Finite Group Databasebeta

← C2×C5×#5688 #14480123

Subgroups — C2×C5×#5688 #14480123

Order 1920.

2336 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#125C211yesmaximal, characteristic, complemented
2C5×#5688C2116yesmaximal, direct product factor
3C2×C5×#896C214yesmaximal
4C2×C5×#892C212yesmaximal
5C2×C5×#888C212yesmaximal, complemented
6C2×C5×#887C214yesmaximal, complemented
7C2×C42×C5×D3C211yesmaximal, characteristic
8C2×C5×#870C211yesmaximal, characteristic, complemented
9C2×C5×#533—31nomaximal, Hall
10C22×C4×C5×D3C2212yes—
11C22×C4×C5×D3C2211yescharacteristic
12C22×C5×Dic6C2214yes—
13C3×C5×#125C2218yescomplemented
14C2×C3×C5×C4⋊C4C2214yes—
15C2×C3×C5×C4⋊C4C2212yes—
16C2×C3×C42×C5C2211yescharacteristic
17C2×C5×(C2×D3)⋊C4C2214yes—
18C2×C5×C4⋊Dic3C2212yes—
19C2×C5×Dic3⋊C4C2214yes—
20C2×C5×Dic3⋊C4C2214yes—
21C2×C4×C5×Dic3C2212yes—
22C2×C4×C5×Dic3C2211yescharacteristic
23C2×C4×C5×Dic3C2214yes—
24C5×#896C22132yes—
25C5×#892C22116yes—
26C5×#888C22116yescomplemented
27C5×#887C22132yescomplemented
28C42×C5×D3C2218yes—
29C5×#870C2218yescomplemented
30C2×#5688C511yesmaximal, characteristic, direct product factor, Hall
31C5×#533—316no—
32C2×C5×C4⋊Q8—32no—
33C2×C5×#125—31no—
34C2×C5×#125D311yescharacteristic, complemented
35C2×C5×C22⋊Q8—34no—
36C2×C4×C5×Q8—34no—
37C2×C5×#117—32no—
38C22×C42×C5—31no—
39C23×C5×D3C2311yescharacteristic
40C22×C3×C4×C5C2312yes—
41C22×C3×C4×C5C2314yes—
42C22×C3×C4×C5C2311yescharacteristic
43C22×C5×Dic3C2314yes—
44C22×C5×Dic3C2311yescharacteristic
45C22×C5×Dic3C2312yes—
46C2×C4×C5×D3—28no—
47C2×C4×C5×D3Q818yescomplemented
48C2×C4×C5×D3C2318yes—
49C2×C4×C5×D3C2314yes—
50C2×C5×Dic6—216no—
51C2×C5×Dic6C23116yes—
52C3×C5×C4⋊C4C23116yes—
53C3×C5×C4⋊C4C2318yes—
54C3×C42×C5C2314yes—
55C5×(C2×D3)⋊C4C23116yes—
56C5×C4⋊Dic3C2318yes—
57C5×Dic3⋊C4C23116yes—
58C5×Dic3⋊C4C23116yes—
59C4×C5×Dic3C2318yes—
60C4×C5×Dic3C2314yes—
61C4×C5×Dic3C23116yes—
62C2×C3×#125C2×C511yescharacteristic, complemented
63#5688C2×C5116yesdirect product factor
64C2×#896C2×C514yes—
65C2×#892C2×C512yes—
66C2×#888C2×C512yescomplemented
67C2×#887C2×C514yescomplemented
68C2×C42×D3C2×C511yescharacteristic
69C2×#870C2×C511yescharacteristic, complemented
70C22×C5×Q8—34no—
71C23×C4×C5—31no—
72C23×C4×C5—32no—
73C5×C4⋊Q8—316no—
74C5×#125—38no—
75C5×#125C2×D318yescomplemented
76C5×C22⋊Q8—332no—
77C4×C5×Q8—332no—
78C5×#117—316no—
79C2×C5×C4⋊C4—34no—
80C2×C5×C4⋊C4—34no—
81C2×C5×C4⋊C4—32no—
82C2×C5×C4⋊C4C2×D312yes—
83C2×C5×C4⋊C4C2×D314yes—
84C2×C5×C22⋊C4—34no—
85C2×C42×C5—32no—
86C2×C42×C5—31no—
87C2×C42×C5—34no—
88C2×C42×C5—38no—
89C2×C42×C5C2×D311yescharacteristic
90C2×#533—31noSylow
91C23×C3×C5C2411yescharacteristic
92C22×C5×D3—24no—
93C22×C5×D3C2×Q814yes—
94C22×C5×D3C2412yes—
95C2×C3×C4×C5—28no—
96C2×C3×C4×C5—24no—
97C2×C3×C4×C5C2×Q814yes—
98C2×C3×C4×C5C2414yes—
99C2×C3×C4×C5C2418yes—
100C2×C3×C4×C5C2412yes—
101C2×C5×Dic3—28no—
102C2×C5×Dic3C2×Q814yes—
103C2×C5×Dic3#4118yes—
104C2×C5×Dic3C2418yes—
105C2×C5×Dic3C2412yes—
106C2×C5×Dic3C2414yes—
107C4×C5×D3—216no—
108C4×C5×D3C2×Q8116yescomplemented
109C5×Dic6—232no—
110C22×C4×D3C22×C512yes—
111C22×C4×D3C22×C511yescharacteristic
112C22×Dic6C22×C514yes—
113C3×#125C22×C518yescomplemented
114C2×C3×C4⋊C4C22×C514yes—
115C2×C3×C4⋊C4C22×C512yes—
116C2×C3×C42C22×C511yescharacteristic
117C2×(C2×D3)⋊C4C22×C514yes—
118C2×C4⋊Dic3C22×C512yes—
119C2×Dic3⋊C4C22×C514yes—
120C2×Dic3⋊C4C22×C514yes—
121C2×C4×Dic3C22×C512yes—
122C2×C4×Dic3C22×C511yescharacteristic
123C2×C4×Dic3C22×C514yes—
124#896C22×C5132yes—
125#892C22×C5116yes—
126#888C22×C5116yescomplemented
127#887C22×C5132yescomplemented
128C42×D3C22×C518yes—
129#870C22×C518yescomplemented
130C24×C5—31no—
131C2×C5×Q8—316no—
132C2×C5×Q8—616no—
133C22×C4×C5—31no—
134C22×C4×C5—32no—
135C22×C4×C5—38no—
136C22×C4×C5—34no—
137C22×C4×C5—34no—
138C22×C4×C5—38no—
139C22×C4×C5—68no—
140C22×C4×C5C22×D312yes—
141C22×C4×C5C22×D314yes—
142C22×C4×C5C22×D311yescharacteristic
143C5×C4⋊C4—38no—
144C5×C4⋊C4—316no—
145C5×C4⋊C4—316no—
146C5×C4⋊C4C22×D318yes—
147C5×C4⋊C4C22×D3116yes—
148C5×C22⋊C4—316no—
149C42×C5—38no—
150C42×C5—34no—
151C42×C5—316no—
152C42×C5C22×D314yes—
153#533—316no—
154C2×C4⋊Q8—32no—
155C2×#125—31no—
156C2×#125C5×D311yescharacteristic, complemented
157C2×C22⋊Q8—34no—
158C2×C4×Q8—34no—
159C2×#117—32no—
160C22×C42—31no—
161C22×C3×C5C22×Q812yes—
162C22×C3×C5C2×#4114yes—
163C22×C3×C5C2511yescharacteristic
164C2×C5×D3—28no—
165C2×C5×D3—24no—
166C2×C5×D3C22×Q814yes—
167C3×C4×C5—28no—
168C3×C4×C5—24no—
169C3×C4×C5C22×Q814yes—
170C5×Dic3—28no—
171C5×Dic3C22×Q814yes—
172C5×Dic3C2×#4118yes—
173C23×D3C23×C511yescharacteristic
174C22×C3×C4C23×C512yes—
175C22×C3×C4C23×C514yes—
176C22×C3×C4C23×C511yescharacteristic
177C22×Dic3C23×C514yes—
178C22×Dic3C23×C511yescharacteristic
179C22×Dic3C23×C512yes—
180C2×C4×D3—28no—
181C2×C4×D3C5×Q818yescomplemented
182C2×C4×D3C23×C518yes—
183C2×C4×D3C23×C514yes—
184C2×Dic6—216no—
185C2×Dic6C23×C5116yes—
186C3×C4⋊C4C23×C5116yes—
187C3×C4⋊C4C23×C518yes—
188C3×C42C23×C514yes—
189(C2×D3)⋊C4C23×C5116yes—
190C4⋊Dic3C23×C518yes—
191Dic3⋊C4C23×C5116yes—
192Dic3⋊C4C23×C5116yes—
193C4×Dic3C23×C518yes—
194C4×Dic3C23×C514yes—
195C4×Dic3C23×C5116yes—
196C23×C5—34no—
197C23×C5—32no—
198C23×C5—64no—
199C23×C5C23×D311yescharacteristic
200C5×Q8—632no—
201C2×C4×C5—24no—
202C2×C4×C5—28no—
203C2×C4×C5—34no—
204C2×C4×C5—38no—
205C2×C4×C5—32no—
206C2×C4×C5—34no—
207C2×C4×C5—316no—
208C2×C4×C5—38no—
209C2×C4×C5—616no—
210C2×C4×C5—68no—
211C2×C4×C5D3×Q814yes—
212C2×C4×C5C23×D314yes—
213C2×C4×C5C23×D318yes—
214C2×C4×C5C23×D312yes—
215C22×Q8—34no—
216C23×C4—31no—
217C23×C4—32no—
218C4⋊Q8—316no—
219#125—38no—
220#125C2×C5×D318yescomplemented
221C22⋊Q8—332no—
222C4×Q8—332no—
223#117—316no—
224C2×C4⋊C4—32no—
225C2×C4⋊C4—34no—
226C2×C4⋊C4—34no—
227C2×C4⋊C4C2×C5×D312yes—
228C2×C4⋊C4C2×C5×D314yes—
229C2×C22⋊C4—34no—
230C2×C42—34no—
231C2×C42—32no—
232C2×C42—31no—
233C2×C42—38no—
234C2×C42C2×C5×D311yescharacteristic
235C2×C3×C5#53314yescomplemented
236C2×C3×C5C23×Q811yescharacteristic
237C2×C3×C5C22×#4112yes—
238C5×D3—24no—
239C23×C3C24×C511yescharacteristic
240C22×D3—24no—
241C22×D3C2×C5×Q814yes—
242C22×D3C24×C512yes—
243C2×C3×C4—28no—
244C2×C3×C4—24no—
245C2×C3×C4C2×C5×Q814yes—
246C2×C3×C4C24×C514yes—
247C2×C3×C4C24×C518yes—
248C2×C3×C4C24×C512yes—
249C2×Dic3—28no—
250C2×Dic3C2×C5×Q814yes—
251C2×Dic3C5×#4118yes—
252C2×Dic3C24×C518yes—
253C2×Dic3C24×C512yes—
254C2×Dic3C24×C514yes—
255C4×D3—216no—
256C4×D3C2×C5×Q8116yescomplemented
257Dic6—232no—
258C22×C5—34no—
259C22×C5—68no—
260C22×C5—64no—
261C22×C5C2×D3×Q812yes—
262C22×C5D3×#4114yes—
263C22×C5C24×D311yescharacteristic
264C4×C5—24no—
265C4×C5—28no—
266C4×C5—34no—
267C4×C5—38no—
268C4×C5—68no—
269C4×C5C2×D3×Q814yes—
270C24—31no—
271C2×Q8—316no—
272C2×Q8—616no—
273C22×C4—31no—
274C22×C4—32no—
275C22×C4—38no—
276C22×C4—34no—
277C22×C4—34no—
278C22×C4—38no—
279C22×C4—68no—
280C22×C4C22×C5×D312yes—
281C22×C4C22×C5×D314yes—
282C22×C4C22×C5×D311yescharacteristic
283C4⋊C4—38no—
284C4⋊C4—316no—
285C4⋊C4—316no—
286C4⋊C4C22×C5×D318yes—
287C4⋊C4C22×C5×D3116yes—
288C22⋊C4—316no—
289C42—38no—
290C42—34no—
291C42—316no—
292C42C22×C5×D314yes—
293C3×C5C2×#53311yescharacteristic, complemented, Hall
294C22×C3C22×C5×Q812yes—
295C22×C3C2×C5×#4114yes—
296C22×C3C25×C511yescharacteristic
297C2×D3—28no—
298C2×D3—24no—
299C2×D3C22×C5×Q814yes—
300C3×C4—28no—
301C3×C4—24no—
302C3×C4C22×C5×Q814yes—
303Dic3—28no—
304Dic3C22×C5×Q814yes—
305Dic3C2×C5×#4118yes—
306C2×C5—64no—
307C2×C5#568814yesdirect product factor
308C2×C5C22×D3×Q811yescharacteristic
309C2×C5C2×D3×#4112yes—
310C23—34no—
311C23—32no—
312C23—64no—
313C23C23×C5×D311yescharacteristic
314Q8—632no—
315C2×C4—24no—
316C2×C4—28no—
317C2×C4—34no—
318C2×C4—38no—
319C2×C4—32no—
320C2×C4—34no—
321C2×C4—316no—
322C2×C4—38no—
323C2×C4—616no—
324C2×C4—68no—
325C2×C4C5×D3×Q814yes—
326C2×C4C23×C5×D314yes—
327C2×C4C23×C5×D318yes—
328C2×C4C23×C5×D312yes—
329C2×C3C5×#53314yescomplemented
330C2×C3C23×C5×Q811yescharacteristic
331C2×C3C22×C5×#4112yes—
332D3—24no—
333C5C2×#568811yescharacteristic, direct product factor, Sylow
334C22—34no—
335C22—68no—
336C22—64no—
337C22C2×C5×D3×Q812yes—
338C22C5×D3×#4114yes—
339C22C24×C5×D311yescharacteristic
340C4—24no—
341C4—28no—
342C4—34no—
343C4—38no—
344C4—68no—
345C4C2×C5×D3×Q814yes—
346C3C2×C5×#53311yescharacteristic, complemented, Sylow
347C2—64no—
348C2C5×#568814yesdirect product factor
349C2C22×C5×D3×Q811yescharacteristic
350C2C2×C5×D3×#4112yes—
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×#533
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×#5688C2direct (N × H)
5. C2×C5×#888C2semidirect
6. C2×C5×#887C2semidirect
8. C2×C5×#870C2semidirect
13. C3×C5×#125C22semidirect
26. C5×#888C22semidirect
27. C5×#887C22semidirect
29. C5×#870C22semidirect
30. C2×#5688C5direct (N × H)
34. C2×C5×#125D3semidirect
47. C2×C4×C5×D3Q8semidirect
62. C2×C3×#125C2×C5semidirect
63. #5688C2×C5direct (N × H)
66. C2×#888C2×C5semidirect
67. C2×#887C2×C5semidirect
69. C2×#870C2×C5semidirect
75. C5×#125C2×D3semidirect
108. C4×C5×D3C2×Q8semidirect
113. C3×#125C22×C5semidirect
126. #888C22×C5semidirect
127. #887C22×C5semidirect
129. #870C22×C5semidirect
156. C2×#125C5×D3semidirect
181. C2×C4×D3C5×Q8semidirect
220. #125C2×C5×D3semidirect
235. C2×C3×C5#533semidirect
256. C4×D3C2×C5×Q8semidirect
293. C3×C5C2×#533semidirect
307. C2×C5#5688direct (N × H)
329. C2×C3C5×#533semidirect
333. C5C2×#5688direct (N × H)
346. C3C2×C5×#533semidirect
348. C2C5×#5688direct (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×#56882
C2×C5×#8962
C2×C5×#8922
C2×C5×#8882
C2×C5×#8872
C2×C42×C5×D32
C2×C5×#8702
C2×C5×#5333
C2×#56885