Finite Group Databasebeta

← C5×#73868 #14322707

Subgroups — C5×#73868 #14322707

Order 1920.

968 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#926C211yesmaximal, characteristic
2C3×C5×#380C211yesmaximal, characteristic
3C5×#5212C211yesmaximal, characteristic
4C5×#4977C211yesmaximal, characteristic, complemented
5C5×#4975C211yesmaximal, characteristic, complemented
6C5×Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
7C5×Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
8C5×#4945C212yesmaximal
9C5×#4944C212yesmaximal
10C5×#4654C212yesmaximal, complemented
11C5×#4653C212yesmaximal, complemented
12C5×#2314—31nomaximal, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C2×C3×C5×C4⋊C4C2211yescharacteristic
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×C3×C5×C22⋊C4C2212yes—
17C2×C5×#217C2212yes—
18C2×C5×#217C2211yescharacteristic
19C2×C5×#217C2211yescharacteristic
20C5×#926C418yescomplemented
21C2×C5×C4⋊Dic3C2211yescharacteristic
22C2×C5×C4⋊Dic3C2211yescharacteristic
23C2×C5×C4⋊Dic3C2211yescharacteristic
24C2×C5×C4⋊Dic3C2211yescharacteristic
25C2×C5×Dic3⋊C4C2212yes—
26C2×C5×Dic3⋊C4C2212yes—
27C2×C4×C5×Dic3C2212yes—
28C2×C4×C5×Dic3C2212yes—
29C2×C4×C5×Dic3C2211yescharacteristic
30C2×C4×C5×Dic3C2211yescharacteristic
31C3×C5×#95C2212yes—
32C5×C3⋊#95C2212yes—
33C5×C3⋊#95C2212yes—
34C5×C3⋊#95C2212yes—
35C5×C3⋊#95C2212yes—
36C5×C3⋊#95C2212yes—
37#73868C511yesmaximal, characteristic, direct product factor, Hall
38C2×C5×#117—31no—
39C5×#388—32no—
40C5×#388—32no—
41C5×#387—32no—
42C5×#387—32no—
43C5×#384—31no—
44C5×#382—31no—
45C5×#380—31no—
46C5×#380D311yescharacteristic
47C4×C5×C4⋊C4—31no—
48C4×C5×C4⋊C4—31no—
49C24×C3×C5C2311yescharacteristic
50C22×C3×C4×C5—22no—
51C22×C3×C4×C5—22no—
52C22×C3×C4×C5C2×C414yes—
53C22×C3×C4×C5C2312yes—
54C22×C3×C4×C5C2312yes—
55C22×C3×C4×C5C2311yescharacteristic
56C22×C3×C4×C5C2311yescharacteristic
57C22×C5×Dic3C2312yes—
58C22×C5×Dic3C2312yes—
59C22×C5×Dic3C2311yescharacteristic
60C22×C5×Dic3C2311yescharacteristic
61C22×C5×Dic3C2311yescharacteristic
62C22×C5×Dic3C2311yescharacteristic
63C3×C5×C4⋊C4—22no—
64C3×C5×C4⋊C4—22no—
65C3×C5×C22⋊C4—42no—
66C5×#217—42no—
67C5×#217C2×C414yes—
68C5×#217C2×C414yes—
69C5×C4⋊Dic3—22no—
70C5×C4⋊Dic3—22no—
71C5×C4⋊Dic3C2×C414yes—
72C5×C4⋊Dic3C2×C414yes—
73C5×Dic3⋊C4—42no—
74C5×Dic3⋊C4—42no—
75C4×C5×Dic3—42no—
76C4×C5×Dic3—42no—
77C4×C5×Dic3C2×C414yes—
78C4×C5×Dic3C2×C414yes—
79C2×#926C2×C511yescharacteristic
80C3×#380C2×C511yescharacteristic
81#5212C2×C511yescharacteristic
82#4977C2×C511yescharacteristic, complemented
83#4975C2×C511yescharacteristic, complemented
84Dic3×C4⋊C4C2×C511yescharacteristic, complemented
85Dic3×C4⋊C4C2×C511yescharacteristic, complemented
86#4945C2×C512yes—
87#4944C2×C512yes—
88#4654C2×C512yescomplemented
89#4653C2×C512yescomplemented
90C23×C4×C5C2×D311yescharacteristic
91C5×#117—38no—
92C2×C5×C4⋊C4—31no—
93C2×C5×C4⋊C4—31no—
94C2×C5×C4⋊C4—31no—
95C2×C5×C4⋊C4—31no—
96C2×C5×C4⋊C4—32no—
97C2×C5×C4⋊C4—32no—
98C2×C5×C4⋊C4C2×D311yescharacteristic
99C2×C5×C4⋊C4C2×D311yescharacteristic
100C2×C5×C22⋊C4—31no—
101C2×C5×C22⋊C4—31no—
102C2×C5×C22⋊C4—32no—
103C2×C5×C22⋊C4C2×D312yes—
104C2×C42×C5—31no—
105C2×C42×C5—31no—
106C2×C42×C5—32no—
107C2×C42×C5—32no—
108C5×#95—32no—
109C5×#95—32no—
110C5×#95—32no—
111C5×#95—32no—
112C5×#95—32no—
113C5×#95C2×D312yes—
114#2314—31noSylow
115C23×C3×C5—21no—
116C23×C3×C5—21no—
117C23×C3×C5—21no—
118C23×C3×C5—21no—
119C23×C3×C5—21no—
120C23×C3×C5—21no—
121C23×C3×C5C22×C412yes—
122C23×C3×C5C2411yescharacteristic
123C2×C3×C4×C5—21no—
124C2×C3×C4×C5—21no—
125C2×C3×C4×C5—28no—
126C2×C3×C4×C5—22no—
127C2×C3×C4×C5—22no—
128C2×C3×C4×C5—22no—
129C2×C3×C4×C5—22no—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5C22×C412yes—
133C2×C3×C4×C5C22×C412yes—
134C2×C3×C4×C5#4112yes—
135C2×C3×C4×C5#4112yes—
136C2×C5×Dic3—21no—
137C2×C5×Dic3—21no—
138C2×C5×Dic3—21no—
139C2×C5×Dic3—21no—
140C2×C5×Dic3—21no—
141C2×C5×Dic3—21no—
142C2×C5×Dic3—21no—
143C2×C5×Dic3—21no—
144C2×C5×Dic3—22no—
145C2×C5×Dic3—22no—
146C2×C5×Dic3—22no—
147C2×C5×Dic3—22no—
148C2×C5×Dic3—22no—
149C2×C5×Dic3—22no—
150C2×C5×Dic3C22×C412yes—
151C2×C5×Dic3C22×C412yes—
152C2×C5×Dic3C22×C412yes—
153C2×C5×Dic3C22×C412yes—
154C23×C3×C4C22×C511yescharacteristic
155C2×C3×C4⋊C4C22×C511yescharacteristic
156C2×C3×C4⋊C4C22×C511yescharacteristic
157C2×C3×C22⋊C4C22×C512yes—
158C2×#217C22×C512yes—
159C2×#217C22×C511yescharacteristic
160C2×#217C22×C511yescharacteristic
161#926C4×C518yescomplemented
162C2×C4⋊Dic3C22×C511yescharacteristic
163C2×C4⋊Dic3C22×C511yescharacteristic
164C2×C4⋊Dic3C22×C511yescharacteristic
165C2×C4⋊Dic3C22×C511yescharacteristic
166C2×Dic3⋊C4C22×C512yes—
167C2×Dic3⋊C4C22×C512yes—
168C2×C4×Dic3C22×C512yes—
169C2×C4×Dic3C22×C512yes—
170C2×C4×Dic3C22×C511yescharacteristic
171C2×C4×Dic3C22×C511yescharacteristic
172C3×#95C22×C512yes—
173C3⋊#95C22×C512yes—
174C3⋊#95C22×C512yes—
175C3⋊#95C22×C512yes—
176C3⋊#95C22×C512yes—
177C3⋊#95C22×C512yes—
178C24×C5C22×D311yescharacteristic
179C22×C4×C5—22no—
180C22×C4×C5—22no—
181C22×C4×C5—31no—
182C22×C4×C5—31no—
183C22×C4×C5—31no—
184C22×C4×C5—31no—
185C22×C4×C5—32no—
186C22×C4×C5—32no—
187C22×C4×C5C4×D314yes—
188C22×C4×C5C22×D312yes—
189C22×C4×C5C22×D312yes—
190C22×C4×C5C22×D311yescharacteristic
191C22×C4×C5C22×D311yescharacteristic
192C5×C4⋊C4—22no—
193C5×C4⋊C4—22no—
194C5×C4⋊C4—34no—
195C5×C4⋊C4—34no—
196C5×C4⋊C4—62no—
197C5×C4⋊C4—62no—
198C5×C4⋊C4—122no—
199C5×C4⋊C4—122no—
200C5×C22⋊C4—34no—
201C5×C22⋊C4—34no—
202C5×C22⋊C4—42no—
203C5×C22⋊C4—122no—
204C42×C5—34no—
205C42×C5—34no—
206C42×C5—122no—
207C42×C5—122no—
208C2×#117—31no—
209#388—32no—
210#388—32no—
211#387—32no—
212#387—32no—
213#384—31no—
214#382—31no—
215#380—31no—
216#380C5×D311yescharacteristic
217C4×C4⋊C4—31no—
218C4×C4⋊C4—31no—
219C22×C3×C5—22no—
220C22×C3×C5—22no—
221C22×C3×C5—22no—
222C22×C3×C5—41no—
223C22×C3×C5—41no—
224C22×C3×C5—41no—
225C22×C3×C5—41no—
226C22×C3×C5C23×C411yescharacteristic
227C22×C3×C5C2×#4111yescharacteristic
228C22×C3×C5C2×#4111yescharacteristic
229C22×C3×C5H(5,𝔽2)11yescharacteristic
230C22×C3×C5H(5,𝔽2)11yescharacteristic
231C22×C3×C52−1+411yescharacteristic
232C22×C3×C52−1+411yescharacteristic
233C3×C4×C5—22no—
234C3×C4×C5—42no—
235C3×C4×C5—42no—
236C3×C4×C5#11714yes—
237C5×Dic3—22no—
238C5×Dic3—22no—
239C5×Dic3—22no—
240C5×Dic3—22no—
241C5×Dic3—42no—
242C5×Dic3—42no—
243C24×C3C23×C511yescharacteristic
244C22×C3×C4—22no—
245C22×C3×C4—22no—
246C22×C3×C4C2×C4×C514yes—
247C22×C3×C4C23×C512yes—
248C22×C3×C4C23×C512yes—
249C22×C3×C4C23×C511yescharacteristic
250C22×C3×C4C23×C511yescharacteristic
251C22×Dic3C23×C512yes—
252C22×Dic3C23×C512yes—
253C22×Dic3C23×C511yescharacteristic
254C22×Dic3C23×C511yescharacteristic
255C22×Dic3C23×C511yescharacteristic
256C22×Dic3C23×C511yescharacteristic
257C3×C4⋊C4—22no—
258C3×C4⋊C4—22no—
259C3×C22⋊C4—42no—
260#217—42no—
261#217C2×C4×C514yes—
262#217C2×C4×C514yes—
263C4⋊Dic3—22no—
264C4⋊Dic3—22no—
265C4⋊Dic3C2×C4×C514yes—
266C4⋊Dic3C2×C4×C514yes—
267Dic3⋊C4—42no—
268Dic3⋊C4—42no—
269C4×Dic3—42no—
270C4×Dic3—42no—
271C4×Dic3C2×C4×C514yes—
272C4×Dic3C2×C4×C514yes—
273C23×C5—21no—
274C23×C5—21no—
275C23×C5—21no—
276C23×C5—21no—
277C23×C5—21no—
278C23×C5—21no—
279C23×C5C2×C4×D312yes—
280C23×C5C23×D311yescharacteristic
281C2×C4×C5—21no—
282C2×C4×C5—21no—
283C2×C4×C5—28no—
284C2×C4×C5—22no—
285C2×C4×C5—22no—
286C2×C4×C5—22no—
287C2×C4×C5—22no—
288C2×C4×C5—22no—
289C2×C4×C5—22no—
290C2×C4×C5—32no—
291C2×C4×C5—32no—
292C2×C4×C5—32no—
293C2×C4×C5—32no—
294C2×C4×C5—61no—
295C2×C4×C5—61no—
296C2×C4×C5—61no—
297C2×C4×C5—61no—
298C2×C4×C5—61no—
299C2×C4×C5—61no—
300C2×C4×C5—61no—
301C2×C4×C5—61no—
302C2×C4×C5—62no—
303C2×C4×C5—62no—
304C2×C4×C5—62no—
305C2×C4×C5—62no—
306C2×C4×C5—62no—
307C2×C4×C5—62no—
308C2×C4×C5C2×C4×D312yes—
309C2×C4×C5C2×C4×D312yes—
310C2×C4×C5#23712yes—
311C2×C4×C5#23712yes—
312C23×C4C2×C5×D311yescharacteristic
313#117—38no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—32no—
319C2×C4⋊C4—32no—
320C2×C4⋊C4C2×C5×D311yescharacteristic
321C2×C4⋊C4C2×C5×D311yescharacteristic
322C2×C22⋊C4—31no—
323C2×C22⋊C4—31no—
324C2×C22⋊C4—32no—
325C2×C22⋊C4C2×C5×D312yes—
326C2×C42—31no—
327C2×C42—31no—
328C2×C42—32no—
329C2×C42—32no—
330#95—32no—
331#95—32no—
332#95—32no—
333#95—32no—
334#95—32no—
335#95C2×C5×D312yes—
336C2×C3×C5—42no—
337C2×C3×C5C2×#11711yescharacteristic
338C2×C3×C5#52011yescharacteristic
339C2×C3×C5#52011yescharacteristic
340C2×C3×C5#54011yescharacteristic
341C2×C3×C5#54111yescharacteristic
342C2×C3×C5#54211yescharacteristic
343C2×C3×C5#54211yescharacteristic
344C23×C3—21no—
345C23×C3—21no—
346C23×C3—21no—
347C23×C3—21no—
348C23×C3—21no—
349C23×C3—21no—
350C23×C3C22×C4×C512yes—
351C23×C3C24×C511yescharacteristic
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—28no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4C22×C4×C512yes—
362C2×C3×C4C22×C4×C512yes—
363C2×C3×C4C5×#4112yes—
364C2×C3×C4C5×#4112yes—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—22no—
374C2×Dic3—22no—
375C2×Dic3—22no—
376C2×Dic3—22no—
377C2×Dic3—22no—
378C2×Dic3—22no—
379C2×Dic3C22×C4×C512yes—
380C2×Dic3C22×C4×C512yes—
381C2×Dic3C22×C4×C512yes—
382C2×Dic3C22×C4×C512yes—
383C22×C5—22no—
384C22×C5—22no—
385C22×C5—22no—
386C22×C5—41no—
387C22×C5—41no—
388C22×C5—41no—
389C22×C5—41no—
390C22×C5C22×C4×D311yescharacteristic
391C22×C5C2×#23711yescharacteristic
392C22×C5C2×#23711yescharacteristic
393C22×C5#100411yescharacteristic
394C22×C5#100711yescharacteristic
395C22×C5#100911yescharacteristic
396C22×C5#101011yescharacteristic
397C4×C5—22no—
398C4×C5—42no—
399C4×C5—42no—
400C4×C5—62no—
401C4×C5—62no—
402C4×C5—62no—
403C4×C5—62no—
404C4×C5—122no—
405C4×C5—122no—
406C4×C5#87714yes—
407C24C22×C5×D311yescharacteristic
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—31no—
411C22×C4—31no—
412C22×C4—31no—
413C22×C4—31no—
414C22×C4—32no—
415C22×C4—32no—
416C22×C4C4×C5×D314yes—
417C22×C4C22×C5×D312yes—
418C22×C4C22×C5×D312yes—
419C22×C4C22×C5×D311yescharacteristic
420C22×C4C22×C5×D311yescharacteristic
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—34no—
424C4⋊C4—34no—
425C4⋊C4—62no—
426C4⋊C4—62no—
427C4⋊C4—122no—
428C4⋊C4—122no—
429C22⋊C4—34no—
430C22⋊C4—34no—
431C22⋊C4—42no—
432C22⋊C4—122no—
433C42—34no—
434C42—34no—
435C42—122no—
436C42—122no—
437C3×C5#231411yescharacteristic, complemented, Hall
438C22×C3—22no—
439C22×C3—22no—
440C22×C3—22no—
441C22×C3—41no—
442C22×C3—41no—
443C22×C3—41no—
444C22×C3—41no—
445C22×C3C23×C4×C511yescharacteristic
446C22×C3C2×C5×#4111yescharacteristic
447C22×C3C2×C5×#4111yescharacteristic
448C22×C3C5×H(5,𝔽2)11yescharacteristic
449C22×C3C5×H(5,𝔽2)11yescharacteristic
450C22×C3C5×2−1+411yescharacteristic
451C22×C3C5×2−1+411yescharacteristic
452C3×C4—22no—
453C3×C4—42no—
454C3×C4—42no—
455C3×C4C5×#11714yes—
456Dic3—22no—
457Dic3—22no—
458Dic3—22no—
459Dic3—22no—
460Dic3—42no—
461Dic3—42no—
462C2×C5—42no—
463C2×C5C2×#87711yescharacteristic
464C2×C5#550411yescharacteristic
465C2×C5#552311yescharacteristic
466C2×C5#560311yescharacteristic
467C2×C5#561911yescharacteristic
468C2×C5#562511yescharacteristic
469C2×C5#564311yescharacteristic
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23C2×C4×C5×D312yes—
477C23C23×C5×D311yescharacteristic
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—28no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—32no—
488C2×C4—32no—
489C2×C4—32no—
490C2×C4—32no—
491C2×C4—61no—
492C2×C4—61no—
493C2×C4—61no—
494C2×C4—61no—
495C2×C4—61no—
496C2×C4—61no—
497C2×C4—61no—
498C2×C4—61no—
499C2×C4—62no—
500C2×C4—62no—
501C2×C4—62no—
502C2×C4—62no—
503C2×C4—62no—
504C2×C4—62no—
505C2×C4C2×C4×C5×D312yes—
506C2×C4C2×C4×C5×D312yes—
507C2×C4C5×#23712yes—
508C2×C4C5×#23712yes—
509C2×C3—42no—
510C2×C3C2×C5×#11711yescharacteristic
511C2×C3C5×#52011yescharacteristic
512C2×C3C5×#52011yescharacteristic
513C2×C3C5×#54011yescharacteristic
514C2×C3C5×#54111yescharacteristic
515C2×C3C5×#54211yescharacteristic
516C2×C3C5×#54211yescharacteristic
517C5#7386811yescharacteristic, direct product factor, Sylow
518C22—22no—
519C22—22no—
520C22—22no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22C22×C4×C5×D311yescharacteristic
526C22C2×C5×#23711yescharacteristic
527C22C2×C5×#23711yescharacteristic
528C22C5×#100411yescharacteristic
529C22C5×#100711yescharacteristic
530C22C5×#100911yescharacteristic
531C22C5×#101011yescharacteristic
532C4—22no—
533C4—42no—
534C4—42no—
535C4—62no—
536C4—62no—
537C4—62no—
538C4—62no—
539C4—122no—
540C4—122no—
541C4C5×#87714yes—
542C3C5×#231411yescharacteristic, complemented, Sylow
543C2—42no—
544C2C2×C5×#87711yescharacteristic
545C2C5×#550411yescharacteristic
546C2C5×#552311yescharacteristic
547C2C5×#560311yescharacteristic
548C2C5×#561911yescharacteristic
549C2C5×#562511yescharacteristic
550C2C5×#564311yescharacteristic
subgroupG/H
centerC23×C5C23×D3
commutatorC22×C3C23×C4×C5
FrattiniC23C23×C5×D3
FittingC3×C5×#380C2
socleC23×C3×C5C24
prime radicalC24×C3×C5C23
cosocleC23×C3C24×C5
hypercenterC5×#380D3
hypocenterC3C5×#2314
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C5 ◁ C5×#380

Complements

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

NH ≅ G/Ntype
4. C5×#4977C2semidirect
5. C5×#4975C2semidirect
6. C5×Dic3×C4⋊C4C2semidirect
7. C5×Dic3×C4⋊C4C2semidirect
10. C5×#4654C2semidirect
11. C5×#4653C2semidirect
20. C5×#926C4semidirect
37. #73868C5direct (N × H)
82. #4977C2×C5semidirect
83. #4975C2×C5semidirect
84. Dic3×C4⋊C4C2×C5semidirect
85. Dic3×C4⋊C4C2×C5semidirect
88. #4654C2×C5semidirect
89. #4653C2×C5semidirect
161. #926C4×C5semidirect
437. C3×C5#2314semidirect
517. C5#73868direct (N × H)
542. C3C5×#2314semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#9262
C3×C5×#3802
C5×#52122
C5×#49772
C5×#49752
C5×Dic3×C4⋊C42
C5×#49452
C5×#49442
C5×#46542
C5×#46532
C5×#23143
#738685