Finite Group Databasebeta

← C2×C5×#5697 #14481097

Subgroups — C2×C5×#5697 #14481097

Order 1920.

2520 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#125C211yesmaximal, characteristic, complemented
2C5×#5697C2116yesmaximal, direct product factor
3C2×C5×#895C214yesmaximal, complemented
4C2×C5×#895C212yesmaximal, complemented
5C2×C5×#894C214yesmaximal, complemented
6C2×C5×#893C212yesmaximal, complemented
7C2×C5×#874C211yesmaximal, characteristic, complemented
8C2×C5×#872C211yesmaximal, characteristic, complemented
9C2×C5×#540—31nomaximal, Hall
10C22×C5×D12C2212yes—
11C22×C5×D12C2214yes—
12C22×C5×D12C2214yes—
13C22×C4×C5×D3C2211yescharacteristic
14C22×C4×C5×D3C2214yes—
15C3×C5×#125C2218yescomplemented
16C2×C3×C5×C4⋊C4C2212yes—
17C2×C3×C5×C4⋊C4C2214yes—
18C2×C3×C42×C5C2211yescharacteristic
19C2×C5×(C2×D3)⋊C4C2212yes—
20C2×C5×(C2×D3)⋊C4C2214yes—
21C2×C5×(C2×D3)⋊C4C2214yes—
22C2×C5×Dic3⋊C4C2212yes—
23C2×C4×C5×Dic3C2211yescharacteristic
24C5×#895C22132yescomplemented
25C5×#895C22116yescomplemented
26C5×#894C22132yescomplemented
27C5×#893C22116yescomplemented
28C5×#874C2218yescomplemented
29C5×#872C2218yescomplemented
30C2×#5697C511yesmaximal, characteristic, direct product factor, Hall
31C5×#540—316no—
32C2×C5×#127—31no—
33C2×C5×#125D311yescharacteristic, complemented
34C2×C5×#123—34no—
35C2×C5×#121—34no—
36C2×C5×#121—32no—
37C2×C4×C5×D4—32no—
38C2×C5×#117—31no—
39C23×C5×D3C2311yescharacteristic
40C23×C5×D3C2314yes—
41C22×C3×C4×C5C2312yes—
42C22×C3×C4×C5C2314yes—
43C22×C3×C4×C5C2311yescharacteristic
44C22×C5×Dic3C2312yes—
45C22×C5×Dic3C2311yescharacteristic
46C2×C5×D12—28no—
47C2×C5×D12—48no—
48C2×C5×D12—48no—
49C2×C5×D12C2318yes—
50C2×C5×D12C23116yes—
51C2×C5×D12C23116yes—
52C2×C4×C5×D3—24no—
53C2×C4×C5×D3—48no—
54C2×C4×C5×D3C2314yes—
55C2×C4×C5×D3C23116yes—
56C3×C5×C4⋊C4C2318yes—
57C3×C5×C4⋊C4C23116yes—
58C3×C42×C5C2314yes—
59C5×(C2×D3)⋊C4C2318yes—
60C5×(C2×D3)⋊C4C23116yes—
61C5×(C2×D3)⋊C4C23116yes—
62C5×Dic3⋊C4C2318yes—
63C4×C5×Dic3C2314yes—
64C2×C3×#125C2×C511yescharacteristic, complemented
65#5697C2×C5116yesdirect product factor
66C2×#895C2×C514yescomplemented
67C2×#895C2×C512yescomplemented
68C2×#894C2×C514yescomplemented
69C2×#893C2×C512yescomplemented
70C2×#874C2×C511yescharacteristic, complemented
71C2×#872C2×C511yescharacteristic, complemented
72C22×C5×D4—32no—
73C22×C5×D4—34no—
74C22×C5×D4—34no—
75C23×C4×C5—31no—
76C23×C4×C5—34no—
77C5×#127—38no—
78C5×#125C2×D318yescomplemented
79C5×#123—332no—
80C5×#121—332no—
81C5×#121—316no—
82C4×C5×D4—316no—
83C5×#117—38no—
84C2×C5×C4⋊C4—32no—
85C2×C5×C4⋊C4C2×D312yes—
86C2×C5×C4⋊C4C2×D314yes—
87C2×C5×C22⋊C4—32no—
88C2×C5×C22⋊C4—34no—
89C2×C5×C22⋊C4—34no—
90C2×C42×C5—31no—
91C2×C42×C5C2×D311yescharacteristic
92C2×#540—31noSylow
93C23×C3×C5C2411yescharacteristic
94C22×C5×D3—22no—
95C22×C5×D3—28no—
96C22×C5×D3—28no—
97C22×C5×D3—24no—
98C22×C5×D3—28no—
99C22×C5×D3C2412yes—
100C22×C5×D3C2418yes—
101C2×C3×C4×C5—28no—
102C2×C3×C4×C5—24no—
103C2×C3×C4×C5#4114yes—
104C2×C3×C4×C5C2414yes—
105C2×C3×C4×C5C2418yes—
106C2×C3×C4×C5C2412yes—
107C2×C5×Dic3—22no—
108C2×C5×Dic3—24no—
109C2×C5×Dic3C2414yes—
110C2×C5×Dic3C2412yes—
111C5×D12—216no—
112C5×D12—416no—
113C5×D12—416no—
114C4×C5×D3—28no—
115C4×C5×D3—416no—
116C22×D12C22×C512yes—
117C22×D12C22×C514yes—
118C22×D12C22×C514yes—
119C22×C4×D3C22×C511yescharacteristic
120C22×C4×D3C22×C514yes—
121C3×#125C22×C518yescomplemented
122C2×C3×C4⋊C4C22×C512yes—
123C2×C3×C4⋊C4C22×C514yes—
124C2×C3×C42C22×C511yescharacteristic
125C2×(C2×D3)⋊C4C22×C512yes—
126C2×(C2×D3)⋊C4C22×C514yes—
127C2×(C2×D3)⋊C4C22×C514yes—
128C2×Dic3⋊C4C22×C512yes—
129C2×C4×Dic3C22×C511yescharacteristic
130#895C22×C5132yescomplemented
131#895C22×C5116yescomplemented
132#894C22×C5132yescomplemented
133#893C22×C5116yescomplemented
134#874C22×C518yescomplemented
135#872C22×C518yescomplemented
136C24×C5—31no—
137C24×C5—34no—
138C2×C5×D4—38no—
139C2×C5×D4—316no—
140C2×C5×D4—316no—
141C2×C5×D4—68no—
142C2×C5×D4—128no—
143C2×C5×D4—128no—
144C22×C4×C5—32no—
145C22×C4×C5—31no—
146C22×C4×C5—34no—
147C22×C4×C5—316no—
148C22×C4×C5—64no—
149C22×C4×C5—128no—
150C22×C4×C5C22×D312yes—
151C22×C4×C5C22×D314yes—
152C22×C4×C5C22×D311yescharacteristic
153C5×C4⋊C4—38no—
154C5×C4⋊C4C22×D318yes—
155C5×C4⋊C4C22×D3116yes—
156C5×C22⋊C4—38no—
157C5×C22⋊C4—316no—
158C5×C22⋊C4—316no—
159C42×C5—34no—
160C42×C5C22×D314yes—
161#540—316no—
162C2×#127—31no—
163C2×#125C5×D311yescharacteristic, complemented
164C2×#123—34no—
165C2×#121—34no—
166C2×#121—32no—
167C2×C4×D4—32no—
168C2×#117—31no—
169C22×C3×C5C2×#4112yes—
170C22×C3×C5H(5,𝔽2)14yes—
171C22×C3×C5C2511yescharacteristic
172C2×C5×D3—22no—
173C2×C5×D3—28no—
174C2×C5×D3—28no—
175C2×C5×D3—24no—
176C2×C5×D3—28no—
177C2×C5×D3—44no—
178C2×C5×D3—416no—
179C3×C4×C5—28no—
180C3×C4×C5—24no—
181C3×C4×C5C2×#4114yes—
182C5×Dic3—22no—
183C5×Dic3—24no—
184C23×D3C23×C511yescharacteristic
185C23×D3C23×C514yes—
186C22×C3×C4C23×C512yes—
187C22×C3×C4C23×C514yes—
188C22×C3×C4C23×C511yescharacteristic
189C22×Dic3C23×C512yes—
190C22×Dic3C23×C511yescharacteristic
191C2×D12—28no—
192C2×D12—48no—
193C2×D12—48no—
194C2×D12C23×C518yes—
195C2×D12C23×C5116yes—
196C2×D12C23×C5116yes—
197C2×C4×D3—24no—
198C2×C4×D3—48no—
199C2×C4×D3C23×C514yes—
200C2×C4×D3C23×C5116yes—
201C3×C4⋊C4C23×C518yes—
202C3×C4⋊C4C23×C5116yes—
203C3×C42C23×C514yes—
204(C2×D3)⋊C4C23×C518yes—
205(C2×D3)⋊C4C23×C5116yes—
206(C2×D3)⋊C4C23×C5116yes—
207Dic3⋊C4C23×C518yes—
208C4×Dic3C23×C514yes—
209C23×C5—32no—
210C23×C5—38no—
211C23×C5—62no—
212C23×C5—68no—
213C23×C5—68no—
214C23×C5—64no—
215C23×C5—68no—
216C23×C5C23×D311yescharacteristic
217C5×D4—616no—
218C5×D4—1216no—
219C5×D4—1216no—
220C2×C4×C5—28no—
221C2×C4×C5—24no—
222C2×C4×C5—34no—
223C2×C4×C5—32no—
224C2×C4×C5—62no—
225C2×C4×C5—64no—
226C2×C4×C5—68no—
227C2×C4×C5—1216no—
228C2×C4×C5#23914yes—
229C2×C4×C5C23×D314yes—
230C2×C4×C5C23×D318yes—
231C2×C4×C5C23×D312yes—
232C22×D4—32no—
233C22×D4—34no—
234C22×D4—34no—
235C23×C4—31no—
236C23×C4—34no—
237#127—38no—
238#125C2×C5×D318yescomplemented
239#123—332no—
240#121—332no—
241#121—316no—
242C4×D4—316no—
243#117—38no—
244C2×C4⋊C4—32no—
245C2×C4⋊C4C2×C5×D312yes—
246C2×C4⋊C4C2×C5×D314yes—
247C2×C22⋊C4—32no—
248C2×C22⋊C4—34no—
249C2×C22⋊C4—34no—
250C2×C42—31no—
251C2×C42C2×C5×D311yescharacteristic
252C2×C3×C5#54014yescomplemented
253C2×C3×C5C22×#4111yescharacteristic
254C2×C3×C5C2×H(5,𝔽2)12yes—
255C5×D3—42no—
256C5×D3—48no—
257C23×C3C24×C511yescharacteristic
258C22×D3—22no—
259C22×D3—28no—
260C22×D3—28no—
261C22×D3—24no—
262C22×D3—28no—
263C22×D3C24×C512yes—
264C22×D3C24×C518yes—
265C2×C3×C4—28no—
266C2×C3×C4—24no—
267C2×C3×C4C5×#4114yes—
268C2×C3×C4C24×C514yes—
269C2×C3×C4C24×C518yes—
270C2×C3×C4C24×C512yes—
271C2×Dic3—22no—
272C2×Dic3—24no—
273C2×Dic3C24×C514yes—
274C2×Dic3C24×C512yes—
275D12—216no—
276D12—416no—
277D12—416no—
278C4×D3—28no—
279C4×D3—416no—
280C22×C5—62no—
281C22×C5—68no—
282C22×C5—68no—
283C22×C5—64no—
284C22×C5—68no—
285C22×C5—124no—
286C22×C5—1216no—
287C22×C5C2×#23912yes—
288C22×C5#100914yes—
289C22×C5C24×D311yescharacteristic
290C4×C5—28no—
291C4×C5—24no—
292C4×C5—62no—
293C4×C5—64no—
294C4×C5C2×#23914yes—
295C24—31no—
296C24—34no—
297C2×D4—38no—
298C2×D4—316no—
299C2×D4—316no—
300C2×D4—68no—
301C2×D4—128no—
302C2×D4—128no—
303C22×C4—32no—
304C22×C4—31no—
305C22×C4—34no—
306C22×C4—316no—
307C22×C4—64no—
308C22×C4—128no—
309C22×C4C22×C5×D312yes—
310C22×C4C22×C5×D314yes—
311C22×C4C22×C5×D311yescharacteristic
312C4⋊C4—38no—
313C4⋊C4C22×C5×D318yes—
314C4⋊C4C22×C5×D3116yes—
315C22⋊C4—38no—
316C22⋊C4—316no—
317C22⋊C4—316no—
318C42—34no—
319C42C22×C5×D314yes—
320C3×C5C2×#54011yescharacteristic, complemented, Hall
321C22×C3C2×C5×#4112yes—
322C22×C3C5×H(5,𝔽2)14yes—
323C22×C3C25×C511yescharacteristic
324C2×D3—22no—
325C2×D3—28no—
326C2×D3—28no—
327C2×D3—24no—
328C2×D3—28no—
329C2×D3—44no—
330C2×D3—416no—
331C3×C4—28no—
332C3×C4—24no—
333C3×C4C2×C5×#4114yes—
334Dic3—22no—
335Dic3—24no—
336C2×C5—122no—
337C2×C5—128no—
338C2×C5#569714yesdirect product factor
339C2×C5C22×#23911yescharacteristic
340C2×C5C2×#100912yes—
341C23—32no—
342C23—38no—
343C23—62no—
344C23—68no—
345C23—68no—
346C23—64no—
347C23—68no—
348C23C23×C5×D311yescharacteristic
349D4—616no—
350D4—1216no—
351D4—1216no—
352C2×C4—28no—
353C2×C4—24no—
354C2×C4—34no—
355C2×C4—32no—
356C2×C4—62no—
357C2×C4—64no—
358C2×C4—68no—
359C2×C4—1216no—
360C2×C4C5×#23914yes—
361C2×C4C23×C5×D314yes—
362C2×C4C23×C5×D318yes—
363C2×C4C23×C5×D312yes—
364C2×C3C5×#54014yescomplemented
365C2×C3C22×C5×#4111yescharacteristic
366C2×C3C2×C5×H(5,𝔽2)12yes—
367D3—42no—
368D3—48no—
369C5C2×#569711yescharacteristic, direct product factor, Sylow
370C22—62no—
371C22—68no—
372C22—68no—
373C22—64no—
374C22—68no—
375C22—124no—
376C22—1216no—
377C22C2×C5×#23912yes—
378C22C5×#100914yes—
379C22C24×C5×D311yescharacteristic
380C4—28no—
381C4—24no—
382C4—62no—
383C4—64no—
384C4C2×C5×#23914yes—
385C3C2×C5×#54011yescharacteristic, complemented, Sylow
386C2—122no—
387C2—128no—
388C2C5×#569714yesdirect product factor
389C2C22×C5×#23911yescharacteristic
390C2C2×C5×#100912yes—
subgroupG/H
centerC23×C5C23×D3
commutator, cosocleC22×C3C25×C5
FrattiniC22C24×C5×D3
FittingC2×C3×C5×#125C2
socleC23×C3×C5C24
hypercenterC2×C5×#125D3
hypocenterC3C2×C5×#540
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C5 ◁ C2×C5×#125

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×#125C2semidirect
2. C5×#5697C2direct (N × H)
3. C2×C5×#895C2semidirect
4. C2×C5×#895C2semidirect
5. C2×C5×#894C2semidirect
6. C2×C5×#893C2semidirect
7. C2×C5×#874C2semidirect
8. C2×C5×#872C2semidirect
15. C3×C5×#125C22semidirect
24. C5×#895C22semidirect
25. C5×#895C22semidirect
26. C5×#894C22semidirect
27. C5×#893C22semidirect
28. C5×#874C22semidirect
29. C5×#872C22semidirect
30. C2×#5697C5direct (N × H)
33. C2×C5×#125D3semidirect
64. C2×C3×#125C2×C5semidirect
65. #5697C2×C5direct (N × H)
66. C2×#895C2×C5semidirect
67. C2×#895C2×C5semidirect
68. C2×#894C2×C5semidirect
69. C2×#893C2×C5semidirect
70. C2×#874C2×C5semidirect
71. C2×#872C2×C5semidirect
78. C5×#125C2×D3semidirect
121. C3×#125C22×C5semidirect
130. #895C22×C5semidirect
131. #895C22×C5semidirect
132. #894C22×C5semidirect
133. #893C22×C5semidirect
134. #874C22×C5semidirect
135. #872C22×C5semidirect
163. C2×#125C5×D3semidirect
238. #125C2×C5×D3semidirect
252. C2×C3×C5#540semidirect
320. C3×C5C2×#540semidirect
338. C2×C5#5697direct (N × H)
364. C2×C3C5×#540semidirect
369. C5C2×#5697direct (N × H)
385. C3C2×C5×#540semidirect
388. C2C5×#5697direct (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
C5×#56972
C2×C5×#8952
C2×C5×#8942
C2×C5×#8932
C2×C5×#8742
C2×C5×#8722
C2×C5×#5403
C2×#56975