Finite Group Databasebeta

← C2×#11752878 #14480548

Subgroups — C2×#11752878 #14480548

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×#1204C211yesmaximal, characteristic, complemented
2C22×#6779C211yesmaximal, characteristic
3C2×#91775C211yesmaximal, characteristic, complemented
4C2×C5×#877C211yesmaximal, characteristic, complemented
5C2×C3×#4032C211yesmaximal, characteristic, complemented
6C2×#91546C211yesmaximal, characteristic, complemented
7C2×#91537C211yesmaximal, characteristic, complemented
8C2×#91469C212yesmaximal, complemented
9C2×#91429C212yesmaximal, complemented
10C2×#91422C212yesmaximal, complemented
11C2×#91353C212yesmaximal, complemented
12#11752878C2116yesmaximal, direct product factor
13C2×#65860—31nomaximal, Hall
14C22×#1220C2211yescharacteristic
15C23×Dic15C2211yescharacteristic
16C23×C5×Dic3C2211yescharacteristic
17C22×C3×C22⋊D5C2211yescharacteristic
18C23×C3×Dic5C2211yescharacteristic
19C22×#1204—28no—
20C22×#1204C2211yescharacteristic
21C22×#1204C2212yes—
22C22×#1204C2212yes—
23C22×#1204C2218yescomplemented
24C22×#1202C2211yescharacteristic
25C22×#1200C2211yescharacteristic
26C2×#6779—28no—
27C2×#6779C2218yes—
28C22×D5×Dic3C2211yescharacteristic
29C2×C3×C5×C22⋊C4C2211yescharacteristic
30C2×#6714C2212yes—
31C2×C4⋊Dic15C2212yes—
32#91775C2218yescomplemented
33C2×C5×#217C2211yescharacteristic
34C2×C5×Dic3⋊C4C2212yes—
35C2×C4×C5×Dic3C2212yes—
36C5×#877C2218yescomplemented
37C2×C3×#685C2212yes—
38C2×C3×C4⋊Dic5C2212yes—
39C3×#4032C2218yescomplemented
40#91546C2218yescomplemented
41C2×Dic3⋊Dic5C2212yes—
42C2×Dic3⋊Dic5C2212yes—
43C2×#6667C2212yes—
44#91537C2218yescomplemented
45C2×#6665C2212yes—
46#91469C22116yescomplemented
47#91429C22116yescomplemented
48#91422C22116yescomplemented
49#91353C22116yescomplemented
50C2×#5648—51nomaximal, Hall
51C22×#709—31no—
52C23×Dic10—31no—
53C2×C5×#117—31no—
54#65860—316no—
55C2×#4043—32no—
56C2×#4043—32no—
57C2×#4032—31no—
58C2×#4032—31no—
59C2×#4032—31no—
60C2×#4032D311yescharacteristic, complemented
61C2×#4021—32no—
62C2×#4015—32no—
63C24×C3×C5C2311yescharacteristic
64C23×D15C2311yescharacteristic
65C23×C3×D5C2311yescharacteristic
66C22×C3×C4×C5C2312yes—
67C2×#1220—24no—
68C2×#1220C2314yes—
69C22×Dic15—24no—
70C22×Dic15C2314yes—
71C22×Dic15C2311yescharacteristic
72C22×Dic15C2312yes—
73C22×C5×Dic3D418yes—
74C22×C5×Dic3C2312yes—
75C22×C5×Dic3C2314yes—
76C22×C5×Dic3C2311yescharacteristic
77C22×C5×Dic3C2311yescharacteristic
78C2×C3×C22⋊D5—24no—
79C2×C3×C22⋊D5C2314yes—
80C22×C3×Dic5—24no—
81C22×C3×Dic5C2311yescharacteristic
82C22×C3×Dic5C2312yes—
83C22×C3×Dic5C2314yes—
84C2×#1204—24no—
85C2×#1204—28no—
86C2×#1204—28no—
87C2×#1204—232no—
88C2×#1204C2318yes—
89C2×#1204C2314yes—
90C2×#1204C2318yes—
91C2×#1202—24no—
92C2×#1202C2314yes—
93C2×#1200—24no—
94C2×#1200C2314yes—
95#6779—232no—
96C2×D5×Dic3—24no—
97C2×D5×Dic3C2314yes—
98C3×C5×C22⋊C4C2314yes—
99#6714C2318yes—
100C4⋊Dic15C2318yes—
101C5×#217C2314yes—
102C5×Dic3⋊C4C2318yes—
103C4×C5×Dic3C2318yes—
104C3×#685C2318yes—
105C3×C4⋊Dic5C2318yes—
106Dic3⋊Dic5C2318yes—
107Dic3⋊Dic5C2318yes—
108#6667C2318yes—
109#6665C2318yes—
110C22×#237—51no—
111C23×Dic6—51no—
112C2×C3×#123—51no—
113C2×#935—51no—
114C2×#929—51no—
115#5648—516no—
116C2×#896—52no—
117C2×#888—52no—
118C2×#886—51no—
119C2×#885—52no—
120C2×#878—52no—
121C2×#877D511yescharacteristic, complemented
122C23×C4×C5—31no—
123C22×C22⋊D5—31no—
124C22×C22⋊D5C2×D311yescharacteristic
125C23×Dic5—31no—
126C23×Dic5C2×D311yescharacteristic
127C2×#709—38no—
128C2×#709—68no—
129C22×D20—31no—
130C22×C4×D5—31no—
131C22×C4×D5—31no—
132C22×Dic10—31no—
133C22×Dic10—32no—
134C22×Dic10—32no—
135C22×Dic10—38no—
136C22×Dic10—68no—
137C5×#117—38no—
138C2×C5×C4⋊C4—32no—
139C2×C5×C22⋊C4—31no—
140C2×C5×C22⋊C4C2×D311yescharacteristic
141C2×C42×C5—32no—
142C2×#685—32no—
143C2×#685—32no—
144C2×#685—32no—
145C2×#685C2×D312yes—
146C2×C4⋊Dic5—32no—
147C2×C4⋊Dic5—32no—
148C2×C4⋊Dic5—32no—
149C2×C4⋊Dic5C2×D312yes—
150#4043—316no—
151#4043—316no—
152#4032—38no—
153#4032—38no—
154#4032—38no—
155#4032C2×D318yescomplemented
156#4021—316no—
157#4015—316no—
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×C3×D5—22no—
167C22×C3×D5—24no—
168C22×C3×D5C2412yes—
169C2×C3×C4×C5—24no—
170C2×C3×C4×C5C2414yes—
171#1220—28no—
172C2×Dic15—22no—
173C2×Dic15—28no—
174C2×Dic15—24no—
175C2×Dic15C2414yes—
176C2×Dic15C2412yes—
177C2×C5×Dic3—24no—
178C2×C5×Dic3C2×D414yes—
179C2×C5×Dic3C2×D4116yes—
180C2×C5×Dic3C2×D414yes—
181C2×C5×Dic3C2414yes—
182C2×C5×Dic3C2412yes—
183C2×C5×Dic3C2412yes—
184C3×C22⋊D5—28no—
185C2×C3×Dic5—22no—
186C2×C3×Dic5—28no—
187C2×C3×Dic5—24no—
188C2×C3×Dic5C2414yes—
189C2×C3×Dic5C2412yes—
190#1204—28no—
191#1204—216no—
192#1204—216no—
193#1202—28no—
194#1200—28no—
195D5×Dic3—28no—
196C22×C3×D4—51no—
197C23×C3×C4—51no—
198C22×#67—51no—
199C22×#67—51no—
200C23×Dic3—51no—
201C23×Dic3—51no—
202C23×Dic3C2×D511yescharacteristic
203C2×#237—58no—
204C2×#237—108no—
205C22×C4×D3—51no—
206C22×Dic6—58no—
207C22×Dic6—52no—
208C22×Dic6—52no—
209C22×Dic6—51no—
210C22×Dic6—108no—
211C3×#123—58no—
212C2×C3×C4⋊C4—52no—
213C2×C3×C22⋊C4—52no—
214C2×C3×C22⋊C4C2×D511yescharacteristic
215C2×#217—52no—
216C2×#217C2×D511yescharacteristic
217#935—58no—
218#929—58no—
219C2×(C2×D3)⋊C4—52no—
220C2×(C2×D3)⋊C4—52no—
221C2×C4⋊Dic3—52no—
222C2×Dic3⋊C4—52no—
223C2×Dic3⋊C4—52no—
224C2×Dic3⋊C4C2×D512yes—
225C2×C4×Dic3C2×D512yes—
226#896—516no—
227#888—516no—
228#886—58no—
229#885—516no—
230#878—516no—
231#877C2×D518yescomplemented
232C24×C5C22×D311yescharacteristic
233C23×D5—31no—
234C23×D5C22×D311yescharacteristic
235C22×C4×C5—38no—
236C22×C4×C5—31no—
237C22×C4×C5—31no—
238C22×C4×C5—34no—
239C22×C4×C5—32no—
240C22×C4×C5C22×D312yes—
241C2×C22⋊D5—24no—
242C2×C22⋊D5—34no—
243C2×C22⋊D5—64no—
244C2×C22⋊D5C22×D314yes—
245C22×Dic5—24no—
246C22×Dic5—34no—
247C22×Dic5—32no—
248C22×Dic5—31no—
249C22×Dic5—64no—
250C22×Dic5C22×D314yes—
251C22×Dic5C22×D312yes—
252C22×Dic5C22×D311yescharacteristic
253#709—632no—
254C2×D20—34no—
255C2×D20—64no—
256C2×C4×D5—34no—
257C2×C4×D5—34no—
258C2×C4×D5—64no—
259C2×C4×D5—64no—
260C2×Dic10—38no—
261C2×Dic10—38no—
262C2×Dic10—34no—
263C2×Dic10—68no—
264C2×Dic10—68no—
265C2×Dic10—632no—
266C2×Dic10—64no—
267C5×C4⋊C4—38no—
268C5×C22⋊C4—34no—
269C5×C22⋊C4C22×D314yes—
270C42×C5—38no—
271#685—38no—
272#685—38no—
273#685—38no—
274#685C22×D318yes—
275C4⋊Dic5—38no—
276C4⋊Dic5—38no—
277C4⋊Dic5—38no—
278C4⋊Dic5C22×D318yes—
279C22×#41—151no—
280C23×Q8—151no—
281#536—1516no—
282C2×C4⋊Q8—152no—
283C2×#124—152no—
284C2×#123—51no—
285C2×#123—151no—
286C2×#123—151no—
287C2×#123—151no—
288C2×C22⋊Q8—152no—
289C2×C22⋊Q8—152no—
290C2×#117—31no—
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×C3×D5—22no—
301C2×C3×D5—24no—
302C2×C3×D5—44no—
303C3×C4×C5—24no—
304Dic15—22no—
305Dic15—24no—
306C3×Dic5—22no—
307C3×Dic5—24no—
308C5×Dic3—24no—
309C5×Dic3C22×D414yes—
310C5×Dic3C22×D414yes—
311C24×C3—51no—
312C24×C3C22×D511yescharacteristic
313C23×D3—51no—
314C2×C3×D4—54no—
315C2×C3×D4—104no—
316C22×C3×C4—51no—
317C22×C3×C4—52no—
318C22×C3×C4—54no—
319C22×C3×C4—104no—
320C22×C3×C4C22×D512yes—
321C2×#67—54no—
322C2×#67—54no—
323C2×#67—104no—
324C2×#67—104no—
325C22×Dic3—54no—
326C22×Dic3—51no—
327C22×Dic3—52no—
328C22×Dic3—54no—
329C22×Dic3—104no—
330C22×Dic3—104no—
331C22×Dic3D2018yes—
332C22×Dic3C22×D512yes—
333C22×Dic3C22×D514yes—
334C22×Dic3C22×D511yescharacteristic
335C22×Dic3C22×D511yescharacteristic
336#237—1032no—
337C2×C4×D3—54no—
338C2×C4×D3—104no—
339C2×Dic6—58no—
340C2×Dic6—58no—
341C2×Dic6—54no—
342C2×Dic6—108no—
343C2×Dic6—108no—
344C2×Dic6—1032no—
345C2×Dic6—104no—
346C3×C4⋊C4—58no—
347C3×C22⋊C4—58no—
348C3×C22⋊C4C22×D514yes—
349#217—58no—
350#217C22×D514yes—
351(C2×D3)⋊C4—58no—
352(C2×D3)⋊C4—58no—
353C4⋊Dic3—58no—
354Dic3⋊C4—58no—
355Dic3⋊C4—58no—
356Dic3⋊C4C22×D518yes—
357C4×Dic3C22×D518yes—
358C23×C5—24no—
359C23×C5D3×D414yes—
360C23×C5C23×D312yes—
361C23×C5C23×D311yescharacteristic
362C22×D5—22no—
363C22×D5—24no—
364C22×D5—32no—
365C22×D5—62no—
366C22×D5—64no—
367C22×D5C23×D312yes—
368C2×C4×C5—24no—
369C2×C4×C5—34no—
370C2×C4×C5—34no—
371C2×C4×C5—316no—
372C2×C4×C5—32no—
373C2×C4×C5—32no—
374C2×C4×C5—34no—
375C2×C4×C5—64no—
376C2×C4×C5C23×D314yes—
377C22⋊D5—28no—
378C22⋊D5—68no—
379C2×Dic5—28no—
380C2×Dic5—24no—
381C2×Dic5—22no—
382C2×Dic5—32no—
383C2×Dic5—34no—
384C2×Dic5—68no—
385C2×Dic5—64no—
386C2×Dic5—62no—
387C2×Dic5C23×D312yes—
388C2×Dic5C23×D314yes—
389D20—68no—
390C4×D5—68no—
391C4×D5—68no—
392Dic10—616no—
393Dic10—616no—
394Dic10—68no—
395C2×#41—158no—
396C2×#41—308no—
397C22×Q8—152no—
398C22×Q8—152no—
399C22×Q8—151no—
400C22×Q8—158no—
401C22×Q8—308no—
402C22×D4—51no—
403C22×D4—151no—
404C22×D4—151no—
405C23×C4—31no—
406C23×C4—51no—
407C23×C4—151no—
408C23×C4—151no—
409C23×C4—151no—
410C4⋊Q8—1516no—
411#124—1516no—
412#123—58no—
413#123—158no—
414#123—158no—
415#123—158no—
416C22⋊Q8—1516no—
417C22⋊Q8—1516no—
418#117—38no—
419C2×C4⋊C4—32no—
420C2×C4⋊C4—52no—
421C2×C4⋊C4—152no—
422C2×C4⋊C4—152no—
423C2×C4⋊C4—152no—
424C2×C22⋊C4—31no—
425C2×C22⋊C4—52no—
426C2×C22⋊C4—152no—
427C2×C22⋊C4—152no—
428C2×C22⋊C4—152no—
429C2×C22⋊C4D3×D511yescharacteristic
430C2×C42—32no—
431C2×C3×C5—24no—
432C2×C3×C5#53614yescomplemented
433C2×C3×C5C23×D411yescharacteristic
434C2×C3×C5C2×2−1+412yes—
435D15—42no—
436C3×D5—42no—
437C23×C3—24no—
438C23×C3—52no—
439C23×C3—102no—
440C23×C3—104no—
441C23×C3C2×D2014yes—
442C23×C3C23×D512yes—
443C23×C3C23×D511yescharacteristic
444C22×D3—52no—
445C22×D3—102no—
446C22×D3—104no—
447C3×D4—108no—
448C2×C3×C4—24no—
449C2×C3×C4—52no—
450C2×C3×C4—54no—
451C2×C3×C4—108no—
452C2×C3×C4—104no—
453C2×C3×C4—102no—
454C2×C3×C4C23×D514yes—
455#67—108no—
456#67—108no—
457C2×Dic3—24no—
458C2×Dic3—52no—
459C2×Dic3—54no—
460C2×Dic3—108no—
461C2×Dic3—104no—
462C2×Dic3—102no—
463C2×Dic3—108no—
464C2×Dic3C2×D2014yes—
465C2×Dic3C2×D2014yes—
466C2×Dic3C2×D20116yes—
467C2×Dic3C23×D514yes—
468C2×Dic3C23×D512yes—
469C2×Dic3C23×D512yes—
470C4×D3—108no—
471Dic6—1016no—
472Dic6—1016no—
473Dic6—108no—
474C22×C5—28no—
475C22×C5—24no—
476C22×C5C2×D3×D414yes—
477C22×C5C2×D3×D412yes—
478C22×C5#101014yes—
479C22×C5C24×D311yescharacteristic
480C2×D5—22no—
481C2×D5—24no—
482C2×D5—44no—
483C2×D5—62no—
484C2×D5—64no—
485C2×D5—124no—
486C4×C5—24no—
487C4×C5—34no—
488C4×C5—34no—
489C4×C5—64no—
490Dic5—24no—
491Dic5—22no—
492Dic5—64no—
493Dic5—62no—
494C24—51no—
495C24—151no—
496C24C2×D3×D511yescharacteristic
497#41—3032no—
498C2×Q8—158no—
499C2×Q8—158no—
500C2×Q8—154no—
501C2×Q8—308no—
502C2×Q8—308no—
503C2×Q8—3032no—
504C2×Q8—304no—
505C2×D4—54no—
506C2×D4—104no—
507C2×D4—154no—
508C2×D4—154no—
509C2×D4—304no—
510C2×D4—304no—
511C22×C4—38no—
512C22×C4—34no—
513C22×C4—31no—
514C22×C4—31no—
515C22×C4—32no—
516C22×C4—54no—
517C22×C4—52no—
518C22×C4—51no—
519C22×C4—104no—
520C22×C4—154no—
521C22×C4—154no—
522C22×C4—154no—
523C22×C4—152no—
524C22×C4—151no—
525C22×C4—304no—
526C22×C4—304no—
527C22×C4—304no—
528C22×C4C2×D3×D512yes—
529C4⋊C4—38no—
530C4⋊C4—58no—
531C4⋊C4—158no—
532C4⋊C4—158no—
533C4⋊C4—158no—
534C22⋊C4—34no—
535C22⋊C4—58no—
536C22⋊C4—158no—
537C22⋊C4—158no—
538C22⋊C4—158no—
539C22⋊C4C2×D3×D514yes—
540C42—38no—
541C3×C5C2×#53611yescharacteristic, complemented, Hall
542C22×C3—28no—
543C22×C3—24no—
544C22×C3—102no—
545C22×C3—104no—
546C22×C3—204no—
547C22×C3C22×D2014yes—
548C22×C3C22×D2012yes—
549C22×C3#415014yes—
550C22×C3C24×D511yescharacteristic
551C2×D3—102no—
552C2×D3—104no—
553C2×D3—204no—
554C3×C4—24no—
555C3×C4—104no—
556C3×C4—102no—
557Dic3—24no—
558Dic3—104no—
559Dic3—102no—
560Dic3C22×D2014yes—
561Dic3C22×D2014yes—
562C2×C5—24no—
563C2×C5#564814yescomplemented
564C2×C5C22×D3×D411yescharacteristic
565C2×C5C2×#101012yes—
566D5—42no—
567D5—122no—
568C23—24no—
569C23—52no—
570C23—102no—
571C23—104no—
572C23—152no—
573C23—302no—
574C23—304no—
575C23D3×D2014yes—
576C23C22×D3×D512yes—
577C23C22×D3×D511yescharacteristic
578Q8—3016no—
579Q8—3016no—
580Q8—308no—
581D4—108no—
582D4—308no—
583D4—308no—
584C2×C4—24no—
585C2×C4—34no—
586C2×C4—34no—
587C2×C4—316no—
588C2×C4—32no—
589C2×C4—32no—
590C2×C4—34no—
591C2×C4—54no—
592C2×C4—52no—
593C2×C4—64no—
594C2×C4—108no—
595C2×C4—104no—
596C2×C4—102no—
597C2×C4—154no—
598C2×C4—152no—
599C2×C4—308no—
600C2×C4—308no—
601C2×C4—302no—
602C2×C4—308no—
603C2×C4—304no—
604C2×C4C22×D3×D514yes—
605C2×C3—24no—
606C2×C3—202no—
607C2×C3#6586014yescomplemented
608C2×C3C23×D2011yescharacteristic
609C2×C3C2×#415012yes—
610D3—202no—
611C5C2×#564811yescharacteristic, complemented, Sylow
612C22—28no—
613C22—24no—
614C22—102no—
615C22—104no—
616C22—204no—
617C22—302no—
618C22—304no—
619C22—604no—
620C22C2×D3×D2014yes—
621C22C2×D3×D2012yes—
622C22#9202014yes—
623C22C23×D3×D511yescharacteristic
624C4—24no—
625C4—34no—
626C4—34no—
627C4—64no—
628C4—104no—
629C4—102no—
630C4—302no—
631C4—304no—
632C3C2×#6586011yescharacteristic, complemented, Sylow
633C2—24no—
634C2—202no—
635C2—602no—
636C2#1175287814yesdirect product factor
637C2C22×D3×D2011yescharacteristic
638C2C2×#9202012yes—
subgroupG/H
centerC23C22×D3×D5
commutator, cosocleC22×C3×C5C25
FrattiniC22C23×D3×D5
FittingC2×C3×C5×C22⋊C4C22
socleC23×C3×C5C24
prime radicalC22×#6779C2
hypercenterC2×C22⋊C4D3×D5
hypocenterC3×C5C2×#536
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C23×#1204C2semidirect
3. C2×#91775C2semidirect
4. C2×C5×#877C2semidirect
5. C2×C3×#4032C2semidirect
6. C2×#91546C2semidirect
7. C2×#91537C2semidirect
8. C2×#91469C2semidirect
9. C2×#91429C2semidirect
10. C2×#91422C2semidirect
11. C2×#91353C2semidirect
12. #11752878C2direct (N × H)
23. C22×#1204C22semidirect
32. #91775C22semidirect
36. C5×#877C22semidirect
39. C3×#4032C22semidirect
40. #91546C22semidirect
44. #91537C22semidirect
46. #91469C22semidirect
47. #91429C22semidirect
48. #91422C22semidirect
49. #91353C22semidirect
60. C2×#4032D3semidirect
121. C2×#877D5semidirect
155. #4032C2×D3semidirect
231. #877C2×D5semidirect
432. C2×C3×C5#536semidirect
541. C3×C5C2×#536semidirect
563. C2×C5#5648semidirect
607. C2×C3#65860semidirect
611. C5C2×#5648semidirect
632. C3C2×#65860semidirect
636. C2#11752878direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#12042
C22×#67792
C2×#917752
C2×C5×#8772
C2×C3×#40322
C2×#915462
C2×#915372
C2×#914692
C2×#914292
C2×#914222
C2×#913532
#117528782
C2×#658603
C2×#56485