Finite Group Databasebeta

← C5×#82235 #14323708

Subgroups — C5×#82235 #14323708

Order 1920.

1112 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×#117C211yesmaximal, characteristic
2C2×C5×#924C211yesmaximal, characteristic
3C2×C4×C5×Dic6C211yesmaximal, characteristic, complemented
4C5×#5212C211yesmaximal, characteristic
5C5×#4978C212yesmaximal, complemented
6C5×#4976C212yesmaximal, complemented
7C5×#4945C212yesmaximal
8C5×#4944C212yesmaximal
9C4×C5×C4⋊Dic3C211yesmaximal, characteristic, complemented
10C5×#4929C211yesmaximal, characteristic, complemented
11C5×#4928C211yesmaximal, characteristic, complemented
12C5×#2323—31nomaximal, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C22×C5×Dic6C2211yescharacteristic
15C3×C5×#117—24no—
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18C2×C3×C5×C22⋊C4C2211yescharacteristic
19C2×C3×C5×C22⋊C4C2211yescharacteristic
20C2×C3×C42×C5C2211yescharacteristic
21C2×C3×C42×C5C2211yescharacteristic
22C2×C5×#217C2212yes—
23C2×C5×#217C2212yes—
24C2×C5×C4⋊Dic3C2211yescharacteristic
25C2×C5×C4⋊Dic3C2211yescharacteristic
26C2×C5×C4⋊Dic3C2211yescharacteristic
27C5×#924C418yescomplemented
28C2×C5×Dic3⋊C4C2212yes—
29C2×C5×Dic3⋊C4C2212yes—
30C2×C5×Dic3⋊C4C2212yes—
31C2×C4×C5×Dic3C2212yes—
32C2×C4×C5×Dic3C2212yes—
33C4×C5×Dic6—24no—
34C5×C3⋊#95C2212yes—
35C5×C3⋊#95C2212yes—
36C5×C3⋊#95C2212yes—
37C5×C3⋊#95C2212yes—
38C5×C3⋊#95C2212yes—
39#82235C511yesmaximal, characteristic, direct product factor, Hall
40C2×C5×C22⋊Q8—31no—
41C2×C4×C5×Q8—31no—
42C2×C5×#117D311yescharacteristic
43C5×#391—31no—
44C5×#389—32no—
45C5×#389—31no—
46C5×#388—32no—
47C5×#388—32no—
48C5×#387—32no—
49C5×#380—31no—
50C4×C5×C4⋊C4—31no—
51C24×C3×C5C2311yescharacteristic
52C22×C3×C4×C5—22no—
53C22×C3×C4×C5—22no—
54C22×C3×C4×C5C2×C414yes—
55C22×C3×C4×C5C2311yescharacteristic
56C22×C3×C4×C5C2311yescharacteristic
57C22×C3×C4×C5C2311yescharacteristic
58C22×C3×C4×C5C2311yescharacteristic
59C22×C3×C4×C5C2311yescharacteristic
60C22×C3×C4×C5C2311yescharacteristic
61C22×C5×Dic3C2312yes—
62C22×C5×Dic3C2312yes—
63C22×C5×Dic3C2312yes—
64C22×C5×Dic3C2312yes—
65C2×C5×Dic6—22no—
66C2×C5×Dic6—22no—
67C2×C5×Dic6C2×C414yes—
68C3×C5×C4⋊C4—22no—
69C3×C5×C4⋊C4—22no—
70C3×C5×C22⋊C4—22no—
71C3×C5×C22⋊C4—22no—
72C3×C42×C5—22no—
73C3×C42×C5D414yes—
74C5×#217—42no—
75C5×#217C2×C418yes—
76C5×C4⋊Dic3—22no—
77C5×C4⋊Dic3—22no—
78C5×C4⋊Dic3C2×C414yes—
79C5×Dic3⋊C4—24no—
80C5×Dic3⋊C4—42no—
81C5×Dic3⋊C4C2×C418yes—
82C4×C5×Dic3—24no—
83C4×C5×Dic3—42no—
84C2×C3×#117C2×C511yescharacteristic
85C2×#924C2×C511yescharacteristic
86C2×C4×Dic6C2×C511yescharacteristic, complemented
87#5212C2×C511yescharacteristic
88#4978C2×C512yescomplemented
89#4976C2×C512yescomplemented
90#4945C2×C512yes—
91#4944C2×C512yes—
92C4×C4⋊Dic3C2×C511yescharacteristic, complemented
93#4929C2×C511yescharacteristic, complemented
94#4928C2×C511yescharacteristic, complemented
95C22×C5×Q8—31no—
96C23×C4×C5C2×D311yescharacteristic
97C5×C22⋊Q8—38no—
98C4×C5×Q8—64no—
99C5×#117—24no—
100C2×C5×C4⋊C4—31no—
101C2×C5×C4⋊C4—32no—
102C2×C5×C4⋊C4—31no—
103C2×C5×C4⋊C4—31no—
104C2×C5×C4⋊C4—32no—
105C2×C5×C4⋊C4—32no—
106C2×C5×C4⋊C4C2×D311yescharacteristic
107C2×C5×C4⋊C4C2×D311yescharacteristic
108C2×C5×C22⋊C4—32no—
109C2×C5×C22⋊C4—32no—
110C2×C5×C22⋊C4C2×D311yescharacteristic
111C2×C5×C22⋊C4C2×D311yescharacteristic
112C2×C42×C5—32no—
113C2×C42×C5—32no—
114C2×C42×C5C2×D311yescharacteristic
115C2×C42×C5C2×D311yescharacteristic
116C5×#95—32no—
117C5×#95—32no—
118C5×#95—32no—
119C5×#95—32no—
120C5×#95—32no—
121#2323—31noSylow
122C23×C3×C5—21no—
123C23×C3×C5—21no—
124C23×C3×C5—21no—
125C23×C3×C5—21no—
126C23×C3×C5—21no—
127C23×C3×C5—21no—
128C23×C3×C5C22×C412yes—
129C23×C3×C5C2411yescharacteristic
130C2×C3×C4×C5—21no—
131C2×C3×C4×C5—21no—
132C2×C3×C4×C5—21no—
133C2×C3×C4×C5—21no—
134C2×C3×C4×C5—21no—
135C2×C3×C4×C5—21no—
136C2×C3×C4×C5—28no—
137C2×C3×C4×C5—21no—
138C2×C3×C4×C5—21no—
139C2×C3×C4×C5—21no—
140C2×C3×C4×C5—21no—
141C2×C3×C4×C5—21no—
142C2×C3×C4×C5—21no—
143C2×C3×C4×C5C22×C412yes—
144C2×C3×C4×C5C22×C412yes—
145C2×C3×C4×C5C2×D412yes—
146C2×C3×C4×C5C2×D412yes—
147C2×C3×C4×C5C2×D412yes—
148C2×C3×C4×C5#4112yes—
149C2×C5×Dic3—22no—
150C2×C5×Dic3—22no—
151C2×C5×Dic3—22no—
152C2×C5×Dic3—22no—
153C2×C5×Dic3—22no—
154C2×C5×Dic3—22no—
155C2×C5×Dic3—22no—
156C2×C5×Dic3—22no—
157C2×C5×Dic3—22no—
158C2×C5×Dic3—22no—
159C2×C5×Dic3C22×C414yes—
160C2×C5×Dic3C22×C414yes—
161C5×Dic6—28no—
162C23×C3×C4C22×C511yescharacteristic
163C22×Dic6C22×C511yescharacteristic
164C3×#117—24no—
165C2×C3×C4⋊C4C22×C511yescharacteristic
166C2×C3×C4⋊C4C22×C511yescharacteristic
167C2×C3×C22⋊C4C22×C511yescharacteristic
168C2×C3×C22⋊C4C22×C511yescharacteristic
169C2×C3×C42C22×C511yescharacteristic
170C2×C3×C42C22×C511yescharacteristic
171C2×#217C22×C512yes—
172C2×#217C22×C512yes—
173C2×C4⋊Dic3C22×C511yescharacteristic
174C2×C4⋊Dic3C22×C511yescharacteristic
175C2×C4⋊Dic3C22×C511yescharacteristic
176#924C4×C518yescomplemented
177C2×Dic3⋊C4C22×C512yes—
178C2×Dic3⋊C4C22×C512yes—
179C2×Dic3⋊C4C22×C512yes—
180C2×C4×Dic3C22×C512yes—
181C2×C4×Dic3C22×C512yes—
182C4×Dic6—24no—
183C3⋊#95C22×C512yes—
184C3⋊#95C22×C512yes—
185C3⋊#95C22×C512yes—
186C3⋊#95C22×C512yes—
187C3⋊#95C22×C512yes—
188C24×C5C22×D311yescharacteristic
189C2×C5×Q8—34no—
190C2×C5×Q8—62no—
191C2×C5×Q8—62no—
192C22×C4×C5—22no—
193C22×C4×C5—22no—
194C22×C4×C5—32no—
195C22×C4×C5—32no—
196C22×C4×C5—32no—
197C22×C4×C5—32no—
198C22×C4×C5C4×D314yes—
199C22×C4×C5C22×D311yescharacteristic
200C22×C4×C5C22×D311yescharacteristic
201C22×C4×C5C22×D311yescharacteristic
202C22×C4×C5C22×D311yescharacteristic
203C22×C4×C5C22×D311yescharacteristic
204C22×C4×C5C22×D311yescharacteristic
205C5×C4⋊C4—22no—
206C5×C4⋊C4—22no—
207C5×C4⋊C4—38no—
208C5×C4⋊C4—34no—
209C5×C4⋊C4—62no—
210C5×C4⋊C4—64no—
211C5×C4⋊C4—62no—
212C5×C4⋊C4—122no—
213C5×C22⋊C4—22no—
214C5×C22⋊C4—22no—
215C5×C22⋊C4—38no—
216C5×C22⋊C4—122no—
217C42×C5—22no—
218C42×C5—64no—
219C42×C5—122no—
220C42×C5#6714yes—
221C2×C22⋊Q8—31no—
222C2×C4×Q8—31no—
223C2×#117C5×D311yescharacteristic
224#391—31no—
225#389—32no—
226#389—31no—
227#388—32no—
228#388—32no—
229#387—32no—
230#380—31no—
231C4×C4⋊C4—31no—
232C22×C3×C5—22no—
233C22×C3×C5—22no—
234C22×C3×C5—22no—
235C22×C3×C5—41no—
236C22×C3×C5—41no—
237C22×C3×C5—41no—
238C22×C3×C5—41no—
239C22×C3×C5C23×C411yescharacteristic
240C22×C3×C5C22×D411yescharacteristic
241C22×C3×C5C2×#4111yescharacteristic
242C22×C3×C5H(5,𝔽2)11yescharacteristic
243C22×C3×C52−1+411yescharacteristic
244C22×C3×C52−1+411yescharacteristic
245C22×C3×C52−1+411yescharacteristic
246C3×C4×C5—22no—
247C3×C4×C5—22no—
248C3×C4×C5—22no—
249C3×C4×C5—41no—
250C3×C4×C5—41no—
251C3×C4×C5C4×D414yes—
252C5×Dic3—24no—
253C5×Dic3—24no—
254C5×Dic3—42no—
255C5×Dic3—42no—
256C24×C3C23×C511yescharacteristic
257C22×C3×C4—22no—
258C22×C3×C4—22no—
259C22×C3×C4C2×C4×C514yes—
260C22×C3×C4C23×C511yescharacteristic
261C22×C3×C4C23×C511yescharacteristic
262C22×C3×C4C23×C511yescharacteristic
263C22×C3×C4C23×C511yescharacteristic
264C22×C3×C4C23×C511yescharacteristic
265C22×C3×C4C23×C511yescharacteristic
266C22×Dic3C23×C512yes—
267C22×Dic3C23×C512yes—
268C22×Dic3C23×C512yes—
269C22×Dic3C23×C512yes—
270C2×Dic6—22no—
271C2×Dic6—22no—
272C2×Dic6C2×C4×C514yes—
273C3×C4⋊C4—22no—
274C3×C4⋊C4—22no—
275C3×C22⋊C4—22no—
276C3×C22⋊C4—22no—
277C3×C42—22no—
278C3×C42C5×D414yes—
279#217—42no—
280#217C2×C4×C518yes—
281C4⋊Dic3—22no—
282C4⋊Dic3—22no—
283C4⋊Dic3C2×C4×C514yes—
284Dic3⋊C4—24no—
285Dic3⋊C4—42no—
286Dic3⋊C4C2×C4×C518yes—
287C4×Dic3—24no—
288C4×Dic3—42no—
289C23×C5—21no—
290C23×C5—21no—
291C23×C5—21no—
292C23×C5—21no—
293C23×C5—21no—
294C23×C5—21no—
295C23×C5C2×C4×D312yes—
296C23×C5C23×D311yescharacteristic
297C5×Q8—68no—
298C2×C4×C5—21no—
299C2×C4×C5—21no—
300C2×C4×C5—21no—
301C2×C4×C5—21no—
302C2×C4×C5—21no—
303C2×C4×C5—21no—
304C2×C4×C5—28no—
305C2×C4×C5—21no—
306C2×C4×C5—21no—
307C2×C4×C5—21no—
308C2×C4×C5—21no—
309C2×C4×C5—21no—
310C2×C4×C5—21no—
311C2×C4×C5—34no—
312C2×C4×C5—34no—
313C2×C4×C5—62no—
314C2×C4×C5—62no—
315C2×C4×C5—62no—
316C2×C4×C5—62no—
317C2×C4×C5—62no—
318C2×C4×C5—62no—
319C2×C4×C5—62no—
320C2×C4×C5—62no—
321C2×C4×C5—62no—
322C2×C4×C5—62no—
323C2×C4×C5C2×C4×D312yes—
324C2×C4×C5C2×C4×D312yes—
325C2×C4×C5#23512yes—
326C2×C4×C5C2×#6712yes—
327C2×C4×C5C2×#6712yes—
328C2×C4×C5C2×#6712yes—
329C22×Q8—31no—
330C23×C4C2×C5×D311yescharacteristic
331C22⋊Q8—38no—
332C4×Q8—64no—
333#117—24no—
334C2×C4⋊C4—31no—
335C2×C4⋊C4—32no—
336C2×C4⋊C4—31no—
337C2×C4⋊C4—32no—
338C2×C4⋊C4—31no—
339C2×C4⋊C4—32no—
340C2×C4⋊C4C2×C5×D311yescharacteristic
341C2×C4⋊C4C2×C5×D311yescharacteristic
342C2×C22⋊C4—32no—
343C2×C22⋊C4—32no—
344C2×C22⋊C4C2×C5×D311yescharacteristic
345C2×C22⋊C4C2×C5×D311yescharacteristic
346C2×C42—32no—
347C2×C42—32no—
348C2×C42C2×C5×D311yescharacteristic
349C2×C42C2×C5×D311yescharacteristic
350#95—32no—
351#95—32no—
352#95—32no—
353#95—32no—
354#95—32no—
355C2×C3×C5—42no—
356C2×C3×C5C2×C4×D411yescharacteristic
357C2×C3×C5#51911yescharacteristic
358C2×C3×C5#52011yescharacteristic
359C2×C3×C5#53611yescharacteristic
360C2×C3×C5#53711yescharacteristic
361C2×C3×C5#54111yescharacteristic
362C2×C3×C5#54211yescharacteristic
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3—21no—
369C23×C3C22×C4×C512yes—
370C23×C3C24×C511yescharacteristic
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—28no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4C22×C4×C512yes—
385C2×C3×C4C22×C4×C512yes—
386C2×C3×C4C2×C5×D412yes—
387C2×C3×C4C2×C5×D412yes—
388C2×C3×C4C2×C5×D412yes—
389C2×C3×C4C5×#4112yes—
390C2×Dic3—22no—
391C2×Dic3—22no—
392C2×Dic3—22no—
393C2×Dic3—22no—
394C2×Dic3—22no—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3—22no—
399C2×Dic3—22no—
400C2×Dic3C22×C4×C514yes—
401C2×Dic3C22×C4×C514yes—
402Dic6—28no—
403C22×C5—22no—
404C22×C5—22no—
405C22×C5—22no—
406C22×C5—41no—
407C22×C5—41no—
408C22×C5—41no—
409C22×C5—41no—
410C22×C5C22×C4×D311yescharacteristic
411C22×C5C2×#23511yescharacteristic
412C22×C5#100911yescharacteristic
413C22×C5#101011yescharacteristic
414C22×C5#101011yescharacteristic
415C22×C5#101011yescharacteristic
416C22×C5C22×#6711yescharacteristic
417C4×C5—22no—
418C4×C5—22no—
419C4×C5—22no—
420C4×C5—41no—
421C4×C5—41no—
422C4×C5—64no—
423C4×C5—64no—
424C4×C5—122no—
425C4×C5—122no—
426C4×C5C4×#6714yes—
427C24C22×C5×D311yescharacteristic
428C2×Q8—34no—
429C2×Q8—62no—
430C2×Q8—62no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—32no—
434C22×C4—32no—
435C22×C4—32no—
436C22×C4—32no—
437C22×C4C4×C5×D314yes—
438C22×C4C22×C5×D311yescharacteristic
439C22×C4C22×C5×D311yescharacteristic
440C22×C4C22×C5×D311yescharacteristic
441C22×C4C22×C5×D311yescharacteristic
442C22×C4C22×C5×D311yescharacteristic
443C22×C4C22×C5×D311yescharacteristic
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—38no—
447C4⋊C4—34no—
448C4⋊C4—62no—
449C4⋊C4—64no—
450C4⋊C4—62no—
451C4⋊C4—122no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—38no—
455C22⋊C4—122no—
456C42—22no—
457C42—64no—
458C42—122no—
459C42C5×#6714yes—
460C3×C5#232311yescharacteristic, complemented, Hall
461C22×C3—22no—
462C22×C3—22no—
463C22×C3—22no—
464C22×C3—41no—
465C22×C3—41no—
466C22×C3—41no—
467C22×C3—41no—
468C22×C3C23×C4×C511yescharacteristic
469C22×C3C22×C5×D411yescharacteristic
470C22×C3C2×C5×#4111yescharacteristic
471C22×C3C5×H(5,𝔽2)11yescharacteristic
472C22×C3C5×2−1+411yescharacteristic
473C22×C3C5×2−1+411yescharacteristic
474C22×C3C5×2−1+411yescharacteristic
475C3×C4—22no—
476C3×C4—22no—
477C3×C4—22no—
478C3×C4—41no—
479C3×C4—41no—
480C3×C4C4×C5×D414yes—
481Dic3—24no—
482Dic3—24no—
483Dic3—42no—
484Dic3—42no—
485C2×C5—42no—
486C2×C5#551611yescharacteristic
487C2×C5#551811yescharacteristic
488C2×C5#552311yescharacteristic
489C2×C5#553411yescharacteristic
490C2×C5C2×C4×#6711yescharacteristic
491C2×C5#582511yescharacteristic
492C2×C5#583311yescharacteristic
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23C2×C4×C5×D312yes—
500C23C23×C5×D311yescharacteristic
501Q8—68no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—28no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—34no—
516C2×C4—34no—
517C2×C4—62no—
518C2×C4—62no—
519C2×C4—62no—
520C2×C4—62no—
521C2×C4—62no—
522C2×C4—62no—
523C2×C4—62no—
524C2×C4—62no—
525C2×C4—62no—
526C2×C4—62no—
527C2×C4C2×C4×C5×D312yes—
528C2×C4C2×C4×C5×D312yes—
529C2×C4C5×#23512yes—
530C2×C4C2×C5×#6712yes—
531C2×C4C2×C5×#6712yes—
532C2×C4C2×C5×#6712yes—
533C2×C3—42no—
534C2×C3C2×C4×C5×D411yescharacteristic
535C2×C3C5×#51911yescharacteristic
536C2×C3C5×#52011yescharacteristic
537C2×C3C5×#53611yescharacteristic
538C2×C3C5×#53711yescharacteristic
539C2×C3C5×#54111yescharacteristic
540C2×C3C5×#54211yescharacteristic
541C5#8223511yescharacteristic, direct product factor, Sylow
542C22—22no—
543C22—22no—
544C22—22no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22C22×C4×C5×D311yescharacteristic
550C22C2×C5×#23511yescharacteristic
551C22C5×#100911yescharacteristic
552C22C5×#101011yescharacteristic
553C22C5×#101011yescharacteristic
554C22C5×#101011yescharacteristic
555C22C22×C5×#6711yescharacteristic
556C4—22no—
557C4—22no—
558C4—22no—
559C4—41no—
560C4—41no—
561C4—64no—
562C4—64no—
563C4—122no—
564C4—122no—
565C4C4×C5×#6714yes—
566C3C5×#232311yescharacteristic, complemented, Sylow
567C2—42no—
568C2C5×#551611yescharacteristic
569C2C5×#551811yescharacteristic
570C2C5×#552311yescharacteristic
571C2C5×#553411yescharacteristic
572C2C2×C4×C5×#6711yescharacteristic
573C2C5×#582511yescharacteristic
574C2C5×#583311yescharacteristic
subgroupG/H
centerC23×C5C23×D3
commutatorC22×C3C23×C4×C5
FrattiniC23C23×C5×D3
FittingC2×C3×C5×#117C2
socleC23×C3×C5C24
prime radicalC24×C3×C5C23
cosocleC23×C3C24×C5
hypercenterC2×C5×#117D3
hypocenterC3C5×#2323
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C5 ◁ C2×C5×#117

Complements

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

NH ≅ G/Ntype
3. C2×C4×C5×Dic6C2semidirect
5. C5×#4978C2semidirect
6. C5×#4976C2semidirect
9. C4×C5×C4⋊Dic3C2semidirect
10. C5×#4929C2semidirect
11. C5×#4928C2semidirect
27. C5×#924C4semidirect
39. #82235C5direct (N × H)
86. C2×C4×Dic6C2×C5semidirect
88. #4978C2×C5semidirect
89. #4976C2×C5semidirect
92. C4×C4⋊Dic3C2×C5semidirect
93. #4929C2×C5semidirect
94. #4928C2×C5semidirect
176. #924C4×C5semidirect
460. C3×C5#2323semidirect
541. C5#82235direct (N × H)
566. C3C5×#2323semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C5×#1172
C2×C5×#9242
C2×C4×C5×Dic62
C5×#52122
C5×#49782
C5×#49762
C5×#49452
C5×#49442
C4×C5×C4⋊Dic32
C5×#49292
C5×#49282
C5×#23233
#822355