Finite Group Databasebeta

← #13035654

Subgroups — #13035654

Order 1280.

962 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65759C211yesmaximal, characteristic, complemented
2C2×#65757C211yesmaximal, characteristic, complemented
3C4×D4×Dic5C211yesmaximal, characteristic
4#10604949C212yesmaximal
5#10604897C211yesmaximal, characteristic, complemented
6#10604897C211yesmaximal, characteristic, complemented
7#10604327C212yesmaximal, complemented
8#10604238C212yesmaximal, complemented
9#10604237C212yesmaximal, complemented
10#10604180C211yesmaximal, characteristic, complemented
11C4×#4002C211yesmaximal, characteristic, complemented
12C2×C4×C5×D4C411yescharacteristic
13C2×D4×Dic5C411yescharacteristic
14C2×D4×Dic5C2211yescharacteristic
15C22×C4×Dic5C2211yescharacteristic
16C22×C4×Dic5C2211yescharacteristic
17C2×C22⋊#170C2212yes—
18C2×C22⋊#170C2212yes—
19C2×C22⋊#170C2212yes—
20C2×C22⋊#170C2212yes—
21#65759—24no—
22#65757—24no—
23C2×#4004C2212yes—
24C2×#4004C2212yes—
25C2×#4002C2212yes—
26C2×#4002C2211yescharacteristic
27C2×#4002C2211yescharacteristic
28C2×#4002C2211yescharacteristic
29C2×#4002C2211yescharacteristic
30C2×C4⋊#170C2212yes—
31C2×C4×#170C2212yes—
32C4×C22⋊Dic5C411yescharacteristic
33C4×C22⋊Dic5C411yescharacteristic
34C4⋊C4×Dic5C411yescharacteristic
35C4⋊C4×Dic5C2211yescharacteristic
36C22⋊C4×Dic5C411yescharacteristic
37C22⋊C4×Dic5C411yescharacteristic
38C22⋊C4×Dic5C2211yescharacteristic
39C22⋊C4×Dic5C2211yescharacteristic
40C4×C4⋊Dic5C411yescharacteristic
41C42×Dic5C2211yescharacteristic
42#64918C2212yes—
43#64918C2212yes—
44#64918C2212yes—
45#21271—51nomaximal, Sylow
46C22×C5×D4C2×C411yescharacteristic
47C23×C4×C5C2×C411yescharacteristic
48C23×C4×C5C2×C411yescharacteristic
49C23×Dic5C2×C411yescharacteristic
50C23×Dic5C2×C411yescharacteristic
51C23×Dic5C2311yescharacteristic
52C23×Dic5C2311yescharacteristic
53C22×#170C2312yes—
54C22×#170C2312yes—
55C22×#170C2312yes—
56C22×#170C2312yes—
57C4×C5×D4—24no—
58C2×C5×C4⋊C4C2×C411yescharacteristic
59C2×C5×C22⋊C4C2×C411yescharacteristic
60C2×C5×C22⋊C4C2×C411yescharacteristic
61C2×C42×C5C2×C411yescharacteristic
62C2×C22⋊Dic5C2×C411yescharacteristic
63C2×C22⋊Dic5C2×C411yescharacteristic
64C2×C22⋊Dic5C2×C411yescharacteristic
65C2×C22⋊Dic5C2×C411yescharacteristic
66D4×Dic5—24no—
67D4×Dic5—24no—
68C2×C4⋊Dic5C2×C411yescharacteristic
69C2×C4⋊Dic5C2×C411yescharacteristic
70C2×C4×Dic5—22no—
71C2×C4×Dic5—22no—
72C2×C4×Dic5—22no—
73C2×C4×Dic5—22no—
74C2×C4×Dic5C2×C411yescharacteristic
75C2×C4×Dic5C2×C411yescharacteristic
76C2×C4×Dic5C2×C411yescharacteristic
77C2×C4×Dic5C2×C411yescharacteristic
78C2×C4×Dic5C2×C411yescharacteristic
79C2×C4×Dic5C2×C411yescharacteristic
80C2×C4×Dic5C2×C411yescharacteristic
81C2×C4×Dic5C2311yescharacteristic
82C2×C4×Dic5C2311yescharacteristic
83C2×C4×Dic5C2311yescharacteristic
84C2×C4×Dic5C2311yescharacteristic
85C2×C4×Dic5C2311yescharacteristic
86C22⋊#170—24no—
87C22⋊#170—24no—
88C22⋊#170—24no—
89C22⋊#170—24no—
90#4004—24no—
91#4004—24no—
92#4002—22no—
93#4002—22no—
94#4002—24no—
95#4002—22no—
96#4002D414yescomplemented
97C4⋊#170—24no—
98C4×#170—24no—
99#3963C2×C411yescharacteristic
100#3963C2×C411yescharacteristic
101#3963C2×C411yescharacteristic
102#3963C2×C411yescharacteristic
103#3963C2×C411yescharacteristic
104#3963C2×C411yescharacteristic
105C2×#432—51no—
106C2×#432—51no—
107C42×D4—51no—
108#1955—52no—
109#1940—52no—
110#1923—52no—
111#1923—52no—
112#1767—51no—
113#1749—51no—
114#1749—51no—
115C4×#97—51no—
116C24×C5C22×C411yescharacteristic
117C24×C5C22×C411yescharacteristic
118C2×C5×D4—22no—
119C2×C5×D4—22no—
120C2×C5×D4—22no—
121C22×C4×C5—22no—
122C22×C4×C5—22no—
123C22×C4×C5—22no—
124C22×C4×C5—22no—
125C22×C4×C5—41no—
126C22×C4×C5—41no—
127C22×C4×C5C22×C411yescharacteristic
128C22×C4×C5C22×C411yescharacteristic
129C22×C4×C5C22×C411yescharacteristic
130C22×C4×C5C22×C411yescharacteristic
131C22×C4×C5C22×C411yescharacteristic
132C22×Dic5—22no—
133C22×Dic5—22no—
134C22×Dic5—22no—
135C22×Dic5—22no—
136C22×Dic5—21no—
137C22×Dic5—21no—
138C22×Dic5—22no—
139C22×Dic5—22no—
140C22×Dic5—21no—
141C22×Dic5—21no—
142C22×Dic5—22no—
143C22×Dic5—22no—
144C22×Dic5—41no—
145C22×Dic5—41no—
146C22×Dic5C22×C411yescharacteristic
147C22×Dic5C22×C411yescharacteristic
148C22×Dic5C22×C411yescharacteristic
149C22×Dic5C22×C411yescharacteristic
150C22×Dic5C22×C411yescharacteristic
151C22×Dic5C22×C411yescharacteristic
152C22×Dic5C22×C411yescharacteristic
153C22×Dic5C2411yescharacteristic
154C2×#170—22no—
155C2×#170—22no—
156C2×#170—22no—
157C2×#170—22no—
158C2×#170—22no—
159C2×#170—22no—
160C2×#170—22no—
161C2×#170—22no—
162C2×#170—22no—
163C2×#170—22no—
164C2×#170C2×D414yes—
165C2×#170#4114yes—
166C5×C4⋊C4—22no—
167C5×C22⋊C4—22no—
168C5×C22⋊C4M1614yes—
169C42×C5—22no—
170C22⋊Dic5—22no—
171C22⋊Dic5—22no—
172C22⋊Dic5—22no—
173C22⋊Dic5M1614yes—
174C4⋊Dic5—22no—
175C4⋊Dic5—22no—
176C4×Dic5—22no—
177C4×Dic5—22no—
178C4×Dic5—22no—
179C4×Dic5—22no—
180C4×Dic5—22no—
181C4×Dic5—21no—
182C4×Dic5—21no—
183C4×Dic5—22no—
184C4×Dic5—21no—
185C4×Dic5—21no—
186C4×Dic5—41no—
187C4×Dic5—41no—
188C4×Dic5C2×D412yes—
189C4×Dic5#4114yes—
190C4×Dic5#4112yes—
191C2×C4×D4—51no—
192C2×C4×D4—51no—
193C2×C4×D4AGL(1,𝔽5)11yescharacteristic
194C22×C42—51no—
195C22×C42—51no—
196#432—104no—
197#432—104no—
198C2×C4⋊C8—52no—
199C2×C4⋊C8—52no—
200C2×C4⋊C8—52no—
201C2×C22⋊C8—52no—
202C2×C22⋊C8—52no—
203C2×C22⋊C8—52no—
204C2×C22⋊C8—52no—
205C2×#97—51no—
206C2×#97—51no—
207C2×#97—51no—
208C2×#97—52no—
209C2×#97—51no—
210C2×C4×C8—52no—
211C4×C4⋊C4—51no—
212C4×C4⋊C4—51no—
213C4×C4⋊C4—51no—
214C4×C22⋊C4—51no—
215C4×C22⋊C4—51no—
216C4×C22⋊C4—51no—
217C4×C22⋊C4—51no—
218C4×C22⋊C4—51no—
219C4×C22⋊C4—51no—
220C43—51no—
221#336—52no—
222#336—52no—
223#336—52no—
224C23×C5—21no—
225C23×C5—21no—
226C23×C5—21no—
227C23×C5—21no—
228C23×C5—21no—
229C23×C5—21no—
230C23×C5—21no—
231C23×C5—21no—
232C23×C5—21no—
233C23×C5—21no—
234C23×C5—21no—
235C23×C5—21no—
236C23×C5C2×M1612yes—
237C23×C5C2×M1612yes—
238C23×C5C23×C411yescharacteristic
239C5×D4—28no—
240C2×C4×C5—21no—
241C2×C4×C5—21no—
242C2×C4×C5—21no—
243C2×C4×C5—21no—
244C2×C4×C5—21no—
245C2×C4×C5—21no—
246C2×C4×C5—21no—
247C2×C4×C5—21no—
248C2×C4×C5—21no—
249C2×C4×C5—21no—
250C2×C4×C5—21no—
251C2×C4×C5—44no—
252C2×C4×C5—44no—
253C2×C4×C5#11712yes—
254C2×C4×C5C4×D412yes—
255C2×C4×C5C2×M1612yes—
256C2×C4×C5C2×M1612yes—
257C2×Dic5—21no—
258C2×Dic5—21no—
259C2×Dic5—21no—
260C2×Dic5—21no—
261C2×Dic5—21no—
262C2×Dic5—21no—
263C2×Dic5—21no—
264C2×Dic5—21no—
265C2×Dic5—21no—
266C2×Dic5—21no—
267C2×Dic5—21no—
268C2×Dic5—21no—
269C2×Dic5—21no—
270C2×Dic5—21no—
271C2×Dic5—21no—
272C2×Dic5—21no—
273C2×Dic5—21no—
274C2×Dic5—44no—
275C2×Dic5—44no—
276C2×Dic5—42no—
277C2×Dic5—42no—
278C2×Dic5—42no—
279C2×Dic5—42no—
280C2×Dic5#11712yes—
281C2×Dic5C4×D412yes—
282C2×Dic5C2×M1612yes—
283C2×Dic5C2×M1612yes—
284C2×Dic5C22×D411yescharacteristic
285C2×Dic5C2×#4111yescharacteristic
286C2×Dic5C2×#4112yes—
287C2×Dic5C2×#4112yes—
288#170—24no—
289#170—24no—
290#170—42no—
291#170—42no—
292C22×D4C2×AGL(1,𝔽5)11yescharacteristic
293C23×C4—51no—
294C23×C4—51no—
295C23×C4—51no—
296C23×C4—51no—
297C23×C4C2×AGL(1,𝔽5)11yescharacteristic
298C23×C4C2×AGL(1,𝔽5)11yescharacteristic
299C22×C8—52no—
300C22×C8—52no—
301C22×C8—52no—
302C22×C8—52no—
303C4×D4—24no—
304C4×D4—104no—
305C4×D4—104no—
306C2×C4⋊C4—51no—
307C2×C4⋊C4—51no—
308C2×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
309C2×C22⋊C4—51no—
310C2×C22⋊C4—51no—
311C2×C22⋊C4—51no—
312C2×C22⋊C4—51no—
313C2×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
314C2×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
315C2×C42—51no—
316C2×C42—51no—
317C2×C42—51no—
318C2×C42—51no—
319C2×C42—51no—
320C2×C42—51no—
321C2×C42—51no—
322C2×C42—51no—
323C2×C42—51no—
324C2×C42—51no—
325C2×C42—51no—
326C2×C42—51no—
327C2×C42—102no—
328C2×C42—102no—
329C2×C42—102no—
330C2×C42—102no—
331C2×C42C2×AGL(1,𝔽5)11yescharacteristic
332C4⋊C8—104no—
333C4⋊C8—104no—
334C4⋊C8—104no—
335C22⋊C8—104no—
336C22⋊C8—104no—
337C22⋊C8—104no—
338C22⋊C8—104no—
339#97—54no—
340#97—104no—
341#97—102no—
342#97—102no—
343#97—102no—
344C4×C8—104no—
345#95—51no—
346#95—51no—
347#95—51no—
348#95—51no—
349#95—51no—
350#95—51no—
351C22×C5—22no—
352C22×C5—22no—
353C22×C5—22no—
354C22×C5—22no—
355C22×C5—22no—
356C22×C5—22no—
357C22×C5—41no—
358C22×C5—41no—
359C22×C5—41no—
360C22×C5—41no—
361C22×C5—41no—
362C22×C5—41no—
363C22×C5—41no—
364C22×C5—41no—
365C22×C5C2×#11711yescharacteristic
366C22×C5C2×C4×D411yescharacteristic
367C22×C5C4×#4111yescharacteristic
368C22×C5C22×M1611yescharacteristic
369C22×C5C22×M1611yescharacteristic
370C22×C5#56811yescharacteristic
371C22×C5#56811yescharacteristic
372C4×C5—22no—
373C4×C5—22no—
374C4×C5—22no—
375C4×C5—22no—
376C4×C5—41no—
377Dic5—22no—
378Dic5—22no—
379Dic5—22no—
380Dic5—22no—
381Dic5—41no—
382Dic5—41no—
383Dic5—41no—
384Dic5#52912yes—
385Dic5#53212yes—
386C24C22×AGL(1,𝔽5)11yescharacteristic
387C24C22×AGL(1,𝔽5)11yescharacteristic
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—51no—
398C22×C4—51no—
399C22×C4—51no—
400C22×C4—51no—
401C22×C4—51no—
402C22×C4—51no—
403C22×C4—51no—
404C22×C4—51no—
405C22×C4—101no—
406C22×C4—101no—
407C22×C4—102no—
408C22×C4—102no—
409C22×C4—101no—
410C22×C4—101no—
411C22×C4—102no—
412C22×C4—102no—
413C22×C4—102no—
414C22×C4—102no—
415C22×C4—102no—
416C22×C4—102no—
417C22×C4—201no—
418C22×C4—201no—
419C22×C4C22×AGL(1,𝔽5)11yescharacteristic
420C22×C4C22×AGL(1,𝔽5)11yescharacteristic
421C22×C4C22×AGL(1,𝔽5)11yescharacteristic
422C22×C4C22×AGL(1,𝔽5)11yescharacteristic
423C22×C4C22×AGL(1,𝔽5)11yescharacteristic
424C2×C8—54no—
425C2×C8—54no—
426C2×C8—102no—
427C2×C8—102no—
428C2×C8—102no—
429C2×C8—102no—
430C2×C8—102no—
431C2×C8—102no—
432C2×C8—102no—
433C2×C8—102no—
434C2×C8—102no—
435C2×C8—102no—
436C4⋊C4—22no—
437C4⋊C4—102no—
438C4⋊C4—102no—
439C22⋊C4—22no—
440C22⋊C4—54no—
441C22⋊C4—102no—
442C22⋊C4—102no—
443C22⋊C4—102no—
444C22⋊C4#70414yes—
445C42—22no—
446C42—54no—
447C42—52no—
448C42—52no—
449C42—102no—
450C42—101no—
451C42—101no—
452C42—101no—
453C42—102no—
454C42—101no—
455C42—102no—
456C42—102no—
457C42—102no—
458C42—102no—
459C42—201no—
460C42—201no—
461C2×C5—42no—
462C2×C5—42no—
463C2×C5#229311yescharacteristic
464C2×C5#292011yescharacteristic
465C2×C5#292311yescharacteristic
466C2×C5D4×M1611yescharacteristic
467C2×C5D4×M1611yescharacteristic
468C2×C5#297011yescharacteristic
469C2×C5#297011yescharacteristic
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23D5⋊M1612yes—
483C23C2×#70412yes—
484C23C23×AGL(1,𝔽5)11yescharacteristic
485D4—28no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—52no—
500C2×C4—51no—
501C2×C4—51no—
502C2×C4—52no—
503C2×C4—52no—
504C2×C4—52no—
505C2×C4—52no—
506C2×C4—52no—
507C2×C4—101no—
508C2×C4—101no—
509C2×C4—101no—
510C2×C4—101no—
511C2×C4—101no—
512C2×C4—101no—
513C2×C4—101no—
514C2×C4—101no—
515C2×C4—101no—
516C2×C4—101no—
517C2×C4—101no—
518C2×C4—101no—
519C2×C4—101no—
520C2×C4—101no—
521C2×C4—101no—
522C2×C4—101no—
523C2×C4—101no—
524C2×C4—202no—
525C2×C4—202no—
526C2×C4—202no—
527C2×C4—202no—
528C2×C4—204no—
529C2×C4—204no—
530C2×C4#413012yes—
531C2×C4D4×AGL(1,𝔽5)12yes—
532C2×C4C2×#70412yes—
533C2×C4C2×#70412yes—
534C8—104no—
535C8—104no—
536C8—202no—
537C8—202no—
538C5#2127111yescharacteristic, complemented, Sylow
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22C2×D5⋊M1611yescharacteristic
554C22C2×#413011yescharacteristic
555C22#6625611yescharacteristic
556C22C2×D4×AGL(1,𝔽5)11yescharacteristic
557C22#6626411yescharacteristic
558C22#41×AGL(1,𝔽5)11yescharacteristic
559C22C22×#70411yescharacteristic
560C4—22no—
561C4—22no—
562C4—22no—
563C4—22no—
564C4—41no—
565C4—52no—
566C4—52no—
567C4—102no—
568C4—102no—
569C4—102no—
570C4—102no—
571C4—201no—
572C4—201no—
573C4—201no—
574C2—42no—
575C2—42no—
576C2#1061714211yescharacteristic
577C2#1061715011yescharacteristic
578C2#1061715611yescharacteristic
579C2#1061717411yescharacteristic
580C2#1061747311yescharacteristic
581C2D4×#70411yescharacteristic
582C2#1061754611yescharacteristic
subgroupG/H
center, FrattiniC23C23×AGL(1,𝔽5)
commutator, socleC23×C5C23×C4
FittingC2×C4×C5×D4C4
prime radicalC22×C5×D4C2×C4
cosocleC22×Dic5C24
hypercenterC2×C4×D4AGL(1,𝔽5)
hypocenterC5#21271
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#65759C2semidirect
2. C2×#65757C2semidirect
5. #10604897C2semidirect
6. #10604897C2semidirect
7. #10604327C2semidirect
8. #10604238C2semidirect
9. #10604237C2semidirect
10. #10604180C2semidirect
11. C4×#4002C2semidirect
96. #4002D4semidirect
538. C5#21271semidirect

Maximal subgroups

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

subgroupindex
C2×#657592
C2×#657572
C4×D4×Dic52
#106049492
#106048972
#106043272
#106042382
#106042372
#106041802
C4×#40022
#212715