Finite Group Databasebeta

← C2×#11754478 #14481720

Subgroups — C2×#11754478 #14481720

Order 1920.

2592 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91813C211yesmaximal, characteristic, complemented
2C22×C5×(C2×D3)⋊C4C211yesmaximal, characteristic
3C2×C3×#4070C211yesmaximal, characteristic, complemented
4C2×#91575C212yesmaximal, complemented
5C2×D3×C22⋊Dic5C212yesmaximal
6#11754478C2116yesmaximal, direct product factor
7C2×#91436C212yesmaximal, complemented
8C2×#91432C212yesmaximal, complemented
9C2×#91417C212yesmaximal, complemented
10C2×#91410C212yesmaximal, complemented
11C2×#65817—31nomaximal, Hall
12C24×C5×D3C2211yescharacteristic
13C23×C3×C4×C5C2211yescharacteristic
14C22×Dic30C2211yescharacteristic, complemented
15C23×C5×Dic3C2211yescharacteristic
16C22×C3×Dic10C2211yescharacteristic, complemented
17C22×#1204C2212yescomplemented
18C22×D3×Dic5C2212yes—
19C22×D3×Dic5C2212yes—
20C2×#6719C2212yes—
21C2×C4⋊Dic15C2211yescharacteristic, complemented
22#91813C2218yescomplemented
23C2×#6712C2212yescomplemented
24C2×C5×(C2×D3)⋊C4C2212yes—
25C2×C5×(C2×D3)⋊C4C2212yes—
26C2×C5×(C2×D3)⋊C4C2218yes—
27C2×C3×C22⋊Dic5C2212yes—
28C2×C3×C4⋊Dic5C2211yescharacteristic, complemented
29C3×#4070C2218yescomplemented
30C2×C3×#683C2212yescomplemented
31#91575C22116yescomplemented
32D3×C22⋊Dic5C22116yes—
33C2×Dic3⋊Dic5C2212yescomplemented
34C2×#6669C2212yescomplemented
35C2×#6668C2212yescomplemented
36C2×(C2×D3)⋊Dic5C2212yes—
37C2×(C2×D3)⋊Dic5C2212yes—
38#91436C22116yescomplemented
39#91432C22116yescomplemented
40#91417C22116yescomplemented
41#91410C22116yescomplemented
42C2×#5634—51nomaximal, Hall
43C22×C5×C22⋊C4—31no—
44C22×C22⋊Dic5—32no—
45C2×#4070—31no—
46C2×#4070—32no—
47C2×#4070D311yescharacteristic, complemented
48#65817—316no—
49C2×#4024—32no—
50C2×#4024—32no—
51C2×#4024—32no—
52C2×#4024—32no—
53C24×C3×C5C2311yescharacteristic
54C23×C5×D3—42no—
55C23×C5×D3—42no—
56C23×C5×D3Q818yescomplemented
57C23×C5×D3C2312yes—
58C23×C5×D3C2314yes—
59C22×C3×C4×C5—42no—
60C22×C3×C4×C5C2311yescharacteristic
61C22×C3×C4×C5C2311yescharacteristic
62C22×C3×C4×C5C2314yes—
63C22×Dic15C2312yes—
64C22×Dic15C2312yes—
65C2×Dic30—42no—
66C2×Dic30C2314yescomplemented
67C22×C5×Dic3—42no—
68C22×C5×Dic3C2312yes—
69C22×C5×Dic3C2314yes—
70C22×C3×Dic5C2312yes—
71C22×C3×Dic5C2312yes—
72C2×C3×Dic10—42no—
73C2×C3×Dic10C2314yescomplemented
74C2×#1204—44no—
75C2×#1204C2318yescomplemented
76C2×D3×Dic5—44no—
77C2×D3×Dic5—44no—
78C2×D3×Dic5C2318yes—
79C2×D3×Dic5C2318yes—
80#6719C2318yes—
81C4⋊Dic15C2314yescomplemented
82#6712C2318yescomplemented
83C5×(C2×D3)⋊C4C2318yes—
84C5×(C2×D3)⋊C4C2318yes—
85C3×C22⋊Dic5C2318yes—
86C3×C4⋊Dic5C2314yescomplemented
87C3×#683C2318yescomplemented
88Dic3⋊Dic5C2318yescomplemented
89#6669C2318yescomplemented
90#6668C2318yescomplemented
91(C2×D3)⋊Dic5C2318yes—
92(C2×D3)⋊Dic5C2318yes—
93C2×C3×C22⋊Q8—51no—
94C2×#946—52no—
95C22×(C2×D3)⋊C4D511yescharacteristic
96C2×#924—51no—
97#5634—516no—
98C2×#897—52no—
99C2×#896—52no—
100C2×#896—52no—
101C2×D3×C22⋊C4—52no—
102C2×#878—52no—
103C25×C5—31no—
104C23×C4×C5—31no—
105C23×C4×C5C2×D311yescharacteristic
106C23×Dic5—32no—
107C23×Dic5—32no—
108C22×Dic10—31no—
109C22×Dic10—32no—
110C22×Dic10C2×D311yescharacteristic, complemented
111C2×C5×C22⋊C4—32no—
112C2×C5×C22⋊C4—32no—
113C2×C5×C22⋊C4—38no—
114C2×C22⋊Dic5—32no—
115C2×C22⋊Dic5—32no—
116C2×C22⋊Dic5—32no—
117C2×C22⋊Dic5—316no—
118C2×C22⋊Dic5C2×D312yes—
119C2×C4⋊Dic5—31no—
120C2×C4⋊Dic5—32no—
121C2×C4⋊Dic5C2×D311yescharacteristic, complemented
122#4070—38no—
123#4070—316no—
124#4070C2×D318yescomplemented
125C2×#683—32no—
126C2×#683—32no—
127C2×#683—32no—
128C2×#683C2×D312yescomplemented
129#4024—316no—
130#4024—316no—
131#4024—316no—
132#4024—316no—
133C2×#543—151noSylow
134C23×C3×C5—22no—
135C23×C3×C5—22no—
136C23×C3×C5C2×Q814yes—
137C23×C3×C5C2412yes—
138C23×C3×C5C2411yescharacteristic
139C22×C5×D3—24no—
140C22×C5×D3—24no—
141C22×C5×D3—416no—
142C22×C5×D3—44no—
143C22×C5×D3—44no—
144C22×C5×D3C2×Q818yes—
145C22×C5×D3C2×Q8116yescomplemented
146C22×C5×D3C2414yes—
147C2×C3×C4×C5—22no—
148C2×C3×C4×C5—22no—
149C2×C3×C4×C5—44no—
150C2×C3×C4×C5C2412yes—
151C2×C3×C4×C5C2412yes—
152C2×Dic15—24no—
153C2×Dic15—24no—
154C2×Dic15C2414yes—
155C2×Dic15C2414yes—
156Dic30—44no—
157C2×C5×Dic3—24no—
158C2×C5×Dic3—44no—
159C2×C5×Dic3C2414yes—
160C2×C3×Dic5—24no—
161C2×C3×Dic5—24no—
162C2×C3×Dic5C2414yes—
163C2×C3×Dic5C2414yes—
164C3×Dic10—44no—
165#1204—48no—
166D3×Dic5—48no—
167D3×Dic5—48no—
168C24×D3C2×D511yescharacteristic
169C22×C3×Q8—51no—
170C23×C3×C4C2×D511yescharacteristic
171C23×Dic3C2×D511yescharacteristic
172C22×C4×D3—52no—
173C22×C4×D3—52no—
174C22×Dic6—51no—
175C22×Dic6—52no—
176C3×C22⋊Q8—58no—
177C2×C3×C4⋊C4—51no—
178C2×C3×C4⋊C4—52no—
179C2×C3×C22⋊C4—52no—
180C2×#217—52no—
181#946—516no—
182C2×(C2×D3)⋊C4—52no—
183C2×(C2×D3)⋊C4—52no—
184C2×(C2×D3)⋊C4C2×D518yes—
185C2×(C2×D3)⋊C4C2×D512yes—
186C2×(C2×D3)⋊C4C2×D512yes—
187C2×C4⋊Dic3—51no—
188C2×C4⋊Dic3—52no—
189#924—58no—
190C2×Dic3⋊C4—52no—
191C2×Dic3⋊C4—52no—
192C2×Dic3⋊C4—52no—
193#897—516no—
194#896—516no—
195#896—516no—
196D3×C22⋊C4—516no—
197#878—516no—
198C24×C5—32no—
199C24×C5—38no—
200C24×C5—34no—
201C24×C5—122no—
202C24×C5—122no—
203C24×C5C22×D311yescharacteristic
204C22×C4×C5—32no—
205C22×C4×C5—34no—
206C22×C4×C5—42no—
207C22×C4×C5—122no—
208C22×C4×C5C22×D311yescharacteristic
209C22×C4×C5C22×D311yescharacteristic
210C22×C4×C5C22×D314yes—
211C22×Dic5—32no—
212C22×Dic5—32no—
213C22×Dic5—38no—
214C22×Dic5—38no—
215C22×Dic5—124no—
216C22×Dic5—124no—
217C22×Dic5C22×D312yes—
218C22×Dic5C22×D312yes—
219C2×Dic10—34no—
220C2×Dic10—38no—
221C2×Dic10—42no—
222C2×Dic10—124no—
223C2×Dic10—122no—
224C2×Dic10C22×D314yescomplemented
225C5×C22⋊C4—38no—
226C5×C22⋊C4—38no—
227C22⋊Dic5—38no—
228C22⋊Dic5—38no—
229C22⋊Dic5—38no—
230C22⋊Dic5C22×D318yes—
231C4⋊Dic5—34no—
232C4⋊Dic5—38no—
233C4⋊Dic5C22×D314yescomplemented
234#683—38no—
235#683—38no—
236#683—38no—
237#683C22×D318yescomplemented
238#543—1516no—
239C2×C22⋊Q8—51no—
240C2×C22⋊Q8—152no—
241C2×C22⋊Q8—152no—
242C2×C22⋊Q8—152no—
243C2×C22⋊Q8—152no—
244C2×C22⋊Q8—152no—
245C2×C22⋊Q8—151no—
246C22×C22⋊C4—31no—
247C22×C22⋊C4—152no—
248C22×C3×C5—28no—
249C22×C3×C5—22no—
250C22×C3×C5—22no—
251C22×C3×C5C22×Q814yes—
252C22×C3×C5C22×Q812yes—
253C22×C3×C5H(5,𝔽2)12yes—
254C22×C3×C5H(5,𝔽2)12yes—
255C22×C3×C5C2511yescharacteristic
256C2×C5×D3—216no—
257C2×C5×D3—24no—
258C2×C5×D3—24no—
259C2×C5×D3—416no—
260C2×C5×D3C22×Q818yes—
261C3×C4×C5—22no—
262C3×C4×C5—22no—
263Dic15—24no—
264Dic15—24no—
265C3×Dic5—24no—
266C3×Dic5—24no—
267C5×Dic3—24no—
268C24×C3C22×D511yescharacteristic
269C23×D3—42no—
270C23×D3—42no—
271C23×D3Dic1018yescomplemented
272C23×D3C22×D512yes—
273C23×D3C22×D514yes—
274C2×C3×Q8—54no—
275C2×C3×Q8—202no—
276C22×C3×C4—42no—
277C22×C3×C4—52no—
278C22×C3×C4—52no—
279C22×C3×C4C22×D511yescharacteristic
280C22×C3×C4C22×D511yescharacteristic
281C22×C3×C4C22×D514yes—
282C22×Dic3—42no—
283C22×Dic3—52no—
284C22×Dic3—52no—
285C22×Dic3C22×D512yes—
286C22×Dic3C22×D514yes—
287C2×C4×D3—58no—
288C2×C4×D3—58no—
289C2×C4×D3—204no—
290C2×C4×D3—204no—
291C2×Dic6—54no—
292C2×Dic6—58no—
293C2×Dic6—204no—
294C2×Dic6—202no—
295C3×C4⋊C4—58no—
296C3×C4⋊C4—54no—
297C3×C22⋊C4—58no—
298#217—58no—
299(C2×D3)⋊C4—58no—
300(C2×D3)⋊C4—58no—
301(C2×D3)⋊C4C22×D518yes—
302(C2×D3)⋊C4C22×D518yes—
303C4⋊Dic3—54no—
304C4⋊Dic3—58no—
305Dic3⋊C4—58no—
306Dic3⋊C4—58no—
307Dic3⋊C4—58no—
308C23×C5—22no—
309C23×C5—22no—
310C23×C5—38no—
311C23×C5—34no—
312C23×C5—316no—
313C23×C5—64no—
314C23×C5—64no—
315C23×C5—1216no—
316C23×C5—124no—
317C23×C5—124no—
318C23×C5D3×Q814yes—
319C23×C5C23×D312yes—
320C23×C5C23×D311yescharacteristic
321C2×C4×C5—22no—
322C2×C4×C5—22no—
323C2×C4×C5—34no—
324C2×C4×C5—44no—
325C2×C4×C5—64no—
326C2×C4×C5—124no—
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×Dic5—128no—
336C2×Dic5—128no—
337C2×Dic5C23×D314yes—
338C2×Dic5C23×D314yes—
339Dic10—44no—
340Dic10—128no—
341Dic10—124no—
342C25—31no—
343C22×Q8—51no—
344C22×Q8—152no—
345C22×Q8—151no—
346C23×C4—31no—
347C23×C4—152no—
348C23×C4—152no—
349C23×C4D3×D511yescharacteristic
350C22⋊Q8—58no—
351C22⋊Q8—1516no—
352C22⋊Q8—1516no—
353C22⋊Q8—1516no—
354C22⋊Q8—1516no—
355C22⋊Q8—1516no—
356C22⋊Q8—158no—
357C2×C4⋊C4—52no—
358C2×C4⋊C4—51no—
359C2×C4⋊C4—152no—
360C2×C4⋊C4—152no—
361C2×C4⋊C4—152no—
362C2×C4⋊C4—152no—
363C2×C4⋊C4—151no—
364C2×C22⋊C4—32no—
365C2×C22⋊C4—32no—
366C2×C22⋊C4—38no—
367C2×C22⋊C4—52no—
368C2×C22⋊C4—1516no—
369C2×C22⋊C4—152no—
370C2×C22⋊C4—152no—
371C2×C22⋊C4—152no—
372C2×C3×C5—24no—
373C2×C3×C5#54314yescomplemented
374C2×C3×C5C23×Q811yescharacteristic
375C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
376C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
377C5×D3—28no—
378C23×C3—22no—
379C23×C3—22no—
380C23×C3C2×Dic1014yes—
381C23×C3C23×D512yes—
382C23×C3C23×D511yescharacteristic
383C22×D3—24no—
384C22×D3—24no—
385C22×D3—416no—
386C22×D3—44no—
387C22×D3—44no—
388C22×D3C2×Dic1018yes—
389C22×D3C2×Dic10116yescomplemented
390C22×D3C23×D514yes—
391C3×Q8—204no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—44no—
395C2×C3×C4—54no—
396C2×C3×C4—54no—
397C2×C3×C4—104no—
398C2×C3×C4—104no—
399C2×C3×C4C23×D512yes—
400C2×C3×C4C23×D512yes—
401C2×Dic3—24no—
402C2×Dic3—44no—
403C2×Dic3—54no—
404C2×Dic3—54no—
405C2×Dic3—104no—
406C2×Dic3—104no—
407C2×Dic3C23×D514yes—
408C4×D3—208no—
409C4×D3—208no—
410Dic6—208no—
411Dic6—204no—
412C22×C5—28no—
413C22×C5—22no—
414C22×C5—22no—
415C22×C5—38no—
416C22×C5—616no—
417C22×C5—64no—
418C22×C5—64no—
419C22×C5—1216no—
420C22×C5#100412yes—
421C22×C5C2×D3×Q814yes—
422C22×C5C2×D3×Q812yes—
423C22×C5#100912yes—
424C22×C5C24×D311yescharacteristic
425C4×C5—22no—
426C4×C5—22no—
427C4×C5—64no—
428Dic5—24no—
429Dic5—24no—
430Dic5—64no—
431Dic5—64no—
432C24—32no—
433C24—38no—
434C24—34no—
435C24—122no—
436C24—122no—
437C24C2×D3×D511yescharacteristic
438C2×Q8—54no—
439C2×Q8—158no—
440C2×Q8—154no—
441C2×Q8—202no—
442C2×Q8—602no—
443C2×Q8—604no—
444C22×C4—32no—
445C22×C4—34no—
446C22×C4—42no—
447C22×C4—52no—
448C22×C4—52no—
449C22×C4—122no—
450C22×C4—158no—
451C22×C4—158no—
452C22×C4—152no—
453C22×C4—152no—
454C22×C4—604no—
455C22×C4—604no—
456C22×C4C2×D3×D511yescharacteristic
457C22×C4C2×D3×D511yescharacteristic
458C22×C4C2×D3×D514yes—
459C4⋊C4—58no—
460C4⋊C4—54no—
461C4⋊C4—158no—
462C4⋊C4—158no—
463C4⋊C4—158no—
464C4⋊C4—158no—
465C4⋊C4—154no—
466C22⋊C4—38no—
467C22⋊C4—38no—
468C22⋊C4—58no—
469C22⋊C4—158no—
470C22⋊C4—158no—
471C22⋊C4—158no—
472C3×C5C2×#54311yescharacteristic, complemented, Hall
473C22×C3—28no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3C22×Dic1014yes—
477C22×C3C22×Dic1012yes—
478C22×C3#414412yes—
479C22×C3#414412yes—
480C22×C3C24×D511yescharacteristic
481C2×D3—216no—
482C2×D3—24no—
483C2×D3—24no—
484C2×D3—416no—
485C2×D3C22×Dic1018yes—
486C3×C4—22no—
487C3×C4—22no—
488C3×C4—104no—
489C3×C4—104no—
490Dic3—24no—
491Dic3—104no—
492Dic3—104no—
493C2×C5—24no—
494C2×C5—68no—
495C2×C5#563414yescomplemented
496C2×C5C2×#100411yescharacteristic
497C2×C5C22×D3×Q811yescharacteristic
498C2×C5C2×#100911yescharacteristic
499C23—22no—
500C23—22no—
501C23—38no—
502C23—34no—
503C23—316no—
504C23—64no—
505C23—64no—
506C23—124no—
507C23—124no—
508C23—1216no—
509C23D3×Dic1014yes—
510C23C22×D3×D512yes—
511C23C22×D3×D511yescharacteristic
512Q8—204no—
513Q8—604no—
514Q8—608no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—34no—
518C2×C4—44no—
519C2×C4—54no—
520C2×C4—54no—
521C2×C4—64no—
522C2×C4—104no—
523C2×C4—104no—
524C2×C4—124no—
525C2×C4—154no—
526C2×C4—154no—
527C2×C4—304no—
528C2×C4—304no—
529C2×C4—608no—
530C2×C4—608no—
531C2×C4C22×D3×D512yes—
532C2×C4C22×D3×D512yes—
533C2×C3—24no—
534C2×C3#6581714yescomplemented
535C2×C3C23×Dic1011yescharacteristic
536C2×C3C2×#414411yescharacteristic
537C2×C3C2×#414411yescharacteristic
538D3—28no—
539C5C2×#563411yescharacteristic, complemented, Sylow
540C22—22no—
541C22—22no—
542C22—28no—
543C22—38no—
544C22—64no—
545C22—64no—
546C22—616no—
547C22—1216no—
548C22C2×D3×Dic1014yes—
549C22C2×D3×Dic1012yes—
550C22#9201812yes—
551C22#9204912yes—
552C22C23×D3×D511yescharacteristic
553C4—22no—
554C4—22no—
555C4—64no—
556C4—104no—
557C4—104no—
558C4—304no—
559C4—304no—
560C3C2×#6581711yescharacteristic, complemented, Sylow
561C2—24no—
562C2—68no—
563C2#1175447814yesdirect product factor
564C2C22×D3×Dic1011yescharacteristic
565C2C2×#9201811yescharacteristic
566C2C2×#9204911yescharacteristic
subgroupG/H
centerC23C22×D3×D5
commutator, cosocleC22×C3×C5C25
FrattiniC22C23×D3×D5
FittingC23×C3×C4×C5C22
socleC23×C3×C5C24
prime radicalC24×C5×D3C22
hypercenterC23×C4D3×D5
hypocenterC3×C5C2×#543
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C2×#91813C2semidirect
3. C2×C3×#4070C2semidirect
4. C2×#91575C2semidirect
6. #11754478C2direct (N × H)
7. C2×#91436C2semidirect
8. C2×#91432C2semidirect
9. C2×#91417C2semidirect
10. C2×#91410C2semidirect
14. C22×Dic30C22semidirect
16. C22×C3×Dic10C22semidirect
17. C22×#1204C22semidirect
21. C2×C4⋊Dic15C22semidirect
22. #91813C22semidirect
23. C2×#6712C22semidirect
28. C2×C3×C4⋊Dic5C22semidirect
29. C3×#4070C22semidirect
30. C2×C3×#683C22semidirect
31. #91575C22semidirect
33. C2×Dic3⋊Dic5C22semidirect
34. C2×#6669C22semidirect
35. C2×#6668C22semidirect
38. #91436C22semidirect
39. #91432C22semidirect
40. #91417C22semidirect
41. #91410C22semidirect
47. C2×#4070D3semidirect
56. C23×C5×D3Q8semidirect
66. C2×Dic30C23semidirect
73. C2×C3×Dic10C23semidirect
75. C2×#1204C23semidirect
81. C4⋊Dic15C23semidirect
82. #6712C23semidirect
86. C3×C4⋊Dic5C23semidirect
87. C3×#683C23semidirect
88. Dic3⋊Dic5C23semidirect
89. #6669C23semidirect
90. #6668C23semidirect
110. C22×Dic10C2×D3semidirect
121. C2×C4⋊Dic5C2×D3semidirect
124. #4070C2×D3semidirect
128. C2×#683C2×D3semidirect
145. C22×C5×D3C2×Q8semidirect
224. C2×Dic10C22×D3semidirect
233. C4⋊Dic5C22×D3semidirect
237. #683C22×D3semidirect
271. C23×D3Dic10semidirect
373. C2×C3×C5#543semidirect
389. C22×D3C2×Dic10semidirect
472. C3×C5C2×#543semidirect
495. C2×C5#5634semidirect
534. C2×C3#65817semidirect
539. C5C2×#5634semidirect
560. C3C2×#65817semidirect
563. C2#11754478direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#918132
C22×C5×(C2×D3)⋊C42
C2×C3×#40702
C2×#915752
C2×D3×C22⋊Dic52
#117544782
C2×#914362
C2×#914322
C2×#914172
C2×#914102
C2×#658173
C2×#56345