Finite Group Databasebeta

← #14331678

Subgroups — #14331678

Order 1920.

1888 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6768C211yesmaximal, characteristic, complemented
2C2×#91782C211yesmaximal, characteristic, complemented
3C2×#91406C212yesmaximal, complemented
4C5×#4991C211yesmaximal, characteristic, complemented
5C3×#65268C211yesmaximal, characteristic, complemented
6#11748932C212yesmaximal, complemented
7#11748929C211yesmaximal, characteristic, complemented
8#11748886C212yesmaximal, complemented
9#11748753C211yesmaximal, characteristic, complemented
10#11748733C212yesmaximal, complemented
11#11748670C211yesmaximal, characteristic, complemented
12#10608620—31nomaximal, Hall
13C22×D60C2211yescharacteristic
14C22×C4×D15C2211yescharacteristic
15C22×C4×D15C2211yescharacteristic
16C22×C5×D12C2211yescharacteristic
17C22×C3×Dic10C2211yescharacteristic
18C22×#1203C2212yes—
19C22×#1203C2212yes—
20C22×D3×Dic5C2212yes—
21C2×#6768—24no—
22C2×#6768—28no—
23C2×C3×C5×C4⋊C4C2211yescharacteristic
24C2×#6714C2211yescharacteristic
25C2×#6714C2211yescharacteristic
26C2×C4×Dic15C2211yescharacteristic
27#91782—24no—
28C2×C5×(C2×D3)⋊C4C2212yes—
29C2×C5×(C2×D3)⋊C4C2212yes—
30C2×C3×#683C2212yes—
31C2×#6667C2212yes—
32C2×#6667C2212yes—
33C2×(C2×D3)⋊Dic5C2212yes—
34C2×Dic3×Dic5C2212yes—
35#91406—28no—
36C5×C3⋊#95C2211yescharacteristic
37C3×#3963C2212yes—
38C3×#3963C2211yescharacteristic
39#90994C2212yes—
40#77169—51nomaximal, Hall
41C22×#711—31no—
42C2×#4039—31no—
43C2×#4027—32no—
44C5×#394—31no—
45#65512—31no—
46#65268D311yescharacteristic, complemented
47#65249—32no—
48#65248—32no—
49C22⋊C4×Dic5—32no—
50#64962—31no—
51#64955—31no—
52C23×D15C2311yescharacteristic
53C23×D15C2311yescharacteristic
54C23×C5×D3C2312yes—
55C22×C3×C4×C5C2311yescharacteristic
56C22×C3×C4×C5C2311yescharacteristic
57C22×C3×C4×C5C2311yescharacteristic
58C22×Dic15C2311yescharacteristic
59C22×Dic15C2311yescharacteristic
60C2×D60—22no—
61C2×D60—22no—
62C2×D60—22no—
63C2×C4×D15—24no—
64C2×C4×D15—24no—
65C2×C4×D15—22no—
66C2×C4×D15D414yescomplemented
67C22×C5×Dic3C2312yes—
68C2×C5×D12—24no—
69C2×C5×D12—22no—
70C22×C3×Dic5C2312yes—
71C22×C3×Dic5C2312yes—
72C2×C3×Dic10—22no—
73C2×C3×Dic10—24no—
74C2×#1203—24no—
75C2×#1203—24no—
76C2×#1203—24no—
77C2×#1203—24no—
78C2×#1203—24no—
79C2×#1203—24no—
80C2×D3×Dic5—24no—
81C2×D3×Dic5—24no—
82C2×D3×Dic5D418yescomplemented
83#6768—232no—
84C3×C5×C4⋊C4—22no—
85#6714—22no—
86#6714—22no—
87C4×Dic15—22no—
88C5×(C2×D3)⋊C4—24no—
89C5×(C2×D3)⋊C4—24no—
90C3×#683—24no—
91#6667—24no—
92#6667—24no—
93(C2×D3)⋊Dic5—24no—
94Dic3×Dic5—24no—
95C22×#239—51no—
96C2×#893—51no—
97C2×#893—52no—
98C3×#395—51no—
99#5231—51no—
100#4991—52no—
101#4991D511yescharacteristic, complemented
102#4671—51no—
103#4671—52no—
104#4666—52no—
105#4666—51no—
106C22×C5×D4—31no—
107C22×C22⋊D5—32no—
108C22×C22⋊D5—32no—
109C23×Dic5—32no—
110C2×#711—64no—
111C2×#711—68no—
112C22×D20—31no—
113C22×C4×D5—31no—
114C22×C4×D5—31no—
115C22×Dic10C2×D311yescharacteristic
116C2×C5×C4⋊C4C2×D311yescharacteristic
117C2×C5×C22⋊C4—32no—
118C2×C5×C22⋊C4—32no—
119C2×C22⋊Dic5—32no—
120C2×#685—31no—
121C2×#685—31no—
122C2×#685—32no—
123C2×#685—32no—
124C2×#683C2×D312yes—
125C2×C4×Dic5—31no—
126C2×C4×Dic5—32no—
127#4039—64no—
128#4027—68no—
129C5×#95—31no—
130#3963—32no—
131#3963C2×D311yescharacteristic
132#3963C2×D312yes—
133#2388—151noSylow
134C23×C3×C5C2411yescharacteristic
135C22×D15—22no—
136C22×D15—21no—
137C22×D15—22no—
138C22×D15—22no—
139C22×D15—22no—
140C22×D15—21no—
141C22×D15—21no—
142C22×D15—21no—
143C22×D15C2×D412yes—
144C22×D15#4112yescomplemented
145C22×C5×D3—22no—
146C22×C5×D3—22no—
147C22×C5×D3—22no—
148C22×C5×D3—22no—
149C22×C5×D3—22no—
150C22×C5×D3—22no—
151C22×C5×D3C2×D414yes—
152C2×C3×C4×C5—21no—
153C2×C3×C4×C5—22no—
154C2×C3×C4×C5—21no—
155C2×C3×C4×C5—22no—
156C2×C3×C4×C5—21no—
157C2×C3×C4×C5C2×D412yes—
158C2×C3×C4×C5#4112yescomplemented
159C2×Dic15—21no—
160C2×Dic15—22no—
161C2×Dic15C2×D414yes—
162C2×Dic15C2×D412yes—
163D60—28no—
164C4×D15—28no—
165C4×D15—44no—
166C2×C5×Dic3—22no—
167C2×C5×Dic3—22no—
168C2×C5×Dic3—22no—
169C5×D12—28no—
170C2×C3×Dic5—22no—
171C2×C3×Dic5—22no—
172C2×C3×Dic5—22no—
173C2×C3×Dic5—22no—
174C2×C3×Dic5C2×D414yes—
175C2×C3×Dic5#4114yescomplemented
176C3×Dic10—28no—
177#1203—216no—
178#1203—216no—
179D3×Dic5—216no—
180C22×C3×Q8—51no—
181C2×#239—108no—
182C2×#239—104no—
183C22×D12—52no—
184C22×D12—52no—
185C22×D12—51no—
186C22×D12C2×D511yescharacteristic
187C22×C4×D3—52no—
188C22×C4×D3—51no—
189C22×C4×D3—51no—
190C2×C3×C4⋊C4—52no—
191C2×C3×C4⋊C4C2×D511yescharacteristic
192C2×(C2×D3)⋊C4—52no—
193C2×(C2×D3)⋊C4—51no—
194C2×(C2×D3)⋊C4—51no—
195C2×(C2×D3)⋊C4—52no—
196C2×(C2×D3)⋊C4—52no—
197C2×(C2×D3)⋊C4C2×D512yes—
198C2×(C2×D3)⋊C4C2×D512yes—
199C2×C4×Dic3—51no—
200C2×C4×Dic3—52no—
201#893—108no—
202#893—104no—
203C3×#95—52no—
204C3×#95—51no—
205C3⋊#95—52no—
206C3⋊#95C2×D511yescharacteristic
207C24×C5—32no—
208C23×D5—31no—
209C23×D5—31no—
210C2×C5×D4—64no—
211C2×C5×D4—62no—
212C22×C4×C5—32no—
213C22×C4×C5C22×D311yescharacteristic
214C22×C4×C5C22×D311yescharacteristic
215C22×C4×C5C22×D311yescharacteristic
216C2×C22⋊D5—64no—
217C2×C22⋊D5—64no—
218C2×C22⋊D5—64no—
219C2×C22⋊D5—64no—
220C2×C22⋊D5—64no—
221C2×C22⋊D5—64no—
222C22×Dic5—31no—
223C22×Dic5—38no—
224C22×Dic5—31no—
225C22×Dic5—64no—
226C22×Dic5—64no—
227C22×Dic5C22×D312yes—
228C22×Dic5C22×D312yes—
229#711—632no—
230C2×D20—62no—
231C2×D20—62no—
232C2×D20—62no—
233C2×C4×D5—34no—
234C2×C4×D5—64no—
235C2×C4×D5—64no—
236C2×C4×D5—62no—
237C2×Dic10—22no—
238C2×Dic10—24no—
239C5×C4⋊C4—22no—
240C5×C22⋊C4—64no—
241C5×C22⋊C4—64no—
242C22⋊Dic5—64no—
243#685—62no—
244#685—62no—
245#685—64no—
246#685—64no—
247#683—24no—
248C4×Dic5—64no—
249C4×Dic5—62no—
250C22×#41—151no—
251C2×C4×D4—152no—
252C2×C4×D4—151no—
253#395—51no—
254#394—31no—
255#394—152no—
256#393—151no—
257#392—152no—
258#392—151no—
259C4×C22⋊C4—151no—
260C4×C22⋊C4—152no—
261C22×C3×C5C22×D412yes—
262C22×C3×C5C22×D411yescharacteristic
263C22×C3×C5C2×#4111yescharacteristic
264C22×C3×C5C2×#4112yes—
265C22×C3×C5C2×#4111yescharacteristic
266C2×D15—22no—
267C2×D15—24no—
268C2×D15—22no—
269C2×D15—22no—
270C2×D15—22no—
271C2×D15—41no—
272C2×D15—42no—
273C2×D15—42no—
274C2×D15—42no—
275C2×D15—41no—
276C2×C5×D3—24no—
277C2×C5×D3—24no—
278C2×C5×D3—24no—
279C2×C5×D3—42no—
280C2×C5×D3—42no—
281C2×C5×D3—42no—
282C2×C5×D3—42no—
283C3×C4×C5—22no—
284C3×C4×C5—22no—
285C3×C4×C5—41no—
286Dic15—41no—
287Dic15#12014yes—
288C3×Dic5—24no—
289C3×Dic5—24no—
290C5×Dic3—42no—
291C23×D3—51no—
292C23×D3—51no—
293C23×D3C22×D512yes—
294C2×C3×Q8—104no—
295C2×C3×Q8—102no—
296C22×C3×C4—52no—
297C22×C3×C4—52no—
298C22×C3×C4C22×D511yescharacteristic
299C22×C3×C4C22×D511yescharacteristic
300C22×C3×C4C22×D511yescharacteristic
301C22×Dic3—51no—
302C22×Dic3—51no—
303C22×Dic3C22×D512yes—
304#239—1032no—
305C2×D12—24no—
306C2×D12—22no—
307C2×D12—102no—
308C2×D12—102no—
309C2×D12—104no—
310C2×D12—104no—
311C2×D12—104no—
312C2×D12—104no—
313C2×D12—104no—
314C2×D12—104no—
315C2×D12—102no—
316C2×C4×D3—58no—
317C2×C4×D3—54no—
318C2×C4×D3—104no—
319C2×C4×D3—104no—
320C2×C4×D3—104no—
321C2×C4×D3—104no—
322C2×C4×D3—102no—
323C3×C4⋊C4—22no—
324C3×C4⋊C4—104no—
325(C2×D3)⋊C4—24no—
326(C2×D3)⋊C4—24no—
327(C2×D3)⋊C4—104no—
328(C2×D3)⋊C4—104no—
329(C2×D3)⋊C4—102no—
330(C2×D3)⋊C4—104no—
331(C2×D3)⋊C4—102no—
332C4×Dic3—104no—
333C4×Dic3—102no—
334C23×C5—34no—
335C23×C5—62no—
336C23×C5—62no—
337C23×C5—62no—
338C23×C5—62no—
339C23×C5—62no—
340C23×C5—62no—
341C23×C5C23×D311yescharacteristic
342C22×D5—32no—
343C22×D5—32no—
344C22×D5—61no—
345C22×D5—62no—
346C22×D5—62no—
347C22×D5—62no—
348C22×D5—61no—
349C22×D5—61no—
350C22×D5—62no—
351C22×D5—61no—
352C5×D4—68no—
353C2×C4×C5—21no—
354C2×C4×C5—22no—
355C2×C4×C5—21no—
356C2×C4×C5—22no—
357C2×C4×C5—21no—
358C2×C4×C5—62no—
359C2×C4×C5—62no—
360C2×C4×C5—62no—
361C2×C4×C5D3×D412yes—
362C2×C4×C5#23912yescomplemented
363C22⋊D5—616no—
364C22⋊D5—616no—
365C2×Dic5—22no—
366C2×Dic5—22no—
367C2×Dic5—22no—
368C2×Dic5—22no—
369C2×Dic5—32no—
370C2×Dic5—34no—
371C2×Dic5—616no—
372C2×Dic5—61no—
373C2×Dic5—62no—
374C2×Dic5D3×D414yes—
375C2×Dic5#23914yescomplemented
376D20—68no—
377C4×D5—68no—
378C4×D5—124no—
379Dic10—28no—
380C2×#41—308no—
381C2×#41—304no—
382C22×Q8—51no—
383C22×D4—31no—
384C22×D4—152no—
385C22×D4—152no—
386C22×D4—151no—
387C23×C4—152no—
388C23×C4—151no—
389C23×C4—151no—
390C4×D4—308no—
391C4×D4—304no—
392C2×C4⋊C4—52no—
393C2×C4⋊C4D3×D511yescharacteristic
394C2×C22⋊C4—32no—
395C2×C22⋊C4—32no—
396C2×C22⋊C4—152no—
397C2×C22⋊C4—152no—
398C2×C22⋊C4—152no—
399C2×C22⋊C4—151no—
400C2×C22⋊C4—151no—
401C2×C42—152no—
402C2×C42—151no—
403#95—31no—
404#95—51no—
405#95—52no—
406#95—152no—
407C2×C3×C5C2×#12011yescharacteristic
408C2×C3×C5#52511yescharacteristic
409C2×C3×C5#52512yes—
410C2×C3×C5#53212yes—
411C2×C3×C5#53211yescharacteristic
412D15—42no—
413D15—42no—
414C5×D3—44no—
415C23×C3C23×D511yescharacteristic
416C22×D3—22no—
417C22×D3—22no—
418C22×D3—22no—
419C22×D3—22no—
420C22×D3—22no—
421C22×D3—22no—
422C22×D3—52no—
423C22×D3—52no—
424C22×D3—102no—
425C22×D3—101no—
426C22×D3—102no—
427C22×D3—101no—
428C22×D3—101no—
429C22×D3—102no—
430C22×D3—101no—
431C22×D3—102no—
432C22×D3D4×D514yes—
433C3×Q8—108no—
434C2×C3×C4—21no—
435C2×C3×C4—22no—
436C2×C3×C4—21no—
437C2×C3×C4—22no—
438C2×C3×C4—21no—
439C2×C3×C4—54no—
440C2×C3×C4—54no—
441C2×C3×C4—102no—
442C2×C3×C4—102no—
443C2×C3×C4—102no—
444C2×C3×C4—102no—
445C2×C3×C4D4×D512yes—
446C2×C3×C4#71112yescomplemented
447C2×Dic3—22no—
448C2×Dic3—22no—
449C2×Dic3—22no—
450C2×Dic3—54no—
451C2×Dic3—52no—
452C2×Dic3—101no—
453C2×Dic3—102no—
454D12—28no—
455D12—108no—
456D12—1016no—
457D12—1016no—
458C4×D3—108no—
459C4×D3—1016no—
460C4×D3—204no—
461C22×C5—64no—
462C22×C5—64no—
463C22×C5—64no—
464C22×C5—122no—
465C22×C5—122no—
466C22×C5—122no—
467C22×C5—122no—
468C22×C5C2×D3×D412yes—
469C22×C5C2×D3×D411yescharacteristic
470C22×C5C2×#23911yescharacteristic
471C22×C5C2×#23912yes—
472C22×C5D3×#4111yescharacteristic
473C2×D5—62no—
474C2×D5—62no—
475C2×D5—62no—
476C2×D5—62no—
477C2×D5—64no—
478C2×D5—121no—
479C2×D5—122no—
480C2×D5—122no—
481C2×D5—121no—
482C2×D5—122no—
483C4×C5—22no—
484C4×C5—22no—
485C4×C5—41no—
486C4×C5—122no—
487Dic5—24no—
488Dic5—24no—
489Dic5—34no—
490Dic5—121no—
491C24—32no—
492C24—151no—
493C24—151no—
494#41—3032no—
495C2×Q8—104no—
496C2×Q8—102no—
497C2×D4—64no—
498C2×D4—62no—
499C2×D4—302no—
500C2×D4—302no—
501C2×D4—304no—
502C2×D4—304no—
503C2×D4—304no—
504C2×D4—304no—
505C2×D4—304no—
506C2×D4—304no—
507C2×D4—302no—
508C22×C4—32no—
509C22×C4—52no—
510C22×C4—52no—
511C22×C4—158no—
512C22×C4—151no—
513C22×C4—154no—
514C22×C4—151no—
515C22×C4—304no—
516C22×C4—304no—
517C22×C4—302no—
518C22×C4—304no—
519C22×C4—304no—
520C22×C4C2×D3×D511yescharacteristic
521C22×C4C2×D3×D511yescharacteristic
522C22×C4C2×D3×D511yescharacteristic
523C4⋊C4—22no—
524C4⋊C4—104no—
525C22⋊C4—64no—
526C22⋊C4—64no—
527C22⋊C4—302no—
528C22⋊C4—304no—
529C22⋊C4—304no—
530C22⋊C4—304no—
531C22⋊C4—302no—
532C42—302no—
533C42—304no—
534C3×C5#238811yescharacteristic, complemented, Hall
535C22×C3C2×D4×D512yes—
536C22×C3C2×D4×D511yescharacteristic
537C22×C3C2×#71111yescharacteristic
538C22×C3D5×#4111yescharacteristic
539C22×C3D5×#4112yes—
540C2×D3—24no—
541C2×D3—24no—
542C2×D3—24no—
543C2×D3—42no—
544C2×D3—42no—
545C2×D3—42no—
546C2×D3—42no—
547C2×D3—102no—
548C2×D3—102no—
549C2×D3—102no—
550C2×D3—104no—
551C2×D3—102no—
552C2×D3—201no—
553C2×D3—202no—
554C2×D3—202no—
555C2×D3—202no—
556C2×D3—201no—
557C3×C4—22no—
558C3×C4—22no—
559C3×C4—41no—
560C3×C4—104no—
561C3×C4—104no—
562Dic3—42no—
563Dic3—54no—
564Dic3—201no—
565C2×C5—124no—
566C2×C5D3×#12011yescharacteristic
567C2×C5#562712yes—
568C2×C5#562711yescharacteristic
569C2×C5#572511yescharacteristic
570C2×C5#572512yes—
571D5—122no—
572D5—122no—
573C23—34no—
574C23—62no—
575C23—62no—
576C23—62no—
577C23—62no—
578C23—62no—
579C23—62no—
580C23—152no—
581C23—152no—
582C23—301no—
583C23—301no—
584C23—301no—
585C23—302no—
586C23—302no—
587C23—302no—
588C23—302no—
589C23—301no—
590C23C22×D3×D511yescharacteristic
591Q8—108no—
592D4—68no—
593D4—308no—
594D4—3016no—
595D4—3016no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—54no—
602C2×C4—54no—
603C2×C4—62no—
604C2×C4—62no—
605C2×C4—62no—
606C2×C4—102no—
607C2×C4—102no—
608C2×C4—102no—
609C2×C4—102no—
610C2×C4—154no—
611C2×C4—152no—
612C2×C4—308no—
613C2×C4—3016no—
614C2×C4—302no—
615C2×C4—301no—
616C2×C4—604no—
617C2×C4#676812yescomplemented
618C2×C4#679012yes—
619C2×C3D5×#12011yescharacteristic
620C2×C3#6593012yes—
621C2×C3#6594011yescharacteristic
622C2×C3#6599012yes—
623C2×C3#6600211yescharacteristic
624D3—44no—
625D3—202no—
626D3—202no—
627C5#7716911yescharacteristic, complemented, Sylow
628C22—64no—
629C22—64no—
630C22—64no—
631C22—122no—
632C22—122no—
633C22—122no—
634C22—122no—
635C22—302no—
636C22—304no—
637C22—302no—
638C22—302no—
639C22—302no—
640C22—601no—
641C22—602no—
642C22—601no—
643C22—602no—
644C22—602no—
645C22C2×#676811yescharacteristic
646C22#9201611yescharacteristic
647C22D3×D4×D512yes—
648C22D5×#23912yes—
649C22C2×#679011yescharacteristic
650C4—22no—
651C4—22no—
652C4—41no—
653C4—104no—
654C4—104no—
655C4—122no—
656C4—154no—
657C4—601no—
658C3#1060862011yescharacteristic, complemented, Sylow
659C2—124no—
660C2—602no—
661C2—602no—
662C2#1175342312yes—
663C2#1175386412yes—
664C2#1175437911yescharacteristic
665C2#1175502011yescharacteristic
666C2#1175517011yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutator, socle, cosocleC23×C3×C5C24
FittingC2×C3×C5×C4⋊C4C22
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5#2388
LCSG ▷ C23×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#6768C2semidirect
2. C2×#91782C2semidirect
3. C2×#91406C2semidirect
4. C5×#4991C2semidirect
5. C3×#65268C2semidirect
6. #11748932C2semidirect
7. #11748929C2semidirect
8. #11748886C2semidirect
9. #11748753C2semidirect
10. #11748733C2semidirect
11. #11748670C2semidirect
46. #65268D3semidirect
66. C2×C4×D15D4semidirect
82. C2×D3×Dic5D4semidirect
101. #4991D5semidirect
144. C22×D15#41semidirect
158. C2×C3×C4×C5#41semidirect
175. C2×C3×Dic5#41semidirect
362. C2×C4×C5#239semidirect
375. C2×Dic5#239semidirect
446. C2×C3×C4#711semidirect
534. C3×C5#2388semidirect
617. C2×C4#6768semidirect
627. C5#77169semidirect
658. C3#10608620semidirect

Maximal subgroups

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

subgroupindex
C22×#67682
C2×#917822
C2×#914062
C5×#49912
C3×#652682
#117489322
#117489292
#117488862
#117487532
#117487332
#117486702
#106086203
#771695