Finite Group Databasebeta

← C2×#11756359 #14479780

Subgroups — C2×#11756359 #14479780

Order 1920.

2464 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
1C2×C3×C5×#121C211yesmaximal, characteristic
2C2×#91825C211yesmaximal, characteristic, complemented
3C2×#91824C212yesmaximal, complemented
4C2×D4×Dic15C212yesmaximal, complemented
5C2×D4×Dic15C211yesmaximal, characteristic, complemented
6C2×#91813C211yesmaximal, characteristic, complemented
7C22×C4×Dic15C211yesmaximal, characteristic, complemented
8#11756359C2116yesmaximal, direct product factor
9C2×#91779C211yesmaximal, characteristic, complemented
10C2×#91776C211yesmaximal, characteristic, complemented
11C2×#91768C212yesmaximal, complemented
12C2×#91766C212yesmaximal, complemented
13C2×#65934—31nomaximal, Hall
14C22×C3×C5×D4C2212yes—
15C22×C3×C5×D4C2211yescharacteristic
16C23×C3×C4×C5C2211yescharacteristic
17C23×Dic15C2212yes—
18C23×Dic15C2211yescharacteristic
19C23×Dic15C2211yescharacteristic
20C22×Dic30C2211yescharacteristic
21C3×C5×#121C2218yes—
22C2×C3×C5×C4⋊C4C2211yescharacteristic
23C2×C3×C5×C22⋊C4C2212yes—
24C2×#6719C2212yes—
25C2×#6719C2212yes—
26C2×#6719C2212yes—
27C2×#6719C2212yes—
28#91825C2218yescomplemented
29#91824C22116yescomplemented
30D4×Dic15C22116yescomplemented
31D4×Dic15C2218yescomplemented
32C2×C4⋊Dic15C2212yes—
33C2×C4⋊Dic15C2211yescharacteristic
34#91813C2218yescomplemented
35C2×#6712C2212yes—
36C2×#6712C2212yes—
37C2×#6712C2212yes—
38C2×C4×Dic15C2212yes—
39C2×C4×Dic15C2211yescharacteristic
40C2×C4×Dic15C2211yescharacteristic
41C2×C4×Dic15C2211yescharacteristic
42C2×C4×Dic15C2211yescharacteristic
43C2×C4×Dic15C2218yescomplemented
44#91779C2218yescomplemented
45#91776C2218yescomplemented
46#91768C22116yescomplemented
47#91766C22116yescomplemented
48C2×#5600—51nomaximal, Hall
49C2×C5×#121D311yescharacteristic
50C2×#4082—31no—
51C2×#4081—32no—
52C2×D4×Dic5—31no—
53C2×D4×Dic5—32no—
54C2×#4070—31no—
55C22×C4×Dic5—31no—
56#65934—316no—
57C2×#4036—31no—
58C2×#4033—31no—
59C2×#4025—32no—
60C2×#4023—32no—
61C24×C3×C5C2312yes—
62C24×C3×C5C2311yescharacteristic
63C2×C3×C5×D4—28no—
64C2×C3×C5×D4—24no—
65C2×C3×C5×D4C2318yes—
66C2×C3×C5×D4C2314yes—
67C22×C3×C4×C5—24no—
68C22×C3×C4×C5C2311yescharacteristic
69C22×C3×C4×C5C2312yes—
70C22×C3×C4×C5C2314yes—
71C22×C3×C4×C5C2311yescharacteristic
72C22×Dic15—24no—
73C22×Dic15—24no—
74C22×Dic15—28no—
75C22×Dic15C2318yes—
76C22×Dic15C2311yescharacteristic
77C22×Dic15C2311yescharacteristic
78C22×Dic15C2311yescharacteristic
79C22×Dic15C2312yes—
80C22×Dic15C2314yes—
81C22×Dic15C2314yes—
82C22×Dic15C2312yes—
83C22×Dic15C2311yescharacteristic
84C2×Dic30—24no—
85C2×Dic30C2314yes—
86C3×C5×C4⋊C4C2314yes—
87C3×C5×C22⋊C4C2318yes—
88#6719C2318yes—
89#6719C2318yes—
90#6719C2318yes—
91#6719C2318yes—
92C4⋊Dic15C2318yes—
93C4⋊Dic15C2314yes—
94#6712C2318yes—
95#6712C2318yes—
96#6712C2318yes—
97C4×Dic15C2314yes—
98C4×Dic15C2318yes—
99C4×Dic15C2314yes—
100C4×Dic15C2314yes—
101C4×Dic15C2314yes—
102C2×C3×#121D511yescharacteristic
103C2×#936—51no—
104C2×#935—52no—
105C2×D4×Dic3—52no—
106C2×D4×Dic3—51no—
107C2×#924—51no—
108C22×C4×Dic3—51no—
109#5600—516no—
110C2×#890—51no—
111C2×#887—51no—
112C2×#879—52no—
113C2×#877—52no—
114C22×C5×D4C2×D312yes—
115C22×C5×D4C2×D311yescharacteristic
116C23×C4×C5C2×D311yescharacteristic
117C23×Dic5—32no—
118C23×Dic5—31no—
119C23×Dic5—31no—
120C22×Dic10—31no—
121C5×#121C2×D318yes—
122C2×C5×C4⋊C4C2×D311yescharacteristic
123C2×C5×C22⋊C4C2×D312yes—
124C2×C22⋊Dic5—32no—
125C2×C22⋊Dic5—32no—
126C2×C22⋊Dic5—32no—
127C2×C22⋊Dic5—32no—
128#4082—38no—
129#4081—316no—
130D4×Dic5—316no—
131D4×Dic5—38no—
132C2×C4⋊Dic5—32no—
133C2×C4⋊Dic5—31no—
134#4070—38no—
135C2×#683—32no—
136C2×#683—32no—
137C2×#683—32no—
138C2×C4×Dic5—31no—
139C2×C4×Dic5—32no—
140C2×C4×Dic5—31no—
141C2×C4×Dic5—31no—
142C2×C4×Dic5—31no—
143C2×C4×Dic5—38no—
144#4036—38no—
145#4033—38no—
146#4025—316no—
147#4023—316no—
148C2×#529—151noSylow
149C23×C3×C5—22no—
150C23×C3×C5—22no—
151C23×C3×C5—24no—
152C23×C3×C5—24no—
153C23×C3×C5—24no—
154C23×C3×C5#4114yes—
155C23×C3×C5C2414yes—
156C23×C3×C5C2412yes—
157C23×C3×C5C2411yescharacteristic
158C3×C5×D4—216no—
159C3×C5×D4—28no—
160C2×C3×C4×C5—22no—
161C2×C3×C4×C5—24no—
162C2×C3×C4×C5—28no—
163C2×C3×C4×C5#4114yes—
164C2×C3×C4×C5C2412yes—
165C2×C3×C4×C5C2414yes—
166C2×C3×C4×C5C2412yes—
167C2×Dic15—24no—
168C2×Dic15—28no—
169C2×Dic15—22no—
170C2×Dic15—22no—
171C2×Dic15—22no—
172C2×Dic15—24no—
173C2×Dic15—28no—
174C2×Dic15—216no—
175C2×Dic15#4114yes—
176C2×Dic15C2412yes—
177C2×Dic15C2412yes—
178C2×Dic15C2412yes—
179C2×Dic15C2414yes—
180C2×Dic15C2414yes—
181C2×Dic15C2412yes—
182Dic30—28no—
183C22×C3×D4C2×D512yes—
184C22×C3×D4C2×D511yescharacteristic
185C23×C3×C4C2×D511yescharacteristic
186C23×Dic3—52no—
187C23×Dic3—51no—
188C23×Dic3—51no—
189C22×Dic6—51no—
190C3×#121C2×D518yes—
191C2×C3×C4⋊C4C2×D511yescharacteristic
192C2×C3×C22⋊C4C2×D512yes—
193C2×#217—52no—
194C2×#217—52no—
195C2×#217—52no—
196C2×#217—52no—
197#936—58no—
198#935—516no—
199D4×Dic3—516no—
200D4×Dic3—58no—
201C2×C4⋊Dic3—52no—
202C2×C4⋊Dic3—51no—
203#924—58no—
204C2×Dic3⋊C4—52no—
205C2×Dic3⋊C4—52no—
206C2×Dic3⋊C4—52no—
207C2×C4×Dic3—51no—
208C2×C4×Dic3—52no—
209C2×C4×Dic3—51no—
210C2×C4×Dic3—51no—
211C2×C4×Dic3—51no—
212C2×C4×Dic3—58no—
213#890—58no—
214#887—58no—
215#879—516no—
216#877—516no—
217C24×C5C22×D312yes—
218C24×C5C22×D311yescharacteristic
219C2×C5×D4—28no—
220C2×C5×D4—24no—
221C2×C5×D4C22×D318yes—
222C2×C5×D4C22×D314yes—
223C22×C4×C5—24no—
224C22×C4×C5C22×D311yescharacteristic
225C22×C4×C5C22×D312yes—
226C22×C4×C5C22×D314yes—
227C22×C4×C5C22×D311yescharacteristic
228C22×Dic5—31no—
229C22×Dic5—32no—
230C22×Dic5—31no—
231C22×Dic5—31no—
232C22×Dic5—31no—
233C22×Dic5—32no—
234C22×Dic5—38no—
235C22×Dic5—34no—
236C22×Dic5—34no—
237C22×Dic5—64no—
238C22×Dic5—68no—
239C22×Dic5—64no—
240C2×Dic10—34no—
241C2×Dic10—64no—
242C5×C4⋊C4C22×D314yes—
243C5×C22⋊C4C22×D318yes—
244C22⋊Dic5—38no—
245C22⋊Dic5—38no—
246C22⋊Dic5—38no—
247C22⋊Dic5—38no—
248C4⋊Dic5—38no—
249C4⋊Dic5—34no—
250#683—38no—
251#683—38no—
252#683—38no—
253C4×Dic5—34no—
254C4×Dic5—38no—
255C4×Dic5—34no—
256C4×Dic5—34no—
257C4×Dic5—34no—
258#529—1516no—
259C2×#126—152no—
260C2×#125—151no—
261C2×#124—151no—
262C2×#123—152no—
263C2×C22⋊Q8—151no—
264C2×#121D1511yescharacteristic
265C2×C4×Q8—151no—
266C2×C4×D4—152no—
267C2×C4×D4—151no—
268C2×#117—152no—
269C22×C42—151no—
270C22×C3×C5—28no—
271C22×C3×C5—22no—
272C22×C3×C5—22no—
273C22×C3×C5—24no—
274C22×C3×C5—24no—
275C22×C3×C5—24no—
276C22×C3×C5—48no—
277C22×C3×C5C2×#4114yes—
278C22×C3×C5C2×#4112yes—
279C22×C3×C5C2×#4112yes—
280C22×C3×C5C2×#4112yes—
281C22×C3×C5C2511yescharacteristic
282C3×C4×C5—22no—
283C3×C4×C5—24no—
284C3×C4×C5C2×#4114yes—
285Dic15—24no—
286Dic15—22no—
287Dic15—22no—
288Dic15—22no—
289Dic15—24no—
290Dic15C2×#4114yes—
291C24×C3C22×D512yes—
292C24×C3C22×D511yescharacteristic
293C2×C3×D4—28no—
294C2×C3×D4—24no—
295C2×C3×D4C22×D518yes—
296C2×C3×D4C22×D514yes—
297C22×C3×C4—24no—
298C22×C3×C4C22×D511yescharacteristic
299C22×C3×C4C22×D514yes—
300C22×C3×C4C22×D512yes—
301C22×C3×C4C22×D511yescharacteristic
302C22×Dic3—51no—
303C22×Dic3—58no—
304C22×Dic3—51no—
305C22×Dic3—51no—
306C22×Dic3—51no—
307C22×Dic3—54no—
308C22×Dic3—54no—
309C22×Dic3—52no—
310C22×Dic3—52no—
311C22×Dic3—104no—
312C22×Dic3—108no—
313C22×Dic3—104no—
314C2×Dic6—54no—
315C2×Dic6—104no—
316C3×C4⋊C4C22×D514yes—
317C3×C22⋊C4C22×D518yes—
318#217—58no—
319#217—58no—
320#217—58no—
321#217—58no—
322C4⋊Dic3—58no—
323C4⋊Dic3—54no—
324Dic3⋊C4—58no—
325Dic3⋊C4—58no—
326Dic3⋊C4—58no—
327C4×Dic3—54no—
328C4×Dic3—58no—
329C4×Dic3—54no—
330C4×Dic3—54no—
331C4×Dic3—54no—
332C23×C5—22no—
333C23×C5—22no—
334C23×C5—24no—
335C23×C5—24no—
336C23×C5—24no—
337C23×C5#23714yes—
338C23×C5C23×D314yes—
339C23×C5C23×D312yes—
340C23×C5C23×D311yescharacteristic
341C5×D4—216no—
342C5×D4—28no—
343C2×C4×C5—22no—
344C2×C4×C5—24no—
345C2×C4×C5—28no—
346C2×C4×C5#23714yes—
347C2×C4×C5C23×D312yes—
348C2×C4×C5C23×D314yes—
349C2×C4×C5C23×D312yes—
350C2×Dic5—34no—
351C2×Dic5—32no—
352C2×Dic5—32no—
353C2×Dic5—32no—
354C2×Dic5—32no—
355C2×Dic5—34no—
356C2×Dic5—34no—
357C2×Dic5—68no—
358C2×Dic5—616no—
359C2×Dic5—64no—
360C2×Dic5—68no—
361C2×Dic5—62no—
362C2×Dic5—62no—
363C2×Dic5—62no—
364C2×Dic5—64no—
365Dic10—68no—
366C22×Q8—151no—
367C22×D4C2×D1512yes—
368C22×D4C2×D1511yescharacteristic
369C23×C4—152no—
370C23×C4—151no—
371C23×C4—151no—
372C23×C4C2×D1511yescharacteristic
373#126—1516no—
374#125—158no—
375#124—158no—
376#123—1516no—
377C22⋊Q8—158no—
378#121C2×D1518yes—
379C4×Q8—158no—
380C4×D4—158no—
381C4×D4—1516no—
382#117—1516no—
383C2×C4⋊C4—152no—
384C2×C4⋊C4—152no—
385C2×C4⋊C4—151no—
386C2×C4⋊C4—152no—
387C2×C4⋊C4—152no—
388C2×C4⋊C4C2×D1511yescharacteristic
389C2×C22⋊C4—152no—
390C2×C22⋊C4—152no—
391C2×C22⋊C4—152no—
392C2×C22⋊C4—152no—
393C2×C22⋊C4C2×D1512yes—
394C2×C42—151no—
395C2×C42—152no—
396C2×C42—151no—
397C2×C42—151no—
398C2×C42—151no—
399C2×C42—158no—
400C2×C3×C5—24no—
401C2×C3×C5—44no—
402C2×C3×C5#52914yescomplemented
403C2×C3×C5C22×#4111yescharacteristic
404C2×C3×C5C22×#4111yescharacteristic
405C2×C3×C5C22×#4111yescharacteristic
406C23×C3—22no—
407C23×C3—22no—
408C23×C3—24no—
409C23×C3—24no—
410C23×C3—24no—
411C23×C3#71114yes—
412C23×C3C23×D514yes—
413C23×C3C23×D512yes—
414C23×C3C23×D511yescharacteristic
415C3×D4—216no—
416C3×D4—28no—
417C2×C3×C4—22no—
418C2×C3×C4—24no—
419C2×C3×C4—28no—
420C2×C3×C4#71114yes—
421C2×C3×C4C23×D512yes—
422C2×C3×C4C23×D514yes—
423C2×C3×C4C23×D512yes—
424C2×Dic3—54no—
425C2×Dic3—52no—
426C2×Dic3—52no—
427C2×Dic3—52no—
428C2×Dic3—52no—
429C2×Dic3—54no—
430C2×Dic3—54no—
431C2×Dic3—108no—
432C2×Dic3—1016no—
433C2×Dic3—104no—
434C2×Dic3—108no—
435C2×Dic3—102no—
436C2×Dic3—102no—
437C2×Dic3—102no—
438C2×Dic3—104no—
439Dic6—108no—
440C22×C5—22no—
441C22×C5—22no—
442C22×C5—28no—
443C22×C5—24no—
444C22×C5—24no—
445C22×C5—24no—
446C22×C5—48no—
447C22×C5C2×#23714yes—
448C22×C5C2×#23712yes—
449C22×C5C2×#23712yes—
450C22×C5D3×#4112yes—
451C22×C5C24×D311yescharacteristic
452C4×C5—22no—
453C4×C5—24no—
454C4×C5C2×#23714yes—
455Dic5—34no—
456Dic5—64no—
457Dic5—62no—
458Dic5—62no—
459Dic5—62no—
460Dic5—64no—
461C24C22×D1512yes—
462C24C22×D1511yescharacteristic
463C2×Q8—154no—
464C2×Q8—304no—
465C2×D4—28no—
466C2×D4—24no—
467C2×D4C22×D1518yes—
468C2×D4C22×D1514yes—
469C22×C4—24no—
470C22×C4—151no—
471C22×C4—154no—
472C22×C4—154no—
473C22×C4—158no—
474C22×C4—151no—
475C22×C4—151no—
476C22×C4—151no—
477C22×C4—152no—
478C22×C4—152no—
479C22×C4—304no—
480C22×C4—308no—
481C22×C4—304no—
482C22×C4C22×D1514yes—
483C22×C4C22×D1511yescharacteristic
484C22×C4C22×D1512yes—
485C22×C4C22×D1511yescharacteristic
486C4⋊C4—158no—
487C4⋊C4—158no—
488C4⋊C4—154no—
489C4⋊C4—158no—
490C4⋊C4—158no—
491C4⋊C4C22×D1514yes—
492C22⋊C4—158no—
493C22⋊C4—158no—
494C22⋊C4—158no—
495C22⋊C4—158no—
496C22⋊C4C22×D1518yes—
497C42—154no—
498C42—154no—
499C42—154no—
500C42—154no—
501C42—158no—
502C3×C5C2×#52911yescharacteristic, complemented, Hall
503C22×C3—28no—
504C22×C3—22no—
505C22×C3—22no—
506C22×C3—24no—
507C22×C3—24no—
508C22×C3—24no—
509C22×C3—48no—
510C22×C3C2×#71114yes—
511C22×C3C2×#71112yes—
512C22×C3C2×#71112yes—
513C22×C3D5×#4112yes—
514C22×C3C24×D511yescharacteristic
515C3×C4—22no—
516C3×C4—24no—
517C3×C4C2×#71114yes—
518Dic3—54no—
519Dic3—104no—
520Dic3—102no—
521Dic3—102no—
522Dic3—102no—
523Dic3—104no—
524C2×C5—24no—
525C2×C5—44no—
526C2×C5#560014yescomplemented
527C2×C5C22×#23711yescharacteristic
528C2×C5C22×#23711yescharacteristic
529C2×C5C2×D3×#4111yescharacteristic
530C23—22no—
531C23—22no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23#681914yes—
536C23C23×D1514yes—
537C23C23×D1512yes—
538C23C23×D1511yescharacteristic
539Q8—308no—
540D4—216no—
541D4—28no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—154no—
546C2×C4—152no—
547C2×C4—152no—
548C2×C4—152no—
549C2×C4—152no—
550C2×C4—154no—
551C2×C4—154no—
552C2×C4—308no—
553C2×C4—3016no—
554C2×C4—308no—
555C2×C4—302no—
556C2×C4—302no—
557C2×C4—304no—
558C2×C4—302no—
559C2×C4—304no—
560C2×C4#681914yes—
561C2×C4C23×D1512yes—
562C2×C4C23×D1514yes—
563C2×C4C23×D1512yes—
564C2×C3—24no—
565C2×C3—44no—
566C2×C3#6593414yescomplemented
567C2×C3C22×#71111yescharacteristic
568C2×C3C22×#71111yescharacteristic
569C2×C3C2×D5×#4111yescharacteristic
570C5C2×#560011yescharacteristic, complemented, Sylow
571C22—28no—
572C22—22no—
573C22—22no—
574C22—24no—
575C22—24no—
576C22—24no—
577C22—48no—
578C22C2×#681914yes—
579C22C2×#681912yes—
580C22C2×#681912yes—
581C22#41×D1512yes—
582C22C24×D1511yescharacteristic
583C4—22no—
584C4—24no—
585C4—154no—
586C4—302no—
587C4—302no—
588C4—304no—
589C4—302no—
590C4—304no—
591C4C2×#681914yes—
592C3C2×#6593411yescharacteristic, complemented, Sylow
593C2—24no—
594C2—44no—
595C2#1175635914yesdirect product factor
596C2C22×#681911yescharacteristic
597C2C22×#681911yescharacteristic
598C2C2×#41×D1511yescharacteristic
subgroupG/H
centerC23C23×D15
commutator, cosocleC22×C3×C5C25
FrattiniC22C24×D15
Fitting, prime radicalC2×C3×C5×#121C2
socleC23×C3×C5C24
hypercenterC2×#121D15
hypocenterC3×C5C2×#529
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
2. C2×#91825C2semidirect
3. C2×#91824C2semidirect
4. C2×D4×Dic15C2semidirect
5. C2×D4×Dic15C2semidirect
6. C2×#91813C2semidirect
7. C22×C4×Dic15C2semidirect
8. #11756359C2direct (N × H)
9. C2×#91779C2semidirect
10. C2×#91776C2semidirect
11. C2×#91768C2semidirect
12. C2×#91766C2semidirect
28. #91825C22semidirect
29. #91824C22semidirect
30. D4×Dic15C22semidirect
31. D4×Dic15C22semidirect
34. #91813C22semidirect
43. C2×C4×Dic15C22semidirect
44. #91779C22semidirect
45. #91776C22semidirect
46. #91768C22semidirect
47. #91766C22semidirect
402. C2×C3×C5#529semidirect
502. C3×C5C2×#529semidirect
526. C2×C5#5600semidirect
566. C2×C3#65934semidirect
570. C5C2×#5600semidirect
592. C3C2×#65934semidirect
595. C2#11756359direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×#1212
C2×#918252
C2×#918242
C2×D4×Dic152
C2×#918132
C22×C4×Dic152
#117563592
C2×#917792
C2×#917762
C2×#917682
C2×#917662
C2×#659343
C2×#56005