Finite Group Databasebeta

← #14326310

Subgroups — #14326310

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
1C22×C5×#217C211yesmaximal, characteristic
2C22×C3×C22⋊Dic5C211yesmaximal, characteristic
3C2×#91528C211yesmaximal, characteristic, complemented
4#11751348C211yesmaximal, characteristic
5#11749511C212yesmaximal, complemented
6#11749506C211yesmaximal, characteristic, complemented
7#11749486C212yesmaximal, complemented
8#11749485C212yesmaximal, complemented
9Dic3×C22⋊Dic5C212yesmaximal, complemented
10Dic5×#217C211yesmaximal, characteristic, complemented
11#11749471C211yesmaximal, characteristic, complemented
12#10613829—31nomaximal, Hall
13C25×C3×C5C2211yescharacteristic
14C23×C5×Dic3C2212yes—
15C23×C3×Dic5C2211yescharacteristic
16C23×C3×Dic5C2211yescharacteristic
17C2×#6719C2211yescharacteristic
18C2×#6719C2211yescharacteristic
19C2×#6719C2212yes—
20C2×#6719C2212yes—
21C2×C5×#217—24no—
22C2×C5×#217C2211yescharacteristic
23C2×C5×#217C2212yes—
24C2×C5×#217C2211yescharacteristic
25C2×C3×C22⋊Dic5C418yescomplemented
26C2×C3×C22⋊Dic5C2211yescharacteristic
27C2×C3×C22⋊Dic5C2211yescharacteristic
28C2×C3×C22⋊Dic5C2211yescharacteristic
29C2×C3×C22⋊Dic5C2211yescharacteristic
30C2×#6668C2212yescomplemented
31C2×#6668C2212yescomplemented
32#91528—24no—
33C2×Dic3×Dic5C2212yescomplemented
34C2×Dic3×Dic5C2212yescomplemented
35#90994C2212yescomplemented
36#90994C2212yescomplemented
37#90994C2212yescomplemented
38#90994C2212yescomplemented
39#82097—51nomaximal, Hall
40C22×C5×C22⋊C4—31no—
41C22×C22⋊Dic5D311yescharacteristic
42C2×#4023—31no—
43#65536—31no—
44#65499—32no—
45C4×C22⋊Dic5—32no—
46#65234—32no—
47#65233—32no—
48#65232—31no—
49#65231—31no—
50C22⋊C4×Dic5—31no—
51C24×C3×C5—22no—
52C24×C3×C5—24no—
53C24×C3×C5—22no—
54C24×C3×C5—42no—
55C24×C3×C5C2×C414yes—
56C24×C3×C5C2311yescharacteristic
57C24×C3×C5C2311yescharacteristic
58C24×C3×C5C2311yescharacteristic
59C22×Dic15C2312yes—
60C22×Dic15C2312yes—
61C22×C5×Dic3—24no—
62C22×C5×Dic3—24no—
63C22×C5×Dic3—42no—
64C22×C5×Dic3C2312yes—
65C22×C5×Dic3C2312yes—
66C22×C3×Dic5—22no—
67C22×C3×Dic5—22no—
68C22×C3×Dic5—42no—
69C22×C3×Dic5C2×C414yescomplemented
70C22×C3×Dic5C2×C414yescomplemented
71C22×C3×Dic5C2311yescharacteristic
72C22×C3×Dic5C2311yescharacteristic
73C22×C3×Dic5C2311yescharacteristic
74C22×C3×Dic5C2311yescharacteristic
75#6719—22no—
76#6719—24no—
77#6719—22no—
78#6719—42no—
79C5×#217—22no—
80C5×#217—24no—
81C5×#217D414yes—
82C3×C22⋊Dic5C2×C414yes—
83C3×C22⋊Dic5C2×C414yes—
84C3×C22⋊Dic5C2×C414yescomplemented
85C3×C22⋊Dic5C2×C414yescomplemented
86#6668—24no—
87#6668—42no—
88Dic3×Dic5—24no—
89Dic3×Dic5—42no—
90C22×C3×C22⋊C4—51no—
91C22×#217D511yescharacteristic
92C2×#926—51no—
93#5247—51no—
94#5212—51no—
95#5211—51no—
96C4×#217—51no—
97#4951—52no—
98#4951—52no—
99#4950—52no—
100Dic3×C22⋊C4—52no—
101C25×C5C2×D311yescharacteristic
102C23×C4×C5—32no—
103C23×Dic5C2×D311yescharacteristic
104C23×Dic5C2×D311yescharacteristic
105C2×C5×C22⋊C4—31no—
106C2×C5×C22⋊C4—31no—
107C2×C5×C22⋊C4—32no—
108C2×C5×C22⋊C4—64no—
109C2×C22⋊Dic5—31no—
110C2×C22⋊Dic5—31no—
111C2×C22⋊Dic5—32no—
112C2×C22⋊Dic5—32no—
113C2×C22⋊Dic5Dic318yescomplemented
114C2×C22⋊Dic5C2×D311yescharacteristic
115C2×C22⋊Dic5C2×D311yescharacteristic
116C2×C22⋊Dic5C2×D311yescharacteristic
117C2×C22⋊Dic5C2×D311yescharacteristic
118C2×#683—32no—
119C2×#683—32no—
120C2×C4×Dic5—32no—
121C2×C4×Dic5—32no—
122#4023—64no—
123#3963—32no—
124#3963—32no—
125#3963—32no—
126#3963—32no—
127#2342—151noSylow
128C23×C3×C5—21no—
129C23×C3×C5—22no—
130C23×C3×C5—21no—
131C23×C3×C5—22no—
132C23×C3×C5—22no—
133C23×C3×C5—28no—
134C23×C3×C5—28no—
135C23×C3×C5—21no—
136C23×C3×C5—22no—
137C23×C3×C5—21no—
138C23×C3×C5—22no—
139C23×C3×C5—44no—
140C23×C3×C5—48no—
141C23×C3×C5—48no—
142C23×C3×C5C22×C412yes—
143C23×C3×C5C22×C412yes—
144C23×C3×C5C22×C412yes—
145C23×C3×C5C2×D412yes—
146C23×C3×C5#4114yes—
147C23×C3×C5#4112yes—
148C23×C3×C5C2411yescharacteristic
149C2×Dic15—22no—
150C2×Dic15—22no—
151C2×Dic15—22no—
152C2×Dic15—22no—
153C2×Dic15—22no—
154C2×Dic15—22no—
155C2×C5×Dic3—22no—
156C2×C5×Dic3—22no—
157C2×C5×Dic3—22no—
158C2×C5×Dic3—22no—
159C2×C5×Dic3—22no—
160C2×C5×Dic3—48no—
161C2×C5×Dic3C2×D414yes—
162C2×C3×Dic5—28no—
163C2×C3×Dic5—21no—
164C2×C3×Dic5—21no—
165C2×C3×Dic5—21no—
166C2×C3×Dic5—21no—
167C2×C3×Dic5—22no—
168C2×C3×Dic5—22no—
169C2×C3×Dic5—44no—
170C2×C3×Dic5C22×C412yes—
171C2×C3×Dic5C22×C412yes—
172C2×C3×Dic5C22×C412yes—
173C2×C3×Dic5C22×C412yes—
174C25×C3C2×D511yescharacteristic
175C23×C3×C4—51no—
176C23×C3×C4—51no—
177C23×Dic3C2×D512yes—
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—24no—
184C2×#217—52no—
185C2×#217—51no—
186C2×#217—51no—
187C2×#217—52no—
188C2×#217C2×D512yes—
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⋊#95—52no—
199C3⋊#95—52no—
200C24×C5—24no—
201C24×C5—22no—
202C24×C5—22no—
203C24×C5—42no—
204C24×C5C2×Dic314yes—
205C24×C5C22×D311yescharacteristic
206C24×C5C22×D311yescharacteristic
207C24×C5C22×D311yescharacteristic
208C22×C4×C5—32no—
209C22×C4×C5—32no—
210C22×C4×C5—64no—
211C22×C4×C5—64no—
212C22×C4×C5—122no—
213C22×Dic5—22no—
214C22×Dic5—22no—
215C22×Dic5—32no—
216C22×Dic5—32no—
217C22×Dic5—42no—
218C22×Dic5C2×Dic314yescomplemented
219C22×Dic5C2×Dic314yescomplemented
220C22×Dic5C22×D311yescharacteristic
221C22×Dic5C22×D311yescharacteristic
222C22×Dic5C22×D311yescharacteristic
223C22×Dic5C22×D311yescharacteristic
224C5×C22⋊C4—34no—
225C5×C22⋊C4—64no—
226C5×C22⋊C4—62no—
227C22⋊Dic5—64no—
228C22⋊Dic5—62no—
229C22⋊Dic5—62no—
230C22⋊Dic5—122no—
231C22⋊Dic5C2×Dic314yes—
232C22⋊Dic5C2×Dic314yes—
233C22⋊Dic5C2×Dic314yescomplemented
234C22⋊Dic5C2×Dic314yescomplemented
235#683—64no—
236#683—122no—
237C4×Dic5—64no—
238C4×Dic5—122no—
239C2×#117—151no—
240C22×C22⋊C4—31no—
241C22×C22⋊C4—51no—
242#388—152no—
243#388—152no—
244#385—152no—
245#381—151no—
246#380—151no—
247#379—151no—
248C4×C22⋊C4—151no—
249C4×C22⋊C4—152no—
250C22×C3×C5—22no—
251C22×C3×C5—22no—
252C22×C3×C5—22no—
253C22×C3×C5—22no—
254C22×C3×C5—24no—
255C22×C3×C5—24no—
256C22×C3×C5—24no—
257C22×C3×C5—24no—
258C22×C3×C5—22no—
259C22×C3×C5—22no—
260C22×C3×C5—22no—
261C22×C3×C5—416no—
262C22×C3×C5—42no—
263C22×C3×C5—42no—
264C22×C3×C5#11714yescomplemented
265C22×C3×C5C4×D414yes—
266C22×C3×C5C23×C411yescharacteristic
267C22×C3×C5C22×D411yescharacteristic
268C22×C3×C5C2×#4111yescharacteristic
269C22×C3×C5C2×#4112yes—
270C22×C3×C5H(5,𝔽2)12yes—
271Dic15—42no—
272Dic15—42no—
273C3×Dic5—22no—
274C3×Dic5—22no—
275C3×Dic5—22no—
276C3×Dic5—22no—
277C5×Dic3—24no—
278C5×Dic3—42no—
279C24×C3—22no—
280C24×C3—24no—
281C24×C3—22no—
282C24×C3—42no—
283C24×C3C4×D514yes—
284C24×C3C22×D511yescharacteristic
285C24×C3C22×D511yescharacteristic
286C24×C3C22×D511yescharacteristic
287C22×C3×C4—54no—
288C22×C3×C4—51no—
289C22×C3×C4—51no—
290C22×C3×C4—51no—
291C22×C3×C4—51no—
292C22×C3×C4—54no—
293C22×C3×C4—102no—
294C22×C3×C4—102no—
295C22×C3×C4—202no—
296C22×Dic3—24no—
297C22×Dic3—24no—
298C22×Dic3—42no—
299C22×Dic3—52no—
300C22×Dic3—52no—
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—24no—
309#217—102no—
310#217—104no—
311#217—102no—
312#217—202no—
313#217C22⋊D514yes—
314C4⋊Dic3—104no—
315C4⋊Dic3—202no—
316C4×Dic3—104no—
317C4×Dic3—202no—
318C23×C5—22no—
319C23×C5—21no—
320C23×C5—21no—
321C23×C5—22no—
322C23×C5—22no—
323C23×C5—28no—
324C23×C5—28no—
325C23×C5—22no—
326C23×C5—21no—
327C23×C5—21no—
328C23×C5—22no—
329C23×C5—44no—
330C23×C5—48no—
331C23×C5—48no—
332C23×C5#23514yes—
333C23×C5D3×D412yes—
334C23×C5#23712yes—
335C23×C5C22×Dic312yes—
336C23×C5C22×Dic312yes—
337C23×C5C22×Dic312yes—
338C23×C5C23×D311yescharacteristic
339C2×C4×C5—34no—
340C2×C4×C5—62no—
341C2×C4×C5—62no—
342C2×C4×C5—62no—
343C2×C4×C5—62no—
344C2×C4×C5—62no—
345C2×C4×C5—128no—
346C2×Dic5—28no—
347C2×Dic5—21no—
348C2×Dic5—21no—
349C2×Dic5—21no—
350C2×Dic5—21no—
351C2×Dic5—22no—
352C2×Dic5—22no—
353C2×Dic5—44no—
354C2×Dic5—62no—
355C2×Dic5—62no—
356C2×Dic5—62no—
357C2×Dic5—62no—
358C2×Dic5—62no—
359C2×Dic5—62no—
360C2×Dic5C22×Dic312yes—
361C2×Dic5C22×Dic312yes—
362C2×Dic5C22×Dic312yes—
363C2×Dic5C22×Dic312yes—
364C25D3×D511yescharacteristic
365C23×C4—32no—
366C23×C4—51no—
367C23×C4—51no—
368#117—304no—
369C2×C4⋊C4—152no—
370C2×C4⋊C4—152no—
371C2×C22⋊C4—31no—
372C2×C22⋊C4—31no—
373C2×C22⋊C4—32no—
374C2×C22⋊C4—58no—
375C2×C22⋊C4—51no—
376C2×C22⋊C4—51no—
377C2×C22⋊C4—51no—
378C2×C22⋊C4—51no—
379C2×C22⋊C4—64no—
380C2×C22⋊C4—151no—
381C2×C22⋊C4—151no—
382C2×C22⋊C4—152no—
383C2×C22⋊C4—152no—
384C2×C42—152no—
385C2×C42—152no—
386#95—152no—
387#95—152no—
388#95—152no—
389#95—152no—
390C2×C3×C5—24no—
391C2×C3×C5—24no—
392C2×C3×C5—42no—
393C2×C3×C5C2×#11711yescharacteristic
394C2×C3×C5C2×C4×D411yescharacteristic
395C2×C3×C5#51811yescharacteristic
396C2×C3×C5#54612yes—
397C2×C3×C5#55112yes—
398C23×C3—21no—
399C23×C3—22no—
400C23×C3—22no—
401C23×C3—22no—
402C23×C3—21no—
403C23×C3—21no—
404C23×C3—22no—
405C23×C3—22no—
406C23×C3—21no—
407C23×C3—28no—
408C23×C3—28no—
409C23×C3—44no—
410C23×C3—48no—
411C23×C3—48no—
412C23×C3C2×C4×D512yes—
413C23×C3C2×C4×D512yes—
414C23×C3C2×C4×D512yes—
415C23×C3#70912yes—
416C23×C3#71114yes—
417C23×C3C2×C22⋊D512yes—
418C23×C3C23×D511yescharacteristic
419C2×C3×C4—52no—
420C2×C3×C4—52no—
421C2×C3×C4—52no—
422C2×C3×C4—52no—
423C2×C3×C4—101no—
424C2×C3×C4—101no—
425C2×C3×C4—102no—
426C2×C3×C4—102no—
427C2×C3×C4—101no—
428C2×C3×C4—101no—
429C2×C3×C4—108no—
430C2×C3×C4—204no—
431C2×Dic3—22no—
432C2×Dic3—22no—
433C2×Dic3—22no—
434C2×Dic3—22no—
435C2×Dic3—22no—
436C2×Dic3—48no—
437C2×Dic3—102no—
438C2×Dic3—102no—
439C2×Dic3—102no—
440C2×Dic3—102no—
441C2×Dic3—102no—
442C2×Dic3—102no—
443C2×Dic3C2×C22⋊D514yes—
444C22×C5—24no—
445C22×C5—24no—
446C22×C5—24no—
447C22×C5—24no—
448C22×C5—22no—
449C22×C5—22no—
450C22×C5—22no—
451C22×C5—22no—
452C22×C5—22no—
453C22×C5—22no—
454C22×C5—22no—
455C22×C5—416no—
456C22×C5—42no—
457C22×C5—42no—
458C22×C5#92614yescomplemented
459C22×C5D4×Dic314yes—
460C22×C5C2×#23512yes—
461C22×C5C2×D3×D411yescharacteristic
462C22×C5C2×#23711yescharacteristic
463C22×C5#100412yes—
464C22×C5C23×Dic311yescharacteristic
465C4×C5—64no—
466C4×C5—122no—
467Dic5—22no—
468Dic5—22no—
469Dic5—22no—
470Dic5—22no—
471Dic5—122no—
472Dic5—122no—
473C24—22no—
474C24—24no—
475C24—22no—
476C24—42no—
477C24D5×Dic314yes—
478C24C2×D3×D511yescharacteristic
479C24C2×D3×D511yescharacteristic
480C24C2×D3×D511yescharacteristic
481C22×C4—32no—
482C22×C4—32no—
483C22×C4—54no—
484C22×C4—51no—
485C22×C4—51no—
486C22×C4—51no—
487C22×C4—51no—
488C22×C4—54no—
489C22×C4—64no—
490C22×C4—64no—
491C22×C4—102no—
492C22×C4—102no—
493C22×C4—122no—
494C22×C4—152no—
495C22×C4—152no—
496C22×C4—202no—
497C4⋊C4—304no—
498C4⋊C4—602no—
499C22⋊C4—34no—
500C22⋊C4—54no—
501C22⋊C4—54no—
502C22⋊C4—54no—
503C22⋊C4—54no—
504C22⋊C4—64no—
505C22⋊C4—62no—
506C22⋊C4—302no—
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—22no—
516C22×C3—24no—
517C22×C3—24no—
518C22×C3—22no—
519C22×C3—24no—
520C22×C3—24no—
521C22×C3—22no—
522C22×C3—22no—
523C22×C3—22no—
524C22×C3—416no—
525C22×C3—42no—
526C22×C3—42no—
527C22×C3#402314yescomplemented
528C22×C3C4×C22⋊D514yes—
529C22×C3C22×C4×D511yescharacteristic
530C22×C3C2×#70911yescharacteristic
531C22×C3C2×#71112yes—
532C22×C3#414412yes—
533C22×C3C22×C22⋊D511yescharacteristic
534C3×C4—102no—
535C3×C4—102no—
536C3×C4—102no—
537C3×C4—102no—
538Dic3—24no—
539Dic3—42no—
540Dic3—202no—
541Dic3—202no—
542C2×C5—24no—
543C2×C5—24no—
544C2×C5—42no—
545C2×C5#549012yes—
546C2×C5#549512yes—
547C2×C5C2×#92611yescharacteristic
548C2×C5C2×D4×Dic311yescharacteristic
549C2×C5#579811yescharacteristic
550C23—21no—
551C23—22no—
552C23—21no—
553C23—22no—
554C23—22no—
555C23—21no—
556C23—28no—
557C23—22no—
558C23—21no—
559C23—22no—
560C23—28no—
561C23—44no—
562C23—48no—
563C23—48no—
564C23C2×D5×Dic312yes—
565C23C2×D5×Dic312yes—
566C23C2×D5×Dic312yes—
567C23#677912yes—
568C23#678214yes—
569C23D3×C22⋊D512yes—
570C23C22×D3×D511yescharacteristic
571C2×C4—34no—
572C2×C4—52no—
573C2×C4—52no—
574C2×C4—52no—
575C2×C4—52no—
576C2×C4—62no—
577C2×C4—62no—
578C2×C4—62no—
579C2×C4—62no—
580C2×C4—62no—
581C2×C4—101no—
582C2×C4—101no—
583C2×C4—101no—
584C2×C4—101no—
585C2×C4—102no—
586C2×C4—102no—
587C2×C4—108no—
588C2×C4—128no—
589C2×C4—204no—
590C2×C4—302no—
591C2×C4—302no—
592C2×C4—302no—
593C2×C4—302no—
594C2×C4—302no—
595C2×C4—302no—
596C2×C3—24no—
597C2×C3—24no—
598C2×C3—42no—
599C2×C3C2×#402311yescharacteristic
600C2×C3#6582011yescharacteristic
601C2×C3#6582912yes—
602C2×C3C2×C4×C22⋊D511yescharacteristic
603C2×C3#6614012yes—
604C5#8209711yescharacteristic, complemented, Sylow
605C22—24no—
606C22—24no—
607C22—22no—
608C22—22no—
609C22—24no—
610C22—24no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—416no—
617C22—42no—
618C22—42no—
619C22#9152814yescomplemented
620C22Dic3×C22⋊D514yes—
621C22C22×D5×Dic311yescharacteristic
622C22C2×#677911yescharacteristic
623C22C2×#678212yes—
624C22C2×D3×C22⋊D511yescharacteristic
625C22#9204912yes—
626C4—64no—
627C4—102no—
628C4—102no—
629C4—102no—
630C4—102no—
631C4—122no—
632C4—602no—
633C4—602no—
634C3#1061382911yescharacteristic, complemented, Sylow
635C2—24no—
636C2—24no—
637C2—42no—
638C2#1175535712yes—
639C2C2×#9152811yescharacteristic
640C2C2×Dic3×C22⋊D511yescharacteristic
641C2#1175539312yes—
642C2#1175540611yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
Fitting, prime radicalC25×C3×C5C22
socle, cosocleC23×C3×C5C24
hypercenterC25D3×D5
hypocenterC3×C5#2342
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C25

Complements

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

NH ≅ G/Ntype
3. C2×#91528C2semidirect
5. #11749511C2semidirect
6. #11749506C2semidirect
7. #11749486C2semidirect
8. #11749485C2semidirect
9. Dic3×C22⋊Dic5C2semidirect
10. Dic5×#217C2semidirect
11. #11749471C2semidirect
25. C2×C3×C22⋊Dic5C4semidirect
30. C2×#6668C22semidirect
31. C2×#6668C22semidirect
33. C2×Dic3×Dic5C22semidirect
34. C2×Dic3×Dic5C22semidirect
35. #90994C22semidirect
36. #90994C22semidirect
37. #90994C22semidirect
38. #90994C22semidirect
69. C22×C3×Dic5C2×C4semidirect
70. C22×C3×Dic5C2×C4semidirect
84. C3×C22⋊Dic5C2×C4semidirect
85. C3×C22⋊Dic5C2×C4semidirect
113. C2×C22⋊Dic5Dic3semidirect
218. C22×Dic5C2×Dic3semidirect
219. C22×Dic5C2×Dic3semidirect
233. C22⋊Dic5C2×Dic3semidirect
234. C22⋊Dic5C2×Dic3semidirect
264. C22×C3×C5#117semidirect
458. C22×C5#926semidirect
512. C3×C5#2342semidirect
527. C22×C3#4023semidirect
604. C5#82097semidirect
619. C22#91528semidirect
634. C3#10613829semidirect

Maximal subgroups

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

subgroupindex
C22×C5×#2172
C22×C3×C22⋊Dic52
C2×#915282
#117513482
#117495112
#117495062
#117494862
#117494852
Dic3×C22⋊Dic52
Dic5×#2172
#117494712
#106138293
#820975