Finite Group Databasebeta

← #14297544

Subgroups — #14297544

Order 1920.

760 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91440C211yesmaximal, characteristic
2#11752424C211yesmaximal, characteristic, complemented
3C3×#65765C211yesmaximal, characteristic, complemented
4C2×#91172C212yesmaximal
5#11746573C212yesmaximal, complemented
6#10604217—31nomaximal, Hall
7C22×C3×C4×D5C2211yescharacteristic
8C22×#1201C2211yescharacteristic
9C22×D3×Dic5C411yescharacteristic, complemented
10C2×#6763C2212yes—
11C2×C4⋊ΓL(1,𝔽16)—21no—
12C2×C3×C22⋊AGL(1,𝔽5)C2212yes—
13C2×C3×C4⋊AGL(1,𝔽5)—21no—
14C2×#6712C411yescharacteristic, complemented
15C2×C5×(C2×D3)⋊C4C411yescharacteristic, complemented
16C2×#6665C2211yescharacteristic
17C2×Dic3×Dic5C411yescharacteristic, complemented
18#91440—21no—
19#91440—21no—
20#91440—21no—
21#91440—21no—
22#91235—21no—
23C3×(C2×AGL(1,𝔽5))⋊C4—21no—
24#91172—24no—
25#69530—51nomaximal, Hall
26C2×#4027—31no—
27C2×#4011—32no—
28#65765—31no—
29#65765D311yescharacteristic, complemented
30#64917—32no—
31C23×C5×D3C2×C411yescharacteristic
32C23×C3×D5C2311yescharacteristic
33C22×ΓL(1,𝔽16)—22no—
34C22×C3×AGL(1,𝔽5)—22no—
35C22×C3×C4×C5C2×C411yescharacteristic
36C22×Dic15C2×C411yescharacteristic
37C22×Dic15C2×C411yescharacteristic
38C22×C5×Dic3C2×C411yescharacteristic
39C22×C3×Dic5C2×C411yescharacteristic
40C2×C3×C4×D5—42no—
41C2×C3×C4×D5D411yescharacteristic
42C2×C3×C4×D5D411yescharacteristic
43C2×C3×C4×D5D411yescharacteristic
44C2×C3×C4×D5D411yescharacteristic
45C2×#1201—21no—
46C2×#1201—21no—
47C2×#1201—21no—
48C2×#1201—21no—
49C2×#1201D412yes—
50C2×#1201Q812yes—
51C2×D3×Dic5—21no—
52C2×D3×Dic5—21no—
53C2×D3×Dic5—42no—
54#6763—22no—
55#6763—22no—
56C4⋊ΓL(1,𝔽16)—42no—
57C3×C22⋊AGL(1,𝔽5)—22no—
58C3×C22⋊AGL(1,𝔽5)—22no—
59C3×C4⋊AGL(1,𝔽5)—42no—
60#6712—21no—
61#6712—21no—
62C5×(C2×D3)⋊C4—21no—
63C5×(C2×D3)⋊C4—21no—
64#6665—21no—
65#6665—21no—
66Dic3×Dic5—21no—
67Dic3×Dic5—21no—
68C2×C4×#67—51no—
69C3×#380—51no—
70#5212—51no—
71C2×#806—52no—
72#4527—52no—
73C22×C22⋊D5—31no—
74C23×Dic5—31no—
75C22×C4×D5C2×D311yescharacteristic
76C2×C22⋊AGL(1,𝔽5)—32no—
77C2×C22⋊AGL(1,𝔽5)C2×D312yes—
78C2×C4⋊AGL(1,𝔽5)—21no—
79C2×C4⋊AGL(1,𝔽5)—61no—
80C2×C5×C22⋊C4—31no—
81C2×#685—31no—
82C2×#683—31no—
83C2×C4×Dic5—31no—
84#4027—61no—
85#4027—61no—
86#4027—61no—
87#4027—61no—
88#4011—64no—
89(C2×AGL(1,𝔽5))⋊C4—21no—
90(C2×AGL(1,𝔽5))⋊C4—61no—
91#1799—151noSylow
92C23×C3×C5C22×C411yescharacteristic
93C22×C5×D3—21no—
94C22×C5×D3—21no—
95C22×C5×D3—41no—
96C22×C5×D3—41no—
97C22×C5×D3C4⋊C412yescomplemented
98C22×C3×D5—41no—
99C22×C3×D5—41no—
100C22×C3×D5C2×D411yescharacteristic
101C22×C3×D5C2×D411yescharacteristic
102C22×C3×D5C2×D411yescharacteristic
103C22×C3×D5C2×Q811yescharacteristic
104C22×C3×D5#4111yescharacteristic
105C22×C3×D5#4111yescharacteristic
106C2×ΓL(1,𝔽16)—42no—
107C2×ΓL(1,𝔽16)—42no—
108C2×ΓL(1,𝔽16)—42no—
109C2×C3×AGL(1,𝔽5)—42no—
110C2×C3×AGL(1,𝔽5)—42no—
111C2×C3×AGL(1,𝔽5)—42no—
112C2×C3×C4×C5—41no—
113C2×C3×C4×C5C22⋊C411yescharacteristic
114C2×C3×C4×C5C22⋊C411yescharacteristic
115C2×Dic15—21no—
116C2×Dic15—21no—
117C2×Dic15—21no—
118C2×Dic15—41no—
119C2×Dic15C4⋊C412yescomplemented
120C2×C5×Dic3—21no—
121C2×C5×Dic3—41no—
122C2×C3×Dic5—22no—
123C2×C3×Dic5C22⋊C411yescharacteristic
124C2×C3×Dic5C22⋊C411yescharacteristic
125C3×C4×D5—22no—
126C3×C4×D5—22no—
127C3×C4×D5—41no—
128C3×C4×D5—41no—
129#1201—22no—
130#1201—22no—
131#1201—22no—
132#1201—22no—
133D3×Dic5—42no—
134D3×Dic5—42no—
135C23×C3×C4—51no—
136C22×#67—51no—
137C22×C4×D3—51no—
138C2×C3×C4⋊C4—101no—
139C2×C3×C22⋊C4—52no—
140C2×#217—51no—
141C2×#217—52no—
142C4×#67—101no—
143C4×#67—101no—
144C4×#67—101no—
145C4×#67—101no—
146C2×(C2×D3)⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
147C2×C4⋊Dic3—101no—
148C2×Dic3⋊C4—51no—
149C2×C4×Dic3—51no—
150C3×#95—101no—
151C3⋊#95—101no—
152#806—104no—
153C24×C5—31no—
154C23×D5C22×D311yescharacteristic
155C22×AGL(1,𝔽5)—22no—
156C22×AGL(1,𝔽5)—62no—
157C22×C4×C5—31no—
158C22×C4×C5C4×D311yescharacteristic
159C2×C22⋊D5—32no—
160C2×C22⋊D5—32no—
161C2×C22⋊D5—61no—
162C2×C22⋊D5—61no—
163C2×C22⋊D5—61no—
164C2×C22⋊D5—61no—
165C22×Dic5—31no—
166C22×Dic5—31no—
167C22×Dic5—61no—
168C22×Dic5—61no—
169C22×Dic5—122no—
170C22×Dic5C4×D311yescharacteristic
171C2×C4×D5—42no—
172C2×C4×D5D1211yescharacteristic
173C2×C4×D5D1211yescharacteristic
174C2×C4×D5#6711yescharacteristic
175C2×C4×D5#6711yescharacteristic
176C22⋊AGL(1,𝔽5)—22no—
177C22⋊AGL(1,𝔽5)—22no—
178C22⋊AGL(1,𝔽5)—62no—
179C22⋊AGL(1,𝔽5)—62no—
180C4⋊AGL(1,𝔽5)—42no—
181C4⋊AGL(1,𝔽5)—122no—
182C5×C22⋊C4—61no—
183C5×C22⋊C4—61no—
184#685—61no—
185#685—61no—
186#683—61no—
187#683—61no—
188C4×Dic5—61no—
189C4×Dic5—61no—
190C2×C4×D4—151no—
191C2×#99—152no—
192#380—51no—
193#380—151no—
194#342—152no—
195C22×C3×C5—21no—
196C22×C3×C5—21no—
197C22×C3×C5C2×C22⋊C411yescharacteristic
198C22×C3×C5C2×C4⋊C411yescharacteristic
199C22×C3×C5#11711yescharacteristic
200C2×C5×D3—22no—
201C2×C5×D3—42no—
202C2×C5×D3—41no—
203C2×C5×D3—41no—
204C2×C5×D3—42no—
205C2×C3×D5—21no—
206C2×C3×D5—21no—
207C2×C3×D5—21no—
208C2×C3×D5—21no—
209C2×C3×D5—41no—
210C2×C3×D5—41no—
211C2×C3×D5—41no—
212C2×C3×D5—41no—
213C2×C3×D5#12111yescharacteristic
214C2×C3×D5#12111yescharacteristic
215C2×C3×D5C22⋊Q811yescharacteristic
216C2×C3×D5C22⋊Q811yescharacteristic
217ΓL(1,𝔽16)—82no—
218C3×AGL(1,𝔽5)—82no—
219C3×C4×C5—21no—
220C3×C4×C5—21no—
221Dic15—22no—
222Dic15—41no—
223C3×Dic5—21no—
224C3×Dic5#9912yescomplemented
225C5×Dic3—41no—
226C24×C3—51no—
227C23×D3C2×AGL(1,𝔽5)11yescharacteristic
228C22×C3×C4—51no—
229C22×C3×C4—51no—
230C22×C3×C4—51no—
231C22×C3×C4—51no—
232C22×C3×C4—51no—
233C22×C3×C4—102no—
234C22×C3×C4—202no—
235C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
236C2×#67—52no—
237C2×#67—52no—
238C2×#67—101no—
239C2×#67—101no—
240C2×#67—101no—
241C2×#67—101no—
242C22×Dic3—51no—
243C22×Dic3—51no—
244C22×Dic3—102no—
245C22×Dic3C2×AGL(1,𝔽5)11yescharacteristic
246C2×C4×D3—101no—
247C2×C4×D3—101no—
248C2×C4×D3—202no—
249C3×C4⋊C4—202no—
250C3×C22⋊C4—102no—
251C3×C22⋊C4—102no—
252#217—101no—
253#217—101no—
254#217—102no—
255#217—102no—
256(C2×D3)⋊C4—21no—
257(C2×D3)⋊C4—21no—
258C4⋊Dic3—202no—
259Dic3⋊C4—101no—
260Dic3⋊C4—101no—
261C4×Dic3—101no—
262C4×Dic3—101no—
263C23×C5—32no—
264C23×C5—61no—
265C23×C5—61no—
266C23×C5—121no—
267C23×C5—121no—
268C23×C5C2×C4×D311yescharacteristic
269C22×D5—41no—
270C22×D5—41no—
271C22×D5C2×D1211yescharacteristic
272C22×D5D3×D411yescharacteristic
273C22×D5#23711yescharacteristic
274C22×D5D3×Q811yescharacteristic
275C22×D5#23911yescharacteristic
276C22×D5C2×#6711yescharacteristic
277C2×AGL(1,𝔽5)—42no—
278C2×AGL(1,𝔽5)—42no—
279C2×AGL(1,𝔽5)—42no—
280C2×AGL(1,𝔽5)—122no—
281C2×AGL(1,𝔽5)—122no—
282C2×AGL(1,𝔽5)—122no—
283C2×C4×C5—41no—
284C2×C4×C5—61no—
285C2×C4×C5—121no—
286C2×C4×C5(C2×D3)⋊C411yescharacteristic
287C2×C4×C5(C2×D3)⋊C411yescharacteristic
288C22⋊D5—62no—
289C22⋊D5—62no—
290C22⋊D5—62no—
291C22⋊D5—62no—
292C2×Dic5—22no—
293C2×Dic5—32no—
294C2×Dic5—61no—
295C2×Dic5—61no—
296C2×Dic5—61no—
297C2×Dic5—122no—
298C2×Dic5—122no—
299C2×Dic5—121no—
300C2×Dic5(C2×D3)⋊C411yescharacteristic
301C2×Dic5(C2×D3)⋊C411yescharacteristic
302C4×D5—22no—
303C4×D5—22no—
304C4×D5—41no—
305C4×D5—41no—
306C22×D4—151no—
307C23×C4—51no—
308C23×C4—151no—
309C4×D4—301no—
310C4×D4—301no—
311C4×D4—301no—
312C4×D4—301no—
313C2×C4⋊C4—101no—
314C2×C4⋊C4—151no—
315C2×C4⋊C4—301no—
316C2×C22⋊C4—31no—
317C2×C22⋊C4—52no—
318C2×C22⋊C4—151no—
319C2×C22⋊C4—152no—
320C2×C42—151no—
321#99—304no—
322#95—101no—
323#95—301no—
324C2×C3×C5—21no—
325C2×C3×C5—21no—
326C2×C3×C5#38011yescharacteristic
327C2×C3×C5C2×#9911yescharacteristic
328C2×C3×C5#41011yescharacteristic
329C3×D5—21no—
330C3×D5—21no—
331C3×D5—21no—
332C3×D5—21no—
333C5×D3—42no—
334C23×C3—51no—
335C23×C3—51no—
336C23×C3—51no—
337C23×C3—51no—
338C23×C3—51no—
339C23×C3—51no—
340C23×C3—201no—
341C23×C3—201no—
342C23×C3C22×AGL(1,𝔽5)11yescharacteristic
343C22×D3—21no—
344C22×D3—21no—
345C22×D3—41no—
346C22×D3—41no—
347C22×D3C4⋊AGL(1,𝔽5)12yescomplemented
348C2×C3×C4—41no—
349C2×C3×C4—51no—
350C2×C3×C4—51no—
351C2×C3×C4—102no—
352C2×C3×C4—102no—
353C2×C3×C4—102no—
354C2×C3×C4—201no—
355C2×C3×C4—201no—
356C2×C3×C4—202no—
357C2×C3×C4—202no—
358C2×C3×C4—202no—
359C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
360C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
361#67—102no—
362#67—102no—
363#67—102no—
364#67—102no—
365C2×Dic3—21no—
366C2×Dic3—41no—
367C2×Dic3—52no—
368C2×Dic3—101no—
369C2×Dic3—101no—
370C2×Dic3—101no—
371C2×Dic3—201no—
372C2×Dic3—202no—
373C2×Dic3—202no—
374C2×Dic3—202no—
375C4×D3—202no—
376C4×D3—202no—
377C22×C5—21no—
378C22×C5—21no—
379C22×C5—62no—
380C22×C5—122no—
381C22×C5—122no—
382C22×C5—121no—
383C22×C5—121no—
384C22×C5D3×C4⋊C411yescharacteristic
385C22×C5#89211yescharacteristic
386C22×C5C2×(C2×D3)⋊C411yescharacteristic
387C2×D5—21no—
388C2×D5—21no—
389C2×D5—21no—
390C2×D5—21no—
391C2×D5—41no—
392C2×D5—41no—
393C2×D5—41no—
394C2×D5—41no—
395C2×D5#89511yescharacteristic
396C2×D5#89711yescharacteristic
397C2×D5#93811yescharacteristic
398C2×D5#94611yescharacteristic
399AGL(1,𝔽5)—82no—
400AGL(1,𝔽5)—242no—
401C4×C5—21no—
402C4×C5—21no—
403C4×C5—121no—
404Dic5—21no—
405Dic5—62no—
406Dic5—121no—
407Dic5#80612yescomplemented
408C24—31no—
409C24—51no—
410C2×D4—152no—
411C2×D4—152no—
412C2×D4—301no—
413C2×D4—301no—
414C2×D4—301no—
415C2×D4—301no—
416C22×C4—31no—
417C22×C4—51no—
418C22×C4—51no—
419C22×C4—51no—
420C22×C4—51no—
421C22×C4—51no—
422C22×C4—102no—
423C22×C4—151no—
424C22×C4—151no—
425C22×C4—202no—
426C22×C4—301no—
427C22×C4—301no—
428C22×C4—302no—
429C22×C4—602no—
430C22×C4D3×AGL(1,𝔽5)11yescharacteristic
431C4⋊C4—202no—
432C4⋊C4—301no—
433C4⋊C4—301no—
434C4⋊C4—602no—
435C22⋊C4—61no—
436C22⋊C4—61no—
437C22⋊C4—102no—
438C22⋊C4—102no—
439C22⋊C4—301no—
440C22⋊C4—301no—
441C22⋊C4—302no—
442C22⋊C4—302no—
443C42—301no—
444C42—301no—
445C3×C5#179911yescharacteristic, complemented, Hall
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—51no—
449C22×C3—51no—
450C22×C3—51no—
451C22×C3—51no—
452C22×C3—101no—
453C22×C3—101no—
454C22×C3—101no—
455C22×C3—101no—
456C22×C3—201no—
457C22×C3—201no—
458C22×C3—201no—
459C22×C3—201no—
460C22×C3C2×C4⋊AGL(1,𝔽5)11yescharacteristic
461C22×C3#413011yescharacteristic
462C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
463C2×D3—22no—
464C2×D3—42no—
465C2×D3—41no—
466C2×D3—41no—
467C2×D3—42no—
468C3×C4—21no—
469C3×C4—21no—
470C3×C4—52no—
471C3×C4—101no—
472C3×C4—402no—
473Dic3—41no—
474Dic3—102no—
475Dic3—201no—
476Dic3—402no—
477C2×C5—21no—
478C2×C5—21no—
479C2×C5—122no—
480C2×C5C2×#80611yescharacteristic
481C2×C5#498711yescharacteristic
482C2×C5#500611yescharacteristic
483D5—21no—
484D5—21no—
485D5—21no—
486D5—21no—
487C23—32no—
488C23—51no—
489C23—51no—
490C23—51no—
491C23—51no—
492C23—51no—
493C23—51no—
494C23—61no—
495C23—61no—
496C23—121no—
497C23—121no—
498C23—201no—
499C23—201no—
500C23C2×D3×AGL(1,𝔽5)11yescharacteristic
501D4—302no—
502D4—302no—
503D4—302no—
504D4—302no—
505C2×C4—41no—
506C2×C4—51no—
507C2×C4—51no—
508C2×C4—61no—
509C2×C4—102no—
510C2×C4—102no—
511C2×C4—102no—
512C2×C4—121no—
513C2×C4—152no—
514C2×C4—201no—
515C2×C4—201no—
516C2×C4—202no—
517C2×C4—202no—
518C2×C4—202no—
519C2×C4—301no—
520C2×C4—301no—
521C2×C4—301no—
522C2×C4—602no—
523C2×C4—602no—
524C2×C4—601no—
525C2×C4—602no—
526C2×C4—602no—
527C2×C4—602no—
528C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
529C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—101no—
533C2×C3—101no—
534C2×C3—101no—
535C2×C3—101no—
536C2×C3#6576511yescharacteristic
537C2×C3D5⋊#9911yescharacteristic
538C2×C3#6577111yescharacteristic
539D3—42no—
540C5#6953011yescharacteristic, complemented, Sylow
541C22—21no—
542C22—21no—
543C22—51no—
544C22—51no—
545C22—51no—
546C22—51no—
547C22—62no—
548C22—101no—
549C22—101no—
550C22—101no—
551C22—101no—
552C22—122no—
553C22—122no—
554C22—121no—
555C22—121no—
556C22—201no—
557C22—201no—
558C22—201no—
559C22—201no—
560C22#9190711yescharacteristic
561C22D3×C4⋊AGL(1,𝔽5)11yescharacteristic
562C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
563C4—21no—
564C4—21no—
565C4—52no—
566C4—101no—
567C4—121no—
568C4—302no—
569C4—402no—
570C4—601no—
571C4—1202no—
572C3#1060421711yescharacteristic, complemented, Sylow
573C2—21no—
574C2—21no—
575C2—101no—
576C2—101no—
577C2—101no—
578C2—101no—
579C2—122no—
580C2#1175181611yescharacteristic
581C2#1175185511yescharacteristic
582C2D5⋊#80611yescharacteristic
subgroupG/H
centerC22C2×(C2×D3)⋊AGL(1,𝔽5)
commutatorC23×C3×C5C22×C4
FrattiniC23C2×D3×AGL(1,𝔽5)
FittingC22×C3×C4×C5C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×#1201C22
cosocleC23×C3×D5C23
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1799
LCSG ▷ C23×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. #11752424C2semidirect
3. C3×#65765C2semidirect
5. #11746573C2semidirect
9. C22×D3×Dic5C4semidirect
14. C2×#6712C4semidirect
15. C2×C5×(C2×D3)⋊C4C4semidirect
17. C2×Dic3×Dic5C4semidirect
29. #65765D3semidirect
97. C22×C5×D3C4⋊C4semidirect
119. C2×Dic15C4⋊C4semidirect
146. C2×(C2×D3)⋊C4AGL(1,𝔽5)semidirect
224. C3×Dic5#99semidirect
347. C22×D3C4⋊AGL(1,𝔽5)semidirect
407. Dic5#806semidirect
445. C3×C5#1799semidirect
540. C5#69530semidirect
572. C3#10604217semidirect

Maximal subgroups

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

subgroupindex
C2×#914402
#117524242
C3×#657652
C2×#911722
#117465732
#106042173
#695305