Finite Group Databasebeta

← D5⋊#1503 #13030201

Subgroups — D5⋊#1503 #13030201

Order 1280.

1194 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D5×#533C211yesmaximal, characteristic
2C2×C4×C4⋊AGL(1,𝔽5)C211yesmaximal, characteristic
3D5⋊#431C211yesmaximal, characteristic
4#10604623C212yesmaximal, complemented
5#10604622C212yesmaximal, complemented
6D5⋊#328C214yesmaximal
7#10604424C214yesmaximal, complemented
8C22×C4⋊AGL(1,𝔽5)—21no—
9C22×C4×AGL(1,𝔽5)—21no—
10C5×#533C411yescharacteristic, complemented
11#66062C2211yescharacteristic
12#66062C2211yescharacteristic
13D5×C4⋊Q8C411yescharacteristic, complemented
14D5×C4⋊Q8C411yescharacteristic, complemented
15#66022—21no—
16D5×#125—21no—
17D5×C22⋊Q8—21no—
18D5×C22⋊Q8C2212yes—
19#65956—21no—
20#65956C412yescomplemented
21#65912—21no—
22#65912C2212yes—
23C4×Q8×D5—21no—
24C4×Q8×D5C412yescomplemented
25D5×#117—21no—
26#65851—21no—
27C2×C42×D5C2211yescharacteristic
28#65804C411yescharacteristic, complemented
29C2×(C2×AGL(1,𝔽5))⋊C4C2212yes—
30#65754—21no—
31D5⋊(C22⋊C8)C2212yes—
32#65748—21no—
33C4×C4⋊AGL(1,𝔽5)—21no—
34C4×C4⋊AGL(1,𝔽5)—21no—
35C4×C4⋊AGL(1,𝔽5)C2212yes—
36C4×C4⋊AGL(1,𝔽5)C2212yes—
37#65680C2212yes—
38D5⋊(C4⋊C8)—21no—
39#65678C2212yes—
40C4×D5⋊C8—21no—
41#64926C2218yes—
42#64869C2218yes—
43C2×#1503—51nomaximal, Sylow
44C23×AGL(1,𝔽5)—22no—
45C2×Q8×D5—21no—
46C2×Q8×D5C2×C412yes—
47C22×C4×D5D411yescharacteristic
48C22×C4×D5D411yescharacteristic
49C22×C4×D5C2311yescharacteristic
50C2×C4⋊AGL(1,𝔽5)—28no—
51C2×C4⋊AGL(1,𝔽5)—22no—
52C2×C4⋊AGL(1,𝔽5)—22no—
53C2×C4×AGL(1,𝔽5)—22no—
54C2×C4×AGL(1,𝔽5)—22no—
55C2×C4×AGL(1,𝔽5)—44no—
56C2×D5⋊C8—22no—
57C5×C4⋊Q8C2×C411yescharacteristic, complemented
58C5×C4⋊Q8C2×C411yescharacteristic, complemented
59C5×#125—21no—
60C5×C22⋊Q8—21no—
61C5×C22⋊Q8C2×C412yes—
62C4×C5×Q8—21no—
63C4×C5×Q8C2×C412yescomplemented
64C5×#117—21no—
65C2×C42×C5C2×C411yescharacteristic
66#4092—21no—
67#4092C2312yes—
68Q8×Dic5—21no—
69Q8×Dic5C2×C412yescomplemented
70#4090—21no—
71#4090C2×C412yescomplemented
72#4072—21no—
73#4070—21no—
74#4070C2×C412yes—
75C2×C4×Dic5C2×C411yescharacteristic
76C2×C4×Dic5D411yescharacteristic
77C2×C4×Dic5D411yescharacteristic
78#4043—21no—
79#4043C2312yes—
80#4042—22no—
81#4042—22no—
82#4038—21no—
83#4038—21no—
84D5×C4⋊C4—21no—
85D5×C4⋊C4—21no—
86D5×C4⋊C4—21no—
87D5×C4⋊C4—22no—
88D5×C4⋊C4—22no—
89D5×C4⋊C4C2×C412yes—
90#4036—21no—
91#4036—21no—
92#4035—22no—
93#4035—22no—
94#4034—21no—
95#4034—21no—
96#4034—21no—
97#4034C2×C412yescomplemented
98#4033—22no—
99#4033—22no—
100#4033—21no—
101#4033C2×C412yescomplemented
102D5×C22⋊C4—22no—
103#4024—22no—
104#4024—22no—
105#4023—22no—
106#4018—22no—
107C42×D5—22no—
108C42×D5C2×C411yescharacteristic
109C42×D5C2×C411yescharacteristic
110C42×D5D411yescharacteristic
111C42×D5D411yescharacteristic
112C42×D5D411yescharacteristic
113C42×D5D411yescharacteristic
114C42×D5C2311yescharacteristic
115C42×D5C2311yescharacteristic
116#4016—21no—
117#4015C2×C411yescharacteristic, complemented
118#4015C2×C411yescharacteristic, complemented
119C4×Dic10—21no—
120C4×Dic10—22no—
121C4×Dic10—22no—
122C4×Dic10C2×C412yescomplemented
123C22⋊#170—22no—
124(C2×AGL(1,𝔽5))⋊C4—22no—
125(C2×AGL(1,𝔽5))⋊C4C2314yes—
126#4004—22no—
127#4003C2314yes—
128C4⋊#170—22no—
129C4×#170—22no—
130C4×#170—22no—
131C2×#533—51no—
132C2×#431—51no—
133C2×C4×C4⋊C4—51no—
134#1503—52no—
135#1503—52no—
136#1503—54no—
137C2×#328—54no—
138C23×D5C2×D411yescharacteristic
139C22×AGL(1,𝔽5)—28no—
140C22×AGL(1,𝔽5)—24no—
141C22×AGL(1,𝔽5)—44no—
142C22×AGL(1,𝔽5)—44no—
143C2×C5×Q8—21no—
144C2×C5×Q8C22×C412yes—
145C22×C4×C5C22⋊C411yescharacteristic
146C22×C4×C5C22⋊C411yescharacteristic
147C22×C4×C5C22×C411yescharacteristic
148C22×Dic5C22⋊C411yescharacteristic
149C22×Dic5C22⋊C411yescharacteristic
150C22×Dic5C2×D411yescharacteristic
151Q8×D5—44no—
152Q8×D5—44no—
153C2×C4×D5—24no—
154C2×C4×D5—22no—
155C2×C4×D5—22no—
156C2×C4×D5—42no—
157C2×C4×D5C22⋊C411yescharacteristic
158C2×C4×D5C22⋊C411yescharacteristic
159C2×C4×D5C22⋊C411yescharacteristic
160C2×C4×D5C22⋊C411yescharacteristic
161C2×C4×D5SD1614yes—
162C2×C4×D5Q1614yes—
163C2×C4×D5C22×C411yescharacteristic
164C2×C4×D5C2×D411yescharacteristic
165C2×C4×D5C2×D411yescharacteristic
166C2×C4×D5C2×D411yescharacteristic
167C2×C4×D5C2×D411yescharacteristic
168C2×C4×D5C2×D411yescharacteristic
169C2×C4×D5C2×D411yescharacteristic
170C2×C4×D5C2411yescharacteristic
171C2×Dic10—21no—
172C2×Dic10—22no—
173C2×Dic10—22no—
174C2×Dic10C22×C412yes—
175C2×#170—24no—
176C4⋊AGL(1,𝔽5)—28no—
177C4⋊AGL(1,𝔽5)—28no—
178C4×AGL(1,𝔽5)—44no—
179C4×AGL(1,𝔽5)—44no—
180D5⋊C8—44no—
181C5×C4⋊C4—21no—
182C5×C4⋊C4—21no—
183C5×C4⋊C4—21no—
184C5×C4⋊C4—22no—
185C5×C4⋊C4—22no—
186C5×C4⋊C4C22×C412yes—
187C5×C22⋊C4—22no—
188C42×C5—22no—
189C42×C5C22⋊C411yescharacteristic
190C42×C5C22⋊C411yescharacteristic
191C42×C5C22×C411yescharacteristic
192C42×C5C22×C411yescharacteristic
193C22⋊Dic5—22no—
194#685—22no—
195#685—22no—
196C4⋊Dic5—21no—
197C4⋊Dic5—21no—
198C4⋊Dic5—21no—
199C4⋊Dic5—22no—
200C4⋊Dic5—22no—
201C4⋊Dic5C22×C412yes—
202#683—22no—
203#683—22no—
204#683—22no—
205#683—22no—
206#683—22no—
207#683—22no—
208#683—22no—
209#683—22no—
210C4×Dic5—22no—
211C4×Dic5—22no—
212C4×Dic5—22no—
213C4×Dic5C22⋊C411yescharacteristic
214C4×Dic5C22⋊C411yescharacteristic
215C4×Dic5C22⋊C411yescharacteristic
216C4×Dic5C22⋊C411yescharacteristic
217C4×Dic5C22⋊C411yescharacteristic
218C4×Dic5C22⋊C411yescharacteristic
219C4×Dic5C22×C411yescharacteristic
220C4×Dic5C22×C411yescharacteristic
221C4×Dic5C2×D411yescharacteristic
222C4×Dic5C2×D411yescharacteristic
223C4×Dic5C2×D411yescharacteristic
224C4×Dic5C2×D411yescharacteristic
225#533—51no—
226#533—51no—
227#533—52no—
228#533—52no—
229#533—51no—
230#533—101no—
231#533—101no—
232#533—101no—
233#533—101no—
234#533AGL(1,𝔽5)11yescharacteristic, complemented
235C2×C4⋊Q8—51no—
236C2×C4⋊Q8—51no—
237C2×#125—101no—
238C2×C22⋊Q8—52no—
239C2×C22⋊Q8—101no—
240C2×C4×Q8—52no—
241C2×C4×Q8—101no—
242C2×#117—101no—
243C22×C4⋊C4—101no—
244C22×C42—51no—
245C22×C42—101no—
246#431—52no—
247#431—52no—
248#431—101no—
249#431—101no—
250C2×C4⋊C8—101no—
251C2×C22⋊C8—52no—
252C2×C4×C8—101no—
253C4×C4⋊C4—52no—
254C4×C4⋊C4—52no—
255C4×C4⋊C4—101no—
256C4×C4⋊C4—101no—
257C2×#95—52no—
258#328—58no—
259#328—58no—
260C23×C5C2×C22⋊C411yescharacteristic
261C22×D5—22no—
262C22×D5—22no—
263C22×D5C2×C22⋊C411yescharacteristic
264C22×D5C2×SD1612yes—
265C22×D5C2×Q1612yes—
266C22×D5C22×D411yescharacteristic
267C2×AGL(1,𝔽5)—28no—
268C2×AGL(1,𝔽5)—44no—
269C2×AGL(1,𝔽5)—416no—
270C2×AGL(1,𝔽5)—44no—
271C5×Q8—42no—
272C5×Q8—42no—
273C2×C4×C5—22no—
274C2×C4×C5—22no—
275C2×C4×C5—22no—
276C2×C4×C5—41no—
277C2×C4×C5#10314yes—
278C2×C4×C5C2×C22⋊C411yescharacteristic
279C2×C4×C5C2×C22⋊C411yescharacteristic
280C2×C4×C5C2×C22⋊C411yescharacteristic
281C2×C4×C5C2×C22⋊C411yescharacteristic
282C2×C4×C5C2×C22⋊C411yescharacteristic
283C2×C4×C5C23×C411yescharacteristic
284C2×Dic5—22no—
285C2×Dic5—22no—
286C2×Dic5—22no—
287C2×Dic5—41no—
288C2×Dic5#10314yes—
289C2×Dic5C2×C22⋊C411yescharacteristic
290C2×Dic5C2×C22⋊C411yescharacteristic
291C2×Dic5C2×C22⋊C411yescharacteristic
292C2×Dic5C2×C22⋊C411yescharacteristic
293C2×Dic5C2×C22⋊C411yescharacteristic
294C2×Dic5C22×D411yescharacteristic
295C4×D5—22no—
296C4×D5—22no—
297C4×D5—22no—
298C4×D5—22no—
299C4×D5—22no—
300C4×D5—41no—
301C4×D5—41no—
302C4×D5—44no—
303C4×D5—44no—
304C4×D5#10314yes—
305C4×D5#10314yes—
306C4×D5#10414yes—
307C4×D5C2×SD1612yes—
308C4×D5C2×SD1612yes—
309C4×D5C2×Q1612yes—
310C4×D5C2×Q1612yes—
311Dic10—44no—
312Dic10—42no—
313Dic10—44no—
314Dic10—42no—
315#170—44no—
316C22×Q8—52no—
317C22×Q8—101no—
318C23×C4—51no—
319C23×C4—51no—
320C23×C4—51no—
321C23×C4—102no—
322C22×C8—102no—
323C4⋊Q8—52no—
324C4⋊Q8—52no—
325C4⋊Q8—51no—
326C4⋊Q8—51no—
327C4⋊Q8—101no—
328C4⋊Q8—101no—
329C4⋊Q8—101no—
330C4⋊Q8—101no—
331C4⋊Q8C2×AGL(1,𝔽5)11yescharacteristic, complemented
332C4⋊Q8C2×AGL(1,𝔽5)11yescharacteristic, complemented
333#125—21no—
334#125—102no—
335#125—102no—
336#125—101no—
337#125—101no—
338#125—101no—
339C22⋊Q8—21no—
340C22⋊Q8—52no—
341C22⋊Q8—52no—
342C22⋊Q8—52no—
343C22⋊Q8—102no—
344C22⋊Q8—102no—
345C22⋊Q8—102no—
346C22⋊Q8—102no—
347C22⋊Q8—101no—
348C22⋊Q8—101no—
349C22⋊Q8—101no—
350C22⋊Q8C2×AGL(1,𝔽5)12yes—
351C4×Q8—21no—
352C4×Q8—52no—
353C4×Q8—52no—
354C4×Q8—52no—
355C4×Q8—102no—
356C4×Q8—102no—
357C4×Q8—102no—
358C4×Q8—101no—
359C4×Q8—102no—
360C4×Q8—101no—
361C4×Q8—101no—
362C4×Q8C2×AGL(1,𝔽5)12yescomplemented
363#117—21no—
364#117—102no—
365#117—102no—
366#117—101no—
367#117—101no—
368#117—101no—
369C2×C4⋊C4—52no—
370C2×C4⋊C4—101no—
371C2×C4⋊C4—101no—
372C2×C4⋊C4—102no—
373C2×C4⋊C4—102no—
374C2×C4⋊C4—101no—
375C2×C4⋊C4—102no—
376C2×C4⋊C4—102no—
377C2×C4⋊C4—108no—
378C2×C22⋊C4—102no—
379C2×C42—51no—
380C2×C42—51no—
381C2×C42—51no—
382C2×C42—51no—
383C2×C42—51no—
384C2×C42—51no—
385C2×C42—51no—
386C2×C42—51no—
387C2×C42—51no—
388C2×C42—51no—
389C2×C42—51no—
390C2×C42—102no—
391C2×C42—102no—
392C2×C42—102no—
393C2×C42—204no—
394C2×C42C2×AGL(1,𝔽5)11yescharacteristic
395C4⋊C8—102no—
396C4⋊C8—102no—
397C22⋊C8—54no—
398C22⋊C8—102no—
399C4×C8—102no—
400C4×C8—102no—
401#95—54no—
402#95—102no—
403C22×C5—21no—
404C22×C5—21no—
405C22×C5C2×#10312yes—
406C22×C5C22×C22⋊C411yescharacteristic
407C2×D5—21no—
408C2×D5—21no—
409C2×D5—24no—
410C2×D5—24no—
411C2×D5#41012yes—
412C2×D5C2×#10312yes—
413C2×D5C2×#10412yes—
414C2×D5C22×SD1611yescharacteristic
415C2×D5C22×Q1611yescharacteristic
416AGL(1,𝔽5)—48no—
417C4×C5—21no—
418C4×C5—21no—
419C4×C5—21no—
420C4×C5—42no—
421C4×C5—42no—
422C4×C5C2×#10312yes—
423C4×C5C2×#10312yes—
424C4×C5C2×#10412yes—
425Dic5—21no—
426Dic5—21no—
427Dic5—21no—
428Dic5—42no—
429Dic5—42no—
430Dic5C2×#10312yes—
431Dic5C2×#10312yes—
432Dic5C2×#10412yes—
433C24—51no—
434C2×Q8—21no—
435C2×Q8—52no—
436C2×Q8—102no—
437C2×Q8—102no—
438C2×Q8—101no—
439C2×Q8—204no—
440C2×Q8—204no—
441C2×Q8C22×AGL(1,𝔽5)12yes—
442C22×C4—51no—
443C22×C4—51no—
444C22×C4—51no—
445C22×C4—51no—
446C22×C4—51no—
447C22×C4—51no—
448C22×C4—54no—
449C22×C4—54no—
450C22×C4—51no—
451C22×C4—51no—
452C22×C4—51no—
453C22×C4—51no—
454C22×C4—51no—
455C22×C4—51no—
456C22×C4—51no—
457C22×C4—51no—
458C22×C4—51no—
459C22×C4—104no—
460C22×C4—102no—
461C22×C4—102no—
462C22×C4—104no—
463C22×C4—108no—
464C22×C4—202no—
465C22×C4—204no—
466C22×C4—204no—
467C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
468C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
469C22×C4C22×AGL(1,𝔽5)11yescharacteristic
470C2×C8—104no—
471C2×C8—204no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—52no—
478C4⋊C4—102no—
479C4⋊C4—102no—
480C4⋊C4—102no—
481C4⋊C4—102no—
482C4⋊C4—102no—
483C4⋊C4—102no—
484C4⋊C4—102no—
485C4⋊C4—102no—
486C4⋊C4—108no—
487C4⋊C4—108no—
488C4⋊C4—101no—
489C4⋊C4—101no—
490C4⋊C4—102no—
491C4⋊C4—102no—
492C4⋊C4—101no—
493C4⋊C4C22×AGL(1,𝔽5)12yes—
494C22⋊C4—22no—
495C22⋊C4—102no—
496C22⋊C4—102no—
497C22⋊C4—102no—
498C42—22no—
499C42—51no—
500C42—51no—
501C42—51no—
502C42—51no—
503C42—51no—
504C42—51no—
505C42—51no—
506C42—51no—
507C42—51no—
508C42—51no—
509C42—51no—
510C42—51no—
511C42—102no—
512C42—102no—
513C42—102no—
514C42—204no—
515C42—204no—
516C42C22⋊AGL(1,𝔽5)11yescharacteristic
517C42C22⋊AGL(1,𝔽5)11yescharacteristic
518C42C22×AGL(1,𝔽5)11yescharacteristic
519C42C22×AGL(1,𝔽5)11yescharacteristic
520C2×C5—22no—
521C2×C5C2×#41011yescharacteristic
522C2×C5C22×#10311yescharacteristic
523C2×C5C22×#10411yescharacteristic
524D5—21no—
525D5—21no—
526D5#150314yescomplemented
527C23—51no—
528C23—51no—
529C23—52no—
530C23—52no—
531C23—102no—
532C23—102no—
533C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
534Q8—42no—
535Q8—42no—
536Q8—202no—
537Q8—204no—
538Q8—204no—
539Q8—202no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—41no—
544C2×C4—51no—
545C2×C4—51no—
546C2×C4—51no—
547C2×C4—54no—
548C2×C4—54no—
549C2×C4—52no—
550C2×C4—52no—
551C2×C4—54no—
552C2×C4—54no—
553C2×C4—52no—
554C2×C4—52no—
555C2×C4—51no—
556C2×C4—51no—
557C2×C4—51no—
558C2×C4—102no—
559C2×C4—102no—
560C2×C4—102no—
561C2×C4—102no—
562C2×C4—102no—
563C2×C4—102no—
564C2×C4—108no—
565C2×C4—102no—
566C2×C4—102no—
567C2×C4—201no—
568C2×C4—201no—
569C2×C4—204no—
570C2×C4—204no—
571C2×C4—201no—
572C2×C4—204no—
573C2×C4—2016no—
574C2×C4—204no—
575C2×C4#400914yes—
576C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
577C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
578C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
579C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
580C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
581C2×C4C23×AGL(1,𝔽5)11yescharacteristic
582C8—204no—
583C5C2×#150311yescharacteristic, complemented, Sylow
584C22—21no—
585C22—21no—
586C22—52no—
587C22—52no—
588C22—52no—
589C22—51no—
590C22—51no—
591C22—104no—
592C22—104no—
593C22—101no—
594C22—101no—
595C22C2×#400912yes—
596C22C22×C22⋊AGL(1,𝔽5)11yescharacteristic
597C4—21no—
598C4—21no—
599C4—21no—
600C4—42no—
601C4—42no—
602C4—52no—
603C4—52no—
604C4—52no—
605C4—101no—
606C4—101no—
607C4—101no—
608C4—208no—
609C4—202no—
610C4—202no—
611C4C2×#400912yes—
612C4C2×#400912yes—
613C4D5⋊#10412yes—
614C2—22no—
615C2—54no—
616C2—101no—
617C2—101no—
618C2C22×#400911yescharacteristic
619C2C2×D5⋊#10411yescharacteristic
620C2D5⋊#41011yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C23×C4
FrattiniC2×C4C23×AGL(1,𝔽5)
FittingC5×#533C4
socleC22×C5C22×C22⋊C4
prime radicalC23×D5C2×D4
cosocleC2×C4×D5C24
hypercenter#533AGL(1,𝔽5)
hypocenterC5C2×#1503
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
4. #10604623C2semidirect
5. #10604622C2semidirect
7. #10604424C2semidirect
10. C5×#533C4semidirect
13. D5×C4⋊Q8C4semidirect
14. D5×C4⋊Q8C4semidirect
20. #65956C4semidirect
24. C4×Q8×D5C4semidirect
28. #65804C4semidirect
57. C5×C4⋊Q8C2×C4semidirect
58. C5×C4⋊Q8C2×C4semidirect
63. C4×C5×Q8C2×C4semidirect
69. Q8×Dic5C2×C4semidirect
71. #4090C2×C4semidirect
97. #4034C2×C4semidirect
101. #4033C2×C4semidirect
117. #4015C2×C4semidirect
118. #4015C2×C4semidirect
122. C4×Dic10C2×C4semidirect
234. #533AGL(1,𝔽5)semidirect
331. C4⋊Q8C2×AGL(1,𝔽5)semidirect
332. C4⋊Q8C2×AGL(1,𝔽5)semidirect
362. C4×Q8C2×AGL(1,𝔽5)semidirect
526. D5#1503semidirect
583. C5C2×#1503semidirect

Maximal subgroups

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

subgroupindex
D5×#5332
C2×C4×C4⋊AGL(1,𝔽5)2
D5⋊#4312
#106046232
#106046222
D5⋊#3282
#106044242
C2×#15035