Finite Group Databasebeta

← C2×#11753780 #14483968

Subgroups — C2×#11753780 #14483968

Order 1920.

2480 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91784C211yesmaximal, characteristic, complemented
2C2×C5×#888C211yesmaximal, characteristic, complemented
3C2×C3×#4038C211yesmaximal, characteristic, complemented
4C2×#91459C212yesmaximal, complemented
5C2×#91455C212yesmaximal, complemented
6C2×#91450C211yesmaximal, characteristic, complemented
7C2×C4×#1202C211yesmaximal, characteristic, complemented
8C2×#91428C211yesmaximal, characteristic, complemented
9C2×#91402C212yesmaximal, complemented
10C2×#91375C212yesmaximal, complemented
11C2×#91347C211yesmaximal, characteristic, complemented
12#11753780C2116yesmaximal, direct product factor
13C2×#66068—31nomaximal, Hall
14C22×D60C2211yescharacteristic
15C22×D60C2212yes—
16C22×C4×D15C2211yescharacteristic
17C22×C5×Dic6C2212yes—
18C22×C3×C4×D5C2211yescharacteristic
19C22×#1202C2211yescharacteristic
20C22×#1202C2212yes—
21C22×#1200C2212yes—
22C2×C3×C5×C4⋊C4C2211yescharacteristic
23C2×#6714C2212yes—
24#91784C2218yescomplemented
25C2×C5×C4⋊Dic3C2211yescharacteristic
26C2×C5×Dic3⋊C4C2212yes—
27C2×C4×C5×Dic3C2211yescharacteristic
28C5×#888C2218yescomplemented
29C2×C3×#685C2212yes—
30C2×C3×C4⋊Dic5C2211yescharacteristic
31C2×C3×C4×Dic5C2212yes—
32C3×#4038C2218yescomplemented
33C2×Dic3⋊Dic5C2211yescharacteristic
34C2×#6667C2211yescharacteristic
35C2×#6667C2212yes—
36C2×#6667C2212yes—
37C2×#6667C2212yes—
38C2×#6665C2211yescharacteristic
39C2×Dic3×Dic5C2212yes—
40#91459C22116yescomplemented
41#91455C22116yescomplemented
42#91450C2218yescomplemented
43C4×#1202C2218yescomplemented
44#91428C2218yescomplemented
45#91402C22116yescomplemented
46#91375C22116yescomplemented
47#91347C2218yescomplemented
48C2×#5532—51nomaximal, Hall
49C2×C5×C4⋊Q8—31no—
50C2×#4093—32no—
51C2×#4093—32no—
52#66068—316no—
53C2×#4041—32no—
54C2×#4041—31no—
55C2×#4041—31no—
56C2×#4038—31no—
57C2×#4038—32no—
58C2×#4038D311yescharacteristic, complemented
59C2×C4×D20—31no—
60C2×C4×D20—31no—
61C23×D15C2312yes—
62C23×D15C2311yescharacteristic
63C23×C3×D5C2311yescharacteristic
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×Dic3C2312yes—
75C22×C5×Dic3C2311yescharacteristic
76C2×C5×Dic6—44no—
77C2×C5×Dic6C2318yes—
78C22×C3×Dic5C2312yes—
79C22×C3×Dic5C2311yescharacteristic
80C2×C3×C4×D5—24no—
81C2×C3×C4×D5C2314yes—
82C2×#1202—24no—
83C2×#1202—44no—
84C2×#1202C2314yes—
85C2×#1202C2318yes—
86C2×#1200—28no—
87C2×#1200C2318yes—
88C3×C5×C4⋊C4C2314yes—
89#6714C2318yes—
90C5×C4⋊Dic3C2314yes—
91C5×Dic3⋊C4C2318yes—
92C4×C5×Dic3C2314yes—
93C3×#685C2318yes—
94C3×C4⋊Dic5C2314yes—
95C3×C4×Dic5C2318yes—
96Dic3⋊Dic5C2314yes—
97#6667C2318yes—
98#6667C2318yes—
99#6667C2314yes—
100#6667C2318yes—
101#6665C2314yes—
102Dic3×Dic5C2318yes—
103C2×C3×#117—51no—
104C2×#930—51no—
105C2×C4×#67—51no—
106#5532—516no—
107C2×#895—51no—
108C2×#893—51no—
109C2×#892—51no—
110C2×#888D511yescharacteristic, complemented
111C2×#885—52no—
112C2×#884—52no—
113C2×#875—52no—
114C2×#872—52no—
115C22×C5×Q8—32no—
116C22×D20—31no—
117C22×D20—31no—
118C22×D20—32no—
119C22×D20—32no—
120C22×C4×D5—31no—
121C22×C4×D5—32no—
122C22×C4×D5C2×D311yescharacteristic
123C5×C4⋊Q8—38no—
124C2×C5×C4⋊C4—32no—
125C2×C5×C4⋊C4—31no—
126C2×C5×C4⋊C4C2×D311yescharacteristic
127C2×C42×C5—31no—
128#4093—316no—
129#4093—316no—
130C2×#685—31no—
131C2×#685—32no—
132C2×#685—32no—
133C2×#685—31no—
134C2×#685—32no—
135C2×#685—32no—
136C2×#685C2×D312yes—
137C2×C4⋊Dic5—31no—
138C2×C4⋊Dic5C2×D311yescharacteristic
139C2×C4×Dic5—32no—
140C2×C4×Dic5C2×D312yes—
141#4041—38no—
142#4041—38no—
143#4041—316no—
144#4038—38no—
145#4038—316no—
146#4038C2×D318yescomplemented
147C4×D20—38no—
148C4×D20—38no—
149C2×#555—151noSylow
150C23×C3×C5C2411yescharacteristic
151C22×D15—24no—
152C22×D15—22no—
153C22×D15—22no—
154C22×D15—24no—
155C22×D15—24no—
156C22×D15—22no—
157C22×D15C2412yes—
158C22×D15C2414yes—
159C22×C3×D5—22no—
160C22×C3×D5—22no—
161C22×C3×D5—22no—
162C22×C3×D5C2412yes—
163C2×C3×C4×C5—24no—
164C2×C3×C4×C5#4114yes—
165C2×C3×C4×C5C2414yes—
166C2×C3×C4×C5C2412yes—
167C2×Dic15—22no—
168C2×Dic15C2412yes—
169D60—28no—
170D60—48no—
171C4×D15—28no—
172C2×C5×Dic3—24no—
173C2×C5×Dic3—22no—
174C2×C5×Dic3#4114yes—
175C2×C5×Dic3C2414yes—
176C2×C5×Dic3C2412yes—
177C2×C5×Dic3C2412yes—
178C5×Dic6—48no—
179C2×C3×Dic5—24no—
180C2×C3×Dic5—22no—
181C2×C3×Dic5C2414yes—
182C2×C3×Dic5C2412yes—
183C3×C4×D5—28no—
184#1202—28no—
185#1202—48no—
186#1200—216no—
187C23×C3×C4—51no—
188C22×#67—51no—
189C22×#67—52no—
190C22×D12—51no—
191C22×D12—52no—
192C22×C4×D3—51no—
193C22×C4×D3—52no—
194C22×Dic6C2×D512yes—
195C3×#117—58no—
196C2×C3×C4⋊C4—51no—
197C2×C3×C4⋊C4C2×D511yescharacteristic
198C2×C3×C22⋊C4—52no—
199C2×C3×C42—52no—
200C2×#217—51no—
201#930—58no—
202C4×#67—58no—
203C2×(C2×D3)⋊C4—52no—
204C2×(C2×D3)⋊C4—52no—
205C2×(C2×D3)⋊C4—51no—
206C2×(C2×D3)⋊C4—52no—
207C2×(C2×D3)⋊C4—52no—
208C2×C4⋊Dic3C2×D511yescharacteristic
209C2×Dic3⋊C4—51no—
210C2×Dic3⋊C4C2×D512yes—
211C2×C4×Dic3—52no—
212C2×C4×Dic3C2×D511yescharacteristic
213#895—58no—
214#893—58no—
215#892—58no—
216#888C2×D518yescomplemented
217#885—516no—
218#884—516no—
219#875—516no—
220#872—516no—
221C23×D5—31no—
222C23×D5—32no—
223C23×D5C22×D311yescharacteristic
224C2×C5×Q8—38no—
225C2×C5×Q8—124no—
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×D312yes—
233C22×Dic5C22×D311yescharacteristic
234C2×D20—34no—
235C2×D20—34no—
236C2×D20—38no—
237C2×D20—38no—
238C2×D20—64no—
239C2×D20—64no—
240C2×D20—124no—
241C2×D20—124no—
242C2×C4×D5—24no—
243C2×C4×D5—34no—
244C2×C4×D5—38no—
245C2×C4×D5—68no—
246C2×C4×D5—64no—
247C2×C4×D5C22×D314yes—
248C5×C4⋊C4—38no—
249C5×C4⋊C4—34no—
250C5×C4⋊C4C22×D314yes—
251C42×C5—34no—
252#685—34no—
253#685—38no—
254#685—38no—
255#685—34no—
256#685—38no—
257#685—38no—
258#685C22×D318yes—
259C4⋊Dic5—34no—
260C4⋊Dic5C22×D314yes—
261C4×Dic5—38no—
262C4×Dic5C22×D318yes—
263#555—1516no—
264C2×C4⋊Q8—31no—
265C2×#124—152no—
266C2×#124—152no—
267C2×#121—152no—
268C2×#121—151no—
269C2×#121—151no—
270C2×C4×D4—151no—
271C2×C4×D4—151no—
272C2×#117—51no—
273C2×#117—152no—
274C2×#117—151no—
275C22×C3×C5C2×#4112yes—
276C22×C3×C5C2×#4112yes—
277C22×C3×C5H(5,𝔽2)12yes—
278C22×C3×C5C2511yescharacteristic
279C2×D15—24no—
280C2×D15—22no—
281C2×D15—22no—
282C2×D15—24no—
283C2×D15—24no—
284C2×D15—22no—
285C2×D15—48no—
286C2×D15—44no—
287C2×C3×D5—22no—
288C2×C3×D5—22no—
289C2×C3×D5—22no—
290C2×C3×D5—44no—
291C3×C4×C5—24no—
292C3×C4×C5C2×#4114yes—
293Dic15—22no—
294C3×Dic5—24no—
295C3×Dic5—22no—
296C5×Dic3—24no—
297C5×Dic3—22no—
298C5×Dic3C2×#4114yes—
299C24×C3—51no—
300C23×D3—51no—
301C23×D3—52no—
302C22×C3×C4—52no—
303C22×C3×C4—51no—
304C22×C3×C4—54no—
305C22×C3×C4—104no—
306C22×C3×C4C22×D512yes—
307C22×C3×C4C22×D511yescharacteristic
308C2×#67—54no—
309C2×#67—58no—
310C2×#67—104no—
311C2×#67—204no—
312C22×Dic3—51no—
313C22×Dic3C22×D512yes—
314C22×Dic3C22×D511yescharacteristic
315C22×Dic3C22×D511yescharacteristic
316C2×D12—54no—
317C2×D12—58no—
318C2×D12—104no—
319C2×D12—204no—
320C2×C4×D3—54no—
321C2×C4×D3—58no—
322C2×C4×D3—104no—
323C2×C4×D3—108no—
324C2×Dic6—44no—
325C2×Dic6C22×D518yes—
326C3×C4⋊C4—54no—
327C3×C4⋊C4C22×D514yes—
328C3×C22⋊C4—58no—
329C3×C42—58no—
330#217—54no—
331(C2×D3)⋊C4—58no—
332(C2×D3)⋊C4—58no—
333(C2×D3)⋊C4—54no—
334(C2×D3)⋊C4—58no—
335(C2×D3)⋊C4—58no—
336C4⋊Dic3C22×D514yes—
337Dic3⋊C4—54no—
338Dic3⋊C4C22×D518yes—
339C4×Dic3—58no—
340C4×Dic3C22×D514yes—
341C23×C5C23×D311yescharacteristic
342C22×D5—22no—
343C22×D5—22no—
344C22×D5—22no—
345C22×D5—32no—
346C22×D5—34no—
347C22×D5—62no—
348C22×D5—64no—
349C22×D5—64no—
350C22×D5—62no—
351C22×D5—62no—
352C22×D5—64no—
353C22×D5C23×D312yes—
354C5×Q8—128no—
355C2×C4×C5—24no—
356C2×C4×C5—34no—
357C2×C4×C5—32no—
358C2×C4×C5—34no—
359C2×C4×C5—32no—
360C2×C4×C5—64no—
361C2×C4×C5—62no—
362C2×C4×C5#23514yes—
363C2×C4×C5C23×D314yes—
364C2×C4×C5C23×D312yes—
365C2×Dic5—22no—
366C2×Dic5—24no—
367C2×Dic5—32no—
368C2×Dic5—62no—
369C2×Dic5C23×D314yes—
370C2×Dic5C23×D312yes—
371D20—68no—
372D20—68no—
373D20—128no—
374D20—128no—
375C4×D5—28no—
376C4×D5—616no—
377C4×D5—68no—
378C22×Q8—32no—
379C22×D4—151no—
380C22×D4—152no—
381C22×D4—151no—
382C22×D4—152no—
383C23×C4—51no—
384C23×C4—152no—
385C23×C4—151no—
386C4⋊Q8—38no—
387#124—1516no—
388#124—1516no—
389#121—1516no—
390#121—158no—
391#121—158no—
392C4×D4—158no—
393C4×D4—158no—
394#117—58no—
395#117—1516no—
396#117—158no—
397C2×C4⋊C4—32no—
398C2×C4⋊C4—31no—
399C2×C4⋊C4—51no—
400C2×C4⋊C4—151no—
401C2×C4⋊C4D3×D511yescharacteristic
402C2×C22⋊C4—52no—
403C2×C22⋊C4—152no—
404C2×C22⋊C4—152no—
405C2×C22⋊C4—151no—
406C2×C22⋊C4—152no—
407C2×C22⋊C4—151no—
408C2×C22⋊C4—152no—
409C2×C42—31no—
410C2×C42—52no—
411C2×C42—152no—
412C2×C3×C5#55514yescomplemented
413C2×C3×C5C22×#4111yescharacteristic
414C2×C3×C5C22×#4111yescharacteristic
415C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
416D15—44no—
417D15—42no—
418C3×D5—42no—
419C23×C3—52no—
420C23×C3—102no—
421C23×C3—102no—
422C23×C3—102no—
423C23×C3C23×D511yescharacteristic
424C22×D3—52no—
425C22×D3—54no—
426C22×D3—102no—
427C22×D3—104no—
428C22×D3—102no—
429C22×D3—104no—
430C22×D3—104no—
431C22×D3—102no—
432C2×C3×C4—24no—
433C2×C3×C4—54no—
434C2×C3×C4—52no—
435C2×C3×C4—104no—
436C2×C3×C4—102no—
437C2×C3×C4—108no—
438C2×C3×C4#71314yes—
439C2×C3×C4C23×D514yes—
440C2×C3×C4C23×D512yes—
441#67—108no—
442#67—208no—
443C2×Dic3—22no—
444C2×Dic3—24no—
445C2×Dic3—52no—
446C2×Dic3—102no—
447C2×Dic3#71314yes—
448C2×Dic3C23×D514yes—
449C2×Dic3C23×D512yes—
450C2×Dic3C23×D512yes—
451D12—108no—
452D12—208no—
453C4×D3—108no—
454C4×D3—1016no—
455Dic6—48no—
456C22×C5C2×#23512yes—
457C22×C5D3×#4112yes—
458C22×C5#100912yes—
459C22×C5C24×D311yescharacteristic
460C2×D5—22no—
461C2×D5—22no—
462C2×D5—22no—
463C2×D5—44no—
464C2×D5—62no—
465C2×D5—64no—
466C2×D5—64no—
467C2×D5—62no—
468C2×D5—64no—
469C2×D5—62no—
470C2×D5—124no—
471C2×D5—128no—
472C4×C5—24no—
473C4×C5—34no—
474C4×C5—64no—
475C4×C5—62no—
476C4×C5C2×#23514yes—
477Dic5—22no—
478Dic5—24no—
479Dic5—62no—
480C24—51no—
481C24—151no—
482C24—152no—
483C2×Q8—38no—
484C2×Q8—124no—
485C2×D4—154no—
486C2×D4—158no—
487C2×D4—154no—
488C2×D4—158no—
489C2×D4—304no—
490C2×D4—304no—
491C2×D4—604no—
492C2×D4—604no—
493C22×C4—31no—
494C22×C4—31no—
495C22×C4—32no—
496C22×C4—51no—
497C22×C4—52no—
498C22×C4—54no—
499C22×C4—104no—
500C22×C4—158no—
501C22×C4—151no—
502C22×C4—154no—
503C22×C4—308no—
504C22×C4—304no—
505C22×C4C2×D3×D512yes—
506C22×C4C2×D3×D511yescharacteristic
507C4⋊C4—38no—
508C4⋊C4—34no—
509C4⋊C4—54no—
510C4⋊C4—154no—
511C4⋊C4C2×D3×D514yes—
512C22⋊C4—58no—
513C22⋊C4—158no—
514C22⋊C4—158no—
515C22⋊C4—154no—
516C22⋊C4—158no—
517C22⋊C4—154no—
518C22⋊C4—158no—
519C42—34no—
520C42—58no—
521C42—158no—
522C3×C5C2×#55511yescharacteristic, complemented, Hall
523C22×C3—102no—
524C22×C3—102no—
525C22×C3—102no—
526C22×C3—204no—
527C22×C3#414412yes—
528C22×C3C2×#71312yes—
529C22×C3C2×#71312yes—
530C22×C3C24×D511yescharacteristic
531C2×D3—102no—
532C2×D3—104no—
533C2×D3—102no—
534C2×D3—104no—
535C2×D3—104no—
536C2×D3—102no—
537C2×D3—204no—
538C2×D3—208no—
539C3×C4—24no—
540C3×C4—104no—
541C3×C4—102no—
542C3×C4C2×#71314yes—
543Dic3—22no—
544Dic3—24no—
545Dic3—102no—
546Dic3C2×#71314yes—
547C2×C5#553214yescomplemented
548C2×C5C22×#23511yescharacteristic
549C2×C5C2×D3×#4111yescharacteristic
550C2×C5C2×#100911yescharacteristic
551D5—42no—
552D5—122no—
553D5—124no—
554C23—52no—
555C23—102no—
556C23—102no—
557C23—102no—
558C23—152no—
559C23—154no—
560C23—302no—
561C23—304no—
562C23—302no—
563C23—304no—
564C23—304no—
565C23—302no—
566C23C22×D3×D511yescharacteristic
567Q8—128no—
568D4—308no—
569D4—308no—
570D4—608no—
571D4—608no—
572C2×C4—24no—
573C2×C4—34no—
574C2×C4—32no—
575C2×C4—32no—
576C2×C4—34no—
577C2×C4—52no—
578C2×C4—54no—
579C2×C4—64no—
580C2×C4—62no—
581C2×C4—102no—
582C2×C4—104no—
583C2×C4—108no—
584C2×C4—152no—
585C2×C4—3016no—
586C2×C4—302no—
587C2×C4—308no—
588C2×C4#677314yes—
589C2×C4C22×D3×D514yes—
590C2×C4C22×D3×D512yes—
591C2×C3—202no—
592C2×C3#6606814yescomplemented
593C2×C3C2×#414411yescharacteristic
594C2×C3C22×#71311yescharacteristic
595C2×C3C22×#71311yescharacteristic
596D3—202no—
597D3—204no—
598C5C2×#553211yescharacteristic, complemented, Sylow
599C22—102no—
600C22—102no—
601C22—102no—
602C22—204no—
603C22—302no—
604C22—304no—
605C22—302no—
606C22—304no—
607C22—304no—
608C22—302no—
609C22—604no—
610C22—608no—
611C22C2×#677312yes—
612C22#9202712yes—
613C22D3×#71312yes—
614C22C23×D3×D511yescharacteristic
615C4—24no—
616C4—34no—
617C4—64no—
618C4—62no—
619C4—102no—
620C4—104no—
621C4—302no—
622C4C2×#677314yes—
623C3C2×#6606811yescharacteristic, complemented, Sylow
624C2—202no—
625C2—602no—
626C2—604no—
627C2#1175378014yesdirect product factor
628C2C22×#677311yescharacteristic
629C2C2×#9202711yescharacteristic
630C2C2×D3×#71311yescharacteristic
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×#555
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×#888C2semidirect
3. C2×C3×#4038C2semidirect
4. C2×#91459C2semidirect
5. C2×#91455C2semidirect
6. C2×#91450C2semidirect
7. C2×C4×#1202C2semidirect
8. C2×#91428C2semidirect
9. C2×#91402C2semidirect
10. C2×#91375C2semidirect
11. C2×#91347C2semidirect
12. #11753780C2direct (N × H)
24. #91784C22semidirect
28. C5×#888C22semidirect
32. C3×#4038C22semidirect
40. #91459C22semidirect
41. #91455C22semidirect
42. #91450C22semidirect
43. C4×#1202C22semidirect
44. #91428C22semidirect
45. #91402C22semidirect
46. #91375C22semidirect
47. #91347C22semidirect
58. C2×#4038D3semidirect
110. C2×#888D5semidirect
146. #4038C2×D3semidirect
216. #888C2×D5semidirect
412. C2×C3×C5#555semidirect
522. C3×C5C2×#555semidirect
547. C2×C5#5532semidirect
592. C2×C3#66068semidirect
598. C5C2×#5532semidirect
623. C3C2×#66068semidirect
627. C2#11753780direct (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×#8882
C2×C3×#40382
C2×#914592
C2×#914552
C2×#914502
C2×C4×#12022
C2×#914282
C2×#914022
C2×#913752
C2×#913472
#117537802
C2×#660683
C2×#55325