Finite Group Databasebeta

← #14331680

Subgroups — #14331680

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×#6769C211yesmaximal, characteristic, complemented
2C2×C5×#893C211yesmaximal, characteristic, complemented
3C2×#91446C212yesmaximal, complemented
4#11751092C211yesmaximal, characteristic, complemented
5C3×#65268C211yesmaximal, characteristic, complemented
6#11748932C211yesmaximal, characteristic, complemented
7#11748929C212yesmaximal, complemented
8#11748881C212yesmaximal, complemented
9#11748746C211yesmaximal, characteristic, complemented
10#11748733C212yesmaximal, complemented
11#11748719C211yesmaximal, characteristic, complemented
12#10614131—31nomaximal, Hall
13C22×D60C2211yescharacteristic
14C22×C5×D12C2211yescharacteristic
15C22×C4×C5×D3C2211yescharacteristic
16C22×C4×C5×D3C2211yescharacteristic
17C22×C3×Dic10C2211yescharacteristic
18C22×#1203C2212yes—
19C22×#1203C2212yes—
20C22×#1200C2212yes—
21C2×#6769—24no—
22C2×#6769—28no—
23C2×C3×C5×C4⋊C4C2211yescharacteristic
24C2×#6714C2212yes—
25C2×#6714C2212yes—
26C2×C5×(C2×D3)⋊C4C2211yescharacteristic
27C2×C5×(C2×D3)⋊C4C2211yescharacteristic
28C2×C4×C5×Dic3C2211yescharacteristic
29C5×#893—24no—
30C2×C3×#683C2212yes—
31C2×#6667C2212yes—
32C2×(C2×D3)⋊Dic5C2212yes—
33C2×(C2×D3)⋊Dic5C2212yes—
34C2×Dic3×Dic5C2212yes—
35#91446—28no—
36#91115C2211yescharacteristic
37C3×#3963C2212yes—
38C3×#3963C2211yescharacteristic
39#90994C2212yes—
40#77169—51nomaximal, Hall
41C22×#709—31no—
42C2×C4×C5×D4—31no—
43C2×C4×C22⋊D5—32no—
44#65512—32no—
45C4×C22⋊Dic5—31no—
46#65280—31no—
47#65268D311yescharacteristic, complemented
48#65249—31no—
49#65248—32no—
50#65236—31no—
51C4×#685—32no—
52C23×D15C2312yes—
53C23×C5×D3C2311yescharacteristic
54C23×C5×D3C2311yescharacteristic
55C22×C3×C4×C5C2311yescharacteristic
56C22×C3×C4×C5C2311yescharacteristic
57C22×C3×C4×C5C2311yescharacteristic
58C22×Dic15C2312yes—
59C2×D60—24no—
60C2×D60—22no—
61C22×C5×Dic3C2311yescharacteristic
62C22×C5×Dic3C2311yescharacteristic
63C2×C5×D12—22no—
64C2×C5×D12—22no—
65C2×C5×D12—22no—
66C2×C4×C5×D3—24no—
67C2×C4×C5×D3—22no—
68C2×C4×C5×D3—24no—
69C2×C4×C5×D3D414yescomplemented
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×#1200—24no—
81C2×#1200—24no—
82C2×#1200D418yescomplemented
83#6769—232no—
84C3×C5×C4⋊C4—22no—
85#6714—24no—
86#6714—24no—
87C5×(C2×D3)⋊C4—22no—
88C5×(C2×D3)⋊C4—22no—
89C4×C5×Dic3—22no—
90C3×#683—24no—
91#6667—24no—
92(C2×D3)⋊Dic5—24no—
93(C2×D3)⋊Dic5—24no—
94Dic3×Dic5—24no—
95C22×#239—51no—
96C2×#893—52no—
97C2×#893D511yescharacteristic, complemented
98C3×#395—51no—
99#5231—51no—
100#4991—51no—
101#4991—52no—
102#4671—52no—
103#4671—51no—
104#4666—51no—
105#4666—52no—
106C22×C5×D4—31no—
107C23×C4×C5—31no—
108C23×C4×C5—31no—
109C22×C22⋊D5—32no—
110C22×C22⋊D5—32no—
111C2×#709—68no—
112C2×#709—64no—
113C22×D20—31no—
114C22×C4×D5—32no—
115C22×Dic10C2×D311yescharacteristic
116C4×C5×D4—64no—
117C2×C5×C4⋊C4C2×D311yescharacteristic
118C2×C5×C22⋊C4—31no—
119C2×C5×C22⋊C4—31no—
120C2×C42×C5—31no—
121C2×C22⋊Dic5—32no—
122C2×C22⋊Dic5—32no—
123C4×C22⋊D5—68no—
124C2×#685—32no—
125C2×#685—32no—
126C2×#685—32no—
127C2×#683C2×D312yes—
128C2×C4×Dic5—32no—
129#3963—31no—
130#3963—32no—
131#3963C2×D312yes—
132#3963C2×D311yescharacteristic
133#2388—151noSylow
134C23×C3×C5C2411yescharacteristic
135C22×D15—22no—
136C22×D15—22no—
137C22×D15—22no—
138C22×D15—22no—
139C22×D15—22no—
140C22×D15—22no—
141C22×D15C2×D414yes—
142C22×C5×D3—22no—
143C22×C5×D3—21no—
144C22×C5×D3—22no—
145C22×C5×D3—22no—
146C22×C5×D3—22no—
147C22×C5×D3—21no—
148C22×C5×D3—21no—
149C22×C5×D3—21no—
150C22×C5×D3C2×D412yes—
151C22×C5×D3#4112yescomplemented
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—22no—
160C2×Dic15—22no—
161C2×Dic15—22no—
162D60—28no—
163C2×C5×Dic3—21no—
164C2×C5×Dic3—22no—
165C2×C5×Dic3C2×D414yes—
166C2×C5×Dic3C2×D412yes—
167C5×D12—28no—
168C4×C5×D3—28no—
169C4×C5×D3—44no—
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—
179#1200—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×D3C2×D511yescharacteristic
189C22×C4×D3C2×D511yescharacteristic
190C2×C3×C4⋊C4—52no—
191C2×C3×C4⋊C4C2×D511yescharacteristic
192C2×(C2×D3)⋊C4—52no—
193C2×(C2×D3)⋊C4—52no—
194C2×(C2×D3)⋊C4—52no—
195C2×(C2×D3)⋊C4—52no—
196C2×(C2×D3)⋊C4—52no—
197C2×(C2×D3)⋊C4C2×D511yescharacteristic
198C2×(C2×D3)⋊C4C2×D511yescharacteristic
199C2×C4×Dic3—52no—
200C2×C4×Dic3C2×D511yescharacteristic
201#893—24no—
202#893—108no—
203C3×#95—52no—
204C3×#95—51no—
205C3⋊#95—51no—
206C3⋊#95—52no—
207C24×C5—31no—
208C24×C5—31no—
209C23×D5—32no—
210C2×C5×D4—62no—
211C2×C5×D4—62no—
212C2×C5×D4—62no—
213C22×C4×C5—31no—
214C22×C4×C5—34no—
215C22×C4×C5—31no—
216C22×C4×C5—64no—
217C22×C4×C5—62no—
218C22×C4×C5—64no—
219C22×C4×C5C22×D311yescharacteristic
220C22×C4×C5C22×D311yescharacteristic
221C22×C4×C5C22×D311yescharacteristic
222C2×C22⋊D5—64no—
223C2×C22⋊D5—64no—
224C2×C22⋊D5—64no—
225C2×C22⋊D5—64no—
226C2×C22⋊D5—64no—
227C2×C22⋊D5—64no—
228C22×Dic5—32no—
229C22×Dic5C22×D312yes—
230C22×Dic5C22×D312yes—
231#709—632no—
232C2×D20—64no—
233C2×D20—62no—
234C2×C4×D5—38no—
235C2×C4×D5—64no—
236C2×C4×D5—64no—
237C2×Dic10—22no—
238C2×Dic10—24no—
239C5×C4⋊C4—22no—
240C5×C22⋊C4—62no—
241C5×C22⋊C4—62no—
242C42×C5—62no—
243C22⋊Dic5—64no—
244C22⋊Dic5—64no—
245#685—64no—
246#685—64no—
247#685—64no—
248#683—24no—
249C4×Dic5—64no—
250C22×#41—151no—
251C2×C4×D4—31no—
252C2×C4×D4—152no—
253#395—51no—
254#394—152no—
255#394—151no—
256#393—151no—
257#392—151no—
258#392—152no—
259C4×C22⋊C4—152no—
260C4×C22⋊C4—151no—
261C22×C3×C5C22×D412yes—
262C22×C3×C5C22×D411yescharacteristic
263C22×C3×C5C2×#4111yescharacteristic
264C22×C3×C5C2×#4111yescharacteristic
265C22×C3×C5C2×#4112yes—
266C2×D15—24no—
267C2×D15—24no—
268C2×D15—24no—
269C2×D15—42no—
270C2×D15—42no—
271C2×D15—42no—
272C2×D15—42no—
273C2×C5×D3—22no—
274C2×C5×D3—24no—
275C2×C5×D3—22no—
276C2×C5×D3—22no—
277C2×C5×D3—22no—
278C2×C5×D3—41no—
279C2×C5×D3—42no—
280C2×C5×D3—42no—
281C2×C5×D3—42no—
282C2×C5×D3—41no—
283C3×C4×C5—22no—
284C3×C4×C5—22no—
285C3×C4×C5—41no—
286Dic15—42no—
287C3×Dic5—24no—
288C3×Dic5—24no—
289C5×Dic3—41no—
290C5×Dic3#12014yes—
291C23×D3—52no—
292C23×D3C22×D511yescharacteristic
293C23×D3C22×D511yescharacteristic
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—52no—
302C22×Dic3C22×D511yescharacteristic
303C22×Dic3C22×D511yescharacteristic
304#239—1032no—
305C2×D12—22no—
306C2×D12—22no—
307C2×D12—22no—
308C2×D12—104no—
309C2×D12—104no—
310C2×D12—104no—
311C2×D12—104no—
312C2×D12—104no—
313C2×D12—104no—
314C2×D12—104no—
315C2×D12—102no—
316C2×C4×D3—24no—
317C2×C4×D3—24no—
318C2×C4×D3—22no—
319C2×C4×D3—58no—
320C2×C4×D3—104no—
321C2×C4×D3—104no—
322C2×C4×D3C22⋊D514yescomplemented
323C3×C4⋊C4—22no—
324C3×C4⋊C4—104no—
325(C2×D3)⋊C4—22no—
326(C2×D3)⋊C4—22no—
327(C2×D3)⋊C4—104no—
328(C2×D3)⋊C4—104no—
329(C2×D3)⋊C4—104no—
330(C2×D3)⋊C4—104no—
331(C2×D3)⋊C4—104no—
332C4×Dic3—22no—
333C4×Dic3—104no—
334C23×C5—32no—
335C23×C5—32no—
336C23×C5—62no—
337C23×C5—61no—
338C23×C5—62no—
339C23×C5—62no—
340C23×C5—62no—
341C23×C5—61no—
342C23×C5—61no—
343C23×C5—61no—
344C23×C5C23×D311yescharacteristic
345C22×D5—34no—
346C22×D5—62no—
347C22×D5—62no—
348C22×D5—62no—
349C22×D5—62no—
350C22×D5—62no—
351C22×D5—62no—
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—34no—
359C2×C4×C5—32no—
360C2×C4×C5—68no—
361C2×C4×C5—61no—
362C2×C4×C5—62no—
363C2×C4×C5—124no—
364C2×C4×C5D3×D412yes—
365C2×C4×C5#23912yescomplemented
366C22⋊D5—616no—
367C22⋊D5—616no—
368C2×Dic5—22no—
369C2×Dic5—22no—
370C2×Dic5—22no—
371C2×Dic5—22no—
372C2×Dic5—62no—
373C2×Dic5—62no—
374C2×Dic5—62no—
375C2×Dic5D3×D414yes—
376C2×Dic5#23914yescomplemented
377D20—68no—
378C4×D5—616no—
379Dic10—28no—
380C2×#41—304no—
381C2×#41—308no—
382C22×Q8—51no—
383C22×D4—31no—
384C22×D4—152no—
385C22×D4—152no—
386C22×D4—151no—
387C23×C4—31no—
388C23×C4—31no—
389C23×C4—152no—
390C4×D4—64no—
391C4×D4—308no—
392C2×C4⋊C4—52no—
393C2×C4⋊C4D3×D511yescharacteristic
394C2×C22⋊C4—31no—
395C2×C22⋊C4—31no—
396C2×C22⋊C4—152no—
397C2×C22⋊C4—152no—
398C2×C22⋊C4—152no—
399C2×C22⋊C4—152no—
400C2×C22⋊C4—152no—
401C2×C42—31no—
402C2×C42—152no—
403#95—51no—
404#95—52no—
405#95—152no—
406#95—151no—
407C2×C3×C5C2×#12011yescharacteristic
408C2×C3×C5#52511yescharacteristic
409C2×C3×C5#52512yes—
410C2×C3×C5#53212yes—
411C2×C3×C5#53211yescharacteristic
412D15—44no—
413C5×D3—42no—
414C5×D3—42no—
415C23×C3C23×D511yescharacteristic
416C22×D3—21no—
417C22×D3—21no—
418C22×D3—21no—
419C22×D3—22no—
420C22×D3—22no—
421C22×D3—22no—
422C22×D3—22no—
423C22×D3—21no—
424C22×D3—54no—
425C22×D3—102no—
426C22×D3—102no—
427C22×D3—102no—
428C22×D3—102no—
429C22×D3—102no—
430C22×D3—102no—
431C22×D3#70912yescomplemented
432C22×D3C2×C22⋊D512yes—
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×C4#70912yescomplemented
446C2×C3×C4C2×C22⋊D512yes—
447C2×Dic3—22no—
448C2×Dic3—21no—
449C2×Dic3—102no—
450C2×Dic3—102no—
451C2×Dic3—102no—
452C2×Dic3D4×D514yes—
453C2×Dic3C2×C22⋊D512yes—
454D12—28no—
455D12—108no—
456D12—1016no—
457D12—1016no—
458C4×D3—28no—
459C4×D3—44no—
460C4×D3—1016no—
461C22×C5—62no—
462C22×C5—62no—
463C22×C5—64no—
464C22×C5—62no—
465C22×C5—62no—
466C22×C5—121no—
467C22×C5—122no—
468C22×C5—122no—
469C22×C5—121no—
470C22×C5—122no—
471C22×C5C2×D3×D412yes—
472C22×C5C2×D3×D411yescharacteristic
473C22×C5C2×#23911yescharacteristic
474C22×C5C2×#23912yes—
475C22×C5D3×#4111yescharacteristic
476C2×D5—64no—
477C2×D5—64no—
478C2×D5—64no—
479C2×D5—122no—
480C2×D5—122no—
481C2×D5—122no—
482C2×D5—122no—
483C4×C5—22no—
484C4×C5—22no—
485C4×C5—34no—
486C4×C5—41no—
487C4×C5—121no—
488Dic5—24no—
489Dic5—24no—
490Dic5—122no—
491C24—31no—
492C24—31no—
493C24—152no—
494#41—3032no—
495C2×Q8—104no—
496C2×Q8—102no—
497C2×D4—62no—
498C2×D4—62no—
499C2×D4—62no—
500C2×D4—304no—
501C2×D4—304no—
502C2×D4—304no—
503C2×D4—304no—
504C2×D4—304no—
505C2×D4—304no—
506C2×D4—304no—
507C2×D4—302no—
508C22×C4—31no—
509C22×C4—34no—
510C22×C4—31no—
511C22×C4—52no—
512C22×C4—52no—
513C22×C4—64no—
514C22×C4—62no—
515C22×C4—64no—
516C22×C4—158no—
517C22×C4—152no—
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—62no—
526C22⋊C4—62no—
527C22⋊C4—304no—
528C22⋊C4—304no—
529C22⋊C4—304no—
530C22⋊C4—304no—
531C22⋊C4—304no—
532C42—62no—
533C42—304no—
534C3×C5#238811yescharacteristic, complemented, Hall
535C22×C3C2×#70911yescharacteristic
536C22×C3C2×#70911yescharacteristic
537C22×C3C2×D4×D512yes—
538C22×C3D5×#4112yes—
539C22×C3C22×C22⋊D511yescharacteristic
540C2×D3—22no—
541C2×D3—22no—
542C2×D3—22no—
543C2×D3—24no—
544C2×D3—22no—
545C2×D3—41no—
546C2×D3—42no—
547C2×D3—42no—
548C2×D3—42no—
549C2×D3—41no—
550C2×D3—104no—
551C2×D3—104no—
552C2×D3—104no—
553C2×D3—202no—
554C2×D3—202no—
555C2×D3—202no—
556C2×D3—202no—
557C3×C4—22no—
558C3×C4—22no—
559C3×C4—41no—
560C3×C4—104no—
561C3×C4—104no—
562Dic3—41no—
563Dic3—202no—
564Dic3#408314yes—
565C2×C5—122no—
566C2×C5—122no—
567C2×C5D3×#12011yescharacteristic
568C2×C5#562711yescharacteristic
569C2×C5#562712yes—
570C2×C5#572511yescharacteristic
571C2×C5#572512yes—
572D5—124no—
573C23—32no—
574C23—32no—
575C23—61no—
576C23—61no—
577C23—61no—
578C23—62no—
579C23—62no—
580C23—62no—
581C23—62no—
582C23—61no—
583C23—154no—
584C23—302no—
585C23—302no—
586C23—302no—
587C23—302no—
588C23—302no—
589C23—302no—
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—34no—
602C2×C4—32no—
603C2×C4—54no—
604C2×C4—54no—
605C2×C4—68no—
606C2×C4—62no—
607C2×C4—61no—
608C2×C4—102no—
609C2×C4—102no—
610C2×C4—102no—
611C2×C4—102no—
612C2×C4—124no—
613C2×C4—3016no—
614C2×C4—302no—
615C2×C4—302no—
616C2×C4—302no—
617C2×C4#676912yescomplemented
618C2×C4D3×C22⋊D512yes—
619C2×C3#6588112yes—
620C2×C3#6588212yes—
621C2×C3#6612511yescharacteristic
622C2×C3C2×#408311yescharacteristic
623C2×C3#6616611yescharacteristic
624D3—42no—
625D3—42no—
626D3—204no—
627C5#7716911yescharacteristic, complemented, Sylow
628C22—62no—
629C22—62no—
630C22—62no—
631C22—64no—
632C22—62no—
633C22—121no—
634C22—122no—
635C22—121no—
636C22—122no—
637C22—122no—
638C22—304no—
639C22—304no—
640C22—304no—
641C22—602no—
642C22—602no—
643C22—602no—
644C22—602no—
645C22C2×#676911yescharacteristic
646C22D3×#70911yescharacteristic
647C22D3×D4×D512yes—
648C22D5×#23912yes—
649C22C2×D3×C22⋊D511yescharacteristic
650C4—22no—
651C4—22no—
652C4—34no—
653C4—41no—
654C4—104no—
655C4—104no—
656C4—121no—
657C4—602no—
658C3#1061413111yescharacteristic, complemented, Sylow
659C2—122no—
660C2—122no—
661C2—604no—
662C2#1175342412yes—
663C2#1175386612yes—
664C2#1175435211yescharacteristic
665C2D3×#408311yescharacteristic
666C2#1175516411yescharacteristic
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×#6769C2semidirect
2. C2×C5×#893C2semidirect
3. C2×#91446C2semidirect
4. #11751092C2semidirect
5. C3×#65268C2semidirect
6. #11748932C2semidirect
7. #11748929C2semidirect
8. #11748881C2semidirect
9. #11748746C2semidirect
10. #11748733C2semidirect
11. #11748719C2semidirect
47. #65268D3semidirect
69. C2×C4×C5×D3D4semidirect
82. C2×#1200D4semidirect
97. C2×#893D5semidirect
151. C22×C5×D3#41semidirect
158. C2×C3×C4×C5#41semidirect
175. C2×C3×Dic5#41semidirect
322. C2×C4×D3C22⋊D5semidirect
365. C2×C4×C5#239semidirect
376. C2×Dic5#239semidirect
431. C22×D3#709semidirect
445. C2×C3×C4#709semidirect
534. C3×C5#2388semidirect
617. C2×C4#6769semidirect
627. C5#77169semidirect
658. C3#10614131semidirect

Maximal subgroups

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

subgroupindex
C22×#67692
C2×C5×#8932
C2×#914462
#117510922
C3×#652682
#117489322
#117489292
#117488812
#117487462
#117487332
#117487192
#106141313
#771695