Finite Group Databasebeta

← #14294708

Subgroups — #14294708

Order 1920.

1360 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×C22⋊AGL(1,𝔽5)C212yesmaximal
2C2×#91473C211yesmaximal, characteristic
3#11746788C211yesmaximal, characteristic, complemented
4C3×#64917C211yesmaximal, characteristic, complemented
5#11746573C212yesmaximal, complemented
6#10604214—31nomaximal, Hall
7C23×D3×D5C2211yescharacteristic
8C22×D3×AGL(1,𝔽5)—22no—
9C22×C4×D15C411yescharacteristic, complemented
10C22×#1200C411yescharacteristic, complemented
11C2×#6763C2212yes—
12C2×C3×C22⋊AGL(1,𝔽5)C2212yes—
13D3×C22⋊AGL(1,𝔽5)—24no—
14D3×C22⋊AGL(1,𝔽5)C418yescomplemented
15C2×(C2×D3)⋊AGL(1,𝔽5)—22no—
16C2×C5×(C2×D3)⋊C4C411yescharacteristic, complemented
17C2×C3×#685C2211yescharacteristic
18C2×(C2×D3)⋊Dic5C411yescharacteristic, complemented
19C2×#6665C2211yescharacteristic
20#91473—22no—
21#91473—22no—
22D3×#342—51nomaximal, Hall
23C22×C22⋊AGL(1,𝔽5)—32no—
24C2×D5×C22⋊C4—31no—
25#64917—31no—
26#64917—32no—
27#64917D311yescharacteristic, complemented
28C23×D15C2×C411yescharacteristic
29C23×C5×D3C2×C411yescharacteristic
30C23×C3×D5C2311yescharacteristic
31C22×D3×D5—42no—
32C22×D3×D5—42no—
33C22×D3×D5C2×C414yes—
34C22×D3×D5D412yes—
35C22×D3×D5D412yes—
36C22×D3×D5D412yes—
37C22×D3×D5Q812yes—
38C22×ΓL(1,𝔽16)—22no—
39C22×C3×AGL(1,𝔽5)—22no—
40C2×D3×AGL(1,𝔽5)—28no—
41C2×D3×AGL(1,𝔽5)—44no—
42C2×D3×AGL(1,𝔽5)—44no—
43C22×C3×C4×C5C2×C411yescharacteristic, complemented
44C22×Dic15C2×C411yescharacteristic, complemented
45C2×C4×D15—21no—
46C2×C4×D15—21no—
47C2×C4×D15—42no—
48C22×C5×Dic3C2×C411yescharacteristic, complemented
49C22×C3×Dic5C2×C411yescharacteristic, complemented
50C2×#1200—21no—
51C2×#1200—21no—
52C2×#1200—42no—
53#6763—22no—
54#6763C2×C414yescomplemented
55C3×C22⋊AGL(1,𝔽5)—22no—
56C3×C22⋊AGL(1,𝔽5)C2×C414yescomplemented
57(C2×D3)⋊AGL(1,𝔽5)—28no—
58C5×(C2×D3)⋊C4—22no—
59C3×#685—22no—
60(C2×D3)⋊Dic5—22no—
61#6665—22no—
62C2×D3×C22⋊C4—52no—
63C2×D3×C22⋊C4—51no—
64C3×#342—51no—
65#4527—52no—
66#4526—51no—
67C24×D5—31no—
68C23×AGL(1,𝔽5)—62no—
69C22×C4×D5—31no—
70C22×C4×D5—31no—
71C2×C22⋊AGL(1,𝔽5)—32no—
72C2×C22⋊AGL(1,𝔽5)—38no—
73C2×C22⋊AGL(1,𝔽5)—64no—
74C2×C22⋊AGL(1,𝔽5)—62no—
75C2×C22⋊AGL(1,𝔽5)C2×D312yes—
76C2×C5×C22⋊C4—31no—
77C2×C22⋊Dic5—31no—
78C2×#685—31no—
79C2×#685C2×D311yescharacteristic
80D5×C22⋊C4—62no—
81D5×C22⋊C4—62no—
82C2×#342—151noSylow
83C23×C3×C5C22×C411yescharacteristic
84C22×D15—22no—
85C22×D15—22no—
86C22×D15C4212yes—
87C22×D15C22⋊C411yescharacteristic
88C22×D15C22⋊C411yescharacteristic
89C22×D15C4⋊C412yes—
90C22×C5×D3—41no—
91C22×C5×D3—41no—
92C22×C5×D3C4212yes—
93C22×C5×D3C22⋊C412yes—
94C22×C5×D3C4⋊C412yes—
95C22×C3×D5—41no—
96C22×C3×D5—41no—
97C22×C3×D5C22×C412yes—
98C22×C3×D5C2×D412yes—
99C22×C3×D5C2×D411yescharacteristic
100C22×C3×D5C2×Q811yescharacteristic
101C2×D3×D5—24no—
102C2×D3×D5—24no—
103C2×D3×D5—48no—
104C2×D3×D5—44no—
105C2×D3×D5—44no—
106C2×D3×D5—42no—
107C2×D3×D5—42no—
108C2×D3×D5C22⋊C414yes—
109C2×D3×D5C22⋊C414yes—
110C2×D3×D5C4⋊C414yes—
111C2×D3×D5C4⋊C414yes—
112C2×ΓL(1,𝔽16)—24no—
113C2×ΓL(1,𝔽16)—42no—
114C2×ΓL(1,𝔽16)—42no—
115C2×C3×AGL(1,𝔽5)—24no—
116C2×C3×AGL(1,𝔽5)—42no—
117C2×C3×AGL(1,𝔽5)—42no—
118D3×AGL(1,𝔽5)—416no—
119C2×C3×C4×C5—21no—
120C2×C3×C4×C5—41no—
121C2×Dic15—21no—
122C2×Dic15—41no—
123C4×D15—42no—
124C4×D15—42no—
125C2×C5×Dic3—21no—
126C2×C5×Dic3—41no—
127C2×C3×Dic5—21no—
128C2×C3×Dic5—41no—
129#1200—42no—
130#1200—42no—
131C24×D3—51no—
132C22×C4×D3—51no—
133C22×C4×D3—51no—
134C22×C4×D3—102no—
135C2×C3×C22⋊C4—52no—
136C2×C3×C22⋊C4—51no—
137C2×#217—52no—
138C2×#217—51no—
139C2×(C2×D3)⋊C4—51no—
140C2×(C2×D3)⋊C4—102no—
141C2×(C2×D3)⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
142D3×C22⋊C4—58no—
143D3×C22⋊C4—102no—
144D3×C22⋊C4—102no—
145D3×C22⋊C4—104no—
146C24×C5—31no—
147C23×D5—31no—
148C23×D5—32no—
149C23×D5—34no—
150C23×D5—32no—
151C23×D5—32no—
152C23×D5—32no—
153C23×D5—122no—
154C23×D5—122no—
155C23×D5C22×D311yescharacteristic
156C22×AGL(1,𝔽5)—22no—
157C22×AGL(1,𝔽5)—68no—
158C22×AGL(1,𝔽5)—62no—
159C22×AGL(1,𝔽5)—124no—
160C22×AGL(1,𝔽5)—124no—
161C22×C4×C5—31no—
162C22×C4×C5C4×D311yescharacteristic, complemented
163C22×Dic5—31no—
164C22×Dic5C4×D311yescharacteristic, complemented
165C2×C4×D5—61no—
166C2×C4×D5—61no—
167C2×C4×D5—61no—
168C2×C4×D5—61no—
169C2×C4×D5—122no—
170C2×C4×D5—122no—
171C22⋊AGL(1,𝔽5)—22no—
172C22⋊AGL(1,𝔽5)—34no—
173C22⋊AGL(1,𝔽5)—68no—
174C22⋊AGL(1,𝔽5)—62no—
175C22⋊AGL(1,𝔽5)C4×D314yescomplemented
176C5×C22⋊C4—62no—
177C22⋊Dic5—62no—
178#685—22no—
179#685—62no—
180C22×C22⋊C4—151no—
181C22×C22⋊C4—152no—
182#342—51no—
183#342—151no—
184#342—152no—
185C22×C3×C5—21no—
186C22×C3×C5—21no—
187C22×C3×C5C2×C4211yescharacteristic
188C22×C3×C5C2×C22⋊C411yescharacteristic
189C22×C3×C5C2×C4⋊C411yescharacteristic
190C2×D15—21no—
191C2×D15—21no—
192C2×D15—24no—
193C2×D15—24no—
194C2×D15#9512yes—
195C2×D15#9512yes—
196C2×D15#9912yes—
197C2×D15#9912yes—
198C2×C5×D3—22no—
199C2×C5×D3—22no—
200C2×C5×D3—42no—
201C2×C5×D3—42no—
202C2×C5×D3#9514yes—
203C2×C3×D5—22no—
204C2×C3×D5—22no—
205C2×C3×D5—42no—
206C2×C3×D5—41no—
207C2×C3×D5—41no—
208C2×C3×D5C2×C22⋊C412yes—
209C2×C3×D5C2×C4⋊C412yes—
210D3×D5—28no—
211D3×D5—28no—
212D3×D5—44no—
213D3×D5—44no—
214ΓL(1,𝔽16)—44no—
215C3×AGL(1,𝔽5)—44no—
216C3×C4×C5—41no—
217Dic15—41no—
218C3×Dic5—41no—
219C5×Dic3—41no—
220C24×C3—51no—
221C23×D3—51no—
222C23×D3—52no—
223C23×D3—54no—
224C23×D3—52no—
225C23×D3—52no—
226C23×D3—52no—
227C23×D3—202no—
228C23×D3—202no—
229C23×D3C2×AGL(1,𝔽5)11yescharacteristic
230C22×C3×C4—51no—
231C22×C3×C4—102no—
232C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic, complemented
233C22×Dic3—51no—
234C22×Dic3—102no—
235C22×Dic3C2×AGL(1,𝔽5)11yescharacteristic, complemented
236C2×C4×D3—101no—
237C2×C4×D3—101no—
238C2×C4×D3—101no—
239C2×C4×D3—101no—
240C2×C4×D3—108no—
241C2×C4×D3—204no—
242C2×C4×D3—204no—
243C2×C4×D3—202no—
244C2×C4×D3—202no—
245C3×C22⋊C4—54no—
246C3×C22⋊C4—102no—
247C3×C22⋊C4—102no—
248#217—54no—
249#217—102no—
250#217—102no—
251(C2×D3)⋊C4—22no—
252(C2×D3)⋊C4—108no—
253(C2×D3)⋊C4—102no—
254C23×C5—32no—
255C23×C5—32no—
256C23×C5—32no—
257C23×C5—121no—
258C23×C5—121no—
259C23×C5C2×C4×D311yescharacteristic
260C22×D5—31no—
261C22×D5—32no—
262C22×D5—31no—
263C22×D5—32no—
264C22×D5—34no—
265C22×D5—34no—
266C22×D5—34no—
267C22×D5—34no—
268C22×D5—41no—
269C22×D5—41no—
270C22×D5—64no—
271C22×D5—64no—
272C22×D5—62no—
273C22×D5—62no—
274C22×D5—122no—
275C22×D5—122no—
276C22×D5—128no—
277C22×D5—124no—
278C22×D5—124no—
279C22×D5C2×C4×D312yes—
280C22×D5D3×D412yes—
281C22×D5D3×D411yescharacteristic
282C22×D5D3×Q811yescharacteristic
283C2×AGL(1,𝔽5)—24no—
284C2×AGL(1,𝔽5)—42no—
285C2×AGL(1,𝔽5)—42no—
286C2×AGL(1,𝔽5)—64no—
287C2×AGL(1,𝔽5)—1216no—
288C2×AGL(1,𝔽5)—122no—
289C2×AGL(1,𝔽5)—122no—
290C2×C4×C5—21no—
291C2×C4×C5—41no—
292C2×C4×C5—61no—
293C2×C4×C5—121no—
294C2×Dic5—21no—
295C2×Dic5—41no—
296C2×Dic5—61no—
297C2×Dic5—121no—
298C4×D5—122no—
299C4×D5—122no—
300C4×D5—122no—
301C4×D5—122no—
302C25—151no—
303C23×C4—151no—
304C23×C4—151no—
305C23×C4—302no—
306C2×C22⋊C4—31no—
307C2×C22⋊C4—52no—
308C2×C22⋊C4—51no—
309C2×C22⋊C4—158no—
310C2×C22⋊C4—152no—
311C2×C22⋊C4—151no—
312C2×C22⋊C4—151no—
313C2×C22⋊C4—302no—
314C2×C22⋊C4—302no—
315C2×C22⋊C4—304no—
316C2×C22⋊C4—302no—
317C2×C3×C5—21no—
318C2×C3×C5—21no—
319C2×C3×C5C2×#9511yescharacteristic
320C2×C3×C5C2×#9911yescharacteristic
321C2×C3×C5C2×#9911yescharacteristic
322D15—22no—
323D15#34214yescomplemented
324C3×D5—22no—
325C3×D5—22no—
326C5×D3—24no—
327C23×C3—52no—
328C23×C3—52no—
329C23×C3—51no—
330C23×C3—51no—
331C23×C3—201no—
332C23×C3—201no—
333C23×C3C22×AGL(1,𝔽5)11yescharacteristic
334C22×D3—41no—
335C22×D3—41no—
336C22×D3—51no—
337C22×D3—52no—
338C22×D3—52no—
339C22×D3—51no—
340C22×D3—54no—
341C22×D3—54no—
342C22×D3—54no—
343C22×D3—54no—
344C22×D3—104no—
345C22×D3—104no—
346C22×D3—102no—
347C22×D3—102no—
348C22×D3—202no—
349C22×D3—202no—
350C22×D3—208no—
351C22×D3—204no—
352C22×D3—204no—
353C22×D3C4×AGL(1,𝔽5)12yes—
354C22×D3C4⋊AGL(1,𝔽5)12yes—
355C22×D3C22⋊AGL(1,𝔽5)12yes—
356C2×C3×C4—21no—
357C2×C3×C4—41no—
358C2×C3×C4—104no—
359C2×C3×C4—101no—
360C2×C3×C4—202no—
361C2×C3×C4—202no—
362C2×C3×C4—201no—
363C2×Dic3—21no—
364C2×Dic3—41no—
365C2×Dic3—104no—
366C2×Dic3—101no—
367C2×Dic3—202no—
368C2×Dic3—202no—
369C2×Dic3—201no—
370C4×D3—202no—
371C4×D3—202no—
372C4×D3—202no—
373C4×D3—202no—
374C4×D3—2016no—
375C22×C5—21no—
376C22×C5—21no—
377C22×C5—34no—
378C22×C5—62no—
379C22×C5—62no—
380C22×C5—122no—
381C22×C5—122no—
382C22×C5C42×D311yescharacteristic
383C22×C5D3×C22⋊C411yescharacteristic
384C22×C5D3×C4⋊C411yescharacteristic
385C2×D5—22no—
386C2×D5—22no—
387C2×D5—32no—
388C2×D5—32no—
389C2×D5—32no—
390C2×D5—32no—
391C2×D5—42no—
392C2×D5—41no—
393C2×D5—41no—
394C2×D5—68no—
395C2×D5—68no—
396C2×D5—64no—
397C2×D5—64no—
398C2×D5—61no—
399C2×D5—61no—
400C2×D5—124no—
401C2×D5—124no—
402C2×D5D3×C22⋊C412yes—
403C2×D5D3×C4⋊C412yes—
404AGL(1,𝔽5)—44no—
405AGL(1,𝔽5)—124no—
406C4×C5—41no—
407C4×C5—121no—
408Dic5—41no—
409Dic5—121no—
410C24—31no—
411C24—51no—
412C24—152no—
413C24—154no—
414C24—152no—
415C24—152no—
416C24—152no—
417C24—151no—
418C24—602no—
419C24—602no—
420C22×C4—31no—
421C22×C4—51no—
422C22×C4—102no—
423C22×C4—151no—
424C22×C4—301no—
425C22×C4—301no—
426C22×C4—301no—
427C22×C4—301no—
428C22×C4—308no—
429C22×C4—302no—
430C22×C4—602no—
431C22×C4—602no—
432C22×C4—604no—
433C22×C4—604no—
434C22×C4D3×AGL(1,𝔽5)11yescharacteristic, complemented
435C22⋊C4—54no—
436C22⋊C4—62no—
437C22⋊C4—102no—
438C22⋊C4—102no—
439C22⋊C4—154no—
440C22⋊C4—302no—
441C22⋊C4—302no—
442C22⋊C4—308no—
443C22⋊C4—302no—
444C3×C5C2×#34211yescharacteristic, complemented, Hall
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—52no—
448C22×C3—52no—
449C22×C3—102no—
450C22×C3—102no—
451C22×C3—202no—
452C22×C3—201no—
453C22×C3—201no—
454C22×C3C2×C4×AGL(1,𝔽5)11yescharacteristic
455C22×C3C2×C4⋊AGL(1,𝔽5)11yescharacteristic
456C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
457C2×D3—22no—
458C2×D3—22no—
459C2×D3—42no—
460C2×D3—42no—
461C2×D3—52no—
462C2×D3—52no—
463C2×D3—52no—
464C2×D3—52no—
465C2×D3—108no—
466C2×D3—108no—
467C2×D3—104no—
468C2×D3—104no—
469C2×D3—101no—
470C2×D3—101no—
471C2×D3—204no—
472C2×D3—204no—
473C2×D3(C2×AGL(1,𝔽5))⋊C414yes—
474C3×C4—41no—
475C3×C4—204no—
476C3×C4—201no—
477Dic3—41no—
478Dic3—204no—
479Dic3—201no—
480C2×C5—21no—
481C2×C5—21no—
482C2×C5—64no—
483C2×C5D3×#9511yescharacteristic
484C2×C5D3×#9911yescharacteristic
485C2×C5D3×#9911yescharacteristic
486D5—22no—
487D5—22no—
488D5—34no—
489D5—62no—
490C23—32no—
491C23—32no—
492C23—32no—
493C23—52no—
494C23—52no—
495C23—51no—
496C23—51no—
497C23—121no—
498C23—121no—
499C23—154no—
500C23—154no—
501C23—154no—
502C23—154no—
503C23—151no—
504C23—152no—
505C23—152no—
506C23—151no—
507C23—201no—
508C23—201no—
509C23—302no—
510C23—302no—
511C23—304no—
512C23—304no—
513C23—602no—
514C23—602no—
515C23—604no—
516C23—608no—
517C23—604no—
518C23C2×D3×AGL(1,𝔽5)11yescharacteristic
519C2×C4—21no—
520C2×C4—41no—
521C2×C4—61no—
522C2×C4—104no—
523C2×C4—101no—
524C2×C4—121no—
525C2×C4—202no—
526C2×C4—202no—
527C2×C4—201no—
528C2×C4—304no—
529C2×C4—301no—
530C2×C4—602no—
531C2×C4—602no—
532C2×C4—602no—
533C2×C4—602no—
534C2×C4—6016no—
535C2×C4—602no—
536C2×C4—602no—
537C2×C4—601no—
538C2×C3—21no—
539C2×C3—21no—
540C2×C3—102no—
541C2×C3—102no—
542C2×C3C2×(C2×AGL(1,𝔽5))⋊C411yescharacteristic
543C2×C3D5⋊#9911yescharacteristic
544C2×C3D5⋊#9911yescharacteristic
545D3—24no—
546D3—54no—
547D3—102no—
548C5D3×#34211yescharacteristic, complemented, Sylow
549C22—21no—
550C22—21no—
551C22—34no—
552C22—52no—
553C22—52no—
554C22—62no—
555C22—62no—
556C22—102no—
557C22—102no—
558C22—122no—
559C22—122no—
560C22—152no—
561C22—152no—
562C22—152no—
563C22—152no—
564C22—202no—
565C22—201no—
566C22—201no—
567C22—304no—
568C22—304no—
569C22—308no—
570C22—308no—
571C22—301no—
572C22—301no—
573C22—604no—
574C22—604no—
575C22C4×D3×AGL(1,𝔽5)11yescharacteristic
576C22D3×C4⋊AGL(1,𝔽5)11yescharacteristic
577C22D3×C22⋊AGL(1,𝔽5)11yescharacteristic
578C4—41no—
579C4—121no—
580C4—204no—
581C4—201no—
582C4—604no—
583C4—601no—
584C3#1060421411yescharacteristic, complemented, Sylow
585C2—21no—
586C2—21no—
587C2—64no—
588C2—102no—
589C2—102no—
590C2—154no—
591C2—302no—
592C2#1175185211yescharacteristic
593C2D3×(C2×AGL(1,𝔽5))⋊C411yescharacteristic
594C2#1175202411yescharacteristic
subgroupG/H
centerC22D3×C22⋊AGL(1,𝔽5)
commutatorC22×C3×C5C2×C42
FrattiniC23C2×D3×AGL(1,𝔽5)
FittingC22×C3×C4×C5C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC23×D3×D5C22
cosocleC23×C3×D5C23
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5C2×#342
LCSG ▷ C22×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. #11746788C2semidirect
4. C3×#64917C2semidirect
5. #11746573C2semidirect
9. C22×C4×D15C4semidirect
10. C22×#1200C4semidirect
14. D3×C22⋊AGL(1,𝔽5)C4semidirect
16. C2×C5×(C2×D3)⋊C4C4semidirect
18. C2×(C2×D3)⋊Dic5C4semidirect
27. #64917D3semidirect
43. C22×C3×C4×C5C2×C4semidirect
44. C22×Dic15C2×C4semidirect
48. C22×C5×Dic3C2×C4semidirect
49. C22×C3×Dic5C2×C4semidirect
54. #6763C2×C4semidirect
56. C3×C22⋊AGL(1,𝔽5)C2×C4semidirect
141. C2×(C2×D3)⋊C4AGL(1,𝔽5)semidirect
162. C22×C4×C5C4×D3semidirect
164. C22×Dic5C4×D3semidirect
175. C22⋊AGL(1,𝔽5)C4×D3semidirect
232. C22×C3×C4C2×AGL(1,𝔽5)semidirect
235. C22×Dic3C2×AGL(1,𝔽5)semidirect
323. D15#342semidirect
434. C22×C4D3×AGL(1,𝔽5)semidirect
444. C3×C5C2×#342semidirect
548. C5D3×#342semidirect
584. C3#10604214semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C22⋊AGL(1,𝔽5)2
C2×#914732
#117467882
C3×#649172
#117465732
#106042143
D3×#3425