Finite Group Databasebeta

← C2×C5×#5711 #14495190

Subgroups — C2×C5×#5711 #14495190

Order 1920.

2688 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#126C211yesmaximal, characteristic, complemented
2C5×#5711C2116yesmaximal, direct product factor
3C2×C5×#895C213yesmaximal, complemented
4C2×C5×#894C213yesmaximal, complemented
5C2×C5×#884C213yesmaximal, complemented
6C2×C5×#882C213yesmaximal, complemented
7C2×C5×#876C211yesmaximal, characteristic, complemented
8C2×C5×#874C211yesmaximal, characteristic, complemented
9C2×C5×#560—31nomaximal, Hall
10C24×C5×D3C2211yescharacteristic
11C22×C5×#67C2213yes—
12C22×C5×D12C2213yes—
13C22×C5×D12C2213yes—
14C22×C5×D12C2213yescomplemented
15C22×C4×C5×D3C2213yescomplemented
16C3×C5×#126C2218yescomplemented
17C2×C3×C5×C4⋊C4C2213yescomplemented
18C2×C3×C5×C22⋊C4C2213yes—
19C2×C3×C42×C5C2211yescharacteristic, complemented
20C2×C5×(C2×D3)⋊C4C2213yes—
21C2×C5×(C2×D3)⋊C4C2213yes—
22C2×C5×(C2×D3)⋊C4C2213yescomplemented
23C2×C5×Dic3⋊C4C2213yescomplemented
24C5×#895C22124yescomplemented
25C5×#894C22124yescomplemented
26C5×#884C22124yescomplemented
27C5×#882C22124yescomplemented
28C5×#876C2218yescomplemented
29C5×#874C2218yescomplemented
30C2×#5711C511yesmaximal, characteristic, direct product factor, Hall
31C5×#560—316no—
32C2×C5×#127—31no—
33C2×C5×#126—31no—
34C2×C5×#126D311yescharacteristic, complemented
35C2×C5×#123—33no—
36C2×C5×#121—33no—
37C2×C5×#121—33no—
38C2×C5×#120—33no—
39C24×C3×C5C2311yescharacteristic
40C23×C5×D3—46no—
41C23×C5×D3C2311yescharacteristic
42C23×C5×D3C2311yescharacteristic
43C23×C5×D3C2313yes—
44C23×C5×D3C2314yes—
45C22×C3×C4×C5C2313yes—
46C22×C3×C4×C5C2313yes—
47C2×C5×#67—46no—
48C2×C5×#67C23112yes—
49C22×C5×Dic3C2313yes—
50C2×C5×D12—46no—
51C2×C5×D12—46no—
52C2×C5×D12—46no—
53C2×C5×D12C23112yes—
54C2×C5×D12C23112yes—
55C2×C5×D12C23112yescomplemented
56C2×C4×C5×D3—46no—
57C2×C4×C5×D3C23112yescomplemented
58C3×C5×C4⋊C4C23112yescomplemented
59C3×C5×C22⋊C4C23112yes—
60C3×C42×C5C2314yescomplemented
61C5×(C2×D3)⋊C4C23112yes—
62C5×(C2×D3)⋊C4C23112yes—
63C5×(C2×D3)⋊C4C23112yescomplemented
64C5×Dic3⋊C4C23112yescomplemented
65C2×C3×#126C2×C511yescharacteristic, complemented
66#5711C2×C5116yesdirect product factor
67C2×#895C2×C513yescomplemented
68C2×#894C2×C513yescomplemented
69C2×#884C2×C513yescomplemented
70C2×#882C2×C513yescomplemented
71C2×#876C2×C511yescharacteristic, complemented
72C2×#874C2×C511yescharacteristic, complemented
73C25×C5—31no—
74C22×C5×D4—33no—
75C22×C5×D4—33no—
76C22×C5×D4—33no—
77C22×C5×D4—33no—
78C23×C4×C5—33no—
79C5×#127—38no—
80C5×#126—38no—
81C5×#126C2×D318yescomplemented
82C5×#123—324no—
83C5×#121—324no—
84C5×#121—324no—
85C5×#120—324no—
86C2×C5×C4⋊C4—33no—
87C2×C5×C4⋊C4C2×D313yescomplemented
88C2×C5×C22⋊C4—33no—
89C2×C5×C22⋊C4—33no—
90C2×C5×C22⋊C4—33no—
91C2×C5×C22⋊C4C2×D313yes—
92C2×C42×C5C2×D311yescharacteristic, complemented
93C2×#560—31noSylow
94C23×C3×C5—26no—
95C23×C3×C5C2412yes—
96C23×C3×C5C2411yescharacteristic
97C22×C5×D3—26no—
98C22×C5×D3—26no—
99C22×C5×D3—26no—
100C22×C5×D3—26no—
101C22×C5×D3—26no—
102C22×C5×D3—412no—
103C22×C5×D3—416no—
104C22×C5×D3C2412yes—
105C22×C5×D3C2412yes—
106C22×C5×D3C2416yes—
107C2×C3×C4×C5—26no—
108C2×C3×C4×C5—26no—
109C2×C3×C4×C5C2416yes—
110C2×C3×C4×C5C2416yes—
111C5×#67—412no—
112C2×C5×Dic3—26no—
113C2×C5×Dic3C2416yes—
114C5×D12—412no—
115C5×D12—412no—
116C5×D12—412no—
117C4×C5×D3—412no—
118C24×D3C22×C511yescharacteristic
119C22×#67C22×C513yes—
120C22×D12C22×C513yes—
121C22×D12C22×C513yes—
122C22×D12C22×C513yescomplemented
123C22×C4×D3C22×C513yescomplemented
124C3×#126C22×C518yescomplemented
125C2×C3×C4⋊C4C22×C513yescomplemented
126C2×C3×C22⋊C4C22×C513yes—
127C2×C3×C42C22×C511yescharacteristic, complemented
128C2×(C2×D3)⋊C4C22×C513yes—
129C2×(C2×D3)⋊C4C22×C513yes—
130C2×(C2×D3)⋊C4C22×C513yescomplemented
131C2×Dic3⋊C4C22×C513yescomplemented
132#895C22×C5124yescomplemented
133#894C22×C5124yescomplemented
134#884C22×C5124yescomplemented
135#882C22×C5124yescomplemented
136#876C22×C518yescomplemented
137#874C22×C518yescomplemented
138C24×C5—31no—
139C24×C5—31no—
140C24×C5—33no—
141C24×C5—34no—
142C24×C5—126no—
143C24×C5C22×D311yescharacteristic
144C2×C5×D4—312no—
145C2×C5×D4—312no—
146C2×C5×D4—312no—
147C2×C5×D4—312no—
148C2×C5×D4—126no—
149C2×C5×D4—126no—
150C2×C5×D4—126no—
151C2×C5×D4—126no—
152C22×C4×C5—33no—
153C22×C4×C5—312no—
154C22×C4×C5—126no—
155C22×C4×C5C22×D313yes—
156C22×C4×C5C22×D313yes—
157C5×C4⋊C4—312no—
158C5×C4⋊C4C22×D3112yescomplemented
159C5×C22⋊C4—312no—
160C5×C22⋊C4—312no—
161C5×C22⋊C4—312no—
162C5×C22⋊C4C22×D3112yes—
163C42×C5C22×D314yescomplemented
164#560—316no—
165C2×#127—31no—
166C2×#126—31no—
167C2×#126C5×D311yescharacteristic, complemented
168C2×#123—33no—
169C2×#121—33no—
170C2×#121—33no—
171C2×#120—33no—
172C22×C3×C5—26no—
173C22×C3×C5—44no—
174C22×C3×C5H(5,𝔽2)16yes—
175C22×C3×C5C2511yescharacteristic
176C2×C5×D3—26no—
177C2×C5×D3—26no—
178C2×C5×D3—26no—
179C2×C5×D3—26no—
180C2×C5×D3—26no—
181C2×C5×D3—416no—
182C2×C5×D3—412no—
183C2×C5×D3—44no—
184C2×C5×D3—44no—
185C3×C4×C5—26no—
186C3×C4×C5—26no—
187C5×Dic3—26no—
188C24×C3C23×C511yescharacteristic
189C23×D3—46no—
190C23×D3C23×C511yescharacteristic
191C23×D3C23×C511yescharacteristic
192C23×D3C23×C513yes—
193C23×D3C23×C514yes—
194C22×C3×C4C23×C513yes—
195C22×C3×C4C23×C513yes—
196C2×#67—46no—
197C2×#67C23×C5112yes—
198C22×Dic3C23×C513yes—
199C2×D12—46no—
200C2×D12—46no—
201C2×D12—46no—
202C2×D12C23×C5112yes—
203C2×D12C23×C5112yes—
204C2×D12C23×C5112yescomplemented
205C2×C4×D3—46no—
206C2×C4×D3C23×C5112yescomplemented
207C3×C4⋊C4C23×C5112yescomplemented
208C3×C22⋊C4C23×C5112yes—
209C3×C42C23×C514yescomplemented
210(C2×D3)⋊C4C23×C5112yes—
211(C2×D3)⋊C4C23×C5112yes—
212(C2×D3)⋊C4C23×C5112yescomplemented
213Dic3⋊C4C23×C5112yescomplemented
214C23×C5—26no—
215C23×C5—32no—
216C23×C5—32no—
217C23×C5—36no—
218C23×C5—66no—
219C23×C5—66no—
220C23×C5—66no—
221C23×C5—66no—
222C23×C5—66no—
223C23×C5—1212no—
224C23×C5—1216no—
225C23×C5C23×D312yes—
226C23×C5C23×D311yescharacteristic
227C5×D4—1212no—
228C5×D4—1212no—
229C5×D4—1212no—
230C5×D4—1212no—
231C2×C4×C5—26no—
232C2×C4×C5—26no—
233C2×C4×C5—36no—
234C2×C4×C5—66no—
235C2×C4×C5—1212no—
236C2×C4×C5C23×D316yes—
237C2×C4×C5C23×D316yes—
238C25—31no—
239C22×D4—33no—
240C22×D4—33no—
241C22×D4—33no—
242C22×D4—33no—
243C23×C4—33no—
244#127—38no—
245#126—38no—
246#126C2×C5×D318yescomplemented
247#123—324no—
248#121—324no—
249#121—324no—
250#120—324no—
251C2×C4⋊C4—33no—
252C2×C4⋊C4C2×C5×D313yescomplemented
253C2×C22⋊C4—33no—
254C2×C22⋊C4—33no—
255C2×C22⋊C4—33no—
256C2×C22⋊C4C2×C5×D313yes—
257C2×C42C2×C5×D311yescharacteristic, complemented
258C2×C3×C5—42no—
259C2×C3×C5#56014yescomplemented
260C2×C3×C5C2×H(5,𝔽2)13yes—
261C5×D3—46no—
262C5×D3—42no—
263C5×D3—42no—
264C23×C3—26no—
265C23×C3C24×C512yes—
266C23×C3C24×C511yescharacteristic
267C22×D3—26no—
268C22×D3—26no—
269C22×D3—26no—
270C22×D3—26no—
271C22×D3—26no—
272C22×D3—412no—
273C22×D3—416no—
274C22×D3C24×C512yes—
275C22×D3C24×C512yes—
276C22×D3C24×C516yes—
277C2×C3×C4—26no—
278C2×C3×C4—26no—
279C2×C3×C4C24×C516yes—
280C2×C3×C4C24×C516yes—
281#67—412no—
282C2×Dic3—26no—
283C2×Dic3C24×C516yes—
284D12—412no—
285D12—412no—
286D12—412no—
287C4×D3—412no—
288C22×C5—26no—
289C22×C5—44no—
290C22×C5—66no—
291C22×C5—66no—
292C22×C5—66no—
293C22×C5—66no—
294C22×C5—66no—
295C22×C5—1216no—
296C22×C5—1212no—
297C22×C5—124no—
298C22×C5—124no—
299C22×C5#100916yes—
300C22×C5C24×D311yescharacteristic
301C4×C5—26no—
302C4×C5—26no—
303C4×C5—66no—
304C24—31no—
305C24—31no—
306C24—33no—
307C24—34no—
308C24—126no—
309C24C22×C5×D311yescharacteristic
310C2×D4—312no—
311C2×D4—312no—
312C2×D4—312no—
313C2×D4—312no—
314C2×D4—126no—
315C2×D4—126no—
316C2×D4—126no—
317C2×D4—126no—
318C22×C4—33no—
319C22×C4—312no—
320C22×C4—126no—
321C22×C4C22×C5×D313yes—
322C22×C4C22×C5×D313yes—
323C4⋊C4—312no—
324C4⋊C4C22×C5×D3112yescomplemented
325C22⋊C4—312no—
326C22⋊C4—312no—
327C22⋊C4—312no—
328C22⋊C4C22×C5×D3112yes—
329C42C22×C5×D314yescomplemented
330C3×C5C2×#56011yescharacteristic, complemented, Hall
331C22×C3—26no—
332C22×C3—44no—
333C22×C3C5×H(5,𝔽2)16yes—
334C22×C3C25×C511yescharacteristic
335C2×D3—26no—
336C2×D3—26no—
337C2×D3—26no—
338C2×D3—26no—
339C2×D3—26no—
340C2×D3—416no—
341C2×D3—412no—
342C2×D3—44no—
343C2×D3—44no—
344C3×C4—26no—
345C3×C4—26no—
346Dic3—26no—
347C2×C5—42no—
348C2×C5—122no—
349C2×C5—122no—
350C2×C5—126no—
351C2×C5#571114yesdirect product factor
352C2×C5C2×#100913yes—
353C23—26no—
354C23—32no—
355C23—32no—
356C23—36no—
357C23—66no—
358C23—66no—
359C23—66no—
360C23—66no—
361C23—66no—
362C23—1212no—
363C23—1216no—
364C23C23×C5×D312yes—
365C23C23×C5×D311yescharacteristic
366D4—1212no—
367D4—1212no—
368D4—1212no—
369D4—1212no—
370C2×C4—26no—
371C2×C4—26no—
372C2×C4—36no—
373C2×C4—66no—
374C2×C4—1212no—
375C2×C4C23×C5×D316yes—
376C2×C4C23×C5×D316yes—
377C2×C3—42no—
378C2×C3C5×#56014yescomplemented
379C2×C3C2×C5×H(5,𝔽2)13yes—
380D3—42no—
381D3—42no—
382D3—46no—
383C5C2×#571111yescharacteristic, direct product factor, Sylow
384C22—26no—
385C22—44no—
386C22—66no—
387C22—66no—
388C22—66no—
389C22—66no—
390C22—66no—
391C22—1216no—
392C22—124no—
393C22—124no—
394C22—1212no—
395C22C5×#100916yes—
396C22C24×C5×D311yescharacteristic
397C4—26no—
398C4—26no—
399C4—66no—
400C3C2×C5×#56011yescharacteristic, complemented, Sylow
401C2—42no—
402C2—122no—
403C2—122no—
404C2—126no—
405C2C5×#571114yesdirect product factor
406C2C2×C5×#100913yes—
subgroupG/H
centerC23×C5C23×D3
commutator, cosocleC22×C3C25×C5
FrattiniC22C24×C5×D3
FittingC2×C3×C5×#126C2
socleC23×C3×C5C24
hypercenterC2×C5×#126D3
hypocenterC3C2×C5×#560
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C5 ◁ C2×C5×#126

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×#126C2semidirect
2. C5×#5711C2direct (N × H)
3. C2×C5×#895C2semidirect
4. C2×C5×#894C2semidirect
5. C2×C5×#884C2semidirect
6. C2×C5×#882C2semidirect
7. C2×C5×#876C2semidirect
8. C2×C5×#874C2semidirect
14. C22×C5×D12C22semidirect
15. C22×C4×C5×D3C22semidirect
16. C3×C5×#126C22semidirect
17. C2×C3×C5×C4⋊C4C22semidirect
19. C2×C3×C42×C5C22semidirect
22. C2×C5×(C2×D3)⋊C4C22semidirect
23. C2×C5×Dic3⋊C4C22semidirect
24. C5×#895C22semidirect
25. C5×#894C22semidirect
26. C5×#884C22semidirect
27. C5×#882C22semidirect
28. C5×#876C22semidirect
29. C5×#874C22semidirect
30. C2×#5711C5direct (N × H)
34. C2×C5×#126D3semidirect
55. C2×C5×D12C23semidirect
57. C2×C4×C5×D3C23semidirect
58. C3×C5×C4⋊C4C23semidirect
60. C3×C42×C5C23semidirect
63. C5×(C2×D3)⋊C4C23semidirect
64. C5×Dic3⋊C4C23semidirect
65. C2×C3×#126C2×C5semidirect
66. #5711C2×C5direct (N × H)
67. C2×#895C2×C5semidirect
68. C2×#894C2×C5semidirect
69. C2×#884C2×C5semidirect
70. C2×#882C2×C5semidirect
71. C2×#876C2×C5semidirect
72. C2×#874C2×C5semidirect
81. C5×#126C2×D3semidirect
87. C2×C5×C4⋊C4C2×D3semidirect
92. C2×C42×C5C2×D3semidirect
122. C22×D12C22×C5semidirect
123. C22×C4×D3C22×C5semidirect
124. C3×#126C22×C5semidirect
125. C2×C3×C4⋊C4C22×C5semidirect
127. C2×C3×C42C22×C5semidirect
130. C2×(C2×D3)⋊C4C22×C5semidirect
131. C2×Dic3⋊C4C22×C5semidirect
132. #895C22×C5semidirect
133. #894C22×C5semidirect
134. #884C22×C5semidirect
135. #882C22×C5semidirect
136. #876C22×C5semidirect
137. #874C22×C5semidirect
158. C5×C4⋊C4C22×D3semidirect
163. C42×C5C22×D3semidirect
167. C2×#126C5×D3semidirect
204. C2×D12C23×C5semidirect
206. C2×C4×D3C23×C5semidirect
207. C3×C4⋊C4C23×C5semidirect
209. C3×C42C23×C5semidirect
212. (C2×D3)⋊C4C23×C5semidirect
213. Dic3⋊C4C23×C5semidirect
246. #126C2×C5×D3semidirect
252. C2×C4⋊C4C2×C5×D3semidirect
257. C2×C42C2×C5×D3semidirect
259. C2×C3×C5#560semidirect
324. C4⋊C4C22×C5×D3semidirect
329. C42C22×C5×D3semidirect
330. C3×C5C2×#560semidirect
351. C2×C5#5711direct (N × H)
378. C2×C3C5×#560semidirect
383. C5C2×#5711direct (N × H)
400. C3C2×C5×#560semidirect
405. C2C5×#5711direct (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×#1262
C5×#57112
C2×C5×#8952
C2×C5×#8942
C2×C5×#8842
C2×C5×#8822
C2×C5×#8762
C2×C5×#8742
C2×C5×#5603
C2×#57115