Finite Group Databasebeta

← C2×C3×#65324 #14431513

Subgroups — C2×C3×#65324 #14431513

Order 1920.

936 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#121C211yesmaximal, characteristic
2C2×C3×#4072C211yesmaximal, characteristic, complemented
3C2×C3×#3964C211yesmaximal, characteristic, complemented
4C2×C3×#3964C211yesmaximal, characteristic, complemented
5C2×C3×C22⋊#168C211yesmaximal, characteristic, complemented
6C3×#65324C218yesmaximal, direct product factor
7C2×C3×#3940C211yesmaximal, characteristic, complemented
8C2×C3×#3939C211yesmaximal, characteristic, complemented
9C2×#65324C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×C5×D4—21no—
11C22×C3×C5×D4C2211yescharacteristic
12C23×C3×C4×C5C2211yescharacteristic
13C3×C5×#121—22no—
14C3×C5×#121C2214yes—
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×C3×C5×C22⋊C4—21no—
17C2×C3×C22⋊Dic5—21no—
18C3×#4072—22no—
19C3×#4072C2214yescomplemented
20C2×C3×C4⋊Dic5C2211yescharacteristic
21C2×C3×C4⋊Dic5C2211yescharacteristic
22C2×C3×C4×Dic5—21no—
23C22×C3×#168C2211yescharacteristic
24C22×C3×#168C2211yescharacteristic
25C3×#3964C2214yescomplemented
26C3×#3964C2214yescomplemented
27C3×C22⋊#168C2214yescomplemented
28C3×#3940C2214yescomplemented
29C3×#3939C2214yescomplemented
30C2×C3×#482—51nomaximal, Hall
31C2×C5×#121C2×C311yescharacteristic
32C2×#4072C2×C311yescharacteristic, complemented
33C2×#3964C2×C311yescharacteristic, complemented
34C2×#3964C2×C311yescharacteristic, complemented
35C2×C22⋊#168C2×C311yescharacteristic, complemented
36#65324C2×C318yesdirect product factor
37C2×#3940C2×C311yescharacteristic, complemented
38C2×#3939C2×C311yescharacteristic, complemented
39C24×C3×C5—21no—
40C24×C3×C5D411yescharacteristic
41C2×C3×C5×D4—24no—
42C2×C3×C5×D4—21no—
43C2×C3×C5×D4—22no—
44C2×C3×C5×D4—22no—
45C2×C3×C5×D4—44no—
46C2×C3×C5×D4C2312yes—
47C22×C3×C4×C5—22no—
48C22×C3×C4×C5—22no—
49C22×C3×C4×C5—21no—
50C22×C3×C4×C5D411yescharacteristic
51C22×C3×C4×C5D412yes—
52C22×C3×C4×C5C2312yes—
53C22×C3×C4×C5C2311yescharacteristic
54C22×C3×Dic5—21no—
55C22×C3×Dic5—21no—
56C3×C5×C4⋊C4—21no—
57C3×C5×C4⋊C4C2312yes—
58C3×C5×C22⋊C4—24no—
59C3×C22⋊Dic5—24no—
60C3×C4⋊Dic5—21no—
61C3×C4⋊Dic5—21no—
62C3×C4⋊Dic5C2312yes—
63C3×C4⋊Dic5C2312yes—
64C3×C4×Dic5—24no—
65C2×C3×#168—22no—
66C2×C3×#168—22no—
67C2×C3×#168C2312yes—
68C2×C3×#168C2312yes—
69C2×C3×#121D511yescharacteristic
70C2×C3×#117—51no—
71C3×#482—58no—
72C2×C3×#107—51no—
73C2×C3×#106—51no—
74C2×C3×#102—51no—
75C2×C3×#102—51no—
76C2×C3×C22⋊C8—51no—
77C22×C5×D4—21no—
78C22×C5×D4C22×C311yescharacteristic
79C23×C4×C5C22×C311yescharacteristic
80C5×#121—22no—
81C5×#121C22×C314yes—
82C2×C5×C4⋊C4C22×C311yescharacteristic
83C2×C5×C22⋊C4—21no—
84C2×C22⋊Dic5—21no—
85#4072—22no—
86#4072C22×C314yescomplemented
87C2×C4⋊Dic5C22×C311yescharacteristic
88C2×C4⋊Dic5C22×C311yescharacteristic
89C2×C4×Dic5—21no—
90C22×#168C22×C311yescharacteristic
91C22×#168C22×C311yescharacteristic
92#3964C22×C314yescomplemented
93#3964C22×C314yescomplemented
94C22⋊#168C22×C314yescomplemented
95#3940C22×C314yescomplemented
96#3939C22×C314yescomplemented
97C2×#482—51noSylow
98C23×C3×C5—22no—
99C23×C3×C5—22no—
100C23×C3×C5—22no—
101C23×C3×C5—22no—
102C23×C3×C5—42no—
103C23×C3×C5—42no—
104C23×C3×C5—42no—
105C23×C3×C5C2×D412yes—
106C23×C3×C5C2×D411yescharacteristic
107C3×C5×D4—22no—
108C3×C5×D4—22no—
109C3×C5×D4—48no—
110C3×C5×D4—42no—
111C2×C3×C4×C5—22no—
112C2×C3×C4×C5—24no—
113C2×C3×C4×C5—22no—
114C2×C3×C4×C5—22no—
115C2×C3×C4×C5—22no—
116C2×C3×C4×C5—42no—
117C2×C3×C4×C5C2×D412yes—
118C2×C3×C4×C5C2×D411yescharacteristic
119C2×C3×C4×C5#4112yes—
120C2×C3×C4×C5#4112yes—
121C2×C3×C4×C5C2411yescharacteristic
122C2×C3×Dic5—22no—
123C2×C3×Dic5—22no—
124C2×C3×Dic5—42no—
125C2×C3×Dic5—42no—
126C3×#168—22no—
127C3×#168—22no—
128C22×C3×D4—21no—
129C22×C3×D4C2×D511yescharacteristic
130C23×C3×C4C2×D511yescharacteristic
131C22×C3×C8—51no—
132C22×C3×C8—51no—
133C3×#121—22no—
134C3×#121C2×D514yes—
135C3×#117—54no—
136C3×#117—102no—
137C2×C3×C4⋊C4—51no—
138C2×C3×C4⋊C4—51no—
139C2×C3×C4⋊C4C2×D511yescharacteristic
140C2×C3×C22⋊C4—21no—
141C2×C3×C22⋊C4—101no—
142C2×C3×C42—101no—
143C3×#107—54no—
144C3×#106—54no—
145C3×#102—54no—
146C3×#102—54no—
147C3×C22⋊C8—54no—
148C24×C5—21no—
149C24×C5C3×D411yescharacteristic
150C2×C5×D4—24no—
151C2×C5×D4—21no—
152C2×C5×D4—22no—
153C2×C5×D4—22no—
154C2×C5×D4—44no—
155C2×C5×D4C23×C312yes—
156C22×C4×C5—22no—
157C22×C4×C5—22no—
158C22×C4×C5—21no—
159C22×C4×C5C3×D411yescharacteristic
160C22×C4×C5C3×D412yes—
161C22×C4×C5C23×C312yes—
162C22×C4×C5C23×C311yescharacteristic
163C22×Dic5—21no—
164C22×Dic5—21no—
165C5×C4⋊C4—21no—
166C5×C4⋊C4C23×C312yes—
167C5×C22⋊C4—24no—
168C22⋊Dic5—24no—
169C4⋊Dic5—21no—
170C4⋊Dic5—21no—
171C4⋊Dic5C23×C312yes—
172C4⋊Dic5C23×C312yes—
173C4×Dic5—24no—
174C2×#168—22no—
175C2×#168—22no—
176C2×#168C23×C312yes—
177C2×#168C23×C312yes—
178C2×#121C3×D511yescharacteristic
179C2×#117—51no—
180#482—58no—
181C2×#107—51no—
182C2×#106—51no—
183C2×#102—51no—
184C2×#102—51no—
185C2×C22⋊C8—51no—
186C22×C3×C5—22no—
187C22×C3×C5—22no—
188C22×C3×C5—22no—
189C22×C3×C5—44no—
190C22×C3×C5—42no—
191C22×C3×C5—42no—
192C22×C3×C5—42no—
193C22×C3×C5—84no—
194C22×C3×C5#12312yes—
195C22×C3×C5#13512yes—
196C22×C3×C5AGL(1,ℤ/8)12yes—
197C22×C3×C5C22×D411yescharacteristic
198C3×C4×C5—22no—
199C3×C4×C5—42no—
200C3×C4×C5#12312yes—
201C3×C4×C5C2×#4111yescharacteristic
202C3×C4×C5C2×#4111yescharacteristic
203C3×Dic5—42no—
204C3×Dic5—42no—
205C24×C3—21no—
206C24×C3C22⋊D511yescharacteristic
207C2×C3×D4—24no—
208C2×C3×D4—22no—
209C2×C3×D4—22no—
210C2×C3×D4—21no—
211C2×C3×D4—44no—
212C2×C3×D4C22×D512yes—
213C22×C3×C4—22no—
214C22×C3×C4—22no—
215C22×C3×C4—21no—
216C22×C3×C4—101no—
217C22×C3×C4—101no—
218C22×C3×C4C22⋊D512yes—
219C22×C3×C4C22⋊D511yescharacteristic
220C22×C3×C4C22×D512yes—
221C22×C3×C4C22×D511yescharacteristic
222C2×C3×C8—52no—
223C2×C3×C8—52no—
224C2×C3×C8—102no—
225C2×C3×C8—102no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—52no—
228C3×C4⋊C4—52no—
229C3×C4⋊C4—101no—
230C3×C4⋊C4—101no—
231C3×C4⋊C4C22×D512yes—
232C3×C22⋊C4—24no—
233C3×C22⋊C4—104no—
234C3×C42—104no—
235C23×C5—22no—
236C23×C5—22no—
237C23×C5—22no—
238C23×C5—22no—
239C23×C5—42no—
240C23×C5—42no—
241C23×C5—42no—
242C23×C5C2×C3×D412yes—
243C23×C5C2×C3×D411yescharacteristic
244C5×D4—22no—
245C5×D4—22no—
246C5×D4—48no—
247C5×D4—42no—
248C2×C4×C5—22no—
249C2×C4×C5—24no—
250C2×C4×C5—22no—
251C2×C4×C5—22no—
252C2×C4×C5—22no—
253C2×C4×C5—42no—
254C2×C4×C5C2×C3×D412yes—
255C2×C4×C5C2×C3×D411yescharacteristic
256C2×C4×C5C3×#4112yes—
257C2×C4×C5C3×#4112yes—
258C2×C4×C5C24×C311yescharacteristic
259C2×Dic5—22no—
260C2×Dic5—22no—
261C2×Dic5—42no—
262C2×Dic5—42no—
263#168—22no—
264#168—22no—
265C22×D4—21no—
266C22×D4C2×C3×D511yescharacteristic
267C23×C4C2×C3×D511yescharacteristic
268C22×C8—51no—
269C22×C8—51no—
270#121—22no—
271#121C2×C3×D514yes—
272#117—54no—
273#117—102no—
274C2×C4⋊C4—51no—
275C2×C4⋊C4—51no—
276C2×C4⋊C4C2×C3×D511yescharacteristic
277C2×C22⋊C4—21no—
278C2×C22⋊C4—101no—
279C2×C42—101no—
280#107—54no—
281#106—54no—
282#102—54no—
283#102—54no—
284C22⋊C8—54no—
285C2×C3×C5—42no—
286C2×C3×C5—82no—
287C2×C3×C5#48214yescomplemented
288C2×C3×C5C2×#12311yescharacteristic
289C2×C3×C5C2×#13511yescharacteristic
290C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
291C23×C3—22no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3C2×C22⋊D512yes—
299C23×C3C2×C22⋊D511yescharacteristic
300C3×D4—22no—
301C3×D4—22no—
302C3×D4—48no—
303C3×D4—42no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—24no—
308C2×C3×C4—22no—
309C2×C3×C4—42no—
310C2×C3×C4—102no—
311C2×C3×C4—102no—
312C2×C3×C4—202no—
313C2×C3×C4—202no—
314C2×C3×C4#71112yes—
315C2×C3×C4#71112yes—
316C2×C3×C4C2×C22⋊D512yes—
317C2×C3×C4C2×C22⋊D511yescharacteristic
318C2×C3×C4C23×D511yescharacteristic
319C3×C8—102no—
320C3×C8—102no—
321C22×C5—22no—
322C22×C5—22no—
323C22×C5—22no—
324C22×C5—44no—
325C22×C5—42no—
326C22×C5—42no—
327C22×C5—42no—
328C22×C5—84no—
329C22×C5C3×#12312yes—
330C22×C5C3×#13512yes—
331C22×C5C3×AGL(1,ℤ/8)12yes—
332C22×C5C22×C3×D411yescharacteristic
333C4×C5—22no—
334C4×C5—42no—
335C4×C5C3×#12312yes—
336C4×C5C2×C3×#4111yescharacteristic
337C4×C5C2×C3×#4111yescharacteristic
338Dic5—42no—
339Dic5—42no—
340C24—21no—
341C24C3×C22⋊D511yescharacteristic
342C2×D4—24no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—21no—
346C2×D4—44no—
347C2×D4C22×C3×D512yes—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—101no—
352C22×C4—101no—
353C22×C4C3×C22⋊D511yescharacteristic
354C22×C4C3×C22⋊D512yes—
355C22×C4C22×C3×D512yes—
356C22×C4C22×C3×D511yescharacteristic
357C2×C8—52no—
358C2×C8—52no—
359C2×C8—102no—
360C2×C8—102no—
361C4⋊C4—21no—
362C4⋊C4—52no—
363C4⋊C4—52no—
364C4⋊C4—101no—
365C4⋊C4—101no—
366C4⋊C4C22×C3×D512yes—
367C22⋊C4—24no—
368C22⋊C4—104no—
369C42—104no—
370C3×C5C2×#48211yescharacteristic, complemented, Hall
371C22×C3—22no—
372C22×C3—22no—
373C22×C3—22no—
374C22×C3—44no—
375C22×C3—42no—
376C22×C3—42no—
377C22×C3—42no—
378C22×C3—84no—
379C22×C3#407812yes—
380C22×C3#408112yes—
381C22×C3#409612yes—
382C22×C3C22×C22⋊D511yescharacteristic
383C3×C4—22no—
384C3×C4—42no—
385C3×C4—202no—
386C3×C4—202no—
387C3×C4#408112yes—
388C3×C4C2×#71111yescharacteristic
389C3×C4C2×#71111yescharacteristic
390C2×C5—42no—
391C2×C5—82no—
392C2×C5C3×#48214yescomplemented
393C2×C5C2×C3×#12311yescharacteristic
394C2×C5C2×C3×#13511yescharacteristic
395C2×C5C2×C3×AGL(1,ℤ/8)11yescharacteristic
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23C2×C3×C22⋊D512yes—
404C23C2×C3×C22⋊D511yescharacteristic
405D4—22no—
406D4—22no—
407D4—48no—
408D4—42no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—42no—
415C2×C4—102no—
416C2×C4—102no—
417C2×C4—202no—
418C2×C4—202no—
419C2×C4C3×#71112yes—
420C2×C4C3×#71112yes—
421C2×C4C2×C3×C22⋊D512yes—
422C2×C4C2×C3×C22⋊D511yescharacteristic
423C2×C4C23×C3×D511yescharacteristic
424C8—102no—
425C8—102no—
426C2×C3—42no—
427C2×C3—82no—
428C2×C3#6532414yesdirect product factor
429C2×C3C2×#407811yescharacteristic
430C2×C3C2×#408111yescharacteristic
431C2×C3C2×#409611yescharacteristic
432C5C2×C3×#48211yescharacteristic, complemented, Sylow
433C22—22no—
434C22—22no—
435C22—22no—
436C22—44no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22—84no—
441C22C3×#407812yes—
442C22C3×#408112yes—
443C22C3×#409612yes—
444C22C22×C3×C22⋊D511yescharacteristic
445C4—22no—
446C4—42no—
447C4—202no—
448C4—202no—
449C4C3×#408112yes—
450C4C2×C3×#71111yescharacteristic
451C4C2×C3×#71111yescharacteristic
452C3C2×#6532411yescharacteristic, direct product factor, Sylow
453C2—42no—
454C2—82no—
455C2C3×#6532414yesdirect product factor
456C2C2×C3×#407811yescharacteristic
457C2C2×C3×#408111yescharacteristic
458C2C2×C3×#409611yescharacteristic
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×#482
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. C2×C3×#4072C2semidirect
3. C2×C3×#3964C2semidirect
4. C2×C3×#3964C2semidirect
5. C2×C3×C22⋊#168C2semidirect
6. C3×#65324C2direct (N × H)
7. C2×C3×#3940C2semidirect
8. C2×C3×#3939C2semidirect
9. C2×#65324C3direct (N × H)
19. C3×#4072C22semidirect
25. C3×#3964C22semidirect
26. C3×#3964C22semidirect
27. C3×C22⋊#168C22semidirect
28. C3×#3940C22semidirect
29. C3×#3939C22semidirect
32. C2×#4072C2×C3semidirect
33. C2×#3964C2×C3semidirect
34. C2×#3964C2×C3semidirect
35. C2×C22⋊#168C2×C3semidirect
36. #65324C2×C3direct (N × H)
37. C2×#3940C2×C3semidirect
38. C2×#3939C2×C3semidirect
86. #4072C22×C3semidirect
92. #3964C22×C3semidirect
93. #3964C22×C3semidirect
94. C22⋊#168C22×C3semidirect
95. #3940C22×C3semidirect
96. #3939C22×C3semidirect
287. C2×C3×C5#482semidirect
370. C3×C5C2×#482semidirect
392. C2×C5C3×#482semidirect
428. C2×C3#65324direct (N × H)
432. C5C2×C3×#482semidirect
452. C3C2×#65324direct (N × H)
455. C2C3×#65324direct (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
C2×C3×#40722
C2×C3×#39642
C2×C3×C22⋊#1682
C3×#653242
C2×C3×#39402
C2×C3×#39392
C2×#653243
C2×C3×#4825