Finite Group Databasebeta

← C2×#11752512 #14481258

Subgroups — C2×#11752512 #14481258

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×C4×Dic30C211yesmaximal, characteristic, complemented
2C2×C5×#872C211yesmaximal, characteristic
3C2×C3×C4×Dic10C211yesmaximal, characteristic, complemented
4C2×#91389C211yesmaximal, characteristic, complemented
5C2×#91382C212yesmaximal, complemented
6C2×#91381C211yesmaximal, characteristic, complemented
7C2×#91378C212yesmaximal, complemented
8C2×#91372C212yesmaximal
9C2×#91369C211yesmaximal, characteristic
10C2×#91367C212yesmaximal
11C2×#91365C211yesmaximal, characteristic
12#11752512C2116yesmaximal, direct product factor
13C2×#65852—31nomaximal, Hall
14C22×Dic30C2211yescharacteristic
15C22×C4×C5×D3C2211yescharacteristic
16C22×C3×Dic10C2211yescharacteristic
17C2×C3×C42×C5C2211yescharacteristic
18C2×C4⋊Dic15C2211yescharacteristic
19C2×#6712C2212yes—
20C2×C4×Dic15C2212yes—
21C4×Dic30C2218yescomplemented
22C2×C5×(C2×D3)⋊C4C2211yescharacteristic
23C2×C5×(C2×D3)⋊C4C2211yescharacteristic
24C2×C5×Dic3⋊C4C2211yescharacteristic
25C2×C5×Dic3⋊C4C2211yescharacteristic
26C2×C4×C5×Dic3C2211yescharacteristic
27C5×#872C2218yes—
28C2×C3×C4⋊Dic5C2211yescharacteristic
29C2×C3×#683C2212yes—
30C2×C3×C4×Dic5C2212yes—
31C3×C4×Dic10C2218yescomplemented
32C2×Dic3⋊Dic5C2212yes—
33C2×Dic3⋊Dic5C2212yes—
34C2×#6669C2212yes—
35C2×#6669C2212yes—
36C2×#6668C2212yes—
37C2×#6668C2212yes—
38C2×(C2×D3)⋊Dic5C2212yes—
39C2×(C2×D3)⋊Dic5C2212yes—
40#91389C2218yescomplemented
41#91382C22116yescomplemented
42#91381C2218yescomplemented
43#91378C22116yescomplemented
44#91372C22116yes—
45#91369C2218yes—
46#91367C22116yes—
47#91365C2218yes—
48C2×#5583—51nomaximal, Hall
49C2×C5×#117—31no—
50C2×#4070—31no—
51C2×#4070—31no—
52#65852—316no—
53C2×#4035—32no—
54C2×#4035—32no—
55C2×#4025—32no—
56C2×#4025—32no—
57C2×#4016—31no—
58C2×#4015—31no—
59C2×C4×Dic10—31no—
60C2×C4×Dic10D311yescharacteristic, complemented
61C23×C5×D3C2311yescharacteristic
62C22×C3×C4×C5C2311yescharacteristic
63C22×C3×C4×C5C2311yescharacteristic
64C22×C3×C4×C5C2311yescharacteristic
65C22×Dic15C2312yes—
66C22×Dic15C2312yes—
67C2×Dic30—24no—
68C2×Dic30C2314yes—
69C22×C5×Dic3C2311yescharacteristic
70C22×C5×Dic3C2311yescharacteristic
71C22×C5×Dic3C2311yescharacteristic
72C2×C4×C5×D3—24no—
73C2×C4×C5×D3C2314yes—
74C22×C3×Dic5C2312yes—
75C22×C3×Dic5C2312yes—
76C2×C3×Dic10—24no—
77C2×C3×Dic10C2314yes—
78C3×C42×C5C2314yes—
79C4⋊Dic15C2314yes—
80#6712C2318yes—
81C4×Dic15C2318yes—
82C5×(C2×D3)⋊C4C2314yes—
83C5×(C2×D3)⋊C4C2314yes—
84C5×Dic3⋊C4C2314yes—
85C5×Dic3⋊C4C2314yes—
86C4×C5×Dic3C2314yes—
87C3×C4⋊Dic5C2314yes—
88C3×#683C2318yes—
89C3×C4×Dic5C2318yes—
90Dic3⋊Dic5C2318yes—
91Dic3⋊Dic5C2318yes—
92#6669C2318yes—
93#6669C2318yes—
94#6668C2318yes—
95#6668C2318yes—
96(C2×D3)⋊Dic5C2318yes—
97(C2×D3)⋊Dic5C2318yes—
98C2×C3×C4×Q8—51no—
99C2×#946—51no—
100C2×#944—51no—
101#5583—516no—
102C2×#898—52no—
103C2×#897—51no—
104C2×#890—51no—
105C2×#889—52no—
106C2×#876—52no—
107C2×#872D511yescharacteristic
108C2×#870—52no—
109C2×C4×Dic6—51no—
110C23×C4×C5—31no—
111C22×Dic10—31no—
112C22×Dic10C2×D311yescharacteristic
113C5×#117—38no—
114C2×C5×C4⋊C4—31no—
115C2×C5×C4⋊C4—31no—
116C2×C5×C22⋊C4—31no—
117C2×C5×C22⋊C4—31no—
118C2×C42×C5—31no—
119C2×C42×C5C2×D311yescharacteristic
120C2×C22⋊Dic5—32no—
121C2×C22⋊Dic5—32no—
122C2×C4⋊Dic5—31no—
123C2×C4⋊Dic5—32no—
124C2×C4⋊Dic5—32no—
125C2×C4⋊Dic5C2×D311yescharacteristic
126#4070—38no—
127#4070—38no—
128C2×#683—32no—
129C2×#683—32no—
130C2×#683—32no—
131C2×#683—32no—
132C2×#683—32no—
133C2×#683C2×D312yes—
134C2×C4×Dic5—32no—
135C2×C4×Dic5C2×D312yes—
136#4035—316no—
137#4035—316no—
138#4025—316no—
139#4025—316no—
140#4016—38no—
141#4015—38no—
142C4×Dic10—38no—
143C4×Dic10C2×D318yescomplemented
144C2×#541—151noSylow
145C23×C3×C5C2411yescharacteristic
146C22×C5×D3—22no—
147C22×C5×D3—22no—
148C22×C5×D3—22no—
149C22×C5×D3C2412yes—
150C2×C3×C4×C5—22no—
151C2×C3×C4×C5—22no—
152C2×C3×C4×C5#4114yes—
153C2×C3×C4×C5C2412yes—
154C2×C3×C4×C5C2412yes—
155C2×C3×C4×C5C2412yes—
156C2×Dic15—24no—
157C2×Dic15—24no—
158C2×Dic15C2414yes—
159C2×Dic15C2414yes—
160Dic30—28no—
161C2×C5×Dic3—22no—
162C2×C5×Dic3—22no—
163C2×C5×Dic3—22no—
164C2×C5×Dic3C2412yes—
165C2×C5×Dic3C2412yes—
166C2×C5×Dic3C2412yes—
167C4×C5×D3—28no—
168C2×C3×Dic5—24no—
169C2×C3×Dic5—24no—
170C2×C3×Dic5C2414yes—
171C2×C3×Dic5C2414yes—
172C3×Dic10—28no—
173C22×C3×Q8—51no—
174C22×C4×D3C2×D511yescharacteristic
175C22×Dic6—51no—
176C3×C4×Q8—58no—
177C2×C3×C4⋊C4—52no—
178C2×C3×C4⋊C4—51no—
179C2×C3×C42—52no—
180C2×C3×C42C2×D511yescharacteristic
181#946—58no—
182#944—58no—
183C2×(C2×D3)⋊C4—52no—
184C2×(C2×D3)⋊C4—52no—
185C2×(C2×D3)⋊C4C2×D511yescharacteristic
186C2×(C2×D3)⋊C4C2×D511yescharacteristic
187C2×C4⋊Dic3—51no—
188C2×C4⋊Dic3—52no—
189C2×C4⋊Dic3—52no—
190C2×Dic3⋊C4—52no—
191C2×Dic3⋊C4—52no—
192C2×Dic3⋊C4—52no—
193C2×Dic3⋊C4—52no—
194C2×Dic3⋊C4—52no—
195C2×Dic3⋊C4C2×D511yescharacteristic
196C2×Dic3⋊C4C2×D511yescharacteristic
197C2×C4×Dic3—52no—
198C2×C4×Dic3C2×D511yescharacteristic
199#898—516no—
200#897—58no—
201#890—58no—
202#889—516no—
203#876—516no—
204#872C2×D518yes—
205#870—516no—
206C4×Dic6—58no—
207C24×C5—31no—
208C22×C4×C5—31no—
209C22×C4×C5—31no—
210C22×C4×C5—31no—
211C22×C4×C5—34no—
212C22×C4×C5—64no—
213C22×C4×C5C22×D311yescharacteristic
214C22×C4×C5C22×D311yescharacteristic
215C22×C4×C5C22×D311yescharacteristic
216C22×Dic5—32no—
217C22×Dic5—32no—
218C22×Dic5C22×D312yes—
219C22×Dic5C22×D312yes—
220C2×Dic10—24no—
221C2×Dic10—34no—
222C2×Dic10—64no—
223C2×Dic10C22×D314yes—
224C5×C4⋊C4—34no—
225C5×C4⋊C4—34no—
226C5×C22⋊C4—34no—
227C5×C22⋊C4—34no—
228C42×C5—34no—
229C42×C5C22×D314yes—
230C22⋊Dic5—38no—
231C22⋊Dic5—38no—
232C4⋊Dic5—34no—
233C4⋊Dic5—38no—
234C4⋊Dic5—38no—
235C4⋊Dic5C22×D314yes—
236#683—38no—
237#683—38no—
238#683—38no—
239#683—38no—
240#683—38no—
241#683C22×D318yes—
242C4×Dic5—38no—
243C4×Dic5C22×D318yes—
244#541—1516no—
245C2×C4⋊Q8—151no—
246C2×#126—152no—
247C2×#126—152no—
248C2×#125—151no—
249C2×#125—152no—
250C2×#125—152no—
251C2×C22⋊Q8—151no—
252C2×C22⋊Q8—151no—
253C2×C4×Q8—51no—
254C2×C4×Q8—151no—
255C2×#117—31no—
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—22no—
265C3×C4×C5—22no—
266C3×C4×C5C2×#4114yes—
267Dic15—24no—
268Dic15—24no—
269C3×Dic5—24no—
270C3×Dic5—24no—
271C5×Dic3—22no—
272C5×Dic3—22no—
273C5×Dic3—22no—
274C23×D3C22×D511yescharacteristic
275C2×C3×Q8—54no—
276C2×C3×Q8—104no—
277C22×C3×C4—52no—
278C22×C3×C4—52no—
279C22×C3×C4C22×D511yescharacteristic
280C22×C3×C4C22×D511yescharacteristic
281C22×C3×C4C22×D511yescharacteristic
282C22×Dic3—52no—
283C22×Dic3—52no—
284C22×Dic3C22×D511yescharacteristic
285C22×Dic3C22×D511yescharacteristic
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—58no—
294C3×C42C22×D514yes—
295(C2×D3)⋊C4—58no—
296(C2×D3)⋊C4—58no—
297(C2×D3)⋊C4C22×D514yes—
298(C2×D3)⋊C4C22×D514yes—
299C4⋊Dic3—54no—
300C4⋊Dic3—58no—
301C4⋊Dic3—58no—
302Dic3⋊C4—58no—
303Dic3⋊C4—58no—
304Dic3⋊C4—58no—
305Dic3⋊C4—58no—
306Dic3⋊C4—58no—
307Dic3⋊C4C22×D514yes—
308Dic3⋊C4C22×D514yes—
309C4×Dic3—58no—
310C4×Dic3C22×D514yes—
311C23×C5—32no—
312C23×C5—62no—
313C23×C5—62no—
314C23×C5—62no—
315C23×C5C23×D311yescharacteristic
316C2×C4×C5—22no—
317C2×C4×C5—22no—
318C2×C4×C5—32no—
319C2×C4×C5—32no—
320C2×C4×C5—32no—
321C2×C4×C5—62no—
322C2×C4×C5—62no—
323C2×C4×C5—62no—
324C2×C4×C5—68no—
325C2×C4×C5#23514yes—
326C2×C4×C5C23×D312yes—
327C2×C4×C5C23×D312yes—
328C2×C4×C5C23×D312yes—
329C2×Dic5—24no—
330C2×Dic5—24no—
331C2×Dic5—34no—
332C2×Dic5—34no—
333C2×Dic5—64no—
334C2×Dic5—64no—
335C2×Dic5C23×D314yes—
336C2×Dic5C23×D314yes—
337Dic10—28no—
338Dic10—68no—
339C22×Q8—51no—
340C22×Q8—151no—
341C23×C4—31no—
342C4⋊Q8—158no—
343#126—1516no—
344#126—1516no—
345#125—158no—
346#125—1516no—
347#125—1516no—
348C22⋊Q8—158no—
349C22⋊Q8—158no—
350C4×Q8—58no—
351C4×Q8—158no—
352#117—38no—
353C2×C4⋊C4—31no—
354C2×C4⋊C4—31no—
355C2×C4⋊C4—51no—
356C2×C4⋊C4—52no—
357C2×C4⋊C4—152no—
358C2×C4⋊C4—152no—
359C2×C4⋊C4—152no—
360C2×C4⋊C4—152no—
361C2×C4⋊C4—152no—
362C2×C4⋊C4—152no—
363C2×C4⋊C4—151no—
364C2×C4⋊C4—152no—
365C2×C22⋊C4—31no—
366C2×C22⋊C4—31no—
367C2×C22⋊C4—152no—
368C2×C22⋊C4—152no—
369C2×C42—31no—
370C2×C42—52no—
371C2×C42—152no—
372C2×C42D3×D511yescharacteristic
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—108no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—54no—
387C2×C3×C4—54no—
388C2×C3×C4—104no—
389C2×C3×C4—104no—
390C2×C3×C4#70914yes—
391C2×C3×C4C23×D512yes—
392C2×C3×C4C23×D512yes—
393C2×C3×C4C23×D512yes—
394C2×Dic3—22no—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3—54no—
398C2×Dic3—54no—
399C2×Dic3—104no—
400C2×Dic3—104no—
401C2×Dic3C23×D512yes—
402C2×Dic3C23×D512yes—
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—22no—
415C4×C5—22no—
416C4×C5—62no—
417C4×C5—62no—
418C4×C5—62no—
419C4×C5C2×#23514yes—
420Dic5—24no—
421Dic5—24no—
422Dic5—64no—
423Dic5—64no—
424C24—31no—
425C2×Q8—54no—
426C2×Q8—104no—
427C2×Q8—154no—
428C2×Q8—304no—
429C22×C4—31no—
430C22×C4—31no—
431C22×C4—31no—
432C22×C4—34no—
433C22×C4—52no—
434C22×C4—52no—
435C22×C4—64no—
436C22×C4—152no—
437C22×C4—152no—
438C22×C4C2×D3×D511yescharacteristic
439C22×C4C2×D3×D511yescharacteristic
440C22×C4C2×D3×D511yescharacteristic
441C4⋊C4—34no—
442C4⋊C4—34no—
443C4⋊C4—54no—
444C4⋊C4—58no—
445C4⋊C4—158no—
446C4⋊C4—158no—
447C4⋊C4—158no—
448C4⋊C4—158no—
449C4⋊C4—158no—
450C4⋊C4—158no—
451C4⋊C4—154no—
452C4⋊C4—158no—
453C22⋊C4—34no—
454C22⋊C4—34no—
455C22⋊C4—158no—
456C22⋊C4—158no—
457C42—34no—
458C42—58no—
459C42—158no—
460C42C2×D3×D514yes—
461C3×C5C2×#54111yescharacteristic, complemented, Hall
462C22×C3C2×#70912yes—
463C22×C3#415012yes—
464C22×C3#415012yes—
465C22×C3C24×D511yescharacteristic
466C2×D3—22no—
467C2×D3—22no—
468C2×D3—22no—
469C2×D3—44no—
470C3×C4—22no—
471C3×C4—22no—
472C3×C4—104no—
473C3×C4—104no—
474C3×C4C2×#70914yes—
475Dic3—22no—
476Dic3—22no—
477Dic3—22no—
478Dic3—104no—
479Dic3—104no—
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—108no—
491Q8—308no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—32no—
495C2×C4—32no—
496C2×C4—32no—
497C2×C4—54no—
498C2×C4—54no—
499C2×C4—62no—
500C2×C4—62no—
501C2×C4—62no—
502C2×C4—68no—
503C2×C4—104no—
504C2×C4—104no—
505C2×C4—154no—
506C2×C4—154no—
507C2×C4—304no—
508C2×C4—304no—
509C2×C4#677114yes—
510C2×C4C22×D3×D512yes—
511C2×C4C22×D3×D512yes—
512C2×C4C22×D3×D512yes—
513C2×C3#6585214yescomplemented
514C2×C3C22×#70911yescharacteristic
515C2×C3C2×#415011yescharacteristic
516C2×C3C2×#415011yescharacteristic
517D3—42no—
518C5C2×#558311yescharacteristic, complemented, Sylow
519C22—62no—
520C22—62no—
521C22—62no—
522C22—124no—
523C22#9200112yes—
524C22#9200412yes—
525C22C2×#677112yes—
526C22C23×D3×D511yescharacteristic
527C4—22no—
528C4—22no—
529C4—62no—
530C4—62no—
531C4—62no—
532C4—104no—
533C4—104no—
534C4—304no—
535C4—304no—
536C4C2×#677114yes—
537C3C2×#6585211yescharacteristic, complemented, Sylow
538C2—122no—
539C2#1175251214yesdirect product factor
540C2C2×#9200111yescharacteristic
541C2C2×#9200411yescharacteristic
542C2C22×#677111yescharacteristic
subgroupG/H
centerC23C22×D3×D5
commutator, cosocleC22×C3×C5C25
FrattiniC22C23×D3×D5
FittingC2×C3×C42×C5C22
socleC23×C3×C5C24
prime radicalC23×C5×D3C23
hypercenterC2×C42D3×D5
hypocenterC3×C5C2×#541
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
1. C2×C4×Dic30C2semidirect
3. C2×C3×C4×Dic10C2semidirect
4. C2×#91389C2semidirect
5. C2×#91382C2semidirect
6. C2×#91381C2semidirect
7. C2×#91378C2semidirect
12. #11752512C2direct (N × H)
21. C4×Dic30C22semidirect
31. C3×C4×Dic10C22semidirect
40. #91389C22semidirect
41. #91382C22semidirect
42. #91381C22semidirect
43. #91378C22semidirect
60. C2×C4×Dic10D3semidirect
143. C4×Dic10C2×D3semidirect
373. C2×C3×C5#541semidirect
461. C3×C5C2×#541semidirect
481. C2×C5#5583semidirect
513. C2×C3#65852semidirect
518. C5C2×#5583semidirect
537. C3C2×#65852semidirect
539. C2#11752512direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×Dic302
C2×C5×#8722
C2×C3×C4×Dic102
C2×#913892
C2×#913822
C2×#913812
C2×#913782
C2×#913722
C2×#913692
C2×#913672
C2×#913652
#117525122
C2×#658523
C2×#55835