Finite Group Databasebeta

← C2×#11755145 #14481252

Subgroups — C2×#11755145 #14481252

Order 1920.

2008 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91833C211yesmaximal, characteristic, complemented
2C2×C5×#946C211yesmaximal, characteristic
3C2×C3×Q8×Dic5C211yesmaximal, characteristic, complemented
4#11755145C2116yesmaximal, direct product factor
5C2×#91384C211yesmaximal, characteristic, complemented
6C2×#91378C212yesmaximal, complemented
7C2×#91372C212yesmaximal
8C2×#91371C211yesmaximal, characteristic
9C2×#91365C211yesmaximal, characteristic
10C2×#91346C212yesmaximal, complemented
11C2×#91341C212yesmaximal
12C2×#91333C211yesmaximal, characteristic, complemented
13C2×#65959—31nomaximal, Hall
14C22×C3×C5×Q8C2211yescharacteristic
15C22×Dic30C2211yescharacteristic
16C22×C4×C5×D3C2211yescharacteristic
17#91833C2218yescomplemented
18C2×#6712C2212yes—
19C2×#6712C2212yes—
20C2×C4×Dic15C2211yescharacteristic
21C5×#946C2218yes—
22C2×C5×(C2×D3)⋊C4C2212yes—
23C2×C5×C4⋊Dic3C2211yescharacteristic
24C2×C5×Dic3⋊C4C2212yes—
25C3×Q8×Dic5C2218yescomplemented
26C2×C3×C4⋊Dic5C2211yescharacteristic
27C2×C3×C4⋊Dic5C2212yes—
28C2×C3×C4×Dic5C2211yescharacteristic
29C2×C3×C4×Dic5C2212yes—
30C2×Dic3⋊Dic5C2211yescharacteristic
31C2×Dic3⋊Dic5C2211yescharacteristic
32C2×Dic3⋊Dic5C2212yes—
33C2×#6669C2212yes—
34C2×#6668C2212yes—
35C2×#6668C2212yes—
36C2×(C2×D3)⋊Dic5C2212yes—
37C2×(C2×D3)⋊Dic5C2211yescharacteristic
38C2×(C2×D3)⋊Dic5C2211yescharacteristic
39C2×Dic3×Dic5C2212yes—
40#91384C2218yescomplemented
41#91378C22116yescomplemented
42#91372C22116yes—
43#91371C2218yes—
44#91365C2218yes—
45#91346C22116yescomplemented
46#91341C22116yes—
47#91333C2218yescomplemented
48C2×#5583—51nomaximal, Hall
49C2×C5×C22⋊Q8—31no—
50C2×Q8×Dic5D311yescharacteristic, complemented
51C2×#4090—31no—
52C2×#4072—31no—
53C2×#4070—31no—
54#65959—316no—
55C2×#4036—31no—
56C2×#4036—32no—
57C2×#4035—32no—
58C2×#4033—31no—
59C2×#4025—32no—
60C2×#4025—32no—
61C23×C5×D3C2311yescharacteristic
62C2×C3×C5×Q8—24no—
63C2×C3×C5×Q8C2314yes—
64C22×C3×C4×C5C2312yes—
65C22×C3×C4×C5C2311yescharacteristic
66C22×Dic15C2312yes—
67C22×Dic15C2311yescharacteristic
68C22×Dic15C2311yescharacteristic
69C2×Dic30—24no—
70C2×Dic30C2314yes—
71C22×C5×Dic3C2311yescharacteristic
72C22×C5×Dic3C2312yes—
73C2×C4×C5×D3—24no—
74C2×C4×C5×D3C2314yes—
75C22×C3×Dic5C2311yescharacteristic
76C22×C3×Dic5C2312yes—
77C22×C3×Dic5C2311yescharacteristic
78#6712C2318yes—
79#6712C2318yes—
80C4×Dic15C2314yes—
81C5×(C2×D3)⋊C4C2318yes—
82C5×C4⋊Dic3C2314yes—
83C5×Dic3⋊C4C2318yes—
84C3×C4⋊Dic5C2318yes—
85C3×C4⋊Dic5C2314yes—
86C3×C4×Dic5C2314yes—
87C3×C4×Dic5C2318yes—
88Dic3⋊Dic5C2314yes—
89Dic3⋊Dic5C2318yes—
90Dic3⋊Dic5C2314yes—
91#6669C2318yes—
92#6668C2318yes—
93#6668C2318yes—
94(C2×D3)⋊Dic5C2314yes—
95(C2×D3)⋊Dic5C2318yes—
96(C2×D3)⋊Dic5C2314yes—
97Dic3×Dic5C2318yes—
98C2×C3×C4×Q8—51no—
99C2×#946D511yescharacteristic
100C2×#944—51no—
101#5583—516no—
102C2×#898—52no—
103C2×#897—51no—
104C2×#890—51no—
105C2×#889—52no—
106C2×#876—52no—
107C2×#872—51no—
108C2×#870—52no—
109C2×C4×Dic6—51no—
110C22×C5×Q8C2×D311yescharacteristic
111C23×C4×C5—31no—
112C22×Dic10—31no—
113C5×C22⋊Q8—38no—
114C2×C5×C4⋊C4—31no—
115C2×C5×C4⋊C4—32no—
116C2×C5×C22⋊C4—32no—
117C2×C22⋊Dic5—32no—
118C2×C22⋊Dic5—31no—
119C2×C22⋊Dic5—31no—
120Q8×Dic5C2×D318yescomplemented
121#4090—38no—
122#4072—38no—
123C2×C4⋊Dic5—32no—
124C2×C4⋊Dic5—31no—
125C2×C4⋊Dic5—31no—
126C2×C4⋊Dic5C2×D312yes—
127C2×C4⋊Dic5C2×D311yescharacteristic
128#4070—38no—
129C2×#683—32no—
130C2×#683—32no—
131C2×#683—32no—
132C2×#683—32no—
133C2×#683—32no—
134C2×C4×Dic5—31no—
135C2×C4×Dic5—32no—
136C2×C4×Dic5C2×D312yes—
137C2×C4×Dic5C2×D311yescharacteristic
138#4036—316no—
139#4036—38no—
140#4035—316no—
141#4033—38no—
142#4025—316no—
143#4025—316no—
144C2×#541—151noSylow
145C23×C3×C5C2411yescharacteristic
146C22×C5×D3—22no—
147C22×C5×D3—22no—
148C22×C5×D3—22no—
149C22×C5×D3C2412yes—
150C3×C5×Q8—28no—
151C2×C3×C4×C5—24no—
152C2×C3×C4×C5#4114yes—
153C2×C3×C4×C5C2414yes—
154C2×C3×C4×C5C2412yes—
155C2×Dic15—24no—
156C2×Dic15—22no—
157C2×Dic15—22no—
158C2×Dic15C2414yes—
159C2×Dic15C2412yes—
160C2×Dic15C2412yes—
161Dic30—28no—
162C2×C5×Dic3—22no—
163C2×C5×Dic3—24no—
164C2×C5×Dic3C2414yes—
165C2×C5×Dic3C2412yes—
166C4×C5×D3—28no—
167C2×C3×Dic5—22no—
168C2×C3×Dic5—24no—
169C2×C3×Dic5—22no—
170C2×C3×Dic5C2412yes—
171C2×C3×Dic5C2414yes—
172C2×C3×Dic5C2412yes—
173C22×C3×Q8C2×D511yescharacteristic
174C22×C4×D3C2×D511yescharacteristic
175C22×Dic6—51no—
176C3×C4×Q8—58no—
177C2×C3×C4⋊C4—52no—
178C2×C3×C4⋊C4—51no—
179C2×C3×C42—51no—
180C2×C3×C42—52no—
181#946C2×D518yes—
182#944—58no—
183C2×(C2×D3)⋊C4—51no—
184C2×(C2×D3)⋊C4—51no—
185C2×(C2×D3)⋊C4—52no—
186C2×(C2×D3)⋊C4C2×D512yes—
187C2×C4⋊Dic3—52no—
188C2×C4⋊Dic3—52no—
189C2×C4⋊Dic3C2×D511yescharacteristic
190C2×Dic3⋊C4—52no—
191C2×Dic3⋊C4—52no—
192C2×Dic3⋊C4—52no—
193C2×Dic3⋊C4—51no—
194C2×Dic3⋊C4—51no—
195C2×Dic3⋊C4—52no—
196C2×Dic3⋊C4C2×D512yes—
197C2×C4×Dic3—51no—
198C2×C4×Dic3—52no—
199#898—516no—
200#897—58no—
201#890—58no—
202#889—516no—
203#876—516no—
204#872—58no—
205#870—516no—
206C4×Dic6—58no—
207C24×C5—31no—
208C2×C5×Q8—24no—
209C2×C5×Q8C22×D314yes—
210C22×C4×C5—32no—
211C22×C4×C5—31no—
212C22×C4×C5—34no—
213C22×C4×C5—64no—
214C22×C4×C5C22×D312yes—
215C22×C4×C5C22×D311yescharacteristic
216C22×Dic5—32no—
217C22×Dic5—31no—
218C22×Dic5—31no—
219C22×Dic5C22×D311yescharacteristic
220C22×Dic5C22×D311yescharacteristic
221C22×Dic5C22×D312yes—
222C2×Dic10—34no—
223C2×Dic10—64no—
224C5×C4⋊C4—38no—
225C5×C4⋊C4—34no—
226C5×C22⋊C4—38no—
227C22⋊Dic5—38no—
228C22⋊Dic5—34no—
229C22⋊Dic5—34no—
230C4⋊Dic5—38no—
231C4⋊Dic5—34no—
232C4⋊Dic5—34no—
233C4⋊Dic5C22×D318yes—
234C4⋊Dic5C22×D314yes—
235#683—38no—
236#683—38no—
237#683—38no—
238#683—38no—
239#683—38no—
240C4×Dic5—34no—
241C4×Dic5—38no—
242C4×Dic5C22×D318yes—
243C4×Dic5C22×D314yes—
244#541—1516no—
245C2×C4⋊Q8—151no—
246C2×#126—152no—
247C2×#126—152no—
248C2×#125—152no—
249C2×#125—152no—
250C2×#125—151no—
251C2×C22⋊Q8—31no—
252C2×C22⋊Q8—151no—
253C2×C4×Q8—51no—
254C2×C4×Q8—151no—
255C2×#117—151no—
256C22×C3×C5C2×#4112yes—
257C22×C3×C52−1+412yes—
258C22×C3×C52−1+412yes—
259C22×C3×C5C2511yescharacteristic
260C2×C5×D3—22no—
261C2×C5×D3—22no—
262C2×C5×D3—22no—
263C2×C5×D3—44no—
264C3×C4×C5—24no—
265C3×C4×C5C2×#4114yes—
266Dic15—24no—
267Dic15—22no—
268Dic15—22no—
269C3×Dic5—22no—
270C3×Dic5—24no—
271C3×Dic5—22no—
272C5×Dic3—22no—
273C5×Dic3—24no—
274C23×D3C22×D511yescharacteristic
275C2×C3×Q8—24no—
276C2×C3×Q8C22×D514yes—
277C22×C3×C4—51no—
278C22×C3×C4—51no—
279C22×C3×C4—52no—
280C22×C3×C4C22×D512yes—
281C22×C3×C4C22×D511yescharacteristic
282C22×Dic3—51no—
283C22×Dic3—51no—
284C22×Dic3—52no—
285C22×Dic3C22×D512yes—
286C22×Dic3C22×D511yescharacteristic
287C2×C4×D3—24no—
288C2×C4×D3C22×D514yes—
289C2×Dic6—54no—
290C2×Dic6—104no—
291C3×C4⋊C4—58no—
292C3×C4⋊C4—54no—
293C3×C42—54no—
294C3×C42—58no—
295(C2×D3)⋊C4—54no—
296(C2×D3)⋊C4—54no—
297(C2×D3)⋊C4—58no—
298(C2×D3)⋊C4C22×D518yes—
299C4⋊Dic3—58no—
300C4⋊Dic3—58no—
301C4⋊Dic3C22×D514yes—
302Dic3⋊C4—58no—
303Dic3⋊C4—58no—
304Dic3⋊C4—58no—
305Dic3⋊C4—54no—
306Dic3⋊C4—54no—
307Dic3⋊C4—58no—
308Dic3⋊C4C22×D518yes—
309C4×Dic3—54no—
310C4×Dic3—58no—
311C23×C5—32no—
312C23×C5—62no—
313C23×C5—62no—
314C23×C5—62no—
315C23×C5C23×D311yescharacteristic
316C5×Q8—28no—
317C2×C4×C5—24no—
318C2×C4×C5—34no—
319C2×C4×C5—32no—
320C2×C4×C5—62no—
321C2×C4×C5—64no—
322C2×C4×C5—68no—
323C2×C4×C5#23514yes—
324C2×C4×C5C23×D314yes—
325C2×C4×C5C23×D312yes—
326C2×Dic5—22no—
327C2×Dic5—22no—
328C2×Dic5—24no—
329C2×Dic5—32no—
330C2×Dic5—34no—
331C2×Dic5—32no—
332C2×Dic5—64no—
333C2×Dic5—62no—
334C2×Dic5—62no—
335C2×Dic5C23×D312yes—
336C2×Dic5C23×D312yes—
337C2×Dic5C23×D314yes—
338Dic10—68no—
339C22×Q8—151no—
340C22×Q8D3×D511yescharacteristic
341C23×C4—31no—
342C4⋊Q8—158no—
343#126—1516no—
344#126—1516no—
345#125—1516no—
346#125—1516no—
347#125—158no—
348C22⋊Q8—38no—
349C22⋊Q8—158no—
350C4×Q8—58no—
351C4×Q8—158no—
352#117—158no—
353C2×C4⋊C4—31no—
354C2×C4⋊C4—32no—
355C2×C4⋊C4—51no—
356C2×C4⋊C4—52no—
357C2×C4⋊C4—151no—
358C2×C4⋊C4—151no—
359C2×C4⋊C4—152no—
360C2×C4⋊C4—152no—
361C2×C4⋊C4—152no—
362C2×C4⋊C4—152no—
363C2×C4⋊C4—152no—
364C2×C4⋊C4—152no—
365C2×C22⋊C4—32no—
366C2×C22⋊C4—151no—
367C2×C22⋊C4—151no—
368C2×C22⋊C4—152no—
369C2×C42—51no—
370C2×C42—52no—
371C2×C42—152no—
372C2×C42—151no—
373C2×C3×C5#54114yescomplemented
374C2×C3×C5C22×#4111yescharacteristic
375C2×C3×C5C2×2−1+411yescharacteristic
376C2×C3×C5C2×2−1+411yescharacteristic
377C5×D3—42no—
378C23×C3C23×D511yescharacteristic
379C22×D3—22no—
380C22×D3—22no—
381C22×D3—22no—
382C22×D3C23×D512yes—
383C3×Q8—28no—
384C2×C3×C4—24no—
385C2×C3×C4—52no—
386C2×C3×C4—52no—
387C2×C3×C4—54no—
388C2×C3×C4—102no—
389C2×C3×C4—104no—
390C2×C3×C4—102no—
391C2×C3×C4#71114yes—
392C2×C3×C4C23×D514yes—
393C2×C3×C4C23×D512yes—
394C2×Dic3—22no—
395C2×Dic3—24no—
396C2×Dic3—52no—
397C2×Dic3—52no—
398C2×Dic3—54no—
399C2×Dic3—104no—
400C2×Dic3—102no—
401C2×Dic3—102no—
402C2×Dic3C23×D514yes—
403C2×Dic3C23×D512yes—
404C4×D3—28no—
405Dic6—108no—
406C22×C5—62no—
407C22×C5—62no—
408C22×C5—62no—
409C22×C5—124no—
410C22×C5C2×#23512yes—
411C22×C5#100712yes—
412C22×C5#101012yes—
413C22×C5C24×D311yescharacteristic
414C4×C5—24no—
415C4×C5—62no—
416C4×C5—64no—
417C4×C5C2×#23514yes—
418Dic5—22no—
419Dic5—22no—
420Dic5—24no—
421Dic5—64no—
422Dic5—62no—
423Dic5—62no—
424C24—31no—
425C2×Q8—24no—
426C2×Q8—154no—
427C2×Q8—304no—
428C2×Q8C2×D3×D514yes—
429C22×C4—32no—
430C22×C4—31no—
431C22×C4—34no—
432C22×C4—51no—
433C22×C4—51no—
434C22×C4—52no—
435C22×C4—64no—
436C22×C4—151no—
437C22×C4—151no—
438C22×C4—152no—
439C22×C4C2×D3×D512yes—
440C22×C4C2×D3×D511yescharacteristic
441C4⋊C4—34no—
442C4⋊C4—38no—
443C4⋊C4—54no—
444C4⋊C4—58no—
445C4⋊C4—154no—
446C4⋊C4—154no—
447C4⋊C4—158no—
448C4⋊C4—158no—
449C4⋊C4—158no—
450C4⋊C4—158no—
451C4⋊C4—158no—
452C4⋊C4—158no—
453C22⋊C4—38no—
454C22⋊C4—154no—
455C22⋊C4—154no—
456C22⋊C4—158no—
457C42—54no—
458C42—58no—
459C42—158no—
460C42—154no—
461C3×C5C2×#54111yescharacteristic, complemented, Hall
462C22×C3C2×#71112yes—
463C22×C3#414712yes—
464C22×C3#415012yes—
465C22×C3C24×D511yescharacteristic
466C2×D3—22no—
467C2×D3—22no—
468C2×D3—22no—
469C2×D3—44no—
470C3×C4—24no—
471C3×C4—102no—
472C3×C4—104no—
473C3×C4—102no—
474C3×C4C2×#71114yes—
475Dic3—22no—
476Dic3—24no—
477Dic3—104no—
478Dic3—102no—
479Dic3—102no—
480C2×C5—122no—
481C2×C5#558314yescomplemented
482C2×C5C22×#23511yescharacteristic
483C2×C5C2×#100711yescharacteristic
484C2×C5C2×#101011yescharacteristic
485C23—32no—
486C23—62no—
487C23—62no—
488C23—62no—
489C23C22×D3×D511yescharacteristic
490Q8—28no—
491Q8—308no—
492C2×C4—24no—
493C2×C4—34no—
494C2×C4—32no—
495C2×C4—52no—
496C2×C4—52no—
497C2×C4—54no—
498C2×C4—64no—
499C2×C4—62no—
500C2×C4—68no—
501C2×C4—102no—
502C2×C4—104no—
503C2×C4—102no—
504C2×C4—152no—
505C2×C4—152no—
506C2×C4—154no—
507C2×C4—302no—
508C2×C4—302no—
509C2×C4—304no—
510C2×C4#678214yes—
511C2×C4C22×D3×D514yes—
512C2×C4C22×D3×D512yes—
513C2×C3#6595914yescomplemented
514C2×C3C22×#71111yescharacteristic
515C2×C3C2×#414711yescharacteristic
516C2×C3C2×#415011yescharacteristic
517D3—42no—
518C5C2×#558311yescharacteristic, complemented, Sylow
519C22—62no—
520C22—62no—
521C22—62no—
522C22—124no—
523C22#9202812yes—
524C22#9203012yes—
525C22C2×#678212yes—
526C22C23×D3×D511yescharacteristic
527C4—24no—
528C4—62no—
529C4—64no—
530C4—102no—
531C4—104no—
532C4—102no—
533C4—304no—
534C4—302no—
535C4—302no—
536C4C2×#678214yes—
537C3C2×#6595911yescharacteristic, complemented, Sylow
538C2—122no—
539C2#1175514514yesdirect product factor
540C2C2×#9202811yescharacteristic
541C2C2×#9203011yescharacteristic
542C2C22×#678211yescharacteristic
subgroupG/H
centerC23C22×D3×D5
commutator, cosocleC22×C3×C5C25
FrattiniC22C23×D3×D5
FittingC22×C3×C5×Q8C22
socleC23×C3×C5C24
prime radicalC23×C5×D3C23
hypercenterC22×Q8D3×D5
hypocenterC3×C5C2×#541
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#91833C2semidirect
3. C2×C3×Q8×Dic5C2semidirect
4. #11755145C2direct (N × H)
5. C2×#91384C2semidirect
6. C2×#91378C2semidirect
10. C2×#91346C2semidirect
12. C2×#91333C2semidirect
17. #91833C22semidirect
25. C3×Q8×Dic5C22semidirect
40. #91384C22semidirect
41. #91378C22semidirect
45. #91346C22semidirect
47. #91333C22semidirect
50. C2×Q8×Dic5D3semidirect
120. Q8×Dic5C2×D3semidirect
373. C2×C3×C5#541semidirect
461. C3×C5C2×#541semidirect
481. C2×C5#5583semidirect
513. C2×C3#65959semidirect
518. C5C2×#5583semidirect
537. C3C2×#65959semidirect
539. C2#11755145direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#918332
C2×C5×#9462
C2×C3×Q8×Dic52
#117551452
C2×#913842
C2×#913782
C2×#913722
C2×#913712
C2×#913652
C2×#913462
C2×#913412
C2×#913332
C2×#659593
C2×#55835