Finite Group Databasebeta

← C2×#11756448 #14481740

Subgroups — C2×#11756448 #14481740

Order 1920.

2160 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#125C211yesmaximal, characteristic, complemented
2#11756448C2116yesmaximal, direct product factor
3C2×#91786C212yesmaximal
4C2×#91785C212yesmaximal
5C2×#91781C211yesmaximal, characteristic
6C2×C4⋊C4×D15C211yesmaximal, characteristic
7C2×#91779C211yesmaximal, characteristic, complemented
8C2×#91778C212yesmaximal, complemented
9C2×#91777C211yesmaximal, characteristic, complemented
10C2×#91777C212yesmaximal, complemented
11C2×#91761C211yesmaximal, characteristic
12C2×#91758C211yesmaximal, characteristic, complemented
13C2×#66023—31nomaximal, Hall
14C22×C4×D15C2212yes—
15C22×C4×D15C2211yescharacteristic
16C22×Dic30C2212yes—
17C22×Dic30C2212yes—
18C3×C5×#125C2218yescomplemented
19C2×C3×C5×C4⋊C4C2211yescharacteristic
20C2×C3×C5×C4⋊C4C2212yes—
21C2×C3×C5×C4⋊C4C2212yes—
22C2×C3×C5×C4⋊C4C2211yescharacteristic
23C2×C3×C42×C5C2211yescharacteristic
24C2×#6714C2211yescharacteristic
25C2×#6714C2211yescharacteristic
26C2×#6714C2212yes—
27C2×C4⋊Dic15C2211yescharacteristic
28C2×C4⋊Dic15C2212yes—
29C2×C4⋊Dic15C2211yescharacteristic
30C2×C4⋊Dic15C2212yes—
31C2×#6712C2211yescharacteristic
32C2×#6712C2211yescharacteristic
33C2×#6712C2212yes—
34C2×#6712C2212yes—
35C2×#6712C2212yes—
36C2×C4×Dic15C2211yescharacteristic
37C2×C4×Dic15C2212yes—
38#91786C22116yes—
39#91785C22116yes—
40#91781C2218yes—
41C4⋊C4×D15C2218yes—
42#91779C2218yescomplemented
43#91778C22116yescomplemented
44#91777C2218yescomplemented
45#91777C22116yescomplemented
46#91761C2218yes—
47#91758C2218yescomplemented
48C2×#5689—51nomaximal, Hall
49C2×C5×#125D311yescharacteristic, complemented
50#66023—316no—
51C2×#4043—32no—
52C2×#4042—32no—
53C2×#4038—31no—
54C2×D5×C4⋊C4—31no—
55C2×#4036—31no—
56C2×#4035—32no—
57C2×#4034—31no—
58C2×#4034—32no—
59C2×#4018—31no—
60C2×#4015—31no—
61C23×D15C2311yescharacteristic
62C22×C3×C4×C5C2312yes—
63C22×C3×C4×C5C2312yes—
64C22×C3×C4×C5C2311yescharacteristic
65C22×C3×C4×C5C2311yescharacteristic
66C22×C3×C4×C5C2311yescharacteristic
67C22×Dic15C2311yescharacteristic
68C22×Dic15C2311yescharacteristic
69C22×Dic15C2312yes—
70C22×Dic15C2312yes—
71C22×Dic15C2311yescharacteristic
72C2×C4×D15—44no—
73C2×C4×D15Q818yescomplemented
74C2×C4×D15C2318yes—
75C2×C4×D15C2314yes—
76C2×Dic30—44no—
77C2×Dic30—44no—
78C2×Dic30C2318yes—
79C2×Dic30C2318yes—
80C3×C5×C4⋊C4C2314yes—
81C3×C5×C4⋊C4C2318yes—
82C3×C5×C4⋊C4C2318yes—
83C3×C5×C4⋊C4C2314yes—
84C3×C42×C5C2314yes—
85#6714C2314yes—
86#6714C2314yes—
87#6714C2318yes—
88C4⋊Dic15C2314yes—
89C4⋊Dic15C2318yes—
90C4⋊Dic15C2314yes—
91C4⋊Dic15C2318yes—
92#6712C2314yes—
93#6712C2314yes—
94#6712C2318yes—
95#6712C2318yes—
96#6712C2318yes—
97C4×Dic15C2314yes—
98C4×Dic15C2318yes—
99C2×C3×#125D511yescharacteristic, complemented
100#5689—516no—
101C2×#897—52no—
102C2×#896—52no—
103C2×#892—51no—
104C2×D3×C4⋊C4—51no—
105C2×#890—51no—
106C2×#889—52no—
107C2×#888—51no—
108C2×#888—52no—
109C2×#872—51no—
110C2×#869—51no—
111C22×C4×D5—31no—
112C22×C4×D5—32no—
113C22×Dic10—32no—
114C22×Dic10—32no—
115C5×#125C2×D318yescomplemented
116C2×C5×C4⋊C4C2×D311yescharacteristic
117C2×C5×C4⋊C4C2×D312yes—
118C2×C5×C4⋊C4C2×D312yes—
119C2×C5×C4⋊C4C2×D311yescharacteristic
120C2×C42×C5C2×D311yescharacteristic
121C2×#685—31no—
122C2×#685—31no—
123C2×#685—32no—
124C2×C4⋊Dic5—31no—
125C2×C4⋊Dic5—32no—
126C2×C4⋊Dic5—31no—
127C2×C4⋊Dic5—32no—
128C2×#683—31no—
129C2×#683—31no—
130C2×#683—32no—
131C2×#683—32no—
132C2×#683—32no—
133C2×C4×Dic5—31no—
134C2×C4×Dic5—32no—
135#4043—316no—
136#4042—316no—
137#4038—38no—
138D5×C4⋊C4—38no—
139#4036—38no—
140#4035—316no—
141#4034—38no—
142#4034—316no—
143#4018—38no—
144#4015—38no—
145C2×#544—151noSylow
146C23×C3×C5C2411yescharacteristic
147C22×D15—22no—
148C22×D15—22no—
149C22×D15C2×Q814yes—
150C22×D15C2412yes—
151C2×C3×C4×C5—22no—
152C2×C3×C4×C5—24no—
153C2×C3×C4×C5—24no—
154C2×C3×C4×C5—22no—
155C2×C3×C4×C5C2×Q814yes—
156C2×C3×C4×C5C2414yes—
157C2×C3×C4×C5C2414yes—
158C2×C3×C4×C5C2412yes—
159C2×C3×C4×C5C2412yes—
160C2×C3×C4×C5C2412yes—
161C2×Dic15—22no—
162C2×Dic15—24no—
163C2×Dic15—24no—
164C2×Dic15—22no—
165C2×Dic15C2×Q814yes—
166C2×Dic15C2412yes—
167C2×Dic15C2412yes—
168C2×Dic15C2414yes—
169C2×Dic15C2414yes—
170C2×Dic15C2412yes—
171C4×D15—48no—
172C4×D15C2×Q8116yescomplemented
173Dic30—48no—
174Dic30—48no—
175C22×C4×D3—51no—
176C22×C4×D3—52no—
177C22×Dic6—52no—
178C22×Dic6—52no—
179C3×#125C2×D518yescomplemented
180C2×C3×C4⋊C4C2×D511yescharacteristic
181C2×C3×C4⋊C4C2×D512yes—
182C2×C3×C4⋊C4C2×D512yes—
183C2×C3×C4⋊C4C2×D511yescharacteristic
184C2×C3×C42C2×D511yescharacteristic
185C2×(C2×D3)⋊C4—51no—
186C2×(C2×D3)⋊C4—51no—
187C2×(C2×D3)⋊C4—52no—
188C2×C4⋊Dic3—51no—
189C2×C4⋊Dic3—52no—
190C2×C4⋊Dic3—51no—
191C2×C4⋊Dic3—52no—
192C2×Dic3⋊C4—51no—
193C2×Dic3⋊C4—51no—
194C2×Dic3⋊C4—52no—
195C2×Dic3⋊C4—52no—
196C2×Dic3⋊C4—52no—
197C2×C4×Dic3—51no—
198C2×C4×Dic3—52no—
199#897—516no—
200#896—516no—
201#892—58no—
202D3×C4⋊C4—58no—
203#890—58no—
204#889—516no—
205#888—58no—
206#888—516no—
207#872—58no—
208#869—58no—
209C23×D5—31no—
210C22×C4×C5C22×D311yescharacteristic
211C22×C4×C5C22×D311yescharacteristic
212C22×C4×C5C22×D312yes—
213C22×C4×C5C22×D312yes—
214C22×C4×C5C22×D311yescharacteristic
215C22×Dic5—31no—
216C22×Dic5—32no—
217C22×Dic5—31no—
218C22×Dic5—32no—
219C22×Dic5—31no—
220C2×C4×D5—38no—
221C2×C4×D5—34no—
222C2×C4×D5—38no—
223C2×C4×D5—124no—
224C2×Dic10—38no—
225C2×Dic10—38no—
226C2×Dic10—124no—
227C2×Dic10—124no—
228C5×C4⋊C4C22×D314yes—
229C5×C4⋊C4C22×D318yes—
230C5×C4⋊C4C22×D318yes—
231C5×C4⋊C4C22×D314yes—
232C42×C5C22×D314yes—
233#685—34no—
234#685—34no—
235#685—38no—
236C4⋊Dic5—34no—
237C4⋊Dic5—38no—
238C4⋊Dic5—34no—
239C4⋊Dic5—38no—
240#683—34no—
241#683—34no—
242#683—38no—
243#683—38no—
244#683—38no—
245C4×Dic5—34no—
246C4×Dic5—38no—
247#544—1516no—
248C2×C4⋊Q8—152no—
249C2×C4⋊Q8—151no—
250C2×C4⋊Q8—151no—
251C2×#125—152no—
252C2×#125—151no—
253C2×#125D1511yescharacteristic, complemented
254C2×C22⋊Q8—152no—
255C2×C22⋊Q8—152no—
256C2×#117—151no—
257C2×#117—151no—
258C22×C4⋊C4—151no—
259C22×C3×C5C22×Q812yes—
260C22×C3×C5H(5,𝔽2)12yes—
261C22×C3×C52−1+412yes—
262C22×C3×C5C2511yescharacteristic
263C2×D15—28no—
264C2×D15—22no—
265C2×D15—22no—
266C2×D15C22×Q814yes—
267C3×C4×C5—22no—
268C3×C4×C5—24no—
269C3×C4×C5—24no—
270C3×C4×C5—22no—
271C3×C4×C5C22×Q814yes—
272Dic15—22no—
273Dic15—24no—
274Dic15—24no—
275Dic15—22no—
276Dic15C22×Q814yes—
277C23×D3—51no—
278C22×C3×C4C22×D512yes—
279C22×C3×C4C22×D512yes—
280C22×C3×C4C22×D511yescharacteristic
281C22×C3×C4C22×D511yescharacteristic
282C22×C3×C4C22×D511yescharacteristic
283C22×Dic3—51no—
284C22×Dic3—51no—
285C22×Dic3—51no—
286C22×Dic3—52no—
287C22×Dic3—52no—
288C2×C4×D3—58no—
289C2×C4×D3—54no—
290C2×C4×D3—58no—
291C2×C4×D3—204no—
292C2×Dic6—58no—
293C2×Dic6—58no—
294C2×Dic6—204no—
295C2×Dic6—204no—
296C3×C4⋊C4C22×D514yes—
297C3×C4⋊C4C22×D518yes—
298C3×C4⋊C4C22×D518yes—
299C3×C4⋊C4C22×D514yes—
300C3×C42C22×D514yes—
301(C2×D3)⋊C4—54no—
302(C2×D3)⋊C4—54no—
303(C2×D3)⋊C4—58no—
304C4⋊Dic3—54no—
305C4⋊Dic3—58no—
306C4⋊Dic3—54no—
307C4⋊Dic3—58no—
308Dic3⋊C4—54no—
309Dic3⋊C4—54no—
310Dic3⋊C4—58no—
311Dic3⋊C4—58no—
312Dic3⋊C4—58no—
313C4×Dic3—54no—
314C4×Dic3—58no—
315C23×C5C23×D311yescharacteristic
316C22×D5—34no—
317C22×D5—32no—
318C22×D5—62no—
319C22×D5—62no—
320C2×C4×C5—22no—
321C2×C4×C5—24no—
322C2×C4×C5—22no—
323C2×C4×C5—24no—
324C2×C4×C5D3×Q814yes—
325C2×C4×C5C23×D314yes—
326C2×C4×C5C23×D314yes—
327C2×C4×C5C23×D312yes—
328C2×C4×C5C23×D312yes—
329C2×C4×C5C23×D312yes—
330C2×Dic5—34no—
331C2×Dic5—32no—
332C2×Dic5—32no—
333C2×Dic5—32no—
334C2×Dic5—34no—
335C2×Dic5—34no—
336C2×Dic5—62no—
337C2×Dic5—64no—
338C2×Dic5—64no—
339C2×Dic5—62no—
340C4×D5—316no—
341C4×D5—128no—
342Dic10—128no—
343Dic10—128no—
344C22×Q8—152no—
345C22×Q8—152no—
346C23×C4—151no—
347C23×C4—152no—
348C4⋊Q8—1516no—
349C4⋊Q8—158no—
350C4⋊Q8—158no—
351#125—1516no—
352#125—158no—
353#125C2×D1518yescomplemented
354C22⋊Q8—1516no—
355C22⋊Q8—1516no—
356#117—158no—
357#117—158no—
358C2×C4⋊C4—152no—
359C2×C4⋊C4—151no—
360C2×C4⋊C4—151no—
361C2×C4⋊C4—151no—
362C2×C4⋊C4—151no—
363C2×C4⋊C4—152no—
364C2×C4⋊C4—152no—
365C2×C4⋊C4—152no—
366C2×C4⋊C4—152no—
367C2×C4⋊C4—158no—
368C2×C4⋊C4C2×D1511yescharacteristic
369C2×C4⋊C4C2×D1511yescharacteristic
370C2×C4⋊C4C2×D1512yes—
371C2×C4⋊C4C2×D1512yes—
372C2×C22⋊C4—151no—
373C2×C22⋊C4—152no—
374C2×C22⋊C4—151no—
375C2×C42—152no—
376C2×C42—151no—
377C2×C42C2×D1511yescharacteristic
378C2×C3×C5#54414yescomplemented
379C2×C3×C5C23×Q811yescharacteristic
380C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
381C2×C3×C5C2×2−1+411yescharacteristic
382D15—24no—
383C23×C3C23×D511yescharacteristic
384C22×D3—54no—
385C22×D3—52no—
386C22×D3—102no—
387C22×D3—102no—
388C2×C3×C4—22no—
389C2×C3×C4—24no—
390C2×C3×C4—24no—
391C2×C3×C4—22no—
392C2×C3×C4Q8×D514yes—
393C2×C3×C4C23×D514yes—
394C2×C3×C4C23×D514yes—
395C2×C3×C4C23×D512yes—
396C2×C3×C4C23×D512yes—
397C2×C3×C4C23×D512yes—
398C2×Dic3—54no—
399C2×Dic3—52no—
400C2×Dic3—52no—
401C2×Dic3—52no—
402C2×Dic3—54no—
403C2×Dic3—54no—
404C2×Dic3—102no—
405C2×Dic3—104no—
406C2×Dic3—104no—
407C2×Dic3—102no—
408C4×D3—516no—
409C4×D3—208no—
410Dic6—208no—
411Dic6—208no—
412C22×C5C2×D3×Q812yes—
413C22×C5#100912yes—
414C22×C5#101012yes—
415C22×C5C24×D311yescharacteristic
416C2×D5—34no—
417C2×D5—68no—
418C2×D5—62no—
419C2×D5—62no—
420C4×C5—22no—
421C4×C5—24no—
422C4×C5—24no—
423C4×C5—22no—
424C4×C5C2×D3×Q814yes—
425Dic5—34no—
426Dic5—62no—
427Dic5—64no—
428Dic5—64no—
429Dic5—62no—
430C24—151no—
431C2×Q8—158no—
432C2×Q8—158no—
433C2×Q8—604no—
434C2×Q8—604no—
435C22×C4—151no—
436C22×C4—158no—
437C22×C4—154no—
438C22×C4—151no—
439C22×C4—151no—
440C22×C4—152no—
441C22×C4—152no—
442C22×C4—158no—
443C22×C4—604no—
444C22×C4C22×D1512yes—
445C22×C4C22×D1511yescharacteristic
446C22×C4C22×D1511yescharacteristic
447C22×C4C22×D1512yes—
448C22×C4C22×D1511yescharacteristic
449C4⋊C4—158no—
450C4⋊C4—154no—
451C4⋊C4—154no—
452C4⋊C4—154no—
453C4⋊C4—154no—
454C4⋊C4—158no—
455C4⋊C4—158no—
456C4⋊C4—158no—
457C4⋊C4—158no—
458C4⋊C4C22×D1514yes—
459C4⋊C4C22×D1514yes—
460C4⋊C4C22×D1518yes—
461C4⋊C4C22×D1518yes—
462C22⋊C4—154no—
463C22⋊C4—158no—
464C22⋊C4—154no—
465C42—158no—
466C42—154no—
467C42C22×D1514yes—
468C3×C5C2×#54411yescharacteristic, complemented, Hall
469C22×C3C2×Q8×D512yes—
470C22×C3#414912yes—
471C22×C3#415012yes—
472C22×C3C24×D511yescharacteristic
473C2×D3—54no—
474C2×D3—108no—
475C2×D3—102no—
476C2×D3—102no—
477C3×C4—22no—
478C3×C4—24no—
479C3×C4—24no—
480C3×C4—22no—
481C3×C4C2×Q8×D514yes—
482Dic3—54no—
483Dic3—102no—
484Dic3—104no—
485Dic3—104no—
486Dic3—102no—
487C2×C5#568914yescomplemented
488C2×C5C22×D3×Q811yescharacteristic
489C2×C5C2×#100911yescharacteristic
490C2×C5C2×#101011yescharacteristic
491D5—64no—
492C23—154no—
493C23—152no—
494C23—302no—
495C23—302no—
496C23C23×D1511yescharacteristic
497Q8—608no—
498Q8—608no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—154no—
504C2×C4—152no—
505C2×C4—1516no—
506C2×C4—152no—
507C2×C4—152no—
508C2×C4—154no—
509C2×C4—154no—
510C2×C4—304no—
511C2×C4—302no—
512C2×C4—304no—
513C2×C4—302no—
514C2×C4—608no—
515C2×C4Q8×D1514yes—
516C2×C4C23×D1514yes—
517C2×C4C23×D1512yes—
518C2×C4C23×D1512yes—
519C2×C4C23×D1514yes—
520C2×C4C23×D1512yes—
521C2×C3#6602314yescomplemented
522C2×C3C22×Q8×D511yescharacteristic
523C2×C3C2×#414911yescharacteristic
524C2×C3C2×#415011yescharacteristic
525D3—104no—
526C5C2×#568911yescharacteristic, complemented, Sylow
527C22—154no—
528C22—302no—
529C22—302no—
530C22—308no—
531C22C2×Q8×D1512yes—
532C22#9210012yes—
533C22#9210112yes—
534C22C24×D1511yescharacteristic
535C4—24no—
536C4—22no—
537C4—24no—
538C4—22no—
539C4—154no—
540C4—304no—
541C4—302no—
542C4—304no—
543C4—302no—
544C4C2×Q8×D1514yes—
545C3C2×#6602311yescharacteristic, complemented, Sylow
546C2—304no—
547C2#1175644814yesdirect product factor
548C2C22×Q8×D1511yescharacteristic
549C2C2×#9210011yescharacteristic
550C2C2×#9210111yescharacteristic
subgroupG/H
centerC23C23×D15
commutator, cosocleC22×C3×C5C25
FrattiniC22C24×D15
FittingC2×C3×C5×#125C2
socleC23×C3×C5C24
prime radicalC23×D15C23
hypercenterC2×#125D15
hypocenterC3×C5C2×#544
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×#125C2semidirect
2. #11756448C2direct (N × H)
7. C2×#91779C2semidirect
8. C2×#91778C2semidirect
9. C2×#91777C2semidirect
10. C2×#91777C2semidirect
12. C2×#91758C2semidirect
18. C3×C5×#125C22semidirect
42. #91779C22semidirect
43. #91778C22semidirect
44. #91777C22semidirect
45. #91777C22semidirect
47. #91758C22semidirect
49. C2×C5×#125D3semidirect
73. C2×C4×D15Q8semidirect
99. C2×C3×#125D5semidirect
115. C5×#125C2×D3semidirect
172. C4×D15C2×Q8semidirect
179. C3×#125C2×D5semidirect
253. C2×#125D15semidirect
353. #125C2×D15semidirect
378. C2×C3×C5#544semidirect
468. C3×C5C2×#544semidirect
487. C2×C5#5689semidirect
521. C2×C3#66023semidirect
526. C5C2×#5689semidirect
545. C3C2×#66023semidirect
547. C2#11756448direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×#1252
#117564482
C2×#917862
C2×#917852
C2×#917812
C2×C4⋊C4×D152
C2×#917792
C2×#917782
C2×#917772
C2×#917612
C2×#917582
C2×#660233
C2×#56895