Finite Group Databasebeta

← #13035954

Subgroups — #13035954

Order 1280.

1414 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#711C211yesmaximal, characteristic
2C2×#65757C211yesmaximal, characteristic, complemented
3C2×C4×#704C211yesmaximal, characteristic, complemented
4#10604947C212yesmaximal, complemented
5#10604897C212yesmaximal, complemented
6#10604331C212yesmaximal, complemented
7#10604241C212yesmaximal, complemented
8#10604235C212yesmaximal, complemented
9#10604177C212yesmaximal, complemented
10C22×#711C411yescharacteristic
11C22×#704C2212yes—
12C2×C4×C5×D4C411yescharacteristic
13C2×D4×Dic5C2211yescharacteristic
14C2×C4×C22⋊D5C411yescharacteristic
15C2×C4×C22⋊D5C411yescharacteristic
16C22×C4×Dic5C2211yescharacteristic
17C22×C4×Dic5C2211yescharacteristic
18C4×#711—24no—
19C4×#711—24no—
20C2×#4038C411yescharacteristic
21C2×#4033C2211yescharacteristic
22C2×#4027C2211yescharacteristic
23C2×#4027C2211yescharacteristic
24C2×#4023C411yescharacteristic
25C2×#4023C411yescharacteristic
26C2×C42×D5C2211yescharacteristic
27C2×C4×Dic10C411yescharacteristic
28C2×C22⋊#170C2214yes—
29C2×C22⋊#170C2212yes—
30#65757—24no—
31C2×#4004C2212yes—
32C2×#4003C2214yes—
33C4×#704—24no—
34C2×#4002C2212yes—
35C2×#4002C2212yes—
36C2×C4×#170C2212yes—
37#64918C2214yes—
38#64918C2214yes—
39#21380—51nomaximal, Sylow
40C22×C5×D4C2×C411yescharacteristic
41C23×C4×C5C2×C411yescharacteristic
42C23×C4×C5C2×C411yescharacteristic
43C22×C22⋊D5C2×C411yescharacteristic
44C22×C22⋊D5C2×C411yescharacteristic
45C23×Dic5C2×C411yescharacteristic
46C23×Dic5C2×C411yescharacteristic
47C23×Dic5C2311yescharacteristic
48C23×Dic5C2311yescharacteristic
49C2×#711—24no—
50C2×#711—24no—
51C2×#711—24no—
52C22×C4×D5C2×C411yescharacteristic
53C22×C4×D5C2×C411yescharacteristic
54C22×C4×D5C2311yescharacteristic
55C22×Dic10C2×C411yescharacteristic
56C2×#704—24no—
57C2×#704—28no—
58C22×#170C2314yes—
59C22×#170C2314yes—
60C4×C5×D4—24no—
61C2×C5×C4⋊C4C2×C411yescharacteristic
62C2×C5×C22⋊C4C2×C411yescharacteristic
63C2×C5×C22⋊C4C2×C411yescharacteristic
64C2×C42×C5C2×C411yescharacteristic
65C2×C22⋊Dic5C2×C411yescharacteristic
66C2×C22⋊Dic5C2×C411yescharacteristic
67D4×Dic5—22no—
68D4×Dic5—22no—
69C4×C22⋊D5—24no—
70C4×C22⋊D5—24no—
71C2×#685C2×C411yescharacteristic
72C2×#685C2×C411yescharacteristic
73C2×C4⋊Dic5C2×C411yescharacteristic
74C2×#683C2×C411yescharacteristic
75C2×#683C2×C411yescharacteristic
76C2×#683C2×C411yescharacteristic
77C2×#683C2×C411yescharacteristic
78C2×C4×Dic5—22no—
79C2×C4×Dic5—22no—
80C2×C4×Dic5C2×C411yescharacteristic
81C2×C4×Dic5C2×C411yescharacteristic
82C2×C4×Dic5C2×C411yescharacteristic
83C2×C4×Dic5C2×C411yescharacteristic
84C2×C4×Dic5C2×C411yescharacteristic
85C2×C4×Dic5D414yes—
86C2×C4×Dic5D414yes—
87C2×C4×Dic5C2311yescharacteristic
88C2×C4×Dic5C2311yescharacteristic
89C2×C4×Dic5C2311yescharacteristic
90C2×C4×Dic5C2311yescharacteristic
91#4038—24no—
92#4033—22no—
93#4033—22no—
94#4027—22no—
95#4027—22no—
96#4027—22no—
97#4027—22no—
98#4023—24no—
99#4023—24no—
100C42×D5—22no—
101C42×D5—22no—
102C4×Dic10—24no—
103C22⋊#170—24no—
104C22⋊#170—28no—
105#4004—24no—
106#4003—28no—
107#4002—24no—
108#4002—24no—
109C4×#170—24no—
110C2×C4×#41—51no—
111C2×#432—51no—
112C2×C4×M16—51no—
113#1984—52no—
114#1954—52no—
115#1870—52no—
116#1870—52no—
117#1749—52no—
118#1749—52no—
119C24×C5C22×C411yescharacteristic
120C24×C5C22×C411yescharacteristic
121C23×D5C22×C411yescharacteristic
122C2×C5×D4—22no—
123C2×C5×D4—22no—
124C2×C5×D4—22no—
125C22×C4×C5—22no—
126C22×C4×C5—42no—
127C22×C4×C5—42no—
128C22×C4×C5C22⋊C414yes—
129C22×C4×C5C22×C411yescharacteristic
130C22×C4×C5C22×C411yescharacteristic
131C22×C4×C5C22×C411yescharacteristic
132C22×C4×C5C22×C411yescharacteristic
133C22×C4×C5C22×C411yescharacteristic
134C2×C22⋊D5—22no—
135C2×C22⋊D5—22no—
136C2×C22⋊D5—22no—
137C2×C22⋊D5—22no—
138C2×C22⋊D5—22no—
139C2×C22⋊D5M1614yescomplemented
140C22×Dic5—22no—
141C22×Dic5—22no—
142C22×Dic5—22no—
143C22×Dic5—22no—
144C22×Dic5—22no—
145C22×Dic5—24no—
146C22×Dic5—21no—
147C22×Dic5—21no—
148C22×Dic5—24no—
149C22×Dic5C22⋊C414yes—
150C22×Dic5C22×C411yescharacteristic
151C22×Dic5C22×C411yescharacteristic
152C22×Dic5C22×C411yescharacteristic
153C22×Dic5C22×C411yescharacteristic
154C22×Dic5C22×C411yescharacteristic
155C22×Dic5C22×C411yescharacteristic
156C22×Dic5C2×D412yes—
157C22×Dic5C2×D412yes—
158C22×Dic5C2411yescharacteristic
159#711—232no—
160C2×C4×D5—22no—
161C2×C4×D5—22no—
162C2×C4×D5—22no—
163C2×C4×D5—22no—
164C2×C4×D5—21no—
165C2×C4×D5—21no—
166C2×C4×D5—24no—
167C2×C4×D5—42no—
168C2×Dic10—22no—
169C2×Dic10—22no—
170C2×Dic10M1614yescomplemented
171#704—216no—
172C2×#170—24no—
173C2×#170—24no—
174C2×#170—24no—
175C2×#170—24no—
176C2×#170—24no—
177C2×#170#4118yescomplemented
178C5×C4⋊C4—22no—
179C5×C22⋊C4—22no—
180C5×C22⋊C4—22no—
181C42×C5—22no—
182C22⋊Dic5—22no—
183C22⋊Dic5—22no—
184#685—22no—
185#685—22no—
186C4⋊Dic5—22no—
187#683—22no—
188#683—22no—
189#683—22no—
190#683—22no—
191C4×Dic5—22no—
192C4×Dic5—22no—
193C4×Dic5—22no—
194C4×Dic5—21no—
195C4×Dic5—21no—
196C4×Dic5—21no—
197C4×Dic5—21no—
198C4×Dic5C22⋊C414yes—
199C4×Dic5C22⋊C414yes—
200C4×Dic5C2×D412yes—
201C4×Dic5C2×D412yes—
202C4×Dic5C2×D412yes—
203C4×Dic5C2×D412yes—
204C22×#41—51no—
205C22×M16—52no—
206C4×#41—104no—
207C4×#41—104no—
208C2×C4×Q8—51no—
209C2×C4×Q8—51no—
210C2×C4×D4—51no—
211C2×C4×D4—51no—
212C2×C4×D4—51no—
213C2×C4×D4—51no—
214C2×C4×D4—51no—
215C2×C4×D4AGL(1,𝔽5)11yescharacteristic
216C2×#117—51no—
217C2×#117—51no—
218C2×#117—51no—
219C22×C42—51no—
220C22×C42—51no—
221C22×C42—51no—
222#432—104no—
223C2×C4⋊C8—52no—
224C2×C22⋊C8—52no—
225C2×C22⋊C8—54no—
226C2×C22⋊C8—54no—
227C4×M16—104no—
228C2×#97—52no—
229C2×#97—52no—
230C2×C4×C8—52no—
231#336—54no—
232#336—54no—
233C23×C5—21no—
234C23×C5—22no—
235C23×C5—22no—
236C23×C5—21no—
237C23×C5—21no—
238C23×C5—21no—
239C23×C5—22no—
240C23×C5—22no—
241C23×C5C2×C22⋊C412yes—
242C23×C5C2×M1612yes—
243C23×C5C23×C411yescharacteristic
244C22×D5—21no—
245C22×D5—22no—
246C22×D5—21no—
247C22×D5—22no—
248C22×D5C2×M1612yes—
249C5×D4—28no—
250C2×C4×C5—21no—
251C2×C4×C5—21no—
252C2×C4×C5—22no—
253C2×C4×C5—22no—
254C2×C4×C5—21no—
255C2×C4×C5—21no—
256C2×C4×C5—22no—
257C2×C4×C5—22no—
258C2×C4×C5—44no—
259C2×C4×C5—44no—
260C2×C4×C5C2×C22⋊C412yes—
261C2×C4×C5C2×C22⋊C412yes—
262C2×C4×C5C2×M1612yes—
263C22⋊D5—28no—
264C22⋊D5—28no—
265C2×Dic5—28no—
266C2×Dic5—21no—
267C2×Dic5—22no—
268C2×Dic5—21no—
269C2×Dic5—21no—
270C2×Dic5—21no—
271C2×Dic5—24no—
272C2×Dic5—24no—
273C2×Dic5—28no—
274C2×Dic5—21no—
275C2×Dic5—21no—
276C2×Dic5—22no—
277C2×Dic5—22no—
278C2×Dic5—42no—
279C2×Dic5—42no—
280C2×Dic5C2×C22⋊C412yes—
281C2×Dic5C2×C22⋊C412yes—
282C2×Dic5C2×C22⋊C412yes—
283C2×Dic5C2×M1612yes—
284C2×Dic5C2×M1612yes—
285C2×Dic5C2×M1612yes—
286C2×Dic5C22×D411yescharacteristic
287C2×Dic5C22×D411yescharacteristic
288C2×Dic5C2×#4114yes—
289C4×D5—28no—
290C4×D5—44no—
291C4×D5—42no—
292C4×D5—42no—
293Dic10—28no—
294#170—28no—
295#170—44no—
296C2×#41—104no—
297C2×#41—104no—
298C2×#41—104no—
299C22×Q8—51no—
300C22×D4—51no—
301C22×D4—51no—
302C22×D4C2×AGL(1,𝔽5)11yescharacteristic
303C23×C4—51no—
304C23×C4—51no—
305C23×C4—51no—
306C23×C4—51no—
307C23×C4—51no—
308C23×C4—51no—
309C23×C4—51no—
310C23×C4C2×AGL(1,𝔽5)11yescharacteristic
311C23×C4C2×AGL(1,𝔽5)11yescharacteristic
312C2×M16—108no—
313C2×M16—104no—
314C22×C8—54no—
315C22×C8—54no—
316C4×Q8—104no—
317C4×Q8—102no—
318C4×Q8—102no—
319C4×D4—24no—
320C4×D4—102no—
321C4×D4—102no—
322C4×D4—104no—
323C4×D4—102no—
324C4×D4—102no—
325C4×D4—102no—
326C4×D4—102no—
327C4×D4—104no—
328#117—104no—
329#117—104no—
330#117—104no—
331C2×C4⋊C4—51no—
332C2×C4⋊C4—51no—
333C2×C4⋊C4—51no—
334C2×C4⋊C4—51no—
335C2×C4⋊C4—51no—
336C2×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
337C2×C22⋊C4—51no—
338C2×C22⋊C4—51no—
339C2×C22⋊C4—51no—
340C2×C22⋊C4—51no—
341C2×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
342C2×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
343C2×C42—54no—
344C2×C42—54no—
345C2×C42—51no—
346C2×C42—51no—
347C2×C42—51no—
348C2×C42—51no—
349C2×C42—51no—
350C2×C42—51no—
351C2×C42—51no—
352C2×C42—51no—
353C2×C42—51no—
354C2×C42—102no—
355C2×C42—102no—
356C2×C42—102no—
357C2×C42—102no—
358C2×C42C2×AGL(1,𝔽5)11yescharacteristic
359C4⋊C8—104no—
360C22⋊C8—108no—
361C22⋊C8—108no—
362C22⋊C8—104no—
363#97—104no—
364#97—104no—
365C4×C8—104no—
366C22×C5—22no—
367C22×C5—22no—
368C22×C5—22no—
369C22×C5—24no—
370C22×C5—22no—
371C22×C5—41no—
372C22×C5—42no—
373C22×C5—42no—
374C22×C5—42no—
375C22×C5—41no—
376C22×C5C22×C22⋊C411yescharacteristic
377C22×C5C4×#4112yes—
378C22×C5C22×M1611yescharacteristic
379C22×C5C22×M1611yescharacteristic
380C22×C5#56812yes—
381C2×D5—22no—
382C2×D5—24no—
383C2×D5—41no—
384C2×D5—42no—
385C2×D5—41no—
386C4×C5—22no—
387C4×C5—22no—
388C4×C5—22no—
389C4×C5—41no—
390C4×C5—41no—
391Dic5—22no—
392Dic5—22no—
393Dic5—22no—
394Dic5—41no—
395Dic5—41no—
396Dic5#40714yes—
397Dic5#52512yes—
398Dic5#53212yes—
399C24—51no—
400C24C22×AGL(1,𝔽5)11yescharacteristic
401C24C22×AGL(1,𝔽5)11yescharacteristic
402#41—1032no—
403C2×Q8—54no—
404C2×Q8—102no—
405C2×Q8—102no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—54no—
410C2×D4—102no—
411C2×D4—102no—
412C2×D4—102no—
413C2×D4—102no—
414C2×D4—102no—
415C22×C4—22no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—51no—
419C22×C4—51no—
420C22×C4—52no—
421C22×C4—54no—
422C22×C4—52no—
423C22×C4—51no—
424C22×C4—51no—
425C22×C4—51no—
426C22×C4—51no—
427C22×C4—51no—
428C22×C4—102no—
429C22×C4—104no—
430C22×C4—101no—
431C22×C4—101no—
432C22×C4—102no—
433C22×C4—102no—
434C22×C4—104no—
435C22×C4—102no—
436C22×C4—102no—
437C22×C4—102no—
438C22×C4—104no—
439C22×C4—101no—
440C22×C4—101no—
441C22×C4—102no—
442C22×C4—102no—
443C22×C4—102no—
444C22×C4—202no—
445C22×C4C22⋊AGL(1,𝔽5)14yes—
446C22×C4C22×AGL(1,𝔽5)11yescharacteristic
447C22×C4C22×AGL(1,𝔽5)11yescharacteristic
448C22×C4C22×AGL(1,𝔽5)11yescharacteristic
449C22×C4C22×AGL(1,𝔽5)11yescharacteristic
450C22×C4C22×AGL(1,𝔽5)11yescharacteristic
451M16—1016no—
452C2×C8—58no—
453C2×C8—104no—
454C2×C8—104no—
455C2×C8—104no—
456C2×C8—104no—
457C2×C8—104no—
458C4⋊C4—22no—
459C4⋊C4—102no—
460C4⋊C4—102no—
461C4⋊C4—102no—
462C4⋊C4—102no—
463C4⋊C4—102no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—102no—
467C22⋊C4—102no—
468C22⋊C4—102no—
469C22⋊C4—102no—
470C42—22no—
471C42—52no—
472C42—52no—
473C42—52no—
474C42—52no—
475C42—54no—
476C42—54no—
477C42—102no—
478C42—102no—
479C42—101no—
480C42—101no—
481C42—101no—
482C42—101no—
483C42—102no—
484C2×C5—42no—
485C2×C5—42no—
486C2×C5#229711yescharacteristic
487C2×C5C2×#40711yescharacteristic
488C2×C5#287911yescharacteristic
489C2×C5#297012yes—
490C2×C5#297112yes—
491D5—42no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—21no—
500C23—52no—
501C23—101no—
502C23—102no—
503C23—102no—
504C23—101no—
505C23D5⋊M1612yes—
506C23C2×C22⋊AGL(1,𝔽5)12yes—
507C23C23×AGL(1,𝔽5)11yescharacteristic
508Q8—108no—
509D4—28no—
510D4—108no—
511D4—108no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—52no—
523C2×C4—52no—
524C2×C4—54no—
525C2×C4—51no—
526C2×C4—51no—
527C2×C4—52no—
528C2×C4—52no—
529C2×C4—52no—
530C2×C4—52no—
531C2×C4—108no—
532C2×C4—104no—
533C2×C4—108no—
534C2×C4—104no—
535C2×C4—108no—
536C2×C4—101no—
537C2×C4—101no—
538C2×C4—101no—
539C2×C4—102no—
540C2×C4—101no—
541C2×C4—102no—
542C2×C4—102no—
543C2×C4—101no—
544C2×C4—101no—
545C2×C4—204no—
546C2×C4—202no—
547C2×C4—202no—
548C2×C4—202no—
549C2×C4—202no—
550C2×C4D5⋊M1612yes—
551C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
552C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
553C8—108no—
554C8—204no—
555C5#2138011yescharacteristic, complemented, Sylow
556C22—24no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—41no—
565C22—41no—
566C22—104no—
567C22—102no—
568C22—202no—
569C22—201no—
570C22—201no—
571C22C2×D5⋊M1611yescharacteristic
572C22C2×D5⋊M1611yescharacteristic
573C22#6625612yes—
574C22#41×AGL(1,𝔽5)12yes—
575C22C22×C22⋊AGL(1,𝔽5)11yescharacteristic
576C4—22no—
577C4—22no—
578C4—22no—
579C4—41no—
580C4—41no—
581C4—54no—
582C4—52no—
583C4—52no—
584C4—102no—
585C4—102no—
586C4—102no—
587C4—201no—
588C4—201no—
589C2—42no—
590C2—42no—
591C2—202no—
592C2#1061715612yes—
593C2#1061715812yes—
594C2D5⋊#40711yescharacteristic
595C2#1061749011yescharacteristic
596C2#1061755511yescharacteristic
subgroupG/H
center, FrattiniC23C23×AGL(1,𝔽5)
commutator, socleC23×C5C23×C4
FittingC2×C4×C5×D4C4
prime radicalC22×#711C4
cosocleC22×Dic5C24
hypercenterC2×C4×D4AGL(1,𝔽5)
hypocenterC5#21380
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#65757C2semidirect
3. C2×C4×#704C2semidirect
4. #10604947C2semidirect
5. #10604897C2semidirect
6. #10604331C2semidirect
7. #10604241C2semidirect
8. #10604235C2semidirect
9. #10604177C2semidirect
139. C2×C22⋊D5M16semidirect
170. C2×Dic10M16semidirect
177. C2×#170#41semidirect
555. C5#21380semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#7112
C2×#657572
C2×C4×#7042
#106049472
#106048972
#106043312
#106042412
#106042352
#106041772
#213805