Finite Group Databasebeta

← C2×C3×#65317 #14426131

Subgroups — C2×C3×#65317 #14426131

Order 1920.

1000 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C5×Q8C211yesmaximal, characteristic, complemented
2C22×C3×#689C211yesmaximal, characteristic, complemented
3C22×C3×#688C211yesmaximal, characteristic
4C2×C3×#4021C211yesmaximal, characteristic
5C3×#65317C218yesmaximal, direct product factor
6C2×C3×#3942C211yesmaximal, characteristic, complemented
7C2×C3×#3941C211yesmaximal, characteristic
8C2×C3×C4⋊#168C211yesmaximal, characteristic, complemented
9C2×#65317C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×C5×Q8C2211yescharacteristic
11C22×C3×D20C2211yescharacteristic
12C22×C3×Dic10C2211yescharacteristic
13C3×C4×C5×Q8—22no—
14C3×C4×C5×Q8C2214yescomplemented
15C2×C3×C5×C4⋊C4—21no—
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17C2×C3×C42×C5—21no—
18C2×C3×C42×C5C2211yescharacteristic
19C2×C3×#689—24no—
20C2×C3×#689C2214yescomplemented
21C2×C3×#688—24no—
22C2×C3×#688C2214yes—
23C2×C3×#685—21no—
24C2×C3×#685—21no—
25C22×C3×#168C2211yescharacteristic
26C22×C3×#168C2211yescharacteristic
27C3×#4021—22no—
28C3×#4021C2214yes—
29C3×#3942C2214yescomplemented
30C3×#3941C2214yes—
31C3×C4⋊#168C2214yescomplemented
32C2×C3×#464—51nomaximal, Hall
33C2×C4×C5×Q8C2×C311yescharacteristic, complemented
34C22×#689C2×C311yescharacteristic, complemented
35C22×#688C2×C311yescharacteristic
36C2×#4021C2×C311yescharacteristic
37#65317C2×C318yesdirect product factor
38C2×#3942C2×C311yescharacteristic, complemented
39C2×#3941C2×C311yescharacteristic
40C2×C4⋊#168C2×C311yescharacteristic, complemented
41C23×C3×D5—21no—
42C2×C3×C5×Q8—21no—
43C2×C3×C5×Q8—22no—
44C2×C3×C5×Q8D414yes—
45C2×C3×C5×Q8C2312yes—
46C22×C3×C4×C5—21no—
47C22×C3×C4×C5—21no—
48C22×C3×C4×C5D411yescharacteristic
49C22×C3×C4×C5D411yescharacteristic
50C22×C3×C4×C5C2311yescharacteristic
51C22×C3×Dic5—21no—
52C2×C3×D20—22no—
53C2×C3×D20—22no—
54C2×C3×D20—21no—
55C2×C3×D20C2312yes—
56C2×C3×Dic10—22no—
57C2×C3×Dic10—22no—
58C2×C3×Dic10—21no—
59C2×C3×Dic10C2312yes—
60C3×C5×C4⋊C4—24no—
61C3×C5×C4⋊C4—21no—
62C3×C5×C4⋊C4C2312yes—
63C3×C42×C5—24no—
64C3×C42×C5D412yes—
65C3×C42×C5C2312yes—
66C3×#689—28no—
67C3×#688—28no—
68C3×#685—24no—
69C3×#685—24no—
70C2×C3×#168—22no—
71C2×C3×#168—22no—
72C2×C3×#168C2312yes—
73C2×C3×#168C2312yes—
74C22×C3×Q16—51no—
75C22×C3×SD16—51no—
76C2×C3×#124—51no—
77C2×C3×C4×Q8D511yescharacteristic, complemented
78C3×#464—58no—
79C2×C3×C4⋊C8—51no—
80C2×C3×#103—51no—
81C2×C3×#102—51no—
82C22×C5×Q8C22×C311yescharacteristic
83C22×D20C22×C311yescharacteristic
84C22×Dic10C22×C311yescharacteristic
85C4×C5×Q8—22no—
86C4×C5×Q8C22×C314yescomplemented
87C2×C5×C4⋊C4—21no—
88C2×C5×C4⋊C4C22×C311yescharacteristic
89C2×C42×C5—21no—
90C2×C42×C5C22×C311yescharacteristic
91C2×#689—24no—
92C2×#689C22×C314yescomplemented
93C2×#688—24no—
94C2×#688C22×C314yes—
95C2×#685—21no—
96C2×#685—21no—
97C22×#168C22×C311yescharacteristic
98C22×#168C22×C311yescharacteristic
99#4021—22no—
100#4021C22×C314yes—
101#3942C22×C314yescomplemented
102#3941C22×C314yes—
103C4⋊#168C22×C314yescomplemented
104C2×#464—51noSylow
105C23×C3×C5C2×D411yescharacteristic
106C22×C3×D5—22no—
107C22×C3×D5—42no—
108C22×C3×D5—42no—
109C22×C3×D5—42no—
110C3×C5×Q8—24no—
111C3×C5×Q8—22no—
112C3×C5×Q8C2×D414yes—
113C2×C3×C4×C5—24no—
114C2×C3×C4×C5—22no—
115C2×C3×C4×C5—22no—
116C2×C3×C4×C5—22no—
117C2×C3×C4×C5—22no—
118C2×C3×C4×C5—42no—
119C2×C3×C4×C5C2×D412yes—
120C2×C3×C4×C5C2×D412yes—
121C2×C3×C4×C5C2×D411yescharacteristic
122C2×C3×C4×C5C2×D412yes—
123C2×C3×C4×C5#4112yes—
124C2×C3×C4×C5C2411yescharacteristic
125C2×C3×Dic5—22no—
126C2×C3×Dic5—42no—
127C3×D20—22no—
128C3×D20—22no—
129C3×D20—42no—
130C3×Dic10—22no—
131C3×Dic10—22no—
132C3×Dic10—42no—
133C3×#168—22no—
134C3×#168—22no—
135C22×C3×Q8—51no—
136C22×C3×Q8C2×D511yescharacteristic
137C22×C3×D4—51no—
138C2×C3×Q16—54no—
139C2×C3×Q16—104no—
140C2×C3×SD16—54no—
141C2×C3×SD16—104no—
142C22×C3×C8—51no—
143C22×C3×C8—51no—
144C3×#124—54no—
145C3×#124—102no—
146C3×C4×Q8—22no—
147C3×C4×Q8C2×D514yescomplemented
148C2×C3×C4⋊C4—21no—
149C2×C3×C4⋊C4C2×D511yescharacteristic
150C2×C3×C22⋊C4—101no—
151C2×C3×C22⋊C4—101no—
152C2×C3×C42—21no—
153C2×C3×C42C2×D511yescharacteristic
154C3×C4⋊C8—54no—
155C3×#103—54no—
156C3×#102—54no—
157C23×D5—21no—
158C2×C5×Q8—21no—
159C2×C5×Q8—22no—
160C2×C5×Q8C3×D414yes—
161C2×C5×Q8C23×C312yes—
162C22×C4×C5—21no—
163C22×C4×C5—21no—
164C22×C4×C5C3×D411yescharacteristic
165C22×C4×C5C3×D411yescharacteristic
166C22×C4×C5C23×C311yescharacteristic
167C22×Dic5—21no—
168C2×D20—22no—
169C2×D20—22no—
170C2×D20—21no—
171C2×D20C23×C312yes—
172C2×Dic10—22no—
173C2×Dic10—22no—
174C2×Dic10—21no—
175C2×Dic10C23×C312yes—
176C5×C4⋊C4—24no—
177C5×C4⋊C4—21no—
178C5×C4⋊C4C23×C312yes—
179C42×C5—24no—
180C42×C5C3×D412yes—
181C42×C5C23×C312yes—
182#689—28no—
183#688—28no—
184#685—24no—
185#685—24no—
186C2×#168—22no—
187C2×#168—22no—
188C2×#168C23×C312yes—
189C2×#168C23×C312yes—
190C22×Q16—51no—
191C22×SD16—51no—
192C2×#124—51no—
193C2×C4×Q8C3×D511yescharacteristic, complemented
194#464—58no—
195C2×C4⋊C8—51no—
196C2×#103—51no—
197C2×#102—51no—
198C22×C3×C5#12112yes—
199C22×C3×C5#13512yes—
200C22×C3×C5#13712yes—
201C22×C3×C5C22×D411yescharacteristic
202C2×C3×D5—42no—
203C2×C3×D5—42no—
204C2×C3×D5—42no—
205C2×C3×D5—84no—
206C3×C4×C5—24no—
207C3×C4×C5—22no—
208C3×C4×C5—22no—
209C3×C4×C5—42no—
210C3×C4×C5#12112yes—
211C3×C4×C5C22×D411yescharacteristic
212C3×C4×C5C2×#4111yescharacteristic
213C3×Dic5—42no—
214C24×C3—101no—
215C2×C3×Q8—22no—
216C2×C3×Q8—21no—
217C2×C3×Q8—52no—
218C2×C3×Q8—101no—
219C2×C3×Q8—102no—
220C2×C3×Q8—102no—
221C2×C3×Q8D2014yes—
222C2×C3×Q8C22×D512yes—
223C2×C3×D4—52no—
224C2×C3×D4—102no—
225C2×C3×D4—102no—
226C2×C3×D4—101no—
227C22×C3×C4—21no—
228C22×C3×C4—21no—
229C22×C3×C4—101no—
230C22×C3×C4C22⋊D511yescharacteristic
231C22×C3×C4C22⋊D511yescharacteristic
232C22×C3×C4C22×D511yescharacteristic
233C3×Q16—108no—
234C3×SD16—108no—
235C2×C3×C8—52no—
236C2×C3×C8—52no—
237C2×C3×C8—102no—
238C2×C3×C8—102no—
239C3×C4⋊C4—24no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4C22×D512yes—
242C3×C22⋊C4—104no—
243C3×C22⋊C4—104no—
244C3×C42—24no—
245C3×C42C22⋊D512yes—
246C3×C42C22×D512yes—
247C23×C5C2×C3×D411yescharacteristic
248C22×D5—22no—
249C22×D5—42no—
250C22×D5—42no—
251C22×D5—42no—
252C5×Q8—24no—
253C5×Q8—22no—
254C5×Q8C2×C3×D414yes—
255C2×C4×C5—24no—
256C2×C4×C5—22no—
257C2×C4×C5—22no—
258C2×C4×C5—22no—
259C2×C4×C5—22no—
260C2×C4×C5—42no—
261C2×C4×C5C2×C3×D412yes—
262C2×C4×C5C2×C3×D412yes—
263C2×C4×C5C2×C3×D412yes—
264C2×C4×C5C2×C3×D411yescharacteristic
265C2×C4×C5C3×#4112yes—
266C2×C4×C5C24×C311yescharacteristic
267C2×Dic5—22no—
268C2×Dic5—42no—
269D20—22no—
270D20—22no—
271D20—42no—
272Dic10—22no—
273Dic10—22no—
274Dic10—42no—
275#168—22no—
276#168—22no—
277C22×Q8—51no—
278C22×Q8C2×C3×D511yescharacteristic
279C22×D4—51no—
280C2×Q16—54no—
281C2×Q16—104no—
282C2×SD16—54no—
283C2×SD16—104no—
284C22×C8—51no—
285C22×C8—51no—
286#124—54no—
287#124—102no—
288C4×Q8—22no—
289C4×Q8C2×C3×D514yescomplemented
290C2×C4⋊C4—21no—
291C2×C4⋊C4C2×C3×D511yescharacteristic
292C2×C22⋊C4—101no—
293C2×C22⋊C4—101no—
294C2×C42—21no—
295C2×C42C2×C3×D511yescharacteristic
296C4⋊C8—54no—
297#103—54no—
298#102—54no—
299C2×C3×C5#46414yescomplemented
300C2×C3×C5C2×#12111yescharacteristic
301C2×C3×C5C2×#13511yescharacteristic
302C2×C3×C5C2×#13711yescharacteristic
303C3×D5—82no—
304C23×C3—102no—
305C23×C3—202no—
306C23×C3—202no—
307C23×C3—202no—
308C23×C3C2×C22⋊D511yescharacteristic
309C3×Q8—22no—
310C3×Q8—24no—
311C3×Q8—102no—
312C3×Q8—102no—
313C3×Q8—202no—
314C3×Q8C2×D2014yes—
315C3×D4—102no—
316C3×D4—102no—
317C3×D4—202no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—24no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—42no—
324C2×C3×C4—102no—
325C2×C3×C4—202no—
326C2×C3×C4C2×D2012yes—
327C2×C3×C4#70912yes—
328C2×C3×C4C2×C22⋊D512yes—
329C2×C3×C4C2×C22⋊D512yes—
330C2×C3×C4C2×C22⋊D511yescharacteristic
331C2×C3×C4C23×D511yescharacteristic
332C3×C8—102no—
333C3×C8—102no—
334C22×C5C3×#12112yes—
335C22×C5C3×#13512yes—
336C22×C5C3×#13712yes—
337C22×C5C22×C3×D411yescharacteristic
338C2×D5—42no—
339C2×D5—42no—
340C2×D5—42no—
341C2×D5—84no—
342C4×C5—22no—
343C4×C5—22no—
344C4×C5—24no—
345C4×C5—42no—
346C4×C5C3×#12112yes—
347C4×C5C22×C3×D411yescharacteristic
348C4×C5C2×C3×#4111yescharacteristic
349Dic5—42no—
350C24—101no—
351C2×Q8—22no—
352C2×Q8—21no—
353C2×Q8—52no—
354C2×Q8—101no—
355C2×Q8—102no—
356C2×Q8—102no—
357C2×Q8C3×D2014yes—
358C2×Q8C22×C3×D512yes—
359C2×D4—52no—
360C2×D4—102no—
361C2×D4—102no—
362C2×D4—101no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—101no—
366C22×C4C3×C22⋊D511yescharacteristic
367C22×C4C3×C22⋊D511yescharacteristic
368C22×C4C22×C3×D511yescharacteristic
369Q16—108no—
370SD16—108no—
371C2×C8—52no—
372C2×C8—52no—
373C2×C8—102no—
374C2×C8—102no—
375C4⋊C4—24no—
376C4⋊C4—21no—
377C4⋊C4C22×C3×D512yes—
378C22⋊C4—104no—
379C22⋊C4—104no—
380C42—24no—
381C42C3×C22⋊D512yes—
382C42C22×C3×D512yes—
383C3×C5C2×#46411yescharacteristic, complemented, Hall
384C22×C3—202no—
385C22×C3—202no—
386C22×C3—202no—
387C22×C3—404no—
388C22×C3#407612yes—
389C22×C3#408812yes—
390C22×C3#409612yes—
391C22×C3C22×C22⋊D511yescharacteristic
392C3×C4—24no—
393C3×C4—22no—
394C3×C4—22no—
395C3×C4—42no—
396C3×C4—202no—
397C3×C4#407612yes—
398C3×C4C22×D2011yescharacteristic
399C3×C4C2×#70911yescharacteristic
400C2×C5C3×#46414yescomplemented
401C2×C5C2×C3×#12111yescharacteristic
402C2×C5C2×C3×#13511yescharacteristic
403C2×C5C2×C3×#13711yescharacteristic
404D5—82no—
405C23—102no—
406C23—202no—
407C23—202no—
408C23—202no—
409C23C2×C3×C22⋊D511yescharacteristic
410Q8—22no—
411Q8—24no—
412Q8—102no—
413Q8—102no—
414Q8—202no—
415Q8C2×C3×D2014yes—
416D4—102no—
417D4—102no—
418D4—202no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—42no—
425C2×C4—102no—
426C2×C4—202no—
427C2×C4C2×C3×D2012yes—
428C2×C4C3×#70912yes—
429C2×C4C2×C3×C22⋊D512yes—
430C2×C4C2×C3×C22⋊D512yes—
431C2×C4C2×C3×C22⋊D511yescharacteristic
432C2×C4C23×C3×D511yescharacteristic
433C8—102no—
434C8—102no—
435C2×C3—402no—
436C2×C3#6531714yesdirect product factor
437C2×C3C2×#407611yescharacteristic
438C2×C3C2×#408811yescharacteristic
439C2×C3C2×#409611yescharacteristic
440C5C2×C3×#46411yescharacteristic, complemented, Sylow
441C22—202no—
442C22—202no—
443C22—202no—
444C22—404no—
445C22C3×#407612yes—
446C22C3×#408812yes—
447C22C3×#409612yes—
448C22C22×C3×C22⋊D511yescharacteristic
449C4—24no—
450C4—22no—
451C4—22no—
452C4—42no—
453C4—202no—
454C4C3×#407612yes—
455C4C22×C3×D2011yescharacteristic
456C4C2×C3×#70911yescharacteristic
457C3C2×#6531711yescharacteristic, direct product factor, Sylow
458C2—402no—
459C2C3×#6531714yesdirect product factor
460C2C2×C3×#407611yescharacteristic
461C2C2×C3×#408811yescharacteristic
462C2C2×C3×#409611yescharacteristic
subgroupG/H
centerC23×C3C2×C22⋊D5
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C4×C5×Q8C2
socleC23×C3×C5C2×D4
prime radicalC22×C3×D20C22
hypercenterC2×C3×C4×Q8D5
hypocenterC5C2×C3×#464
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×C5×Q8C2semidirect
2. C22×C3×#689C2semidirect
5. C3×#65317C2direct (N × H)
6. C2×C3×#3942C2semidirect
8. C2×C3×C4⋊#168C2semidirect
9. C2×#65317C3direct (N × H)
14. C3×C4×C5×Q8C22semidirect
20. C2×C3×#689C22semidirect
29. C3×#3942C22semidirect
31. C3×C4⋊#168C22semidirect
33. C2×C4×C5×Q8C2×C3semidirect
34. C22×#689C2×C3semidirect
37. #65317C2×C3direct (N × H)
38. C2×#3942C2×C3semidirect
40. C2×C4⋊#168C2×C3semidirect
77. C2×C3×C4×Q8D5semidirect
86. C4×C5×Q8C22×C3semidirect
92. C2×#689C22×C3semidirect
101. #3942C22×C3semidirect
103. C4⋊#168C22×C3semidirect
147. C3×C4×Q8C2×D5semidirect
193. C2×C4×Q8C3×D5semidirect
289. C4×Q8C2×C3×D5semidirect
299. C2×C3×C5#464semidirect
383. C3×C5C2×#464semidirect
400. C2×C5C3×#464semidirect
436. C2×C3#65317direct (N × H)
440. C5C2×C3×#464semidirect
457. C3C2×#65317direct (N × H)
459. C2C3×#65317direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×C5×Q82
C22×C3×#6892
C22×C3×#6882
C2×C3×#40212
C3×#653172
C2×C3×#39422
C2×C3×#39412
C2×C3×C4⋊#1682
C2×#653173
C2×C3×#4645