Finite Group Databasebeta

← C2×#11752606 #14484740

Subgroups — C2×#11752606 #14484740

Order 1920.

2480 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D60C211yesmaximal, characteristic, complemented
2C2×C4×C5×Dic6C211yesmaximal, characteristic, complemented
3C2×C3×#4021C211yesmaximal, characteristic, complemented
4C2×#91458C211yesmaximal, characteristic
5C2×#91455C212yesmaximal, complemented
6C2×Dic3×D20C211yesmaximal, characteristic, complemented
7C2×#91413C212yesmaximal, complemented
8C2×#91396C212yesmaximal, complemented
9C2×#91374C211yesmaximal, characteristic, complemented
10C2×#91354C212yesmaximal, complemented
11C2×Dic3×Dic10C211yesmaximal, characteristic, complemented
12#11752606C2116yesmaximal, direct product factor
13C2×#65919—31nomaximal, Hall
14C22×D60C2211yescharacteristic
15C22×C4×D15C2212yes—
16C22×C5×Dic6C2211yescharacteristic
17C22×C3×D20C2211yescharacteristic
18C22×C3×Dic10C2211yescharacteristic
19C22×#1202C2212yes—
20C22×#1202C2212yes—
21C22×D5×Dic3C2212yes—
22C2×C3×C42×C5C2211yescharacteristic
23C2×#6714C2212yes—
24C2×C4⋊Dic15C2211yescharacteristic
25C4×D60C2218yescomplemented
26C2×C5×C4⋊Dic3C2211yescharacteristic
27C2×C5×Dic3⋊C4C2211yescharacteristic
28C2×C5×Dic3⋊C4C2211yescharacteristic
29C2×C4×C5×Dic3C2211yescharacteristic
30C2×C4×C5×Dic3C2211yescharacteristic
31C4×C5×Dic6C2218yescomplemented
32C2×C3×#685C2212yes—
33C2×C3×#685C2212yes—
34C3×#4021C2218yescomplemented
35C2×#6668C2212yes—
36C2×#6667C2212yes—
37C2×#6667C2212yes—
38C2×#6665C2212yes—
39C2×Dic3×Dic5C2212yes—
40#91458C2218yes—
41#91455C22116yescomplemented
42Dic3×D20C2218yescomplemented
43#91413C22116yescomplemented
44#91396C22116yescomplemented
45#91374C2218yescomplemented
46#91354C22116yescomplemented
47Dic3×Dic10C2218yescomplemented
48C2×#5681—51nomaximal, Hall
49C2×C4×C5×Q8—31no—
50C2×#4093—32no—
51#65919—316no—
52C2×#4040—32no—
53C2×#4040—32no—
54C2×#4039—32no—
55C2×#4021—31no—
56C2×#4021D311yescharacteristic, complemented
57C2×#4020—31no—
58C2×C4×D20—31no—
59C2×C4×D20—31no—
60C2×C4×Dic10—31no—
61C23×D15C2312yes—
62C23×C3×D5C2312yes—
63C22×C3×C4×C5C2311yescharacteristic
64C22×C3×C4×C5C2311yescharacteristic
65C22×C3×C4×C5C2311yescharacteristic
66C22×Dic15C2312yes—
67C2×D60—24no—
68C2×D60C2314yes—
69C2×C4×D15—44no—
70C2×C4×D15C2318yes—
71C22×C5×Dic3C2311yescharacteristic
72C22×C5×Dic3C2311yescharacteristic
73C22×C5×Dic3C2311yescharacteristic
74C22×C5×Dic3C2311yescharacteristic
75C2×C5×Dic6—24no—
76C2×C5×Dic6C2314yes—
77C22×C3×Dic5C2312yes—
78C2×C3×D20—24no—
79C2×C3×D20C2314yes—
80C2×C3×Dic10—24no—
81C2×C3×Dic10C2314yes—
82C2×#1202—28no—
83C2×#1202—44no—
84C2×#1202C2318yes—
85C2×#1202C2318yes—
86C2×D5×Dic3—44no—
87C2×D5×Dic3C2318yes—
88C3×C42×C5C2314yes—
89#6714C2318yes—
90C4⋊Dic15C2314yes—
91C5×C4⋊Dic3C2314yes—
92C5×Dic3⋊C4C2314yes—
93C5×Dic3⋊C4C2314yes—
94C4×C5×Dic3C2314yes—
95C4×C5×Dic3C2314yes—
96C3×#685C2318yes—
97C3×#685C2318yes—
98#6668C2318yes—
99#6667C2318yes—
100#6667C2318yes—
101#6665C2318yes—
102Dic3×Dic5C2318yes—
103C2×C3×#124—51no—
104C2×#947—51no—
105C2×Q8×Dic3—51no—
106C2×#940—51no—
107C2×D4×Dic3—51no—
108#5681—516no—
109C2×#886—52no—
110C2×#885—52no—
111C2×#883—52no—
112C2×#67⋊C4—52no—
113C2×C4×D12—51no—
114C2×C4×Dic6D511yescharacteristic, complemented
115C22×C5×Q8—31no—
116C22×D20—32no—
117C22×D20—31no—
118C22×D20—32no—
119C22×D20C2×D311yescharacteristic
120C22×C4×D5—32no—
121C22×C4×D5—32no—
122C22×Dic10C2×D311yescharacteristic
123C4×C5×Q8—38no—
124C2×C5×C4⋊C4—31no—
125C2×C5×C4⋊C4—31no—
126C2×C5×C4⋊C4—31no—
127C2×C42×C5—31no—
128C2×C42×C5—31no—
129C2×C42×C5C2×D311yescharacteristic
130#4093—316no—
131C2×#685—32no—
132C2×#685—32no—
133C2×#685—32no—
134C2×#685—32no—
135C2×#685C2×D312yes—
136C2×#685C2×D312yes—
137C2×C4⋊Dic5—31no—
138C2×#683—32no—
139C2×C4×Dic5—32no—
140#4040—316no—
141#4040—316no—
142#4039—316no—
143#4021—38no—
144#4021C2×D318yescomplemented
145#4020—38no—
146C4×D20—38no—
147C4×D20—38no—
148C4×Dic10—38no—
149C2×#559—151noSylow
150C23×C3×C5C2411yescharacteristic
151C22×D15—24no—
152C22×D15—24no—
153C22×D15—24no—
154C22×D15C2414yes—
155C22×C3×D5—24no—
156C22×C3×D5—24no—
157C22×C3×D5—24no—
158C22×C3×D5C2414yes—
159C2×C3×C4×C5—22no—
160C2×C3×C4×C5—22no—
161C2×C3×C4×C5#4114yes—
162C2×C3×C4×C5C2412yes—
163C2×C3×C4×C5C2412yes—
164C2×C3×C4×C5C2412yes—
165C2×Dic15—24no—
166C2×Dic15C2414yes—
167D60—28no—
168C4×D15—48no—
169C2×C5×Dic3—22no—
170C2×C5×Dic3—22no—
171C2×C5×Dic3—22no—
172C2×C5×Dic3#4114yes—
173C2×C5×Dic3C2412yes—
174C2×C5×Dic3C2412yes—
175C2×C5×Dic3C2412yes—
176C2×C5×Dic3C2412yes—
177C5×Dic6—28no—
178C2×C3×Dic5—24no—
179C2×C3×Dic5C2414yes—
180C3×D20—28no—
181C3×Dic10—28no—
182#1202—216no—
183#1202—48no—
184D5×Dic3—48no—
185C22×C3×Q8—51no—
186C22×C3×D4—51no—
187C22×#67—52no—
188C22×#67—52no—
189C23×Dic3—52no—
190C22×D12—51no—
191C22×C4×D3—52no—
192C22×Dic6C2×D511yescharacteristic
193C3×#124—58no—
194C2×C3×C22⋊C4—52no—
195C2×C3×C22⋊C4—52no—
196C2×C3×C42C2×D511yescharacteristic
197C2×#217—52no—
198#947—58no—
199Q8×Dic3—58no—
200#940—58no—
201D4×Dic3—58no—
202C2×(C2×D3)⋊C4—52no—
203C2×(C2×D3)⋊C4—52no—
204C2×(C2×D3)⋊C4—52no—
205C2×C4⋊Dic3—51no—
206C2×C4⋊Dic3—52no—
207C2×C4⋊Dic3C2×D511yescharacteristic
208C2×Dic3⋊C4C2×D511yescharacteristic
209C2×Dic3⋊C4C2×D511yescharacteristic
210C2×C4×Dic3—52no—
211C2×C4×Dic3C2×D511yescharacteristic
212C2×C4×Dic3C2×D511yescharacteristic
213#886—516no—
214#885—516no—
215#883—516no—
216#67⋊C4—516no—
217C4×D12—58no—
218C4×Dic6C2×D518yescomplemented
219C23×D5—32no—
220C23×D5C22×D312yes—
221C2×C5×Q8—34no—
222C2×C5×Q8—64no—
223C22×C4×C5—31no—
224C22×C4×C5—31no—
225C22×C4×C5—31no—
226C22×C4×C5—31no—
227C22×C4×C5C22×D311yescharacteristic
228C22×C4×C5C22×D311yescharacteristic
229C22×C4×C5C22×D311yescharacteristic
230C22×Dic5—32no—
231C22×Dic5C22×D312yes—
232C2×D20—24no—
233C2×D20—38no—
234C2×D20—34no—
235C2×D20—38no—
236C2×D20—68no—
237C2×D20—64no—
238C2×D20—124no—
239C2×D20C22×D314yes—
240C2×C4×D5—38no—
241C2×C4×D5—38no—
242C2×C4×D5—124no—
243C2×C4×D5—124no—
244C2×Dic10—24no—
245C2×Dic10C22×D314yes—
246C5×C4⋊C4—34no—
247C5×C4⋊C4—34no—
248C5×C4⋊C4—34no—
249C42×C5—34no—
250C42×C5—34no—
251C42×C5C22×D314yes—
252#685—38no—
253#685—38no—
254#685—38no—
255#685—38no—
256#685C22×D318yes—
257#685C22×D318yes—
258C4⋊Dic5—34no—
259#683—38no—
260C4×Dic5—38no—
261#559—1516no—
262C2×#127—151no—
263C2×#124—51no—
264C2×#124—151no—
265C2×#124—152no—
266C2×#123—152no—
267C2×#123—152no—
268C2×C4×Q8—31no—
269C2×C4×Q8—151no—
270C2×C4×D4—151no—
271C2×C4×D4—152no—
272C2×C4×D4—151no—
273C22×C3×C5C2×#4112yes—
274C22×C3×C5C2×#4112yes—
275C22×C3×C5H(5,𝔽2)12yes—
276C22×C3×C5C2511yescharacteristic
277C2×D15—24no—
278C2×D15—24no—
279C2×D15—24no—
280C2×D15—48no—
281C2×C3×D5—24no—
282C2×C3×D5—24no—
283C2×C3×D5—24no—
284C2×C3×D5—48no—
285C3×C4×C5—22no—
286C3×C4×C5—22no—
287C3×C4×C5C2×#4114yes—
288Dic15—24no—
289C3×Dic5—24no—
290C5×Dic3—22no—
291C5×Dic3—22no—
292C5×Dic3—22no—
293C5×Dic3C2×#4114yes—
294C24×C3—52no—
295C23×D3—52no—
296C2×C3×Q8—54no—
297C2×C3×Q8—104no—
298C2×C3×D4—54no—
299C2×C3×D4—104no—
300C22×C3×C4—52no—
301C22×C3×C4C22×D511yescharacteristic
302C22×C3×C4C22×D511yescharacteristic
303C22×C3×C4C22×D511yescharacteristic
304C2×#67—58no—
305C2×#67—58no—
306C2×#67—108no—
307C2×#67—204no—
308C22×Dic3—52no—
309C22×Dic3—58no—
310C22×Dic3—204no—
311C22×Dic3C22×D511yescharacteristic
312C22×Dic3C22×D511yescharacteristic
313C22×Dic3C22×D511yescharacteristic
314C22×Dic3C22×D511yescharacteristic
315C2×D12—54no—
316C2×D12—104no—
317C2×C4×D3—58no—
318C2×C4×D3—204no—
319C2×Dic6—24no—
320C2×Dic6C22×D514yes—
321C3×C22⋊C4—58no—
322C3×C22⋊C4—58no—
323C3×C42C22×D514yes—
324#217—58no—
325(C2×D3)⋊C4—58no—
326(C2×D3)⋊C4—58no—
327(C2×D3)⋊C4—58no—
328C4⋊Dic3—54no—
329C4⋊Dic3—58no—
330C4⋊Dic3C22×D514yes—
331Dic3⋊C4C22×D514yes—
332Dic3⋊C4C22×D514yes—
333C4×Dic3—58no—
334C4×Dic3C22×D514yes—
335C4×Dic3C22×D514yes—
336C23×C5C23×D311yescharacteristic
337C22×D5—24no—
338C22×D5—24no—
339C22×D5—24no—
340C22×D5—34no—
341C22×D5—64no—
342C22×D5—64no—
343C22×D5—64no—
344C22×D5C23×D314yes—
345C5×Q8—68no—
346C2×C4×C5—22no—
347C2×C4×C5—22no—
348C2×C4×C5—34no—
349C2×C4×C5—32no—
350C2×C4×C5—32no—
351C2×C4×C5—32no—
352C2×C4×C5—32no—
353C2×C4×C5—62no—
354C2×C4×C5—62no—
355C2×C4×C5—62no—
356C2×C4×C5#23714yes—
357C2×C4×C5C23×D312yes—
358C2×C4×C5C23×D312yes—
359C2×C4×C5C23×D312yes—
360C2×Dic5—24no—
361C2×Dic5—34no—
362C2×Dic5—64no—
363C2×Dic5C23×D314yes—
364D20—28no—
365D20—616no—
366D20—68no—
367D20—128no—
368C4×D5—128no—
369C4×D5—128no—
370Dic10—28no—
371C22×Q8—31no—
372C22×Q8—51no—
373C22×D4—51no—
374C22×D4—152no—
375C22×D4—152no—
376C22×D4—151no—
377C23×C4—152no—
378C23×C4—152no—
379#127—158no—
380#124—58no—
381#124—158no—
382#124—1516no—
383#123—1516no—
384#123—1516no—
385C4×Q8—38no—
386C4×Q8—158no—
387C4×D4—158no—
388C4×D4—1516no—
389C4×D4—158no—
390C2×C4⋊C4—31no—
391C2×C4⋊C4—31no—
392C2×C4⋊C4—31no—
393C2×C4⋊C4—152no—
394C2×C4⋊C4—151no—
395C2×C22⋊C4—52no—
396C2×C22⋊C4—52no—
397C2×C22⋊C4—152no—
398C2×C22⋊C4—152no—
399C2×C22⋊C4—152no—
400C2×C22⋊C4—152no—
401C2×C42—31no—
402C2×C42—31no—
403C2×C42—152no—
404C2×C42D3×D511yescharacteristic
405C2×C3×C5#55914yescomplemented
406C2×C3×C5C22×#4111yescharacteristic
407C2×C3×C5C22×#4111yescharacteristic
408C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
409D15—44no—
410C3×D5—44no—
411C23×C3—54no—
412C23×C3—104no—
413C23×C3—104no—
414C23×C3—104no—
415C23×C3C23×D511yescharacteristic
416C22×D3—54no—
417C22×D3—104no—
418C22×D3—104no—
419C22×D3—104no—
420C3×Q8—108no—
421C3×D4—108no—
422C2×C3×C4—22no—
423C2×C3×C4—22no—
424C2×C3×C4—54no—
425C2×C3×C4—104no—
426C2×C3×C4#71314yes—
427C2×C3×C4C23×D512yes—
428C2×C3×C4C23×D512yes—
429C2×C3×C4C23×D512yes—
430#67—1016no—
431#67—208no—
432C2×Dic3—22no—
433C2×Dic3—22no—
434C2×Dic3—22no—
435C2×Dic3—54no—
436C2×Dic3—104no—
437C2×Dic3—208no—
438C2×Dic3#70914yes—
439C2×Dic3C23×D512yes—
440C2×Dic3C23×D512yes—
441C2×Dic3C23×D512yes—
442C2×Dic3C23×D512yes—
443D12—108no—
444C4×D3—208no—
445Dic6—28no—
446C22×C5C2×#23712yes—
447C22×C5D3×#4112yes—
448C22×C5#100912yes—
449C22×C5C24×D311yescharacteristic
450C2×D5—24no—
451C2×D5—24no—
452C2×D5—24no—
453C2×D5—48no—
454C2×D5—64no—
455C2×D5—64no—
456C2×D5—64no—
457C2×D5—128no—
458C4×C5—22no—
459C4×C5—22no—
460C4×C5—34no—
461C4×C5—62no—
462C4×C5—62no—
463C4×C5—62no—
464C4×C5C2×#23714yes—
465Dic5—24no—
466Dic5—64no—
467C24—52no—
468C24—152no—
469C2×Q8—34no—
470C2×Q8—54no—
471C2×Q8—64no—
472C2×Q8—104no—
473C2×D4—54no—
474C2×D4—104no—
475C2×D4—158no—
476C2×D4—158no—
477C2×D4—154no—
478C2×D4—308no—
479C2×D4—304no—
480C2×D4—604no—
481C22×C4—31no—
482C22×C4—31no—
483C22×C4—31no—
484C22×C4—31no—
485C22×C4—52no—
486C22×C4—158no—
487C22×C4—152no—
488C22×C4—158no—
489C22×C4—604no—
490C22×C4—604no—
491C22×C4C2×D3×D511yescharacteristic
492C22×C4C2×D3×D511yescharacteristic
493C22×C4C2×D3×D511yescharacteristic
494C4⋊C4—34no—
495C4⋊C4—34no—
496C4⋊C4—34no—
497C4⋊C4—158no—
498C4⋊C4—154no—
499C22⋊C4—58no—
500C22⋊C4—58no—
501C22⋊C4—158no—
502C22⋊C4—158no—
503C22⋊C4—158no—
504C22⋊C4—158no—
505C42—34no—
506C42—34no—
507C42—158no—
508C42C2×D3×D514yes—
509C3×C5C2×#55911yescharacteristic, complemented, Hall
510C22×C3—104no—
511C22×C3—104no—
512C22×C3—104no—
513C22×C3—208no—
514C22×C3C2×#70912yes—
515C22×C3C2×#71312yes—
516C22×C3#414912yes—
517C22×C3C24×D511yescharacteristic
518C2×D3—104no—
519C2×D3—104no—
520C2×D3—104no—
521C2×D3—208no—
522C3×C4—22no—
523C3×C4—22no—
524C3×C4—104no—
525C3×C4C2×#71314yes—
526Dic3—22no—
527Dic3—22no—
528Dic3—22no—
529Dic3—104no—
530Dic3C2×#70914yes—
531C2×C5#568114yescomplemented
532C2×C5C22×#23711yescharacteristic
533C2×C5C2×D3×#4111yescharacteristic
534C2×C5C2×#100911yescharacteristic
535D5—44no—
536D5—124no—
537C23—54no—
538C23—104no—
539C23—104no—
540C23—104no—
541C23—154no—
542C23—304no—
543C23—304no—
544C23—304no—
545C23C22×D3×D511yescharacteristic
546Q8—68no—
547Q8—108no—
548D4—108no—
549D4—3016no—
550D4—308no—
551D4—608no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—34no—
555C2×C4—32no—
556C2×C4—32no—
557C2×C4—32no—
558C2×C4—32no—
559C2×C4—54no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—104no—
564C2×C4—154no—
565C2×C4—304no—
566C2×C4—608no—
567C2×C4—608no—
568C2×C4#676614yes—
569C2×C4C22×D3×D512yes—
570C2×C4C22×D3×D512yes—
571C2×C4C22×D3×D512yes—
572C2×C3—204no—
573C2×C3#6591914yescomplemented
574C2×C3C22×#70911yescharacteristic
575C2×C3C22×#71311yescharacteristic
576C2×C3C2×#414911yescharacteristic
577D3—204no—
578C5C2×#568111yescharacteristic, complemented, Sylow
579C22—104no—
580C22—104no—
581C22—104no—
582C22—208no—
583C22—304no—
584C22—304no—
585C22—304no—
586C22—608no—
587C22C2×#676612yes—
588C22D3×#70912yes—
589C22#9201912yes—
590C22C23×D3×D511yescharacteristic
591C4—22no—
592C4—22no—
593C4—34no—
594C4—62no—
595C4—62no—
596C4—62no—
597C4—104no—
598C4—304no—
599C4C2×#676614yes—
600C3C2×#6591911yescharacteristic, complemented, Sylow
601C2—204no—
602C2—604no—
603C2#1175260614yesdirect product factor
604C2C22×#676611yescharacteristic
605C2C2×D3×#70911yescharacteristic
606C2C2×#9201911yescharacteristic
subgroupG/H
centerC23C22×D3×D5
commutator, cosocleC22×C3×C5C25
FrattiniC22C23×D3×D5
FittingC2×C3×C42×C5C22
socleC23×C3×C5C24
prime radicalC2×#91458C2
hypercenterC2×C42D3×D5
hypocenterC3×C5C2×#559
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×D60C2semidirect
2. C2×C4×C5×Dic6C2semidirect
3. C2×C3×#4021C2semidirect
5. C2×#91455C2semidirect
6. C2×Dic3×D20C2semidirect
7. C2×#91413C2semidirect
8. C2×#91396C2semidirect
9. C2×#91374C2semidirect
10. C2×#91354C2semidirect
11. C2×Dic3×Dic10C2semidirect
12. #11752606C2direct (N × H)
25. C4×D60C22semidirect
31. C4×C5×Dic6C22semidirect
34. C3×#4021C22semidirect
41. #91455C22semidirect
42. Dic3×D20C22semidirect
43. #91413C22semidirect
44. #91396C22semidirect
45. #91374C22semidirect
46. #91354C22semidirect
47. Dic3×Dic10C22semidirect
56. C2×#4021D3semidirect
114. C2×C4×Dic6D5semidirect
144. #4021C2×D3semidirect
218. C4×Dic6C2×D5semidirect
405. C2×C3×C5#559semidirect
509. C3×C5C2×#559semidirect
531. C2×C5#5681semidirect
573. C2×C3#65919semidirect
578. C5C2×#5681semidirect
600. C3C2×#65919semidirect
603. C2#11752606direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×D602
C2×C4×C5×Dic62
C2×C3×#40212
C2×#914582
C2×#914552
C2×Dic3×D202
C2×#914132
C2×#913962
C2×#913742
C2×#913542
C2×Dic3×Dic102
#117526062
C2×#659193
C2×#56815