Finite Group Databasebeta

← C5×#85311 #14511049

Subgroups — C5×#85311 #14511049

Order 1920.

674 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 674 conjugacy classes of subgroups — too many to draw. Only its 84 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×C4×C5×S4C211yesmaximal, characteristic
2C2×C5×A4⋊Q8C211yesmaximal, characteristic, complemented
3C5×A4×C4⋊C4C211yesmaximal, characteristic, complemented
4C5×(C2×S4)⋊C4C211yesmaximal, characteristic
5C5×(C2×S4)⋊C4C211yesmaximal, characteristic
6C5×SL(2,ℤ/4)⋊C4C211yesmaximal, characteristic, complemented
7C5×SL(2,ℤ/4)⋊C4C211yesmaximal, characteristic, complemented
8C5×#2449—31nomaximal, Hall
9C22×C5×S4C2211yescharacteristic
10C2×C4×C5×A4C2211yescharacteristic
11C2×C4×C5×A4C2211yescharacteristic
12C2×C4×C5×A4C2211yescharacteristic
13C2×C5×SL(2,ℤ/4)C2211yescharacteristic
14C2×C5×SL(2,ℤ/4)C2211yescharacteristic
15C2×C5×SL(2,ℤ/4)C2211yescharacteristic
16C4×C5×S4—22no—
17C5×A4⋊Q8—22no—
18C5×#896—41nomaximal
19#85311C511yesmaximal, characteristic, direct product factor, Hall
20C2×C5×C22⋊Q8—31no—
21C2×C5×C22⋊Q8—31no—
22C2×C4×C5×D4—31no—
23C22×C5×C4⋊C4D311yescharacteristic, complemented
24C5×#399—31no—
25C5×#398—31no—
26C5×#396—31no—
27C5×#395—31no—
28C5×#389—31no—
29C5×#387—31no—
30C5×#386—31no—
31C5×#386—31no—
32C5×#385—31no—
33C5×#385—31no—
34C5×#385—31no—
35C22×C5×A4C2311yescharacteristic
36C2×C5×S4—21no—
37C2×C5×S4—21no—
38C2×C5×S4Q812yescomplemented
39C2×C4×C5×D3—41no—
40C2×C5×Dic6—41no—
41C4×C5×A4—21no—
42C4×C5×A4—21no—
43C4×C5×A4—21no—
44C5×SL(2,ℤ/4)—21no—
45C5×SL(2,ℤ/4)—21no—
46C5×SL(2,ℤ/4)D412yes—
47C3×C5×C4⋊C4—41no—
48C5×(C2×D3)⋊C4—41no—
49C5×(C2×D3)⋊C4—41no—
50C5×Dic3⋊C4—41no—
51C5×Dic3⋊C4—41no—
52C2×C4×S4C2×C511yescharacteristic
53C2×A4⋊Q8C2×C511yescharacteristic, complemented
54A4×C4⋊C4C2×C511yescharacteristic, complemented
55(C2×S4)⋊C4C2×C511yescharacteristic
56(C2×S4)⋊C4C2×C511yescharacteristic
57SL(2,ℤ/4)⋊C4C2×C511yescharacteristic, complemented
58SL(2,ℤ/4)⋊C4C2×C511yescharacteristic, complemented
59C22×C5×Q8—31no—
60C22×C5×D4—31no—
61C23×C4×C5—31no—
62C23×C4×C5C2×D311yescharacteristic
63C23×C4×C5C2×D311yescharacteristic
64C23×C4×C5C2×D311yescharacteristic
65C5×C22⋊Q8—62no—
66C5×C22⋊Q8—62no—
67C5×C22⋊Q8—61no—
68C5×C22⋊Q8—61no—
69C5×C22⋊Q8—61no—
70C5×C22⋊Q8—61no—
71C4×C5×D4—62no—
72C4×C5×D4—62no—
73C2×C5×C4⋊C4—31no—
74C2×C5×C4⋊C4—31no—
75C2×C5×C4⋊C4—31no—
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×C4⋊C4—31no—
82C2×C5×C4⋊C4—31no—
83C2×C5×C4⋊C4—31no—
84C2×C5×C4⋊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×C42×C5—31no—
94C5×#95—31no—
95C5×#95—31no—
96C5×#95—31no—
97C5×#95—31no—
98C5×#95—31no—
99C5×#95—31no—
100C5×#95—31no—
101C5×#95—31no—
102#2449—31noSylow
103C22×C5×D3—41no—
104C2×C5×A4C2×D411yescharacteristic
105C2×C5×A4C2×Q811yescharacteristic
106C2×C5×A4#4111yescharacteristic
107C5×S4—22no—
108C2×C3×C4×C5—41no—
109C2×C3×C4×C5—41no—
110C2×C3×C4×C5—41no—
111C2×C5×Dic3—41no—
112C2×C5×Dic3—41no—
113C2×C5×Dic3—41no—
114C4×C5×D3—82no—
115C5×Dic6—82no—
116C22×S4C22×C511yescharacteristic
117C2×C4×A4C22×C511yescharacteristic
118C2×C4×A4C22×C511yescharacteristic
119C2×C4×A4C22×C511yescharacteristic
120C2×SL(2,ℤ/4)C22×C511yescharacteristic
121C2×SL(2,ℤ/4)C22×C511yescharacteristic
122C2×SL(2,ℤ/4)C22×C511yescharacteristic
123C4×S4—22no—
124A4⋊Q8—22no—
125#896—41no—
126C24×C5—31no—
127C24×C5C22×D311yescharacteristic
128C2×C5×Q8—62no—
129C2×C5×Q8—61no—
130C2×C5×Q8—61no—
131C2×C5×Q8—121no—
132C2×C5×D4—32no—
133C2×C5×D4—32no—
134C2×C5×D4—61no—
135C2×C5×D4—61no—
136C2×C5×D4—61no—
137C2×C5×D4—61no—
138C22×C4×C5—21no—
139C22×C4×C5—21no—
140C22×C4×C5—21no—
141C22×C4×C5—31no—
142C22×C4×C5—31no—
143C22×C4×C5—31no—
144C22×C4×C5—31no—
145C22×C4×C5—31no—
146C22×C4×C5—31no—
147C22×C4×C5—31no—
148C22×C4×C5—31no—
149C22×C4×C5—31no—
150C22×C4×C5—31no—
151C22×C4×C5—31no—
152C22×C4×C5—31no—
153C22×C4×C5—31no—
154C22×C4×C5—61no—
155C22×C4×C5—62no—
156C22×C4×C5—61no—
157C22×C4×C5—61no—
158C22×C4×C5—61no—
159C22×C4×C5—61no—
160C22×C4×C5—121no—
161C5×C4⋊C4—31no—
162C5×C4⋊C4—31no—
163C5×C4⋊C4—31no—
164C5×C4⋊C4—61no—
165C5×C4⋊C4—62no—
166C5×C4⋊C4—62no—
167C5×C4⋊C4—62no—
168C5×C4⋊C4—62no—
169C5×C4⋊C4—62no—
170C5×C4⋊C4—61no—
171C5×C4⋊C4—61no—
172C5×C4⋊C4—61no—
173C5×C4⋊C4—61no—
174C5×C4⋊C4—121no—
175C5×C4⋊C4S411yescharacteristic, complemented
176C5×C22⋊C4—32no—
177C5×C22⋊C4—32no—
178C5×C22⋊C4—62no—
179C5×C22⋊C4—62no—
180C5×C22⋊C4—61no—
181C5×C22⋊C4—61no—
182C5×C22⋊C4—61no—
183C5×C22⋊C4—61no—
184C5×C22⋊C4—61no—
185C5×C22⋊C4—61no—
186C5×C22⋊C4—61no—
187C5×C22⋊C4—61no—
188C5×C22⋊C4—121no—
189C42×C5—62no—
190C2×C22⋊Q8—31no—
191C2×C22⋊Q8—31no—
192C2×C4×D4—31no—
193C22×C4⋊C4C5×D311yescharacteristic, complemented
194#399—31no—
195#398—31no—
196#396—31no—
197#395—31no—
198#389—31no—
199#387—31no—
200#386—31no—
201#386—31no—
202#385—31no—
203#385—31no—
204#385—31no—
205C22×C3×C5—41no—
206C2×C5×D3—42no—
207C2×C5×D3—81no—
208C2×C5×D3—81no—
209C5×A4C22⋊Q811yescharacteristic, complemented
210C3×C4×C5—81no—
211C3×C4×C5—81no—
212C3×C4×C5—81no—
213C5×Dic3—42no—
214C5×Dic3—81no—
215C5×Dic3—81no—
216C22×A4C23×C511yescharacteristic
217C2×S4—21no—
218C2×S4—21no—
219C2×S4C5×Q812yescomplemented
220C2×C4×D3—41no—
221C2×Dic6—41no—
222C4×A4—21no—
223C4×A4—21no—
224C4×A4—21no—
225SL(2,ℤ/4)—21no—
226SL(2,ℤ/4)—21no—
227SL(2,ℤ/4)C5×D412yes—
228C3×C4⋊C4—41no—
229(C2×D3)⋊C4—41no—
230(C2×D3)⋊C4—41no—
231Dic3⋊C4—41no—
232Dic3⋊C4—41no—
233C23×C5—31no—
234C23×C5—31no—
235C23×C5—31no—
236C23×C5—31no—
237C23×C5—32no—
238C23×C5—61no—
239C23×C5—61no—
240C23×C5—61no—
241C23×C5—61no—
242C23×C5—61no—
243C23×C5—61no—
244C23×C5#23511yescharacteristic
245C23×C5D3×D411yescharacteristic
246C23×C5D3×Q811yescharacteristic
247C5×Q8—122no—
248C5×Q8—122no—
249C5×D4—62no—
250C5×D4—62no—
251C5×D4—62no—
252C5×D4—62no—
253C2×C4×C5—32no—
254C2×C4×C5—32no—
255C2×C4×C5—32no—
256C2×C4×C5—32no—
257C2×C4×C5—31no—
258C2×C4×C5—31no—
259C2×C4×C5—31no—
260C2×C4×C5—61no—
261C2×C4×C5—61no—
262C2×C4×C5—61no—
263C2×C4×C5—61no—
264C2×C4×C5—61no—
265C2×C4×C5—61no—
266C2×C4×C5—61no—
267C2×C4×C5—61no—
268C2×C4×C5—61no—
269C2×C4×C5—61no—
270C2×C4×C5—61no—
271C2×C4×C5—61no—
272C2×C4×C5—61no—
273C2×C4×C5—61no—
274C2×C4×C5—61no—
275C2×C4×C5—61no—
276C2×C4×C5—61no—
277C2×C4×C5—61no—
278C2×C4×C5—61no—
279C2×C4×C5—61no—
280C2×C4×C5—61no—
281C2×C4×C5—61no—
282C2×C4×C5—61no—
283C2×C4×C5—61no—
284C2×C4×C5—61no—
285C2×C4×C5—61no—
286C2×C4×C5—61no—
287C2×C4×C5—61no—
288C2×C4×C5—61no—
289C2×C4×C5—122no—
290C2×C4×C5—122no—
291C2×C4×C5C2×S411yescharacteristic
292C2×C4×C5C2×S411yescharacteristic
293C2×C4×C5C2×S411yescharacteristic
294C22×Q8—31no—
295C22×D4—31no—
296C23×C4—31no—
297C23×C4C2×C5×D311yescharacteristic
298C23×C4C2×C5×D311yescharacteristic
299C23×C4C2×C5×D311yescharacteristic
300C22⋊Q8—62no—
301C22⋊Q8—62no—
302C22⋊Q8—61no—
303C22⋊Q8—61no—
304C22⋊Q8—61no—
305C22⋊Q8—61no—
306C4×D4—62no—
307C4×D4—62no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—31no—
320C2×C22⋊C4—31no—
321C2×C22⋊C4—31no—
322C2×C22⋊C4—31no—
323C2×C22⋊C4—31no—
324C2×C22⋊C4—31no—
325C2×C22⋊C4—31no—
326C2×C22⋊C4—31no—
327C2×C22⋊C4—31no—
328C2×C42—31no—
329#95—31no—
330#95—31no—
331#95—31no—
332#95—31no—
333#95—31no—
334#95—31no—
335#95—31no—
336#95—31no—
337C2×C3×C5—41no—
338C2×C3×C5—41no—
339C2×C3×C5—41no—
340C5×D3—82no—
341C22×D3—41no—
342C2×A4C2×C5×D411yescharacteristic
343C2×A4C2×C5×Q811yescharacteristic
344C2×A4C5×#4111yescharacteristic
345S4—22no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×Dic3—41no—
350C2×Dic3—41no—
351C2×Dic3—41no—
352C4×D3—82no—
353Dic6—82no—
354C22×C5—31no—
355C22×C5—31no—
356C22×C5—31no—
357C22×C5—31no—
358C22×C5—31no—
359C22×C5—31no—
360C22×C5—31no—
361C22×C5—31no—
362C22×C5—31no—
363C22×C5—61no—
364C22×C5—62no—
365C22×C5—62no—
366C22×C5—62no—
367C22×C5—121no—
368C22×C5—121no—
369C22×C5—121no—
370C22×C5—121no—
371C22×C5#89611yescharacteristic, complemented
372C22×C5C22×S411yescharacteristic
373C4×C5—21no—
374C4×C5—21no—
375C4×C5—21no—
376C4×C5—62no—
377C4×C5—62no—
378C4×C5—61no—
379C4×C5—62no—
380C4×C5—61no—
381C4×C5—61no—
382C4×C5—121no—
383C4×C5—121no—
384C4×C5—121no—
385C4×C5—121no—
386C24—31no—
387C24C22×C5×D311yescharacteristic
388C2×Q8—62no—
389C2×Q8—61no—
390C2×Q8—61no—
391C2×Q8—121no—
392C2×D4—32no—
393C2×D4—32no—
394C2×D4—61no—
395C2×D4—61no—
396C2×D4—61no—
397C2×D4—61no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—31no—
402C22×C4—31no—
403C22×C4—31no—
404C22×C4—31no—
405C22×C4—31no—
406C22×C4—31no—
407C22×C4—31no—
408C22×C4—31no—
409C22×C4—31no—
410C22×C4—31no—
411C22×C4—31no—
412C22×C4—31no—
413C22×C4—31no—
414C22×C4—61no—
415C22×C4—62no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—121no—
421C4⋊C4—31no—
422C4⋊C4—31no—
423C4⋊C4—31no—
424C4⋊C4—61no—
425C4⋊C4—62no—
426C4⋊C4—62no—
427C4⋊C4—62no—
428C4⋊C4—62no—
429C4⋊C4—61no—
430C4⋊C4—61no—
431C4⋊C4—62no—
432C4⋊C4—61no—
433C4⋊C4—61no—
434C4⋊C4—121no—
435C4⋊C4C5×S411yescharacteristic, complemented
436C22⋊C4—32no—
437C22⋊C4—32no—
438C22⋊C4—62no—
439C22⋊C4—61no—
440C22⋊C4—61no—
441C22⋊C4—62no—
442C22⋊C4—61no—
443C22⋊C4—61no—
444C22⋊C4—61no—
445C22⋊C4—61no—
446C22⋊C4—61no—
447C22⋊C4—61no—
448C22⋊C4—121no—
449C42—62no—
450C3×C5—41noHall
451C22×C3—41no—
452C2×D3—42no—
453C2×D3—81no—
454C2×D3—81no—
455A4C5×C22⋊Q811yescharacteristic, complemented
456C3×C4—81no—
457C3×C4—81no—
458C3×C4—81no—
459Dic3—42no—
460Dic3—81no—
461Dic3—81no—
462C2×C5—31no—
463C2×C5—31no—
464C2×C5—31no—
465C2×C5—31no—
466C2×C5—122no—
467C2×C5#591211yescharacteristic
468C2×C5D4×S411yescharacteristic
469C2×C5Q8×S411yescharacteristic
470C23—31no—
471C23—31no—
472C23—31no—
473C23—31no—
474C23—32no—
475C23—61no—
476C23—61no—
477C23—61no—
478C23—61no—
479C23—61no—
480C23—61no—
481C23C5×#23511yescharacteristic
482C23C5×D3×D411yescharacteristic
483C23C5×D3×Q811yescharacteristic
484Q8—122no—
485Q8—122no—
486D4—62no—
487D4—62no—
488D4—62no—
489D4—62no—
490C2×C4—32no—
491C2×C4—32no—
492C2×C4—32no—
493C2×C4—32no—
494C2×C4—31no—
495C2×C4—31no—
496C2×C4—31no—
497C2×C4—61no—
498C2×C4—61no—
499C2×C4—61no—
500C2×C4—61no—
501C2×C4—61no—
502C2×C4—61no—
503C2×C4—61no—
504C2×C4—61no—
505C2×C4—61no—
506C2×C4—61no—
507C2×C4—61no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—61no—
524C2×C4—61no—
525C2×C4—61no—
526C2×C4—122no—
527C2×C4—122no—
528C2×C4C2×C5×S411yescharacteristic
529C2×C4C2×C5×S411yescharacteristic
530C2×C4C2×C5×S411yescharacteristic
531C2×C3—41no—
532C2×C3—41no—
533C2×C3—41no—
534D3—82no—
535C5#8531111yescharacteristic, direct product factor, Sylow
536C22—31no—
537C22—31no—
538C22—31no—
539C22—31no—
540C22—31no—
541C22—31no—
542C22—31no—
543C22—31no—
544C22—31no—
545C22—61no—
546C22—62no—
547C22—62no—
548C22—62no—
549C22—121no—
550C22—121no—
551C22—121no—
552C22—121no—
553C22C5×#89611yescharacteristic, complemented
554C22C22×C5×S411yescharacteristic
555C4—21no—
556C4—21no—
557C4—21no—
558C4—62no—
559C4—62no—
560C4—61no—
561C4—61no—
562C4—61no—
563C4—62no—
564C4—121no—
565C4—121no—
566C4—121no—
567C4—121no—
568C3—41noSylow
569C2—31no—
570C2—31no—
571C2—31no—
572C2—31no—
573C2—122no—
574C2C5×#591211yescharacteristic
575C2C5×D4×S411yescharacteristic
576C2C5×Q8×S411yescharacteristic
subgroupG/H
centerC22×C5C22×S4
commutator, cosocleC22×A4C23×C5
FrattiniC22C22×C5×S4
FittingC22×C5×C4⋊C4D3
socleC24×C5C22×D3
prime radicalC22×C5×S4C22
hypercenterC5×C4⋊C4S4
hypocenterA4C5×C22⋊Q8
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22×C5 ◁ C5×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C5×A4⋊Q8C2semidirect
3. C5×A4×C4⋊C4C2semidirect
6. C5×SL(2,ℤ/4)⋊C4C2semidirect
7. C5×SL(2,ℤ/4)⋊C4C2semidirect
19. #85311C5direct (N × H)
23. C22×C5×C4⋊C4D3semidirect
38. C2×C5×S4Q8semidirect
53. C2×A4⋊Q8C2×C5semidirect
54. A4×C4⋊C4C2×C5semidirect
57. SL(2,ℤ/4)⋊C4C2×C5semidirect
58. SL(2,ℤ/4)⋊C4C2×C5semidirect
175. C5×C4⋊C4S4semidirect
193. C22×C4⋊C4C5×D3semidirect
209. C5×A4C22⋊Q8semidirect
219. C2×S4C5×Q8semidirect
371. C22×C5#896semidirect
435. C4⋊C4C5×S4semidirect
455. A4C5×C22⋊Q8semidirect
535. C5#85311direct (N × H)
553. C22C5×#896semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C5×S42
C2×C5×A4⋊Q82
C5×A4×C4⋊C42
C5×(C2×S4)⋊C42
C5×SL(2,ℤ/4)⋊C42
C5×#24493
C5×#8964
#853115