Finite Group Databasebeta

← C2×#11753556 #14483965

Subgroups — C2×#11753556 #14483965

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×#892C211yesmaximal, characteristic, complemented
3C2×C3×#4034C211yesmaximal, characteristic, complemented
4C2×#91460C211yesmaximal, characteristic, complemented
5C2×C4×#1203C211yesmaximal, characteristic, complemented
6C2×#91420C212yesmaximal, complemented
7C2×#91418C211yesmaximal, characteristic, complemented
8C2×#91403C212yesmaximal, complemented
9C2×#91374C212yesmaximal, complemented
10C2×#91347C211yesmaximal, characteristic, complemented
11C2×#91326C212yesmaximal, complemented
12#11753556C2116yesmaximal, direct product factor
13C2×#65866—31nomaximal, Hall
14C22×D60C2211yescharacteristic
15C22×D60C2212yes—
16C22×C4×D15C2211yescharacteristic
17C22×C4×C5×D3C2211yescharacteristic
18C22×C3×Dic10C2212yes—
19C22×#1203C2211yescharacteristic
20C22×#1203C2212yes—
21C22×#1200C2212yes—
22C2×C3×C5×C4⋊C4C2211yescharacteristic
23C2×#6714C2212yes—
24#91784C2218yescomplemented
25C2×C5×(C2×D3)⋊C4C2212yes—
26C2×C5×C4⋊Dic3C2211yescharacteristic
27C2×C4×C5×Dic3C2212yes—
28C5×#892C2218yescomplemented
29C2×C3×C4⋊Dic5C2211yescharacteristic
30C2×C3×#683C2212yes—
31C2×C3×C4×Dic5C2211yescharacteristic
32C3×#4034C2218yescomplemented
33C2×#6668C2211yescharacteristic
34C2×#6667C2211yescharacteristic
35C2×#6667C2212yes—
36C2×#6667C2212yes—
37C2×#6667C2212yes—
38C2×(C2×D3)⋊Dic5C2211yescharacteristic
39C2×Dic3×Dic5C2212yes—
40#91460C2218yescomplemented
41C4×#1203C2218yescomplemented
42#91420C22116yescomplemented
43#91418C2218yescomplemented
44#91403C22116yescomplemented
45#91374C22116yescomplemented
46#91347C2218yescomplemented
47#91326C22116yescomplemented
48C2×#5734—51nomaximal, Hall
49C2×C5×#117—31no—
50C2×#4076—31no—
51C2×C4×C22⋊D5—31no—
52#65866—316no—
53C2×#4041—31no—
54C2×#4039—31no—
55C2×#4038—31no—
56C2×#4034D311yescharacteristic, complemented
57C2×#4031—32no—
58C2×#4030—32no—
59C2×#4021—32no—
60C2×#4018—32no—
61C23×D15C2312yes—
62C23×D15C2311yescharacteristic
63C23×C5×D3C2311yescharacteristic
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—
75C2×C4×C5×D3—24no—
76C2×C4×C5×D3C2314yes—
77C22×C3×Dic5C2312yes—
78C22×C3×Dic5C2311yescharacteristic
79C22×C3×Dic5C2311yescharacteristic
80C2×C3×Dic10—44no—
81C2×C3×Dic10C2318yes—
82C2×#1203—24no—
83C2×#1203—44no—
84C2×#1203C2314yes—
85C2×#1203C2318yes—
86C2×#1200—28no—
87C2×#1200C2318yes—
88C3×C5×C4⋊C4C2314yes—
89#6714C2318yes—
90C5×(C2×D3)⋊C4C2318yes—
91C5×C4⋊Dic3C2314yes—
92C4×C5×Dic3C2318yes—
93C3×C4⋊Dic5C2314yes—
94C3×#683C2318yes—
95C3×C4×Dic5C2314yes—
96#6668C2314yes—
97#6667C2318yes—
98#6667C2318yes—
99#6667C2318yes—
100#6667C2314yes—
101(C2×D3)⋊Dic5C2314yes—
102Dic3×Dic5C2318yes—
103C2×C3×C4⋊Q8—51no—
104C2×#947—52no—
105C2×#947—52no—
106#5734—516no—
107C2×#895—51no—
108C2×#895—51no—
109C2×#895—52no—
110C2×#892—51no—
111C2×#892—52no—
112C2×#892D511yescharacteristic, complemented
113C2×C4×D12—51no—
114C2×C4×D12—51no—
115C23×C4×C5—31no—
116C22×C22⋊D5—31no—
117C22×C22⋊D5—32no—
118C22×D20—31no—
119C22×D20—32no—
120C22×C4×D5—31no—
121C22×C4×D5—32no—
122C22×Dic10C2×D312yes—
123C5×#117—38no—
124C2×C5×C4⋊C4—31no—
125C2×C5×C4⋊C4C2×D311yescharacteristic
126C2×C5×C22⋊C4—32no—
127C2×C42×C5—32no—
128C2×C22⋊Dic5—31no—
129#4076—38no—
130C4×C22⋊D5—38no—
131C2×#685—32no—
132C2×#685—32no—
133C2×#685—31no—
134C2×#685—32no—
135C2×#685—32no—
136C2×C4⋊Dic5C2×D311yescharacteristic
137C2×#683—31no—
138C2×#683C2×D312yes—
139C2×C4×Dic5—32no—
140C2×C4×Dic5C2×D311yescharacteristic
141#4041—38no—
142#4039—38no—
143#4038—38no—
144#4034C2×D318yescomplemented
145#4031—316no—
146#4030—316no—
147#4021—316no—
148#4018—316no—
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×C5×D3—22no—
160C22×C5×D3—22no—
161C22×C5×D3—22no—
162C22×C5×D3C2412yes—
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×Dic3C2414yes—
175C2×C5×Dic3C2412yes—
176C4×C5×D3—28no—
177C2×C3×Dic5—24no—
178C2×C3×Dic5—22no—
179C2×C3×Dic5#4114yes—
180C2×C3×Dic5C2414yes—
181C2×C3×Dic5C2412yes—
182C2×C3×Dic5C2412yes—
183C3×Dic10—48no—
184#1203—28no—
185#1203—48no—
186#1200—216no—
187C22×C3×Q8—52no—
188C22×D12—51no—
189C22×D12—51no—
190C22×D12—52no—
191C22×D12—52no—
192C22×C4×D3—51no—
193C22×C4×D3—52no—
194C22×C4×D3C2×D511yescharacteristic
195C3×C4⋊Q8—58no—
196C2×C3×C4⋊C4—52no—
197C2×C3×C4⋊C4—51no—
198C2×C3×C4⋊C4C2×D511yescharacteristic
199C2×C3×C42—51no—
200#947—516no—
201#947—516no—
202C2×(C2×D3)⋊C4—51no—
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×(C2×D3)⋊C4C2×D512yes—
209C2×C4⋊Dic3—51no—
210C2×C4⋊Dic3C2×D511yescharacteristic
211C2×C4×Dic3—52no—
212C2×C4×Dic3C2×D512yes—
213#895—58no—
214#895—58no—
215#895—516no—
216#892—58no—
217#892—516no—
218#892C2×D518yescomplemented
219C4×D12—58no—
220C4×D12—58no—
221C24×C5—31no—
222C23×D5—31no—
223C23×D5—32no—
224C22×C4×C5—31no—
225C22×C4×C5—32no—
226C22×C4×C5—34no—
227C22×C4×C5—64no—
228C22×C4×C5C22×D312yes—
229C22×C4×C5C22×D311yescharacteristic
230C2×C22⋊D5—34no—
231C2×C22⋊D5—38no—
232C2×C22⋊D5—64no—
233C2×C22⋊D5—124no—
234C22×Dic5—31no—
235C22×Dic5C22×D311yescharacteristic
236C22×Dic5C22×D312yes—
237C22×Dic5C22×D311yescharacteristic
238C2×D20—34no—
239C2×D20—38no—
240C2×D20—64no—
241C2×D20—124no—
242C2×C4×D5—34no—
243C2×C4×D5—38no—
244C2×C4×D5—64no—
245C2×C4×D5—68no—
246C2×Dic10—44no—
247C2×Dic10C22×D318yes—
248C5×C4⋊C4—34no—
249C5×C4⋊C4C22×D314yes—
250C5×C22⋊C4—38no—
251C42×C5—38no—
252C22⋊Dic5—34no—
253#685—38no—
254#685—38no—
255#685—34no—
256#685—38no—
257#685—38no—
258C4⋊Dic5C22×D314yes—
259#683—34no—
260#683C22×D318yes—
261C4×Dic5—38no—
262C4×Dic5C22×D314yes—
263#555—1516no—
264C2×C4⋊Q8—51no—
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—31no—
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×C5×D3—22no—
288C2×C5×D3—22no—
289C2×C5×D3—22no—
290C2×C5×D3—44no—
291C3×C4×C5—24no—
292C3×C4×C5C2×#4114yes—
293Dic15—22no—
294C3×Dic5—24no—
295C3×Dic5—22no—
296C3×Dic5C2×#4114yes—
297C5×Dic3—24no—
298C5×Dic3—22no—
299C23×D3—51no—
300C23×D3—52no—
301C23×D3C22×D511yescharacteristic
302C2×C3×Q8—58no—
303C2×C3×Q8—204no—
304C22×C3×C4—51no—
305C22×C3×C4—52no—
306C22×C3×C4—51no—
307C22×C3×C4C22×D512yes—
308C22×C3×C4C22×D511yescharacteristic
309C22×Dic3—51no—
310C22×Dic3C22×D512yes—
311C22×Dic3C22×D511yescharacteristic
312C2×D12—54no—
313C2×D12—54no—
314C2×D12—58no—
315C2×D12—58no—
316C2×D12—104no—
317C2×D12—104no—
318C2×D12—204no—
319C2×D12—204no—
320C2×C4×D3—24no—
321C2×C4×D3—54no—
322C2×C4×D3—58no—
323C2×C4×D3—108no—
324C2×C4×D3—104no—
325C2×C4×D3C22×D514yes—
326C3×C4⋊C4—58no—
327C3×C4⋊C4—54no—
328C3×C4⋊C4C22×D514yes—
329C3×C42—54no—
330(C2×D3)⋊C4—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—
336(C2×D3)⋊C4C22×D518yes—
337C4⋊Dic3—54no—
338C4⋊Dic3C22×D514yes—
339C4×Dic3—58no—
340C4×Dic3C22×D518yes—
341C23×C5—32no—
342C23×C5—62no—
343C23×C5—62no—
344C23×C5—62no—
345C23×C5C23×D311yescharacteristic
346C22×D5—32no—
347C22×D5—34no—
348C22×D5—62no—
349C22×D5—64no—
350C22×D5—62no—
351C22×D5—64no—
352C22×D5—64no—
353C22×D5—62no—
354C2×C4×C5—24no—
355C2×C4×C5—34no—
356C2×C4×C5—32no—
357C2×C4×C5—64no—
358C2×C4×C5—62no—
359C2×C4×C5—68no—
360C2×C4×C5#23914yes—
361C2×C4×C5C23×D314yes—
362C2×C4×C5C23×D312yes—
363C22⋊D5—68no—
364C22⋊D5—128no—
365C2×Dic5—22no—
366C2×Dic5—24no—
367C2×Dic5—32no—
368C2×Dic5—62no—
369C2×Dic5#23914yes—
370C2×Dic5C23×D314yes—
371C2×Dic5C23×D312yes—
372C2×Dic5C23×D312yes—
373D20—68no—
374D20—128no—
375C4×D5—68no—
376C4×D5—616no—
377Dic10—48no—
378C22×Q8—52no—
379C22×D4—151no—
380C22×D4—152no—
381C22×D4—151no—
382C22×D4—152no—
383C23×C4—31no—
384C23×C4—152no—
385C23×C4—151no—
386C4⋊Q8—58no—
387#124—1516no—
388#124—1516no—
389#121—1516no—
390#121—158no—
391#121—158no—
392C4×D4—158no—
393C4×D4—158no—
394#117—38no—
395#117—1516no—
396#117—158no—
397C2×C4⋊C4—31no—
398C2×C4⋊C4—52no—
399C2×C4⋊C4—51no—
400C2×C4⋊C4—151no—
401C2×C4⋊C4D3×D511yescharacteristic
402C2×C22⋊C4—32no—
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—32no—
410C2×C42—51no—
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—
418C5×D3—42no—
419C23×C3C23×D511yescharacteristic
420C22×D3—22no—
421C22×D3—22no—
422C22×D3—22no—
423C22×D3—52no—
424C22×D3—54no—
425C22×D3—102no—
426C22×D3—104no—
427C22×D3—104no—
428C22×D3—102no—
429C22×D3—104no—
430C22×D3—102no—
431C22×D3C23×D512yes—
432C3×Q8—208no—
433C2×C3×C4—24no—
434C2×C3×C4—54no—
435C2×C3×C4—52no—
436C2×C3×C4—54no—
437C2×C3×C4—52no—
438C2×C3×C4—104no—
439C2×C3×C4—102no—
440C2×C3×C4#70914yes—
441C2×C3×C4C23×D514yes—
442C2×C3×C4C23×D512yes—
443C2×Dic3—22no—
444C2×Dic3—24no—
445C2×Dic3—52no—
446C2×Dic3—102no—
447C2×Dic3C23×D514yes—
448C2×Dic3C23×D512yes—
449D12—108no—
450D12—108no—
451D12—208no—
452D12—208no—
453C4×D3—28no—
454C4×D3—1016no—
455C4×D3—108no—
456C22×C5—62no—
457C22×C5—62no—
458C22×C5—62no—
459C22×C5—124no—
460C22×C5#100412yes—
461C22×C5C2×#23912yes—
462C22×C5C2×#23912yes—
463C22×C5C24×D311yescharacteristic
464C2×D5—62no—
465C2×D5—64no—
466C2×D5—62no—
467C2×D5—64no—
468C2×D5—64no—
469C2×D5—62no—
470C2×D5—124no—
471C2×D5—128no—
472C4×C5—24no—
473C4×C5—64no—
474C4×C5—62no—
475C4×C5C2×#23914yes—
476Dic5—22no—
477Dic5—24no—
478Dic5—62no—
479Dic5C2×#23914yes—
480C24—31no—
481C24—151no—
482C24—152no—
483C2×Q8—58no—
484C2×Q8—204no—
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—32no—
495C22×C4—34no—
496C22×C4—51no—
497C22×C4—51no—
498C22×C4—52no—
499C22×C4—64no—
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—34no—
508C4⋊C4—58no—
509C4⋊C4—54no—
510C4⋊C4—154no—
511C4⋊C4C2×D3×D514yes—
512C22⋊C4—38no—
513C22⋊C4—158no—
514C22⋊C4—158no—
515C22⋊C4—154no—
516C22⋊C4—158no—
517C22⋊C4—154no—
518C22⋊C4—158no—
519C42—38no—
520C42—54no—
521C42—158no—
522C3×C5C2×#55511yescharacteristic, complemented, Hall
523C22×C3C2×#70912yes—
524C22×C3D5×#4112yes—
525C22×C3#414912yes—
526C22×C3C24×D511yescharacteristic
527C2×D3—22no—
528C2×D3—22no—
529C2×D3—22no—
530C2×D3—44no—
531C2×D3—102no—
532C2×D3—104no—
533C2×D3—104no—
534C2×D3—102no—
535C2×D3—104no—
536C2×D3—102no—
537C2×D3—204no—
538C2×D3—208no—
539C3×C4—24no—
540C3×C4—54no—
541C3×C4—104no—
542C3×C4—102no—
543C3×C4C2×#70914yes—
544Dic3—22no—
545Dic3—24no—
546Dic3—102no—
547C2×C5—122no—
548C2×C5#573414yescomplemented
549C2×C5C2×#100411yescharacteristic
550C2×C5C22×#23911yescharacteristic
551C2×C5C22×#23911yescharacteristic
552D5—122no—
553D5—124no—
554C23—32no—
555C23—62no—
556C23—62no—
557C23—62no—
558C23—152no—
559C23—154no—
560C23—302no—
561C23—304no—
562C23—302no—
563C23—304no—
564C23—304no—
565C23—302no—
566C23C22×D3×D511yescharacteristic
567Q8—208no—
568D4—308no—
569D4—308no—
570D4—608no—
571D4—608no—
572C2×C4—24no—
573C2×C4—32no—
574C2×C4—34no—
575C2×C4—54no—
576C2×C4—52no—
577C2×C4—52no—
578C2×C4—54no—
579C2×C4—62no—
580C2×C4—64no—
581C2×C4—68no—
582C2×C4—104no—
583C2×C4—102no—
584C2×C4—152no—
585C2×C4—3016no—
586C2×C4—302no—
587C2×C4—308no—
588C2×C4#676914yes—
589C2×C4C22×D3×D514yes—
590C2×C4C22×D3×D512yes—
591C2×C3#6586614yescomplemented
592C2×C3C22×#70911yescharacteristic
593C2×C3C2×D5×#4111yescharacteristic
594C2×C3C2×#414911yescharacteristic
595D3—42no—
596D3—202no—
597D3—204no—
598C5C2×#573411yescharacteristic, complemented, Sylow
599C22—62no—
600C22—62no—
601C22—62no—
602C22—124no—
603C22—302no—
604C22—304no—
605C22—302no—
606C22—304no—
607C22—304no—
608C22—302no—
609C22—604no—
610C22—608no—
611C22C2×#676912yes—
612C22#9202612yes—
613C22D5×#23912yes—
614C22C23×D3×D511yescharacteristic
615C4—24no—
616C4—54no—
617C4—62no—
618C4—64no—
619C4—104no—
620C4—102no—
621C4—302no—
622C4C2×#676914yes—
623C3C2×#6586611yescharacteristic, complemented, Sylow
624C2—122no—
625C2—602no—
626C2—604no—
627C2#1175355614yesdirect product factor
628C2C22×#676911yescharacteristic
629C2C2×#9202611yescharacteristic
630C2C2×D5×#23911yescharacteristic
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×#892C2semidirect
3. C2×C3×#4034C2semidirect
4. C2×#91460C2semidirect
5. C2×C4×#1203C2semidirect
6. C2×#91420C2semidirect
7. C2×#91418C2semidirect
8. C2×#91403C2semidirect
9. C2×#91374C2semidirect
10. C2×#91347C2semidirect
11. C2×#91326C2semidirect
12. #11753556C2direct (N × H)
24. #91784C22semidirect
28. C5×#892C22semidirect
32. C3×#4034C22semidirect
40. #91460C22semidirect
41. C4×#1203C22semidirect
42. #91420C22semidirect
43. #91418C22semidirect
44. #91403C22semidirect
45. #91374C22semidirect
46. #91347C22semidirect
47. #91326C22semidirect
56. C2×#4034D3semidirect
112. C2×#892D5semidirect
144. #4034C2×D3semidirect
218. #892C2×D5semidirect
412. C2×C3×C5#555semidirect
522. C3×C5C2×#555semidirect
548. C2×C5#5734semidirect
591. C2×C3#65866semidirect
598. C5C2×#5734semidirect
623. C3C2×#65866semidirect
627. C2#11753556direct (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×#8922
C2×C3×#40342
C2×#914602
C2×C4×#12032
C2×#914202
C2×#914182
C2×#914032
C2×#913742
C2×#913472
C2×#913262
#117535562
C2×#658663
C2×#57345