Finite Group Databasebeta

← C2×C3×#65071 #14427982

Subgroups — C2×C3×#65071 #14427982

Order 1920.

1192 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#4084C211yesmaximal, characteristic, complemented
2C22×C3×#686C211yesmaximal, characteristic, complemented
3C22×C3×#676C211yesmaximal, characteristic, complemented
4C2×C3×#4041C211yesmaximal, characteristic, complemented
5C2×C3×C5×#102C211yesmaximal, characteristic, complemented
6C2×C3×#3953C211yesmaximal, characteristic, complemented
7C2×C3×#3940C211yesmaximal, characteristic, complemented
8C3×#65071C218yesmaximal, direct product factor
9C2×#65071C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×C5×D4C2211yescharacteristic
11C22×C3×C22⋊D5—21no—
12C22×C3×D20—21no—
13C22×C3×D20C2211yescharacteristic
14C22×C3×C4×D5C2211yescharacteristic
15C22×C3×C5×C8C2211yescharacteristic
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17C2×C3×C22⋊Dic5—21no—
18C3×#4084—22no—
19C3×#4084C2214yescomplemented
20C2×C3×#686—24no—
21C2×C3×#686C2214yescomplemented
22C2×C3×#685—21no—
23C2×C3×C4⋊Dic5C2211yescharacteristic
24C22×C3×#168C2211yescharacteristic
25C2×C3×#676—24no—
26C2×C3×#676C2214yescomplemented
27C3×#4041—22no—
28C3×#4041C2214yescomplemented
29C3×C5×#102C2214yescomplemented
30C3×#3953C2214yescomplemented
31C3×#3940C2214yescomplemented
32C2×C3×#469—51nomaximal, Hall
33C2×#4084C2×C311yescharacteristic, complemented
34C22×#686C2×C311yescharacteristic, complemented
35C22×#676C2×C311yescharacteristic, complemented
36C2×#4041C2×C311yescharacteristic, complemented
37C2×C5×#102C2×C311yescharacteristic, complemented
38C2×#3953C2×C311yescharacteristic, complemented
39C2×#3940C2×C311yescharacteristic, complemented
40#65071C2×C318yesdirect product factor
41C24×C3×C5—21no—
42C23×C3×D5—21no—
43C23×C3×D5D411yescharacteristic
44C2×C3×C5×D4—22no—
45C2×C3×C5×D4—21no—
46C2×C3×C5×D4—22no—
47C2×C3×C5×D4C2312yes—
48C22×C3×C4×C5—21no—
49C22×C3×C4×C5C2311yescharacteristic
50C2×C3×C22⋊D5—24no—
51C2×C3×C22⋊D5—44no—
52C22×C3×Dic5—21no—
53C22×C3×Dic5D411yescharacteristic
54C2×C3×D20—22no—
55C2×C3×D20—22no—
56C2×C3×D20—24no—
57C2×C3×D20—21no—
58C2×C3×D20—44no—
59C2×C3×D20C2312yes—
60C2×C3×C4×D5—22no—
61C2×C3×C4×D5—22no—
62C2×C3×C4×D5D412yes—
63C2×C3×C4×D5C2312yes—
64C2×C3×C5×C8—22no—
65C2×C3×C5×C8C2312yes—
66C3×C5×C4⋊C4—21no—
67C3×C5×C4⋊C4C2312yes—
68C3×C22⋊Dic5—24no—
69C3×#686—28no—
70C3×#685—24no—
71C3×C4⋊Dic5—21no—
72C3×C4⋊Dic5C2312yes—
73C2×C3×#168D414yescomplemented
74C2×C3×#168C2312yes—
75C3×#676—28no—
76C22×C3×D8—51no—
77C22×C3×M16—51no—
78C2×C3×#121—51no—
79C2×C3×#121—51no—
80C3×#469—58no—
81C2×C3×#106—51no—
82C2×C3×#102—51no—
83C2×C3×#102D511yescharacteristic, complemented
84C22×C5×D4C22×C311yescharacteristic
85C22×C22⋊D5—21no—
86C22×D20—21no—
87C22×D20C22×C311yescharacteristic
88C22×C4×D5C22×C311yescharacteristic
89C22×C5×C8C22×C311yescharacteristic
90C2×C5×C4⋊C4C22×C311yescharacteristic
91C2×C22⋊Dic5—21no—
92#4084—22no—
93#4084C22×C314yescomplemented
94C2×#686—24no—
95C2×#686C22×C314yescomplemented
96C2×#685—21no—
97C2×C4⋊Dic5C22×C311yescharacteristic
98C22×#168C22×C311yescharacteristic
99C2×#676—24no—
100C2×#676C22×C314yescomplemented
101#4041—22no—
102#4041C22×C314yescomplemented
103C5×#102C22×C314yescomplemented
104#3953C22×C314yescomplemented
105#3940C22×C314yescomplemented
106C2×#469—51noSylow
107C23×C3×C5—22no—
108C23×C3×C5—42no—
109C23×C3×C5—42no—
110C23×C3×C5—42no—
111C23×C3×C5C2×D411yescharacteristic
112C22×C3×D5—22no—
113C22×C3×D5—22no—
114C22×C3×D5—22no—
115C22×C3×D5—22no—
116C22×C3×D5—42no—
117C22×C3×D5—42no—
118C22×C3×D5—42no—
119C22×C3×D5C2×D412yes—
120C3×C5×D4—22no—
121C3×C5×D4—22no—
122C3×C5×D4—42no—
123C2×C3×C4×C5—22no—
124C2×C3×C4×C5—42no—
125C2×C3×C4×C5C2×D411yescharacteristic
126C2×C3×C4×C5C2×D412yes—
127C2×C3×C4×C5#4112yes—
128C2×C3×C4×C5C2411yescharacteristic
129C3×C22⋊D5—48no—
130C2×C3×Dic5—22no—
131C2×C3×Dic5—22no—
132C2×C3×Dic5—42no—
133C2×C3×Dic5C2×D412yes—
134C3×D20—22no—
135C3×D20—22no—
136C3×D20—48no—
137C3×D20—42no—
138C3×C4×D5—24no—
139C3×C4×D5—22no—
140C3×C4×D5—22no—
141C3×C5×C8—22no—
142C3×#168C2×D414yescomplemented
143C22×C3×D4—51no—
144C22×C3×D4—101no—
145C22×C3×D4—101no—
146C22×C3×D4C2×D511yescharacteristic
147C23×C3×C4—51no—
148C2×C3×D8—54no—
149C2×C3×D8—104no—
150C2×C3×M16—54no—
151C2×C3×M16—104no—
152C22×C3×C8—51no—
153C22×C3×C8C2×D511yescharacteristic
154C3×#121—54no—
155C3×#121—54no—
156C3×#121—102no—
157C3×#121—102no—
158C2×C3×C4⋊C4—51no—
159C2×C3×C4⋊C4C2×D511yescharacteristic
160C2×C3×C22⋊C4—101no—
161C2×C3×C22⋊C4—101no—
162C3×#106—54no—
163C3×#102—54no—
164C3×#102C2×D514yescomplemented
165C24×C5—21no—
166C23×D5—21no—
167C23×D5C3×D411yescharacteristic
168C2×C5×D4—22no—
169C2×C5×D4—21no—
170C2×C5×D4—22no—
171C2×C5×D4C23×C312yes—
172C22×C4×C5—21no—
173C22×C4×C5C23×C311yescharacteristic
174C2×C22⋊D5—24no—
175C2×C22⋊D5—44no—
176C22×Dic5—21no—
177C22×Dic5C3×D411yescharacteristic
178C2×D20—22no—
179C2×D20—22no—
180C2×D20—24no—
181C2×D20—21no—
182C2×D20—44no—
183C2×D20C23×C312yes—
184C2×C4×D5—22no—
185C2×C4×D5—22no—
186C2×C4×D5C3×D412yes—
187C2×C4×D5C23×C312yes—
188C2×C5×C8—22no—
189C2×C5×C8C23×C312yes—
190C5×C4⋊C4—21no—
191C5×C4⋊C4C23×C312yes—
192C22⋊Dic5—24no—
193#686—28no—
194#685—24no—
195C4⋊Dic5—21no—
196C4⋊Dic5C23×C312yes—
197C2×#168C3×D414yescomplemented
198C2×#168C23×C312yes—
199#676—28no—
200C22×D8—51no—
201C22×M16—51no—
202C2×#121—51no—
203C2×#121—51no—
204#469—58no—
205C2×#106—51no—
206C2×#102—51no—
207C2×#102C3×D511yescharacteristic, complemented
208C22×C3×C5—42no—
209C22×C3×C5—42no—
210C22×C3×C5—42no—
211C22×C3×C5—84no—
212C22×C3×C5#12112yes—
213C22×C3×C5AGL(1,ℤ/8)12yes—
214C22×C3×C5AGL(1,ℤ/8)12yes—
215C22×C3×C5C22×D411yescharacteristic
216C2×C3×D5—22no—
217C2×C3×D5—22no—
218C2×C3×D5—22no—
219C2×C3×D5—44no—
220C2×C3×D5—42no—
221C2×C3×D5—42no—
222C2×C3×D5—42no—
223C2×C3×D5—84no—
224C3×C4×C5—42no—
225C3×C4×C5#12112yes—
226C3×C4×C5C22×D411yescharacteristic
227C3×C4×C5C2×#4111yescharacteristic
228C3×Dic5—22no—
229C3×Dic5—42no—
230C24×C3—21no—
231C24×C3—51no—
232C24×C3—101no—
233C2×C3×D4—22no—
234C2×C3×D4—22no—
235C2×C3×D4—21no—
236C2×C3×D4—52no—
237C2×C3×D4—104no—
238C2×C3×D4—104no—
239C2×C3×D4—102no—
240C2×C3×D4—102no—
241C2×C3×D4—101no—
242C2×C3×D4—204no—
243C2×C3×D4—204no—
244C2×C3×D4C22×D512yes—
245C22×C3×C4—21no—
246C22×C3×C4—51no—
247C22×C3×C4—52no—
248C22×C3×C4—52no—
249C22×C3×C4—102no—
250C22×C3×C4—102no—
251C22×C3×C4—101no—
252C22×C3×C4C22×D511yescharacteristic
253C3×D8—108no—
254C3×M16—108no—
255C2×C3×C8—22no—
256C2×C3×C8—54no—
257C2×C3×C8—52no—
258C2×C3×C8C22×D512yes—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—52no—
261C3×C4⋊C4—101no—
262C3×C4⋊C4C22×D512yes—
263C3×C22⋊C4—104no—
264C3×C22⋊C4—104no—
265C23×C5—22no—
266C23×C5—42no—
267C23×C5—42no—
268C23×C5—42no—
269C23×C5C2×C3×D411yescharacteristic
270C22×D5—22no—
271C22×D5—22no—
272C22×D5—22no—
273C22×D5—22no—
274C22×D5—42no—
275C22×D5—42no—
276C22×D5—42no—
277C22×D5C2×C3×D412yes—
278C5×D4—22no—
279C5×D4—22no—
280C5×D4—42no—
281C2×C4×C5—22no—
282C2×C4×C5—42no—
283C2×C4×C5C2×C3×D411yescharacteristic
284C2×C4×C5C2×C3×D412yes—
285C2×C4×C5C3×#4112yes—
286C2×C4×C5C24×C311yescharacteristic
287C22⋊D5—48no—
288C2×Dic5—22no—
289C2×Dic5—22no—
290C2×Dic5—42no—
291C2×Dic5C2×C3×D412yes—
292D20—22no—
293D20—22no—
294D20—48no—
295D20—42no—
296C4×D5—24no—
297C4×D5—22no—
298C4×D5—22no—
299C5×C8—22no—
300#168C2×C3×D414yescomplemented
301C22×D4—51no—
302C22×D4—101no—
303C22×D4—101no—
304C22×D4C2×C3×D511yescharacteristic
305C23×C4—51no—
306C2×D8—54no—
307C2×D8—104no—
308C2×M16—54no—
309C2×M16—104no—
310C22×C8—51no—
311C22×C8C2×C3×D511yescharacteristic
312#121—54no—
313#121—54no—
314#121—102no—
315#121—102no—
316C2×C4⋊C4—51no—
317C2×C4⋊C4C2×C3×D511yescharacteristic
318C2×C22⋊C4—101no—
319C2×C22⋊C4—101no—
320#106—54no—
321#102—54no—
322#102C2×C3×D514yescomplemented
323C2×C3×C5—82no—
324C2×C3×C5#46914yescomplemented
325C2×C3×C5C2×#12111yescharacteristic
326C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
327C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
328C3×D5—42no—
329C3×D5—82no—
330C23×C3—22no—
331C23×C3—42no—
332C23×C3—42no—
333C23×C3—42no—
334C23×C3—52no—
335C23×C3—102no—
336C23×C3—102no—
337C23×C3—102no—
338C23×C3—102no—
339C23×C3—202no—
340C23×C3—202no—
341C23×C3—202no—
342C23×C3D4×D511yescharacteristic
343C3×D4—22no—
344C3×D4—22no—
345C3×D4—42no—
346C3×D4—102no—
347C3×D4—102no—
348C3×D4—208no—
349C3×D4—208no—
350C3×D4—202no—
351C2×C3×C4—22no—
352C2×C3×C4—42no—
353C2×C3×C4—52no—
354C2×C3×C4—102no—
355C2×C3×C4—102no—
356C2×C3×C4—102no—
357C2×C3×C4—104no—
358C2×C3×C4—102no—
359C2×C3×C4—202no—
360C2×C3×C4#70912yes—
361C2×C3×C4D4×D512yes—
362C2×C3×C4D4×D511yescharacteristic
363C2×C3×C4C23×D511yescharacteristic
364C3×C8—22no—
365C3×C8—54no—
366C22×C5—42no—
367C22×C5—42no—
368C22×C5—42no—
369C22×C5—84no—
370C22×C5C3×#12112yes—
371C22×C5C3×AGL(1,ℤ/8)12yes—
372C22×C5C3×AGL(1,ℤ/8)12yes—
373C22×C5C22×C3×D411yescharacteristic
374C2×D5—22no—
375C2×D5—22no—
376C2×D5—22no—
377C2×D5—44no—
378C2×D5—42no—
379C2×D5—42no—
380C2×D5—42no—
381C2×D5—84no—
382C4×C5—42no—
383C4×C5C3×#12112yes—
384C4×C5C22×C3×D411yescharacteristic
385C4×C5C2×C3×#4111yescharacteristic
386Dic5—22no—
387Dic5—42no—
388C24—21no—
389C24—51no—
390C24—101no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—21no—
394C2×D4—52no—
395C2×D4—104no—
396C2×D4—104no—
397C2×D4—102no—
398C2×D4—102no—
399C2×D4—101no—
400C2×D4—204no—
401C2×D4—204no—
402C2×D4C22×C3×D512yes—
403C22×C4—21no—
404C22×C4—51no—
405C22×C4—52no—
406C22×C4—52no—
407C22×C4—102no—
408C22×C4—102no—
409C22×C4—101no—
410C22×C4C22×C3×D511yescharacteristic
411D8—108no—
412M16—108no—
413C2×C8—22no—
414C2×C8—54no—
415C2×C8—52no—
416C2×C8C22×C3×D512yes—
417C4⋊C4—21no—
418C4⋊C4—52no—
419C4⋊C4—101no—
420C4⋊C4C22×C3×D512yes—
421C22⋊C4—104no—
422C22⋊C4—104no—
423C3×C5C2×#46911yescharacteristic, complemented, Hall
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3—84no—
428C22×C3—102no—
429C22×C3—102no—
430C22×C3—102no—
431C22×C3—204no—
432C22×C3—202no—
433C22×C3—202no—
434C22×C3—202no—
435C22×C3—404no—
436C22×C3#403012yes—
437C22×C3#405712yes—
438C22×C3#406012yes—
439C22×C3C2×D4×D511yescharacteristic
440C3×C4—42no—
441C3×C4—102no—
442C3×C4—202no—
443C3×C4#403012yes—
444C3×C4C2×#70911yescharacteristic
445C3×C4C2×D4×D511yescharacteristic
446C2×C5—82no—
447C2×C5C3×#46914yescomplemented
448C2×C5C2×C3×#12111yescharacteristic
449C2×C5C2×C3×AGL(1,ℤ/8)11yescharacteristic
450C2×C5C2×C3×AGL(1,ℤ/8)11yescharacteristic
451D5—42no—
452D5—82no—
453C23—22no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—52no—
458C23—102no—
459C23—102no—
460C23—102no—
461C23—102no—
462C23—202no—
463C23—202no—
464C23—202no—
465C23C3×D4×D511yescharacteristic
466D4—22no—
467D4—22no—
468D4—42no—
469D4—102no—
470D4—102no—
471D4—208no—
472D4—208no—
473D4—202no—
474C2×C4—22no—
475C2×C4—42no—
476C2×C4—52no—
477C2×C4—102no—
478C2×C4—102no—
479C2×C4—102no—
480C2×C4—104no—
481C2×C4—102no—
482C2×C4—202no—
483C2×C4C3×#70912yes—
484C2×C4C3×D4×D512yes—
485C2×C4C3×D4×D511yescharacteristic
486C2×C4C23×C3×D511yescharacteristic
487C8—22no—
488C8—54no—
489C2×C3—82no—
490C2×C3—202no—
491C2×C3—402no—
492C2×C3#6507114yesdirect product factor
493C2×C3C2×#403011yescharacteristic
494C2×C3C2×#405711yescharacteristic
495C2×C3C2×#406011yescharacteristic
496C5C2×C3×#46911yescharacteristic, complemented, Sylow
497C22—42no—
498C22—42no—
499C22—42no—
500C22—84no—
501C22—102no—
502C22—102no—
503C22—102no—
504C22—204no—
505C22—202no—
506C22—202no—
507C22—202no—
508C22—404no—
509C22C3×#403012yes—
510C22C3×#405712yes—
511C22C3×#406012yes—
512C22C2×C3×D4×D511yescharacteristic
513C4—42no—
514C4—102no—
515C4—202no—
516C4C3×#403012yes—
517C4C2×C3×#70911yescharacteristic
518C4C2×C3×D4×D511yescharacteristic
519C3C2×#6507111yescharacteristic, direct product factor, Sylow
520C2—82no—
521C2—202no—
522C2—402no—
523C2C3×#6507114yesdirect product factor
524C2C2×C3×#403011yescharacteristic
525C2C2×C3×#405711yescharacteristic
526C2C2×C3×#406011yescharacteristic
subgroupG/H
centerC23×C3D4×D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C5×#102C2
socleC23×C3×C5C2×D4
hypercenterC2×C3×#102D5
hypocenterC5C2×C3×#469
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C2×C3×#102

Complements

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

NH ≅ G/Ntype
1. C2×C3×#4084C2semidirect
2. C22×C3×#686C2semidirect
3. C22×C3×#676C2semidirect
4. C2×C3×#4041C2semidirect
5. C2×C3×C5×#102C2semidirect
6. C2×C3×#3953C2semidirect
7. C2×C3×#3940C2semidirect
8. C3×#65071C2direct (N × H)
9. C2×#65071C3direct (N × H)
19. C3×#4084C22semidirect
21. C2×C3×#686C22semidirect
26. C2×C3×#676C22semidirect
28. C3×#4041C22semidirect
29. C3×C5×#102C22semidirect
30. C3×#3953C22semidirect
31. C3×#3940C22semidirect
33. C2×#4084C2×C3semidirect
34. C22×#686C2×C3semidirect
35. C22×#676C2×C3semidirect
36. C2×#4041C2×C3semidirect
37. C2×C5×#102C2×C3semidirect
38. C2×#3953C2×C3semidirect
39. C2×#3940C2×C3semidirect
40. #65071C2×C3direct (N × H)
73. C2×C3×#168D4semidirect
83. C2×C3×#102D5semidirect
93. #4084C22×C3semidirect
95. C2×#686C22×C3semidirect
100. C2×#676C22×C3semidirect
102. #4041C22×C3semidirect
103. C5×#102C22×C3semidirect
104. #3953C22×C3semidirect
105. #3940C22×C3semidirect
142. C3×#168C2×D4semidirect
164. C3×#102C2×D5semidirect
197. C2×#168C3×D4semidirect
207. C2×#102C3×D5semidirect
300. #168C2×C3×D4semidirect
322. #102C2×C3×D5semidirect
324. C2×C3×C5#469semidirect
423. C3×C5C2×#469semidirect
447. C2×C5C3×#469semidirect
492. C2×C3#65071direct (N × H)
496. C5C2×C3×#469semidirect
519. C3C2×#65071direct (N × H)
523. C2C3×#65071direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#40842
C22×C3×#6862
C22×C3×#6762
C2×C3×#40412
C2×C3×C5×#1022
C2×C3×#39532
C2×C3×#39402
C3×#650712
C2×#650713
C2×C3×#4695