Finite Group Databasebeta

← #13034498

Subgroups — #13034498

Order 1280.

1370 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D5C211yesmaximal, characteristic
2#10617048C211yesmaximal, characteristic
3D5⋊#431C211yesmaximal, characteristic
4#10604973C212yesmaximal
5C4×#4009C214yesmaximal, complemented
6#10604451C214yesmaximal, complemented
7#10604283C212yesmaximal
8C22×Q8×D5C2211yescharacteristic
9C2×#4130—21no—
10C2×C4×C5×Q8C411yescharacteristic, complemented
11C2×Q8×Dic5C411yescharacteristic, complemented
12C4×Q8×D5—21no—
13C4×Q8×D5—21no—
14C4×Q8×D5—21no—
15C4×Q8×D5—21no—
16C4×Q8×D5C412yescomplemented
17C4×Q8×D5C412yescomplemented
18C4×Q8×D5C2212yes—
19C4×Q8×D5C2212yes—
20C2×D5×C4⋊C4—21no—
21C2×D5×C4⋊C4C2211yescharacteristic
22C2×#4033—21no—
23C2×#4033C411yescharacteristic, complemented
24C2×C42×D5—21no—
25C2×C42×D5C2211yescharacteristic
26C2×C4×Dic10—21no—
27C2×C4×Dic10C411yescharacteristic, complemented
28C2×#4009C2218yes—
29#65765—21no—
30#65765C2212yes—
31#65754—21no—
32D5⋊(C22⋊C8)C2212yes—
33#65748—21no—
34#65689—21no—
35#65688C2212yes—
36C4×C4⋊AGL(1,𝔽5)—21no—
37C4×C4⋊AGL(1,𝔽5)C2212yes—
38#65680—21no—
39D5⋊(C4⋊C8)C2212yes—
40#65678—21no—
41C4×D5⋊C8C2212yes—
42#64874C2218yes—
43#20734—51nomaximal, Sylow
44C22×C5×Q8C2×C411yescharacteristic
45C2×Q8×D5—28no—
46C2×Q8×D5—21no—
47C2×Q8×D5—21no—
48C2×Q8×D5C2×C412yes—
49C2×Q8×D5C2312yes—
50C22×C4×D5—21no—
51C22×C4×D5—21no—
52C22×C4×D5D411yescharacteristic
53C22×C4×D5D411yescharacteristic
54C22×C4×D5C2311yescharacteristic
55C22×Dic10—21no—
56C22×Dic10C2×C411yescharacteristic
57C2×C22⋊AGL(1,𝔽5)—22no—
58#4130—44no—
59C2×C4⋊AGL(1,𝔽5)—22no—
60C2×C4⋊AGL(1,𝔽5)—22no—
61C2×C4⋊AGL(1,𝔽5)C2314yes—
62C2×C4×AGL(1,𝔽5)—22no—
63C2×D5⋊C8C2314yes—
64C4×C5×Q8—21no—
65C4×C5×Q8—21no—
66C4×C5×Q8C2×C412yes—
67C4×C5×Q8C2×C412yes—
68C2×C5×C4⋊C4—21no—
69C2×C5×C4⋊C4C2×C411yescharacteristic
70C2×C42×C5—21no—
71C2×C42×C5C2×C411yescharacteristic
72Q8×Dic5—21no—
73Q8×Dic5—21no—
74Q8×Dic5C2×C412yes—
75Q8×Dic5C2×C412yes—
76C2×C4⋊Dic5—21no—
77C2×C4⋊Dic5C2×C411yescharacteristic
78C2×#683—21no—
79C2×#683—21no—
80C2×#683—21no—
81C2×C4×Dic5—21no—
82C2×C4×Dic5—21no—
83C2×C4×Dic5—21no—
84C2×C4×Dic5C2×C411yescharacteristic
85C2×C4×Dic5D411yescharacteristic
86C2×C4×Dic5D411yescharacteristic
87D5×C4⋊C4—22no—
88D5×C4⋊C4—22no—
89D5×C4⋊C4—22no—
90D5×C4⋊C4—22no—
91D5×C4⋊C4—21no—
92D5×C4⋊C4—21no—
93D5×C4⋊C4C2×C412yes—
94D5×C4⋊C4C2312yes—
95#4033—22no—
96#4033—22no—
97#4033—21no—
98#4033—21no—
99#4033—22no—
100#4033—22no—
101#4033C2×C412yes—
102#4033C2×C412yes—
103C42×D5—22no—
104C42×D5—22no—
105C42×D5—22no—
106C42×D5—22no—
107C42×D5C2×C411yescharacteristic
108C42×D5C2×C411yescharacteristic
109C42×D5D411yescharacteristic
110C42×D5D411yescharacteristic
111C42×D5D411yescharacteristic
112C42×D5D411yescharacteristic
113C42×D5C2311yescharacteristic
114C42×D5C2311yescharacteristic
115C4×Dic10—21no—
116C4×Dic10—21no—
117C4×Dic10—22no—
118C4×Dic10—22no—
119C4×Dic10—22no—
120C4×Dic10—22no—
121C4×Dic10C2×C412yes—
122C4×Dic10C2×C412yes—
123C22⋊#170—22no—
124#4009—216no—
125(C2×AGL(1,𝔽5))⋊C4—22no—
126(C2×AGL(1,𝔽5))⋊C4—22no—
127#4004—22no—
128#4003—22no—
129C4⋊#170—22no—
130C4×#170—22no—
131C4×#170—22no—
132C22×C4×Q8—51no—
133C2×#431—51no—
134#2285—51no—
135#1824—52no—
136#1803—54no—
137#1787—52no—
138C4×#103—54no—
139C23×D5C2×D411yescharacteristic
140C22×AGL(1,𝔽5)—24no—
141C2×C5×Q8—24no—
142C2×C5×Q8—21no—
143C2×C5×Q8C22×C412yes—
144C22×C4×C5—21no—
145C22×C4×C5—21no—
146C22×C4×C5C22⋊C411yescharacteristic
147C22×C4×C5C22⋊C411yescharacteristic
148C22×C4×C5C22×C411yescharacteristic
149C22×Dic5—21no—
150C22×Dic5—21no—
151C22×Dic5C22⋊C411yescharacteristic
152C22×Dic5C22⋊C411yescharacteristic
153C22×Dic5C2×D411yescharacteristic
154Q8×D5—28no—
155Q8×D5—28no—
156Q8×D5—24no—
157Q8×D5—24no—
158Q8×D5—24no—
159Q8×D5#4118yes—
160C2×C4×D5—28no—
161C2×C4×D5—22no—
162C2×C4×D5—22no—
163C2×C4×D5—24no—
164C2×C4×D5—22no—
165C2×C4×D5—22no—
166C2×C4×D5—42no—
167C2×C4×D5—44no—
168C2×C4×D5C22⋊C411yescharacteristic
169C2×C4×D5C22⋊C411yescharacteristic
170C2×C4×D5C22⋊C411yescharacteristic
171C2×C4×D5C22⋊C411yescharacteristic
172C2×C4×D5SD1614yes—
173C2×C4×D5Q1614yes—
174C2×C4×D5C22×C411yescharacteristic
175C2×C4×D5C2×D411yescharacteristic
176C2×C4×D5C2×D411yescharacteristic
177C2×C4×D5C2×D411yescharacteristic
178C2×C4×D5C2×D411yescharacteristic
179C2×C4×D5C2×D411yescharacteristic
180C2×C4×D5C2×D411yescharacteristic
181C2×C4×D5C2411yescharacteristic
182C2×Dic10—28no—
183C2×Dic10—21no—
184C2×Dic10—22no—
185C2×Dic10—22no—
186C2×Dic10—24no—
187C2×Dic10C22×C412yes—
188C22⋊AGL(1,𝔽5)—44no—
189C2×#170—24no—
190C4⋊AGL(1,𝔽5)—28no—
191C4⋊AGL(1,𝔽5)—44no—
192C4⋊AGL(1,𝔽5)—44no—
193C4×AGL(1,𝔽5)—44no—
194D5⋊C8—28no—
195C5×C4⋊C4—22no—
196C5×C4⋊C4—22no—
197C5×C4⋊C4—21no—
198C5×C4⋊C4C22×C412yes—
199C42×C5—22no—
200C42×C5—22no—
201C42×C5C22⋊C411yescharacteristic
202C42×C5C22⋊C411yescharacteristic
203C42×C5C22×C411yescharacteristic
204C42×C5C22×C411yescharacteristic
205C4⋊Dic5—21no—
206C4⋊Dic5—22no—
207C4⋊Dic5—22no—
208C4⋊Dic5C22×C412yes—
209#683—22no—
210#683—22no—
211#683—22no—
212#683—22no—
213#683—22no—
214#683—22no—
215C4×Dic5—22no—
216C4×Dic5—22no—
217C4×Dic5—22no—
218C4×Dic5—22no—
219C4×Dic5—22no—
220C4×Dic5—22no—
221C4×Dic5C22⋊C411yescharacteristic
222C4×Dic5C22⋊C411yescharacteristic
223C4×Dic5C22⋊C411yescharacteristic
224C4×Dic5C22⋊C411yescharacteristic
225C4×Dic5C22⋊C411yescharacteristic
226C4×Dic5C22⋊C411yescharacteristic
227C4×Dic5C22×C411yescharacteristic
228C4×Dic5C22×C411yescharacteristic
229C4×Dic5C2×D411yescharacteristic
230C4×Dic5C2×D411yescharacteristic
231C4×Dic5C2×D411yescharacteristic
232C4×Dic5C2×D411yescharacteristic
233C23×Q8—51no—
234C2×C4×Q8—51no—
235C2×C4×Q8—52no—
236C2×C4×Q8—52no—
237C2×C4×Q8—52no—
238C2×C4×Q8—52no—
239C2×C4×Q8—51no—
240C2×C4×Q8—51no—
241C2×C4×Q8—101no—
242C2×C4×Q8—101no—
243C2×C4×Q8—101no—
244C2×C4×Q8—101no—
245C2×C4×Q8—101no—
246C2×C4×Q8—101no—
247C2×C4×Q8AGL(1,𝔽5)11yescharacteristic, complemented
248C2×#117—101no—
249C22×C4⋊C4—51no—
250C22×C4⋊C4—101no—
251C22×C42—51no—
252C22×C42—101no—
253#431—101no—
254#431—101no—
255#431—101no—
256#431—101no—
257C2×C4⋊C8—52no—
258C2×#103—58no—
259C2×C22⋊C8—52no—
260C2×C4×C8—52no—
261#384—52no—
262#382—101no—
263#380—52no—
264#380—101no—
265C4×C4⋊C4—52no—
266C4×C4⋊C4—101no—
267#340—58no—
268C23×C5C2×C22⋊C411yescharacteristic
269C22×D5—22no—
270C22×D5—22no—
271C22×D5C2×C22⋊C411yescharacteristic
272C22×D5C2×SD1612yes—
273C22×D5C2×Q1612yes—
274C22×D5C22×D411yescharacteristic
275C2×AGL(1,𝔽5)—44no—
276C2×AGL(1,𝔽5)—44no—
277C2×AGL(1,𝔽5)—44no—
278C5×Q8—24no—
279C5×Q8—22no—
280C5×Q8#11714yescomplemented
281C2×C4×C5—24no—
282C2×C4×C5—22no—
283C2×C4×C5—22no—
284C2×C4×C5—22no—
285C2×C4×C5—41no—
286C2×C4×C5—42no—
287C2×C4×C5#10314yescomplemented
288C2×C4×C5C2×C22⋊C411yescharacteristic
289C2×C4×C5C2×C22⋊C411yescharacteristic
290C2×C4×C5C2×C22⋊C411yescharacteristic
291C2×C4×C5C2×C22⋊C411yescharacteristic
292C2×C4×C5C2×C22⋊C411yescharacteristic
293C2×C4×C5C23×C411yescharacteristic
294C2×Dic5—24no—
295C2×Dic5—22no—
296C2×Dic5—22no—
297C2×Dic5—22no—
298C2×Dic5—41no—
299C2×Dic5—42no—
300C2×Dic5#10314yescomplemented
301C2×Dic5C2×C22⋊C411yescharacteristic
302C2×Dic5C2×C22⋊C411yescharacteristic
303C2×Dic5C2×C22⋊C411yescharacteristic
304C2×Dic5C2×C22⋊C411yescharacteristic
305C2×Dic5C2×C22⋊C411yescharacteristic
306C2×Dic5C22×D411yescharacteristic
307C4×D5—28no—
308C4×D5—28no—
309C4×D5—22no—
310C4×D5—22no—
311C4×D5—22no—
312C4×D5—22no—
313C4×D5—22no—
314C4×D5—41no—
315C4×D5—41no—
316C4×D5—44no—
317C4×D5—44no—
318C4×D5#10314yescomplemented
319C4×D5#10314yescomplemented
320C4×D5C2×SD1612yes—
321C4×D5C2×SD1612yes—
322C4×D5C2×Q1612yes—
323C4×D5C2×Q1612yes—
324C4×D5C2×#4114yes—
325Dic10—28no—
326Dic10—24no—
327Dic10—22no—
328Dic10—28no—
329Dic10#11714yescomplemented
330#170—44no—
331C22×Q8—52no—
332C22×Q8—52no—
333C22×Q8—51no—
334C22×Q8—101no—
335C22×Q8—101no—
336C22×Q8—101no—
337C22×Q8—108no—
338C22×Q8C2×AGL(1,𝔽5)11yescharacteristic
339C23×C4—51no—
340C23×C4—51no—
341C23×C4—51no—
342C23×C4—101no—
343C23×C4—101no—
344C22×C8—54no—
345C4×Q8—21no—
346C4×Q8—21no—
347C4×Q8—52no—
348C4×Q8—52no—
349C4×Q8—52no—
350C4×Q8—52no—
351C4×Q8—52no—
352C4×Q8—52no—
353C4×Q8—102no—
354C4×Q8—102no—
355C4×Q8—102no—
356C4×Q8—102no—
357C4×Q8—102no—
358C4×Q8—102no—
359C4×Q8—102no—
360C4×Q8—102no—
361C4×Q8—101no—
362C4×Q8—101no—
363C4×Q8—101no—
364C4×Q8—101no—
365C4×Q8—101no—
366C4×Q8—101no—
367C4×Q8C2×AGL(1,𝔽5)12yes—
368C4×Q8C2×AGL(1,𝔽5)12yes—
369#117—204no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—51no—
372C2×C4⋊C4—52no—
373C2×C4⋊C4—52no—
374C2×C4⋊C4—54no—
375C2×C4⋊C4—101no—
376C2×C4⋊C4—101no—
377C2×C4⋊C4—102no—
378C2×C4⋊C4—102no—
379C2×C4⋊C4—102no—
380C2×C4⋊C4—102no—
381C2×C4⋊C4—101no—
382C2×C4⋊C4—101no—
383C2×C4⋊C4—101no—
384C2×C4⋊C4—101no—
385C2×C4⋊C4—102no—
386C2×C4⋊C4—102no—
387C2×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
388C2×C22⋊C4—102no—
389C2×C42—21no—
390C2×C42—51no—
391C2×C42—51no—
392C2×C42—51no—
393C2×C42—51no—
394C2×C42—51no—
395C2×C42—51no—
396C2×C42—51no—
397C2×C42—51no—
398C2×C42—51no—
399C2×C42—51no—
400C2×C42—51no—
401C2×C42—102no—
402C2×C42—102no—
403C2×C42—102no—
404C2×C42—102no—
405C2×C42—101no—
406C2×C42—101no—
407C2×C42—101no—
408C2×C42—102no—
409C2×C42C2×AGL(1,𝔽5)11yescharacteristic
410C4⋊C8—102no—
411C4⋊C8—102no—
412#103—1016no—
413C22⋊C8—102no—
414C22⋊C8—102no—
415C4×C8—102no—
416C4×C8—102no—
417#95—102no—
418#95—102no—
419C22×C5—21no—
420C22×C5—21no—
421C22×C5C2×#10312yes—
422C22×C5C22×C22⋊C411yescharacteristic
423C2×D5—21no—
424C2×D5—21no—
425C2×D5—24no—
426C2×D5—24no—
427C2×D5C2×#10312yes—
428C2×D5#52512yes—
429C2×D5C22×SD1611yescharacteristic
430C2×D5C22×Q1611yescharacteristic
431C2×D5#57512yes—
432AGL(1,𝔽5)—84no—
433C4×C5—21no—
434C4×C5—21no—
435C4×C5—24no—
436C4×C5—21no—
437C4×C5—42no—
438C4×C5C2×#10312yes—
439C4×C5C2×#10312yes—
440C4×C5C2×#11712yes—
441Dic5—21no—
442Dic5—21no—
443Dic5—21no—
444Dic5—24no—
445Dic5—42no—
446Dic5C2×#10312yes—
447Dic5C2×#10312yes—
448Dic5#52512yes—
449C24—51no—
450C2×Q8—24no—
451C2×Q8—21no—
452C2×Q8—58no—
453C2×Q8—52no—
454C2×Q8—102no—
455C2×Q8—102no—
456C2×Q8—104no—
457C2×Q8—108no—
458C2×Q8—104no—
459C2×Q8—104no—
460C2×Q8—108no—
461C2×Q8—108no—
462C2×Q8—104no—
463C2×Q8—101no—
464C2×Q8C22×AGL(1,𝔽5)12yes—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—51no—
468C22×C4—51no—
469C22×C4—51no—
470C22×C4—51no—
471C22×C4—51no—
472C22×C4—51no—
473C22×C4—54no—
474C22×C4—54no—
475C22×C4—51no—
476C22×C4—51no—
477C22×C4—51no—
478C22×C4—51no—
479C22×C4—51no—
480C22×C4—51no—
481C22×C4—51no—
482C22×C4—51no—
483C22×C4—51no—
484C22×C4—104no—
485C22×C4—102no—
486C22×C4—102no—
487C22×C4—108no—
488C22×C4—102no—
489C22×C4—102no—
490C22×C4—101no—
491C22×C4—101no—
492C22×C4—104no—
493C22×C4—202no—
494C22×C4—204no—
495C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
496C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
497C22×C4C22×AGL(1,𝔽5)11yescharacteristic
498C2×C8—104no—
499C2×C8—108no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—52no—
504C4⋊C4—102no—
505C4⋊C4—102no—
506C4⋊C4—102no—
507C4⋊C4—102no—
508C4⋊C4—102no—
509C4⋊C4—102no—
510C4⋊C4—108no—
511C4⋊C4—101no—
512C4⋊C4—102no—
513C4⋊C4—102no—
514C4⋊C4—204no—
515C4⋊C4—204no—
516C4⋊C4C22×AGL(1,𝔽5)12yes—
517C22⋊C4—204no—
518C42—22no—
519C42—22no—
520C42—51no—
521C42—51no—
522C42—51no—
523C42—51no—
524C42—51no—
525C42—51no—
526C42—51no—
527C42—51no—
528C42—51no—
529C42—51no—
530C42—51no—
531C42—51no—
532C42—102no—
533C42—102no—
534C42—102no—
535C42—102no—
536C42—102no—
537C42—102no—
538C42—204no—
539C42C22⋊AGL(1,𝔽5)11yescharacteristic
540C42C22⋊AGL(1,𝔽5)11yescharacteristic
541C42C22×AGL(1,𝔽5)11yescharacteristic
542C42C22×AGL(1,𝔽5)11yescharacteristic
543C2×C5—22no—
544C2×C5#228011yescharacteristic
545C2×C5C22×#10311yescharacteristic
546C2×C5#289611yescharacteristic
547D5—22no—
548D5#310312yes—
549D5#310412yes—
550C23—51no—
551C23—51no—
552C23—52no—
553C23—52no—
554C23—102no—
555C23—102no—
556C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
557Q8—24no—
558Q8—22no—
559Q8—54no—
560Q8—108no—
561Q8—108no—
562Q8—104no—
563Q8—102no—
564Q8#413014yescomplemented
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—51no—
572C2×C4—51no—
573C2×C4—51no—
574C2×C4—54no—
575C2×C4—54no—
576C2×C4—52no—
577C2×C4—52no—
578C2×C4—54no—
579C2×C4—54no—
580C2×C4—52no—
581C2×C4—52no—
582C2×C4—51no—
583C2×C4—51no—
584C2×C4—51no—
585C2×C4—102no—
586C2×C4—102no—
587C2×C4—102no—
588C2×C4—102no—
589C2×C4—108no—
590C2×C4—108no—
591C2×C4—102no—
592C2×C4—104no—
593C2×C4—102no—
594C2×C4—102no—
595C2×C4—102no—
596C2×C4—201no—
597C2×C4—201no—
598C2×C4—204no—
599C2×C4—204no—
600C2×C4—201no—
601C2×C4—202no—
602C2×C4—204no—
603C2×C4—204no—
604C2×C4—204no—
605C2×C4#400914yescomplemented
606C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
607C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
608C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
609C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
610C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
611C2×C4C23×AGL(1,𝔽5)11yescharacteristic
612C8—204no—
613C5#2073411yescharacteristic, complemented, Sylow
614C22—21no—
615C22—21no—
616C22—52no—
617C22—51no—
618C22—51no—
619C22—52no—
620C22—52no—
621C22—104no—
622C22—104no—
623C22—101no—
624C22—101no—
625C22C2×#400912yes—
626C22C22×C22⋊AGL(1,𝔽5)11yescharacteristic
627C4—24no—
628C4—21no—
629C4—21no—
630C4—21no—
631C4—42no—
632C4—52no—
633C4—52no—
634C4—52no—
635C4—101no—
636C4—104no—
637C4—101no—
638C4—101no—
639C4—202no—
640C4—404no—
641C4C2×#400912yes—
642C4C2×#400912yes—
643C4C2×#413012yes—
644C2—22no—
645C2—52no—
646C2—52no—
647C2—102no—
648C2#1061748011yescharacteristic
649C2C22×#400911yescharacteristic
650C2#1061753311yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C23×C4
FrattiniC2×C4C23×AGL(1,𝔽5)
FittingC2×C4×C5×Q8C4
socleC22×C5C22×C22⋊C4
prime radicalC23×D5C2×D4
cosocleC2×C4×D5C24
hypercenterC2×C4×Q8AGL(1,𝔽5)
hypocenterC5#20734
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
5. C4×#4009C2semidirect
6. #10604451C2semidirect
10. C2×C4×C5×Q8C4semidirect
11. C2×Q8×Dic5C4semidirect
16. C4×Q8×D5C4semidirect
17. C4×Q8×D5C4semidirect
23. C2×#4033C4semidirect
27. C2×C4×Dic10C4semidirect
247. C2×C4×Q8AGL(1,𝔽5)semidirect
280. C5×Q8#117semidirect
287. C2×C4×C5#103semidirect
300. C2×Dic5#103semidirect
318. C4×D5#103semidirect
319. C4×D5#103semidirect
329. Dic10#117semidirect
564. Q8#4130semidirect
605. C2×C4#4009semidirect
613. C5#20734semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Q8×D52
#106170482
D5⋊#4312
#106049732
C4×#40092
#106044512
#106042832
#207345