Finite Group Databasebeta

← C5×#76197 #14323702

Subgroups — C5×#76197 #14323702

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×C5×#897C211yesmaximal, characteristic
2C2×C5×#887C211yesmaximal, characteristic, complemented
3C2×C5×#872C211yesmaximal, characteristic
4C3×C5×#389C211yesmaximal, characteristic, complemented
5C5×#4990C212yesmaximal
6C5×#4987C211yesmaximal, characteristic
7C5×Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
8C5×#4972C212yesmaximal, complemented
9C5×#4929C211yesmaximal, characteristic, complemented
10C5×#4669C212yesmaximal
11C5×#4654C212yesmaximal, complemented
12C5×#2323—31nomaximal, Hall
13C22×C4×C5×D3C2211yescharacteristic
14C22×C5×Dic6C2211yescharacteristic
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×C3×C5×C4⋊C4C2212yes—
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18C2×C3×C42×C5C2211yescharacteristic
19C2×C5×(C2×D3)⋊C4C2211yescharacteristic
20C2×C5×(C2×D3)⋊C4C2211yescharacteristic
21C2×C5×(C2×D3)⋊C4C2212yes—
22C2×C5×(C2×D3)⋊C4C2212yes—
23C2×C5×C4⋊Dic3C2211yescharacteristic
24C2×C5×C4⋊Dic3C2212yes—
25C2×C5×Dic3⋊C4C2211yescharacteristic
26C2×C5×Dic3⋊C4C2212yes—
27C2×C5×Dic3⋊C4C2211yescharacteristic
28C2×C4×C5×Dic3C2212yes—
29C2×C4×C5×Dic3C2212yes—
30C2×C4×C5×Dic3C2211yescharacteristic
31C5×#897C418yescomplemented
32C5×#887—24no—
33C5×#872—24no—
34C3×C5×#95C2212yes—
35C5×C3⋊#95C2212yes—
36C5×C3⋊#95C2212yes—
37C5×C3⋊#95C2212yes—
38C5×C3⋊#95C2212yes—
39#76197C511yesmaximal, characteristic, direct product factor, Hall
40C2×C5×C22⋊Q8—31no—
41C2×C4×C5×Q8—31no—
42C2×C5×#117—31no—
43C5×#391—31no—
44C5×#389—32no—
45C5×#389D311yescharacteristic, complemented
46C5×#388—32no—
47C5×#388—32no—
48C5×#387—32no—
49C5×#380—31no—
50C4×C5×C4⋊C4—31no—
51C23×C5×D3C2311yescharacteristic
52C22×C3×C4×C5C2312yes—
53C22×C3×C4×C5C2311yescharacteristic
54C22×C3×C4×C5C2311yescharacteristic
55C22×C3×C4×C5C2312yes—
56C22×C3×C4×C5C2311yescharacteristic
57C22×C5×Dic3C2312yes—
58C22×C5×Dic3C2312yes—
59C22×C5×Dic3C2311yescharacteristic
60C22×C5×Dic3C2311yescharacteristic
61C22×C5×Dic3C2311yescharacteristic
62C2×C4×C5×D3—22no—
63C2×C4×C5×D3—22no—
64C2×C4×C5×D3C2×C414yes—
65C2×C5×Dic6—22no—
66C2×C5×Dic6—22no—
67C2×C5×Dic6C2×C414yes—
68C3×C5×C4⋊C4—22no—
69C3×C5×C4⋊C4—42no—
70C3×C5×C4⋊C4C2×C414yes—
71C3×C42×C5—22no—
72C5×(C2×D3)⋊C4—22no—
73C5×(C2×D3)⋊C4—22no—
74C5×(C2×D3)⋊C4—42no—
75C5×(C2×D3)⋊C4C2×C418yes—
76C5×C4⋊Dic3—22no—
77C5×C4⋊Dic3C2×C418yes—
78C5×Dic3⋊C4—22no—
79C5×Dic3⋊C4—22no—
80C5×Dic3⋊C4—24no—
81C4×C5×Dic3—24no—
82C4×C5×Dic3—42no—
83C4×C5×Dic3D414yes—
84C2×#897C2×C511yescharacteristic
85C2×#887C2×C511yescharacteristic, complemented
86C2×#872C2×C511yescharacteristic
87C3×#389C2×C511yescharacteristic, complemented
88#4990C2×C512yes—
89#4987C2×C511yescharacteristic
90Dic3×C4⋊C4C2×C511yescharacteristic, complemented
91#4972C2×C512yescomplemented
92#4929C2×C511yescharacteristic, complemented
93#4669C2×C512yes—
94#4654C2×C512yescomplemented
95C22×C5×Q8—31no—
96C23×C4×C5—31no—
97C5×C22⋊Q8—38no—
98C4×C5×Q8—64no—
99C5×#117—64no—
100C2×C5×C4⋊C4—32no—
101C2×C5×C4⋊C4—31no—
102C2×C5×C4⋊C4—31no—
103C2×C5×C4⋊C4—32no—
104C2×C5×C4⋊C4—31no—
105C2×C5×C4⋊C4C2×D311yescharacteristic
106C2×C5×C4⋊C4C2×D312yes—
107C2×C5×C4⋊C4C2×D311yescharacteristic
108C2×C5×C22⋊C4—32no—
109C2×C5×C22⋊C4—31no—
110C2×C5×C22⋊C4—31no—
111C2×C5×C22⋊C4—32no—
112C2×C42×C5—31no—
113C2×C42×C5—32no—
114C2×C42×C5—32no—
115C2×C42×C5C2×D311yescharacteristic
116C5×#95—32no—
117C5×#95—32no—
118C5×#95—32no—
119C5×#95—32no—
120C5×#95C2×D312yes—
121#2323—31noSylow
122C23×C3×C5C2411yescharacteristic
123C22×C5×D3—21no—
124C22×C5×D3—21no—
125C22×C5×D3—21no—
126C22×C5×D3—21no—
127C22×C5×D3—21no—
128C22×C5×D3—21no—
129C22×C5×D3C22×C412yes—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5—21no—
134C2×C3×C4×C5—21no—
135C2×C3×C4×C5—22no—
136C2×C3×C4×C5—21no—
137C2×C3×C4×C5—21no—
138C2×C3×C4×C5—21no—
139C2×C3×C4×C5—22no—
140C2×C3×C4×C5—21no—
141C2×C3×C4×C5C22×C414yes—
142C2×C3×C4×C5C22×C412yes—
143C2×C3×C4×C5C2×D412yes—
144C2×C3×C4×C5#4112yes—
145C2×C5×Dic3—22no—
146C2×C5×Dic3—21no—
147C2×C5×Dic3—22no—
148C2×C5×Dic3—21no—
149C2×C5×Dic3—21no—
150C2×C5×Dic3—21no—
151C2×C5×Dic3—21no—
152C2×C5×Dic3—21no—
153C2×C5×Dic3—22no—
154C2×C5×Dic3—22no—
155C2×C5×Dic3—22no—
156C2×C5×Dic3C22×C412yes—
157C2×C5×Dic3C22×C414yes—
158C2×C5×Dic3C2×D412yes—
159C2×C5×Dic3C2×D412yes—
160C4×C5×D3—28no—
161C5×Dic6—28no—
162C22×C4×D3C22×C511yescharacteristic
163C22×Dic6C22×C511yescharacteristic
164C2×C3×C4⋊C4C22×C511yescharacteristic
165C2×C3×C4⋊C4C22×C512yes—
166C2×C3×C4⋊C4C22×C511yescharacteristic
167C2×C3×C42C22×C511yescharacteristic
168C2×(C2×D3)⋊C4C22×C511yescharacteristic
169C2×(C2×D3)⋊C4C22×C511yescharacteristic
170C2×(C2×D3)⋊C4C22×C512yes—
171C2×(C2×D3)⋊C4C22×C512yes—
172C2×C4⋊Dic3C22×C511yescharacteristic
173C2×C4⋊Dic3C22×C512yes—
174C2×Dic3⋊C4C22×C511yescharacteristic
175C2×Dic3⋊C4C22×C512yes—
176C2×Dic3⋊C4C22×C511yescharacteristic
177C2×C4×Dic3C22×C512yes—
178C2×C4×Dic3C22×C512yes—
179C2×C4×Dic3C22×C511yescharacteristic
180#897C4×C518yescomplemented
181#887—24no—
182#872—24no—
183C3×#95C22×C512yes—
184C3⋊#95C22×C512yes—
185C3⋊#95C22×C512yes—
186C3⋊#95C22×C512yes—
187C3⋊#95C22×C512yes—
188C24×C5—31no—
189C2×C5×Q8—34no—
190C2×C5×Q8—62no—
191C2×C5×Q8—62no—
192C22×C4×C5—31no—
193C22×C4×C5—31no—
194C22×C4×C5—32no—
195C22×C4×C5—31no—
196C22×C4×C5—34no—
197C22×C4×C5—32no—
198C22×C4×C5—62no—
199C22×C4×C5—62no—
200C22×C4×C5C22×D312yes—
201C22×C4×C5C22×D311yescharacteristic
202C22×C4×C5C22×D311yescharacteristic
203C22×C4×C5C22×D312yes—
204C22×C4×C5C22×D311yescharacteristic
205C5×C4⋊C4—22no—
206C5×C4⋊C4—38no—
207C5×C4⋊C4—42no—
208C5×C4⋊C4—62no—
209C5×C4⋊C4—62no—
210C5×C4⋊C4—64no—
211C5×C4⋊C4—62no—
212C5×C4⋊C4C4×D314yes—
213C5×C22⋊C4—38no—
214C5×C22⋊C4—62no—
215C5×C22⋊C4—62no—
216C5×C22⋊C4—122no—
217C42×C5—22no—
218C42×C5—34no—
219C42×C5—64no—
220C42×C5—122no—
221C2×C22⋊Q8—31no—
222C2×C4×Q8—31no—
223C2×#117—31no—
224#391—31no—
225#389—32no—
226#389C5×D311yescharacteristic, complemented
227#388—32no—
228#388—32no—
229#387—32no—
230#380—31no—
231C4×C4⋊C4—31no—
232C22×C3×C5C23×C411yescharacteristic
233C22×C3×C5C22×D411yescharacteristic
234C22×C3×C5C2×#4111yescharacteristic
235C22×C3×C5H(5,𝔽2)11yescharacteristic
236C22×C3×C52−1+411yescharacteristic
237C22×C3×C52−1+411yescharacteristic
238C22×C3×C52−1+411yescharacteristic
239C2×C5×D3—22no—
240C2×C5×D3—22no—
241C2×C5×D3—22no—
242C2×C5×D3—41no—
243C2×C5×D3—41no—
244C2×C5×D3—41no—
245C2×C5×D3—41no—
246C3×C4×C5—24no—
247C3×C4×C5—41no—
248C3×C4×C5—42no—
249C3×C4×C5—41no—
250C3×C4×C5C4×D414yes—
251C5×Dic3—24no—
252C5×Dic3—22no—
253C5×Dic3—22no—
254C5×Dic3—22no—
255C5×Dic3—42no—
256C23×D3C23×C511yescharacteristic
257C22×C3×C4C23×C511yescharacteristic
258C22×C3×C4C23×C511yescharacteristic
259C22×C3×C4C23×C512yes—
260C22×C3×C4C23×C512yes—
261C22×C3×C4C23×C511yescharacteristic
262C22×Dic3C23×C512yes—
263C22×Dic3C23×C512yes—
264C22×Dic3C23×C511yescharacteristic
265C22×Dic3C23×C511yescharacteristic
266C22×Dic3C23×C511yescharacteristic
267C2×C4×D3—22no—
268C2×C4×D3—22no—
269C2×C4×D3C2×C4×C514yes—
270C2×Dic6—22no—
271C2×Dic6—22no—
272C2×Dic6C2×C4×C514yes—
273C3×C4⋊C4—22no—
274C3×C4⋊C4—42no—
275C3×C4⋊C4C2×C4×C514yes—
276C3×C42—22no—
277(C2×D3)⋊C4—22no—
278(C2×D3)⋊C4—22no—
279(C2×D3)⋊C4—42no—
280(C2×D3)⋊C4C2×C4×C518yes—
281C4⋊Dic3—22no—
282C4⋊Dic3C2×C4×C518yes—
283Dic3⋊C4—22no—
284Dic3⋊C4—22no—
285Dic3⋊C4—24no—
286C4×Dic3—24no—
287C4×Dic3—42no—
288C4×Dic3C5×D414yes—
289C23×C5—32no—
290C23×C5—61no—
291C23×C5—61no—
292C23×C5—61no—
293C23×C5—61no—
294C23×C5—61no—
295C23×C5—61no—
296C23×C5C23×D311yescharacteristic
297C5×Q8—68no—
298C2×C4×C5—22no—
299C2×C4×C5—22no—
300C2×C4×C5—21no—
301C2×C4×C5—21no—
302C2×C4×C5—22no—
303C2×C4×C5—22no—
304C2×C4×C5—21no—
305C2×C4×C5—21no—
306C2×C4×C5—22no—
307C2×C4×C5—21no—
308C2×C4×C5—21no—
309C2×C4×C5—34no—
310C2×C4×C5—32no—
311C2×C4×C5—32no—
312C2×C4×C5—32no—
313C2×C4×C5—62no—
314C2×C4×C5—61no—
315C2×C4×C5—61no—
316C2×C4×C5—61no—
317C2×C4×C5—62no—
318C2×C4×C5—61no—
319C2×C4×C5—61no—
320C2×C4×C5—61no—
321C2×C4×C5—68no—
322C2×C4×C5—62no—
323C2×C4×C5—62no—
324C2×C4×C5—62no—
325C2×C4×C5C2×C4×D314yes—
326C2×C4×C5C2×C4×D312yes—
327C2×C4×C5D3×D412yes—
328C2×C4×C5#23912yes—
329C22×Q8—31no—
330C23×C4—31no—
331C22⋊Q8—38no—
332C4×Q8—64no—
333#117—64no—
334C2×C4⋊C4—32no—
335C2×C4⋊C4—31no—
336C2×C4⋊C4—31no—
337C2×C4⋊C4—32no—
338C2×C4⋊C4—31no—
339C2×C4⋊C4C2×C5×D311yescharacteristic
340C2×C4⋊C4C2×C5×D312yes—
341C2×C4⋊C4C2×C5×D311yescharacteristic
342C2×C22⋊C4—32no—
343C2×C22⋊C4—31no—
344C2×C22⋊C4—31no—
345C2×C22⋊C4—32no—
346C2×C42—31no—
347C2×C42—32no—
348C2×C42—32no—
349C2×C42C2×C5×D311yescharacteristic
350#95—32no—
351#95—32no—
352#95—32no—
353#95—32no—
354#95C2×C5×D312yes—
355C2×C3×C5C2×C4×D411yescharacteristic
356C2×C3×C5#51911yescharacteristic
357C2×C3×C5#52011yescharacteristic
358C2×C3×C5#53611yescharacteristic
359C2×C3×C5#53711yescharacteristic
360C2×C3×C5#54111yescharacteristic
361C2×C3×C5#54211yescharacteristic
362C5×D3—42no—
363C23×C3C24×C511yescharacteristic
364C22×D3—21no—
365C22×D3—21no—
366C22×D3—21no—
367C22×D3—21no—
368C22×D3—21no—
369C22×D3—21no—
370C22×D3C22×C4×C512yes—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—22no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—22no—
382C2×C3×C4C22×C4×C514yes—
383C2×C3×C4C22×C4×C512yes—
384C2×C3×C4C2×C5×D412yes—
385C2×C3×C4C5×#4112yes—
386C2×Dic3—22no—
387C2×Dic3—21no—
388C2×Dic3—22no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—22no—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3C22×C4×C512yes—
398C2×Dic3C22×C4×C514yes—
399C2×Dic3C2×C5×D412yes—
400C2×Dic3C2×C5×D412yes—
401C4×D3—28no—
402Dic6—28no—
403C22×C5—62no—
404C22×C5—62no—
405C22×C5—62no—
406C22×C5—121no—
407C22×C5—121no—
408C22×C5—121no—
409C22×C5—121no—
410C22×C5C22×C4×D311yescharacteristic
411C22×C5C2×D3×D411yescharacteristic
412C22×C5#100411yescharacteristic
413C22×C5C2×#23911yescharacteristic
414C22×C5#100711yescharacteristic
415C22×C5#101011yescharacteristic
416C22×C5#101011yescharacteristic
417C4×C5—24no—
418C4×C5—41no—
419C4×C5—42no—
420C4×C5—41no—
421C4×C5—62no—
422C4×C5—64no—
423C4×C5—62no—
424C4×C5—62no—
425C4×C5—122no—
426C4×C5#89314yes—
427C24—31no—
428C2×Q8—34no—
429C2×Q8—62no—
430C2×Q8—62no—
431C22×C4—31no—
432C22×C4—32no—
433C22×C4—31no—
434C22×C4—31no—
435C22×C4—34no—
436C22×C4—32no—
437C22×C4—62no—
438C22×C4—62no—
439C22×C4C22×C5×D311yescharacteristic
440C22×C4C22×C5×D311yescharacteristic
441C22×C4C22×C5×D312yes—
442C22×C4C22×C5×D312yes—
443C22×C4C22×C5×D311yescharacteristic
444C4⋊C4—22no—
445C4⋊C4—38no—
446C4⋊C4—42no—
447C4⋊C4—62no—
448C4⋊C4—62no—
449C4⋊C4—64no—
450C4⋊C4—62no—
451C4⋊C4C4×C5×D314yes—
452C22⋊C4—38no—
453C22⋊C4—62no—
454C22⋊C4—62no—
455C22⋊C4—122no—
456C42—22no—
457C42—34no—
458C42—64no—
459C42—122no—
460C3×C5#232311yescharacteristic, complemented, Hall
461C22×C3C23×C4×C511yescharacteristic
462C22×C3C22×C5×D411yescharacteristic
463C22×C3C2×C5×#4111yescharacteristic
464C22×C3C5×H(5,𝔽2)11yescharacteristic
465C22×C3C5×2−1+411yescharacteristic
466C22×C3C5×2−1+411yescharacteristic
467C22×C3C5×2−1+411yescharacteristic
468C2×D3—22no—
469C2×D3—22no—
470C2×D3—22no—
471C2×D3—41no—
472C2×D3—41no—
473C2×D3—41no—
474C2×D3—41no—
475C3×C4—24no—
476C3×C4—41no—
477C3×C4—42no—
478C3×C4—41no—
479C3×C4C4×C5×D414yes—
480Dic3—24no—
481Dic3—22no—
482Dic3—22no—
483Dic3—22no—
484Dic3—42no—
485C2×C5—122no—
486C2×C5C2×#89311yescharacteristic
487C2×C5#554611yescharacteristic
488C2×C5#557211yescharacteristic
489C2×C5#560811yescharacteristic
490C2×C5#562811yescharacteristic
491C2×C5#569811yescharacteristic
492C2×C5#573311yescharacteristic
493C23—32no—
494C23—61no—
495C23—61no—
496C23—61no—
497C23—61no—
498C23—61no—
499C23—61no—
500C23C23×C5×D311yescharacteristic
501Q8—68no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—34no—
514C2×C4—32no—
515C2×C4—32no—
516C2×C4—32no—
517C2×C4—62no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—62no—
522C2×C4—61no—
523C2×C4—61no—
524C2×C4—61no—
525C2×C4—68no—
526C2×C4—62no—
527C2×C4—62no—
528C2×C4—62no—
529C2×C4C2×C4×C5×D314yes—
530C2×C4C2×C4×C5×D312yes—
531C2×C4C5×D3×D412yes—
532C2×C4C5×#23912yes—
533C2×C3C2×C4×C5×D411yescharacteristic
534C2×C3C5×#51911yescharacteristic
535C2×C3C5×#52011yescharacteristic
536C2×C3C5×#53611yescharacteristic
537C2×C3C5×#53711yescharacteristic
538C2×C3C5×#54111yescharacteristic
539C2×C3C5×#54211yescharacteristic
540D3—42no—
541C5#7619711yescharacteristic, direct product factor, Sylow
542C22—62no—
543C22—62no—
544C22—62no—
545C22—121no—
546C22—121no—
547C22—121no—
548C22—121no—
549C22C22×C4×C5×D311yescharacteristic
550C22C2×C5×D3×D411yescharacteristic
551C22C5×#100411yescharacteristic
552C22C2×C5×#23911yescharacteristic
553C22C5×#100711yescharacteristic
554C22C5×#101011yescharacteristic
555C22C5×#101011yescharacteristic
556C4—24no—
557C4—41no—
558C4—42no—
559C4—41no—
560C4—62no—
561C4—64no—
562C4—62no—
563C4—62no—
564C4—122no—
565C4C5×#89314yes—
566C3C5×#232311yescharacteristic, complemented, Sylow
567C2—122no—
568C2C2×C5×#89311yescharacteristic
569C2C5×#554611yescharacteristic
570C2C5×#557211yescharacteristic
571C2C5×#560811yescharacteristic
572C2C5×#562811yescharacteristic
573C2C5×#569811yescharacteristic
574C2C5×#573311yescharacteristic
subgroupG/H
centerC23×C5C23×D3
commutatorC22×C3C23×C4×C5
FrattiniC23C23×C5×D3
FittingC3×C5×#389C2
socleC23×C3×C5C24
prime radicalC23×C5×D3C23
cosocleC23×C3C24×C5
hypercenterC5×#389D3
hypocenterC3C5×#2323
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C5 ◁ C5×#389

Complements

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

NH ≅ G/Ntype
2. C2×C5×#887C2semidirect
4. C3×C5×#389C2semidirect
7. C5×Dic3×C4⋊C4C2semidirect
8. C5×#4972C2semidirect
9. C5×#4929C2semidirect
11. C5×#4654C2semidirect
31. C5×#897C4semidirect
39. #76197C5direct (N × H)
45. C5×#389D3semidirect
85. C2×#887C2×C5semidirect
87. C3×#389C2×C5semidirect
90. Dic3×C4⋊C4C2×C5semidirect
91. #4972C2×C5semidirect
92. #4929C2×C5semidirect
94. #4654C2×C5semidirect
180. #897C4×C5semidirect
226. #389C5×D3semidirect
460. C3×C5#2323semidirect
541. C5#76197direct (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×C5×#8972
C2×C5×#8872
C2×C5×#8722
C3×C5×#3892
C5×#49902
C5×#49872
C5×Dic3×C4⋊C42
C5×#49722
C5×#49292
C5×#46692
C5×#46542
C5×#23233
#761975