Finite Group Databasebeta

← #14323441

Subgroups — #14323441

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
1C22×#6719C211yesmaximal, characteristic, complemented
2C2×C3×#4099C211yesmaximal, characteristic
3C2×#91559C211yesmaximal, characteristic, complemented
4C5×#5247C211yesmaximal, characteristic, complemented
5#11749518C212yesmaximal, complemented
6#11749515C211yesmaximal, characteristic, complemented
7#11749500C211yesmaximal, characteristic, complemented
8#11749489C212yesmaximal, complemented
9#11749487C211yesmaximal, characteristic, complemented
10#11749486C212yesmaximal, complemented
11#11749485C212yesmaximal, complemented
12#10605552—31nomaximal, Hall
13C25×C3×C5C2211yescharacteristic
14C23×Dic15C2211yescharacteristic
15C23×Dic15C2211yescharacteristic
16C22×C3×C22⋊D5C2211yescharacteristic
17C22×C3×C22⋊D5C2212yes—
18C22×D5×Dic3C2211yescharacteristic, complemented
19C2×#6719—24no—
20C2×#6719C2212yes—
21C2×#6719C2212yes—
22C2×C5×#217C2211yescharacteristic
23C2×C5×#217C2211yescharacteristic
24C2×C5×#217C2212yes—
25C2×C5×#217C2212yes—
26C3×#4099C418yescomplemented
27C2×C3×C22⋊Dic5C2212yes—
28C2×C3×C22⋊Dic5C2211yescharacteristic
29#91559—24no—
30C2×#6668C2211yescharacteristic, complemented
31C2×#6668C2212yescomplemented
32C2×#6665C2211yescharacteristic, complemented
33C2×#6665C2212yescomplemented
34C2×Dic3×Dic5C2211yescharacteristic, complemented
35C2×Dic3×Dic5C2212yescomplemented
36#90994C2212yescomplemented
37#90994C2212yescomplemented
38#90994C2212yescomplemented
39#82578—51nomaximal, Hall
40C2×#4099D311yescharacteristic
41C22×C22⋊Dic5—31no—
42C2×#4027—31no—
43C5×#379—31no—
44#65247—31no—
45#65246—32no—
46#65245—32no—
47#65235—31no—
48#65234—32no—
49#65233—32no—
50C22⋊C4×Dic5—31no—
51C24×C3×C5—22no—
52C24×C3×C5—22no—
53C24×C3×C5—42no—
54C24×C3×C5—42no—
55C24×C3×C5C2×C414yescomplemented
56C24×C3×C5C2312yes—
57C24×C3×C5C2311yescharacteristic
58C23×C3×D5C2311yescharacteristic
59C22×Dic15—22no—
60C22×Dic15—42no—
61C22×Dic15—42no—
62C22×Dic15D414yescomplemented
63C22×Dic15C2312yes—
64C22×Dic15C2311yescharacteristic
65C22×Dic15C2311yescharacteristic
66C22×C5×Dic3C2311yescharacteristic
67C22×C5×Dic3C2312yes—
68C22×C5×Dic3C2311yescharacteristic
69C2×C3×C22⋊D5—22no—
70C2×C3×C22⋊D5—22no—
71C2×C3×C22⋊D5—42no—
72C2×C3×C22⋊D5—42no—
73C2×C3×C22⋊D5C2×C414yes—
74C2×C3×C22⋊D5C2×C418yescomplemented
75C22×C3×Dic5C2311yescharacteristic
76C22×C3×Dic5C2312yes—
77C2×D5×Dic3—22no—
78C2×D5×Dic3—42no—
79#6719—24no—
80#6719—24no—
81C5×#217—22no—
82C5×#217—22no—
83C5×#217—42no—
84C5×#217—42no—
85C3×C22⋊Dic5C2×C418yescomplemented
86C3×C22⋊Dic5C2×C414yescomplemented
87#6668—22no—
88#6668—42no—
89#6665—22no—
90#6665—42no—
91Dic3×Dic5—22no—
92Dic3×Dic5—42no—
93C2×C3×#120—51no—
94C22×#217—51no—
95C2×D4×Dic3—51no—
96#5247D511yescharacteristic, complemented
97#5221—52no—
98#5220—51no—
99#5220—52no—
100#4951—52no—
101#4951—52no—
102#4950—51no—
103Dic3×C22⋊C4—51no—
104C25×C5C2×D311yescharacteristic
105C22×C22⋊D5C2×D311yescharacteristic
106C22×C22⋊D5C2×D312yes—
107C23×Dic5—31no—
108C23×Dic5—31no—
109C22×C4×D5—31no—
110C2×C5×C22⋊C4—31no—
111C2×C5×C22⋊C4—31no—
112C2×C5×C22⋊C4—32no—
113C2×C5×C22⋊C4—32no—
114#4099Dic318yescomplemented
115C2×C22⋊Dic5—32no—
116C2×C22⋊Dic5—32no—
117C2×C22⋊Dic5—64no—
118C2×C22⋊Dic5C2×D311yescharacteristic
119C2×C22⋊Dic5C2×D312yes—
120C2×#685—31no—
121C2×#685—32no—
122C2×#683—31no—
123C2×#683—32no—
124C2×C4×Dic5—31no—
125C2×C4×Dic5—32no—
126#4027—64no—
127#3963—32no—
128#3963—32no—
129#3963—32no—
130#2321—151noSylow
131C23×C3×C5—22no—
132C23×C3×C5—22no—
133C23×C3×C5—28no—
134C23×C3×C5—22no—
135C23×C3×C5—22no—
136C23×C3×C5—22no—
137C23×C3×C5—22no—
138C23×C3×C5—22no—
139C23×C3×C5—22no—
140C23×C3×C5—48no—
141C23×C3×C5—44no—
142C23×C3×C5—44no—
143C23×C3×C5—48no—
144C23×C3×C5C22×C414yes—
145C23×C3×C5C22×C412yes—
146C23×C3×C5C2×D412yes—
147C23×C3×C5#4112yes—
148C23×C3×C5C2411yescharacteristic
149C22×C3×D5—21no—
150C22×C3×D5—21no—
151C22×C3×D5—22no—
152C22×C3×D5—22no—
153C22×C3×D5C22×C412yes—
154C2×Dic15—22no—
155C2×Dic15—22no—
156C2×Dic15—22no—
157C2×Dic15—22no—
158C2×Dic15—22no—
159C2×Dic15—44no—
160C2×Dic15—44no—
161C2×Dic15C2×D412yes—
162C2×Dic15C2×D412yes—
163C2×C5×Dic3—21no—
164C2×C5×Dic3—21no—
165C2×C5×Dic3—22no—
166C2×C5×Dic3—22no—
167C2×C5×Dic3—22no—
168C2×C5×Dic3—22no—
169C2×C5×Dic3—22no—
170C3×C22⋊D5—28no—
171C3×C22⋊D5—48no—
172C2×C3×Dic5—21no—
173C2×C3×Dic5—21no—
174C2×C3×Dic5—22no—
175C2×C3×Dic5—22no—
176C2×C3×Dic5C22×C412yes—
177C2×C3×Dic5C22×C414yes—
178D5×Dic3—44no—
179C25×C3C2×D511yescharacteristic
180C22×C3×D4—51no—
181C22×C3×D4—52no—
182C23×Dic3—51no—
183C23×Dic3—51no—
184C23×Dic3—51no—
185C3×#120—58no—
186C2×C3×C22⋊C4—51no—
187C2×C3×C22⋊C4—52no—
188C2×#217—52no—
189C2×#217—52no—
190C2×#217—51no—
191C2×#217—52no—
192C2×#217—104no—
193C2×#217C2×D511yescharacteristic
194C2×#217C2×D511yescharacteristic
195C2×#217C2×D512yes—
196C2×#217C2×D512yes—
197D4×Dic3—104no—
198C2×C4⋊Dic3—51no—
199C2×C4⋊Dic3—52no—
200C2×C4×Dic3—51no—
201C2×C4×Dic3—52no—
202C3⋊#95—52no—
203C3⋊#95—52no—
204C3⋊#95—52no—
205C24×C5—22no—
206C24×C5—22no—
207C24×C5—42no—
208C24×C5—42no—
209C24×C5C2×Dic314yescomplemented
210C24×C5C22×D312yes—
211C24×C5C22×D311yescharacteristic
212C23×D5C22×D311yescharacteristic
213C22×C4×C5—31no—
214C22×C4×C5—32no—
215C22×C4×C5—31no—
216C2×C22⋊D5—22no—
217C2×C22⋊D5—22no—
218C2×C22⋊D5—42no—
219C2×C22⋊D5—42no—
220C2×C22⋊D5C2×Dic314yes—
221C2×C22⋊D5C2×Dic318yescomplemented
222C22×Dic5—31no—
223C22×Dic5—31no—
224C22×Dic5—34no—
225C22×Dic5—32no—
226C22×Dic5—62no—
227C22×Dic5—122no—
228C22×Dic5—122no—
229C22×Dic5C22×D311yescharacteristic
230C22×Dic5C22×D312yes—
231C2×C4×D5—62no—
232C2×C4×D5—122no—
233C5×C22⋊C4—62no—
234C5×C22⋊C4—62no—
235C5×C22⋊C4—122no—
236C5×C22⋊C4—122no—
237C22⋊Dic5—64no—
238C22⋊Dic5—64no—
239C22⋊Dic5C2×Dic318yescomplemented
240C22⋊Dic5C2×Dic314yescomplemented
241#685—62no—
242#685—122no—
243#683—62no—
244#683—122no—
245C4×Dic5—62no—
246C4×Dic5—122no—
247C2×#120—51no—
248C2×C4×D4—151no—
249C22×C22⋊C4—151no—
250#390—152no—
251#388—152no—
252#388—152no—
253#386—152no—
254#386—151no—
255#385—151no—
256#379—31no—
257C4×C22⋊C4—151no—
258C22×C3×C5—24no—
259C22×C3×C5—22no—
260C22×C3×C5—22no—
261C22×C3×C5—24no—
262C22×C3×C5—24no—
263C22×C3×C5—24no—
264C22×C3×C5—24no—
265C22×C3×C5—416no—
266C22×C3×C5—42no—
267C22×C3×C5—42no—
268C22×C3×C5—42no—
269C22×C3×C5—42no—
270C22×C3×C5C4×D414yes—
271C22×C3×C5C23×C411yescharacteristic
272C22×C3×C5C22×D411yescharacteristic
273C22×C3×C5C2×#4111yescharacteristic
274C22×C3×C5H(5,𝔽2)12yes—
275C22×C3×C5H(5,𝔽2)12yes—
276C2×C3×D5—22no—
277C2×C3×D5—24no—
278C2×C3×D5—41no—
279C2×C3×D5—41no—
280C2×C3×D5—42no—
281Dic15—22no—
282Dic15—22no—
283Dic15—42no—
284C3×Dic5—22no—
285C3×Dic5—24no—
286C5×Dic3—41no—
287C5×Dic3—41no—
288C5×Dic3—42no—
289C24×C3—22no—
290C24×C3—22no—
291C24×C3—42no—
292C24×C3—42no—
293C24×C3—51no—
294C24×C3C4×D514yescomplemented
295C24×C3C22×D512yes—
296C24×C3C22×D511yescharacteristic
297C2×C3×D4—54no—
298C2×C3×D4—58no—
299C2×C3×D4—102no—
300C2×C3×D4—102no—
301C2×C3×D4—202no—
302C2×C3×D4—202no—
303C22×C3×C4—51no—
304C22×C3×C4—52no—
305C22×Dic3—51no—
306C22×Dic3—51no—
307C22×Dic3—54no—
308C22×Dic3—52no—
309C22×Dic3—102no—
310C22×Dic3—102no—
311C22×Dic3—202no—
312C22×Dic3—202no—
313C22×Dic3—202no—
314C22×Dic3C22×D512yes—
315C22×Dic3C22×D511yescharacteristic
316C22×Dic3C22×D511yescharacteristic
317C3×C22⋊C4—58no—
318C3×C22⋊C4—54no—
319#217—22no—
320#217—22no—
321#217—42no—
322#217—42no—
323#217—102no—
324#217—104no—
325#217—104no—
326#217—202no—
327C4⋊Dic3—102no—
328C4⋊Dic3—202no—
329C4×Dic3—102no—
330C4×Dic3—202no—
331C23×C5—22no—
332C23×C5—22no—
333C23×C5—28no—
334C23×C5—22no—
335C23×C5—22no—
336C23×C5—22no—
337C23×C5—22no—
338C23×C5—22no—
339C23×C5—22no—
340C23×C5—48no—
341C23×C5—44no—
342C23×C5—44no—
343C23×C5—48no—
344C23×C5D3×D412yes—
345C23×C5#23712yes—
346C23×C5C22×Dic314yes—
347C23×C5C22×Dic312yes—
348C23×C5C23×D311yescharacteristic
349C22×D5—21no—
350C22×D5—21no—
351C22×D5—22no—
352C22×D5—22no—
353C22×D5C22×Dic312yes—
354C2×C4×C5—61no—
355C2×C4×C5—61no—
356C2×C4×C5—62no—
357C2×C4×C5—62no—
358C2×C4×C5—62no—
359C2×C4×C5—62no—
360C2×C4×C5—62no—
361C22⋊D5—28no—
362C22⋊D5—48no—
363C2×Dic5—21no—
364C2×Dic5—21no—
365C2×Dic5—22no—
366C2×Dic5—22no—
367C2×Dic5—32no—
368C2×Dic5—32no—
369C2×Dic5—62no—
370C2×Dic5—62no—
371C2×Dic5—62no—
372C2×Dic5—62no—
373C2×Dic5—62no—
374C2×Dic5—124no—
375C2×Dic5—124no—
376C2×Dic5C22×Dic312yes—
377C2×Dic5C22×Dic314yes—
378C4×D5—124no—
379C25D3×D511yescharacteristic
380C22×D4—51no—
381C22×D4—52no—
382C23×C4—151no—
383C23×C4—151no—
384C23×C4—151no—
385#120—58no—
386C4×D4—304no—
387C2×C4⋊C4—151no—
388C2×C4⋊C4—152no—
389C2×C22⋊C4—31no—
390C2×C22⋊C4—31no—
391C2×C22⋊C4—32no—
392C2×C22⋊C4—32no—
393C2×C22⋊C4—51no—
394C2×C22⋊C4—52no—
395C2×C22⋊C4—151no—
396C2×C22⋊C4—152no—
397C2×C22⋊C4—152no—
398C2×C22⋊C4—152no—
399C2×C22⋊C4—304no—
400C2×C42—151no—
401C2×C42—152no—
402#95—152no—
403#95—152no—
404#95—152no—
405C2×C3×C5—24no—
406C2×C3×C5—44no—
407C2×C3×C5C2×C4×D411yescharacteristic
408C2×C3×C5#51812yes—
409C2×C3×C5#53411yescharacteristic
410C2×C3×C5#53511yescharacteristic
411C2×C3×C5#53812yes—
412C3×D5—42no—
413C23×C3—22no—
414C23×C3—22no—
415C23×C3—22no—
416C23×C3—22no—
417C23×C3—22no—
418C23×C3—22no—
419C23×C3—22no—
420C23×C3—22no—
421C23×C3—28no—
422C23×C3—48no—
423C23×C3—44no—
424C23×C3—44no—
425C23×C3—48no—
426C23×C3—52no—
427C23×C3—101no—
428C23×C3—101no—
429C23×C3—102no—
430C23×C3—102no—
431C23×C3C2×C4×D514yes—
432C23×C3C2×C4×D512yes—
433C23×C3D4×D512yes—
434C23×C3#71112yes—
435C23×C3C23×D511yescharacteristic
436C3×D4—108no—
437C3×D4—208no—
438C2×C3×C4—52no—
439C2×C3×C4—54no—
440C2×C3×C4—101no—
441C2×C3×C4—101no—
442C2×C3×C4—102no—
443C2×C3×C4—102no—
444C2×Dic3—21no—
445C2×Dic3—21no—
446C2×Dic3—22no—
447C2×Dic3—22no—
448C2×Dic3—22no—
449C2×Dic3—22no—
450C2×Dic3—22no—
451C2×Dic3—52no—
452C2×Dic3—52no—
453C2×Dic3—102no—
454C2×Dic3—102no—
455C2×Dic3—102no—
456C2×Dic3—102no—
457C2×Dic3—102no—
458C2×Dic3—204no—
459C2×Dic3—204no—
460C2×Dic3—204no—
461C22×C5—24no—
462C22×C5—22no—
463C22×C5—22no—
464C22×C5—24no—
465C22×C5—24no—
466C22×C5—24no—
467C22×C5—24no—
468C22×C5—416no—
469C22×C5—42no—
470C22×C5—42no—
471C22×C5—42no—
472C22×C5—42no—
473C22×C5D4×Dic314yes—
474C22×C5C2×D3×D411yescharacteristic
475C22×C5C2×#23711yescharacteristic
476C22×C5#100412yes—
477C22×C5#100412yes—
478C22×C5C23×Dic311yescharacteristic
479C2×D5—22no—
480C2×D5—24no—
481C2×D5—41no—
482C2×D5—41no—
483C2×D5—42no—
484C4×C5—121no—
485C4×C5—121no—
486C4×C5—122no—
487Dic5—22no—
488Dic5—24no—
489Dic5—62no—
490Dic5—62no—
491Dic5—122no—
492C24—22no—
493C24—22no—
494C24—42no—
495C24—42no—
496C24—51no—
497C24D5×Dic314yescomplemented
498C24C2×D3×D512yes—
499C24C2×D3×D511yescharacteristic
500C2×D4—54no—
501C2×D4—58no—
502C2×D4—102no—
503C2×D4—102no—
504C2×D4—202no—
505C2×D4—202no—
506C22×C4—32no—
507C22×C4—31no—
508C22×C4—31no—
509C22×C4—51no—
510C22×C4—52no—
511C22×C4—151no—
512C22×C4—151no—
513C22×C4—154no—
514C22×C4—152no—
515C22×C4—302no—
516C22×C4—302no—
517C22×C4—602no—
518C22×C4—602no—
519C22×C4—602no—
520C4⋊C4—302no—
521C4⋊C4—602no—
522C22⋊C4—58no—
523C22⋊C4—54no—
524C22⋊C4—62no—
525C22⋊C4—62no—
526C22⋊C4—122no—
527C22⋊C4—122no—
528C22⋊C4—302no—
529C22⋊C4—304no—
530C22⋊C4—304no—
531C22⋊C4—602no—
532C42—302no—
533C42—602no—
534C3×C5#232111yescharacteristic, complemented, Hall
535C22×C3—24no—
536C22×C3—22no—
537C22×C3—22no—
538C22×C3—24no—
539C22×C3—24no—
540C22×C3—24no—
541C22×C3—24no—
542C22×C3—416no—
543C22×C3—42no—
544C22×C3—42no—
545C22×C3—42no—
546C22×C3—42no—
547C22×C3—102no—
548C22×C3—104no—
549C22×C3—201no—
550C22×C3—201no—
551C22×C3—202no—
552C22×C3#402714yes—
553C22×C3C22×C4×D511yescharacteristic
554C22×C3C2×D4×D511yescharacteristic
555C22×C3C2×#71111yescharacteristic
556C22×C3#414412yes—
557C22×C3#414412yes—
558C3×C4—102no—
559C3×C4—104no—
560Dic3—41no—
561Dic3—41no—
562Dic3—42no—
563Dic3—102no—
564Dic3—102no—
565Dic3—202no—
566C2×C5—24no—
567C2×C5—44no—
568C2×C5#558911yescharacteristic
569C2×C5#559211yescharacteristic
570C2×C5#559312yes—
571C2×C5C2×D4×Dic311yescharacteristic
572C2×C5#579812yes—
573D5—42no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—28no—
583C23—48no—
584C23—44no—
585C23—48no—
586C23—44no—
587C23—52no—
588C23—101no—
589C23—101no—
590C23—102no—
591C23—102no—
592C23C2×D5×Dic314yes—
593C23C2×D5×Dic312yes—
594C23#678012yes—
595C23#679012yes—
596C23C22×D3×D511yescharacteristic
597D4—108no—
598D4—208no—
599C2×C4—52no—
600C2×C4—54no—
601C2×C4—62no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—62no—
605C2×C4—62no—
606C2×C4—62no—
607C2×C4—62no—
608C2×C4—101no—
609C2×C4—101no—
610C2×C4—102no—
611C2×C4—102no—
612C2×C4—152no—
613C2×C4—152no—
614C2×C4—302no—
615C2×C4—302no—
616C2×C4—302no—
617C2×C4—302no—
618C2×C4—302no—
619C2×C4—604no—
620C2×C4—604no—
621C2×C4—604no—
622C2×C3—24no—
623C2×C3—44no—
624C2×C3—202no—
625C2×C3C2×#402711yescharacteristic
626C2×C3#6582012yes—
627C2×C3#6592311yescharacteristic
628C2×C3#6592611yescharacteristic
629C2×C3#6592712yes—
630C5#8257811yescharacteristic, complemented, Sylow
631C22—22no—
632C22—24no—
633C22—24no—
634C22—22no—
635C22—24no—
636C22—24no—
637C22—24no—
638C22—416no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—102no—
644C22—104no—
645C22—201no—
646C22—202no—
647C22—201no—
648C22#9155914yes—
649C22C22×D5×Dic311yescharacteristic
650C22C2×#678011yescharacteristic
651C22C2×#679011yescharacteristic
652C22#9204912yes—
653C22#9204912yes—
654C4—102no—
655C4—104no—
656C4—121no—
657C4—122no—
658C4—121no—
659C4—302no—
660C4—302no—
661C4—602no—
662C3#1060555211yescharacteristic, complemented, Sylow
663C2—24no—
664C2—44no—
665C2—202no—
666C2C2×#9155911yescharacteristic
667C2#1175540011yescharacteristic
668C2#1175540612yes—
669C2#1175540812yes—
670C2#1175544211yescharacteristic
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC25×C3×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC2×C3×#4099C2
hypercenterC25D3×D5
hypocenterC3×C5#2321
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C25

Complements

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

NH ≅ G/Ntype
1. C22×#6719C2semidirect
3. C2×#91559C2semidirect
4. C5×#5247C2semidirect
5. #11749518C2semidirect
6. #11749515C2semidirect
7. #11749500C2semidirect
8. #11749489C2semidirect
9. #11749487C2semidirect
10. #11749486C2semidirect
11. #11749485C2semidirect
18. C22×D5×Dic3C22semidirect
26. C3×#4099C4semidirect
30. C2×#6668C22semidirect
31. C2×#6668C22semidirect
32. C2×#6665C22semidirect
33. C2×#6665C22semidirect
34. C2×Dic3×Dic5C22semidirect
35. C2×Dic3×Dic5C22semidirect
36. #90994C22semidirect
37. #90994C22semidirect
38. #90994C22semidirect
55. C24×C3×C5C2×C4semidirect
62. C22×Dic15D4semidirect
74. C2×C3×C22⋊D5C2×C4semidirect
85. C3×C22⋊Dic5C2×C4semidirect
86. C3×C22⋊Dic5C2×C4semidirect
96. #5247D5semidirect
114. #4099Dic3semidirect
209. C24×C5C2×Dic3semidirect
221. C2×C22⋊D5C2×Dic3semidirect
239. C22⋊Dic5C2×Dic3semidirect
240. C22⋊Dic5C2×Dic3semidirect
294. C24×C3C4×D5semidirect
497. C24D5×Dic3semidirect
534. C3×C5#2321semidirect
630. C5#82578semidirect
662. C3#10605552semidirect

Maximal subgroups

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

subgroupindex
C22×#67192
C2×C3×#40992
C2×#915592
C5×#52472
#117495182
#117495152
#117495002
#117494892
#117494872
#117494862
#117494852
#106055523
#825785