Finite Group Databasebeta

← #14327320

Subgroups — #14327320

Order 1920.

1320 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#1220C211yesmaximal, characteristic
2C2×#91815C211yesmaximal, characteristic, complemented
3C2×C4⋊C4×D15C211yesmaximal, characteristic, complemented
4C3×C5×#380C211yesmaximal, characteristic, complemented
5#11751088C211yesmaximal, characteristic, complemented
6#11751078C211yesmaximal, characteristic, complemented
7C4⋊C4×Dic15C211yesmaximal, characteristic, complemented
8#11751057C212yesmaximal
9#11751045C212yesmaximal, complemented
10#11750767C212yesmaximal, complemented
11#11750753C212yesmaximal, complemented
12#10605743—31nomaximal, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C22×#1220C2211yescharacteristic
15C22×C4×D15C2212yes—
16C22×C4×D15C2211yescharacteristic
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18C2×C3×C5×C4⋊C4C2211yescharacteristic
19C2×C3×C5×C22⋊C4C2212yes—
20C2×#6719C2212yes—
21C2×#6719C2211yescharacteristic
22C4×#1220C418yescomplemented
23C2×#6714C2212yes—
24C2×#6714C2211yescharacteristic
25#91815—24no—
26C2×C4⋊Dic15C2211yescharacteristic
27C2×C4⋊Dic15C2211yescharacteristic
28C2×#6712C2212yes—
29C2×#6712C2211yescharacteristic
30C2×C4×Dic15C2212yes—
31C2×C4×Dic15C2212yes—
32C2×C4×Dic15C2211yescharacteristic
33C4⋊C4×D15—24no—
34C3×C5×#95C2212yes—
35#91115C2212yes—
36#91115C2212yes—
37#91115C2212yes—
38#91115C2212yes—
39#74012—51nomaximal, Hall
40C2×C4×C22⋊D5—31no—
41C2×#4072—31no—
42C2×D5×C4⋊C4—31no—
43C5×#380D311yescharacteristic, complemented
44#65276—31no—
45#65266—31no—
46C4⋊C4×Dic5—31no—
47#65245—32no—
48#65233—32no—
49#64955—32no—
50#64941—32no—
51C24×C3×C5C2311yescharacteristic
52C23×D15C2311yescharacteristic
53C22×C3×C4×C5—22no—
54C22×C3×C4×C5—22no—
55C22×C3×C4×C5C2×C414yescomplemented
56C22×C3×C4×C5C2312yes—
57C22×C3×C4×C5C2312yes—
58C22×C3×C4×C5C2311yescharacteristic
59C22×C3×C4×C5C2311yescharacteristic
60C2×#1220—22no—
61C2×#1220—22no—
62C2×#1220C2×C414yes—
63C22×Dic15C2312yes—
64C22×Dic15C2311yescharacteristic
65C22×Dic15C2311yescharacteristic
66C22×Dic15C2311yescharacteristic
67C22×Dic15C2312yes—
68C2×C4×D15—22no—
69C2×C4×D15—22no—
70C2×C4×D15—24no—
71C2×C4×D15—24no—
72C2×C4×D15—42no—
73C2×C4×D15C2×C414yes—
74C3×C5×C4⋊C4—22no—
75C3×C5×C4⋊C4—22no—
76C3×C5×C22⋊C4—42no—
77#6719—24no—
78#6719C2×C414yescomplemented
79#6714—24no—
80#6714C2×C414yes—
81C4⋊Dic15—22no—
82C4⋊Dic15D414yescomplemented
83#6712—24no—
84#6712C2×C414yescomplemented
85C4×Dic15—24no—
86C4×Dic15—42no—
87C4×Dic15C2×C414yescomplemented
88C2×C4×#67—51no—
89C2×#926—51no—
90C2×D3×C4⋊C4—51no—
91C3×#380D511yescharacteristic, complemented
92#4987—51no—
93#4977—51no—
94Dic3×C4⋊C4—51no—
95#4956—52no—
96#4944—52no—
97#4666—52no—
98#4652—52no—
99C23×C4×C5C2×D311yescharacteristic
100C22×C22⋊D5—31no—
101C22×C4×D5—31no—
102C22×C4×D5—32no—
103C2×C5×C4⋊C4C2×D311yescharacteristic
104C2×C5×C4⋊C4C2×D311yescharacteristic
105C2×C5×C22⋊C4C2×D312yes—
106C2×C22⋊Dic5—31no—
107C2×C22⋊Dic5—32no—
108C4×C22⋊D5—38no—
109C2×#685—31no—
110C2×#685—32no—
111#4072—64no—
112C2×C4⋊Dic5—31no—
113C2×C4⋊Dic5—31no—
114C2×#683—31no—
115C2×#683—32no—
116C2×C4×Dic5—31no—
117C2×C4×Dic5—32no—
118C2×C4×Dic5—32no—
119D5×C4⋊C4—64no—
120C5×#95C2×D312yes—
121#3963—32no—
122#3963—32no—
123#3963—32no—
124#3963—32no—
125#2352—151noSylow
126C23×C3×C5—21no—
127C23×C3×C5—21no—
128C23×C3×C5—21no—
129C23×C3×C5—21no—
130C23×C3×C5—21no—
131C23×C3×C5—21no—
132C23×C3×C5C22×C412yes—
133C23×C3×C5C2411yescharacteristic
134C22×D15—21no—
135C22×D15—21no—
136C22×D15—21no—
137C22×D15—21no—
138C22×D15C22×C412yes—
139C22×D15#4112yes—
140C22×D15#4112yes—
141C2×C3×C4×C5—21no—
142C2×C3×C4×C5—21no—
143C2×C3×C4×C5—28no—
144C2×C3×C4×C5—22no—
145C2×C3×C4×C5—22no—
146C2×C3×C4×C5—22no—
147C2×C3×C4×C5—22no—
148C2×C3×C4×C5—22no—
149C2×C3×C4×C5—22no—
150C2×C3×C4×C5C22×C412yes—
151C2×C3×C4×C5C22×C412yes—
152C2×C3×C4×C5C2×D412yes—
153C2×C3×C4×C5#4112yes—
154#1220—28no—
155C2×Dic15—21no—
156C2×Dic15—21no—
157C2×Dic15—22no—
158C2×Dic15—21no—
159C2×Dic15—21no—
160C2×Dic15—21no—
161C2×Dic15—22no—
162C2×Dic15—21no—
163C2×Dic15—22no—
164C2×Dic15—22no—
165C2×Dic15—22no—
166C2×Dic15C22×C412yes—
167C2×Dic15C22×C412yes—
168C2×Dic15C22×C412yes—
169C2×Dic15C2×D414yes—
170C4×D15—28no—
171C4×D15—48no—
172C23×C3×C4C2×D511yescharacteristic
173C22×#67—51no—
174C22×C4×D3—51no—
175C22×C4×D3—52no—
176C2×C3×C4⋊C4C2×D511yescharacteristic
177C2×C3×C4⋊C4C2×D511yescharacteristic
178C2×C3×C22⋊C4C2×D512yes—
179C2×#217—51no—
180C2×#217—52no—
181C4×#67—58no—
182C2×(C2×D3)⋊C4—51no—
183C2×(C2×D3)⋊C4—52no—
184#926—104no—
185C2×C4⋊Dic3—51no—
186C2×C4⋊Dic3—51no—
187C2×Dic3⋊C4—51no—
188C2×Dic3⋊C4—52no—
189C2×C4×Dic3—51no—
190C2×C4×Dic3—52no—
191C2×C4×Dic3—52no—
192D3×C4⋊C4—104no—
193C3×#95C2×D512yes—
194C3⋊#95—52no—
195C3⋊#95—52no—
196C3⋊#95—52no—
197C3⋊#95—52no—
198C24×C5C22×D311yescharacteristic
199C23×D5—31no—
200C22×C4×C5—22no—
201C22×C4×C5—22no—
202C22×C4×C5C4×D314yescomplemented
203C22×C4×C5C22×D312yes—
204C22×C4×C5C22×D312yes—
205C22×C4×C5C22×D311yescharacteristic
206C22×C4×C5C22×D311yescharacteristic
207C2×C22⋊D5—34no—
208C2×C22⋊D5—62no—
209C2×C22⋊D5—62no—
210C22×Dic5—31no—
211C22×Dic5—32no—
212C22×Dic5—31no—
213C22×Dic5—31no—
214C22×Dic5—32no—
215C2×C4×D5—34no—
216C2×C4×D5—62no—
217C2×C4×D5—62no—
218C2×C4×D5—64no—
219C2×C4×D5—64no—
220C2×C4×D5—122no—
221C5×C4⋊C4—22no—
222C5×C4⋊C4—22no—
223C5×C22⋊C4—42no—
224C22⋊Dic5—34no—
225C22⋊Dic5—64no—
226#685—34no—
227#685—64no—
228C4⋊Dic5—34no—
229C4⋊Dic5—62no—
230#683—34no—
231#683—64no—
232C4×Dic5—34no—
233C4×Dic5—64no—
234C4×Dic5—122no—
235C2×C4×D4—151no—
236C2×#117—151no—
237C22×C4⋊C4—151no—
238#388—152no—
239#387—152no—
240#386—152no—
241#382—151no—
242#380—151no—
243#380D1511yescharacteristic, complemented
244C4×C4⋊C4—151no—
245C4×C22⋊C4—152no—
246C22×C3×C5—22no—
247C22×C3×C5—22no—
248C22×C3×C5—22no—
249C22×C3×C5—41no—
250C22×C3×C5—41no—
251C22×C3×C5—41no—
252C22×C3×C5—41no—
253C22×C3×C5C23×C411yescharacteristic
254C22×C3×C5C22×D411yescharacteristic
255C22×C3×C5C2×#4111yescharacteristic
256C22×C3×C5C2×#4111yescharacteristic
257C22×C3×C5C2×#4111yescharacteristic
258C22×C3×C5H(5,𝔽2)11yescharacteristic
259C22×C3×C52−1+411yescharacteristic
260C2×D15—22no—
261C2×D15—22no—
262C2×D15—22no—
263C2×D15—22no—
264C2×D15—22no—
265C2×D15—22no—
266C2×D15#11714yescomplemented
267C3×C4×C5—22no—
268C3×C4×C5—42no—
269C3×C4×C5—42no—
270C3×C4×C5C4×D414yes—
271Dic15—22no—
272Dic15—22no—
273Dic15—22no—
274Dic15—24no—
275Dic15—42no—
276C24×C3C22×D511yescharacteristic
277C23×D3—51no—
278C22×C3×C4—22no—
279C22×C3×C4—22no—
280C22×C3×C4C4×D514yescomplemented
281C22×C3×C4C22×D512yes—
282C22×C3×C4C22×D512yes—
283C22×C3×C4C22×D511yescharacteristic
284C22×C3×C4C22×D511yescharacteristic
285C2×#67—54no—
286C2×#67—102no—
287C2×#67—102no—
288C22×Dic3—51no—
289C22×Dic3—51no—
290C22×Dic3—51no—
291C22×Dic3—52no—
292C22×Dic3—52no—
293C2×C4×D3—54no—
294C2×C4×D3—102no—
295C2×C4×D3—102no—
296C2×C4×D3—104no—
297C2×C4×D3—104no—
298C2×C4×D3—202no—
299C3×C4⋊C4—22no—
300C3×C4⋊C4—22no—
301C3×C22⋊C4—42no—
302#217—54no—
303#217—104no—
304(C2×D3)⋊C4—54no—
305(C2×D3)⋊C4—104no—
306C4⋊Dic3—54no—
307C4⋊Dic3—102no—
308Dic3⋊C4—54no—
309Dic3⋊C4—104no—
310C4×Dic3—54no—
311C4×Dic3—104no—
312C4×Dic3—202no—
313C23×C5—21no—
314C23×C5—21no—
315C23×C5—21no—
316C23×C5—21no—
317C23×C5—21no—
318C23×C5—21no—
319C23×C5C2×C4×D312yes—
320C23×C5C23×D311yescharacteristic
321C22×D5—32no—
322C22×D5—32no—
323C22×D5—32no—
324C22×D5—61no—
325C22×D5—61no—
326C22×D5—61no—
327C22×D5—61no—
328C2×C4×C5—21no—
329C2×C4×C5—21no—
330C2×C4×C5—28no—
331C2×C4×C5—22no—
332C2×C4×C5—22no—
333C2×C4×C5—22no—
334C2×C4×C5—22no—
335C2×C4×C5—22no—
336C2×C4×C5—22no—
337C2×C4×C5C2×C4×D312yes—
338C2×C4×C5C2×C4×D312yes—
339C2×C4×C5D3×D412yes—
340C2×C4×C5#23712yes—
341C22⋊D5—68no—
342C2×Dic5—32no—
343C2×Dic5—32no—
344C2×Dic5—32no—
345C2×Dic5—34no—
346C2×Dic5—61no—
347C2×Dic5—61no—
348C2×Dic5—61no—
349C2×Dic5—62no—
350C2×Dic5—61no—
351C2×Dic5—62no—
352C2×Dic5—61no—
353C2×Dic5—61no—
354C2×Dic5—62no—
355C2×Dic5—62no—
356C2×Dic5—62no—
357C4×D5—68no—
358C4×D5—128no—
359C22×D4—151no—
360C23×C4—151no—
361C23×C4—152no—
362C23×C4C2×D1511yescharacteristic
363C4×D4—158no—
364#117—304no—
365C2×C4⋊C4—151no—
366C2×C4⋊C4—151no—
367C2×C4⋊C4—152no—
368C2×C4⋊C4—151no—
369C2×C4⋊C4—304no—
370C2×C4⋊C4C2×D1511yescharacteristic
371C2×C4⋊C4C2×D1511yescharacteristic
372C2×C22⋊C4—151no—
373C2×C22⋊C4—151no—
374C2×C22⋊C4—152no—
375C2×C22⋊C4—152no—
376C2×C22⋊C4C2×D1512yes—
377C2×C42—151no—
378C2×C42—152no—
379C2×C42—152no—
380#95—152no—
381#95—152no—
382#95—152no—
383#95—152no—
384#95C2×D1512yes—
385C2×C3×C5—42no—
386C2×C3×C5C2×#11711yescharacteristic
387C2×C3×C5C2×C4×D411yescharacteristic
388C2×C3×C5#52011yescharacteristic
389C2×C3×C5#54611yescharacteristic
390C2×C3×C5#54811yescharacteristic
391C2×C3×C5#55211yescharacteristic
392C2×C3×C5#55311yescharacteristic
393D15—24no—
394C23×C3—21no—
395C23×C3—21no—
396C23×C3—21no—
397C23×C3—21no—
398C23×C3—21no—
399C23×C3—21no—
400C23×C3C2×C4×D512yes—
401C23×C3C23×D511yescharacteristic
402C22×D3—52no—
403C22×D3—52no—
404C22×D3—52no—
405C22×D3—101no—
406C22×D3—101no—
407C22×D3—101no—
408C22×D3—101no—
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—28no—
412C2×C3×C4—22no—
413C2×C3×C4—22no—
414C2×C3×C4—22no—
415C2×C3×C4—22no—
416C2×C3×C4—22no—
417C2×C3×C4—22no—
418C2×C3×C4C2×C4×D512yes—
419C2×C3×C4C2×C4×D512yes—
420C2×C3×C4D4×D512yes—
421C2×C3×C4#71112yes—
422#67—108no—
423C2×Dic3—52no—
424C2×Dic3—52no—
425C2×Dic3—52no—
426C2×Dic3—54no—
427C2×Dic3—101no—
428C2×Dic3—101no—
429C2×Dic3—101no—
430C2×Dic3—102no—
431C2×Dic3—101no—
432C2×Dic3—102no—
433C2×Dic3—101no—
434C2×Dic3—101no—
435C2×Dic3—102no—
436C2×Dic3—102no—
437C2×Dic3—102no—
438C4×D3—108no—
439C4×D3—208no—
440C22×C5—22no—
441C22×C5—22no—
442C22×C5—22no—
443C22×C5—41no—
444C22×C5—41no—
445C22×C5—41no—
446C22×C5—41no—
447C22×C5C22×C4×D311yescharacteristic
448C22×C5C2×D3×D411yescharacteristic
449C22×C5C2×#23711yescharacteristic
450C22×C5#100411yescharacteristic
451C22×C5#100711yescharacteristic
452C22×C5D3×#4111yescharacteristic
453C22×C5D3×#4111yescharacteristic
454C2×D5—34no—
455C2×D5—62no—
456C2×D5—62no—
457C2×D5—62no—
458C2×D5—62no—
459C2×D5—62no—
460C2×D5—62no—
461C4×C5—22no—
462C4×C5—42no—
463C4×C5—42no—
464C4×C5#67⋊C414yes—
465Dic5—62no—
466Dic5—62no—
467Dic5—62no—
468Dic5—64no—
469Dic5—122no—
470C24—151no—
471C24C22×D1511yescharacteristic
472C2×D4—154no—
473C2×D4—302no—
474C2×D4—302no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—151no—
478C22×C4—151no—
479C22×C4—151no—
480C22×C4—152no—
481C22×C4—154no—
482C22×C4—152no—
483C22×C4—304no—
484C22×C4—304no—
485C22×C4—302no—
486C22×C4—302no—
487C22×C4—602no—
488C22×C4C4×D1514yescomplemented
489C22×C4C22×D1512yes—
490C22×C4C22×D1512yes—
491C22×C4C22×D1511yescharacteristic
492C22×C4C22×D1511yescharacteristic
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—154no—
496C4⋊C4—154no—
497C4⋊C4—304no—
498C4⋊C4—302no—
499C22⋊C4—42no—
500C22⋊C4—154no—
501C22⋊C4—154no—
502C22⋊C4—304no—
503C22⋊C4—304no—
504C42—154no—
505C42—304no—
506C42—602no—
507C3×C5#235211yescharacteristic, complemented, Hall
508C22×C3—22no—
509C22×C3—22no—
510C22×C3—22no—
511C22×C3—41no—
512C22×C3—41no—
513C22×C3—41no—
514C22×C3—41no—
515C22×C3C22×C4×D511yescharacteristic
516C22×C3C2×D4×D511yescharacteristic
517C22×C3C2×#71111yescharacteristic
518C22×C3#414411yescharacteristic
519C22×C3#414711yescharacteristic
520C22×C3D5×#4111yescharacteristic
521C22×C3D5×#4111yescharacteristic
522C2×D3—54no—
523C2×D3—102no—
524C2×D3—102no—
525C2×D3—102no—
526C2×D3—102no—
527C2×D3—102no—
528C2×D3—102no—
529C3×C4—22no—
530C3×C4—42no—
531C3×C4—42no—
532C3×C4#402714yes—
533Dic3—102no—
534Dic3—102no—
535Dic3—102no—
536Dic3—104no—
537Dic3—202no—
538C2×C5—42no—
539C2×C5C2×#67⋊C411yescharacteristic
540C2×C5#550411yescharacteristic
541C2×C5D3×#11711yescharacteristic
542C2×C5#561211yescharacteristic
543C2×C5#561411yescharacteristic
544C2×C5#563311yescharacteristic
545C2×C5#563911yescharacteristic
546D5—64no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—152no—
554C23—152no—
555C23—152no—
556C23—301no—
557C23—301no—
558C23—301no—
559C23—301no—
560C23C2×C4×D1512yes—
561C23C23×D1511yescharacteristic
562D4—308no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—28no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—152no—
573C2×C4—152no—
574C2×C4—152no—
575C2×C4—154no—
576C2×C4—301no—
577C2×C4—301no—
578C2×C4—301no—
579C2×C4—302no—
580C2×C4—301no—
581C2×C4—302no—
582C2×C4—301no—
583C2×C4—301no—
584C2×C4—308no—
585C2×C4—302no—
586C2×C4—302no—
587C2×C4—302no—
588C2×C4—608no—
589C2×C4C2×C4×D1512yes—
590C2×C4C2×C4×D1512yes—
591C2×C4D4×D1512yes—
592C2×C4#681912yes—
593C2×C3—42no—
594C2×C3C2×#402711yescharacteristic
595C2×C3#6583811yescharacteristic
596C2×C3D5×#11711yescharacteristic
597C2×C3#6594611yescharacteristic
598C2×C3#6594811yescharacteristic
599C2×C3#6596711yescharacteristic
600C2×C3#6597311yescharacteristic
601D3—104no—
602C5#7401211yescharacteristic, complemented, Sylow
603C22—22no—
604C22—22no—
605C22—22no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—154no—
611C22—302no—
612C22—302no—
613C22—302no—
614C22—302no—
615C22—302no—
616C22—302no—
617C22C22×C4×D1511yescharacteristic
618C22C2×D4×D1511yescharacteristic
619C22C2×#681911yescharacteristic
620C22#9209511yescharacteristic
621C22#9209811yescharacteristic
622C22#41×D1511yescharacteristic
623C22#41×D1511yescharacteristic
624C4—22no—
625C4—42no—
626C4—42no—
627C4—304no—
628C4—302no—
629C4—302no—
630C4—302no—
631C4—602no—
632C4#9177014yes—
633C3#1060574311yescharacteristic, complemented, Sylow
634C2—42no—
635C2—304no—
636C2C2×#9177011yescharacteristic
637C2#1175626311yescharacteristic
638C2D15×#11711yescharacteristic
639C2#1175637111yescharacteristic
640C2#1175637311yescharacteristic
641C2#1175639211yescharacteristic
642C2#1175639811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D15
commutatorC22×C3×C5C23×C4
FittingC3×C5×#380C2
socle, cosocleC23×C3×C5C24
prime radicalC22×#1220C22
hypercenter#380D15
hypocenterC3×C5#2352
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#91815C2semidirect
3. C2×C4⋊C4×D15C2semidirect
4. C3×C5×#380C2semidirect
5. #11751088C2semidirect
6. #11751078C2semidirect
7. C4⋊C4×Dic15C2semidirect
9. #11751045C2semidirect
10. #11750767C2semidirect
11. #11750753C2semidirect
22. C4×#1220C4semidirect
43. C5×#380D3semidirect
55. C22×C3×C4×C5C2×C4semidirect
78. #6719C2×C4semidirect
82. C4⋊Dic15D4semidirect
84. #6712C2×C4semidirect
87. C4×Dic15C2×C4semidirect
91. C3×#380D5semidirect
202. C22×C4×C5C4×D3semidirect
243. #380D15semidirect
266. C2×D15#117semidirect
280. C22×C3×C4C4×D5semidirect
488. C22×C4C4×D15semidirect
507. C3×C5#2352semidirect
602. C5#74012semidirect
633. C3#10605743semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#12202
C2×#918152
C2×C4⋊C4×D152
C3×C5×#3802
#117510882
#117510782
C4⋊C4×Dic152
#117510572
#117510452
#117507672
#117507532
#106057433
#740125