Finite Group Databasebeta

← #14327414

Subgroups — #14327414

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
2C22×C3×C4⋊Dic5C211yesmaximal, characteristic, complemented
3C2×#91347C211yesmaximal, characteristic, complemented
4C5×#5212C211yesmaximal, characteristic, complemented
5#11749494C212yesmaximal
6Dic5×#217C212yesmaximal, complemented
7#11748799C211yesmaximal, characteristic, complemented
8#11748699C211yesmaximal, characteristic, complemented
9#11748678C212yesmaximal, complemented
10#11748674C212yesmaximal, complemented
11#11748623C211yesmaximal, characteristic, complemented
12#10605614—31nomaximal, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C22×#1220C2211yescharacteristic
15C22×C4×D15C2211yescharacteristic
16C23×C3×Dic5C2212yes—
17C2×#6719C2211yescharacteristic
18C4×#1220C418yescomplemented
19C2×#6714C2211yescharacteristic
20C2×#6712C2211yescharacteristic
21C2×C4×Dic15C2211yescharacteristic
22C2×C5×#217C2212yes—
23C2×C5×C4⋊Dic3C2211yescharacteristic
24C2×C5×C4⋊Dic3C2211yescharacteristic
25C2×C3×C4⋊Dic5—24no—
26C2×C3×C4⋊Dic5C2211yescharacteristic
27C2×C3×C4⋊Dic5C2212yes—
28C2×C3×C4⋊Dic5C2211yescharacteristic
29C2×#6667C2212yes—
30C2×#6667C2212yes—
31C2×Dic3×Dic5C2212yes—
32C2×Dic3×Dic5C2212yes—
33#91347—24no—
34C5×C3⋊#95C2212yes—
35#90994C2212yes—
36#90994C2212yes—
37#90994C2212yes—
38#90994C2212yes—
39#82161—51nomaximal, Hall
40C2×C4×C22⋊D5—31no—
41C22×C4⋊Dic5D311yescharacteristic, complemented
42C2×#4038—31no—
43C5×#380—31no—
44#65276—31no—
45#65266—31no—
46C4⋊C4×Dic5—31no—
47#65247—32no—
48C22⋊C4×Dic5—32no—
49#64959—32no—
50#64944—32no—
51C24×C3×C5C2311yescharacteristic
52C23×D15C2311yescharacteristic
53C22×C3×C4×C5—22no—
54C22×C3×C4×C5—22no—
55C22×C3×C4×C5C2×C414yes—
56C22×C3×C4×C5C2311yescharacteristic
57C22×C3×C4×C5C2311yescharacteristic
58C2×#1220—22no—
59C2×#1220—22no—
60C2×#1220C2×C414yes—
61C22×Dic15C2311yescharacteristic
62C22×Dic15C2311yescharacteristic
63C22×Dic15C2311yescharacteristic
64C2×C4×D15—22no—
65C2×C4×D15—22no—
66C2×C4×D15C2×C414yescomplemented
67C22×C5×Dic3C2312yes—
68C22×C5×Dic3C2312yes—
69C22×C3×Dic5—24no—
70C22×C3×Dic5—24no—
71C22×C3×Dic5—42no—
72C22×C3×Dic5C2312yes—
73C22×C3×Dic5C2312yes—
74#6719C2×C414yes—
75#6714C2×C414yescomplemented
76#6712C2×C414yescomplemented
77C4×Dic15C2×C414yescomplemented
78C5×#217—24no—
79C5×C4⋊Dic3—22no—
80C5×C4⋊Dic3—22no—
81C3×C4⋊Dic5—22no—
82C3×C4⋊Dic5—24no—
83C3×C4⋊Dic5D414yescomplemented
84#6667—24no—
85#6667—42no—
86Dic3×Dic5—24no—
87Dic3×Dic5—42no—
88C22×C3×C4⋊C4—51no—
89C2×C4×#67—51no—
90C2×#892—51no—
91#5213—52no—
92#5212D511yescharacteristic, complemented
93C4×#217—52no—
94#4990—52no—
95#4987—51no—
96#4978—52no—
97#4977—51no—
98Dic3×C4⋊C4—51no—
99C23×C4×C5C2×D311yescharacteristic
100C22×C22⋊D5—31no—
101C23×Dic5C2×D312yes—
102C22×C4×D5—31no—
103C2×C5×C4⋊C4—31no—
104C2×C5×C4⋊C4—31no—
105C2×C5×C22⋊C4—32no—
106C2×C22⋊Dic5—31no—
107C4×C22⋊D5—38no—
108C2×#685—31no—
109C2×#685—32no—
110C2×#685—32no—
111C2×C4⋊Dic5—24no—
112C2×C4⋊Dic5C2×D312yes—
113C2×C4⋊Dic5C2×D311yescharacteristic
114C2×C4⋊Dic5C2×D311yescharacteristic
115C2×#683—31no—
116C2×C4×Dic5—31no—
117C2×C4×Dic5—32no—
118C2×C4×Dic5—32no—
119#4038—64no—
120C5×#95—32no—
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×C5C22×C412yes—
131C23×C3×C5#4112yes—
132C23×C3×C5#4112yes—
133C23×C3×C5C2411yescharacteristic
134C22×D15—21no—
135C22×D15—21no—
136C22×D15—21no—
137C22×D15—21no—
138C22×D15—21no—
139C22×D15—21no—
140C22×D15C22×C412yes—
141C2×C3×C4×C5—21no—
142C2×C3×C4×C5—21no—
143C2×C3×C4×C5—28no—
144C2×C3×C4×C5C22×C412yes—
145C2×C3×C4×C5C22×C412yes—
146C2×C3×C4×C5C2×D412yes—
147C2×C3×C4×C5#4112yes—
148#1220—28no—
149C2×Dic15—21no—
150C2×Dic15—21no—
151C2×Dic15—21no—
152C2×Dic15—21no—
153C2×Dic15—21no—
154C2×Dic15—21no—
155C2×Dic15C22×C412yes—
156C2×Dic15C22×C412yes—
157C2×Dic15C22×C412yes—
158C4×D15—28no—
159C2×C5×Dic3—22no—
160C2×C5×Dic3—22no—
161C2×C5×Dic3—22no—
162C2×C5×Dic3—22no—
163C2×C5×Dic3—22no—
164C2×C5×Dic3—22no—
165C2×C3×Dic5—22no—
166C2×C3×Dic5—22no—
167C2×C3×Dic5—22no—
168C2×C3×Dic5—22no—
169C2×C3×Dic5—22no—
170C2×C3×Dic5—48no—
171C2×C3×Dic5C2×D414yes—
172C23×C3×C4—52no—
173C23×C3×C4C2×D511yescharacteristic
174C22×#67—51no—
175C22×C4×D3—51no—
176C2×C3×C4⋊C4—51no—
177C2×C3×C4⋊C4—51no—
178C2×C3×C4⋊C4—52no—
179C2×C3×C4⋊C4—104no—
180C2×#217—51no—
181C2×#217C2×D512yes—
182C4×#67—58no—
183C2×(C2×D3)⋊C4—51no—
184C2×(C2×D3)⋊C4—52no—
185C2×(C2×D3)⋊C4—52no—
186C2×C4⋊Dic3C2×D511yescharacteristic
187C2×C4⋊Dic3C2×D511yescharacteristic
188C2×Dic3⋊C4—51no—
189C2×C4×Dic3—51no—
190C2×C4×Dic3—52no—
191C2×C4×Dic3—52no—
192#892—104no—
193C3⋊#95—52no—
194C3⋊#95—52no—
195C3⋊#95—52no—
196C3⋊#95—52no—
197C3⋊#95C2×D512yes—
198C24×C5C22×D311yescharacteristic
199C23×D5—31no—
200C22×C4×C5—22no—
201C22×C4×C5—22no—
202C22×C4×C5—32no—
203C22×C4×C5—32no—
204C22×C4×C5C4×D314yes—
205C22×C4×C5C22×D311yescharacteristic
206C22×C4×C5C22×D311yescharacteristic
207C2×C22⋊D5—34no—
208C2×C22⋊D5—62no—
209C2×C22⋊D5—62no—
210C22×Dic5—24no—
211C22×Dic5—24no—
212C22×Dic5—31no—
213C22×Dic5—31no—
214C22×Dic5—31no—
215C22×Dic5—42no—
216C22×Dic5C22×D312yes—
217C22×Dic5C22×D312yes—
218C2×C4×D5—34no—
219C2×C4×D5—62no—
220C2×C4×D5—62no—
221C5×C4⋊C4—62no—
222C5×C4⋊C4—62no—
223C5×C22⋊C4—64no—
224C22⋊Dic5—34no—
225#685—34no—
226#685—64no—
227#685—122no—
228C4⋊Dic5—22no—
229C4⋊Dic5—24no—
230C4⋊Dic5#6714yescomplemented
231#683—34no—
232C4×Dic5—34no—
233C4×Dic5—64no—
234C4×Dic5—122no—
235C2×C4×D4—151no—
236C2×#117—151no—
237C22×C4⋊C4—51no—
238#388—152no—
239#387—152no—
240#386—152no—
241#382—151no—
242#380—31no—
243#380—151no—
244C4×C4⋊C4—151no—
245C4×C22⋊C4—152no—
246C22×C3×C5—22no—
247C22×C3×C5—22no—
248C22×C3×C5—22no—
249C22×C3×C5—22no—
250C22×C3×C5—22no—
251C22×C3×C5—22no—
252C22×C3×C5#11714yescomplemented
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—41no—
264C2×D15—41no—
265C2×D15—41no—
266C2×D15—41no—
267C3×C4×C5—22no—
268C3×C4×C5C4×D414yes—
269Dic15—22no—
270Dic15—22no—
271Dic15—22no—
272C3×Dic5—24no—
273C3×Dic5—42no—
274C5×Dic3—42no—
275C5×Dic3—42no—
276C24×C3C22×D511yescharacteristic
277C23×D3—51no—
278C22×C3×C4—22no—
279C22×C3×C4—22no—
280C22×C3×C4—52no—
281C22×C3×C4—52no—
282C22×C3×C4—104no—
283C22×C3×C4—104no—
284C22×C3×C4—202no—
285C22×C3×C4C4×D514yes—
286C22×C3×C4C22×D511yescharacteristic
287C22×C3×C4C22×D511yescharacteristic
288C2×#67—54no—
289C2×#67—102no—
290C2×#67—102no—
291C22×Dic3—51no—
292C22×Dic3—51no—
293C22×Dic3—51no—
294C22×Dic3C22×D512yes—
295C22×Dic3C22×D512yes—
296C2×C4×D3—54no—
297C2×C4×D3—102no—
298C2×C4×D3—102no—
299C3×C4⋊C4—54no—
300C3×C4⋊C4—104no—
301C3×C4⋊C4—102no—
302#217—24no—
303#217—54no—
304(C2×D3)⋊C4—54no—
305(C2×D3)⋊C4—104no—
306(C2×D3)⋊C4—202no—
307C4⋊Dic3—22no—
308C4⋊Dic3—22no—
309Dic3⋊C4—54no—
310C4×Dic3—54no—
311C4×Dic3—104no—
312C4×Dic3—202no—
313C23×C5—21no—
314C23×C5—21no—
315C23×C5—21no—
316C23×C5—21no—
317C23×C5C2×C4×D312yes—
318C23×C5#23712yes—
319C23×C5#23912yes—
320C23×C5C23×D311yescharacteristic
321C22×D5—32no—
322C22×D5—61no—
323C22×D5—61no—
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—62no—
332C2×C4×C5—62no—
333C2×C4×C5—62no—
334C2×C4×C5—62no—
335C2×C4×C5—62no—
336C2×C4×C5—62no—
337C2×C4×C5C2×C4×D312yes—
338C2×C4×C5C2×C4×D312yes—
339C2×C4×C5#23512yes—
340C2×C4×C5C2×#6712yes—
341C22⋊D5—68no—
342C2×Dic5—22no—
343C2×Dic5—22no—
344C2×Dic5—22no—
345C2×Dic5—22no—
346C2×Dic5—22no—
347C2×Dic5—32no—
348C2×Dic5—32no—
349C2×Dic5—32no—
350C2×Dic5—48no—
351C2×Dic5—61no—
352C2×Dic5—61no—
353C2×Dic5—61no—
354C2×Dic5—61no—
355C2×Dic5—61no—
356C2×Dic5—61no—
357C2×Dic5C2×#6714yes—
358C4×D5—68no—
359C22×D4—151no—
360C23×C4—52no—
361C23×C4—151no—
362C23×C4D3×D511yescharacteristic
363C4×D4—158no—
364#117—304no—
365C2×C4⋊C4—31no—
366C2×C4⋊C4—31no—
367C2×C4⋊C4—51no—
368C2×C4⋊C4—51no—
369C2×C4⋊C4—52no—
370C2×C4⋊C4—104no—
371C2×C4⋊C4—151no—
372C2×C22⋊C4—32no—
373C2×C22⋊C4—151no—
374C2×C22⋊C4—151no—
375C2×C22⋊C4—152no—
376C2×C22⋊C4—152no—
377C2×C42—151no—
378C2×C42—152no—
379C2×C42—152no—
380#95—32no—
381#95—152no—
382#95—152no—
383#95—152no—
384#95—152no—
385C2×C3×C5—24no—
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—42no—
394C23×C3—21no—
395C23×C3—21no—
396C23×C3—21no—
397C23×C3—21no—
398C23×C3C2×C4×D512yes—
399C23×C3#71112yes—
400C23×C3#71312yes—
401C23×C3C23×D511yescharacteristic
402C22×D3—52no—
403C22×D3—101no—
404C22×D3—101no—
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—54no—
413C2×C3×C4—102no—
414C2×C3×C4—102no—
415C2×C3×C4—102no—
416C2×C3×C4—102no—
417C2×C3×C4—102no—
418C2×C3×C4—208no—
419C2×C3×C4C2×C4×D512yes—
420C2×C3×C4C2×C4×D512yes—
421C2×C3×C4D4×D512yes—
422C2×C3×C4#71112yes—
423#67—108no—
424C2×Dic3—22no—
425C2×Dic3—22no—
426C2×Dic3—22no—
427C2×Dic3—22no—
428C2×Dic3—22no—
429C2×Dic3—22no—
430C2×Dic3—52no—
431C2×Dic3—52no—
432C2×Dic3—52no—
433C2×Dic3—101no—
434C2×Dic3—101no—
435C2×Dic3—101no—
436C2×Dic3—101no—
437C2×Dic3—101no—
438C2×Dic3—101no—
439C4×D3—108no—
440C22×C5—22no—
441C22×C5—22no—
442C22×C5—22no—
443C22×C5—22no—
444C22×C5—22no—
445C22×C5—22no—
446C22×C5#89214yescomplemented
447C22×C5C22×C4×D311yescharacteristic
448C22×C5C2×#23511yescharacteristic
449C22×C5C2×#23711yescharacteristic
450C22×C5#100411yescharacteristic
451C22×C5C2×#23911yescharacteristic
452C22×C5#100711yescharacteristic
453C22×C5C22×#6711yescharacteristic
454C2×D5—62no—
455C2×D5—62no—
456C2×D5—62no—
457C2×D5—121no—
458C2×D5—121no—
459C2×D5—121no—
460C2×D5—121no—
461C4×C5—22no—
462C4×C5—122no—
463C4×C5—122no—
464C4×C5C4×#6714yes—
465Dic5—24no—
466Dic5—42no—
467Dic5—62no—
468Dic5—62no—
469Dic5—62no—
470C24—151no—
471C24C2×D3×D511yescharacteristic
472C2×D4—154no—
473C2×D4—302no—
474C2×D4—302no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—32no—
478C22×C4—32no—
479C22×C4—52no—
480C22×C4—52no—
481C22×C4—104no—
482C22×C4—104no—
483C22×C4—151no—
484C22×C4—151no—
485C22×C4—151no—
486C22×C4—154no—
487C22×C4—202no—
488C22×C4—302no—
489C22×C4—302no—
490C22×C4#120014yes—
491C22×C4C2×D3×D511yescharacteristic
492C22×C4C2×D3×D511yescharacteristic
493C4⋊C4—54no—
494C4⋊C4—62no—
495C4⋊C4—62no—
496C4⋊C4—104no—
497C4⋊C4—102no—
498C4⋊C4—154no—
499C22⋊C4—64no—
500C22⋊C4—154no—
501C22⋊C4—154no—
502C22⋊C4—304no—
503C22⋊C4—602no—
504C42—154no—
505C42—304no—
506C42—602no—
507C3×C5#235211yescharacteristic, complemented, Hall
508C22×C3—22no—
509C22×C3—22no—
510C22×C3—22no—
511C22×C3—22no—
512C22×C3—22no—
513C22×C3—22no—
514C22×C3#403814yescomplemented
515C22×C3C22×C4×D511yescharacteristic
516C22×C3C2×D4×D511yescharacteristic
517C22×C3C2×#71111yescharacteristic
518C22×C3C2×#71111yescharacteristic
519C22×C3C2×#71311yescharacteristic
520C22×C3#414911yescharacteristic
521C22×C3#415011yescharacteristic
522C2×D3—102no—
523C2×D3—102no—
524C2×D3—102no—
525C2×D3—201no—
526C2×D3—201no—
527C2×D3—201no—
528C2×D3—201no—
529C3×C4—22no—
530C3×C4—104no—
531C3×C4—202no—
532C3×C4#402714yes—
533Dic3—42no—
534Dic3—42no—
535Dic3—102no—
536Dic3—102no—
537Dic3—102no—
538C2×C5—24no—
539C2×C5C2×#89211yescharacteristic
540C2×C5#550411yescharacteristic
541C2×C5#551311yescharacteristic
542C2×C5#551411yescharacteristic
543C2×C5C2×C4×#6711yescharacteristic
544C2×C5#580611yescharacteristic
545C2×C5#581711yescharacteristic
546D5—122no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—152no—
552C23—301no—
553C23—301no—
554C23—301no—
555C23—301no—
556C23—301no—
557C23—301no—
558C23#676612yes—
559C23#676812yes—
560C23C2×#120012yes—
561C23C22×D3×D511yescharacteristic
562D4—308no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—28no—
566C2×C4—54no—
567C2×C4—62no—
568C2×C4—62no—
569C2×C4—62no—
570C2×C4—62no—
571C2×C4—62no—
572C2×C4—62no—
573C2×C4—102no—
574C2×C4—102no—
575C2×C4—102no—
576C2×C4—102no—
577C2×C4—102no—
578C2×C4—152no—
579C2×C4—152no—
580C2×C4—152no—
581C2×C4—208no—
582C2×C4—301no—
583C2×C4—301no—
584C2×C4—301no—
585C2×C4—301no—
586C2×C4—301no—
587C2×C4—301no—
588C2×C4—308no—
589C2×C4#678212yes—
590C2×C4C2×#120012yes—
591C2×C4C2×#120012yes—
592C2×C4D5×#6712yes—
593C2×C3—24no—
594C2×C3C2×#402711yescharacteristic
595C2×C3C2×#403811yescharacteristic
596C2×C3#6585711yescharacteristic
597C2×C3#6593211yescharacteristic
598C2×C3#6595211yescharacteristic
599C2×C3#6596411yescharacteristic
600C2×C3#6597611yescharacteristic
601D3—202no—
602C5#8216111yescharacteristic, complemented, Sylow
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—302no—
610C22—302no—
611C22—302no—
612C22—601no—
613C22—601no—
614C22—601no—
615C22—601no—
616C22#9134714yescomplemented
617C22C2×#676611yescharacteristic
618C22C2×#676811yescharacteristic
619C22#9200411yescharacteristic
620C22#9201711yescharacteristic
621C22C2×#678211yescharacteristic
622C22C22×#120011yescharacteristic
623C22C2×D5×#6711yescharacteristic
624C4—22no—
625C4—104no—
626C4—122no—
627C4—122no—
628C4—202no—
629C4—302no—
630C4—302no—
631C4—302no—
632C4#9155214yes—
633C3#1060561411yescharacteristic, complemented, Sylow
634C2—24no—
635C2—602no—
636C2C2×#9134711yescharacteristic
637C2#1175421011yescharacteristic
638C2#1175430811yescharacteristic
639C2#1175431911yescharacteristic
640C2#1175449511yescharacteristic
641C2#1175449611yescharacteristic
642C2C2×#9155211yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC23×C3×C4×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC22×#1220C22
hypercenterC23×C4D3×D5
hypocenterC3×C5#2352
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. C22×C3×C4⋊Dic5C2semidirect
3. C2×#91347C2semidirect
4. C5×#5212C2semidirect
6. Dic5×#217C2semidirect
7. #11748799C2semidirect
8. #11748699C2semidirect
9. #11748678C2semidirect
10. #11748674C2semidirect
11. #11748623C2semidirect
18. C4×#1220C4semidirect
41. C22×C4⋊Dic5D3semidirect
66. C2×C4×D15C2×C4semidirect
75. #6714C2×C4semidirect
76. #6712C2×C4semidirect
77. C4×Dic15C2×C4semidirect
83. C3×C4⋊Dic5D4semidirect
92. #5212D5semidirect
230. C4⋊Dic5#67semidirect
252. C22×C3×C5#117semidirect
446. C22×C5#892semidirect
507. C3×C5#2352semidirect
514. C22×C3#4038semidirect
602. C5#82161semidirect
616. C22#91347semidirect
633. C3#10605614semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#12202
C22×C3×C4⋊Dic52
C2×#913472
C5×#52122
#117494942
Dic5×#2172
#117487992
#117486992
#117486782
#117486742
#117486232
#106056143
#821615