Finite Group Databasebeta

← C2×#11756321 #14481084

Subgroups — C2×#11756321 #14481084

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×C3×C4×C5×D4C211yesmaximal, characteristic, complemented
2C2×#91829C211yesmaximal, characteristic, complemented
3C2×#91827C211yesmaximal, characteristic, complemented
4C2×#91819C212yesmaximal, complemented
5C2×#91818C212yesmaximal, complemented
6#11756321C2116yesmaximal, direct product factor
7C2×#91784C211yesmaximal, characteristic, complemented
8C2×#91779C211yesmaximal, characteristic, complemented
9C2×#91773C212yesmaximal, complemented
10C2×#91772C212yesmaximal, complemented
11C2×C4×D60C211yesmaximal, characteristic, complemented
12C2×#91761C211yesmaximal, characteristic, complemented
13C2×#65896—31nomaximal, Hall
14C22×C3×C5×D4C2211yescharacteristic
15C23×C3×C4×C5C2212yes—
16C22×#1220C2212yes—
17C22×#1220C2212yes—
18C22×#1220C2212yes—
19C22×D60C2211yescharacteristic
20C22×D60C2212yes—
21C22×C4×D15C2211yescharacteristic
22C22×C4×D15C2212yes—
23C3×C4×C5×D4C2218yescomplemented
24C2×C3×C5×C4⋊C4C2211yescharacteristic
25C2×C3×C5×C22⋊C4C2212yes—
26C2×C3×C42×C5C2211yescharacteristic
27C2×#6719C2212yes—
28#91829C2218yescomplemented
29#91827C2218yescomplemented
30#91819C22116yescomplemented
31#91818C22116yescomplemented
32C2×#6714C2212yes—
33C2×#6714C2212yes—
34C2×#6714C2211yescharacteristic
35C2×#6714C2211yescharacteristic
36C2×C4⋊Dic15C2212yes—
37C2×C4⋊Dic15C2211yescharacteristic
38C2×#6712C2211yescharacteristic
39C2×#6712C2212yes—
40C2×#6712C2211yescharacteristic
41C2×C4×Dic15C2211yescharacteristic
42#91784C2218yescomplemented
43#91779C2218yescomplemented
44#91773C22116yescomplemented
45#91772C22116yescomplemented
46C4×D60C2218yescomplemented
47#91761C2218yescomplemented
48C2×#5562—51nomaximal, Hall
49C2×C4×C5×D4D311yescharacteristic, complemented
50C2×#4086—31no—
51C2×#4084—31no—
52C2×#4076—32no—
53C2×#4075—32no—
54#65896—316no—
55C2×#4041—31no—
56C2×#4036—31no—
57C2×#4030—32no—
58C2×#4029—32no—
59C2×C4×D20—31no—
60C2×#4018—31no—
61C24×C3×C5C2312yes—
62C23×D15C2311yescharacteristic
63C23×D15C2312yes—
64C2×C3×C5×D4—24no—
65C2×C3×C5×D4C2314yes—
66C22×C3×C4×C5—44no—
67C22×C3×C4×C5C2312yes—
68C22×C3×C4×C5C2311yescharacteristic
69C22×C3×C4×C5C2311yescharacteristic
70C22×C3×C4×C5C2318yes—
71C22×C3×C4×C5C2311yescharacteristic
72C2×#1220—44no—
73C2×#1220—44no—
74C2×#1220—44no—
75C2×#1220C2318yes—
76C2×#1220C2318yes—
77C2×#1220C2318yes—
78C22×Dic15C2311yescharacteristic
79C22×Dic15C2311yescharacteristic
80C22×Dic15C2311yescharacteristic
81C22×Dic15C2312yes—
82C2×D60—24no—
83C2×D60—44no—
84C2×D60C2314yes—
85C2×D60C2318yes—
86C2×C4×D15—24no—
87C2×C4×D15—44no—
88C2×C4×D15C2314yes—
89C2×C4×D15C2318yes—
90C3×C5×C4⋊C4C2314yes—
91C3×C5×C22⋊C4C2318yes—
92C3×C42×C5C2314yes—
93#6719C2318yes—
94#6714C2314yes—
95#6714C2314yes—
96#6714C2318yes—
97#6714C2318yes—
98C4⋊Dic15C2314yes—
99C4⋊Dic15C2318yes—
100#6712C2314yes—
101#6712C2314yes—
102#6712C2318yes—
103C4×Dic15C2314yes—
104C2×C3×C4×D4D511yescharacteristic, complemented
105C2×#940—51no—
106C2×#938—51no—
107C2×#930—52no—
108C2×#929—52no—
109#5562—516no—
110C2×#895—51no—
111C2×#890—51no—
112C2×#884—52no—
113C2×#883—52no—
114C2×C4×D12—51no—
115C2×#872—51no—
116C22×C5×D4C2×D311yescharacteristic
117C23×C4×C5C2×D312yes—
118C22×C22⋊D5—32no—
119C22×C22⋊D5—32no—
120C22×C22⋊D5—32no—
121C22×D20—31no—
122C22×D20—32no—
123C22×C4×D5—31no—
124C22×C4×D5—32no—
125C4×C5×D4C2×D318yescomplemented
126C2×C5×C4⋊C4C2×D311yescharacteristic
127C2×C5×C22⋊C4C2×D312yes—
128C2×C42×C5C2×D311yescharacteristic
129C2×C22⋊Dic5—32no—
130#4086—38no—
131#4084—38no—
132#4076—316no—
133#4075—316no—
134C2×#685—31no—
135C2×#685—31no—
136C2×#685—32no—
137C2×#685—32no—
138C2×C4⋊Dic5—31no—
139C2×C4⋊Dic5—32no—
140C2×#683—31no—
141C2×#683—31no—
142C2×#683—32no—
143C2×C4×Dic5—31no—
144#4041—38no—
145#4036—38no—
146#4030—316no—
147#4029—316no—
148C4×D20—38no—
149#4018—38no—
150C2×#540—151noSylow
151C23×C3×C5—24no—
152C23×C3×C5—24no—
153C23×C3×C5—24no—
154C23×C3×C5C2414yes—
155C23×C3×C5C2411yescharacteristic
156C22×D15—22no—
157C22×D15—24no—
158C22×D15—24no—
159C22×D15—22no—
160C22×D15—24no—
161C22×D15—22no—
162C22×D15C2412yes—
163C22×D15C2414yes—
164C3×C5×D4—28no—
165C2×C3×C4×C5—22no—
166C2×C3×C4×C5—24no—
167C2×C3×C4×C5—22no—
168C2×C3×C4×C5—48no—
169C2×C3×C4×C5#4114yes—
170C2×C3×C4×C5C2414yes—
171C2×C3×C4×C5C2412yes—
172C2×C3×C4×C5C2412yes—
173C2×C3×C4×C5C2412yes—
174#1220—48no—
175#1220—48no—
176#1220—48no—
177C2×Dic15—22no—
178C2×Dic15—24no—
179C2×Dic15—22no—
180C2×Dic15—22no—
181C2×Dic15C2412yes—
182C2×Dic15C2412yes—
183C2×Dic15C2412yes—
184C2×Dic15C2414yes—
185D60—28no—
186D60—48no—
187C4×D15—28no—
188C4×D15—48no—
189C22×C3×D4C2×D511yescharacteristic
190C23×C3×C4C2×D512yes—
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×D4C2×D518yescomplemented
199C2×C3×C4⋊C4C2×D511yescharacteristic
200C2×C3×C22⋊C4C2×D512yes—
201C2×C3×C42C2×D511yescharacteristic
202C2×#217—52no—
203#940—58no—
204#938—58no—
205#930—516no—
206#929—516no—
207C2×(C2×D3)⋊C4—51no—
208C2×(C2×D3)⋊C4—51no—
209C2×(C2×D3)⋊C4—52no—
210C2×(C2×D3)⋊C4—52no—
211C2×C4⋊Dic3—51no—
212C2×C4⋊Dic3—52no—
213C2×Dic3⋊C4—51no—
214C2×Dic3⋊C4—51no—
215C2×Dic3⋊C4—52no—
216C2×C4×Dic3—51no—
217#895—58no—
218#890—58no—
219#884—516no—
220#883—516no—
221C4×D12—58no—
222#872—58no—
223C24×C5C22×D312yes—
224C23×D5—31no—
225C23×D5—32no—
226C2×C5×D4—24no—
227C2×C5×D4C22×D314yes—
228C22×C4×C5—44no—
229C22×C4×C5C22×D311yescharacteristic
230C22×C4×C5C22×D311yescharacteristic
231C22×C4×C5C22×D312yes—
232C22×C4×C5C22×D318yes—
233C22×C4×C5C22×D311yescharacteristic
234C2×C22⋊D5—38no—
235C2×C22⋊D5—38no—
236C2×C22⋊D5—38no—
237C2×C22⋊D5—124no—
238C2×C22⋊D5—124no—
239C2×C22⋊D5—124no—
240C22×Dic5—31no—
241C22×Dic5—32no—
242C22×Dic5—31no—
243C22×Dic5—31no—
244C2×D20—34no—
245C2×D20—38no—
246C2×D20—64no—
247C2×D20—124no—
248C2×C4×D5—34no—
249C2×C4×D5—38no—
250C2×C4×D5—64no—
251C2×C4×D5—124no—
252C5×C4⋊C4C22×D314yes—
253C5×C22⋊C4C22×D318yes—
254C42×C5C22×D314yes—
255C22⋊Dic5—38no—
256#685—34no—
257#685—34no—
258#685—38no—
259#685—38no—
260C4⋊Dic5—34no—
261C4⋊Dic5—38no—
262#683—34no—
263#683—34no—
264#683—38no—
265C4×Dic5—34no—
266#540—1516no—
267C2×#127—151no—
268C2×#125—151no—
269C2×#123—152no—
270C2×#123—152no—
271C2×#121—151no—
272C2×#121—152no—
273C2×#121—151no—
274C2×#121—152no—
275C2×C4×D4—151no—
276C2×C4×D4D1511yescharacteristic, complemented
277C2×#117—151no—
278C22×C3×C5—24no—
279C22×C3×C5—24no—
280C22×C3×C5—24no—
281C22×C3×C5—48no—
282C22×C3×C5C2×#4112yes—
283C22×C3×C5H(5,𝔽2)12yes—
284C22×C3×C5H(5,𝔽2)12yes—
285C22×C3×C5C2511yescharacteristic
286C2×D15—22no—
287C2×D15—24no—
288C2×D15—24no—
289C2×D15—22no—
290C2×D15—24no—
291C2×D15—22no—
292C2×D15—44no—
293C2×D15—48no—
294C3×C4×C5—22no—
295C3×C4×C5—24no—
296C3×C4×C5—22no—
297C3×C4×C5C2×#4114yes—
298Dic15—22no—
299Dic15—24no—
300Dic15—22no—
301Dic15—22no—
302C24×C3C22×D512yes—
303C23×D3—51no—
304C23×D3—52no—
305C2×C3×D4—24no—
306C2×C3×D4C22×D514yes—
307C22×C3×C4—44no—
308C22×C3×C4C22×D512yes—
309C22×C3×C4C22×D511yescharacteristic
310C22×C3×C4C22×D511yescharacteristic
311C22×C3×C4C22×D518yes—
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×Dic3—51no—
321C22×Dic3—51no—
322C22×Dic3—52no—
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⋊C4C22×D518yes—
333C3×C42C22×D514yes—
334#217—58no—
335(C2×D3)⋊C4—54no—
336(C2×D3)⋊C4—54no—
337(C2×D3)⋊C4—58no—
338(C2×D3)⋊C4—58no—
339C4⋊Dic3—54no—
340C4⋊Dic3—58no—
341Dic3⋊C4—54no—
342Dic3⋊C4—54no—
343Dic3⋊C4—58no—
344C4×Dic3—54no—
345C23×C5—24no—
346C23×C5—24no—
347C23×C5—24no—
348C23×C5C23×D314yes—
349C23×C5C23×D311yescharacteristic
350C22×D5—32no—
351C22×D5—34no—
352C22×D5—62no—
353C22×D5—64no—
354C22×D5—62no—
355C22×D5—64no—
356C22×D5—64no—
357C22×D5—62no—
358C5×D4—28no—
359C2×C4×C5—24no—
360C2×C4×C5—22no—
361C2×C4×C5—22no—
362C2×C4×C5—48no—
363C2×C4×C5#23514yes—
364C2×C4×C5C23×D314yes—
365C2×C4×C5C23×D312yes—
366C2×C4×C5C23×D312yes—
367C2×C4×C5C23×D312yes—
368C22⋊D5—128no—
369C22⋊D5—128no—
370C22⋊D5—128no—
371C2×Dic5—32no—
372C2×Dic5—32no—
373C2×Dic5—32no—
374C2×Dic5—34no—
375C2×Dic5—62no—
376C2×Dic5—62no—
377C2×Dic5—64no—
378C2×Dic5—62no—
379D20—68no—
380D20—128no—
381C4×D5—68no—
382C4×D5—128no—
383C22×D4—152no—
384C22×D4—152no—
385C22×D4—152no—
386C22×D4—151no—
387C22×D4—152no—
388C22×D4C2×D1511yescharacteristic
389C23×C4—151no—
390C23×C4—152no—
391C23×C4C2×D1512yes—
392#127—158no—
393#125—158no—
394#123—1516no—
395#123—1516no—
396#121—158no—
397#121—1516no—
398#121—158no—
399#121—1516no—
400C4×D4—158no—
401C4×D4C2×D1518yescomplemented
402#117—158no—
403C2×C4⋊C4—151no—
404C2×C4⋊C4—151no—
405C2×C4⋊C4—151no—
406C2×C4⋊C4—152no—
407C2×C4⋊C4—152no—
408C2×C4⋊C4C2×D1511yescharacteristic
409C2×C22⋊C4—152no—
410C2×C22⋊C4—151no—
411C2×C22⋊C4—151no—
412C2×C22⋊C4—152no—
413C2×C22⋊C4—152no—
414C2×C22⋊C4C2×D1512yes—
415C2×C42—151no—
416C2×C42C2×D1511yescharacteristic
417C2×C3×C5—44no—
418C2×C3×C5#54014yescomplemented
419C2×C3×C5C22×#4111yescharacteristic
420C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
421C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
422D15—42no—
423D15—44no—
424C23×C3—24no—
425C23×C3—24no—
426C23×C3—24no—
427C23×C3C23×D514yes—
428C23×C3C23×D511yescharacteristic
429C22×D3—52no—
430C22×D3—54no—
431C22×D3—102no—
432C22×D3—104no—
433C22×D3—102no—
434C22×D3—104no—
435C22×D3—104no—
436C22×D3—102no—
437C3×D4—28no—
438C2×C3×C4—24no—
439C2×C3×C4—22no—
440C2×C3×C4—22no—
441C2×C3×C4—48no—
442C2×C3×C4#70914yes—
443C2×C3×C4C23×D514yes—
444C2×C3×C4C23×D512yes—
445C2×C3×C4C23×D512yes—
446C2×C3×C4C23×D512yes—
447#67—208no—
448#67—208no—
449#67—208no—
450C2×Dic3—52no—
451C2×Dic3—52no—
452C2×Dic3—52no—
453C2×Dic3—54no—
454C2×Dic3—102no—
455C2×Dic3—102no—
456C2×Dic3—104no—
457C2×Dic3—102no—
458D12—108no—
459D12—208no—
460C4×D3—108no—
461C4×D3—208no—
462C22×C5—24no—
463C22×C5—24no—
464C22×C5—24no—
465C22×C5—48no—
466C22×C5C2×#23512yes—
467C22×C5#100412yes—
468C22×C5#100912yes—
469C22×C5C24×D311yescharacteristic
470C2×D5—62no—
471C2×D5—64no—
472C2×D5—62no—
473C2×D5—64no—
474C2×D5—64no—
475C2×D5—62no—
476C2×D5—124no—
477C2×D5—128no—
478C4×C5—24no—
479C4×C5—22no—
480C4×C5—22no—
481C4×C5C2×#23514yes—
482Dic5—62no—
483Dic5—62no—
484Dic5—64no—
485Dic5—62no—
486C24—151no—
487C24—152no—
488C24C22×D1512yes—
489C2×D4—24no—
490C2×D4—158no—
491C2×D4—158no—
492C2×D4—158no—
493C2×D4—154no—
494C2×D4—158no—
495C2×D4—304no—
496C2×D4—604no—
497C2×D4—604no—
498C2×D4—604no—
499C2×D4—604no—
500C2×D4C22×D1514yes—
501C22×C4—44no—
502C22×C4—151no—
503C22×C4—151no—
504C22×C4—151no—
505C22×C4—152no—
506C22×C4—154no—
507C22×C4—158no—
508C22×C4—304no—
509C22×C4—604no—
510C22×C4C22×D1511yescharacteristic
511C22×C4C22×D1511yescharacteristic
512C22×C4C22×D1512yes—
513C22×C4C22×D1518yes—
514C22×C4C22×D1511yescharacteristic
515C4⋊C4—154no—
516C4⋊C4—154no—
517C4⋊C4—154no—
518C4⋊C4—158no—
519C4⋊C4—158no—
520C4⋊C4C22×D1514yes—
521C22⋊C4—158no—
522C22⋊C4—154no—
523C22⋊C4—154no—
524C22⋊C4—158no—
525C22⋊C4—158no—
526C22⋊C4C22×D1518yes—
527C42—154no—
528C42C22×D1514yes—
529C3×C5C2×#54011yescharacteristic, complemented, Hall
530C22×C3—24no—
531C22×C3—24no—
532C22×C3—24no—
533C22×C3—48no—
534C22×C3C2×#70912yes—
535C22×C3#414412yes—
536C22×C3#414912yes—
537C22×C3C24×D511yescharacteristic
538C2×D3—102no—
539C2×D3—104no—
540C2×D3—102no—
541C2×D3—104no—
542C2×D3—104no—
543C2×D3—102no—
544C2×D3—204no—
545C2×D3—208no—
546C3×C4—24no—
547C3×C4—22no—
548C3×C4—22no—
549C3×C4C2×#70914yes—
550Dic3—102no—
551Dic3—102no—
552Dic3—104no—
553Dic3—102no—
554C2×C5—44no—
555C2×C5#556214yescomplemented
556C2×C5C22×#23511yescharacteristic
557C2×C5C2×#100411yescharacteristic
558C2×C5C2×#100911yescharacteristic
559D5—122no—
560D5—124no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—152no—
565C23—154no—
566C23—302no—
567C23—304no—
568C23—302no—
569C23—304no—
570C23—304no—
571C23—302no—
572C23C23×D1514yes—
573C23C23×D1511yescharacteristic
574D4—28no—
575D4—308no—
576D4—608no—
577D4—608no—
578D4—608no—
579D4—608no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—48no—
584C2×C4—152no—
585C2×C4—152no—
586C2×C4—152no—
587C2×C4—154no—
588C2×C4—302no—
589C2×C4—302no—
590C2×C4—304no—
591C2×C4—302no—
592C2×C4—308no—
593C2×C4—608no—
594C2×C4#681714yes—
595C2×C4C23×D1512yes—
596C2×C4C23×D1512yes—
597C2×C4C23×D1514yes—
598C2×C4C23×D1512yes—
599C2×C3—44no—
600C2×C3#6589614yescomplemented
601C2×C3C22×#70911yescharacteristic
602C2×C3C2×#414411yescharacteristic
603C2×C3C2×#414911yescharacteristic
604D3—202no—
605D3—204no—
606C5C2×#556211yescharacteristic, complemented, Sylow
607C22—24no—
608C22—24no—
609C22—24no—
610C22—48no—
611C22—302no—
612C22—304no—
613C22—302no—
614C22—304no—
615C22—304no—
616C22—302no—
617C22—604no—
618C22—608no—
619C22C2×#681712yes—
620C22#9209512yes—
621C22#9210012yes—
622C22C24×D1511yescharacteristic
623C4—22no—
624C4—24no—
625C4—22no—
626C4—302no—
627C4—302no—
628C4—304no—
629C4—302no—
630C4C2×#681714yes—
631C3C2×#6589611yescharacteristic, complemented, Sylow
632C2—44no—
633C2—602no—
634C2—604no—
635C2#1175632114yesdirect product factor
636C2C22×#681711yescharacteristic
637C2C2×#9209511yescharacteristic
638C2C2×#9210011yescharacteristic
subgroupG/H
centerC23C23×D15
commutator, cosocleC22×C3×C5C25
FrattiniC22C24×D15
FittingC2×C3×C4×C5×D4C2
socleC23×C3×C5C24
hypercenterC2×C4×D4D15
hypocenterC3×C5C2×#540
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×C5×D4C2semidirect
2. C2×#91829C2semidirect
3. C2×#91827C2semidirect
4. C2×#91819C2semidirect
5. C2×#91818C2semidirect
6. #11756321C2direct (N × H)
7. C2×#91784C2semidirect
8. C2×#91779C2semidirect
9. C2×#91773C2semidirect
10. C2×#91772C2semidirect
11. C2×C4×D60C2semidirect
12. C2×#91761C2semidirect
23. C3×C4×C5×D4C22semidirect
28. #91829C22semidirect
29. #91827C22semidirect
30. #91819C22semidirect
31. #91818C22semidirect
42. #91784C22semidirect
43. #91779C22semidirect
44. #91773C22semidirect
45. #91772C22semidirect
46. C4×D60C22semidirect
47. #91761C22semidirect
49. C2×C4×C5×D4D3semidirect
104. C2×C3×C4×D4D5semidirect
125. C4×C5×D4C2×D3semidirect
198. C3×C4×D4C2×D5semidirect
276. C2×C4×D4D15semidirect
401. C4×D4C2×D15semidirect
418. C2×C3×C5#540semidirect
529. C3×C5C2×#540semidirect
555. C2×C5#5562semidirect
600. C2×C3#65896semidirect
606. C5C2×#5562semidirect
631. C3C2×#65896semidirect
635. C2#11756321direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×C5×D42
C2×#918292
C2×#918272
C2×#918192
C2×#918182
#117563212
C2×#917842
C2×#917792
C2×#917732
C2×#917722
C2×C4×D602
C2×#917612
C2×#658963
C2×#55625