Finite Group Databasebeta

← C3×#10610431 #14425312

Subgroups — C3×#10610431 #14425312

Order 1920.

1524 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×#4144C211yesmaximal, characteristic, complemented
2C3×#66139C211yesmaximal, characteristic, complemented
3C3×#66082C212yesmaximal, complemented
4C2×C3×C5×#100C211yesmaximal, characteristic, complemented
5C3×#65428C212yesmaximal, complemented
6C3×#65036C214yesmaximal, complemented
7C3×#65035C214yesmaximal, complemented
8#10610431C311yesmaximal, characteristic, direct product factor, Hall
9C22×C3×C5×D4C2211yescharacteristic
10C22×C3×C22⋊D5—22no—
11C3×#4144—22no—
12C3×#4144—24no—
13C3×#4144C2214yes—
14C2×C3×#711—22no—
15C2×C3×D4×D5—22no—
16C2×C3×#709C2212yes—
17C2×C3×#709C2211yescharacteristic, complemented
18C2×C3×C5×M16C2212yescomplemented
19C3×#4099—22no—
20C3×#4086C2212yescomplemented
21C3×#4084—22no—
22C3×#4082C2212yescomplemented
23C3×#4072C2211yescharacteristic, complemented
24C3×#4055—22no—
25C3×#4055C2214yescomplemented
26C3×#4054—22no—
27C3×#4054C2214yescomplemented
28C3×#4050—24no—
29C3×C5×#100C2214yescomplemented
30C3×#3957C2218yescomplemented
31C3×#3019—51nomaximal, Hall
32C2×#4144C2×C311yescharacteristic, complemented
33#66139C2×C311yescharacteristic, complemented
34#66082C2×C312yescomplemented
35C2×C5×#100C2×C311yescharacteristic, complemented
36#65428C2×C312yescomplemented
37#65036C2×C314yescomplemented
38#65035C2×C314yescomplemented
39C24×C3×C5D412yes—
40C23×C3×D5—22no—
41C2×C3×C5×D4—22no—
42C2×C3×C5×D4—22no—
43C2×C3×C5×D4D412yes—
44C2×C3×C5×D4C2312yes—
45C22×C3×C4×C5C2311yescharacteristic
46C2×C3×C22⋊D5—24no—
47C2×C3×C22⋊D5—28no—
48C2×C3×C22⋊D5—22no—
49C2×C3×C22⋊D5—28no—
50C2×C3×C22⋊D5—21no—
51C2×C3×C22⋊D5—24no—
52C22×C3×Dic5—22no—
53C3×#711—28no—
54C3×#711—28no—
55C3×D4×D5—28no—
56C3×D4×D5—28no—
57C3×#709—22no—
58C3×#709—22no—
59C3×#709—22no—
60C3×#709—22no—
61C3×#709D414yescomplemented
62C3×#709C2314yes—
63C2×C3×D20D412yescomplemented
64C2×C3×D20C2311yescharacteristic, complemented
65C2×C3×C4×D5—22no—
66C2×C3×C4×D5—21no—
67C2×C3×Dic10D412yescomplemented
68C2×C3×Dic10C2311yescharacteristic, complemented
69C3×C5×M16—22no—
70C3×C5×M16C2314yescomplemented
71C2×C3×C5×C8—22no—
72C3×C22⋊Dic5—24no—
73C3×C22⋊Dic5—21no—
74C3×C4⋊Dic5—21no—
75C3×C4×Dic5C2312yescomplemented
76C3×Dic20—24no—
77C3×D40—24no—
78C3×#677—24no—
79C3×#677—24no—
80C2×C3×H(5,𝔽2)—51no—
81C3×#575—52no—
82C3×#535—51no—
83C3×#455—54no—
84C3×#453—54no—
85C3×#421—52no—
86C2×C3×#100D511yescharacteristic, complemented
87C22×C5×D4C22×C311yescharacteristic
88C22×C22⋊D5—22no—
89#4144—22no—
90#4144—24no—
91#4144C22×C314yes—
92C2×#711—22no—
93C2×D4×D5—22no—
94C2×#709C22×C312yes—
95C2×#709C22×C311yescharacteristic, complemented
96C2×C5×M16C22×C312yescomplemented
97#4099—22no—
98#4086C22×C312yescomplemented
99#4084—22no—
100#4082C22×C312yescomplemented
101#4072C22×C311yescharacteristic, complemented
102#4055—22no—
103#4055C22×C314yescomplemented
104#4054—22no—
105#4054C22×C314yescomplemented
106#4050—24no—
107C5×#100C22×C314yescomplemented
108#3957C22×C318yescomplemented
109#3019—51noSylow
110C23×C3×C5—22no—
111C23×C3×C5—22no—
112C23×C3×C5—82no—
113C23×C3×C5C2×D414yes—
114C23×C3×C5C2×D411yescharacteristic
115C22×C3×D5—22no—
116C22×C3×D5—28no—
117C22×C3×D5—21no—
118C22×C3×D5—84no—
119C3×C5×D4—24no—
120C3×C5×D4—24no—
121C2×C3×C4×C5C2×D411yescharacteristic
122C2×C3×C4×C5C2×D412yes—
123C2×C3×C4×C5C2×D412yes—
124C2×C3×C4×C5C2411yescharacteristic
125C3×C22⋊D5—216no—
126C3×C22⋊D5—24no—
127C3×C22⋊D5—24no—
128C3×C22⋊D5—41no—
129C3×C22⋊D5—41no—
130C3×C22⋊D5—84no—
131C2×C3×Dic5—28no—
132C2×C3×Dic5—22no—
133C2×C3×Dic5—22no—
134C2×C3×Dic5—21no—
135C3×D20—22no—
136C3×D20—22no—
137C3×D20C2×D414yes—
138C3×C4×D5—24no—
139C3×C4×D5—24no—
140C3×C4×D5—42no—
141C3×Dic10—22no—
142C3×Dic10—22no—
143C3×Dic10C2×D414yes—
144C3×C5×C8—24no—
145C3×H(5,𝔽2)—54no—
146C3×H(5,𝔽2)—104no—
147C3×H(5,𝔽2)—102no—
148C2×C3×#41—52no—
149C2×C3×#41—51no—
150C2×C3×#41—102no—
151C22×C3×D4—102no—
152C22×C3×D4—102no—
153C22×C3×D4C2×D511yescharacteristic
154C3×#137—54no—
155C3×#137—102no—
156C3×AGL(1,ℤ/8)—54no—
157C3×AGL(1,ℤ/8)—102no—
158C3×#135—104no—
159C2×C3×M16C2×D512yescomplemented
160C3×#127—52no—
161C3×#124—52no—
162C3×#121—102no—
163C3×#120—102no—
164C3×#117—51no—
165C3×#104—58no—
166C3×#100C2×D514yescomplemented
167C24×C5C3×D412yes—
168C23×D5—22no—
169C2×C5×D4—22no—
170C2×C5×D4—22no—
171C2×C5×D4C3×D412yes—
172C2×C5×D4C23×C312yes—
173C22×C4×C5C23×C311yescharacteristic
174C2×C22⋊D5—24no—
175C2×C22⋊D5—28no—
176C2×C22⋊D5—22no—
177C2×C22⋊D5—28no—
178C2×C22⋊D5—21no—
179C2×C22⋊D5—24no—
180C22×Dic5—22no—
181#711—28no—
182#711—28no—
183D4×D5—28no—
184D4×D5—28no—
185#709—22no—
186#709—22no—
187#709—22no—
188#709—22no—
189#709C3×D414yescomplemented
190#709C23×C314yes—
191C2×D20C3×D412yescomplemented
192C2×D20C23×C311yescharacteristic, complemented
193C2×C4×D5—22no—
194C2×C4×D5—21no—
195C2×Dic10C3×D412yescomplemented
196C2×Dic10C23×C311yescharacteristic, complemented
197C5×M16—22no—
198C5×M16C23×C314yescomplemented
199C2×C5×C8—22no—
200C22⋊Dic5—21no—
201C22⋊Dic5—24no—
202C4⋊Dic5—21no—
203C4×Dic5C23×C312yescomplemented
204Dic20—24no—
205D40—24no—
206#677—24no—
207#677—24no—
208C2×H(5,𝔽2)—51no—
209#575—52no—
210#535—51no—
211#455—54no—
212#453—54no—
213#421—52no—
214C2×#100C3×D511yescharacteristic, complemented
215C22×C3×C5—24no—
216C22×C3×C5—41no—
217C22×C3×C5—44no—
218C22×C3×C5—44no—
219C22×C3×C5—82no—
220C22×C3×C5#12011yescharacteristic
221C22×C3×C5#12011yescharacteristic
222C22×C3×C5C22×D411yescharacteristic
223C2×C3×D5—24no—
224C2×C3×D5—44no—
225C2×C3×D5—42no—
226C2×C3×D5—41no—
227C2×C3×D5—88no—
228C3×C4×C5#12012yes—
229C3×C4×C5C22×D412yes—
230C3×Dic5—24no—
231C3×Dic5—42no—
232C3×Dic5—41no—
233C24×C3—102no—
234C24×C3D2012yes—
235C3×#41—54no—
236C3×#41—54no—
237C3×#41—108no—
238C3×#41—108no—
239C3×#41—102no—
240C3×#41—102no—
241C3×#41—102no—
242C3×#41—102no—
243C2×C3×Q8—52no—
244C2×C3×Q8—51no—
245C2×C3×D4—22no—
246C2×C3×D4—22no—
247C2×C3×D4—52no—
248C2×C3×D4—51no—
249C2×C3×D4—108no—
250C2×C3×D4—108no—
251C2×C3×D4—104no—
252C2×C3×D4—108no—
253C2×C3×D4—102no—
254C2×C3×D4—108no—
255C2×C3×D4—101no—
256C2×C3×D4—104no—
257C2×C3×D4D2012yes—
258C2×C3×D4C22×D512yes—
259C22×C3×C4—102no—
260C22×C3×C4—102no—
261C22×C3×C4—101no—
262C22×C3×C4C22×D511yescharacteristic
263C3×Q16—104no—
264C3×SD16—104no—
265C3×SD16—104no—
266C3×D8—104no—
267C3×M16—22no—
268C3×M16C22×D514yescomplemented
269C2×C3×C8—22no—
270C3×C4⋊C4—101no—
271C3×C22⋊C4—101no—
272C3×C22⋊C4—104no—
273C3×C42—52no—
274C23×C5—22no—
275C23×C5—22no—
276C23×C5—82no—
277C23×C5C2×C3×D414yes—
278C23×C5C2×C3×D411yescharacteristic
279C22×D5—22no—
280C22×D5—28no—
281C22×D5—21no—
282C22×D5—84no—
283C5×D4—24no—
284C5×D4—24no—
285C2×C4×C5C2×C3×D411yescharacteristic
286C2×C4×C5C2×C3×D412yes—
287C2×C4×C5C2×C3×D412yes—
288C2×C4×C5C24×C311yescharacteristic
289C22⋊D5—216no—
290C22⋊D5—24no—
291C22⋊D5—24no—
292C22⋊D5—41no—
293C22⋊D5—41no—
294C22⋊D5—84no—
295C2×Dic5—28no—
296C2×Dic5—22no—
297C2×Dic5—22no—
298C2×Dic5—21no—
299D20—22no—
300D20—22no—
301D20C2×C3×D414yes—
302C4×D5—24no—
303C4×D5—24no—
304C4×D5—42no—
305Dic10—22no—
306Dic10—22no—
307Dic10C2×C3×D414yes—
308C5×C8—24no—
309H(5,𝔽2)—54no—
310H(5,𝔽2)—104no—
311H(5,𝔽2)—102no—
312C2×#41—52no—
313C2×#41—51no—
314C2×#41—102no—
315C22×D4—102no—
316C22×D4—102no—
317C22×D4C2×C3×D511yescharacteristic
318#137—54no—
319#137—102no—
320AGL(1,ℤ/8)—54no—
321AGL(1,ℤ/8)—102no—
322#135—104no—
323C2×M16C2×C3×D512yescomplemented
324#127—52no—
325#124—52no—
326#121—102no—
327#120—102no—
328#117—51no—
329#104—58no—
330#100C2×C3×D514yescomplemented
331C2×C3×C5—21no—
332C2×C3×C5—21no—
333C2×C3×C5—21no—
334C2×C3×C5—44no—
335C2×C3×C5C2×#12011yescharacteristic
336C3×D5—44no—
337C3×D5—81no—
338C23×C3—22no—
339C23×C3—22no—
340C23×C3—82no—
341C23×C3—108no—
342C23×C3—102no—
343C23×C3—101no—
344C23×C3—404no—
345C23×C3C2×D2014yes—
346C23×C3C2×D2011yescharacteristic
347C3×Q8—54no—
348C3×Q8—102no—
349C3×Q8—102no—
350C3×D4—24no—
351C3×D4—24no—
352C3×D4—54no—
353C3×D4—1016no—
354C3×D4—104no—
355C3×D4—104no—
356C3×D4—102no—
357C3×D4—102no—
358C3×D4—201no—
359C3×D4—201no—
360C3×D4—404no—
361C2×C3×C4—108no—
362C2×C3×C4—102no—
363C2×C3×C4—104no—
364C2×C3×C4—104no—
365C2×C3×C4—102no—
366C2×C3×C4—101no—
367C2×C3×C4—202no—
368C2×C3×C4C2×D2011yescharacteristic
369C2×C3×C4D4×D512yes—
370C2×C3×C4D4×D512yes—
371C2×C3×C4C23×D511yescharacteristic
372C3×C8—24no—
373C22×C5—24no—
374C22×C5—41no—
375C22×C5—44no—
376C22×C5—44no—
377C22×C5—82no—
378C22×C5C3×#12011yescharacteristic
379C22×C5C3×#12011yescharacteristic
380C22×C5C22×C3×D411yescharacteristic
381C2×D5—24no—
382C2×D5—44no—
383C2×D5—42no—
384C2×D5—41no—
385C2×D5—88no—
386C4×C5C3×#12012yes—
387C4×C5C22×C3×D412yes—
388Dic5—24no—
389Dic5—42no—
390Dic5—41no—
391C24—102no—
392C24C3×D2012yes—
393#41—54no—
394#41—54no—
395#41—108no—
396#41—108no—
397#41—102no—
398#41—102no—
399#41—102no—
400#41—102no—
401C2×Q8—52no—
402C2×Q8—51no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—52no—
406C2×D4—51no—
407C2×D4—108no—
408C2×D4—108no—
409C2×D4—104no—
410C2×D4—108no—
411C2×D4—102no—
412C2×D4—108no—
413C2×D4—101no—
414C2×D4—104no—
415C2×D4C3×D2012yes—
416C2×D4C22×C3×D512yes—
417C22×C4—102no—
418C22×C4—102no—
419C22×C4—101no—
420C22×C4C22×C3×D511yescharacteristic
421Q16—104no—
422SD16—104no—
423SD16—104no—
424D8—104no—
425M16—22no—
426M16C22×C3×D514yescomplemented
427C2×C8—22no—
428C4⋊C4—101no—
429C22⋊C4—101no—
430C22⋊C4—104no—
431C42—52no—
432C3×C5#301911yescharacteristic, complemented, Hall
433C22×C3—24no—
434C22×C3—41no—
435C22×C3—44no—
436C22×C3—44no—
437C22×C3—82no—
438C22×C3—104no—
439C22×C3—204no—
440C22×C3—202no—
441C22×C3—201no—
442C22×C3—408no—
443C22×C3#402811yescharacteristic
444C22×C3#402811yescharacteristic
445C22×C3C22×D2011yescharacteristic
446C3×C4—104no—
447C3×C4—202no—
448C3×C4—201no—
449C3×C4#402812yes—
450C3×C4C2×D4×D512yes—
451C2×C5—21no—
452C2×C5—21no—
453C2×C5—21no—
454C2×C5—44no—
455C2×C5C2×C3×#12011yescharacteristic
456D5—44no—
457D5—81no—
458C23—22no—
459C23—22no—
460C23—82no—
461C23—108no—
462C23—102no—
463C23—101no—
464C23—404no—
465C23C2×C3×D2014yes—
466C23C2×C3×D2011yescharacteristic
467Q8—54no—
468Q8—102no—
469Q8—102no—
470D4—24no—
471D4—24no—
472D4—54no—
473D4—1016no—
474D4—104no—
475D4—104no—
476D4—102no—
477D4—102no—
478D4—201no—
479D4—201no—
480D4—404no—
481C2×C4—108no—
482C2×C4—102no—
483C2×C4—104no—
484C2×C4—104no—
485C2×C4—102no—
486C2×C4—101no—
487C2×C4—202no—
488C2×C4C2×C3×D2011yescharacteristic
489C2×C4C3×D4×D512yes—
490C2×C4C3×D4×D512yes—
491C2×C4C23×C3×D511yescharacteristic
492C8—24no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—44no—
497C2×C3—204no—
498C2×C3—401no—
499C2×C3C2×#402811yescharacteristic
500C5C3×#301911yescharacteristic, complemented, Sylow
501C22—24no—
502C22—41no—
503C22—44no—
504C22—44no—
505C22—82no—
506C22—104no—
507C22—204no—
508C22—202no—
509C22—201no—
510C22—408no—
511C22C3×#402811yescharacteristic
512C22C3×#402811yescharacteristic
513C22C22×C3×D2011yescharacteristic
514C4—104no—
515C4—202no—
516C4—201no—
517C4C3×#402812yes—
518C4C2×C3×D4×D512yes—
519C3#1061043111yescharacteristic, direct product factor, Sylow
520C2—21no—
521C2—21no—
522C2—21no—
523C2—44no—
524C2—204no—
525C2—401no—
526C2C2×C3×#402811yescharacteristic
subgroupG/H
centerC2×C3C2×#4028
commutator, cosocleC2×C4×C5C24×C3
FrattiniC2×C4C23×C3×D5
FittingC2×C3×C5×#100C2
socleC2×C3×C5C2×#120
hypercenterC2×C3×#100D5
hypocenterC5C3×#3019
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#100

Complements

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

NH ≅ G/Ntype
1. C2×C3×#4144C2semidirect
2. C3×#66139C2semidirect
3. C3×#66082C2semidirect
4. C2×C3×C5×#100C2semidirect
5. C3×#65428C2semidirect
6. C3×#65036C2semidirect
7. C3×#65035C2semidirect
8. #10610431C3direct (N × H)
17. C2×C3×#709C22semidirect
18. C2×C3×C5×M16C22semidirect
20. C3×#4086C22semidirect
22. C3×#4082C22semidirect
23. C3×#4072C22semidirect
25. C3×#4055C22semidirect
27. C3×#4054C22semidirect
29. C3×C5×#100C22semidirect
30. C3×#3957C22semidirect
32. C2×#4144C2×C3semidirect
33. #66139C2×C3semidirect
34. #66082C2×C3semidirect
35. C2×C5×#100C2×C3semidirect
36. #65428C2×C3semidirect
37. #65036C2×C3semidirect
38. #65035C2×C3semidirect
61. C3×#709D4semidirect
63. C2×C3×D20D4semidirect
64. C2×C3×D20C23semidirect
67. C2×C3×Dic10D4semidirect
68. C2×C3×Dic10C23semidirect
70. C3×C5×M16C23semidirect
75. C3×C4×Dic5C23semidirect
86. C2×C3×#100D5semidirect
95. C2×#709C22×C3semidirect
96. C2×C5×M16C22×C3semidirect
98. #4086C22×C3semidirect
100. #4082C22×C3semidirect
101. #4072C22×C3semidirect
103. #4055C22×C3semidirect
105. #4054C22×C3semidirect
107. C5×#100C22×C3semidirect
108. #3957C22×C3semidirect
159. C2×C3×M16C2×D5semidirect
166. C3×#100C2×D5semidirect
189. #709C3×D4semidirect
191. C2×D20C3×D4semidirect
192. C2×D20C23×C3semidirect
195. C2×Dic10C3×D4semidirect
196. C2×Dic10C23×C3semidirect
198. C5×M16C23×C3semidirect
203. C4×Dic5C23×C3semidirect
214. C2×#100C3×D5semidirect
268. C3×M16C22×D5semidirect
323. C2×M16C2×C3×D5semidirect
330. #100C2×C3×D5semidirect
426. M16C22×C3×D5semidirect
432. C3×C5#3019semidirect
500. C5C3×#3019semidirect
519. C3#10610431direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#41442
C3×#661392
C3×#660822
C2×C3×C5×#1002
C3×#654282
C3×#650362
C3×#650352
#106104313
C3×#30195