Finite Group Databasebeta

← C5×#76303 #14327996

Subgroups — C5×#76303 #14327996

Order 1920.

1120 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×D4×Dic3C211yesmaximal, characteristic
2C3×C5×#390C211yesmaximal, characteristic
3C5×#5221C211yesmaximal, characteristic
4C5×#4977C211yesmaximal, characteristic, complemented
5C5×Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
6C5×#4945C212yesmaximal, complemented
7C5×#4944C212yesmaximal, complemented
8C5×#4943C212yesmaximal, complemented
9C5×Dic3×C22⋊C4C212yesmaximal, complemented
10C4×C5×C4⋊Dic3C211yesmaximal, characteristic, complemented
11C4×C5×Dic3⋊C4C211yesmaximal, characteristic, complemented
12C5×#2357—31nomaximal, Hall
13C22×C3×C5×D4C2211yescharacteristic
14C23×C5×Dic3C2212yes—
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×C3×C5×C22⋊C4C2212yes—
17C2×C3×C5×C22⋊C4C2212yes—
18C2×C3×C42×C5C2211yescharacteristic
19C2×C5×#217C2212yes—
20C2×C5×#217C2212yes—
21C2×C5×#217C2212yes—
22C5×D4×Dic3C418yescomplemented
23C2×C5×C4⋊Dic3C2211yescharacteristic
24C2×C5×C4⋊Dic3C2211yescharacteristic
25C2×C5×Dic3⋊C4C2211yescharacteristic
26C2×C5×Dic3⋊C4C2211yescharacteristic
27C2×C4×C5×Dic3C2211yescharacteristic
28C2×C4×C5×Dic3C2212yes—
29C2×C4×C5×Dic3C2212yes—
30C2×C4×C5×Dic3C2211yescharacteristic
31C5×C3⋊#95C2211yescharacteristic
32C5×C3⋊#95C2211yescharacteristic
33C5×C3⋊#95C2212yes—
34C5×C3⋊#95C2212yes—
35C5×C3⋊#95C2212yes—
36C5×C3⋊#95C2212yes—
37#76303C511yesmaximal, characteristic, direct product factor, Hall
38C2×C4×C5×D4—31no—
39C5×#390—31no—
40C5×#390D311yescharacteristic
41C5×#388—32no—
42C5×#388—32no—
43C5×#382—31no—
44C5×#381—32no—
45C4×C5×C4⋊C4—31no—
46C4×C5×C4⋊C4—31no—
47C4×C5×C4⋊C4—31no—
48C4×C5×C22⋊C4—32no—
49C24×C3×C5C2312yes—
50C2×C3×C5×D4—22no—
51C2×C3×C5×D4—22no—
52C2×C3×C5×D4C2×C414yes—
53C22×C3×C4×C5C2312yes—
54C22×C3×C4×C5C2311yescharacteristic
55C22×C3×C4×C5C2311yescharacteristic
56C22×C3×C4×C5C2311yescharacteristic
57C22×C5×Dic3—42no—
58C22×C5×Dic3—42no—
59C22×C5×Dic3C2×C418yes—
60C22×C5×Dic3C2311yescharacteristic
61C22×C5×Dic3C2312yes—
62C22×C5×Dic3C2312yes—
63C22×C5×Dic3C2311yescharacteristic
64C22×C5×Dic3C2311yescharacteristic
65C22×C5×Dic3C2311yescharacteristic
66C3×C5×C4⋊C4—22no—
67C3×C5×C22⋊C4—24no—
68C3×C5×C22⋊C4—42no—
69C3×C42×C5—22no—
70C5×#217—24no—
71C5×#217—42no—
72C5×#217C2×C418yes—
73C5×C4⋊Dic3—22no—
74C5×C4⋊Dic3C2×C414yes—
75C5×Dic3⋊C4—22no—
76C5×Dic3⋊C4—22no—
77C4×C5×Dic3—22no—
78C4×C5×Dic3—42no—
79C4×C5×Dic3—42no—
80C4×C5×Dic3C2×C414yes—
81C2×D4×Dic3C2×C511yescharacteristic
82C3×#390C2×C511yescharacteristic
83#5221C2×C511yescharacteristic
84#4977C2×C511yescharacteristic, complemented
85Dic3×C4⋊C4C2×C511yescharacteristic, complemented
86#4945C2×C512yescomplemented
87#4944C2×C512yescomplemented
88#4943C2×C512yescomplemented
89Dic3×C22⋊C4C2×C512yescomplemented
90C4×C4⋊Dic3C2×C511yescharacteristic, complemented
91C4×Dic3⋊C4C2×C511yescharacteristic, complemented
92C22×C5×D4C2×D311yescharacteristic
93C23×C4×C5—32no—
94C4×C5×D4—38no—
95C2×C5×C4⋊C4—31no—
96C2×C5×C4⋊C4—31no—
97C2×C5×C4⋊C4—31no—
98C2×C5×C4⋊C4—31no—
99C2×C5×C4⋊C4C2×D311yescharacteristic
100C2×C5×C22⋊C4—32no—
101C2×C5×C22⋊C4—32no—
102C2×C5×C22⋊C4—32no—
103C2×C5×C22⋊C4C2×D312yes—
104C2×C5×C22⋊C4C2×D312yes—
105C2×C42×C5—31no—
106C2×C42×C5—31no—
107C2×C42×C5—32no—
108C2×C42×C5—32no—
109C2×C42×C5C2×D311yescharacteristic
110C5×#95—31no—
111C5×#95—32no—
112C5×#95—31no—
113C5×#95—32no—
114C5×#95—32no—
115C5×#95—32no—
116#2357—31noSylow
117C23×C3×C5—22no—
118C23×C3×C5—22no—
119C23×C3×C5—22no—
120C23×C3×C5—22no—
121C23×C3×C5—22no—
122C23×C3×C5—22no—
123C23×C3×C5C22×C414yes—
124C23×C3×C5C2411yescharacteristic
125C3×C5×D4—28no—
126C2×C3×C4×C5—21no—
127C2×C3×C4×C5—21no—
128C2×C3×C4×C5—21no—
129C2×C3×C4×C5—21no—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5—21no—
134C2×C3×C4×C5C22×C412yes—
135C2×C3×C4×C5#4112yes—
136C2×C3×C4×C5#4112yes—
137C2×C3×C4×C5#4112yes—
138C2×C5×Dic3—21no—
139C2×C5×Dic3—21no—
140C2×C5×Dic3—21no—
141C2×C5×Dic3—21no—
142C2×C5×Dic3—22no—
143C2×C5×Dic3—22no—
144C2×C5×Dic3—21no—
145C2×C5×Dic3—21no—
146C2×C5×Dic3—21no—
147C2×C5×Dic3—21no—
148C2×C5×Dic3—22no—
149C2×C5×Dic3—22no—
150C2×C5×Dic3—22no—
151C2×C5×Dic3—21no—
152C2×C5×Dic3—48no—
153C2×C5×Dic3C22×C412yes—
154C2×C5×Dic3C22×C412yes—
155C2×C5×Dic3C22×C414yes—
156C2×C5×Dic3#4112yes—
157C22×C3×D4C22×C511yescharacteristic
158C23×Dic3C22×C512yes—
159C2×C3×C4⋊C4C22×C511yescharacteristic
160C2×C3×C22⋊C4C22×C512yes—
161C2×C3×C22⋊C4C22×C512yes—
162C2×C3×C42C22×C511yescharacteristic
163C2×#217C22×C512yes—
164C2×#217C22×C512yes—
165C2×#217C22×C512yes—
166D4×Dic3C4×C518yescomplemented
167C2×C4⋊Dic3C22×C511yescharacteristic
168C2×C4⋊Dic3C22×C511yescharacteristic
169C2×Dic3⋊C4C22×C511yescharacteristic
170C2×Dic3⋊C4C22×C511yescharacteristic
171C2×C4×Dic3C22×C511yescharacteristic
172C2×C4×Dic3C22×C512yes—
173C2×C4×Dic3C22×C512yes—
174C2×C4×Dic3C22×C511yescharacteristic
175C3⋊#95C22×C511yescharacteristic
176C3⋊#95C22×C511yescharacteristic
177C3⋊#95C22×C512yes—
178C3⋊#95C22×C512yes—
179C3⋊#95C22×C512yes—
180C3⋊#95C22×C512yes—
181C24×C5C22×D312yes—
182C2×C5×D4—22no—
183C2×C5×D4—22no—
184C2×C5×D4C4×D314yes—
185C22×C4×C5—31no—
186C22×C4×C5—32no—
187C22×C4×C5—31no—
188C22×C4×C5—31no—
189C22×C4×C5—38no—
190C22×C4×C5—31no—
191C22×C4×C5—32no—
192C22×C4×C5—122no—
193C22×C4×C5—122no—
194C22×C4×C5C22×D312yes—
195C22×C4×C5C22×D311yescharacteristic
196C22×C4×C5C22×D311yescharacteristic
197C22×C4×C5C22×D311yescharacteristic
198C5×C4⋊C4—22no—
199C5×C4⋊C4—34no—
200C5×C4⋊C4—62no—
201C5×C4⋊C4—62no—
202C5×C4⋊C4—62no—
203C5×C22⋊C4—24no—
204C5×C22⋊C4—38no—
205C5×C22⋊C4—42no—
206C5×C22⋊C4—64no—
207C5×C22⋊C4—122no—
208C42×C5—22no—
209C42×C5—34no—
210C42×C5—62no—
211C42×C5—122no—
212C42×C5—122no—
213C2×C4×D4—31no—
214#390—31no—
215#390C5×D311yescharacteristic
216#388—32no—
217#388—32no—
218#382—31no—
219#381—32no—
220C4×C4⋊C4—31no—
221C4×C4⋊C4—31no—
222C4×C4⋊C4—31no—
223C4×C22⋊C4—32no—
224C22×C3×C5—24no—
225C22×C3×C5—24no—
226C22×C3×C5—24no—
227C22×C3×C5—42no—
228C22×C3×C5—42no—
229C22×C3×C5—42no—
230C22×C3×C5—42no—
231C22×C3×C5C23×C411yescharacteristic
232C22×C3×C5C2×#4111yescharacteristic
233C22×C3×C5C2×#4111yescharacteristic
234C22×C3×C5C2×#4111yescharacteristic
235C22×C3×C5C2×#4111yescharacteristic
236C22×C3×C5H(5,𝔽2)11yescharacteristic
237C22×C3×C5H(5,𝔽2)11yescharacteristic
238C3×C4×C5—22no—
239C3×C4×C5—41no—
240C3×C4×C5—42no—
241C3×C4×C5#11714yes—
242C5×Dic3—22no—
243C5×Dic3—22no—
244C5×Dic3—24no—
245C5×Dic3—22no—
246C5×Dic3—41no—
247C5×Dic3—42no—
248C24×C3C23×C512yes—
249C2×C3×D4—22no—
250C2×C3×D4—22no—
251C2×C3×D4C2×C4×C514yes—
252C22×C3×C4C23×C512yes—
253C22×C3×C4C23×C511yescharacteristic
254C22×C3×C4C23×C511yescharacteristic
255C22×C3×C4C23×C511yescharacteristic
256C22×Dic3—42no—
257C22×Dic3—42no—
258C22×Dic3C2×C4×C518yes—
259C22×Dic3C23×C511yescharacteristic
260C22×Dic3C23×C512yes—
261C22×Dic3C23×C511yescharacteristic
262C22×Dic3C23×C512yes—
263C22×Dic3C23×C511yescharacteristic
264C22×Dic3C23×C511yescharacteristic
265C3×C4⋊C4—22no—
266C3×C22⋊C4—24no—
267C3×C22⋊C4—42no—
268C3×C42—22no—
269#217—24no—
270#217—42no—
271#217C2×C4×C518yes—
272C4⋊Dic3—22no—
273C4⋊Dic3C2×C4×C514yes—
274Dic3⋊C4—22no—
275Dic3⋊C4—22no—
276C4×Dic3—22no—
277C4×Dic3—42no—
278C4×Dic3—42no—
279C4×Dic3C2×C4×C514yes—
280C23×C5—22no—
281C23×C5—22no—
282C23×C5—22no—
283C23×C5—22no—
284C23×C5—22no—
285C23×C5—22no—
286C23×C5C2×C4×D314yes—
287C23×C5C23×D311yescharacteristic
288C5×D4—28no—
289C2×C4×C5—21no—
290C2×C4×C5—21no—
291C2×C4×C5—21no—
292C2×C4×C5—22no—
293C2×C4×C5—22no—
294C2×C4×C5—22no—
295C2×C4×C5—21no—
296C2×C4×C5—21no—
297C2×C4×C5—32no—
298C2×C4×C5—32no—
299C2×C4×C5—34no—
300C2×C4×C5—32no—
301C2×C4×C5—61no—
302C2×C4×C5—61no—
303C2×C4×C5—61no—
304C2×C4×C5—62no—
305C2×C4×C5—61no—
306C2×C4×C5—61no—
307C2×C4×C5—62no—
308C2×C4×C5—61no—
309C2×C4×C5—61no—
310C2×C4×C5—62no—
311C2×C4×C5—61no—
312C2×C4×C5—62no—
313C2×C4×C5—62no—
314C2×C4×C5—61no—
315C2×C4×C5—128no—
316C2×C4×C5C2×C4×D312yes—
317C2×C4×C5#23712yes—
318C2×C4×C5#23712yes—
319C2×C4×C5#23712yes—
320C22×D4C2×C5×D311yescharacteristic
321C23×C4—32no—
322C4×D4—38no—
323C2×C4⋊C4—31no—
324C2×C4⋊C4—31no—
325C2×C4⋊C4—31no—
326C2×C4⋊C4—31no—
327C2×C4⋊C4C2×C5×D311yescharacteristic
328C2×C22⋊C4—32no—
329C2×C22⋊C4—32no—
330C2×C22⋊C4—32no—
331C2×C22⋊C4C2×C5×D312yes—
332C2×C22⋊C4C2×C5×D312yes—
333C2×C42—31no—
334C2×C42—31no—
335C2×C42—32no—
336C2×C42—32no—
337C2×C42C2×C5×D311yescharacteristic
338#95—31no—
339#95—32no—
340#95—31no—
341#95—32no—
342#95—32no—
343#95—32no—
344C2×C3×C5—44no—
345C2×C3×C5C2×#11711yescharacteristic
346C2×C3×C5C4×#4111yescharacteristic
347C2×C3×C5#51811yescharacteristic
348C2×C3×C5#55311yescharacteristic
349C2×C3×C5#55311yescharacteristic
350C2×C3×C5#55911yescharacteristic
351C2×C3×C5#55911yescharacteristic
352C23×C3—22no—
353C23×C3—22no—
354C23×C3—22no—
355C23×C3—22no—
356C23×C3—22no—
357C23×C3—22no—
358C23×C3C22×C4×C514yes—
359C23×C3C24×C511yescharacteristic
360C3×D4—28no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—21no—
369C2×C3×C4C22×C4×C512yes—
370C2×C3×C4C5×#4112yes—
371C2×C3×C4C5×#4112yes—
372C2×C3×C4C5×#4112yes—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—22no—
378C2×Dic3—22no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—21no—
387C2×Dic3—48no—
388C2×Dic3C22×C4×C512yes—
389C2×Dic3C22×C4×C512yes—
390C2×Dic3C22×C4×C514yes—
391C2×Dic3C5×#4112yes—
392C22×C5—24no—
393C22×C5—24no—
394C22×C5—24no—
395C22×C5—42no—
396C22×C5—42no—
397C22×C5—42no—
398C22×C5—42no—
399C22×C5C22×C4×D311yescharacteristic
400C22×C5C2×#23711yescharacteristic
401C22×C5C2×#23711yescharacteristic
402C22×C5C2×#23711yescharacteristic
403C22×C5#100411yescharacteristic
404C22×C5D3×#4111yescharacteristic
405C22×C5#100911yescharacteristic
406C4×C5—22no—
407C4×C5—42no—
408C4×C5—41no—
409C4×C5—62no—
410C4×C5—64no—
411C4×C5—62no—
412C4×C5—62no—
413C4×C5—122no—
414C4×C5—121no—
415C4×C5#87714yes—
416C24C22×C5×D312yes—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4C4×C5×D314yes—
420C22×C4—31no—
421C22×C4—32no—
422C22×C4—31no—
423C22×C4—31no—
424C22×C4—38no—
425C22×C4—31no—
426C22×C4—32no—
427C22×C4—122no—
428C22×C4—122no—
429C22×C4C22×C5×D312yes—
430C22×C4C22×C5×D311yescharacteristic
431C22×C4C22×C5×D311yescharacteristic
432C22×C4C22×C5×D311yescharacteristic
433C4⋊C4—22no—
434C4⋊C4—34no—
435C4⋊C4—62no—
436C4⋊C4—62no—
437C4⋊C4—62no—
438C22⋊C4—24no—
439C22⋊C4—38no—
440C22⋊C4—42no—
441C22⋊C4—64no—
442C22⋊C4—122no—
443C42—22no—
444C42—34no—
445C42—62no—
446C42—122no—
447C42—122no—
448C3×C5#235711yescharacteristic, complemented, Hall
449C22×C3—24no—
450C22×C3—24no—
451C22×C3—24no—
452C22×C3—42no—
453C22×C3—42no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3C23×C4×C511yescharacteristic
457C22×C3C2×C5×#4111yescharacteristic
458C22×C3C2×C5×#4111yescharacteristic
459C22×C3C2×C5×#4111yescharacteristic
460C22×C3C2×C5×#4111yescharacteristic
461C22×C3C5×H(5,𝔽2)11yescharacteristic
462C22×C3C5×H(5,𝔽2)11yescharacteristic
463C3×C4—22no—
464C3×C4—41no—
465C3×C4—42no—
466C3×C4C5×#11714yes—
467Dic3—22no—
468Dic3—22no—
469Dic3—24no—
470Dic3—22no—
471Dic3—41no—
472Dic3—42no—
473C2×C5—44no—
474C2×C5C2×#87711yescharacteristic
475C2×C5C4×#23711yescharacteristic
476C2×C5#554511yescharacteristic
477C2×C5#561411yescharacteristic
478C2×C5#561811yescharacteristic
479C2×C5#568111yescharacteristic
480C2×C5#571311yescharacteristic
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23C2×C4×C5×D314yes—
488C23C23×C5×D311yescharacteristic
489D4—28no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—32no—
499C2×C4—32no—
500C2×C4—34no—
501C2×C4—32no—
502C2×C4—61no—
503C2×C4—61no—
504C2×C4—61no—
505C2×C4—62no—
506C2×C4—61no—
507C2×C4—61no—
508C2×C4—62no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—62no—
512C2×C4—61no—
513C2×C4—62no—
514C2×C4—62no—
515C2×C4—61no—
516C2×C4—128no—
517C2×C4C2×C4×C5×D312yes—
518C2×C4C5×#23712yes—
519C2×C4C5×#23712yes—
520C2×C4C5×#23712yes—
521C2×C3—44no—
522C2×C3C2×C5×#11711yescharacteristic
523C2×C3C4×C5×#4111yescharacteristic
524C2×C3C5×#51811yescharacteristic
525C2×C3C5×#55311yescharacteristic
526C2×C3C5×#55311yescharacteristic
527C2×C3C5×#55911yescharacteristic
528C2×C3C5×#55911yescharacteristic
529C5#7630311yescharacteristic, direct product factor, Sylow
530C22—24no—
531C22—24no—
532C22—24no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22C22×C4×C5×D311yescharacteristic
538C22C2×C5×#23711yescharacteristic
539C22C2×C5×#23711yescharacteristic
540C22C2×C5×#23711yescharacteristic
541C22C5×#100411yescharacteristic
542C22C5×D3×#4111yescharacteristic
543C22C5×#100911yescharacteristic
544C4—22no—
545C4—42no—
546C4—41no—
547C4—62no—
548C4—62no—
549C4—64no—
550C4—62no—
551C4—122no—
552C4—121no—
553C4C5×#87714yes—
554C3C5×#235711yescharacteristic, complemented, Sylow
555C2—44no—
556C2C2×C5×#87711yescharacteristic
557C2C4×C5×#23711yescharacteristic
558C2C5×#554511yescharacteristic
559C2C5×#561411yescharacteristic
560C2C5×#561811yescharacteristic
561C2C5×#568111yescharacteristic
562C2C5×#571311yescharacteristic
subgroupG/H
centerC23×C5C23×D3
commutatorC22×C3C23×C4×C5
FrattiniC23C23×C5×D3
FittingC3×C5×#390C2
socleC23×C3×C5C24
prime radicalC22×C3×C5×D4C22
cosocleC23×C3C24×C5
hypercenterC5×#390D3
hypocenterC3C5×#2357
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C5 ◁ C5×#390

Complements

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

NH ≅ G/Ntype
4. C5×#4977C2semidirect
5. C5×Dic3×C4⋊C4C2semidirect
6. C5×#4945C2semidirect
7. C5×#4944C2semidirect
8. C5×#4943C2semidirect
9. C5×Dic3×C22⋊C4C2semidirect
10. C4×C5×C4⋊Dic3C2semidirect
11. C4×C5×Dic3⋊C4C2semidirect
22. C5×D4×Dic3C4semidirect
37. #76303C5direct (N × H)
84. #4977C2×C5semidirect
85. Dic3×C4⋊C4C2×C5semidirect
86. #4945C2×C5semidirect
87. #4944C2×C5semidirect
88. #4943C2×C5semidirect
89. Dic3×C22⋊C4C2×C5semidirect
90. C4×C4⋊Dic3C2×C5semidirect
91. C4×Dic3⋊C4C2×C5semidirect
166. D4×Dic3C4×C5semidirect
448. C3×C5#2357semidirect
529. C5#76303direct (N × H)
554. C3C5×#2357semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C5×D4×Dic32
C3×C5×#3902
C5×#52212
C5×#49772
C5×Dic3×C4⋊C42
C5×#49452
C5×#49442
C5×#49432
C5×Dic3×C22⋊C42
C4×C5×C4⋊Dic32
C4×C5×Dic3⋊C42
C5×#23573
#763035