Finite Group Databasebeta

← C2×C3×#65483 #14478267

Subgroups — C2×C3×#65483 #14478267

Order 1920.

736 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#135C211yesmaximal, characteristic, complemented
2C2×C3×#4050C211yesmaximal, characteristic, complemented
3C3×#65483C218yesmaximal, direct product factor
4C2×C3×#3961C211yesmaximal, characteristic, complemented
5C2×C3×#3960C211yesmaximal, characteristic, complemented
6C2×C3×#3959C211yesmaximal, characteristic, complemented
7C2×C3×#3958C211yesmaximal, characteristic, complemented
8C22×C3×#672C211yesmaximal, characteristic, complemented
9C2×#65483C311yesmaximal, characteristic, direct product factor, Hall
10C2×C3×C5×#41—21no—
11C2×C3×#709—21no—
12C3×C5×#135—22no—
13C3×C5×#135C2214yescomplemented
14C2×C3×C5×Q16C2211yescharacteristic
15C2×C3×C5×SD16—21no—
16C2×C3×C5×D8C2211yescharacteristic
17C22×C3×C5×C8C2211yescharacteristic
18C2×C3×Dic20C2211yescharacteristic
19C3×#4050—22no—
20C3×#4050C2214yescomplemented
21C2×C3×D40C2211yescharacteristic
22C2×C3×#677—21no—
23C3×#3961C2214yescomplemented
24C3×#3960C2214yescomplemented
25C3×#3959C2214yescomplemented
26C3×#3958C2214yescomplemented
27C2×C3×#672C2211yescharacteristic
28C2×C3×#672C2211yescharacteristic
29C2×C3×#672C2214yescomplemented
30C2×C3×#508—51nomaximal, Hall
31C2×C5×#135C2×C311yescharacteristic, complemented
32C2×#4050C2×C311yescharacteristic, complemented
33#65483C2×C318yesdirect product factor
34C2×#3961C2×C311yescharacteristic, complemented
35C2×#3960C2×C311yescharacteristic, complemented
36C2×#3959C2×C311yescharacteristic, complemented
37C2×#3958C2×C311yescharacteristic, complemented
38C22×#672C2×C311yescharacteristic, complemented
39C3×C5×#41—24no—
40C3×C5×#41—42no—
41C2×C3×C5×Q8—21no—
42C2×C3×C5×D4—21no—
43C2×C3×C5×D4—41no—
44C22×C3×C4×C5—41no—
45C22×C3×C4×C5D411yescharacteristic
46C2×C3×C22⋊D5—41no—
47C3×#709—24no—
48C3×#709—42no—
49C2×C3×D20—21no—
50C2×C3×C4×D5—41no—
51C2×C3×Dic10—21no—
52C3×C5×Q16—21no—
53C3×C5×Q16C2312yes—
54C3×C5×SD16—24no—
55C3×C5×D8—21no—
56C3×C5×D8C2312yes—
57C2×C3×C5×C8D411yescharacteristic
58C2×C3×C5×C8D412yes—
59C2×C3×C5×C8C2311yescharacteristic
60C2×C3×C5×C8C2312yes—
61C3×Dic20—21no—
62C3×Dic20C2312yes—
63C3×D40—21no—
64C3×D40C2312yes—
65C3×#677—24no—
66C3×#672C2312yes—
67C3×#672C2312yes—
68C2×C3×#135—51no—
69C2×C3×#135D511yescharacteristic, complemented
70C3×#508—58no—
71C2×C3×Q32—51no—
72C2×C3×SD32—51no—
73C2×C3×SD32—51no—
74C2×C3×D16—51no—
75C22×C3×C16—51no—
76C2×C5×#41—21no—
77C2×#709—21no—
78C5×#135—22no—
79C5×#135C22×C314yescomplemented
80C2×C5×Q16C22×C311yescharacteristic
81C2×C5×SD16—21no—
82C2×C5×D8C22×C311yescharacteristic
83C22×C5×C8C22×C311yescharacteristic
84C2×Dic20C22×C311yescharacteristic
85#4050—22no—
86#4050C22×C314yescomplemented
87C2×D40C22×C311yescharacteristic
88C2×#677—21no—
89#3961C22×C314yescomplemented
90#3960C22×C314yescomplemented
91#3959C22×C314yescomplemented
92#3958C22×C314yescomplemented
93C2×#672C22×C311yescharacteristic
94C2×#672C22×C311yescharacteristic
95C2×#672C22×C314yescomplemented
96C2×#508—51noSylow
97C23×C3×C5—41no—
98C23×C3×C5D811yescharacteristic
99C22×C3×D5—41no—
100C3×C5×Q8—22no—
101C3×C5×Q8—41no—
102C3×C5×D4—22no—
103C3×C5×D4—44no—
104C3×C5×D4—41no—
105C2×C3×C4×C5—41no—
106C2×C3×C4×C5—44no—
107C2×C3×C4×C5D812yes—
108C2×C3×C4×C5D811yescharacteristic
109C2×C3×C4×C5C2×D411yescharacteristic
110C2×C3×C4×C5C2×D412yes—
111C3×C22⋊D5—44no—
112C2×C3×Dic5—41no—
113C3×D20—22no—
114C3×D20—41no—
115C3×C4×D5—44no—
116C3×Dic10—22no—
117C3×Dic10—41no—
118C3×C5×C8C2×D412yes—
119C3×C5×C8C2×D411yescharacteristic
120C3×C5×C8C2411yescharacteristic
121C2×C3×#41—21no—
122C2×C3×#41—101no—
123C3×#135—22no—
124C3×#135—54no—
125C3×#135—102no—
126C3×#135C2×D514yescomplemented
127C2×C3×Q16—51no—
128C2×C3×Q16C2×D511yescharacteristic
129C2×C3×SD16—21no—
130C2×C3×SD16—101no—
131C2×C3×D8—51no—
132C2×C3×D8C2×D511yescharacteristic
133C22×C3×C8C2×D511yescharacteristic
134C3×Q32—54no—
135C3×SD32—54no—
136C3×SD32—54no—
137C3×D16—54no—
138C2×C3×C16—51no—
139C2×C3×C16—51no—
140C2×C3×C16—54no—
141C5×#41—24no—
142C5×#41—42no—
143C2×C5×Q8—21no—
144C2×C5×D4—21no—
145C2×C5×D4—41no—
146C22×C4×C5—41no—
147C22×C4×C5C3×D411yescharacteristic
148C2×C22⋊D5—41no—
149#709—24no—
150#709—42no—
151C2×D20—21no—
152C2×C4×D5—41no—
153C2×Dic10—21no—
154C5×Q16—21no—
155C5×Q16C23×C312yes—
156C5×SD16—24no—
157C5×D8—21no—
158C5×D8C23×C312yes—
159C2×C5×C8C3×D411yescharacteristic
160C2×C5×C8C3×D412yes—
161C2×C5×C8C23×C311yescharacteristic
162C2×C5×C8C23×C312yes—
163Dic20—21no—
164Dic20C23×C312yes—
165D40—21no—
166D40C23×C312yes—
167#677—24no—
168#672C23×C312yes—
169#672C23×C312yes—
170C2×#135—51no—
171C2×#135C3×D511yescharacteristic, complemented
172#508—58no—
173C2×Q32—51no—
174C2×SD32—51no—
175C2×SD32—51no—
176C2×D16—51no—
177C22×C16—51no—
178C22×C3×C5—22no—
179C22×C3×C5—42no—
180C22×C3×C5—82no—
181C22×C3×C5C2×D812yes—
182C22×C3×C5C2×D811yescharacteristic
183C2×C3×D5—42no—
184C2×C3×D5—82no—
185C3×C4×C5—42no—
186C3×C4×C5C2×D811yescharacteristic
187C3×C4×C5C2×D812yes—
188C3×C4×C5C22×D411yescharacteristic
189C3×Dic5—42no—
190C3×#41—24no—
191C3×#41—42no—
192C3×#41—104no—
193C3×#41—202no—
194C2×C3×Q8—21no—
195C2×C3×Q8—101no—
196C2×C3×D4—21no—
197C2×C3×D4—41no—
198C2×C3×D4—101no—
199C2×C3×D4—201no—
200C22×C3×C4—41no—
201C22×C3×C4—201no—
202C22×C3×C4C22⋊D511yescharacteristic
203C3×Q16—21no—
204C3×Q16—52no—
205C3×Q16—101no—
206C3×Q16C22×D512yes—
207C3×SD16—24no—
208C3×SD16—104no—
209C3×D8—21no—
210C3×D8—52no—
211C3×D8—101no—
212C3×D8C22×D512yes—
213C2×C3×C8C22⋊D511yescharacteristic
214C2×C3×C8C22⋊D512yes—
215C2×C3×C8C22×D511yescharacteristic
216C2×C3×C8C22×D512yes—
217C3×C16—52no—
218C3×C16—52no—
219C23×C5—41no—
220C23×C5C3×D811yescharacteristic
221C22×D5—41no—
222C5×Q8—22no—
223C5×Q8—41no—
224C5×D4—22no—
225C5×D4—44no—
226C5×D4—41no—
227C2×C4×C5—41no—
228C2×C4×C5—44no—
229C2×C4×C5C3×D811yescharacteristic
230C2×C4×C5C3×D812yes—
231C2×C4×C5C2×C3×D411yescharacteristic
232C2×C4×C5C2×C3×D412yes—
233C22⋊D5—44no—
234C2×Dic5—41no—
235D20—22no—
236D20—41no—
237C4×D5—44no—
238Dic10—22no—
239Dic10—41no—
240C5×C8C2×C3×D412yes—
241C5×C8C2×C3×D411yescharacteristic
242C5×C8C24×C311yescharacteristic
243C2×#41—21no—
244C2×#41—101no—
245#135—22no—
246#135—54no—
247#135—102no—
248#135C2×C3×D514yescomplemented
249C2×Q16—51no—
250C2×Q16C2×C3×D511yescharacteristic
251C2×SD16—21no—
252C2×SD16—101no—
253C2×D8—51no—
254C2×D8C2×C3×D511yescharacteristic
255C22×C8C2×C3×D511yescharacteristic
256Q32—54no—
257SD32—54no—
258SD32—54no—
259D16—54no—
260C2×C16—51no—
261C2×C16—51no—
262C2×C16—54no—
263C2×C3×C5—22no—
264C2×C3×C5—82no—
265C2×C3×C5#50812yescomplemented
266C2×C3×C5C22×D811yescharacteristic
267C3×D5—82no—
268C23×C3—41no—
269C23×C3—201no—
270C23×C3#68611yescharacteristic
271C3×Q8—22no—
272C3×Q8—41no—
273C3×Q8—102no—
274C3×Q8—201no—
275C3×D4—22no—
276C3×D4—44no—
277C3×D4—41no—
278C3×D4—102no—
279C3×D4—204no—
280C3×D4—201no—
281C2×C3×C4—41no—
282C2×C3×C4—44no—
283C2×C3×C4—201no—
284C2×C3×C4—204no—
285C2×C3×C4#68612yes—
286C2×C3×C4#68611yescharacteristic
287C2×C3×C4C2×C22⋊D511yescharacteristic
288C2×C3×C4C2×C22⋊D512yes—
289C3×C8C2×C22⋊D511yescharacteristic
290C3×C8C2×C22⋊D512yes—
291C3×C8C23×D511yescharacteristic
292C22×C5—22no—
293C22×C5—42no—
294C22×C5—82no—
295C22×C5C2×C3×D812yes—
296C22×C5C2×C3×D811yescharacteristic
297C2×D5—42no—
298C2×D5—82no—
299C4×C5—42no—
300C4×C5C2×C3×D811yescharacteristic
301C4×C5C2×C3×D812yes—
302C4×C5C22×C3×D411yescharacteristic
303Dic5—42no—
304#41—24no—
305#41—42no—
306#41—104no—
307#41—202no—
308C2×Q8—21no—
309C2×Q8—101no—
310C2×D4—21no—
311C2×D4—41no—
312C2×D4—101no—
313C2×D4—201no—
314C22×C4—41no—
315C22×C4—201no—
316C22×C4C3×C22⋊D511yescharacteristic
317Q16—21no—
318Q16—52no—
319Q16—101no—
320Q16C22×C3×D512yes—
321SD16—24no—
322SD16—104no—
323D8—21no—
324D8—52no—
325D8—101no—
326D8C22×C3×D512yes—
327C2×C8C3×C22⋊D511yescharacteristic
328C2×C8C3×C22⋊D512yes—
329C2×C8C22×C3×D511yescharacteristic
330C2×C8C22×C3×D512yes—
331C16—52no—
332C16—52no—
333C3×C5C2×#50811yescharacteristic, complemented, Hall
334C22×C3—22no—
335C22×C3—42no—
336C22×C3—82no—
337C22×C3—202no—
338C22×C3—402no—
339C22×C3C2×#68612yes—
340C22×C3C2×#68611yescharacteristic
341C3×C4—42no—
342C3×C4—202no—
343C3×C4C2×#68612yes—
344C3×C4C2×#68611yescharacteristic
345C3×C4C22×C22⋊D511yescharacteristic
346C2×C5—22no—
347C2×C5—82no—
348C2×C5C3×#50812yescomplemented
349C2×C5C22×C3×D811yescharacteristic
350D5—82no—
351C23—41no—
352C23—201no—
353C23C3×#68611yescharacteristic
354Q8—22no—
355Q8—41no—
356Q8—102no—
357Q8—201no—
358D4—22no—
359D4—44no—
360D4—41no—
361D4—102no—
362D4—204no—
363D4—201no—
364C2×C4—41no—
365C2×C4—44no—
366C2×C4—201no—
367C2×C4—204no—
368C2×C4C3×#68611yescharacteristic
369C2×C4C3×#68612yes—
370C2×C4C2×C3×C22⋊D511yescharacteristic
371C2×C4C2×C3×C22⋊D512yes—
372C8C2×C3×C22⋊D511yescharacteristic
373C8C2×C3×C22⋊D512yes—
374C8C23×C3×D511yescharacteristic
375C2×C3—22no—
376C2×C3—82no—
377C2×C3—402no—
378C2×C3#6548312yesdirect product factor
379C2×C3C22×#68611yescharacteristic
380C5C2×C3×#50811yescharacteristic, complemented, Sylow
381C22—22no—
382C22—42no—
383C22—82no—
384C22—202no—
385C22—402no—
386C22C2×C3×#68612yes—
387C22C2×C3×#68611yescharacteristic
388C4—42no—
389C4—202no—
390C4C2×C3×#68612yes—
391C4C2×C3×#68611yescharacteristic
392C4C22×C3×C22⋊D511yescharacteristic
393C3C2×#6548311yescharacteristic, direct product factor, Sylow
394C2—22no—
395C2—82no—
396C2—402no—
397C2C3×#6548312yesdirect product factor
398C2C22×C3×#68611yescharacteristic
subgroupG/H
centerC2×C3×C4#686
commutator, cosocleC5×C8C24×C3
FrattiniC8C23×C3×D5
FittingC2×C3×C5×#135C2
socleC22×C3×C5C2×D8
hypercenterC2×C3×#135D5
hypocenterC5C2×C3×#508
LCSG ▷ C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ C2×C3×#135

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×#135C2semidirect
2. C2×C3×#4050C2semidirect
3. C3×#65483C2direct (N × H)
4. C2×C3×#3961C2semidirect
5. C2×C3×#3960C2semidirect
6. C2×C3×#3959C2semidirect
7. C2×C3×#3958C2semidirect
8. C22×C3×#672C2semidirect
9. C2×#65483C3direct (N × H)
13. C3×C5×#135C22semidirect
20. C3×#4050C22semidirect
23. C3×#3961C22semidirect
24. C3×#3960C22semidirect
25. C3×#3959C22semidirect
26. C3×#3958C22semidirect
29. C2×C3×#672C22semidirect
31. C2×C5×#135C2×C3semidirect
32. C2×#4050C2×C3semidirect
33. #65483C2×C3direct (N × H)
34. C2×#3961C2×C3semidirect
35. C2×#3960C2×C3semidirect
36. C2×#3959C2×C3semidirect
37. C2×#3958C2×C3semidirect
38. C22×#672C2×C3semidirect
69. C2×C3×#135D5semidirect
79. C5×#135C22×C3semidirect
86. #4050C22×C3semidirect
89. #3961C22×C3semidirect
90. #3960C22×C3semidirect
91. #3959C22×C3semidirect
92. #3958C22×C3semidirect
95. C2×#672C22×C3semidirect
126. C3×#135C2×D5semidirect
171. C2×#135C3×D5semidirect
248. #135C2×C3×D5semidirect
265. C2×C3×C5#508semidirect
333. C3×C5C2×#508semidirect
348. C2×C5C3×#508semidirect
378. C2×C3#65483direct (N × H)
380. C5C2×C3×#508semidirect
393. C3C2×#65483direct (N × H)
397. C2C3×#65483direct (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×#1352
C2×C3×#40502
C3×#654832
C2×C3×#39612
C2×C3×#39602
C2×C3×#39592
C2×C3×#39582
C22×C3×#6722
C2×#654833
C2×C3×#5085