Finite Group Databasebeta

← C2×C5×#5204 #14429430

Subgroups — C2×C5×#5204 #14429430

Order 1920.

896 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#137C211yesmaximal, characteristic
2C2×C5×#951C211yesmaximal, characteristic
3C2×C5×#948C211yesmaximal, characteristic
4C22×C5×#216C211yesmaximal, characteristic, complemented
5C2×C5×#836C211yesmaximal, characteristic, complemented
6C5×#5204C218yesmaximal, direct product factor
7C2×C5×#824C211yesmaximal, characteristic, complemented
8C2×C5×#822C211yesmaximal, characteristic, complemented
9C2×C5×#473—31nomaximal, Hall
10C2×C3×C5×#41C2211yescharacteristic
11C22×C3×C5×Q8C2211yescharacteristic
12C22×C5×Dic6C2211yescharacteristic
13C3×C5×#137—22no—
14C3×C5×#137C2214yes—
15C2×C3×C5×Q16—21no—
16C2×C3×C5×SD16—21no—
17C2×C3×C5×M16C2211yescharacteristic
18C5×#951—22no—
19C5×#951C2214yes—
20C5×#948—22no—
21C5×#948C2214yes—
22C2×C5×#216—21no—
23C2×C5×#216—21no—
24C2×C5×#216—21no—
25C2×C5×#216—22no—
26C2×C5×#216C2214yescomplemented
27C2×C5×#214—21no—
28C2×C5×#208—21no—
29C2×C5×#208C2211yescharacteristic
30C2×C5×#208C2211yescharacteristic
31C22×C5×C3⋊C8—21no—
32C22×C5×C3⋊C8C2211yescharacteristic
33C5×#836C2214yescomplemented
34C5×#824C2214yescomplemented
35C5×#822C2214yescomplemented
36C2×#5204C511yesmaximal, characteristic, direct product factor, Hall
37C2×C5×#137—31no—
38C2×C5×#137D311yescharacteristic
39C22×C5×Q16—31no—
40C2×C5×#131—31no—
41C5×#473—38no—
42C2×C5×#108—31no—
43C2×C5×#101—31no—
44C2×C5×#101—31no—
45C3×C5×#41—21no—
46C3×C5×#41—21no—
47C3×C5×#41D412yes—
48C3×C5×#41C2312yes—
49C2×C3×C5×Q8—21no—
50C2×C3×C5×Q8—21no—
51C2×C3×C5×Q8—21no—
52C2×C3×C5×Q8—21no—
53C2×C3×C5×Q8—21no—
54C2×C3×C5×Q8D411yescharacteristic
55C2×C3×C5×Q8C2312yes—
56C2×C3×C5×D4—21no—
57C2×C3×C5×D4D411yescharacteristic
58C22×C3×C4×C5—21no—
59C22×C3×C4×C5—21no—
60C22×C3×C4×C5C2311yescharacteristic
61C22×C5×Dic3—21no—
62C2×C5×Dic6—21no—
63C2×C5×Dic6—21no—
64C2×C5×Dic6—21no—
65C2×C5×Dic6—21no—
66C2×C5×Dic6—21no—
67C2×C5×Dic6C2312yes—
68C3×C5×Q16—24no—
69C3×C5×SD16—24no—
70C3×C5×M16—21no—
71C3×C5×M16C2312yes—
72C2×C3×C5×C8—21no—
73C5×#216—24no—
74C5×#216—24no—
75C5×#216—24no—
76C5×#214—24no—
77C5×#208—21no—
78C5×#208—21no—
79C5×#208—24no—
80C5×#208C2312yes—
81C5×#208C2312yes—
82C2×C5×C3⋊C8—21no—
83C2×C5×C3⋊C8—21no—
84C2×C5×C3⋊C8—24no—
85C2×C5×C3⋊C8D411yescharacteristic
86C2×C5×C3⋊C8D411yescharacteristic
87C2×C5×C3⋊C8D412yes—
88C2×C5×C3⋊C8C2312yes—
89C2×C3×#137C2×C511yescharacteristic
90C2×#951C2×C511yescharacteristic
91C2×#948C2×C511yescharacteristic
92C22×#216C2×C511yescharacteristic, complemented
93C2×#836C2×C511yescharacteristic, complemented
94#5204C2×C518yesdirect product factor
95C2×#824C2×C511yescharacteristic, complemented
96C2×#822C2×C511yescharacteristic, complemented
97C2×C5×#41C2×D311yescharacteristic
98C22×C5×Q8—31no—
99C22×C5×Q8C2×D311yescharacteristic
100C5×#137—22no—
101C5×#137—34no—
102C5×#137—62no—
103C5×#137C2×D314yes—
104C2×C5×Q16—21no—
105C2×C5×Q16—34no—
106C2×C5×Q16—61no—
107C2×C5×Q16—61no—
108C2×C5×Q16—62no—
109C2×C5×Q16—61no—
110C2×C5×SD16—21no—
111C2×C5×SD16—61no—
112C5×#131—34no—
113C5×#131—62no—
114C2×C5×M16—31no—
115C2×C5×M16—31no—
116C2×C5×M16—61no—
117C2×C5×M16C2×D311yescharacteristic
118C22×C5×C8—31no—
119C22×C5×C8—61no—
120C5×#108—34no—
121C5×#101—34no—
122C5×#101—34no—
123C2×#473—31noSylow
124C23×C3×C5—21no—
125C23×C3×C5#4111yescharacteristic
126C3×C5×Q8—21no—
127C3×C5×Q8—22no—
128C3×C5×Q8—22no—
129C3×C5×Q8—41no—
130C3×C5×Q8—41no—
131C3×C5×Q8C2×D412yes—
132C3×C5×D4—22no—
133C3×C5×D4—22no—
134C3×C5×D4—21no—
135C3×C5×D4C2×D412yes—
136C2×C3×C4×C5—21no—
137C2×C3×C4×C5—22no—
138C2×C3×C4×C5—22no—
139C2×C3×C4×C5—22no—
140C2×C3×C4×C5—41no—
141C2×C3×C4×C5—41no—
142C2×C3×C4×C5C2×D411yescharacteristic
143C2×C3×C4×C5C2×D411yescharacteristic
144C2×C3×C4×C5C2×D411yescharacteristic
145C2×C3×C4×C5C2×D411yescharacteristic
146C2×C3×C4×C5#4111yescharacteristic
147C2×C3×C4×C5C2411yescharacteristic
148C2×C5×Dic3—22no—
149C2×C5×Dic3—41no—
150C2×C5×Dic3—41no—
151C5×Dic6—22no—
152C5×Dic6—22no—
153C5×Dic6—41no—
154C5×Dic6—41no—
155C3×C5×C8—22no—
156C5×C3⋊C8—22no—
157C5×C3⋊C8—22no—
158C5×C3⋊C8C2×D412yes—
159C5×C3⋊C8C2×D412yes—
160C2×C3×#41C22×C511yescharacteristic
161C22×C3×Q8C22×C511yescharacteristic
162C22×Dic6C22×C511yescharacteristic
163C3×#137—22no—
164C3×#137C22×C514yes—
165C2×C3×Q16—21no—
166C2×C3×SD16—21no—
167C2×C3×M16C22×C511yescharacteristic
168#951—22no—
169#951C22×C514yes—
170#948—22no—
171#948C22×C514yes—
172C2×#216—21no—
173C2×#216—21no—
174C2×#216—21no—
175C2×#216—22no—
176C2×#216C22×C514yescomplemented
177C2×#214—21no—
178C2×#208—21no—
179C2×#208C22×C511yescharacteristic
180C2×#208C22×C511yescharacteristic
181C22×C3⋊C8—21no—
182C22×C3⋊C8C22×C511yescharacteristic
183#836C22×C514yescomplemented
184#824C22×C514yescomplemented
185#822C22×C514yescomplemented
186C5×#41—21no—
187C5×#41—21no—
188C5×#41#6712yes—
189C5×#41C22×D312yes—
190C2×C5×Q8—21no—
191C2×C5×Q8—21no—
192C2×C5×Q8—21no—
193C2×C5×Q8—21no—
194C2×C5×Q8—21no—
195C2×C5×Q8—32no—
196C2×C5×Q8—61no—
197C2×C5×Q8—61no—
198C2×C5×Q8—61no—
199C2×C5×Q8—61no—
200C2×C5×Q8—61no—
201C2×C5×Q8#6711yescharacteristic
202C2×C5×Q8C22×D312yes—
203C2×C5×D4—21no—
204C2×C5×D4#6711yescharacteristic
205C22×C4×C5—21no—
206C22×C4×C5—21no—
207C22×C4×C5—61no—
208C22×C4×C5C22×D311yescharacteristic
209C5×Q16—24no—
210C5×Q16—64no—
211C5×Q16—64no—
212C5×Q16—64no—
213C5×SD16—24no—
214C5×SD16—64no—
215C5×M16—21no—
216C5×M16—32no—
217C5×M16—32no—
218C5×M16—64no—
219C5×M16—61no—
220C5×M16—61no—
221C5×M16C22×D312yes—
222C2×C5×C8—21no—
223C2×C5×C8—31no—
224C2×C5×C8—31no—
225C2×C5×C8—32no—
226C2×C5×C8—32no—
227C2×C5×C8—64no—
228C2×C5×C8—61no—
229C2×C5×C8—61no—
230C2×#137—31no—
231C2×#137C5×D311yescharacteristic
232C22×Q16—31no—
233C2×#131—31no—
234#473—38no—
235C2×#108—31no—
236C2×#101—31no—
237C2×#101—31no—
238C22×C3×C5—22no—
239C22×C3×C5—22no—
240C22×C3×C5—42no—
241C22×C3×C5#12111yescharacteristic
242C22×C3×C5#12111yescharacteristic
243C22×C3×C5C2×#4111yescharacteristic
244C3×C4×C5—22no—
245C3×C4×C5—42no—
246C3×C4×C5#12111yescharacteristic
247C3×C4×C5#12111yescharacteristic
248C3×C4×C5C22×D411yescharacteristic
249C3×C4×C5C22×D411yescharacteristic
250C5×Dic3—42no—
251C3×#41—21no—
252C3×#41—21no—
253C3×#41C5×D412yes—
254C3×#41C23×C512yes—
255C2×C3×Q8—21no—
256C2×C3×Q8—21no—
257C2×C3×Q8—21no—
258C2×C3×Q8—21no—
259C2×C3×Q8—21no—
260C2×C3×Q8C5×D411yescharacteristic
261C2×C3×Q8C23×C512yes—
262C2×C3×D4—21no—
263C2×C3×D4C5×D411yescharacteristic
264C22×C3×C4—21no—
265C22×C3×C4—21no—
266C22×C3×C4C23×C511yescharacteristic
267C22×Dic3—21no—
268C2×Dic6—21no—
269C2×Dic6—21no—
270C2×Dic6—21no—
271C2×Dic6—21no—
272C2×Dic6—21no—
273C2×Dic6C23×C512yes—
274C3×Q16—24no—
275C3×SD16—24no—
276C3×M16—21no—
277C3×M16C23×C512yes—
278C2×C3×C8—21no—
279#216—24no—
280#216—24no—
281#216—24no—
282#214—24no—
283#208—21no—
284#208—21no—
285#208—24no—
286#208C23×C512yes—
287#208C23×C512yes—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—24no—
291C2×C3⋊C8C5×D411yescharacteristic
292C2×C3⋊C8C5×D411yescharacteristic
293C2×C3⋊C8C5×D412yes—
294C2×C3⋊C8C23×C512yes—
295C23×C5—21no—
296C23×C5#23711yescharacteristic
297C5×Q8—21no—
298C5×Q8—22no—
299C5×Q8—22no—
300C5×Q8—41no—
301C5×Q8—41no—
302C5×Q8—62no—
303C5×Q8—62no—
304C5×Q8—121no—
305C5×Q8—121no—
306C5×Q8C2×#6712yes—
307C5×D4—22no—
308C5×D4—22no—
309C5×D4—21no—
310C5×D4C2×#6712yes—
311C2×C4×C5—21no—
312C2×C4×C5—22no—
313C2×C4×C5—22no—
314C2×C4×C5—22no—
315C2×C4×C5—41no—
316C2×C4×C5—41no—
317C2×C4×C5—62no—
318C2×C4×C5—121no—
319C2×C4×C5—121no—
320C2×C4×C5D3×D411yescharacteristic
321C2×C4×C5D3×D411yescharacteristic
322C2×C4×C5#23711yescharacteristic
323C2×C4×C5C2×#6711yescharacteristic
324C2×C4×C5C2×#6711yescharacteristic
325C2×C4×C5C23×D311yescharacteristic
326C5×C8—22no—
327C5×C8—32no—
328C5×C8—32no—
329C5×C8—62no—
330C5×C8—62no—
331C2×#41C2×C5×D311yescharacteristic
332C22×Q8—31no—
333C22×Q8C2×C5×D311yescharacteristic
334#137—22no—
335#137—34no—
336#137—62no—
337#137C2×C5×D314yes—
338C2×Q16—21no—
339C2×Q16—34no—
340C2×Q16—61no—
341C2×Q16—61no—
342C2×Q16—62no—
343C2×Q16—61no—
344C2×SD16—21no—
345C2×SD16—61no—
346#131—34no—
347#131—62no—
348C2×M16—31no—
349C2×M16—31no—
350C2×M16—61no—
351C2×M16C2×C5×D311yescharacteristic
352C22×C8—31no—
353C22×C8—61no—
354#108—34no—
355#101—34no—
356#101—34no—
357C2×C3×C5—21no—
358C2×C3×C5—21no—
359C2×C3×C5—42no—
360C2×C3×C5#47312yescomplemented
361C2×C3×C5C2×#12111yescharacteristic
362C23×C3—21no—
363C23×C3C5×#4111yescharacteristic
364C3×Q8—21no—
365C3×Q8—22no—
366C3×Q8—22no—
367C3×Q8—41no—
368C3×Q8—41no—
369C3×Q8C2×C5×D412yes—
370C3×D4—22no—
371C3×D4—22no—
372C3×D4—21no—
373C3×D4C2×C5×D412yes—
374C2×C3×C4—21no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4C2×C5×D411yescharacteristic
381C2×C3×C4C2×C5×D411yescharacteristic
382C2×C3×C4C2×C5×D411yescharacteristic
383C2×C3×C4C2×C5×D411yescharacteristic
384C2×C3×C4C5×#4111yescharacteristic
385C2×C3×C4C24×C511yescharacteristic
386C2×Dic3—22no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389Dic6—22no—
390Dic6—22no—
391Dic6—41no—
392Dic6—41no—
393C3×C8—22no—
394C3⋊C8—22no—
395C3⋊C8—22no—
396C3⋊C8C2×C5×D412yes—
397C3⋊C8C2×C5×D412yes—
398C22×C5—22no—
399C22×C5—22no—
400C22×C5—42no—
401C22×C5#93911yescharacteristic
402C22×C5#93911yescharacteristic
403C22×C5C2×#23711yescharacteristic
404C4×C5—22no—
405C4×C5—42no—
406C4×C5—122no—
407C4×C5#93911yescharacteristic
408C4×C5#93911yescharacteristic
409C4×C5C2×D3×D411yescharacteristic
410C4×C5C22×#6711yescharacteristic
411#41—21no—
412#41—21no—
413#41C5×#6712yes—
414#41C22×C5×D312yes—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—32no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8C5×#6711yescharacteristic
427C2×Q8C22×C5×D312yes—
428C2×D4—21no—
429C2×D4C5×#6711yescharacteristic
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—61no—
433C22×C4C22×C5×D311yescharacteristic
434Q16—24no—
435Q16—64no—
436Q16—64no—
437Q16—64no—
438SD16—24no—
439SD16—64no—
440M16—21no—
441M16—32no—
442M16—32no—
443M16—64no—
444M16—61no—
445M16—61no—
446M16C22×C5×D312yes—
447C2×C8—21no—
448C2×C8—31no—
449C2×C8—31no—
450C2×C8—32no—
451C2×C8—32no—
452C2×C8—64no—
453C2×C8—61no—
454C2×C8—61no—
455C3×C5C2×#47311yescharacteristic, complemented, Hall
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—42no—
459C22×C3C5×#12111yescharacteristic
460C22×C3C5×#12111yescharacteristic
461C22×C3C2×C5×#4111yescharacteristic
462C3×C4—22no—
463C3×C4—42no—
464C3×C4C5×#12111yescharacteristic
465C3×C4C5×#12111yescharacteristic
466C3×C4C22×C5×D411yescharacteristic
467C3×C4C22×C5×D411yescharacteristic
468Dic3—42no—
469C2×C5—21no—
470C2×C5—21no—
471C2×C5—42no—
472C2×C5#520412yesdirect product factor
473C2×C5C2×#93911yescharacteristic
474C23—21no—
475C23C5×#23711yescharacteristic
476Q8—21no—
477Q8—22no—
478Q8—22no—
479Q8—41no—
480Q8—41no—
481Q8—62no—
482Q8—62no—
483Q8—121no—
484Q8—121no—
485Q8C2×C5×#6712yes—
486D4—22no—
487D4—22no—
488D4—21no—
489D4C2×C5×#6712yes—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—62no—
497C2×C4—121no—
498C2×C4—121no—
499C2×C4C5×D3×D411yescharacteristic
500C2×C4C5×D3×D411yescharacteristic
501C2×C4C5×#23711yescharacteristic
502C2×C4C2×C5×#6711yescharacteristic
503C2×C4C2×C5×#6711yescharacteristic
504C2×C4C23×C5×D311yescharacteristic
505C8—22no—
506C8—32no—
507C8—32no—
508C8—62no—
509C8—62no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—42no—
513C2×C3C5×#47312yescomplemented
514C2×C3C2×C5×#12111yescharacteristic
515C5C2×#520411yescharacteristic, direct product factor, Sylow
516C22—22no—
517C22—22no—
518C22—42no—
519C22C5×#93911yescharacteristic
520C22C5×#93911yescharacteristic
521C22C2×C5×#23711yescharacteristic
522C4—22no—
523C4—42no—
524C4—122no—
525C4C5×#93911yescharacteristic
526C4C5×#93911yescharacteristic
527C4C2×C5×D3×D411yescharacteristic
528C4C22×C5×#6711yescharacteristic
529C3C2×C5×#47311yescharacteristic, complemented, Sylow
530C2—21no—
531C2—21no—
532C2—42no—
533C2C5×#520412yesdirect product factor
534C2C2×C5×#93911yescharacteristic
subgroupG/H
centerC22×C5#939
commutator, cosocleC2×C3×C4C24×C5
FrattiniC2×C4C23×C5×D3
FittingC2×C3×C5×#137C2
socleC22×C3×C5#121
prime radicalC2×C3×C5×#41C22
hypercenterC2×C5×#137D3
hypocenterC3C2×C5×#473
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C5×#137

Complements

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

NH ≅ G/Ntype
4. C22×C5×#216C2semidirect
5. C2×C5×#836C2semidirect
6. C5×#5204C2direct (N × H)
7. C2×C5×#824C2semidirect
8. C2×C5×#822C2semidirect
26. C2×C5×#216C22semidirect
33. C5×#836C22semidirect
34. C5×#824C22semidirect
35. C5×#822C22semidirect
36. C2×#5204C5direct (N × H)
92. C22×#216C2×C5semidirect
93. C2×#836C2×C5semidirect
94. #5204C2×C5direct (N × H)
95. C2×#824C2×C5semidirect
96. C2×#822C2×C5semidirect
176. C2×#216C22×C5semidirect
183. #836C22×C5semidirect
184. #824C22×C5semidirect
185. #822C22×C5semidirect
360. C2×C3×C5#473semidirect
455. C3×C5C2×#473semidirect
472. C2×C5#5204direct (N × H)
513. C2×C3C5×#473semidirect
515. C5C2×#5204direct (N × H)
529. C3C2×C5×#473semidirect
533. C2C5×#5204direct (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×#1372
C2×C5×#9512
C2×C5×#9482
C22×C5×#2162
C2×C5×#8362
C5×#52042
C2×C5×#8242
C2×C5×#8222
C2×C5×#4733
C2×#52045