Finite Group Databasebeta

← #14331615

Subgroups — #14331615

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×#6817C211yesmaximal, characteristic, complemented
2C2×C3×C4×C5×D4C211yesmaximal, characteristic, complemented
3C2×C4×#1220C212yesmaximal, complemented
4#11751324C212yesmaximal, complemented
5C4×#6719C211yesmaximal, characteristic, complemented
6#11751092C211yesmaximal, characteristic, complemented
7#11751080C211yesmaximal, characteristic, complemented
8#11751061C211yesmaximal, characteristic, complemented
9#11751060C212yesmaximal, complemented
10#11751048C211yesmaximal, characteristic, complemented
11C4×#6714C212yesmaximal, complemented
12#10614131—31nomaximal, Hall
13C22×C3×C5×D4C2211yescharacteristic
14C23×C3×C4×C5C2211yescharacteristic
15C23×C3×C4×C5C2211yescharacteristic
16C22×#1220C2212yes—
17C22×#1220C2212yes—
18C2×#6817—24no—
19C2×#6817—28no—
20C22×D60C2211yescharacteristic
21C22×C4×D15C2212yes—
22C22×Dic30C2211yescharacteristic
23C3×C4×C5×D4—24no—
24C2×C3×C5×C4⋊C4C2211yescharacteristic
25C2×C3×C5×C22⋊C4C2211yescharacteristic
26C2×C3×C5×C22⋊C4C2211yescharacteristic
27C2×C3×C42×C5C2211yescharacteristic
28C2×#6719C2212yes—
29C2×#6719C2212yes—
30C4×#1220—28no—
31C2×#6714C2212yes—
32C2×#6714C2212yes—
33C2×#6714C2212yes—
34C2×#6712C2212yes—
35C2×C4×Dic15C2212yes—
36#91115C2212yes—
37#91115C2212yes—
38#91115C2211yescharacteristic
39#91115C2211yescharacteristic
40#82400—51nomaximal, Hall
41C22×#709—31no—
42C2×C4×C5×D4D311yescharacteristic, complemented
43C2×C4×C22⋊D5—32no—
44#65512—32no—
45C4×C22⋊Dic5—31no—
46#65280—31no—
47#65268—31no—
48#65249—31no—
49#65248—32no—
50#65236—31no—
51C4×#685—32no—
52C24×C3×C5C2311yescharacteristic
53C24×C3×C5C2311yescharacteristic
54C23×D15C2312yes—
55C2×C3×C5×D4—22no—
56C2×C3×C5×D4—22no—
57C2×C3×C5×D4—22no—
58C22×C3×C4×C5—24no—
59C22×C3×C4×C5—24no—
60C22×C3×C4×C5—22no—
61C22×C3×C4×C5D414yescomplemented
62C22×C3×C4×C5C2311yescharacteristic
63C22×C3×C4×C5C2311yescharacteristic
64C22×C3×C4×C5C2311yescharacteristic
65C22×C3×C4×C5C2311yescharacteristic
66C22×C3×C4×C5C2311yescharacteristic
67C2×#1220—24no—
68C2×#1220—24no—
69C2×#1220—24no—
70C2×#1220—24no—
71C2×#1220—24no—
72C2×#1220—24no—
73C22×Dic15C2312yes—
74C22×Dic15C2312yes—
75C22×Dic15C2312yes—
76#6817—232no—
77C2×D60—24no—
78C2×D60—22no—
79C2×C4×D15—24no—
80C2×C4×D15—24no—
81C2×C4×D15D418yescomplemented
82C2×Dic30—22no—
83C2×Dic30—24no—
84C3×C5×C4⋊C4—22no—
85C3×C5×C22⋊C4—22no—
86C3×C5×C22⋊C4—22no—
87C3×C42×C5—22no—
88#6719—24no—
89#6719—24no—
90#6714—24no—
91#6714—24no—
92#6714—24no—
93#6712—24no—
94C4×Dic15—24no—
95C22×#235—51no—
96C2×C3×C4×D4D511yescharacteristic, complemented
97C2×C4×#67—52no—
98#5223—52no—
99C4×#217—51no—
100#4991—51no—
101#4979—51no—
102#4960—51no—
103#4959—52no—
104#4947—51no—
105C4×(C2×D3)⋊C4—52no—
106C22×C5×D4C2×D311yescharacteristic
107C23×C4×C5C2×D311yescharacteristic
108C23×C4×C5C2×D311yescharacteristic
109C22×C22⋊D5—32no—
110C22×C22⋊D5—32no—
111C2×#709—68no—
112C2×#709—64no—
113C22×D20—31no—
114C22×C4×D5—32no—
115C22×Dic10—31no—
116C4×C5×D4—24no—
117C2×C5×C4⋊C4C2×D311yescharacteristic
118C2×C5×C22⋊C4C2×D311yescharacteristic
119C2×C5×C22⋊C4C2×D311yescharacteristic
120C2×C42×C5C2×D311yescharacteristic
121C2×C22⋊Dic5—32no—
122C2×C22⋊Dic5—32no—
123C4×C22⋊D5—68no—
124C2×#685—32no—
125C2×#685—32no—
126C2×#685—32no—
127C2×#683—32no—
128C2×C4×Dic5—32no—
129#3963—31no—
130#3963—32no—
131#3963—32no—
132#3963—31no—
133#2388—151noSylow
134C23×C3×C5—22no—
135C23×C3×C5—21no—
136C23×C3×C5—22no—
137C23×C3×C5—22no—
138C23×C3×C5—22no—
139C23×C3×C5—21no—
140C23×C3×C5—21no—
141C23×C3×C5—21no—
142C23×C3×C5C2×D412yes—
143C23×C3×C5#4112yescomplemented
144C23×C3×C5C2411yescharacteristic
145C22×D15—22no—
146C22×D15—22no—
147C22×D15—22no—
148C22×D15—22no—
149C22×D15—22no—
150C22×D15—22no—
151C22×D15C2×D414yes—
152C3×C5×D4—28no—
153C2×C3×C4×C5—21no—
154C2×C3×C4×C5—22no—
155C2×C3×C4×C5—21no—
156C2×C3×C4×C5—28no—
157C2×C3×C4×C5—22no—
158C2×C3×C4×C5—21no—
159C2×C3×C4×C5—21no—
160C2×C3×C4×C5—22no—
161C2×C3×C4×C5—44no—
162C2×C3×C4×C5C2×D412yes—
163C2×C3×C4×C5C2×D414yes—
164C2×C3×C4×C5C2×D412yes—
165C2×C3×C4×C5#4112yescomplemented
166#1220—216no—
167#1220—216no—
168C2×Dic15—22no—
169C2×Dic15—22no—
170C2×Dic15—22no—
171C2×Dic15—22no—
172C2×Dic15—22no—
173C2×Dic15—22no—
174C2×Dic15—22no—
175C2×Dic15C2×D414yes—
176C2×Dic15#4114yescomplemented
177D60—28no—
178C4×D15—216no—
179Dic30—28no—
180C22×C3×D4C2×D511yescharacteristic
181C23×C3×C4C2×D511yescharacteristic
182C23×C3×C4C2×D511yescharacteristic
183C22×#67—52no—
184C22×#67—52no—
185C2×#235—108no—
186C2×#235—104no—
187C22×D12—51no—
188C22×C4×D3—52no—
189C22×Dic6—51no—
190C3×C4×D4—24no—
191C2×C3×C4⋊C4C2×D511yescharacteristic
192C2×C3×C22⋊C4C2×D511yescharacteristic
193C2×C3×C22⋊C4C2×D511yescharacteristic
194C2×C3×C42C2×D511yescharacteristic
195C2×#217—52no—
196C2×#217—52no—
197C4×#67—108no—
198C2×(C2×D3)⋊C4—52no—
199C2×(C2×D3)⋊C4—52no—
200C2×(C2×D3)⋊C4—52no—
201C2×Dic3⋊C4—52no—
202C2×C4×Dic3—52no—
203C3⋊#95—52no—
204C3⋊#95—52no—
205C3⋊#95—51no—
206C3⋊#95—51no—
207C24×C5C22×D311yescharacteristic
208C24×C5C22×D311yescharacteristic
209C23×D5—32no—
210C2×C5×D4—22no—
211C2×C5×D4—22no—
212C2×C5×D4—22no—
213C22×C4×C5—24no—
214C22×C4×C5—24no—
215C22×C4×C5—22no—
216C22×C4×C5#6714yescomplemented
217C22×C4×C5C22×D311yescharacteristic
218C22×C4×C5C22×D311yescharacteristic
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×Dic5—32no—
230C22×Dic5—32no—
231#709—632no—
232C2×D20—64no—
233C2×D20—62no—
234C2×C4×D5—38no—
235C2×C4×D5—64no—
236C2×C4×D5—64no—
237C2×Dic10—64no—
238C2×Dic10—62no—
239C5×C4⋊C4—22no—
240C5×C22⋊C4—22no—
241C5×C22⋊C4—22no—
242C42×C5—22no—
243C22⋊Dic5—64no—
244C22⋊Dic5—64no—
245#685—64no—
246#685—64no—
247#685—64no—
248#683—64no—
249C4×Dic5—64no—
250C22×#41—151no—
251C2×C4×D4—152no—
252C2×C4×D4D1511yescharacteristic, complemented
253#395—151no—
254#394—151no—
255#394—152no—
256#393—151no—
257#392—152no—
258#392—151no—
259C4×C22⋊C4—151no—
260C4×C22⋊C4—152no—
261C22×C3×C5—22no—
262C22×C3×C5—24no—
263C22×C3×C5—22no—
264C22×C3×C5—22no—
265C22×C3×C5—22no—
266C22×C3×C5—41no—
267C22×C3×C5—42no—
268C22×C3×C5—42no—
269C22×C3×C5—42no—
270C22×C3×C5—41no—
271C22×C3×C5C22×D412yes—
272C22×C3×C5C22×D411yescharacteristic
273C22×C3×C5C2×#4111yescharacteristic
274C22×C3×C5C2×#4111yescharacteristic
275C22×C3×C5C2×#4112yes—
276C2×D15—24no—
277C2×D15—24no—
278C2×D15—24no—
279C2×D15—42no—
280C2×D15—42no—
281C2×D15—42no—
282C2×D15—42no—
283C3×C4×C5—22no—
284C3×C4×C5—22no—
285C3×C4×C5—41no—
286C3×C4×C5—41no—
287C3×C4×C5#12014yes—
288Dic15—24no—
289Dic15—24no—
290Dic15—42no—
291C24×C3C22×D511yescharacteristic
292C24×C3C22×D511yescharacteristic
293C23×D3—52no—
294C2×C3×D4—22no—
295C2×C3×D4—22no—
296C2×C3×D4—22no—
297C22×C3×C4—24no—
298C22×C3×C4—24no—
299C22×C3×C4—22no—
300C22×C3×C4C22⋊D514yescomplemented
301C22×C3×C4C22×D511yescharacteristic
302C22×C3×C4C22×D511yescharacteristic
303C22×C3×C4C22×D511yescharacteristic
304C22×C3×C4C22×D511yescharacteristic
305C22×C3×C4C22×D511yescharacteristic
306C2×#67—104no—
307C2×#67—104no—
308C2×#67—104no—
309C2×#67—104no—
310C2×#67—104no—
311C2×#67—104no—
312C22×Dic3—52no—
313C22×Dic3—52no—
314C22×Dic3—52no—
315#235—1032no—
316C2×D12—104no—
317C2×D12—102no—
318C2×C4×D3—58no—
319C2×C4×D3—104no—
320C2×C4×D3—104no—
321C2×Dic6—104no—
322C2×Dic6—102no—
323C3×C4⋊C4—22no—
324C3×C22⋊C4—22no—
325C3×C22⋊C4—22no—
326C3×C42—22no—
327#217—104no—
328#217—104no—
329(C2×D3)⋊C4—104no—
330(C2×D3)⋊C4—104no—
331(C2×D3)⋊C4—104no—
332Dic3⋊C4—104no—
333C4×Dic3—104no—
334C23×C5—21no—
335C23×C5—21no—
336C23×C5—21no—
337C23×C5—22no—
338C23×C5—22no—
339C23×C5—22no—
340C23×C5—22no—
341C23×C5—21no—
342C23×C5#23512yescomplemented
343C23×C5C2×#6712yes—
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—28no—
353C2×C4×C5—21no—
354C2×C4×C5—22no—
355C2×C4×C5—21no—
356C2×C4×C5—28no—
357C2×C4×C5—22no—
358C2×C4×C5—22no—
359C2×C4×C5—21no—
360C2×C4×C5—21no—
361C2×C4×C5—44no—
362C2×C4×C5#23512yescomplemented
363C2×C4×C5D3×D414yes—
364C2×C4×C5C2×#6712yes—
365C2×C4×C5C2×#6712yes—
366C22⋊D5—616no—
367C22⋊D5—616no—
368C2×Dic5—34no—
369C2×Dic5—34no—
370C2×Dic5—62no—
371C2×Dic5—62no—
372C2×Dic5—62no—
373C2×Dic5—62no—
374C2×Dic5—62no—
375C2×Dic5—62no—
376C2×Dic5—62no—
377D20—68no—
378C4×D5—616no—
379Dic10—68no—
380C2×#41—308no—
381C2×#41—304no—
382C22×Q8—151no—
383C22×D4—152no—
384C22×D4—152no—
385C22×D4—151no—
386C22×D4C2×D1511yescharacteristic
387C23×C4—152no—
388C23×C4C2×D1511yescharacteristic
389C23×C4C2×D1511yescharacteristic
390C4×D4—24no—
391C4×D4—308no—
392C2×C4⋊C4—152no—
393C2×C4⋊C4C2×D1511yescharacteristic
394C2×C22⋊C4—152no—
395C2×C22⋊C4—152no—
396C2×C22⋊C4—152no—
397C2×C22⋊C4—152no—
398C2×C22⋊C4—152no—
399C2×C22⋊C4C2×D1511yescharacteristic
400C2×C22⋊C4C2×D1511yescharacteristic
401C2×C42—152no—
402C2×C42C2×D1511yescharacteristic
403#95—151no—
404#95—152no—
405#95—152no—
406#95—151no—
407C2×C3×C5—42no—
408C2×C3×C5—42no—
409C2×C3×C5C2×#12011yescharacteristic
410C2×C3×C5#52511yescharacteristic
411C2×C3×C5#52512yes—
412C2×C3×C5#53212yes—
413C2×C3×C5#53211yescharacteristic
414D15—44no—
415C23×C3—21no—
416C23×C3—21no—
417C23×C3—21no—
418C23×C3—22no—
419C23×C3—22no—
420C23×C3—22no—
421C23×C3—22no—
422C23×C3—21no—
423C23×C3#70912yescomplemented
424C23×C3C2×C22⋊D512yes—
425C23×C3C23×D511yescharacteristic
426C22×D3—54no—
427C22×D3—102no—
428C22×D3—102no—
429C22×D3—102no—
430C22×D3—102no—
431C22×D3—102no—
432C22×D3—102no—
433C3×D4—28no—
434C2×C3×C4—21no—
435C2×C3×C4—22no—
436C2×C3×C4—21no—
437C2×C3×C4—28no—
438C2×C3×C4—22no—
439C2×C3×C4—22no—
440C2×C3×C4—21no—
441C2×C3×C4—21no—
442C2×C3×C4—44no—
443C2×C3×C4#70912yescomplemented
444C2×C3×C4D4×D514yes—
445C2×C3×C4C2×C22⋊D512yes—
446C2×C3×C4C2×C22⋊D512yes—
447#67—1016no—
448#67—1016no—
449C2×Dic3—54no—
450C2×Dic3—54no—
451C2×Dic3—102no—
452C2×Dic3—102no—
453C2×Dic3—102no—
454C2×Dic3—102no—
455C2×Dic3—102no—
456C2×Dic3—102no—
457C2×Dic3—102no—
458D12—108no—
459C4×D3—1016no—
460Dic6—108no—
461C22×C5—22no—
462C22×C5—22no—
463C22×C5—22no—
464C22×C5—24no—
465C22×C5—22no—
466C22×C5—41no—
467C22×C5—42no—
468C22×C5—42no—
469C22×C5—42no—
470C22×C5—41no—
471C22×C5C2×#23511yescharacteristic
472C22×C5C2×#23511yescharacteristic
473C22×C5C2×D3×D412yes—
474C22×C5D3×#4112yes—
475C22×C5C22×#6711yescharacteristic
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—41no—
486C4×C5—41no—
487C4×C5#93714yes—
488Dic5—64no—
489Dic5—64no—
490Dic5—122no—
491C24—152no—
492C24C22×D1511yescharacteristic
493C24C22×D1511yescharacteristic
494#41—3032no—
495C2×Q8—304no—
496C2×Q8—302no—
497C2×D4—22no—
498C2×D4—22no—
499C2×D4—22no—
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—24no—
509C22×C4—24no—
510C22×C4—22no—
511C22×C4—152no—
512C22×C4—152no—
513C22×C4—158no—
514C22×C4—152no—
515C22×C4—304no—
516C22×C4—304no—
517C22×C4#122014yescomplemented
518C22×C4C22×D1511yescharacteristic
519C22×C4C22×D1511yescharacteristic
520C22×C4C22×D1511yescharacteristic
521C22×C4C22×D1511yescharacteristic
522C22×C4C22×D1511yescharacteristic
523C4⋊C4—22no—
524C4⋊C4—304no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—304no—
528C22⋊C4—304no—
529C22⋊C4—304no—
530C22⋊C4—304no—
531C22⋊C4—304no—
532C42—22no—
533C42—304no—
534C3×C5#238811yescharacteristic, complemented, Hall
535C22×C3—22no—
536C22×C3—22no—
537C22×C3—22no—
538C22×C3—24no—
539C22×C3—22no—
540C22×C3—41no—
541C22×C3—42no—
542C22×C3—42no—
543C22×C3—42no—
544C22×C3—41no—
545C22×C3C2×#70911yescharacteristic
546C22×C3C2×#70911yescharacteristic
547C22×C3C2×D4×D512yes—
548C22×C3D5×#4112yes—
549C22×C3C22×C22⋊D511yescharacteristic
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—41no—
561C3×C4#408314yes—
562Dic3—104no—
563Dic3—104no—
564Dic3—202no—
565C2×C5—42no—
566C2×C5—42no—
567C2×C5#554712yes—
568C2×C5#554812yes—
569C2×C5#579111yescharacteristic
570C2×C5C2×#93711yescharacteristic
571C2×C5#583211yescharacteristic
572D5—124no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—21no—
581C23—154no—
582C23—302no—
583C23—302no—
584C23—302no—
585C23—302no—
586C23—302no—
587C23—302no—
588C23#681712yescomplemented
589C23C2×#122012yes—
590C23C23×D1511yescharacteristic
591Q8—308no—
592D4—28no—
593D4—308no—
594D4—3016no—
595D4—3016no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—28no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—44no—
605C2×C4—154no—
606C2×C4—154no—
607C2×C4—302no—
608C2×C4—302no—
609C2×C4—302no—
610C2×C4—302no—
611C2×C4—3016no—
612C2×C4—302no—
613C2×C4—302no—
614C2×C4—302no—
615C2×C4#681712yescomplemented
616C2×C4D4×D1514yes—
617C2×C4C2×#122012yes—
618C2×C4C2×#122012yes—
619C2×C3—42no—
620C2×C3—42no—
621C2×C3#6588112yes—
622C2×C3#6588212yes—
623C2×C3#6612511yescharacteristic
624C2×C3C2×#408311yescharacteristic
625C2×C3#6616611yescharacteristic
626D3—204no—
627C5#8240011yescharacteristic, complemented, Sylow
628C22—22no—
629C22—22no—
630C22—22no—
631C22—24no—
632C22—22no—
633C22—41no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—41no—
638C22—304no—
639C22—304no—
640C22—304no—
641C22—602no—
642C22—602no—
643C22—602no—
644C22—602no—
645C22C2×#681711yescharacteristic
646C22C2×#681711yescharacteristic
647C22C2×D4×D1512yes—
648C22#41×D1512yes—
649C22C22×#122011yescharacteristic
650C4—22no—
651C4—22no—
652C4—41no—
653C4—41no—
654C4—304no—
655C4—304no—
656C4—602no—
657C4#9182614yes—
658C3#1061413111yescharacteristic, complemented, Sylow
659C2—42no—
660C2—42no—
661C2—604no—
662C2#1175630612yes—
663C2#1175630712yes—
664C2#1175655011yescharacteristic
665C2C2×#9182611yescharacteristic
666C2#1175659111yescharacteristic
subgroupG/H
centerC22×C4C22×D15
commutator, socle, cosocleC23×C3×C5C24
FrattiniC23C23×D15
FittingC2×C3×C4×C5×D4C2
hypercenterC2×C4×D4D15
hypocenterC3×C5#2388
LCSG ▷ C23×C3×C5 ▷ C3×C5
UCSC1 ◁ C22×C4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C22×#6817C2semidirect
2. C2×C3×C4×C5×D4C2semidirect
3. C2×C4×#1220C2semidirect
4. #11751324C2semidirect
5. C4×#6719C2semidirect
6. #11751092C2semidirect
7. #11751080C2semidirect
8. #11751061C2semidirect
9. #11751060C2semidirect
10. #11751048C2semidirect
11. C4×#6714C2semidirect
42. C2×C4×C5×D4D3semidirect
61. C22×C3×C4×C5D4semidirect
81. C2×C4×D15D4semidirect
96. C2×C3×C4×D4D5semidirect
143. C23×C3×C5#41semidirect
165. C2×C3×C4×C5#41semidirect
176. C2×Dic15#41semidirect
216. C22×C4×C5#67semidirect
252. C2×C4×D4D15semidirect
300. C22×C3×C4C22⋊D5semidirect
342. C23×C5#235semidirect
362. C2×C4×C5#235semidirect
423. C23×C3#709semidirect
443. C2×C3×C4#709semidirect
517. C22×C4#1220semidirect
534. C3×C5#2388semidirect
588. C23#6817semidirect
615. C2×C4#6817semidirect
627. C5#82400semidirect
658. C3#10614131semidirect

Maximal subgroups

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

subgroupindex
C22×#68172
C2×C3×C4×C5×D42
C2×C4×#12202
#117513242
C4×#67192
#117510922
#117510802
#117510612
#117510602
#117510482
C4×#67142
#106141313
#824005