Finite Group Databasebeta

← #14326271

Subgroups — #14326271

Order 1920.

1584 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×#91815C211yesmaximal, characteristic, complemented
2C22×C3×#685C211yesmaximal, characteristic
3C2×D5×#217C211yesmaximal, characteristic
4C5×#5211C211yesmaximal, characteristic, complemented
5#11749506C211yesmaximal, characteristic, complemented
6#11749488C211yesmaximal, characteristic
7Dic5×#217C211yesmaximal, characteristic, complemented
8#11748900C212yesmaximal, complemented
9#11748884C212yesmaximal, complemented
10#11748791C212yesmaximal, complemented
11#11748772C212yesmaximal, complemented
12#10605853—31nomaximal, Hall
13C24×C3×D5C2211yescharacteristic
14C23×C3×C4×C5C2211yescharacteristic
15C23×C3×Dic5C2211yescharacteristic
16C22×D5×Dic3C2212yes—
17C2×#6719C2211yescharacteristic
18C2×#6719C2211yescharacteristic
19#91815—24no—
20C2×C4⋊Dic15C2212yescomplemented
21C2×C4×Dic15C2212yescomplemented
22C2×C5×#217C2211yescharacteristic
23C2×C5×#217C2211yescharacteristic
24C2×C3×#685C418yescomplemented
25C2×C3×#685C2211yescharacteristic
26C2×C3×#685C2211yescharacteristic
27C2×C3×#685C2211yescharacteristic
28C2×C3×#685C2211yescharacteristic
29D5×#217—24no—
30C2×#6668C2212yescomplemented
31C2×#6665C2212yes—
32C2×#6665C2212yes—
33C2×#6665C2212yes—
34C2×Dic3×Dic5C2212yescomplemented
35C5×C3⋊#95C2212yescomplemented
36C5×C3⋊#95C2212yescomplemented
37#90994C2212yescomplemented
38#90994C2212yescomplemented
39#82097—51nomaximal, Hall
40C22×#685D311yescharacteristic
41C2×#4072—31no—
42C2×D5×C22⋊C4—31no—
43C5×#381—31no—
44#65244—31no—
45#65231—31no—
46C22⋊C4×Dic5—31no—
47#64959—32no—
48#64958—32no—
49#64957—32no—
50#64955—32no—
51C24×C3×C5C2311yescharacteristic
52C23×C3×D5—22no—
53C23×C3×D5—22no—
54C23×C3×D5—24no—
55C23×C3×D5—42no—
56C23×C3×D5C2×C414yes—
57C23×C3×D5C2311yescharacteristic
58C23×C3×D5C2311yescharacteristic
59C22×C3×C4×C5—22no—
60C22×C3×C4×C5—22no—
61C22×C3×C4×C5C2×C414yescomplemented
62C22×C3×C4×C5C2311yescharacteristic
63C22×C3×C4×C5C2311yescharacteristic
64C22×Dic15C2312yes—
65C22×Dic15C2312yes—
66C22×C5×Dic3C2312yes—
67C22×C5×Dic3C2312yes—
68C22×C3×Dic5—42no—
69C22×C3×Dic5C2×C414yescomplemented
70C22×C3×Dic5C2311yescharacteristic
71C22×C3×Dic5C2311yescharacteristic
72C2×D5×Dic3—24no—
73C2×D5×Dic3—24no—
74C2×D5×Dic3—42no—
75#6719—22no—
76#6719D414yes—
77C4⋊Dic15—24no—
78C4×Dic15—24no—
79C5×#217—22no—
80C5×#217—22no—
81C3×#685C2×C414yes—
82C3×#685C2×C414yes—
83C3×#685C2×C414yescomplemented
84C3×#685C2×C414yescomplemented
85#6668—42no—
86#6665—24no—
87#6665—24no—
88#6665—42no—
89Dic3×Dic5—42no—
90C22×C3×C22⋊C4—51no—
91C22×#217—51no—
92C2×#926—51no—
93#5247—51no—
94#5212—51no—
95#5211D511yescharacteristic, complemented
96C4×#217—51no—
97#4951—52no—
98#4951—52no—
99#4950—52no—
100Dic3×C22⋊C4—52no—
101C24×D5C2×D311yescharacteristic
102C23×C4×C5C2×D311yescharacteristic
103C23×Dic5C2×D311yescharacteristic
104C22×C4×D5—32no—
105C2×C5×C22⋊C4—31no—
106C2×C5×C22⋊C4—31no—
107C2×C22⋊Dic5—31no—
108C2×C22⋊Dic5—31no—
109C2×#685—32no—
110C2×#685—32no—
111C2×#685—32no—
112C2×#685Dic318yescomplemented
113C2×#685C2×D311yescharacteristic
114C2×#685C2×D311yescharacteristic
115C2×#685C2×D311yescharacteristic
116C2×#685C2×D311yescharacteristic
117#4072—64no—
118C2×C4⋊Dic5—32no—
119C2×#683—32no—
120C2×C4×Dic5—32no—
121C2×C4×Dic5—32no—
122D5×C22⋊C4—64no—
123C5×#95—32no—
124C5×#95—32no—
125#3963—32no—
126#3963—32no—
127#2342—151noSylow
128C23×C3×C5—22no—
129C23×C3×C5—22no—
130C23×C3×C5C22×C412yes—
131C23×C3×C5C2×D412yes—
132C23×C3×C5#4112yes—
133C23×C3×C5C2411yescharacteristic
134C22×C3×D5—21no—
135C22×C3×D5—21no—
136C22×C3×D5—22no—
137C22×C3×D5—28no—
138C22×C3×D5—21no—
139C22×C3×D5—22no—
140C22×C3×D5—21no—
141C22×C3×D5—22no—
142C22×C3×D5—28no—
143C22×C3×D5—44no—
144C22×C3×D5—48no—
145C22×C3×D5—48no—
146C22×C3×D5C22×C412yes—
147C22×C3×D5C22×C412yes—
148C22×C3×D5#4114yes—
149C2×C3×C4×C5—28no—
150C2×C3×C4×C5—21no—
151C2×C3×C4×C5—21no—
152C2×C3×C4×C5—21no—
153C2×C3×C4×C5—21no—
154C2×C3×C4×C5C22×C412yes—
155C2×C3×C4×C5C22×C412yes—
156C2×Dic15—22no—
157C2×Dic15—22no—
158C2×Dic15—22no—
159C2×Dic15—22no—
160C2×Dic15—22no—
161C2×Dic15C2×D414yes—
162C2×C5×Dic3—22no—
163C2×C5×Dic3—22no—
164C2×C5×Dic3—22no—
165C2×C5×Dic3—22no—
166C2×C5×Dic3—22no—
167C2×C5×Dic3—22no—
168C2×C3×Dic5—22no—
169C2×C3×Dic5—22no—
170C2×C3×Dic5—44no—
171C2×C3×Dic5C22×C412yes—
172C2×C3×Dic5C22×C412yes—
173D5×Dic3—48no—
174C25×C3—51no—
175C23×C3×C4—51no—
176C23×C3×C4C2×D511yescharacteristic
177C23×Dic3—52no—
178C2×C3×C22⋊C4—58no—
179C2×C3×C22⋊C4—51no—
180C2×C3×C22⋊C4—51no—
181C2×C3×C22⋊C4—51no—
182C2×C3×C22⋊C4—51no—
183C2×#217—51no—
184C2×#217—51no—
185C2×#217—52no—
186C2×#217—52no—
187C2×#217—52no—
188C2×#217—104no—
189C2×#217C2×D511yescharacteristic
190C2×#217C2×D511yescharacteristic
191#926—104no—
192C2×C4⋊Dic3—52no—
193C2×C4⋊Dic3—52no—
194C2×C4×Dic3—52no—
195C2×C4×Dic3—52no—
196C3⋊#95—52no—
197C3⋊#95—52no—
198C3⋊#95C2×D512yescomplemented
199C3⋊#95C2×D512yescomplemented
200C24×C5C22×D311yescharacteristic
201C23×D5—22no—
202C23×D5—22no—
203C23×D5—24no—
204C23×D5—42no—
205C23×D5C2×Dic314yes—
206C23×D5C22×D311yescharacteristic
207C23×D5C22×D311yescharacteristic
208C22×C4×C5—22no—
209C22×C4×C5—22no—
210C22×C4×C5—32no—
211C22×C4×C5—32no—
212C22×C4×C5C2×Dic314yescomplemented
213C22×C4×C5C22×D311yescharacteristic
214C22×C4×C5C22×D311yescharacteristic
215C22×Dic5—32no—
216C22×Dic5—32no—
217C22×Dic5—42no—
218C22×Dic5C2×Dic314yescomplemented
219C22×Dic5C22×D311yescharacteristic
220C22×Dic5C22×D311yescharacteristic
221C2×C4×D5—64no—
222C2×C4×D5—64no—
223C2×C4×D5—122no—
224C5×C22⋊C4—62no—
225C5×C22⋊C4—62no—
226C22⋊Dic5—34no—
227C22⋊Dic5—62no—
228#685—64no—
229#685—64no—
230#685—122no—
231#685C2×Dic314yes—
232#685C2×Dic314yes—
233#685C2×Dic314yescomplemented
234#685C2×Dic314yescomplemented
235C4⋊Dic5—64no—
236#683—122no—
237C4×Dic5—64no—
238C4×Dic5—122no—
239C2×#117—151no—
240C22×C22⋊C4—51no—
241C22×C22⋊C4—151no—
242#388—152no—
243#388—152no—
244#385—152no—
245#381—31no—
246#380—151no—
247#379—151no—
248C4×C22⋊C4—152no—
249C4×C22⋊C4—151no—
250C22×C3×C5—22no—
251C22×C3×C5—22no—
252C22×C3×C5—24no—
253C22×C3×C5—24no—
254C22×C3×C5C4×D414yes—
255C22×C3×C5C23×C411yescharacteristic
256C22×C3×C5C22×D411yescharacteristic
257C22×C3×C5C2×#4111yescharacteristic
258C22×C3×C5C2×#4112yes—
259C22×C3×C5H(5,𝔽2)12yes—
260C2×C3×D5—22no—
261C2×C3×D5—22no—
262C2×C3×D5—24no—
263C2×C3×D5—24no—
264C2×C3×D5—22no—
265C2×C3×D5—22no—
266C2×C3×D5—22no—
267C2×C3×D5—416no—
268C2×C3×D5—42no—
269C2×C3×D5—42no—
270C2×C3×D5#11714yescomplemented
271C3×C4×C5—22no—
272C3×C4×C5—22no—
273Dic15—24no—
274Dic15—42no—
275C3×Dic5—22no—
276C3×Dic5—22no—
277C5×Dic3—42no—
278C5×Dic3—42no—
279C24×C3—51no—
280C24×C3—51no—
281C24×C3—54no—
282C24×C3—104no—
283C24×C3—102no—
284C24×C3—102no—
285C24×C3—202no—
286C24×C3C22×D511yescharacteristic
287C22×C3×C4—22no—
288C22×C3×C4—22no—
289C22×C3×C4—51no—
290C22×C3×C4—51no—
291C22×C3×C4—54no—
292C22×C3×C4—202no—
293C22×C3×C4C4×D514yescomplemented
294C22×C3×C4C22×D511yescharacteristic
295C22×C3×C4C22×D511yescharacteristic
296C22×Dic3—52no—
297C22×Dic3—52no—
298C22×Dic3—104no—
299C22×Dic3—104no—
300C22×Dic3—202no—
301C22×Dic3C22×D512yes—
302C22×Dic3C22×D512yes—
303C3×C22⋊C4—54no—
304C3×C22⋊C4—54no—
305C3×C22⋊C4—54no—
306C3×C22⋊C4—54no—
307#217—22no—
308#217—22no—
309#217—54no—
310#217—104no—
311#217—104no—
312#217—102no—
313#217—202no—
314C4⋊Dic3—104no—
315C4⋊Dic3—202no—
316C4×Dic3—104no—
317C4×Dic3—202no—
318C23×C5—22no—
319C23×C5—22no—
320C23×C5D3×D412yes—
321C23×C5#23712yes—
322C23×C5C22×Dic312yes—
323C23×C5C23×D311yescharacteristic
324C22×D5—21no—
325C22×D5—21no—
326C22×D5—22no—
327C22×D5—28no—
328C22×D5—22no—
329C22×D5—21no—
330C22×D5—21no—
331C22×D5—22no—
332C22×D5—28no—
333C22×D5—44no—
334C22×D5—48no—
335C22×D5—48no—
336C22×D5#23514yes—
337C22×D5C22×Dic312yes—
338C22×D5C22×Dic312yes—
339C2×C4×C5—28no—
340C2×C4×C5—21no—
341C2×C4×C5—21no—
342C2×C4×C5—21no—
343C2×C4×C5—21no—
344C2×C4×C5—62no—
345C2×C4×C5—62no—
346C2×C4×C5—62no—
347C2×C4×C5—62no—
348C2×C4×C5—62no—
349C2×C4×C5—62no—
350C2×C4×C5C22×Dic312yes—
351C2×C4×C5C22×Dic312yes—
352C2×Dic5—22no—
353C2×Dic5—22no—
354C2×Dic5—34no—
355C2×Dic5—44no—
356C2×Dic5—62no—
357C2×Dic5—62no—
358C2×Dic5—62no—
359C2×Dic5—62no—
360C2×Dic5—62no—
361C2×Dic5C22×Dic312yes—
362C2×Dic5C22×Dic312yes—
363C4×D5—128no—
364C25—51no—
365C23×C4—51no—
366C23×C4—152no—
367C23×C4D3×D511yescharacteristic
368#117—304no—
369C2×C4⋊C4—152no—
370C2×C4⋊C4—152no—
371C2×C22⋊C4—31no—
372C2×C22⋊C4—31no—
373C2×C22⋊C4—58no—
374C2×C22⋊C4—51no—
375C2×C22⋊C4—51no—
376C2×C22⋊C4—51no—
377C2×C22⋊C4—51no—
378C2×C22⋊C4—151no—
379C2×C22⋊C4—152no—
380C2×C22⋊C4—152no—
381C2×C22⋊C4—151no—
382C2×C22⋊C4—152no—
383C2×C22⋊C4—304no—
384C2×C42—152no—
385C2×C42—152no—
386#95—32no—
387#95—32no—
388#95—152no—
389#95—152no—
390C2×C3×C5—24no—
391C2×C3×C5C2×#11711yescharacteristic
392C2×C3×C5C2×C4×D411yescharacteristic
393C2×C3×C5#51811yescharacteristic
394C2×C3×C5#54612yes—
395C2×C3×C5#55112yes—
396C3×D5—24no—
397C3×D5—42no—
398C23×C3—22no—
399C23×C3—22no—
400C23×C3—52no—
401C23×C3—54no—
402C23×C3—52no—
403C23×C3—108no—
404C23×C3—101no—
405C23×C3—101no—
406C23×C3—102no—
407C23×C3—101no—
408C23×C3—101no—
409C23×C3—102no—
410C23×C3—102no—
411C23×C3—108no—
412C23×C3—204no—
413C23×C3—208no—
414C23×C3—208no—
415C23×C3C2×C4×D512yes—
416C23×C3D4×D512yes—
417C23×C3#71112yes—
418C23×C3C23×D511yescharacteristic
419C2×C3×C4—21no—
420C2×C3×C4—21no—
421C2×C3×C4—21no—
422C2×C3×C4—21no—
423C2×C3×C4—28no—
424C2×C3×C4—52no—
425C2×C3×C4—52no—
426C2×C3×C4—102no—
427C2×C3×C4—102no—
428C2×C3×C4—204no—
429C2×C3×C4C2×C4×D512yes—
430C2×C3×C4C2×C4×D512yes—
431C2×Dic3—22no—
432C2×Dic3—22no—
433C2×Dic3—22no—
434C2×Dic3—22no—
435C2×Dic3—22no—
436C2×Dic3—22no—
437C2×Dic3—54no—
438C2×Dic3—102no—
439C2×Dic3—102no—
440C2×Dic3—102no—
441C2×Dic3—102no—
442C2×Dic3—102no—
443C2×Dic3—208no—
444C22×C5—24no—
445C22×C5—24no—
446C22×C5—22no—
447C22×C5—22no—
448C22×C5D4×Dic314yes—
449C22×C5C2×#23512yes—
450C22×C5C2×D3×D411yescharacteristic
451C22×C5C2×#23711yescharacteristic
452C22×C5#100412yes—
453C22×C5C23×Dic311yescharacteristic
454C2×D5—24no—
455C2×D5—24no—
456C2×D5—22no—
457C2×D5—22no—
458C2×D5—22no—
459C2×D5—22no—
460C2×D5—22no—
461C2×D5—416no—
462C2×D5—42no—
463C2×D5—42no—
464C2×D5#92614yescomplemented
465C4×C5—22no—
466C4×C5—22no—
467C4×C5—122no—
468C4×C5—122no—
469Dic5—22no—
470Dic5—22no—
471Dic5—64no—
472Dic5—122no—
473C24—51no—
474C24—51no—
475C24—54no—
476C24—104no—
477C24—102no—
478C24—102no—
479C24—202no—
480C24C2×D3×D511yescharacteristic
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—32no—
484C22×C4—32no—
485C22×C4—51no—
486C22×C4—51no—
487C22×C4—54no—
488C22×C4—152no—
489C22×C4—152no—
490C22×C4—202no—
491C22×C4—304no—
492C22×C4—304no—
493C22×C4—602no—
494C22×C4D5×Dic314yescomplemented
495C22×C4C2×D3×D511yescharacteristic
496C22×C4C2×D3×D511yescharacteristic
497C4⋊C4—304no—
498C4⋊C4—602no—
499C22⋊C4—54no—
500C22⋊C4—54no—
501C22⋊C4—54no—
502C22⋊C4—54no—
503C22⋊C4—62no—
504C22⋊C4—62no—
505C22⋊C4—154no—
506C22⋊C4—304no—
507C22⋊C4—304no—
508C22⋊C4—302no—
509C22⋊C4—602no—
510C42—304no—
511C42—602no—
512C3×C5#234211yescharacteristic, complemented, Hall
513C22×C3—22no—
514C22×C3—22no—
515C22×C3—24no—
516C22×C3—24no—
517C22×C3—54no—
518C22×C3—102no—
519C22×C3—102no—
520C22×C3—104no—
521C22×C3—104no—
522C22×C3—102no—
523C22×C3—102no—
524C22×C3—102no—
525C22×C3—2016no—
526C22×C3—202no—
527C22×C3—202no—
528C22×C3#402714yes—
529C22×C3C22×C4×D511yescharacteristic
530C22×C3C2×D4×D511yescharacteristic
531C22×C3C2×#71111yescharacteristic
532C22×C3D5×#4112yes—
533C22×C3#414912yes—
534C3×C4—22no—
535C3×C4—22no—
536C3×C4—102no—
537C3×C4—102no—
538Dic3—42no—
539Dic3—42no—
540Dic3—104no—
541Dic3—202no—
542C2×C5—24no—
543C2×C5#549012yes—
544C2×C5#549512yes—
545C2×C5C2×#92611yescharacteristic
546C2×C5C2×D4×Dic311yescharacteristic
547C2×C5#579811yescharacteristic
548D5—24no—
549D5—42no—
550C23—22no—
551C23—22no—
552C23—54no—
553C23—52no—
554C23—52no—
555C23—108no—
556C23—102no—
557C23—101no—
558C23—101no—
559C23—102no—
560C23—102no—
561C23—101no—
562C23—101no—
563C23—108no—
564C23—204no—
565C23—208no—
566C23—208no—
567C23C2×D5×Dic312yes—
568C23#678012yes—
569C23#679012yes—
570C23C22×D3×D511yescharacteristic
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—28no—
576C2×C4—52no—
577C2×C4—52no—
578C2×C4—62no—
579C2×C4—62no—
580C2×C4—62no—
581C2×C4—62no—
582C2×C4—62no—
583C2×C4—62no—
584C2×C4—102no—
585C2×C4—102no—
586C2×C4—154no—
587C2×C4—204no—
588C2×C4—302no—
589C2×C4—302no—
590C2×C4—302no—
591C2×C4—302no—
592C2×C4—302no—
593C2×C4—608no—
594C2×C4C2×D5×Dic312yes—
595C2×C4C2×D5×Dic312yes—
596C2×C3—24no—
597C2×C3—104no—
598C2×C3—202no—
599C2×C3C2×#402711yescharacteristic
600C2×C3D5×#11711yescharacteristic
601C2×C3#6585511yescharacteristic
602C2×C3#6598812yes—
603C2×C3#6599212yes—
604C5#8209711yescharacteristic, complemented, Sylow
605C22—22no—
606C22—24no—
607C22—24no—
608C22—22no—
609C22—54no—
610C22—102no—
611C22—104no—
612C22—104no—
613C22—102no—
614C22—102no—
615C22—102no—
616C22—102no—
617C22—2016no—
618C22—202no—
619C22—202no—
620C22#9155914yes—
621C22D5×#23512yes—
622C22#9201712yes—
623C22C22×D5×Dic311yescharacteristic
624C22C2×#678011yescharacteristic
625C22C2×#679011yescharacteristic
626C4—22no—
627C4—22no—
628C4—102no—
629C4—102no—
630C4—122no—
631C4—122no—
632C4—304no—
633C4—602no—
634C3#1060585311yescharacteristic, complemented, Sylow
635C2—24no—
636C2—104no—
637C2—202no—
638C2D5×#92611yescharacteristic
639C2#1175440112yes—
640C2#1175448811yescharacteristic
641C2#1175466412yes—
642C2C2×#9155911yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC23×C3×C4×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC24×C3×D5C22
hypercenterC23×C4D3×D5
hypocenterC3×C5#2342
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C2×#91815C2semidirect
4. C5×#5211C2semidirect
5. #11749506C2semidirect
7. Dic5×#217C2semidirect
8. #11748900C2semidirect
9. #11748884C2semidirect
10. #11748791C2semidirect
11. #11748772C2semidirect
20. C2×C4⋊Dic15C22semidirect
21. C2×C4×Dic15C22semidirect
24. C2×C3×#685C4semidirect
30. C2×#6668C22semidirect
34. C2×Dic3×Dic5C22semidirect
35. C5×C3⋊#95C22semidirect
36. C5×C3⋊#95C22semidirect
37. #90994C22semidirect
38. #90994C22semidirect
61. C22×C3×C4×C5C2×C4semidirect
69. C22×C3×Dic5C2×C4semidirect
83. C3×#685C2×C4semidirect
84. C3×#685C2×C4semidirect
95. #5211D5semidirect
112. C2×#685Dic3semidirect
198. C3⋊#95C2×D5semidirect
199. C3⋊#95C2×D5semidirect
212. C22×C4×C5C2×Dic3semidirect
218. C22×Dic5C2×Dic3semidirect
233. #685C2×Dic3semidirect
234. #685C2×Dic3semidirect
270. C2×C3×D5#117semidirect
293. C22×C3×C4C4×D5semidirect
464. C2×D5#926semidirect
494. C22×C4D5×Dic3semidirect
512. C3×C5#2342semidirect
604. C5#82097semidirect
634. C3#10605853semidirect

Maximal subgroups

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

subgroupindex
C2×#918152
C22×C3×#6852
C2×D5×#2172
C5×#52112
#117495062
#117494882
Dic5×#2172
#117489002
#117488842
#117487912
#117487722
#106058533
#820975