Finite Group Databasebeta

← C2×C3×#65331 #14427892

Subgroups — C2×C3×#65331 #14427892

Order 1920.

1088 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#121C211yesmaximal, characteristic
2C22×C3×#687C211yesmaximal, characteristic, complemented
3C2×C3×#4070C211yesmaximal, characteristic, complemented
4C22×C3×#681C211yesmaximal, characteristic, complemented
5C2×C3×#3964C211yesmaximal, characteristic, complemented
6C3×#65331C218yesmaximal, direct product factor
7C2×C3×#3942C211yesmaximal, characteristic, complemented
8C2×C3×#3939C211yesmaximal, characteristic, complemented
9C2×#65331C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×C5×D4—21no—
11C22×C3×C5×D4C2211yescharacteristic
12C23×C3×C4×C5C2211yescharacteristic
13C22×C3×Dic10C2211yescharacteristic
14C3×C5×#121—22no—
15C3×C5×#121C2214yes—
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17C2×C3×C5×C22⋊C4—21no—
18C2×C3×C22⋊Dic5—21no—
19C2×C3×#687—24no—
20C2×C3×#687C2214yescomplemented
21C2×C3×C4⋊Dic5C2211yescharacteristic
22C3×#4070—22no—
23C3×#4070C2214yescomplemented
24C2×C3×#683—21no—
25C2×C3×#681—24no—
26C2×C3×#681C2214yescomplemented
27C22×C3×#168C2211yescharacteristic
28C22×C3×#168C2211yescharacteristic
29C3×#3964C2214yescomplemented
30C3×#3942C2214yescomplemented
31C3×#3939C2214yescomplemented
32C2×C3×#468—51nomaximal, Hall
33C2×C5×#121C2×C311yescharacteristic
34C22×#687C2×C311yescharacteristic, complemented
35C2×#4070C2×C311yescharacteristic, complemented
36C22×#681C2×C311yescharacteristic, complemented
37C2×#3964C2×C311yescharacteristic, complemented
38#65331C2×C318yesdirect product factor
39C2×#3942C2×C311yescharacteristic, complemented
40C2×#3939C2×C311yescharacteristic, complemented
41C24×C3×C5—21no—
42C24×C3×C5D411yescharacteristic
43C2×C3×C5×D4—24no—
44C2×C3×C5×D4—22no—
45C2×C3×C5×D4—21no—
46C2×C3×C5×D4—22no—
47C2×C3×C5×D4—44no—
48C2×C3×C5×D4C2312yes—
49C22×C3×C4×C5—22no—
50C22×C3×C4×C5—22no—
51C22×C3×C4×C5—21no—
52C22×C3×C4×C5D411yescharacteristic
53C22×C3×C4×C5D412yes—
54C22×C3×C4×C5C2312yes—
55C22×C3×C4×C5C2311yescharacteristic
56C22×C3×Dic5—21no—
57C22×C3×Dic5—21no—
58C2×C3×Dic10—22no—
59C2×C3×Dic10—22no—
60C2×C3×Dic10—21no—
61C2×C3×Dic10C2312yes—
62C3×C5×C4⋊C4—21no—
63C3×C5×C4⋊C4C2312yes—
64C3×C5×C22⋊C4—24no—
65C3×C22⋊Dic5—24no—
66C3×#687—28no—
67C3×C4⋊Dic5—21no—
68C3×C4⋊Dic5C2312yes—
69C3×#683—24no—
70C3×#681—28no—
71C2×C3×#168—22no—
72C2×C3×#168D414yescomplemented
73C2×C3×#168C2312yes—
74C2×C3×#168C2312yes—
75C22×C3×SD16—51no—
76C22×C3×M16—51no—
77C2×C3×C22⋊Q8—51no—
78C2×C3×#121D511yescharacteristic
79C3×#468—58no—
80C2×C3×#107—51no—
81C2×C3×#103—51no—
82C2×C3×#102—51no—
83C22×C5×D4—21no—
84C22×C5×D4C22×C311yescharacteristic
85C23×C4×C5C22×C311yescharacteristic
86C22×Dic10C22×C311yescharacteristic
87C5×#121—22no—
88C5×#121C22×C314yes—
89C2×C5×C4⋊C4C22×C311yescharacteristic
90C2×C5×C22⋊C4—21no—
91C2×C22⋊Dic5—21no—
92C2×#687—24no—
93C2×#687C22×C314yescomplemented
94C2×C4⋊Dic5C22×C311yescharacteristic
95#4070—22no—
96#4070C22×C314yescomplemented
97C2×#683—21no—
98C2×#681—24no—
99C2×#681C22×C314yescomplemented
100C22×#168C22×C311yescharacteristic
101C22×#168C22×C311yescharacteristic
102#3964C22×C314yescomplemented
103#3942C22×C314yescomplemented
104#3939C22×C314yescomplemented
105C2×#468—51noSylow
106C23×C3×C5—22no—
107C23×C3×C5—22no—
108C23×C3×C5—22no—
109C23×C3×C5—22no—
110C23×C3×C5—42no—
111C23×C3×C5—42no—
112C23×C3×C5—42no—
113C23×C3×C5C2×D412yes—
114C23×C3×C5C2×D411yescharacteristic
115C3×C5×D4—22no—
116C3×C5×D4—22no—
117C3×C5×D4—48no—
118C3×C5×D4—42no—
119C2×C3×C4×C5—22no—
120C2×C3×C4×C5—24no—
121C2×C3×C4×C5—22no—
122C2×C3×C4×C5—22no—
123C2×C3×C4×C5—22no—
124C2×C3×C4×C5—42no—
125C2×C3×C4×C5C2×D412yes—
126C2×C3×C4×C5C2×D411yescharacteristic
127C2×C3×C4×C5C2×D412yes—
128C2×C3×C4×C5#4112yes—
129C2×C3×C4×C5C2411yescharacteristic
130C2×C3×Dic5—22no—
131C2×C3×Dic5—22no—
132C2×C3×Dic5—42no—
133C2×C3×Dic5—42no—
134C3×Dic10—22no—
135C3×Dic10—22no—
136C3×Dic10—42no—
137C3×#168—22no—
138C3×#168C2×D414yescomplemented
139C22×C3×Q8—51no—
140C22×C3×D4—21no—
141C22×C3×D4C2×D511yescharacteristic
142C23×C3×C4C2×D511yescharacteristic
143C2×C3×SD16—54no—
144C2×C3×SD16—104no—
145C2×C3×M16—54no—
146C2×C3×M16—104no—
147C22×C3×C8—51no—
148C22×C3×C8—51no—
149C3×C22⋊Q8—54no—
150C3×C22⋊Q8—102no—
151C3×#121—22no—
152C3×#121C2×D514yes—
153C2×C3×C4⋊C4—51no—
154C2×C3×C4⋊C4—101no—
155C2×C3×C4⋊C4C2×D511yescharacteristic
156C2×C3×C22⋊C4—21no—
157C2×C3×C22⋊C4—101no—
158C3×#107—54no—
159C3×#103—54no—
160C3×#102—54no—
161C24×C5—21no—
162C24×C5C3×D411yescharacteristic
163C2×C5×D4—24no—
164C2×C5×D4—22no—
165C2×C5×D4—21no—
166C2×C5×D4—22no—
167C2×C5×D4—44no—
168C2×C5×D4C23×C312yes—
169C22×C4×C5—22no—
170C22×C4×C5—22no—
171C22×C4×C5—21no—
172C22×C4×C5C3×D411yescharacteristic
173C22×C4×C5C3×D412yes—
174C22×C4×C5C23×C312yes—
175C22×C4×C5C23×C311yescharacteristic
176C22×Dic5—21no—
177C22×Dic5—21no—
178C2×Dic10—22no—
179C2×Dic10—22no—
180C2×Dic10—21no—
181C2×Dic10C23×C312yes—
182C5×C4⋊C4—21no—
183C5×C4⋊C4C23×C312yes—
184C5×C22⋊C4—24no—
185C22⋊Dic5—24no—
186#687—28no—
187C4⋊Dic5—21no—
188C4⋊Dic5C23×C312yes—
189#683—24no—
190#681—28no—
191C2×#168—22no—
192C2×#168C3×D414yescomplemented
193C2×#168C23×C312yes—
194C2×#168C23×C312yes—
195C22×SD16—51no—
196C22×M16—51no—
197C2×C22⋊Q8—51no—
198C2×#121C3×D511yescharacteristic
199#468—58no—
200C2×#107—51no—
201C2×#103—51no—
202C2×#102—51no—
203C22×C3×C5—22no—
204C22×C3×C5—22no—
205C22×C3×C5—22no—
206C22×C3×C5—44no—
207C22×C3×C5—42no—
208C22×C3×C5—42no—
209C22×C3×C5—42no—
210C22×C3×C5—84no—
211C22×C3×C5#12112yes—
212C22×C3×C5AGL(1,ℤ/8)12yes—
213C22×C3×C5#13712yes—
214C22×C3×C5C22×D411yescharacteristic
215C3×C4×C5—22no—
216C3×C4×C5—42no—
217C3×C4×C5#12112yes—
218C3×C4×C5C22×D411yescharacteristic
219C3×C4×C5C2×#4111yescharacteristic
220C3×Dic5—42no—
221C3×Dic5—42no—
222C24×C3—21no—
223C24×C3C22⋊D511yescharacteristic
224C2×C3×Q8—52no—
225C2×C3×Q8—101no—
226C2×C3×Q8—102no—
227C2×C3×Q8—102no—
228C2×C3×D4—24no—
229C2×C3×D4—22no—
230C2×C3×D4—22no—
231C2×C3×D4—21no—
232C2×C3×D4—44no—
233C2×C3×D4C22×D512yes—
234C22×C3×C4—22no—
235C22×C3×C4—22no—
236C22×C3×C4—21no—
237C22×C3×C4—101no—
238C22×C3×C4—101no—
239C22×C3×C4C22⋊D512yes—
240C22×C3×C4C22⋊D511yescharacteristic
241C22×C3×C4C22×D512yes—
242C22×C3×C4C22×D511yescharacteristic
243C3×SD16—108no—
244C3×M16—108no—
245C2×C3×C8—54no—
246C2×C3×C8—52no—
247C2×C3×C8—52no—
248C2×C3×C8—102no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—52no—
251C3×C4⋊C4—104no—
252C3×C4⋊C4—101no—
253C3×C4⋊C4C22×D512yes—
254C3×C22⋊C4—24no—
255C3×C22⋊C4—104no—
256C23×C5—22no—
257C23×C5—22no—
258C23×C5—22no—
259C23×C5—22no—
260C23×C5—42no—
261C23×C5—42no—
262C23×C5—42no—
263C23×C5C2×C3×D412yes—
264C23×C5C2×C3×D411yescharacteristic
265C5×D4—22no—
266C5×D4—22no—
267C5×D4—48no—
268C5×D4—42no—
269C2×C4×C5—22no—
270C2×C4×C5—24no—
271C2×C4×C5—22no—
272C2×C4×C5—22no—
273C2×C4×C5—22no—
274C2×C4×C5—42no—
275C2×C4×C5C2×C3×D412yes—
276C2×C4×C5C2×C3×D412yes—
277C2×C4×C5C2×C3×D411yescharacteristic
278C2×C4×C5C3×#4112yes—
279C2×C4×C5C24×C311yescharacteristic
280C2×Dic5—22no—
281C2×Dic5—22no—
282C2×Dic5—42no—
283C2×Dic5—42no—
284Dic10—22no—
285Dic10—22no—
286Dic10—42no—
287#168—22no—
288#168C2×C3×D414yescomplemented
289C22×Q8—51no—
290C22×D4—21no—
291C22×D4C2×C3×D511yescharacteristic
292C23×C4C2×C3×D511yescharacteristic
293C2×SD16—54no—
294C2×SD16—104no—
295C2×M16—54no—
296C2×M16—104no—
297C22×C8—51no—
298C22×C8—51no—
299C22⋊Q8—54no—
300C22⋊Q8—102no—
301#121—22no—
302#121C2×C3×D514yes—
303C2×C4⋊C4—51no—
304C2×C4⋊C4—101no—
305C2×C4⋊C4C2×C3×D511yescharacteristic
306C2×C22⋊C4—21no—
307C2×C22⋊C4—101no—
308#107—54no—
309#103—54no—
310#102—54no—
311C2×C3×C5—42no—
312C2×C3×C5—82no—
313C2×C3×C5#46814yescomplemented
314C2×C3×C5C2×#12111yescharacteristic
315C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
316C2×C3×C5C2×#13711yescharacteristic
317C23×C3—22no—
318C23×C3—22no—
319C23×C3—22no—
320C23×C3—22no—
321C23×C3—42no—
322C23×C3—42no—
323C23×C3—42no—
324C23×C3C2×C22⋊D512yes—
325C23×C3C2×C22⋊D511yescharacteristic
326C3×Q8—102no—
327C3×Q8—102no—
328C3×Q8—202no—
329C3×D4—22no—
330C3×D4—22no—
331C3×D4—48no—
332C3×D4—42no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—24no—
337C2×C3×C4—22no—
338C2×C3×C4—42no—
339C2×C3×C4—102no—
340C2×C3×C4—102no—
341C2×C3×C4—202no—
342C2×C3×C4—202no—
343C2×C3×C4D4×D512yes—
344C2×C3×C4#71112yes—
345C2×C3×C4C2×C22⋊D512yes—
346C2×C3×C4C2×C22⋊D511yescharacteristic
347C2×C3×C4C23×D511yescharacteristic
348C3×C8—54no—
349C3×C8—102no—
350C22×C5—22no—
351C22×C5—22no—
352C22×C5—22no—
353C22×C5—44no—
354C22×C5—42no—
355C22×C5—42no—
356C22×C5—42no—
357C22×C5—84no—
358C22×C5C3×#12112yes—
359C22×C5C3×AGL(1,ℤ/8)12yes—
360C22×C5C3×#13712yes—
361C22×C5C22×C3×D411yescharacteristic
362C4×C5—22no—
363C4×C5—42no—
364C4×C5C3×#12112yes—
365C4×C5C22×C3×D411yescharacteristic
366C4×C5C2×C3×#4111yescharacteristic
367Dic5—42no—
368Dic5—42no—
369C24—21no—
370C24C3×C22⋊D511yescharacteristic
371C2×Q8—52no—
372C2×Q8—101no—
373C2×Q8—102no—
374C2×Q8—102no—
375C2×D4—24no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—21no—
379C2×D4—44no—
380C2×D4C22×C3×D512yes—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—101no—
385C22×C4—101no—
386C22×C4C3×C22⋊D511yescharacteristic
387C22×C4C3×C22⋊D512yes—
388C22×C4C22×C3×D512yes—
389C22×C4C22×C3×D511yescharacteristic
390SD16—108no—
391M16—108no—
392C2×C8—54no—
393C2×C8—52no—
394C2×C8—52no—
395C2×C8—102no—
396C4⋊C4—21no—
397C4⋊C4—52no—
398C4⋊C4—101no—
399C4⋊C4—104no—
400C4⋊C4C22×C3×D512yes—
401C22⋊C4—24no—
402C22⋊C4—104no—
403C3×C5C2×#46811yescharacteristic, complemented, Hall
404C22×C3—22no—
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—44no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—42no—
411C22×C3—84no—
412C22×C3#407812yes—
413C22×C3#408512yes—
414C22×C3#409712yes—
415C22×C3C22×C22⋊D511yescharacteristic
416C3×C4—22no—
417C3×C4—42no—
418C3×C4—202no—
419C3×C4—202no—
420C3×C4#408512yes—
421C3×C4C2×D4×D511yescharacteristic
422C3×C4C2×#71111yescharacteristic
423C2×C5—42no—
424C2×C5—82no—
425C2×C5C3×#46814yescomplemented
426C2×C5C2×C3×#12111yescharacteristic
427C2×C5C2×C3×AGL(1,ℤ/8)11yescharacteristic
428C2×C5C2×C3×#13711yescharacteristic
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23C2×C3×C22⋊D512yes—
437C23C2×C3×C22⋊D511yescharacteristic
438Q8—102no—
439Q8—102no—
440Q8—202no—
441D4—22no—
442D4—22no—
443D4—48no—
444D4—42no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—42no—
451C2×C4—102no—
452C2×C4—102no—
453C2×C4—202no—
454C2×C4—202no—
455C2×C4C3×D4×D512yes—
456C2×C4C3×#71112yes—
457C2×C4C2×C3×C22⋊D512yes—
458C2×C4C2×C3×C22⋊D511yescharacteristic
459C2×C4C23×C3×D511yescharacteristic
460C8—54no—
461C8—102no—
462C2×C3—42no—
463C2×C3—82no—
464C2×C3#6533114yesdirect product factor
465C2×C3C2×#407811yescharacteristic
466C2×C3C2×#408511yescharacteristic
467C2×C3C2×#409711yescharacteristic
468C5C2×C3×#46811yescharacteristic, complemented, Sylow
469C22—22no—
470C22—22no—
471C22—22no—
472C22—44no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—84no—
477C22C3×#407812yes—
478C22C3×#408512yes—
479C22C3×#409712yes—
480C22C22×C3×C22⋊D511yescharacteristic
481C4—22no—
482C4—42no—
483C4—202no—
484C4—202no—
485C4C3×#408512yes—
486C4C2×C3×D4×D511yescharacteristic
487C4C2×C3×#71111yescharacteristic
488C3C2×#6533111yescharacteristic, direct product factor, Sylow
489C2—42no—
490C2—82no—
491C2C3×#6533114yesdirect product factor
492C2C2×C3×#407811yescharacteristic
493C2C2×C3×#408511yescharacteristic
494C2C2×C3×#409711yescharacteristic
subgroupG/H
centerC23×C3C2×C22⋊D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
Fitting, prime radicalC2×C3×C5×#121C2
socleC23×C3×C5C2×D4
hypercenterC2×C3×#121D5
hypocenterC5C2×C3×#468
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C2×C3×#121

Complements

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

NH ≅ G/Ntype
2. C22×C3×#687C2semidirect
3. C2×C3×#4070C2semidirect
4. C22×C3×#681C2semidirect
5. C2×C3×#3964C2semidirect
6. C3×#65331C2direct (N × H)
7. C2×C3×#3942C2semidirect
8. C2×C3×#3939C2semidirect
9. C2×#65331C3direct (N × H)
20. C2×C3×#687C22semidirect
23. C3×#4070C22semidirect
26. C2×C3×#681C22semidirect
29. C3×#3964C22semidirect
30. C3×#3942C22semidirect
31. C3×#3939C22semidirect
34. C22×#687C2×C3semidirect
35. C2×#4070C2×C3semidirect
36. C22×#681C2×C3semidirect
37. C2×#3964C2×C3semidirect
38. #65331C2×C3direct (N × H)
39. C2×#3942C2×C3semidirect
40. C2×#3939C2×C3semidirect
72. C2×C3×#168D4semidirect
93. C2×#687C22×C3semidirect
96. #4070C22×C3semidirect
99. C2×#681C22×C3semidirect
102. #3964C22×C3semidirect
103. #3942C22×C3semidirect
104. #3939C22×C3semidirect
138. C3×#168C2×D4semidirect
192. C2×#168C3×D4semidirect
288. #168C2×C3×D4semidirect
313. C2×C3×C5#468semidirect
403. C3×C5C2×#468semidirect
425. C2×C5C3×#468semidirect
464. C2×C3#65331direct (N × H)
468. C5C2×C3×#468semidirect
488. C3C2×#65331direct (N × H)
491. C2C3×#65331direct (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×#1212
C22×C3×#6872
C2×C3×#40702
C22×C3×#6812
C2×C3×#39642
C3×#653312
C2×C3×#39422
C2×C3×#39392
C2×#653313
C2×C3×#4685