Finite Group Databasebeta

← C3×C5×#2121 #14275155

Subgroups — C3×C5×#2121 #14275155

Order 1920.

560 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#534C211yesmaximal, characteristic
2C2×C3×C5×#100C211yesmaximal, characteristic, complemented
3C2×C3×C5×#99C211yesmaximal, characteristic, complemented
4C3×C5×#352C212yesmaximal, complemented
5C3×C5×AΣL(1,𝔽16)C212yesmaximal, complemented
6C5×#2121C311yesmaximal, characteristic, direct product factor, Hall
7C22×C3×C5×D4C411yescharacteristic, complemented
8C22×C3×C5×D4C2211yescharacteristic
9C2×C3×C5×M16—21no—
10C3×C5×#123—21no—
11C3×C5×#123C411yescharacteristic, complemented
12C3×C5×#123C2211yescharacteristic
13C3×C5×#121—21no—
14C3×C5×#121—21no—
15C3×C5×#120—21no—
16C3×C5×#120C411yescharacteristic, complemented
17C3×C5×#120C2211yescharacteristic
18C2×C3×C5×C22⋊C4—21no—
19C2×C3×C5×C22⋊C4C411yescharacteristic, complemented
20C3×C5×#100—21no—
21C3×C5×#100C2212yes—
22C3×C5×#99—21no—
23C3×C5×#99C2212yes—
24C3×#2121C511yesmaximal, characteristic, direct product factor, Hall
25C5×#534C2×C311yescharacteristic
26C2×C5×#100C2×C311yescharacteristic, complemented
27C2×C5×#99C2×C311yescharacteristic, complemented
28C5×#352C2×C312yescomplemented
29C5×AΣL(1,𝔽16)C2×C312yescomplemented
30C24×C3×C5C2×C411yescharacteristic
31C24×C3×C5C2×C411yescharacteristic, complemented
32C24×C3×C5D411yescharacteristic
33C2×C3×C5×D4—21no—
34C2×C3×C5×D4—21no—
35C2×C3×C5×D4—21no—
36C2×C3×C5×D4—21no—
37C2×C3×C5×D4—41no—
38C2×C3×C5×D4—41no—
39C2×C3×C5×D4—41no—
40C2×C3×C5×D4—41no—
41C2×C3×C5×D4C2×C411yescharacteristic
42C2×C3×C5×D4D411yescharacteristic
43C2×C3×C5×D4D411yescharacteristic
44C2×C3×C5×D4C2311yescharacteristic
45C22×C3×C4×C5—21no—
46C22×C3×C4×C5—41no—
47C22×C3×C4×C5C2×C411yescharacteristic, complemented
48C22×C3×C4×C5D411yescharacteristic
49C3×C5×M16—22no—
50C3×C5×M16—41no—
51C2×C3×C5×C8—41no—
52C3×C5×C4⋊C4—21no—
53C3×C5×C4⋊C4—21no—
54C3×C5×C22⋊C4—21no—
55C3×C5×C22⋊C4—21no—
56C3×C5×C22⋊C4—21no—
57C3×C5×C22⋊C4—21no—
58C3×C5×C22⋊C4—21no—
59C3×C5×C22⋊C4—22no—
60C3×C5×C22⋊C4—41no—
61C3×C5×C22⋊C4C2×C411yescharacteristic, complemented
62C3×C5×C22⋊C4C2×C411yescharacteristic, complemented
63C3×#534C2×C511yescharacteristic
64C2×C3×#100C2×C511yescharacteristic, complemented
65C2×C3×#99C2×C511yescharacteristic, complemented
66C3×#352C2×C512yescomplemented
67C3×AΣL(1,𝔽16)C2×C512yescomplemented
68C22×C5×D4C3×C411yescharacteristic, complemented
69C22×C5×D4C22×C311yescharacteristic
70C2×C5×M16—21no—
71C5×#123—21no—
72C5×#123C3×C411yescharacteristic, complemented
73C5×#123C22×C311yescharacteristic
74C5×#121—21no—
75C5×#121—21no—
76C5×#120—21no—
77C5×#120C3×C411yescharacteristic, complemented
78C5×#120C22×C311yescharacteristic
79C2×C5×C22⋊C4—21no—
80C2×C5×C22⋊C4C3×C411yescharacteristic, complemented
81C5×#100—21no—
82C5×#100C22×C312yes—
83C5×#99—21no—
84C5×#99C22×C312yes—
85#2121C3×C511yescharacteristic, direct product factor, Sylow
86C23×C3×C5—21no—
87C23×C3×C5—21no—
88C23×C3×C5—21no—
89C23×C3×C5—21no—
90C23×C3×C5—21no—
91C23×C3×C5—41no—
92C23×C3×C5—81no—
93C23×C3×C5—81no—
94C23×C3×C5—81no—
95C23×C3×C5C22⋊C411yescharacteristic
96C23×C3×C5C22⋊C411yescharacteristic
97C23×C3×C5C22⋊C411yescharacteristic
98C23×C3×C5C22×C411yescharacteristic
99C23×C3×C5C2×D411yescharacteristic
100C3×C5×D4—41no—
101C3×C5×D4—41no—
102C3×C5×D4—41no—
103C3×C5×D4—41no—
104C3×C5×D4—42no—
105C3×C5×D4—41no—
106C3×C5×D4—41no—
107C3×C5×D4—81no—
108C3×C5×D4—81no—
109C2×C3×C4×C5—21no—
110C2×C3×C4×C5—21no—
111C2×C3×C4×C5—21no—
112C2×C3×C4×C5—21no—
113C2×C3×C4×C5—21no—
114C2×C3×C4×C5—41no—
115C2×C3×C4×C5—42no—
116C2×C3×C4×C5—81no—
117C2×C3×C4×C5—82no—
118C2×C3×C4×C5C22⋊C411yescharacteristic
119C2×C3×C4×C5C2×D411yescharacteristic
120C3×C5×C8—42no—
121C22×C3×D4C4×C511yescharacteristic, complemented
122C22×C3×D4C22×C511yescharacteristic
123C2×C3×M16—21no—
124C3×#123—21no—
125C3×#123C4×C511yescharacteristic, complemented
126C3×#123C22×C511yescharacteristic
127C3×#121—21no—
128C3×#121—21no—
129C3×#120—21no—
130C3×#120C4×C511yescharacteristic, complemented
131C3×#120C22×C511yescharacteristic
132C2×C3×C22⋊C4—21no—
133C2×C3×C22⋊C4C4×C511yescharacteristic, complemented
134C3×#100—21no—
135C3×#100C22×C512yes—
136C3×#99—21no—
137C3×#99C22×C512yes—
138C24×C5C2×C3×C411yescharacteristic
139C24×C5C2×C3×C411yescharacteristic, complemented
140C24×C5C3×D411yescharacteristic
141C2×C5×D4—21no—
142C2×C5×D4—21no—
143C2×C5×D4—21no—
144C2×C5×D4—21no—
145C2×C5×D4—41no—
146C2×C5×D4—41no—
147C2×C5×D4—41no—
148C2×C5×D4—41no—
149C2×C5×D4C2×C3×C411yescharacteristic
150C2×C5×D4C3×D411yescharacteristic
151C2×C5×D4C3×D411yescharacteristic
152C2×C5×D4C23×C311yescharacteristic
153C22×C4×C5—21no—
154C22×C4×C5—41no—
155C22×C4×C5C2×C3×C411yescharacteristic, complemented
156C22×C4×C5C3×D411yescharacteristic
157C5×M16—22no—
158C5×M16—41no—
159C2×C5×C8—41no—
160C5×C4⋊C4—21no—
161C5×C4⋊C4—21no—
162C5×C22⋊C4—21no—
163C5×C22⋊C4—21no—
164C5×C22⋊C4—21no—
165C5×C22⋊C4—21no—
166C5×C22⋊C4—21no—
167C5×C22⋊C4—22no—
168C5×C22⋊C4—41no—
169C5×C22⋊C4C2×C3×C411yescharacteristic, complemented
170C5×C22⋊C4C2×C3×C411yescharacteristic, complemented
171#534C2×C3×C511yescharacteristic
172C2×#100C2×C3×C511yescharacteristic, complemented
173C2×#99C2×C3×C511yescharacteristic, complemented
174#352C2×C3×C512yescomplemented
175AΣL(1,𝔽16)C2×C3×C512yescomplemented
176C22×C3×C5—21no—
177C22×C3×C5—21no—
178C22×C3×C5—21no—
179C22×C3×C5—21no—
180C22×C3×C5—41no—
181C22×C3×C5—41no—
182C22×C3×C5—41no—
183C22×C3×C5—41no—
184C22×C3×C5—41no—
185C22×C3×C5—42no—
186C22×C3×C5—41no—
187C22×C3×C5—41no—
188C22×C3×C5—41no—
189C22×C3×C5—81no—
190C22×C3×C5—81no—
191C22×C3×C5—81no—
192C22×C3×C5—82no—
193C22×C3×C5#9911yescharacteristic
194C22×C3×C5#9911yescharacteristic
195C22×C3×C5C2×C22⋊C411yescharacteristic
196C3×C4×C5—21no—
197C3×C4×C5—21no—
198C3×C4×C5—41no—
199C3×C4×C5—41no—
200C3×C4×C5—41no—
201C3×C4×C5—82no—
202C24×C3C2×C4×C511yescharacteristic
203C24×C3C2×C4×C511yescharacteristic, complemented
204C24×C3C5×D411yescharacteristic
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—41no—
210C2×C3×D4—41no—
211C2×C3×D4—41no—
212C2×C3×D4—41no—
213C2×C3×D4C2×C4×C511yescharacteristic
214C2×C3×D4C5×D411yescharacteristic
215C2×C3×D4C5×D411yescharacteristic
216C2×C3×D4C23×C511yescharacteristic
217C22×C3×C4—21no—
218C22×C3×C4—41no—
219C22×C3×C4C2×C4×C511yescharacteristic, complemented
220C22×C3×C4C5×D411yescharacteristic
221C3×M16—22no—
222C3×M16—41no—
223C2×C3×C8—41no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C22⋊C4—21no—
227C3×C22⋊C4—21no—
228C3×C22⋊C4—21no—
229C3×C22⋊C4—21no—
230C3×C22⋊C4—21no—
231C3×C22⋊C4—22no—
232C3×C22⋊C4—41no—
233C3×C22⋊C4C2×C4×C511yescharacteristic, complemented
234C3×C22⋊C4C2×C4×C511yescharacteristic, complemented
235C23×C5—21no—
236C23×C5—21no—
237C23×C5—21no—
238C23×C5—21no—
239C23×C5—21no—
240C23×C5—41no—
241C23×C5—81no—
242C23×C5—81no—
243C23×C5—81no—
244C23×C5C3×C22⋊C411yescharacteristic
245C23×C5C3×C22⋊C411yescharacteristic
246C23×C5C3×C22⋊C411yescharacteristic
247C23×C5C22×C3×C411yescharacteristic
248C23×C5C2×C3×D411yescharacteristic
249C5×D4—41no—
250C5×D4—41no—
251C5×D4—41no—
252C5×D4—41no—
253C5×D4—42no—
254C5×D4—41no—
255C5×D4—41no—
256C5×D4—81no—
257C5×D4—81no—
258C2×C4×C5—21no—
259C2×C4×C5—21no—
260C2×C4×C5—21no—
261C2×C4×C5—21no—
262C2×C4×C5—21no—
263C2×C4×C5—41no—
264C2×C4×C5—42no—
265C2×C4×C5—81no—
266C2×C4×C5—82no—
267C2×C4×C5C3×C22⋊C411yescharacteristic
268C2×C4×C5C2×C3×D411yescharacteristic
269C5×C8—42no—
270C22×D4C3×C4×C511yescharacteristic, complemented
271C22×D4C22×C3×C511yescharacteristic
272C2×M16—21no—
273#123—21no—
274#123C3×C4×C511yescharacteristic, complemented
275#123C22×C3×C511yescharacteristic
276#121—21no—
277#121—21no—
278#120—21no—
279#120C3×C4×C511yescharacteristic, complemented
280#120C22×C3×C511yescharacteristic
281C2×C22⋊C4—21no—
282C2×C22⋊C4C3×C4×C511yescharacteristic, complemented
283#100—21no—
284#100C22×C3×C512yes—
285#99—21no—
286#99C22×C3×C512yes—
287C2×C3×C5—21no—
288C2×C3×C5—21no—
289C2×C3×C5—21no—
290C2×C3×C5—41no—
291C2×C3×C5—41no—
292C2×C3×C5—41no—
293C2×C3×C5—41no—
294C2×C3×C5—81no—
295C2×C3×C5C2×#9911yescharacteristic
296C23×C3—21no—
297C23×C3—21no—
298C23×C3—21no—
299C23×C3—21no—
300C23×C3—21no—
301C23×C3—41no—
302C23×C3—81no—
303C23×C3—81no—
304C23×C3—81no—
305C23×C3C5×C22⋊C411yescharacteristic
306C23×C3C5×C22⋊C411yescharacteristic
307C23×C3C5×C22⋊C411yescharacteristic
308C23×C3C22×C4×C511yescharacteristic
309C23×C3C2×C5×D411yescharacteristic
310C3×D4—41no—
311C3×D4—41no—
312C3×D4—41no—
313C3×D4—41no—
314C3×D4—42no—
315C3×D4—41no—
316C3×D4—41no—
317C3×D4—81no—
318C3×D4—81no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—41no—
325C2×C3×C4—42no—
326C2×C3×C4—81no—
327C2×C3×C4—82no—
328C2×C3×C4C5×C22⋊C411yescharacteristic
329C2×C3×C4C2×C5×D411yescharacteristic
330C3×C8—42no—
331C22×C5—21no—
332C22×C5—21no—
333C22×C5—21no—
334C22×C5—21no—
335C22×C5—41no—
336C22×C5—41no—
337C22×C5—41no—
338C22×C5—41no—
339C22×C5—41no—
340C22×C5—42no—
341C22×C5—41no—
342C22×C5—41no—
343C22×C5—41no—
344C22×C5—81no—
345C22×C5—81no—
346C22×C5—81no—
347C22×C5—82no—
348C22×C5C3×#9911yescharacteristic
349C22×C5C3×#9911yescharacteristic
350C22×C5C2×C3×C22⋊C411yescharacteristic
351C4×C5—21no—
352C4×C5—21no—
353C4×C5—41no—
354C4×C5—41no—
355C4×C5—41no—
356C4×C5—82no—
357C24C2×C3×C4×C511yescharacteristic
358C24C2×C3×C4×C511yescharacteristic, complemented
359C24C3×C5×D411yescharacteristic
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4C2×C3×C4×C511yescharacteristic
369C2×D4C3×C5×D411yescharacteristic
370C2×D4C3×C5×D411yescharacteristic
371C2×D4C23×C3×C511yescharacteristic
372C22×C4—21no—
373C22×C4—41no—
374C22×C4C2×C3×C4×C511yescharacteristic, complemented
375C22×C4C3×C5×D411yescharacteristic
376M16—22no—
377M16—41no—
378C2×C8—41no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C22⋊C4—21no—
382C22⋊C4—21no—
383C22⋊C4—21no—
384C22⋊C4—21no—
385C22⋊C4—21no—
386C22⋊C4—22no—
387C22⋊C4—41no—
388C22⋊C4C2×C3×C4×C511yescharacteristic, complemented
389C22⋊C4C2×C3×C4×C511yescharacteristic, complemented
390C3×C5#212111yescharacteristic, direct product factor, Hall
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—41no—
396C22×C3—41no—
397C22×C3—41no—
398C22×C3—41no—
399C22×C3—41no—
400C22×C3—42no—
401C22×C3—41no—
402C22×C3—41no—
403C22×C3—41no—
404C22×C3—81no—
405C22×C3—81no—
406C22×C3—81no—
407C22×C3—82no—
408C22×C3C5×#9911yescharacteristic
409C22×C3C5×#9911yescharacteristic
410C22×C3C2×C5×C22⋊C411yescharacteristic
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—41no—
414C3×C4—41no—
415C3×C4—41no—
416C3×C4—82no—
417C2×C5—21no—
418C2×C5—21no—
419C2×C5—21no—
420C2×C5—41no—
421C2×C5—41no—
422C2×C5—41no—
423C2×C5—41no—
424C2×C5—81no—
425C2×C5C2×C3×#9911yescharacteristic
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—41no—
432C23—81no—
433C23—81no—
434C23—81no—
435C23C3×C5×C22⋊C411yescharacteristic
436C23C3×C5×C22⋊C411yescharacteristic
437C23C3×C5×C22⋊C411yescharacteristic
438C23C22×C3×C4×C511yescharacteristic
439C23C2×C3×C5×D411yescharacteristic
440D4—41no—
441D4—41no—
442D4—41no—
443D4—41no—
444D4—42no—
445D4—41no—
446D4—41no—
447D4—81no—
448D4—81no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—81no—
457C2×C4—82no—
458C2×C4C3×C5×C22⋊C411yescharacteristic
459C2×C4C2×C3×C5×D411yescharacteristic
460C8—42no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—41no—
465C2×C3—41no—
466C2×C3—41no—
467C2×C3—41no—
468C2×C3—81no—
469C2×C3C2×C5×#9911yescharacteristic
470C5C3×#212111yescharacteristic, direct product factor, Sylow
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22—42no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22—82no—
488C22C3×C5×#9911yescharacteristic
489C22C3×C5×#9911yescharacteristic
490C22C2×C3×C5×C22⋊C411yescharacteristic
491C4—21no—
492C4—21no—
493C4—41no—
494C4—41no—
495C4—41no—
496C4—82no—
497C3C5×#212111yescharacteristic, direct product factor, Sylow
498C2—21no—
499C2—21no—
500C2—21no—
501C2—41no—
502C2—41no—
503C2—41no—
504C2—41no—
505C2—81no—
506C2C2×C3×C5×#9911yescharacteristic
subgroupG/H
center, socleC2×C3×C5C2×#99
commutatorC23C22×C3×C4×C5
Frattini, cosocleC2×D4C23×C3×C5
prime radicalC3×C5×#534C2
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×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×#352C2semidirect
5. C3×C5×AΣL(1,𝔽16)C2semidirect
6. C5×#2121C3direct (N × H)
7. C22×C3×C5×D4C4semidirect
11. C3×C5×#123C4semidirect
16. C3×C5×#120C4semidirect
19. C2×C3×C5×C22⋊C4C4semidirect
24. C3×#2121C5direct (N × H)
26. C2×C5×#100C2×C3semidirect
27. C2×C5×#99C2×C3semidirect
28. C5×#352C2×C3semidirect
29. C5×AΣL(1,𝔽16)C2×C3semidirect
31. C24×C3×C5C2×C4semidirect
47. C22×C3×C4×C5C2×C4semidirect
61. C3×C5×C22⋊C4C2×C4semidirect
62. C3×C5×C22⋊C4C2×C4semidirect
64. C2×C3×#100C2×C5semidirect
65. C2×C3×#99C2×C5semidirect
66. C3×#352C2×C5semidirect
67. C3×AΣL(1,𝔽16)C2×C5semidirect
68. C22×C5×D4C3×C4semidirect
72. C5×#123C3×C4semidirect
77. C5×#120C3×C4semidirect
80. C2×C5×C22⋊C4C3×C4semidirect
85. #2121C3×C5direct (N × H)
121. C22×C3×D4C4×C5semidirect
125. C3×#123C4×C5semidirect
130. C3×#120C4×C5semidirect
133. C2×C3×C22⋊C4C4×C5semidirect
139. C24×C5C2×C3×C4semidirect
155. C22×C4×C5C2×C3×C4semidirect
169. C5×C22⋊C4C2×C3×C4semidirect
170. C5×C22⋊C4C2×C3×C4semidirect
172. C2×#100C2×C3×C5semidirect
173. C2×#99C2×C3×C5semidirect
174. #352C2×C3×C5semidirect
175. AΣL(1,𝔽16)C2×C3×C5semidirect
203. C24×C3C2×C4×C5semidirect
219. C22×C3×C4C2×C4×C5semidirect
233. C3×C22⋊C4C2×C4×C5semidirect
234. C3×C22⋊C4C2×C4×C5semidirect
270. C22×D4C3×C4×C5semidirect
274. #123C3×C4×C5semidirect
279. #120C3×C4×C5semidirect
282. C2×C22⋊C4C3×C4×C5semidirect
358. C24C2×C3×C4×C5semidirect
374. C22×C4C2×C3×C4×C5semidirect
388. C22⋊C4C2×C3×C4×C5semidirect
389. C22⋊C4C2×C3×C4×C5semidirect
390. C3×C5#2121direct (N × H)
470. C5C3×#2121direct (N × H)
497. C3C5×#2121direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5342
C2×C3×C5×#1002
C2×C3×C5×#992
C3×C5×#3522
C3×C5×AΣL(1,𝔽16)2
C5×#21213
C3×#21215