Finite Group Databasebeta

← C2×C5×#5730 #14483808

Subgroups — C2×C5×#5730 #14483808

Order 1920.

2400 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×C5×#946C214yesmaximal, complemented
3C5×#5730C2116yesmaximal, direct product factor
4C2×C5×#897C212yesmaximal, complemented
5C2×C5×#893C212yesmaximal
6C2×C5×D3×C4⋊C4C212yesmaximal, complemented
7C2×C5×#890C212yesmaximal, complemented
8C2×C4×C5×D12C211yesmaximal, characteristic
9C2×C4×C5×Dic6C211yesmaximal, characteristic, complemented
10C2×C5×#554—31nomaximal, Hall
11C22×C3×C5×Q8C2212yes—
12C22×C5×D12C2211yescharacteristic
13C22×C4×C5×D3C2212yes—
14C22×C4×C5×D3C2214yes—
15C22×C5×Dic6C2211yescharacteristic
16C3×C5×C4⋊Q8C2218yescomplemented
17C2×C3×C5×C4⋊C4C2212yes—
18C2×C3×C5×C4⋊C4C2212yes—
19C2×C3×C42×C5C2211yescharacteristic
20C5×#946C22132yescomplemented
21C2×C5×(C2×D3)⋊C4C2212yes—
22C2×C5×(C2×D3)⋊C4C2214yes—
23C2×C5×C4⋊Dic3C2211yescharacteristic
24C2×C5×C4⋊Dic3C2211yescharacteristic
25C2×C5×C4⋊Dic3C2214yes—
26C2×C5×Dic3⋊C4C2212yes—
27C2×C5×Dic3⋊C4C2214yes—
28C2×C4×C5×Dic3C2212yes—
29C5×#897C22116yescomplemented
30C5×#893C22116yes—
31C5×D3×C4⋊C4C22116yescomplemented
32C5×#890C22116yescomplemented
33C4×C5×D12C2218yes—
34C4×C5×Dic6C2218yescomplemented
35C2×#5730C511yesmaximal, characteristic, direct product factor, Hall
36C5×#554—316no—
37C2×C5×C4⋊Q8D311yescharacteristic, complemented
38C2×C5×#125—32no—
39C2×C5×C22⋊Q8—34no—
40C2×C5×C22⋊Q8—32no—
41C2×C4×C5×Q8—31no—
42C2×C4×C5×D4—32no—
43C2×C4×C5×D4—31no—
44C22×C5×C4⋊C4—32no—
45C23×C5×D3C2312yes—
46C2×C3×C5×Q8—44no—
47C2×C3×C5×Q8C2318yes—
48C22×C3×C4×C5C2311yescharacteristic
49C22×C3×C4×C5C2314yes—
50C22×C3×C4×C5C2311yescharacteristic
51C22×C3×C4×C5C2311yescharacteristic
52C22×C5×Dic3C2312yes—
53C22×C5×Dic3C2312yes—
54C22×C5×Dic3C2312yes—
55C2×C5×D12Q818yescomplemented
56C2×C5×D12C2314yes—
57C2×C4×C5×D3—28no—
58C2×C4×C5×D3—48no—
59C2×C4×C5×D3C2318yes—
60C2×C4×C5×D3C23116yes—
61C2×C5×Dic6—24no—
62C2×C5×Dic6C2314yes—
63C3×C5×C4⋊C4C2318yes—
64C3×C5×C4⋊C4C2318yes—
65C3×C42×C5C2314yes—
66C5×(C2×D3)⋊C4C2318yes—
67C5×(C2×D3)⋊C4C23116yes—
68C5×C4⋊Dic3C2314yes—
69C5×C4⋊Dic3C2314yes—
70C5×C4⋊Dic3C23116yes—
71C5×Dic3⋊C4C2318yes—
72C5×Dic3⋊C4C23116yes—
73C4×C5×Dic3C2318yes—
74C2×C3×C4⋊Q8C2×C511yescharacteristic, complemented
75C2×#946C2×C514yescomplemented
76#5730C2×C5116yesdirect product factor
77C2×#897C2×C512yescomplemented
78C2×#893C2×C512yes—
79C2×D3×C4⋊C4C2×C512yescomplemented
80C2×#890C2×C512yescomplemented
81C2×C4×D12C2×C511yescharacteristic
82C2×C4×Dic6C2×C511yescharacteristic, complemented
83C22×C5×Q8—31no—
84C22×C5×Q8C2×D312yes—
85C22×C5×D4—31no—
86C23×C4×C5—32no—
87C23×C4×C5—34no—
88C5×C4⋊Q8C2×D318yescomplemented
89C5×#125—316no—
90C5×C22⋊Q8—332no—
91C5×C22⋊Q8—316no—
92C4×C5×Q8—38no—
93C4×C5×D4—316no—
94C4×C5×D4—38no—
95C2×C5×C4⋊C4—34no—
96C2×C5×C4⋊C4—32no—
97C2×C5×C4⋊C4—31no—
98C2×C5×C4⋊C4—31no—
99C2×C5×C4⋊C4—34no—
100C2×C5×C4⋊C4—316no—
101C2×C5×C4⋊C4C2×D312yes—
102C2×C5×C4⋊C4C2×D312yes—
103C2×C5×C22⋊C4—34no—
104C2×C5×C22⋊C4—32no—
105C2×C42×C5—32no—
106C2×C42×C5C2×D311yescharacteristic
107C2×#554—31noSylow
108C23×C3×C5C2411yescharacteristic
109C22×C5×D3—24no—
110C22×C5×D3—24no—
111C22×C5×D3C2×Q818yes—
112C22×C5×D3C2414yes—
113C3×C5×Q8—48no—
114C2×C3×C4×C5—28no—
115C2×C3×C4×C5—22no—
116C2×C3×C4×C5C2×Q814yes—
117C2×C3×C4×C5#4114yes—
118C2×C3×C4×C5C2412yes—
119C2×C3×C4×C5C2418yes—
120C2×C3×C4×C5C2412yes—
121C2×C3×C4×C5C2412yes—
122C2×C5×Dic3—24no—
123C2×C5×Dic3—24no—
124C2×C5×Dic3—24no—
125C2×C5×Dic3C2414yes—
126C2×C5×Dic3C2414yes—
127C2×C5×Dic3C2414yes—
128C5×D12C2×Q8116yescomplemented
129C4×C5×D3—216no—
130C4×C5×D3—416no—
131C5×Dic6—28no—
132C22×C3×Q8C22×C512yes—
133C22×D12C22×C511yescharacteristic
134C22×C4×D3C22×C512yes—
135C22×C4×D3C22×C514yes—
136C22×Dic6C22×C511yescharacteristic
137C3×C4⋊Q8C22×C518yescomplemented
138C2×C3×C4⋊C4C22×C512yes—
139C2×C3×C4⋊C4C22×C512yes—
140C2×C3×C42C22×C511yescharacteristic
141#946C22×C5132yescomplemented
142C2×(C2×D3)⋊C4C22×C512yes—
143C2×(C2×D3)⋊C4C22×C514yes—
144C2×C4⋊Dic3C22×C511yescharacteristic
145C2×C4⋊Dic3C22×C511yescharacteristic
146C2×C4⋊Dic3C22×C514yes—
147C2×Dic3⋊C4C22×C512yes—
148C2×Dic3⋊C4C22×C514yes—
149C2×C4×Dic3C22×C512yes—
150#897C22×C5116yescomplemented
151#893C22×C5116yes—
152D3×C4⋊C4C22×C5116yescomplemented
153#890C22×C5116yescomplemented
154C4×D12C22×C518yes—
155C4×Dic6C22×C518yescomplemented
156C24×C5—32no—
157C2×C5×Q8—34no—
158C2×C5×Q8—44no—
159C2×C5×Q8—64no—
160C2×C5×Q8C22×D318yes—
161C2×C5×D4—38no—
162C2×C5×D4—34no—
163C22×C4×C5—32no—
164C22×C4×C5—32no—
165C22×C4×C5—32no—
166C22×C4×C5—38no—
167C22×C4×C5—316no—
168C22×C4×C5—68no—
169C22×C4×C5—128no—
170C22×C4×C5C22×D311yescharacteristic
171C22×C4×C5C22×D314yes—
172C22×C4×C5C22×D311yescharacteristic
173C22×C4×C5C22×D311yescharacteristic
174C5×C4⋊C4—34no—
175C5×C4⋊C4—316no—
176C5×C4⋊C4—34no—
177C5×C4⋊C4—38no—
178C5×C4⋊C4—316no—
179C5×C4⋊C4C22×D318yes—
180C5×C4⋊C4C22×D318yes—
181C5×C22⋊C4—316no—
182C5×C22⋊C4—38no—
183C42×C5—38no—
184C42×C5C22×D314yes—
185#554—316no—
186C2×C4⋊Q8C5×D311yescharacteristic, complemented
187C2×#125—32no—
188C2×C22⋊Q8—34no—
189C2×C22⋊Q8—32no—
190C2×C4×Q8—31no—
191C2×C4×D4—32no—
192C2×C4×D4—31no—
193C22×C4⋊C4—32no—
194C22×C3×C5C22×Q812yes—
195C22×C3×C5C2×#4112yes—
196C22×C3×C5H(5,𝔽2)12yes—
197C22×C3×C5C2511yescharacteristic
198C2×C5×D3—216no—
199C2×C5×D3—24no—
200C2×C5×D3—24no—
201C2×C5×D3C22×Q818yes—
202C3×C4×C5—28no—
203C3×C4×C5—22no—
204C3×C4×C5C22×Q814yes—
205C3×C4×C5C2×#4114yes—
206C5×Dic3—24no—
207C5×Dic3—24no—
208C5×Dic3—24no—
209C23×D3C23×C512yes—
210C2×C3×Q8—44no—
211C2×C3×Q8C23×C518yes—
212C22×C3×C4C23×C511yescharacteristic
213C22×C3×C4C23×C514yes—
214C22×C3×C4C23×C511yescharacteristic
215C22×C3×C4C23×C511yescharacteristic
216C22×Dic3C23×C512yes—
217C22×Dic3C23×C512yes—
218C22×Dic3C23×C512yes—
219C2×D12C5×Q818yescomplemented
220C2×D12C23×C514yes—
221C2×C4×D3—28no—
222C2×C4×D3—48no—
223C2×C4×D3C23×C518yes—
224C2×C4×D3C23×C5116yes—
225C2×Dic6—24no—
226C2×Dic6C23×C514yes—
227C3×C4⋊C4C23×C518yes—
228C3×C4⋊C4C23×C518yes—
229C3×C42C23×C514yes—
230(C2×D3)⋊C4C23×C518yes—
231(C2×D3)⋊C4C23×C5116yes—
232C4⋊Dic3C23×C514yes—
233C4⋊Dic3C23×C514yes—
234C4⋊Dic3C23×C5116yes—
235Dic3⋊C4C23×C518yes—
236Dic3⋊C4C23×C5116yes—
237C4×Dic3C23×C518yes—
238C23×C5—38no—
239C23×C5—34no—
240C23×C5—64no—
241C23×C5—64no—
242C23×C5C23×D311yescharacteristic
243C5×Q8—48no—
244C5×Q8—68no—
245C5×D4—316no—
246C2×C4×C5—28no—
247C2×C4×C5—22no—
248C2×C4×C5—34no—
249C2×C4×C5—34no—
250C2×C4×C5—34no—
251C2×C4×C5—64no—
252C2×C4×C5—64no—
253C2×C4×C5—64no—
254C2×C4×C5—616no—
255C2×C4×C5—1216no—
256C2×C4×C5D3×Q814yes—
257C2×C4×C5#23914yes—
258C2×C4×C5C23×D312yes—
259C2×C4×C5C23×D318yes—
260C2×C4×C5C23×D312yes—
261C2×C4×C5C23×D312yes—
262C22×Q8—31no—
263C22×Q8C2×C5×D312yes—
264C22×D4—31no—
265C23×C4—32no—
266C23×C4—34no—
267C4⋊Q8C2×C5×D318yescomplemented
268#125—316no—
269C22⋊Q8—332no—
270C22⋊Q8—316no—
271C4×Q8—38no—
272C4×D4—316no—
273C4×D4—38no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—34no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—32no—
278C2×C4⋊C4—34no—
279C2×C4⋊C4—316no—
280C2×C4⋊C4C2×C5×D312yes—
281C2×C4⋊C4C2×C5×D312yes—
282C2×C22⋊C4—34no—
283C2×C22⋊C4—32no—
284C2×C42—32no—
285C2×C42C2×C5×D311yescharacteristic
286C2×C3×C5#55414yescomplemented
287C2×C3×C5C23×Q811yescharacteristic
288C2×C3×C5C22×#4111yescharacteristic
289C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
290C5×D3—28no—
291C23×C3C24×C511yescharacteristic
292C22×D3—24no—
293C22×D3—24no—
294C22×D3C2×C5×Q818yes—
295C22×D3C24×C514yes—
296C3×Q8—48no—
297C2×C3×C4—28no—
298C2×C3×C4—22no—
299C2×C3×C4C2×C5×Q814yes—
300C2×C3×C4C5×#4114yes—
301C2×C3×C4C24×C512yes—
302C2×C3×C4C24×C518yes—
303C2×C3×C4C24×C512yes—
304C2×C3×C4C24×C512yes—
305C2×Dic3—24no—
306C2×Dic3—24no—
307C2×Dic3—24no—
308C2×Dic3C24×C514yes—
309C2×Dic3C24×C514yes—
310C2×Dic3C24×C514yes—
311D12C2×C5×Q8116yescomplemented
312C4×D3—216no—
313C4×D3—416no—
314Dic6—28no—
315C22×C5—38no—
316C22×C5—64no—
317C22×C5—64no—
318C22×C5—616no—
319C22×C5#100412yes—
320C22×C5C2×D3×Q812yes—
321C22×C5C2×#23912yes—
322C22×C5C24×D311yescharacteristic
323C4×C5—28no—
324C4×C5—22no—
325C4×C5—64no—
326C4×C5—64no—
327C4×C5—64no—
328C4×C5C2×D3×Q814yes—
329C4×C5C2×#23914yes—
330C24—32no—
331C2×Q8—34no—
332C2×Q8—44no—
333C2×Q8—64no—
334C2×Q8C22×C5×D318yes—
335C2×D4—38no—
336C2×D4—34no—
337C22×C4—32no—
338C22×C4—32no—
339C22×C4—32no—
340C22×C4—38no—
341C22×C4—316no—
342C22×C4—68no—
343C22×C4—128no—
344C22×C4C22×C5×D311yescharacteristic
345C22×C4C22×C5×D314yes—
346C22×C4C22×C5×D311yescharacteristic
347C22×C4C22×C5×D311yescharacteristic
348C4⋊C4—34no—
349C4⋊C4—316no—
350C4⋊C4—34no—
351C4⋊C4—38no—
352C4⋊C4—316no—
353C4⋊C4C22×C5×D318yes—
354C4⋊C4C22×C5×D318yes—
355C22⋊C4—316no—
356C22⋊C4—38no—
357C42—38no—
358C42C22×C5×D314yes—
359C3×C5C2×#55411yescharacteristic, complemented, Hall
360C22×C3C22×C5×Q812yes—
361C22×C3C2×C5×#4112yes—
362C22×C3C5×H(5,𝔽2)12yes—
363C22×C3C25×C511yescharacteristic
364C2×D3—216no—
365C2×D3—24no—
366C2×D3—24no—
367C2×D3C22×C5×Q818yes—
368C3×C4—28no—
369C3×C4—22no—
370C3×C4C22×C5×Q814yes—
371C3×C4C2×C5×#4114yes—
372Dic3—24no—
373Dic3—24no—
374Dic3—24no—
375C2×C5—68no—
376C2×C5#573014yesdirect product factor
377C2×C5C2×#100411yescharacteristic
378C2×C5C22×D3×Q811yescharacteristic
379C2×C5C22×#23911yescharacteristic
380C23—38no—
381C23—34no—
382C23—64no—
383C23—64no—
384C23C23×C5×D311yescharacteristic
385Q8—48no—
386Q8—68no—
387D4—316no—
388C2×C4—28no—
389C2×C4—22no—
390C2×C4—34no—
391C2×C4—34no—
392C2×C4—34no—
393C2×C4—64no—
394C2×C4—64no—
395C2×C4—64no—
396C2×C4—616no—
397C2×C4—1216no—
398C2×C4C5×D3×Q814yes—
399C2×C4C5×#23914yes—
400C2×C4C23×C5×D312yes—
401C2×C4C23×C5×D318yes—
402C2×C4C23×C5×D312yes—
403C2×C4C23×C5×D312yes—
404C2×C3C5×#55414yescomplemented
405C2×C3C23×C5×Q811yescharacteristic
406C2×C3C22×C5×#4111yescharacteristic
407C2×C3C2×C5×H(5,𝔽2)11yescharacteristic
408D3—28no—
409C5C2×#573011yescharacteristic, direct product factor, Sylow
410C22—38no—
411C22—64no—
412C22—64no—
413C22—616no—
414C22C5×#100412yes—
415C22C2×C5×D3×Q812yes—
416C22C2×C5×#23912yes—
417C22C24×C5×D311yescharacteristic
418C4—28no—
419C4—22no—
420C4—64no—
421C4—64no—
422C4—64no—
423C4C2×C5×D3×Q814yes—
424C4C2×C5×#23914yes—
425C3C2×C5×#55411yescharacteristic, complemented, Sylow
426C2—68no—
427C2C5×#573014yesdirect product factor
428C2C2×C5×#100411yescharacteristic
429C2C22×C5×D3×Q811yescharacteristic
430C2C22×C5×#23911yescharacteristic
subgroupG/H
centerC23×C5C23×D3
commutator, cosocleC22×C3C25×C5
FrattiniC22C24×C5×D3
FittingC2×C3×C5×C4⋊Q8C2
socleC23×C3×C5C24
prime radicalC22×C5×D12C22
hypercenterC2×C5×C4⋊Q8D3
hypocenterC3C2×C5×#554
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C5 ◁ C2×C5×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×C4⋊Q8C2semidirect
2. C2×C5×#946C2semidirect
3. C5×#5730C2direct (N × H)
4. C2×C5×#897C2semidirect
6. C2×C5×D3×C4⋊C4C2semidirect
7. C2×C5×#890C2semidirect
9. C2×C4×C5×Dic6C2semidirect
16. C3×C5×C4⋊Q8C22semidirect
20. C5×#946C22semidirect
29. C5×#897C22semidirect
31. C5×D3×C4⋊C4C22semidirect
32. C5×#890C22semidirect
34. C4×C5×Dic6C22semidirect
35. C2×#5730C5direct (N × H)
37. C2×C5×C4⋊Q8D3semidirect
55. C2×C5×D12Q8semidirect
74. C2×C3×C4⋊Q8C2×C5semidirect
75. C2×#946C2×C5semidirect
76. #5730C2×C5direct (N × H)
77. C2×#897C2×C5semidirect
79. C2×D3×C4⋊C4C2×C5semidirect
80. C2×#890C2×C5semidirect
82. C2×C4×Dic6C2×C5semidirect
88. C5×C4⋊Q8C2×D3semidirect
128. C5×D12C2×Q8semidirect
137. C3×C4⋊Q8C22×C5semidirect
141. #946C22×C5semidirect
150. #897C22×C5semidirect
152. D3×C4⋊C4C22×C5semidirect
153. #890C22×C5semidirect
155. C4×Dic6C22×C5semidirect
186. C2×C4⋊Q8C5×D3semidirect
219. C2×D12C5×Q8semidirect
267. C4⋊Q8C2×C5×D3semidirect
286. C2×C3×C5#554semidirect
311. D12C2×C5×Q8semidirect
359. C3×C5C2×#554semidirect
376. C2×C5#5730direct (N × H)
404. C2×C3C5×#554semidirect
409. C5C2×#5730direct (N × H)
425. C3C2×C5×#554semidirect
427. C2C5×#5730direct (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×C4⋊Q82
C2×C5×#9462
C5×#57302
C2×C5×#8972
C2×C5×#8932
C2×C5×D3×C4⋊C42
C2×C5×#8902
C2×C4×C5×D122
C2×C4×C5×Dic62
C2×C5×#5543
C2×#57305