Finite Group Databasebeta

← C2×C3×C5×AΣL(1,𝔽16) #14275152

Subgroups — C2×C3×C5×AΣL(1,𝔽16) #14275152

Order 1920.

876 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#120C211yesmaximal, characteristic
2C2×C3×C5×#100C211yesmaximal, characteristic, complemented
3C2×C3×C5×#99C211yesmaximal, characteristic, complemented
4C3×C5×AΣL(1,𝔽16)C214yesmaximal, direct product factor
5C2×C5×AΣL(1,𝔽16)C311yesmaximal, characteristic, direct product factor, Hall
6C25×C3×C5C411yescharacteristic, complemented
7C22×C3×C5×D4—21no—
8C22×C3×C5×D4C2211yescharacteristic
9C2×C3×C5×M16—21no—
10C3×C5×#120—21no—
11C3×C5×#120—21no—
12C3×C5×#120C412yescomplemented
13C3×C5×#120C2212yes—
14C2×C3×C5×C22⋊C4—21no—
15C2×C3×C5×C22⋊C4—21no—
16C2×C3×C5×C22⋊C4C411yescharacteristic, complemented
17C3×C5×#100—21no—
18C3×C5×#100C2212yescomplemented
19C3×C5×#99—21no—
20C3×C5×#99C2212yescomplemented
21C2×C3×AΣL(1,𝔽16)C511yesmaximal, characteristic, direct product factor, Hall
22C2×C5×#120C2×C311yescharacteristic
23C2×C5×#100C2×C311yescharacteristic, complemented
24C2×C5×#99C2×C311yescharacteristic, complemented
25C5×AΣL(1,𝔽16)C2×C314yesdirect product factor
26C24×C3×C5—21no—
27C24×C3×C5—21no—
28C24×C3×C5—44no—
29C24×C3×C5—41no—
30C24×C3×C5—41no—
31C24×C3×C5C2×C411yescharacteristic
32C24×C3×C5C2×C412yescomplemented
33C24×C3×C5D411yescharacteristic
34C2×C3×C5×D4—22no—
35C2×C3×C5×D4—22no—
36C2×C3×C5×D4—41no—
37C2×C3×C5×D4—41no—
38C2×C3×C5×D4—44no—
39C2×C3×C5×D4C2×C411yescharacteristic
40C2×C3×C5×D4D411yescharacteristic
41C2×C3×C5×D4D411yescharacteristic
42C2×C3×C5×D4C2311yescharacteristic
43C22×C3×C4×C5—21no—
44C22×C3×C4×C5—41no—
45C22×C3×C4×C5D411yescharacteristic
46C3×C5×M16—22no—
47C3×C5×M16—41no—
48C2×C3×C5×C8—41no—
49C3×C5×C22⋊C4—21no—
50C3×C5×C22⋊C4—22no—
51C3×C5×C22⋊C4—22no—
52C3×C5×C22⋊C4—22no—
53C3×C5×C22⋊C4—41no—
54C3×C5×C22⋊C4C2×C412yescomplemented
55C2×C3×#120C2×C511yescharacteristic
56C2×C3×#100C2×C511yescharacteristic, complemented
57C2×C3×#99C2×C511yescharacteristic, complemented
58C3×AΣL(1,𝔽16)C2×C514yesdirect product factor
59C25×C5C3×C411yescharacteristic, complemented
60C22×C5×D4—21no—
61C22×C5×D4C22×C311yescharacteristic
62C2×C5×M16—21no—
63C5×#120—21no—
64C5×#120—21no—
65C5×#120C3×C412yescomplemented
66C5×#120C22×C312yes—
67C2×C5×C22⋊C4—21no—
68C2×C5×C22⋊C4—21no—
69C2×C5×C22⋊C4C3×C411yescharacteristic, complemented
70C5×#100—21no—
71C5×#100C22×C312yescomplemented
72C5×#99—21no—
73C5×#99C22×C312yescomplemented
74C2×AΣL(1,𝔽16)C3×C511yescharacteristic, direct product factor, Sylow
75C23×C3×C5—24no—
76C23×C3×C5—21no—
77C23×C3×C5—21no—
78C23×C3×C5—21no—
79C23×C3×C5—22no—
80C23×C3×C5—21no—
81C23×C3×C5—42no—
82C23×C3×C5—44no—
83C23×C3×C5—42no—
84C23×C3×C5—42no—
85C23×C3×C5—44no—
86C23×C3×C5—42no—
87C23×C3×C5—42no—
88C23×C3×C5—48no—
89C23×C3×C5—44no—
90C23×C3×C5—41no—
91C23×C3×C5—41no—
92C23×C3×C5—44no—
93C23×C3×C5C22⋊C411yescharacteristic
94C23×C3×C5C22⋊C411yescharacteristic
95C23×C3×C5C22⋊C411yescharacteristic
96C23×C3×C5C22×C411yescharacteristic
97C23×C3×C5C2×D411yescharacteristic
98C3×C5×D4—41no—
99C3×C5×D4—41no—
100C3×C5×D4—41no—
101C3×C5×D4—41no—
102C3×C5×D4—44no—
103C3×C5×D4—44no—
104C2×C3×C4×C5—21no—
105C2×C3×C4×C5—21no—
106C2×C3×C4×C5—22no—
107C2×C3×C4×C5—42no—
108C2×C3×C4×C5—42no—
109C2×C3×C4×C5—82no—
110C2×C3×C4×C5C22⋊C411yescharacteristic
111C2×C3×C4×C5C2×D411yescharacteristic
112C3×C5×C8—42no—
113C25×C3C4×C511yescharacteristic, complemented
114C22×C3×D4—21no—
115C22×C3×D4C22×C511yescharacteristic
116C2×C3×M16—21no—
117C3×#120—21no—
118C3×#120—21no—
119C3×#120C4×C512yescomplemented
120C3×#120C22×C512yes—
121C2×C3×C22⋊C4—21no—
122C2×C3×C22⋊C4—21no—
123C2×C3×C22⋊C4C4×C511yescharacteristic, complemented
124C3×#100—21no—
125C3×#100C22×C512yescomplemented
126C3×#99—21no—
127C3×#99C22×C512yescomplemented
128C24×C5—21no—
129C24×C5—21no—
130C24×C5—44no—
131C24×C5—41no—
132C24×C5—41no—
133C24×C5C2×C3×C411yescharacteristic
134C24×C5C2×C3×C412yescomplemented
135C24×C5C3×D411yescharacteristic
136C2×C5×D4—22no—
137C2×C5×D4—22no—
138C2×C5×D4—41no—
139C2×C5×D4—41no—
140C2×C5×D4—44no—
141C2×C5×D4C2×C3×C411yescharacteristic
142C2×C5×D4C3×D411yescharacteristic
143C2×C5×D4C3×D411yescharacteristic
144C2×C5×D4C23×C311yescharacteristic
145C22×C4×C5—21no—
146C22×C4×C5—41no—
147C22×C4×C5C3×D411yescharacteristic
148C5×M16—22no—
149C5×M16—41no—
150C2×C5×C8—41no—
151C5×C22⋊C4—21no—
152C5×C22⋊C4—22no—
153C5×C22⋊C4—22no—
154C5×C22⋊C4—22no—
155C5×C22⋊C4—41no—
156C5×C22⋊C4C2×C3×C412yescomplemented
157C2×#120C2×C3×C511yescharacteristic
158C2×#100C2×C3×C511yescharacteristic, complemented
159C2×#99C2×C3×C511yescharacteristic, complemented
160AΣL(1,𝔽16)C2×C3×C514yesdirect product factor
161C22×C3×C5—22no—
162C22×C3×C5—24no—
163C22×C3×C5—21no—
164C22×C3×C5—21no—
165C22×C3×C5—21no—
166C22×C3×C5—21no—
167C22×C3×C5—41no—
168C22×C3×C5—41no—
169C22×C3×C5—41no—
170C22×C3×C5—41no—
171C22×C3×C5—42no—
172C22×C3×C5—44no—
173C22×C3×C5—42no—
174C22×C3×C5—44no—
175C22×C3×C5—48no—
176C22×C3×C5—44no—
177C22×C3×C5—42no—
178C22×C3×C5—44no—
179C22×C3×C5—42no—
180C22×C3×C5—42no—
181C22×C3×C5—81no—
182C22×C3×C5#9911yescharacteristic
183C22×C3×C5#9911yescharacteristic
184C22×C3×C5C2×C22⋊C411yescharacteristic
185C3×C4×C5—21no—
186C3×C4×C5—21no—
187C3×C4×C5—42no—
188C3×C4×C5—82no—
189C24×C3—21no—
190C24×C3—21no—
191C24×C3—44no—
192C24×C3—41no—
193C24×C3—41no—
194C24×C3C2×C4×C511yescharacteristic
195C24×C3C2×C4×C512yescomplemented
196C24×C3C5×D411yescharacteristic
197C2×C3×D4—22no—
198C2×C3×D4—22no—
199C2×C3×D4—41no—
200C2×C3×D4—41no—
201C2×C3×D4—44no—
202C2×C3×D4C2×C4×C511yescharacteristic
203C2×C3×D4C5×D411yescharacteristic
204C2×C3×D4C5×D411yescharacteristic
205C2×C3×D4C23×C511yescharacteristic
206C22×C3×C4—21no—
207C22×C3×C4—41no—
208C22×C3×C4C5×D411yescharacteristic
209C3×M16—22no—
210C3×M16—41no—
211C2×C3×C8—41no—
212C3×C22⋊C4—21no—
213C3×C22⋊C4—22no—
214C3×C22⋊C4—22no—
215C3×C22⋊C4—22no—
216C3×C22⋊C4—41no—
217C3×C22⋊C4C2×C4×C512yescomplemented
218C23×C5—24no—
219C23×C5—21no—
220C23×C5—21no—
221C23×C5—21no—
222C23×C5—22no—
223C23×C5—21no—
224C23×C5—42no—
225C23×C5—44no—
226C23×C5—42no—
227C23×C5—42no—
228C23×C5—44no—
229C23×C5—42no—
230C23×C5—42no—
231C23×C5—44no—
232C23×C5—48no—
233C23×C5—44no—
234C23×C5—41no—
235C23×C5—41no—
236C23×C5C3×C22⋊C411yescharacteristic
237C23×C5C3×C22⋊C411yescharacteristic
238C23×C5C3×C22⋊C411yescharacteristic
239C23×C5C22×C3×C411yescharacteristic
240C23×C5C2×C3×D411yescharacteristic
241C5×D4—41no—
242C5×D4—41no—
243C5×D4—41no—
244C5×D4—41no—
245C5×D4—44no—
246C5×D4—44no—
247C2×C4×C5—21no—
248C2×C4×C5—21no—
249C2×C4×C5—22no—
250C2×C4×C5—42no—
251C2×C4×C5—42no—
252C2×C4×C5—82no—
253C2×C4×C5C3×C22⋊C411yescharacteristic
254C2×C4×C5C2×C3×D411yescharacteristic
255C5×C8—42no—
256C25C3×C4×C511yescharacteristic, complemented
257C22×D4—21no—
258C22×D4C22×C3×C511yescharacteristic
259C2×M16—21no—
260#120—21no—
261#120—21no—
262#120C3×C4×C512yescomplemented
263#120C22×C3×C512yes—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4C3×C4×C511yescharacteristic, complemented
267#100—21no—
268#100C22×C3×C512yescomplemented
269#99—21no—
270#99C22×C3×C512yescomplemented
271C2×C3×C5—21no—
272C2×C3×C5—21no—
273C2×C3×C5—41no—
274C2×C3×C5—44no—
275C2×C3×C5—41no—
276C2×C3×C5—41no—
277C2×C3×C5—41no—
278C2×C3×C5AΣL(1,𝔽16)12yesdirect product factor
279C2×C3×C5C2×#9911yescharacteristic
280C23×C3—24no—
281C23×C3—22no—
282C23×C3—21no—
283C23×C3—21no—
284C23×C3—21no—
285C23×C3—21no—
286C23×C3—42no—
287C23×C3—44no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—44no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3—44no—
294C23×C3—48no—
295C23×C3—44no—
296C23×C3—41no—
297C23×C3—41no—
298C23×C3C5×C22⋊C411yescharacteristic
299C23×C3C5×C22⋊C411yescharacteristic
300C23×C3C5×C22⋊C411yescharacteristic
301C23×C3C22×C4×C511yescharacteristic
302C23×C3C2×C5×D411yescharacteristic
303C3×D4—41no—
304C3×D4—41no—
305C3×D4—41no—
306C3×D4—41no—
307C3×D4—44no—
308C3×D4—44no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—22no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—82no—
315C2×C3×C4C5×C22⋊C411yescharacteristic
316C2×C3×C4C2×C5×D411yescharacteristic
317C3×C8—42no—
318C22×C5—22no—
319C22×C5—24no—
320C22×C5—21no—
321C22×C5—21no—
322C22×C5—21no—
323C22×C5—21no—
324C22×C5—41no—
325C22×C5—41no—
326C22×C5—41no—
327C22×C5—41no—
328C22×C5—42no—
329C22×C5—44no—
330C22×C5—42no—
331C22×C5—44no—
332C22×C5—44no—
333C22×C5—48no—
334C22×C5—42no—
335C22×C5—42no—
336C22×C5—44no—
337C22×C5—42no—
338C22×C5—81no—
339C22×C5C3×#9911yescharacteristic
340C22×C5C3×#9911yescharacteristic
341C22×C5C2×C3×C22⋊C411yescharacteristic
342C4×C5—21no—
343C4×C5—21no—
344C4×C5—42no—
345C4×C5—82no—
346C24—21no—
347C24—21no—
348C24—44no—
349C24—41no—
350C24—41no—
351C24C2×C3×C4×C511yescharacteristic
352C24C2×C3×C4×C512yescomplemented
353C24C3×C5×D411yescharacteristic
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—44no—
359C2×D4C2×C3×C4×C511yescharacteristic
360C2×D4C3×C5×D411yescharacteristic
361C2×D4C3×C5×D411yescharacteristic
362C2×D4C23×C3×C511yescharacteristic
363C22×C4—21no—
364C22×C4—41no—
365C22×C4C3×C5×D411yescharacteristic
366M16—22no—
367M16—41no—
368C2×C8—41no—
369C22⋊C4—21no—
370C22⋊C4—22no—
371C22⋊C4—22no—
372C22⋊C4—22no—
373C22⋊C4—41no—
374C22⋊C4C2×C3×C4×C512yescomplemented
375C3×C5C2×AΣL(1,𝔽16)11yescharacteristic, direct product factor, Hall
376C22×C3—22no—
377C22×C3—24no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—41no—
383C22×C3—41no—
384C22×C3—41no—
385C22×C3—41no—
386C22×C3—42no—
387C22×C3—44no—
388C22×C3—42no—
389C22×C3—44no—
390C22×C3—44no—
391C22×C3—48no—
392C22×C3—42no—
393C22×C3—42no—
394C22×C3—44no—
395C22×C3—42no—
396C22×C3—81no—
397C22×C3C5×#9911yescharacteristic
398C22×C3C5×#9911yescharacteristic
399C22×C3C2×C5×C22⋊C411yescharacteristic
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—42no—
403C3×C4—82no—
404C2×C5—21no—
405C2×C5—21no—
406C2×C5—41no—
407C2×C5—44no—
408C2×C5—41no—
409C2×C5—41no—
410C2×C5—41no—
411C2×C5C3×AΣL(1,𝔽16)12yesdirect product factor
412C2×C5C2×C3×#9911yescharacteristic
413C23—24no—
414C23—22no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—42no—
420C23—44no—
421C23—42no—
422C23—42no—
423C23—44no—
424C23—42no—
425C23—42no—
426C23—44no—
427C23—48no—
428C23—44no—
429C23—41no—
430C23—41no—
431C23C3×C5×C22⋊C411yescharacteristic
432C23C3×C5×C22⋊C411yescharacteristic
433C23C3×C5×C22⋊C411yescharacteristic
434C23C22×C3×C4×C511yescharacteristic
435C23C2×C3×C5×D411yescharacteristic
436D4—41no—
437D4—41no—
438D4—41no—
439D4—41no—
440D4—44no—
441D4—44no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—82no—
448C2×C4C3×C5×C22⋊C411yescharacteristic
449C2×C4C2×C3×C5×D411yescharacteristic
450C8—42no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—41no—
454C2×C3—44no—
455C2×C3—41no—
456C2×C3—41no—
457C2×C3—41no—
458C2×C3C5×AΣL(1,𝔽16)12yesdirect product factor
459C2×C3C2×C5×#9911yescharacteristic
460C5C2×C3×AΣL(1,𝔽16)11yescharacteristic, direct product factor, Sylow
461C22—22no—
462C22—24no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—42no—
472C22—44no—
473C22—42no—
474C22—44no—
475C22—44no—
476C22—48no—
477C22—42no—
478C22—44no—
479C22—42no—
480C22—42no—
481C22—81no—
482C22C3×C5×#9911yescharacteristic
483C22C3×C5×#9911yescharacteristic
484C22C2×C3×C5×C22⋊C411yescharacteristic
485C4—21no—
486C4—21no—
487C4—42no—
488C4—82no—
489C3C2×C5×AΣL(1,𝔽16)11yescharacteristic, direct product factor, Sylow
490C2—21no—
491C2—21no—
492C2—41no—
493C2—44no—
494C2—41no—
495C2—41no—
496C2—41no—
497C2C3×C5×AΣL(1,𝔽16)12yesdirect product factor
498C2C2×C3×C5×#9911yescharacteristic
subgroupG/H
center, socleC22×C3×C5#99
commutatorC23C22×C3×C4×C5
Frattini, cosocleC2×D4C23×C3×C5
prime radicalC2×C3×C5×#120C2
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3×C5 ◁ C23×C3×C5 ◁ C22×C3×C5×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×C5×#100C2semidirect
3. C2×C3×C5×#99C2semidirect
4. C3×C5×AΣL(1,𝔽16)C2direct (N × H)
5. C2×C5×AΣL(1,𝔽16)C3direct (N × H)
6. C25×C3×C5C4semidirect
12. C3×C5×#120C4semidirect
16. C2×C3×C5×C22⋊C4C4semidirect
18. C3×C5×#100C22semidirect
20. C3×C5×#99C22semidirect
21. C2×C3×AΣL(1,𝔽16)C5direct (N × H)
23. C2×C5×#100C2×C3semidirect
24. C2×C5×#99C2×C3semidirect
25. C5×AΣL(1,𝔽16)C2×C3direct (N × H)
32. C24×C3×C5C2×C4semidirect
54. C3×C5×C22⋊C4C2×C4semidirect
56. C2×C3×#100C2×C5semidirect
57. C2×C3×#99C2×C5semidirect
58. C3×AΣL(1,𝔽16)C2×C5direct (N × H)
59. C25×C5C3×C4semidirect
65. C5×#120C3×C4semidirect
69. C2×C5×C22⋊C4C3×C4semidirect
71. C5×#100C22×C3semidirect
73. C5×#99C22×C3semidirect
74. C2×AΣL(1,𝔽16)C3×C5direct (N × H)
113. C25×C3C4×C5semidirect
119. C3×#120C4×C5semidirect
123. C2×C3×C22⋊C4C4×C5semidirect
125. C3×#100C22×C5semidirect
127. C3×#99C22×C5semidirect
134. C24×C5C2×C3×C4semidirect
156. C5×C22⋊C4C2×C3×C4semidirect
158. C2×#100C2×C3×C5semidirect
159. C2×#99C2×C3×C5semidirect
160. AΣL(1,𝔽16)C2×C3×C5direct (N × H)
195. C24×C3C2×C4×C5semidirect
217. C3×C22⋊C4C2×C4×C5semidirect
256. C25C3×C4×C5semidirect
262. #120C3×C4×C5semidirect
266. C2×C22⋊C4C3×C4×C5semidirect
268. #100C22×C3×C5semidirect
270. #99C22×C3×C5semidirect
278. C2×C3×C5AΣL(1,𝔽16)direct (N × H)
352. C24C2×C3×C4×C5semidirect
374. C22⋊C4C2×C3×C4×C5semidirect
375. C3×C5C2×AΣL(1,𝔽16)direct (N × H)
411. C2×C5C3×AΣL(1,𝔽16)direct (N × H)
458. C2×C3C5×AΣL(1,𝔽16)direct (N × H)
460. C5C2×C3×AΣL(1,𝔽16)direct (N × H)
489. C3C2×C5×AΣL(1,𝔽16)direct (N × H)
497. C2C3×C5×AΣL(1,𝔽16)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×#1202
C2×C3×C5×#1002
C2×C3×C5×#992
C3×C5×AΣL(1,𝔽16)2
C2×C5×AΣL(1,𝔽16)3
C2×C3×AΣL(1,𝔽16)5