Finite Group Databasebeta

← #14480637

Subgroups — #14480637

Order 1920.

3112 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D15C212yesmaximal, complemented
2C3×C5×#537C211yesmaximal, characteristic, complemented
3#11756592C212yesmaximal, complemented
4#11756387C214yesmaximal, complemented
5#11756386C214yesmaximal, complemented
6#11756366C214yesmaximal, complemented
7#11756365C214yesmaximal, complemented
8#11756364C214yesmaximal, complemented
9#11756355C214yesmaximal, complemented
10#11756269C211yesmaximal, characteristic, complemented
11C2×#91781C211yesmaximal, characteristic, complemented
12#10618264—31nomaximal, Hall
13C2×C3×C5×#41C2212yescomplemented
14#41×D15—216no—
15C2×#6821C2212yes—
16C2×Q8×D15C2212yescomplemented
17C2×#6819C2212yes—
18C2×#6819C2214yescomplemented
19C2×D4×D15C2214yes—
20C2×D4×D15C2212yescomplemented
21C2×#6817C2214yes—
22C2×#6817C2212yescomplemented
23C22×C4×D15C2211yescharacteristic
24C22×C4×D15C2214yescomplemented
25C3×C5×C22⋊Q8C2214yescomplemented
26C3×C5×#121C2214yescomplemented
27C3×C5×#121C2214yescomplemented
28C2×C3×C5×C4⋊C4C2211yescharacteristic, complemented
29#91835C2214yescomplemented
30#91829C2214yescomplemented
31#91828C2218yescomplemented
32#91827C2214yescomplemented
33#91826C2218yescomplemented
34D4×Dic15C2214yescomplemented
35#91819C2214yescomplemented
36#91819C2214yescomplemented
37C2×#6714C2212yes—
38C2×C4⋊Dic15C2211yescharacteristic, complemented
39#91813C2214yescomplemented
40C2×C4×Dic15C2212yescomplemented
41#91786C2214yescomplemented
42#91784C2214yescomplemented
43#91783C2218yescomplemented
44#91782C2214yescomplemented
45#91781C2214yescomplemented
46#91781C2214yescomplemented
47#91777C2214yescomplemented
48#91774C2218yescomplemented
49#91772C2218yescomplemented
50#91771C2218yescomplemented
51#91770C2218yescomplemented
52#86363—51nomaximal, Hall
53C2×D5×#41—32no—
54C5×#537D311yescharacteristic, complemented
55#66167—32no—
56#65962—34no—
57#65961—34no—
58#65941—34no—
59#65940—34no—
60#65939—34no—
61#65930—34no—
62#65844—31no—
63C2×#4038—31no—
64C23×D15C2314yes—
65C23×D15C2311yescharacteristic
66C3×C5×#41—28no—
67C2×C3×C5×Q8C2312yes—
68C2×C3×C5×D4C2312yes—
69C2×C3×C5×D4C2314yes—
70C2×C3×C5×D4C2314yes—
71C22×C3×C4×C5C2314yes—
72C22×C3×C4×C5C2312yes—
73C22×C3×C4×C5C2311yescharacteristic
74C2×#1220C2314yes—
75C2×#1220C2314yes—
76C2×#1220C2314yes—
77C2×#1220C2318yes—
78C22×Dic15C2314yes—
79C22×Dic15C2312yes—
80C22×Dic15C2311yescharacteristic
81#6821—28no—
82Q8×D15—28no—
83#6819—216no—
84#6819—28no—
85D4×D15—216no—
86D4×D15—28no—
87#6817—28no—
88#6817—216no—
89C2×D60C2314yes—
90C2×D60C2312yes—
91C2×D60C2314yes—
92C2×C4×D15—216no—
93C2×C4×D15D418yescomplemented
94C2×C4×D15C2314yes—
95C2×C4×D15C2314yes—
96C2×C4×D15C2314yes—
97C2×C4×D15C2312yes—
98C2×C4×D15C2312yes—
99C2×Dic30C2314yes—
100C2×Dic30C2312yes—
101C3×C5×C4⋊C4C2314yes—
102C3×C5×C4⋊C4C2314yes—
103C3×C5×C22⋊C4C2318yes—
104#6719C2318yes—
105#6714C2318yes—
106#6714C2318yes—
107#6714C2318yes—
108C4⋊Dic15C2314yes—
109C4⋊Dic15C2314yes—
110#6712C2318yes—
111C4×Dic15C2314yes—
112C4×Dic15C2314yes—
113C2×D3×#41—52no—
114C3×#537D511yescharacteristic, complemented
115#5833—52no—
116#5628—54no—
117#5627—54no—
118#5607—54no—
119#5606—54no—
120#5605—54no—
121#5596—54no—
122#5510—51no—
123C2×#892—51no—
124C2×C5×#41C2×D312yescomplemented
125D5×#41—616no—
126C2×#713—32no—
127C2×Q8×D5—32no—
128C2×#711—34no—
129C2×#711—32no—
130C2×D4×D5—34no—
131C2×D4×D5—32no—
132C2×#709—32no—
133C2×#709—34no—
134C22×C4×D5—31no—
135C22×C4×D5—34no—
136C5×C22⋊Q8C2×D314yescomplemented
137C5×#121C2×D314yescomplemented
138C5×#121C2×D314yescomplemented
139C2×C5×C4⋊C4C2×D311yescharacteristic, complemented
140#4092—34no—
141#4086—34no—
142#4085—38no—
143#4084—34no—
144#4083—38no—
145D4×Dic5—34no—
146#4076—34no—
147#4076—34no—
148C2×#685—32no—
149C2×C4⋊Dic5—31no—
150#4070—34no—
151C2×C4×Dic5—32no—
152#4043—34no—
153#4041—34no—
154#4040—38no—
155#4039—34no—
156#4038—34no—
157#4038—34no—
158#4034—34no—
159#4031—38no—
160#4029—38no—
161#4028—38no—
162#4027—38no—
163#3449—151noSylow
164C23×C3×C5C2414yes—
165C23×C3×C5C2411yescharacteristic
166C22×D15—28no—
167C22×D15—41no—
168C22×D15—41no—
169C22×D15—44no—
170C22×D15—44no—
171C22×D15C2×D414yes—
172C22×D15C2414yes—
173C22×D15C2412yes—
174C3×C5×Q8—24no—
175C3×C5×D4—28no—
176C3×C5×D4—24no—
177C3×C5×D4—44no—
178C2×C3×C4×C5—28no—
179C2×C3×C4×C5—42no—
180C2×C3×C4×C5C2×D414yes—
181C2×C3×C4×C5C2414yes—
182C2×C3×C4×C5C2414yes—
183C2×C3×C4×C5C2412yes—
184#1220—28no—
185#1220—28no—
186#1220—28no—
187#1220—48no—
188C2×Dic15—28no—
189C2×Dic15—42no—
190C2×Dic15C2×D414yes—
191C2×Dic15C2414yes—
192C2×Dic15C2414yes—
193C2×Dic15C2411yescharacteristic
194C2×Dic15C2411yescharacteristic
195D60—28no—
196D60—24no—
197D60—44no—
198C4×D15—28no—
199C4×D15—28no—
200C4×D15—28no—
201C4×D15C2×D418yes—
202C4×D15C2×D418yes—
203Dic30—24no—
204Dic30—28no—
205C2×C3×#41C2×D512yescomplemented
206D3×#41—1016no—
207C2×#239—52no—
208C2×D3×Q8—52no—
209C2×#237—52no—
210C2×#237—54no—
211C2×D3×D4—54no—
212C2×D3×D4—52no—
213C2×#235—52no—
214C2×#235—54no—
215C22×C4×D3—51no—
216C22×C4×D3—54no—
217C3×C22⋊Q8C2×D514yescomplemented
218C3×#121C2×D514yescomplemented
219C3×#121C2×D514yescomplemented
220C2×C3×C4⋊C4C2×D511yescharacteristic, complemented
221#946—54no—
222#940—54no—
223#939—58no—
224#938—54no—
225#937—58no—
226D4×Dic3—54no—
227#930—54no—
228#930—54no—
229C2×(C2×D3)⋊C4—52no—
230C2×C4⋊Dic3—51no—
231#924—54no—
232C2×C4×Dic3—52no—
233#897—54no—
234#895—54no—
235#894—58no—
236#893—54no—
237#892—54no—
238#892—54no—
239#888—54no—
240#885—58no—
241#883—58no—
242#882—58no—
243#67⋊C4—58no—
244C23×D5—31no—
245C23×D5—34no—
246C5×#41—28no—
247C2×C5×Q8C22×D312yes—
248C2×C5×D4C22×D312yes—
249C2×C5×D4C22×D314yes—
250C2×C5×D4C22×D314yes—
251C22×C4×C5C22×D314yes—
252C22×C4×C5C22×D312yes—
253C22×C4×C5C22×D311yescharacteristic
254C2×C22⋊D5—34no—
255C2×C22⋊D5—34no—
256C2×C22⋊D5—34no—
257C2×C22⋊D5—38no—
258C22×Dic5—31no—
259C22×Dic5—34no—
260C22×Dic5—32no—
261#713—68no—
262Q8×D5—68no—
263#711—68no—
264#711—616no—
265D4×D5—68no—
266D4×D5—616no—
267#709—616no—
268#709—68no—
269C2×D20—34no—
270C2×D20—32no—
271C2×D20—34no—
272C2×C4×D5—38no—
273C2×C4×D5—32no—
274C2×C4×D5—32no—
275C2×C4×D5—34no—
276C2×C4×D5—34no—
277C2×C4×D5—34no—
278C2×C4×D5—616no—
279C2×Dic10—34no—
280C2×Dic10—32no—
281C5×C4⋊C4C22×D314yes—
282C5×C4⋊C4C22×D314yes—
283C5×C22⋊C4C22×D318yes—
284C22⋊Dic5—38no—
285#685—38no—
286#685—38no—
287#685—38no—
288C4⋊Dic5—34no—
289C4⋊Dic5—34no—
290#683—38no—
291C4×Dic5—34no—
292C4×Dic5—34no—
293C22×#41—152no—
294#537—152no—
295#537—151no—
296#537D1511yescharacteristic, complemented
297#536—154no—
298#535—154no—
299#534—154no—
300#532—154no—
301#525—154no—
302#525—154no—
303C2×#117—151no—
304C22×C3×C5—21no—
305C22×C3×C5—21no—
306C22×C3×C5—24no—
307C22×C3×C5—24no—
308C22×C3×C5—24no—
309C22×C3×C5C22×D412yes—
310C22×C3×C5C2511yescharacteristic
311C2×D15—22no—
312C2×D15—22no—
313C2×D15—24no—
314C2×D15—24no—
315C2×D15—24no—
316C2×D15—44no—
317C2×D15—416no—
318C2×D15C22×D414yes—
319C3×C4×C5—24no—
320C3×C4×C5—24no—
321C3×C4×C5C22×D414yes—
322Dic15—24no—
323Dic15—24no—
324Dic15C22×D412yes—
325Dic15C22×D412yes—
326C23×D3—51no—
327C23×D3—54no—
328C3×#41—28no—
329C2×C3×Q8C22×D512yes—
330C2×C3×D4C22×D512yes—
331C2×C3×D4C22×D514yes—
332C2×C3×D4C22×D514yes—
333C22×C3×C4C22×D514yes—
334C22×C3×C4C22×D512yes—
335C22×C3×C4C22×D511yescharacteristic
336C2×#67—54no—
337C2×#67—54no—
338C2×#67—54no—
339C2×#67—58no—
340C22×Dic3—51no—
341C22×Dic3—54no—
342C22×Dic3—52no—
343#239—108no—
344D3×Q8—108no—
345#237—108no—
346#237—1016no—
347D3×D4—108no—
348D3×D4—1016no—
349#235—1016no—
350#235—108no—
351C2×D12—54no—
352C2×D12—52no—
353C2×D12—54no—
354C2×C4×D3—58no—
355C2×C4×D3—52no—
356C2×C4×D3—52no—
357C2×C4×D3—54no—
358C2×C4×D3—54no—
359C2×C4×D3—54no—
360C2×C4×D3—1016no—
361C2×Dic6—54no—
362C2×Dic6—52no—
363C3×C4⋊C4C22×D514yes—
364C3×C4⋊C4C22×D514yes—
365C3×C22⋊C4C22×D518yes—
366#217—58no—
367(C2×D3)⋊C4—58no—
368(C2×D3)⋊C4—58no—
369(C2×D3)⋊C4—58no—
370C4⋊Dic3—54no—
371C4⋊Dic3—54no—
372Dic3⋊C4—58no—
373C4×Dic3—54no—
374C4×Dic3—54no—
375C23×C5C23×D314yes—
376C23×C5C23×D311yescharacteristic
377C22×D5—32no—
378C22×D5—34no—
379C22×D5—34no—
380C22×D5—68no—
381C22×D5—121no—
382C22×D5—124no—
383C22×D5—121no—
384C22×D5—124no—
385C5×Q8—24no—
386C5×D4—28no—
387C5×D4—24no—
388C5×D4—44no—
389C2×C4×C5—28no—
390C2×C4×C5—42no—
391C2×C4×C5D3×D414yes—
392C2×C4×C5C23×D314yes—
393C2×C4×C5C23×D314yes—
394C2×C4×C5C23×D312yes—
395C22⋊D5—68no—
396C22⋊D5—68no—
397C22⋊D5—68no—
398C22⋊D5—128no—
399C2×Dic5—34no—
400C2×Dic5—31no—
401C2×Dic5—31no—
402C2×Dic5—34no—
403C2×Dic5—34no—
404C2×Dic5—68no—
405C2×Dic5—122no—
406D20—68no—
407D20—64no—
408D20—124no—
409C4×D5—38no—
410C4×D5—38no—
411C4×D5—68no—
412C4×D5—68no—
413C4×D5—68no—
414Dic10—68no—
415Dic10—64no—
416C2×#41—152no—
417C2×#41—152no—
418C2×#41—154no—
419C2×#41—154no—
420C2×#41—152no—
421C2×#41—3016no—
422C2×#41C2×D1512yescomplemented
423C22×Q8—152no—
424C22×D4—154no—
425C22×D4—152no—
426C23×C4—151no—
427C23×C4—154no—
428C4⋊Q8—154no—
429#127—154no—
430#124—158no—
431#123—158no—
432#123—158no—
433C22⋊Q8—154no—
434C22⋊Q8—154no—
435C22⋊Q8—154no—
436C22⋊Q8C2×D1514yescomplemented
437#121—154no—
438#121—158no—
439#121—154no—
440#121—154no—
441#121—154no—
442#121C2×D1514yescomplemented
443#121C2×D1514yescomplemented
444#120—158no—
445#120—158no—
446C4×D4—154no—
447C4×D4—154no—
448C4×D4—158no—
449#117—154no—
450#117—154no—
451C2×C4⋊C4—151no—
452C2×C4⋊C4C2×D1511yescharacteristic, complemented
453C2×C22⋊C4—152no—
454C2×C42—152no—
455C2×C3×C5—22no—
456C2×C3×C5—44no—
457C2×C3×C5C23×D411yescharacteristic
458C2×C3×C5#58511yescharacteristic
459C2×C3×C5#58511yescharacteristic
460D15—24no—
461D15—44no—
462C23×C3C23×D514yes—
463C23×C3C23×D511yescharacteristic
464C22×D3—52no—
465C22×D3—54no—
466C22×D3—54no—
467C22×D3—108no—
468C22×D3—201no—
469C22×D3—204no—
470C22×D3—201no—
471C22×D3—204no—
472C3×Q8—24no—
473C3×D4—28no—
474C3×D4—24no—
475C3×D4—44no—
476C2×C3×C4—28no—
477C2×C3×C4—42no—
478C2×C3×C4D4×D514yes—
479C2×C3×C4C23×D514yes—
480C2×C3×C4C23×D514yes—
481C2×C3×C4C23×D512yes—
482#67—108no—
483#67—108no—
484#67—108no—
485#67—208no—
486C2×Dic3—54no—
487C2×Dic3—51no—
488C2×Dic3—51no—
489C2×Dic3—54no—
490C2×Dic3—54no—
491C2×Dic3—108no—
492C2×Dic3—202no—
493D12—108no—
494D12—104no—
495D12—204no—
496C4×D3—58no—
497C4×D3—58no—
498C4×D3—108no—
499C4×D3—108no—
500C4×D3—108no—
501Dic6—108no—
502Dic6—104no—
503C22×C5—21no—
504C22×C5—21no—
505C22×C5—24no—
506C22×C5—24no—
507C22×C5—24no—
508C22×C5C2×D3×D412yes—
509C22×C5C24×D311yescharacteristic
510C2×D5—34no—
511C2×D5—62no—
512C2×D5—62no—
513C2×D5—64no—
514C2×D5—64no—
515C2×D5—64no—
516C2×D5—124no—
517C2×D5—1216no—
518C4×C5—24no—
519C4×C5—24no—
520C4×C5C2×D3×D414yes—
521Dic5—32no—
522Dic5—32no—
523Dic5—64no—
524Dic5—64no—
525C24—151no—
526C24—154no—
527#41—28no—
528#41—3016no—
529#41—3016no—
530#41—308no—
531#41—308no—
532#41—308no—
533C2×Q8—152no—
534C2×Q8—154no—
535C2×Q8—308no—
536C2×Q8C22×D1512yes—
537C2×D4—154no—
538C2×D4—154no—
539C2×D4—154no—
540C2×D4—158no—
541C2×D4—154no—
542C2×D4—152no—
543C2×D4—154no—
544C2×D4—308no—
545C2×D4—3016no—
546C2×D4C22×D1514yes—
547C2×D4C22×D1512yes—
548C2×D4C22×D1514yes—
549C22×C4—151no—
550C22×C4—158no—
551C22×C4—152no—
552C22×C4—152no—
553C22×C4—154no—
554C22×C4—152no—
555C22×C4—154no—
556C22×C4—154no—
557C22×C4—154no—
558C22×C4—3016no—
559C22×C4C22×D1514yes—
560C22×C4C22×D1512yes—
561C22×C4C22×D1511yescharacteristic
562C4⋊C4—154no—
563C4⋊C4—158no—
564C4⋊C4—154no—
565C4⋊C4C22×D1514yes—
566C4⋊C4C22×D1514yes—
567C22⋊C4—158no—
568C22⋊C4—158no—
569C22⋊C4—158no—
570C22⋊C4—158no—
571C22⋊C4C22×D1518yes—
572C42—154no—
573C42—154no—
574C3×C5#344911yescharacteristic, complemented, Hall
575C22×C3—21no—
576C22×C3—21no—
577C22×C3—24no—
578C22×C3—24no—
579C22×C3—24no—
580C22×C3C2×D4×D512yes—
581C22×C3C24×D511yescharacteristic
582C2×D3—54no—
583C2×D3—102no—
584C2×D3—102no—
585C2×D3—104no—
586C2×D3—104no—
587C2×D3—104no—
588C2×D3—204no—
589C2×D3—2016no—
590C3×C4—24no—
591C3×C4—24no—
592C3×C4C2×D4×D514yes—
593Dic3—52no—
594Dic3—52no—
595Dic3—104no—
596Dic3—104no—
597C2×C5—22no—
598C2×C5—44no—
599C2×C5C22×D3×D411yescharacteristic
600C2×C5#596611yescharacteristic
601C2×C5#596811yescharacteristic
602D5—64no—
603D5—124no—
604C23—152no—
605C23—154no—
606C23—154no—
607C23—308no—
608C23—601no—
609C23—601no—
610C23—604no—
611C23—604no—
612C23C23×D1514yes—
613C23C23×D1511yescharacteristic
614Q8—24no—
615Q8—308no—
616Q8—304no—
617D4—28no—
618D4—24no—
619D4—44no—
620D4—308no—
621D4—308no—
622D4—308no—
623D4—308no—
624D4—304no—
625D4—604no—
626D4—608no—
627C2×C4—28no—
628C2×C4—42no—
629C2×C4—154no—
630C2×C4—151no—
631C2×C4—151no—
632C2×C4—158no—
633C2×C4—158no—
634C2×C4—154no—
635C2×C4—154no—
636C2×C4—308no—
637C2×C4—308no—
638C2×C4—308no—
639C2×C4—308no—
640C2×C4—602no—
641C2×C4D4×D1514yes—
642C2×C4C23×D1514yes—
643C2×C4C23×D1514yes—
644C2×C4C23×D1512yes—
645C2×C3—22no—
646C2×C3—44no—
647C2×C3C22×D4×D511yescharacteristic
648C2×C3#6628511yescharacteristic
649C2×C3#6628711yescharacteristic
650D3—104no—
651D3—204no—
652C5#8636311yescharacteristic, complemented, Sylow
653C22—21no—
654C22—21no—
655C22—24no—
656C22—24no—
657C22—24no—
658C22—154no—
659C22—302no—
660C22—302no—
661C22—304no—
662C22—304no—
663C22—304no—
664C22—604no—
665C22—6016no—
666C22C2×D4×D1512yes—
667C22C24×D1511yescharacteristic
668C4—24no—
669C4—24no—
670C4—152no—
671C4—152no—
672C4—304no—
673C4—304no—
674C4C2×D4×D1514yes—
675C3#1061826411yescharacteristic, complemented, Sylow
676C2—22no—
677C2—44no—
678C2—304no—
679C2—604no—
680C2C22×D4×D1511yescharacteristic
681C2#1175714811yescharacteristic
682C2#1175715011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D15
commutator, socle, cosocleC22×C3×C5C25
FittingC3×C5×#537C2
hypercenter#537D15
hypocenterC3×C5#3449
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#41×D15C2semidirect
2. C3×C5×#537C2semidirect
3. #11756592C2semidirect
4. #11756387C2semidirect
5. #11756386C2semidirect
6. #11756366C2semidirect
7. #11756365C2semidirect
8. #11756364C2semidirect
9. #11756355C2semidirect
10. #11756269C2semidirect
11. C2×#91781C2semidirect
13. C2×C3×C5×#41C22semidirect
16. C2×Q8×D15C22semidirect
18. C2×#6819C22semidirect
20. C2×D4×D15C22semidirect
22. C2×#6817C22semidirect
24. C22×C4×D15C22semidirect
25. C3×C5×C22⋊Q8C22semidirect
26. C3×C5×#121C22semidirect
27. C3×C5×#121C22semidirect
28. C2×C3×C5×C4⋊C4C22semidirect
29. #91835C22semidirect
30. #91829C22semidirect
31. #91828C22semidirect
32. #91827C22semidirect
33. #91826C22semidirect
34. D4×Dic15C22semidirect
35. #91819C22semidirect
36. #91819C22semidirect
38. C2×C4⋊Dic15C22semidirect
39. #91813C22semidirect
40. C2×C4×Dic15C22semidirect
41. #91786C22semidirect
42. #91784C22semidirect
43. #91783C22semidirect
44. #91782C22semidirect
45. #91781C22semidirect
46. #91781C22semidirect
47. #91777C22semidirect
48. #91774C22semidirect
49. #91772C22semidirect
50. #91771C22semidirect
51. #91770C22semidirect
54. C5×#537D3semidirect
93. C2×C4×D15D4semidirect
114. C3×#537D5semidirect
124. C2×C5×#41C2×D3semidirect
136. C5×C22⋊Q8C2×D3semidirect
137. C5×#121C2×D3semidirect
138. C5×#121C2×D3semidirect
139. C2×C5×C4⋊C4C2×D3semidirect
205. C2×C3×#41C2×D5semidirect
217. C3×C22⋊Q8C2×D5semidirect
218. C3×#121C2×D5semidirect
219. C3×#121C2×D5semidirect
220. C2×C3×C4⋊C4C2×D5semidirect
296. #537D15semidirect
422. C2×#41C2×D15semidirect
436. C22⋊Q8C2×D15semidirect
442. #121C2×D15semidirect
443. #121C2×D15semidirect
452. C2×C4⋊C4C2×D15semidirect
574. C3×C5#3449semidirect
652. C5#86363semidirect
675. C3#10618264semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D152
C3×C5×#5372
#117565922
#117563872
#117563862
#117563662
#117563652
#117563642
#117563552
#117562692
C2×#917812
#106182643
#863635