Finite Group Databasebeta

← #14425736

Subgroups — #14425736

Order 1920.

1600 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
1C2×#92095C211yesmaximal, characteristic, complemented
2C3×#66138C211yesmaximal, characteristic, complemented
3#11755427C212yesmaximal, complemented
4C2×C5×#834C211yesmaximal, characteristic, complemented
5C2×#90997C212yesmaximal, complemented
6#11749185C214yesmaximal, complemented
7#11749179C214yesmaximal, complemented
8#10610417—31nomaximal, Hall
9C22×C3×C5×D4C2211yescharacteristic
10C22×#1220—21no—
11C22×#1220C2212yes—
12#92095—22no—
13#92095—24no—
14#92095C2214yes—
15C2×#6819—22no—
16C2×D4×D15—22no—
17C2×#6817—21no—
18C22×C3×C22⋊D5C2211yescharacteristic, complemented
19C2×C5×#217C2212yescomplemented
20C3×#4099—21no—
21C2×C3×C22⋊Dic5C2211yescharacteristic, complemented
22C3×#4085—22no—
23C3×#4083C2212yescomplemented
24C3×#4081C2212yescomplemented
25C3×#4075—21no—
26#91572—22no—
27#91569—24no—
28#91564C2214yescomplemented
29#91546C2214yescomplemented
30#91537—22no—
31C5×#834C2214yescomplemented
32#90997C2218yescomplemented
33#83375—51nomaximal, Hall
34C2×#4144—31no—
35#66138D311yescharacteristic, complemented
36#65827—32no—
37C2×C5×#99—31no—
38C2×#3938—32no—
39#65031—34no—
40#65030—34no—
41C24×C3×C5D411yescharacteristic, complemented
42C24×C3×C5C2311yescharacteristic
43C23×D15—21no—
44C23×D15D412yescomplemented
45C23×C3×D5C2311yescharacteristic, complemented
46C2×C3×C5×D4—22no—
47C2×C3×C5×D4—22no—
48C2×C3×C5×D4D412yescomplemented
49C2×C3×C5×D4C2312yes—
50C22×C3×C4×C5D411yescharacteristic, complemented
51C2×#1220—22no—
52C2×#1220—24no—
53C2×#1220—22no—
54C2×#1220—22no—
55C2×#1220—22no—
56C2×#1220—22no—
57C2×#1220—24no—
58C2×#1220D414yescomplemented
59C2×#1220C2314yes—
60C22×Dic15—21no—
61C22×Dic15D412yescomplemented
62#6819—28no—
63#6819—28no—
64D4×D15—28no—
65D4×D15—28no—
66#6817—24no—
67#6817—24no—
68C2×D60—21no—
69C2×C4×D15—22no—
70C2×Dic30—21no—
71C22×C5×Dic3—22no—
72C2×C3×C22⋊D5—21no—
73C2×C3×C22⋊D5—21no—
74C2×C3×C22⋊D5—22no—
75C2×C3×C22⋊D5—22no—
76C2×C3×C22⋊D5—22no—
77C22×C3×Dic5—21no—
78C22×C3×Dic5C2311yescharacteristic, complemented
79C2×#1202—24no—
80C5×#217—22no—
81C5×#217C2314yescomplemented
82C3×C22⋊Dic5—22no—
83C3×C22⋊Dic5—21no—
84C3×C22⋊Dic5C2312yescomplemented
85C3×#685—22no—
86C3×#683—22no—
87Dic3⋊Dic5—24no—
88#6667—24no—
89#6665—24no—
90C2×#1004—51no—
91C3×#534—51no—
92#5804—52no—
93C2×#834D511yescharacteristic, complemented
94#5053—54no—
95#5032—54no—
96C2×#806—52no—
97C22×C5×D4C2×D311yescharacteristic
98C22×C22⋊D5—32no—
99C22×C22⋊D5—61no—
100C22×C22⋊D5C2×D311yescharacteristic, complemented
101#4144—34no—
102#4144—64no—
103#4144—62no—
104C2×#711—62no—
105C2×D4×D5—62no—
106C2×#709—61no—
107C2×C5×C22⋊C4—32no—
108#4099—21no—
109C2×C22⋊Dic5C2×D311yescharacteristic, complemented
110#4085—22no—
111#4083C2×D312yescomplemented
112#4081C2×D312yescomplemented
113#4076—64no—
114#4075—21no—
115#4032—34no—
116#4032—62no—
117#4028—34no—
118#4028—62no—
119C5×#99—34no—
120#3938—38no—
121#3026—151noSylow
122C23×C3×C5—21no—
123C23×C3×C5—21no—
124C23×C3×C5—81no—
125C23×C3×C5—81no—
126C23×C3×C5C2×D412yes—
127C23×C3×C5C2×D411yescharacteristic
128C23×C3×C5C2×D411yescharacteristic
129C23×C3×C5C2×D412yes—
130C23×C3×C5C2×D412yes—
131C23×C3×C5C2411yescharacteristic
132C22×D15—22no—
133C22×D15—24no—
134C22×D15—22no—
135C22×D15—82no—
136C22×D15—82no—
137C22×D15C2×D414yes—
138C22×C3×D5—22no—
139C22×C3×D5—21no—
140C22×C3×D5—81no—
141C3×C5×D4—24no—
142C3×C5×D4—24no—
143C2×C3×C4×C5—21no—
144C2×C3×C4×C5—21no—
145C2×C3×C4×C5C2×D412yes—
146#1220—28no—
147#1220—24no—
148#1220—24no—
149#1220—24no—
150#1220—24no—
151C2×Dic15—24no—
152C2×Dic15—22no—
153C2×Dic15—22no—
154C2×Dic15C2×D414yes—
155D60—24no—
156C4×D15—28no—
157Dic30—24no—
158C2×C5×Dic3—24no—
159C2×C5×Dic3—82no—
160C3×C22⋊D5—41no—
161C3×C22⋊D5—42no—
162C3×C22⋊D5—41no—
163C3×C22⋊D5—82no—
164C2×C3×Dic5—22no—
165C2×C3×Dic5—21no—
166C2×C3×Dic5—22no—
167C2×C3×Dic5—81no—
168#1202—84no—
169C22×C3×D4—51no—
170C22×C3×D4C2×D511yescharacteristic
171C22×#67—52no—
172C22×#67—101no—
173#1004—54no—
174#1004—104no—
175#1004—102no—
176C2×#237—102no—
177C2×D3×D4—102no—
178C2×#235—101no—
179C3×#123—52no—
180C3×#123—101no—
181C3×#121—102no—
182C3×#120—52no—
183C3×#120—101no—
184C2×C3×C22⋊C4—51no—
185#953—54no—
186C2×#217C2×D512yescomplemented
187#939—104no—
188#937—102no—
189#935—102no—
190#929—54no—
191#834C2×D514yescomplemented
192#806—58no—
193C24×C5#6711yescharacteristic, complemented
194C24×C5C22×D311yescharacteristic
195C23×D5—32no—
196C23×D5—61no—
197C23×D5C22×D311yescharacteristic, complemented
198C2×C5×D4—22no—
199C2×C5×D4—22no—
200C2×C5×D4#6712yescomplemented
201C2×C5×D4C22×D312yes—
202C22×C4×C5—62no—
203C22×C4×C5#6711yescharacteristic, complemented
204C2×C22⋊D5—22no—
205C2×C22⋊D5—22no—
206C2×C22⋊D5—21no—
207C2×C22⋊D5—21no—
208C2×C22⋊D5—22no—
209C2×C22⋊D5—34no—
210C2×C22⋊D5—34no—
211C2×C22⋊D5—62no—
212C2×C22⋊D5—64no—
213C2×C22⋊D5—62no—
214C2×C22⋊D5—62no—
215C2×C22⋊D5—62no—
216C2×C22⋊D5—64no—
217C2×C22⋊D5—62no—
218C22×Dic5—21no—
219C22×Dic5—32no—
220C22×Dic5—61no—
221C22×Dic5C22×D311yescharacteristic, complemented
222#711—68no—
223#711—68no—
224D4×D5—68no—
225D4×D5—68no—
226#709—64no—
227#709—64no—
228C2×D20—61no—
229C2×D20—64no—
230C2×C4×D5—62no—
231C2×Dic10—61no—
232C5×C22⋊C4—34no—
233C5×C22⋊C4—62no—
234C22⋊Dic5—21no—
235C22⋊Dic5—22no—
236C22⋊Dic5C22×D312yescomplemented
237#685—22no—
238#685—64no—
239#685—64no—
240C4⋊Dic5—64no—
241#683—22no—
242C2×H(5,𝔽2)—151no—
243#534—51no—
244#534—152no—
245#458—154no—
246B(4,𝔽2)—154no—
247C2×#99—31no—
248C2×#99—152no—
249C22×C3×C5—22no—
250C22×C3×C5—41no—
251C22×C3×C5—42no—
252C22×C3×C5—41no—
253C22×C3×C5—42no—
254C22×C3×C5—41no—
255C22×C3×C5—42no—
256C22×C3×C5—42no—
257C22×C3×C5—81no—
258C22×C3×C5#12011yescharacteristic
259C22×C3×C5#12012yes—
260C22×C3×C5#12011yescharacteristic
261C22×C3×C5C22×D411yescharacteristic
262C22×C3×C5C22×D412yes—
263C2×D15—24no—
264C2×D15—44no—
265C2×D15—44no—
266C2×D15—82no—
267C2×D15—84no—
268C2×C3×D5—44no—
269C2×C3×D5—42no—
270C2×C3×D5—41no—
271C3×C4×C5—22no—
272Dic15—24no—
273C3×Dic5—42no—
274C3×Dic5—41no—
275C5×Dic3—44no—
276C24×C3—51no—
277C24×C3D2011yescharacteristic, complemented
278C24×C3C22×D511yescharacteristic
279C23×D3—52no—
280C23×D3—101no—
281C2×C3×D4—22no—
282C2×C3×D4—22no—
283C2×C3×D4—101no—
284C2×C3×D4—101no—
285C2×C3×D4—102no—
286C2×C3×D4—102no—
287C2×C3×D4—102no—
288C2×C3×D4D2012yescomplemented
289C2×C3×D4C22×D512yes—
290C22×C3×C4—51no—
291C22×C3×C4—101no—
292C22×C3×C4D2011yescharacteristic, complemented
293C2×#67—54no—
294C2×#67—54no—
295C2×#67—104no—
296C2×#67—102no—
297C2×#67—104no—
298C2×#67—102no—
299C2×#67—104no—
300C2×#67—102no—
301C2×#67—102no—
302C2×#67—102no—
303C22×Dic3—22no—
304C22×Dic3—52no—
305C22×Dic3—101no—
306#237—108no—
307#237—108no—
308D3×D4—108no—
309D3×D4—108no—
310#235—104no—
311#235—104no—
312C2×D12—101no—
313C2×C4×D3—102no—
314C2×Dic6—101no—
315C3×C4⋊C4—102no—
316C3×C22⋊C4—52no—
317C3×C22⋊C4—101no—
318C3×C22⋊C4—102no—
319C3×C22⋊C4—102no—
320#217—22no—
321#217—104no—
322#217C22×D514yescomplemented
323(C2×D3)⋊C4—104no—
324Dic3⋊C4—104no—
325C23×C5—21no—
326C23×C5—21no—
327C23×C5—81no—
328C23×C5—81no—
329C23×C5D3×D412yes—
330C23×C5D3×D412yes—
331C23×C5C2×#6712yes—
332C23×C5C2×#6711yescharacteristic
333C23×C5C2×#6711yescharacteristic
334C23×C5C23×D311yescharacteristic
335C22×D5—22no—
336C22×D5—21no—
337C22×D5—34no—
338C22×D5—62no—
339C22×D5—64no—
340C22×D5—62no—
341C22×D5—81no—
342C22×D5—242no—
343C22×D5—242no—
344C5×D4—24no—
345C5×D4—24no—
346C2×C4×C5—21no—
347C2×C4×C5—21no—
348C2×C4×C5—64no—
349C2×C4×C5—242no—
350C2×C4×C5C2×#6712yes—
351C22⋊D5—41no—
352C22⋊D5—42no—
353C22⋊D5—41no—
354C22⋊D5—68no—
355C22⋊D5—64no—
356C22⋊D5—64no—
357C22⋊D5—64no—
358C22⋊D5—64no—
359C22⋊D5—82no—
360C2×Dic5—22no—
361C2×Dic5—21no—
362C2×Dic5—22no—
363C2×Dic5—34no—
364C2×Dic5—64no—
365C2×Dic5—62no—
366C2×Dic5—62no—
367C2×Dic5—81no—
368D20—64no—
369D20—244no—
370C4×D5—68no—
371Dic10—64no—
372H(5,𝔽2)—154no—
373H(5,𝔽2)—304no—
374H(5,𝔽2)—302no—
375C2×#41—302no—
376C2×#41—301no—
377C22×D4—51no—
378C22×D4—152no—
379C22×D4—302no—
380C22×D4—301no—
381C22×D4D3×D511yescharacteristic
382#123—52no—
383#123—101no—
384#123—154no—
385#123—302no—
386#121—102no—
387#121—304no—
388#120—52no—
389#120—101no—
390#120—154no—
391#120—302no—
392C2×C22⋊C4—32no—
393C2×C22⋊C4—51no—
394#99—34no—
395#99—158no—
396C2×C3×C5—22no—
397C2×C3×C5—21no—
398C2×C3×C5—21no—
399C2×C3×C5—22no—
400C2×C3×C5—21no—
401C2×C3×C5—42no—
402C2×C3×C5C2×#12011yescharacteristic
403D15—44no—
404C3×D5—81no—
405C23×C3—21no—
406C23×C3—21no—
407C23×C3—81no—
408C23×C3—81no—
409C23×C3—101no—
410C23×C3—102no—
411C23×C3—401no—
412C23×C3C2×D2012yes—
413C23×C3C2×D2011yescharacteristic
414C23×C3C2×D2011yescharacteristic
415C23×C3D4×D512yes—
416C23×C3D4×D512yes—
417C23×C3C23×D511yescharacteristic
418C22×D3—54no—
419C22×D3—102no—
420C22×D3—104no—
421C22×D3—102no—
422C22×D3—402no—
423C22×D3—402no—
424C3×D4—24no—
425C3×D4—24no—
426C3×D4—201no—
427C3×D4—202no—
428C3×D4—201no—
429C3×D4—402no—
430C2×C3×C4—21no—
431C2×C3×C4—21no—
432C2×C3×C4—102no—
433C2×C3×C4—101no—
434C2×C3×C4—102no—
435C2×C3×C4—401no—
436C2×C3×C4C2×D2012yes—
437#67—108no—
438#67—104no—
439#67—104no—
440#67—104no—
441#67—104no—
442#67—404no—
443C2×Dic3—24no—
444C2×Dic3—54no—
445C2×Dic3—82no—
446C2×Dic3—104no—
447C2×Dic3—102no—
448C2×Dic3—102no—
449D12—104no—
450C4×D3—108no—
451Dic6—104no—
452C22×C5—22no—
453C22×C5—41no—
454C22×C5—42no—
455C22×C5—41no—
456C22×C5—42no—
457C22×C5—41no—
458C22×C5—42no—
459C22×C5—42no—
460C22×C5—81no—
461C22×C5#93711yescharacteristic
462C22×C5#93712yes—
463C22×C5#93711yescharacteristic
464C22×C5C2×D3×D412yes—
465C22×C5C22×#6711yescharacteristic
466C2×D5—44no—
467C2×D5—42no—
468C2×D5—41no—
469C2×D5—64no—
470C2×D5—124no—
471C2×D5—124no—
472C2×D5—242no—
473C2×D5—244no—
474C4×C5—22no—
475C4×C5—124no—
476Dic5—42no—
477Dic5—41no—
478Dic5—64no—
479C24—51no—
480C24—152no—
481C24—301no—
482C24#120211yescharacteristic, complemented
483C24C2×D3×D511yescharacteristic
484#41—304no—
485#41—308no—
486#41—308no—
487#41—304no—
488C2×Q8—301no—
489C2×D4—22no—
490C2×D4—22no—
491C2×D4—101no—
492C2×D4—101no—
493C2×D4—102no—
494C2×D4—102no—
495C2×D4—102no—
496C2×D4—154no—
497C2×D4—154no—
498C2×D4—301no—
499C2×D4—304no—
500C2×D4—308no—
501C2×D4—308no—
502C2×D4—302no—
503C2×D4—304no—
504C2×D4—302no—
505C2×D4—302no—
506C2×D4—304no—
507C2×D4—302no—
508C2×D4—302no—
509C2×D4#120212yescomplemented
510C2×D4C2×D3×D512yes—
511C22×C4—51no—
512C22×C4—62no—
513C22×C4—101no—
514C22×C4—152no—
515C22×C4—302no—
516C22×C4—301no—
517C22×C4#120211yescharacteristic, complemented
518C4⋊C4—102no—
519C4⋊C4—304no—
520C22⋊C4—34no—
521C22⋊C4—52no—
522C22⋊C4—62no—
523C22⋊C4—101no—
524C22⋊C4—102no—
525C22⋊C4—102no—
526C22⋊C4—304no—
527C22⋊C4—304no—
528C3×C5#302611yescharacteristic, complemented, Hall
529C22×C3—22no—
530C22×C3—42no—
531C22×C3—41no—
532C22×C3—42no—
533C22×C3—41no—
534C22×C3—41no—
535C22×C3—42no—
536C22×C3—42no—
537C22×C3—81no—
538C22×C3—204no—
539C22×C3—202no—
540C22×C3—201no—
541C22×C3#402811yescharacteristic
542C22×C3#402812yes—
543C22×C3#402811yescharacteristic
544C22×C3C22×D2011yescharacteristic
545C22×C3C2×D4×D512yes—
546C2×D3—104no—
547C2×D3—204no—
548C2×D3—204no—
549C2×D3—402no—
550C2×D3—404no—
551C3×C4—22no—
552C3×C4—202no—
553C3×C4—201no—
554Dic3—44no—
555Dic3—104no—
556C2×C5—21no—
557C2×C5—22no—
558C2×C5—21no—
559C2×C5—22no—
560C2×C5—21no—
561C2×C5—42no—
562C2×C5C2×#93711yescharacteristic
563D5—81no—
564D5—124no—
565C23—21no—
566C23—21no—
567C23—81no—
568C23—81no—
569C23—101no—
570C23—102no—
571C23—154no—
572C23—304no—
573C23—302no—
574C23—302no—
575C23—401no—
576C23—1202no—
577C23—1202no—
578C23C2×#120212yes—
579C23C2×#120211yescharacteristic
580C23C2×#120211yescharacteristic
581C23#679012yes—
582C23#679012yes—
583C23C22×D3×D511yescharacteristic
584Q8—304no—
585D4—24no—
586D4—24no—
587D4—201no—
588D4—201no—
589D4—202no—
590D4—304no—
591D4—308no—
592D4—304no—
593D4—304no—
594D4—304no—
595D4—304no—
596D4—402no—
597D4—1204no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—64no—
601C2×C4—102no—
602C2×C4—101no—
603C2×C4—102no—
604C2×C4—154no—
605C2×C4—242no—
606C2×C4—308no—
607C2×C4—304no—
608C2×C4—302no—
609C2×C4—302no—
610C2×C4—401no—
611C2×C4C2×#120212yes—
612C2×C3—22no—
613C2×C3—21no—
614C2×C3—21no—
615C2×C3—22no—
616C2×C3—21no—
617C2×C3—42no—
618C2×C3—401no—
619C2×C3C2×#402811yescharacteristic
620D3—204no—
621C5#8337511yescharacteristic, complemented, Sylow
622C22—22no—
623C22—42no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—81no—
631C22—202no—
632C22—204no—
633C22—201no—
634C22—304no—
635C22—604no—
636C22—604no—
637C22—1202no—
638C22—1204no—
639C22#9157211yescharacteristic
640C22#9157211yescharacteristic
641C22#9157212yes—
642C22C22×#120211yescharacteristic
643C22C2×#679012yes—
644C4—22no—
645C4—124no—
646C4—202no—
647C4—201no—
648C4—304no—
649C3#1061041711yescharacteristic, complemented, Sylow
650C2—22no—
651C2—21no—
652C2—21no—
653C2—21no—
654C2—22no—
655C2—42no—
656C2—401no—
657C2—604no—
658C2C2×#9157211yescharacteristic
subgroupG/H
centerC2C2×#91572
commutator, cosocleC23×C3×C5C24
FrattiniC23C22×D3×D5
FittingC22×C3×C5×D4C22
socleC2×C3×C5C2×#120
hypercenterC22×D4D3×D5
hypocenterC3×C5#3026
LCSG ▷ C23×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C24 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×#92095C2semidirect
2. C3×#66138C2semidirect
3. #11755427C2semidirect
4. C2×C5×#834C2semidirect
5. C2×#90997C2semidirect
6. #11749185C2semidirect
7. #11749179C2semidirect
18. C22×C3×C22⋊D5C22semidirect
19. C2×C5×#217C22semidirect
21. C2×C3×C22⋊Dic5C22semidirect
23. C3×#4083C22semidirect
24. C3×#4081C22semidirect
28. #91564C22semidirect
29. #91546C22semidirect
31. C5×#834C22semidirect
32. #90997C22semidirect
35. #66138D3semidirect
41. C24×C3×C5D4semidirect
44. C23×D15D4semidirect
45. C23×C3×D5C23semidirect
48. C2×C3×C5×D4D4semidirect
50. C22×C3×C4×C5D4semidirect
58. C2×#1220D4semidirect
61. C22×Dic15D4semidirect
78. C22×C3×Dic5C23semidirect
81. C5×#217C23semidirect
84. C3×C22⋊Dic5C23semidirect
93. C2×#834D5semidirect
100. C22×C22⋊D5C2×D3semidirect
109. C2×C22⋊Dic5C2×D3semidirect
111. #4083C2×D3semidirect
112. #4081C2×D3semidirect
186. C2×#217C2×D5semidirect
191. #834C2×D5semidirect
193. C24×C5#67semidirect
197. C23×D5C22×D3semidirect
200. C2×C5×D4#67semidirect
203. C22×C4×C5#67semidirect
221. C22×Dic5C22×D3semidirect
236. C22⋊Dic5C22×D3semidirect
277. C24×C3D20semidirect
288. C2×C3×D4D20semidirect
292. C22×C3×C4D20semidirect
322. #217C22×D5semidirect
482. C24#1202semidirect
509. C2×D4#1202semidirect
517. C22×C4#1202semidirect
528. C3×C5#3026semidirect
621. C5#83375semidirect
649. C3#10610417semidirect

Maximal subgroups

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

subgroupindex
C2×#920952
C3×#661382
#117554272
C2×C5×#8342
C2×#909972
#117491852
#117491792
#106104173
#833755