Finite Group Databasebeta

← C2×#11754317 #14480545

Subgroups — C2×#11754317 #14480545

Order 1920.

3024 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
1C23×C3×Dic10C211yesmaximal, characteristic, complemented
2C22×#6782C211yesmaximal, characteristic
3C2×#91818C211yesmaximal, characteristic, complemented
4C2×C5×#929C211yesmaximal, characteristic, complemented
5C2×#91546C211yesmaximal, characteristic, complemented
6C2×#91539C211yesmaximal, characteristic, complemented
7C2×#91528C211yesmaximal, characteristic, complemented
8C2×#91411C212yesmaximal, complemented
9#11754317C2116yesmaximal, direct product factor
10C2×#91410C212yesmaximal, complemented
11C2×#91327C212yesmaximal, complemented
12C2×#91326C212yesmaximal, complemented
13C2×#65982—31nomaximal, Hall
14C23×C3×C4×C5C2211yescharacteristic
15C22×#1220C2211yescharacteristic
16C22×C5×#67C2211yescharacteristic
17C23×C3×Dic5C2211yescharacteristic
18C23×C3×Dic5C2211yescharacteristic
19C22×C3×Dic10—28no—
20C22×C3×Dic10C2212yes—
21C22×C3×Dic10C2212yes—
22C22×C3×Dic10C2218yescomplemented
23C22×#1204C2211yescharacteristic
24C22×#1203C2211yescharacteristic
25C22×#1200C2211yescharacteristic
26C2×#6782—28no—
27C2×#6782C2218yes—
28C22×D3×Dic5C2211yescharacteristic
29C2×#6719C2211yescharacteristic
30#91818C2218yescomplemented
31C2×#6714C2212yes—
32C2×#6712C2212yes—
33C2×C5×#217C2211yescharacteristic
34C5×#929C2218yescomplemented
35C2×C5×(C2×D3)⋊C4C2212yes—
36C2×C5×Dic3⋊C4C2212yes—
37#91546C2218yescomplemented
38C2×Dic3⋊Dic5C2212yes—
39C2×#6669C2212yes—
40C2×#6668C2212yes—
41C2×#6667C2212yes—
42#91539C2218yescomplemented
43C2×(C2×D3)⋊Dic5C2212yes—
44#91528C2218yescomplemented
45C2×Dic3×Dic5C2212yes—
46#91411C22116yescomplemented
47#91410C22116yescomplemented
48#91327C22116yescomplemented
49#91326C22116yescomplemented
50C2×#5816—51nomaximal, Hall
51C22×#711—31no—
52C23×Dic10D311yescharacteristic, complemented
53C2×C5×#123—31no—
54C2×#4081—31no—
55C2×#4075—31no—
56#65982—316no—
57C2×#4042—32no—
58C2×#4034—32no—
59C2×#4032—31no—
60C2×#4031—32no—
61C2×#4024—32no—
62C2×#4023—31no—
63C24×C3×C5C2311yescharacteristic
64C23×D15C2311yescharacteristic
65C23×C5×D3C2311yescharacteristic
66C22×C3×C4×C5—24no—
67C22×C3×C4×C5C2312yes—
68C22×C3×C4×C5C2314yes—
69C2×#1220—24no—
70C2×#1220C2314yes—
71C22×Dic15C2312yes—
72C22×Dic15C2311yescharacteristic
73C2×C5×#67—24no—
74C2×C5×#67C2314yes—
75C22×C5×Dic3C2311yescharacteristic
76C22×C5×Dic3C2312yes—
77C22×C3×Dic5—24no—
78C22×C3×Dic5D418yes—
79C22×C3×Dic5C2312yes—
80C22×C3×Dic5C2314yes—
81C22×C3×Dic5C2311yescharacteristic
82C22×C3×Dic5C2311yescharacteristic
83C22×C3×Dic5C2314yes—
84C2×C3×Dic10—28no—
85C2×C3×Dic10—28no—
86C2×C3×Dic10—232no—
87C2×C3×Dic10C2318yes—
88C2×C3×Dic10C2318yes—
89C2×#1204—24no—
90C2×#1204C2314yes—
91C2×#1203—24no—
92C2×#1203C2314yes—
93C2×#1200—24no—
94C2×#1200C2314yes—
95#6782—232no—
96C2×D3×Dic5—24no—
97C2×D3×Dic5C2314yes—
98#6719C2314yes—
99#6714C2318yes—
100#6712C2318yes—
101C5×#217C2314yes—
102C5×(C2×D3)⋊C4C2318yes—
103C5×Dic3⋊C4C2318yes—
104Dic3⋊Dic5C2318yes—
105#6669C2318yes—
106#6668C2318yes—
107#6667C2318yes—
108(C2×D3)⋊Dic5C2318yes—
109Dic3×Dic5C2318yes—
110C23×C3×Q8—51no—
111C22×#235—51no—
112#5816—516no—
113C2×#947—52no—
114C2×#946—52no—
115C2×#946—52no—
116C2×#944—52no—
117C2×#929—51no—
118C2×#929—51no—
119C2×#929—51no—
120C2×#929D511yescharacteristic, complemented
121C2×#926—51no—
122C22×C5×D4—31no—
123C23×C4×C5C2×D311yescharacteristic
124C22×C22⋊D5—31no—
125C22×C22⋊D5—31no—
126C23×Dic5—31no—
127C23×Dic5C2×D311yescharacteristic
128C23×Dic5C2×D311yescharacteristic
129C2×#711—38no—
130C2×#711—68no—
131C22×C4×D5—31no—
132C22×Dic10—28no—
133C22×Dic10—31no—
134C22×Dic10C2×D312yes—
135C22×Dic10C2×D312yes—
136C22×Dic10C2×D318yescomplemented
137C5×#123—38no—
138C2×C5×C4⋊C4—32no—
139C2×C5×C22⋊C4—31no—
140C2×C5×C22⋊C4—32no—
141C2×C22⋊Dic5—31no—
142C2×C22⋊Dic5—32no—
143#4081—38no—
144#4075—38no—
145C2×#685—32no—
146C2×#685—32no—
147C2×C4⋊Dic5—32no—
148C2×#683—32no—
149C2×#683—32no—
150C2×#683—32no—
151C2×C4×Dic5—32no—
152#4042—316no—
153#4034—316no—
154#4032—38no—
155#4031—316no—
156#4024—316no—
157#4023—38no—
158C2×#536—151noSylow
159C23×C3×C5—24no—
160C23×C3×C5C2×D414yes—
161C23×C3×C5C2412yes—
162C23×C3×C5C2411yescharacteristic
163C22×D15—22no—
164C22×D15—24no—
165C22×D15C2412yes—
166C22×C5×D3—22no—
167C22×C5×D3—24no—
168C22×C5×D3C2412yes—
169C2×C3×C4×C5—28no—
170C2×C3×C4×C5—24no—
171C2×C3×C4×C5C2414yes—
172#1220—28no—
173C2×Dic15—22no—
174C2×Dic15—24no—
175C2×Dic15C2414yes—
176C2×Dic15C2412yes—
177C5×#67—28no—
178C2×C5×Dic3—22no—
179C2×C5×Dic3—24no—
180C2×C5×Dic3C2414yes—
181C2×C5×Dic3C2412yes—
182C2×C3×Dic5—28no—
183C2×C3×Dic5—24no—
184C2×C3×Dic5C2×D414yes—
185C2×C3×Dic5C2×D4116yes—
186C2×C3×Dic5C2×D414yes—
187C2×C3×Dic5C2414yes—
188C2×C3×Dic5C2412yes—
189C2×C3×Dic5C2412yes—
190C3×Dic10—216no—
191C3×Dic10—216no—
192#1204—28no—
193#1203—28no—
194#1200—28no—
195D3×Dic5—28no—
196C22×C3×Q8—52no—
197C22×C3×Q8—52no—
198C22×C3×Q8—58no—
199C22×C3×Q8—108no—
200C23×C3×C4—51no—
201C23×C3×C4—51no—
202C23×C3×C4C2×D511yescharacteristic
203C22×#67—51no—
204C22×#67C2×D511yescharacteristic
205C2×#235—58no—
206C2×#235—108no—
207C22×D12—51no—
208C22×C4×D3—51no—
209C22×C4×D3—51no—
210C22×Dic6—51no—
211C2×#217—51no—
212C2×#217C2×D511yescharacteristic
213#947—516no—
214#946—516no—
215#946—516no—
216#944—516no—
217#929—58no—
218#929—58no—
219#929—58no—
220#929C2×D518yescomplemented
221C2×(C2×D3)⋊C4—52no—
222C2×(C2×D3)⋊C4—52no—
223C2×(C2×D3)⋊C4—52no—
224C2×(C2×D3)⋊C4C2×D512yes—
225#926—58no—
226C2×C4⋊Dic3—52no—
227C2×Dic3⋊C4—52no—
228C2×Dic3⋊C4—52no—
229C2×Dic3⋊C4—52no—
230C2×Dic3⋊C4C2×D512yes—
231C2×C4×Dic3—52no—
232C24×C5—31no—
233C24×C5C22×D311yescharacteristic
234C23×D5—31no—
235C2×C5×D4—34no—
236C2×C5×D4—64no—
237C22×C4×C5—24no—
238C22×C4×C5—32no—
239C22×C4×C5—31no—
240C22×C4×C5C22×D312yes—
241C22×C4×C5C22×D314yes—
242C2×C22⋊D5—34no—
243C2×C22⋊D5—34no—
244C2×C22⋊D5—64no—
245C2×C22⋊D5—64no—
246C22×Dic5—24no—
247C22×Dic5—32no—
248C22×Dic5—31no—
249C22×Dic5—34no—
250C22×Dic5—64no—
251C22×Dic5#6718yes—
252C22×Dic5C22×D312yes—
253C22×Dic5C22×D314yes—
254C22×Dic5C22×D311yescharacteristic
255C22×Dic5C22×D311yescharacteristic
256C22×Dic5C22×D314yes—
257#711—632no—
258C2×C4×D5—34no—
259C2×C4×D5—64no—
260C2×Dic10—28no—
261C2×Dic10—28no—
262C2×Dic10—232no—
263C2×Dic10—34no—
264C2×Dic10—64no—
265C2×Dic10C22×D318yes—
266C2×Dic10C22×D318yes—
267C5×C4⋊C4—38no—
268C5×C22⋊C4—34no—
269C5×C22⋊C4—38no—
270C22⋊Dic5—34no—
271C22⋊Dic5—38no—
272#685—38no—
273#685—38no—
274C4⋊Dic5—38no—
275#683—38no—
276#683—38no—
277#683—38no—
278C4×Dic5—38no—
279C22×#41—151no—
280C23×Q8—51no—
281#536—1516no—
282C2×C4⋊Q8—152no—
283C2×#124—152no—
284C2×#123—31no—
285C2×#123—151no—
286C2×#123—151no—
287C2×#123—151no—
288C2×C22⋊Q8—152no—
289C2×C22⋊Q8—152no—
290C2×#117—151no—
291C22×C3×C5—28no—
292C22×C3×C5—24no—
293C22×C3×C5C22×D414yes—
294C22×C3×C5C22×D412yes—
295C22×C3×C52−1+414yes—
296C22×C3×C5C2511yescharacteristic
297C2×D15—22no—
298C2×D15—24no—
299C2×D15—44no—
300C2×C5×D3—22no—
301C2×C5×D3—24no—
302C2×C5×D3—44no—
303C3×C4×C5—24no—
304Dic15—22no—
305Dic15—24no—
306C3×Dic5—24no—
307C3×Dic5C22×D414yes—
308C3×Dic5C22×D414yes—
309C5×Dic3—22no—
310C5×Dic3—24no—
311C24×C3C22×D511yescharacteristic
312C23×D3—51no—
313C23×D3C22×D511yescharacteristic
314C2×C3×Q8—58no—
315C2×C3×Q8—58no—
316C2×C3×Q8—108no—
317C2×C3×Q8—108no—
318C2×C3×Q8—1032no—
319C22×C3×C4—24no—
320C22×C3×C4—58no—
321C22×C3×C4—51no—
322C22×C3×C4—51no—
323C22×C3×C4—54no—
324C22×C3×C4—52no—
325C22×C3×C4—54no—
326C22×C3×C4—104no—
327C22×C3×C4C22×D512yes—
328C22×C3×C4C22×D514yes—
329C2×#67—24no—
330C2×#67—54no—
331C2×#67—104no—
332C2×#67C22×D514yes—
333C22×Dic3—52no—
334C22×Dic3—51no—
335C22×Dic3C22×D512yes—
336C22×Dic3C22×D511yescharacteristic
337#235—1032no—
338C2×D12—54no—
339C2×D12—104no—
340C2×C4×D3—54no—
341C2×C4×D3—54no—
342C2×C4×D3—104no—
343C2×C4×D3—104no—
344C2×Dic6—54no—
345C2×Dic6—104no—
346#217—54no—
347#217C22×D514yes—
348(C2×D3)⋊C4—58no—
349(C2×D3)⋊C4—58no—
350(C2×D3)⋊C4—58no—
351(C2×D3)⋊C4C22×D518yes—
352C4⋊Dic3—58no—
353Dic3⋊C4—58no—
354Dic3⋊C4—58no—
355Dic3⋊C4—58no—
356Dic3⋊C4C22×D518yes—
357C4×Dic3—58no—
358C23×C5—24no—
359C23×C5—32no—
360C23×C5—64no—
361C23×C5—62no—
362C23×C5C2×#6714yes—
363C23×C5C23×D312yes—
364C23×C5C23×D311yescharacteristic
365C22×D5—32no—
366C22×D5—64no—
367C22×D5—62no—
368C5×D4—68no—
369C2×C4×C5—28no—
370C2×C4×C5—24no—
371C2×C4×C5—34no—
372C2×C4×C5—32no—
373C2×C4×C5—64no—
374C2×C4×C5—62no—
375C2×C4×C5C23×D314yes—
376C22⋊D5—68no—
377C22⋊D5—68no—
378C2×Dic5—28no—
379C2×Dic5—24no—
380C2×Dic5—34no—
381C2×Dic5—32no—
382C2×Dic5—64no—
383C2×Dic5—62no—
384C2×Dic5—68no—
385C2×Dic5C2×#6714yes—
386C2×Dic5C2×#6714yes—
387C2×Dic5C2×#67116yes—
388C2×Dic5C23×D314yes—
389C2×Dic5C23×D312yes—
390C2×Dic5C23×D312yes—
391C4×D5—68no—
392Dic10—216no—
393Dic10—216no—
394Dic10—68no—
395C2×#41—158no—
396C2×#41—308no—
397C22×Q8—52no—
398C22×Q8—52no—
399C22×Q8—58no—
400C22×Q8—108no—
401C22×Q8—151no—
402C22×D4—31no—
403C22×D4—151no—
404C22×D4—151no—
405C23×C4—51no—
406C23×C4—51no—
407C23×C4—151no—
408C23×C4—151no—
409C23×C4D3×D511yescharacteristic
410C4⋊Q8—1516no—
411#124—1516no—
412#123—38no—
413#123—158no—
414#123—158no—
415#123—158no—
416C22⋊Q8—1516no—
417C22⋊Q8—1516no—
418#117—158no—
419C2×C4⋊C4—32no—
420C2×C4⋊C4—152no—
421C2×C4⋊C4—152no—
422C2×C4⋊C4—152no—
423C2×C4⋊C4—152no—
424C2×C22⋊C4—31no—
425C2×C22⋊C4—32no—
426C2×C22⋊C4—152no—
427C2×C22⋊C4—152no—
428C2×C22⋊C4—151no—
429C2×C22⋊C4—152no—
430C2×C42—152no—
431C2×C3×C5—24no—
432C2×C3×C5#53614yescomplemented
433C2×C3×C5C23×D411yescharacteristic
434C2×C3×C5C2×2−1+412yes—
435D15—42no—
436C5×D3—42no—
437C23×C3—24no—
438C23×C3D4×D514yes—
439C23×C3C23×D512yes—
440C23×C3C23×D511yescharacteristic
441C22×D3—22no—
442C22×D3—24no—
443C22×D3—52no—
444C22×D3—102no—
445C22×D3—104no—
446C22×D3C23×D512yes—
447C3×Q8—1016no—
448C3×Q8—1016no—
449C2×C3×C4—28no—
450C2×C3×C4—24no—
451C2×C3×C4—54no—
452C2×C3×C4—54no—
453C2×C3×C4—516no—
454C2×C3×C4—52no—
455C2×C3×C4—52no—
456C2×C3×C4—54no—
457C2×C3×C4—108no—
458C2×C3×C4—104no—
459C2×C3×C4C23×D514yes—
460#67—28no—
461#67—108no—
462C2×Dic3—22no—
463C2×Dic3—24no—
464C2×Dic3—54no—
465C2×Dic3—52no—
466C2×Dic3—102no—
467C2×Dic3—104no—
468C2×Dic3C23×D514yes—
469C2×Dic3C23×D512yes—
470D12—108no—
471C4×D3—108no—
472C4×D3—108no—
473Dic6—108no—
474C22×C5—24no—
475C22×C5—28no—
476C22×C5—64no—
477C22×C5—62no—
478C22×C5—124no—
479C22×C5#100714yes—
480C22×C5C22×#6714yes—
481C22×C5C22×#6712yes—
482C22×C5C24×D311yescharacteristic
483C2×D5—64no—
484C2×D5—62no—
485C2×D5—124no—
486C4×C5—24no—
487C4×C5—64no—
488C4×C5—62no—
489Dic5—24no—
490Dic5—64no—
491Dic5—62no—
492Dic5C22×#6714yes—
493Dic5C22×#6714yes—
494C24—31no—
495C24—151no—
496C24C2×D3×D511yescharacteristic
497#41—3032no—
498C2×Q8—58no—
499C2×Q8—58no—
500C2×Q8—108no—
501C2×Q8—108no—
502C2×Q8—1032no—
503C2×Q8—154no—
504C2×Q8—304no—
505C2×D4—34no—
506C2×D4—64no—
507C2×D4—154no—
508C2×D4—154no—
509C2×D4—304no—
510C2×D4—304no—
511C22×C4—24no—
512C22×C4—32no—
513C22×C4—31no—
514C22×C4—58no—
515C22×C4—54no—
516C22×C4—51no—
517C22×C4—51no—
518C22×C4—54no—
519C22×C4—52no—
520C22×C4—104no—
521C22×C4—154no—
522C22×C4—154no—
523C22×C4—152no—
524C22×C4—151no—
525C22×C4—304no—
526C22×C4—304no—
527C22×C4C2×D3×D514yes—
528C22×C4C2×D3×D512yes—
529C4⋊C4—38no—
530C4⋊C4—158no—
531C4⋊C4—158no—
532C4⋊C4—158no—
533C4⋊C4—158no—
534C22⋊C4—34no—
535C22⋊C4—38no—
536C22⋊C4—158no—
537C22⋊C4—158no—
538C22⋊C4—154no—
539C22⋊C4—158no—
540C42—158no—
541C3×C5C2×#53611yescharacteristic, complemented, Hall
542C22×C3—28no—
543C22×C3—24no—
544C22×C3C2×D4×D514yes—
545C22×C3C2×D4×D512yes—
546C22×C3#415014yes—
547C22×C3C24×D511yescharacteristic
548C2×D3—22no—
549C2×D3—24no—
550C2×D3—44no—
551C2×D3—102no—
552C2×D3—104no—
553C2×D3—204no—
554C3×C4—24no—
555C3×C4—54no—
556C3×C4—54no—
557C3×C4—104no—
558Dic3—22no—
559Dic3—24no—
560Dic3—102no—
561Dic3—104no—
562C2×C5—24no—
563C2×C5—122no—
564C2×C5#581614yescomplemented
565C2×C5C2×#100712yes—
566C2×C5C23×#6711yescharacteristic
567D5—122no—
568C23—24no—
569C23—32no—
570C23—64no—
571C23—62no—
572C23—152no—
573C23—304no—
574C23—302no—
575C23D5×#6714yes—
576C23C22×D3×D512yes—
577C23C22×D3×D511yescharacteristic
578Q8—1016no—
579Q8—1016no—
580Q8—308no—
581D4—68no—
582D4—308no—
583D4—308no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—34no—
587C2×C4—32no—
588C2×C4—54no—
589C2×C4—54no—
590C2×C4—516no—
591C2×C4—52no—
592C2×C4—52no—
593C2×C4—54no—
594C2×C4—64no—
595C2×C4—62no—
596C2×C4—108no—
597C2×C4—104no—
598C2×C4—154no—
599C2×C4—152no—
600C2×C4—308no—
601C2×C4—308no—
602C2×C4—304no—
603C2×C4—302no—
604C2×C4C22×D3×D514yes—
605C2×C3—24no—
606C2×C3#6598214yescomplemented
607C2×C3C22×D4×D511yescharacteristic
608C2×C3C2×#415012yes—
609D3—42no—
610D3—202no—
611C5C2×#581611yescharacteristic, complemented, Sylow
612C22—28no—
613C22—24no—
614C22—62no—
615C22—64no—
616C22—124no—
617C22—302no—
618C22—304no—
619C22—604no—
620C22#9200414yes—
621C22C2×D5×#6714yes—
622C22C2×D5×#6712yes—
623C22C23×D3×D511yescharacteristic
624C4—24no—
625C4—54no—
626C4—54no—
627C4—62no—
628C4—64no—
629C4—104no—
630C4—302no—
631C4—304no—
632C3C2×#6598211yescharacteristic, complemented, Sylow
633C2—24no—
634C2—122no—
635C2—602no—
636C2#1175431714yesdirect product factor
637C2C2×#9200412yes—
638C2C22×D5×#6711yescharacteristic
subgroupG/H
centerC23C22×D3×D5
commutator, cosocleC22×C3×C5C25
FrattiniC22C23×D3×D5
FittingC23×C3×C4×C5C22
socleC23×C3×C5C24
prime radicalC22×#6782C2
hypercenterC23×C4D3×D5
hypocenterC3×C5C2×#536
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. C23×C3×Dic10C2semidirect
3. C2×#91818C2semidirect
4. C2×C5×#929C2semidirect
5. C2×#91546C2semidirect
6. C2×#91539C2semidirect
7. C2×#91528C2semidirect
8. C2×#91411C2semidirect
9. #11754317C2direct (N × H)
10. C2×#91410C2semidirect
11. C2×#91327C2semidirect
12. C2×#91326C2semidirect
22. C22×C3×Dic10C22semidirect
30. #91818C22semidirect
34. C5×#929C22semidirect
37. #91546C22semidirect
42. #91539C22semidirect
44. #91528C22semidirect
46. #91411C22semidirect
47. #91410C22semidirect
48. #91327C22semidirect
49. #91326C22semidirect
52. C23×Dic10D3semidirect
120. C2×#929D5semidirect
136. C22×Dic10C2×D3semidirect
220. #929C2×D5semidirect
432. C2×C3×C5#536semidirect
541. C3×C5C2×#536semidirect
564. C2×C5#5816semidirect
606. C2×C3#65982semidirect
611. C5C2×#5816semidirect
632. C3C2×#65982semidirect
636. C2#11754317direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×Dic102
C22×#67822
C2×#918182
C2×C5×#9292
C2×#915462
C2×#915392
C2×#915282
C2×#914112
#117543172
C2×#914102
C2×#913272
C2×#913262
C2×#659823
C2×#58165