Finite Group Databasebeta

← C2×#11754468 #14481718

Subgroups — C2×#11754468 #14481718

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
2C2×C5×#924C211yesmaximal, characteristic, complemented
3C22×C3×#685C211yesmaximal, characteristic
4C2×#91574C212yesmaximal, complemented
5C2×D5×#217C212yesmaximal
6#11754468C2116yesmaximal, direct product factor
7C2×#91431C212yesmaximal, complemented
8C2×#91430C212yesmaximal, complemented
9C2×#91422C212yesmaximal, complemented
10C2×#91421C212yesmaximal, complemented
11C2×#65968—31nomaximal, Hall
12C24×C3×D5C2211yescharacteristic
13C23×C3×C4×C5C2211yescharacteristic
14C22×Dic30C2211yescharacteristic, complemented
15C22×C5×Dic6C2211yescharacteristic, complemented
16C23×C3×Dic5C2211yescharacteristic
17C22×#1204C2212yescomplemented
18C22×D5×Dic3C2212yes—
19C22×D5×Dic3C2212yes—
20C2×#6719C2212yes—
21C2×C4⋊Dic15C2211yescharacteristic, complemented
22#91813C2218yescomplemented
23C2×#6712C2212yescomplemented
24C2×C5×#217C2212yes—
25C2×C5×C4⋊Dic3C2211yescharacteristic, complemented
26C5×#924C2218yescomplemented
27C2×C5×Dic3⋊C4C2212yescomplemented
28C2×C3×#685C2212yes—
29C2×C3×#685C2212yes—
30C2×C3×#685C2218yes—
31#91574C22116yescomplemented
32D5×#217C22116yes—
33C2×Dic3⋊Dic5C2212yescomplemented
34C2×#6669C2212yescomplemented
35C2×#6668C2212yescomplemented
36C2×#6665C2212yes—
37C2×#6665C2212yes—
38#91431C22116yescomplemented
39#91430C22116yescomplemented
40#91422C22116yescomplemented
41#91421C22116yescomplemented
42C2×#5483—51nomaximal, Hall
43C2×C5×C22⋊Q8—31no—
44C2×#4092—32no—
45C22×#685D311yescharacteristic
46C2×#4070—31no—
47#65968—316no—
48C2×#4043—32no—
49C2×#4042—32no—
50C2×#4042—32no—
51C2×D5×C22⋊C4—32no—
52C2×#4024—32no—
53C24×C3×C5C2311yescharacteristic
54C23×C3×D5—42no—
55C23×C3×D5—42no—
56C23×C3×D5Q818yescomplemented
57C23×C3×D5C2312yes—
58C23×C3×D5C2314yes—
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×Dic3C2312yes—
68C22×C5×Dic3C2312yes—
69C2×C5×Dic6—42no—
70C2×C5×Dic6C2314yescomplemented
71C22×C3×Dic5—42no—
72C22×C3×Dic5C2312yes—
73C22×C3×Dic5C2314yes—
74C2×#1204—44no—
75C2×#1204C2318yescomplemented
76C2×D5×Dic3—44no—
77C2×D5×Dic3—44no—
78C2×D5×Dic3C2318yes—
79C2×D5×Dic3C2318yes—
80#6719C2318yes—
81C4⋊Dic15C2314yescomplemented
82#6712C2318yescomplemented
83C5×#217C2318yes—
84C5×C4⋊Dic3C2314yescomplemented
85C5×Dic3⋊C4C2318yescomplemented
86C3×#685C2318yes—
87C3×#685C2318yes—
88Dic3⋊Dic5C2318yescomplemented
89#6669C2318yescomplemented
90#6668C2318yescomplemented
91#6665C2318yes—
92#6665C2318yes—
93C22×C3×C22⋊C4—51no—
94C22×#217—52no—
95C2×#924—51no—
96C2×#924—52no—
97C2×#924D511yescharacteristic, complemented
98#5483—516no—
99C2×#878—52no—
100C2×#878—52no—
101C2×#878—52no—
102C2×#878—52no—
103C24×D5C2×D311yescharacteristic
104C22×C5×Q8—31no—
105C23×C4×C5C2×D311yescharacteristic
106C23×Dic5C2×D311yescharacteristic
107C22×C4×D5—32no—
108C22×C4×D5—32no—
109C22×Dic10—31no—
110C22×Dic10—32no—
111C5×C22⋊Q8—38no—
112C2×C5×C4⋊C4—31no—
113C2×C5×C4⋊C4—32no—
114C2×C5×C22⋊C4—32no—
115C2×C22⋊Dic5—32no—
116#4092—316no—
117C2×#685—32no—
118C2×#685—32no—
119C2×#685C2×D312yes—
120C2×#685C2×D312yes—
121C2×#685C2×D318yes—
122C2×C4⋊Dic5—31no—
123C2×C4⋊Dic5—32no—
124#4070—38no—
125C2×#683—32no—
126C2×#683—32no—
127C2×#683—32no—
128#4043—316no—
129#4042—316no—
130#4042—316no—
131D5×C22⋊C4—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×C3×D5—24no—
140C22×C3×D5—24no—
141C22×C3×D5—416no—
142C22×C3×D5—44no—
143C22×C3×D5—44no—
144C22×C3×D5C2×Q818yes—
145C22×C3×D5C2×Q8116yescomplemented
146C22×C3×D5C2414yes—
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—24no—
159C2×C5×Dic3C2414yes—
160C2×C5×Dic3C2414yes—
161C5×Dic6—44no—
162C2×C3×Dic5—24no—
163C2×C3×Dic5—44no—
164C2×C3×Dic5C2414yes—
165#1204—48no—
166D5×Dic3—48no—
167D5×Dic3—48no—
168C25×C3—51no—
169C23×C3×C4—51no—
170C23×C3×C4C2×D511yescharacteristic
171C23×Dic3—52no—
172C23×Dic3—52no—
173C22×Dic6—51no—
174C22×Dic6—52no—
175C22×Dic6C2×D511yescharacteristic, complemented
176C2×C3×C22⋊C4—52no—
177C2×C3×C22⋊C4—52no—
178C2×C3×C22⋊C4—58no—
179C2×#217—52no—
180C2×#217—52no—
181C2×#217—52no—
182C2×#217—516no—
183C2×#217C2×D512yes—
184C2×C4⋊Dic3—51no—
185C2×C4⋊Dic3—52no—
186C2×C4⋊Dic3C2×D511yescharacteristic, complemented
187#924—58no—
188#924—516no—
189#924C2×D518yescomplemented
190C2×Dic3⋊C4—52no—
191C2×Dic3⋊C4—52no—
192C2×Dic3⋊C4—52no—
193C2×Dic3⋊C4C2×D512yescomplemented
194#878—516no—
195#878—516no—
196#878—516no—
197#878—516no—
198C24×C5C22×D311yescharacteristic
199C23×D5—42no—
200C23×D5—42no—
201C23×D5Dic618yescomplemented
202C23×D5C22×D312yes—
203C23×D5C22×D314yes—
204C2×C5×Q8—34no—
205C2×C5×Q8—122no—
206C22×C4×C5—32no—
207C22×C4×C5—32no—
208C22×C4×C5—42no—
209C22×C4×C5C22×D311yescharacteristic
210C22×C4×C5C22×D311yescharacteristic
211C22×C4×C5C22×D314yes—
212C22×Dic5—32no—
213C22×Dic5—32no—
214C22×Dic5—42no—
215C22×Dic5C22×D312yes—
216C22×Dic5C22×D314yes—
217C2×C4×D5—38no—
218C2×C4×D5—38no—
219C2×C4×D5—124no—
220C2×C4×D5—124no—
221C2×Dic10—34no—
222C2×Dic10—38no—
223C2×Dic10—124no—
224C2×Dic10—122no—
225C5×C4⋊C4—38no—
226C5×C4⋊C4—34no—
227C5×C22⋊C4—38no—
228C22⋊Dic5—38no—
229#685—38no—
230#685—38no—
231#685C22×D318yes—
232#685C22×D318yes—
233C4⋊Dic5—34no—
234C4⋊Dic5—38no—
235#683—38no—
236#683—38no—
237#683—38no—
238#543—1516no—
239C2×C22⋊Q8—31no—
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—51no—
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×C3×D5—216no—
257C2×C3×D5—24no—
258C2×C3×D5—24no—
259C2×C3×D5—416no—
260C2×C3×D5C22×Q818yes—
261C3×C4×C5—22no—
262C3×C4×C5—22no—
263Dic15—24no—
264Dic15—24no—
265C3×Dic5—24no—
266C5×Dic3—24no—
267C5×Dic3—24no—
268C24×C3—52no—
269C24×C3—58no—
270C24×C3—54no—
271C24×C3—202no—
272C24×C3—202no—
273C24×C3C22×D511yescharacteristic
274C22×C3×C4—42no—
275C22×C3×C4—52no—
276C22×C3×C4—54no—
277C22×C3×C4—202no—
278C22×C3×C4C22×D511yescharacteristic
279C22×C3×C4C22×D511yescharacteristic
280C22×C3×C4C22×D514yes—
281C22×Dic3—52no—
282C22×Dic3—52no—
283C22×Dic3—58no—
284C22×Dic3—58no—
285C22×Dic3—204no—
286C22×Dic3—204no—
287C22×Dic3C22×D512yes—
288C22×Dic3C22×D512yes—
289C2×Dic6—42no—
290C2×Dic6—54no—
291C2×Dic6—58no—
292C2×Dic6—204no—
293C2×Dic6—202no—
294C2×Dic6C22×D514yescomplemented
295C3×C22⋊C4—58no—
296C3×C22⋊C4—58no—
297#217—58no—
298#217—58no—
299#217—58no—
300#217C22×D518yes—
301C4⋊Dic3—54no—
302C4⋊Dic3—58no—
303C4⋊Dic3C22×D514yescomplemented
304Dic3⋊C4—58no—
305Dic3⋊C4—58no—
306Dic3⋊C4—58no—
307Dic3⋊C4C22×D518yescomplemented
308C23×C5—22no—
309C23×C5—22no—
310C23×C5C2×Dic614yes—
311C23×C5C23×D312yes—
312C23×C5C23×D311yescharacteristic
313C22×D5—24no—
314C22×D5—24no—
315C22×D5—416no—
316C22×D5—44no—
317C22×D5—44no—
318C22×D5C2×Dic618yes—
319C22×D5C2×Dic6116yescomplemented
320C22×D5C23×D314yes—
321C5×Q8—124no—
322C2×C4×C5—22no—
323C2×C4×C5—22no—
324C2×C4×C5—34no—
325C2×C4×C5—34no—
326C2×C4×C5—44no—
327C2×C4×C5—64no—
328C2×C4×C5—64no—
329C2×C4×C5C23×D312yes—
330C2×C4×C5C23×D312yes—
331C2×Dic5—24no—
332C2×Dic5—34no—
333C2×Dic5—34no—
334C2×Dic5—44no—
335C2×Dic5—64no—
336C2×Dic5—64no—
337C2×Dic5C23×D314yes—
338C4×D5—128no—
339C4×D5—128no—
340Dic10—128no—
341Dic10—124no—
342C25—51no—
343C22×Q8—31no—
344C22×Q8—152no—
345C22×Q8—151no—
346C23×C4—51no—
347C23×C4—152no—
348C23×C4—152no—
349C23×C4D3×D511yescharacteristic
350C22⋊Q8—38no—
351C22⋊Q8—1516no—
352C22⋊Q8—1516no—
353C22⋊Q8—1516no—
354C22⋊Q8—1516no—
355C22⋊Q8—1516no—
356C22⋊Q8—158no—
357C2×C4⋊C4—32no—
358C2×C4⋊C4—31no—
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—52no—
366C2×C22⋊C4—52no—
367C2×C22⋊C4—58no—
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
377C3×D5—28no—
378C23×C3—22no—
379C23×C3—22no—
380C23×C3—58no—
381C23×C3—54no—
382C23×C3—516no—
383C23×C3—104no—
384C23×C3—104no—
385C23×C3—2016no—
386C23×C3—204no—
387C23×C3—204no—
388C23×C3Q8×D514yes—
389C23×C3C23×D512yes—
390C23×C3C23×D511yescharacteristic
391C2×C3×C4—22no—
392C2×C3×C4—22no—
393C2×C3×C4—44no—
394C2×C3×C4—54no—
395C2×C3×C4—104no—
396C2×C3×C4—204no—
397C2×C3×C4C23×D512yes—
398C2×C3×C4C23×D512yes—
399C2×Dic3—24no—
400C2×Dic3—24no—
401C2×Dic3—54no—
402C2×Dic3—54no—
403C2×Dic3—104no—
404C2×Dic3—104no—
405C2×Dic3—208no—
406C2×Dic3—208no—
407C2×Dic3C23×D514yes—
408C2×Dic3C23×D514yes—
409Dic6—44no—
410Dic6—208no—
411Dic6—204no—
412C22×C5—22no—
413C22×C5—22no—
414C22×C5—28no—
415C22×C5C22×Dic614yes—
416C22×C5C22×Dic612yes—
417C22×C5#100412yes—
418C22×C5#100412yes—
419C22×C5C24×D311yescharacteristic
420C2×D5—24no—
421C2×D5—24no—
422C2×D5—216no—
423C2×D5—416no—
424C2×D5C22×Dic618yes—
425C4×C5—22no—
426C4×C5—22no—
427C4×C5—64no—
428C4×C5—64no—
429Dic5—24no—
430Dic5—64no—
431Dic5—64no—
432C24—52no—
433C24—58no—
434C24—54no—
435C24—202no—
436C24—202no—
437C24C2×D3×D511yescharacteristic
438C2×Q8—34no—
439C2×Q8—122no—
440C2×Q8—158no—
441C2×Q8—154no—
442C2×Q8—602no—
443C2×Q8—604no—
444C22×C4—32no—
445C22×C4—32no—
446C22×C4—42no—
447C22×C4—52no—
448C22×C4—54no—
449C22×C4—158no—
450C22×C4—158no—
451C22×C4—152no—
452C22×C4—152no—
453C22×C4—202no—
454C22×C4—604no—
455C22×C4—604no—
456C22×C4C2×D3×D511yescharacteristic
457C22×C4C2×D3×D511yescharacteristic
458C22×C4C2×D3×D514yes—
459C4⋊C4—38no—
460C4⋊C4—34no—
461C4⋊C4—158no—
462C4⋊C4—158no—
463C4⋊C4—158no—
464C4⋊C4—158no—
465C4⋊C4—154no—
466C22⋊C4—38no—
467C22⋊C4—58no—
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×C3—58no—
477C22×C3—1016no—
478C22×C3—104no—
479C22×C3—104no—
480C22×C3—2016no—
481C22×C3#414412yes—
482C22×C3C2×Q8×D514yes—
483C22×C3C2×Q8×D512yes—
484C22×C3#414912yes—
485C22×C3C24×D511yescharacteristic
486C3×C4—22no—
487C3×C4—22no—
488C3×C4—104no—
489Dic3—24no—
490Dic3—24no—
491Dic3—104no—
492Dic3—104no—
493C2×C5—24no—
494C2×C5#548314yescomplemented
495C2×C5C23×Dic611yescharacteristic
496C2×C5C2×#100411yescharacteristic
497C2×C5C2×#100411yescharacteristic
498D5—28no—
499C23—22no—
500C23—22no—
501C23—58no—
502C23—54no—
503C23—516no—
504C23—104no—
505C23—104no—
506C23—204no—
507C23—204no—
508C23—2016no—
509C23D5×Dic614yes—
510C23C22×D3×D512yes—
511C23C22×D3×D511yescharacteristic
512Q8—124no—
513Q8—604no—
514Q8—608no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—34no—
518C2×C4—34no—
519C2×C4—44no—
520C2×C4—54no—
521C2×C4—64no—
522C2×C4—64no—
523C2×C4—104no—
524C2×C4—154no—
525C2×C4—154no—
526C2×C4—204no—
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—108no—
535C2×C3#6596814yescomplemented
536C2×C3C2×#414411yescharacteristic
537C2×C3C22×Q8×D511yescharacteristic
538C2×C3C2×#414911yescharacteristic
539C5C2×#548311yescharacteristic, complemented, Sylow
540C22—22no—
541C22—22no—
542C22—28no—
543C22—58no—
544C22—104no—
545C22—104no—
546C22—1016no—
547C22—2016no—
548C22C2×D5×Dic614yes—
549C22C2×D5×Dic612yes—
550C22#9201712yes—
551C22#9204912yes—
552C22C23×D3×D511yescharacteristic
553C4—22no—
554C4—22no—
555C4—64no—
556C4—64no—
557C4—104no—
558C4—304no—
559C4—304no—
560C3C2×#6596811yescharacteristic, complemented, Sylow
561C2—24no—
562C2—108no—
563C2#1175446814yesdirect product factor
564C2C22×D5×Dic611yescharacteristic
565C2C2×#9201711yescharacteristic
566C2C2×#9204911yescharacteristic
subgroupG/H
centerC23C22×D3×D5
commutator, cosocleC22×C3×C5C25
FrattiniC22C23×D3×D5
FittingC23×C3×C4×C5C22
socleC23×C3×C5C24
prime radicalC24×C3×D5C22
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
2. C2×C5×#924C2semidirect
4. C2×#91574C2semidirect
6. #11754468C2direct (N × H)
7. C2×#91431C2semidirect
8. C2×#91430C2semidirect
9. C2×#91422C2semidirect
10. C2×#91421C2semidirect
14. C22×Dic30C22semidirect
15. C22×C5×Dic6C22semidirect
17. C22×#1204C22semidirect
21. C2×C4⋊Dic15C22semidirect
22. #91813C22semidirect
23. C2×#6712C22semidirect
25. C2×C5×C4⋊Dic3C22semidirect
26. C5×#924C22semidirect
27. C2×C5×Dic3⋊C4C22semidirect
31. #91574C22semidirect
33. C2×Dic3⋊Dic5C22semidirect
34. C2×#6669C22semidirect
35. C2×#6668C22semidirect
38. #91431C22semidirect
39. #91430C22semidirect
40. #91422C22semidirect
41. #91421C22semidirect
56. C23×C3×D5Q8semidirect
66. C2×Dic30C23semidirect
70. C2×C5×Dic6C23semidirect
75. C2×#1204C23semidirect
81. C4⋊Dic15C23semidirect
82. #6712C23semidirect
84. C5×C4⋊Dic3C23semidirect
85. C5×Dic3⋊C4C23semidirect
88. Dic3⋊Dic5C23semidirect
89. #6669C23semidirect
90. #6668C23semidirect
97. C2×#924D5semidirect
145. C22×C3×D5C2×Q8semidirect
175. C22×Dic6C2×D5semidirect
186. C2×C4⋊Dic3C2×D5semidirect
189. #924C2×D5semidirect
193. C2×Dic3⋊C4C2×D5semidirect
201. C23×D5Dic6semidirect
294. C2×Dic6C22×D5semidirect
303. C4⋊Dic3C22×D5semidirect
307. Dic3⋊C4C22×D5semidirect
319. C22×D5C2×Dic6semidirect
373. C2×C3×C5#543semidirect
472. C3×C5C2×#543semidirect
494. C2×C5#5483semidirect
535. C2×C3#65968semidirect
539. C5C2×#5483semidirect
560. C3C2×#65968semidirect
563. C2#11754468direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#918132
C2×C5×#9242
C22×C3×#6852
C2×#915742
C2×D5×#2172
#117544682
C2×#914312
C2×#914302
C2×#914222
C2×#914212
C2×#659683
C2×#54835