Finite Group Databasebeta

← #13028150

Subgroups — #13028150

Order 1280.

2166 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D5×C4⋊C4C211yesmaximal, characteristic
2C2×#65765C211yesmaximal, characteristic
3C2×C4×C22⋊AGL(1,𝔽5)C211yesmaximal, characteristic
4#10604909C212yesmaximal
5#10604909C212yesmaximal
6#10604344C214yesmaximal, complemented
7#10604343C214yesmaximal, complemented
8C23×C4×D5C2211yescharacteristic
9C23×C4×D5C2211yescharacteristic
10C23×C4×D5C2211yescharacteristic
11C22×C22⋊AGL(1,𝔽5)C2212yes—
12C22×C22⋊AGL(1,𝔽5)C2212yes—
13C22×C4⋊AGL(1,𝔽5)C2212yes—
14C22×C4×AGL(1,𝔽5)C2212yes—
15C22×C5×C4⋊C4C411yescharacteristic, complemented
16C22×C4⋊Dic5C411yescharacteristic, complemented
17C22×#683C411yescharacteristic, complemented
18C22×#683C411yescharacteristic, complemented
19C2×D5×C4⋊C4—24no—
20C2×D5×C4⋊C4—24no—
21C2×D5×C4⋊C4C412yescomplemented
22C2×D5×C4⋊C4C412yescomplemented
23C2×D5×C4⋊C4C2212yes—
24C2×D5×C4⋊C4C2212yes—
25#65765—24no—
26C4×C22⋊AGL(1,𝔽5)—24no—
27C2×(C2×AGL(1,𝔽5))⋊C4C2212yes—
28C2×(C2×AGL(1,𝔽5))⋊C4C2218yes—
29C2×(C2×AGL(1,𝔽5))⋊C4C2212yes—
30C2×(C2×AGL(1,𝔽5))⋊C4C2218yes—
31#13923—51nomaximal, Sylow
32C24×D5C2311yescharacteristic
33C23×AGL(1,𝔽5)C2314yes—
34C23×AGL(1,𝔽5)C2314yes—
35C23×C4×C5C2×C411yescharacteristic
36C23×C4×C5C2×C411yescharacteristic
37C23×C4×C5C2×C411yescharacteristic
38C23×Dic5C2×C411yescharacteristic
39C23×Dic5C2×C411yescharacteristic
40C23×Dic5C2×C411yescharacteristic
41C22×C4×D5—22no—
42C22×C4×D5—22no—
43C22×C4×D5—28no—
44C22×C4×D5—22no—
45C22×C4×D5—22no—
46C22×C4×D5—44no—
47C22×C4×D5—44no—
48C22×C4×D5C2×C412yes—
49C22×C4×D5C2×C412yes—
50C22×C4×D5C2×C411yescharacteristic
51C22×C4×D5C2×C411yescharacteristic
52C22×C4×D5D414yes—
53C22×C4×D5D414yes—
54C22×C4×D5C2312yes—
55C22×C4×D5C2311yescharacteristic
56C22×C4×D5C2311yescharacteristic
57C22×C4×D5C2312yes—
58C2×C22⋊AGL(1,𝔽5)—24no—
59C2×C22⋊AGL(1,𝔽5)—24no—
60C2×C22⋊AGL(1,𝔽5)—24no—
61C2×C22⋊AGL(1,𝔽5)—28no—
62C2×C22⋊AGL(1,𝔽5)Q818yes—
63C2×C4⋊AGL(1,𝔽5)—24no—
64C2×C4⋊AGL(1,𝔽5)—44no—
65C2×C4×AGL(1,𝔽5)—24no—
66C2×C4×AGL(1,𝔽5)—44no—
67C2×C5×C4⋊C4—24no—
68C2×C5×C4⋊C4C2×C412yes—
69C2×C5×C4⋊C4C2×C412yes—
70C2×C4⋊Dic5—24no—
71C2×C4⋊Dic5C2×C412yes—
72C2×C4⋊Dic5C2×C412yes—
73C2×#683—24no—
74C2×#683—24no—
75C2×#683C2×C412yes—
76C2×#683C2×C412yes—
77C2×#683C2×C412yes—
78C2×#683C2×C412yes—
79D5×C4⋊C4—28no—
80D5×C4⋊C4—28no—
81D5×C4⋊C4—24no—
82D5×C4⋊C4—24no—
83D5×C4⋊C4—24no—
84D5×C4⋊C4—24no—
85(C2×AGL(1,𝔽5))⋊C4—24no—
86(C2×AGL(1,𝔽5))⋊C4—24no—
87(C2×AGL(1,𝔽5))⋊C4—216no—
88(C2×AGL(1,𝔽5))⋊C4—48no—
89C23×C4⋊C4—51no—
90C2×#380—51no—
91C2×C4×C22⋊C4—51no—
92#1441—54no—
93#1436—54no—
94C22⋊#95—52no—
95C22⋊#95—52no—
96C24×C5C22×C411yescharacteristic
97C23×D5—24no—
98C23×D5—24no—
99C23×D5C22×C411yescharacteristic
100C23×D5C2×D412yes—
101C23×D5C2×D412yes—
102C23×D5C2×Q814yes—
103C23×D5#4114yes—
104C23×D5C2411yescharacteristic
105C22×AGL(1,𝔽5)—24no—
106C22×AGL(1,𝔽5)—24no—
107C22×AGL(1,𝔽5)—28no—
108C22×AGL(1,𝔽5)—28no—
109C22×AGL(1,𝔽5)—24no—
110C22×AGL(1,𝔽5)—28no—
111C22×AGL(1,𝔽5)—28no—
112C22×AGL(1,𝔽5)—28no—
113C22×AGL(1,𝔽5)C2×Q818yes—
114C22×C4×C5—22no—
115C22×C4×C5—24no—
116C22×C4×C5—22no—
117C22×C4×C5—42no—
118C22×C4×C5—42no—
119C22×C4×C5C22⋊C414yes—
120C22×C4×C5C22×C412yes—
121C22×C4×C5C22×C411yescharacteristic
122C22×C4×C5C22×C411yescharacteristic
123C22×C4×C5C22×C412yes—
124C22×Dic5—22no—
125C22×Dic5—24no—
126C22×Dic5—22no—
127C22×Dic5—42no—
128C22×Dic5—42no—
129C22×Dic5C22⋊C414yes—
130C22×Dic5C22×C412yes—
131C22×Dic5C22×C411yescharacteristic
132C22×Dic5C22×C411yescharacteristic
133C22×Dic5C22×C412yes—
134C2×C4×D5—216no—
135C2×C4×D5—216no—
136C2×C4×D5—24no—
137C2×C4×D5—22no—
138C2×C4×D5—22no—
139C2×C4×D5—24no—
140C2×C4×D5—24no—
141C2×C4×D5—24no—
142C2×C4×D5—24no—
143C2×C4×D5—24no—
144C2×C4×D5—24no—
145C2×C4×D5—22no—
146C2×C4×D5—22no—
147C2×C4×D5—28no—
148C2×C4×D5—48no—
149C2×C4×D5—48no—
150C2×C4×D5—42no—
151C2×C4×D5—42no—
152C2×C4×D5—44no—
153C2×C4×D5—48no—
154C2×C4×D5—48no—
155C2×C4×D5C22⋊C414yes—
156C2×C4×D5C22⋊C414yes—
157C2×C4×D5C2×D412yes—
158C2×C4×D5C2×D412yes—
159C2×C4×D5C2×D412yes—
160C2×C4×D5C2×D412yes—
161C2×C4×D5#4118yes—
162C22⋊AGL(1,𝔽5)—216no—
163C22⋊AGL(1,𝔽5)—48no—
164C4⋊AGL(1,𝔽5)—48no—
165C4×AGL(1,𝔽5)—48no—
166C5×C4⋊C4—24no—
167C5×C4⋊C4—24no—
168C4⋊Dic5—24no—
169C4⋊Dic5—24no—
170#683—24no—
171#683—24no—
172#683—24no—
173#683—24no—
174C24×C4—51no—
175C24×C4—51no—
176C24×C4—51no—
177C22×C4⋊C4—51no—
178C22×C4⋊C4—51no—
179C22×C4⋊C4—51no—
180C22×C4⋊C4—52no—
181C22×C4⋊C4—52no—
182C22×C4⋊C4—52no—
183C22×C4⋊C4—52no—
184C22×C4⋊C4—52no—
185C22×C4⋊C4—104no—
186C22×C4⋊C4—104no—
187C22×C4⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
188C22×C22⋊C4—52no—
189C22×C22⋊C4—52no—
190C22×C42—52no—
191#380—104no—
192C4×C22⋊C4—104no—
193C2×#95—58no—
194C2×#95—52no—
195C2×#95—52no—
196C2×#95—58no—
197C23×C5—22no—
198C23×C5—22no—
199C23×C5C2×C22⋊C412yes—
200C23×C5C4×Q814yes—
201C23×C5C23×C411yescharacteristic
202C22×D5—22no—
203C22×D5—22no—
204C22×D5—24no—
205C22×D5—24no—
206C22×D5—24no—
207C22×D5—24no—
208C22×D5—216no—
209C22×D5—216no—
210C22×D5C2×C22⋊C412yes—
211C22×D5C4×Q814yes—
212C22×D5C22⋊Q818yes—
213C22×D5#12414yes—
214C22×D5C4⋊Q814yes—
215C22×D5C22×D411yescharacteristic
216C22×D5C22×D411yescharacteristic
217C22×D5C22×Q812yes—
218C22×D5C2×#4112yes—
219C22×D5C2×#4114yes—
220C22×D5H(5,𝔽2)14yes—
221C2×AGL(1,𝔽5)—28no—
222C2×AGL(1,𝔽5)—216no—
223C2×AGL(1,𝔽5)—44no—
224C2×AGL(1,𝔽5)—44no—
225C2×AGL(1,𝔽5)—48no—
226C2×AGL(1,𝔽5)—44no—
227C2×AGL(1,𝔽5)—44no—
228C2×AGL(1,𝔽5)—44no—
229C2×AGL(1,𝔽5)—416no—
230C2×C4×C5—28no—
231C2×C4×C5—22no—
232C2×C4×C5—22no—
233C2×C4×C5—22no—
234C2×C4×C5—22no—
235C2×C4×C5—22no—
236C2×C4×C5—24no—
237C2×C4×C5—44no—
238C2×C4×C5—44no—
239C2×C4×C5C2×C22⋊C412yes—
240C2×C4×C5C2×C22⋊C412yes—
241C2×C4×C5#11714yes—
242C2×Dic5—28no—
243C2×Dic5—22no—
244C2×Dic5—22no—
245C2×Dic5—22no—
246C2×Dic5—22no—
247C2×Dic5—22no—
248C2×Dic5—24no—
249C2×Dic5—44no—
250C2×Dic5—44no—
251C2×Dic5C2×C22⋊C412yes—
252C2×Dic5C2×C22⋊C412yes—
253C2×Dic5#11714yes—
254C4×D5—216no—
255C4×D5—28no—
256C4×D5—28no—
257C4×D5—24no—
258C4×D5—24no—
259C4×D5—24no—
260C4×D5—24no—
261C4×D5—44no—
262C4×D5—44no—
263C4×D5—48no—
264C4×D5—44no—
265C4×D5—44no—
266C25—51no—
267C23×C4—51no—
268C23×C4—51no—
269C23×C4—52no—
270C23×C4—51no—
271C23×C4—54no—
272C23×C4—54no—
273C23×C4—51no—
274C23×C4—51no—
275C23×C4—54no—
276C23×C4—51no—
277C23×C4—51no—
278C23×C4—52no—
279C23×C4—52no—
280C23×C4—52no—
281C23×C4—54no—
282C23×C4—108no—
283C23×C4—102no—
284C23×C4—102no—
285C23×C4—102no—
286C23×C4—102no—
287C23×C4—204no—
288C23×C4—204no—
289C23×C4C2×AGL(1,𝔽5)11yescharacteristic
290C23×C4C2×AGL(1,𝔽5)11yescharacteristic
291C23×C4C2×AGL(1,𝔽5)11yescharacteristic
292C2×C4⋊C4—24no—
293C2×C4⋊C4—52no—
294C2×C4⋊C4—52no—
295C2×C4⋊C4—52no—
296C2×C4⋊C4—52no—
297C2×C4⋊C4—52no—
298C2×C4⋊C4—52no—
299C2×C4⋊C4—104no—
300C2×C4⋊C4—104no—
301C2×C4⋊C4—104no—
302C2×C4⋊C4—104no—
303C2×C4⋊C4—108no—
304C2×C4⋊C4—108no—
305C2×C4⋊C4—104no—
306C2×C4⋊C4—104no—
307C2×C4⋊C4—104no—
308C2×C4⋊C4—104no—
309C2×C4⋊C4—204no—
310C2×C4⋊C4C2×AGL(1,𝔽5)12yes—
311C2×C4⋊C4C2×AGL(1,𝔽5)12yes—
312C2×C22⋊C4—58no—
313C2×C22⋊C4—104no—
314C2×C22⋊C4—104no—
315C2×C22⋊C4—108no—
316C2×C22⋊C4—104no—
317C2×C42—104no—
318C2×C42—204no—
319#95—104no—
320#95—104no—
321#95—1016no—
322#95—208no—
323C22×C5—22no—
324C22×C5—28no—
325C22×C5—22no—
326C22×C5#39114yes—
327C22×C5C22×C22⋊C411yescharacteristic
328C22×C5C2×#11712yes—
329C22×C5C2×C4×Q812yes—
330C22×C5#51812yes—
331C2×D5—22no—
332C2×D5—28no—
333C2×D5—22no—
334C2×D5—216no—
335C2×D5—28no—
336C2×D5—28no—
337C2×D5#39114yes—
338C2×D5C2×C22⋊Q812yes—
339C2×D5#52512yes—
340C2×D5#52512yes—
341C2×D5C2×#12411yescharacteristic
342C2×D5C2×C4⋊Q811yescharacteristic
343C2×D5#53412yes—
344C2×D5#53412yes—
345C2×D5#55318yes—
346C2×D5#55418yes—
347AGL(1,𝔽5)—48no—
348AGL(1,𝔽5)—84no—
349C4×C5—22no—
350C4×C5—22no—
351C4×C5—24no—
352C4×C5—42no—
353Dic5—22no—
354Dic5—22no—
355Dic5—24no—
356Dic5—42no—
357C24—51no—
358C24—54no—
359C24—54no—
360C24—52no—
361C24—52no—
362C24—51no—
363C24—104no—
364C24—104no—
365C24C22×AGL(1,𝔽5)11yescharacteristic
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—52no—
372C22×C4—51no—
373C22×C4—51no—
374C22×C4—54no—
375C22×C4—58no—
376C22×C4—52no—
377C22×C4—52no—
378C22×C4—52no—
379C22×C4—52no—
380C22×C4—54no—
381C22×C4—54no—
382C22×C4—58no—
383C22×C4—52no—
384C22×C4—104no—
385C22×C4—104no—
386C22×C4—102no—
387C22×C4—102no—
388C22×C4—104no—
389C22×C4—104no—
390C22×C4—104no—
391C22×C4—1016no—
392C22×C4—1016no—
393C22×C4—108no—
394C22×C4—102no—
395C22×C4—102no—
396C22×C4—104no—
397C22×C4—104no—
398C22×C4—102no—
399C22×C4—104no—
400C22×C4—104no—
401C22×C4—108no—
402C22×C4—108no—
403C22×C4—104no—
404C22×C4—102no—
405C22×C4—104no—
406C22×C4—108no—
407C22×C4—108no—
408C22×C4—108no—
409C22×C4—202no—
410C22×C4—202no—
411C22×C4—204no—
412C22×C4—208no—
413C22×C4—208no—
414C22×C4—208no—
415C22×C4—208no—
416C22×C4—202no—
417C22×C4—202no—
418C22×C4C22⋊AGL(1,𝔽5)14yes—
419C22×C4C22×AGL(1,𝔽5)12yes—
420C22×C4C22×AGL(1,𝔽5)12yes—
421C22×C4C22×AGL(1,𝔽5)11yescharacteristic
422C22×C4C22×AGL(1,𝔽5)11yescharacteristic
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—104no—
426C4⋊C4—104no—
427C4⋊C4—104no—
428C4⋊C4—104no—
429C4⋊C4—104no—
430C4⋊C4—104no—
431C4⋊C4—208no—
432C22⋊C4—1016no—
433C22⋊C4—208no—
434C42—208no—
435C2×C5—24no—
436C2×C5#228011yescharacteristic
437C2×C5C2×#39111yescharacteristic
438C2×C5#230811yescharacteristic
439C2×C5#234514yes—
440D5—24no—
441D5#257914yes—
442D5#258312yes—
443D5#259112yes—
444C23—22no—
445C23—22no—
446C23—54no—
447C23—54no—
448C23—52no—
449C23—52no—
450C23—51no—
451C23—51no—
452C23—54no—
453C23—52no—
454C23—58no—
455C23—54no—
456C23—54no—
457C23—104no—
458C23—104no—
459C23—104no—
460C23—104no—
461C23—1016no—
462C23—1016no—
463C23—102no—
464C23—102no—
465C23Q8×AGL(1,𝔽5)14yes—
466C23C2×C22⋊AGL(1,𝔽5)12yes—
467C23C23×AGL(1,𝔽5)11yescharacteristic
468C2×C4—28no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—54no—
478C2×C4—52no—
479C2×C4—52no—
480C2×C4—104no—
481C2×C4—104no—
482C2×C4—108no—
483C2×C4—108no—
484C2×C4—104no—
485C2×C4—104no—
486C2×C4—1016no—
487C2×C4—108no—
488C2×C4—102no—
489C2×C4—102no—
490C2×C4—102no—
491C2×C4—102no—
492C2×C4—108no—
493C2×C4—1016no—
494C2×C4—104no—
495C2×C4—102no—
496C2×C4—204no—
497C2×C4—204no—
498C2×C4—204no—
499C2×C4—204no—
500C2×C4—208no—
501C2×C4—204no—
502C2×C4—204no—
503C2×C4—204no—
504C2×C4—208no—
505C2×C4—204no—
506C2×C4—204no—
507C2×C4—204no—
508C2×C4—2016no—
509C2×C4—204no—
510C2×C4#413014yes—
511C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
512C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
513C5#1392311yescharacteristic, complemented, Sylow
514C22—28no—
515C22—22no—
516C22—22no—
517C22—58no—
518C22—58no—
519C22—52no—
520C22—52no—
521C22—52no—
522C22—52no—
523C22—52no—
524C22—51no—
525C22—51no—
526C22—54no—
527C22—1016no—
528C22—108no—
529C22—108no—
530C22—108no—
531C22—102no—
532C22—102no—
533C22(C2×AGL(1,𝔽5))⋊Q814yes—
534C22C2×#413012yes—
535C22#6625812yes—
536C22C2×Q8×AGL(1,𝔽5)12yes—
537C22C22×C22⋊AGL(1,𝔽5)11yescharacteristic
538C4—24no—
539C4—22no—
540C4—22no—
541C4—42no—
542C4—104no—
543C4—102no—
544C4—102no—
545C4—208no—
546C4—202no—
547C4—404no—
548C2—24no—
549C2—54no—
550C2—52no—
551C2—52no—
552C2—104no—
553C2#1061710414yes—
554C2#1061748011yescharacteristic
555C2#1061750611yescharacteristic
556C2C2×(C2×AGL(1,𝔽5))⋊Q811yescharacteristic
subgroupG/H
center, FrattiniC23C23×AGL(1,𝔽5)
commutator, socleC23×C5C23×C4
FittingC22×C5×C4⋊C4C4
prime radicalC24×D5C23
cosocleC23×D5C24
hypercenterC22×C4⋊C4AGL(1,𝔽5)
hypocenterC5#13923
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
6. #10604344C2semidirect
7. #10604343C2semidirect
15. C22×C5×C4⋊C4C4semidirect
16. C22×C4⋊Dic5C4semidirect
17. C22×#683C4semidirect
18. C22×#683C4semidirect
21. C2×D5×C4⋊C4C4semidirect
22. C2×D5×C4⋊C4C4semidirect
187. C22×C4⋊C4AGL(1,𝔽5)semidirect
513. C5#13923semidirect

Maximal subgroups

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

subgroupindex
C22×D5×C4⋊C42
C2×#657652
C2×C4×C22⋊AGL(1,𝔽5)2
#106049092
#106043442
#106043432
#139235