Finite Group Databasebeta

← C2×C3×#65155 #14431528

Subgroups — C2×C3×#65155 #14431528

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×#4041C211yesmaximal, characteristic
2C2×C3×#4038C211yesmaximal, characteristic, complemented
3C2×C3×C5×#106C211yesmaximal, characteristic, complemented
4C2×C3×#3953C211yesmaximal, characteristic, complemented
5C2×C3×#3952C211yesmaximal, characteristic, complemented
6C2×C3×#3941C211yesmaximal, characteristic, complemented
7C2×C3×#3939C211yesmaximal, characteristic, complemented
8C3×#65155C218yesmaximal, direct product factor
9C2×#65155C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×D20—21no—
11C22×C3×D20C2211yescharacteristic
12C22×C3×C4×D5C2211yescharacteristic
13C22×C3×C5×C8C2211yescharacteristic
14C2×C3×C5×C4⋊C4C2211yescharacteristic
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×C3×#685—21no—
17C2×C3×#685—21no—
18C2×C3×C4⋊Dic5C2211yescharacteristic
19C2×C3×C4×Dic5—21no—
20C22×C3×#168C2211yescharacteristic
21C3×#4041—22no—
22C3×#4041C2214yes—
23C3×#4038—22no—
24C3×#4038C2214yescomplemented
25C3×C5×#106C2214yescomplemented
26C3×#3953C2214yescomplemented
27C3×#3952C2214yescomplemented
28C3×#3941C2214yescomplemented
29C3×#3939C2214yescomplemented
30C2×C3×#482—51nomaximal, Hall
31C2×#4041C2×C311yescharacteristic
32C2×#4038C2×C311yescharacteristic, complemented
33C2×C5×#106C2×C311yescharacteristic, complemented
34C2×#3953C2×C311yescharacteristic, complemented
35C2×#3952C2×C311yescharacteristic, complemented
36C2×#3941C2×C311yescharacteristic, complemented
37C2×#3939C2×C311yescharacteristic, complemented
38#65155C2×C318yesdirect product factor
39C23×C3×D5—21no—
40C23×C3×D5D411yescharacteristic
41C22×C3×C4×C5—21no—
42C22×C3×C4×C5—21no—
43C22×C3×C4×C5C2311yescharacteristic
44C22×C3×Dic5—21no—
45C22×C3×Dic5D411yescharacteristic
46C2×C3×D20—22no—
47C2×C3×D20—22no—
48C2×C3×D20—24no—
49C2×C3×D20—21no—
50C2×C3×D20—44no—
51C2×C3×D20C2312yes—
52C2×C3×C4×D5—22no—
53C2×C3×C4×D5—22no—
54C2×C3×C4×D5D412yes—
55C2×C3×C4×D5C2312yes—
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×#685—24no—
64C3×C4⋊Dic5—21no—
65C3×C4⋊Dic5C2312yes—
66C3×C4×Dic5—24no—
67C2×C3×#168—22no—
68C2×C3×#168C2312yes—
69C2×C3×#121—51no—
70C2×C3×#117—51no—
71C3×#482—58no—
72C2×C3×#107—51no—
73C2×C3×#106D511yescharacteristic, complemented
74C2×C3×#102—51no—
75C2×C3×#102—51no—
76C2×C3×C22⋊C8—51no—
77C22×D20—21no—
78C22×D20C22×C311yescharacteristic
79C22×C4×D5C22×C311yescharacteristic
80C22×C5×C8C22×C311yescharacteristic
81C2×C5×C4⋊C4C22×C311yescharacteristic
82C2×C5×C4⋊C4C22×C311yescharacteristic
83C2×#685—21no—
84C2×#685—21no—
85C2×C4⋊Dic5C22×C311yescharacteristic
86C2×C4×Dic5—21no—
87C22×#168C22×C311yescharacteristic
88#4041—22no—
89#4041C22×C314yes—
90#4038—22no—
91#4038C22×C314yescomplemented
92C5×#106C22×C314yescomplemented
93#3953C22×C314yescomplemented
94#3952C22×C314yescomplemented
95#3941C22×C314yescomplemented
96#3939C22×C314yescomplemented
97C2×#482—51noSylow
98C23×C3×C5C2×D411yescharacteristic
99C22×C3×D5—22no—
100C22×C3×D5—22no—
101C22×C3×D5—22no—
102C22×C3×D5—22no—
103C22×C3×D5—42no—
104C22×C3×D5—42no—
105C22×C3×D5—42no—
106C22×C3×D5C2×D412yes—
107C2×C3×C4×C5—22no—
108C2×C3×C4×C5—22no—
109C2×C3×C4×C5—42no—
110C2×C3×C4×C5—42no—
111C2×C3×C4×C5C2×D411yescharacteristic
112C2×C3×C4×C5#4112yes—
113C2×C3×C4×C5#4112yes—
114C2×C3×C4×C5C2411yescharacteristic
115C2×C3×Dic5—22no—
116C2×C3×Dic5—22no—
117C2×C3×Dic5—42no—
118C2×C3×Dic5C2×D412yes—
119C3×D20—22no—
120C3×D20—22no—
121C3×D20—48no—
122C3×D20—42no—
123C3×C4×D5—24no—
124C3×C4×D5—22no—
125C3×C4×D5—22no—
126C3×C5×C8—22no—
127C3×#168—22no—
128C22×C3×D4—51no—
129C22×C3×D4—101no—
130C23×C3×C4—51no—
131C22×C3×C8—51no—
132C22×C3×C8C2×D511yescharacteristic
133C3×#121—54no—
134C3×#121—102no—
135C3×#117—54no—
136C3×#117—102no—
137C2×C3×C4⋊C4—51no—
138C2×C3×C4⋊C4C2×D511yescharacteristic
139C2×C3×C4⋊C4C2×D511yescharacteristic
140C2×C3×C22⋊C4—101no—
141C2×C3×C22⋊C4—101no—
142C2×C3×C42—101no—
143C3×#107—54no—
144C3×#106C2×D514yescomplemented
145C3×#102—54no—
146C3×#102—54no—
147C3×C22⋊C8—54no—
148C23×D5—21no—
149C23×D5C3×D411yescharacteristic
150C22×C4×C5—21no—
151C22×C4×C5—21no—
152C22×C4×C5C23×C311yescharacteristic
153C22×Dic5—21no—
154C22×Dic5C3×D411yescharacteristic
155C2×D20—22no—
156C2×D20—22no—
157C2×D20—24no—
158C2×D20—21no—
159C2×D20—44no—
160C2×D20C23×C312yes—
161C2×C4×D5—22no—
162C2×C4×D5—22no—
163C2×C4×D5C3×D412yes—
164C2×C4×D5C23×C312yes—
165C2×C5×C8—22no—
166C2×C5×C8C23×C312yes—
167C5×C4⋊C4—21no—
168C5×C4⋊C4—21no—
169C5×C4⋊C4C23×C312yes—
170C5×C4⋊C4C23×C312yes—
171#685—24no—
172#685—24no—
173C4⋊Dic5—21no—
174C4⋊Dic5C23×C312yes—
175C4×Dic5—24no—
176C2×#168—22no—
177C2×#168C23×C312yes—
178C2×#121—51no—
179C2×#117—51no—
180#482—58no—
181C2×#107—51no—
182C2×#106C3×D511yescharacteristic, complemented
183C2×#102—51no—
184C2×#102—51no—
185C2×C22⋊C8—51no—
186C22×C3×C5#12312yes—
187C22×C3×C5#13512yes—
188C22×C3×C5AGL(1,ℤ/8)12yes—
189C22×C3×C5C22×D411yescharacteristic
190C2×C3×D5—22no—
191C2×C3×D5—22no—
192C2×C3×D5—22no—
193C2×C3×D5—44no—
194C2×C3×D5—42no—
195C2×C3×D5—42no—
196C2×C3×D5—42no—
197C2×C3×D5—84no—
198C3×C4×C5—42no—
199C3×C4×C5—42no—
200C3×C4×C5#12312yes—
201C3×C4×C5C2×#4111yescharacteristic
202C3×C4×C5C2×#4111yescharacteristic
203C3×Dic5—22no—
204C3×Dic5—42no—
205C24×C3—51no—
206C24×C3—101no—
207C2×C3×D4—52no—
208C2×C3×D4—104no—
209C2×C3×D4—102no—
210C2×C3×D4—101no—
211C2×C3×D4—102no—
212C2×C3×D4—204no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—51no—
216C22×C3×C4—52no—
217C22×C3×C4—52no—
218C22×C3×C4—102no—
219C22×C3×C4—102no—
220C22×C3×C4—101no—
221C22×C3×C4C22×D511yescharacteristic
222C2×C3×C8—22no—
223C2×C3×C8—52no—
224C2×C3×C8—102no—
225C2×C3×C8C22×D512yes—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—52no—
229C3×C4⋊C4—101no—
230C3×C4⋊C4C22×D512yes—
231C3×C4⋊C4C22×D512yes—
232C3×C22⋊C4—104no—
233C3×C22⋊C4—104no—
234C3×C42—104no—
235C23×C5C2×C3×D411yescharacteristic
236C22×D5—22no—
237C22×D5—22no—
238C22×D5—22no—
239C22×D5—22no—
240C22×D5—42no—
241C22×D5—42no—
242C22×D5—42no—
243C22×D5C2×C3×D412yes—
244C2×C4×C5—22no—
245C2×C4×C5—22no—
246C2×C4×C5—42no—
247C2×C4×C5—42no—
248C2×C4×C5C2×C3×D411yescharacteristic
249C2×C4×C5C3×#4112yes—
250C2×C4×C5C3×#4112yes—
251C2×C4×C5C24×C311yescharacteristic
252C2×Dic5—22no—
253C2×Dic5—22no—
254C2×Dic5—42no—
255C2×Dic5C2×C3×D412yes—
256D20—22no—
257D20—22no—
258D20—48no—
259D20—42no—
260C4×D5—24no—
261C4×D5—22no—
262C4×D5—22no—
263C5×C8—22no—
264#168—22no—
265C22×D4—51no—
266C22×D4—101no—
267C23×C4—51no—
268C22×C8—51no—
269C22×C8C2×C3×D511yescharacteristic
270#121—54no—
271#121—102no—
272#117—54no—
273#117—102no—
274C2×C4⋊C4—51no—
275C2×C4⋊C4C2×C3×D511yescharacteristic
276C2×C4⋊C4C2×C3×D511yescharacteristic
277C2×C22⋊C4—101no—
278C2×C22⋊C4—101no—
279C2×C42—101no—
280#107—54no—
281#106C2×C3×D514yescomplemented
282#102—54no—
283#102—54no—
284C22⋊C8—54no—
285C2×C3×C5#48214yescomplemented
286C2×C3×C5C2×#12311yescharacteristic
287C2×C3×C5C2×#13511yescharacteristic
288C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
289C3×D5—42no—
290C3×D5—82no—
291C23×C3—52no—
292C23×C3—102no—
293C23×C3—102no—
294C23×C3—102no—
295C23×C3—102no—
296C23×C3—202no—
297C23×C3—202no—
298C23×C3—202no—
299C23×C3D4×D511yescharacteristic
300C3×D4—102no—
301C3×D4—102no—
302C3×D4—208no—
303C3×D4—202no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—52no—
309C2×C3×C4—102no—
310C2×C3×C4—102no—
311C2×C3×C4—102no—
312C2×C3×C4—104no—
313C2×C3×C4—102no—
314C2×C3×C4—202no—
315C2×C3×C4#70912yes—
316C2×C3×C4D4×D511yescharacteristic
317C2×C3×C4#71312yes—
318C2×C3×C4C23×D511yescharacteristic
319C3×C8—22no—
320C3×C8—102no—
321C22×C5C3×#12312yes—
322C22×C5C3×#13512yes—
323C22×C5C3×AGL(1,ℤ/8)12yes—
324C22×C5C22×C3×D411yescharacteristic
325C2×D5—22no—
326C2×D5—22no—
327C2×D5—22no—
328C2×D5—44no—
329C2×D5—42no—
330C2×D5—42no—
331C2×D5—42no—
332C2×D5—84no—
333C4×C5—42no—
334C4×C5—42no—
335C4×C5C3×#12312yes—
336C4×C5C2×C3×#4111yescharacteristic
337C4×C5C2×C3×#4111yescharacteristic
338Dic5—22no—
339Dic5—42no—
340C24—51no—
341C24—101no—
342C2×D4—52no—
343C2×D4—104no—
344C2×D4—102no—
345C2×D4—101no—
346C2×D4—102no—
347C2×D4—204no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—51no—
351C22×C4—52no—
352C22×C4—52no—
353C22×C4—102no—
354C22×C4—102no—
355C22×C4—101no—
356C22×C4C22×C3×D511yescharacteristic
357C2×C8—22no—
358C2×C8—52no—
359C2×C8—102no—
360C2×C8C22×C3×D512yes—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—52no—
364C4⋊C4—101no—
365C4⋊C4C22×C3×D512yes—
366C4⋊C4C22×C3×D512yes—
367C22⋊C4—104no—
368C22⋊C4—104no—
369C42—104no—
370C3×C5C2×#48211yescharacteristic, complemented, Hall
371C22×C3—102no—
372C22×C3—102no—
373C22×C3—102no—
374C22×C3—204no—
375C22×C3—202no—
376C22×C3—202no—
377C22×C3—202no—
378C22×C3—404no—
379C22×C3#404012yes—
380C22×C3#406012yes—
381C22×C3#406212yes—
382C22×C3C2×D4×D511yescharacteristic
383C3×C4—42no—
384C3×C4—42no—
385C3×C4—102no—
386C3×C4—202no—
387C3×C4#404012yes—
388C3×C4C2×#70911yescharacteristic
389C3×C4C2×#71311yescharacteristic
390C2×C5C3×#48214yescomplemented
391C2×C5C2×C3×#12311yescharacteristic
392C2×C5C2×C3×#13511yescharacteristic
393C2×C5C2×C3×AGL(1,ℤ/8)11yescharacteristic
394D5—42no—
395D5—82no—
396C23—52no—
397C23—102no—
398C23—102no—
399C23—102no—
400C23—102no—
401C23—202no—
402C23—202no—
403C23—202no—
404C23C3×D4×D511yescharacteristic
405D4—102no—
406D4—102no—
407D4—208no—
408D4—202no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—52no—
414C2×C4—102no—
415C2×C4—102no—
416C2×C4—102no—
417C2×C4—104no—
418C2×C4—102no—
419C2×C4—202no—
420C2×C4C3×#70912yes—
421C2×C4C3×D4×D511yescharacteristic
422C2×C4C3×#71312yes—
423C2×C4C23×C3×D511yescharacteristic
424C8—22no—
425C8—102no—
426C2×C3—202no—
427C2×C3—402no—
428C2×C3#6515514yesdirect product factor
429C2×C3C2×#404011yescharacteristic
430C2×C3C2×#406011yescharacteristic
431C2×C3C2×#406211yescharacteristic
432C5C2×C3×#48211yescharacteristic, complemented, Sylow
433C22—102no—
434C22—102no—
435C22—102no—
436C22—204no—
437C22—202no—
438C22—202no—
439C22—202no—
440C22—404no—
441C22C3×#404012yes—
442C22C3×#406012yes—
443C22C3×#406212yes—
444C22C2×C3×D4×D511yescharacteristic
445C4—42no—
446C4—42no—
447C4—102no—
448C4—202no—
449C4C3×#404012yes—
450C4C2×C3×#70911yescharacteristic
451C4C2×C3×#71311yescharacteristic
452C3C2×#6515511yescharacteristic, direct product factor, Sylow
453C2—202no—
454C2—402no—
455C2C3×#6515514yesdirect product factor
456C2C2×C3×#404011yescharacteristic
457C2C2×C3×#406011yescharacteristic
458C2C2×C3×#406211yescharacteristic
subgroupG/H
centerC23×C3D4×D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C5×#106C2
socleC23×C3×C5C2×D4
prime radicalC2×C3×#4041C2
hypercenterC2×C3×#106D5
hypocenterC5C2×C3×#482
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
2. C2×C3×#4038C2semidirect
3. C2×C3×C5×#106C2semidirect
4. C2×C3×#3953C2semidirect
5. C2×C3×#3952C2semidirect
6. C2×C3×#3941C2semidirect
7. C2×C3×#3939C2semidirect
8. C3×#65155C2direct (N × H)
9. C2×#65155C3direct (N × H)
24. C3×#4038C22semidirect
25. C3×C5×#106C22semidirect
26. C3×#3953C22semidirect
27. C3×#3952C22semidirect
28. C3×#3941C22semidirect
29. C3×#3939C22semidirect
32. C2×#4038C2×C3semidirect
33. C2×C5×#106C2×C3semidirect
34. C2×#3953C2×C3semidirect
35. C2×#3952C2×C3semidirect
36. C2×#3941C2×C3semidirect
37. C2×#3939C2×C3semidirect
38. #65155C2×C3direct (N × H)
73. C2×C3×#106D5semidirect
91. #4038C22×C3semidirect
92. C5×#106C22×C3semidirect
93. #3953C22×C3semidirect
94. #3952C22×C3semidirect
95. #3941C22×C3semidirect
96. #3939C22×C3semidirect
144. C3×#106C2×D5semidirect
182. C2×#106C3×D5semidirect
281. #106C2×C3×D5semidirect
285. C2×C3×C5#482semidirect
370. C3×C5C2×#482semidirect
390. C2×C5C3×#482semidirect
428. C2×C3#65155direct (N × H)
432. C5C2×C3×#482semidirect
452. C3C2×#65155direct (N × H)
455. C2C3×#65155direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#40412
C2×C3×#40382
C2×C3×C5×#1062
C2×C3×#39532
C2×C3×#39522
C2×C3×#39412
C2×C3×#39392
C3×#651552
C2×#651553
C2×C3×#4825