Finite Group Databasebeta

← C2×C5×#5698 #14481208

Subgroups — C2×C5×#5698 #14481208

Order 1920.

2008 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#125C211yesmaximal, characteristic, complemented
2C5×#5698C2116yesmaximal, direct product factor
3C2×C5×#898C214yesmaximal
4C2×C5×#897C212yesmaximal
5C2×C5×#890C214yesmaximal, complemented
6C2×C5×#887C212yesmaximal, complemented
7C2×C5×#872C211yesmaximal, characteristic
8C2×C5×#869C211yesmaximal, characteristic, complemented
9C2×C5×#541—31nomaximal, Hall
10C22×C4×C5×D3C2211yescharacteristic
11C22×C5×Dic6C2212yes—
12C3×C5×#125C2218yescomplemented
13C2×C3×C5×C4⋊C4C2212yes—
14C2×C3×C5×C4⋊C4C2214yes—
15C2×C3×C42×C5C2211yescharacteristic
16C2×C5×(C2×D3)⋊C4C2212yes—
17C2×C5×(C2×D3)⋊C4C2214yes—
18C2×C5×C4⋊Dic3C2214yes—
19C2×C5×C4⋊Dic3C2214yes—
20C2×C5×Dic3⋊C4C2212yes—
21C2×C5×Dic3⋊C4C2214yes—
22C2×C4×C5×Dic3C2211yescharacteristic
23C2×C4×C5×Dic3C2214yes—
24C5×#898C22132yes—
25C5×#897C22116yes—
26C5×#890C22132yescomplemented
27C5×#887C22116yescomplemented
28C5×#872C2218yes—
29C5×#869C2218yescomplemented
30C2×#5698C511yesmaximal, characteristic, direct product factor, Hall
31C5×#541—316no—
32C2×C5×C4⋊Q8—31no—
33C2×C5×#126—34no—
34C2×C5×#125—34no—
35C2×C5×#125D311yescharacteristic, complemented
36C2×C5×C22⋊Q8—32no—
37C2×C4×C5×Q8—32no—
38C2×C5×#117—31no—
39C23×C5×D3C2311yescharacteristic
40C22×C3×C4×C5C2314yes—
41C22×C3×C4×C5C2312yes—
42C22×C3×C4×C5C2311yescharacteristic
43C22×C5×Dic3C2312yes—
44C22×C5×Dic3C2314yes—
45C22×C5×Dic3C2311yescharacteristic
46C2×C4×C5×D3—24no—
47C2×C4×C5×D3C2314yes—
48C2×C5×Dic6—28no—
49C2×C5×Dic6C2318yes—
50C3×C5×C4⋊C4C2318yes—
51C3×C5×C4⋊C4C23116yes—
52C3×C42×C5C2314yes—
53C5×(C2×D3)⋊C4C2318yes—
54C5×(C2×D3)⋊C4C23116yes—
55C5×C4⋊Dic3C23116yes—
56C5×C4⋊Dic3C23116yes—
57C5×Dic3⋊C4C2318yes—
58C5×Dic3⋊C4C23116yes—
59C4×C5×Dic3C2314yes—
60C4×C5×Dic3C23116yes—
61C2×C3×#125C2×C511yescharacteristic, complemented
62#5698C2×C5116yesdirect product factor
63C2×#898C2×C514yes—
64C2×#897C2×C512yes—
65C2×#890C2×C514yescomplemented
66C2×#887C2×C512yescomplemented
67C2×#872C2×C511yescharacteristic
68C2×#869C2×C511yescharacteristic, complemented
69C22×C5×Q8—32no—
70C23×C4×C5—31no—
71C5×C4⋊Q8—38no—
72C5×#126—332no—
73C5×#125—332no—
74C5×#125C2×D318yescomplemented
75C5×C22⋊Q8—316no—
76C4×C5×Q8—316no—
77C5×#117—38no—
78C2×C5×C4⋊C4—32no—
79C2×C5×C4⋊C4—34no—
80C2×C5×C4⋊C4—34no—
81C2×C5×C4⋊C4—34no—
82C2×C5×C4⋊C4C2×D312yes—
83C2×C5×C4⋊C4C2×D314yes—
84C2×C5×C22⋊C4—32no—
85C2×C5×C22⋊C4—34no—
86C2×C42×C5—31no—
87C2×C42×C5—34no—
88C2×C42×C5C2×D311yescharacteristic
89C2×#541—31noSylow
90C23×C3×C5C2411yescharacteristic
91C22×C5×D3—22no—
92C22×C5×D3—24no—
93C22×C5×D3C2412yes—
94C2×C3×C4×C5—28no—
95C2×C3×C4×C5—24no—
96C2×C3×C4×C5#4114yes—
97C2×C3×C4×C5C2418yes—
98C2×C3×C4×C5C2414yes—
99C2×C3×C4×C5C2412yes—
100C2×C5×Dic3—28no—
101C2×C5×Dic3—22no—
102C2×C5×Dic3—24no—
103C2×C5×Dic3C2414yes—
104C2×C5×Dic3C2418yes—
105C2×C5×Dic3C2412yes—
106C4×C5×D3—28no—
107C5×Dic6—216no—
108C22×C4×D3C22×C511yescharacteristic
109C22×Dic6C22×C512yes—
110C3×#125C22×C518yescomplemented
111C2×C3×C4⋊C4C22×C512yes—
112C2×C3×C4⋊C4C22×C514yes—
113C2×C3×C42C22×C511yescharacteristic
114C2×(C2×D3)⋊C4C22×C512yes—
115C2×(C2×D3)⋊C4C22×C514yes—
116C2×C4⋊Dic3C22×C514yes—
117C2×C4⋊Dic3C22×C514yes—
118C2×Dic3⋊C4C22×C512yes—
119C2×Dic3⋊C4C22×C514yes—
120C2×C4×Dic3C22×C511yescharacteristic
121C2×C4×Dic3C22×C514yes—
122#898C22×C5132yes—
123#897C22×C5116yes—
124#890C22×C5132yescomplemented
125#887C22×C5116yescomplemented
126#872C22×C518yes—
127#869C22×C518yescomplemented
128C24×C5—31no—
129C2×C5×Q8—38no—
130C2×C5×Q8—68no—
131C22×C4×C5—32no—
132C22×C4×C5—34no—
133C22×C4×C5—31no—
134C22×C4×C5—34no—
135C22×C4×C5—64no—
136C22×C4×C5C22×D314yes—
137C22×C4×C5C22×D312yes—
138C22×C4×C5C22×D311yescharacteristic
139C5×C4⋊C4—38no—
140C5×C4⋊C4—316no—
141C5×C4⋊C4—316no—
142C5×C4⋊C4—316no—
143C5×C4⋊C4C22×D318yes—
144C5×C4⋊C4C22×D3116yes—
145C5×C22⋊C4—38no—
146C5×C22⋊C4—316no—
147C42×C5—34no—
148C42×C5—316no—
149C42×C5C22×D314yes—
150#541—316no—
151C2×C4⋊Q8—31no—
152C2×#126—34no—
153C2×#125—34no—
154C2×#125C5×D311yescharacteristic, complemented
155C2×C22⋊Q8—32no—
156C2×C4×Q8—32no—
157C2×#117—31no—
158C22×C3×C5C2×#4112yes—
159C22×C3×C52−1+414yes—
160C22×C3×C5C2511yescharacteristic
161C2×C5×D3—22no—
162C2×C5×D3—24no—
163C2×C5×D3—44no—
164C3×C4×C5—28no—
165C3×C4×C5—24no—
166C3×C4×C5C2×#4114yes—
167C5×Dic3—28no—
168C5×Dic3—22no—
169C5×Dic3—24no—
170C23×D3C23×C511yescharacteristic
171C22×C3×C4C23×C514yes—
172C22×C3×C4C23×C512yes—
173C22×C3×C4C23×C511yescharacteristic
174C22×Dic3C23×C512yes—
175C22×Dic3C23×C514yes—
176C22×Dic3C23×C511yescharacteristic
177C2×C4×D3—24no—
178C2×C4×D3C23×C514yes—
179C2×Dic6—28no—
180C2×Dic6C23×C518yes—
181C3×C4⋊C4C23×C518yes—
182C3×C4⋊C4C23×C5116yes—
183C3×C42C23×C514yes—
184(C2×D3)⋊C4C23×C518yes—
185(C2×D3)⋊C4C23×C5116yes—
186C4⋊Dic3C23×C5116yes—
187C4⋊Dic3C23×C5116yes—
188Dic3⋊C4C23×C518yes—
189Dic3⋊C4C23×C5116yes—
190C4×Dic3C23×C514yes—
191C4×Dic3C23×C5116yes—
192C23×C5—32no—
193C23×C5—62no—
194C23×C5—64no—
195C23×C5C23×D311yescharacteristic
196C5×Q8—616no—
197C2×C4×C5—28no—
198C2×C4×C5—24no—
199C2×C4×C5—34no—
200C2×C4×C5—38no—
201C2×C4×C5—32no—
202C2×C4×C5—62no—
203C2×C4×C5—68no—
204C2×C4×C5—64no—
205C2×C4×C5—68no—
206C2×C4×C5#23914yes—
207C2×C4×C5C23×D318yes—
208C2×C4×C5C23×D314yes—
209C2×C4×C5C23×D312yes—
210C22×Q8—32no—
211C23×C4—31no—
212C4⋊Q8—38no—
213#126—332no—
214#125—332no—
215#125C2×C5×D318yescomplemented
216C22⋊Q8—316no—
217C4×Q8—316no—
218#117—38no—
219C2×C4⋊C4—32no—
220C2×C4⋊C4—34no—
221C2×C4⋊C4—34no—
222C2×C4⋊C4—34no—
223C2×C4⋊C4C2×C5×D312yes—
224C2×C4⋊C4C2×C5×D314yes—
225C2×C22⋊C4—32no—
226C2×C22⋊C4—34no—
227C2×C42—31no—
228C2×C42—34no—
229C2×C42C2×C5×D311yescharacteristic
230C2×C3×C5#54114yescomplemented
231C2×C3×C5C22×#4111yescharacteristic
232C2×C3×C5C2×2−1+412yes—
233C5×D3—42no—
234C23×C3C24×C511yescharacteristic
235C22×D3—22no—
236C22×D3—24no—
237C22×D3C24×C512yes—
238C2×C3×C4—28no—
239C2×C3×C4—24no—
240C2×C3×C4C5×#4114yes—
241C2×C3×C4C24×C518yes—
242C2×C3×C4C24×C514yes—
243C2×C3×C4C24×C512yes—
244C2×Dic3—28no—
245C2×Dic3—22no—
246C2×Dic3—24no—
247C2×Dic3C24×C514yes—
248C2×Dic3C24×C518yes—
249C2×Dic3C24×C512yes—
250C4×D3—28no—
251Dic6—216no—
252C22×C5—62no—
253C22×C5—64no—
254C22×C5—124no—
255C22×C5C2×#23912yes—
256C22×C5#101014yes—
257C22×C5C24×D311yescharacteristic
258C4×C5—28no—
259C4×C5—24no—
260C4×C5—62no—
261C4×C5—68no—
262C4×C5—64no—
263C4×C5C2×#23914yes—
264C24—31no—
265C2×Q8—38no—
266C2×Q8—68no—
267C22×C4—32no—
268C22×C4—34no—
269C22×C4—31no—
270C22×C4—34no—
271C22×C4—64no—
272C22×C4C22×C5×D314yes—
273C22×C4C22×C5×D312yes—
274C22×C4C22×C5×D311yescharacteristic
275C4⋊C4—38no—
276C4⋊C4—316no—
277C4⋊C4—316no—
278C4⋊C4—316no—
279C4⋊C4C22×C5×D318yes—
280C4⋊C4C22×C5×D3116yes—
281C22⋊C4—38no—
282C22⋊C4—316no—
283C42—34no—
284C42—316no—
285C42C22×C5×D314yes—
286C3×C5C2×#54111yescharacteristic, complemented, Hall
287C22×C3C2×C5×#4112yes—
288C22×C3C5×2−1+414yes—
289C22×C3C25×C511yescharacteristic
290C2×D3—22no—
291C2×D3—24no—
292C2×D3—44no—
293C3×C4—28no—
294C3×C4—24no—
295C3×C4C2×C5×#4114yes—
296Dic3—28no—
297Dic3—22no—
298Dic3—24no—
299C2×C5—122no—
300C2×C5#569814yesdirect product factor
301C2×C5C22×#23911yescharacteristic
302C2×C5C2×#101012yes—
303C23—32no—
304C23—62no—
305C23—64no—
306C23C23×C5×D311yescharacteristic
307Q8—616no—
308C2×C4—28no—
309C2×C4—24no—
310C2×C4—34no—
311C2×C4—38no—
312C2×C4—32no—
313C2×C4—62no—
314C2×C4—68no—
315C2×C4—64no—
316C2×C4—68no—
317C2×C4C5×#23914yes—
318C2×C4C23×C5×D318yes—
319C2×C4C23×C5×D314yes—
320C2×C4C23×C5×D312yes—
321C2×C3C5×#54114yescomplemented
322C2×C3C22×C5×#4111yescharacteristic
323C2×C3C2×C5×2−1+412yes—
324D3—42no—
325C5C2×#569811yescharacteristic, direct product factor, Sylow
326C22—62no—
327C22—64no—
328C22—124no—
329C22C2×C5×#23912yes—
330C22C5×#101014yes—
331C22C24×C5×D311yescharacteristic
332C4—28no—
333C4—24no—
334C4—62no—
335C4—68no—
336C4—64no—
337C4C2×C5×#23914yes—
338C3C2×C5×#54111yescharacteristic, complemented, Sylow
339C2—122no—
340C2C5×#569814yesdirect product factor
341C2C22×C5×#23911yescharacteristic
342C2C2×C5×#101012yes—
subgroupG/H
centerC23×C5C23×D3
commutator, cosocleC22×C3C25×C5
FrattiniC22C24×C5×D3
FittingC2×C3×C5×#125C2
socleC23×C3×C5C24
prime radicalC23×C5×D3C23
hypercenterC2×C5×#125D3
hypocenterC3C2×C5×#541
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×#5698C2direct (N × H)
5. C2×C5×#890C2semidirect
6. C2×C5×#887C2semidirect
8. C2×C5×#869C2semidirect
12. C3×C5×#125C22semidirect
26. C5×#890C22semidirect
27. C5×#887C22semidirect
29. C5×#869C22semidirect
30. C2×#5698C5direct (N × H)
35. C2×C5×#125D3semidirect
61. C2×C3×#125C2×C5semidirect
62. #5698C2×C5direct (N × H)
65. C2×#890C2×C5semidirect
66. C2×#887C2×C5semidirect
68. C2×#869C2×C5semidirect
74. C5×#125C2×D3semidirect
110. C3×#125C22×C5semidirect
124. #890C22×C5semidirect
125. #887C22×C5semidirect
127. #869C22×C5semidirect
154. C2×#125C5×D3semidirect
215. #125C2×C5×D3semidirect
230. C2×C3×C5#541semidirect
286. C3×C5C2×#541semidirect
300. C2×C5#5698direct (N × H)
321. C2×C3C5×#541semidirect
325. C5C2×#5698direct (N × H)
338. C3C2×C5×#541semidirect
340. C2C5×#5698direct (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×#56982
C2×C5×#8982
C2×C5×#8972
C2×C5×#8902
C2×C5×#8872
C2×C5×#8722
C2×C5×#8692
C2×C5×#5413
C2×#56985