Finite Group Databasebeta

← C2×#11755201 #14484096

Subgroups — C2×#11755201 #14484096

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×#91836C211yesmaximal, characteristic
2C2×C5×Q8×Dic3C211yesmaximal, characteristic, complemented
3C2×C3×#4090C211yesmaximal, characteristic, complemented
4#11755201C2116yesmaximal, direct product factor
5C2×C4×#1204C211yesmaximal, characteristic, complemented
6C2×#91419C212yesmaximal, complemented
7C2×#91418C211yesmaximal, characteristic
8C2×#91403C212yesmaximal, complemented
9C2×#91380C212yesmaximal, complemented
10C2×#91374C211yesmaximal, characteristic
11C2×#91337C212yesmaximal, complemented
12C2×#91331C211yesmaximal, characteristic, complemented
13C2×#65902—31nomaximal, Hall
14C22×C3×C5×Q8C2211yescharacteristic
15C22×D60C2211yescharacteristic
16C22×C3×Dic10C2211yescharacteristic
17C22×#1204C2211yescharacteristic
18C22×#1200C2212yes—
19#91836C2218yes—
20C2×#6714C2212yes—
21C2×#6714C2212yes—
22C2×C4×Dic15C2211yescharacteristic
23C5×Q8×Dic3C2218yescomplemented
24C2×C5×C4⋊Dic3C2211yescharacteristic
25C2×C5×C4⋊Dic3C2212yes—
26C2×C4×C5×Dic3C2211yescharacteristic
27C2×C4×C5×Dic3C2212yes—
28C3×#4090C2218yescomplemented
29C2×C3×#683C2212yes—
30C2×C3×#683C2212yes—
31C2×C3×C4×Dic5C2211yescharacteristic
32C2×Dic3⋊Dic5C2211yescharacteristic
33C2×#6669C2211yescharacteristic
34C2×#6668C2212yes—
35C2×#6668C2211yescharacteristic
36C2×#6667C2212yes—
37C2×#6667C2212yes—
38C2×#6667C2212yes—
39C2×Dic3×Dic5C2212yes—
40C4×#1204C2218yescomplemented
41#91419C22116yescomplemented
42#91418C2218yes—
43#91403C22116yescomplemented
44#91380C22116yescomplemented
45#91374C2218yes—
46#91337C22116yescomplemented
47#91331C2218yescomplemented
48C2×#5732—51nomaximal, Hall
49C2×C4×C5×Q8—31no—
50C2×#4093—31no—
51C2×#4090D311yescharacteristic, complemented
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×Dic15C2311yescharacteristic
67C22×Dic15C2311yescharacteristic
68C2×D60—24no—
69C2×D60C2314yes—
70C22×C5×Dic3C2311yescharacteristic
71C22×C5×Dic3C2312yes—
72C22×C5×Dic3C2311yescharacteristic
73C22×C3×Dic5C2312yes—
74C22×C3×Dic5C2311yescharacteristic
75C22×C3×Dic5C2311yescharacteristic
76C2×C3×Dic10—24no—
77C2×C3×Dic10C2314yes—
78C2×#1204—24no—
79C2×#1204C2314yes—
80C2×#1200—28no—
81C2×#1200C2318yes—
82#6714C2318yes—
83#6714C2318yes—
84C4×Dic15C2314yes—
85C5×C4⋊Dic3C2314yes—
86C5×C4⋊Dic3C2318yes—
87C4×C5×Dic3C2318yes—
88C4×C5×Dic3C2314yes—
89C3×#683C2318yes—
90C3×#683C2318yes—
91C3×C4×Dic5C2314yes—
92Dic3⋊Dic5C2314yes—
93#6669C2314yes—
94#6668C2314yes—
95#6668C2318yes—
96#6667C2318yes—
97#6667C2318yes—
98#6667C2318yes—
99Dic3×Dic5C2318yes—
100C2×C3×C4⋊Q8—51no—
101C2×#947—51no—
102C2×#947—51no—
103C2×Q8×Dic3—51no—
104C2×Q8×Dic3D511yescharacteristic, complemented
105#5732—516no—
106C2×#898—52no—
107C2×#898—52no—
108C2×#897—52no—
109C2×#892—52no—
110C2×C4×D12—51no—
111C2×C4×Dic6—51no—
112C22×C5×Q8C2×D311yescharacteristic
113C22×D20—31no—
114C22×C4×D5—32no—
115C22×Dic10—31no—
116C22×Dic10C2×D311yescharacteristic
117C4×C5×Q8—38no—
118C2×C5×C4⋊C4—31no—
119C2×C5×C4⋊C4—32no—
120C2×C42×C5—31no—
121C2×C42×C5—32no—
122#4093—38no—
123#4090C2×D318yescomplemented
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×#683C2×D312yes—
134C2×#683C2×D312yes—
135C2×C4×Dic5—31no—
136C2×C4×Dic5—32no—
137C2×C4×Dic5C2×D311yescharacteristic
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—24no—
154C2×C3×C4×C5#4114yes—
155C2×C3×C4×C5C2414yes—
156C2×C3×C4×C5C2412yes—
157C2×Dic15—22no—
158C2×Dic15—22no—
159C2×Dic15C2412yes—
160C2×Dic15C2412yes—
161D60—28no—
162C2×C5×Dic3—24no—
163C2×C5×Dic3—22no—
164C2×C5×Dic3—22no—
165C2×C5×Dic3C2414yes—
166C2×C5×Dic3C2412yes—
167C2×C5×Dic3C2412yes—
168C2×C3×Dic5—24no—
169C2×C3×Dic5—22no—
170C2×C3×Dic5#4114yes—
171C2×C3×Dic5C2414yes—
172C2×C3×Dic5C2412yes—
173C2×C3×Dic5C2412yes—
174C3×Dic10—28no—
175#1204—28no—
176#1200—216no—
177C22×C3×Q8—51no—
178C22×C3×Q8C2×D511yescharacteristic
179C22×D12—51no—
180C22×C4×D3—52no—
181C22×Dic6—51no—
182C3×C4⋊Q8—58no—
183C2×C3×C4⋊C4—52no—
184C2×C3×C4⋊C4—52no—
185C2×C3×C42—51no—
186#947—58no—
187#947—58no—
188Q8×Dic3—58no—
189Q8×Dic3C2×D518yescomplemented
190C2×(C2×D3)⋊C4—52no—
191C2×(C2×D3)⋊C4—52no—
192C2×(C2×D3)⋊C4—52no—
193C2×(C2×D3)⋊C4—52no—
194C2×(C2×D3)⋊C4—52no—
195C2×C4⋊Dic3—51no—
196C2×C4⋊Dic3—52no—
197C2×C4⋊Dic3C2×D511yescharacteristic
198C2×C4⋊Dic3C2×D512yes—
199C2×Dic3⋊C4—51no—
200C2×Dic3⋊C4—51no—
201C2×C4×Dic3—51no—
202C2×C4×Dic3—52no—
203C2×C4×Dic3C2×D511yescharacteristic
204C2×C4×Dic3C2×D512yes—
205#898—516no—
206#898—516no—
207#897—516no—
208#892—516no—
209C4×D12—58no—
210C4×Dic6—58no—
211C23×D5—32no—
212C2×C5×Q8—24no—
213C2×C5×Q8C22×D314yes—
214C22×C4×C5—31no—
215C22×C4×C5—31no—
216C22×C4×C5—32no—
217C22×C4×C5C22×D312yes—
218C22×C4×C5C22×D311yescharacteristic
219C22×Dic5—31no—
220C22×Dic5—31no—
221C22×Dic5C22×D312yes—
222C22×Dic5C22×D311yescharacteristic
223C22×Dic5C22×D311yescharacteristic
224C2×D20—34no—
225C2×D20—64no—
226C2×C4×D5—38no—
227C2×C4×D5—68no—
228C2×Dic10—24no—
229C2×Dic10—34no—
230C2×Dic10—64no—
231C2×Dic10C22×D314yes—
232C5×C4⋊C4—34no—
233C5×C4⋊C4—38no—
234C42×C5—34no—
235C42×C5—38no—
236#685—38no—
237#685—38no—
238#685—38no—
239#685—38no—
240#685—38no—
241C4⋊Dic5—34no—
242#683—34no—
243#683—38no—
244#683—34no—
245#683C22×D318yes—
246#683C22×D318yes—
247C4×Dic5—34no—
248C4×Dic5—38no—
249C4×Dic5C22×D314yes—
250#556—1516no—
251C2×C4⋊Q8—51no—
252C2×#126—152no—
253C2×#126—152no—
254C2×#124—151no—
255C2×#124—151no—
256C2×C22⋊Q8—152no—
257C2×C4×Q8—31no—
258C2×C4×Q8—151no—
259C2×C4×Q8—151no—
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—24no—
271C3×C4×C5C2×#4114yes—
272Dic15—22no—
273Dic15—22no—
274C3×Dic5—24no—
275C3×Dic5—22no—
276C3×Dic5C2×#4114yes—
277C5×Dic3—24no—
278C5×Dic3—22no—
279C5×Dic3—22no—
280C23×D3—52no—
281C2×C3×Q8—24no—
282C2×C3×Q8—54no—
283C2×C3×Q8—104no—
284C2×C3×Q8C22×D514yes—
285C22×C3×C4—51no—
286C22×C3×C4—52no—
287C22×C3×C4—51no—
288C22×C3×C4C22×D512yes—
289C22×C3×C4C22×D511yescharacteristic
290C22×Dic3—51no—
291C22×Dic3—51no—
292C22×Dic3C22×D512yes—
293C22×Dic3C22×D511yescharacteristic
294C22×Dic3C22×D511yescharacteristic
295C2×D12—54no—
296C2×D12—104no—
297C2×C4×D3—58no—
298C2×C4×D3—108no—
299C2×Dic6—54no—
300C2×Dic6—104no—
301C3×C4⋊C4—58no—
302C3×C4⋊C4—58no—
303C3×C42—54no—
304(C2×D3)⋊C4—58no—
305(C2×D3)⋊C4—58no—
306(C2×D3)⋊C4—58no—
307(C2×D3)⋊C4—58no—
308(C2×D3)⋊C4—58no—
309C4⋊Dic3—54no—
310C4⋊Dic3—58no—
311C4⋊Dic3C22×D514yes—
312C4⋊Dic3C22×D518yes—
313Dic3⋊C4—54no—
314Dic3⋊C4—54no—
315C4×Dic3—54no—
316C4×Dic3—58no—
317C4×Dic3C22×D514yes—
318C4×Dic3C22×D518yes—
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—34no—
327C2×C4×C5—32no—
328C2×C4×C5—32no—
329C2×C4×C5—62no—
330C2×C4×C5—64no—
331C2×C4×C5—62no—
332C2×C4×C5#23714yes—
333C2×C4×C5C23×D314yes—
334C2×C4×C5C23×D312yes—
335C2×Dic5—24no—
336C2×Dic5—22no—
337C2×Dic5—32no—
338C2×Dic5—32no—
339C2×Dic5—62no—
340C2×Dic5—62no—
341C2×Dic5#23914yes—
342C2×Dic5C23×D314yes—
343C2×Dic5C23×D312yes—
344C2×Dic5C23×D312yes—
345D20—68no—
346C4×D5—616no—
347Dic10—28no—
348Dic10—68no—
349C22×Q8—51no—
350C22×Q8—151no—
351C22×Q8D3×D511yescharacteristic
352C22×D4—151no—
353C23×C4—152no—
354C4⋊Q8—58no—
355#126—1516no—
356#126—1516no—
357#124—158no—
358#124—158no—
359C22⋊Q8—1516no—
360C4×Q8—38no—
361C4×Q8—158no—
362C4×Q8—158no—
363C4×D4—158no—
364#117—1516no—
365C2×C4⋊C4—31no—
366C2×C4⋊C4—32no—
367C2×C4⋊C4—52no—
368C2×C4⋊C4—52no—
369C2×C4⋊C4—151no—
370C2×C4⋊C4—152no—
371C2×C4⋊C4—151no—
372C2×C4⋊C4—151no—
373C2×C22⋊C4—152no—
374C2×C22⋊C4—152no—
375C2×C22⋊C4—152no—
376C2×C22⋊C4—152no—
377C2×C22⋊C4—152no—
378C2×C42—32no—
379C2×C42—31no—
380C2×C42—51no—
381C2×C42—152no—
382C2×C42—151no—
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—
394C3×Q8—108no—
395C2×C3×C4—24no—
396C2×C3×C4—54no—
397C2×C3×C4—52no—
398C2×C3×C4—54no—
399C2×C3×C4—52no—
400C2×C3×C4—104no—
401C2×C3×C4—102no—
402C2×C3×C4#70914yes—
403C2×C3×C4C23×D514yes—
404C2×C3×C4C23×D512yes—
405C2×Dic3—22no—
406C2×Dic3—22no—
407C2×Dic3—24no—
408C2×Dic3—52no—
409C2×Dic3—52no—
410C2×Dic3—102no—
411C2×Dic3—102no—
412C2×Dic3C23×D514yes—
413C2×Dic3C23×D512yes—
414C2×Dic3C23×D512yes—
415D12—108no—
416C4×D3—1016no—
417Dic6—108no—
418C22×C5C2×#23712yes—
419C22×C5C2×#23912yes—
420C22×C5#100712yes—
421C22×C5C24×D311yescharacteristic
422C2×D5—64no—
423C2×D5—64no—
424C2×D5—64no—
425C2×D5—128no—
426C4×C5—24no—
427C4×C5—62no—
428C4×C5—64no—
429C4×C5—62no—
430C4×C5C2×#23714yes—
431Dic5—24no—
432Dic5—22no—
433Dic5—62no—
434Dic5—62no—
435Dic5C2×#23914yes—
436C24—152no—
437C2×Q8—24no—
438C2×Q8—54no—
439C2×Q8—104no—
440C2×Q8—154no—
441C2×Q8—304no—
442C2×Q8C2×D3×D514yes—
443C2×D4—154no—
444C2×D4—304no—
445C22×C4—31no—
446C22×C4—32no—
447C22×C4—31no—
448C22×C4—51no—
449C22×C4—51no—
450C22×C4—52no—
451C22×C4—158no—
452C22×C4—151no—
453C22×C4—151no—
454C22×C4—308no—
455C22×C4C2×D3×D512yes—
456C22×C4C2×D3×D511yescharacteristic
457C4⋊C4—34no—
458C4⋊C4—38no—
459C4⋊C4—58no—
460C4⋊C4—58no—
461C4⋊C4—154no—
462C4⋊C4—158no—
463C4⋊C4—154no—
464C4⋊C4—154no—
465C22⋊C4—158no—
466C22⋊C4—158no—
467C22⋊C4—158no—
468C22⋊C4—158no—
469C22⋊C4—158no—
470C42—38no—
471C42—34no—
472C42—54no—
473C42—158no—
474C42—154no—
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—54no—
486C3×C4—104no—
487C3×C4—102no—
488C3×C4C2×#70914yes—
489Dic3—22no—
490Dic3—22no—
491Dic3—24no—
492Dic3—102no—
493Dic3—102no—
494C2×C5#573214yescomplemented
495C2×C5C22×#23711yescharacteristic
496C2×C5C22×#23911yescharacteristic
497C2×C5C2×#100711yescharacteristic
498D5—124no—
499C23—154no—
500C23—304no—
501C23—304no—
502C23—304no—
503C23C22×D3×D511yescharacteristic
504Q8—28no—
505Q8—108no—
506Q8—308no—
507D4—308no—
508C2×C4—24no—
509C2×C4—32no—
510C2×C4—34no—
511C2×C4—32no—
512C2×C4—54no—
513C2×C4—52no—
514C2×C4—52no—
515C2×C4—54no—
516C2×C4—62no—
517C2×C4—64no—
518C2×C4—62no—
519C2×C4—102no—
520C2×C4—104no—
521C2×C4—152no—
522C2×C4—152no—
523C2×C4—3016no—
524C2×C4—302no—
525C2×C4—302no—
526C2×C4#677914yes—
527C2×C4C22×D3×D514yes—
528C2×C4C22×D3×D512yes—
529C2×C3#6590214yescomplemented
530C2×C3C22×#70911yescharacteristic
531C2×C3C2×#414711yescharacteristic
532C2×C3C2×D5×#4111yescharacteristic
533D3—204no—
534C5C2×#573211yescharacteristic, complemented, Sylow
535C22—304no—
536C22—304no—
537C22—304no—
538C22—608no—
539C22#9202912yes—
540C22D5×#23912yes—
541C22C2×#677912yes—
542C22C23×D3×D511yescharacteristic
543C4—24no—
544C4—54no—
545C4—62no—
546C4—64no—
547C4—62no—
548C4—104no—
549C4—102no—
550C4—302no—
551C4—302no—
552C4C2×#677914yes—
553C3C2×#6590211yescharacteristic, complemented, Sylow
554C2—604no—
555C2#1175520114yesdirect product factor
556C2C2×#9202911yescharacteristic
557C2C2×D5×#23911yescharacteristic
558C2C22×#677911yescharacteristic
subgroupG/H
centerC23C22×D3×D5
commutator, cosocleC22×C3×C5C25
FrattiniC22C23×D3×D5
FittingC22×C3×C5×Q8C22
socleC23×C3×C5C24
prime radicalC22×D60C22
hypercenterC22×Q8D3×D5
hypocenterC3×C5C2×#556
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C5×Q8×Dic3C2semidirect
3. C2×C3×#4090C2semidirect
4. #11755201C2direct (N × H)
5. C2×C4×#1204C2semidirect
6. C2×#91419C2semidirect
8. C2×#91403C2semidirect
9. C2×#91380C2semidirect
11. C2×#91337C2semidirect
12. C2×#91331C2semidirect
23. C5×Q8×Dic3C22semidirect
28. C3×#4090C22semidirect
40. C4×#1204C22semidirect
41. #91419C22semidirect
43. #91403C22semidirect
44. #91380C22semidirect
46. #91337C22semidirect
47. #91331C22semidirect
51. C2×#4090D3semidirect
104. C2×Q8×Dic3D5semidirect
123. #4090C2×D3semidirect
189. Q8×Dic3C2×D5semidirect
383. C2×C3×C5#556semidirect
475. C3×C5C2×#556semidirect
494. C2×C5#5732semidirect
529. C2×C3#65902semidirect
534. C5C2×#5732semidirect
553. C3C2×#65902semidirect
555. C2#11755201direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#918362
C2×C5×Q8×Dic32
C2×C3×#40902
#117552012
C2×C4×#12042
C2×#914192
C2×#914182
C2×#914032
C2×#913802
C2×#913742
C2×#913372
C2×#913312
C2×#659023
C2×#57325