Finite Group Databasebeta

← C2×C3×#65102 #14427739

Subgroups — C2×C3×#65102 #14427739

Order 1920.

1024 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4092C211yesmaximal, characteristic
2C22×C3×#689C211yesmaximal, characteristic, complemented
3C22×C3×C8×D5C211yesmaximal, characteristic
4C2×C3×#4043C211yesmaximal, characteristic
5C2×C3×C5×#103C211yesmaximal, characteristic, complemented
6C2×C3×#3948C211yesmaximal, characteristic, complemented
7C2×C3×#3939C211yesmaximal, characteristic, complemented
8C3×#65102C218yesmaximal, direct product factor
9C2×#65102C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×C5×Q8C2211yescharacteristic
11C22×C3×C4×D5C2211yescharacteristic
12C22×C3×Dic10C2211yescharacteristic
13C22×C3×C5×C8C2211yescharacteristic
14C2×C3×C5×C4⋊C4C2211yescharacteristic
15C3×#4092—22no—
16C3×#4092C2214yes—
17C2×C3×#689—24no—
18C2×C3×#689C2214yescomplemented
19C2×C3×#685—21no—
20C2×C3×#685—21no—
21C2×C3×C4⋊Dic5—21no—
22C2×C3×C4⋊Dic5C2211yescharacteristic
23C2×C3×#683—21no—
24C22×C3×#168C2211yescharacteristic
25C2×C3×C8×D5—24no—
26C2×C3×C8×D5C2214yes—
27C3×#4043—22no—
28C3×#4043C2214yes—
29C3×C5×#103C2214yescomplemented
30C3×#3948C2214yescomplemented
31C3×#3939C2214yescomplemented
32C2×C3×#467—51nomaximal, Hall
33C2×#4092C2×C311yescharacteristic
34C22×#689C2×C311yescharacteristic, complemented
35C22×C8×D5C2×C311yescharacteristic
36C2×#4043C2×C311yescharacteristic
37C2×C5×#103C2×C311yescharacteristic, complemented
38C2×#3948C2×C311yescharacteristic, complemented
39C2×#3939C2×C311yescharacteristic, complemented
40#65102C2×C318yesdirect product factor
41C23×C3×D5D411yescharacteristic
42C2×C3×C5×Q8—21no—
43C2×C3×C5×Q8—22no—
44C2×C3×C5×Q8—22no—
45C2×C3×C5×Q8C2312yes—
46C22×C3×C4×C5—21no—
47C22×C3×C4×C5—21no—
48C22×C3×C4×C5C2311yescharacteristic
49C22×C3×Dic5—21no—
50C22×C3×Dic5—21no—
51C22×C3×Dic5D411yescharacteristic
52C2×C3×C4×D5—22no—
53C2×C3×C4×D5—22no—
54C2×C3×C4×D5D412yes—
55C2×C3×C4×D5C2312yes—
56C2×C3×Dic10—22no—
57C2×C3×Dic10—22no—
58C2×C3×Dic10—21no—
59C2×C3×Dic10C2312yes—
60C2×C3×C5×C8—22no—
61C2×C3×C5×C8C2312yes—
62C3×C5×C4⋊C4—21no—
63C3×C5×C4⋊C4C2312yes—
64C3×#689—28no—
65C3×#685—24no—
66C3×#685—24no—
67C3×C4⋊Dic5—21no—
68C3×C4⋊Dic5—24no—
69C3×C4⋊Dic5C2312yes—
70C3×#683—24no—
71C2×C3×#168D414yes—
72C2×C3×#168C2312yes—
73C3×C8×D5—28no—
74C22×C3×Q16—51no—
75C23×C3×C8—51no—
76C2×C3×C22⋊Q8—51no—
77C2×C3×C22⋊Q8—51no—
78C3×#467—58no—
79C2×C3×#107—51no—
80C2×C3×#103—51no—
81C2×C3×#103D511yescharacteristic, complemented
82C22×C5×Q8C22×C311yescharacteristic
83C22×C4×D5C22×C311yescharacteristic
84C22×Dic10C22×C311yescharacteristic
85C22×C5×C8C22×C311yescharacteristic
86C2×C5×C4⋊C4C22×C311yescharacteristic
87#4092—22no—
88#4092C22×C314yes—
89C2×#689—24no—
90C2×#689C22×C314yescomplemented
91C2×#685—21no—
92C2×#685—21no—
93C2×C4⋊Dic5—21no—
94C2×C4⋊Dic5C22×C311yescharacteristic
95C2×#683—21no—
96C22×#168C22×C311yescharacteristic
97C2×C8×D5—24no—
98C2×C8×D5C22×C314yes—
99#4043—22no—
100#4043C22×C314yes—
101C5×#103C22×C314yescomplemented
102#3948C22×C314yescomplemented
103#3939C22×C314yescomplemented
104C2×#467—51noSylow
105C23×C3×C5C2×D411yescharacteristic
106C22×C3×D5—22no—
107C22×C3×D5—22no—
108C22×C3×D5Q1614yescomplemented
109C22×C3×D5C2×D412yes—
110C3×C5×Q8—22no—
111C3×C5×Q8—22no—
112C3×C5×Q8—42no—
113C2×C3×C4×C5—22no—
114C2×C3×C4×C5—22no—
115C2×C3×C4×C5—42no—
116C2×C3×C4×C5—42no—
117C2×C3×C4×C5C2×D411yescharacteristic
118C2×C3×C4×C5C2×D412yes—
119C2×C3×C4×C5#4112yes—
120C2×C3×C4×C5C2411yescharacteristic
121C2×C3×Dic5—22no—
122C2×C3×Dic5—22no—
123C2×C3×Dic5—22no—
124C2×C3×Dic5—42no—
125C2×C3×Dic5—42no—
126C2×C3×Dic5C2×D412yes—
127C3×C4×D5—24no—
128C3×C4×D5—22no—
129C3×C4×D5—22no—
130C3×Dic10—22no—
131C3×Dic10—22no—
132C3×Dic10—42no—
133C3×C5×C8—22no—
134C3×#168C2×D414yes—
135C22×C3×Q8—51no—
136C22×C3×Q8C2×D511yescharacteristic
137C23×C3×C4—51no—
138C2×C3×Q16—54no—
139C2×C3×Q16—104no—
140C22×C3×C8—51no—
141C22×C3×C8—54no—
142C22×C3×C8—104no—
143C22×C3×C8C2×D511yescharacteristic
144C3×C22⋊Q8—54no—
145C3×C22⋊Q8—54no—
146C3×C22⋊Q8—102no—
147C3×C22⋊Q8—102no—
148C2×C3×C4⋊C4—51no—
149C2×C3×C4⋊C4—101no—
150C2×C3×C4⋊C4—101no—
151C2×C3×C4⋊C4C2×D511yescharacteristic
152C2×C3×C22⋊C4—101no—
153C2×C3×C22⋊C4—101no—
154C3×#107—54no—
155C3×#103—54no—
156C3×#103C2×D514yescomplemented
157C23×D5C3×D411yescharacteristic
158C2×C5×Q8—21no—
159C2×C5×Q8—22no—
160C2×C5×Q8—22no—
161C2×C5×Q8C23×C312yes—
162C22×C4×C5—21no—
163C22×C4×C5—21no—
164C22×C4×C5C23×C311yescharacteristic
165C22×Dic5—21no—
166C22×Dic5—21no—
167C22×Dic5C3×D411yescharacteristic
168C2×C4×D5—22no—
169C2×C4×D5—22no—
170C2×C4×D5C3×D412yes—
171C2×C4×D5C23×C312yes—
172C2×Dic10—22no—
173C2×Dic10—22no—
174C2×Dic10—21no—
175C2×Dic10C23×C312yes—
176C2×C5×C8—22no—
177C2×C5×C8C23×C312yes—
178C5×C4⋊C4—21no—
179C5×C4⋊C4C23×C312yes—
180#689—28no—
181#685—24no—
182#685—24no—
183C4⋊Dic5—21no—
184C4⋊Dic5—24no—
185C4⋊Dic5C23×C312yes—
186#683—24no—
187C2×#168C3×D414yes—
188C2×#168C23×C312yes—
189C8×D5—28no—
190C22×Q16—51no—
191C23×C8—51no—
192C2×C22⋊Q8—51no—
193C2×C22⋊Q8—51no—
194#467—58no—
195C2×#107—51no—
196C2×#103—51no—
197C2×#103C3×D511yescharacteristic, complemented
198C22×C3×C5#12112yes—
199C22×C3×C5C2×Q1612yes—
200C22×C3×C5#13512yes—
201C22×C3×C5C22×D411yescharacteristic
202C2×C3×D5—22no—
203C2×C3×D5—22no—
204C2×C3×D5—28no—
205C2×C3×D5C2×Q1614yescomplemented
206C3×C4×C5—42no—
207C3×C4×C5—42no—
208C3×C4×C5#12112yes—
209C3×C4×C5C22×D411yescharacteristic
210C3×C4×C5C2×#4111yescharacteristic
211C3×Dic5—22no—
212C3×Dic5—42no—
213C3×Dic5—42no—
214C24×C3—51no—
215C2×C3×Q8—21no—
216C2×C3×Q8—22no—
217C2×C3×Q8—22no—
218C2×C3×Q8—52no—
219C2×C3×Q8—101no—
220C2×C3×Q8—102no—
221C2×C3×Q8—102no—
222C2×C3×Q8C22×D512yes—
223C22×C3×C4—21no—
224C22×C3×C4—21no—
225C22×C3×C4—51no—
226C22×C3×C4—52no—
227C22×C3×C4—52no—
228C22×C3×C4—102no—
229C22×C3×C4—102no—
230C22×C3×C4—101no—
231C22×C3×C4—101no—
232C22×C3×C4C22×D511yescharacteristic
233C3×Q16—108no—
234C2×C3×C8—22no—
235C2×C3×C8—54no—
236C2×C3×C8—52no—
237C2×C3×C8—108no—
238C2×C3×C8C22×D512yes—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—52no—
241C3×C4⋊C4—104no—
242C3×C4⋊C4—104no—
243C3×C4⋊C4—101no—
244C3×C4⋊C4C22×D512yes—
245C3×C22⋊C4—104no—
246C3×C22⋊C4—104no—
247C23×C5C2×C3×D411yescharacteristic
248C22×D5—22no—
249C22×D5—22no—
250C22×D5C3×Q1614yescomplemented
251C22×D5C2×C3×D412yes—
252C5×Q8—22no—
253C5×Q8—22no—
254C5×Q8—42no—
255C2×C4×C5—22no—
256C2×C4×C5—22no—
257C2×C4×C5—42no—
258C2×C4×C5—42no—
259C2×C4×C5C2×C3×D411yescharacteristic
260C2×C4×C5C2×C3×D412yes—
261C2×C4×C5C3×#4112yes—
262C2×C4×C5C24×C311yescharacteristic
263C2×Dic5—22no—
264C2×Dic5—22no—
265C2×Dic5—22no—
266C2×Dic5—42no—
267C2×Dic5—42no—
268C2×Dic5C2×C3×D412yes—
269C4×D5—24no—
270C4×D5—22no—
271C4×D5—22no—
272Dic10—22no—
273Dic10—22no—
274Dic10—42no—
275C5×C8—22no—
276#168C2×C3×D414yes—
277C22×Q8—51no—
278C22×Q8C2×C3×D511yescharacteristic
279C23×C4—51no—
280C2×Q16—54no—
281C2×Q16—104no—
282C22×C8—51no—
283C22×C8—54no—
284C22×C8—104no—
285C22×C8C2×C3×D511yescharacteristic
286C22⋊Q8—54no—
287C22⋊Q8—54no—
288C22⋊Q8—102no—
289C22⋊Q8—102no—
290C2×C4⋊C4—51no—
291C2×C4⋊C4—101no—
292C2×C4⋊C4—101no—
293C2×C4⋊C4C2×C3×D511yescharacteristic
294C2×C22⋊C4—101no—
295C2×C22⋊C4—101no—
296#107—54no—
297#103—54no—
298#103C2×C3×D514yescomplemented
299C2×C3×C5#46714yescomplemented
300C2×C3×C5C2×#12111yescharacteristic
301C2×C3×C5C22×Q1611yescharacteristic
302C2×C3×C5C2×#13511yescharacteristic
303C3×D5—24no—
304C23×C3—52no—
305C23×C3—54no—
306C23×C3—102no—
307C23×C3—102no—
308C23×C3D4×D511yescharacteristic
309C3×Q8—22no—
310C3×Q8—22no—
311C3×Q8—42no—
312C3×Q8—102no—
313C3×Q8—102no—
314C3×Q8—202no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—52no—
320C2×C3×C4—102no—
321C2×C3×C4—102no—
322C2×C3×C4—102no—
323C2×C3×C4—104no—
324C2×C3×C4—102no—
325C2×C3×C4—102no—
326C2×C3×C4—202no—
327C2×C3×C4—202no—
328C2×C3×C4#70912yes—
329C2×C3×C4D4×D512yes—
330C2×C3×C4D4×D511yescharacteristic
331C2×C3×C4C23×D511yescharacteristic
332C3×C8—22no—
333C3×C8—54no—
334C22×C5C3×#12112yes—
335C22×C5C2×C3×Q1612yes—
336C22×C5C3×#13512yes—
337C22×C5C22×C3×D411yescharacteristic
338C2×D5—22no—
339C2×D5—22no—
340C2×D5—28no—
341C2×D5C2×C3×Q1614yescomplemented
342C4×C5—42no—
343C4×C5—42no—
344C4×C5C3×#12112yes—
345C4×C5C22×C3×D411yescharacteristic
346C4×C5C2×C3×#4111yescharacteristic
347Dic5—22no—
348Dic5—42no—
349Dic5—42no—
350C24—51no—
351C2×Q8—21no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—52no—
355C2×Q8—101no—
356C2×Q8—102no—
357C2×Q8—102no—
358C2×Q8C22×C3×D512yes—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—51no—
362C22×C4—52no—
363C22×C4—52no—
364C22×C4—102no—
365C22×C4—102no—
366C22×C4—101no—
367C22×C4—101no—
368C22×C4C22×C3×D511yescharacteristic
369Q16—108no—
370C2×C8—22no—
371C2×C8—54no—
372C2×C8—52no—
373C2×C8—108no—
374C2×C8C22×C3×D512yes—
375C4⋊C4—21no—
376C4⋊C4—52no—
377C4⋊C4—104no—
378C4⋊C4—104no—
379C4⋊C4—101no—
380C4⋊C4C22×C3×D512yes—
381C22⋊C4—104no—
382C22⋊C4—104no—
383C3×C5C2×#46711yescharacteristic, complemented, Hall
384C22×C3—54no—
385C22×C3—102no—
386C22×C3—102no—
387C22×C3—108no—
388C22×C3#403012yes—
389C22×C3#406212yes—
390C22×C3D5×Q1612yes—
391C22×C3C2×D4×D511yescharacteristic
392C3×C4—42no—
393C3×C4—42no—
394C3×C4—102no—
395C3×C4—202no—
396C3×C4—202no—
397C3×C4#403012yes—
398C3×C4C2×#70911yescharacteristic
399C3×C4C2×D4×D511yescharacteristic
400C2×C5C3×#46714yescomplemented
401C2×C5C2×C3×#12111yescharacteristic
402C2×C5C22×C3×Q1611yescharacteristic
403C2×C5C2×C3×#13511yescharacteristic
404D5—24no—
405C23—52no—
406C23—54no—
407C23—102no—
408C23—102no—
409C23C3×D4×D511yescharacteristic
410Q8—22no—
411Q8—22no—
412Q8—42no—
413Q8—102no—
414Q8—102no—
415Q8—202no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—52no—
421C2×C4—102no—
422C2×C4—102no—
423C2×C4—102no—
424C2×C4—104no—
425C2×C4—102no—
426C2×C4—102no—
427C2×C4—202no—
428C2×C4—202no—
429C2×C4C3×#70912yes—
430C2×C4C3×D4×D512yes—
431C2×C4C3×D4×D511yescharacteristic
432C2×C4C23×C3×D511yescharacteristic
433C8—22no—
434C8—54no—
435C2×C3—104no—
436C2×C3#6510214yesdirect product factor
437C2×C3C2×#403011yescharacteristic
438C2×C3C2×#406211yescharacteristic
439C2×C3C2×D5×Q1611yescharacteristic
440C5C2×C3×#46711yescharacteristic, complemented, Sylow
441C22—54no—
442C22—108no—
443C22—102no—
444C22—102no—
445C22C3×#403012yes—
446C22C3×#406212yes—
447C22C3×D5×Q1612yes—
448C22C2×C3×D4×D511yescharacteristic
449C4—42no—
450C4—42no—
451C4—102no—
452C4—202no—
453C4—202no—
454C4C3×#403012yes—
455C4C2×C3×#70911yescharacteristic
456C4C2×C3×D4×D511yescharacteristic
457C3C2×#6510211yescharacteristic, direct product factor, Sylow
458C2—104no—
459C2C3×#6510214yesdirect product factor
460C2C2×C3×#403011yescharacteristic
461C2C2×C3×#406211yescharacteristic
462C2C2×C3×D5×Q1611yescharacteristic
subgroupG/H
centerC23×C3D4×D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C5×#103C2
socleC23×C3×C5C2×D4
prime radicalC23×C3×D5D4
hypercenterC2×C3×#103D5
hypocenterC5C2×C3×#467
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C2×C3×#103

Complements

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

NH ≅ G/Ntype
2. C22×C3×#689C2semidirect
5. C2×C3×C5×#103C2semidirect
6. C2×C3×#3948C2semidirect
7. C2×C3×#3939C2semidirect
8. C3×#65102C2direct (N × H)
9. C2×#65102C3direct (N × H)
18. C2×C3×#689C22semidirect
29. C3×C5×#103C22semidirect
30. C3×#3948C22semidirect
31. C3×#3939C22semidirect
34. C22×#689C2×C3semidirect
37. C2×C5×#103C2×C3semidirect
38. C2×#3948C2×C3semidirect
39. C2×#3939C2×C3semidirect
40. #65102C2×C3direct (N × H)
81. C2×C3×#103D5semidirect
90. C2×#689C22×C3semidirect
101. C5×#103C22×C3semidirect
102. #3948C22×C3semidirect
103. #3939C22×C3semidirect
108. C22×C3×D5Q16semidirect
156. C3×#103C2×D5semidirect
197. C2×#103C3×D5semidirect
205. C2×C3×D5C2×Q16semidirect
250. C22×D5C3×Q16semidirect
298. #103C2×C3×D5semidirect
299. C2×C3×C5#467semidirect
341. C2×D5C2×C3×Q16semidirect
383. C3×C5C2×#467semidirect
400. C2×C5C3×#467semidirect
436. C2×C3#65102direct (N × H)
440. C5C2×C3×#467semidirect
457. C3C2×#65102direct (N × H)
459. C2C3×#65102direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#40922
C22×C3×#6892
C22×C3×C8×D52
C2×C3×#40432
C2×C3×C5×#1032
C2×C3×#39482
C2×C3×#39392
C3×#651022
C2×#651023
C2×C3×#4675