Finite Group Databasebeta

← #13034430

Subgroups — #13034430

Order 1280.

1418 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#4023C211yesmaximal, characteristic
2C2×#65697C212yesmaximal, complemented
3#10605025C211yesmaximal, characteristic
4#10605025C211yesmaximal, characteristic
5#10604900C211yesmaximal, characteristic, complemented
6#10604899C211yesmaximal, characteristic, complemented
7#10604897C211yesmaximal, characteristic, complemented
8C4×C22⋊#170C211yesmaximal, characteristic, complemented
9#10604229C212yesmaximal, complemented
10#10604228C212yesmaximal, complemented
11#10604224C212yesmaximal, complemented
12C24×Dic5C2211yescharacteristic
13C22×C5×C22⋊C4C411yescharacteristic
14C22×C22⋊Dic5C411yescharacteristic
15C22×#683C411yescharacteristic
16C22×#683C411yescharacteristic
17C22×C4×Dic5C2211yescharacteristic
18C22×C4×Dic5C2211yescharacteristic
19C2×#4023—28no—
20C2×#4023C411yescharacteristic
21C2×#4023C411yescharacteristic
22C2×#4023C411yescharacteristic
23C2×#4023C411yescharacteristic
24C2×#4023C2211yescharacteristic
25C2×#4023C2211yescharacteristic
26C2×#4023C2211yescharacteristic
27C2×#4023C2211yescharacteristic
28C2×C22⋊#170C2211yescharacteristic
29C2×C22⋊#170C2211yescharacteristic
30C2×C22⋊#170C2211yescharacteristic
31C2×C22⋊#170C2211yescharacteristic
32C2×C22⋊#170C2212yes—
33C2×C22⋊#170C2212yes—
34C2×C22⋊#170C2212yes—
35C2×C22⋊#170C2212yes—
36C2×#4004C2212yescomplemented
37C2×#4002C2212yescomplemented
38C2×C4⋊#170C2212yescomplemented
39C2×C4×#170C2212yescomplemented
40#65697—28no—
41#64918C2212yescomplemented
42#64918C2212yescomplemented
43#64918C2212yescomplemented
44#64918C2212yescomplemented
45#20716—51nomaximal, Sylow
46C25×C5C2×C411yescharacteristic
47C23×C4×C5C2×C411yescharacteristic
48C23×C4×C5C2×C411yescharacteristic
49C23×Dic5—24no—
50C23×Dic5—24no—
51C23×Dic5—22no—
52C23×Dic5—22no—
53C23×Dic5C2×C411yescharacteristic
54C23×Dic5C2×C411yescharacteristic
55C23×Dic5C2×C411yescharacteristic
56C23×Dic5C2×C411yescharacteristic
57C23×Dic5C2×C411yescharacteristic
58C23×Dic5C2311yescharacteristic
59C23×Dic5C2311yescharacteristic
60C23×Dic5C2311yescharacteristic
61C22×#170C2312yes—
62C22×#170C2312yes—
63C22×#170C2312yes—
64C22×#170C2312yes—
65C2×C5×C22⋊C4—24no—
66C2×C5×C22⋊C4C2×C411yescharacteristic
67C2×C5×C22⋊C4C2×C411yescharacteristic
68C2×C5×C22⋊C4C2×C411yescharacteristic
69C2×C5×C22⋊C4C2×C411yescharacteristic
70C2×C22⋊Dic5—24no—
71C2×C22⋊Dic5C2×C411yescharacteristic
72C2×C22⋊Dic5C2×C411yescharacteristic
73C2×C22⋊Dic5C2×C411yescharacteristic
74C2×C22⋊Dic5C2×C411yescharacteristic
75C2×#683—24no—
76C2×#683—24no—
77C2×#683C2×C411yescharacteristic
78C2×#683C2×C411yescharacteristic
79C2×#683C2×C411yescharacteristic
80C2×#683C2×C411yescharacteristic
81C2×#683C2×C411yescharacteristic
82C2×#683C2×C411yescharacteristic
83C2×#683C2×C411yescharacteristic
84C2×#683C2×C411yescharacteristic
85C2×C4×Dic5—24no—
86C2×C4×Dic5—24no—
87C2×C4×Dic5C2×C411yescharacteristic
88C2×C4×Dic5C2×C411yescharacteristic
89C2×C4×Dic5C2×C411yescharacteristic
90C2×C4×Dic5C2×C411yescharacteristic
91C2×C4×Dic5C2311yescharacteristic
92C2×C4×Dic5C2311yescharacteristic
93C2×C4×Dic5C2311yescharacteristic
94C2×C4×Dic5C2311yescharacteristic
95#4023—24no—
96#4023—24no—
97#4023—24no—
98#4023—24no—
99#4023—24no—
100#4023—24no—
101#4023—24no—
102#4023D418yes—
103C22⋊#170—24no—
104C22⋊#170—24no—
105C22⋊#170—22no—
106C22⋊#170—22no—
107C22⋊#170—22no—
108C22⋊#170—22no—
109C22⋊#170—24no—
110C22⋊#170—24no—
111#4004—24no—
112#4002—24no—
113C4⋊#170—24no—
114C4×#170—24no—
115C22×#117—51no—
116C2×#433—52no—
117#1818—52no—
118#1798—52no—
119#1797—51no—
120#1797—51no—
121#1779—51no—
122#1779—51no—
123#1750—52no—
124#1749—51no—
125C4×C22⋊C8—51no—
126C24×C5—22no—
127C24×C5—22no—
128C24×C5—22no—
129C24×C5—22no—
130C24×C5—22no—
131C24×C5—41no—
132C24×C5—41no—
133C24×C5C22×C411yescharacteristic
134C24×C5C22×C411yescharacteristic
135C24×C5C22×C411yescharacteristic
136C22×C4×C5—22no—
137C22×C4×C5—22no—
138C22×C4×C5—22no—
139C22×C4×C5—22no—
140C22×C4×C5—41no—
141C22×C4×C5—41no—
142C22×C4×C5C22×C411yescharacteristic
143C22×C4×C5C22×C411yescharacteristic
144C22×C4×C5C22×C411yescharacteristic
145C22×C4×C5C22×C411yescharacteristic
146C22×Dic5—216no—
147C22×Dic5—216no—
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—21no—
160C22×Dic5—21no—
161C22×Dic5—22no—
162C22×Dic5—22no—
163C22×Dic5—21no—
164C22×Dic5—21no—
165C22×Dic5—22no—
166C22×Dic5—22no—
167C22×Dic5—22no—
168C22×Dic5—22no—
169C22×Dic5—41no—
170C22×Dic5—41no—
171C22×Dic5C22×C411yescharacteristic
172C22×Dic5C22×C411yescharacteristic
173C22×Dic5C22×C411yescharacteristic
174C22×Dic5C22×C411yescharacteristic
175C22×Dic5C22×C411yescharacteristic
176C22×Dic5C22×C411yescharacteristic
177C22×Dic5C22×C411yescharacteristic
178C22×Dic5C2×D414yes—
179C22×Dic5#4114yes—
180C22×Dic5#4112yes—
181C22×Dic5#4112yes—
182C22×Dic5C2411yescharacteristic
183C2×#170—22no—
184C2×#170—22no—
185C2×#170—22no—
186C2×#170—22no—
187C2×#170—22no—
188C2×#170—22no—
189C2×#170—22no—
190C2×#170—22no—
191C2×#170—22no—
192C2×#170—22no—
193C2×#170—22no—
194C2×#170—22no—
195C5×C22⋊C4—22no—
196C5×C22⋊C4—22no—
197C5×C22⋊C4—22no—
198C5×C22⋊C4C22⋊C414yes—
199C22⋊Dic5—22no—
200C22⋊Dic5—22no—
201C22⋊Dic5—22no—
202C22⋊Dic5C22⋊C414yes—
203#683—22no—
204#683—22no—
205#683—22no—
206#683—22no—
207#683—22no—
208#683—22no—
209#683C22⋊C414yes—
210#683C22⋊C414yes—
211C4×Dic5—22no—
212C4×Dic5—22no—
213C4×Dic5—22no—
214C4×Dic5—22no—
215C4×Dic5—22no—
216C4×Dic5—22no—
217C4×Dic5C2×D414yes—
218C4×Dic5C2×D414yes—
219C24×C4—51no—
220C2×#117—51no—
221C2×#117—51no—
222C2×#117—51no—
223C2×#117—51no—
224C2×#117—51no—
225C2×#117—51no—
226C2×#117—51no—
227C2×#117—51no—
228C2×#117—108no—
229C22×C4⋊C4—51no—
230C22×C4⋊C4—51no—
231C22×C22⋊C4—51no—
232C22×C22⋊C4AGL(1,𝔽5)11yescharacteristic
233C22×C42—51no—
234C22×C42—51no—
235#433—108no—
236C2×C4⋊C8—52no—
237C2×C4⋊C8—52no—
238C2×C22⋊C8—51no—
239C2×C22⋊C8—51no—
240C2×C22⋊C8—51no—
241C2×C22⋊C8—51no—
242C2×C22⋊C8—52no—
243C2×C22⋊C8—52no—
244C2×C22⋊C8—52no—
245C2×C22⋊C8—52no—
246C2×#97—52no—
247C2×C4×C8—52no—
248#336—52no—
249#336—52no—
250#336—52no—
251#336—52no—
252C23×C5—21no—
253C23×C5—21no—
254C23×C5—21no—
255C23×C5—21no—
256C23×C5—28no—
257C23×C5—21no—
258C23×C5—21no—
259C23×C5—21no—
260C23×C5—21no—
261C23×C5—21no—
262C23×C5—21no—
263C23×C5—21no—
264C23×C5—21no—
265C23×C5—21no—
266C23×C5—21no—
267C23×C5—21no—
268C23×C5—21no—
269C23×C5—44no—
270C23×C5—44no—
271C23×C5—44no—
272C23×C5—44no—
273C23×C5—44no—
274C23×C5—44no—
275C23×C5C2×C22⋊C412yes—
276C23×C5#11712yes—
277C23×C5#11712yes—
278C23×C5#13112yes—
279C23×C5#13112yes—
280C23×C5C23×C411yescharacteristic
281C2×C4×C5—21no—
282C2×C4×C5—21no—
283C2×C4×C5—21no—
284C2×C4×C5—21no—
285C2×C4×C5—21no—
286C2×C4×C5—21no—
287C2×C4×C5—21no—
288C2×C4×C5—21no—
289C2×C4×C5—21no—
290C2×C4×C5—21no—
291C2×C4×C5—44no—
292C2×C4×C5—44no—
293C2×C4×C5C2×C22⋊C412yes—
294C2×C4×C5C2×C22⋊C412yes—
295C2×Dic5—28no—
296C2×Dic5—28no—
297C2×Dic5—28no—
298C2×Dic5—21no—
299C2×Dic5—21no—
300C2×Dic5—21no—
301C2×Dic5—21no—
302C2×Dic5—21no—
303C2×Dic5—21no—
304C2×Dic5—21no—
305C2×Dic5—21no—
306C2×Dic5—21no—
307C2×Dic5—21no—
308C2×Dic5—24no—
309C2×Dic5—24no—
310C2×Dic5—24no—
311C2×Dic5—24no—
312C2×Dic5—24no—
313C2×Dic5—24no—
314C2×Dic5—21no—
315C2×Dic5—21no—
316C2×Dic5—21no—
317C2×Dic5—21no—
318C2×Dic5—21no—
319C2×Dic5—21no—
320C2×Dic5—44no—
321C2×Dic5—44no—
322C2×Dic5C2×C22⋊C412yes—
323C2×Dic5C2×C22⋊C412yes—
324C2×Dic5C2×C22⋊C412yes—
325C2×Dic5#11712yes—
326C2×Dic5#11712yes—
327C2×Dic5C22×D412yes—
328C2×Dic5C2×#4111yescharacteristic
329C2×Dic5C2×#4111yescharacteristic
330C2×Dic5C2×#4112yes—
331#170—42no—
332#170—42no—
333#170—42no—
334#170—42no—
335C25C2×AGL(1,𝔽5)11yescharacteristic
336C23×C4—51no—
337C23×C4—51no—
338C23×C4—51no—
339C23×C4—51no—
340C23×C4—51no—
341C23×C4—51no—
342C23×C4—51no—
343C23×C4—51no—
344C23×C4—104no—
345C23×C4—104no—
346C23×C4—102no—
347C23×C4—102no—
348C23×C4C2×AGL(1,𝔽5)11yescharacteristic
349C23×C4C2×AGL(1,𝔽5)11yescharacteristic
350C22×C8—52no—
351C22×C8—52no—
352C22×C8—52no—
353C22×C8—52no—
354#117—58no—
355#117—104no—
356#117—104no—
357#117—104no—
358#117—104no—
359#117—104no—
360#117—104no—
361#117—104no—
362C2×C4⋊C4—51no—
363C2×C4⋊C4—51no—
364C2×C4⋊C4—51no—
365C2×C4⋊C4—51no—
366C2×C4⋊C4—51no—
367C2×C4⋊C4—51no—
368C2×C4⋊C4—51no—
369C2×C4⋊C4—51no—
370C2×C4⋊C4—104no—
371C2×C4⋊C4—104no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—51no—
374C2×C22⋊C4—51no—
375C2×C22⋊C4—51no—
376C2×C22⋊C4—51no—
377C2×C22⋊C4—104no—
378C2×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
379C2×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
380C2×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
381C2×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
382C2×C42—51no—
383C2×C42—51no—
384C2×C42—51no—
385C2×C42—51no—
386C2×C42—51no—
387C2×C42—51no—
388C2×C42—51no—
389C2×C42—51no—
390C2×C42—104no—
391C2×C42—104no—
392C4⋊C8—104no—
393C4⋊C8—104no—
394C22⋊C8—102no—
395C22⋊C8—102no—
396C22⋊C8—102no—
397C22⋊C8—102no—
398C22⋊C8—104no—
399C22⋊C8—104no—
400C22⋊C8—104no—
401C22⋊C8—104no—
402#97—104no—
403C4×C8—104no—
404C22×C5—22no—
405C22×C5—22no—
406C22×C5—22no—
407C22×C5—22no—
408C22×C5—22no—
409C22×C5—22no—
410C22×C5—22no—
411C22×C5—22no—
412C22×C5—22no—
413C22×C5—22no—
414C22×C5—22no—
415C22×C5—22no—
416C22×C5—416no—
417C22×C5—41no—
418C22×C5—41no—
419C22×C5—41no—
420C22×C5—41no—
421C22×C5—41no—
422C22×C5—41no—
423C22×C5—41no—
424C22×C5—41no—
425C22×C5#43314yescomplemented
426C22×C5C22×C22⋊C411yescharacteristic
427C22×C5C2×#11711yescharacteristic
428C22×C5C2×#11711yescharacteristic
429C22×C5C2×#13111yescharacteristic
430C22×C5C2×#13111yescharacteristic
431C22×C5#56811yescharacteristic
432C22×C5#56811yescharacteristic
433C4×C5—22no—
434C4×C5—22no—
435C4×C5—41no—
436C4×C5—41no—
437Dic5—22no—
438Dic5—22no—
439Dic5—22no—
440Dic5—22no—
441Dic5—22no—
442Dic5—41no—
443Dic5—41no—
444Dic5#52512yes—
445Dic5#52512yes—
446C24—22no—
447C24—22no—
448C24—22no—
449C24—22no—
450C24—22no—
451C24—41no—
452C24—41no—
453C24C22×AGL(1,𝔽5)11yescharacteristic
454C24C22×AGL(1,𝔽5)11yescharacteristic
455C24C22×AGL(1,𝔽5)11yescharacteristic
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—51no—
463C22×C4—54no—
464C22×C4—52no—
465C22×C4—52no—
466C22×C4—54no—
467C22×C4—51no—
468C22×C4—51no—
469C22×C4—51no—
470C22×C4—51no—
471C22×C4—51no—
472C22×C4—51no—
473C22×C4—51no—
474C22×C4—102no—
475C22×C4—102no—
476C22×C4—102no—
477C22×C4—102no—
478C22×C4—102no—
479C22×C4—101no—
480C22×C4—101no—
481C22×C4—101no—
482C22×C4—101no—
483C22×C4—102no—
484C22×C4—102no—
485C22×C4—102no—
486C22×C4—102no—
487C22×C4—102no—
488C22×C4—102no—
489C22×C4—102no—
490C22×C4—1016no—
491C22×C4—1016no—
492C22×C4—102no—
493C22×C4—102no—
494C22×C4—102no—
495C22×C4—102no—
496C22×C4—102no—
497C22×C4—201no—
498C22×C4—201no—
499C22×C4C22×AGL(1,𝔽5)11yescharacteristic
500C22×C4C22×AGL(1,𝔽5)11yescharacteristic
501C22×C4C22×AGL(1,𝔽5)11yescharacteristic
502C22×C4C22×AGL(1,𝔽5)11yescharacteristic
503C2×C8—102no—
504C2×C8—102no—
505C2×C8—102no—
506C2×C8—102no—
507C2×C8—102no—
508C2×C8—102no—
509C2×C8—102no—
510C2×C8—102no—
511C2×C8—102no—
512C2×C8—102no—
513C2×C8—102no—
514C2×C8—102no—
515C4⋊C4—54no—
516C4⋊C4—54no—
517C4⋊C4—102no—
518C4⋊C4—102no—
519C4⋊C4—102no—
520C4⋊C4—102no—
521C4⋊C4—102no—
522C4⋊C4—102no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—54no—
527C22⋊C4—102no—
528C22⋊C4—102no—
529C22⋊C4—102no—
530C22⋊C4C22⋊AGL(1,𝔽5)14yes—
531C42—54no—
532C42—54no—
533C42—102no—
534C42—102no—
535C42—102no—
536C42—102no—
537C42—102no—
538C42—102no—
539C2×C5—24no—
540C2×C5—42no—
541C2×C5—42no—
542C2×C5#228011yescharacteristic
543C2×C5#288011yescharacteristic
544C2×C5#288011yescharacteristic
545C2×C5C2×#43311yescharacteristic
546C2×C5#292111yescharacteristic
547C2×C5#292411yescharacteristic
548C2×C5#292411yescharacteristic
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—28no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23#413012yes—
573C23#413012yes—
574C23#413112yes—
575C23#413112yes—
576C23C2×C22⋊AGL(1,𝔽5)12yes—
577C23C23×AGL(1,𝔽5)11yescharacteristic
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—51no—
591C2×C4—51no—
592C2×C4—52no—
593C2×C4—52no—
594C2×C4—52no—
595C2×C4—52no—
596C2×C4—52no—
597C2×C4—52no—
598C2×C4—52no—
599C2×C4—104no—
600C2×C4—104no—
601C2×C4—104no—
602C2×C4—104no—
603C2×C4—104no—
604C2×C4—104no—
605C2×C4—108no—
606C2×C4—108no—
607C2×C4—108no—
608C2×C4—101no—
609C2×C4—101no—
610C2×C4—101no—
611C2×C4—101no—
612C2×C4—101no—
613C2×C4—101no—
614C2×C4—101no—
615C2×C4—101no—
616C2×C4—101no—
617C2×C4—101no—
618C2×C4—101no—
619C2×C4—101no—
620C2×C4—101no—
621C2×C4—101no—
622C2×C4—101no—
623C2×C4—101no—
624C2×C4—204no—
625C2×C4—204no—
626C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
627C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
628C8—202no—
629C8—202no—
630C8—202no—
631C8—202no—
632C5#2071611yescharacteristic, complemented, Sylow
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—416no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22#6569714yescomplemented
655C22C2×#413011yescharacteristic
656C22C2×#413011yescharacteristic
657C22C2×#413111yescharacteristic
658C22C2×#413111yescharacteristic
659C22#6625611yescharacteristic
660C22#6625611yescharacteristic
661C22C22×C22⋊AGL(1,𝔽5)11yescharacteristic
662C4—22no—
663C4—22no—
664C4—41no—
665C4—41no—
666C4—52no—
667C4—52no—
668C4—102no—
669C4—102no—
670C4—102no—
671C4—102no—
672C4—102no—
673C4—201no—
674C4—201no—
675C2—24no—
676C2—42no—
677C2—42no—
678C2C2×#6569711yescharacteristic
679C2#1061705411yescharacteristic
680C2#1061705511yescharacteristic
681C2#1061705511yescharacteristic
682C2#1061748011yescharacteristic
683C2#1061749211yescharacteristic
684C2#1061749211yescharacteristic
subgroupG/H
center, FrattiniC23C23×AGL(1,𝔽5)
commutator, socleC23×C5C23×C4
FittingC22×C5×C22⋊C4C4
prime radicalC25×C5C2×C4
cosocleC22×Dic5C24
hypercenterC22×C22⋊C4AGL(1,𝔽5)
hypocenterC5#20716
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#65697C2semidirect
5. #10604900C2semidirect
6. #10604899C2semidirect
7. #10604897C2semidirect
8. C4×C22⋊#170C2semidirect
9. #10604229C2semidirect
10. #10604228C2semidirect
11. #10604224C2semidirect
36. C2×#4004C22semidirect
37. C2×#4002C22semidirect
38. C2×C4⋊#170C22semidirect
39. C2×C4×#170C22semidirect
41. #64918C22semidirect
42. #64918C22semidirect
43. #64918C22semidirect
44. #64918C22semidirect
425. C22×C5#433semidirect
632. C5#20716semidirect
654. C22#65697semidirect

Maximal subgroups

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

subgroupindex
C22×#40232
C2×#656972
#106050252
#106049002
#106048992
#106048972
C4×C22⋊#1702
#106042292
#106042282
#106042242
#207165