Finite Group Databasebeta

← C2×#11751046 #14320455

Subgroups — C2×#11751046 #14320455

Order 1920.

1856 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×C22⋊C4C211yesmaximal, characteristic
2C22×#6719C211yesmaximal, characteristic
3C22×#6719C211yesmaximal, characteristic
4C22×#6712C211yesmaximal, characteristic, complemented
5C22×C4×Dic15C211yesmaximal, characteristic, complemented
6C2×#91115C211yesmaximal, characteristic, complemented
7C2×#91115C211yesmaximal, characteristic, complemented
8#11751046C218yesmaximal, direct product factor
9C2×#65234—31nomaximal, Hall
10C25×C3×C5C2211yescharacteristic
11C23×C3×C4×C5C2211yescharacteristic
12C23×C3×C4×C5C2211yescharacteristic
13C23×Dic15C2211yescharacteristic
14C23×Dic15C2211yescharacteristic
15C23×Dic15C2211yescharacteristic
16C23×Dic15C2211yescharacteristic
17C2×C3×C5×C22⋊C4—24no—
18C2×C3×C5×C22⋊C4C2214yes—
19C2×#6719—24no—
20C2×#6719C418yescomplemented
21C2×#6719C2214yes—
22C2×#6719C2214yes—
23C2×#6712—24no—
24C2×#6712C2214yescomplemented
25C2×C4×Dic15—24no—
26C2×C4×Dic15C2214yescomplemented
27#91115C2214yescomplemented
28#91115C2214yescomplemented
29C2×#4945—51nomaximal, Hall
30C22×C5×C22⋊C4D311yescharacteristic
31C22×C22⋊Dic5—31no—
32C22×C22⋊Dic5—31no—
33C22×#683—31no—
34C22×C4×Dic5—31no—
35C2×#3963—31no—
36C2×#3963—31no—
37#65234—38no—
38C24×C3×C5—22no—
39C24×C3×C5—22no—
40C24×C3×C5—22no—
41C24×C3×C5—22no—
42C24×C3×C5—22no—
43C24×C3×C5—22no—
44C24×C3×C5C2×C414yes—
45C24×C3×C5C2312yes—
46C24×C3×C5C2311yescharacteristic
47C22×C3×C4×C5—22no—
48C22×C3×C4×C5—22no—
49C22×C3×C4×C5—22no—
50C22×C3×C4×C5—22no—
51C22×C3×C4×C5—22no—
52C22×C3×C4×C5D414yes—
53C22×C3×C4×C5C2312yes—
54C22×C3×C4×C5C2312yes—
55C22×Dic15—22no—
56C22×Dic15—22no—
57C22×Dic15—22no—
58C22×Dic15—22no—
59C22×Dic15—22no—
60C22×Dic15—22no—
61C22×Dic15—22no—
62C22×Dic15—22no—
63C22×Dic15—22no—
64C22×Dic15C2×C414yes—
65C22×Dic15C2×C414yes—
66C22×Dic15D414yes—
67C22×Dic15C2312yes—
68C22×Dic15C2312yes—
69C22×Dic15C2312yes—
70C22×Dic15C2312yes—
71C3×C5×C22⋊C4—28no—
72#6719—28no—
73#6719C2×C4116yescomplemented
74#6712—28no—
75C4×Dic15—28no—
76C22×C3×C22⋊C4D511yescharacteristic
77C22×#217—51no—
78C22×#217—51no—
79C22×Dic3⋊C4—51no—
80C22×C4×Dic3—51no—
81C2×C3⋊#95—51no—
82C2×C3⋊#95—51no—
83#4945—58no—
84C25×C5C2×D311yescharacteristic
85C23×C4×C5C2×D311yescharacteristic
86C23×C4×C5C2×D311yescharacteristic
87C23×Dic5—31no—
88C23×Dic5—31no—
89C23×Dic5—31no—
90C23×Dic5—31no—
91C2×C5×C22⋊C4—24no—
92C2×C5×C22⋊C4C2×D314yes—
93C2×C22⋊Dic5—38no—
94C2×C22⋊Dic5—34no—
95C2×C22⋊Dic5—34no—
96C2×C22⋊Dic5—64no—
97C2×#683—34no—
98C2×#683—64no—
99C2×C4×Dic5—34no—
100C2×C4×Dic5—64no—
101#3963—34no—
102#3963—34no—
103C2×#388—151noSylow
104C23×C3×C5—28no—
105C23×C3×C5—28no—
106C23×C3×C5—28no—
107C23×C3×C5—22no—
108C23×C3×C5—22no—
109C23×C3×C5—22no—
110C23×C3×C5—22no—
111C23×C3×C5—22no—
112C23×C3×C5—22no—
113C23×C3×C5—44no—
114C23×C3×C5—44no—
115C23×C3×C5—44no—
116C23×C3×C5—44no—
117C23×C3×C5C22×C414yes—
118C23×C3×C5C22×C412yes—
119C23×C3×C5C2×D412yes—
120C23×C3×C5C2×D412yes—
121C23×C3×C5#4112yes—
122C23×C3×C5#4112yes—
123C23×C3×C5#4112yes—
124C23×C3×C5#4112yes—
125C23×C3×C5C2411yescharacteristic
126C2×C3×C4×C5—28no—
127C2×C3×C4×C5—22no—
128C2×C3×C4×C5—22no—
129C2×C3×C4×C5—22no—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5—44no—
133C2×C3×C4×C5C2×D414yes—
134C2×Dic15—28no—
135C2×Dic15—28no—
136C2×Dic15—28no—
137C2×Dic15—22no—
138C2×Dic15—22no—
139C2×Dic15—22no—
140C2×Dic15—22no—
141C2×Dic15—22no—
142C2×Dic15—22no—
143C2×Dic15—22no—
144C2×Dic15—22no—
145C2×Dic15—22no—
146C2×Dic15—44no—
147C2×Dic15C22×C414yes—
148C2×Dic15C22×C414yes—
149C2×Dic15C2×D414yes—
150C25×C3C2×D511yescharacteristic
151C23×C3×C4C2×D511yescharacteristic
152C23×C3×C4C2×D511yescharacteristic
153C23×Dic3—51no—
154C23×Dic3—51no—
155C23×Dic3—51no—
156C23×Dic3—51no—
157C2×C3×C22⋊C4—24no—
158C2×C3×C22⋊C4C2×D514yes—
159C2×#217—58no—
160C2×#217—54no—
161C2×#217—54no—
162C2×#217—104no—
163C2×Dic3⋊C4—54no—
164C2×Dic3⋊C4—104no—
165C2×C4×Dic3—54no—
166C2×C4×Dic3—104no—
167C3⋊#95—54no—
168C3⋊#95—54no—
169C24×C5—22no—
170C24×C5—22no—
171C24×C5—22no—
172C24×C5—22no—
173C24×C5—22no—
174C24×C5—22no—
175C24×C5C4×D314yes—
176C24×C5C22×D312yes—
177C24×C5C22×D311yescharacteristic
178C22×C4×C5—22no—
179C22×C4×C5—22no—
180C22×C4×C5—22no—
181C22×C4×C5—22no—
182C22×C4×C5—22no—
183C22×C4×C5#6714yes—
184C22×C4×C5C22×D312yes—
185C22×C4×C5C22×D312yes—
186C22×Dic5—34no—
187C22×Dic5—34no—
188C22×Dic5—34no—
189C22×Dic5—32no—
190C22×Dic5—32no—
191C22×Dic5—32no—
192C22×Dic5—32no—
193C22×Dic5—62no—
194C22×Dic5—62no—
195C22×Dic5—62no—
196C22×Dic5—62no—
197C22×Dic5—62no—
198C22×Dic5—62no—
199C22×Dic5—62no—
200C22×Dic5—62no—
201C22×Dic5—62no—
202C5×C22⋊C4—28no—
203C22⋊Dic5—316no—
204C22⋊Dic5—68no—
205#683—68no—
206C4×Dic5—68no—
207C22×C4⋊C4—151no—
208C22×C22⋊C4—151no—
209C22×C22⋊C4—151no—
210C22×C22⋊C4D1511yescharacteristic
211C22×C42—151no—
212#388—158no—
213C2×#95—151no—
214C2×#95—151no—
215C22×C3×C5—24no—
216C22×C3×C5—24no—
217C22×C3×C5—24no—
218C22×C3×C5—42no—
219C22×C3×C5—42no—
220C22×C3×C5—42no—
221C22×C3×C5—42no—
222C22×C3×C5—416no—
223C22×C3×C5#11714yes—
224C22×C3×C5C4×D414yes—
225C22×C3×C5C4×D414yes—
226C22×C3×C5#12114yes—
227C22×C3×C5#12314yes—
228C22×C3×C5#12414yes—
229C22×C3×C5#12614yes—
230C22×C3×C5C23×C411yescharacteristic
231C22×C3×C5C22×D411yescharacteristic
232C22×C3×C5C22×D411yescharacteristic
233C22×C3×C5C2×#4111yescharacteristic
234C22×C3×C5C2×#4111yescharacteristic
235C22×C3×C5C2×#4111yescharacteristic
236C22×C3×C5C2×#4111yescharacteristic
237C3×C4×C5—24no—
238C3×C4×C5—42no—
239Dic15—24no—
240Dic15—24no—
241Dic15—24no—
242Dic15—42no—
243C24×C3—22no—
244C24×C3—22no—
245C24×C3—22no—
246C24×C3—22no—
247C24×C3—22no—
248C24×C3—22no—
249C24×C3C4×D514yes—
250C24×C3C22×D512yes—
251C24×C3C22×D511yescharacteristic
252C22×C3×C4—22no—
253C22×C3×C4—22no—
254C22×C3×C4—22no—
255C22×C3×C4—22no—
256C22×C3×C4—22no—
257C22×C3×C4C22⋊D514yes—
258C22×C3×C4C22×D512yes—
259C22×C3×C4C22×D512yes—
260C22×Dic3—54no—
261C22×Dic3—54no—
262C22×Dic3—54no—
263C22×Dic3—52no—
264C22×Dic3—52no—
265C22×Dic3—52no—
266C22×Dic3—52no—
267C22×Dic3—102no—
268C22×Dic3—102no—
269C22×Dic3—102no—
270C22×Dic3—102no—
271C22×Dic3—102no—
272C22×Dic3—102no—
273C22×Dic3—102no—
274C22×Dic3—102no—
275C22×Dic3—102no—
276C3×C22⋊C4—28no—
277#217—516no—
278#217—108no—
279Dic3⋊C4—108no—
280C4×Dic3—108no—
281C23×C5—28no—
282C23×C5—28no—
283C23×C5—28no—
284C23×C5—22no—
285C23×C5—22no—
286C23×C5—22no—
287C23×C5—22no—
288C23×C5—22no—
289C23×C5—22no—
290C23×C5—44no—
291C23×C5—44no—
292C23×C5—44no—
293C23×C5—44no—
294C23×C5C2×C4×D314yes—
295C23×C5C2×C4×D312yes—
296C23×C5#23512yes—
297C23×C5D3×D412yes—
298C23×C5#23712yes—
299C23×C5#23712yes—
300C23×C5#23712yes—
301C23×C5C2×#6712yes—
302C23×C5C23×D311yescharacteristic
303C2×C4×C5—28no—
304C2×C4×C5—22no—
305C2×C4×C5—22no—
306C2×C4×C5—22no—
307C2×C4×C5—22no—
308C2×C4×C5—22no—
309C2×C4×C5—44no—
310C2×C4×C5C2×#6714yes—
311C2×Dic5—34no—
312C2×Dic5—34no—
313C2×Dic5—34no—
314C2×Dic5—68no—
315C2×Dic5—68no—
316C2×Dic5—68no—
317C2×Dic5—62no—
318C2×Dic5—62no—
319C2×Dic5—62no—
320C2×Dic5—62no—
321C2×Dic5—62no—
322C2×Dic5—62no—
323C2×Dic5—62no—
324C2×Dic5—62no—
325C2×Dic5—62no—
326C2×Dic5—124no—
327C25C2×D1511yescharacteristic
328C23×C4—151no—
329C23×C4—151no—
330C23×C4—151no—
331C23×C4—151no—
332C23×C4C2×D1511yescharacteristic
333C23×C4C2×D1511yescharacteristic
334C2×C4⋊C4—154no—
335C2×C4⋊C4—304no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—158no—
338C2×C22⋊C4—154no—
339C2×C22⋊C4—154no—
340C2×C22⋊C4—304no—
341C2×C22⋊C4C2×D1514yes—
342C2×C42—154no—
343C2×C42—304no—
344#95—154no—
345#95—154no—
346C2×C3×C5—44no—
347C2×C3×C5#38818yescomplemented
348C2×C3×C5C2×#11711yescharacteristic
349C2×C3×C5C2×C4×D411yescharacteristic
350C2×C3×C5C2×C4×D411yescharacteristic
351C2×C3×C5C2×#12111yescharacteristic
352C2×C3×C5C2×#12311yescharacteristic
353C2×C3×C5C2×#12411yescharacteristic
354C2×C3×C5C2×#12611yescharacteristic
355C23×C3—22no—
356C23×C3—22no—
357C23×C3—22no—
358C23×C3—22no—
359C23×C3—22no—
360C23×C3—22no—
361C23×C3—28no—
362C23×C3—28no—
363C23×C3—28no—
364C23×C3—44no—
365C23×C3—44no—
366C23×C3—44no—
367C23×C3—44no—
368C23×C3C2×C4×D514yes—
369C23×C3C2×C4×D512yes—
370C23×C3#70912yes—
371C23×C3D4×D512yes—
372C23×C3#71112yes—
373C23×C3#71112yes—
374C23×C3#71112yes—
375C23×C3C2×C22⋊D512yes—
376C23×C3C23×D511yescharacteristic
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—28no—
380C2×C3×C4—22no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—44no—
384C2×C3×C4C2×C22⋊D514yes—
385C2×Dic3—54no—
386C2×Dic3—54no—
387C2×Dic3—54no—
388C2×Dic3—102no—
389C2×Dic3—102no—
390C2×Dic3—102no—
391C2×Dic3—102no—
392C2×Dic3—102no—
393C2×Dic3—102no—
394C2×Dic3—108no—
395C2×Dic3—108no—
396C2×Dic3—108no—
397C2×Dic3—102no—
398C2×Dic3—102no—
399C2×Dic3—102no—
400C2×Dic3—204no—
401C22×C5—24no—
402C22×C5—24no—
403C22×C5—24no—
404C22×C5—416no—
405C22×C5—42no—
406C22×C5—42no—
407C22×C5—42no—
408C22×C5—42no—
409C22×C5#87714yes—
410C22×C5#87914yes—
411C22×C5#67⋊C414yes—
412C22×C5#88314yes—
413C22×C5C4×#6714yes—
414C22×C5#93614yes—
415C22×C5#93914yes—
416C22×C5C22×C4×D311yescharacteristic
417C22×C5C2×#23511yescharacteristic
418C22×C5C2×D3×D411yescharacteristic
419C22×C5C2×#23711yescharacteristic
420C22×C5C2×#23711yescharacteristic
421C22×C5C2×#23711yescharacteristic
422C22×C5C22×#6711yescharacteristic
423C4×C5—24no—
424C4×C5—42no—
425Dic5—64no—
426Dic5—64no—
427Dic5—64no—
428Dic5—122no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24C4×D1514yes—
436C24C22×D1512yes—
437C24C22×D1511yescharacteristic
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—154no—
444C22×C4—154no—
445C22×C4—154no—
446C22×C4—152no—
447C22×C4—152no—
448C22×C4—152no—
449C22×C4—152no—
450C22×C4—302no—
451C22×C4—302no—
452C22×C4—302no—
453C22×C4—302no—
454C22×C4—302no—
455C22×C4—302no—
456C22×C4—302no—
457C22×C4—302no—
458C22×C4—302no—
459C22×C4#122014yes—
460C22×C4C22×D1512yes—
461C22×C4C22×D1512yes—
462C4⋊C4—308no—
463C22⋊C4—28no—
464C22⋊C4—1516no—
465C22⋊C4—308no—
466C42—308no—
467C3×C5C2×#38811yescharacteristic, complemented, Hall
468C22×C3—24no—
469C22×C3—24no—
470C22×C3—24no—
471C22×C3—42no—
472C22×C3—42no—
473C22×C3—42no—
474C22×C3—42no—
475C22×C3—416no—
476C22×C3#402314yes—
477C22×C3#402514yes—
478C22×C3#402714yes—
479C22×C3#402914yes—
480C22×C3C4×C22⋊D514yes—
481C22×C3#408214yes—
482C22×C3#408514yes—
483C22×C3C22×C4×D511yescharacteristic
484C22×C3C2×#70911yescharacteristic
485C22×C3C2×D4×D511yescharacteristic
486C22×C3C2×#71111yescharacteristic
487C22×C3C2×#71111yescharacteristic
488C22×C3C2×#71111yescharacteristic
489C22×C3C22×C22⋊D511yescharacteristic
490C3×C4—24no—
491C3×C4—42no—
492Dic3—104no—
493Dic3—104no—
494Dic3—104no—
495Dic3—202no—
496C2×C5—44no—
497C2×C5#494518yescomplemented
498C2×C5C2×#87711yescharacteristic
499C2×C5C2×#87911yescharacteristic
500C2×C5C2×#67⋊C411yescharacteristic
501C2×C5C2×#88311yescharacteristic
502C2×C5C2×C4×#6711yescharacteristic
503C2×C5C2×#93611yescharacteristic
504C2×C5C2×#93911yescharacteristic
505C23—28no—
506C23—28no—
507C23—28no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23C2×C4×D1514yes—
519C23C2×C4×D1512yes—
520C23#681712yes—
521C23D4×D1512yes—
522C23#681912yes—
523C23#681912yes—
524C23#681912yes—
525C23C2×#122012yes—
526C23C23×D1511yescharacteristic
527C2×C4—28no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—44no—
534C2×C4—154no—
535C2×C4—154no—
536C2×C4—154no—
537C2×C4—308no—
538C2×C4—308no—
539C2×C4—308no—
540C2×C4—302no—
541C2×C4—302no—
542C2×C4—302no—
543C2×C4—302no—
544C2×C4—302no—
545C2×C4—302no—
546C2×C4—302no—
547C2×C4—302no—
548C2×C4—302no—
549C2×C4—604no—
550C2×C4C2×#122014yes—
551C2×C3—44no—
552C2×C3#6523418yescomplemented
553C2×C3C2×#402311yescharacteristic
554C2×C3C2×#402511yescharacteristic
555C2×C3C2×#402711yescharacteristic
556C2×C3C2×#402911yescharacteristic
557C2×C3C2×C4×C22⋊D511yescharacteristic
558C2×C3C2×#408211yescharacteristic
559C2×C3C2×#408511yescharacteristic
560C5C2×#494511yescharacteristic, complemented, Sylow
561C22—24no—
562C22—24no—
563C22—24no—
564C22—42no—
565C22—42no—
566C22—416no—
567C22—42no—
568C22—42no—
569C22#9176614yes—
570C22#9176814yes—
571C22#9177014yes—
572C22#9177214yes—
573C22C4×#122014yes—
574C22#9182514yes—
575C22#9182814yes—
576C22C22×C4×D1511yescharacteristic
577C22C2×#681711yescharacteristic
578C22C2×D4×D1511yescharacteristic
579C22C2×#681911yescharacteristic
580C22C2×#681911yescharacteristic
581C22C2×#681911yescharacteristic
582C22C22×#122011yescharacteristic
583C4—24no—
584C4—42no—
585C4—304no—
586C4—304no—
587C4—304no—
588C4—602no—
589C3C2×#6523411yescharacteristic, complemented, Sylow
590C2—44no—
591C2#1175104618yesdirect product factor
592C2C2×#9176611yescharacteristic
593C2C2×#9176811yescharacteristic
594C2C2×#9177011yescharacteristic
595C2C2×#9177211yescharacteristic
596C2C2×C4×#122011yescharacteristic
597C2C2×#9182511yescharacteristic
598C2C2×#9182811yescharacteristic
subgroupG/H
centerC24C22×D15
commutatorC22×C3×C5C23×C4
FrattiniC23C23×D15
FittingC22×C3×C5×C22⋊C4C2
socleC24×C3×C5C23
prime radicalC25×C3×C5C22
cosocleC23×C3×C5C24
hypercenterC22×C22⋊C4D15
hypocenterC3×C5C2×#388
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C24 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C22×#6712C2semidirect
5. C22×C4×Dic15C2semidirect
6. C2×#91115C2semidirect
7. C2×#91115C2semidirect
8. #11751046C2direct (N × H)
20. C2×#6719C4semidirect
24. C2×#6712C22semidirect
26. C2×C4×Dic15C22semidirect
27. #91115C22semidirect
28. #91115C22semidirect
73. #6719C2×C4semidirect
347. C2×C3×C5#388semidirect
467. C3×C5C2×#388semidirect
497. C2×C5#4945semidirect
552. C2×C3#65234semidirect
560. C5C2×#4945semidirect
589. C3C2×#65234semidirect
591. C2#11751046direct (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×C22⋊C42
C22×#67192
C22×#67122
C22×C4×Dic152
C2×#911152
#117510462
C2×#652343
C2×#49455