Finite Group Databasebeta

← C5×#85309 #14511047

Subgroups — C5×#85309 #14511047

Order 1920.

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

normal subgroups only

This group has 708 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×C4⋊S4C211yesmaximal, characteristic
2C2×C4×C5×S4C211yesmaximal, characteristic, complemented
3C5×A4×C4⋊C4C211yesmaximal, characteristic, complemented
4C5×(C2×S4)⋊C4C211yesmaximal, characteristic, complemented
5C5×(C2×S4)⋊C4C211yesmaximal, characteristic, complemented
6C5×(C2×S4)⋊C4C211yesmaximal, characteristic, complemented
7C5×SL(2,ℤ/4)⋊C4C211yesmaximal, characteristic, complemented
8C5×#2443—31nomaximal, Hall
9C22×C5×S4C2211yescharacteristic
10C22×C5×S4C2211yescharacteristic
11C2×C4×C5×A4C2211yescharacteristic
12C2×C4×C5×A4C2211yescharacteristic
13C2×C4×C5×A4C2211yescharacteristic
14C2×C5×SL(2,ℤ/4)C2211yescharacteristic
15C2×C5×SL(2,ℤ/4)C2211yescharacteristic
16C5×C4⋊S4—22no—
17C4×C5×S4—22no—
18C5×#894—41nomaximal
19#85309C511yesmaximal, characteristic, direct product factor, Hall
20C2×C5×#123—31no—
21C2×C5×#121—31no—
22C2×C4×C5×D4—31no—
23C22×C5×C4⋊C4D311yescharacteristic, complemented
24C5×#399—31no—
25C5×#398—31no—
26C5×#397—31no—
27C5×#394—31no—
28C5×#388—31no—
29C5×#387—31no—
30C5×#386—31no—
31C5×#386—31no—
32C5×#386—31no—
33C5×#385—31no—
34C5×#385—31no—
35C22×C5×A4C2311yescharacteristic
36C2×C5×S4—21no—
37C2×C5×S4—21no—
38C2×C5×S4—21no—
39C2×C5×S4—21no—
40C2×C5×S4—21no—
41C2×C5×S4D412yes—
42C2×C5×D12—41no—
43C2×C4×C5×D3—41no—
44C4×C5×A4—21no—
45C4×C5×A4—21no—
46C4×C5×A4—21no—
47C5×SL(2,ℤ/4)—21no—
48C5×SL(2,ℤ/4)—21no—
49C3×C5×C4⋊C4—41no—
50C5×(C2×D3)⋊C4—41no—
51C5×(C2×D3)⋊C4—41no—
52C5×(C2×D3)⋊C4—41no—
53C5×Dic3⋊C4—41no—
54C2×C4⋊S4C2×C511yescharacteristic
55C2×C4×S4C2×C511yescharacteristic, complemented
56A4×C4⋊C4C2×C511yescharacteristic, complemented
57(C2×S4)⋊C4C2×C511yescharacteristic, complemented
58(C2×S4)⋊C4C2×C511yescharacteristic, complemented
59(C2×S4)⋊C4C2×C511yescharacteristic, complemented
60SL(2,ℤ/4)⋊C4C2×C511yescharacteristic, complemented
61C22×C5×D4—31no—
62C22×C5×D4—31no—
63C22×C5×D4—31no—
64C23×C4×C5—31no—
65C23×C4×C5C2×D311yescharacteristic
66C23×C4×C5C2×D311yescharacteristic
67C23×C4×C5C2×D311yescharacteristic
68C5×#123—61no—
69C5×#123—61no—
70C5×#123—61no—
71C5×#123—61no—
72C5×#121—62no—
73C5×#121—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×C4⋊C4—31no—
82C2×C5×C4⋊C4—31no—
83C2×C5×C4⋊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×C42×C5—31no—
96C5×#95—31no—
97C5×#95—31no—
98C5×#95—31no—
99C5×#95—31no—
100C5×#95—31no—
101C5×#95—31no—
102C5×#95—31no—
103C5×#95—31no—
104#2443—31noSylow
105C22×C5×D3—41no—
106C22×C5×D3—41no—
107C2×C5×A4C2×D411yescharacteristic
108C2×C5×A4#4111yescharacteristic
109C2×C5×A4#4111yescharacteristic
110C5×S4—22no—
111C5×S4—41no—
112C2×C3×C4×C5—41no—
113C2×C3×C4×C5—41no—
114C2×C3×C4×C5—41no—
115C2×C5×Dic3—41no—
116C2×C5×Dic3—41no—
117C5×D12—82no—
118C4×C5×D3—82no—
119C22×S4C22×C511yescharacteristic
120C22×S4C22×C511yescharacteristic
121C2×C4×A4C22×C511yescharacteristic
122C2×C4×A4C22×C511yescharacteristic
123C2×C4×A4C22×C511yescharacteristic
124C2×SL(2,ℤ/4)C22×C511yescharacteristic
125C2×SL(2,ℤ/4)C22×C511yescharacteristic
126C4⋊S4—22no—
127C4×S4—22no—
128#894—41no—
129C24×C5—31no—
130C24×C5—31no—
131C24×C5C22×D311yescharacteristic
132C2×C5×D4—32no—
133C2×C5×D4—32no—
134C2×C5×D4—61no—
135C2×C5×D4—61no—
136C2×C5×D4—61no—
137C2×C5×D4—61no—
138C2×C5×D4—61no—
139C2×C5×D4—62no—
140C2×C5×D4—61no—
141C2×C5×D4—61no—
142C2×C5×D4—61no—
143C2×C5×D4—61no—
144C2×C5×D4—61no—
145C2×C5×D4—61no—
146C2×C5×D4—61no—
147C2×C5×D4—121no—
148C22×C4×C5—21no—
149C22×C4×C5—21no—
150C22×C4×C5—21no—
151C22×C4×C5—31no—
152C22×C4×C5—31no—
153C22×C4×C5—31no—
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—61no—
164C22×C4×C5—61no—
165C22×C4×C5—61no—
166C22×C4×C5—61no—
167C22×C4×C5—62no—
168C22×C4×C5—61no—
169C22×C4×C5—121no—
170C5×C4⋊C4—31no—
171C5×C4⋊C4—31no—
172C5×C4⋊C4—31no—
173C5×C4⋊C4—61no—
174C5×C4⋊C4—62no—
175C5×C4⋊C4—62no—
176C5×C4⋊C4—61no—
177C5×C4⋊C4—61no—
178C5×C4⋊C4—121no—
179C5×C4⋊C4S411yescharacteristic, complemented
180C5×C22⋊C4—62no—
181C5×C22⋊C4—62no—
182C5×C22⋊C4—62no—
183C5×C22⋊C4—61no—
184C5×C22⋊C4—61no—
185C5×C22⋊C4—61no—
186C5×C22⋊C4—61no—
187C5×C22⋊C4—62no—
188C5×C22⋊C4—61no—
189C5×C22⋊C4—62no—
190C5×C22⋊C4—61no—
191C5×C22⋊C4—61no—
192C5×C22⋊C4—61no—
193C5×C22⋊C4—61no—
194C5×C22⋊C4—61no—
195C5×C22⋊C4—121no—
196C42×C5—62no—
197C2×#123—31no—
198C2×#121—31no—
199C2×C4×D4—31no—
200C22×C4⋊C4C5×D311yescharacteristic, complemented
201#399—31no—
202#398—31no—
203#397—31no—
204#394—31no—
205#388—31no—
206#387—31no—
207#386—31no—
208#386—31no—
209#386—31no—
210#385—31no—
211#385—31no—
212C22×C3×C5—41no—
213C2×C5×D3—42no—
214C2×C5×D3—81no—
215C2×C5×D3—81no—
216C2×C5×D3—81no—
217C2×C5×D3—81no—
218C2×C5×D3—81no—
219C5×A4#12311yescharacteristic, complemented
220C3×C4×C5—81no—
221C3×C4×C5—81no—
222C3×C4×C5—81no—
223C5×Dic3—81no—
224C5×Dic3—81no—
225C22×A4C23×C511yescharacteristic
226C2×S4—21no—
227C2×S4—21no—
228C2×S4—21no—
229C2×S4—21no—
230C2×S4—21no—
231C2×S4C5×D412yes—
232C2×D12—41no—
233C2×C4×D3—41no—
234C4×A4—21no—
235C4×A4—21no—
236C4×A4—21no—
237SL(2,ℤ/4)—21no—
238SL(2,ℤ/4)—21no—
239C3×C4⋊C4—41no—
240(C2×D3)⋊C4—41no—
241(C2×D3)⋊C4—41no—
242(C2×D3)⋊C4—41no—
243Dic3⋊C4—41no—
244C23×C5—31no—
245C23×C5—31no—
246C23×C5—31no—
247C23×C5—31no—
248C23×C5—32no—
249C23×C5—61no—
250C23×C5—61no—
251C23×C5—61no—
252C23×C5—61no—
253C23×C5—61no—
254C23×C5—61no—
255C23×C5—61no—
256C23×C5—61no—
257C23×C5—61no—
258C23×C5—61no—
259C23×C5—61no—
260C23×C5—61no—
261C23×C5—61no—
262C23×C5#23511yescharacteristic
263C23×C5D3×D411yescharacteristic
264C23×C5#23911yescharacteristic
265C5×D4—62no—
266C5×D4—62no—
267C5×D4—62no—
268C5×D4—62no—
269C5×D4—122no—
270C5×D4—122no—
271C5×D4—121no—
272C5×D4—121no—
273C5×D4—121no—
274C5×D4—121no—
275C2×C4×C5—32no—
276C2×C4×C5—32no—
277C2×C4×C5—31no—
278C2×C4×C5—31no—
279C2×C4×C5—31no—
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—61no—
290C2×C4×C5—61no—
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—122no—
309C2×C4×C5—122no—
310C2×C4×C5C2×S411yescharacteristic
311C2×C4×C5C2×S411yescharacteristic
312C2×C4×C5C2×S411yescharacteristic
313C22×D4—31no—
314C22×D4—31no—
315C22×D4—31no—
316C23×C4—31no—
317C23×C4C2×C5×D311yescharacteristic
318C23×C4C2×C5×D311yescharacteristic
319C23×C4C2×C5×D311yescharacteristic
320#123—61no—
321#123—61no—
322#123—61no—
323#123—61no—
324#121—62no—
325#121—62no—
326C4×D4—62no—
327C4×D4—62no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—31no—
332C2×C4⋊C4—31no—
333C2×C4⋊C4—31no—
334C2×C4⋊C4—31no—
335C2×C4⋊C4—31no—
336C2×C22⋊C4—31no—
337C2×C22⋊C4—31no—
338C2×C22⋊C4—31no—
339C2×C22⋊C4—31no—
340C2×C22⋊C4—31no—
341C2×C22⋊C4—31no—
342C2×C22⋊C4—31no—
343C2×C22⋊C4—31no—
344C2×C22⋊C4—31no—
345C2×C22⋊C4—31no—
346C2×C22⋊C4—31no—
347C2×C42—31no—
348#95—31no—
349#95—31no—
350#95—31no—
351#95—31no—
352#95—31no—
353#95—31no—
354#95—31no—
355#95—31no—
356C2×C3×C5—41no—
357C2×C3×C5—41no—
358C2×C3×C5—41no—
359C5×D3—82no—
360C5×D3—161no—
361C22×D3—41no—
362C22×D3—41no—
363C2×A4C2×C5×D411yescharacteristic
364C2×A4C5×#4111yescharacteristic
365C2×A4C5×#4111yescharacteristic
366S4—22no—
367S4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373D12—82no—
374C4×D3—82no—
375C22×C5—31no—
376C22×C5—31no—
377C22×C5—31no—
378C22×C5—31no—
379C22×C5—31no—
380C22×C5—31no—
381C22×C5—31no—
382C22×C5—31no—
383C22×C5—31no—
384C22×C5—61no—
385C22×C5—62no—
386C22×C5—62no—
387C22×C5—62no—
388C22×C5—121no—
389C22×C5—121no—
390C22×C5—121no—
391C22×C5—121no—
392C22×C5—121no—
393C22×C5—121no—
394C22×C5—121no—
395C22×C5—121no—
396C22×C5—121no—
397C22×C5—121no—
398C22×C5—121no—
399C22×C5#89411yescharacteristic, complemented
400C22×C5C22×S411yescharacteristic
401C4×C5—21no—
402C4×C5—21no—
403C4×C5—21no—
404C4×C5—62no—
405C4×C5—61no—
406C4×C5—61no—
407C4×C5—61no—
408C4×C5—121no—
409C4×C5—121no—
410C4×C5—121no—
411C4×C5—121no—
412C4×C5—121no—
413C24—31no—
414C24—31no—
415C24C22×C5×D311yescharacteristic
416C2×D4—32no—
417C2×D4—32no—
418C2×D4—61no—
419C2×D4—61no—
420C2×D4—61no—
421C2×D4—61no—
422C2×D4—61no—
423C2×D4—62no—
424C2×D4—61no—
425C2×D4—61no—
426C2×D4—61no—
427C2×D4—61no—
428C2×D4—61no—
429C2×D4—61no—
430C2×D4—61no—
431C2×D4—121no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—31no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—31no—
441C22×C4—31no—
442C22×C4—31no—
443C22×C4—31no—
444C22×C4—31no—
445C22×C4—31no—
446C22×C4—31no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—62no—
452C22×C4—61no—
453C22×C4—121no—
454C4⋊C4—31no—
455C4⋊C4—31no—
456C4⋊C4—31no—
457C4⋊C4—61no—
458C4⋊C4—62no—
459C4⋊C4—62no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—121no—
463C4⋊C4C5×S411yescharacteristic, complemented
464C22⋊C4—62no—
465C22⋊C4—61no—
466C22⋊C4—61no—
467C22⋊C4—61no—
468C22⋊C4—62no—
469C22⋊C4—62no—
470C22⋊C4—62no—
471C22⋊C4—61no—
472C22⋊C4—62no—
473C22⋊C4—61no—
474C22⋊C4—61no—
475C22⋊C4—61no—
476C22⋊C4—61no—
477C22⋊C4—61no—
478C22⋊C4—61no—
479C22⋊C4—121no—
480C42—62no—
481C3×C5—41noHall
482C22×C3—41no—
483C2×D3—42no—
484C2×D3—81no—
485C2×D3—81no—
486C2×D3—81no—
487C2×D3—81no—
488C2×D3—81no—
489A4C5×#12311yescharacteristic, complemented
490C3×C4—81no—
491C3×C4—81no—
492C3×C4—81no—
493Dic3—81no—
494Dic3—81no—
495C2×C5—31no—
496C2×C5—31no—
497C2×C5—31no—
498C2×C5—31no—
499C2×C5—122no—
500C2×C5—241no—
501C2×C5#591211yescharacteristic
502C2×C5D4×S411yescharacteristic
503C2×C5#591911yescharacteristic
504C23—31no—
505C23—31no—
506C23—31no—
507C23—31no—
508C23—32no—
509C23—61no—
510C23—61no—
511C23—61no—
512C23—61no—
513C23—61no—
514C23—61no—
515C23—61no—
516C23—61no—
517C23—61no—
518C23—61no—
519C23—61no—
520C23—61no—
521C23—61no—
522C23C5×#23511yescharacteristic
523C23C5×D3×D411yescharacteristic
524C23C5×#23911yescharacteristic
525D4—62no—
526D4—62no—
527D4—62no—
528D4—62no—
529D4—122no—
530D4—122no—
531D4—121no—
532D4—121no—
533D4—121no—
534D4—121no—
535C2×C4—32no—
536C2×C4—32no—
537C2×C4—31no—
538C2×C4—31no—
539C2×C4—31no—
540C2×C4—61no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—122no—
569C2×C4—122no—
570C2×C4C2×C5×S411yescharacteristic
571C2×C4C2×C5×S411yescharacteristic
572C2×C4C2×C5×S411yescharacteristic
573C2×C3—41no—
574C2×C3—41no—
575C2×C3—41no—
576D3—82no—
577D3—161no—
578C5#8530911yescharacteristic, direct product factor, Sylow
579C22—31no—
580C22—31no—
581C22—31no—
582C22—31no—
583C22—31no—
584C22—31no—
585C22—31no—
586C22—31no—
587C22—31no—
588C22—61no—
589C22—62no—
590C22—62no—
591C22—62no—
592C22—121no—
593C22—121no—
594C22—121no—
595C22—121no—
596C22—121no—
597C22—121no—
598C22—121no—
599C22—121no—
600C22—121no—
601C22—121no—
602C22—121no—
603C22C5×#89411yescharacteristic, complemented
604C22C22×C5×S411yescharacteristic
605C4—21no—
606C4—21no—
607C4—21no—
608C4—62no—
609C4—61no—
610C4—61no—
611C4—61no—
612C4—121no—
613C4—121no—
614C4—121no—
615C4—121no—
616C4—121no—
617C3—41noSylow
618C2—31no—
619C2—31no—
620C2—31no—
621C2—31no—
622C2—122no—
623C2—241no—
624C2C5×#591211yescharacteristic
625C2C5×D4×S411yescharacteristic
626C2C5×#591911yescharacteristic
subgroupG/H
centerC22×C5C22×S4
commutator, cosocleC22×A4C23×C5
FrattiniC22C22×C5×S4
FittingC22×C5×C4⋊C4D3
socleC24×C5C22×D3
prime radicalC2×C5×C4⋊S4C2
hypercenterC5×C4⋊C4S4
hypocenterA4C5×#123
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×C4×C5×S4C2semidirect
3. C5×A4×C4⋊C4C2semidirect
4. C5×(C2×S4)⋊C4C2semidirect
5. C5×(C2×S4)⋊C4C2semidirect
6. C5×(C2×S4)⋊C4C2semidirect
7. C5×SL(2,ℤ/4)⋊C4C2semidirect
19. #85309C5direct (N × H)
23. C22×C5×C4⋊C4D3semidirect
55. C2×C4×S4C2×C5semidirect
56. A4×C4⋊C4C2×C5semidirect
57. (C2×S4)⋊C4C2×C5semidirect
58. (C2×S4)⋊C4C2×C5semidirect
59. (C2×S4)⋊C4C2×C5semidirect
60. SL(2,ℤ/4)⋊C4C2×C5semidirect
179. C5×C4⋊C4S4semidirect
200. C22×C4⋊C4C5×D3semidirect
219. C5×A4#123semidirect
399. C22×C5#894semidirect
463. C4⋊C4C5×S4semidirect
489. A4C5×#123semidirect
578. C5#85309direct (N × H)
603. C22C5×#894semidirect

Maximal subgroups

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

subgroupindex
C2×C5×C4⋊S42
C2×C4×C5×S42
C5×A4×C4⋊C42
C5×(C2×S4)⋊C42
C5×SL(2,ℤ/4)⋊C42
C5×#24433
C5×#8944
#853095