Finite Group Databasebeta

← C2×#11756327 #14484002

Subgroups — C2×#11756327 #14484002

Order 1920.

2224 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C5×Q8C211yesmaximal, characteristic, complemented
2C2×#91836C211yesmaximal, characteristic
3C2×#91833C211yesmaximal, characteristic, complemented
4#11756327C2116yesmaximal, direct product factor
5C2×#91787C212yesmaximal, complemented
6C2×#91785C212yesmaximal, complemented
7C2×#91782C211yesmaximal, characteristic
8C2×#91776C211yesmaximal, characteristic, complemented
9C2×#91765C212yesmaximal, complemented
10C2×#91764C211yesmaximal, characteristic
11C2×#91761C212yesmaximal, complemented
12C2×C4×Dic30C211yesmaximal, characteristic, complemented
13C2×#65902—31nomaximal, Hall
14C22×C3×C5×Q8C2211yescharacteristic
15C22×D60C2211yescharacteristic
16C22×C4×D15C2212yes—
17C22×Dic30C2211yescharacteristic
18C22×Dic30C2211yescharacteristic
19C3×C4×C5×Q8C2218yescomplemented
20C2×C3×C5×C4⋊C4C2211yescharacteristic
21C2×C3×C5×C4⋊C4C2212yes—
22C2×C3×C42×C5C2211yescharacteristic
23C2×C3×C42×C5C2212yes—
24#91836C2218yes—
25#91833C2218yescomplemented
26C2×#6714C2212yes—
27C2×#6714C2212yes—
28C2×#6714C2212yes—
29C2×#6714C2212yes—
30C2×#6714C2212yes—
31C2×C4⋊Dic15C2211yescharacteristic
32C2×#6712C2212yes—
33C2×#6712C2212yes—
34C2×#6712C2211yescharacteristic
35C2×#6712C2211yescharacteristic
36C2×#6712C2212yes—
37C2×C4×Dic15C2211yescharacteristic
38C2×C4×Dic15C2211yescharacteristic
39C2×C4×Dic15C2212yes—
40#91787C22116yescomplemented
41#91785C22116yescomplemented
42#91782C2218yes—
43#91776C2218yescomplemented
44#91765C22116yescomplemented
45#91764C2218yes—
46#91761C22116yescomplemented
47C4×Dic30C2218yescomplemented
48C2×#5568—51nomaximal, Hall
49C2×C4×C5×Q8D311yescharacteristic, complemented
50C2×#4093—31no—
51C2×#4090—31no—
52#65902—316no—
53C2×#4044—32no—
54C2×#4042—32no—
55C2×#4039—31no—
56C2×#4033—31no—
57C2×#4022—32no—
58C2×#4021—31no—
59C2×#4018—32no—
60C2×C4×Dic10—31no—
61C23×D15C2312yes—
62C2×C3×C5×Q8—24no—
63C2×C3×C5×Q8C2314yes—
64C22×C3×C4×C5C2312yes—
65C22×C3×C4×C5C2311yescharacteristic
66C22×C3×C4×C5C2312yes—
67C22×C3×C4×C5C2311yescharacteristic
68C22×C3×C4×C5C2311yescharacteristic
69C22×Dic15C2311yescharacteristic
70C22×Dic15C2311yescharacteristic
71C22×Dic15C2311yescharacteristic
72C22×Dic15C2312yes—
73C22×Dic15C2311yescharacteristic
74C2×D60—24no—
75C2×D60C2314yes—
76C2×C4×D15—28no—
77C2×C4×D15C2318yes—
78C2×Dic30—24no—
79C2×Dic30—24no—
80C2×Dic30C2314yes—
81C2×Dic30C2314yes—
82C3×C5×C4⋊C4C2314yes—
83C3×C5×C4⋊C4C2318yes—
84C3×C42×C5C2314yes—
85C3×C42×C5C2318yes—
86#6714C2318yes—
87#6714C2318yes—
88#6714C2318yes—
89#6714C2318yes—
90#6714C2318yes—
91C4⋊Dic15C2314yes—
92#6712C2314yes—
93#6712C2314yes—
94#6712C2318yes—
95#6712C2318yes—
96#6712C2318yes—
97C4×Dic15C2314yes—
98C4×Dic15C2314yes—
99C4×Dic15C2318yes—
100C2×C3×C4×Q8D511yescharacteristic, complemented
101C2×#947—51no—
102C2×#944—51no—
103#5568—516no—
104C2×#898—52no—
105C2×#896—52no—
106C2×#893—51no—
107C2×#887—51no—
108C2×#876—52no—
109C2×#875—51no—
110C2×#872—52no—
111C2×C4×Dic6—51no—
112C22×C5×Q8C2×D311yescharacteristic
113C22×D20—31no—
114C22×C4×D5—32no—
115C22×Dic10—31no—
116C22×Dic10—31no—
117C4×C5×Q8C2×D318yescomplemented
118C2×C5×C4⋊C4C2×D311yescharacteristic
119C2×C5×C4⋊C4C2×D312yes—
120C2×C42×C5C2×D311yescharacteristic
121C2×C42×C5C2×D312yes—
122#4093—38no—
123#4090—38no—
124C2×#685—32no—
125C2×#685—32no—
126C2×#685—32no—
127C2×#685—32no—
128C2×#685—32no—
129C2×C4⋊Dic5—31no—
130C2×#683—31no—
131C2×#683—31no—
132C2×#683—32no—
133C2×#683—32no—
134C2×#683—32no—
135C2×C4×Dic5—31no—
136C2×C4×Dic5—31no—
137C2×C4×Dic5—32no—
138#4044—316no—
139#4042—316no—
140#4039—38no—
141#4033—38no—
142#4022—316no—
143#4021—38no—
144#4018—316no—
145C4×Dic10—38no—
146C2×#556—151noSylow
147C23×C3×C5C2411yescharacteristic
148C22×D15—24no—
149C22×D15—24no—
150C22×D15—24no—
151C22×D15C2414yes—
152C3×C5×Q8—28no—
153C2×C3×C4×C5—22no—
154C2×C3×C4×C5—24no—
155C2×C3×C4×C5—24no—
156C2×C3×C4×C5—22no—
157C2×C3×C4×C5#4114yes—
158C2×C3×C4×C5C2414yes—
159C2×C3×C4×C5C2412yes—
160C2×C3×C4×C5C2414yes—
161C2×C3×C4×C5C2412yes—
162C2×C3×C4×C5C2412yes—
163C2×Dic15—24no—
164C2×Dic15—22no—
165C2×Dic15—22no—
166C2×Dic15—22no—
167C2×Dic15#4114yes—
168C2×Dic15C2412yes—
169C2×Dic15C2412yes—
170C2×Dic15C2412yes—
171C2×Dic15C2414yes—
172C2×Dic15C2412yes—
173D60—28no—
174C4×D15—216no—
175Dic30—28no—
176Dic30—28no—
177C22×C3×Q8C2×D511yescharacteristic
178C22×D12—51no—
179C22×C4×D3—52no—
180C22×Dic6—51no—
181C22×Dic6—51no—
182C3×C4×Q8C2×D518yescomplemented
183C2×C3×C4⋊C4C2×D511yescharacteristic
184C2×C3×C4⋊C4C2×D512yes—
185C2×C3×C42C2×D511yescharacteristic
186C2×C3×C42C2×D512yes—
187#947—58no—
188#944—58no—
189C2×(C2×D3)⋊C4—52no—
190C2×(C2×D3)⋊C4—52no—
191C2×(C2×D3)⋊C4—52no—
192C2×(C2×D3)⋊C4—52no—
193C2×(C2×D3)⋊C4—52no—
194C2×C4⋊Dic3—51no—
195C2×Dic3⋊C4—51no—
196C2×Dic3⋊C4—51no—
197C2×Dic3⋊C4—52no—
198C2×Dic3⋊C4—52no—
199C2×Dic3⋊C4—52no—
200C2×C4×Dic3—51no—
201C2×C4×Dic3—51no—
202C2×C4×Dic3—52no—
203#898—516no—
204#896—516no—
205#893—58no—
206#887—58no—
207#876—516no—
208#875—58no—
209#872—516no—
210C4×Dic6—58no—
211C23×D5—32no—
212C2×C5×Q8—24no—
213C2×C5×Q8C22×D314yes—
214C22×C4×C5C22×D311yescharacteristic
215C22×C4×C5C22×D311yescharacteristic
216C22×C4×C5C22×D312yes—
217C22×C4×C5C22×D312yes—
218C22×C4×C5C22×D311yescharacteristic
219C22×Dic5—31no—
220C22×Dic5—32no—
221C22×Dic5—31no—
222C22×Dic5—31no—
223C22×Dic5—31no—
224C2×D20—34no—
225C2×D20—64no—
226C2×C4×D5—38no—
227C2×C4×D5—68no—
228C2×Dic10—34no—
229C2×Dic10—34no—
230C2×Dic10—64no—
231C2×Dic10—64no—
232C5×C4⋊C4C22×D314yes—
233C5×C4⋊C4C22×D318yes—
234C42×C5C22×D314yes—
235C42×C5C22×D318yes—
236#685—38no—
237#685—38no—
238#685—38no—
239#685—38no—
240#685—38no—
241C4⋊Dic5—34no—
242#683—34no—
243#683—34no—
244#683—38no—
245#683—38no—
246#683—38no—
247C4×Dic5—34no—
248C4×Dic5—34no—
249C4×Dic5—38no—
250#556—1516no—
251C2×C4⋊Q8—151no—
252C2×#126—152no—
253C2×#126—152no—
254C2×#124—151no—
255C2×#124—151no—
256C2×C22⋊Q8—152no—
257C2×C4×Q8—151no—
258C2×C4×Q8—151no—
259C2×C4×Q8D1511yescharacteristic, complemented
260C2×C4×D4—151no—
261C2×#117—152no—
262C22×C3×C5C2×#4112yes—
263C22×C3×C5C2×#4112yes—
264C22×C3×C52−1+412yes—
265C22×C3×C5C2511yescharacteristic
266C2×D15—24no—
267C2×D15—24no—
268C2×D15—24no—
269C2×D15—48no—
270C3×C4×C5—22no—
271C3×C4×C5—24no—
272C3×C4×C5—24no—
273C3×C4×C5—22no—
274C3×C4×C5C2×#4114yes—
275Dic15—24no—
276Dic15—22no—
277Dic15—22no—
278Dic15—22no—
279Dic15C2×#4114yes—
280C23×D3—52no—
281C2×C3×Q8—24no—
282C2×C3×Q8C22×D514yes—
283C22×C3×C4C22×D512yes—
284C22×C3×C4C22×D511yescharacteristic
285C22×C3×C4C22×D512yes—
286C22×C3×C4C22×D511yescharacteristic
287C22×C3×C4C22×D511yescharacteristic
288C22×Dic3—51no—
289C22×Dic3—51no—
290C22×Dic3—51no—
291C22×Dic3—52no—
292C22×Dic3—51no—
293C2×D12—54no—
294C2×D12—104no—
295C2×C4×D3—58no—
296C2×C4×D3—108no—
297C2×Dic6—54no—
298C2×Dic6—54no—
299C2×Dic6—104no—
300C2×Dic6—104no—
301C3×C4⋊C4C22×D514yes—
302C3×C4⋊C4C22×D518yes—
303C3×C42C22×D514yes—
304C3×C42C22×D518yes—
305(C2×D3)⋊C4—58no—
306(C2×D3)⋊C4—58no—
307(C2×D3)⋊C4—58no—
308(C2×D3)⋊C4—58no—
309(C2×D3)⋊C4—58no—
310C4⋊Dic3—54no—
311Dic3⋊C4—54no—
312Dic3⋊C4—54no—
313Dic3⋊C4—58no—
314Dic3⋊C4—58no—
315Dic3⋊C4—58no—
316C4×Dic3—54no—
317C4×Dic3—54no—
318C4×Dic3—58no—
319C23×C5C23×D311yescharacteristic
320C22×D5—34no—
321C22×D5—64no—
322C22×D5—64no—
323C22×D5—64no—
324C5×Q8—28no—
325C2×C4×C5—24no—
326C2×C4×C5—22no—
327C2×C4×C5—22no—
328C2×C4×C5—24no—
329C2×C4×C5#23514yes—
330C2×C4×C5C23×D314yes—
331C2×C4×C5C23×D312yes—
332C2×C4×C5C23×D314yes—
333C2×C4×C5C23×D312yes—
334C2×C4×C5C23×D312yes—
335C2×Dic5—34no—
336C2×Dic5—32no—
337C2×Dic5—32no—
338C2×Dic5—32no—
339C2×Dic5—34no—
340C2×Dic5—32no—
341C2×Dic5—64no—
342C2×Dic5—62no—
343C2×Dic5—62no—
344C2×Dic5—62no—
345D20—68no—
346C4×D5—616no—
347Dic10—68no—
348Dic10—68no—
349C22×Q8—151no—
350C22×Q8—151no—
351C22×Q8C2×D1511yescharacteristic
352C22×D4—151no—
353C23×C4—152no—
354C4⋊Q8—158no—
355#126—1516no—
356#126—1516no—
357#124—158no—
358#124—158no—
359C22⋊Q8—1516no—
360C4×Q8—158no—
361C4×Q8—158no—
362C4×Q8C2×D1518yescomplemented
363C4×D4—158no—
364#117—1516no—
365C2×C4⋊C4—151no—
366C2×C4⋊C4—152no—
367C2×C4⋊C4—152no—
368C2×C4⋊C4—151no—
369C2×C4⋊C4—151no—
370C2×C4⋊C4—152no—
371C2×C4⋊C4C2×D1511yescharacteristic
372C2×C4⋊C4C2×D1512yes—
373C2×C22⋊C4—152no—
374C2×C22⋊C4—152no—
375C2×C22⋊C4—152no—
376C2×C22⋊C4—152no—
377C2×C22⋊C4—152no—
378C2×C42—152no—
379C2×C42—151no—
380C2×C42—151no—
381C2×C42C2×D1512yes—
382C2×C42C2×D1511yescharacteristic
383C2×C3×C5#55614yescomplemented
384C2×C3×C5C22×#4111yescharacteristic
385C2×C3×C5C22×#4111yescharacteristic
386C2×C3×C5C2×2−1+411yescharacteristic
387D15—44no—
388C23×C3C23×D511yescharacteristic
389C22×D3—54no—
390C22×D3—104no—
391C22×D3—104no—
392C22×D3—104no—
393C3×Q8—28no—
394C2×C3×C4—24no—
395C2×C3×C4—22no—
396C2×C3×C4—22no—
397C2×C3×C4—24no—
398C2×C3×C4#70914yes—
399C2×C3×C4C23×D514yes—
400C2×C3×C4C23×D512yes—
401C2×C3×C4C23×D514yes—
402C2×C3×C4C23×D512yes—
403C2×C3×C4C23×D512yes—
404C2×Dic3—54no—
405C2×Dic3—52no—
406C2×Dic3—52no—
407C2×Dic3—52no—
408C2×Dic3—54no—
409C2×Dic3—52no—
410C2×Dic3—104no—
411C2×Dic3—102no—
412C2×Dic3—102no—
413C2×Dic3—102no—
414D12—108no—
415C4×D3—1016no—
416Dic6—108no—
417Dic6—108no—
418C22×C5C2×#23512yes—
419C22×C5#100712yes—
420C22×C5D3×#4112yes—
421C22×C5C24×D311yescharacteristic
422C2×D5—64no—
423C2×D5—64no—
424C2×D5—64no—
425C2×D5—128no—
426C4×C5—24no—
427C4×C5—22no—
428C4×C5—22no—
429C4×C5—24no—
430C4×C5C2×#23514yes—
431Dic5—34no—
432Dic5—64no—
433Dic5—62no—
434Dic5—62no—
435Dic5—62no—
436C24—152no—
437C2×Q8—24no—
438C2×Q8—154no—
439C2×Q8—154no—
440C2×Q8—304no—
441C2×Q8—304no—
442C2×Q8C22×D1514yes—
443C2×D4—154no—
444C2×D4—304no—
445C22×C4—151no—
446C22×C4—151no—
447C22×C4—151no—
448C22×C4—152no—
449C22×C4—151no—
450C22×C4—158no—
451C22×C4—308no—
452C22×C4C22×D1512yes—
453C22×C4C22×D1511yescharacteristic
454C22×C4C22×D1512yes—
455C22×C4C22×D1511yescharacteristic
456C22×C4C22×D1511yescharacteristic
457C4⋊C4—154no—
458C4⋊C4—158no—
459C4⋊C4—158no—
460C4⋊C4—154no—
461C4⋊C4—154no—
462C4⋊C4—158no—
463C4⋊C4C22×D1514yes—
464C4⋊C4C22×D1518yes—
465C22⋊C4—158no—
466C22⋊C4—158no—
467C22⋊C4—158no—
468C22⋊C4—158no—
469C22⋊C4—158no—
470C42—158no—
471C42—154no—
472C42—154no—
473C42C22×D1518yes—
474C42C22×D1514yes—
475C3×C5C2×#55611yescharacteristic, complemented, Hall
476C22×C3C2×#70912yes—
477C22×C3#414712yes—
478C22×C3D5×#4112yes—
479C22×C3C24×D511yescharacteristic
480C2×D3—104no—
481C2×D3—104no—
482C2×D3—104no—
483C2×D3—208no—
484C3×C4—24no—
485C3×C4—22no—
486C3×C4—22no—
487C3×C4—24no—
488C3×C4C2×#70914yes—
489Dic3—54no—
490Dic3—104no—
491Dic3—102no—
492Dic3—102no—
493Dic3—102no—
494C2×C5#556814yescomplemented
495C2×C5C22×#23511yescharacteristic
496C2×C5C2×#100711yescharacteristic
497C2×C5C2×D3×#4111yescharacteristic
498D5—124no—
499C23—154no—
500C23—304no—
501C23—304no—
502C23—304no—
503C23C23×D1511yescharacteristic
504Q8—28no—
505Q8—308no—
506Q8—308no—
507D4—308no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—154no—
513C2×C4—152no—
514C2×C4—152no—
515C2×C4—152no—
516C2×C4—154no—
517C2×C4—152no—
518C2×C4—3016no—
519C2×C4—304no—
520C2×C4—302no—
521C2×C4—302no—
522C2×C4—302no—
523C2×C4#681714yes—
524C2×C4C23×D1514yes—
525C2×C4C23×D1512yes—
526C2×C4C23×D1514yes—
527C2×C4C23×D1512yes—
528C2×C4C23×D1512yes—
529C2×C3#6590214yescomplemented
530C2×C3C22×#70911yescharacteristic
531C2×C3C2×#414711yescharacteristic
532C2×C3C2×D5×#4111yescharacteristic
533D3—204no—
534C5C2×#556811yescharacteristic, complemented, Sylow
535C22—304no—
536C22—304no—
537C22—304no—
538C22—608no—
539C22C2×#681712yes—
540C22#9209812yes—
541C22#41×D1512yes—
542C22C24×D1511yescharacteristic
543C4—24no—
544C4—22no—
545C4—24no—
546C4—22no—
547C4—154no—
548C4—304no—
549C4—302no—
550C4—302no—
551C4—302no—
552C4C2×#681714yes—
553C3C2×#6590211yescharacteristic, complemented, Sylow
554C2—604no—
555C2#1175632714yesdirect product factor
556C2C22×#681711yescharacteristic
557C2C2×#9209811yescharacteristic
558C2C2×#41×D1511yescharacteristic
subgroupG/H
centerC23C23×D15
commutator, cosocleC22×C3×C5C25
FrattiniC22C24×D15
FittingC2×C3×C4×C5×Q8C2
socleC23×C3×C5C24
prime radicalC22×D60C22
hypercenterC2×C4×Q8D15
hypocenterC3×C5C2×#556
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×C5×Q8C2semidirect
3. C2×#91833C2semidirect
4. #11756327C2direct (N × H)
5. C2×#91787C2semidirect
6. C2×#91785C2semidirect
8. C2×#91776C2semidirect
9. C2×#91765C2semidirect
11. C2×#91761C2semidirect
12. C2×C4×Dic30C2semidirect
19. C3×C4×C5×Q8C22semidirect
25. #91833C22semidirect
40. #91787C22semidirect
41. #91785C22semidirect
43. #91776C22semidirect
44. #91765C22semidirect
46. #91761C22semidirect
47. C4×Dic30C22semidirect
49. C2×C4×C5×Q8D3semidirect
100. C2×C3×C4×Q8D5semidirect
117. C4×C5×Q8C2×D3semidirect
182. C3×C4×Q8C2×D5semidirect
259. C2×C4×Q8D15semidirect
362. C4×Q8C2×D15semidirect
383. C2×C3×C5#556semidirect
475. C3×C5C2×#556semidirect
494. C2×C5#5568semidirect
529. C2×C3#65902semidirect
534. C5C2×#5568semidirect
553. C3C2×#65902semidirect
555. C2#11756327direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×C5×Q82
C2×#918362
C2×#918332
#117563272
C2×#917872
C2×#917852
C2×#917822
C2×#917762
C2×#917652
C2×#917642
C2×#917612
C2×C4×Dic302
C2×#659023
C2×#55685