Finite Group Databasebeta

← C2×C3×C5×#478 #14276106

Subgroups — C2×C3×C5×#478 #14276106

Order 1920.

896 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#125C211yesmaximal, characteristic, complemented
2C2×C3×C4×C5×D4C211yesmaximal, characteristic
3C3×C5×#478C218yesmaximal, direct product factor
4C2×C3×C5×#107C211yesmaximal, characteristic, complemented
5C2×C3×C5×#106C211yesmaximal, characteristic, complemented
6C2×C3×C5×C4⋊C8C211yesmaximal, characteristic, complemented
7C2×C3×C5×#102C211yesmaximal, characteristic
8C2×C3×C5×#102C211yesmaximal, characteristic
9C2×C5×#478C311yesmaximal, characteristic, direct product factor, Hall
10C22×C3×C5×D4C2211yescharacteristic
11C23×C3×C4×C5—21no—
12C22×C3×C5×C8C2211yescharacteristic
13C22×C3×C5×C8C2211yescharacteristic
14C3×C5×#125—22no—
15C3×C5×#125C2214yescomplemented
16C3×C4×C5×D4—22no—
17C3×C4×C5×D4C2214yes—
18C2×C3×C5×C4⋊C4—21no—
19C2×C3×C5×C4⋊C4—21no—
20C2×C3×C5×C4⋊C4C2211yescharacteristic
21C2×C3×C5×C4⋊C4C2211yescharacteristic
22C2×C3×C5×C4⋊C4C2211yescharacteristic
23C2×C3×C5×C22⋊C4—21no—
24C2×C3×C42×C5C2211yescharacteristic
25C3×C5×#107C2214yescomplemented
26C3×C5×#106C2214yescomplemented
27C3×C5×C4⋊C8C2214yescomplemented
28C3×C5×#102C2214yes—
29C3×C5×#102C2214yes—
30C2×C3×#478C511yesmaximal, characteristic, direct product factor, Hall
31C2×C5×#125C2×C311yescharacteristic, complemented
32C2×C4×C5×D4C2×C311yescharacteristic
33C5×#478C2×C318yesdirect product factor
34C2×C5×#107C2×C311yescharacteristic, complemented
35C2×C5×#106C2×C311yescharacteristic, complemented
36C2×C5×C4⋊C8C2×C311yescharacteristic, complemented
37C2×C5×#102C2×C311yescharacteristic
38C2×C5×#102C2×C311yescharacteristic
39C24×C3×C5—21no—
40C2×C3×C5×D4—21no—
41C2×C3×C5×D4—22no—
42C2×C3×C5×D4Q814yes—
43C2×C3×C5×D4C2312yes—
44C22×C3×C4×C5—21no—
45C22×C3×C4×C5—21no—
46C22×C3×C4×C5—21no—
47C22×C3×C4×C5—24no—
48C22×C3×C4×C5—44no—
49C22×C3×C4×C5D411yescharacteristic
50C22×C3×C4×C5D411yescharacteristic
51C22×C3×C4×C5C2311yescharacteristic
52C2×C3×C5×C8—22no—
53C2×C3×C5×C8—22no—
54C2×C3×C5×C8C2312yes—
55C2×C3×C5×C8C2312yes—
56C3×C5×C4⋊C4—24no—
57C3×C5×C4⋊C4—24no—
58C3×C5×C4⋊C4—21no—
59C3×C5×C4⋊C4—21no—
60C3×C5×C4⋊C4—21no—
61C3×C5×C4⋊C4C2312yes—
62C3×C5×C4⋊C4C2312yes—
63C3×C5×C4⋊C4C2312yes—
64C3×C5×C22⋊C4—24no—
65C3×C42×C5D412yes—
66C3×C42×C5C2312yes—
67C2×C3×#125C2×C511yescharacteristic, complemented
68C2×C3×C4×D4C2×C511yescharacteristic
69C3×#478C2×C518yesdirect product factor
70C2×C3×#107C2×C511yescharacteristic, complemented
71C2×C3×#106C2×C511yescharacteristic, complemented
72C2×C3×C4⋊C8C2×C511yescharacteristic, complemented
73C2×C3×#102C2×C511yescharacteristic
74C2×C3×#102C2×C511yescharacteristic
75C22×C5×D4C22×C311yescharacteristic
76C23×C4×C5—21no—
77C22×C5×C8C22×C311yescharacteristic
78C22×C5×C8C22×C311yescharacteristic
79C5×#125—22no—
80C5×#125C22×C314yescomplemented
81C4×C5×D4—22no—
82C4×C5×D4C22×C314yes—
83C2×C5×C4⋊C4—21no—
84C2×C5×C4⋊C4—21no—
85C2×C5×C4⋊C4C22×C311yescharacteristic
86C2×C5×C4⋊C4C22×C311yescharacteristic
87C2×C5×C4⋊C4C22×C311yescharacteristic
88C2×C5×C22⋊C4—21no—
89C2×C42×C5C22×C311yescharacteristic
90C5×#107C22×C314yescomplemented
91C5×#106C22×C314yescomplemented
92C5×C4⋊C8C22×C314yescomplemented
93C5×#102C22×C314yes—
94C5×#102C22×C314yes—
95C2×#478C3×C511yescharacteristic, direct product factor, Sylow
96C23×C3×C5—24no—
97C23×C3×C5—22no—
98C23×C3×C5—42no—
99C23×C3×C5—42no—
100C23×C3×C5C2×D411yescharacteristic
101C3×C5×D4—24no—
102C3×C5×D4—22no—
103C3×C5×D4C2×Q814yes—
104C2×C3×C4×C5—22no—
105C2×C3×C4×C5—22no—
106C2×C3×C4×C5—22no—
107C2×C3×C4×C5—22no—
108C2×C3×C4×C5—22no—
109C2×C3×C4×C5—48no—
110C2×C3×C4×C5—42no—
111C2×C3×C4×C5—42no—
112C2×C3×C4×C5—42no—
113C2×C3×C4×C5C2×D412yes—
114C2×C3×C4×C5C2×D412yes—
115C2×C3×C4×C5C2×D411yescharacteristic
116C2×C3×C4×C5C2×Q812yes—
117C2×C3×C4×C5#4112yes—
118C2×C3×C4×C5C2411yescharacteristic
119C3×C5×C8—22no—
120C3×C5×C8—22no—
121C22×C3×D4C22×C511yescharacteristic
122C23×C3×C4—21no—
123C22×C3×C8C22×C511yescharacteristic
124C22×C3×C8C22×C511yescharacteristic
125C3×#125—22no—
126C3×#125C22×C514yescomplemented
127C3×C4×D4—22no—
128C3×C4×D4C22×C514yes—
129C2×C3×C4⋊C4—21no—
130C2×C3×C4⋊C4—21no—
131C2×C3×C4⋊C4C22×C511yescharacteristic
132C2×C3×C4⋊C4C22×C511yescharacteristic
133C2×C3×C4⋊C4C22×C511yescharacteristic
134C2×C3×C22⋊C4—21no—
135C2×C3×C42C22×C511yescharacteristic
136C3×#107C22×C514yescomplemented
137C3×#106C22×C514yescomplemented
138C3×C4⋊C8C22×C514yescomplemented
139C3×#102C22×C514yes—
140C3×#102C22×C514yes—
141C24×C5—21no—
142C2×C5×D4—21no—
143C2×C5×D4—22no—
144C2×C5×D4C3×Q814yes—
145C2×C5×D4C23×C312yes—
146C22×C4×C5—21no—
147C22×C4×C5—21no—
148C22×C4×C5—21no—
149C22×C4×C5—24no—
150C22×C4×C5—44no—
151C22×C4×C5C3×D411yescharacteristic
152C22×C4×C5C3×D411yescharacteristic
153C22×C4×C5C23×C311yescharacteristic
154C2×C5×C8—22no—
155C2×C5×C8—22no—
156C2×C5×C8C23×C312yes—
157C2×C5×C8C23×C312yes—
158C5×C4⋊C4—24no—
159C5×C4⋊C4—24no—
160C5×C4⋊C4—21no—
161C5×C4⋊C4—21no—
162C5×C4⋊C4—21no—
163C5×C4⋊C4C23×C312yes—
164C5×C4⋊C4C23×C312yes—
165C5×C4⋊C4C23×C312yes—
166C5×C22⋊C4—24no—
167C42×C5C3×D412yes—
168C42×C5C23×C312yes—
169C2×#125C2×C3×C511yescharacteristic, complemented
170C2×C4×D4C2×C3×C511yescharacteristic
171#478C2×C3×C518yesdirect product factor
172C2×#107C2×C3×C511yescharacteristic, complemented
173C2×#106C2×C3×C511yescharacteristic, complemented
174C2×C4⋊C8C2×C3×C511yescharacteristic, complemented
175C2×#102C2×C3×C511yescharacteristic
176C2×#102C2×C3×C511yescharacteristic
177C22×C3×C5—24no—
178C22×C3×C5—42no—
179C22×C3×C5—42no—
180C22×C3×C5—48no—
181C22×C3×C5C22⋊Q812yes—
182C22×C3×C5#13512yes—
183C22×C3×C5AGL(1,ℤ/8)12yes—
184C22×C3×C5C22×D411yescharacteristic
185C3×C4×C5—22no—
186C3×C4×C5—22no—
187C3×C4×C5—42no—
188C3×C4×C5—42no—
189C3×C4×C5—42no—
190C3×C4×C5C22⋊Q812yes—
191C3×C4×C5C22×Q811yescharacteristic
192C3×C4×C5C2×#4111yescharacteristic
193C24×C3—21no—
194C2×C3×D4—21no—
195C2×C3×D4—22no—
196C2×C3×D4C5×Q814yes—
197C2×C3×D4C23×C512yes—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—24no—
202C22×C3×C4—44no—
203C22×C3×C4C5×D411yescharacteristic
204C22×C3×C4C5×D411yescharacteristic
205C22×C3×C4C23×C511yescharacteristic
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C2×C3×C8C23×C512yes—
209C2×C3×C8C23×C512yes—
210C3×C4⋊C4—24no—
211C3×C4⋊C4—24no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4C23×C512yes—
216C3×C4⋊C4C23×C512yes—
217C3×C4⋊C4C23×C512yes—
218C3×C22⋊C4—24no—
219C3×C42C5×D412yes—
220C3×C42C23×C512yes—
221C23×C5—24no—
222C23×C5—22no—
223C23×C5—42no—
224C23×C5—42no—
225C23×C5C2×C3×D411yescharacteristic
226C5×D4—24no—
227C5×D4—22no—
228C5×D4C2×C3×Q814yes—
229C2×C4×C5—22no—
230C2×C4×C5—22no—
231C2×C4×C5—22no—
232C2×C4×C5—22no—
233C2×C4×C5—22no—
234C2×C4×C5—48no—
235C2×C4×C5—42no—
236C2×C4×C5—42no—
237C2×C4×C5—42no—
238C2×C4×C5C2×C3×D412yes—
239C2×C4×C5C2×C3×D412yes—
240C2×C4×C5C2×C3×D411yescharacteristic
241C2×C4×C5C2×C3×Q812yes—
242C2×C4×C5C3×#4112yes—
243C2×C4×C5C24×C311yescharacteristic
244C5×C8—22no—
245C5×C8—22no—
246C22×D4C22×C3×C511yescharacteristic
247C23×C4—21no—
248C22×C8C22×C3×C511yescharacteristic
249C22×C8C22×C3×C511yescharacteristic
250#125—22no—
251#125C22×C3×C514yescomplemented
252C4×D4—22no—
253C4×D4C22×C3×C514yes—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4C22×C3×C511yescharacteristic
257C2×C4⋊C4C22×C3×C511yescharacteristic
258C2×C4⋊C4C22×C3×C511yescharacteristic
259C2×C22⋊C4—21no—
260C2×C42C22×C3×C511yescharacteristic
261#107C22×C3×C514yescomplemented
262#106C22×C3×C514yescomplemented
263C4⋊C8C22×C3×C514yescomplemented
264#102C22×C3×C514yes—
265#102C22×C3×C514yes—
266C2×C3×C5—44no—
267C2×C3×C5#47814yesdirect product factor
268C2×C3×C5C2×C22⋊Q811yescharacteristic
269C2×C3×C5C2×#13511yescharacteristic
270C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
271C23×C3—24no—
272C23×C3—22no—
273C23×C3—42no—
274C23×C3—42no—
275C23×C3C2×C5×D411yescharacteristic
276C3×D4—24no—
277C3×D4—22no—
278C3×D4C2×C5×Q814yes—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—48no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4C2×C5×D412yes—
289C2×C3×C4C2×C5×D412yes—
290C2×C3×C4C2×C5×D411yescharacteristic
291C2×C3×C4C2×C5×Q812yes—
292C2×C3×C4C5×#4112yes—
293C2×C3×C4C24×C511yescharacteristic
294C3×C8—22no—
295C3×C8—22no—
296C22×C5—24no—
297C22×C5—42no—
298C22×C5—42no—
299C22×C5—48no—
300C22×C5C3×C22⋊Q812yes—
301C22×C5C3×#13512yes—
302C22×C5C3×AGL(1,ℤ/8)12yes—
303C22×C5C22×C3×D411yescharacteristic
304C4×C5—22no—
305C4×C5—22no—
306C4×C5—42no—
307C4×C5—42no—
308C4×C5—42no—
309C4×C5C3×C22⋊Q812yes—
310C4×C5C22×C3×Q811yescharacteristic
311C4×C5C2×C3×#4111yescharacteristic
312C24—21no—
313C2×D4—21no—
314C2×D4—22no—
315C2×D4C3×C5×Q814yes—
316C2×D4C23×C3×C512yes—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—24no—
321C22×C4—44no—
322C22×C4C3×C5×D411yescharacteristic
323C22×C4C3×C5×D411yescharacteristic
324C22×C4C23×C3×C511yescharacteristic
325C2×C8—22no—
326C2×C8—22no—
327C2×C8C23×C3×C512yes—
328C2×C8C23×C3×C512yes—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4C23×C3×C512yes—
335C4⋊C4C23×C3×C512yes—
336C4⋊C4C23×C3×C512yes—
337C22⋊C4—24no—
338C42C3×C5×D412yes—
339C42C23×C3×C512yes—
340C3×C5C2×#47811yescharacteristic, direct product factor, Hall
341C22×C3—24no—
342C22×C3—48no—
343C22×C3—42no—
344C22×C3—42no—
345C22×C3C5×C22⋊Q812yes—
346C22×C3C5×#13512yes—
347C22×C3C5×AGL(1,ℤ/8)12yes—
348C22×C3C22×C5×D411yescharacteristic
349C3×C4—22no—
350C3×C4—22no—
351C3×C4—42no—
352C3×C4—42no—
353C3×C4—42no—
354C3×C4C5×C22⋊Q812yes—
355C3×C4C22×C5×Q811yescharacteristic
356C3×C4C2×C5×#4111yescharacteristic
357C2×C5—44no—
358C2×C5C3×#47814yesdirect product factor
359C2×C5C2×C3×C22⋊Q811yescharacteristic
360C2×C5C2×C3×#13511yescharacteristic
361C2×C5C2×C3×AGL(1,ℤ/8)11yescharacteristic
362C23—22no—
363C23—24no—
364C23—42no—
365C23—42no—
366C23C2×C3×C5×D411yescharacteristic
367D4—24no—
368D4—22no—
369D4C2×C3×C5×Q814yes—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—48no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4C2×C3×C5×D412yes—
380C2×C4C2×C3×C5×D412yes—
381C2×C4C2×C3×C5×D411yescharacteristic
382C2×C4C2×C3×C5×Q812yes—
383C2×C4C3×C5×#4112yes—
384C2×C4C24×C3×C511yescharacteristic
385C8—22no—
386C8—22no—
387C2×C3—44no—
388C2×C3C5×#47814yesdirect product factor
389C2×C3C2×C5×C22⋊Q811yescharacteristic
390C2×C3C2×C5×#13511yescharacteristic
391C2×C3C2×C5×AGL(1,ℤ/8)11yescharacteristic
392C5C2×C3×#47811yescharacteristic, direct product factor, Sylow
393C22—24no—
394C22—48no—
395C22—42no—
396C22—42no—
397C22C3×C5×C22⋊Q812yes—
398C22C3×C5×#13512yes—
399C22C3×C5×AGL(1,ℤ/8)12yes—
400C22C22×C3×C5×D411yescharacteristic
401C4—22no—
402C4—22no—
403C4—42no—
404C4—42no—
405C4—42no—
406C4C3×C5×C22⋊Q812yes—
407C4C22×C3×C5×Q811yescharacteristic
408C4C2×C3×C5×#4111yescharacteristic
409C3C2×C5×#47811yescharacteristic, direct product factor, Sylow
410C2—44no—
411C2C3×C5×#47814yesdirect product factor
412C2C2×C3×C5×C22⋊Q811yescharacteristic
413C2C2×C3×C5×#13511yescharacteristic
414C2C2×C3×C5×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
center, socleC23×C3×C5C2×D4
commutator, Frattini, cosocleC2×C4C24×C3×C5
prime radicalC22×C3×C5×D4C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3×C5 ◁ C2×C3×C42×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×#125C2semidirect
3. C3×C5×#478C2direct (N × H)
4. C2×C3×C5×#107C2semidirect
5. C2×C3×C5×#106C2semidirect
6. C2×C3×C5×C4⋊C8C2semidirect
9. C2×C5×#478C3direct (N × H)
15. C3×C5×#125C22semidirect
25. C3×C5×#107C22semidirect
26. C3×C5×#106C22semidirect
27. C3×C5×C4⋊C8C22semidirect
30. C2×C3×#478C5direct (N × H)
31. C2×C5×#125C2×C3semidirect
33. C5×#478C2×C3direct (N × H)
34. C2×C5×#107C2×C3semidirect
35. C2×C5×#106C2×C3semidirect
36. C2×C5×C4⋊C8C2×C3semidirect
67. C2×C3×#125C2×C5semidirect
69. C3×#478C2×C5direct (N × H)
70. C2×C3×#107C2×C5semidirect
71. C2×C3×#106C2×C5semidirect
72. C2×C3×C4⋊C8C2×C5semidirect
80. C5×#125C22×C3semidirect
90. C5×#107C22×C3semidirect
91. C5×#106C22×C3semidirect
92. C5×C4⋊C8C22×C3semidirect
95. C2×#478C3×C5direct (N × H)
126. C3×#125C22×C5semidirect
136. C3×#107C22×C5semidirect
137. C3×#106C22×C5semidirect
138. C3×C4⋊C8C22×C5semidirect
169. C2×#125C2×C3×C5semidirect
171. #478C2×C3×C5direct (N × H)
172. C2×#107C2×C3×C5semidirect
173. C2×#106C2×C3×C5semidirect
174. C2×C4⋊C8C2×C3×C5semidirect
251. #125C22×C3×C5semidirect
261. #107C22×C3×C5semidirect
262. #106C22×C3×C5semidirect
263. C4⋊C8C22×C3×C5semidirect
267. C2×C3×C5#478direct (N × H)
340. C3×C5C2×#478direct (N × H)
358. C2×C5C3×#478direct (N × H)
388. C2×C3C5×#478direct (N × H)
392. C5C2×C3×#478direct (N × H)
409. C3C2×C5×#478direct (N × H)
411. C2C3×C5×#478direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×#1252
C2×C3×C4×C5×D42
C3×C5×#4782
C2×C3×C5×#1072
C2×C3×C5×#1062
C2×C3×C5×C4⋊C82
C2×C3×C5×#1022
C2×C5×#4783
C2×C3×#4785