Finite Group Databasebeta

← C4×C5×2−1+4⋊C3 #14511319

Subgroups — C4×C5×2−1+4⋊C3 #14511319

Order 1920.

744 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×2−1+4⋊C3C211yesmaximal, characteristic
2C5×#520⋊C3C211yesmaximal, characteristic, complemented
3C5×#519⋊C3C212yesmaximal, complemented
4C4×C5×#41⋊C3C211yesmaximal, characteristic, complemented
5C2×C4×C5×SL(2,𝔽3)C212yesmaximal, complemented
6C4×C5×2−1+4C311yesmaximal, characteristic, complemented, Hall
7C5×2−1+4⋊C3C414yesdirect product factor
8C2×C5×#41⋊C3C2211yescharacteristic
9C22×C5×SL(2,𝔽3)C2212yes—
10C3×C4×C5×D4—41nomaximal
11C4×C5×SL(2,𝔽3)C2211yescharacteristic
12C4×C5×SL(2,𝔽3)C2212yes—
13C4×C5×SL(2,𝔽3)C2211yescharacteristic
14C4×2−1+4⋊C3C511yesmaximal, characteristic, direct product factor, Hall
15C2×C5×2−1+4C2×C311yescharacteristic
16C5×#520—31no—
17C5×#520—32no—
18C5×#520C2×C311yescharacteristic, complemented
19C5×#519—31no—
20C5×#519C2×C312yescomplemented
21C4×C5×#41—31no—
22C4×C5×#41—32no—
23C4×C5×#41C2×C311yescharacteristic, complemented
24C2×C4×C5×Q8—31no—
25C2×C4×C5×Q8C2×C312yescomplemented
26C2×C3×C5×D4—41no—
27C22×C3×C4×C5—42no—
28C5×#41⋊C3C2×C412yescomplemented
29C2×C5×SL(2,𝔽3)C2×C414yescomplemented
30C2×C5×SL(2,𝔽3)C2311yescharacteristic
31C3×C5×C4⋊C4—41no—
32C3×C5×C22⋊C4—42no—
33C3×C42×C5—41no—
34C2×2−1+4⋊C3C2×C511yescharacteristic
35#520⋊C3C2×C511yescharacteristic, complemented
36#519⋊C3C2×C512yescomplemented
37C4×#41⋊C3C2×C511yescharacteristic, complemented
38C2×C4×SL(2,𝔽3)C2×C512yescomplemented
39C5×2−1+4—34no—
40C5×2−1+4C3×C414yescomplemented
41C2×C5×#41—31no—
42C2×C5×#41—32no—
43C2×C5×#41C22×C311yescharacteristic
44C22×C5×Q8—31no—
45C22×C5×Q8C22×C312yes—
46C4×C5×Q8—32no—
47C4×C5×Q8—31no—
48C4×C5×Q8—32no—
49C4×C5×Q8—32no—
50C4×C5×Q8—31no—
51C4×C5×Q8—32no—
52C4×C5×Q8—32no—
53C4×C5×Q8C22×C311yescharacteristic
54C4×C5×Q8C22×C312yes—
55C4×C5×Q8C22×C311yescharacteristic
56C4×C5×D4—34no—
57C4×C5×D4—31no—
58C4×C5×D4—32no—
59C4×C5×D4—31no—
60C4×C5×D4—31no—
61C4×C5×D4—32no—
62C4×C5×D4—32no—
63C4×C5×D4A411yescharacteristic
64C5×#117—34no—
65C5×#117—31no—
66C5×#117—32no—
67C5×#117—32no—
68C5×#117—31no—
69C2×C5×C4⋊C4—32no—
70C2×C5×C4⋊C4—32no—
71C2×C5×C4⋊C4—31no—
72C2×C42×C5—32no—
73C2×C42×C5—32no—
74C2×C42×C5—31no—
75C4×2−1+4C3×C511yescharacteristic, complemented, Sylow
76C23×C3×C5—42no—
77C3×C5×D4—44no—
78C2×C3×C4×C5—41no—
79C2×C3×C4×C5—41no—
80C2×C3×C4×C5—41no—
81C2×C3×C4×C5—42no—
82C2×C3×C4×C5—84no—
83C5×SL(2,𝔽3)C22×C411yescharacteristic
842−1+4⋊C3C4×C514yesdirect product factor
85C2×#41⋊C3C22×C511yescharacteristic
86C22×SL(2,𝔽3)C22×C512yes—
87C3×C4×D4—41no—
88C4×SL(2,𝔽3)C22×C511yescharacteristic
89C4×SL(2,𝔽3)C22×C512yes—
90C4×SL(2,𝔽3)C22×C511yescharacteristic
91C5×#41—38no—
92C5×#41—34no—
93C5×#41—34no—
94C5×#41—38no—
95C5×#41—32no—
96C5×#41C2×C3×C412yescomplemented
97C2×C5×Q8—34no—
98C2×C5×Q8—34no—
99C2×C5×Q8—31no—
100C2×C5×Q8—32no—
101C2×C5×Q8—34no—
102C2×C5×Q8C2×C3×C414yescomplemented
103C2×C5×Q8C23×C311yescharacteristic
104C2×C5×D4—31no—
105C2×C5×D4—32no—
106C2×C5×D4C2×A411yescharacteristic
107C22×C4×C5—32no—
108C22×C4×C5—32no—
109C22×C4×C5—31no—
110C22×C4×C5—32no—
111C22×C4×C5—32no—
112C22×C4×C5—31no—
113C22×C4×C5—31no—
114C22×C4×C5C2×A412yes—
115C5×C4⋊C4—31no—
116C5×C4⋊C4—32no—
117C5×C4⋊C4—31no—
118C5×C4⋊C4—34no—
119C5×C4⋊C4—32no—
120C5×C4⋊C4—31no—
121C5×C4⋊C4—32no—
122C5×C4⋊C4C2×A411yescharacteristic
123C5×C22⋊C4—32no—
124C5×C22⋊C4—32no—
125C5×C22⋊C4—32no—
126C5×C22⋊C4C2×A412yes—
127C42×C5—34no—
128C42×C5—32no—
129C42×C5—31no—
130C42×C5—32no—
131C42×C5—31no—
132C42×C5—32no—
133C42×C5—31no—
134C42×C5C2×A411yescharacteristic
135C2×2−1+4C2×C3×C511yescharacteristic
136#520—31no—
137#520—32no—
138#520C2×C3×C511yescharacteristic, complemented
139#519—31no—
140#519C2×C3×C512yescomplemented
141C4×#41—31no—
142C4×#41—32no—
143C4×#41C2×C3×C511yescharacteristic, complemented
144C2×C4×Q8—31no—
145C2×C4×Q8C2×C3×C512yescomplemented
146C22×C3×C5—44no—
147C22×C3×C5—41no—
148C22×C3×C5—82no—
149C22×C3×C5—82no—
150C3×C4×C5—42no—
151C3×C4×C5—42no—
152C3×C4×C5—81no—
153C3×C4×C5—82no—
154C2×C3×D4—41no—
155C22×C3×C4—42no—
156#41⋊C3C2×C4×C512yescomplemented
157C2×SL(2,𝔽3)C2×C4×C514yescomplemented
158C2×SL(2,𝔽3)C23×C511yescharacteristic
159C3×C4⋊C4—41no—
160C3×C22⋊C4—42no—
161C3×C42—41no—
162C23×C5—31no—
163C23×C5C22×A412yes—
164C5×Q8—34no—
165C5×Q8—31no—
166C5×Q8—34no—
167C5×Q8—34no—
168C5×Q8C22×C3×C411yescharacteristic
169C5×D4—32no—
170C5×D4—32no—
171C5×D4—38no—
172C5×D4C4×A414yes—
173C2×C4×C5—24no—
174C2×C4×C5—38no—
175C2×C4×C5—34no—
176C2×C4×C5—34no—
177C2×C4×C5—32no—
178C2×C4×C5—31no—
179C2×C4×C5—32no—
180C2×C4×C5—32no—
181C2×C4×C5—31no—
182C2×C4×C5—32no—
183C2×C4×C5—31no—
184C2×C4×C5—62no—
185C2×C4×C5—121no—
186C2×C4×C5—122no—
187C2×C4×C5—122no—
188C2×C4×C5C22×A412yes—
189C2×C4×C5C22×A411yescharacteristic
190C2×C4×C5C22×A411yescharacteristic
191C2×C4×C5C22×A411yescharacteristic
1922−1+4—34no—
1932−1+4C3×C4×C514yescomplemented
194C2×#41—31no—
195C2×#41—32no—
196C2×#41C22×C3×C511yescharacteristic
197C22×Q8—31no—
198C22×Q8C22×C3×C512yes—
199C4×Q8—32no—
200C4×Q8—31no—
201C4×Q8—32no—
202C4×Q8—32no—
203C4×Q8—31no—
204C4×Q8—32no—
205C4×Q8—32no—
206C4×Q8C22×C3×C511yescharacteristic
207C4×Q8C22×C3×C512yes—
208C4×Q8C22×C3×C511yescharacteristic
209C4×D4—34no—
210C4×D4—31no—
211C4×D4—32no—
212C4×D4—31no—
213C4×D4—31no—
214C4×D4—32no—
215C4×D4—32no—
216C4×D4C5×A411yescharacteristic
217#117—34no—
218#117—31no—
219#117—32no—
220#117—32no—
221#117—31no—
222C2×C4⋊C4—32no—
223C2×C4⋊C4—32no—
224C2×C4⋊C4—31no—
225C2×C42—32no—
226C2×C42—32no—
227C2×C42—31no—
228C2×C3×C5—41no—
229C2×C3×C5—41no—
230C2×C3×C5—41no—
231C2×C3×C5—84no—
232C23×C3—42no—
233C3×D4—44no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—42no—
238C2×C3×C4—84no—
239SL(2,𝔽3)C22×C4×C511yescharacteristic
240C22×C5—22no—
241C22×C5—22no—
242C22×C5—32no—
243C22×C5—61no—
244C22×C5—61no—
245C22×C5C2×C4×A414yes—
246C22×C5C23×A411yescharacteristic
247C4×C5—22no—
248C4×C5—21no—
249C4×C5—34no—
250C4×C5—31no—
251C4×C5—31no—
252C4×C5—61no—
253C4×C5—61no—
254C4×C5—62no—
255C4×C5C2×C4×A412yes—
256C4×C52−1+4⋊C312yesdirect product factor
257#41—38no—
258#41—34no—
259#41—34no—
260#41—38no—
261#41—32no—
262#41C2×C3×C4×C512yescomplemented
263C2×Q8—34no—
264C2×Q8—34no—
265C2×Q8—31no—
266C2×Q8—32no—
267C2×Q8—34no—
268C2×Q8C2×C3×C4×C514yescomplemented
269C2×Q8C23×C3×C511yescharacteristic
270C2×D4—31no—
271C2×D4—32no—
272C2×D4C2×C5×A411yescharacteristic
273C22×C4—32no—
274C22×C4—32no—
275C22×C4—31no—
276C22×C4—32no—
277C22×C4—32no—
278C22×C4—31no—
279C22×C4—31no—
280C22×C4C2×C5×A412yes—
281C4⋊C4—31no—
282C4⋊C4—32no—
283C4⋊C4—31no—
284C4⋊C4—34no—
285C4⋊C4—32no—
286C4⋊C4—31no—
287C4⋊C4—32no—
288C4⋊C4C2×C5×A411yescharacteristic
289C22⋊C4—32no—
290C22⋊C4—32no—
291C22⋊C4—32no—
292C22⋊C4C2×C5×A412yes—
293C42—34no—
294C42—31no—
295C42—32no—
296C42—31no—
297C42—32no—
298C42—32no—
299C42—31no—
300C42C2×C5×A411yescharacteristic
301C3×C5—41noHall
302C22×C3—44no—
303C22×C3—41no—
304C22×C3—82no—
305C22×C3—82no—
306C3×C4—42no—
307C3×C4—42no—
308C3×C4—81no—
309C3×C4—82no—
310C2×C5—24no—
311C2×C5—62no—
312C2×C5C22×C4×A411yescharacteristic
313C2×C5C2×2−1+4⋊C311yescharacteristic
314C2×C5#594511yescharacteristic
315C23—31no—
316C23C22×C5×A412yes—
317Q8—34no—
318Q8—31no—
319Q8—34no—
320Q8—34no—
321Q8C22×C3×C4×C511yescharacteristic
322D4—32no—
323D4—32no—
324D4—38no—
325D4C4×C5×A414yes—
326C2×C4—24no—
327C2×C4—38no—
328C2×C4—34no—
329C2×C4—34no—
330C2×C4—31no—
331C2×C4—32no—
332C2×C4—32no—
333C2×C4—32no—
334C2×C4—31no—
335C2×C4—32no—
336C2×C4—31no—
337C2×C4—62no—
338C2×C4—121no—
339C2×C4—122no—
340C2×C4—122no—
341C2×C4C22×C5×A412yes—
342C2×C4C22×C5×A411yescharacteristic
343C2×C4C22×C5×A411yescharacteristic
344C2×C4C22×C5×A411yescharacteristic
345C2×C3—41no—
346C2×C3—41no—
347C2×C3—41no—
348C2×C3—84no—
349C5C4×2−1+4⋊C311yescharacteristic, direct product factor, Sylow
350C22—22no—
351C22—22no—
352C22—32no—
353C22—61no—
354C22—61no—
355C22C2×C4×C5×A414yes—
356C22C23×C5×A411yescharacteristic
357C4—22no—
358C4—21no—
359C4—31no—
360C4—34no—
361C4—31no—
362C4—61no—
363C4—61no—
364C4—62no—
365C4C2×C4×C5×A412yes—
366C4C5×2−1+4⋊C312yesdirect product factor
367C3—41noSylow
368C2—24no—
369C2—62no—
370C2C22×C4×C5×A411yescharacteristic
371C2C2×C5×2−1+4⋊C311yescharacteristic
372C2C5×#594511yescharacteristic
subgroupG/H
centerC2×C4×C5C22×A4
commutator, hypocenterQ8C22×C3×C4×C5
FrattiniC22C23×C5×A4
FittingC4×C5×2−1+4C3
socleC22×C5C23×A4
prime radicalC2×C5×2−1+4⋊C3C2
cosocleC2×Q8C23×C3×C5
hypercenterC4×C5×D4A4
LCSG ▷ Q8
UCSC1 ◁ C2×C4×C5 ◁ C4×C5×D4

Complements

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

NH ≅ G/Ntype
2. C5×#520⋊C3C2semidirect
3. C5×#519⋊C3C2semidirect
4. C4×C5×#41⋊C3C2semidirect
5. C2×C4×C5×SL(2,𝔽3)C2semidirect
6. C4×C5×2−1+4C3semidirect
7. C5×2−1+4⋊C3C4direct (N × H)
14. C4×2−1+4⋊C3C5direct (N × H)
18. C5×#520C2×C3semidirect
20. C5×#519C2×C3semidirect
23. C4×C5×#41C2×C3semidirect
25. C2×C4×C5×Q8C2×C3semidirect
28. C5×#41⋊C3C2×C4semidirect
29. C2×C5×SL(2,𝔽3)C2×C4semidirect
35. #520⋊C3C2×C5semidirect
36. #519⋊C3C2×C5semidirect
37. C4×#41⋊C3C2×C5semidirect
38. C2×C4×SL(2,𝔽3)C2×C5semidirect
40. C5×2−1+4C3×C4semidirect
75. C4×2−1+4C3×C5semidirect
84. 2−1+4⋊C3C4×C5direct (N × H)
96. C5×#41C2×C3×C4semidirect
102. C2×C5×Q8C2×C3×C4semidirect
138. #520C2×C3×C5semidirect
140. #519C2×C3×C5semidirect
143. C4×#41C2×C3×C5semidirect
145. C2×C4×Q8C2×C3×C5semidirect
156. #41⋊C3C2×C4×C5semidirect
157. C2×SL(2,𝔽3)C2×C4×C5semidirect
193. 2−1+4C3×C4×C5semidirect
256. C4×C52−1+4⋊C3direct (N × H)
262. #41C2×C3×C4×C5semidirect
268. C2×Q8C2×C3×C4×C5semidirect
349. C5C4×2−1+4⋊C3direct (N × H)
366. C4C5×2−1+4⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×2−1+4⋊C32
C5×#520⋊C32
C5×#519⋊C32
C4×C5×#41⋊C32
C2×C4×C5×SL(2,𝔽3)2
C4×C5×2−1+43
C3×C4×C5×D44
C4×2−1+4⋊C35