Finite Group Databasebeta

← C2×C3×#65403 #14427598

Subgroups — C2×C3×#65403 #14427598

Order 1920.

1128 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
1C23×C3×C5×C8C211yesmaximal, characteristic, complemented
2C2×C3×#4076C211yesmaximal, characteristic
3C2×C3×#4070C211yesmaximal, characteristic, complemented
4C22×C3×#677C211yesmaximal, characteristic, complemented
5C3×#65403C218yesmaximal, direct product factor
6C2×C3×#3953C211yesmaximal, characteristic, complemented
7C2×C3×#3949C211yesmaximal, characteristic, complemented
8C2×C3×#3948C211yesmaximal, characteristic, complemented
9C2×#65403C311yesmaximal, characteristic, direct product factor, Hall
10C23×C3×C4×C5C2211yescharacteristic
11C22×C3×C22⋊D5—21no—
12C22×C3×D20C2211yescharacteristic
13C22×C3×Dic10C2211yescharacteristic
14C22×C3×C5×C8—24no—
15C22×C3×C5×C8C2211yescharacteristic
16C22×C3×C5×C8C2211yescharacteristic
17C22×C3×C5×C8C2214yescomplemented
18C2×C3×C22⋊Dic5—21no—
19C3×#4076—22no—
20C3×#4076C2214yes—
21C2×C3×#685—21no—
22C2×C3×C4⋊Dic5C2211yescharacteristic
23C2×C3×C4⋊Dic5C2211yescharacteristic
24C3×#4070—22no—
25C3×#4070C2214yescomplemented
26C2×C3×#683—21no—
27C2×C3×#677—24no—
28C2×C3×#677C2214yescomplemented
29C3×#3953C2214yescomplemented
30C3×#3949C2214yescomplemented
31C3×#3948C2214yescomplemented
32C2×C3×#465—51nomaximal, Hall
33C23×C5×C8C2×C311yescharacteristic, complemented
34C2×#4076C2×C311yescharacteristic
35C2×#4070C2×C311yescharacteristic, complemented
36C22×#677C2×C311yescharacteristic, complemented
37#65403C2×C318yesdirect product factor
38C2×#3953C2×C311yescharacteristic, complemented
39C2×#3949C2×C311yescharacteristic, complemented
40C2×#3948C2×C311yescharacteristic, complemented
41C24×C3×C5D411yescharacteristic
42C23×C3×D5—21no—
43C22×C3×C4×C5—22no—
44C22×C3×C4×C5—22no—
45C22×C3×C4×C5D411yescharacteristic
46C22×C3×C4×C5D412yes—
47C22×C3×C4×C5C2312yes—
48C22×C3×C4×C5C2311yescharacteristic
49C2×C3×C22⋊D5—24no—
50C2×C3×C22⋊D5—44no—
51C22×C3×Dic5—21no—
52C22×C3×Dic5—21no—
53C22×C3×Dic5—21no—
54C2×C3×D20—22no—
55C2×C3×D20—22no—
56C2×C3×D20—21no—
57C2×C3×D20C2312yes—
58C2×C3×Dic10—22no—
59C2×C3×Dic10—22no—
60C2×C3×Dic10—21no—
61C2×C3×Dic10C2312yes—
62C2×C3×C5×C8—22no—
63C2×C3×C5×C8—28no—
64C2×C3×C5×C8D414yescomplemented
65C2×C3×C5×C8C2312yes—
66C2×C3×C5×C8C2312yes—
67C3×C22⋊Dic5—24no—
68C3×#685—24no—
69C3×C4⋊Dic5—21no—
70C3×C4⋊Dic5—21no—
71C3×C4⋊Dic5C2312yes—
72C3×C4⋊Dic5C2312yes—
73C3×#683—24no—
74C3×#677—28no—
75C22×C3×SD16—51no—
76C23×C3×C8D511yescharacteristic, complemented
77C2×C3×C22⋊Q8—51no—
78C2×C3×#121—51no—
79C3×#465—58no—
80C2×C3×#106—51no—
81C2×C3×#103—51no—
82C2×C3×#102—51no—
83C23×C4×C5C22×C311yescharacteristic
84C22×C22⋊D5—21no—
85C22×D20C22×C311yescharacteristic
86C22×Dic10C22×C311yescharacteristic
87C22×C5×C8—24no—
88C22×C5×C8C22×C311yescharacteristic
89C22×C5×C8C22×C311yescharacteristic
90C22×C5×C8C22×C314yescomplemented
91C2×C22⋊Dic5—21no—
92#4076—22no—
93#4076C22×C314yes—
94C2×#685—21no—
95C2×C4⋊Dic5C22×C311yescharacteristic
96C2×C4⋊Dic5C22×C311yescharacteristic
97#4070—22no—
98#4070C22×C314yescomplemented
99C2×#683—21no—
100C2×#677—24no—
101C2×#677C22×C314yescomplemented
102#3953C22×C314yescomplemented
103#3949C22×C314yescomplemented
104#3948C22×C314yescomplemented
105C2×#465—51noSylow
106C23×C3×C5—22no—
107C23×C3×C5—22no—
108C23×C3×C5SD1614yescomplemented
109C23×C3×C5C2×D412yes—
110C23×C3×C5C2×D411yescharacteristic
111C22×C3×D5—22no—
112C22×C3×D5—42no—
113C22×C3×D5—42no—
114C22×C3×D5—42no—
115C2×C3×C4×C5—22no—
116C2×C3×C4×C5—24no—
117C2×C3×C4×C5—22no—
118C2×C3×C4×C5—22no—
119C2×C3×C4×C5C2×D412yes—
120C2×C3×C4×C5C2×D411yescharacteristic
121C2×C3×C4×C5C2×D412yes—
122C2×C3×C4×C5#4112yes—
123C2×C3×C4×C5C2411yescharacteristic
124C3×C22⋊D5—48no—
125C2×C3×Dic5—22no—
126C2×C3×Dic5—22no—
127C2×C3×Dic5—22no—
128C2×C3×Dic5—42no—
129C2×C3×Dic5—42no—
130C2×C3×Dic5—42no—
131C3×D20—22no—
132C3×D20—22no—
133C3×D20—42no—
134C3×Dic10—22no—
135C3×Dic10—22no—
136C3×Dic10—42no—
137C3×C5×C8—22no—
138C3×C5×C8C2×D414yescomplemented
139C22×C3×Q8—51no—
140C22×C3×D4—51no—
141C22×C3×D4—101no—
142C23×C3×C4C2×D511yescharacteristic
143C2×C3×SD16—54no—
144C2×C3×SD16—104no—
145C22×C3×C8—24no—
146C22×C3×C8C2×D511yescharacteristic
147C22×C3×C8C2×D511yescharacteristic
148C22×C3×C8C2×D514yescomplemented
149C3×C22⋊Q8—54no—
150C3×C22⋊Q8—102no—
151C3×#121—54no—
152C3×#121—102no—
153C2×C3×C4⋊C4—51no—
154C2×C3×C4⋊C4—51no—
155C2×C3×C4⋊C4—101no—
156C2×C3×C22⋊C4—101no—
157C2×C3×C22⋊C4—101no—
158C3×#106—54no—
159C3×#103—54no—
160C3×#102—54no—
161C24×C5C3×D411yescharacteristic
162C23×D5—21no—
163C22×C4×C5—22no—
164C22×C4×C5—22no—
165C22×C4×C5C3×D411yescharacteristic
166C22×C4×C5C3×D412yes—
167C22×C4×C5C23×C312yes—
168C22×C4×C5C23×C311yescharacteristic
169C2×C22⋊D5—24no—
170C2×C22⋊D5—44no—
171C22×Dic5—21no—
172C22×Dic5—21no—
173C22×Dic5—21no—
174C2×D20—22no—
175C2×D20—22no—
176C2×D20—21no—
177C2×D20C23×C312yes—
178C2×Dic10—22no—
179C2×Dic10—22no—
180C2×Dic10—21no—
181C2×Dic10C23×C312yes—
182C2×C5×C8—22no—
183C2×C5×C8—28no—
184C2×C5×C8C3×D414yescomplemented
185C2×C5×C8C23×C312yes—
186C2×C5×C8C23×C312yes—
187C22⋊Dic5—24no—
188#685—24no—
189C4⋊Dic5—21no—
190C4⋊Dic5—21no—
191C4⋊Dic5C23×C312yes—
192C4⋊Dic5C23×C312yes—
193#683—24no—
194#677—28no—
195C22×SD16—51no—
196C23×C8C3×D511yescharacteristic, complemented
197C2×C22⋊Q8—51no—
198C2×#121—51no—
199#465—58no—
200C2×#106—51no—
201C2×#103—51no—
202C2×#102—51no—
203C22×C3×C5—22no—
204C22×C3×C5—22no—
205C22×C3×C5—28no—
206C22×C3×C5#12112yes—
207C22×C3×C5C2×SD1612yes—
208C22×C3×C5C2×SD1614yescomplemented
209C22×C3×C5#13512yes—
210C22×C3×C5C22×D411yescharacteristic
211C2×C3×D5—42no—
212C2×C3×D5—42no—
213C2×C3×D5—42no—
214C2×C3×D5—84no—
215C3×C4×C5—22no—
216C3×C4×C5#12112yes—
217C3×C4×C5C22×D411yescharacteristic
218C3×C4×C5C2×#4111yescharacteristic
219C3×Dic5—42no—
220C3×Dic5—42no—
221C3×Dic5—42no—
222C24×C3—101no—
223C24×C3D2011yescharacteristic
224C2×C3×Q8—52no—
225C2×C3×Q8—101no—
226C2×C3×Q8—102no—
227C2×C3×Q8—102no—
228C2×C3×D4—52no—
229C2×C3×D4—104no—
230C2×C3×D4—101no—
231C2×C3×D4—102no—
232C2×C3×D4—102no—
233C2×C3×D4—204no—
234C22×C3×C4—22no—
235C22×C3×C4—22no—
236C22×C3×C4—101no—
237C22×C3×C4—101no—
238C22×C3×C4—101no—
239C22×C3×C4D2012yes—
240C22×C3×C4D2011yescharacteristic
241C22×C3×C4C22×D512yes—
242C22×C3×C4C22×D511yescharacteristic
243C3×SD16—108no—
244C2×C3×C8—22no—
245C2×C3×C8—28no—
246C2×C3×C8C22⋊D514yescomplemented
247C2×C3×C8C22×D512yes—
248C2×C3×C8C22×D512yes—
249C3×C4⋊C4—52no—
250C3×C4⋊C4—52no—
251C3×C4⋊C4—104no—
252C3×C4⋊C4—101no—
253C3×C4⋊C4—101no—
254C3×C22⋊C4—104no—
255C3×C22⋊C4—104no—
256C23×C5—22no—
257C23×C5—22no—
258C23×C5C3×SD1614yescomplemented
259C23×C5C2×C3×D412yes—
260C23×C5C2×C3×D411yescharacteristic
261C22×D5—22no—
262C22×D5—42no—
263C22×D5—42no—
264C22×D5—42no—
265C2×C4×C5—22no—
266C2×C4×C5—24no—
267C2×C4×C5—22no—
268C2×C4×C5—22no—
269C2×C4×C5C2×C3×D412yes—
270C2×C4×C5C2×C3×D411yescharacteristic
271C2×C4×C5C2×C3×D412yes—
272C2×C4×C5C3×#4112yes—
273C2×C4×C5C24×C311yescharacteristic
274C22⋊D5—48no—
275C2×Dic5—22no—
276C2×Dic5—22no—
277C2×Dic5—22no—
278C2×Dic5—42no—
279C2×Dic5—42no—
280C2×Dic5—42no—
281D20—22no—
282D20—22no—
283D20—42no—
284Dic10—22no—
285Dic10—22no—
286Dic10—42no—
287C5×C8—22no—
288C5×C8C2×C3×D414yescomplemented
289C22×Q8—51no—
290C22×D4—51no—
291C22×D4—101no—
292C23×C4C2×C3×D511yescharacteristic
293C2×SD16—54no—
294C2×SD16—104no—
295C22×C8—24no—
296C22×C8C2×C3×D511yescharacteristic
297C22×C8C2×C3×D511yescharacteristic
298C22×C8C2×C3×D514yescomplemented
299C22⋊Q8—54no—
300C22⋊Q8—102no—
301#121—54no—
302#121—102no—
303C2×C4⋊C4—51no—
304C2×C4⋊C4—51no—
305C2×C4⋊C4—101no—
306C2×C22⋊C4—101no—
307C2×C22⋊C4—101no—
308#106—54no—
309#103—54no—
310#102—54no—
311C2×C3×C5—24no—
312C2×C3×C5#46514yescomplemented
313C2×C3×C5C2×#12111yescharacteristic
314C2×C3×C5C22×SD1611yescharacteristic
315C2×C3×C5C2×#13511yescharacteristic
316C3×D5—82no—
317C23×C3—22no—
318C23×C3—22no—
319C23×C3—102no—
320C23×C3—202no—
321C23×C3—202no—
322C23×C3—202no—
323C23×C3#67714yescomplemented
324C23×C3C2×D2012yes—
325C23×C3C2×D2011yescharacteristic
326C3×Q8—102no—
327C3×Q8—102no—
328C3×Q8—202no—
329C3×D4—102no—
330C3×D4—102no—
331C3×D4—208no—
332C3×D4—202no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—24no—
337C2×C3×C4—102no—
338C2×C3×C4—102no—
339C2×C3×C4—102no—
340C2×C3×C4—202no—
341C2×C3×C4—202no—
342C2×C3×C4—202no—
343C2×C3×C4C2×D2012yes—
344C2×C3×C4C2×D2011yescharacteristic
345C2×C3×C4#70912yes—
346C2×C3×C4C2×C22⋊D512yes—
347C2×C3×C4C23×D511yescharacteristic
348C3×C8—22no—
349C3×C8C2×C22⋊D514yescomplemented
350C22×C5—22no—
351C22×C5—28no—
352C22×C5—22no—
353C22×C5C3×#12112yes—
354C22×C5C2×C3×SD1612yes—
355C22×C5C2×C3×SD1614yescomplemented
356C22×C5C3×#13512yes—
357C22×C5C22×C3×D411yescharacteristic
358C2×D5—42no—
359C2×D5—42no—
360C2×D5—42no—
361C2×D5—84no—
362C4×C5—22no—
363C4×C5C3×#12112yes—
364C4×C5C22×C3×D411yescharacteristic
365C4×C5C2×C3×#4111yescharacteristic
366Dic5—42no—
367Dic5—42no—
368Dic5—42no—
369C24—101no—
370C24C3×D2011yescharacteristic
371C2×Q8—52no—
372C2×Q8—101no—
373C2×Q8—102no—
374C2×Q8—102no—
375C2×D4—52no—
376C2×D4—104no—
377C2×D4—101no—
378C2×D4—102no—
379C2×D4—102no—
380C2×D4—204no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—101no—
384C22×C4—101no—
385C22×C4—101no—
386C22×C4C3×D2011yescharacteristic
387C22×C4C3×D2012yes—
388C22×C4C22×C3×D512yes—
389C22×C4C22×C3×D511yescharacteristic
390SD16—108no—
391C2×C8—22no—
392C2×C8—28no—
393C2×C8C3×C22⋊D514yescomplemented
394C2×C8C22×C3×D512yes—
395C2×C8C22×C3×D512yes—
396C4⋊C4—52no—
397C4⋊C4—52no—
398C4⋊C4—101no—
399C4⋊C4—101no—
400C4⋊C4—104no—
401C22⋊C4—104no—
402C22⋊C4—104no—
403C3×C5C2×#46511yescharacteristic, complemented, Hall
404C22×C3—22no—
405C22×C3—22no—
406C22×C3—28no—
407C22×C3—202no—
408C22×C3—202no—
409C22×C3—202no—
410C22×C3—404no—
411C22×C3C2×#67712yes—
412C22×C3C2×#67714yescomplemented
413C22×C3#405012yes—
414C22×C3#407612yes—
415C22×C3C22×D2011yescharacteristic
416C3×C4—22no—
417C3×C4—202no—
418C3×C4—202no—
419C3×C4—202no—
420C3×C4#407612yes—
421C3×C4C2×#70911yescharacteristic
422C3×C4C22×C22⋊D511yescharacteristic
423C2×C5—24no—
424C2×C5C3×#46514yescomplemented
425C2×C5C2×C3×#12111yescharacteristic
426C2×C5C22×C3×SD1611yescharacteristic
427C2×C5C2×C3×#13511yescharacteristic
428D5—82no—
429C23—22no—
430C23—22no—
431C23—102no—
432C23—202no—
433C23—202no—
434C23—202no—
435C23C3×#67714yescomplemented
436C23C2×C3×D2012yes—
437C23C2×C3×D2011yescharacteristic
438Q8—102no—
439Q8—102no—
440Q8—202no—
441D4—102no—
442D4—102no—
443D4—208no—
444D4—202no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—102no—
450C2×C4—102no—
451C2×C4—102no—
452C2×C4—202no—
453C2×C4—202no—
454C2×C4—202no—
455C2×C4C2×C3×D2012yes—
456C2×C4C2×C3×D2011yescharacteristic
457C2×C4C3×#70912yes—
458C2×C4C2×C3×C22⋊D512yes—
459C2×C4C23×C3×D511yescharacteristic
460C8—22no—
461C8C2×C3×C22⋊D514yescomplemented
462C2×C3—24no—
463C2×C3—402no—
464C2×C3#6540314yesdirect product factor
465C2×C3C22×#67711yescharacteristic
466C2×C3C2×#405011yescharacteristic
467C2×C3C2×#407611yescharacteristic
468C5C2×C3×#46511yescharacteristic, complemented, Sylow
469C22—22no—
470C22—28no—
471C22—22no—
472C22—202no—
473C22—202no—
474C22—202no—
475C22—404no—
476C22C2×C3×#67712yes—
477C22C2×C3×#67714yescomplemented
478C22C3×#405012yes—
479C22C3×#407612yes—
480C22C22×C3×D2011yescharacteristic
481C4—22no—
482C4—202no—
483C4—202no—
484C4—202no—
485C4C3×#407612yes—
486C4C2×C3×#70911yescharacteristic
487C4C22×C3×C22⋊D511yescharacteristic
488C3C2×#6540311yescharacteristic, direct product factor, Sylow
489C2—24no—
490C2—402no—
491C2C3×#6540314yesdirect product factor
492C2C22×C3×#67711yescharacteristic
493C2C2×C3×#405011yescharacteristic
494C2C2×C3×#407611yescharacteristic
subgroupG/H
centerC23×C3C2×D20
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC23×C3×C5×C8C2
socleC23×C3×C5C2×D4
prime radicalC2×C3×#4076C2
hypercenterC23×C3×C8D5
hypocenterC5C2×C3×#465
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C23×C3×C8

Complements

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

NH ≅ G/Ntype
1. C23×C3×C5×C8C2semidirect
3. C2×C3×#4070C2semidirect
4. C22×C3×#677C2semidirect
5. C3×#65403C2direct (N × H)
6. C2×C3×#3953C2semidirect
7. C2×C3×#3949C2semidirect
8. C2×C3×#3948C2semidirect
9. C2×#65403C3direct (N × H)
17. C22×C3×C5×C8C22semidirect
25. C3×#4070C22semidirect
28. C2×C3×#677C22semidirect
29. C3×#3953C22semidirect
30. C3×#3949C22semidirect
31. C3×#3948C22semidirect
33. C23×C5×C8C2×C3semidirect
35. C2×#4070C2×C3semidirect
36. C22×#677C2×C3semidirect
37. #65403C2×C3direct (N × H)
38. C2×#3953C2×C3semidirect
39. C2×#3949C2×C3semidirect
40. C2×#3948C2×C3semidirect
64. C2×C3×C5×C8D4semidirect
76. C23×C3×C8D5semidirect
90. C22×C5×C8C22×C3semidirect
98. #4070C22×C3semidirect
101. C2×#677C22×C3semidirect
102. #3953C22×C3semidirect
103. #3949C22×C3semidirect
104. #3948C22×C3semidirect
108. C23×C3×C5SD16semidirect
138. C3×C5×C8C2×D4semidirect
148. C22×C3×C8C2×D5semidirect
184. C2×C5×C8C3×D4semidirect
196. C23×C8C3×D5semidirect
208. C22×C3×C5C2×SD16semidirect
246. C2×C3×C8C22⋊D5semidirect
258. C23×C5C3×SD16semidirect
288. C5×C8C2×C3×D4semidirect
298. C22×C8C2×C3×D5semidirect
312. C2×C3×C5#465semidirect
323. C23×C3#677semidirect
349. C3×C8C2×C22⋊D5semidirect
355. C22×C5C2×C3×SD16semidirect
393. C2×C8C3×C22⋊D5semidirect
403. C3×C5C2×#465semidirect
412. C22×C3C2×#677semidirect
424. C2×C5C3×#465semidirect
435. C23C3×#677semidirect
461. C8C2×C3×C22⋊D5semidirect
464. C2×C3#65403direct (N × H)
468. C5C2×C3×#465semidirect
477. C22C2×C3×#677semidirect
488. C3C2×#65403direct (N × H)
491. C2C3×#65403direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C5×C82
C2×C3×#40762
C2×C3×#40702
C22×C3×#6772
C3×#654032
C2×C3×#39532
C2×C3×#39492
C2×C3×#39482
C2×#654033
C2×C3×#4655