Finite Group Databasebeta

← #12021707

Subgroups — #12021707

Order 1280.

1710 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#4033C211yesmaximal, characteristic
2C2×C4×#683C211yesmaximal, characteristic
3C2×#64939C211yesmaximal, characteristic
4C5×#1436C211yesmaximal, characteristic
5#10602357C211yesmaximal, characteristic
6#10602329C211yesmaximal, characteristic
7#10600217C212yesmaximal
8#10600214C212yesmaximal
9#10600198C211yesmaximal, characteristic
10#10600197C212yesmaximal
11#10600197C212yesmaximal
12C23×Dic10C2211yescharacteristic
13C22×C5×C4⋊C4C2211yescharacteristic
14C22×C42×C5C2211yescharacteristic
15C22×#683C2212yes—
16C22×#683C2211yescharacteristic
17C22×#683C2211yescharacteristic
18C22×#683C2212yes—
19C22×C4×Dic5C2211yescharacteristic
20C22×C4×Dic5C2211yescharacteristic
21C22×C4×Dic5C2211yescharacteristic
22C22×C4×Dic5C2211yescharacteristic
23C2×#4033—24no—
24C2×#4033—24no—
25C2×#4033C418yescomplemented
26C2×C5×#95C2212yes—
27C2×C5×#95C2211yescharacteristic
28C2×C5×#95C2212yes—
29C2×#3963C2212yes—
30C2×#3963C2212yes—
31C2×#3963C2212yes—
32C2×#3963C2212yes—
33C2×#3963C2212yes—
34C2×#3963C2211yescharacteristic
35C2×#3963C2212yes—
36C2×#3963C2212yes—
37C2×#3963C2212yes—
38C4×#683—24no—
39#64939—24no—
40#13925—51nomaximal, Sylow
41C23×C4×C5C2312yes—
42C23×C4×C5C2311yescharacteristic
43C23×C4×C5C2312yes—
44C23×C4×C5C2311yescharacteristic
45C23×C4×C5C2311yescharacteristic
46C23×Dic5C2312yes—
47C23×Dic5C2311yescharacteristic
48C23×Dic5C2312yes—
49C23×Dic5C2311yescharacteristic
50C23×Dic5C2311yescharacteristic
51C23×Dic5C2311yescharacteristic
52C22×Dic10—24no—
53C22×Dic10—22no—
54C22×Dic10—22no—
55C22×Dic10—24no—
56C22×Dic10C2×C414yes—
57C2×C5×C4⋊C4—22no—
58C2×C5×C4⋊C4—22no—
59C2×C5×C4⋊C4C2×C414yes—
60C2×C42×C5—22no—
61C2×C42×C5—42no—
62C2×#683—22no—
63C2×#683—22no—
64C2×#683—22no—
65C2×#683—22no—
66C2×#683—24no—
67C2×#683—24no—
68C2×#683—24no—
69C2×#683—24no—
70C2×#683—24no—
71C2×#683C2×C414yes—
72C2×#683C2×C414yes—
73C2×#683Q818yescomplemented
74C2×C4×Dic5—22no—
75C2×C4×Dic5—22no—
76C2×C4×Dic5—22no—
77C2×C4×Dic5—22no—
78C2×C4×Dic5—22no—
79C2×C4×Dic5—42no—
80C2×C4×Dic5C2×C414yes—
81C2×C4×Dic5C2×C414yes—
82C2×C4×Dic5C2×C414yes—
83C2×C4×Dic5D414yes—
84C2×C4×Dic5D414yes—
85#4033—232no—
86C5×#95—22no—
87C5×#95—24no—
88C5×#95—42no—
89#3963—22no—
90#3963—24no—
91#3963—24no—
92#3963—24no—
93#3963—24no—
94#3963—24no—
95#3963—42no—
96#3963—42no—
97#3963—42no—
98C22×C4×Q8—51no—
99C2×#382—51no—
100C2×C4×C4⋊C4—51no—
101#1446—52no—
102#1445—52no—
103#1436—51no—
104#1436—51no—
105#1436—51no—
106#1436D511yescharacteristic
107C4⋊#95—52no—
108C4⋊#95—52no—
109C24×C5C2411yescharacteristic
110C22×C4×C5—21no—
111C22×C4×C5—21no—
112C22×C4×C5—22no—
113C22×C4×C5—22no—
114C22×C4×C5—22no—
115C22×C4×C5—22no—
116C22×C4×C5—21no—
117C22×C4×C5—21no—
118C22×C4×C5—22no—
119C22×C4×C5—22no—
120C22×C4×C5—22no—
121C22×C4×C5—22no—
122C22×C4×C5—22no—
123C22×C4×C5—22no—
124C22×C4×C5—22no—
125C22×C4×C5—22no—
126C22×C4×C5—22no—
127C22×C4×C5—22no—
128C22×C4×C5—22no—
129C22×C4×C5—22no—
130C22×C4×C5—22no—
131C22×C4×C5—22no—
132C22×C4×C5C22×C412yes—
133C22×C4×C5C22×C412yes—
134C22×C4×C5C22×C412yes—
135C22×C4×C5C2×D412yes—
136C22×C4×C5C2×D412yes—
137C22×C4×C5C2×Q814yes—
138C22×C4×C5#4114yes—
139C22×Dic5—21no—
140C22×Dic5—21no—
141C22×Dic5—22no—
142C22×Dic5—22no—
143C22×Dic5—22no—
144C22×Dic5—22no—
145C22×Dic5—21no—
146C22×Dic5—21no—
147C22×Dic5—22no—
148C22×Dic5—22no—
149C22×Dic5—22no—
150C22×Dic5—22no—
151C22×Dic5—22no—
152C22×Dic5—22no—
153C22×Dic5—22no—
154C22×Dic5—22no—
155C22×Dic5—22no—
156C22×Dic5—22no—
157C22×Dic5—22no—
158C22×Dic5—22no—
159C22×Dic5—22no—
160C22×Dic5—22no—
161C22×Dic5C22×C412yes—
162C22×Dic5C22×C412yes—
163C22×Dic5C22×C412yes—
164C22×Dic5C22×C412yes—
165C22×Dic5C2×D412yes—
166C22×Dic5C2×D412yes—
167C22×Dic5C2×D412yes—
168C22×Dic5C2×D412yes—
169C22×Dic5C2×Q814yes—
170C22×Dic5C2×Q814yes—
171C22×Dic5#4114yes—
172C22×Dic5#4114yes—
173C2×Dic10—216no—
174C2×Dic10—216no—
175C2×Dic10—28no—
176C2×Dic10—28no—
177C2×Dic10—28no—
178C5×C4⋊C4—28no—
179C42×C5—44no—
180#683—28no—
181#683—28no—
182#683—216no—
183#683—48no—
184C4×Dic5—28no—
185C4×Dic5—28no—
186C4×Dic5—44no—
187C4×Dic5C22⋊C4116yes—
188C23×Q8—51no—
189C2×C4×Q8—58no—
190C2×C4×Q8—104no—
191C2×C4×Q8—104no—
192C22×C4⋊C4—51no—
193C22×C4⋊C4—51no—
194C22×C4⋊C4—52no—
195C22×C4⋊C4—52no—
196C22×C4⋊C4C2×D511yescharacteristic
197C22×C42—51no—
198C22×C42—51no—
199C22×C42—51no—
200C22×C42—51no—
201C22×C42C2×D511yescharacteristic
202#382—104no—
203C4×C4⋊C4—104no—
204C2×#95—52no—
205C2×#95—52no—
206C2×#95—52no—
207C2×#95—51no—
208C2×#95—52no—
209C2×#95—52no—
210C2×#95—52no—
211C2×#95—52no—
212C2×#95—52no—
213C2×#95C2×D512yes—
214C2×#95C2×D511yescharacteristic
215C2×#95C2×D512yes—
216C23×C5C23×C411yescharacteristic
217C23×C5C22×D411yescharacteristic
218C23×C5C22×D411yescharacteristic
219C23×C5C22×Q812yes—
220C23×C5C2×#4112yes—
221C23×C5C2×#4112yes—
222C23×C5C2×#4112yes—
223C23×C5H(5,𝔽2)12yes—
224C23×C52−1+412yes—
225C2×C4×C5—22no—
226C2×C4×C5—22no—
227C2×C4×C5—24no—
228C2×C4×C5—24no—
229C2×C4×C5—24no—
230C2×C4×C5—22no—
231C2×C4×C5—22no—
232C2×C4×C5—24no—
233C2×C4×C5—22no—
234C2×C4×C5—24no—
235C2×C4×C5—24no—
236C2×C4×C5—24no—
237C2×C4×C5—24no—
238C2×C4×C5—42no—
239C2×C4×C5—42no—
240C2×C4×C5—41no—
241C2×C4×C5—41no—
242C2×C4×C5—42no—
243C2×C4×C5—42no—
244C2×C4×C5—42no—
245C2×C4×C5—42no—
246C2×C4×C5—42no—
247C2×C4×C5—42no—
248C2×C4×C5—42no—
249C2×C4×C5—42no—
250C2×C4×C5—42no—
251C2×C4×C5—42no—
252C2×C4×C5C2×C22⋊C414yes—
253C2×C4×C5#11714yes—
254C2×Dic5—22no—
255C2×Dic5—22no—
256C2×Dic5—22no—
257C2×Dic5—22no—
258C2×Dic5—24no—
259C2×Dic5—24no—
260C2×Dic5—24no—
261C2×Dic5—24no—
262C2×Dic5—22no—
263C2×Dic5—24no—
264C2×Dic5—24no—
265C2×Dic5—24no—
266C2×Dic5—24no—
267C2×Dic5—42no—
268C2×Dic5—42no—
269C2×Dic5—41no—
270C2×Dic5—41no—
271C2×Dic5—42no—
272C2×Dic5—42no—
273C2×Dic5—42no—
274C2×Dic5—42no—
275C2×Dic5—42no—
276C2×Dic5—42no—
277C2×Dic5—42no—
278C2×Dic5—42no—
279C2×Dic5—42no—
280C2×Dic5—42no—
281C2×Dic5C2×C22⋊C414yes—
282C2×Dic5C2×C22⋊C414yes—
283C2×Dic5#11714yes—
284C2×Dic5C4×Q818yes—
285C2×Dic5C22⋊Q818yes—
286C2×Dic5#12414yes—
287C2×Dic5C4⋊Q814yes—
288Dic10—232no—
289C22×Q8—54no—
290C22×Q8—102no—
291C22×Q8—102no—
292C22×Q8—104no—
293C22×Q8—104no—
294C23×C4—51no—
295C23×C4—51no—
296C23×C4—51no—
297C23×C4—51no—
298C23×C4—52no—
299C23×C4—52no—
300C23×C4C22×D512yes—
301C23×C4C22×D511yescharacteristic
302C23×C4C22×D512yes—
303C23×C4C22×D511yescharacteristic
304C23×C4C22×D511yescharacteristic
305C4×Q8—1032no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—54no—
309C2×C4⋊C4—54no—
310C2×C4⋊C4—58no—
311C2×C4⋊C4—102no—
312C2×C4⋊C4—102no—
313C2×C4⋊C4—104no—
314C2×C4⋊C4—104no—
315C2×C4⋊C4—102no—
316C2×C4⋊C4—102no—
317C2×C4⋊C4—104no—
318C2×C4⋊C4—104no—
319C2×C4⋊C4—104no—
320C2×C4⋊C4C4×D514yes—
321C2×C42—22no—
322C2×C42—42no—
323C2×C42—54no—
324C2×C42—54no—
325C2×C42—54no—
326C2×C42—54no—
327C2×C42—54no—
328C2×C42—102no—
329C2×C42—102no—
330C2×C42—102no—
331C2×C42—102no—
332C2×C42—102no—
333C2×C42—202no—
334#95—24no—
335#95—22no—
336#95—42no—
337#95—104no—
338#95—104no—
339#95—104no—
340#95—104no—
341#95—104no—
342#95—102no—
343#95—202no—
344#95—202no—
345#95—202no—
346C22×C5C22×C22⋊C411yescharacteristic
347C22×C5C2×#11711yescharacteristic
348C22×C5C2×#11711yescharacteristic
349C22×C5C2×C4×Q812yes—
350C22×C5#52012yes—
351C22×C5C2×C22⋊Q812yes—
352C22×C5#52512yes—
353C22×C5#52512yes—
354C22×C5C2×#12411yescharacteristic
355C22×C5C2×C4⋊Q811yescharacteristic
356C22×C5#53511yescharacteristic
357C22×C5#53611yescharacteristic
358C22×C5#53712yes—
359C22×C5#55412yes—
360C22×C5#55412yes—
361C22×C5#55612yes—
362C22×C5#55612yes—
363C22×C5#55712yes—
364C22×C5#55712yes—
365C22×C5#55912yes—
366C22×C5#55912yes—
367C4×C5—24no—
368C4×C5—24no—
369C4×C5—42no—
370C4×C5—44no—
371C4×C5—82no—
372Dic5—24no—
373Dic5—24no—
374Dic5—42no—
375Dic5—44no—
376Dic5—82no—
377Dic5#39118yes—
378C24C23×D511yescharacteristic
379C2×Q8—108no—
380C2×Q8—108no—
381C2×Q8—1016no—
382C2×Q8—1016no—
383C2×Q8—108no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—52no—
407C22×C4—52no—
408C22×C4—52no—
409C22×C4—54no—
410C22×C4—54no—
411C22×C4—52no—
412C22×C4—52no—
413C22×C4—52no—
414C22×C4—52no—
415C22×C4—54no—
416C22×C4—52no—
417C22×C4—54no—
418C22×C4—102no—
419C22×C4—102no—
420C22×C4—101no—
421C22×C4—101no—
422C22×C4—102no—
423C22×C4—102no—
424C22×C4—102no—
425C22×C4—101no—
426C22×C4—101no—
427C22×C4—102no—
428C22×C4—102no—
429C22×C4—102no—
430C22×C4—102no—
431C22×C4—102no—
432C22×C4—102no—
433C22×C4—102no—
434C22×C4—102no—
435C22×C4—102no—
436C22×C4—102no—
437C22×C4—102no—
438C22×C4—102no—
439C22×C4—102no—
440C22×C4C2×C4×D512yes—
441C22×C4C2×C4×D512yes—
442C22×C4C2×C4×D512yes—
443C22×C4#70914yes—
444C22×C4D4×D512yes—
445C22×C4D4×D512yes—
446C22×C4Q8×D514yes—
447C4⋊C4—28no—
448C4⋊C4—108no—
449C4⋊C4—1016no—
450C4⋊C4—108no—
451C4⋊C4—208no—
452C42—44no—
453C42—516no—
454C42—108no—
455C42—108no—
456C42—204no—
457C2×C5#228011yescharacteristic
458C2×C5C2×#39111yescharacteristic
459C2×C5#231111yescharacteristic
460C2×C5#235512yes—
461C2×C5#235512yes—
462C2×C5#258012yes—
463C2×C5#258112yes—
464C2×C5#258412yes—
465C2×C5#259212yes—
466C23C22×C4×D511yescharacteristic
467C23C2×#70912yes—
468C23C2×D4×D511yescharacteristic
469C23C2×D4×D511yescharacteristic
470C23#414412yes—
471C23C2×Q8×D512yes—
472C23D5×#4112yes—
473C23D5×#4112yes—
474C23#415012yes—
475Q8—1032no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—54no—
504C2×C4—54no—
505C2×C4—58no—
506C2×C4—58no—
507C2×C4—54no—
508C2×C4—54no—
509C2×C4—54no—
510C2×C4—102no—
511C2×C4—102no—
512C2×C4—102no—
513C2×C4—102no—
514C2×C4—104no—
515C2×C4—104no—
516C2×C4—104no—
517C2×C4—104no—
518C2×C4—102no—
519C2×C4—104no—
520C2×C4—104no—
521C2×C4—104no—
522C2×C4—104no—
523C2×C4—202no—
524C2×C4—202no—
525C2×C4—202no—
526C2×C4—201no—
527C2×C4—202no—
528C2×C4—201no—
529C2×C4—202no—
530C2×C4—202no—
531C2×C4—202no—
532C2×C4—202no—
533C2×C4—202no—
534C2×C4—202no—
535C2×C4—202no—
536C2×C4—202no—
537C2×C4#401814yes—
538C2×C4D5×C22⋊C414yes—
539C5#1392511yescharacteristic, complemented, Sylow
540C22C2×#401811yescharacteristic
541C22C2×D5×C22⋊C411yescharacteristic
542C22#6584512yes—
543C22D5×#11711yescharacteristic
544C22#6586712yes—
545C22#6588012yes—
546C22#6588112yes—
547C22#6588112yes—
548C22#6589312yes—
549C22#6590112yes—
550C22C4×Q8×D512yes—
551C22#6592611yescharacteristic
552C22#6593112yes—
553C22D5×C22⋊Q812yes—
554C22#6597012yes—
555C22#6598211yescharacteristic
556C22#6599812yes—
557C22D5×#12411yescharacteristic
558C22#6601912yes—
559C22#6603612yes—
560C22D5×C4⋊Q811yescharacteristic
561C4—24no—
562C4—24no—
563C4—42no—
564C4—44no—
565C4—58no—
566C4—82no—
567C4—104no—
568C4—104no—
569C4—202no—
570C4—204no—
571C4—402no—
572C2#1060529211yescharacteristic
573C2#1060537212yes—
574C2#1060621812yes—
575C2#1060639311yescharacteristic
576C2#1060669412yes—
577C2#1060682112yes—
578C2#1060735012yes—
579C2#1060781112yes—
580C2D5×#39111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC5×#1436C2
socle, prime radical, cosocleC24×C5C24
hypercenter#1436D5
hypocenterC5#13925
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ #1436

Complements

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

NH ≅ G/Ntype
25. C2×#4033C4semidirect
73. C2×#683Q8semidirect
539. C5#13925semidirect

Maximal subgroups

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

subgroupindex
C22×#40332
C2×C4×#6832
C2×#649392
C5×#14362
#106023572
#106023292
#106002172
#106002142
#106001982
#106001972
#139255