Finite Group Databasebeta

← C2×C3×#65151 #14431468

Subgroups — C2×C3×#65151 #14431468

Order 1920.

920 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4043C211yesmaximal, characteristic
2C2×C3×D5×C4⋊C4C211yesmaximal, characteristic
3C2×C3×C5×#106C211yesmaximal, characteristic, complemented
4C2×C3×#3952C211yesmaximal, characteristic
5C2×C3×#3948C211yesmaximal, characteristic, complemented
6C2×C3×#3942C211yesmaximal, characteristic, complemented
7C2×C3×#3940C211yesmaximal, characteristic, complemented
8C3×#65151C218yesmaximal, direct product factor
9C2×#65151C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×C4×D5—21no—
11C22×C3×C4×D5C2211yescharacteristic
12C22×C3×Dic10C2211yescharacteristic
13C22×C3×C5×C8C2211yescharacteristic
14C2×C3×C5×C4⋊C4C2211yescharacteristic
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×C3×#685—21no—
17C2×C3×C4⋊Dic5—21no—
18C2×C3×C4⋊Dic5C2211yescharacteristic
19C2×C3×#683—21no—
20C22×C3×#168C2211yescharacteristic
21C3×#4043—22no—
22C3×#4043C2214yes—
23C3×D5×C4⋊C4—22no—
24C3×D5×C4⋊C4C2214yes—
25C3×C5×#106C2214yescomplemented
26C3×#3952C2214yes—
27C3×#3948C2214yescomplemented
28C3×#3942C2214yescomplemented
29C3×#3940C2214yescomplemented
30C2×C3×#483—51nomaximal, Hall
31C2×#4043C2×C311yescharacteristic
32C2×D5×C4⋊C4C2×C311yescharacteristic
33C2×C5×#106C2×C311yescharacteristic, complemented
34C2×#3952C2×C311yescharacteristic
35C2×#3948C2×C311yescharacteristic, complemented
36C2×#3942C2×C311yescharacteristic, complemented
37C2×#3940C2×C311yescharacteristic, complemented
38#65151C2×C318yesdirect product factor
39C23×C3×D5D411yescharacteristic
40C22×C3×C4×C5—21no—
41C22×C3×C4×C5—21no—
42C22×C3×C4×C5C2311yescharacteristic
43C22×C3×Dic5—21no—
44C22×C3×Dic5—21no—
45C22×C3×Dic5D411yescharacteristic
46C2×C3×C4×D5—22no—
47C2×C3×C4×D5—22no—
48C2×C3×C4×D5—24no—
49C2×C3×C4×D5—44no—
50C2×C3×C4×D5D412yes—
51C2×C3×C4×D5C2312yes—
52C2×C3×Dic10—22no—
53C2×C3×Dic10—22no—
54C2×C3×Dic10—21no—
55C2×C3×Dic10C2312yes—
56C2×C3×C5×C8—22no—
57C2×C3×C5×C8C2312yes—
58C3×C5×C4⋊C4—21no—
59C3×C5×C4⋊C4—21no—
60C3×C5×C4⋊C4C2312yes—
61C3×C5×C4⋊C4C2312yes—
62C3×#685—24no—
63C3×C4⋊Dic5—21no—
64C3×C4⋊Dic5—24no—
65C3×C4⋊Dic5C2312yes—
66C3×#683—24no—
67C2×C3×#168—22no—
68C2×C3×#168C2312yes—
69C2×C3×C22⋊Q8—51no—
70C22×C3×C4⋊C4—51no—
71C3×#483—58no—
72C2×C3×#106—51no—
73C2×C3×#106D511yescharacteristic, complemented
74C2×C3×#103—51no—
75C2×C3×#103—51no—
76C2×C3×C22⋊C8—51no—
77C22×C4×D5—21no—
78C22×C4×D5C22×C311yescharacteristic
79C22×Dic10C22×C311yescharacteristic
80C22×C5×C8C22×C311yescharacteristic
81C2×C5×C4⋊C4C22×C311yescharacteristic
82C2×C5×C4⋊C4C22×C311yescharacteristic
83C2×#685—21no—
84C2×C4⋊Dic5—21no—
85C2×C4⋊Dic5C22×C311yescharacteristic
86C2×#683—21no—
87C22×#168C22×C311yescharacteristic
88#4043—22no—
89#4043C22×C314yes—
90D5×C4⋊C4—22no—
91D5×C4⋊C4C22×C314yes—
92C5×#106C22×C314yescomplemented
93#3952C22×C314yes—
94#3948C22×C314yescomplemented
95#3942C22×C314yescomplemented
96#3940C22×C314yescomplemented
97C2×#483—51noSylow
98C23×C3×C5C2×D411yescharacteristic
99C22×C3×D5—22no—
100C22×C3×D5—22no—
101C22×C3×D5SD1614yes—
102C22×C3×D5C2×D412yes—
103C2×C3×C4×C5—22no—
104C2×C3×C4×C5—22no—
105C2×C3×C4×C5—42no—
106C2×C3×C4×C5—42no—
107C2×C3×C4×C5C2×D411yescharacteristic
108C2×C3×C4×C5#4112yes—
109C2×C3×C4×C5#4112yes—
110C2×C3×C4×C5C2411yescharacteristic
111C2×C3×Dic5—22no—
112C2×C3×Dic5—22no—
113C2×C3×Dic5—22no—
114C2×C3×Dic5—42no—
115C2×C3×Dic5—42no—
116C2×C3×Dic5C2×D412yes—
117C3×C4×D5—24no—
118C3×C4×D5—22no—
119C3×C4×D5—22no—
120C3×C4×D5—48no—
121C3×Dic10—22no—
122C3×Dic10—22no—
123C3×Dic10—42no—
124C3×C5×C8—22no—
125C3×#168—22no—
126C22×C3×Q8—51no—
127C23×C3×C4—51no—
128C23×C3×C4—101no—
129C22×C3×C8—51no—
130C22×C3×C8C2×D511yescharacteristic
131C3×C22⋊Q8—54no—
132C3×C22⋊Q8—102no—
133C2×C3×C4⋊C4—51no—
134C2×C3×C4⋊C4—54no—
135C2×C3×C4⋊C4—102no—
136C2×C3×C4⋊C4—101no—
137C2×C3×C4⋊C4—101no—
138C2×C3×C4⋊C4C2×D511yescharacteristic
139C2×C3×C4⋊C4C2×D511yescharacteristic
140C2×C3×C22⋊C4—101no—
141C3×#106—54no—
142C3×#106C2×D514yescomplemented
143C3×#103—54no—
144C3×#103—54no—
145C3×C22⋊C8—54no—
146C23×D5C3×D411yescharacteristic
147C22×C4×C5—21no—
148C22×C4×C5—21no—
149C22×C4×C5C23×C311yescharacteristic
150C22×Dic5—21no—
151C22×Dic5—21no—
152C22×Dic5C3×D411yescharacteristic
153C2×C4×D5—22no—
154C2×C4×D5—22no—
155C2×C4×D5—24no—
156C2×C4×D5—44no—
157C2×C4×D5C3×D412yes—
158C2×C4×D5C23×C312yes—
159C2×Dic10—22no—
160C2×Dic10—22no—
161C2×Dic10—21no—
162C2×Dic10C23×C312yes—
163C2×C5×C8—22no—
164C2×C5×C8C23×C312yes—
165C5×C4⋊C4—21no—
166C5×C4⋊C4—21no—
167C5×C4⋊C4C23×C312yes—
168C5×C4⋊C4C23×C312yes—
169#685—24no—
170C4⋊Dic5—21no—
171C4⋊Dic5—24no—
172C4⋊Dic5C23×C312yes—
173#683—24no—
174C2×#168—22no—
175C2×#168C23×C312yes—
176C2×C22⋊Q8—51no—
177C22×C4⋊C4—51no—
178#483—58no—
179C2×#106—51no—
180C2×#106C3×D511yescharacteristic, complemented
181C2×#103—51no—
182C2×#103—51no—
183C2×C22⋊C8—51no—
184C22×C3×C5#12312yes—
185C22×C3×C5C2×SD1612yes—
186C22×C3×C5#13712yes—
187C22×C3×C5C22×D411yescharacteristic
188C2×C3×D5—22no—
189C2×C3×D5—22no—
190C2×C3×D5—28no—
191C2×C3×D5C2×SD1614yes—
192C3×C4×C5—42no—
193C3×C4×C5—42no—
194C3×C4×C5#12312yes—
195C3×C4×C5C2×#4111yescharacteristic
196C3×C4×C5C2×#4111yescharacteristic
197C3×Dic5—22no—
198C3×Dic5—42no—
199C3×Dic5—42no—
200C24×C3—51no—
201C2×C3×Q8—52no—
202C2×C3×Q8—101no—
203C2×C3×Q8—102no—
204C2×C3×Q8—102no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—51no—
208C22×C3×C4—52no—
209C22×C3×C4—52no—
210C22×C3×C4—104no—
211C22×C3×C4—102no—
212C22×C3×C4—102no—
213C22×C3×C4—101no—
214C22×C3×C4—101no—
215C22×C3×C4—204no—
216C22×C3×C4C22×D511yescharacteristic
217C2×C3×C8—22no—
218C2×C3×C8—52no—
219C2×C3×C8—102no—
220C2×C3×C8C22×D512yes—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—52no—
224C3×C4⋊C4—104no—
225C3×C4⋊C4—104no—
226C3×C4⋊C4—101no—
227C3×C4⋊C4C22×D512yes—
228C3×C4⋊C4C22×D512yes—
229C3×C22⋊C4—104no—
230C23×C5C2×C3×D411yescharacteristic
231C22×D5—22no—
232C22×D5—22no—
233C22×D5C3×SD1614yes—
234C22×D5C2×C3×D412yes—
235C2×C4×C5—22no—
236C2×C4×C5—22no—
237C2×C4×C5—42no—
238C2×C4×C5—42no—
239C2×C4×C5C2×C3×D411yescharacteristic
240C2×C4×C5C3×#4112yes—
241C2×C4×C5C3×#4112yes—
242C2×C4×C5C24×C311yescharacteristic
243C2×Dic5—22no—
244C2×Dic5—22no—
245C2×Dic5—22no—
246C2×Dic5—42no—
247C2×Dic5—42no—
248C2×Dic5C2×C3×D412yes—
249C4×D5—24no—
250C4×D5—22no—
251C4×D5—22no—
252C4×D5—48no—
253Dic10—22no—
254Dic10—22no—
255Dic10—42no—
256C5×C8—22no—
257#168—22no—
258C22×Q8—51no—
259C23×C4—51no—
260C23×C4—101no—
261C22×C8—51no—
262C22×C8C2×C3×D511yescharacteristic
263C22⋊Q8—54no—
264C22⋊Q8—102no—
265C2×C4⋊C4—51no—
266C2×C4⋊C4—54no—
267C2×C4⋊C4—102no—
268C2×C4⋊C4—101no—
269C2×C4⋊C4—101no—
270C2×C4⋊C4C2×C3×D511yescharacteristic
271C2×C4⋊C4C2×C3×D511yescharacteristic
272C2×C22⋊C4—101no—
273#106—54no—
274#106C2×C3×D514yescomplemented
275#103—54no—
276#103—54no—
277C22⋊C8—54no—
278C2×C3×C5#48314yescomplemented
279C2×C3×C5C2×#12311yescharacteristic
280C2×C3×C5C22×SD1611yescharacteristic
281C2×C3×C5C2×#13711yescharacteristic
282C3×D5—24no—
283C23×C3—52no—
284C23×C3—54no—
285C23×C3—102no—
286C23×C3—102no—
287C23×C3D4×D511yescharacteristic
288C3×Q8—102no—
289C3×Q8—102no—
290C3×Q8—202no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—52no—
296C2×C3×C4—102no—
297C2×C3×C4—102no—
298C2×C3×C4—102no—
299C2×C3×C4—104no—
300C2×C3×C4—102no—
301C2×C3×C4—102no—
302C2×C3×C4—208no—
303C2×C3×C4—202no—
304C2×C3×C4—202no—
305C2×C3×C4#70912yes—
306C2×C3×C4D4×D511yescharacteristic
307C2×C3×C4#71312yes—
308C2×C3×C4C23×D511yescharacteristic
309C3×C8—22no—
310C3×C8—102no—
311C22×C5C3×#12312yes—
312C22×C5C2×C3×SD1612yes—
313C22×C5C3×#13712yes—
314C22×C5C22×C3×D411yescharacteristic
315C2×D5—22no—
316C2×D5—22no—
317C2×D5—28no—
318C2×D5C2×C3×SD1614yes—
319C4×C5—42no—
320C4×C5—42no—
321C4×C5C3×#12312yes—
322C4×C5C2×C3×#4111yescharacteristic
323C4×C5C2×C3×#4111yescharacteristic
324Dic5—22no—
325Dic5—42no—
326Dic5—42no—
327C24—51no—
328C2×Q8—52no—
329C2×Q8—101no—
330C2×Q8—102no—
331C2×Q8—102no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—51no—
335C22×C4—52no—
336C22×C4—52no—
337C22×C4—104no—
338C22×C4—102no—
339C22×C4—102no—
340C22×C4—101no—
341C22×C4—101no—
342C22×C4—204no—
343C22×C4C22×C3×D511yescharacteristic
344C2×C8—22no—
345C2×C8—52no—
346C2×C8—102no—
347C2×C8C22×C3×D512yes—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—52no—
351C4⋊C4—104no—
352C4⋊C4—104no—
353C4⋊C4—101no—
354C4⋊C4C22×C3×D512yes—
355C4⋊C4C22×C3×D512yes—
356C22⋊C4—104no—
357C3×C5C2×#48311yescharacteristic, complemented, Hall
358C22×C3—54no—
359C22×C3—102no—
360C22×C3—102no—
361C22×C3—108no—
362C22×C3#404012yes—
363C22×C3D5×SD1612yes—
364C22×C3#406112yes—
365C22×C3C2×D4×D511yescharacteristic
366C3×C4—42no—
367C3×C4—42no—
368C3×C4—102no—
369C3×C4—202no—
370C3×C4—202no—
371C3×C4#404012yes—
372C3×C4C2×#70911yescharacteristic
373C3×C4C2×#71311yescharacteristic
374C2×C5C3×#48314yescomplemented
375C2×C5C2×C3×#12311yescharacteristic
376C2×C5C22×C3×SD1611yescharacteristic
377C2×C5C2×C3×#13711yescharacteristic
378D5—24no—
379C23—52no—
380C23—54no—
381C23—102no—
382C23—102no—
383C23C3×D4×D511yescharacteristic
384Q8—102no—
385Q8—102no—
386Q8—202no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—52no—
392C2×C4—102no—
393C2×C4—102no—
394C2×C4—102no—
395C2×C4—104no—
396C2×C4—102no—
397C2×C4—102no—
398C2×C4—208no—
399C2×C4—202no—
400C2×C4—202no—
401C2×C4C3×#70912yes—
402C2×C4C3×D4×D511yescharacteristic
403C2×C4C3×#71312yes—
404C2×C4C23×C3×D511yescharacteristic
405C8—22no—
406C8—102no—
407C2×C3—104no—
408C2×C3#6515114yesdirect product factor
409C2×C3C2×#404011yescharacteristic
410C2×C3C2×D5×SD1611yescharacteristic
411C2×C3C2×#406111yescharacteristic
412C5C2×C3×#48311yescharacteristic, complemented, Sylow
413C22—54no—
414C22—108no—
415C22—102no—
416C22—102no—
417C22C3×#404012yes—
418C22C3×D5×SD1612yes—
419C22C3×#406112yes—
420C22C2×C3×D4×D511yescharacteristic
421C4—42no—
422C4—42no—
423C4—102no—
424C4—202no—
425C4—202no—
426C4C3×#404012yes—
427C4C2×C3×#70911yescharacteristic
428C4C2×C3×#71311yescharacteristic
429C3C2×#6515111yescharacteristic, direct product factor, Sylow
430C2—104no—
431C2C3×#6515114yesdirect product factor
432C2C2×C3×#404011yescharacteristic
433C2C2×C3×D5×SD1611yescharacteristic
434C2C2×C3×#406111yescharacteristic
subgroupG/H
centerC23×C3D4×D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C5×#106C2
socleC23×C3×C5C2×D4
prime radicalC23×C3×D5D4
hypercenterC2×C3×#106D5
hypocenterC5C2×C3×#483
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C2×C3×#106

Complements

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

NH ≅ G/Ntype
3. C2×C3×C5×#106C2semidirect
5. C2×C3×#3948C2semidirect
6. C2×C3×#3942C2semidirect
7. C2×C3×#3940C2semidirect
8. C3×#65151C2direct (N × H)
9. C2×#65151C3direct (N × H)
25. C3×C5×#106C22semidirect
27. C3×#3948C22semidirect
28. C3×#3942C22semidirect
29. C3×#3940C22semidirect
33. C2×C5×#106C2×C3semidirect
35. C2×#3948C2×C3semidirect
36. C2×#3942C2×C3semidirect
37. C2×#3940C2×C3semidirect
38. #65151C2×C3direct (N × H)
73. C2×C3×#106D5semidirect
92. C5×#106C22×C3semidirect
94. #3948C22×C3semidirect
95. #3942C22×C3semidirect
96. #3940C22×C3semidirect
142. C3×#106C2×D5semidirect
180. C2×#106C3×D5semidirect
274. #106C2×C3×D5semidirect
278. C2×C3×C5#483semidirect
357. C3×C5C2×#483semidirect
374. C2×C5C3×#483semidirect
408. C2×C3#65151direct (N × H)
412. C5C2×C3×#483semidirect
429. C3C2×#65151direct (N × H)
431. C2C3×#65151direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#40432
C2×C3×D5×C4⋊C42
C2×C3×C5×#1062
C2×C3×#39522
C2×C3×#39482
C2×C3×#39422
C2×C3×#39402
C3×#651512
C2×#651513
C2×C3×#4835