Finite Group Databasebeta

← #14299998

Subgroups — #14299998

Order 1920.

724 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×AGL(1,𝔽5)×D12C211yesmaximal, characteristic
2C3×#65712C211yesmaximal, characteristic, complemented
3C2×#91160C211yesmaximal, characteristic, complemented
4C2×#91151C211yesmaximal, characteristic
5D5×#809C211yesmaximal, characteristic
6#11746745C211yesmaximal, characteristic, complemented
7(C2×AGL(1,𝔽5))⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10603957—31nomaximal, Hall
9C22×D3×AGL(1,𝔽5)—21no—
10C2×D5×D12C2211yescharacteristic
11C2×C4⋊ΓL(1,𝔽16)C2211yescharacteristic
12C2×D5⋊(C3⋊C8)C2211yescharacteristic
13C2×C3×C4⋊AGL(1,𝔽5)—21no—
14C2×C3×C4⋊AGL(1,𝔽5)C2211yescharacteristic
15C2×C3×C4×AGL(1,𝔽5)C2211yescharacteristic
16C2×(C2×D3)⋊AGL(1,𝔽5)—21no—
17AGL(1,𝔽5)×D12—22no—
18AGL(1,𝔽5)×D12—22no—
19C3×D5×C4⋊C4C2211yescharacteristic
20C2×D5×C3⋊C8C2211yescharacteristic
21C3×(C2×AGL(1,𝔽5))⋊C4—21no—
22#91160—22no—
23#91151—22no—
24#91093C411yescharacteristic, complemented
25C5×#809C411yescharacteristic, complemented
26#90968C411yescharacteristic, complemented
27#90966C411yescharacteristic, complemented
28#69625—51nomaximal, Hall
29C2×D4×AGL(1,𝔽5)—31no—
30C2×#4007—31no—
31C2×#3994—31no—
32#65712D311yescharacteristic, complemented
33D5×#102—31no—
34(C2×AGL(1,𝔽5))⋊C8—31no—
35#64874—31no—
36C22×D3×D5—21no—
37C22×ΓL(1,𝔽16)—21no—
38C22×C3×AGL(1,𝔽5)—21no—
39C22×C3×AGL(1,𝔽5)D411yescharacteristic
40C22×C3×AGL(1,𝔽5)D411yescharacteristic
41C2×D3×AGL(1,𝔽5)—24no—
42C2×D3×AGL(1,𝔽5)—42no—
43C2×D3×AGL(1,𝔽5)—42no—
44C2×D60C2×C411yescharacteristic
45C2×C5×D12C2×C411yescharacteristic
46C2×C3×C4×D5—21no—
47C2×C3×C4×D5C2311yescharacteristic
48C2×#1203—21no—
49D5×D12—21no—
50D5×D12—21no—
51D5×D12D412yes—
52D5×D12Q812yes—
53C2×C3⋊#170—21no—
54C4⋊ΓL(1,𝔽16)—21no—
55C4⋊ΓL(1,𝔽16)D412yes—
56D5⋊(C3⋊C8)—21no—
57D5⋊(C3⋊C8)—21no—
58C3×C4⋊AGL(1,𝔽5)—21no—
59C3×C4⋊AGL(1,𝔽5)—21no—
60C3×C4⋊AGL(1,𝔽5)—42no—
61C3×C4×AGL(1,𝔽5)—21no—
62C3×C4×AGL(1,𝔽5)—21no—
63(C2×D3)⋊AGL(1,𝔽5)—24no—
64C3×C5×C4⋊C4C2×C411yescharacteristic, complemented
65C2×C3⋊#168C2×C411yescharacteristic, complemented
66C2×C5×C3⋊C8C2×C411yescharacteristic, complemented
67C3×C4⋊Dic5C2×C411yescharacteristic, complemented
68C3×#683—21no—
69D5×C3⋊C8—21no—
70D5×C3⋊C8—21no—
71C2×C4×D12—51no—
72C3×#389—51no—
73C2×#809—51no—
74C2×#809—51no—
75C2×#807—51no—
76#4529—51no—
77#4524—51no—
78C23×AGL(1,𝔽5)—61no—
79C2×D4×D5—31no—
80C2×C22⋊AGL(1,𝔽5)—61no—
81D4×AGL(1,𝔽5)—62no—
82D4×AGL(1,𝔽5)—62no—
83C2×C4⋊AGL(1,𝔽5)—21no—
84C2×C4⋊AGL(1,𝔽5)—31no—
85C2×C4⋊AGL(1,𝔽5)C2×D311yescharacteristic
86C2×C4×AGL(1,𝔽5)C2×D311yescharacteristic
87C2×D5⋊C8—31no—
88C2×C8×D5—31no—
89D5×C4⋊C4C2×D311yescharacteristic
90#4007—62no—
91(C2×AGL(1,𝔽5))⋊C4—21no—
92#3994—62no—
93C5×#102—31no—
94#3964—31no—
95#3953—31no—
96#3941—31no—
97#1862—151noSylow
98C22×D15—21no—
99C22×C5×D3—21no—
100C22×C3×D5C2×D411yescharacteristic
101C2×D3×D5—22no—
102C2×D3×D5—22no—
103C2×D3×D5—41no—
104C2×D3×D5—41no—
105C2×D3×D5—41no—
106C2×D3×D5—41no—
107C2×ΓL(1,𝔽16)—22no—
108C2×ΓL(1,𝔽16)—41no—
109C2×ΓL(1,𝔽16)—41no—
110C2×C3×AGL(1,𝔽5)—21no—
111C2×C3×AGL(1,𝔽5)—21no—
112C2×C3×AGL(1,𝔽5)—21no—
113C2×C3×AGL(1,𝔽5)—21no—
114C2×C3×AGL(1,𝔽5)—21no—
115C2×C3×AGL(1,𝔽5)—41no—
116C2×C3×AGL(1,𝔽5)—41no—
117C2×C3×AGL(1,𝔽5)—41no—
118C2×C3×AGL(1,𝔽5)D812yes—
119D3×AGL(1,𝔽5)—48no—
120C2×C3×C4×C5—21no—
121C2×C3×C4×C5C22×C411yescharacteristic
122D60—21no—
123D60C4⋊C412yescomplemented
124C5×D12—21no—
125C5×D12C4⋊C412yescomplemented
126C2×C3×Dic5—21no—
127C2×C3×Dic5#4111yescharacteristic
128C3×C4×D5—41no—
129C3×C4×D5—41no—
130C3×C4×D5C2×D411yescharacteristic
131C3×C4×D5C2×D411yescharacteristic
132C3×C4×D5C2×Q811yescharacteristic
133C3×C4×D5#4111yescharacteristic
134#1203—22no—
135#1203—22no—
136C3⋊#170—41no—
137C3⋊#168—21no—
138C5×C3⋊C8—21no—
139C22×D12—51no—
140C22×C4×D3—101no—
141C2×C3×C4⋊C4—51no—
142C2×C3×C4⋊C4—51no—
143C2×C3×C4⋊C4—101no—
144C2×C3×C42—51no—
145C2×(C2×D3)⋊C4—101no—
146C2×C4⋊Dic3—51no—
147C22×C3⋊C8—51no—
148C22×C3⋊C8—51no—
149C4×D12—102no—
150C4×D12—102no—
151C3×#95—101no—
152#809—51no—
153#809—51no—
154#809—51no—
155#809—102no—
156#809AGL(1,𝔽5)11yescharacteristic, complemented
157#807—102no—
158C23×D5—61no—
159C22×AGL(1,𝔽5)—21no—
160C22×AGL(1,𝔽5)—64no—
161C22×AGL(1,𝔽5)—61no—
162C22×AGL(1,𝔽5)—122no—
163C22×AGL(1,𝔽5)—122no—
164C22×AGL(1,𝔽5)#6711yescharacteristic
165C22×AGL(1,𝔽5)#6711yescharacteristic
166C2×C5×D4—31no—
167C2×C22⋊D5—61no—
168D4×D5—32no—
169D4×D5—32no—
170D4×D5—61no—
171D4×D5—61no—
172C2×D20—31no—
173C2×C4×D5—21no—
174C2×C4×D5C22×D311yescharacteristic
175C22⋊AGL(1,𝔽5)—64no—
176C2×#170—61no—
177C4⋊AGL(1,𝔽5)—21no—
178C4⋊AGL(1,𝔽5)—21no—
179C4⋊AGL(1,𝔽5)—32no—
180C4⋊AGL(1,𝔽5)—42no—
181C4⋊AGL(1,𝔽5)—61no—
182C4×AGL(1,𝔽5)—21no—
183C4×AGL(1,𝔽5)—21no—
184D5⋊C8—61no—
185D5⋊C8—61no—
186C2×C5×C8—31no—
187C5×C4⋊C4C4×D311yescharacteristic, complemented
188C4⋊Dic5C4×D311yescharacteristic, complemented
189#683—21no—
190C2×#168—31no—
191C8×D5—61no—
192C8×D5—61no—
193C2×C4×D4—151no—
194C2×#107—151no—
195C2×#102—151no—
196C2×#102—151no—
197#389—51no—
198#340—151no—
199#336—151no—
200C22×C3×C5C4×D411yescharacteristic
201C2×D15—22no—
202C2×D15—41no—
203C2×D15—41no—
204C2×C5×D3—22no—
205C2×C5×D3—41no—
206C2×C5×D3—41no—
207C2×C3×D5#12011yescharacteristic
208C2×C3×D5C22⋊Q811yescharacteristic
209C2×C3×D5C2×D811yescharacteristic
210C2×C3×D5#13511yescharacteristic
211C2×C3×D5AGL(1,ℤ/8)11yescharacteristic
212C2×C3×D5#13711yescharacteristic
213D3×D5—42no—
214D3×D5—42no—
215D3×D5—42no—
216D3×D5—42no—
217ΓL(1,𝔽16)—42no—
218C3×AGL(1,𝔽5)—22no—
219C3×AGL(1,𝔽5)—41no—
220C3×AGL(1,𝔽5)—81no—
221C3×C4×C5—41no—
222C3×C4×C5C2×C4⋊C411yescharacteristic
223C3×C4×C5C4×D411yescharacteristic
224C3×Dic5—41no—
225C3×Dic5C22⋊Q811yescharacteristic
226C3×Dic5#12311yescharacteristic
227C23×D3—101no—
228C22×C3×C4—51no—
229C22×C3×C4—51no—
230C22×C3×C4—51no—
231C22×C3×C4—101no—
232C22×C3×C4—101no—
233C22×Dic3—101no—
234C2×D12—51no—
235C2×D12—52no—
236C2×D12—52no—
237C2×D12—101no—
238C2×D12—101no—
239C2×D12—101no—
240C2×D12C2×AGL(1,𝔽5)11yescharacteristic
241C2×C4×D3—104no—
242C2×C4×D3—202no—
243C2×C4×D3—202no—
244C3×C4⋊C4—51no—
245C3×C4⋊C4—101no—
246C3×C4⋊C4—101no—
247C3×C4⋊C4—101no—
248C3×C4⋊C4—202no—
249C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic, complemented
250C3×C42—101no—
251C3×C42—101no—
252(C2×D3)⋊C4—104no—
253C4⋊Dic3—52no—
254C4⋊Dic3—101no—
255C2×C3⋊C8—51no—
256C2×C3⋊C8—101no—
257C2×C3⋊C8—101no—
258C2×C3⋊C8—101no—
259C2×C3⋊C8—101no—
260C2×C3⋊C8—101no—
261C2×C3⋊C8C2×AGL(1,𝔽5)11yescharacteristic, complemented
262C23×C5—61no—
263C22×D5—61no—
264C22×D5—62no—
265C22×D5—62no—
266C22×D5—121no—
267C22×D5—121no—
268C22×D5—121no—
269C22×D5—121no—
270C22×D5C2×#6711yescharacteristic
271C2×AGL(1,𝔽5)—21no—
272C2×AGL(1,𝔽5)—21no—
273C2×AGL(1,𝔽5)—21no—
274C2×AGL(1,𝔽5)—21no—
275C2×AGL(1,𝔽5)—21no—
276C2×AGL(1,𝔽5)—41no—
277C2×AGL(1,𝔽5)—41no—
278C2×AGL(1,𝔽5)—41no—
279C2×AGL(1,𝔽5)—62no—
280C2×AGL(1,𝔽5)—128no—
281C2×AGL(1,𝔽5)—121no—
282C2×AGL(1,𝔽5)—121no—
283C2×AGL(1,𝔽5)#21312yes—
284C5×D4—32no—
285C5×D4—61no—
286C2×C4×C5—21no—
287C2×C4×C5C2×C4×D311yescharacteristic
288C22⋊D5—62no—
289C22⋊D5—62no—
290C2×Dic5—21no—
291C2×Dic5#23511yescharacteristic
292D20—32no—
293D20—61no—
294C4×D5—41no—
295C4×D5—41no—
296C4×D5D3×D411yescharacteristic
297C4×D5D3×D411yescharacteristic
298C4×D5D3×Q811yescharacteristic
299C4×D5#23911yescharacteristic
300#170—121no—
301C5×C8—61no—
302#168—61no—
303C22×D4—151no—
304C23×C4—301no—
305C22×C8—151no—
306C22×C8—151no—
307C4×D4—302no—
308C4×D4—302no—
309C2×C4⋊C4—51no—
310C2×C4⋊C4—51no—
311C2×C4⋊C4—101no—
312C2×C4⋊C4—151no—
313C2×C22⋊C4—301no—
314C2×C42—51no—
315#107—302no—
316#102—31no—
317#102—151no—
318#102—151no—
319#102—151no—
320#102—302no—
321#95—101no—
322C2×C3×C5#38511yescharacteristic
323C2×C3×C5C4×D811yescharacteristic
324C2×C3×C5#44011yescharacteristic
325D15—42no—
326C3×D5#44711yescharacteristic
327C3×D5#45211yescharacteristic
328C3×D5#47411yescharacteristic
329C3×D5#47811yescharacteristic
330C5×D3—42no—
331C23×C3—51no—
332C22×D3—21no—
333C22×D3—102no—
334C22×D3—102no—
335C22×D3—101no—
336C22×D3—201no—
337C22×D3—201no—
338C22×D3—201no—
339C22×D3—201no—
340C2×C3×C4—21no—
341C2×C3×C4—51no—
342C2×C3×C4—51no—
343C2×C3×C4—51no—
344C2×C3×C4—51no—
345C2×C3×C4—51no—
346C2×C3×C4—52no—
347C2×C3×C4—101no—
348C2×C3×C4—101no—
349C2×C3×C4—101no—
350C2×C3×C4—101no—
351C2×C3×C4—101no—
352C2×C3×C4—101no—
353C2×C3×C4—201no—
354C2×C3×C4—201no—
355C2×C3×C4—201no—
356C2×C3×C4—201no—
357C2×C3×C4—201no—
358C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
359C2×Dic3—102no—
360C2×Dic3—201no—
361C2×Dic3—201no—
362D12—21no—
363D12—52no—
364D12—102no—
365D12—102no—
366D12—101no—
367D12C4⋊AGL(1,𝔽5)12yescomplemented
368C4×D3—208no—
369C3⋊C8—21no—
370C3⋊C8—101no—
371C3⋊C8—201no—
372C22×C5—62no—
373C22×C5—121no—
374C22×C5—121no—
375C22×C5C4×#6711yescharacteristic
376C2×D5—62no—
377C2×D5—121no—
378C2×D5—121no—
379C2×D5—122no—
380C2×D5—122no—
381C2×D5—122no—
382C2×D5—122no—
383C2×D5C2×#21311yescharacteristic
384C2×D5#93711yescharacteristic
385C2×D5#94211yescharacteristic
386C2×D5#94611yescharacteristic
387C2×D5#94911yescharacteristic
388C2×D5#95011yescharacteristic
389AGL(1,𝔽5)—22no—
390AGL(1,𝔽5)—41no—
391AGL(1,𝔽5)—81no—
392AGL(1,𝔽5)—122no—
393C4×C5—41no—
394C4×C5D3×C4⋊C411yescharacteristic
395C4×C5#89311yescharacteristic
396Dic5—41no—
397Dic5#89411yescharacteristic
398Dic5#89611yescharacteristic
399C24—301no—
400C2×D4—31no—
401C2×D4—152no—
402C2×D4—152no—
403C2×D4—151no—
404C2×D4—301no—
405C2×D4—301no—
406C2×D4—301no—
407C22×C4—51no—
408C22×C4—51no—
409C22×C4—51no—
410C22×C4—101no—
411C22×C4—101no—
412C22×C4—304no—
413C22×C4—301no—
414C22×C4—602no—
415C22×C4—602no—
416C2×C8—31no—
417C2×C8—151no—
418C2×C8—301no—
419C2×C8—301no—
420C2×C8—301no—
421C2×C8—301no—
422C2×C8—301no—
423C4⋊C4—51no—
424C4⋊C4—101no—
425C4⋊C4—101no—
426C4⋊C4—101no—
427C4⋊C4—152no—
428C4⋊C4—202no—
429C4⋊C4—301no—
430C4⋊C4D3×AGL(1,𝔽5)11yescharacteristic, complemented
431C22⋊C4—304no—
432C42—101no—
433C42—101no—
434C3×C5#186211yescharacteristic, complemented, Hall
435C22×C3—51no—
436C22×C3—51no—
437C22×C3—51no—
438C22×C3—51no—
439C22×C3—51no—
440C22×C3—51no—
441C22×C3D4×AGL(1,𝔽5)11yescharacteristic
442C2×D3—22no—
443C2×D3—41no—
444C2×D3—41no—
445C2×D3—102no—
446C2×D3—202no—
447C2×D3—202no—
448C2×D3—202no—
449C2×D3—202no—
450C2×D3—201no—
451C2×D3—201no—
452C3×C4—41no—
453C3×C4—51no—
454C3×C4—51no—
455C3×C4—102no—
456C3×C4—201no—
457C3×C4—201no—
458C3×C4—401no—
459C3×C4C2×C4⋊AGL(1,𝔽5)11yescharacteristic
460C3×C4D4×AGL(1,𝔽5)11yescharacteristic
461Dic3—202no—
462C2×C5—122no—
463C2×C5#498911yescharacteristic
464C2×C5C4×#21311yescharacteristic
465C2×C5#502611yescharacteristic
466D5—122no—
467D5#503611yescharacteristic
468D5#504711yescharacteristic
469D5#506611yescharacteristic
470D5#508711yescharacteristic
471C23—51no—
472C23—61no—
473C23—302no—
474C23—302no—
475C23—301no—
476C23—601no—
477C23—601no—
478C23—601no—
479C23—601no—
480D4—32no—
481D4—61no—
482D4—152no—
483D4—302no—
484D4—302no—
485D4—301no—
486C2×C4—21no—
487C2×C4—51no—
488C2×C4—51no—
489C2×C4—51no—
490C2×C4—51no—
491C2×C4—51no—
492C2×C4—52no—
493C2×C4—101no—
494C2×C4—101no—
495C2×C4—101no—
496C2×C4—101no—
497C2×C4—101no—
498C2×C4—101no—
499C2×C4—201no—
500C2×C4—201no—
501C2×C4—201no—
502C2×C4—201no—
503C2×C4—201no—
504C2×C4—302no—
505C2×C4—608no—
506C2×C4—601no—
507C2×C4—601no—
508C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
509C8—61no—
510C8—301no—
511C8—601no—
512C2×C3—51no—
513C2×C3—51no—
514C2×C3—51no—
515C2×C3—51no—
516C2×C3#6569911yescharacteristic
517C2×C3D8×AGL(1,𝔽5)11yescharacteristic
518C2×C3#6573511yescharacteristic
519D3—42no—
520D3—202no—
521C5#6962511yescharacteristic, complemented, Sylow
522C22—51no—
523C22—51no—
524C22—51no—
525C22—51no—
526C22—51no—
527C22—51no—
528C22—62no—
529C22—121no—
530C22—121no—
531C22—302no—
532C22—602no—
533C22—602no—
534C22—602no—
535C22—602no—
536C22—601no—
537C22—601no—
538C22AGL(1,𝔽5)×#6711yescharacteristic
539C4—41no—
540C4—51no—
541C4—51no—
542C4—102no—
543C4—201no—
544C4—201no—
545C4—401no—
546C4—602no—
547C4D3×C4⋊AGL(1,𝔽5)11yescharacteristic
548C4#9192111yescharacteristic
549C3#1060395711yescharacteristic, complemented, Sylow
550C2—51no—
551C2—51no—
552C2—51no—
553C2—51no—
554C2—122no—
555C2—602no—
556C2#1175193211yescharacteristic
557C2AGL(1,𝔽5)×#21311yescharacteristic
558C2#1175196411yescharacteristic
subgroupG/H
centerC22AGL(1,𝔽5)×#67
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC3×C5×C4⋊C4C2×C4
socleC22×C3×C5C4×D4
prime radicalC2×D5×D12C22
cosocleC2×C3×C4×D5C23
hypercenterC4⋊C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1862
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#65712C2semidirect
3. C2×#91160C2semidirect
6. #11746745C2semidirect
7. (C2×AGL(1,𝔽5))⋊(C3⋊C8)C2semidirect
24. #91093C4semidirect
25. C5×#809C4semidirect
26. #90968C4semidirect
27. #90966C4semidirect
32. #65712D3semidirect
64. C3×C5×C4⋊C4C2×C4semidirect
65. C2×C3⋊#168C2×C4semidirect
66. C2×C5×C3⋊C8C2×C4semidirect
67. C3×C4⋊Dic5C2×C4semidirect
123. D60C4⋊C4semidirect
125. C5×D12C4⋊C4semidirect
156. #809AGL(1,𝔽5)semidirect
187. C5×C4⋊C4C4×D3semidirect
188. C4⋊Dic5C4×D3semidirect
249. C3×C4⋊C4C2×AGL(1,𝔽5)semidirect
261. C2×C3⋊C8C2×AGL(1,𝔽5)semidirect
367. D12C4⋊AGL(1,𝔽5)semidirect
430. C4⋊C4D3×AGL(1,𝔽5)semidirect
434. C3×C5#1862semidirect
521. C5#69625semidirect
549. C3#10603957semidirect

Maximal subgroups

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

subgroupindex
C2×AGL(1,𝔽5)×D122
C3×#657122
C2×#911602
C2×#911512
D5×#8092
#117467452
(C2×AGL(1,𝔽5))⋊(C3⋊C8)2
#106039573
#696255