Finite Group Databasebeta

← C2×#11753821 #14481155

Subgroups — C2×#11753821 #14481155

Order 1920.

2520 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91784C211yesmaximal, characteristic, complemented
2C2×C5×#890C211yesmaximal, characteristic, complemented
3C2×C3×#4039C211yesmaximal, characteristic, complemented
4C2×#91465C211yesmaximal, characteristic, complemented
5C2×#91459C212yesmaximal, complemented
6C2×#91458C211yesmaximal, characteristic, complemented
7C2×#91450C212yesmaximal, complemented
8C2×#91418C211yesmaximal, characteristic, complemented
9C2×#91386C211yesmaximal, characteristic, complemented
10C2×#91356C212yesmaximal, complemented
11C2×#91348C212yesmaximal, complemented
12#11753821C2116yesmaximal, direct product factor
13C2×#66032—31nomaximal, Hall
14C22×D60C2211yescharacteristic
15C22×D60C2212yes—
16C22×C4×D15C2211yescharacteristic
17C22×C3×D20C2211yescharacteristic
18C22×C3×C4×D5C2212yes—
19C22×#1202C2212yes—
20C22×#1202C2212yes—
21C22×#1202C2212yes—
22C22×#1200C2212yes—
23C2×C3×C5×C4⋊C4C2211yescharacteristic
24C2×#6714C2212yes—
25#91784C2218yescomplemented
26C2×C5×C4⋊Dic3C2212yes—
27C2×C5×C4⋊Dic3C2211yescharacteristic
28C2×C5×Dic3⋊C4C2212yes—
29C2×C4×C5×Dic3C2211yescharacteristic
30C5×#890C2218yescomplemented
31C2×C3×#685C2212yes—
32C2×C3×C4×Dic5C2211yescharacteristic
33C3×#4039C2218yescomplemented
34C2×#6669C2211yescharacteristic
35C2×#6669C2211yescharacteristic
36C2×#6667C2211yescharacteristic
37C2×#6667C2211yescharacteristic
38C2×#6667C2212yes—
39C2×#6665C2212yes—
40#91465C2218yescomplemented
41#91459C22116yescomplemented
42#91458C2218yescomplemented
43#91450C22116yescomplemented
44#91418C2218yescomplemented
45#91386C2218yescomplemented
46#91356C22116yescomplemented
47#91348C22116yescomplemented
48C2×#5562—51nomaximal, Hall
49C2×C5×#125—31no—
50#66032—316no—
51C2×#4041—32no—
52C2×#4041—31no—
53C2×#4041—32no—
54C2×#4041—31no—
55C2×#4040—32no—
56C2×#4040—32no—
57C2×#4039—31no—
58C2×#4039D311yescharacteristic, complemented
59C2×#4020—31no—
60C2×#4018—31no—
61C23×D15C2312yes—
62C23×D15C2311yescharacteristic
63C23×C3×D5C2312yes—
64C22×C3×C4×C5C2312yes—
65C22×C3×C4×C5C2311yescharacteristic
66C22×Dic15C2311yescharacteristic
67C2×D60—24no—
68C2×D60—44no—
69C2×D60C2314yes—
70C2×D60C2318yes—
71C2×C4×D15—24no—
72C2×C4×D15C2314yes—
73C22×C5×Dic3C2311yescharacteristic
74C22×C5×Dic3C2311yescharacteristic
75C22×C5×Dic3C2312yes—
76C22×C3×Dic5C2311yescharacteristic
77C22×C3×Dic5C2311yescharacteristic
78C2×C3×D20—24no—
79C2×C3×D20C2314yes—
80C2×C3×C4×D5—44no—
81C2×C3×C4×D5C2318yes—
82C2×#1202—44no—
83C2×#1202—44no—
84C2×#1202—44no—
85C2×#1202C2318yes—
86C2×#1202C2318yes—
87C2×#1202C2318yes—
88C2×#1200—44no—
89C2×#1200C2318yes—
90C3×C5×C4⋊C4C2314yes—
91#6714C2318yes—
92C5×C4⋊Dic3C2314yes—
93C5×C4⋊Dic3C2318yes—
94C5×Dic3⋊C4C2318yes—
95C4×C5×Dic3C2314yes—
96C3×#685C2318yes—
97C3×C4×Dic5C2314yes—
98#6669C2314yes—
99#6669C2314yes—
100#6667C2318yes—
101#6667C2314yes—
102#6667C2314yes—
103#6665C2318yes—
104C2×C3×C4×D4—51no—
105C2×#940—51no—
106C2×#938—51no—
107C2×#930—52no—
108C2×#929—52no—
109#5562—516no—
110C2×#895—51no—
111C2×#890D511yescharacteristic, complemented
112C2×#884—52no—
113C2×#883—52no—
114C2×C4×D12—51no—
115C2×#872—51no—
116C22×D20—31no—
117C22×D20—32no—
118C22×D20—32no—
119C22×D20—32no—
120C22×D20—32no—
121C22×D20C2×D311yescharacteristic
122C22×C4×D5—31no—
123C22×C4×D5—32no—
124C22×C4×D5C2×D312yes—
125C5×#125—38no—
126C2×C5×C4⋊C4—31no—
127C2×C5×C4⋊C4—32no—
128C2×C5×C4⋊C4—32no—
129C2×C5×C4⋊C4C2×D311yescharacteristic
130C2×C42×C5—31no—
131C2×#685—32no—
132C2×#685—32no—
133C2×#685—31no—
134C2×#685—31no—
135C2×#685—32no—
136C2×#685C2×D312yes—
137C2×#683—31no—
138C2×#683—31no—
139C2×C4×Dic5C2×D311yescharacteristic
140#4041—38no—
141#4041—316no—
142#4041—38no—
143#4041—316no—
144#4040—316no—
145#4040—316no—
146#4039—38no—
147#4039C2×D318yescomplemented
148#4020—38no—
149#4018—38no—
150C2×#540—151noSylow
151C23×C3×C5C2411yescharacteristic
152C22×D15—22no—
153C22×D15—24no—
154C22×D15—24no—
155C22×D15—22no—
156C22×D15—24no—
157C22×D15—22no—
158C22×D15C2412yes—
159C22×D15C2414yes—
160C22×C3×D5—24no—
161C22×C3×D5—24no—
162C22×C3×D5—24no—
163C22×C3×D5C2414yes—
164C2×C3×C4×C5—24no—
165C2×C3×C4×C5#4114yes—
166C2×C3×C4×C5C2414yes—
167C2×C3×C4×C5C2412yes—
168C2×Dic15—22no—
169C2×Dic15C2412yes—
170D60—28no—
171D60—48no—
172C4×D15—28no—
173C2×C5×Dic3—24no—
174C2×C5×Dic3—22no—
175C2×C5×Dic3—22no—
176C2×C5×Dic3C2414yes—
177C2×C5×Dic3C2412yes—
178C2×C5×Dic3C2412yes—
179C2×C3×Dic5—22no—
180C2×C3×Dic5—22no—
181C2×C3×Dic5C2412yes—
182C2×C3×Dic5C2412yes—
183C3×D20—28no—
184C3×C4×D5—48no—
185#1202—48no—
186#1202—48no—
187#1202—48no—
188#1200—48no—
189C22×C3×D4—51no—
190C23×C3×C4—52no—
191C22×#67—52no—
192C22×#67—52no—
193C22×#67—52no—
194C22×D12—51no—
195C22×D12—52no—
196C22×C4×D3—51no—
197C22×C4×D3—52no—
198C3×C4×D4—58no—
199C2×C3×C4⋊C4C2×D511yescharacteristic
200C2×C3×C22⋊C4—52no—
201C2×C3×C42—51no—
202C2×#217—52no—
203#940—58no—
204#938—58no—
205#930—516no—
206#929—516no—
207C2×(C2×D3)⋊C4—52no—
208C2×(C2×D3)⋊C4—51no—
209C2×(C2×D3)⋊C4—51no—
210C2×(C2×D3)⋊C4—52no—
211C2×C4⋊Dic3C2×D511yescharacteristic
212C2×C4⋊Dic3C2×D512yes—
213C2×Dic3⋊C4—51no—
214C2×Dic3⋊C4—51no—
215C2×Dic3⋊C4C2×D512yes—
216C2×C4×Dic3C2×D511yescharacteristic
217#895—58no—
218#890C2×D518yescomplemented
219#884—516no—
220#883—516no—
221C4×D12—58no—
222#872—58no—
223C23×D5—31no—
224C23×D5—32no—
225C23×D5C22×D312yes—
226C22×C4×C5—31no—
227C22×C4×C5—31no—
228C22×C4×C5—32no—
229C22×C4×C5C22×D312yes—
230C22×C4×C5C22×D311yescharacteristic
231C22×Dic5—31no—
232C22×Dic5C22×D311yescharacteristic
233C22×Dic5C22×D311yescharacteristic
234C2×D20—24no—
235C2×D20—34no—
236C2×D20—38no—
237C2×D20—38no—
238C2×D20—38no—
239C2×D20—38no—
240C2×D20—64no—
241C2×D20—124no—
242C2×D20—124no—
243C2×D20—124no—
244C2×D20—124no—
245C2×D20C22×D314yes—
246C2×C4×D5—34no—
247C2×C4×D5—38no—
248C2×C4×D5—44no—
249C2×C4×D5—64no—
250C2×C4×D5—124no—
251C2×C4×D5C22×D318yes—
252C5×C4⋊C4—38no—
253C5×C4⋊C4—34no—
254C5×C4⋊C4—38no—
255C5×C4⋊C4C22×D314yes—
256C42×C5—34no—
257#685—38no—
258#685—38no—
259#685—34no—
260#685—34no—
261#685—38no—
262#685C22×D318yes—
263#683—34no—
264#683—34no—
265C4×Dic5C22×D314yes—
266#540—1516no—
267C2×#127—151no—
268C2×#125—31no—
269C2×#123—152no—
270C2×#123—152no—
271C2×#121—152no—
272C2×#121—152no—
273C2×#121—151no—
274C2×#121—151no—
275C2×C4×D4—51no—
276C2×C4×D4—151no—
277C2×#117—151no—
278C22×C3×C5C2×#4112yes—
279C22×C3×C5H(5,𝔽2)12yes—
280C22×C3×C5H(5,𝔽2)12yes—
281C22×C3×C5C2511yescharacteristic
282C2×D15—22no—
283C2×D15—24no—
284C2×D15—24no—
285C2×D15—22no—
286C2×D15—24no—
287C2×D15—22no—
288C2×D15—44no—
289C2×D15—48no—
290C2×C3×D5—24no—
291C2×C3×D5—24no—
292C2×C3×D5—24no—
293C2×C3×D5—48no—
294C3×C4×C5—24no—
295C3×C4×C5C2×#4114yes—
296Dic15—22no—
297C3×Dic5—22no—
298C3×Dic5—22no—
299C5×Dic3—24no—
300C5×Dic3—22no—
301C5×Dic3—22no—
302C24×C3—52no—
303C23×D3—51no—
304C23×D3—52no—
305C2×C3×D4—54no—
306C2×C3×D4—104no—
307C22×C3×C4—51no—
308C22×C3×C4—51no—
309C22×C3×C4—58no—
310C22×C3×C4—204no—
311C22×C3×C4C22×D512yes—
312C22×C3×C4C22×D511yescharacteristic
313C2×#67—58no—
314C2×#67—58no—
315C2×#67—58no—
316C2×#67—204no—
317C2×#67—204no—
318C2×#67—204no—
319C22×Dic3—51no—
320C22×Dic3C22×D511yescharacteristic
321C22×Dic3C22×D511yescharacteristic
322C22×Dic3C22×D512yes—
323C2×D12—54no—
324C2×D12—58no—
325C2×D12—104no—
326C2×D12—204no—
327C2×C4×D3—54no—
328C2×C4×D3—58no—
329C2×C4×D3—104no—
330C2×C4×D3—204no—
331C3×C4⋊C4C22×D514yes—
332C3×C22⋊C4—58no—
333C3×C42—54no—
334#217—58no—
335(C2×D3)⋊C4—58no—
336(C2×D3)⋊C4—54no—
337(C2×D3)⋊C4—54no—
338(C2×D3)⋊C4—58no—
339C4⋊Dic3C22×D514yes—
340C4⋊Dic3C22×D518yes—
341Dic3⋊C4—54no—
342Dic3⋊C4—54no—
343Dic3⋊C4C22×D518yes—
344C4×Dic3C22×D514yes—
345C23×C5C23×D311yescharacteristic
346C22×D5—24no—
347C22×D5—24no—
348C22×D5—24no—
349C22×D5—32no—
350C22×D5—34no—
351C22×D5—62no—
352C22×D5—64no—
353C22×D5—62no—
354C22×D5—64no—
355C22×D5—64no—
356C22×D5—62no—
357C22×D5C23×D314yes—
358C2×C4×C5—24no—
359C2×C4×C5—32no—
360C2×C4×C5—32no—
361C2×C4×C5—34no—
362C2×C4×C5—62no—
363C2×C4×C5—64no—
364C2×C4×C5—62no—
365C2×C4×C5#23514yes—
366C2×C4×C5C23×D314yes—
367C2×C4×C5C23×D312yes—
368C2×Dic5—22no—
369C2×Dic5—22no—
370C2×Dic5—32no—
371C2×Dic5—62no—
372C2×Dic5C23×D312yes—
373C2×Dic5C23×D312yes—
374D20—28no—
375D20—68no—
376D20—128no—
377D20—128no—
378D20—128no—
379D20—128no—
380C4×D5—48no—
381C4×D5—68no—
382C4×D5—128no—
383C22×D4—51no—
384C22×D4—152no—
385C22×D4—152no—
386C22×D4—152no—
387C22×D4—151no—
388C22×D4—152no—
389C23×C4—52no—
390C23×C4—152no—
391C23×C4—151no—
392#127—158no—
393#125—38no—
394#123—1516no—
395#123—1516no—
396#121—1516no—
397#121—1516no—
398#121—158no—
399#121—158no—
400C4×D4—58no—
401C4×D4—158no—
402#117—158no—
403C2×C4⋊C4—31no—
404C2×C4⋊C4—32no—
405C2×C4⋊C4—32no—
406C2×C4⋊C4—151no—
407C2×C4⋊C4—151no—
408C2×C4⋊C4D3×D511yescharacteristic
409C2×C22⋊C4—52no—
410C2×C22⋊C4—151no—
411C2×C22⋊C4—151no—
412C2×C22⋊C4—152no—
413C2×C22⋊C4—152no—
414C2×C22⋊C4—152no—
415C2×C42—31no—
416C2×C42—51no—
417C2×C3×C5#54014yescomplemented
418C2×C3×C5C22×#4111yescharacteristic
419C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
420C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
421D15—42no—
422D15—44no—
423C3×D5—44no—
424C23×C3—54no—
425C23×C3—104no—
426C23×C3—104no—
427C23×C3—104no—
428C23×C3C23×D511yescharacteristic
429C22×D3—52no—
430C22×D3—54no—
431C22×D3—102no—
432C22×D3—104no—
433C22×D3—104no—
434C22×D3—102no—
435C22×D3—104no—
436C22×D3—102no—
437C3×D4—108no—
438C2×C3×C4—24no—
439C2×C3×C4—52no—
440C2×C3×C4—52no—
441C2×C3×C4—102no—
442C2×C3×C4—102no—
443C2×C3×C4—208no—
444C2×C3×C4#71314yes—
445C2×C3×C4C23×D514yes—
446C2×C3×C4C23×D512yes—
447#67—208no—
448#67—208no—
449#67—208no—
450C2×Dic3—24no—
451C2×Dic3—22no—
452C2×Dic3—22no—
453C2×Dic3—52no—
454C2×Dic3—102no—
455C2×Dic3C23×D512yes—
456C2×Dic3C23×D512yes—
457C2×Dic3C23×D514yes—
458D12—108no—
459D12—208no—
460C4×D3—108no—
461C4×D3—208no—
462C22×C5C2×#23512yes—
463C22×C5#100412yes—
464C22×C5#100912yes—
465C22×C5C24×D311yescharacteristic
466C2×D5—24no—
467C2×D5—24no—
468C2×D5—24no—
469C2×D5—48no—
470C2×D5—62no—
471C2×D5—64no—
472C2×D5—64no—
473C2×D5—62no—
474C2×D5—64no—
475C2×D5—62no—
476C2×D5—124no—
477C2×D5—128no—
478C4×C5—24no—
479C4×C5—62no—
480C4×C5—64no—
481C4×C5—62no—
482C4×C5C2×#23514yes—
483Dic5—22no—
484Dic5—22no—
485Dic5—62no—
486C24—52no—
487C24—151no—
488C24—152no—
489C2×D4—54no—
490C2×D4—104no—
491C2×D4—158no—
492C2×D4—158no—
493C2×D4—158no—
494C2×D4—154no—
495C2×D4—158no—
496C2×D4—304no—
497C2×D4—604no—
498C2×D4—604no—
499C2×D4—604no—
500C2×D4—604no—
501C22×C4—31no—
502C22×C4—31no—
503C22×C4—32no—
504C22×C4—51no—
505C22×C4—51no—
506C22×C4—58no—
507C22×C4—158no—
508C22×C4—151no—
509C22×C4—154no—
510C22×C4—204no—
511C22×C4—304no—
512C22×C4—604no—
513C22×C4C2×D3×D512yes—
514C22×C4C2×D3×D511yescharacteristic
515C4⋊C4—34no—
516C4⋊C4—38no—
517C4⋊C4—38no—
518C4⋊C4—154no—
519C4⋊C4—154no—
520C4⋊C4C2×D3×D514yes—
521C22⋊C4—58no—
522C22⋊C4—154no—
523C22⋊C4—154no—
524C22⋊C4—158no—
525C22⋊C4—158no—
526C22⋊C4—158no—
527C42—34no—
528C42—54no—
529C3×C5C2×#54011yescharacteristic, complemented, Hall
530C22×C3—104no—
531C22×C3—104no—
532C22×C3—104no—
533C22×C3—208no—
534C22×C3C2×#71312yes—
535C22×C3#414912yes—
536C22×C3#414912yes—
537C22×C3C24×D511yescharacteristic
538C2×D3—102no—
539C2×D3—104no—
540C2×D3—104no—
541C2×D3—102no—
542C2×D3—104no—
543C2×D3—102no—
544C2×D3—204no—
545C2×D3—208no—
546C3×C4—24no—
547C3×C4—102no—
548C3×C4—102no—
549C3×C4C2×#71314yes—
550Dic3—24no—
551Dic3—22no—
552Dic3—22no—
553Dic3—102no—
554C2×C5#556214yescomplemented
555C2×C5C22×#23511yescharacteristic
556C2×C5C2×#100411yescharacteristic
557C2×C5C2×#100911yescharacteristic
558D5—44no—
559D5—122no—
560D5—124no—
561C23—54no—
562C23—104no—
563C23—104no—
564C23—104no—
565C23—152no—
566C23—154no—
567C23—302no—
568C23—304no—
569C23—302no—
570C23—304no—
571C23—304no—
572C23—302no—
573C23C22×D3×D511yescharacteristic
574D4—108no—
575D4—308no—
576D4—608no—
577D4—608no—
578D4—608no—
579D4—608no—
580C2×C4—24no—
581C2×C4—32no—
582C2×C4—32no—
583C2×C4—34no—
584C2×C4—52no—
585C2×C4—52no—
586C2×C4—62no—
587C2×C4—64no—
588C2×C4—62no—
589C2×C4—102no—
590C2×C4—102no—
591C2×C4—152no—
592C2×C4—208no—
593C2×C4—302no—
594C2×C4—308no—
595C2×C4—608no—
596C2×C4#677314yes—
597C2×C4C22×D3×D514yes—
598C2×C4C22×D3×D512yes—
599C2×C3—204no—
600C2×C3#6603214yescomplemented
601C2×C3C22×#71311yescharacteristic
602C2×C3C2×#414911yescharacteristic
603C2×C3C2×#414911yescharacteristic
604D3—202no—
605D3—204no—
606C5C2×#556211yescharacteristic, complemented, Sylow
607C22—104no—
608C22—104no—
609C22—104no—
610C22—208no—
611C22—302no—
612C22—304no—
613C22—302no—
614C22—304no—
615C22—304no—
616C22—302no—
617C22—604no—
618C22—608no—
619C22C2×#677312yes—
620C22#9202612yes—
621C22#9203512yes—
622C22C23×D3×D511yescharacteristic
623C4—24no—
624C4—62no—
625C4—64no—
626C4—62no—
627C4—102no—
628C4—102no—
629C4—302no—
630C4C2×#677314yes—
631C3C2×#6603211yescharacteristic, complemented, Sylow
632C2—204no—
633C2—602no—
634C2—604no—
635C2#1175382114yesdirect product factor
636C2C22×#677311yescharacteristic
637C2C2×#9202611yescharacteristic
638C2C2×#9203511yescharacteristic
subgroupG/H
centerC23C22×D3×D5
commutator, cosocleC22×C3×C5C25
FrattiniC22C23×D3×D5
FittingC2×C3×C5×C4⋊C4C22
socleC23×C3×C5C24
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5C2×#540
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#91784C2semidirect
2. C2×C5×#890C2semidirect
3. C2×C3×#4039C2semidirect
4. C2×#91465C2semidirect
5. C2×#91459C2semidirect
6. C2×#91458C2semidirect
7. C2×#91450C2semidirect
8. C2×#91418C2semidirect
9. C2×#91386C2semidirect
10. C2×#91356C2semidirect
11. C2×#91348C2semidirect
12. #11753821C2direct (N × H)
25. #91784C22semidirect
30. C5×#890C22semidirect
33. C3×#4039C22semidirect
40. #91465C22semidirect
41. #91459C22semidirect
42. #91458C22semidirect
43. #91450C22semidirect
44. #91418C22semidirect
45. #91386C22semidirect
46. #91356C22semidirect
47. #91348C22semidirect
58. C2×#4039D3semidirect
111. C2×#890D5semidirect
147. #4039C2×D3semidirect
218. #890C2×D5semidirect
417. C2×C3×C5#540semidirect
529. C3×C5C2×#540semidirect
554. C2×C5#5562semidirect
600. C2×C3#66032semidirect
606. C5C2×#5562semidirect
631. C3C2×#66032semidirect
635. C2#11753821direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#917842
C2×C5×#8902
C2×C3×#40392
C2×#914652
C2×#914592
C2×#914582
C2×#914502
C2×#914182
C2×#913862
C2×#913562
C2×#913482
#117538212
C2×#660323
C2×#55625