Finite Group Databasebeta

← C2×C3×C5×#385 #14275320

Subgroups — C2×C3×C5×#385 #14275320

Order 1920.

2464 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×C3×C4×C5C211yesmaximal, characteristic
2C22×C3×C5×C4⋊C4C212yesmaximal, complemented
3C22×C3×C5×C22⋊C4C211yesmaximal, characteristic
4C22×C3×C5×C22⋊C4C211yesmaximal, characteristic
5C3×C5×#385C218yesmaximal, direct product factor
6C2×C3×C5×#95C212yesmaximal, complemented
7C2×C5×#385C311yesmaximal, characteristic, direct product factor, Hall
8C25×C3×C5C2211yescharacteristic
9C23×C3×C4×C5—24no—
10C23×C3×C4×C5—28no—
11C23×C3×C4×C5C2214yes—
12C23×C3×C4×C5C2211yescharacteristic
13C23×C3×C4×C5C2211yescharacteristic
14C23×C3×C4×C5C2212yes—
15C23×C3×C4×C5C2212yes—
16C2×C3×C5×C4⋊C4—28no—
17C2×C3×C5×C4⋊C4C2218yescomplemented
18C2×C3×C5×C22⋊C4—24no—
19C2×C3×C5×C22⋊C4C418yescomplemented
20C2×C3×C5×C22⋊C4C2214yes—
21C2×C3×C5×C22⋊C4C2214yes—
22C3×C5×#95C2218yescomplemented
23C2×C3×#385C511yesmaximal, characteristic, direct product factor, Hall
24C24×C4×C5C2×C311yescharacteristic
25C22×C5×C4⋊C4C2×C312yescomplemented
26C22×C5×C22⋊C4C2×C311yescharacteristic
27C22×C5×C22⋊C4C2×C311yescharacteristic
28C5×#385C2×C318yesdirect product factor
29C2×C5×#95C2×C312yescomplemented
30C24×C3×C5—24no—
31C24×C3×C5—24no—
32C24×C3×C5C2×C414yes—
33C24×C3×C5D414yes—
34C24×C3×C5Q814yes—
35C24×C3×C5C2312yes—
36C24×C3×C5C2311yescharacteristic
37C22×C3×C4×C5—232no—
38C22×C3×C4×C5—24no—
39C22×C3×C4×C5—28no—
40C22×C3×C4×C5—24no—
41C22×C3×C4×C5—24no—
42C22×C3×C4×C5—24no—
43C22×C3×C4×C5—216no—
44C22×C3×C4×C5—22no—
45C22×C3×C4×C5—22no—
46C22×C3×C4×C5—24no—
47C22×C3×C4×C5—24no—
48C22×C3×C4×C5C2×C418yescomplemented
49C22×C3×C4×C5D418yes—
50C22×C3×C4×C5C2312yes—
51C22×C3×C4×C5C2312yes—
52C22×C3×C4×C5C2314yes—
53C22×C3×C4×C5C2314yes—
54C3×C5×C4⋊C4—216no—
55C3×C5×C22⋊C4—28no—
56C3×C5×C22⋊C4C2×C4116yescomplemented
57C24×C3×C4C2×C511yescharacteristic
58C22×C3×C4⋊C4C2×C512yescomplemented
59C22×C3×C22⋊C4C2×C511yescharacteristic
60C22×C3×C22⋊C4C2×C511yescharacteristic
61C3×#385C2×C518yesdirect product factor
62C2×C3×#95C2×C512yescomplemented
63C25×C5C22×C311yescharacteristic
64C23×C4×C5—24no—
65C23×C4×C5—28no—
66C23×C4×C5C22×C311yescharacteristic
67C23×C4×C5C22×C314yes—
68C23×C4×C5C22×C311yescharacteristic
69C23×C4×C5C22×C312yes—
70C23×C4×C5C22×C312yes—
71C2×C5×C4⋊C4—28no—
72C2×C5×C4⋊C4C22×C318yescomplemented
73C2×C5×C22⋊C4—24no—
74C2×C5×C22⋊C4C3×C418yescomplemented
75C2×C5×C22⋊C4C22×C314yes—
76C2×C5×C22⋊C4C22×C314yes—
77C5×#95C22×C318yescomplemented
78C2×#385C3×C511yescharacteristic, direct product factor, Sylow
79C23×C3×C5—216no—
80C23×C3×C5—28no—
81C23×C3×C5—216no—
82C23×C3×C5—28no—
83C23×C3×C5—24no—
84C23×C3×C5—24no—
85C23×C3×C5C4⋊C4116yescomplemented
86C23×C3×C5C22×C414yes—
87C23×C3×C5C22×C412yes—
88C23×C3×C5C2×D414yes—
89C23×C3×C5C2×D412yes—
90C23×C3×C5C2×D414yes—
91C23×C3×C5C2×Q814yes—
92C23×C3×C5C2×Q812yes—
93C23×C3×C5#4114yes—
94C23×C3×C5C2411yescharacteristic
95C2×C3×C4×C5—232no—
96C2×C3×C4×C5—216no—
97C2×C3×C4×C5—216no—
98C2×C3×C4×C5—24no—
99C2×C3×C4×C5—24no—
100C2×C3×C4×C5—24no—
101C2×C3×C4×C5—24no—
102C2×C3×C4×C5—22no—
103C2×C3×C4×C5—22no—
104C2×C3×C4×C5—24no—
105C2×C3×C4×C5—24no—
106C2×C3×C4×C5—44no—
107C2×C3×C4×C5—44no—
108C2×C3×C4×C5C22×C418yescomplemented
109C2×C3×C4×C5C2×D418yes—
110C25×C3C22×C511yescharacteristic
111C23×C3×C4—24no—
112C23×C3×C4—28no—
113C23×C3×C4C22×C514yes—
114C23×C3×C4C22×C511yescharacteristic
115C23×C3×C4C22×C511yescharacteristic
116C23×C3×C4C22×C512yes—
117C23×C3×C4C22×C512yes—
118C2×C3×C4⋊C4—28no—
119C2×C3×C4⋊C4C22×C518yescomplemented
120C2×C3×C22⋊C4—24no—
121C2×C3×C22⋊C4C4×C518yescomplemented
122C2×C3×C22⋊C4C22×C514yes—
123C2×C3×C22⋊C4C22×C514yes—
124C3×#95C22×C518yescomplemented
125C24×C5—24no—
126C24×C5—24no—
127C24×C5C2×C3×C414yes—
128C24×C5C3×D414yes—
129C24×C5C3×Q814yes—
130C24×C5C23×C312yes—
131C24×C5C23×C311yescharacteristic
132C22×C4×C5—232no—
133C22×C4×C5—24no—
134C22×C4×C5—28no—
135C22×C4×C5—24no—
136C22×C4×C5—24no—
137C22×C4×C5—24no—
138C22×C4×C5—216no—
139C22×C4×C5—22no—
140C22×C4×C5—22no—
141C22×C4×C5—24no—
142C22×C4×C5—24no—
143C22×C4×C5C2×C3×C418yescomplemented
144C22×C4×C5C3×D418yes—
145C22×C4×C5C23×C312yes—
146C22×C4×C5C23×C312yes—
147C22×C4×C5C23×C314yes—
148C22×C4×C5C23×C314yes—
149C5×C4⋊C4—216no—
150C5×C22⋊C4—28no—
151C5×C22⋊C4C2×C3×C4116yescomplemented
152C24×C4C2×C3×C511yescharacteristic
153C22×C4⋊C4C2×C3×C512yescomplemented
154C22×C22⋊C4C2×C3×C511yescharacteristic
155C22×C22⋊C4C2×C3×C511yescharacteristic
156#385C2×C3×C518yesdirect product factor
157C2×#95C2×C3×C512yescomplemented
158C22×C3×C5—28no—
159C22×C3×C5—24no—
160C22×C3×C5—28no—
161C22×C3×C5—24no—
162C22×C3×C5—232no—
163C22×C3×C5C2×C4⋊C414yes—
164C22×C3×C5C2×C4⋊C418yescomplemented
165C22×C3×C5C4×D418yes—
166C22×C3×C5#12014yes—
167C22×C3×C5C22⋊Q818yes—
168C22×C3×C5#12314yes—
169C22×C3×C5C23×C411yescharacteristic
170C22×C3×C5C22×D411yescharacteristic
171C22×C3×C5C22×D412yes—
172C22×C3×C5C22×Q811yescharacteristic
173C22×C3×C5C2×#4112yes—
174C3×C4×C5—28no—
175C3×C4×C5—28no—
176C3×C4×C5—42no—
177C3×C4×C5—42no—
178C24×C3—24no—
179C24×C3—24no—
180C24×C3C2×C4×C514yes—
181C24×C3C5×D414yes—
182C24×C3C5×Q814yes—
183C24×C3C23×C512yes—
184C24×C3C23×C511yescharacteristic
185C22×C3×C4—232no—
186C22×C3×C4—24no—
187C22×C3×C4—28no—
188C22×C3×C4—24no—
189C22×C3×C4—24no—
190C22×C3×C4—24no—
191C22×C3×C4—216no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4—24no—
195C22×C3×C4—24no—
196C22×C3×C4C2×C4×C518yescomplemented
197C22×C3×C4C5×D418yes—
198C22×C3×C4C23×C512yes—
199C22×C3×C4C23×C512yes—
200C22×C3×C4C23×C514yes—
201C22×C3×C4C23×C514yes—
202C3×C4⋊C4—216no—
203C3×C22⋊C4—28no—
204C3×C22⋊C4C2×C4×C5116yescomplemented
205C23×C5—216no—
206C23×C5—28no—
207C23×C5—216no—
208C23×C5—28no—
209C23×C5—24no—
210C23×C5—24no—
211C23×C5C3×C4⋊C4116yescomplemented
212C23×C5C22×C3×C414yes—
213C23×C5C22×C3×C412yes—
214C23×C5C2×C3×D414yes—
215C23×C5C2×C3×D412yes—
216C23×C5C2×C3×D414yes—
217C23×C5C2×C3×Q814yes—
218C23×C5C2×C3×Q812yes—
219C23×C5C3×#4114yes—
220C23×C5C24×C311yescharacteristic
221C2×C4×C5—232no—
222C2×C4×C5—216no—
223C2×C4×C5—216no—
224C2×C4×C5—24no—
225C2×C4×C5—24no—
226C2×C4×C5—24no—
227C2×C4×C5—24no—
228C2×C4×C5—22no—
229C2×C4×C5—22no—
230C2×C4×C5—24no—
231C2×C4×C5—24no—
232C2×C4×C5—44no—
233C2×C4×C5—44no—
234C2×C4×C5C22×C3×C418yescomplemented
235C2×C4×C5C2×C3×D418yes—
236C25C22×C3×C511yescharacteristic
237C23×C4—24no—
238C23×C4—28no—
239C23×C4C22×C3×C511yescharacteristic
240C23×C4C22×C3×C511yescharacteristic
241C23×C4C22×C3×C514yes—
242C23×C4C22×C3×C512yes—
243C23×C4C22×C3×C512yes—
244C2×C4⋊C4—28no—
245C2×C4⋊C4C22×C3×C518yescomplemented
246C2×C22⋊C4—24no—
247C2×C22⋊C4C3×C4×C518yescomplemented
248C2×C22⋊C4C22×C3×C514yes—
249C2×C22⋊C4C22×C3×C514yes—
250#95C22×C3×C518yescomplemented
251C2×C3×C5—28no—
252C2×C3×C5#38518yesdirect product factor
253C2×C3×C5C22×C4⋊C411yescharacteristic
254C2×C3×C5C2×C4×D412yes—
255C2×C3×C5C2×#12011yescharacteristic
256C2×C3×C5C2×C22⋊Q812yes—
257C2×C3×C5C2×#12311yescharacteristic
258C23×C3—216no—
259C23×C3—28no—
260C23×C3—216no—
261C23×C3—28no—
262C23×C3—24no—
263C23×C3—24no—
264C23×C3C5×C4⋊C4116yescomplemented
265C23×C3C22×C4×C514yes—
266C23×C3C22×C4×C512yes—
267C23×C3C2×C5×D414yes—
268C23×C3C2×C5×D412yes—
269C23×C3C2×C5×D414yes—
270C23×C3C2×C5×Q814yes—
271C23×C3C2×C5×Q812yes—
272C23×C3C5×#4114yes—
273C23×C3C24×C511yescharacteristic
274C2×C3×C4—232no—
275C2×C3×C4—216no—
276C2×C3×C4—216no—
277C2×C3×C4—24no—
278C2×C3×C4—24no—
279C2×C3×C4—24no—
280C2×C3×C4—24no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—24no—
284C2×C3×C4—24no—
285C2×C3×C4—44no—
286C2×C3×C4—44no—
287C2×C3×C4C22×C4×C518yescomplemented
288C2×C3×C4C2×C5×D418yes—
289C22×C5—232no—
290C22×C5—28no—
291C22×C5—24no—
292C22×C5—28no—
293C22×C5—24no—
294C22×C5C2×C3×C4⋊C414yes—
295C22×C5C2×C3×C4⋊C418yescomplemented
296C22×C5C3×C4×D418yes—
297C22×C5C3×#12014yes—
298C22×C5C3×C22⋊Q818yes—
299C22×C5C3×#12314yes—
300C22×C5C23×C3×C411yescharacteristic
301C22×C5C22×C3×D411yescharacteristic
302C22×C5C22×C3×D412yes—
303C22×C5C22×C3×Q811yescharacteristic
304C22×C5C2×C3×#4112yes—
305C4×C5—28no—
306C4×C5—28no—
307C4×C5—42no—
308C4×C5—42no—
309C24—24no—
310C24—24no—
311C24C2×C3×C4×C514yes—
312C24C3×C5×D414yes—
313C24C3×C5×Q814yes—
314C24C23×C3×C512yes—
315C24C23×C3×C511yescharacteristic
316C22×C4—232no—
317C22×C4—24no—
318C22×C4—28no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—216no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4C2×C3×C4×C518yescomplemented
328C22×C4C3×C5×D418yes—
329C22×C4C23×C3×C512yes—
330C22×C4C23×C3×C512yes—
331C22×C4C23×C3×C514yes—
332C22×C4C23×C3×C514yes—
333C4⋊C4—216no—
334C22⋊C4—28no—
335C22⋊C4C2×C3×C4×C5116yescomplemented
336C3×C5C2×#38511yescharacteristic, direct product factor, Hall
337C22×C3—232no—
338C22×C3—28no—
339C22×C3—24no—
340C22×C3—28no—
341C22×C3—24no—
342C22×C3C2×C5×C4⋊C414yes—
343C22×C3C2×C5×C4⋊C418yescomplemented
344C22×C3C4×C5×D418yes—
345C22×C3C5×#12014yes—
346C22×C3C5×C22⋊Q818yes—
347C22×C3C5×#12314yes—
348C22×C3C23×C4×C511yescharacteristic
349C22×C3C22×C5×D411yescharacteristic
350C22×C3C22×C5×D412yes—
351C22×C3C22×C5×Q811yescharacteristic
352C22×C3C2×C5×#4112yes—
353C3×C4—28no—
354C3×C4—28no—
355C3×C4—42no—
356C3×C4—42no—
357C2×C5—28no—
358C2×C5C3×#38518yesdirect product factor
359C2×C5C22×C3×C4⋊C411yescharacteristic
360C2×C5C2×C3×C4×D412yes—
361C2×C5C2×C3×#12011yescharacteristic
362C2×C5C2×C3×C22⋊Q812yes—
363C2×C5C2×C3×#12311yescharacteristic
364C23—216no—
365C23—28no—
366C23—24no—
367C23—216no—
368C23—24no—
369C23—28no—
370C23C3×C5×C4⋊C4116yescomplemented
371C23C22×C3×C4×C514yes—
372C23C22×C3×C4×C512yes—
373C23C2×C3×C5×D414yes—
374C23C2×C3×C5×D412yes—
375C23C2×C3×C5×D414yes—
376C23C2×C3×C5×Q814yes—
377C23C2×C3×C5×Q812yes—
378C23C3×C5×#4114yes—
379C23C24×C3×C511yescharacteristic
380C2×C4—232no—
381C2×C4—216no—
382C2×C4—216no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4C22×C3×C4×C518yescomplemented
394C2×C4C2×C3×C5×D418yes—
395C2×C3—28no—
396C2×C3C5×#38518yesdirect product factor
397C2×C3C22×C5×C4⋊C411yescharacteristic
398C2×C3C2×C4×C5×D412yes—
399C2×C3C2×C5×#12011yescharacteristic
400C2×C3C2×C5×C22⋊Q812yes—
401C2×C3C2×C5×#12311yescharacteristic
402C5C2×C3×#38511yescharacteristic, direct product factor, Sylow
403C22—28no—
404C22—24no—
405C22—28no—
406C22—24no—
407C22—232no—
408C22C2×C3×C5×C4⋊C414yes—
409C22C2×C3×C5×C4⋊C418yescomplemented
410C22C3×C4×C5×D418yes—
411C22C3×C5×#12014yes—
412C22C3×C5×C22⋊Q818yes—
413C22C3×C5×#12314yes—
414C22C23×C3×C4×C511yescharacteristic
415C22C22×C3×C5×D411yescharacteristic
416C22C22×C3×C5×D412yes—
417C22C22×C3×C5×Q811yescharacteristic
418C22C2×C3×C5×#4112yes—
419C4—28no—
420C4—28no—
421C4—42no—
422C4—42no—
423C3C2×C5×#38511yescharacteristic, direct product factor, Sylow
424C2—28no—
425C2C3×C5×#38518yesdirect product factor
426C2C22×C3×C5×C4⋊C411yescharacteristic
427C2C2×C3×C4×C5×D412yes—
428C2C2×C3×C5×#12011yescharacteristic
429C2C2×C3×C5×C22⋊Q812yes—
430C2C2×C3×C5×#12311yescharacteristic
subgroupG/H
center, socleC24×C3×C5C23
commutatorC22C23×C3×C4×C5
Frattini, cosocleC23C24×C3×C5
prime radicalC25×C3×C5C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24×C3×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C3×C5×C4⋊C4C2semidirect
5. C3×C5×#385C2direct (N × H)
6. C2×C3×C5×#95C2semidirect
7. C2×C5×#385C3direct (N × H)
17. C2×C3×C5×C4⋊C4C22semidirect
19. C2×C3×C5×C22⋊C4C4semidirect
22. C3×C5×#95C22semidirect
23. C2×C3×#385C5direct (N × H)
25. C22×C5×C4⋊C4C2×C3semidirect
28. C5×#385C2×C3direct (N × H)
29. C2×C5×#95C2×C3semidirect
48. C22×C3×C4×C5C2×C4semidirect
56. C3×C5×C22⋊C4C2×C4semidirect
58. C22×C3×C4⋊C4C2×C5semidirect
61. C3×#385C2×C5direct (N × H)
62. C2×C3×#95C2×C5semidirect
72. C2×C5×C4⋊C4C22×C3semidirect
74. C2×C5×C22⋊C4C3×C4semidirect
77. C5×#95C22×C3semidirect
78. C2×#385C3×C5direct (N × H)
85. C23×C3×C5C4⋊C4semidirect
108. C2×C3×C4×C5C22×C4semidirect
119. C2×C3×C4⋊C4C22×C5semidirect
121. C2×C3×C22⋊C4C4×C5semidirect
124. C3×#95C22×C5semidirect
143. C22×C4×C5C2×C3×C4semidirect
151. C5×C22⋊C4C2×C3×C4semidirect
153. C22×C4⋊C4C2×C3×C5semidirect
156. #385C2×C3×C5direct (N × H)
157. C2×#95C2×C3×C5semidirect
164. C22×C3×C5C2×C4⋊C4semidirect
196. C22×C3×C4C2×C4×C5semidirect
204. C3×C22⋊C4C2×C4×C5semidirect
211. C23×C5C3×C4⋊C4semidirect
234. C2×C4×C5C22×C3×C4semidirect
245. C2×C4⋊C4C22×C3×C5semidirect
247. C2×C22⋊C4C3×C4×C5semidirect
250. #95C22×C3×C5semidirect
252. C2×C3×C5#385direct (N × H)
264. C23×C3C5×C4⋊C4semidirect
287. C2×C3×C4C22×C4×C5semidirect
295. C22×C5C2×C3×C4⋊C4semidirect
327. C22×C4C2×C3×C4×C5semidirect
335. C22⋊C4C2×C3×C4×C5semidirect
336. C3×C5C2×#385direct (N × H)
343. C22×C3C2×C5×C4⋊C4semidirect
358. C2×C5C3×#385direct (N × H)
370. C23C3×C5×C4⋊C4semidirect
393. C2×C4C22×C3×C4×C5semidirect
396. C2×C3C5×#385direct (N × H)
402. C5C2×C3×#385direct (N × H)
409. C22C2×C3×C5×C4⋊C4semidirect
423. C3C2×C5×#385direct (N × H)
425. C2C3×C5×#385direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C3×C4×C52
C22×C3×C5×C4⋊C42
C22×C3×C5×C22⋊C42
C3×C5×#3852
C2×C3×C5×#952
C2×C5×#3853
C2×C3×#3855