Finite Group Databasebeta

← C5×#72098 #14307980

Subgroups — C5×#72098 #14307980

Order 1920.

812 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
2C5×#5833C211yesmaximal, characteristic, complemented
3C2×C5×#932C211yesmaximal, characteristic, complemented
4C2×C5×#837C211yesmaximal, characteristic, complemented
5C5×#5221C211yesmaximal, characteristic, complemented
6C2×C5×#825C211yesmaximal, characteristic, complemented
7C5×#4554C211yesmaximal, characteristic, complemented
8C5×#2006—31nomaximal, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C22×C3×C5×D4C2211yescharacteristic
11C2×C5×#235C2211yescharacteristic
12C3×C5×AGL(1,ℤ/8)—22no—
13C3×C5×AGL(1,ℤ/8)—22no—
14C2×C3×C5×SD16—21no—
15C2×C3×C5×D8—21no—
16C2×C3×C5×M16C2211yescharacteristic
17C2×C5×#217—22no—
18C5×#946—21no—
19C5×#939—22no—
20C5×#938—21no—
21C2×C5×#214—21no—
22C5×#932—22no—
23C5×#932—22no—
24C2×C5×#213—21no—
25C5×#930—21no—
26C2×C5×C4⋊Dic3C2211yescharacteristic
27C5×#924—21no—
28C2×C4×C5×Dic3C2211yescharacteristic
29C2×C5×#208C2211yescharacteristic
30C5×#837—22no—
31C5×#825—22no—
32#72098C511yesmaximal, characteristic, direct product factor, Hall
33C2×C5×AGL(1,ℤ/8)—31no—
34C2×C5×AGL(1,ℤ/8)D311yescharacteristic, complemented
35C5×#537—31no—
36C2×C5×#104—31no—
37C2×C5×#104—31no—
38C5×#390—31no—
39C5×#343—31no—
40C24×C3×C5—21no—
41C3×C5×#41—21no—
42C3×C5×#41—21no—
43C3×C5×#41—21no—
44C3×C5×#41—21no—
45C2×C3×C5×Q8D411yescharacteristic
46C2×C3×C5×D4—21no—
47C2×C3×C5×D4—21no—
48C2×C3×C5×D4—21no—
49C2×C3×C5×D4—21no—
50C2×C3×C5×D4—21no—
51C2×C3×C5×D4—21no—
52C2×C3×C5×D4D411yescharacteristic
53C2×C3×C5×D4D412yescomplemented
54C22×C3×C4×C5—21no—
55C22×C3×C4×C5C2311yescharacteristic
56C2×C5×#67—21no—
57C2×C5×#67—22no—
58C22×C5×Dic3—21no—
59C22×C5×Dic3D412yes—
60C5×#235—21no—
61C5×#235—21no—
62C5×#235—21no—
63C5×#235—21no—
64C2×C5×D12D411yescharacteristic
65C2×C4×C5×D3—21no—
66C2×C5×Dic6D411yescharacteristic
67C3×C5×SD16—24no—
68C3×C5×D8—24no—
69C3×C5×M16—21no—
70C3×C5×M16D412yescomplemented
71C2×C3×C5×C8—21no—
72C5×#217—22no—
73C5×#217—44no—
74C5×#214—24no—
75C5×#213—24no—
76C5×(C2×D3)⋊C4—22no—
77C5×C4⋊Dic3—21no—
78C5×C4⋊Dic3—21no—
79C5×Dic3⋊C4—22no—
80C4×C5×Dic3—21no—
81C4×C5×Dic3—21no—
82C5×#208—21no—
83C5×#208D412yescomplemented
84C2×C5×C3⋊C8—21no—
85C2×C3×AGL(1,ℤ/8)C2×C511yescharacteristic, complemented
86#5833C2×C511yescharacteristic, complemented
87C2×#932C2×C511yescharacteristic, complemented
88C2×#837C2×C511yescharacteristic, complemented
89#5221C2×C511yescharacteristic, complemented
90C2×#825C2×C511yescharacteristic, complemented
91#4554C2×C511yescharacteristic, complemented
92C2×C5×#41—31no—
93C2×C5×#41C2×D311yescharacteristic
94C22×C5×D4C2×D311yescharacteristic
95C5×AGL(1,ℤ/8)—22no—
96C5×AGL(1,ℤ/8)—22no—
97C5×AGL(1,ℤ/8)—62no—
98C5×AGL(1,ℤ/8)—62no—
99C2×C5×SD16—21no—
100C2×C5×SD16—61no—
101C2×C5×D8—21no—
102C2×C5×D8—61no—
103C2×C5×M16—31no—
104C2×C5×M16C2×D311yescharacteristic
105C5×C22⋊Q8—61no—
106C5×C22⋊Q8—61no—
107C5×#121—61no—
108C5×#121—62no—
109C5×#121—61no—
110C2×C5×C4⋊C4—31no—
111C2×C5×C22⋊C4—62no—
112C2×C42×C5—31no—
113C5×#104—62no—
114C5×#104—62no—
115#2006—31noSylow
116C23×C3×C5—21no—
117C23×C3×C5—22no—
118C23×C3×C5—42no—
119C23×C3×C5—42no—
120C23×C3×C5—41no—
121C23×C3×C5—41no—
122C23×C3×C5C2×D411yescharacteristic
123C22×C5×D3—21no—
124C3×C5×Q8—21no—
125C3×C5×Q8—21no—
126C3×C5×D4—22no—
127C3×C5×D4—21no—
128C3×C5×D4—22no—
129C3×C5×D4—21no—
130C3×C5×D4—41no—
131C3×C5×D4—41no—
132C3×C5×D4—41no—
133C3×C5×D4—41no—
134C2×C3×C4×C5—21no—
135C2×C3×C4×C5—41no—
136C2×C3×C4×C5—41no—
137C2×C3×C4×C5C2×D411yescharacteristic
138C2×C3×C4×C5C2×D411yescharacteristic
139C2×C3×C4×C5C2×D411yescharacteristic
140C2×C3×C4×C5C2×D411yescharacteristic
141C2×C3×C4×C5C2×D411yescharacteristic
142C2×C3×C4×C5#4111yescharacteristic
143C5×#67—41no—
144C5×#67—41no—
145C5×#67—82no—
146C2×C5×Dic3—22no—
147C2×C5×Dic3—21no—
148C2×C5×Dic3—22no—
149C2×C5×Dic3—41no—
150C2×C5×Dic3—41no—
151C2×C5×Dic3—42no—
152C5×D12—21no—
153C5×D12—21no—
154C4×C5×D3—41no—
155C4×C5×D3—41no—
156C5×Dic6—21no—
157C5×Dic6—21no—
158C3×C5×C8—22no—
159C5×C3⋊C8—22no—
160C2×C3×#41C22×C511yescharacteristic
161C22×C3×D4C22×C511yescharacteristic
162C2×#235C22×C511yescharacteristic
163C3×AGL(1,ℤ/8)—22no—
164C3×AGL(1,ℤ/8)—22no—
165C2×C3×SD16—21no—
166C2×C3×D8—21no—
167C2×C3×M16C22×C511yescharacteristic
168C2×#217—22no—
169#946—21no—
170#939—22no—
171#938—21no—
172C2×#214—21no—
173#932—22no—
174#932—22no—
175C2×#213—21no—
176#930—21no—
177C2×C4⋊Dic3C22×C511yescharacteristic
178#924—21no—
179C2×C4×Dic3C22×C511yescharacteristic
180C2×#208C22×C511yescharacteristic
181#837—22no—
182#825—22no—
183C24×C5—21no—
184C5×#41—21no—
185C5×#41—21no—
186C5×#41—21no—
187C5×#41—21no—
188C5×#41—61no—
189C5×#41—61no—
190C5×#41—61no—
191C5×#41—61no—
192C2×C5×Q8—31no—
193C2×C5×Q8#6711yescharacteristic
194C2×C5×D4—21no—
195C2×C5×D4—21no—
196C2×C5×D4—21no—
197C2×C5×D4—21no—
198C2×C5×D4—21no—
199C2×C5×D4—21no—
200C2×C5×D4—31no—
201C2×C5×D4—61no—
202C2×C5×D4—62no—
203C2×C5×D4#6711yescharacteristic
204C2×C5×D4#6712yescomplemented
205C22×C4×C5—21no—
206C22×C4×C5—32no—
207C22×C4×C5—61no—
208C22×C4×C5—61no—
209C22×C4×C5C22×D311yescharacteristic
210C5×SD16—24no—
211C5×SD16—64no—
212C5×D8—24no—
213C5×D8—64no—
214C5×M16—21no—
215C5×M16—32no—
216C5×M16—61no—
217C5×M16#6712yescomplemented
218C2×C5×C8—21no—
219C2×C5×C8—61no—
220C5×C4⋊C4—62no—
221C5×C4⋊C4—61no—
222C5×C4⋊C4—61no—
223C5×C22⋊C4—62no—
224C5×C22⋊C4—62no—
225C5×C22⋊C4—124no—
226C42×C5—61no—
227C42×C5—61no—
228C2×AGL(1,ℤ/8)—31no—
229C2×AGL(1,ℤ/8)C5×D311yescharacteristic, complemented
230#537—31no—
231C2×#104—31no—
232C2×#104—31no—
233#390—31no—
234#343—31no—
235C22×C3×C5—21no—
236C22×C3×C5—21no—
237C22×C3×C5—44no—
238C22×C3×C5—41no—
239C22×C3×C5—42no—
240C22×C3×C5—41no—
241C22×C3×C5—41no—
242C22×C3×C5—82no—
243C22×C3×C5—81no—
244C22×C3×C5—81no—
245C22×C3×C5#12011yescharacteristic
246C22×C3×C5#12111yescharacteristic
247C22×C3×C5#12711yescharacteristic
248C2×C5×D3—41no—
249C2×C5×D3—41no—
250C2×C5×D3—41no—
251C3×C4×C5—41no—
252C3×C4×C5#12011yescharacteristic
253C3×C4×C5#12011yescharacteristic
254C3×C4×C5#12111yescharacteristic
255C3×C4×C5#12111yescharacteristic
256C5×Dic3—42no—
257C5×Dic3—41no—
258C5×Dic3—42no—
259C24×C3—21no—
260C3×#41—21no—
261C3×#41—21no—
262C3×#41—21no—
263C3×#41—21no—
264C2×C3×Q8C5×D411yescharacteristic
265C2×C3×D4—21no—
266C2×C3×D4—21no—
267C2×C3×D4—21no—
268C2×C3×D4—21no—
269C2×C3×D4—21no—
270C2×C3×D4—21no—
271C2×C3×D4C5×D411yescharacteristic
272C2×C3×D4C5×D412yescomplemented
273C22×C3×C4—21no—
274C22×C3×C4C23×C511yescharacteristic
275C2×#67—21no—
276C2×#67—22no—
277C22×Dic3—21no—
278C22×Dic3C5×D412yes—
279#235—21no—
280#235—21no—
281#235—21no—
282#235—21no—
283C2×D12C5×D411yescharacteristic
284C2×C4×D3—21no—
285C2×Dic6C5×D411yescharacteristic
286C3×SD16—24no—
287C3×D8—24no—
288C3×M16—21no—
289C3×M16C5×D412yescomplemented
290C2×C3×C8—21no—
291#217—22no—
292#217—44no—
293#214—24no—
294#213—24no—
295(C2×D3)⋊C4—22no—
296C4⋊Dic3—21no—
297C4⋊Dic3—21no—
298Dic3⋊C4—22no—
299C4×Dic3—21no—
300C4×Dic3—21no—
301#208—21no—
302#208C5×D412yescomplemented
303C2×C3⋊C8—21no—
304C23×C5—21no—
305C23×C5—22no—
306C23×C5—42no—
307C23×C5—42no—
308C23×C5—41no—
309C23×C5—41no—
310C23×C5—61no—
311C23×C5D3×D411yescharacteristic
312C5×Q8—21no—
313C5×Q8—21no—
314C5×Q8—61no—
315C5×Q8—61no—
316C5×D4—22no—
317C5×D4—21no—
318C5×D4—21no—
319C5×D4—22no—
320C5×D4—41no—
321C5×D4—41no—
322C5×D4—41no—
323C5×D4—41no—
324C5×D4—61no—
325C5×D4—61no—
326C5×D4—121no—
327C5×D4—121no—
328C5×D4—242no—
329C2×C4×C5—21no—
330C2×C4×C5—41no—
331C2×C4×C5—41no—
332C2×C4×C5—62no—
333C2×C4×C5—62no—
334C2×C4×C5—61no—
335C2×C4×C5—121no—
336C2×C4×C5—121no—
337C2×C4×C5—121no—
338C2×C4×C5—121no—
339C2×C4×C5—122no—
340C2×C4×C5D3×D411yescharacteristic
341C2×C4×C5D3×D411yescharacteristic
342C2×C4×C5#23711yescharacteristic
343C2×C4×C5C2×#6711yescharacteristic
344C2×C4×C5C2×#6711yescharacteristic
345C2×C4×C5C2×#6711yescharacteristic
346C5×C8—22no—
347C5×C8—62no—
348C2×#41—31no—
349C2×#41C2×C5×D311yescharacteristic
350C22×D4C2×C5×D311yescharacteristic
351AGL(1,ℤ/8)—22no—
352AGL(1,ℤ/8)—22no—
353AGL(1,ℤ/8)—62no—
354AGL(1,ℤ/8)—62no—
355C2×SD16—21no—
356C2×SD16—61no—
357C2×D8—21no—
358C2×D8—61no—
359C2×M16—31no—
360C2×M16C2×C5×D311yescharacteristic
361C22⋊Q8—61no—
362C22⋊Q8—61no—
363#121—61no—
364#121—62no—
365#121—61no—
366C2×C4⋊C4—31no—
367C2×C22⋊C4—62no—
368C2×C42—31no—
369#104—62no—
370#104—62no—
371C2×C3×C5—21no—
372C2×C3×C5—21no—
373C2×C3×C5—81no—
374C2×C3×C5—82no—
375C2×C3×C5#39211yescharacteristic
376C2×C3×C5#45311yescharacteristic
377C2×C3×C5#45411yescharacteristic
378C5×D3—81no—
379C23×C3—21no—
380C23×C3—22no—
381C23×C3—42no—
382C23×C3—42no—
383C23×C3—41no—
384C23×C3—41no—
385C23×C3C2×C5×D411yescharacteristic
386C22×D3—21no—
387C3×Q8—21no—
388C3×Q8—21no—
389C3×D4—22no—
390C3×D4—21no—
391C3×D4—22no—
392C3×D4—21no—
393C3×D4—41no—
394C3×D4—41no—
395C3×D4—41no—
396C3×D4—41no—
397C2×C3×C4—21no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4C2×C5×D411yescharacteristic
401C2×C3×C4C2×C5×D411yescharacteristic
402C2×C3×C4C2×C5×D411yescharacteristic
403C2×C3×C4C2×C5×D411yescharacteristic
404C2×C3×C4C2×C5×D411yescharacteristic
405C2×C3×C4C5×#4111yescharacteristic
406#67—41no—
407#67—41no—
408#67—82no—
409C2×Dic3—22no—
410C2×Dic3—21no—
411C2×Dic3—22no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—42no—
415D12—21no—
416D12—21no—
417C4×D3—41no—
418C4×D3—41no—
419Dic6—21no—
420Dic6—21no—
421C3×C8—22no—
422C3⋊C8—22no—
423C22×C5—21no—
424C22×C5—21no—
425C22×C5—44no—
426C22×C5—41no—
427C22×C5—41no—
428C22×C5—41no—
429C22×C5—42no—
430C22×C5—82no—
431C22×C5—81no—
432C22×C5—81no—
433C22×C5—121no—
434C22×C5—121no—
435C22×C5—121no—
436C22×C5#93711yescharacteristic
437C22×C5#93911yescharacteristic
438C22×C5#94011yescharacteristic
439C4×C5—41no—
440C4×C5—122no—
441C4×C5—122no—
442C4×C5—121no—
443C4×C5#93711yescharacteristic
444C4×C5#93811yescharacteristic
445C4×C5#93911yescharacteristic
446C4×C5#95311yescharacteristic
447C24—21no—
448#41—21no—
449#41—21no—
450#41—21no—
451#41—21no—
452#41—61no—
453#41—61no—
454#41—61no—
455#41—61no—
456C2×Q8—31no—
457C2×Q8C5×#6711yescharacteristic
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—31no—
465C2×D4—61no—
466C2×D4—62no—
467C2×D4C5×#6711yescharacteristic
468C2×D4C5×#6712yescomplemented
469C22×C4—21no—
470C22×C4—32no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4C22×C5×D311yescharacteristic
474SD16—24no—
475SD16—64no—
476D8—24no—
477D8—64no—
478M16—21no—
479M16—32no—
480M16—61no—
481M16C5×#6712yescomplemented
482C2×C8—21no—
483C2×C8—61no—
484C4⋊C4—62no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C22⋊C4—62no—
488C22⋊C4—62no—
489C22⋊C4—124no—
490C42—61no—
491C42—61no—
492C3×C5#200611yescharacteristic, complemented, Hall
493C22×C3—21no—
494C22×C3—21no—
495C22×C3—44no—
496C22×C3—41no—
497C22×C3—42no—
498C22×C3—41no—
499C22×C3—41no—
500C22×C3—82no—
501C22×C3—81no—
502C22×C3—81no—
503C22×C3C5×#12011yescharacteristic
504C22×C3C5×#12111yescharacteristic
505C22×C3C5×#12711yescharacteristic
506C2×D3—41no—
507C2×D3—41no—
508C2×D3—41no—
509C3×C4—41no—
510C3×C4C5×#12011yescharacteristic
511C3×C4C5×#12011yescharacteristic
512C3×C4C5×#12111yescharacteristic
513C3×C4C5×#12111yescharacteristic
514Dic3—42no—
515Dic3—41no—
516Dic3—42no—
517C2×C5—21no—
518C2×C5—21no—
519C2×C5—82no—
520C2×C5—81no—
521C2×C5—241no—
522C2×C5#519811yescharacteristic
523C2×C5#520111yescharacteristic
524C2×C5#522311yescharacteristic
525C23—21no—
526C23—22no—
527C23—42no—
528C23—42no—
529C23—41no—
530C23—41no—
531C23—61no—
532C23C5×D3×D411yescharacteristic
533Q8—21no—
534Q8—21no—
535Q8—61no—
536Q8—61no—
537D4—22no—
538D4—21no—
539D4—21no—
540D4—22no—
541D4—41no—
542D4—41no—
543D4—41no—
544D4—41no—
545D4—61no—
546D4—61no—
547D4—121no—
548D4—121no—
549D4—242no—
550C2×C4—21no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—61no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—122no—
561C2×C4C5×D3×D411yescharacteristic
562C2×C4C5×D3×D411yescharacteristic
563C2×C4C5×#23711yescharacteristic
564C2×C4C2×C5×#6711yescharacteristic
565C2×C4C2×C5×#6711yescharacteristic
566C2×C4C2×C5×#6711yescharacteristic
567C8—22no—
568C8—62no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—82no—
572C2×C3—81no—
573C2×C3C5×#39211yescharacteristic
574C2×C3C5×#45311yescharacteristic
575C2×C3C5×#45411yescharacteristic
576D3—81no—
577C5#7209811yescharacteristic, direct product factor, Sylow
578C22—21no—
579C22—21no—
580C22—44no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—42no—
585C22—82no—
586C22—81no—
587C22—81no—
588C22—121no—
589C22—121no—
590C22—121no—
591C22C5×#93711yescharacteristic
592C22C5×#93911yescharacteristic
593C22C5×#94011yescharacteristic
594C4—41no—
595C4—122no—
596C4—122no—
597C4—121no—
598C4C5×#93711yescharacteristic
599C4C5×#93811yescharacteristic
600C4C5×#93911yescharacteristic
601C4C5×#95311yescharacteristic
602C3C5×#200611yescharacteristic, complemented, Sylow
603C2—21no—
604C2—21no—
605C2—82no—
606C2—81no—
607C2—241no—
608C2C5×#519811yescharacteristic
609C2C5×#520111yescharacteristic
610C2C5×#522311yescharacteristic
subgroupG/H
centerC22×C5#937
commutator, cosocleC22×C3×C4C23×C5
FrattiniC22×C4C22×C5×D3
FittingC2×C3×C5×AGL(1,ℤ/8)C2
socleC22×C3×C5#120
hypercenterC2×C5×AGL(1,ℤ/8)D3
hypocenterC3C5×#2006
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C5×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×AGL(1,ℤ/8)C2semidirect
2. C5×#5833C2semidirect
3. C2×C5×#932C2semidirect
4. C2×C5×#837C2semidirect
5. C5×#5221C2semidirect
6. C2×C5×#825C2semidirect
7. C5×#4554C2semidirect
32. #72098C5direct (N × H)
34. C2×C5×AGL(1,ℤ/8)D3semidirect
53. C2×C3×C5×D4D4semidirect
70. C3×C5×M16D4semidirect
83. C5×#208D4semidirect
85. C2×C3×AGL(1,ℤ/8)C2×C5semidirect
86. #5833C2×C5semidirect
87. C2×#932C2×C5semidirect
88. C2×#837C2×C5semidirect
89. #5221C2×C5semidirect
90. C2×#825C2×C5semidirect
91. #4554C2×C5semidirect
204. C2×C5×D4#67semidirect
217. C5×M16#67semidirect
229. C2×AGL(1,ℤ/8)C5×D3semidirect
272. C2×C3×D4C5×D4semidirect
289. C3×M16C5×D4semidirect
302. #208C5×D4semidirect
468. C2×D4C5×#67semidirect
481. M16C5×#67semidirect
492. C3×C5#2006semidirect
577. C5#72098direct (N × H)
602. C3C5×#2006semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C5×AGL(1,ℤ/8)2
C5×#58332
C2×C5×#9322
C2×C5×#8372
C5×#52212
C2×C5×#8252
C5×#45542
C5×#20063
#720985