Finite Group Databasebeta

← C2×#11753767 #14483963

Subgroups — C2×#11753767 #14483963

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×#91781C211yesmaximal, characteristic, complemented
2C2×C5×#893C211yesmaximal, characteristic, complemented
3C2×C3×#4038C211yesmaximal, characteristic, complemented
4C2×#91469C211yesmaximal, characteristic, complemented
5C2×#91466C211yesmaximal, characteristic, complemented
6C2×#91464C212yesmaximal, complemented
7C2×#91454C212yesmaximal, complemented
8C2×#91449C211yesmaximal, characteristic, complemented
9C2×C4×#1202C211yesmaximal, characteristic, complemented
10C2×#91371C212yesmaximal, complemented
11C2×#91350C212yesmaximal, complemented
12#11753767C2116yesmaximal, direct product factor
13C2×#65897—31nomaximal, Hall
14C22×C4×D15C2211yescharacteristic
15C22×C5×D12C2211yescharacteristic
16C22×C4×C5×D3C2212yes—
17C22×C3×C4×D5C2211yescharacteristic
18C22×#1204C2212yes—
19C22×#1203C2212yes—
20C22×#1202C2211yescharacteristic
21C22×#1201C2212yes—
22C2×C3×C5×C4⋊C4C2211yescharacteristic
23C2×#6714C2212yes—
24C2×C4⋊Dic15C2211yescharacteristic
25C2×C4×Dic15C2212yes—
26#91781C2218yescomplemented
27C2×C5×(C2×D3)⋊C4C2212yes—
28C2×C4×C5×Dic3C2211yescharacteristic
29C5×#893C2218yescomplemented
30C2×C3×#685C2212yes—
31C2×C3×C4⋊Dic5C2211yescharacteristic
32C2×C3×C4×Dic5C2212yes—
33C3×#4038C2218yescomplemented
34C2×Dic3⋊Dic5C2211yescharacteristic
35C2×Dic3⋊Dic5C2212yes—
36C2×#6667C2211yescharacteristic
37C2×(C2×D3)⋊Dic5C2212yes—
38C2×(C2×D3)⋊Dic5C2212yes—
39C2×#6665C2211yescharacteristic
40#91469C2218yescomplemented
41#91466C2218yescomplemented
42#91464C22116yescomplemented
43#91454C22116yescomplemented
44#91449C2218yescomplemented
45C4×#1202C2218yescomplemented
46#91371C22116yescomplemented
47#91350C22116yescomplemented
48C2×#5532—51nomaximal, Hall
49C2×C4×C5×D4—31no—
50C2×#4084—31no—
51C2×#4084—31no—
52C2×#4076—32no—
53C2×#4072—32no—
54#65897—316no—
55C2×#4038—31no—
56C2×#4038D311yescharacteristic, complemented
57C2×#4031—32no—
58C2×#4031—32no—
59C2×C4×D20—31no—
60C2×#4015—31no—
61C23×D15C2311yescharacteristic
62C23×C5×D3C2312yes—
63C23×C3×D5C2311yescharacteristic
64C22×C3×C4×C5C2312yes—
65C22×C3×C4×C5C2311yescharacteristic
66C22×Dic15C2312yes—
67C22×Dic15C2311yescharacteristic
68C2×C4×D15—24no—
69C2×C4×D15C2314yes—
70C22×C5×Dic3C2311yescharacteristic
71C22×C5×Dic3C2311yescharacteristic
72C2×C5×D12—24no—
73C2×C5×D12C2314yes—
74C2×C4×C5×D3—28no—
75C2×C4×C5×D3C2318yes—
76C22×C3×Dic5C2312yes—
77C22×C3×Dic5C2311yescharacteristic
78C2×C3×C4×D5—24no—
79C2×C3×C4×D5C2314yes—
80C2×#1204—44no—
81C2×#1204C2318yes—
82C2×#1203—44no—
83C2×#1203C2318yes—
84C2×#1202—24no—
85C2×#1202C2314yes—
86C2×#1201—44no—
87C2×#1201C2318yes—
88C3×C5×C4⋊C4C2314yes—
89#6714C2318yes—
90C4⋊Dic15C2314yes—
91C4×Dic15C2318yes—
92C5×(C2×D3)⋊C4C2318yes—
93C4×C5×Dic3C2314yes—
94C3×#685C2318yes—
95C3×C4⋊Dic5C2314yes—
96C3×C4×Dic5C2318yes—
97Dic3⋊Dic5C2314yes—
98Dic3⋊Dic5C2318yes—
99#6667C2314yes—
100(C2×D3)⋊Dic5C2318yes—
101(C2×D3)⋊Dic5C2318yes—
102#6665C2314yes—
103C2×C3×#117—51no—
104C2×#930—51no—
105C2×C4×#67—51no—
106#5532—516no—
107C2×#895—51no—
108C2×#893D511yescharacteristic, complemented
109C2×#892—51no—
110C2×#888—51no—
111C2×#885—52no—
112C2×#884—52no—
113C2×#875—52no—
114C2×#872—52no—
115C22×C5×D4—31no—
116C23×C4×C5—32no—
117C22×C22⋊D5—32no—
118C22×C22⋊D5—32no—
119C22×D20—31no—
120C22×C4×D5—31no—
121C22×C4×D5C2×D311yescharacteristic
122C22×Dic10—32no—
123C4×C5×D4—38no—
124C2×C5×C4⋊C4C2×D311yescharacteristic
125C2×C5×C22⋊C4—32no—
126C2×C42×C5—31no—
127C2×C22⋊Dic5—32no—
128C2×C22⋊Dic5—32no—
129#4084—38no—
130#4084—38no—
131#4076—316no—
132C2×#685—31no—
133C2×#685—32no—
134C2×#685—31no—
135C2×#685C2×D312yes—
136#4072—316no—
137C2×C4⋊Dic5—31no—
138C2×C4⋊Dic5—32no—
139C2×C4⋊Dic5—31no—
140C2×C4⋊Dic5C2×D311yescharacteristic
141C2×C4×Dic5—32no—
142C2×C4×Dic5C2×D312yes—
143#4038—38no—
144#4038C2×D318yescomplemented
145#4031—316no—
146#4031—316no—
147C4×D20—38no—
148#4015—38no—
149C2×#555—151noSylow
150C23×C3×C5C2411yescharacteristic
151C22×D15—22no—
152C22×D15—22no—
153C22×D15—22no—
154C22×D15C2412yes—
155C22×C5×D3—24no—
156C22×C5×D3—24no—
157C22×C5×D3—24no—
158C22×C5×D3C2414yes—
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×Dic15—24no—
169C2×Dic15C2412yes—
170C2×Dic15C2414yes—
171C4×D15—28no—
172C2×C5×Dic3—22no—
173C2×C5×Dic3#4114yes—
174C2×C5×Dic3C2412yes—
175C2×C5×Dic3C2412yes—
176C5×D12—28no—
177C4×C5×D3—216no—
178C2×C3×Dic5—22no—
179C2×C3×Dic5—24no—
180C2×C3×Dic5C2412yes—
181C2×C3×Dic5C2414yes—
182C3×C4×D5—28no—
183#1204—48no—
184#1203—48no—
185#1202—28no—
186#1201—48no—
187C23×C3×C4—51no—
188C22×#67—52no—
189C22×#67—51no—
190C22×D12—52no—
191C22×D12C2×D511yescharacteristic
192C22×C4×D3—51no—
193C22×C4×D3C2×D512yes—
194C22×Dic6—52no—
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—52no—
206C2×(C2×D3)⋊C4—51no—
207C2×(C2×D3)⋊C4C2×D512yes—
208C2×C4⋊Dic3—51no—
209C2×Dic3⋊C4—51no—
210C2×Dic3⋊C4—52no—
211C2×C4×Dic3—52no—
212C2×C4×Dic3C2×D511yescharacteristic
213#895—58no—
214#893C2×D518yescomplemented
215#892—58no—
216#888—58no—
217#885—516no—
218#884—516no—
219#875—516no—
220#872—516no—
221C24×C5—32no—
222C23×D5—31no—
223C23×D5C22×D311yescharacteristic
224C2×C5×D4—34no—
225C2×C5×D4—64no—
226C22×C4×C5—31no—
227C22×C4×C5—31no—
228C22×C4×C5—38no—
229C22×C4×C5—68no—
230C22×C4×C5C22×D312yes—
231C22×C4×C5C22×D311yescharacteristic
232C2×C22⋊D5—38no—
233C2×C22⋊D5—38no—
234C2×C22⋊D5—124no—
235C2×C22⋊D5—124no—
236C22×Dic5—31no—
237C22×Dic5—32no—
238C22×Dic5C22×D311yescharacteristic
239C22×Dic5C22×D312yes—
240C2×D20—34no—
241C2×D20—64no—
242C2×C4×D5—24no—
243C2×C4×D5—34no—
244C2×C4×D5—64no—
245C2×C4×D5C22×D314yes—
246C2×Dic10—38no—
247C2×Dic10—124no—
248C5×C4⋊C4C22×D314yes—
249C5×C22⋊C4—38no—
250C42×C5—34no—
251C22⋊Dic5—38no—
252C22⋊Dic5—38no—
253#685—34no—
254#685—38no—
255#685—34no—
256#685C22×D318yes—
257C4⋊Dic5—34no—
258C4⋊Dic5—38no—
259C4⋊Dic5—34no—
260C4⋊Dic5C22×D314yes—
261C4×Dic5—38no—
262C4×Dic5C22×D318yes—
263#555—1516no—
264C2×C4⋊Q8—151no—
265C2×#124—152no—
266C2×#124—152no—
267C2×#121—152no—
268C2×#121—151no—
269C2×#121—151no—
270C2×C4×D4—31no—
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—22no—
280C2×D15—22no—
281C2×D15—22no—
282C2×D15—44no—
283C2×C5×D3—24no—
284C2×C5×D3—24no—
285C2×C5×D3—24no—
286C2×C5×D3—48no—
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—
294Dic15—24no—
295C3×Dic5—22no—
296C3×Dic5—24no—
297C5×Dic3—22no—
298C5×Dic3C2×#4114yes—
299C24×C3—51no—
300C23×D3—51no—
301C23×D3C22×D512yes—
302C22×C3×C4—51no—
303C22×C3×C4—52no—
304C22×C3×C4—54no—
305C22×C3×C4—104no—
306C22×C3×C4C22×D512yes—
307C22×C3×C4C22×D511yescharacteristic
308C2×#67—58no—
309C2×#67—54no—
310C2×#67—104no—
311C2×#67—204no—
312C22×Dic3—51no—
313C22×Dic3—52no—
314C22×Dic3C22×D511yescharacteristic
315C22×Dic3C22×D511yescharacteristic
316C2×D12—24no—
317C2×D12—58no—
318C2×D12—204no—
319C2×D12C22×D514yes—
320C2×C4×D3—28no—
321C2×C4×D3—54no—
322C2×C4×D3—104no—
323C2×C4×D3C22×D518yes—
324C2×Dic6—58no—
325C2×Dic6—204no—
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—58no—
334(C2×D3)⋊C4—54no—
335(C2×D3)⋊C4C22×D518yes—
336C4⋊Dic3—54no—
337Dic3⋊C4—54no—
338Dic3⋊C4—58no—
339C4×Dic3—58no—
340C4×Dic3C22×D514yes—
341C23×C5—34no—
342C23×C5—64no—
343C23×C5—64no—
344C23×C5—64no—
345C23×C5C23×D311yescharacteristic
346C22×D5—22no—
347C22×D5—22no—
348C22×D5—22no—
349C22×D5—32no—
350C22×D5—62no—
351C22×D5—62no—
352C22×D5—62no—
353C22×D5C23×D312yes—
354C5×D4—68no—
355C2×C4×C5—24no—
356C2×C4×C5—34no—
357C2×C4×C5—32no—
358C2×C4×C5—32no—
359C2×C4×C5—616no—
360C2×C4×C5—62no—
361C2×C4×C5#23514yes—
362C2×C4×C5C23×D314yes—
363C2×C4×C5C23×D312yes—
364C22⋊D5—128no—
365C22⋊D5—128no—
366C2×Dic5—22no—
367C2×Dic5—24no—
368C2×Dic5—32no—
369C2×Dic5—34no—
370C2×Dic5—62no—
371C2×Dic5—64no—
372C2×Dic5C23×D312yes—
373C2×Dic5C23×D314yes—
374D20—68no—
375C4×D5—28no—
376C4×D5—68no—
377Dic10—128no—
378C22×Q8—152no—
379C22×D4—31no—
380C22×D4—151no—
381C22×D4—152no—
382C22×D4—152no—
383C23×C4—32no—
384C23×C4—51no—
385C23×C4—151no—
386C4⋊Q8—158no—
387#124—1516no—
388#124—1516no—
389#121—1516no—
390#121—158no—
391#121—158no—
392C4×D4—38no—
393C4×D4—158no—
394#117—58no—
395#117—1516no—
396#117—158no—
397C2×C4⋊C4—51no—
398C2×C4⋊C4—151no—
399C2×C4⋊C4—152no—
400C2×C4⋊C4—151no—
401C2×C4⋊C4D3×D511yescharacteristic
402C2×C22⋊C4—32no—
403C2×C22⋊C4—52no—
404C2×C22⋊C4—151no—
405C2×C22⋊C4—151no—
406C2×C22⋊C4—152no—
407C2×C22⋊C4—152no—
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—42no—
417C3×D5—42no—
418C5×D3—44no—
419C23×C3—52no—
420C23×C3—102no—
421C23×C3—102no—
422C23×C3—102no—
423C23×C3C23×D511yescharacteristic
424C22×D3—24no—
425C22×D3—24no—
426C22×D3—24no—
427C22×D3—52no—
428C22×D3—102no—
429C22×D3—102no—
430C22×D3—102no—
431C22×D3C23×D514yes—
432C2×C3×C4—24no—
433C2×C3×C4—52no—
434C2×C3×C4—54no—
435C2×C3×C4—102no—
436C2×C3×C4—104no—
437C2×C3×C4—108no—
438C2×C3×C4#70914yes—
439C2×C3×C4C23×D514yes—
440C2×C3×C4C23×D512yes—
441#67—108no—
442#67—208no—
443C2×Dic3—22no—
444C2×Dic3—52no—
445C2×Dic3—54no—
446C2×Dic3—102no—
447C2×Dic3—104no—
448C2×Dic3#71114yes—
449C2×Dic3C23×D512yes—
450C2×Dic3C23×D512yes—
451D12—28no—
452D12—208no—
453C4×D3—216no—
454C4×D3—108no—
455Dic6—208no—
456C22×C5—64no—
457C22×C5—64no—
458C22×C5—64no—
459C22×C5—128no—
460C22×C5C2×#23512yes—
461C22×C5D3×#4112yes—
462C22×C5#100912yes—
463C22×C5C24×D311yescharacteristic
464C2×D5—22no—
465C2×D5—22no—
466C2×D5—22no—
467C2×D5—44no—
468C2×D5—62no—
469C2×D5—62no—
470C2×D5—62no—
471C2×D5—124no—
472C4×C5—24no—
473C4×C5—34no—
474C4×C5—62no—
475C4×C5C2×#23514yes—
476Dic5—22no—
477Dic5—24no—
478Dic5—62no—
479Dic5—64no—
480C24—32no—
481C24—51no—
482C24—151no—
483C2×Q8—158no—
484C2×Q8—604no—
485C2×D4—34no—
486C2×D4—64no—
487C2×D4—154no—
488C2×D4—158no—
489C2×D4—158no—
490C2×D4—304no—
491C2×D4—604no—
492C2×D4—604no—
493C22×C4—31no—
494C22×C4—31no—
495C22×C4—38no—
496C22×C4—51no—
497C22×C4—52no—
498C22×C4—54no—
499C22×C4—68no—
500C22×C4—104no—
501C22×C4—151no—
502C22×C4—152no—
503C22×C4—154no—
504C22×C4—304no—
505C22×C4C2×D3×D512yes—
506C22×C4C2×D3×D511yescharacteristic
507C4⋊C4—54no—
508C4⋊C4—154no—
509C4⋊C4—158no—
510C4⋊C4—154no—
511C4⋊C4C2×D3×D514yes—
512C22⋊C4—38no—
513C22⋊C4—58no—
514C22⋊C4—154no—
515C22⋊C4—154no—
516C22⋊C4—158no—
517C22⋊C4—158no—
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×C3C2×#70912yes—
528C22×C3C2×#71112yes—
529C22×C3#414912yes—
530C22×C3C24×D511yescharacteristic
531C2×D3—24no—
532C2×D3—24no—
533C2×D3—24no—
534C2×D3—48no—
535C2×D3—102no—
536C2×D3—102no—
537C2×D3—102no—
538C2×D3—204no—
539C3×C4—24no—
540C3×C4—102no—
541C3×C4—104no—
542C3×C4C2×#70914yes—
543Dic3—22no—
544Dic3—102no—
545Dic3—104no—
546Dic3C2×#71114yes—
547C2×C5—124no—
548C2×C5#553214yescomplemented
549C2×C5C22×#23511yescharacteristic
550C2×C5C2×D3×#4111yescharacteristic
551C2×C5C2×#100911yescharacteristic
552D5—42no—
553D5—122no—
554C23—34no—
555C23—52no—
556C23—64no—
557C23—64no—
558C23—64no—
559C23—102no—
560C23—102no—
561C23—102no—
562C23—152no—
563C23—302no—
564C23—302no—
565C23—302no—
566C23C22×D3×D511yescharacteristic
567Q8—608no—
568D4—68no—
569D4—308no—
570D4—608no—
571D4—608no—
572C2×C4—24no—
573C2×C4—34no—
574C2×C4—32no—
575C2×C4—32no—
576C2×C4—52no—
577C2×C4—54no—
578C2×C4—616no—
579C2×C4—62no—
580C2×C4—102no—
581C2×C4—104no—
582C2×C4—108no—
583C2×C4—152no—
584C2×C4—154no—
585C2×C4—302no—
586C2×C4—304no—
587C2×C4—308no—
588C2×C4#677114yes—
589C2×C4C22×D3×D514yes—
590C2×C4C22×D3×D512yes—
591C2×C3—202no—
592C2×C3#6589714yescomplemented
593C2×C3C22×#70911yescharacteristic
594C2×C3C22×#71111yescharacteristic
595C2×C3C2×#414911yescharacteristic
596D3—44no—
597D3—202no—
598C5C2×#553211yescharacteristic, complemented, Sylow
599C22—64no—
600C22—64no—
601C22—64no—
602C22—102no—
603C22—102no—
604C22—102no—
605C22—128no—
606C22—204no—
607C22—302no—
608C22—302no—
609C22—302no—
610C22—604no—
611C22C2×#677112yes—
612C22D3×#71112yes—
613C22#9203512yes—
614C22C23×D3×D511yescharacteristic
615C4—24no—
616C4—34no—
617C4—62no—
618C4—102no—
619C4—104no—
620C4—302no—
621C4—304no—
622C4C2×#677114yes—
623C3C2×#6589711yescharacteristic, complemented, Sylow
624C2—124no—
625C2—202no—
626C2—602no—
627C2#1175376714yesdirect product factor
628C2C22×#677111yescharacteristic
629C2C2×D3×#71111yescharacteristic
630C2C2×#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×#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×#91781C2semidirect
2. C2×C5×#893C2semidirect
3. C2×C3×#4038C2semidirect
4. C2×#91469C2semidirect
5. C2×#91466C2semidirect
6. C2×#91464C2semidirect
7. C2×#91454C2semidirect
8. C2×#91449C2semidirect
9. C2×C4×#1202C2semidirect
10. C2×#91371C2semidirect
11. C2×#91350C2semidirect
12. #11753767C2direct (N × H)
26. #91781C22semidirect
29. C5×#893C22semidirect
33. C3×#4038C22semidirect
40. #91469C22semidirect
41. #91466C22semidirect
42. #91464C22semidirect
43. #91454C22semidirect
44. #91449C22semidirect
45. C4×#1202C22semidirect
46. #91371C22semidirect
47. #91350C22semidirect
56. C2×#4038D3semidirect
108. C2×#893D5semidirect
144. #4038C2×D3semidirect
214. #893C2×D5semidirect
412. C2×C3×C5#555semidirect
522. C3×C5C2×#555semidirect
548. C2×C5#5532semidirect
592. C2×C3#65897semidirect
598. C5C2×#5532semidirect
623. C3C2×#65897semidirect
627. C2#11753767direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#917812
C2×C5×#8932
C2×C3×#40382
C2×#914692
C2×#914662
C2×#914642
C2×#914542
C2×#914492
C2×C4×#12022
C2×#913712
C2×#913502
#117537672
C2×#658973
C2×#55325