Finite Group Databasebeta

← #14323457

Subgroups — #14323457

Order 1920.

1616 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91563C211yesmaximal, characteristic
2C2×#67⋊Dic5C211yesmaximal, characteristic, complemented
3C22×#6665C211yesmaximal, characteristic, complemented
4C22⋊C4×Dic15C211yesmaximal, characteristic, complemented
5C5×#4958C211yesmaximal, characteristic, complemented
6C3×#65244C211yesmaximal, characteristic, complemented
7#11749512C211yesmaximal, characteristic, complemented
8#11748915C212yesmaximal, complemented
9#11748911C212yesmaximal, complemented
10#11748710C212yesmaximal, complemented
11#11748708C212yesmaximal, complemented
12#10606606—31nomaximal, Hall
13C24×C3×D5C2211yescharacteristic
14C23×Dic15C2211yescharacteristic
15C22×C5×#67C2211yescharacteristic
16C23×C5×Dic3C2211yescharacteristic
17C22×#1201C2212yes—
18C22×D3×Dic5C2211yescharacteristic, complemented
19C2×C3×C5×C22⋊C4C2211yescharacteristic
20C2×#6719C2211yescharacteristic
21C2×C4×Dic15C2212yescomplemented
22C2×C5×(C2×D3)⋊C4C2212yescomplemented
23C2×C3×C22⋊Dic5C2211yescharacteristic
24C2×C3×#685C2212yes—
25C2×C3×#685C2212yes—
26#91563C418yescomplemented
27#67⋊Dic5—24no—
28C2×Dic3⋊Dic5C2211yescharacteristic, complemented
29C2×#6669C2212yescomplemented
30C2×(C2×D3)⋊Dic5C2211yescharacteristic, complemented
31C2×#6665—24no—
32C2×#6665C2212yes—
33C2×#6665C2212yes—
34C2×#6665C2212yes—
35C2×Dic3×Dic5C2211yescharacteristic, complemented
36#91115C2212yescomplemented
37C5×C3⋊#95C2212yescomplemented
38#90994C2212yescomplemented
39#73821—51nomaximal, Hall
40C2×#4083—31no—
41C2×D4×Dic5—31no—
42C22×#685—31no—
43C5×#386—31no—
44#65499—31no—
45#65247—32no—
46#65246—32no—
47#65244D311yescharacteristic, complemented
48C22⋊C4×Dic5—31no—
49#64959—32no—
50#64958—32no—
51C24×C3×C5C2311yescharacteristic
52C23×C5×D3C2311yescharacteristic
53C23×C3×D5—22no—
54C23×C3×D5—22no—
55C23×C3×D5—42no—
56C23×C3×D5—42no—
57C23×C3×D5C2×C414yescomplemented
58C23×C3×D5C2312yes—
59C22×C3×C4×C5C2312yes—
60C22×Dic15—42no—
61C22×Dic15D414yescomplemented
62C22×Dic15C2312yes—
63C22×Dic15C2311yescharacteristic
64C22×Dic15C2311yescharacteristic
65C2×C5×#67—22no—
66C2×C5×#67—22no—
67C2×C5×#67C2×C414yes—
68C22×C5×Dic3—22no—
69C22×C5×Dic3—42no—
70C22×C5×Dic3C2312yes—
71C22×C5×Dic3C2311yescharacteristic
72C22×C3×Dic5C2311yescharacteristic
73C22×C3×Dic5C2311yescharacteristic
74C2×#1201—42no—
75C2×#1201—42no—
76C2×#1201C2×C418yescomplemented
77C2×D3×Dic5—22no—
78C2×D3×Dic5—42no—
79C3×C5×C22⋊C4—22no—
80#6719C2×C414yescomplemented
81C4×Dic15—42no—
82C5×(C2×D3)⋊C4—42no—
83C3×C22⋊Dic5—22no—
84C3×#685—42no—
85C3×#685—42no—
86Dic3⋊Dic5—22no—
87#6669—42no—
88(C2×D3)⋊Dic5—22no—
89#6665—24no—
90#6665—24no—
91#6665C2×C418yescomplemented
92Dic3×Dic5—22no—
93C2×#953—51no—
94C22×#217—51no—
95C2×#67⋊C4—51no—
96C3×#379—51no—
97#4958D511yescharacteristic, complemented
98#4957—52no—
99#4956—52no—
100#4946—51no—
101#4945—52no—
102#4944—52no—
103Dic3×C22⋊C4—51no—
104C24×D5C2×D311yescharacteristic
105C22×C5×D4—31no—
106C23×C4×C5—31no—
107C22×C22⋊D5—32no—
108C23×Dic5—31no—
109C23×Dic5—31no—
110C2×C5×C22⋊C4—32no—
111C2×C5×C22⋊C4C2×D311yescharacteristic
112C2×C22⋊Dic5—31no—
113C2×C22⋊Dic5—31no—
114C2×C22⋊Dic5C2×D311yescharacteristic
115#4083—38no—
116D4×Dic5—64no—
117C2×#685—32no—
118C2×#685—32no—
119C2×#685—32no—
120C2×#685—64no—
121C2×#685C2×D312yes—
122C2×#685C2×D312yes—
123C2×C4⋊Dic5—31no—
124C2×#683—32no—
125C2×C4×Dic5—31no—
126C2×C4×Dic5—32no—
127C5×#95—32no—
128#3963—32no—
129#3963—32no—
130#2321—151noSylow
131C23×C3×C5—22no—
132C23×C3×C5—22no—
133C23×C3×C5C22×C412yes—
134C23×C3×C5C2×D412yes—
135C23×C3×C5#4112yes—
136C23×C3×C5C2411yescharacteristic
137C22×C5×D3—21no—
138C22×C5×D3—21no—
139C22×C5×D3—22no—
140C22×C5×D3—22no—
141C22×C5×D3C22×C412yes—
142C22×C3×D5—28no—
143C22×C3×D5—22no—
144C22×C3×D5—22no—
145C22×C3×D5—22no—
146C22×C3×D5—22no—
147C22×C3×D5—22no—
148C22×C3×D5—22no—
149C22×C3×D5—48no—
150C22×C3×D5—44no—
151C22×C3×D5—44no—
152C22×C3×D5—48no—
153C22×C3×D5C22×C414yes—
154C2×C3×C4×C5—22no—
155C2×C3×C4×C5—22no—
156C2×C3×C4×C5—22no—
157C2×Dic15—22no—
158C2×Dic15—22no—
159C2×Dic15—22no—
160C2×Dic15—22no—
161C2×Dic15—44no—
162C2×Dic15C22×C414yes—
163C2×Dic15C2×D412yes—
164C2×Dic15C2×D412yes—
165C5×#67—28no—
166C2×C5×Dic3—21no—
167C2×C5×Dic3—21no—
168C2×C5×Dic3—22no—
169C2×C5×Dic3—22no—
170C2×C5×Dic3—22no—
171C2×C5×Dic3—44no—
172C2×C5×Dic3C22×C412yes—
173C2×C3×Dic5—21no—
174C2×C3×Dic5—21no—
175C2×C3×Dic5—22no—
176C2×C3×Dic5—22no—
177#1201—48no—
178D3×Dic5—44no—
179C25×C3—51no—
180C22×#67—52no—
181C22×#67C2×D511yescharacteristic
182C23×Dic3—51no—
183C23×Dic3C2×D511yescharacteristic
184C22×C4×D3—51no—
185C2×C3×C22⋊C4—51no—
186C2×C3×C22⋊C4—52no—
187C2×C3×C22⋊C4—52no—
188C2×C3×C22⋊C4C2×D511yescharacteristic
189#953—58no—
190C2×#217—51no—
191C2×#217—52no—
192C2×#217—52no—
193C2×#217—52no—
194C2×#217—104no—
195C2×(C2×D3)⋊C4—51no—
196C2×(C2×D3)⋊C4C2×D512yescomplemented
197C2×Dic3⋊C4—51no—
198C2×Dic3⋊C4—52no—
199C2×C4×Dic3—51no—
200C2×C4×Dic3—52no—
201#67⋊C4—104no—
202C3⋊#95—52no—
203C3⋊#95—52no—
204C3⋊#95C2×D512yescomplemented
205C24×C5—31no—
206C24×C5C22×D311yescharacteristic
207C23×D5—22no—
208C23×D5—22no—
209C23×D5—42no—
210C23×D5—42no—
211C23×D5C4×D314yescomplemented
212C23×D5C22×D312yes—
213C2×C5×D4—34no—
214C2×C5×D4—62no—
215C2×C5×D4—62no—
216C22×C4×C5—31no—
217C22×C4×C5—32no—
218C22×C4×C5—62no—
219C22×C4×C5—122no—
220C22×C4×C5C22×D312yes—
221C2×C22⋊D5—38no—
222C2×C22⋊D5—122no—
223C2×C22⋊D5—122no—
224C22×Dic5—31no—
225C22×Dic5—32no—
226C22×Dic5—31no—
227C22×Dic5—34no—
228C22×Dic5—62no—
229C22×Dic5—122no—
230C22×Dic5—122no—
231C22×Dic5C22×D311yescharacteristic
232C22×Dic5C22×D311yescharacteristic
233C5×C22⋊C4—22no—
234C5×C22⋊C4—122no—
235C22⋊Dic5—22no—
236C22⋊Dic5—34no—
237C22⋊Dic5—62no—
238#685—38no—
239#685—42no—
240#685—42no—
241#685—64no—
242#685—64no—
243C4⋊Dic5—62no—
244#683—122no—
245C4×Dic5—62no—
246C4×Dic5—122no—
247C2×#120—151no—
248C2×C4×D4—151no—
249C22×C22⋊C4—151no—
250#390—152no—
251#388—152no—
252#388—152no—
253#386—31no—
254#386—152no—
255#385—151no—
256#379—51no—
257C4×C22⋊C4—151no—
258C22×C3×C5—24no—
259C22×C3×C5—22no—
260C22×C3×C5—22no—
261C22×C3×C5—24no—
262C22×C3×C5C4×D414yes—
263C22×C3×C5C23×C411yescharacteristic
264C22×C3×C5C22×D411yescharacteristic
265C22×C3×C5C2×#4111yescharacteristic
266C22×C3×C5H(5,𝔽2)12yes—
267C22×C3×C5H(5,𝔽2)12yes—
268C2×C5×D3—22no—
269C2×C5×D3—24no—
270C2×C5×D3—41no—
271C2×C5×D3—41no—
272C2×C5×D3—42no—
273C2×C3×D5—24no—
274C2×C3×D5—24no—
275C2×C3×D5—24no—
276C2×C3×D5—416no—
277C2×C3×D5—42no—
278C2×C3×D5—42no—
279C2×C3×D5—42no—
280C2×C3×D5—42no—
281C3×C4×C5—42no—
282Dic15—24no—
283Dic15—22no—
284Dic15—22no—
285C3×Dic5—41no—
286C3×Dic5—41no—
287C5×Dic3—22no—
288C5×Dic3—42no—
289C24×C3—52no—
290C24×C3—54no—
291C24×C3—102no—
292C24×C3—102no—
293C24×C3—202no—
294C24×C3—202no—
295C24×C3C22×D511yescharacteristic
296C23×D3C22×D511yescharacteristic
297C22×C3×C4—51no—
298C22×C3×C4—51no—
299C22×C3×C4C22×D512yes—
300C2×#67—22no—
301C2×#67—22no—
302C2×#67—58no—
303C2×#67—202no—
304C2×#67—202no—
305C2×#67C4×D514yes—
306C22×Dic3—22no—
307C22×Dic3—42no—
308C22×Dic3—52no—
309C22×Dic3—51no—
310C22×Dic3—51no—
311C22×Dic3—54no—
312C22×Dic3—202no—
313C22×Dic3C22×D512yes—
314C22×Dic3C22×D511yescharacteristic
315C2×C4×D3—102no—
316C2×C4×D3—202no—
317C3×C22⋊C4—22no—
318C3×C22⋊C4—102no—
319C3×C22⋊C4—202no—
320C3×C22⋊C4—202no—
321#217—54no—
322#217—58no—
323#217—104no—
324#217—104no—
325(C2×D3)⋊C4—42no—
326(C2×D3)⋊C4—102no—
327Dic3⋊C4—102no—
328Dic3⋊C4—202no—
329C4×Dic3—102no—
330C4×Dic3—202no—
331C23×C5—22no—
332C23×C5—22no—
333C23×C5—32no—
334C23×C5—61no—
335C23×C5—61no—
336C23×C5—62no—
337C23×C5—62no—
338C23×C5C2×C4×D312yes—
339C23×C5D3×D412yes—
340C23×C5#23712yes—
341C23×C5C23×D311yescharacteristic
342C22×D5—28no—
343C22×D5—22no—
344C22×D5—22no—
345C22×D5—22no—
346C22×D5—22no—
347C22×D5—22no—
348C22×D5—22no—
349C22×D5—48no—
350C22×D5—44no—
351C22×D5—44no—
352C22×D5—48no—
353C22×D5C2×C4×D314yes—
354C5×D4—68no—
355C2×C4×C5—22no—
356C2×C4×C5—22no—
357C2×C4×C5—22no—
358C2×C4×C5—32no—
359C2×C4×C5—61no—
360C2×C4×C5—61no—
361C2×C4×C5—62no—
362C2×C4×C5—62no—
363C2×C4×C5—62no—
364C2×C4×C5—124no—
365C22⋊D5—128no—
366C2×Dic5—21no—
367C2×Dic5—21no—
368C2×Dic5—22no—
369C2×Dic5—22no—
370C2×Dic5—34no—
371C2×Dic5—32no—
372C2×Dic5—32no—
373C2×Dic5—62no—
374C2×Dic5—62no—
375C2×Dic5—62no—
376C2×Dic5—62no—
377C2×Dic5—124no—
378C2×Dic5—124no—
379C25—51no—
380C22×D4—31no—
381C22×D4—152no—
382C23×C4—31no—
383C23×C4—151no—
384C23×C4—151no—
385#120—158no—
386C4×D4—304no—
387C2×C4⋊C4—151no—
388C2×C4⋊C4—152no—
389C2×C22⋊C4—32no—
390C2×C22⋊C4—51no—
391C2×C22⋊C4—52no—
392C2×C22⋊C4—52no—
393C2×C22⋊C4—152no—
394C2×C22⋊C4—151no—
395C2×C22⋊C4—152no—
396C2×C22⋊C4—152no—
397C2×C22⋊C4—151no—
398C2×C22⋊C4—304no—
399C2×C22⋊C4D3×D511yescharacteristic
400C2×C42—151no—
401C2×C42—152no—
402#95—32no—
403#95—152no—
404#95—152no—
405C2×C3×C5—24no—
406C2×C3×C5C2×C4×D411yescharacteristic
407C2×C3×C5#51812yes—
408C2×C3×C5#53411yescharacteristic
409C2×C3×C5#53511yescharacteristic
410C2×C3×C5#53812yes—
411C3×D5—44no—
412C5×D3—42no—
413C23×C3—22no—
414C23×C3—22no—
415C23×C3—54no—
416C23×C3—102no—
417C23×C3—102no—
418C23×C3—102no—
419C23×C3—102no—
420C23×C3—102no—
421C23×C3—102no—
422C23×C3—108no—
423C23×C3—208no—
424C23×C3—204no—
425C23×C3—208no—
426C23×C3—204no—
427C23×C3C2×C4×D512yes—
428C23×C3D4×D512yes—
429C23×C3#71112yes—
430C23×C3C23×D511yescharacteristic
431C22×D3—21no—
432C22×D3—21no—
433C22×D3—22no—
434C22×D3—22no—
435C22×D3C2×C4×D512yes—
436C2×C3×C4—22no—
437C2×C3×C4—22no—
438C2×C3×C4—22no—
439C2×C3×C4—101no—
440C2×C3×C4—101no—
441C2×C3×C4—102no—
442C2×C3×C4—102no—
443#67—28no—
444#67—208no—
445C2×Dic3—21no—
446C2×Dic3—21no—
447C2×Dic3—22no—
448C2×Dic3—22no—
449C2×Dic3—22no—
450C2×Dic3—44no—
451C2×Dic3—54no—
452C2×Dic3—52no—
453C2×Dic3—52no—
454C2×Dic3—102no—
455C2×Dic3—102no—
456C2×Dic3—102no—
457C2×Dic3—102no—
458C2×Dic3—204no—
459C2×Dic3C2×C4×D512yes—
460C4×D3—204no—
461C22×C5—24no—
462C22×C5—22no—
463C22×C5—22no—
464C22×C5—24no—
465C22×C5—62no—
466C22×C5—64no—
467C22×C5—121no—
468C22×C5—121no—
469C22×C5—122no—
470C22×C5#67⋊C414yes—
471C22×C5C22×C4×D311yescharacteristic
472C22×C5C2×D3×D411yescharacteristic
473C22×C5C2×#23711yescharacteristic
474C22×C5#100412yes—
475C22×C5#100412yes—
476C2×D5—24no—
477C2×D5—24no—
478C2×D5—24no—
479C2×D5—416no—
480C2×D5—42no—
481C2×D5—42no—
482C2×D5—42no—
483C2×D5—42no—
484C4×C5—42no—
485C4×C5—62no—
486C4×C5—122no—
487Dic5—41no—
488Dic5—41no—
489Dic5—64no—
490Dic5—62no—
491Dic5—62no—
492C24—31no—
493C24—52no—
494C24—54no—
495C24—102no—
496C24—102no—
497C24—202no—
498C24—202no—
499C24C2×D3×D511yescharacteristic
500C2×D4—34no—
501C2×D4—62no—
502C2×D4—62no—
503C2×D4—158no—
504C2×D4—602no—
505C2×D4—602no—
506C22×C4—31no—
507C22×C4—32no—
508C22×C4—51no—
509C22×C4—51no—
510C22×C4—62no—
511C22×C4—122no—
512C22×C4—152no—
513C22×C4—151no—
514C22×C4—151no—
515C22×C4—154no—
516C22×C4—302no—
517C22×C4—602no—
518C22×C4—602no—
519C22×C4C2×D3×D512yes—
520C4⋊C4—302no—
521C4⋊C4—602no—
522C22⋊C4—22no—
523C22⋊C4—102no—
524C22⋊C4—122no—
525C22⋊C4—158no—
526C22⋊C4—154no—
527C22⋊C4—202no—
528C22⋊C4—202no—
529C22⋊C4—304no—
530C22⋊C4—304no—
531C22⋊C4—302no—
532C42—302no—
533C42—602no—
534C3×C5#232111yescharacteristic, complemented, Hall
535C22×C3—24no—
536C22×C3—22no—
537C22×C3—24no—
538C22×C3—22no—
539C22×C3—104no—
540C22×C3—104no—
541C22×C3—104no—
542C22×C3—2016no—
543C22×C3—202no—
544C22×C3—202no—
545C22×C3—202no—
546C22×C3—202no—
547C22×C3#402714yes—
548C22×C3C22×C4×D511yescharacteristic
549C22×C3C2×D4×D511yescharacteristic
550C22×C3C2×#71111yescharacteristic
551C22×C3#414412yes—
552C22×C3#414912yes—
553C2×D3—22no—
554C2×D3—24no—
555C2×D3—41no—
556C2×D3—42no—
557C2×D3—41no—
558C3×C4—42no—
559C3×C4—201no—
560C3×C4—201no—
561Dic3—22no—
562Dic3—42no—
563Dic3—104no—
564Dic3—102no—
565Dic3—102no—
566C2×C5—24no—
567C2×C5—122no—
568C2×C5C2×#67⋊C411yescharacteristic
569C2×C5#548612yes—
570C2×C5#558911yescharacteristic
571C2×C5#559211yescharacteristic
572C2×C5#559312yes—
573D5—44no—
574C23—22no—
575C23—22no—
576C23—32no—
577C23—54no—
578C23—61no—
579C23—62no—
580C23—62no—
581C23—61no—
582C23—108no—
583C23—102no—
584C23—102no—
585C23—102no—
586C23—102no—
587C23—102no—
588C23—102no—
589C23—208no—
590C23—204no—
591C23—204no—
592C23—208no—
593C23C4×D3×D512yes—
594C23#677712yes—
595C23#678012yes—
596C23C22×D3×D511yescharacteristic
597D4—68no—
598D4—608no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—32no—
603C2×C4—61no—
604C2×C4—61no—
605C2×C4—62no—
606C2×C4—62no—
607C2×C4—62no—
608C2×C4—101no—
609C2×C4—101no—
610C2×C4—102no—
611C2×C4—102no—
612C2×C4—124no—
613C2×C4—154no—
614C2×C4—152no—
615C2×C4—152no—
616C2×C4—302no—
617C2×C4—302no—
618C2×C4—302no—
619C2×C4—302no—
620C2×C4—604no—
621C2×C4—604no—
622C2×C3—24no—
623C2×C3—204no—
624C2×C3C2×#402711yescharacteristic
625C2×C3#6587912yes—
626C2×C3#6592611yescharacteristic
627C2×C3#6595112yes—
628C2×C3#6598711yescharacteristic
629D3—42no—
630C5#7382111yescharacteristic, complemented, Sylow
631C22—22no—
632C22—22no—
633C22—24no—
634C22—24no—
635C22—62no—
636C22—64no—
637C22—104no—
638C22—104no—
639C22—104no—
640C22—121no—
641C22—121no—
642C22—122no—
643C22—2016no—
644C22—202no—
645C22—202no—
646C22—202no—
647C22—202no—
648C22#9144214yes—
649C22C2×C4×D3×D511yescharacteristic
650C22C2×#677711yescharacteristic
651C22#9202512yes—
652C22#9202612yes—
653C22C2×#678011yescharacteristic
654C4—42no—
655C4—62no—
656C4—122no—
657C4—201no—
658C4—201no—
659C4—302no—
660C4—302no—
661C4—304no—
662C3#1060660611yescharacteristic, complemented, Sylow
663C2—24no—
664C2—122no—
665C2—204no—
666C2#1175308311yescharacteristic
667C2#1175320012yes—
668C2#1175327511yescharacteristic
669C2C2×#9144211yescharacteristic
670C2#1175474512yes—
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C5×C22⋊C4C22
socle, cosocleC23×C3×C5C24
prime radicalC2×#91563C2
hypercenterC2×C22⋊C4D3×D5
hypocenterC3×C5#2321
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#67⋊Dic5C2semidirect
3. C22×#6665C2semidirect
4. C22⋊C4×Dic15C2semidirect
5. C5×#4958C2semidirect
6. C3×#65244C2semidirect
7. #11749512C2semidirect
8. #11748915C2semidirect
9. #11748911C2semidirect
10. #11748710C2semidirect
11. #11748708C2semidirect
18. C22×D3×Dic5C22semidirect
21. C2×C4×Dic15C22semidirect
22. C2×C5×(C2×D3)⋊C4C22semidirect
26. #91563C4semidirect
28. C2×Dic3⋊Dic5C22semidirect
29. C2×#6669C22semidirect
30. C2×(C2×D3)⋊Dic5C22semidirect
35. C2×Dic3×Dic5C22semidirect
36. #91115C22semidirect
37. C5×C3⋊#95C22semidirect
38. #90994C22semidirect
47. #65244D3semidirect
57. C23×C3×D5C2×C4semidirect
61. C22×Dic15D4semidirect
76. C2×#1201C2×C4semidirect
80. #6719C2×C4semidirect
91. #6665C2×C4semidirect
97. #4958D5semidirect
196. C2×(C2×D3)⋊C4C2×D5semidirect
204. C3⋊#95C2×D5semidirect
211. C23×D5C4×D3semidirect
534. C3×C5#2321semidirect
630. C5#73821semidirect
662. C3#10606606semidirect

Maximal subgroups

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

subgroupindex
C2×#915632
C2×#67⋊Dic52
C22×#66652
C22⋊C4×Dic152
C5×#49582
C3×#652442
#117495122
#117489152
#117489112
#117487102
#117487082
#106066063
#738215