Finite Group Databasebeta

← C2×#11756270 #14483792

Subgroups — C2×#11756270 #14483792

Order 1920.

2400 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C5×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×C4×#1220C211yesmaximal, characteristic
3C2×C4×#1220C212yesmaximal
4C2×#91813C212yesmaximal, complemented
5C2×#91813C211yesmaximal, characteristic, complemented
6#11756270C2116yesmaximal, direct product factor
7C2×#91786C211yesmaximal, characteristic, complemented
8C2×#91785C212yesmaximal, complemented
9C2×C4⋊C4×D15C211yesmaximal, characteristic, complemented
10C2×#91778C212yesmaximal, complemented
11C2×#91777C211yesmaximal, characteristic, complemented
12C2×#91776C211yesmaximal, characteristic, complemented
13C2×#65845—31nomaximal, Hall
14C23×C3×C4×C5C2211yescharacteristic
15C23×C3×C4×C5C2212yes—
16C22×#1220C2211yescharacteristic
17C22×C4×D15C2211yescharacteristic
18C22×C4×D15C2212yes—
19C22×Dic30C2211yescharacteristic
20C22×Dic30C2212yes—
21C2×C3×C5×C4⋊C4C2211yescharacteristic
22C2×C3×C5×C4⋊C4C2211yescharacteristic
23C2×C3×C5×C4⋊C4C2212yes—
24C2×C3×C5×C4⋊C4C2218yescomplemented
25C2×#6719C2212yes—
26C2×#6719C2211yescharacteristic
27C4×#1220C2218yes—
28C4×#1220C22116yes—
29C2×#6714C2211yescharacteristic
30C2×#6714C2212yes—
31C2×C4⋊Dic15C2211yescharacteristic
32C2×C4⋊Dic15C2212yes—
33#91813C22116yescomplemented
34#91813C2218yescomplemented
35C2×#6712C2212yes—
36C2×#6712C2211yescharacteristic
37C2×#6712C2212yes—
38C2×#6712C2212yes—
39C2×#6712C2212yes—
40C2×C4×Dic15C2211yescharacteristic
41C2×C4×Dic15C2212yes—
42#91786C2218yescomplemented
43#91785C22116yescomplemented
44C4⋊C4×D15C2218yescomplemented
45#91778C22116yescomplemented
46#91777C2218yescomplemented
47#91776C2218yescomplemented
48C2×#5511—51nomaximal, Hall
49C22×C5×C4⋊C4D311yescharacteristic, complemented
50C2×C4×C22⋊D5—31no—
51C2×C4×C22⋊D5—32no—
52C2×#4070—31no—
53C2×#4070—32no—
54#65845—316no—
55C2×#4043—31no—
56C2×#4042—32no—
57C2×D5×C4⋊C4—31no—
58C2×#4035—32no—
59C2×#4034—31no—
60C2×#4033—31no—
61C24×C3×C5C2311yescharacteristic
62C23×D15C2311yescharacteristic
63C22×C3×C4×C5—24no—
64C22×C3×C4×C5—44no—
65C22×C3×C4×C5C2312yes—
66C22×C3×C4×C5C2311yescharacteristic
67C22×C3×C4×C5C2312yes—
68C22×C3×C4×C5C2314yes—
69C22×C3×C4×C5C2318yes—
70C22×C3×C4×C5C2311yescharacteristic
71C2×#1220Q818yescomplemented
72C2×#1220C2314yes—
73C22×Dic15C2312yes—
74C22×Dic15C2311yescharacteristic
75C22×Dic15C2311yescharacteristic
76C22×Dic15C2312yes—
77C22×Dic15C2311yescharacteristic
78C2×C4×D15—24no—
79C2×C4×D15—44no—
80C2×C4×D15C2314yes—
81C2×C4×D15C2318yes—
82C2×Dic30—24no—
83C2×Dic30—44no—
84C2×Dic30C2314yes—
85C2×Dic30C2318yes—
86C3×C5×C4⋊C4C2314yes—
87C3×C5×C4⋊C4C2314yes—
88C3×C5×C4⋊C4C2318yes—
89#6719C2318yes—
90#6719C2314yes—
91#6714C2314yes—
92#6714C2318yes—
93C4⋊Dic15C2314yes—
94C4⋊Dic15C2318yes—
95#6712C2318yes—
96#6712C2318yes—
97#6712C2314yes—
98#6712C2318yes—
99#6712C2318yes—
100C4×Dic15C2314yes—
101C4×Dic15C2318yes—
102C22×C3×C4⋊C4D511yescharacteristic, complemented
103C2×C4×#67—51no—
104C2×C4×#67—52no—
105C2×#924—52no—
106C2×#924—51no—
107#5511—516no—
108C2×#897—51no—
109C2×#896—52no—
110C2×D3×C4⋊C4—51no—
111C2×#889—52no—
112C2×#888—51no—
113C2×#887—51no—
114C23×C4×C5C2×D311yescharacteristic
115C23×C4×C5C2×D312yes—
116C22×C22⋊D5—31no—
117C22×C4×D5—31no—
118C22×C4×D5—32no—
119C22×Dic10—31no—
120C22×Dic10—32no—
121C2×C5×C4⋊C4C2×D311yescharacteristic
122C2×C5×C4⋊C4C2×D311yescharacteristic
123C2×C5×C4⋊C4C2×D312yes—
124C2×C5×C4⋊C4C2×D318yescomplemented
125C2×C22⋊Dic5—31no—
126C2×C22⋊Dic5—32no—
127C4×C22⋊D5—38no—
128C4×C22⋊D5—316no—
129C2×#685—31no—
130C2×#685—32no—
131C2×C4⋊Dic5—31no—
132C2×C4⋊Dic5—32no—
133#4070—316no—
134#4070—38no—
135C2×#683—32no—
136C2×#683—32no—
137C2×#683—31no—
138C2×#683—32no—
139C2×#683—32no—
140C2×C4×Dic5—31no—
141C2×C4×Dic5—32no—
142#4043—38no—
143#4042—316no—
144D5×C4⋊C4—38no—
145#4035—316no—
146#4034—38no—
147#4033—38no—
148C2×#554—151noSylow
149C23×C3×C5—22no—
150C23×C3×C5—22no—
151C23×C3×C5C2×Q814yes—
152C23×C3×C5C2412yes—
153C23×C3×C5C2411yescharacteristic
154C22×D15—22no—
155C22×D15—22no—
156C22×D15C2×Q814yes—
157C22×D15C2412yes—
158C2×C3×C4×C5—22no—
159C2×C3×C4×C5—24no—
160C2×C3×C4×C5—24no—
161C2×C3×C4×C5—28no—
162C2×C3×C4×C5—48no—
163C2×C3×C4×C5#4114yes—
164C2×C3×C4×C5C2414yes—
165C2×C3×C4×C5C2412yes—
166C2×C3×C4×C5C2414yes—
167C2×C3×C4×C5C2412yes—
168#1220C2×Q8116yescomplemented
169C2×Dic15—24no—
170C2×Dic15—22no—
171C2×Dic15—22no—
172C2×Dic15—24no—
173C2×Dic15C2×Q814yes—
174C2×Dic15C2414yes—
175C2×Dic15C2412yes—
176C2×Dic15C2412yes—
177C2×Dic15C2414yes—
178C2×Dic15C2412yes—
179C4×D15—28no—
180C4×D15—48no—
181Dic30—28no—
182Dic30—48no—
183C23×C3×C4C2×D511yescharacteristic
184C23×C3×C4C2×D512yes—
185C22×#67—51no—
186C22×C4×D3—51no—
187C22×C4×D3—52no—
188C22×Dic6—51no—
189C22×Dic6—52no—
190C2×C3×C4⋊C4C2×D511yescharacteristic
191C2×C3×C4⋊C4C2×D511yescharacteristic
192C2×C3×C4⋊C4C2×D512yes—
193C2×C3×C4⋊C4C2×D518yescomplemented
194C2×#217—52no—
195C2×#217—51no—
196C4×#67—58no—
197C4×#67—516no—
198C2×(C2×D3)⋊C4—51no—
199C2×(C2×D3)⋊C4—52no—
200C2×C4⋊Dic3—51no—
201C2×C4⋊Dic3—52no—
202#924—516no—
203#924—58no—
204C2×Dic3⋊C4—52no—
205C2×Dic3⋊C4—52no—
206C2×Dic3⋊C4—51no—
207C2×Dic3⋊C4—52no—
208C2×Dic3⋊C4—52no—
209C2×C4×Dic3—51no—
210C2×C4×Dic3—52no—
211#897—58no—
212#896—516no—
213D3×C4⋊C4—58no—
214#889—516no—
215#888—58no—
216#887—58no—
217C24×C5C22×D311yescharacteristic
218C23×D5—31no—
219C22×C4×C5—24no—
220C22×C4×C5—44no—
221C22×C4×C5C22×D312yes—
222C22×C4×C5C22×D312yes—
223C22×C4×C5C22×D311yescharacteristic
224C22×C4×C5C22×D314yes—
225C22×C4×C5C22×D318yes—
226C22×C4×C5C22×D311yescharacteristic
227C2×C22⋊D5—38no—
228C2×C22⋊D5—34no—
229C22×Dic5—31no—
230C22×Dic5—31no—
231C22×Dic5—32no—
232C22×Dic5—32no—
233C22×Dic5—31no—
234C2×C4×D5—34no—
235C2×C4×D5—38no—
236C2×C4×D5—64no—
237C2×C4×D5—124no—
238C2×Dic10—34no—
239C2×Dic10—38no—
240C2×Dic10—64no—
241C2×Dic10—124no—
242C5×C4⋊C4C22×D314yes—
243C5×C4⋊C4C22×D314yes—
244C5×C4⋊C4C22×D318yes—
245C22⋊Dic5—38no—
246C22⋊Dic5—34no—
247#685—34no—
248#685—38no—
249C4⋊Dic5—34no—
250C4⋊Dic5—38no—
251#683—38no—
252#683—38no—
253#683—34no—
254#683—38no—
255#683—38no—
256C4×Dic5—34no—
257C4×Dic5—38no—
258#554—1516no—
259C2×C4⋊Q8—151no—
260C2×#125—152no—
261C2×C22⋊Q8—152no—
262C2×C22⋊Q8—151no—
263C2×C22⋊Q8—152no—
264C2×C22⋊Q8—151no—
265C2×C4×Q8—151no—
266C2×C4×D4—152no—
267C2×C4×D4—151no—
268C22×C4⋊C4—151no—
269C22×C4⋊C4D1511yescharacteristic, complemented
270C22×C3×C5—28no—
271C22×C3×C5—22no—
272C22×C3×C5—22no—
273C22×C3×C5C22×Q814yes—
274C22×C3×C5C22×Q812yes—
275C22×C3×C5C2×#4112yes—
276C22×C3×C5H(5,𝔽2)12yes—
277C22×C3×C5C2511yescharacteristic
278C2×D15—28no—
279C2×D15—22no—
280C2×D15—22no—
281C2×D15C22×Q814yes—
282C3×C4×C5—22no—
283C3×C4×C5—24no—
284C3×C4×C5—24no—
285C3×C4×C5C2×#4114yes—
286Dic15—24no—
287Dic15—22no—
288Dic15—22no—
289Dic15—24no—
290Dic15C22×Q814yes—
291C24×C3C22×D511yescharacteristic
292C23×D3—51no—
293C22×C3×C4—24no—
294C22×C3×C4—44no—
295C22×C3×C4C22×D512yes—
296C22×C3×C4C22×D511yescharacteristic
297C22×C3×C4C22×D512yes—
298C22×C3×C4C22×D514yes—
299C22×C3×C4C22×D518yes—
300C22×C3×C4C22×D511yescharacteristic
301C2×#67—58no—
302C2×#67—54no—
303C22×Dic3—51no—
304C22×Dic3—52no—
305C22×Dic3—51no—
306C22×Dic3—51no—
307C22×Dic3—52no—
308C2×C4×D3—54no—
309C2×C4×D3—58no—
310C2×C4×D3—104no—
311C2×C4×D3—204no—
312C2×Dic6—54no—
313C2×Dic6—58no—
314C2×Dic6—104no—
315C2×Dic6—204no—
316C3×C4⋊C4C22×D514yes—
317C3×C4⋊C4C22×D514yes—
318C3×C4⋊C4C22×D518yes—
319#217—58no—
320#217—54no—
321(C2×D3)⋊C4—54no—
322(C2×D3)⋊C4—58no—
323C4⋊Dic3—54no—
324C4⋊Dic3—58no—
325Dic3⋊C4—58no—
326Dic3⋊C4—58no—
327Dic3⋊C4—54no—
328Dic3⋊C4—58no—
329Dic3⋊C4—58no—
330C4×Dic3—54no—
331C4×Dic3—58no—
332C23×C5—22no—
333C23×C5—22no—
334C23×C5D3×Q814yes—
335C23×C5C23×D312yes—
336C23×C5C23×D311yescharacteristic
337C22×D5—34no—
338C22×D5—32no—
339C22×D5—62no—
340C22×D5—62no—
341C2×C4×C5—22no—
342C2×C4×C5—24no—
343C2×C4×C5—24no—
344C2×C4×C5—28no—
345C2×C4×C5—48no—
346C2×C4×C5#23514yes—
347C2×C4×C5C23×D314yes—
348C2×C4×C5C23×D312yes—
349C2×C4×C5C23×D314yes—
350C2×C4×C5C23×D312yes—
351C22⋊D5—316no—
352C2×Dic5—34no—
353C2×Dic5—32no—
354C2×Dic5—34no—
355C2×Dic5—32no—
356C2×Dic5—32no—
357C2×Dic5—34no—
358C2×Dic5—64no—
359C2×Dic5—62no—
360C2×Dic5—62no—
361C2×Dic5—64no—
362C4×D5—68no—
363C4×D5—128no—
364Dic10—68no—
365Dic10—128no—
366C22×Q8—151no—
367C22×Q8—152no—
368C22×D4—151no—
369C23×C4—151no—
370C23×C4—152no—
371C23×C4C2×D1511yescharacteristic
372C23×C4C2×D1512yes—
373C4⋊Q8—158no—
374#125—1516no—
375C22⋊Q8—1516no—
376C22⋊Q8—158no—
377C22⋊Q8—1516no—
378C22⋊Q8—158no—
379C4×Q8—158no—
380C4×D4—1516no—
381C4×D4—158no—
382C2×C4⋊C4—152no—
383C2×C4⋊C4—151no—
384C2×C4⋊C4—151no—
385C2×C4⋊C4—152no—
386C2×C4⋊C4—152no—
387C2×C4⋊C4—152no—
388C2×C4⋊C4—152no—
389C2×C4⋊C4—158no—
390C2×C4⋊C4C2×D1511yescharacteristic
391C2×C4⋊C4C2×D1512yes—
392C2×C4⋊C4C2×D1511yescharacteristic
393C2×C4⋊C4C2×D1518yescomplemented
394C2×C22⋊C4—152no—
395C2×C22⋊C4—151no—
396C2×C22⋊C4—152no—
397C2×C22⋊C4—151no—
398C2×C42—152no—
399C2×C42—151no—
400C2×C3×C5—24no—
401C2×C3×C5#55414yescomplemented
402C2×C3×C5C23×Q811yescharacteristic
403C2×C3×C5C22×#4111yescharacteristic
404C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
405D15—24no—
406C23×C3—22no—
407C23×C3—22no—
408C23×C3Q8×D514yes—
409C23×C3C23×D512yes—
410C23×C3C23×D511yescharacteristic
411C22×D3—54no—
412C22×D3—52no—
413C22×D3—102no—
414C22×D3—102no—
415C2×C3×C4—22no—
416C2×C3×C4—24no—
417C2×C3×C4—24no—
418C2×C3×C4—28no—
419C2×C3×C4—48no—
420C2×C3×C4#70914yes—
421C2×C3×C4C23×D514yes—
422C2×C3×C4C23×D512yes—
423C2×C3×C4C23×D514yes—
424C2×C3×C4C23×D512yes—
425#67—516no—
426C2×Dic3—54no—
427C2×Dic3—52no—
428C2×Dic3—54no—
429C2×Dic3—52no—
430C2×Dic3—52no—
431C2×Dic3—54no—
432C2×Dic3—104no—
433C2×Dic3—102no—
434C2×Dic3—102no—
435C2×Dic3—104no—
436C4×D3—108no—
437C4×D3—208no—
438Dic6—108no—
439Dic6—208no—
440C22×C5—22no—
441C22×C5—22no—
442C22×C5—28no—
443C22×C5C2×#23512yes—
444C22×C5#100412yes—
445C22×C5C2×D3×Q814yes—
446C22×C5C2×D3×Q812yes—
447C22×C5C24×D311yescharacteristic
448C2×D5—34no—
449C2×D5—62no—
450C2×D5—62no—
451C2×D5—68no—
452C4×C5—22no—
453C4×C5—24no—
454C4×C5—24no—
455C4×C5C2×#23514yes—
456Dic5—34no—
457Dic5—64no—
458Dic5—62no—
459Dic5—62no—
460Dic5—64no—
461C24—151no—
462C24C22×D1511yescharacteristic
463C2×Q8—154no—
464C2×Q8—158no—
465C2×Q8—304no—
466C2×Q8—604no—
467C2×D4—158no—
468C2×D4—154no—
469C22×C4—24no—
470C22×C4—44no—
471C22×C4—151no—
472C22×C4—152no—
473C22×C4—151no—
474C22×C4—151no—
475C22×C4—152no—
476C22×C4—154no—
477C22×C4—158no—
478C22×C4—304no—
479C22×C4—604no—
480C22×C4C22×D1514yes—
481C22×C4C22×D1512yes—
482C22×C4C22×D1511yescharacteristic
483C22×C4C22×D1512yes—
484C22×C4C22×D1518yes—
485C22×C4C22×D1511yescharacteristic
486C4⋊C4—158no—
487C4⋊C4—154no—
488C4⋊C4—154no—
489C4⋊C4—158no—
490C4⋊C4—158no—
491C4⋊C4—158no—
492C4⋊C4—158no—
493C4⋊C4C22×D1514yes—
494C4⋊C4C22×D1518yes—
495C4⋊C4C22×D1514yes—
496C22⋊C4—158no—
497C22⋊C4—154no—
498C22⋊C4—158no—
499C22⋊C4—154no—
500C42—158no—
501C42—154no—
502C3×C5C2×#55411yescharacteristic, complemented, Hall
503C22×C3—28no—
504C22×C3—22no—
505C22×C3—22no—
506C22×C3C2×#70912yes—
507C22×C3#414412yes—
508C22×C3C2×Q8×D514yes—
509C22×C3C2×Q8×D512yes—
510C22×C3C24×D511yescharacteristic
511C2×D3—54no—
512C2×D3—108no—
513C2×D3—102no—
514C2×D3—102no—
515C3×C4—22no—
516C3×C4—24no—
517C3×C4—24no—
518C3×C4C2×#70914yes—
519Dic3—54no—
520Dic3—104no—
521Dic3—102no—
522Dic3—102no—
523Dic3—104no—
524C2×C5—24no—
525C2×C5#551114yescomplemented
526C2×C5C22×#23511yescharacteristic
527C2×C5C2×#100411yescharacteristic
528C2×C5C22×D3×Q811yescharacteristic
529D5—64no—
530C23—22no—
531C23—22no—
532C23—154no—
533C23—152no—
534C23—302no—
535C23—302no—
536C23Q8×D1514yes—
537C23C23×D1512yes—
538C23C23×D1511yescharacteristic
539Q8—308no—
540Q8—608no—
541D4—1516no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—28no—
546C2×C4—48no—
547C2×C4—154no—
548C2×C4—152no—
549C2×C4—154no—
550C2×C4—152no—
551C2×C4—152no—
552C2×C4—154no—
553C2×C4—302no—
554C2×C4—304no—
555C2×C4—304no—
556C2×C4—302no—
557C2×C4—308no—
558C2×C4—608no—
559C2×C4#681714yes—
560C2×C4C23×D1514yes—
561C2×C4C23×D1512yes—
562C2×C4C23×D1514yes—
563C2×C4C23×D1512yes—
564C2×C3—24no—
565C2×C3#6584514yescomplemented
566C2×C3C22×#70911yescharacteristic
567C2×C3C2×#414411yescharacteristic
568C2×C3C22×Q8×D511yescharacteristic
569D3—104no—
570C5C2×#551111yescharacteristic, complemented, Sylow
571C22—22no—
572C22—22no—
573C22—28no—
574C22—154no—
575C22—302no—
576C22—302no—
577C22—308no—
578C22C2×#681712yes—
579C22#9209512yes—
580C22C2×Q8×D1514yes—
581C22C2×Q8×D1512yes—
582C22C24×D1511yescharacteristic
583C4—22no—
584C4—24no—
585C4—24no—
586C4—154no—
587C4—302no—
588C4—304no—
589C4—304no—
590C4—302no—
591C4C2×#681714yes—
592C3C2×#6584511yescharacteristic, complemented, Sylow
593C2—24no—
594C2—304no—
595C2#1175627014yesdirect product factor
596C2C22×#681711yescharacteristic
597C2C2×#9209511yescharacteristic
598C2C22×Q8×D1511yescharacteristic
subgroupG/H
centerC23C23×D15
commutator, cosocleC22×C3×C5C25
FrattiniC22C24×D15
FittingC22×C3×C5×C4⋊C4C2
socleC23×C3×C5C24
prime radicalC22×#1220C22
hypercenterC22×C4⋊C4D15
hypocenterC3×C5C2×#554
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C5×C4⋊C4C2semidirect
4. C2×#91813C2semidirect
5. C2×#91813C2semidirect
6. #11756270C2direct (N × H)
7. C2×#91786C2semidirect
8. C2×#91785C2semidirect
9. C2×C4⋊C4×D15C2semidirect
10. C2×#91778C2semidirect
11. C2×#91777C2semidirect
12. C2×#91776C2semidirect
24. C2×C3×C5×C4⋊C4C22semidirect
33. #91813C22semidirect
34. #91813C22semidirect
42. #91786C22semidirect
43. #91785C22semidirect
44. C4⋊C4×D15C22semidirect
45. #91778C22semidirect
46. #91777C22semidirect
47. #91776C22semidirect
49. C22×C5×C4⋊C4D3semidirect
71. C2×#1220Q8semidirect
102. C22×C3×C4⋊C4D5semidirect
124. C2×C5×C4⋊C4C2×D3semidirect
168. #1220C2×Q8semidirect
193. C2×C3×C4⋊C4C2×D5semidirect
269. C22×C4⋊C4D15semidirect
393. C2×C4⋊C4C2×D15semidirect
401. C2×C3×C5#554semidirect
502. C3×C5C2×#554semidirect
525. C2×C5#5511semidirect
565. C2×C3#65845semidirect
570. C5C2×#5511semidirect
592. C3C2×#65845semidirect
595. C2#11756270direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C5×C4⋊C42
C2×C4×#12202
C2×#918132
#117562702
C2×#917862
C2×#917852
C2×C4⋊C4×D152
C2×#917782
C2×#917772
C2×#917762
C2×#658453
C2×#55115