Finite Group Databasebeta

← C3×C5×#2003 #14275037

Subgroups — C3×C5×#2003 #14275037

Order 1920.

764 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×AGL(1,ℤ/8)C211yesmaximal, characteristic
2C3×C5×#536C211yesmaximal, characteristic, complemented
3C2×C3×C5×#124C211yesmaximal, characteristic, complemented
4C2×C3×C5×#104C211yesmaximal, characteristic, complemented
5C3×C5×#418C211yesmaximal, characteristic, complemented
6C2×C3×C5×#101C211yesmaximal, characteristic, complemented
7C3×C5×#339C211yesmaximal, characteristic, complemented
8C5×#2003C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C22×C3×C5×Q8C2211yescharacteristic
11C22×C3×C5×D4C2211yescharacteristic
12C3×C5×AGL(1,ℤ/8)—22no—
13C3×C5×AGL(1,ℤ/8)—22no—
14C2×C3×C5×SD16—21no—
15C2×C3×C5×D8—21no—
16C2×C3×C5×M16C2211yescharacteristic
17C2×C3×C5×M16C2211yescharacteristic
18C3×C5×C4⋊Q8—21no—
19C3×C5×#124—22no—
20C3×C5×#124—21no—
21C3×C5×#124—22no—
22C3×C5×#123—22no—
23C3×C5×C22⋊Q8—22no—
24C3×C5×#117C2211yescharacteristic
25C2×C3×C5×C22⋊C4—22no—
26C2×C3×C42×C5C2211yescharacteristic
27C3×C5×#104—22no—
28C3×C5×#102—22no—
29C3×C5×#101—22no—
30C3×#2003C511yesmaximal, characteristic, direct product factor, Hall
31C2×C5×AGL(1,ℤ/8)C2×C311yescharacteristic
32C5×#536C2×C311yescharacteristic, complemented
33C2×C5×#124C2×C311yescharacteristic, complemented
34C2×C5×#104C2×C311yescharacteristic, complemented
35C5×#418C2×C311yescharacteristic, complemented
36C2×C5×#101C2×C311yescharacteristic, complemented
37C5×#339C2×C311yescharacteristic, complemented
38C24×C3×C5—21no—
39C3×C5×#41—21no—
40C3×C5×#41—21no—
41C3×C5×#41—21no—
42C3×C5×#41—21no—
43C2×C3×C5×Q8—21no—
44C2×C3×C5×Q8—21no—
45C2×C3×C5×Q8—21no—
46C2×C3×C5×Q8—21no—
47C2×C3×C5×Q8—21no—
48C2×C3×C5×Q8—21no—
49C2×C3×C5×Q8D411yescharacteristic
50C2×C3×C5×D4—21no—
51C2×C3×C5×D4—21no—
52C2×C3×C5×D4—21no—
53C2×C3×C5×D4—21no—
54C2×C3×C5×D4—21no—
55C2×C3×C5×D4—21no—
56C2×C3×C5×D4D412yes—
57C2×C3×C5×D4D411yescharacteristic
58C22×C3×C4×C5—21no—
59C22×C3×C4×C5—21no—
60C22×C3×C4×C5D412yes—
61C22×C3×C4×C5C2311yescharacteristic
62C3×C5×SD16—24no—
63C3×C5×D8—24no—
64C3×C5×M16—21no—
65C3×C5×M16—21no—
66C3×C5×M16—21no—
67C3×C5×M16D412yes—
68C2×C3×C5×C8—21no—
69C2×C3×C5×C8—21no—
70C3×C5×C4⋊C4—24no—
71C3×C5×C4⋊C4D412yes—
72C3×C5×C22⋊C4—28no—
73C3×C5×C22⋊C4—24no—
74C3×C5×C22⋊C4—21no—
75C3×C42×C5—21no—
76C3×C42×C5—21no—
77C3×C42×C5D412yes—
78C2×C3×AGL(1,ℤ/8)C2×C511yescharacteristic
79C3×#536C2×C511yescharacteristic, complemented
80C2×C3×#124C2×C511yescharacteristic, complemented
81C2×C3×#104C2×C511yescharacteristic, complemented
82C3×#418C2×C511yescharacteristic, complemented
83C2×C3×#101C2×C511yescharacteristic, complemented
84C3×#339C2×C511yescharacteristic, complemented
85C2×C5×#41C22×C311yescharacteristic
86C22×C5×Q8C22×C311yescharacteristic
87C22×C5×D4C22×C311yescharacteristic
88C5×AGL(1,ℤ/8)—22no—
89C5×AGL(1,ℤ/8)—22no—
90C2×C5×SD16—21no—
91C2×C5×D8—21no—
92C2×C5×M16C22×C311yescharacteristic
93C2×C5×M16C22×C311yescharacteristic
94C5×C4⋊Q8—21no—
95C5×#124—22no—
96C5×#124—21no—
97C5×#124—22no—
98C5×#123—22no—
99C5×C22⋊Q8—22no—
100C5×#117C22×C311yescharacteristic
101C2×C5×C22⋊C4—22no—
102C2×C42×C5C22×C311yescharacteristic
103C5×#104—22no—
104C5×#102—22no—
105C5×#101—22no—
106#2003C3×C511yescharacteristic, direct product factor, Sylow
107C23×C3×C5—22no—
108C23×C3×C5—21no—
109C23×C3×C5—44no—
110C23×C3×C5—41no—
111C23×C3×C5—41no—
112C23×C3×C5C2×D411yescharacteristic
113C3×C5×Q8—21no—
114C3×C5×Q8—21no—
115C3×C5×Q8—41no—
116C3×C5×Q8—41no—
117C3×C5×Q8—41no—
118C3×C5×Q8—41no—
119C3×C5×D4—21no—
120C3×C5×D4—22no—
121C3×C5×D4—22no—
122C3×C5×D4—21no—
123C3×C5×D4—41no—
124C3×C5×D4—41no—
125C3×C5×D4—41no—
126C3×C5×D4—41no—
127C2×C3×C4×C5—22no—
128C2×C3×C4×C5—22no—
129C2×C3×C4×C5—22no—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5—21no—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5—42no—
134C2×C3×C4×C5—41no—
135C2×C3×C4×C5—41no—
136C2×C3×C4×C5C2×D411yescharacteristic
137C2×C3×C4×C5C2×D411yescharacteristic
138C2×C3×C4×C5C2×D411yescharacteristic
139C2×C3×C4×C5C2×D411yescharacteristic
140C2×C3×C4×C5C2×D411yescharacteristic
141C2×C3×C4×C5#4111yescharacteristic
142C3×C5×C8—22no—
143C3×C5×C8—41no—
144C2×C3×#41C22×C511yescharacteristic
145C22×C3×Q8C22×C511yescharacteristic
146C22×C3×D4C22×C511yescharacteristic
147C3×AGL(1,ℤ/8)—22no—
148C3×AGL(1,ℤ/8)—22no—
149C2×C3×SD16—21no—
150C2×C3×D8—21no—
151C2×C3×M16C22×C511yescharacteristic
152C2×C3×M16C22×C511yescharacteristic
153C3×C4⋊Q8—21no—
154C3×#124—22no—
155C3×#124—21no—
156C3×#124—22no—
157C3×#123—22no—
158C3×C22⋊Q8—22no—
159C3×#117C22×C511yescharacteristic
160C2×C3×C22⋊C4—22no—
161C2×C3×C42C22×C511yescharacteristic
162C3×#104—22no—
163C3×#102—22no—
164C3×#101—22no—
165C24×C5—21no—
166C5×#41—21no—
167C5×#41—21no—
168C5×#41—21no—
169C5×#41—21no—
170C2×C5×Q8—21no—
171C2×C5×Q8—21no—
172C2×C5×Q8—21no—
173C2×C5×Q8—21no—
174C2×C5×Q8—21no—
175C2×C5×Q8—21no—
176C2×C5×Q8C3×D411yescharacteristic
177C2×C5×D4—21no—
178C2×C5×D4—21no—
179C2×C5×D4—21no—
180C2×C5×D4—21no—
181C2×C5×D4—21no—
182C2×C5×D4—21no—
183C2×C5×D4C3×D411yescharacteristic
184C2×C5×D4C3×D412yes—
185C22×C4×C5—21no—
186C22×C4×C5—21no—
187C22×C4×C5C3×D412yes—
188C22×C4×C5C23×C311yescharacteristic
189C5×SD16—24no—
190C5×D8—24no—
191C5×M16—21no—
192C5×M16—21no—
193C5×M16—21no—
194C5×M16C3×D412yes—
195C2×C5×C8—21no—
196C2×C5×C8—21no—
197C5×C4⋊C4—24no—
198C5×C4⋊C4C3×D412yes—
199C5×C22⋊C4—28no—
200C5×C22⋊C4—24no—
201C5×C22⋊C4—21no—
202C42×C5—21no—
203C42×C5—21no—
204C42×C5C3×D412yes—
205C2×AGL(1,ℤ/8)C2×C3×C511yescharacteristic
206#536C2×C3×C511yescharacteristic, complemented
207C2×#124C2×C3×C511yescharacteristic, complemented
208C2×#104C2×C3×C511yescharacteristic, complemented
209#418C2×C3×C511yescharacteristic, complemented
210C2×#101C2×C3×C511yescharacteristic, complemented
211#339C2×C3×C511yescharacteristic, complemented
212C22×C3×C5—22no—
213C22×C3×C5—44no—
214C22×C3×C5—41no—
215C22×C3×C5—42no—
216C22×C3×C5—42no—
217C22×C3×C5—82no—
218C22×C3×C5—82no—
219C22×C3×C5#12011yescharacteristic
220C22×C3×C5#12011yescharacteristic
221C22×C3×C5#12111yescharacteristic
222C3×C4×C5—42no—
223C3×C4×C5—42no—
224C3×C4×C5—41no—
225C3×C4×C5—42no—
226C3×C4×C5#12011yescharacteristic
227C3×C4×C5#12111yescharacteristic
228C3×C4×C5#12111yescharacteristic
229C3×C4×C5#12711yescharacteristic
230C24×C3—21no—
231C3×#41—21no—
232C3×#41—21no—
233C3×#41—21no—
234C3×#41—21no—
235C2×C3×Q8—21no—
236C2×C3×Q8—21no—
237C2×C3×Q8—21no—
238C2×C3×Q8—21no—
239C2×C3×Q8—21no—
240C2×C3×Q8—21no—
241C2×C3×Q8C5×D411yescharacteristic
242C2×C3×D4—21no—
243C2×C3×D4—21no—
244C2×C3×D4—21no—
245C2×C3×D4—21no—
246C2×C3×D4—21no—
247C2×C3×D4—21no—
248C2×C3×D4C5×D411yescharacteristic
249C2×C3×D4C5×D412yes—
250C22×C3×C4—21no—
251C22×C3×C4—21no—
252C22×C3×C4C5×D412yes—
253C22×C3×C4C23×C511yescharacteristic
254C3×SD16—24no—
255C3×D8—24no—
256C3×M16—21no—
257C3×M16—21no—
258C3×M16—21no—
259C3×M16C5×D412yes—
260C2×C3×C8—21no—
261C2×C3×C8—21no—
262C3×C4⋊C4—24no—
263C3×C4⋊C4C5×D412yes—
264C3×C22⋊C4—28no—
265C3×C22⋊C4—24no—
266C3×C22⋊C4—21no—
267C3×C42—21no—
268C3×C42—21no—
269C3×C42C5×D412yes—
270C23×C5—21no—
271C23×C5—22no—
272C23×C5—44no—
273C23×C5—41no—
274C23×C5—41no—
275C23×C5C2×C3×D411yescharacteristic
276C5×Q8—21no—
277C5×Q8—21no—
278C5×Q8—41no—
279C5×Q8—41no—
280C5×Q8—41no—
281C5×Q8—41no—
282C5×D4—21no—
283C5×D4—22no—
284C5×D4—22no—
285C5×D4—21no—
286C5×D4—41no—
287C5×D4—41no—
288C5×D4—41no—
289C5×D4—41no—
290C2×C4×C5—22no—
291C2×C4×C5—22no—
292C2×C4×C5—22no—
293C2×C4×C5—22no—
294C2×C4×C5—21no—
295C2×C4×C5—22no—
296C2×C4×C5—42no—
297C2×C4×C5—41no—
298C2×C4×C5—41no—
299C2×C4×C5C2×C3×D411yescharacteristic
300C2×C4×C5C2×C3×D411yescharacteristic
301C2×C4×C5C2×C3×D411yescharacteristic
302C2×C4×C5C2×C3×D411yescharacteristic
303C2×C4×C5C2×C3×D411yescharacteristic
304C2×C4×C5C3×#4111yescharacteristic
305C5×C8—22no—
306C5×C8—41no—
307C2×#41C22×C3×C511yescharacteristic
308C22×Q8C22×C3×C511yescharacteristic
309C22×D4C22×C3×C511yescharacteristic
310AGL(1,ℤ/8)—22no—
311AGL(1,ℤ/8)—22no—
312C2×SD16—21no—
313C2×D8—21no—
314C2×M16C22×C3×C511yescharacteristic
315C2×M16C22×C3×C511yescharacteristic
316C4⋊Q8—21no—
317#124—22no—
318#124—21no—
319#124—22no—
320#123—22no—
321C22⋊Q8—22no—
322#117C22×C3×C511yescharacteristic
323C2×C22⋊C4—22no—
324C2×C42C22×C3×C511yescharacteristic
325#104—22no—
326#102—22no—
327#101—22no—
328C2×C3×C5—21no—
329C2×C3×C5—21no—
330C2×C3×C5—81no—
331C2×C3×C5—82no—
332C2×C3×C5#39211yescharacteristic
333C2×C3×C5#45412yes—
334C23×C3—21no—
335C23×C3—22no—
336C23×C3—44no—
337C23×C3—41no—
338C23×C3—41no—
339C23×C3C2×C5×D411yescharacteristic
340C3×Q8—21no—
341C3×Q8—21no—
342C3×Q8—41no—
343C3×Q8—41no—
344C3×Q8—41no—
345C3×Q8—41no—
346C3×D4—21no—
347C3×D4—22no—
348C3×D4—22no—
349C3×D4—21no—
350C3×D4—41no—
351C3×D4—41no—
352C3×D4—41no—
353C3×D4—41no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—21no—
359C2×C3×C4—22no—
360C2×C3×C4—42no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4C2×C5×D411yescharacteristic
364C2×C3×C4C2×C5×D411yescharacteristic
365C2×C3×C4C2×C5×D411yescharacteristic
366C2×C3×C4C2×C5×D411yescharacteristic
367C2×C3×C4C2×C5×D411yescharacteristic
368C2×C3×C4C5×#4111yescharacteristic
369C3×C8—22no—
370C3×C8—41no—
371C22×C5—22no—
372C22×C5—44no—
373C22×C5—41no—
374C22×C5—42no—
375C22×C5—42no—
376C22×C5—82no—
377C22×C5—82no—
378C22×C5C3×#12011yescharacteristic
379C22×C5C3×#12011yescharacteristic
380C22×C5C3×#12111yescharacteristic
381C4×C5—42no—
382C4×C5—42no—
383C4×C5—41no—
384C4×C5—42no—
385C4×C5C3×#12011yescharacteristic
386C4×C5C3×#12111yescharacteristic
387C4×C5C3×#12111yescharacteristic
388C4×C5C3×#12711yescharacteristic
389C24—21no—
390#41—21no—
391#41—21no—
392#41—21no—
393#41—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8C3×C5×D411yescharacteristic
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4C3×C5×D411yescharacteristic
408C2×D4C3×C5×D412yes—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4C3×C5×D412yes—
412C22×C4C23×C3×C511yescharacteristic
413SD16—24no—
414D8—24no—
415M16—21no—
416M16—21no—
417M16—21no—
418M16C3×C5×D412yes—
419C2×C8—21no—
420C2×C8—21no—
421C4⋊C4—24no—
422C4⋊C4C3×C5×D412yes—
423C22⋊C4—28no—
424C22⋊C4—24no—
425C22⋊C4—21no—
426C42—21no—
427C42—21no—
428C42C3×C5×D412yes—
429C3×C5#200311yescharacteristic, direct product factor, Hall
430C22×C3—22no—
431C22×C3—44no—
432C22×C3—41no—
433C22×C3—42no—
434C22×C3—42no—
435C22×C3—82no—
436C22×C3—82no—
437C22×C3C5×#12011yescharacteristic
438C22×C3C5×#12011yescharacteristic
439C22×C3C5×#12111yescharacteristic
440C3×C4—42no—
441C3×C4—42no—
442C3×C4—41no—
443C3×C4—42no—
444C3×C4C5×#12011yescharacteristic
445C3×C4C5×#12111yescharacteristic
446C3×C4C5×#12111yescharacteristic
447C3×C4C5×#12711yescharacteristic
448C2×C5—21no—
449C2×C5—21no—
450C2×C5—82no—
451C2×C5—81no—
452C2×C5C3×#39211yescharacteristic
453C2×C5C3×#45412yes—
454C23—21no—
455C23—22no—
456C23—44no—
457C23—41no—
458C23—41no—
459C23C2×C3×C5×D411yescharacteristic
460Q8—21no—
461Q8—21no—
462Q8—41no—
463Q8—41no—
464Q8—41no—
465Q8—41no—
466D4—21no—
467D4—22no—
468D4—22no—
469D4—21no—
470D4—41no—
471D4—41no—
472D4—41no—
473D4—41no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4C2×C3×C5×D411yescharacteristic
484C2×C4C2×C3×C5×D411yescharacteristic
485C2×C4C2×C3×C5×D411yescharacteristic
486C2×C4C2×C3×C5×D411yescharacteristic
487C2×C4C2×C3×C5×D411yescharacteristic
488C2×C4C3×C5×#4111yescharacteristic
489C8—22no—
490C8—41no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—82no—
494C2×C3—81no—
495C2×C3C5×#39211yescharacteristic
496C2×C3C5×#45412yes—
497C5C3×#200311yescharacteristic, direct product factor, Sylow
498C22—22no—
499C22—44no—
500C22—42no—
501C22—41no—
502C22—42no—
503C22—82no—
504C22—82no—
505C22C3×C5×#12011yescharacteristic
506C22C3×C5×#12011yescharacteristic
507C22C3×C5×#12111yescharacteristic
508C4—42no—
509C4—42no—
510C4—42no—
511C4—41no—
512C4C3×C5×#12011yescharacteristic
513C4C3×C5×#12111yescharacteristic
514C4C3×C5×#12111yescharacteristic
515C4C3×C5×#12711yescharacteristic
516C3C5×#200311yescharacteristic, direct product factor, Sylow
517C2—21no—
518C2—21no—
519C2—82no—
520C2—81no—
521C2C3×C5×#39211yescharacteristic
522C2C3×C5×#45412yes—
subgroupG/H
center, socleC22×C3×C5#120
commutator, Frattini, cosocleC22×C4C23×C3×C5
prime radicalC2×C3×C5×AGL(1,ℤ/8)C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3×C5 ◁ C22×C3×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×C5×#536C2semidirect
3. C2×C3×C5×#124C2semidirect
4. C2×C3×C5×#104C2semidirect
5. C3×C5×#418C2semidirect
6. C2×C3×C5×#101C2semidirect
7. C3×C5×#339C2semidirect
8. C5×#2003C3direct (N × H)
30. C3×#2003C5direct (N × H)
32. C5×#536C2×C3semidirect
33. C2×C5×#124C2×C3semidirect
34. C2×C5×#104C2×C3semidirect
35. C5×#418C2×C3semidirect
36. C2×C5×#101C2×C3semidirect
37. C5×#339C2×C3semidirect
79. C3×#536C2×C5semidirect
80. C2×C3×#124C2×C5semidirect
81. C2×C3×#104C2×C5semidirect
82. C3×#418C2×C5semidirect
83. C2×C3×#101C2×C5semidirect
84. C3×#339C2×C5semidirect
106. #2003C3×C5direct (N × H)
206. #536C2×C3×C5semidirect
207. C2×#124C2×C3×C5semidirect
208. C2×#104C2×C3×C5semidirect
209. #418C2×C3×C5semidirect
210. C2×#101C2×C3×C5semidirect
211. #339C2×C3×C5semidirect
429. C3×C5#2003direct (N × H)
497. C5C3×#2003direct (N × H)
516. C3C5×#2003direct (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×AGL(1,ℤ/8)2
C3×C5×#5362
C2×C3×C5×#1242
C2×C3×C5×#1042
C3×C5×#4182
C2×C3×C5×#1012
C3×C5×#3392
C5×#20033
C3×#20035