752876C214yesmaximal, complemented
11#11752875C214yesmaximal, complemented
12#10617636—31nomaximal, Hall
13C22×#1220C2212yes—
14C23×C5×Dic3C2211yescharacteristic
15C22×C3×C22⋊D5C2212yes—
16C22×#1204C2212yescomplemented
17C22×#1202C2212yescomplemented
18C22×#1200C2212yescomplemented
19C2×#6779—216no—
20C2×#6779C2214yes—
21C2×#6779C2214yes—
22C2×#6779C2214yescomplemented
23C2×#6779C2214yescomplemented
24C22×D5×Dic3C2212yescomplemented
25C2×C3×C5×C22⋊C4C2211yescharacteristic
26#91819C2214yescomplemented
27#91775C2214yescomplemented
28#91771C2214yescomplemented
29C2×C5×#217C2211yescharacteristic
30C2×C5×Dic3⋊C4C2212yescomplemented
31C2×C4×C5×Dic3C2212yescomplemented
32C5×#877C2214yescomplemented
33C5×#877C2214yescomplemented
34C3×#4076C2214yescomplemented
35C3×#4032C2214yescomplemented
36C3×#4028C2214yescomplemented
37#91572C2214yescomplemented
38#91569C2218yescomplemented
39#91564C2214yescomplemented
40#91546C2214yescomplemented
41#91537C2214yescomplemented
42#91469C2214yescomplemented
43#91459C2218yescomplemented
44#91458C2214yescomplemented
45#91429C2218yescomplemented
46#91428C2214yescomplemented
47Dic3×D20C2214yescomplemented
48#91422C2218yescomplemented
49#91409C2218yescomplemented
50#91395C2218yescomplemented
51#91353C2218yescomplemented
52#86294—51nomaximal, Hall
53C22×#709—32no—
54C2×C5×#117—31no—
55#65860—34no—
56#65859—34no—
57#65858—34no—
58#65858—34no—
59#65844—34no—
60#65827—32no—
61#65827—31no—
62#65827D311yescharacteristic, complemented
63#65811—34no—
64C24×C3×C5C2311yescharacteristic
65C23×D15C2312yes—
66C23×C3×D5C2312yes—
67C22×C3×C4×C5C2312yes—
68C2×#1220—28no—
69C2×#1220C2314yes—
70C2×#1220C2314yes—
71C22×Dic15C2312yes—
72C2×D60C2314yes—
73C22×C5×Dic3D418yescomplemented
74C22×C5×Dic3C2312yes—
75C22×C5×Dic3C2312yes—
76C22×C5×Dic3C2312yes—
77C22×C5×Dic3C2311yescharacteristic
78C22×C5×Dic3C2311yescharacteristic
79C2×C3×C22⋊D5—28no—
80C2×C3×C22⋊D5C2314yes—
81C2×C3×C22⋊D5C2314yes—
82C22×C3×Dic5C2312yes—
83C2×C3×D20C2314yes—
84C2×#1204—28no—
85C2×#1204C2314yes—
86C2×#1204C2314yes—
87C2×#1202—28no—
88C2×#1202C2314yes—
89C2×#1202C2314yes—
90C2×#1202C2318yes—
91C2×#1200—28no—
92C2×#1200C2314yes—
93C2×#1200C2314yes—
94#6779—216no—
95#6779—216no—
96#6779—216no—
97#6779—216no—
98C2×D5×Dic3—28no—
99C2×D5×Dic3C2314yes—
100C2×D5×Dic3C2314yes—
101C3×C5×C22⋊C4C2314yes—
102#6714C2318yes—
103C4⋊Dic15C2314yes—
104C5×#217C2314yes—
105C5×Dic3⋊C4C2318yes—
106C4×C5×Dic3C2314yes—
107C4×C5×Dic3C2314yes—
108C3×#685C2318yes—
109C3×C4⋊Dic5C2314yes—
110Dic3⋊Dic5C2318yes—
111#6667C2318yes—
112#6665C2318yes—
113C22×#237—52no—
114C3×#534—51no—
115#5804—52no—
116#5656—54no—
117#5648—54no—
118#5597—54no—
119#5596—54no—
120#5592—54no—
121#5586—54no—
122#5493—51no—
123C2×#877D511yescharacteristic, complemented
124C23×C4×C5—31no—
125C22×C22⋊D5—32no—
126C22×C22⋊D5C2×D312yes—
127C2×#709—34no—
128C2×#709—34no—
129C2×#709—34no—
130C2×#709—34no—
131C2×#709—616no—
132C22×D20—32no—
133C22×C4×D5—32no—
134C22×C4×D5—32no—
135C22×Dic10—32no—
136C5×#117—34no—
137C5×#117—34no—
138C2×C5×C4⋊C4—32no—
139C2×C5×C22⋊C4—31no—
140C2×C5×C22⋊C4C2×D311yescharacteristic
141C2×C42×C5—32no—
142#4076—34no—
143#4076—38no—
144#4076C2×D314yescomplemented
145#4043—38no—
146#4043—38no—
147#4041—38no—
148#4041—38no—
149#4032—34no—
150#4032—34no—
151#4032—34no—
152#4032C2×D314yescomplemented
153#4028—34no—
154#4028—34no—
155#4028—34no—
156#4028C2×D314yescomplemented
157#4021—38no—
158#4020—34no—
159C4×D20—34no—
160C4×D20—38no—
161C4×D20—34no—
162#4015—34no—
163#3449—151noSylow
164C23×C3×C5—42no—
165C23×C3×C5C2×D414yes—
166C23×C3×C5C2412yes—
167C23×C3×C5C2411yescharacteristic
168C22×D15—24no—
169C22×D15—44no—
170C22×D15C2414yes—
171C22×C3×D5—24no—
172C22×C3×D5—44no—
173C22×C3×D5C2414yes—
174C2×C3×C4×C5—42no—
175C2×C3×C4×C5C2414yes—
176#1220—28no—
177#1220—28no—
178C2×Dic15—24no—
179C2×Dic15C2414yes—
180D60—44no—
181C2×C5×Dic3—42no—
182C2×C5×Dic3C2×D418yes—
183C2×C5×Dic3C2×D414yes—
184C2×C5×Dic3C2×D414yes—
185C2×C5×Dic3C2×D418yes—
186C2×C5×Dic3C2414yes—
187C2×C5×Dic3C2411yescharacteristic
188C2×C5×Dic3C2411yescharacteristic
189C2×C5×Dic3C2412yes—
190C3×C22⋊D5—28no—
191C3×C22⋊D5—28no—
192C2×C3×Dic5—24no—
193C2×C3×Dic5C2414yes—
194C3×D20—44no—
195#1204—28no—
196#1204—28no—
197#1202—28no—
198#1202—28no—
199#1202—48no—
200#1200—28no—
201#1200—28no—
202D5×Dic3—28no—
203D5×Dic3—28no—
204C22×C3×D4—52no—
205C22×#67—52no—
206C22×#67—52no—
207C23×Dic3—52no—
208C23×Dic3C2×D511yescharacteristic
209C2×#237—54no—
210C2×#237—54no—
211C2×#237—54no—
212C2×#237—54no—
213C2×#237—1016no—
214C22×C4×D3—52no—
215C22×Dic6—52no—
216C3×#123—54no—
217C3×#121—54no—
218C3×#120—54no—
219C2×C3×C22⋊C4C2×D511yescharacteristic
220#953—54no—
221C2×#217C2×D511yescharacteristic
222#940—54no—
223#939—58no—
224#939—58no—
225#937—54no—
226#935—54no—
227D4×Dic3—54no—
228#930—54no—
229#929—54no—
230C2×Dic3⋊C4C2×D512yescomplemented
231C2×C4×Dic3C2×D512yescomplemented
232#896—58no—
233#893—54no—
234#888—54no—
235#886—54no—
236#885—58no—
237#884—58no—
238#882—54no—
239#67⋊C4—58no—
240#878—58no—
241#877C2×D514yescomplemented
242#877C2×D514yescomplemented
243C24×C5C22×D311yescharacteristic
244C23×D5—32no—
245C23×D5C22×D312yes—
246C22×C4×C5—38no—
247C22×C4×C5—31no—
248C22×C4×C5—31no—
249C22×C4×C5—32no—
250C22×C4×C5—32no—
251C22×C4×C5—32no—
252C22×C4×C5C22×D312yes—
253C2×C22⋊D5—28no—
254C2×C22⋊D5—34no—
255C2×C22⋊D5—34no—
256C2×C22⋊D5—68no—
257C2×C22⋊D5C22×D314yes—
258C2×C22⋊D5C22×D314yes—
259C22×Dic5—32no—
260C22×Dic5C22×D312yes—
261#709—616no—
262#709—616no—
263#709—616no—
264#709—616no—
265C2×D20—34no—
266C2×D20—34no—
267C2×D20—38no—
268C2×D20—34no—
269C2×D20—68no—
270C2×D20C22×D314yes—
271C2×C4×D5—34no—
272C2×C4×D5—34no—
273C2×C4×D5—34no—
274C2×C4×D5—34no—
275C2×C4×D5—68no—
276C2×C4×D5—68no—
277C2×Dic10—34no—
278C2×Dic10—34no—
279C2×Dic10—68no—
280C5×C4⋊C4—38no—
281C5×C22⋊C4—34no—
282C5×C22⋊C4C22×D314yes—
283C42×C5—34no—
284C42×C5—34no—
285#685—38no—
286#685—38no—
287#685—38no—
288#685C22×D318yes—
289C4⋊Dic5—34no—
290C4⋊Dic5—38no—
291C4⋊Dic5C22×D314yes—
292C22×#41—152no—
293#537—154no—
294#536—154no—
295#535—154no—
296#534—51no—
297#534—152no—
298#534—151no—
299#532—154no—
300#525—154no—
301#525—154no—
302C2×#117—31no—
303C22×C3×C5—22no—
304C22×C3×C5—24no—
305C22×C3×C5—44no—
306C22×C3×C5C22×D414yes—
307C22×C3×C5C22×D412yes—
308C22×C3×C5C2511yescharacteristic
309C2×D15—24no—
310C2×D15—28no—
311C2×D15—48no—
312C2×C3×D5—24no—
313C2×C3×D5—28no—
314C2×C3×D5—48no—
315C3×C4×C5—24no—
316Dic15—24no—
317C3×Dic5—24no—
318C5×Dic3—24no—
319C5×Dic3C22×D412yes—
320C5×Dic3C22×D414yes—
321C5×Dic3C22×D412yes—
322C24×C3—52no—
323C24×C3C22×D511yescharacteristic
324C23×D3—52no—
325C2×C3×D4—54no—
326C2×C3×D4—54no—
327C2×C3×D4—54no—
328C2×C3×D4—108no—
329C22×C3×C4—52no—
330C22×C3×C4C22×D512yes—
331C2×#67—54no—
332C2×#67—54no—
333C2×#67—54no—
334C2×#67—54no—
335C2×#67—58no—
336C2×#67—108no—
337C2×#67—108no—
338C22×Dic3—52no—
339C22×Dic3—54no—
340C22×Dic3—54no—
341C22×Dic3—108no—
342C22×Dic3D2018yescomplemented
343C22×Dic3C22×D512yes—
344C22×Dic3C22×D512yes—
345C22×Dic3C22×D512yes—
346C22×Dic3C22×D511yescharacteristic
347C22×Dic3C22×D511yescharacteristic
348#237—1016no—
349#237—1016no—
350#237—1016no—
351#237—1016no—
352C2×D12—54no—
353C2×C4×D3—54no—
354C2×C4×D3—54no—
355C2×C4×D3—108no—
356C2×Dic6—54no—
357C2×Dic6—54no—
358C2×Dic6—108no—
359C3×C4⋊C4—54no—
360C3×C22⋊C4—58no—
361C3×C22⋊C4C22×D514yes—
362#217—58no—
363#217C22×D514yes—
364(C2×D3)⋊C4—58no—
365(C2×D3)⋊C4—58no—
366C4⋊Dic3—54no—
367Dic3⋊C4—58no—
368Dic3⋊C4C22×D518yes—
369C4×Dic3C22×D514yes—
370C4×Dic3C22×D514yes—
371C23×C5—42no—
372C23×C5D3×D414yes—
373C23×C5C23×D312yes—
374C23×C5C23×D311yescharacteristic
375C22×D5—24no—
376C22×D5—34no—
377C22×D5—44no—
378C22×D5—64no—
379C22×D5—124no—
380C22×D5C23×D314yes—
381C2×C4×C5—38no—
382C2×C4×C5—38no—
383C2×C4×C5—34no—
384C2×C4×C5—34no—
385C2×C4×C5—31no—
386C2×C4×C5—31no—
387C2×C4×C5—32no—
388C2×C4×C5—34no—
389C2×C4×C5—42no—
390C2×C4×C5—122no—
391C2×C4×C5C23×D314yes—
392C22⋊D5—28no—
393C22⋊D5—28no—
394C22⋊D5—68no—
395C22⋊D5—68no—
396C2×Dic5—24no—
397C2×Dic5—34no—
398C2×Dic5—64no—
399C2×Dic5C23×D314yes—
400D20—44no—
401D20—68no—
402D20—68no—
403D20—124no—
404D20—128no—
405C4×D5—68no—
406C4×D5—68no—
407C4×D5—68no—
408C4×D5—68no—
409Dic10—68no—
410Dic10—68no—
411C2×#41—154no—
412C2×#41—154no—
413C2×#41—154no—
414C2×#41—154no—
415C2×#41—3016no—
416C22×Q8—152no—
417C22×D4—52no—
418C22×D4—152no—
419C22×D4—152no—
420C23×C4—31no—
421C23×C4—152no—
422C23×C4—152no—
423C4⋊Q8—154no—
424#127—154no—
425#124—158no—
426#123—54no—
427#123—154no—
428#123—154no—
429#123—154no—
430C22⋊Q8—158no—
431C22⋊Q8—158no—
432#121—54no—
433#121—158no—
434#121—158no—
435#121—158no—
436#121—154no—
437#120—54no—
438#120—154no—
439#120—154no—
440#120—154no—
441C4×D4—158no—
442C4×D4—154no—
443C4×D4—154no—
444#117—34no—
445#117—34no—
446C2×C4⋊C4—32no—
447C2×C22⋊C4—31no—
448C2×C22⋊C4D3×D511yescharacteristic
449C2×C42—32no—
450C2×C3×C5—24no—
451C2×C3×C5—22no—
452C2×C3×C5C23×D411yescharacteristic
453C2×C3×C5#58512yes—
454D15—44no—
455C3×D5—44no—
456C23×C3—42no—
457C23×C3—54no—
458C23×C3—104no—
459C23×C3—204no—
460C23×C3C2×D2014yes—
461C23×C3C23×D512yes—
462C23×C3C23×D511yescharacteristic
463C22×D3—54no—
464C22×D3—104no—
465C22×D3—204no—
466C3×D4—108no—
467C3×D4—108no—
468C3×D4—204no—
469C2×C3×C4—42no—
470C2×C3×C4—54no—
471C2×C3×C4—104no—
472C2×C3×C4C23×D514yes—
473#67—108no—
474#67—108no—
475#67—108no—
476#67—108no—
477#67—208no—
478C2×Dic3—42no—
479C2×Dic3—54no—
480C2×Dic3—104no—
481C2×Dic3—108no—
482C2×Dic3—108no—
483C2×Dic3C2×D2018yes—
484C2×Dic3C2×D2018yes—
485C2×Dic3C2×D2014yes—
486C2×Dic3C2×D2014yes—
487C2×Dic3C23×D514yes—
488C2×Dic3C23×D511yescharacteristic
489C2×Dic3C23×D511yescharacteristic
490C2×Dic3C23×D512yes—
491D12—204no—
492C4×D3—108no—
493C4×D3—108no—
494Dic6—108no—
495Dic6—108no—
496C22×C5—22no—
497C22×C5—24no—
498C22×C5—44no—
499C22×C5C2×D3×D414yes—
500C22×C5C2×D3×D412yes—
501C22×C5C24×D311yescharacteristic
502C2×D5—24no—
503C2×D5—28no—
504C2×D5—48no—
505C2×D5—64no—
506C2×D5—68no—
507C2×D5—128no—
508C4×C5—24no—
509C4×C5—32no—
510C4×C5—34no—
511C4×C5—32no—
512C4×C5—64no—
513Dic5—24no—
514Dic5—64no—
515C24—52no—
516C24—152no—
517C24C2×D3×D511yescharacteristic
518#41—3016no—
519#41—3016no—
520#41—3016no—
521#41—3016no—
522C2×Q8—154no—
523C2×Q8—154no—
524C2×Q8—308no—
525C2×D4—54no—
526C2×D4—54no—
527C2×D4—54no—
528C2×D4—108no—
529C2×D4—154no—
530C2×D4—154no—
531C2×D4—158no—
532C2×D4—154no—
533C2×D4—154no—
534C2×D4—154no—
535C2×D4—308no—
536C2×D4—308no—
537C22×C4—38no—
538C22×C4—31no—
539C22×C4—31no—
540C22×C4—32no—
541C22×C4—32no—
542C22×C4—32no—
543C22×C4—52no—
544C22×C4—154no—
545C22×C4—154no—
546C22×C4—154no—
547C22×C4—154no—
548C22×C4—152no—
549C22×C4—308no—
550C22×C4—308no—
551C22×C4C2×D3×D512yes—
552C4⋊C4—38no—
553C4⋊C4—54no—
554C4⋊C4—158no—
555C4⋊C4—154no—
556C22⋊C4—34no—
557C22⋊C4—58no—
558C22⋊C4—158no—
559C22⋊C4—158no—
560C22⋊C4—158no—
561C22⋊C4C2×D3×D514yes—
562C42—34no—
563C42—34no—
564C3×C5#344911yescharacteristic, complemented, Hall
565C22×C3—22no—
566C22×C3—24no—
567C22×C3—44no—
568C22×C3—104no—
569C22×C3—108no—
570C22×C3—208no—
571C22×C3C22×D2014yes—
572C22×C3C22×D2012yes—
573C22×C3C24×D511yescharacteristic
574C2×D3—104no—
575C2×D3—108no—
576C2×D3—208no—
577C3×C4—24no—
578C3×C4—104no—
579Dic3—24no—
580Dic3—104no—
581Dic3C22×D2012yes—
582Dic3C22×D2014yes—
583Dic3C22×D2012yes—
584C2×C5—24no—
585C2×C5—22no—
586C2×C5C22×D3×D411yescharacteristic
587C2×C5#596612yes—
588D5—44no—
589D5—124no—
590C23—42no—
591C23—54no—
592C23—104no—
593C23—154no—
594C23—204no—
595C23—304no—
596C23—604no—
597C23D3×D2014yes—
598C23C22×D3×D512yes—
599C23C22×D3×D511yescharacteristic
600Q8—308no—
601Q8—308no—
602D4—108no—
603D4—108no—
604D4—204no—
605D4—308no—
606D4—308no—
607D4—308no—
608D4—308no—
609D4—604no—
610D4—608no—
611C2×C4—38no—
612C2×C4—38no—
613C2×C4—34no—
614C2×C4—34no—
615C2×C4—31no—
616C2×C4—31no—
617C2×C4—32no—
618C2×C4—34no—
619C2×C4—42no—
620C2×C4—54no—
621C2×C4—104no—
622C2×C4—122no—
623C2×C4—154no—
624C2×C4—308no—
625C2×C4—308no—
626C2×C4—308no—
627C2×C4—308no—
628C2×C4—304no—
629C2×C4C22×D3×D514yes—
630C2×C3—24no—
631C2×C3—22no—
632C2×C3—204no—
633C2×C3C23×D2011yescharacteristic
634C2×C3#6628312yes—
635D3—204no—
636C5#8629411yescharacteristic, complemented, Sylow
637C22—22no—
638C22—24no—
639C22—44no—
640C22—104no—
641C22—108no—
642C22—208no—
643C22—304no—
644C22—308no—
645C22—608no—
646C22C2×D3×D2014yes—
647C22C2×D3×D2012yes—
648C22C23×D3×D511yescharacteristic
649C4—24no—
650C4—32no—
651C4—34no—
652C4—32no—
653C4—64no—
654C4—104no—
655C4—304no—
656C3#1061763611yescharacteristic, complemented, Sylow
657C2—24no—
658C2—22no—
659C2—204no—
660C2—604no—
661C2C22×D3×D2011yescharacteristic
662C2#1175703512yes—
subgroupG/H
center, FrattiniC22C23×D3×D5
commutator, socle, cosocleC22×C3×C5C25
FittingC2×C3×C5×C22⋊C4C22
hypercenterC2×C22⋊C4D3×D5
hypocenterC3×C5#3449
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#6779C2semidirect
2. #11756252C2semidirect
3. C2×C5×#877C2semidirect
4. C3×#65827C2semidirect
5. #11755427C2semidirect
6. #11754357C2semidirect
7. #11754356C2semidirect
8. #11752878C2semidirect
9. #11752877C2semidirect
10. #11752876C2semidirect
11. #11752875C2semidirect
16. C22×#1204C22semidirect
17. C22×#1202C22semidirect
18. C22×#1200C22semidirect
22. C2×#6779C22semidirect
23. C2×#6779C22semidirect
24. C22×D5×Dic3C22semidirect
26. #91819C22semidirect
27. #91775C22semidirect
28. #91771C22semidirect
30. C2×C5×Dic3⋊C4C22semidirect
31. C2×C4×C5×Dic3C22semidirect
32. C5×#877C22semidirect
33. C5×#877C22semidirect
34. C3×#4076C22semidirect
35. C3×#4032C22semidirect
36. C3×#4028C22semidirect
37. #91572C22semidirect
38. #91569C22semidirect
39. #91564C22semidirect
40. #91546C22semidirect
41. #91537C22semidirect
42. #91469C22semidirect
43. #91459C22semidirect
44. #91458C22semidirect
45. #91429C22semidirect
46. #91428C22semidirect
47. Dic3×D20C22semidirect
48. #91422C22semidirect
49. #91409C22semidirect
50. #91395C22semidirect
51. #91353C22semidirect
62. #65827D3semidirect
73. C22×C5×Dic3D4semidirect
123. C2×#877D5semidirect
144. #4076C2×D3semidirect
152. #4032C2×D3semidirect
156. #4028C2×D3semidirect
230. C2×Dic3⋊C4C2×D5semidirect
231. C2×C4×Dic3C2×D5semidirect
241. #877C2×D5semidirect
242. #877C2×D5semidirect
342. C22×Dic3D20semidirect
564. C3×C5#3449semidirect
636. C5#86294semidirect
656. C3#10617636semidirect

Maximal subgroups

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

subgroupindex
C22×#67792
#117562522
C2×C5×#8772
C3×#658272
#117554272
#117543572
#117543562
#117528782
#117528772
#117528762
#117528752
#106176363
#862945