Finite Group Databasebeta

← C3×#10602447 #14301619

Subgroups — C3×#10602447 #14301619

Order 1920.

1156 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#518C211yesmaximal, characteristic, complemented
2C2×C3×#4083C211yesmaximal, characteristic
3C2×C3×#3966C211yesmaximal, characteristic, complemented
4C3×#65244C212yesmaximal, complemented
5C3×#64746C212yesmaximal, complemented
6#10602447C311yesmaximal, characteristic, direct product factor, Hall
7C24×C3×D5C2211yescharacteristic
8C22×C3×C5×D4C2211yescharacteristic
9C22×C3×C22⋊D5—21no—
10C3×C4×C5×D4—24no—
11C3×C5×#117—21no—
12C2×C3×C5×C22⋊C4—21no—
13C2×C3×C5×C22⋊C4C2212yes—
14C2×C3×C22⋊Dic5C2212yes—
15C2×C3×C22⋊Dic5C2211yescharacteristic
16C3×#4083—22no—
17C3×#4083C414yescomplemented
18C2×C3×#685—21no—
19C2×C3×#685—24no—
20C3×#3966—22no—
21C3×#1889—51nomaximal, Hall
22C5×#518C2×C311yescharacteristic, complemented
23C2×#4083C2×C311yescharacteristic
24C2×#3966C2×C311yescharacteristic, complemented
25#65244C2×C312yescomplemented
26#64746C2×C312yescomplemented
27C24×C3×C5D411yescharacteristic
28C24×C3×C5C2311yescharacteristic
29C23×C3×D5—21no—
30C23×C3×D5—22no—
31C23×C3×D5—21no—
32C23×C3×D5—22no—
33C23×C3×D5—44no—
34C23×C3×D5C2×C412yescomplemented
35C2×C3×C5×D4—22no—
36C2×C3×C5×D4—22no—
37C2×C3×C5×D4C2×C412yes—
38C2×C3×C5×D4D412yes—
39C22×C3×C4×C5—22no—
40C22×C3×C4×C5D411yescharacteristic
41C22×C3×C4×C5D414yescomplemented
42C2×C3×C22⋊D5—24no—
43C2×C3×C22⋊D5—44no—
44C22×C3×Dic5—21no—
45C22×C3×Dic5—22no—
46C3×C5×C4⋊C4—22no—
47C3×C5×C22⋊C4—22no—
48C3×C5×C22⋊C4—22no—
49C3×C5×C22⋊C4—22no—
50C3×C42×C5—22no—
51C3×C22⋊Dic5—21no—
52C3×C22⋊Dic5—22no—
53C3×C22⋊Dic5—22no—
54C3×C22⋊Dic5C2×C412yescomplemented
55C3×#685—24no—
56C3×#685—48no—
57C2×C3×#120—51no—
58C3×#518D511yescharacteristic, complemented
59C2×C3×#99—51no—
60C3×#379—52no—
61C3×#342—52no—
62C24×D5C22×C311yescharacteristic
63C22×C5×D4C22×C311yescharacteristic
64C22×C22⋊D5—21no—
65C4×C5×D4—24no—
66C5×#117—21no—
67C2×C5×C22⋊C4—21no—
68C2×C5×C22⋊C4C22×C312yes—
69C2×C22⋊Dic5C22×C312yes—
70C2×C22⋊Dic5C22×C311yescharacteristic
71#4083—22no—
72#4083C3×C414yescomplemented
73C2×#685—21no—
74C2×#685—24no—
75#3966—22no—
76#1889—51noSylow
77C23×C3×C5—21no—
78C23×C3×C5—21no—
79C23×C3×C5—42no—
80C23×C3×C5—42no—
81C23×C3×C5C22⋊C412yescomplemented
82C23×C3×C5C22×C411yescharacteristic
83C23×C3×C5C2×D411yescharacteristic
84C23×C3×C5C2×D412yes—
85C23×C3×C5C2×D411yescharacteristic
86C23×C3×C5#4112yes—
87C22×C3×D5—28no—
88C22×C3×D5—24no—
89C22×C3×D5—22no—
90C22×C3×D5—24no—
91C22×C3×D5—44no—
92C22×C3×D5—44no—
93C22×C3×D5—48no—
94C22×C3×D5—44no—
95C22×C3×D5—42no—
96C22×C3×D5—41no—
97C22×C3×D5—42no—
98C22×C3×D5—41no—
99C3×C5×D4—24no—
100C3×C5×D4—24no—
101C2×C3×C4×C5—21no—
102C2×C3×C4×C5—21no—
103C2×C3×C4×C5—24no—
104C2×C3×C4×C5—44no—
105C2×C3×C4×C5—82no—
106C2×C3×C4×C5C22⋊C412yescomplemented
107C3×C22⋊D5—48no—
108C2×C3×Dic5—22no—
109C2×C3×Dic5—42no—
110C2×C3×Dic5—44no—
111C2×C3×Dic5—42no—
112C25×C3—51no—
113C22×C3×D4—101no—
114C22×C3×D4C2×D511yescharacteristic
115C3×#120—54no—
116C3×#120—102no—
117C3×C4×D4—24no—
118C3×#117—21no—
119C2×C3×C22⋊C4—21no—
120C2×C3×C22⋊C4—51no—
121C2×C3×C22⋊C4—52no—
122C2×C3×C22⋊C4—101no—
123C2×C3×C22⋊C4—104no—
124C2×C3×C22⋊C4C2×D512yes—
125C3×#99—102no—
126C24×C5C3×D411yescharacteristic
127C24×C5C23×C311yescharacteristic
128C23×D5—21no—
129C23×D5—22no—
130C23×D5—21no—
131C23×D5—22no—
132C23×D5—44no—
133C23×D5C2×C3×C412yescomplemented
134C2×C5×D4—22no—
135C2×C5×D4—22no—
136C2×C5×D4C2×C3×C412yes—
137C2×C5×D4C3×D412yes—
138C22×C4×C5—22no—
139C22×C4×C5C3×D411yescharacteristic
140C22×C4×C5C3×D414yescomplemented
141C2×C22⋊D5—24no—
142C2×C22⋊D5—44no—
143C22×Dic5—21no—
144C22×Dic5—22no—
145C5×C4⋊C4—22no—
146C5×C22⋊C4—22no—
147C5×C22⋊C4—22no—
148C5×C22⋊C4—22no—
149C42×C5—22no—
150C22⋊Dic5—21no—
151C22⋊Dic5—22no—
152C22⋊Dic5—22no—
153C22⋊Dic5C2×C3×C412yescomplemented
154#685—24no—
155#685—48no—
156C2×#120—51no—
157#518C3×D511yescharacteristic, complemented
158C2×#99—51no—
159#379—52no—
160#342—52no—
161C22×C3×C5—24no—
162C22×C3×C5—22no—
163C22×C3×C5—22no—
164C22×C3×C5—41no—
165C22×C3×C5—41no—
166C22×C3×C5—42no—
167C22×C3×C5—42no—
168C22×C3×C5—42no—
169C22×C3×C5—81no—
170C22×C3×C5C2×C22⋊C411yescharacteristic
171C22×C3×C5C4×D412yes—
172C22×C3×C5#12011yescharacteristic
173C22×C3×C5#12112yes—
174C22×C3×C5#12311yescharacteristic
175C2×C3×D5—24no—
176C2×C3×D5—28no—
177C2×C3×D5—42no—
178C2×C3×D5—48no—
179C2×C3×D5—44no—
180C2×C3×D5—44no—
181C2×C3×D5—42no—
182C2×C3×D5—44no—
183C3×C4×C5—22no—
184C3×C4×C5—44no—
185C3×Dic5—42no—
186C3×Dic5—82no—
187C24×C3—52no—
188C24×C3—102no—
189C24×C3—101no—
190C24×C3—101no—
191C24×C3—102no—
192C24×C3—204no—
193C24×C3C22⋊D511yescharacteristic
194C24×C3C22×D511yescharacteristic
195C2×C3×D4—22no—
196C2×C3×D4—22no—
197C2×C3×D4—104no—
198C2×C3×D4—204no—
199C2×C3×D4C4×D512yes—
200C2×C3×D4D2012yes—
201C22×C3×C4—22no—
202C22×C3×C4—101no—
203C22×C3×C4—102no—
204C22×C3×C4C22⋊D511yescharacteristic
205C22×C3×C4C22⋊D514yescomplemented
206C3×C4⋊C4—22no—
207C3×C22⋊C4—22no—
208C3×C22⋊C4—22no—
209C3×C22⋊C4—22no—
210C3×C22⋊C4—52no—
211C3×C22⋊C4—101no—
212C3×C22⋊C4—104no—
213C3×C22⋊C4—102no—
214C3×C22⋊C4—102no—
215C3×C22⋊C4—208no—
216C3×C42—22no—
217C23×C5—21no—
218C23×C5—21no—
219C23×C5—42no—
220C23×C5—42no—
221C23×C5C3×C22⋊C412yescomplemented
222C23×C5C22×C3×C411yescharacteristic
223C23×C5C2×C3×D411yescharacteristic
224C23×C5C2×C3×D411yescharacteristic
225C23×C5C2×C3×D412yes—
226C23×C5C3×#4112yes—
227C22×D5—28no—
228C22×D5—24no—
229C22×D5—22no—
230C22×D5—24no—
231C22×D5—44no—
232C22×D5—44no—
233C22×D5—48no—
234C22×D5—44no—
235C22×D5—42no—
236C22×D5—41no—
237C22×D5—42no—
238C22×D5—41no—
239C5×D4—24no—
240C5×D4—24no—
241C2×C4×C5—21no—
242C2×C4×C5—21no—
243C2×C4×C5—24no—
244C2×C4×C5—44no—
245C2×C4×C5—82no—
246C2×C4×C5C3×C22⋊C412yescomplemented
247C22⋊D5—48no—
248C2×Dic5—22no—
249C2×Dic5—42no—
250C2×Dic5—44no—
251C2×Dic5—42no—
252C25—51no—
253C22×D4—101no—
254C22×D4C2×C3×D511yescharacteristic
255#120—54no—
256#120—102no—
257C4×D4—24no—
258#117—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—51no—
261C2×C22⋊C4—52no—
262C2×C22⋊C4—101no—
263C2×C22⋊C4—104no—
264C2×C22⋊C4C2×C3×D512yes—
265#99—102no—
266C2×C3×C5—21no—
267C2×C3×C5—22no—
268C2×C3×C5—22no—
269C2×C3×C5—21no—
270C2×C3×C5—42no—
271C2×C3×C5#38611yescharacteristic
272C2×C3×C5B(4,𝔽2)12yes—
273C3×D5—44no—
274C23×C3—21no—
275C23×C3—21no—
276C23×C3—42no—
277C23×C3—42no—
278C23×C3—108no—
279C23×C3—104no—
280C23×C3—102no—
281C23×C3—104no—
282C23×C3—204no—
283C23×C3—202no—
284C23×C3—204no—
285C23×C3—204no—
286C23×C3—208no—
287C23×C3—201no—
288C23×C3—201no—
289C23×C3—202no—
290C23×C3#68512yescomplemented
291C23×C3C2×C4×D511yescharacteristic
292C23×C3C2×D2011yescharacteristic
293C23×C3#70912yes—
294C23×C3C2×C22⋊D511yescharacteristic
295C23×C3C2×C22⋊D512yes—
296C3×D4—24no—
297C3×D4—24no—
298C3×D4—208no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—24no—
302C2×C3×C4—44no—
303C2×C3×C4—82no—
304C2×C3×C4—102no—
305C2×C3×C4—202no—
306C2×C3×C4—204no—
307C2×C3×C4—202no—
308C2×C3×C4#68512yescomplemented
309C22×C5—24no—
310C22×C5—22no—
311C22×C5—22no—
312C22×C5—41no—
313C22×C5—41no—
314C22×C5—42no—
315C22×C5—42no—
316C22×C5—42no—
317C22×C5—81no—
318C22×C5C2×C3×C22⋊C411yescharacteristic
319C22×C5C3×C4×D412yes—
320C22×C5C3×#12011yescharacteristic
321C22×C5C3×#12112yes—
322C22×C5C3×#12311yescharacteristic
323C2×D5—24no—
324C2×D5—28no—
325C2×D5—42no—
326C2×D5—48no—
327C2×D5—44no—
328C2×D5—44no—
329C2×D5—42no—
330C2×D5—44no—
331C4×C5—22no—
332C4×C5—44no—
333Dic5—42no—
334Dic5—82no—
335C24—52no—
336C24—102no—
337C24—101no—
338C24—101no—
339C24—102no—
340C24—204no—
341C24C3×C22⋊D511yescharacteristic
342C24C22×C3×D511yescharacteristic
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—104no—
346C2×D4—204no—
347C2×D4C3×C4×D512yes—
348C2×D4C3×D2012yes—
349C22×C4—22no—
350C22×C4—101no—
351C22×C4—102no—
352C22×C4C3×C22⋊D511yescharacteristic
353C22×C4C3×C22⋊D514yescomplemented
354C4⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—22no—
358C22⋊C4—52no—
359C22⋊C4—101no—
360C22⋊C4—104no—
361C22⋊C4—102no—
362C22⋊C4—102no—
363C22⋊C4—208no—
364C42—22no—
365C3×C5#188911yescharacteristic, complemented, Hall
366C22×C3—24no—
367C22×C3—22no—
368C22×C3—22no—
369C22×C3—41no—
370C22×C3—41no—
371C22×C3—42no—
372C22×C3—42no—
373C22×C3—42no—
374C22×C3—81no—
375C22×C3—104no—
376C22×C3—108no—
377C22×C3—202no—
378C22×C3—208no—
379C22×C3—204no—
380C22×C3—204no—
381C22×C3—202no—
382C22×C3—204no—
383C22×C3C2×#68511yescharacteristic
384C22×C3C4×C22⋊D512yes—
385C22×C3#407511yescharacteristic
386C22×C3#407612yes—
387C22×C3#409911yescharacteristic
388C3×C4—22no—
389C3×C4—44no—
390C3×C4—202no—
391C3×C4—402no—
392C2×C5—21no—
393C2×C5—22no—
394C2×C5—22no—
395C2×C5—21no—
396C2×C5—42no—
397C2×C5C3×#38611yescharacteristic
398C2×C5C3×B(4,𝔽2)12yes—
399D5—44no—
400C23—21no—
401C23—21no—
402C23—42no—
403C23—42no—
404C23—108no—
405C23—104no—
406C23—102no—
407C23—104no—
408C23—204no—
409C23—202no—
410C23—204no—
411C23—204no—
412C23—208no—
413C23—201no—
414C23—201no—
415C23—202no—
416C23C3×#68512yescomplemented
417C23C2×C3×C4×D511yescharacteristic
418C23C2×C3×D2011yescharacteristic
419C23C3×#70912yes—
420C23C2×C3×C22⋊D511yescharacteristic
421C23C2×C3×C22⋊D512yes—
422D4—24no—
423D4—24no—
424D4—208no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—24no—
428C2×C4—44no—
429C2×C4—82no—
430C2×C4—102no—
431C2×C4—202no—
432C2×C4—204no—
433C2×C4—202no—
434C2×C4C3×#68512yescomplemented
435C2×C3—21no—
436C2×C3—22no—
437C2×C3—21no—
438C2×C3—22no—
439C2×C3—42no—
440C2×C3—204no—
441C2×C3#6550211yescharacteristic
442C2×C3#6553312yes—
443C5C3×#188911yescharacteristic, complemented, Sylow
444C22—24no—
445C22—22no—
446C22—22no—
447C22—41no—
448C22—41no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—81no—
453C22—104no—
454C22—108no—
455C22—202no—
456C22—208no—
457C22—204no—
458C22—204no—
459C22—202no—
460C22—204no—
461C22C2×C3×#68511yescharacteristic
462C22C3×C4×C22⋊D512yes—
463C22C3×#407511yescharacteristic
464C22C3×#407612yes—
465C22C3×#409911yescharacteristic
466C4—22no—
467C4—44no—
468C4—202no—
469C4—402no—
470C3#1060244711yescharacteristic, direct product factor, Sylow
471C2—21no—
472C2—22no—
473C2—21no—
474C2—22no—
475C2—42no—
476C2—204no—
477C2C3×#6550211yescharacteristic
478C2C3×#6553312yes—
subgroupG/H
centerC22×C3#4099
commutatorC23×C5C22×C3×C4
FrattiniC24C22×C3×D5
FittingC3×C5×#518C2
socleC22×C3×C5#120
prime radicalC2×C3×#4083C2
cosocleC24×C5C23×C3
hypercenterC3×#518D5
hypocenterC5C3×#1889
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C24×C3 ◁ C3×#518

Complements

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

NH ≅ G/Ntype
1. C3×C5×#518C2semidirect
3. C2×C3×#3966C2semidirect
4. C3×#65244C2semidirect
5. C3×#64746C2semidirect
6. #10602447C3direct (N × H)
17. C3×#4083C4semidirect
22. C5×#518C2×C3semidirect
24. C2×#3966C2×C3semidirect
25. #65244C2×C3semidirect
26. #64746C2×C3semidirect
34. C23×C3×D5C2×C4semidirect
41. C22×C3×C4×C5D4semidirect
54. C3×C22⋊Dic5C2×C4semidirect
58. C3×#518D5semidirect
72. #4083C3×C4semidirect
81. C23×C3×C5C22⋊C4semidirect
106. C2×C3×C4×C5C22⋊C4semidirect
133. C23×D5C2×C3×C4semidirect
140. C22×C4×C5C3×D4semidirect
153. C22⋊Dic5C2×C3×C4semidirect
157. #518C3×D5semidirect
205. C22×C3×C4C22⋊D5semidirect
221. C23×C5C3×C22⋊C4semidirect
246. C2×C4×C5C3×C22⋊C4semidirect
290. C23×C3#685semidirect
308. C2×C3×C4#685semidirect
353. C22×C4C3×C22⋊D5semidirect
365. C3×C5#1889semidirect
416. C23C3×#685semidirect
434. C2×C4C3×#685semidirect
443. C5C3×#1889semidirect
470. C3#10602447direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5182
C2×C3×#40832
C2×C3×#39662
C3×#652442
C3×#647462
#106024473
C3×#18895