Finite Group Databasebeta

← C2×#11754601 #14481149

Subgroups — C2×#11754601 #14481149

Order 1920.

2520 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91815C211yesmaximal, characteristic, complemented
2C2×C5×#930C211yesmaximal, characteristic, complemented
3C2×C3×#4076C211yesmaximal, characteristic, complemented
4C2×#91565C212yesmaximal, complemented
5C2×#91549C212yesmaximal, complemented
6#11754601C2116yesmaximal, direct product factor
7C2×#91463C211yesmaximal, characteristic, complemented
8C2×#91457C212yesmaximal, complemented
9C2×#91435C211yesmaximal, characteristic, complemented
10C2×#91434C211yesmaximal, characteristic, complemented
11C2×#91414C212yesmaximal, complemented
12C2×#91384C211yesmaximal, characteristic, complemented
13C2×#65953—31nomaximal, Hall
14C23×C3×C4×C5C2211yescharacteristic
15C22×C5×#67C2212yes—
16C22×C5×D12C2211yescharacteristic
17C22×C3×C22⋊D5C2212yes—
18C22×C3×D20C2211yescharacteristic
19C22×#1201C2212yes—
20C22×#1201C2212yes—
21C22×D3×Dic5C2212yes—
22C22×D5×Dic3C2212yes—
23C2×#6719C2211yescharacteristic
24C2×#6719C2211yescharacteristic
25#91815C2218yescomplemented
26C2×C4⋊Dic15C2211yescharacteristic
27C2×C4⋊Dic15C2211yescharacteristic
28C2×C4×Dic15C2211yescharacteristic
29C2×C4×Dic15C2211yescharacteristic
30C5×#930C2218yescomplemented
31C2×C5×(C2×D3)⋊C4C2212yes—
32C2×C5×C4⋊Dic3C2211yescharacteristic
33C3×#4076C2218yescomplemented
34C2×C3×#685C2212yes—
35C2×C3×C4⋊Dic5C2211yescharacteristic
36#91565C22116yescomplemented
37#91549C22116yescomplemented
38C2×Dic3⋊Dic5C2212yes—
39C2×#6668C2212yes—
40C2×(C2×D3)⋊Dic5C2212yes—
41C2×#6665C2212yes—
42#91463C2218yescomplemented
43#91457C22116yescomplemented
44#91435C2218yescomplemented
45#91434C2218yescomplemented
46#91414C22116yescomplemented
47#91384C2218yescomplemented
48C2×#5619—51nomaximal, Hall
49C2×C5×#121—31no—
50C2×#4086—31no—
51C2×#4085—32no—
52C2×#4084—32no—
53C2×D4×Dic5—31no—
54C2×#4076D311yescharacteristic, complemented
55C2×#4072—31no—
56#65953—316no—
57C2×#4039—31no—
58C2×#4036—31no—
59C2×#4032—32no—
60C2×#4029—32no—
61C24×C3×C5C2311yescharacteristic
62C23×C5×D3C2312yes—
63C23×C3×D5C2312yes—
64C22×C3×C4×C5—24no—
65C22×C3×C4×C5C2311yescharacteristic
66C22×C3×C4×C5C2314yes—
67C22×C3×C4×C5C2311yescharacteristic
68C22×Dic15C2311yescharacteristic
69C22×Dic15C2311yescharacteristic
70C22×Dic15C2311yescharacteristic
71C22×Dic15C2311yescharacteristic
72C2×C5×#67—44no—
73C2×C5×#67C2318yes—
74C22×C5×Dic3C2312yes—
75C2×C5×D12—24no—
76C2×C5×D12C2314yes—
77C2×C3×C22⋊D5—44no—
78C2×C3×C22⋊D5C2318yes—
79C22×C3×Dic5C2312yes—
80C2×C3×D20—24no—
81C2×C3×D20C2314yes—
82C2×#1201—44no—
83C2×#1201—44no—
84C2×#1201C2318yes—
85C2×#1201C2318yes—
86C2×D3×Dic5—44no—
87C2×D3×Dic5C2318yes—
88C2×D5×Dic3—44no—
89C2×D5×Dic3C2318yes—
90#6719C2314yes—
91#6719C2314yes—
92C4⋊Dic15C2314yes—
93C4⋊Dic15C2314yes—
94C4×Dic15C2314yes—
95C4×Dic15C2314yes—
96C5×(C2×D3)⋊C4C2318yes—
97C5×C4⋊Dic3C2314yes—
98C3×#685C2318yes—
99C3×C4⋊Dic5C2314yes—
100Dic3⋊Dic5C2318yes—
101#6668C2318yes—
102(C2×D3)⋊Dic5C2318yes—
103#6665C2318yes—
104C2×C3×#121—51no—
105C2×#940—51no—
106C2×#939—52no—
107C2×#938—52no—
108C2×D4×Dic3—51no—
109C2×#930D511yescharacteristic, complemented
110C2×#926—51no—
111#5619—516no—
112C2×#893—51no—
113C2×#890—51no—
114C2×#886—52no—
115C2×#883—52no—
116C22×C5×D4—32no—
117C22×C5×D4—31no—
118C23×C4×C5C2×D311yescharacteristic
119C22×C22⋊D5—32no—
120C22×C22⋊D5—32no—
121C22×C22⋊D5C2×D312yes—
122C23×Dic5—32no—
123C22×D20C2×D311yescharacteristic
124C22×C4×D5—32no—
125C5×#121—38no—
126C2×C5×C4⋊C4—31no—
127C2×C5×C22⋊C4—32no—
128C2×C22⋊Dic5—31no—
129C2×C22⋊Dic5—31no—
130C2×C22⋊Dic5—32no—
131#4086—38no—
132#4085—316no—
133#4084—316no—
134D4×Dic5—38no—
135#4076C2×D318yescomplemented
136C2×#685—32no—
137C2×#685C2×D312yes—
138#4072—38no—
139C2×C4⋊Dic5—31no—
140C2×C4⋊Dic5—31no—
141C2×C4⋊Dic5—32no—
142C2×C4⋊Dic5C2×D311yescharacteristic
143C2×#683—32no—
144C2×C4×Dic5—31no—
145C2×C4×Dic5—31no—
146#4039—38no—
147#4036—38no—
148#4032—316no—
149#4029—316no—
150C2×#540—151noSylow
151C23×C3×C5—22no—
152C23×C3×C5—22no—
153C23×C3×C5—22no—
154C23×C3×C5C2412yes—
155C23×C3×C5C2411yescharacteristic
156C22×C5×D3—24no—
157C22×C5×D3—24no—
158C22×C5×D3—24no—
159C22×C5×D3C2414yes—
160C22×C3×D5—24no—
161C22×C3×D5—24no—
162C22×C3×D5—24no—
163C22×C3×D5C2414yes—
164C2×C3×C4×C5—22no—
165C2×C3×C4×C5—28no—
166C2×C3×C4×C5#4114yes—
167C2×C3×C4×C5C2412yes—
168C2×C3×C4×C5C2412yes—
169C2×Dic15—22no—
170C2×Dic15—22no—
171C2×Dic15—22no—
172C2×Dic15—22no—
173C2×Dic15C2412yes—
174C2×Dic15C2412yes—
175C2×Dic15C2412yes—
176C2×Dic15C2412yes—
177C5×#67—48no—
178C2×C5×Dic3—24no—
179C2×C5×Dic3C2414yes—
180C5×D12—28no—
181C3×C22⋊D5—48no—
182C2×C3×Dic5—24no—
183C2×C3×Dic5C2414yes—
184C3×D20—28no—
185#1201—48no—
186#1201—48no—
187D3×Dic5—48no—
188D5×Dic3—48no—
189C22×C3×D4—52no—
190C22×C3×D4—51no—
191C23×C3×C4C2×D511yescharacteristic
192C22×#67—52no—
193C22×#67—52no—
194C22×#67C2×D512yes—
195C23×Dic3—52no—
196C22×D12C2×D511yescharacteristic
197C22×C4×D3—52no—
198C3×#121—58no—
199C2×C3×C4⋊C4—51no—
200C2×C3×C22⋊C4—52no—
201C2×#217—51no—
202C2×#217—51no—
203C2×#217—52no—
204#940—58no—
205#939—516no—
206#938—516no—
207D4×Dic3—58no—
208#930C2×D518yescomplemented
209C2×(C2×D3)⋊C4—52no—
210C2×(C2×D3)⋊C4C2×D512yes—
211#926—58no—
212C2×C4⋊Dic3—51no—
213C2×C4⋊Dic3—51no—
214C2×C4⋊Dic3—52no—
215C2×C4⋊Dic3C2×D511yescharacteristic
216C2×Dic3⋊C4—52no—
217C2×C4×Dic3—51no—
218C2×C4×Dic3—51no—
219#893—58no—
220#890—58no—
221#886—516no—
222#883—516no—
223C24×C5—32no—
224C24×C5C22×D311yescharacteristic
225C23×D5C22×D312yes—
226C2×C5×D4—38no—
227C2×C5×D4—34no—
228C2×C5×D4—64no—
229C2×C5×D4—124no—
230C22×C4×C5—24no—
231C22×C4×C5—32no—
232C22×C4×C5C22×D311yescharacteristic
233C22×C4×C5C22×D314yes—
234C22×C4×C5C22×D311yescharacteristic
235C2×C22⋊D5—38no—
236C2×C22⋊D5—38no—
237C2×C22⋊D5—44no—
238C2×C22⋊D5—124no—
239C2×C22⋊D5—124no—
240C2×C22⋊D5C22×D318yes—
241C22×Dic5—31no—
242C22×Dic5—31no—
243C22×Dic5—31no—
244C22×Dic5—31no—
245C22×Dic5—38no—
246C22×Dic5—124no—
247C22×Dic5C22×D312yes—
248C2×D20—24no—
249C2×D20C22×D314yes—
250C2×C4×D5—38no—
251C2×C4×D5—124no—
252C5×C4⋊C4—34no—
253C5×C22⋊C4—38no—
254C22⋊Dic5—34no—
255C22⋊Dic5—34no—
256C22⋊Dic5—38no—
257#685—38no—
258#685C22×D318yes—
259C4⋊Dic5—34no—
260C4⋊Dic5—34no—
261C4⋊Dic5—38no—
262C4⋊Dic5C22×D314yes—
263#683—38no—
264C4×Dic5—34no—
265C4×Dic5—34no—
266#540—1516no—
267C2×#127—151no—
268C2×#125—151no—
269C2×#123—152no—
270C2×#123—152no—
271C2×#121—31no—
272C2×#121—51no—
273C2×#121—152no—
274C2×#121—152no—
275C2×C4×D4—151no—
276C2×C4×D4—151no—
277C2×#117—151no—
278C22×C3×C5—22no—
279C22×C3×C5—22no—
280C22×C3×C5—22no—
281C22×C3×C5—44no—
282C22×C3×C5C2×#4112yes—
283C22×C3×C5H(5,𝔽2)12yes—
284C22×C3×C5H(5,𝔽2)12yes—
285C22×C3×C5C2511yescharacteristic
286C2×C5×D3—24no—
287C2×C5×D3—24no—
288C2×C5×D3—24no—
289C2×C5×D3—48no—
290C2×C3×D5—24no—
291C2×C3×D5—24no—
292C2×C3×D5—24no—
293C2×C3×D5—48no—
294C3×C4×C5—22no—
295C3×C4×C5C2×#4114yes—
296Dic15—22no—
297Dic15—22no—
298Dic15—22no—
299Dic15—22no—
300C3×Dic5—24no—
301C5×Dic3—24no—
302C24×C3—52no—
303C24×C3C22×D511yescharacteristic
304C23×D3C22×D512yes—
305C2×C3×D4—58no—
306C2×C3×D4—54no—
307C2×C3×D4—104no—
308C2×C3×D4—204no—
309C22×C3×C4—24no—
310C22×C3×C4—52no—
311C22×C3×C4C22×D511yescharacteristic
312C22×C3×C4C22×D514yes—
313C22×C3×C4C22×D511yescharacteristic
314C2×#67—44no—
315C2×#67—58no—
316C2×#67—58no—
317C2×#67—204no—
318C2×#67—204no—
319C2×#67C22×D518yes—
320C22×Dic3—51no—
321C22×Dic3—51no—
322C22×Dic3—51no—
323C22×Dic3—51no—
324C22×Dic3—58no—
325C22×Dic3—204no—
326C22×Dic3C22×D512yes—
327C2×D12—24no—
328C2×D12C22×D514yes—
329C2×C4×D3—58no—
330C2×C4×D3—204no—
331C3×C4⋊C4—54no—
332C3×C22⋊C4—58no—
333#217—54no—
334#217—54no—
335#217—58no—
336(C2×D3)⋊C4—58no—
337(C2×D3)⋊C4C22×D518yes—
338C4⋊Dic3—54no—
339C4⋊Dic3—54no—
340C4⋊Dic3—58no—
341C4⋊Dic3C22×D514yes—
342Dic3⋊C4—58no—
343C4×Dic3—54no—
344C4×Dic3—54no—
345C23×C5—22no—
346C23×C5—22no—
347C23×C5—22no—
348C23×C5—34no—
349C23×C5—64no—
350C23×C5—64no—
351C23×C5—64no—
352C23×C5C23×D312yes—
353C23×C5C23×D311yescharacteristic
354C22×D5—24no—
355C22×D5—24no—
356C22×D5—24no—
357C22×D5C23×D314yes—
358C5×D4—68no—
359C5×D4—128no—
360C2×C4×C5—22no—
361C2×C4×C5—28no—
362C2×C4×C5—34no—
363C2×C4×C5—64no—
364C2×C4×C5#23714yes—
365C2×C4×C5C23×D312yes—
366C2×C4×C5C23×D312yes—
367C22⋊D5—48no—
368C22⋊D5—128no—
369C22⋊D5—128no—
370C2×Dic5—24no—
371C2×Dic5—32no—
372C2×Dic5—32no—
373C2×Dic5—32no—
374C2×Dic5—32no—
375C2×Dic5—62no—
376C2×Dic5—62no—
377C2×Dic5—62no—
378C2×Dic5—62no—
379C2×Dic5—128no—
380C2×Dic5C23×D314yes—
381D20—28no—
382C4×D5—128no—
383C22×D4—32no—
384C22×D4—31no—
385C22×D4—52no—
386C22×D4—51no—
387C22×D4—152no—
388C22×D4—152no—
389C23×C4—152no—
390C23×C4—152no—
391C23×C4D3×D511yescharacteristic
392#127—158no—
393#125—158no—
394#123—1516no—
395#123—1516no—
396#121—38no—
397#121—58no—
398#121—1516no—
399#121—1516no—
400C4×D4—158no—
401C4×D4—158no—
402#117—158no—
403C2×C4⋊C4—31no—
404C2×C4⋊C4—51no—
405C2×C4⋊C4—152no—
406C2×C4⋊C4—152no—
407C2×C4⋊C4—151no—
408C2×C4⋊C4—151no—
409C2×C22⋊C4—32no—
410C2×C22⋊C4—52no—
411C2×C22⋊C4—152no—
412C2×C22⋊C4—152no—
413C2×C22⋊C4—151no—
414C2×C22⋊C4—151no—
415C2×C42—151no—
416C2×C42—151no—
417C2×C3×C5—42no—
418C2×C3×C5#54014yescomplemented
419C2×C3×C5C22×#4111yescharacteristic
420C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
421C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
422C3×D5—44no—
423C5×D3—44no—
424C23×C3—22no—
425C23×C3—22no—
426C23×C3—22no—
427C23×C3—54no—
428C23×C3—104no—
429C23×C3—104no—
430C23×C3—104no—
431C23×C3C23×D512yes—
432C23×C3C23×D511yescharacteristic
433C22×D3—24no—
434C22×D3—24no—
435C22×D3—24no—
436C22×D3C23×D514yes—
437C3×D4—108no—
438C3×D4—208no—
439C2×C3×C4—22no—
440C2×C3×C4—28no—
441C2×C3×C4—54no—
442C2×C3×C4—104no—
443C2×C3×C4#71114yes—
444C2×C3×C4C23×D512yes—
445C2×C3×C4C23×D512yes—
446#67—48no—
447#67—208no—
448#67—208no—
449C2×Dic3—24no—
450C2×Dic3—52no—
451C2×Dic3—52no—
452C2×Dic3—52no—
453C2×Dic3—52no—
454C2×Dic3—102no—
455C2×Dic3—102no—
456C2×Dic3—102no—
457C2×Dic3—102no—
458C2×Dic3—208no—
459C2×Dic3C23×D514yes—
460D12—28no—
461C4×D3—208no—
462C22×C5—22no—
463C22×C5—22no—
464C22×C5—22no—
465C22×C5—44no—
466C22×C5—64no—
467C22×C5—64no—
468C22×C5—64no—
469C22×C5—128no—
470C22×C5C2×#23712yes—
471C22×C5#100412yes—
472C22×C5#100912yes—
473C22×C5C24×D311yescharacteristic
474C2×D5—24no—
475C2×D5—24no—
476C2×D5—24no—
477C2×D5—48no—
478C4×C5—22no—
479C4×C5—64no—
480C4×C5C2×#23714yes—
481Dic5—24no—
482Dic5—62no—
483Dic5—62no—
484Dic5—62no—
485Dic5—62no—
486C24—32no—
487C24—52no—
488C24C2×D3×D511yescharacteristic
489C2×D4—38no—
490C2×D4—34no—
491C2×D4—58no—
492C2×D4—54no—
493C2×D4—64no—
494C2×D4—104no—
495C2×D4—124no—
496C2×D4—158no—
497C2×D4—158no—
498C2×D4—204no—
499C2×D4—604no—
500C2×D4—604no—
501C22×C4—24no—
502C22×C4—32no—
503C22×C4—52no—
504C22×C4—158no—
505C22×C4—158no—
506C22×C4—151no—
507C22×C4—151no—
508C22×C4—151no—
509C22×C4—151no—
510C22×C4—604no—
511C22×C4—604no—
512C22×C4C2×D3×D514yes—
513C22×C4C2×D3×D511yescharacteristic
514C22×C4C2×D3×D511yescharacteristic
515C4⋊C4—34no—
516C4⋊C4—54no—
517C4⋊C4—158no—
518C4⋊C4—158no—
519C4⋊C4—154no—
520C4⋊C4—154no—
521C22⋊C4—38no—
522C22⋊C4—58no—
523C22⋊C4—158no—
524C22⋊C4—158no—
525C22⋊C4—154no—
526C22⋊C4—154no—
527C42—154no—
528C42—154no—
529C3×C5C2×#54011yescharacteristic, complemented, Hall
530C22×C3—22no—
531C22×C3—22no—
532C22×C3—22no—
533C22×C3—44no—
534C22×C3—104no—
535C22×C3—104no—
536C22×C3—104no—
537C22×C3—208no—
538C22×C3C2×#71112yes—
539C22×C3#414412yes—
540C22×C3#414912yes—
541C22×C3C24×D511yescharacteristic
542C2×D3—24no—
543C2×D3—24no—
544C2×D3—24no—
545C2×D3—48no—
546C3×C4—22no—
547C3×C4—104no—
548C3×C4C2×#71114yes—
549Dic3—24no—
550Dic3—102no—
551Dic3—102no—
552Dic3—102no—
553Dic3—102no—
554C2×C5—42no—
555C2×C5—124no—
556C2×C5#561914yescomplemented
557C2×C5C22×#23711yescharacteristic
558C2×C5C2×#100411yescharacteristic
559C2×C5C2×#100911yescharacteristic
560D5—44no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—34no—
565C23—54no—
566C23—64no—
567C23—64no—
568C23—64no—
569C23—104no—
570C23—104no—
571C23—104no—
572C23C22×D3×D512yes—
573C23C22×D3×D511yescharacteristic
574D4—68no—
575D4—108no—
576D4—128no—
577D4—208no—
578D4—608no—
579D4—608no—
580C2×C4—22no—
581C2×C4—28no—
582C2×C4—34no—
583C2×C4—54no—
584C2×C4—64no—
585C2×C4—104no—
586C2×C4—152no—
587C2×C4—152no—
588C2×C4—152no—
589C2×C4—152no—
590C2×C4—302no—
591C2×C4—302no—
592C2×C4—302no—
593C2×C4—302no—
594C2×C4—608no—
595C2×C4—608no—
596C2×C4#678014yes—
597C2×C4C22×D3×D512yes—
598C2×C4C22×D3×D512yes—
599C2×C3—42no—
600C2×C3—204no—
601C2×C3#6595314yescomplemented
602C2×C3C22×#71111yescharacteristic
603C2×C3C2×#414411yescharacteristic
604C2×C3C2×#414911yescharacteristic
605D3—44no—
606C5C2×#561911yescharacteristic, complemented, Sylow
607C22—22no—
608C22—22no—
609C22—22no—
610C22—44no—
611C22—64no—
612C22—64no—
613C22—64no—
614C22—104no—
615C22—104no—
616C22—104no—
617C22—128no—
618C22—208no—
619C22#9201712yes—
620C22#9201812yes—
621C22C2×#678012yes—
622C22C23×D3×D511yescharacteristic
623C4—22no—
624C4—64no—
625C4—104no—
626C4—302no—
627C4—302no—
628C4—302no—
629C4—302no—
630C4C2×#678014yes—
631C3C2×#6595311yescharacteristic, complemented, Sylow
632C2—42no—
633C2—124no—
634C2—204no—
635C2#1175460114yesdirect product factor
636C2C2×#9201711yescharacteristic
637C2C2×#9201811yescharacteristic
638C2C22×#678011yescharacteristic
subgroupG/H
centerC23C22×D3×D5
commutator, cosocleC22×C3×C5C25
FrattiniC22C23×D3×D5
FittingC23×C3×C4×C5C22
socleC23×C3×C5C24
hypercenterC23×C4D3×D5
hypocenterC3×C5C2×#540
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×#91815C2semidirect
2. C2×C5×#930C2semidirect
3. C2×C3×#4076C2semidirect
4. C2×#91565C2semidirect
5. C2×#91549C2semidirect
6. #11754601C2direct (N × H)
7. C2×#91463C2semidirect
8. C2×#91457C2semidirect
9. C2×#91435C2semidirect
10. C2×#91434C2semidirect
11. C2×#91414C2semidirect
12. C2×#91384C2semidirect
25. #91815C22semidirect
30. C5×#930C22semidirect
33. C3×#4076C22semidirect
36. #91565C22semidirect
37. #91549C22semidirect
42. #91463C22semidirect
43. #91457C22semidirect
44. #91435C22semidirect
45. #91434C22semidirect
46. #91414C22semidirect
47. #91384C22semidirect
54. C2×#4076D3semidirect
109. C2×#930D5semidirect
135. #4076C2×D3semidirect
208. #930C2×D5semidirect
418. C2×C3×C5#540semidirect
529. C3×C5C2×#540semidirect
556. C2×C5#5619semidirect
601. C2×C3#65953semidirect
606. C5C2×#5619semidirect
631. C3C2×#65953semidirect
635. C2#11754601direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#918152
C2×C5×#9302
C2×C3×#40762
C2×#915652
C2×#915492
#117546012
C2×#914632
C2×#914572
C2×#914352
C2×#914342
C2×#914142
C2×#913842
C2×#659533
C2×#56195