Finite Group Databasebeta

← C2×#11753815 #14481153

Subgroups — C2×#11753815 #14481153

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×#91779C211yesmaximal, characteristic, complemented
2C2×C5×#895C211yesmaximal, characteristic, complemented
3C2×C3×#4039C211yesmaximal, characteristic, complemented
4C2×#91463C211yesmaximal, characteristic, complemented
5C2×#91457C212yesmaximal, complemented
6C2×#91456C211yesmaximal, characteristic, complemented
7C2×#91453C212yesmaximal, complemented
8C2×#91449C212yesmaximal, complemented
9C2×#91447C212yesmaximal, complemented
10C2×Dic5×D12C211yesmaximal, characteristic, complemented
11C2×#91371C211yesmaximal, characteristic, complemented
12#11753815C2116yesmaximal, direct product factor
13C2×#65953—31nomaximal, Hall
14C22×C5×D12C2211yescharacteristic
15C22×C5×D12C2212yes—
16C22×C4×C5×D3C2211yescharacteristic
17C22×C3×D20C2211yescharacteristic
18C22×C3×C4×D5C2212yes—
19C22×#1201C2212yes—
20C22×#1201C2212yes—
21C22×#1201C2212yes—
22C22×D3×Dic5C2212yes—
23C2×C3×C5×C4⋊C4C2211yescharacteristic
24C2×C4⋊Dic15C2212yes—
25C2×C4⋊Dic15C2211yescharacteristic
26C2×#6712C2212yes—
27C2×C4×Dic15C2211yescharacteristic
28#91779C2218yescomplemented
29C2×C5×(C2×D3)⋊C4C2212yes—
30C5×#895C2218yescomplemented
31C2×C3×#685C2212yes—
32C2×C3×C4×Dic5C2211yescharacteristic
33C3×#4039C2218yescomplemented
34C2×Dic3⋊Dic5C2211yescharacteristic
35C2×Dic3⋊Dic5C2211yescharacteristic
36C2×(C2×D3)⋊Dic5C2211yescharacteristic
37C2×(C2×D3)⋊Dic5C2211yescharacteristic
38C2×(C2×D3)⋊Dic5C2212yes—
39C2×#6665C2212yes—
40#91463C2218yescomplemented
41#91457C22116yescomplemented
42#91456C2218yescomplemented
43#91453C22116yescomplemented
44#91449C22116yescomplemented
45#91447C22116yescomplemented
46Dic5×D12C2218yescomplemented
47#91371C2218yescomplemented
48C2×#5562—51nomaximal, Hall
49C2×C5×#121—31no—
50C2×#4086—31no—
51C2×#4085—32no—
52C2×#4084—32no—
53C2×D4×Dic5—31no—
54C2×#4076—31no—
55C2×#4072—31no—
56#65953—316no—
57C2×#4039D311yescharacteristic, complemented
58C2×#4036—31no—
59C2×#4032—32no—
60C2×#4029—32no—
61C23×C5×D3C2312yes—
62C23×C5×D3C2311yescharacteristic
63C23×C3×D5C2312yes—
64C22×C3×C4×C5C2312yes—
65C22×C3×C4×C5C2311yescharacteristic
66C22×Dic15C2312yes—
67C22×Dic15C2311yescharacteristic
68C22×Dic15C2311yescharacteristic
69C22×C5×Dic3C2311yescharacteristic
70C2×C5×D12—24no—
71C2×C5×D12—44no—
72C2×C5×D12C2314yes—
73C2×C5×D12C2318yes—
74C2×C4×C5×D3—24no—
75C2×C4×C5×D3C2314yes—
76C22×C3×Dic5C2311yescharacteristic
77C22×C3×Dic5C2311yescharacteristic
78C2×C3×D20—24no—
79C2×C3×D20C2314yes—
80C2×C3×C4×D5—44no—
81C2×C3×C4×D5C2318yes—
82C2×#1201—44no—
83C2×#1201—44no—
84C2×#1201—44no—
85C2×#1201C2318yes—
86C2×#1201C2318yes—
87C2×#1201C2318yes—
88C2×D3×Dic5—44no—
89C2×D3×Dic5C2318yes—
90C3×C5×C4⋊C4C2314yes—
91C4⋊Dic15C2314yes—
92C4⋊Dic15C2318yes—
93#6712C2318yes—
94C4×Dic15C2314yes—
95C5×(C2×D3)⋊C4C2318yes—
96C3×#685C2318yes—
97C3×C4×Dic5C2314yes—
98Dic3⋊Dic5C2314yes—
99Dic3⋊Dic5C2314yes—
100(C2×D3)⋊Dic5C2318yes—
101(C2×D3)⋊Dic5C2314yes—
102(C2×D3)⋊Dic5C2314yes—
103#6665C2318yes—
104C2×C3×C4×D4—51no—
105C2×#940—51no—
106C2×#938—51no—
107C2×#930—52no—
108C2×#929—52no—
109#5562—516no—
110C2×#895D511yescharacteristic, complemented
111C2×#890—51no—
112C2×#884—52no—
113C2×#883—52no—
114C2×C4×D12—51no—
115C2×#872—51no—
116C22×C5×D4—31no—
117C22×C5×D4—32no—
118C23×C4×C5—31no—
119C22×C22⋊D5—32no—
120C22×C22⋊D5—32no—
121C22×C22⋊D5—32no—
122C23×Dic5—32no—
123C22×D20C2×D311yescharacteristic
124C22×C4×D5C2×D312yes—
125C5×#121—38no—
126C2×C5×C4⋊C4C2×D311yescharacteristic
127C2×C5×C22⋊C4—32no—
128C2×C22⋊Dic5—32no—
129C2×C22⋊Dic5—31no—
130C2×C22⋊Dic5—31no—
131#4086—38no—
132#4085—316no—
133#4084—316no—
134D4×Dic5—38no—
135#4076—38no—
136C2×#685—32no—
137C2×#685C2×D312yes—
138#4072—38no—
139C2×C4⋊Dic5—32no—
140C2×C4⋊Dic5—31no—
141C2×C4⋊Dic5—31no—
142C2×C4⋊Dic5—31no—
143C2×#683—32no—
144C2×C4×Dic5—31no—
145C2×C4×Dic5C2×D311yescharacteristic
146#4039C2×D318yescomplemented
147#4036—38no—
148#4032—316no—
149#4029—316no—
150C2×#540—151noSylow
151C23×C3×C5C2411yescharacteristic
152C22×C5×D3—22no—
153C22×C5×D3—24no—
154C22×C5×D3—24no—
155C22×C5×D3—22no—
156C22×C5×D3—24no—
157C22×C5×D3—22no—
158C22×C5×D3C2412yes—
159C22×C5×D3C2414yes—
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—24no—
169C2×Dic15—22no—
170C2×Dic15—22no—
171C2×Dic15C2414yes—
172C2×Dic15C2412yes—
173C2×Dic15C2412yes—
174C2×C5×Dic3—22no—
175C2×C5×Dic3C2412yes—
176C5×D12—28no—
177C5×D12—48no—
178C4×C5×D3—28no—
179C2×C3×Dic5—22no—
180C2×C3×Dic5—22no—
181C2×C3×Dic5C2412yes—
182C2×C3×Dic5C2412yes—
183C3×D20—28no—
184C3×C4×D5—48no—
185#1201—48no—
186#1201—48no—
187#1201—48no—
188D3×Dic5—48no—
189C22×C3×D4—51no—
190C23×C3×C4—52no—
191C22×#67—52no—
192C22×#67—52no—
193C22×#67—52no—
194C22×D12C2×D511yescharacteristic
195C22×D12C2×D512yes—
196C22×C4×D3—52no—
197C22×C4×D3C2×D511yescharacteristic
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—51no—
208C2×(C2×D3)⋊C4—51no—
209C2×(C2×D3)⋊C4—52no—
210C2×(C2×D3)⋊C4C2×D512yes—
211C2×C4⋊Dic3—51no—
212C2×C4⋊Dic3—52no—
213C2×Dic3⋊C4—52no—
214C2×Dic3⋊C4—51no—
215C2×Dic3⋊C4—51no—
216C2×C4×Dic3—51no—
217#895C2×D518yescomplemented
218#890—58no—
219#884—516no—
220#883—516no—
221C4×D12—58no—
222#872—58no—
223C24×C5—31no—
224C24×C5—32no—
225C23×D5C22×D312yes—
226C2×C5×D4—34no—
227C2×C5×D4—38no—
228C2×C5×D4—64no—
229C2×C5×D4—124no—
230C22×C4×C5—31no—
231C22×C4×C5—34no—
232C22×C4×C5—64no—
233C22×C4×C5C22×D312yes—
234C22×C4×C5C22×D311yescharacteristic
235C2×C22⋊D5—38no—
236C2×C22⋊D5—38no—
237C2×C22⋊D5—38no—
238C2×C22⋊D5—124no—
239C2×C22⋊D5—124no—
240C2×C22⋊D5—124no—
241C22×Dic5—32no—
242C22×Dic5—31no—
243C22×Dic5—31no—
244C22×Dic5—38no—
245C22×Dic5—124no—
246C22×Dic5C22×D311yescharacteristic
247C22×Dic5C22×D311yescharacteristic
248C2×D20—24no—
249C2×D20C22×D314yes—
250C2×C4×D5—44no—
251C2×C4×D5C22×D318yes—
252C5×C4⋊C4C22×D314yes—
253C5×C22⋊C4—38no—
254C22⋊Dic5—38no—
255C22⋊Dic5—34no—
256C22⋊Dic5—34no—
257#685—38no—
258#685C22×D318yes—
259C4⋊Dic5—38no—
260C4⋊Dic5—34no—
261C4⋊Dic5—34no—
262C4⋊Dic5—34no—
263#683—38no—
264C4×Dic5—34no—
265C4×Dic5C22×D314yes—
266#540—1516no—
267C2×#127—151no—
268C2×#125—151no—
269C2×#123—152no—
270C2×#123—152no—
271C2×#121—31no—
272C2×#121—152no—
273C2×#121—151no—
274C2×#121—152no—
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×C5×D3—22no—
283C2×C5×D3—24no—
284C2×C5×D3—24no—
285C2×C5×D3—22no—
286C2×C5×D3—24no—
287C2×C5×D3—22no—
288C2×C5×D3—44no—
289C2×C5×D3—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—24no—
297Dic15—22no—
298Dic15—22no—
299C3×Dic5—22no—
300C3×Dic5—22no—
301C5×Dic3—22no—
302C24×C3—52no—
303C23×D3C22×D511yescharacteristic
304C23×D3C22×D512yes—
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×Dic3—51no—
321C22×Dic3—52no—
322C22×Dic3C22×D511yescharacteristic
323C2×D12—24no—
324C2×D12—44no—
325C2×D12C22×D514yes—
326C2×D12C22×D518yes—
327C2×C4×D3—24no—
328C2×C4×D3—58no—
329C2×C4×D3—204no—
330C2×C4×D3C22×D514yes—
331C3×C4⋊C4C22×D514yes—
332C3×C22⋊C4—58no—
333C3×C42—54no—
334#217—58no—
335(C2×D3)⋊C4—54no—
336(C2×D3)⋊C4—54no—
337(C2×D3)⋊C4—58no—
338(C2×D3)⋊C4C22×D518yes—
339C4⋊Dic3—54no—
340C4⋊Dic3—58no—
341Dic3⋊C4—58no—
342Dic3⋊C4—54no—
343Dic3⋊C4—54no—
344C4×Dic3—54no—
345C23×C5—32no—
346C23×C5—34no—
347C23×C5—62no—
348C23×C5—64no—
349C23×C5—64no—
350C23×C5—62no—
351C23×C5—64no—
352C23×C5—62no—
353C23×C5C23×D311yescharacteristic
354C22×D5—24no—
355C22×D5—24no—
356C22×D5—24no—
357C22×D5C23×D314yes—
358C5×D4—68no—
359C5×D4—128no—
360C2×C4×C5—24no—
361C2×C4×C5—32no—
362C2×C4×C5—62no—
363C2×C4×C5—68no—
364C2×C4×C5#23514yes—
365C2×C4×C5C23×D314yes—
366C2×C4×C5C23×D312yes—
367C22⋊D5—128no—
368C22⋊D5—128no—
369C22⋊D5—128no—
370C2×Dic5—22no—
371C2×Dic5—22no—
372C2×Dic5—32no—
373C2×Dic5—32no—
374C2×Dic5—34no—
375C2×Dic5—62no—
376C2×Dic5—64no—
377C2×Dic5—62no—
378C2×Dic5—128no—
379C2×Dic5C23×D312yes—
380C2×Dic5C23×D312yes—
381D20—28no—
382C4×D5—48no—
383C22×D4—31no—
384C22×D4—32no—
385C22×D4—51no—
386C22×D4—152no—
387C22×D4—152no—
388C22×D4—152no—
389C23×C4—31no—
390C23×C4—52no—
391C23×C4—152no—
392#127—158no—
393#125—158no—
394#123—1516no—
395#123—1516no—
396#121—38no—
397#121—1516no—
398#121—158no—
399#121—1516no—
400C4×D4—58no—
401C4×D4—158no—
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⋊C4D3×D511yescharacteristic
409C2×C22⋊C4—32no—
410C2×C22⋊C4—52no—
411C2×C22⋊C4—152no—
412C2×C22⋊C4—151no—
413C2×C22⋊C4—151no—
414C2×C22⋊C4—152no—
415C2×C42—51no—
416C2×C42—151no—
417C2×C3×C5#54014yescomplemented
418C2×C3×C5C22×#4111yescharacteristic
419C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
420C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
421C3×D5—44no—
422C5×D3—42no—
423C5×D3—44no—
424C23×C3—54no—
425C23×C3—104no—
426C23×C3—104no—
427C23×C3—104no—
428C23×C3C23×D511yescharacteristic
429C22×D3—22no—
430C22×D3—24no—
431C22×D3—22no—
432C22×D3—24no—
433C22×D3—24no—
434C22×D3—22no—
435C22×D3C23×D512yes—
436C22×D3C23×D514yes—
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#71114yes—
445C2×C3×C4C23×D514yes—
446C2×C3×C4C23×D512yes—
447#67—208no—
448#67—208no—
449#67—208no—
450C2×Dic3—22no—
451C2×Dic3—52no—
452C2×Dic3—52no—
453C2×Dic3—54no—
454C2×Dic3—102no—
455C2×Dic3—104no—
456C2×Dic3—102no—
457C2×Dic3C23×D512yes—
458D12—28no—
459D12—48no—
460C4×D3—28no—
461C4×D3—208no—
462C22×C5—62no—
463C22×C5—64no—
464C22×C5—64no—
465C22×C5—62no—
466C22×C5—64no—
467C22×C5—62no—
468C22×C5—124no—
469C22×C5—128no—
470C22×C5C2×#23512yes—
471C22×C5#100412yes—
472C22×C5#100912yes—
473C22×C5C24×D311yescharacteristic
474C2×D5—24no—
475C2×D5—24no—
476C2×D5—24no—
477C2×D5—48no—
478C4×C5—24no—
479C4×C5—62no—
480C4×C5C2×#23514yes—
481Dic5—22no—
482Dic5—22no—
483Dic5—62no—
484Dic5—64no—
485Dic5—62no—
486C24—32no—
487C24—31no—
488C24—52no—
489C2×D4—34no—
490C2×D4—38no—
491C2×D4—54no—
492C2×D4—64no—
493C2×D4—104no—
494C2×D4—124no—
495C2×D4—158no—
496C2×D4—158no—
497C2×D4—158no—
498C2×D4—604no—
499C2×D4—604no—
500C2×D4—604no—
501C22×C4—31no—
502C22×C4—34no—
503C22×C4—51no—
504C22×C4—51no—
505C22×C4—58no—
506C22×C4—64no—
507C22×C4—158no—
508C22×C4—151no—
509C22×C4—151no—
510C22×C4—152no—
511C22×C4—204no—
512C22×C4—604no—
513C22×C4C2×D3×D512yes—
514C22×C4C2×D3×D511yescharacteristic
515C4⋊C4—154no—
516C4⋊C4—154no—
517C4⋊C4—154no—
518C4⋊C4—158no—
519C4⋊C4—158no—
520C4⋊C4C2×D3×D514yes—
521C22⋊C4—38no—
522C22⋊C4—58no—
523C22⋊C4—158no—
524C22⋊C4—154no—
525C22⋊C4—154no—
526C22⋊C4—158no—
527C42—54no—
528C42—154no—
529C3×C5C2×#54011yescharacteristic, complemented, Hall
530C22×C3—104no—
531C22×C3—104no—
532C22×C3—104no—
533C22×C3—208no—
534C22×C3C2×#71112yes—
535C22×C3#414412yes—
536C22×C3#414912yes—
537C22×C3C24×D511yescharacteristic
538C2×D3—22no—
539C2×D3—24no—
540C2×D3—22no—
541C2×D3—24no—
542C2×D3—24no—
543C2×D3—22no—
544C2×D3—44no—
545C2×D3—48no—
546C3×C4—24no—
547C3×C4—102no—
548C3×C4—102no—
549C3×C4C2×#71114yes—
550Dic3—22no—
551Dic3—102no—
552Dic3—104no—
553Dic3—102no—
554C2×C5—122no—
555C2×C5—124no—
556C2×C5#556214yescomplemented
557C2×C5C22×#23511yescharacteristic
558C2×C5C2×#100411yescharacteristic
559C2×C5C2×#100911yescharacteristic
560D5—44no—
561C23—34no—
562C23—32no—
563C23—54no—
564C23—62no—
565C23—64no—
566C23—62no—
567C23—64no—
568C23—64no—
569C23—62no—
570C23—104no—
571C23—104no—
572C23—104no—
573C23C22×D3×D511yescharacteristic
574D4—68no—
575D4—108no—
576D4—128no—
577D4—608no—
578D4—608no—
579D4—608no—
580C2×C4—24no—
581C2×C4—32no—
582C2×C4—52no—
583C2×C4—52no—
584C2×C4—62no—
585C2×C4—68no—
586C2×C4—102no—
587C2×C4—102no—
588C2×C4—152no—
589C2×C4—152no—
590C2×C4—154no—
591C2×C4—208no—
592C2×C4—302no—
593C2×C4—304no—
594C2×C4—302no—
595C2×C4—608no—
596C2×C4#677214yes—
597C2×C4C22×D3×D514yes—
598C2×C4C22×D3×D512yes—
599C2×C3—204no—
600C2×C3#6595314yescomplemented
601C2×C3C22×#71111yescharacteristic
602C2×C3C2×#414411yescharacteristic
603C2×C3C2×#414911yescharacteristic
604D3—42no—
605D3—44no—
606C5C2×#556211yescharacteristic, complemented, Sylow
607C22—62no—
608C22—64no—
609C22—62no—
610C22—64no—
611C22—64no—
612C22—62no—
613C22—104no—
614C22—104no—
615C22—104no—
616C22—124no—
617C22—128no—
618C22—208no—
619C22C2×#677212yes—
620C22#9202512yes—
621C22#9203512yes—
622C22C23×D3×D511yescharacteristic
623C4—24no—
624C4—62no—
625C4—102no—
626C4—102no—
627C4—302no—
628C4—304no—
629C4—302no—
630C4C2×#677214yes—
631C3C2×#6595311yescharacteristic, complemented, Sylow
632C2—122no—
633C2—124no—
634C2—204no—
635C2#1175381514yesdirect product factor
636C2C22×#677211yescharacteristic
637C2C2×#9202511yescharacteristic
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×#91779C2semidirect
2. C2×C5×#895C2semidirect
3. C2×C3×#4039C2semidirect
4. C2×#91463C2semidirect
5. C2×#91457C2semidirect
6. C2×#91456C2semidirect
7. C2×#91453C2semidirect
8. C2×#91449C2semidirect
9. C2×#91447C2semidirect
10. C2×Dic5×D12C2semidirect
11. C2×#91371C2semidirect
12. #11753815C2direct (N × H)
28. #91779C22semidirect
30. C5×#895C22semidirect
33. C3×#4039C22semidirect
40. #91463C22semidirect
41. #91457C22semidirect
42. #91456C22semidirect
43. #91453C22semidirect
44. #91449C22semidirect
45. #91447C22semidirect
46. Dic5×D12C22semidirect
47. #91371C22semidirect
57. C2×#4039D3semidirect
110. C2×#895D5semidirect
146. #4039C2×D3semidirect
217. #895C2×D5semidirect
417. C2×C3×C5#540semidirect
529. C3×C5C2×#540semidirect
556. C2×C5#5562semidirect
600. C2×C3#65953semidirect
606. C5C2×#5562semidirect
631. C3C2×#65953semidirect
635. C2#11753815direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#917792
C2×C5×#8952
C2×C3×#40392
C2×#914632
C2×#914572
C2×#914562
C2×#914532
C2×#914492
C2×#914472
C2×Dic5×D122
C2×#913712
#117538152
C2×#659533
C2×#55625