Finite Group Databasebeta

← C2×C5×#5732 #14484006

Subgroups — C2×C5×#5732 #14484006

Order 1920.

2224 conjugacy classes of subgroups.

The subgroup lattice has 424 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×#947C212yesmaximal
3C2×C5×Q8×Dic3C212yesmaximal, complemented
4C5×#5732C2116yesmaximal, direct product factor
5C2×C5×#898C214yesmaximal, complemented
6C2×C5×#897C212yesmaximal, complemented
7C2×C5×#892C212yesmaximal, complemented
8C2×C4×C5×D12C211yesmaximal, characteristic
9C2×C4×C5×Dic6C211yesmaximal, characteristic, complemented
10C2×C5×#556—31nomaximal, Hall
11C22×C3×C5×Q8C2212yes—
12C22×C5×D12C2211yescharacteristic
13C22×C4×C5×D3C2212yes—
14C22×C5×Dic6C2211yescharacteristic
15C3×C5×C4⋊Q8C2218yescomplemented
16C2×C3×C5×C4⋊C4C2212yes—
17C2×C3×C5×C4⋊C4C2212yes—
18C2×C3×C42×C5C2211yescharacteristic
19C5×#947C22116yes—
20C5×Q8×Dic3C22116yescomplemented
21C2×C5×(C2×D3)⋊C4C2212yes—
22C2×C5×(C2×D3)⋊C4C2214yes—
23C2×C5×(C2×D3)⋊C4C2214yes—
24C2×C5×C4⋊Dic3C2211yescharacteristic
25C2×C5×C4⋊Dic3C2211yescharacteristic
26C2×C5×C4⋊Dic3C2214yes—
27C2×C5×Dic3⋊C4C2212yes—
28C2×C4×C5×Dic3C2212yes—
29C2×C4×C5×Dic3C2214yes—
30C5×#898C22132yescomplemented
31C5×#897C22116yescomplemented
32C5×#892C22116yescomplemented
33C4×C5×D12C2218yes—
34C4×C5×Dic6C2218yescomplemented
35C2×#5732C511yesmaximal, characteristic, direct product factor, Hall
36C5×#556—316no—
37C2×C5×C4⋊Q8D311yescharacteristic, complemented
38C2×C5×#126—34no—
39C2×C5×#124—32no—
40C2×C5×C22⋊Q8—32no—
41C2×C4×C5×Q8—32no—
42C2×C4×C5×Q8—31no—
43C2×C4×C5×D4—31no—
44C2×C5×#117—32no—
45C23×C5×D3C2312yes—
46C2×C3×C5×Q8—28no—
47C2×C3×C5×Q8C2318yes—
48C22×C3×C4×C5C2314yes—
49C22×C3×C4×C5C2311yescharacteristic
50C22×C3×C4×C5C2311yescharacteristic
51C22×C3×C4×C5C2311yescharacteristic
52C22×C5×Dic3C2312yes—
53C22×C5×Dic3C2312yes—
54C22×C5×Dic3C2312yes—
55C2×C5×D12—24no—
56C2×C5×D12C2314yes—
57C2×C4×C5×D3—28no—
58C2×C4×C5×D3C2318yes—
59C2×C5×Dic6—24no—
60C2×C5×Dic6C2314yes—
61C3×C5×C4⋊C4C2318yes—
62C3×C5×C4⋊C4C2318yes—
63C3×C42×C5C2314yes—
64C5×(C2×D3)⋊C4C2318yes—
65C5×(C2×D3)⋊C4C23116yes—
66C5×(C2×D3)⋊C4C23116yes—
67C5×C4⋊Dic3C2314yes—
68C5×C4⋊Dic3C2314yes—
69C5×C4⋊Dic3C23116yes—
70C5×Dic3⋊C4C2318yes—
71C4×C5×Dic3C2318yes—
72C4×C5×Dic3C23116yes—
73C2×C3×C4⋊Q8C2×C511yescharacteristic, complemented
74C2×#947C2×C512yes—
75C2×Q8×Dic3C2×C512yescomplemented
76#5732C2×C5116yesdirect product factor
77C2×#898C2×C514yescomplemented
78C2×#897C2×C512yescomplemented
79C2×#892C2×C512yescomplemented
80C2×C4×D12C2×C511yescharacteristic
81C2×C4×Dic6C2×C511yescharacteristic, complemented
82C22×C5×Q8—31no—
83C22×C5×Q8C2×D312yes—
84C22×C5×D4—31no—
85C23×C4×C5—32no—
86C5×C4⋊Q8C2×D318yescomplemented
87C5×#126—332no—
88C5×#124—316no—
89C5×C22⋊Q8—316no—
90C4×C5×Q8—316no—
91C4×C5×Q8—38no—
92C4×C5×D4—38no—
93C5×#117—316no—
94C2×C5×C4⋊C4—32no—
95C2×C5×C4⋊C4—31no—
96C2×C5×C4⋊C4—31no—
97C2×C5×C4⋊C4—34no—
98C2×C5×C4⋊C4C2×D312yes—
99C2×C5×C4⋊C4C2×D312yes—
100C2×C5×C22⋊C4—34no—
101C2×C5×C22⋊C4—32no—
102C2×C5×C22⋊C4—34no—
103C2×C42×C5—34no—
104C2×C42×C5—32no—
105C2×C42×C5C2×D311yescharacteristic
106C2×#556—31noSylow
107C23×C3×C5C2411yescharacteristic
108C22×C5×D3—24no—
109C22×C5×D3—24no—
110C22×C5×D3—24no—
111C22×C5×D3C2414yes—
112C3×C5×Q8—216no—
113C2×C3×C4×C5—28no—
114C2×C3×C4×C5—22no—
115C2×C3×C4×C5#4114yes—
116C2×C3×C4×C5#4114yes—
117C2×C3×C4×C5C2418yes—
118C2×C3×C4×C5C2412yes—
119C2×C3×C4×C5C2412yes—
120C2×C3×C4×C5C2412yes—
121C2×C5×Dic3—24no—
122C2×C5×Dic3—24no—
123C2×C5×Dic3—24no—
124C2×C5×Dic3C2414yes—
125C2×C5×Dic3C2414yes—
126C2×C5×Dic3C2414yes—
127C5×D12—28no—
128C4×C5×D3—216no—
129C5×Dic6—28no—
130C22×C3×Q8C22×C512yes—
131C22×D12C22×C511yescharacteristic
132C22×C4×D3C22×C512yes—
133C22×Dic6C22×C511yescharacteristic
134C3×C4⋊Q8C22×C518yescomplemented
135C2×C3×C4⋊C4C22×C512yes—
136C2×C3×C4⋊C4C22×C512yes—
137C2×C3×C42C22×C511yescharacteristic
138#947C22×C5116yes—
139Q8×Dic3C22×C5116yescomplemented
140C2×(C2×D3)⋊C4C22×C512yes—
141C2×(C2×D3)⋊C4C22×C514yes—
142C2×(C2×D3)⋊C4C22×C514yes—
143C2×C4⋊Dic3C22×C511yescharacteristic
144C2×C4⋊Dic3C22×C511yescharacteristic
145C2×C4⋊Dic3C22×C514yes—
146C2×Dic3⋊C4C22×C512yes—
147C2×C4×Dic3C22×C512yes—
148C2×C4×Dic3C22×C514yes—
149#898C22×C5132yescomplemented
150#897C22×C5116yescomplemented
151#892C22×C5116yescomplemented
152C4×D12C22×C518yes—
153C4×Dic6C22×C518yescomplemented
154C24×C5—32no—
155C2×C5×Q8—28no—
156C2×C5×Q8—34no—
157C2×C5×Q8—64no—
158C2×C5×Q8C22×D318yes—
159C2×C5×D4—34no—
160C2×C5×D4—64no—
161C22×C4×C5—32no—
162C22×C4×C5—32no—
163C22×C4×C5—32no—
164C22×C4×C5—38no—
165C22×C4×C5—68no—
166C22×C4×C5C22×D314yes—
167C22×C4×C5C22×D311yescharacteristic
168C22×C4×C5C22×D311yescharacteristic
169C22×C4×C5C22×D311yescharacteristic
170C5×C4⋊C4—34no—
171C5×C4⋊C4—34no—
172C5×C4⋊C4—316no—
173C5×C4⋊C4—38no—
174C5×C4⋊C4C22×D318yes—
175C5×C4⋊C4C22×D318yes—
176C5×C22⋊C4—316no—
177C5×C22⋊C4—38no—
178C5×C22⋊C4—316no—
179C42×C5—316no—
180C42×C5—38no—
181C42×C5C22×D314yes—
182#556—316no—
183C2×C4⋊Q8C5×D311yescharacteristic, complemented
184C2×#126—34no—
185C2×#124—32no—
186C2×C22⋊Q8—32no—
187C2×C4×Q8—32no—
188C2×C4×Q8—31no—
189C2×C4×D4—31no—
190C2×#117—32no—
191C22×C3×C5C2×#4112yes—
192C22×C3×C5C2×#4112yes—
193C22×C3×C52−1+412yes—
194C22×C3×C5C2511yescharacteristic
195C2×C5×D3—24no—
196C2×C5×D3—24no—
197C2×C5×D3—24no—
198C2×C5×D3—48no—
199C3×C4×C5—28no—
200C3×C4×C5—22no—
201C3×C4×C5C2×#4114yes—
202C3×C4×C5C2×#4114yes—
203C5×Dic3—24no—
204C5×Dic3—24no—
205C5×Dic3—24no—
206C23×D3C23×C512yes—
207C2×C3×Q8—28no—
208C2×C3×Q8C23×C518yes—
209C22×C3×C4C23×C514yes—
210C22×C3×C4C23×C511yescharacteristic
211C22×C3×C4C23×C511yescharacteristic
212C22×C3×C4C23×C511yescharacteristic
213C22×Dic3C23×C512yes—
214C22×Dic3C23×C512yes—
215C22×Dic3C23×C512yes—
216C2×D12—24no—
217C2×D12C23×C514yes—
218C2×C4×D3—28no—
219C2×C4×D3C23×C518yes—
220C2×Dic6—24no—
221C2×Dic6C23×C514yes—
222C3×C4⋊C4C23×C518yes—
223C3×C4⋊C4C23×C518yes—
224C3×C42C23×C514yes—
225(C2×D3)⋊C4C23×C518yes—
226(C2×D3)⋊C4C23×C5116yes—
227(C2×D3)⋊C4C23×C5116yes—
228C4⋊Dic3C23×C514yes—
229C4⋊Dic3C23×C514yes—
230C4⋊Dic3C23×C5116yes—
231Dic3⋊C4C23×C518yes—
232C4×Dic3C23×C518yes—
233C4×Dic3C23×C5116yes—
234C23×C5—34no—
235C23×C5—64no—
236C23×C5—64no—
237C23×C5—64no—
238C23×C5C23×D311yescharacteristic
239C5×Q8—216no—
240C5×Q8—68no—
241C5×D4—68no—
242C2×C4×C5—28no—
243C2×C4×C5—22no—
244C2×C4×C5—34no—
245C2×C4×C5—34no—
246C2×C4×C5—34no—
247C2×C4×C5—64no—
248C2×C4×C5—64no—
249C2×C4×C5—64no—
250C2×C4×C5—616no—
251C2×C4×C5#23714yes—
252C2×C4×C5#23914yes—
253C2×C4×C5C23×D318yes—
254C2×C4×C5C23×D312yes—
255C2×C4×C5C23×D312yes—
256C2×C4×C5C23×D312yes—
257C22×Q8—31no—
258C22×Q8C2×C5×D312yes—
259C22×D4—31no—
260C23×C4—32no—
261C4⋊Q8C2×C5×D318yescomplemented
262#126—332no—
263#124—316no—
264C22⋊Q8—316no—
265C4×Q8—316no—
266C4×Q8—38no—
267C4×D4—38no—
268#117—316no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—34no—
272C2×C4⋊C4—32no—
273C2×C4⋊C4C2×C5×D312yes—
274C2×C4⋊C4C2×C5×D312yes—
275C2×C22⋊C4—34no—
276C2×C22⋊C4—32no—
277C2×C22⋊C4—34no—
278C2×C42—34no—
279C2×C42—32no—
280C2×C42C2×C5×D311yescharacteristic
281C2×C3×C5#55614yescomplemented
282C2×C3×C5C22×#4111yescharacteristic
283C2×C3×C5C22×#4111yescharacteristic
284C2×C3×C5C2×2−1+411yescharacteristic
285C5×D3—44no—
286C23×C3C24×C511yescharacteristic
287C22×D3—24no—
288C22×D3—24no—
289C22×D3—24no—
290C22×D3C24×C514yes—
291C3×Q8—216no—
292C2×C3×C4—28no—
293C2×C3×C4—22no—
294C2×C3×C4C5×#4114yes—
295C2×C3×C4C5×#4114yes—
296C2×C3×C4C24×C518yes—
297C2×C3×C4C24×C512yes—
298C2×C3×C4C24×C512yes—
299C2×C3×C4C24×C512yes—
300C2×Dic3—24no—
301C2×Dic3—24no—
302C2×Dic3—24no—
303C2×Dic3C24×C514yes—
304C2×Dic3C24×C514yes—
305C2×Dic3C24×C514yes—
306D12—28no—
307C4×D3—216no—
308Dic6—28no—
309C22×C5—64no—
310C22×C5—64no—
311C22×C5—64no—
312C22×C5—128no—
313C22×C5C2×#23712yes—
314C22×C5C2×#23912yes—
315C22×C5#100712yes—
316C22×C5C24×D311yescharacteristic
317C4×C5—28no—
318C4×C5—22no—
319C4×C5—64no—
320C4×C5—64no—
321C4×C5—64no—
322C4×C5C2×#23714yes—
323C4×C5C2×#23914yes—
324C24—32no—
325C2×Q8—28no—
326C2×Q8—34no—
327C2×Q8—64no—
328C2×Q8C22×C5×D318yes—
329C2×D4—34no—
330C2×D4—64no—
331C22×C4—32no—
332C22×C4—32no—
333C22×C4—32no—
334C22×C4—38no—
335C22×C4—68no—
336C22×C4C22×C5×D314yes—
337C22×C4C22×C5×D311yescharacteristic
338C22×C4C22×C5×D311yescharacteristic
339C22×C4C22×C5×D311yescharacteristic
340C4⋊C4—34no—
341C4⋊C4—34no—
342C4⋊C4—316no—
343C4⋊C4—38no—
344C4⋊C4C22×C5×D318yes—
345C4⋊C4C22×C5×D318yes—
346C22⋊C4—316no—
347C22⋊C4—38no—
348C22⋊C4—316no—
349C42—316no—
350C42—38no—
351C42C22×C5×D314yes—
352C3×C5C2×#55611yescharacteristic, complemented, Hall
353C22×C3C2×C5×#4112yes—
354C22×C3C2×C5×#4112yes—
355C22×C3C5×2−1+412yes—
356C22×C3C25×C511yescharacteristic
357C2×D3—24no—
358C2×D3—24no—
359C2×D3—24no—
360C2×D3—48no—
361C3×C4—28no—
362C3×C4—22no—
363C3×C4C2×C5×#4114yes—
364C3×C4C2×C5×#4114yes—
365Dic3—24no—
366Dic3—24no—
367Dic3—24no—
368C2×C5—124no—
369C2×C5#573214yesdirect product factor
370C2×C5C22×#23711yescharacteristic
371C2×C5C22×#23911yescharacteristic
372C2×C5C2×#100711yescharacteristic
373C23—34no—
374C23—64no—
375C23—64no—
376C23—64no—
377C23C23×C5×D311yescharacteristic
378Q8—216no—
379Q8—68no—
380D4—68no—
381C2×C4—28no—
382C2×C4—22no—
383C2×C4—34no—
384C2×C4—34no—
385C2×C4—34no—
386C2×C4—64no—
387C2×C4—64no—
388C2×C4—64no—
389C2×C4—616no—
390C2×C4C5×#23714yes—
391C2×C4C5×#23914yes—
392C2×C4C23×C5×D318yes—
393C2×C4C23×C5×D312yes—
394C2×C4C23×C5×D312yes—
395C2×C4C23×C5×D312yes—
396C2×C3C5×#55614yescomplemented
397C2×C3C22×C5×#4111yescharacteristic
398C2×C3C22×C5×#4111yescharacteristic
399C2×C3C2×C5×2−1+411yescharacteristic
400D3—44no—
401C5C2×#573211yescharacteristic, direct product factor, Sylow
402C22—64no—
403C22—64no—
404C22—64no—
405C22—128no—
406C22C2×C5×#23712yes—
407C22C2×C5×#23912yes—
408C22C5×#100712yes—
409C22C24×C5×D311yescharacteristic
410C4—28no—
411C4—22no—
412C4—64no—
413C4—64no—
414C4—64no—
415C4C2×C5×#23714yes—
416C4C2×C5×#23914yes—
417C3C2×C5×#55611yescharacteristic, complemented, Sylow
418C2—124no—
419C2C5×#573214yesdirect product factor
420C2C22×C5×#23711yescharacteristic
421C2C22×C5×#23911yescharacteristic
422C2C2×C5×#100711yescharacteristic
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×#556
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
3. C2×C5×Q8×Dic3C2semidirect
4. C5×#5732C2direct (N × H)
5. C2×C5×#898C2semidirect
6. C2×C5×#897C2semidirect
7. C2×C5×#892C2semidirect
9. C2×C4×C5×Dic6C2semidirect
15. C3×C5×C4⋊Q8C22semidirect
20. C5×Q8×Dic3C22semidirect
30. C5×#898C22semidirect
31. C5×#897C22semidirect
32. C5×#892C22semidirect
34. C4×C5×Dic6C22semidirect
35. C2×#5732C5direct (N × H)
37. C2×C5×C4⋊Q8D3semidirect
73. C2×C3×C4⋊Q8C2×C5semidirect
75. C2×Q8×Dic3C2×C5semidirect
76. #5732C2×C5direct (N × H)
77. C2×#898C2×C5semidirect
78. C2×#897C2×C5semidirect
79. C2×#892C2×C5semidirect
81. C2×C4×Dic6C2×C5semidirect
86. C5×C4⋊Q8C2×D3semidirect
134. C3×C4⋊Q8C22×C5semidirect
139. Q8×Dic3C22×C5semidirect
149. #898C22×C5semidirect
150. #897C22×C5semidirect
151. #892C22×C5semidirect
153. C4×Dic6C22×C5semidirect
183. C2×C4⋊Q8C5×D3semidirect
261. C4⋊Q8C2×C5×D3semidirect
281. C2×C3×C5#556semidirect
352. C3×C5C2×#556semidirect
369. C2×C5#5732direct (N × H)
396. C2×C3C5×#556semidirect
401. C5C2×#5732direct (N × H)
417. C3C2×C5×#556semidirect
419. C2C5×#5732direct (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×#9472
C2×C5×Q8×Dic32
C5×#57322
C2×C5×#8982
C2×C5×#8972
C2×C5×#8922
C2×C4×C5×D122
C2×C4×C5×Dic62
C2×C5×#5563
C2×#57325