Finite Group Databasebeta

← C5×#85298 #14511036

Subgroups — C5×#85298 #14511036

Order 1920.

754 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×GL(2,ℤ/4)C211yesmaximal, characteristic
2C2×C4×C5×S4C211yesmaximal, characteristic, complemented
3C5×A4×C22⋊C4C211yesmaximal, characteristic, complemented
4C5×#5432C211yesmaximal, characteristic, complemented
5C5×(C2×S4)⋊C4C211yesmaximal, characteristic, complemented
6C5×C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
7C5×SL(2,ℤ/4)⋊C4C211yesmaximal, characteristic, complemented
8C5×#2418—31nomaximal, Hall
9C23×C5×A4C2211yescharacteristic
10C22×C5×S4C2211yescharacteristic
11C2×C4×C5×A4C2211yescharacteristic
12C2×C4×C5×A4C2211yescharacteristic
13C5×GL(2,ℤ/4)—22no—
14C2×C5×SL(2,ℤ/4)C2211yescharacteristic
15C2×C5×SL(2,ℤ/4)C2211yescharacteristic
16C2×C5×SL(2,ℤ/4)C2211yescharacteristic
17C4×C5×S4—22no—
18C5×#883—41nomaximal
19#85298C511yesmaximal, characteristic, direct product factor, Hall
20C2×C5×#123—31no—
21C2×C5×#120—31no—
22C2×C4×C5×D4—31no—
23C22×C5×C22⋊C4D311yescharacteristic, complemented
24C5×#399—31no—
25C5×#397—31no—
26C5×#394—31no—
27C5×#394—31no—
28C5×#388—31no—
29C5×#388—31no—
30C5×#386—31no—
31C5×#386—31no—
32C5×#385—31no—
33C5×#380—31no—
34C5×#379—31no—
35C22×C5×A4—21no—
36C22×C5×A4—21no—
37C22×C5×A4—21no—
38C22×C5×A4C2311yescharacteristic
39C2×C5×S4—21no—
40C2×C5×S4—21no—
41C2×C5×S4D412yes—
42C2×C5×#67—41no—
43C2×C4×C5×D3—41no—
44C4×C5×A4—21no—
45C4×C5×A4—21no—
46C5×SL(2,ℤ/4)—21no—
47C5×SL(2,ℤ/4)—21no—
48C5×SL(2,ℤ/4)—21no—
49C3×C5×C22⋊C4—41no—
50C5×#217—41no—
51C5×(C2×D3)⋊C4—41no—
52C5×C4⋊Dic3—41no—
53C5×Dic3⋊C4—41no—
54C2×GL(2,ℤ/4)C2×C511yescharacteristic
55C2×C4×S4C2×C511yescharacteristic, complemented
56A4×C22⋊C4C2×C511yescharacteristic, complemented
57#5432C2×C511yescharacteristic, complemented
58(C2×S4)⋊C4C2×C511yescharacteristic, complemented
59C4⋊SL(2,ℤ/4)C2×C511yescharacteristic, complemented
60SL(2,ℤ/4)⋊C4C2×C511yescharacteristic, complemented
61C25×C5C2×D311yescharacteristic
62C22×C5×D4—31no—
63C22×C5×D4—31no—
64C22×C5×D4—31no—
65C23×C4×C5—31no—
66C23×C4×C5C2×D311yescharacteristic
67C23×C4×C5C2×D311yescharacteristic
68C5×#123—61no—
69C5×#123—61no—
70C5×#123—61no—
71C5×#123—61no—
72C5×#120—62no—
73C5×#120—62no—
74C4×C5×D4—62no—
75C4×C5×D4—62no—
76C2×C5×C4⋊C4—31no—
77C2×C5×C4⋊C4—31no—
78C2×C5×C4⋊C4—31no—
79C2×C5×C4⋊C4—31no—
80C2×C5×C4⋊C4—31no—
81C2×C5×C22⋊C4—31no—
82C2×C5×C22⋊C4—31no—
83C2×C5×C22⋊C4—31no—
84C2×C5×C22⋊C4—31no—
85C2×C5×C22⋊C4—31no—
86C2×C5×C22⋊C4—31no—
87C2×C5×C22⋊C4—31no—
88C2×C5×C22⋊C4—31no—
89C2×C5×C22⋊C4—31no—
90C2×C5×C22⋊C4—31no—
91C2×C5×C22⋊C4—31no—
92C2×C5×C22⋊C4—31no—
93C2×C5×C22⋊C4—31no—
94C2×C5×C22⋊C4—31no—
95C2×C5×C22⋊C4—31no—
96C2×C5×C22⋊C4—31no—
97C2×C42×C5—31no—
98C5×#95—31no—
99C5×#95—31no—
100C5×#95—31no—
101C5×#95—31no—
102C5×#95—31no—
103C5×#95—31no—
104#2418—31noSylow
105C23×C3×C5—41no—
106C22×C5×D3—41no—
107C2×C5×A4—41no—
108C2×C5×A4C2×D411yescharacteristic
109C2×C5×A4#4111yescharacteristic
110C2×C5×A4#4111yescharacteristic
111C5×S4—22no—
112C2×C3×C4×C5—41no—
113C2×C3×C4×C5—41no—
114C5×#67—82no—
115C2×C5×Dic3—41no—
116C2×C5×Dic3—41no—
117C2×C5×Dic3—41no—
118C4×C5×D3—82no—
119C23×A4C22×C511yescharacteristic
120C22×S4C22×C511yescharacteristic
121C2×C4×A4C22×C511yescharacteristic
122C2×C4×A4C22×C511yescharacteristic
123GL(2,ℤ/4)—22no—
124C2×SL(2,ℤ/4)C22×C511yescharacteristic
125C2×SL(2,ℤ/4)C22×C511yescharacteristic
126C2×SL(2,ℤ/4)C22×C511yescharacteristic
127C4×S4—22no—
128#883—41no—
129C24×C5—21no—
130C24×C5—21no—
131C24×C5—21no—
132C24×C5—31no—
133C24×C5—31no—
134C24×C5—31no—
135C24×C5—61no—
136C24×C5—61no—
137C24×C5—61no—
138C24×C5C22×D311yescharacteristic
139C2×C5×D4—32no—
140C2×C5×D4—32no—
141C2×C5×D4—61no—
142C2×C5×D4—61no—
143C2×C5×D4—61no—
144C2×C5×D4—61no—
145C2×C5×D4—62no—
146C2×C5×D4—62no—
147C2×C5×D4—61no—
148C2×C5×D4—62no—
149C2×C5×D4—61no—
150C2×C5×D4—121no—
151C2×C5×D4—121no—
152C22×C4×C5—21no—
153C22×C4×C5—21no—
154C22×C4×C5—31no—
155C22×C4×C5—31no—
156C22×C4×C5—31no—
157C22×C4×C5—31no—
158C22×C4×C5—31no—
159C22×C4×C5—31no—
160C22×C4×C5—31no—
161C22×C4×C5—31no—
162C22×C4×C5—31no—
163C22×C4×C5—31no—
164C22×C4×C5—31no—
165C22×C4×C5—61no—
166C22×C4×C5—61no—
167C22×C4×C5—61no—
168C22×C4×C5—62no—
169C22×C4×C5—61no—
170C22×C4×C5—121no—
171C5×C4⋊C4—62no—
172C5×C4⋊C4—61no—
173C5×C4⋊C4—61no—
174C5×C4⋊C4—61no—
175C5×C4⋊C4—61no—
176C5×C4⋊C4—121no—
177C5×C4⋊C4—121no—
178C5×C22⋊C4—31no—
179C5×C22⋊C4—31no—
180C5×C22⋊C4—31no—
181C5×C22⋊C4—61no—
182C5×C22⋊C4—62no—
183C5×C22⋊C4—62no—
184C5×C22⋊C4—61no—
185C5×C22⋊C4—61no—
186C5×C22⋊C4—62no—
187C5×C22⋊C4—62no—
188C5×C22⋊C4—61no—
189C5×C22⋊C4—61no—
190C5×C22⋊C4—61no—
191C5×C22⋊C4—61no—
192C5×C22⋊C4—61no—
193C5×C22⋊C4—61no—
194C5×C22⋊C4—61no—
195C5×C22⋊C4—61no—
196C5×C22⋊C4—121no—
197C5×C22⋊C4—121no—
198C5×C22⋊C4—121no—
199C5×C22⋊C4S411yescharacteristic, complemented
200C42×C5—62no—
201C2×#123—31no—
202C2×#120—31no—
203C2×C4×D4—31no—
204C22×C22⋊C4C5×D311yescharacteristic, complemented
205#399—31no—
206#397—31no—
207#394—31no—
208#394—31no—
209#388—31no—
210#388—31no—
211#386—31no—
212#386—31no—
213#385—31no—
214#380—31no—
215#379—31no—
216C22×C3×C5—41no—
217C22×C3×C5—81no—
218C22×C3×C5—81no—
219C22×C3×C5—81no—
220C2×C5×D3—42no—
221C2×C5×D3—81no—
222C2×C5×D3—81no—
223C5×A4#12311yescharacteristic, complemented
224C3×C4×C5—81no—
225C3×C4×C5—81no—
226C5×Dic3—81no—
227C5×Dic3—81no—
228C5×Dic3—81no—
229C22×A4—21no—
230C22×A4—21no—
231C22×A4—21no—
232C22×A4C23×C511yescharacteristic
233C2×S4—21no—
234C2×S4—21no—
235C2×S4C5×D412yes—
236C2×#67—41no—
237C2×C4×D3—41no—
238C4×A4—21no—
239C4×A4—21no—
240SL(2,ℤ/4)—21no—
241SL(2,ℤ/4)—21no—
242SL(2,ℤ/4)—21no—
243C3×C22⋊C4—41no—
244#217—41no—
245(C2×D3)⋊C4—41no—
246C4⋊Dic3—41no—
247Dic3⋊C4—41no—
248C23×C5—31no—
249C23×C5—31no—
250C23×C5—31no—
251C23×C5—31no—
252C23×C5—32no—
253C23×C5—32no—
254C23×C5—31no—
255C23×C5—41no—
256C23×C5—61no—
257C23×C5—61no—
258C23×C5—61no—
259C23×C5—61no—
260C23×C5—61no—
261C23×C5—61no—
262C23×C5—61no—
263C23×C5—61no—
264C23×C5—61no—
265C23×C5—61no—
266C23×C5—61no—
267C23×C5—61no—
268C23×C5—61no—
269C23×C5—61no—
270C23×C5—61no—
271C23×C5—61no—
272C23×C5—61no—
273C23×C5—121no—
274C23×C5—121no—
275C23×C5—121no—
276C23×C5—122no—
277C23×C5#23511yescharacteristic
278C23×C5D3×D411yescharacteristic
279C23×C5#23711yescharacteristic
280C23×C5C2×S411yescharacteristic
281C5×D4—62no—
282C5×D4—62no—
283C5×D4—62no—
284C5×D4—62no—
285C5×D4—122no—
286C5×D4—124no—
287C5×D4—122no—
288C2×C4×C5—32no—
289C2×C4×C5—31no—
290C2×C4×C5—31no—
291C2×C4×C5—61no—
292C2×C4×C5—61no—
293C2×C4×C5—61no—
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—61no—
303C2×C4×C5—61no—
304C2×C4×C5—61no—
305C2×C4×C5—61no—
306C2×C4×C5—61no—
307C2×C4×C5—61no—
308C2×C4×C5—61no—
309C2×C4×C5—61no—
310C2×C4×C5—61no—
311C2×C4×C5—61no—
312C2×C4×C5—61no—
313C2×C4×C5—61no—
314C2×C4×C5—61no—
315C2×C4×C5—61no—
316C2×C4×C5—61no—
317C2×C4×C5—61no—
318C2×C4×C5—61no—
319C2×C4×C5—122no—
320C2×C4×C5—122no—
321C2×C4×C5C2×S411yescharacteristic
322C2×C4×C5C2×S411yescharacteristic
323C25C2×C5×D311yescharacteristic
324C22×D4—31no—
325C22×D4—31no—
326C22×D4—31no—
327C23×C4—31no—
328C23×C4C2×C5×D311yescharacteristic
329C23×C4C2×C5×D311yescharacteristic
330#123—61no—
331#123—61no—
332#123—61no—
333#123—61no—
334#120—62no—
335#120—62no—
336C4×D4—62no—
337C4×D4—62no—
338C2×C4⋊C4—31no—
339C2×C4⋊C4—31no—
340C2×C4⋊C4—31no—
341C2×C4⋊C4—31no—
342C2×C4⋊C4—31no—
343C2×C22⋊C4—31no—
344C2×C22⋊C4—31no—
345C2×C22⋊C4—31no—
346C2×C22⋊C4—31no—
347C2×C22⋊C4—31no—
348C2×C22⋊C4—31no—
349C2×C22⋊C4—31no—
350C2×C22⋊C4—31no—
351C2×C22⋊C4—31no—
352C2×C22⋊C4—31no—
353C2×C22⋊C4—31no—
354C2×C22⋊C4—31no—
355C2×C22⋊C4—31no—
356C2×C22⋊C4—31no—
357C2×C22⋊C4—31no—
358C2×C22⋊C4—31no—
359C2×C42—31no—
360#95—31no—
361#95—31no—
362#95—31no—
363#95—31no—
364#95—31no—
365#95—31no—
366C2×C3×C5—41no—
367C2×C3×C5—41no—
368C2×C3×C5—41no—
369C2×C3×C5—161no—
370C5×D3—82no—
371C23×C3—41no—
372C22×D3—41no—
373C2×A4—41no—
374C2×A4C2×C5×D411yescharacteristic
375C2×A4C5×#4111yescharacteristic
376C2×A4C5×#4111yescharacteristic
377S4—22no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380#67—82no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C4×D3—82no—
385C22×C5—21no—
386C22×C5—21no—
387C22×C5—21no—
388C22×C5—31no—
389C22×C5—31no—
390C22×C5—31no—
391C22×C5—31no—
392C22×C5—31no—
393C22×C5—31no—
394C22×C5—31no—
395C22×C5—31no—
396C22×C5—31no—
397C22×C5—61no—
398C22×C5—62no—
399C22×C5—62no—
400C22×C5—62no—
401C22×C5—62no—
402C22×C5—61no—
403C22×C5—62no—
404C22×C5—61no—
405C22×C5—61no—
406C22×C5—121no—
407C22×C5—121no—
408C22×C5—121no—
409C22×C5—121no—
410C22×C5—121no—
411C22×C5—121no—
412C22×C5—121no—
413C22×C5—121no—
414C22×C5—121no—
415C22×C5—121no—
416C22×C5#88311yescharacteristic, complemented
417C22×C5C22×S411yescharacteristic
418C4×C5—21no—
419C4×C5—21no—
420C4×C5—62no—
421C4×C5—61no—
422C4×C5—61no—
423C4×C5—121no—
424C4×C5—121no—
425C4×C5—121no—
426C4×C5—121no—
427C4×C5—121no—
428C4×C5—121no—
429C24—21no—
430C24—21no—
431C24—21no—
432C24—31no—
433C24—31no—
434C24—31no—
435C24—61no—
436C24—61no—
437C24—61no—
438C24C22×C5×D311yescharacteristic
439C2×D4—32no—
440C2×D4—32no—
441C2×D4—61no—
442C2×D4—61no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4—62no—
446C2×D4—62no—
447C2×D4—61no—
448C2×D4—62no—
449C2×D4—61no—
450C2×D4—121no—
451C2×D4—121no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—31no—
455C22×C4—31no—
456C22×C4—31no—
457C22×C4—31no—
458C22×C4—31no—
459C22×C4—31no—
460C22×C4—31no—
461C22×C4—31no—
462C22×C4—31no—
463C22×C4—31no—
464C22×C4—31no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—62no—
469C22×C4—61no—
470C22×C4—121no—
471C4⋊C4—62no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—121no—
477C4⋊C4—121no—
478C22⋊C4—31no—
479C22⋊C4—31no—
480C22⋊C4—31no—
481C22⋊C4—61no—
482C22⋊C4—62no—
483C22⋊C4—62no—
484C22⋊C4—61no—
485C22⋊C4—62no—
486C22⋊C4—61no—
487C22⋊C4—61no—
488C22⋊C4—61no—
489C22⋊C4—61no—
490C22⋊C4—62no—
491C22⋊C4—61no—
492C22⋊C4—61no—
493C22⋊C4—61no—
494C22⋊C4—61no—
495C22⋊C4—61no—
496C22⋊C4—121no—
497C22⋊C4—121no—
498C22⋊C4—121no—
499C22⋊C4C5×S411yescharacteristic, complemented
500C42—62no—
501C3×C5—41noHall
502C22×C3—41no—
503C22×C3—81no—
504C22×C3—81no—
505C22×C3—81no—
506C2×D3—42no—
507C2×D3—81no—
508C2×D3—81no—
509A4C5×#12311yescharacteristic, complemented
510C3×C4—81no—
511C3×C4—81no—
512Dic3—81no—
513Dic3—81no—
514Dic3—81no—
515C2×C5—31no—
516C2×C5—31no—
517C2×C5—31no—
518C2×C5—31no—
519C2×C5—41no—
520C2×C5—121no—
521C2×C5—122no—
522C2×C5#591211yescharacteristic
523C2×C5D4×S411yescharacteristic
524C2×C5#591411yescharacteristic
525C23—31no—
526C23—31no—
527C23—31no—
528C23—31no—
529C23—32no—
530C23—32no—
531C23—31no—
532C23—41no—
533C23—61no—
534C23—61no—
535C23—61no—
536C23—61no—
537C23—61no—
538C23—61no—
539C23—61no—
540C23—61no—
541C23—61no—
542C23—61no—
543C23—61no—
544C23—61no—
545C23—61no—
546C23—61no—
547C23—61no—
548C23—61no—
549C23—61no—
550C23—121no—
551C23—121no—
552C23—121no—
553C23—122no—
554C23C5×#23511yescharacteristic
555C23C5×D3×D411yescharacteristic
556C23C5×#23711yescharacteristic
557C23C2×C5×S411yescharacteristic
558D4—62no—
559D4—62no—
560D4—62no—
561D4—62no—
562D4—122no—
563D4—124no—
564D4—122no—
565C2×C4—32no—
566C2×C4—31no—
567C2×C4—31no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—122no—
597C2×C4—122no—
598C2×C4C2×C5×S411yescharacteristic
599C2×C4C2×C5×S411yescharacteristic
600C2×C3—41no—
601C2×C3—41no—
602C2×C3—41no—
603C2×C3—161no—
604D3—82no—
605C5#8529811yescharacteristic, direct product factor, Sylow
606C22—21no—
607C22—21no—
608C22—21no—
609C22—31no—
610C22—31no—
611C22—31no—
612C22—31no—
613C22—31no—
614C22—31no—
615C22—31no—
616C22—31no—
617C22—31no—
618C22—61no—
619C22—62no—
620C22—62no—
621C22—62no—
622C22—62no—
623C22—61no—
624C22—62no—
625C22—61no—
626C22—61no—
627C22—121no—
628C22—121no—
629C22—121no—
630C22—121no—
631C22—121no—
632C22—121no—
633C22—121no—
634C22—121no—
635C22—121no—
636C22—121no—
637C22C5×#88311yescharacteristic, complemented
638C22C22×C5×S411yescharacteristic
639C4—21no—
640C4—21no—
641C4—62no—
642C4—61no—
643C4—61no—
644C4—121no—
645C4—121no—
646C4—121no—
647C4—121no—
648C4—121no—
649C4—121no—
650C3—41noSylow
651C2—31no—
652C2—31no—
653C2—31no—
654C2—31no—
655C2—41no—
656C2—121no—
657C2—122no—
658C2C5×#591211yescharacteristic
659C2C5×D4×S411yescharacteristic
660C2C5×#591411yescharacteristic
subgroupG/H
centerC22×C5C22×S4
commutator, cosocleC22×A4C23×C5
FrattiniC22C22×C5×S4
FittingC22×C5×C22⋊C4D3
socleC24×C5C22×D3
prime radicalC2×C5×GL(2,ℤ/4)C2
hypercenterC5×C22⋊C4S4
hypocenterA4C5×#123
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22×C5 ◁ C5×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C5×S4C2semidirect
3. C5×A4×C22⋊C4C2semidirect
4. C5×#5432C2semidirect
5. C5×(C2×S4)⋊C4C2semidirect
6. C5×C4⋊SL(2,ℤ/4)C2semidirect
7. C5×SL(2,ℤ/4)⋊C4C2semidirect
19. #85298C5direct (N × H)
23. C22×C5×C22⋊C4D3semidirect
55. C2×C4×S4C2×C5semidirect
56. A4×C22⋊C4C2×C5semidirect
57. #5432C2×C5semidirect
58. (C2×S4)⋊C4C2×C5semidirect
59. C4⋊SL(2,ℤ/4)C2×C5semidirect
60. SL(2,ℤ/4)⋊C4C2×C5semidirect
199. C5×C22⋊C4S4semidirect
204. C22×C22⋊C4C5×D3semidirect
223. C5×A4#123semidirect
416. C22×C5#883semidirect
499. C22⋊C4C5×S4semidirect
509. A4C5×#123semidirect
605. C5#85298direct (N × H)
637. C22C5×#883semidirect

Maximal subgroups

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

subgroupindex
C2×C5×GL(2,ℤ/4)2
C2×C4×C5×S42
C5×A4×C22⋊C42
C5×#54322
C5×(C2×S4)⋊C42
C5×C4⋊SL(2,ℤ/4)2
C5×SL(2,ℤ/4)⋊C42
C5×#24183
C5×#8834
#852985