Finite Group Databasebeta

← C2×C5×#5699 #14495377

Subgroups — C2×C5×#5699 #14495377

Order 1920.

2048 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#125C211yesmaximal, characteristic, complemented
2C5×#5699C2116yesmaximal, direct product factor
3C2×C5×#898C212yesmaximal, complemented
4C2×C5×#898C212yesmaximal, complemented
5C2×C5×#897C212yesmaximal, complemented
6C2×C5×#896C212yesmaximal, complemented
7C2×C5×#894C212yesmaximal
8C2×C5×#888C212yesmaximal, complemented
9C2×C5×#875C211yesmaximal, characteristic
10C2×C5×#870C211yesmaximal, characteristic, complemented
11C2×C5×#563—31nomaximal, Hall
12C22×C5×D12C2211yescharacteristic
13C22×C4×C5×D3C2212yes—
14C22×C5×Dic6C2211yescharacteristic
15C22×C5×Dic6C2212yes—
16C3×C5×#125C2218yescomplemented
17C2×C3×C5×C4⋊C4C2212yes—
18C2×C3×C5×C4⋊C4C2211yescharacteristic
19C2×C3×C5×C4⋊C4C2212yes—
20C2×C3×C5×C4⋊C4C2211yescharacteristic
21C2×C3×C42×C5C2211yescharacteristic
22C2×C5×(C2×D3)⋊C4C2212yes—
23C2×C5×(C2×D3)⋊C4C2212yes—
24C2×C5×(C2×D3)⋊C4C2212yes—
25C2×C5×(C2×D3)⋊C4C2212yes—
26C2×C5×(C2×D3)⋊C4C2212yes—
27C2×C5×C4⋊Dic3C2211yescharacteristic
28C2×C5×C4⋊Dic3C2211yescharacteristic
29C2×C5×C4⋊Dic3C2212yes—
30C2×C5×Dic3⋊C4C2212yes—
31C2×C5×Dic3⋊C4C2212yes—
32C2×C5×Dic3⋊C4C2212yes—
33C2×C4×C5×Dic3C2212yes—
34C5×#898C22116yescomplemented
35C5×#898C22116yescomplemented
36C5×#897C22116yescomplemented
37C5×#896C22116yescomplemented
38C5×#894C22116yes—
39C5×#888C22116yescomplemented
40C5×#875C2218yes—
41C5×#870C2218yescomplemented
42C2×#5699C511yesmaximal, characteristic, direct product factor, Hall
43C5×#563—316no—
44C2×C5×C4⋊Q8—32no—
45C2×C5×#126—32no—
46C2×C5×#126—32no—
47C2×C5×#125—31no—
48C2×C5×#125D311yescharacteristic, complemented
49C2×C5×#124—31no—
50C2×C5×#123—32no—
51C2×C5×C22⋊Q8—32no—
52C2×C5×C22⋊Q8—32no—
53C23×C5×D3C2312yes—
54C22×C3×C4×C5C2312yes—
55C22×C3×C4×C5C2311yescharacteristic
56C22×C3×C4×C5C2311yescharacteristic
57C22×C3×C4×C5C2312yes—
58C22×C3×C4×C5C2311yescharacteristic
59C22×C5×Dic3C2312yes—
60C22×C5×Dic3C2312yes—
61C22×C5×Dic3C2312yes—
62C2×C5×D12—42no—
63C2×C5×D12C2314yes—
64C2×C4×C5×D3—44no—
65C2×C4×C5×D3C2318yes—
66C2×C5×Dic6—42no—
67C2×C5×Dic6—44no—
68C2×C5×Dic6C2314yes—
69C2×C5×Dic6C2318yes—
70C3×C5×C4⋊C4C2318yes—
71C3×C5×C4⋊C4C2314yes—
72C3×C5×C4⋊C4C2318yes—
73C3×C5×C4⋊C4C2314yes—
74C3×C42×C5C2314yes—
75C5×(C2×D3)⋊C4C2318yes—
76C5×(C2×D3)⋊C4C2318yes—
77C5×(C2×D3)⋊C4C2318yes—
78C5×(C2×D3)⋊C4C2318yes—
79C5×(C2×D3)⋊C4C2318yes—
80C5×C4⋊Dic3C2314yes—
81C5×C4⋊Dic3C2314yes—
82C5×C4⋊Dic3C2318yes—
83C5×Dic3⋊C4C2318yes—
84C5×Dic3⋊C4C2318yes—
85C5×Dic3⋊C4C2318yes—
86C4×C5×Dic3C2318yes—
87C2×C3×#125C2×C511yescharacteristic, complemented
88#5699C2×C5116yesdirect product factor
89C2×#898C2×C512yescomplemented
90C2×#898C2×C512yescomplemented
91C2×#897C2×C512yescomplemented
92C2×#896C2×C512yescomplemented
93C2×#894C2×C512yes—
94C2×#888C2×C512yescomplemented
95C2×#875C2×C511yescharacteristic
96C2×#870C2×C511yescharacteristic, complemented
97C22×C5×Q8—32no—
98C22×C5×Q8—31no—
99C22×C5×D4—31no—
100C23×C4×C5—32no—
101C5×C4⋊Q8—316no—
102C5×#126—316no—
103C5×#126—316no—
104C5×#125—38no—
105C5×#125C2×D318yescomplemented
106C5×#124—38no—
107C5×#123—316no—
108C5×C22⋊Q8—316no—
109C5×C22⋊Q8—316no—
110C2×C5×C4⋊C4—32no—
111C2×C5×C4⋊C4—32no—
112C2×C5×C4⋊C4—32no—
113C2×C5×C4⋊C4—31no—
114C2×C5×C4⋊C4—32no—
115C2×C5×C4⋊C4—31no—
116C2×C5×C4⋊C4C2×D311yescharacteristic
117C2×C5×C4⋊C4C2×D312yes—
118C2×C5×C4⋊C4C2×D312yes—
119C2×C5×C4⋊C4C2×D311yescharacteristic
120C2×C5×C22⋊C4—32no—
121C2×C5×C22⋊C4—32no—
122C2×C5×C22⋊C4—32no—
123C2×C5×C22⋊C4—32no—
124C2×C5×C22⋊C4—32no—
125C2×C42×C5—32no—
126C2×C42×C5C2×D311yescharacteristic
127C2×#563—31noSylow
128C23×C3×C5C2411yescharacteristic
129C22×C5×D3—24no—
130C22×C5×D3—24no—
131C22×C5×D3—24no—
132C22×C5×D3C2414yes—
133C2×C3×C4×C5—24no—
134C2×C3×C4×C5—22no—
135C2×C3×C4×C5—22no—
136C2×C3×C4×C5—24no—
137C2×C3×C4×C5—22no—
138C2×C3×C4×C5C2414yes—
139C2×C3×C4×C5C2412yes—
140C2×C3×C4×C5C2412yes—
141C2×C3×C4×C5C2414yes—
142C2×C3×C4×C5C2412yes—
143C2×C5×Dic3—24no—
144C2×C5×Dic3—24no—
145C2×C5×Dic3—24no—
146C2×C5×Dic3C2414yes—
147C2×C5×Dic3C2414yes—
148C2×C5×Dic3C2414yes—
149C5×D12—44no—
150C4×C5×D3—48no—
151C5×Dic6—44no—
152C5×Dic6—48no—
153C22×D12C22×C511yescharacteristic
154C22×C4×D3C22×C512yes—
155C22×Dic6C22×C511yescharacteristic
156C22×Dic6C22×C512yes—
157C3×#125C22×C518yescomplemented
158C2×C3×C4⋊C4C22×C512yes—
159C2×C3×C4⋊C4C22×C511yescharacteristic
160C2×C3×C4⋊C4C22×C512yes—
161C2×C3×C4⋊C4C22×C511yescharacteristic
162C2×C3×C42C22×C511yescharacteristic
163C2×(C2×D3)⋊C4C22×C512yes—
164C2×(C2×D3)⋊C4C22×C512yes—
165C2×(C2×D3)⋊C4C22×C512yes—
166C2×(C2×D3)⋊C4C22×C512yes—
167C2×(C2×D3)⋊C4C22×C512yes—
168C2×C4⋊Dic3C22×C511yescharacteristic
169C2×C4⋊Dic3C22×C511yescharacteristic
170C2×C4⋊Dic3C22×C512yes—
171C2×Dic3⋊C4C22×C512yes—
172C2×Dic3⋊C4C22×C512yes—
173C2×Dic3⋊C4C22×C512yes—
174C2×C4×Dic3C22×C512yes—
175#898C22×C5116yescomplemented
176#898C22×C5116yescomplemented
177#897C22×C5116yescomplemented
178#896C22×C5116yescomplemented
179#894C22×C5116yes—
180#888C22×C5116yescomplemented
181#875C22×C518yes—
182#870C22×C518yescomplemented
183C24×C5—32no—
184C2×C5×Q8—38no—
185C2×C5×Q8—34no—
186C2×C5×Q8—124no—
187C2×C5×Q8—122no—
188C2×C5×D4—34no—
189C2×C5×D4—122no—
190C22×C4×C5—32no—
191C22×C4×C5—32no—
192C22×C4×C5—32no—
193C22×C4×C5—38no—
194C22×C4×C5—124no—
195C22×C4×C5C22×D312yes—
196C22×C4×C5C22×D311yescharacteristic
197C22×C4×C5C22×D311yescharacteristic
198C22×C4×C5C22×D311yescharacteristic
199C22×C4×C5C22×D312yes—
200C5×C4⋊C4—34no—
201C5×C4⋊C4—38no—
202C5×C4⋊C4—38no—
203C5×C4⋊C4—38no—
204C5×C4⋊C4—38no—
205C5×C4⋊C4—34no—
206C5×C4⋊C4C22×D314yes—
207C5×C4⋊C4C22×D318yes—
208C5×C4⋊C4C22×D318yes—
209C5×C4⋊C4C22×D314yes—
210C5×C22⋊C4—38no—
211C5×C22⋊C4—38no—
212C5×C22⋊C4—38no—
213C5×C22⋊C4—38no—
214C5×C22⋊C4—38no—
215C42×C5—38no—
216C42×C5C22×D314yes—
217#563—316no—
218C2×C4⋊Q8—32no—
219C2×#126—32no—
220C2×#126—32no—
221C2×#125—31no—
222C2×#125C5×D311yescharacteristic, complemented
223C2×#124—31no—
224C2×#123—32no—
225C2×C22⋊Q8—32no—
226C2×C22⋊Q8—32no—
227C22×C3×C5H(5,𝔽2)12yes—
228C22×C3×C52−1+412yes—
229C22×C3×C52−1+412yes—
230C22×C3×C5C2511yescharacteristic
231C2×C5×D3—24no—
232C2×C5×D3—24no—
233C2×C5×D3—24no—
234C2×C5×D3—48no—
235C3×C4×C5—24no—
236C3×C4×C5—22no—
237C3×C4×C5—22no—
238C3×C4×C5—24no—
239C3×C4×C5—22no—
240C5×Dic3—24no—
241C5×Dic3—24no—
242C5×Dic3—24no—
243C23×D3C23×C512yes—
244C22×C3×C4C23×C512yes—
245C22×C3×C4C23×C511yescharacteristic
246C22×C3×C4C23×C511yescharacteristic
247C22×C3×C4C23×C512yes—
248C22×C3×C4C23×C511yescharacteristic
249C22×Dic3C23×C512yes—
250C22×Dic3C23×C512yes—
251C22×Dic3C23×C512yes—
252C2×D12—42no—
253C2×D12C23×C514yes—
254C2×C4×D3—44no—
255C2×C4×D3C23×C518yes—
256C2×Dic6—42no—
257C2×Dic6—44no—
258C2×Dic6C23×C514yes—
259C2×Dic6C23×C518yes—
260C3×C4⋊C4C23×C518yes—
261C3×C4⋊C4C23×C514yes—
262C3×C4⋊C4C23×C518yes—
263C3×C4⋊C4C23×C514yes—
264C3×C42C23×C514yes—
265(C2×D3)⋊C4C23×C518yes—
266(C2×D3)⋊C4C23×C518yes—
267(C2×D3)⋊C4C23×C518yes—
268(C2×D3)⋊C4C23×C518yes—
269(C2×D3)⋊C4C23×C518yes—
270C4⋊Dic3C23×C514yes—
271C4⋊Dic3C23×C514yes—
272C4⋊Dic3C23×C518yes—
273Dic3⋊C4C23×C518yes—
274Dic3⋊C4C23×C518yes—
275Dic3⋊C4C23×C518yes—
276C4×Dic3C23×C518yes—
277C23×C5—34no—
278C23×C5—64no—
279C23×C5—64no—
280C23×C5—64no—
281C23×C5C23×D311yescharacteristic
282C5×Q8—128no—
283C5×Q8—124no—
284C5×D4—124no—
285C2×C4×C5—22no—
286C2×C4×C5—24no—
287C2×C4×C5—22no—
288C2×C4×C5—24no—
289C2×C4×C5—22no—
290C2×C4×C5—34no—
291C2×C4×C5—34no—
292C2×C4×C5—34no—
293C2×C4×C5—64no—
294C2×C4×C5—64no—
295C2×C4×C5—64no—
296C2×C4×C5—128no—
297C2×C4×C5C23×D314yes—
298C2×C4×C5C23×D312yes—
299C2×C4×C5C23×D312yes—
300C2×C4×C5C23×D312yes—
301C2×C4×C5C23×D314yes—
302C22×Q8—32no—
303C22×Q8—31no—
304C22×D4—31no—
305C23×C4—32no—
306C4⋊Q8—316no—
307#126—316no—
308#126—316no—
309#125—38no—
310#125C2×C5×D318yescomplemented
311#124—38no—
312#123—316no—
313C22⋊Q8—316no—
314C22⋊Q8—316no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—32no—
317C2×C4⋊C4—32no—
318C2×C4⋊C4—32no—
319C2×C4⋊C4—32no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4C2×C5×D311yescharacteristic
322C2×C4⋊C4C2×C5×D312yes—
323C2×C4⋊C4C2×C5×D312yes—
324C2×C4⋊C4C2×C5×D311yescharacteristic
325C2×C22⋊C4—32no—
326C2×C22⋊C4—32no—
327C2×C22⋊C4—32no—
328C2×C22⋊C4—32no—
329C2×C22⋊C4—32no—
330C2×C42—32no—
331C2×C42C2×C5×D311yescharacteristic
332C2×C3×C5#56314yescomplemented
333C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
334C2×C3×C5C2×2−1+411yescharacteristic
335C2×C3×C5C2×2−1+411yescharacteristic
336C5×D3—44no—
337C23×C3C24×C511yescharacteristic
338C22×D3—24no—
339C22×D3—24no—
340C22×D3—24no—
341C22×D3C24×C514yes—
342C2×C3×C4—24no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—24no—
346C2×C3×C4—22no—
347C2×C3×C4C24×C514yes—
348C2×C3×C4C24×C512yes—
349C2×C3×C4C24×C512yes—
350C2×C3×C4C24×C514yes—
351C2×C3×C4C24×C512yes—
352C2×Dic3—24no—
353C2×Dic3—24no—
354C2×Dic3—24no—
355C2×Dic3C24×C514yes—
356C2×Dic3C24×C514yes—
357C2×Dic3C24×C514yes—
358D12—44no—
359C4×D3—48no—
360Dic6—44no—
361Dic6—48no—
362C22×C5—64no—
363C22×C5—64no—
364C22×C5—64no—
365C22×C5—128no—
366C22×C5#100712yes—
367C22×C5#100912yes—
368C22×C5#101012yes—
369C22×C5C24×D311yescharacteristic
370C4×C5—22no—
371C4×C5—24no—
372C4×C5—22no—
373C4×C5—24no—
374C4×C5—22no—
375C4×C5—64no—
376C4×C5—64no—
377C4×C5—64no—
378C24—32no—
379C2×Q8—38no—
380C2×Q8—34no—
381C2×Q8—124no—
382C2×Q8—122no—
383C2×D4—34no—
384C2×D4—122no—
385C22×C4—32no—
386C22×C4—32no—
387C22×C4—32no—
388C22×C4—38no—
389C22×C4—124no—
390C22×C4C22×C5×D312yes—
391C22×C4C22×C5×D311yescharacteristic
392C22×C4C22×C5×D311yescharacteristic
393C22×C4C22×C5×D311yescharacteristic
394C22×C4C22×C5×D312yes—
395C4⋊C4—34no—
396C4⋊C4—38no—
397C4⋊C4—38no—
398C4⋊C4—38no—
399C4⋊C4—38no—
400C4⋊C4—34no—
401C4⋊C4C22×C5×D314yes—
402C4⋊C4C22×C5×D318yes—
403C4⋊C4C22×C5×D318yes—
404C4⋊C4C22×C5×D314yes—
405C22⋊C4—38no—
406C22⋊C4—38no—
407C22⋊C4—38no—
408C22⋊C4—38no—
409C22⋊C4—38no—
410C42—38no—
411C42C22×C5×D314yes—
412C3×C5C2×#56311yescharacteristic, complemented, Hall
413C22×C3C5×H(5,𝔽2)12yes—
414C22×C3C5×2−1+412yes—
415C22×C3C5×2−1+412yes—
416C22×C3C25×C511yescharacteristic
417C2×D3—24no—
418C2×D3—24no—
419C2×D3—24no—
420C2×D3—48no—
421C3×C4—24no—
422C3×C4—22no—
423C3×C4—22no—
424C3×C4—24no—
425C3×C4—22no—
426Dic3—24no—
427Dic3—24no—
428Dic3—24no—
429C2×C5—124no—
430C2×C5#569914yesdirect product factor
431C2×C5C2×#100711yescharacteristic
432C2×C5C2×#100911yescharacteristic
433C2×C5C2×#101011yescharacteristic
434C23—34no—
435C23—64no—
436C23—64no—
437C23—64no—
438C23C23×C5×D311yescharacteristic
439Q8—128no—
440Q8—124no—
441D4—124no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—34no—
448C2×C4—34no—
449C2×C4—34no—
450C2×C4—64no—
451C2×C4—64no—
452C2×C4—64no—
453C2×C4—128no—
454C2×C4C23×C5×D314yes—
455C2×C4C23×C5×D312yes—
456C2×C4C23×C5×D312yes—
457C2×C4C23×C5×D312yes—
458C2×C4C23×C5×D314yes—
459C2×C3C5×#56314yescomplemented
460C2×C3C2×C5×H(5,𝔽2)11yescharacteristic
461C2×C3C2×C5×2−1+411yescharacteristic
462C2×C3C2×C5×2−1+411yescharacteristic
463D3—44no—
464C5C2×#569911yescharacteristic, direct product factor, Sylow
465C22—64no—
466C22—64no—
467C22—64no—
468C22—128no—
469C22C5×#100712yes—
470C22C5×#100912yes—
471C22C5×#101012yes—
472C22C24×C5×D311yescharacteristic
473C4—22no—
474C4—24no—
475C4—22no—
476C4—24no—
477C4—22no—
478C4—64no—
479C4—64no—
480C4—64no—
481C3C2×C5×#56311yescharacteristic, complemented, Sylow
482C2—124no—
483C2C5×#569914yesdirect product factor
484C2C2×C5×#100711yescharacteristic
485C2C2×C5×#100911yescharacteristic
486C2C2×C5×#101011yescharacteristic
subgroupG/H
centerC23×C5C23×D3
commutator, cosocleC22×C3C25×C5
FrattiniC22C24×C5×D3
FittingC2×C3×C5×#125C2
socleC23×C3×C5C24
prime radicalC22×C5×D12C22
hypercenterC2×C5×#125D3
hypocenterC3C2×C5×#563
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×#5699C2direct (N × H)
3. C2×C5×#898C2semidirect
4. C2×C5×#898C2semidirect
5. C2×C5×#897C2semidirect
6. C2×C5×#896C2semidirect
8. C2×C5×#888C2semidirect
10. C2×C5×#870C2semidirect
16. C3×C5×#125C22semidirect
34. C5×#898C22semidirect
35. C5×#898C22semidirect
36. C5×#897C22semidirect
37. C5×#896C22semidirect
39. C5×#888C22semidirect
41. C5×#870C22semidirect
42. C2×#5699C5direct (N × H)
48. C2×C5×#125D3semidirect
87. C2×C3×#125C2×C5semidirect
88. #5699C2×C5direct (N × H)
89. C2×#898C2×C5semidirect
90. C2×#898C2×C5semidirect
91. C2×#897C2×C5semidirect
92. C2×#896C2×C5semidirect
94. C2×#888C2×C5semidirect
96. C2×#870C2×C5semidirect
105. C5×#125C2×D3semidirect
157. C3×#125C22×C5semidirect
175. #898C22×C5semidirect
176. #898C22×C5semidirect
177. #897C22×C5semidirect
178. #896C22×C5semidirect
180. #888C22×C5semidirect
182. #870C22×C5semidirect
222. C2×#125C5×D3semidirect
310. #125C2×C5×D3semidirect
332. C2×C3×C5#563semidirect
412. C3×C5C2×#563semidirect
430. C2×C5#5699direct (N × H)
459. C2×C3C5×#563semidirect
464. C5C2×#5699direct (N × H)
481. C3C2×C5×#563semidirect
483. C2C5×#5699direct (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×#56992
C2×C5×#8982
C2×C5×#8972
C2×C5×#8962
C2×C5×#8942
C2×C5×#8882
C2×C5×#8752
C2×C5×#8702
C2×C5×#5633
C2×#56995