Finite Group Databasebeta

← #13031034

Subgroups — #13031034

Order 1280.

1018 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65956C211yesmaximal, characteristic
2C2×#65754C211yesmaximal, characteristic
3C2×#65758C211yesmaximal, characteristic
4#10604971C212yesmaximal, complemented
5#10604518C214yesmaximal, complemented
6#10604517C214yesmaximal, complemented
7#10604278C212yesmaximal, complemented
8C22×#704—21no—
9C2×C5×C22⋊Q8C411yescharacteristic
10C2×Q8×Dic5C411yescharacteristic
11C2×#4090C2211yescharacteristic
12C2×#4070C411yescharacteristic
13C22×C4×Dic5C2211yescharacteristic
14#65956—21no—
15#65956—21no—
16#65956—21no—
17#65956—21no—
18#65956C412yes—
19#65956C412yes—
20#65956C2212yes—
21#65956C2212yes—
22C2×#4035—21no—
23C2×#4034C2211yescharacteristic
24C2×#4033—21no—
25C2×#4033C411yescharacteristic
26C2×#4024—21no—
27C2×#4023—21no—
28C2×C22⋊#170—21no—
29#65758—21no—
30#65758—21no—
31#65758C2212yes—
32#65758C2212yes—
33#65754—21no—
34#65754—21no—
35#65754C2212yes—
36#65754C2212yes—
37C2×#4004C2212yes—
38C2×#4004C2212yes—
39C2×C4⋊#170—21no—
40C2×C4×#170—21no—
41#64931C2218yescomplemented
42#64873C2218yescomplemented
43#17708—51nomaximal, Sylow
44C22×C5×Q8C2×C411yescharacteristic
45C23×C4×C5C2×C411yescharacteristic
46C23×Dic5D411yescharacteristic
47C23×Dic5D411yescharacteristic
48C22×Dic10—21no—
49C22×Dic10C2×C411yescharacteristic
50C2×#704—22no—
51C2×#704—22no—
52C2×#704—44no—
53C22×#170—22no—
54C22×#170—22no—
55C5×C22⋊Q8—21no—
56C5×C22⋊Q8—21no—
57C5×C22⋊Q8C2×C412yes—
58C5×C22⋊Q8C2×C412yes—
59C2×C5×C4⋊C4—21no—
60C2×C5×C4⋊C4C2×C411yescharacteristic
61C2×C5×C22⋊C4—21no—
62C2×C22⋊Dic5—21no—
63Q8×Dic5—21no—
64Q8×Dic5—21no—
65Q8×Dic5C2×C412yes—
66Q8×Dic5C2×C412yes—
67#4090—21no—
68#4090—21no—
69#4090C2×C412yes—
70#4090C2312yes—
71C2×C4⋊Dic5—21no—
72C2×C4⋊Dic5C2×C411yescharacteristic
73#4070—21no—
74#4070—21no—
75#4070C2×C412yes—
76#4070C2×C412yes—
77C2×#683—21no—
78C2×#683—21no—
79C2×#683—21no—
80C2×#683—21no—
81C2×#683—21no—
82C2×C4×Dic5—21no—
83C2×C4×Dic5—21no—
84C2×C4×Dic5C2×C411yescharacteristic
85C2×C4×Dic5C2×C411yescharacteristic
86C2×C4×Dic5C2×C411yescharacteristic
87C2×C4×Dic5D411yescharacteristic
88C2×C4×Dic5D411yescharacteristic
89C2×C4×Dic5D411yescharacteristic
90C2×C4×Dic5D411yescharacteristic
91C2×C4×Dic5D411yescharacteristic
92C2×C4×Dic5D411yescharacteristic
93C2×C4×Dic5C2311yescharacteristic
94C2×C4×Dic5C2311yescharacteristic
95C2×C4×Dic5C2311yescharacteristic
96#4035—22no—
97#4035—22no—
98#4035—22no—
99#4035—22no—
100#4034—21no—
101#4034—21no—
102#4034C2×C412yes—
103#4034C2312yes—
104#4033—22no—
105#4033—22no—
106#4033—21no—
107#4033—21no—
108#4033—22no—
109#4033—22no—
110#4033C2×C412yes—
111#4033C2×C412yes—
112#4024—22no—
113#4024—22no—
114#4024—22no—
115#4024—22no—
116#4023—22no—
117#4023—22no—
118#4023—22no—
119#4023—22no—
120C22⋊#170—22no—
121C22⋊#170—22no—
122#4004—22no—
123#4004—22no—
124#4004C2314yes—
125#4004C2314yes—
126C4⋊#170—22no—
127C4⋊#170—22no—
128C4×#170—22no—
129C4×#170—22no—
130C2×#533—51no—
131C2×#431—51no—
132C2×#423—51no—
133#1550—54no—
134#1549—52no—
135#1548—52no—
136#1542—54no—
137C24×C5C22⋊C411yescharacteristic
138C2×C5×Q8—21no—
139C2×C5×Q8—42no—
140C2×C5×Q8C22×C412yes—
141C22×C4×C5—21no—
142C22×C4×C5—21no—
143C22×C4×C5—42no—
144C22×C4×C5C22⋊C411yescharacteristic
145C22×C4×C5C22⋊C411yescharacteristic
146C22×C4×C5C22⋊C411yescharacteristic
147C22×C4×C5C22×C411yescharacteristic
148C22×C4×C5C22×C411yescharacteristic
149C22×C4×C5C22×C411yescharacteristic
150C22×Dic5—21no—
151C22×Dic5—21no—
152C22×Dic5—42no—
153C22×Dic5C22⋊C411yescharacteristic
154C22×Dic5C22⋊C411yescharacteristic
155C22×Dic5C22⋊C411yescharacteristic
156C22×Dic5C22⋊C411yescharacteristic
157C22×Dic5C22⋊C411yescharacteristic
158C22×Dic5C22⋊C411yescharacteristic
159C22×Dic5SD1614yes—
160C22×Dic5Q1614yes—
161C22×Dic5C2×D411yescharacteristic
162C22×Dic5C2×D411yescharacteristic
163C22×Dic5C2×D411yescharacteristic
164C22×Dic5C2×D411yescharacteristic
165C22×Dic5C2×D411yescharacteristic
166C22×Dic5C2×D411yescharacteristic
167C2×Dic10—21no—
168C2×Dic10—22no—
169C2×Dic10—22no—
170C2×Dic10—44no—
171C2×Dic10—42no—
172C2×Dic10C22×C412yes—
173#704—44no—
174#704—44no—
175C2×#170—24no—
176C2×#170—24no—
177C2×#170—44no—
178C2×#170—44no—
179C5×C4⋊C4—22no—
180C5×C4⋊C4—22no—
181C5×C4⋊C4—21no—
182C5×C4⋊C4C22×C412yes—
183C5×C22⋊C4—22no—
184C5×C22⋊C4—22no—
185C22⋊Dic5—22no—
186C22⋊Dic5—22no—
187C4⋊Dic5—21no—
188C4⋊Dic5—22no—
189C4⋊Dic5—22no—
190C4⋊Dic5C22×C412yes—
191#683—22no—
192#683—22no—
193#683—22no—
194#683—22no—
195#683—22no—
196#683—22no—
197#683—22no—
198#683—22no—
199#683—22no—
200#683—22no—
201C4×Dic5—22no—
202C4×Dic5—22no—
203C4×Dic5—22no—
204C4×Dic5—22no—
205C4×Dic5C22⋊C411yescharacteristic
206C4×Dic5C22⋊C411yescharacteristic
207C4×Dic5C22⋊C411yescharacteristic
208C4×Dic5C22⋊C411yescharacteristic
209C4×Dic5C22⋊C411yescharacteristic
210C4×Dic5C22⋊C411yescharacteristic
211C4×Dic5C22×C411yescharacteristic
212C4×Dic5C22×C411yescharacteristic
213C4×Dic5C22×C411yescharacteristic
214C4×Dic5C2×D411yescharacteristic
215C4×Dic5C2×D411yescharacteristic
216C4×Dic5C2×D411yescharacteristic
217C4×Dic5C2×D411yescharacteristic
218C4×Dic5C2×D411yescharacteristic
219C4×Dic5C2×D411yescharacteristic
220C4×Dic5C2411yescharacteristic
221C22×M16—101no—
222#533—52no—
223#533—52no—
224#533—52no—
225#533—52no—
226#533—101no—
227#533—101no—
228#533—101no—
229#533—101no—
230C2×C4⋊Q8—51no—
231C2×C4⋊Q8—51no—
232C2×#125—101no—
233C2×C22⋊Q8—51no—
234C2×C22⋊Q8—101no—
235C2×C22⋊Q8AGL(1,𝔽5)11yescharacteristic
236C2×C4×Q8—51no—
237C2×C4×Q8—51no—
238C2×C4×Q8—101no—
239C2×#117—101no—
240C22×C42—51no—
241#431—52no—
242#431—52no—
243#431—101no—
244#431—101no—
245#423—52no—
246#423—52no—
247#423—101no—
248#423—101no—
249C2×C4⋊C8—52no—
250C2×C4⋊C8—52no—
251C2×C4⋊C8—101no—
252C2×C22⋊C8—101no—
253C2×C4×C8—101no—
254#333—58no—
255#332—58no—
256C23×C5—41no—
257C23×C5—41no—
258C23×C5#10314yes—
259C23×C5C2×C22⋊C411yescharacteristic
260C23×C5C2×C22⋊C411yescharacteristic
261C23×C5C2×C22⋊C411yescharacteristic
262C5×Q8—42no—
263C5×Q8—42no—
264C2×C4×C5—22no—
265C2×C4×C5—22no—
266C2×C4×C5—22no—
267C2×C4×C5—22no—
268C2×C4×C5—22no—
269C2×C4×C5—41no—
270C2×C4×C5—41no—
271C2×C4×C5—41no—
272C2×C4×C5—42no—
273C2×C4×C5—42no—
274C2×C4×C5C2×C22⋊C411yescharacteristic
275C2×C4×C5C2×C22⋊C411yescharacteristic
276C2×C4×C5C2×C22⋊C411yescharacteristic
277C2×C4×C5C23×C411yescharacteristic
278C2×Dic5—22no—
279C2×Dic5—22no—
280C2×Dic5—22no—
281C2×Dic5—22no—
282C2×Dic5—22no—
283C2×Dic5—41no—
284C2×Dic5—41no—
285C2×Dic5—41no—
286C2×Dic5—42no—
287C2×Dic5—42no—
288C2×Dic5#10314yes—
289C2×Dic5#10314yes—
290C2×Dic5#10314yes—
291C2×Dic5C2×C22⋊C411yescharacteristic
292C2×Dic5C2×C22⋊C411yescharacteristic
293C2×Dic5C2×C22⋊C411yescharacteristic
294C2×Dic5C2×C22⋊C411yescharacteristic
295C2×Dic5C2×C22⋊C411yescharacteristic
296C2×Dic5C2×C22⋊C411yescharacteristic
297C2×Dic5C2×SD1612yes—
298C2×Dic5C2×SD1612yes—
299C2×Dic5C2×SD1612yes—
300C2×Dic5C2×Q1612yes—
301C2×Dic5C2×Q1612yes—
302C2×Dic5C2×Q1612yes—
303C2×Dic5C22×D411yescharacteristic
304C2×Dic5C22×D411yescharacteristic
305Dic10—44no—
306Dic10—42no—
307Dic10—44no—
308Dic10—42no—
309#170—44no—
310#170—44no—
311C22×Q8—51no—
312C22×Q8—101no—
313C22×Q8C2×AGL(1,𝔽5)11yescharacteristic
314C23×C4—51no—
315C23×C4—51no—
316C23×C4C2×AGL(1,𝔽5)11yescharacteristic
317C2×M16—102no—
318C2×M16—102no—
319C2×M16—204no—
320C22×C8—102no—
321C22×C8—102no—
322C4⋊Q8—52no—
323C4⋊Q8—52no—
324C4⋊Q8—52no—
325C4⋊Q8—52no—
326C4⋊Q8—101no—
327C4⋊Q8—101no—
328C4⋊Q8—101no—
329C4⋊Q8—101no—
330#125—102no—
331#125—102no—
332#125—102no—
333#125—102no—
334C22⋊Q8—21no—
335C22⋊Q8—21no—
336C22⋊Q8—52no—
337C22⋊Q8—52no—
338C22⋊Q8—102no—
339C22⋊Q8—102no—
340C22⋊Q8—102no—
341C22⋊Q8—102no—
342C22⋊Q8—101no—
343C22⋊Q8—101no—
344C22⋊Q8C2×AGL(1,𝔽5)12yes—
345C22⋊Q8C2×AGL(1,𝔽5)12yes—
346C4×Q8—52no—
347C4×Q8—52no—
348C4×Q8—52no—
349C4×Q8—52no—
350C4×Q8—101no—
351C4×Q8—101no—
352C4×Q8—102no—
353C4×Q8—102no—
354C4×Q8—102no—
355C4×Q8—102no—
356C4×Q8—101no—
357C4×Q8—101no—
358#117—102no—
359#117—102no—
360#117—102no—
361#117—102no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—51no—
364C2×C4⋊C4—101no—
365C2×C4⋊C4—101no—
366C2×C4⋊C4—101no—
367C2×C4⋊C4—101no—
368C2×C4⋊C4—101no—
369C2×C4⋊C4—101no—
370C2×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
371C2×C22⋊C4—21no—
372C2×C22⋊C4—101no—
373C2×C42—51no—
374C2×C42—51no—
375C2×C42—51no—
376C2×C42—51no—
377C2×C42—51no—
378C2×C42—51no—
379C2×C42—51no—
380C2×C42—51no—
381C2×C42—51no—
382C2×C42—51no—
383C2×C42—51no—
384C2×C42—51no—
385C2×C42—101no—
386C2×C42—101no—
387C4⋊C8—54no—
388C4⋊C8—54no—
389C4⋊C8—102no—
390C4⋊C8—102no—
391C4⋊C8—102no—
392C4⋊C8—102no—
393C22⋊C8—102no—
394C22⋊C8—102no—
395C4×C8—102no—
396C4×C8—102no—
397C22×C5—21no—
398C22×C5—21no—
399C22×C5—21no—
400C22×C5—21no—
401C22×C5—21no—
402C22×C5—21no—
403C22×C5—44no—
404C22×C5C2×#10312yes—
405C22×C5C2×#10312yes—
406C22×C5C2×#10312yes—
407C22×C5C22×C22⋊C411yescharacteristic
408C4×C5—21no—
409C4×C5—21no—
410C4×C5—21no—
411C4×C5—42no—
412C4×C5—42no—
413C4×C5#41312yes—
414Dic5—21no—
415Dic5—21no—
416Dic5—21no—
417Dic5—42no—
418Dic5—42no—
419Dic5C2×#10312yes—
420Dic5C2×#10312yes—
421Dic5C2×#10312yes—
422Dic5C22×SD1611yescharacteristic
423Dic5C22×Q1611yescharacteristic
424Dic5#57512yes—
425C24C22⋊AGL(1,𝔽5)11yescharacteristic
426C2×Q8—21no—
427C2×Q8—42no—
428C2×Q8—52no—
429C2×Q8—102no—
430C2×Q8—102no—
431C2×Q8—101no—
432C2×Q8—202no—
433C2×Q8—204no—
434C2×Q8C22×AGL(1,𝔽5)12yes—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—42no—
438C22×C4—51no—
439C22×C4—51no—
440C22×C4—51no—
441C22×C4—51no—
442C22×C4—51no—
443C22×C4—51no—
444C22×C4—54no—
445C22×C4—54no—
446C22×C4—51no—
447C22×C4—51no—
448C22×C4—51no—
449C22×C4—51no—
450C22×C4—51no—
451C22×C4—51no—
452C22×C4—101no—
453C22×C4—101no—
454C22×C4—202no—
455C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
456C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
457C22×C4C22⋊AGL(1,𝔽5)11yescharacteristic
458C22×C4C22×AGL(1,𝔽5)11yescharacteristic
459C22×C4C22×AGL(1,𝔽5)11yescharacteristic
460C22×C4C22×AGL(1,𝔽5)11yescharacteristic
461M16—204no—
462M16—204no—
463C2×C8—104no—
464C2×C8—104no—
465C2×C8—204no—
466C2×C8—204no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—52no—
471C4⋊C4—102no—
472C4⋊C4—102no—
473C4⋊C4—102no—
474C4⋊C4—102no—
475C4⋊C4—102no—
476C4⋊C4—102no—
477C4⋊C4—102no—
478C4⋊C4—102no—
479C4⋊C4—102no—
480C4⋊C4—102no—
481C4⋊C4—101no—
482C4⋊C4—102no—
483C4⋊C4—102no—
484C4⋊C4C22×AGL(1,𝔽5)12yes—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—102no—
488C22⋊C4—102no—
489C42—51no—
490C42—51no—
491C42—51no—
492C42—51no—
493C42—51no—
494C42—51no—
495C42—51no—
496C42—51no—
497C42—51no—
498C42—51no—
499C42—51no—
500C42—51no—
501C42—51no—
502C42—51no—
503C42—51no—
504C42—51no—
505C42—102no—
506C42—102no—
507C42—102no—
508C42—102no—
509C2×C5—22no—
510C2×C5—22no—
511C2×C5—22no—
512C2×C5C2×#41311yescharacteristic
513C2×C5C22×#10311yescharacteristic
514C2×C5#289611yescharacteristic
515C23—41no—
516C23—41no—
517C23#400914yes—
518C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
519C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
520C23C2×C22⋊AGL(1,𝔽5)11yescharacteristic
521Q8—42no—
522Q8—42no—
523Q8—202no—
524Q8—204no—
525Q8—204no—
526Q8—202no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—51no—
538C2×C4—51no—
539C2×C4—51no—
540C2×C4—51no—
541C2×C4—51no—
542C2×C4—52no—
543C2×C4—52no—
544C2×C4—52no—
545C2×C4—54no—
546C2×C4—54no—
547C2×C4—54no—
548C2×C4—52no—
549C2×C4—52no—
550C2×C4—52no—
551C2×C4—51no—
552C2×C4—51no—
553C2×C4—51no—
554C2×C4—102no—
555C2×C4—102no—
556C2×C4—102no—
557C2×C4—102no—
558C2×C4—102no—
559C2×C4—201no—
560C2×C4—201no—
561C2×C4—201no—
562C2×C4—202no—
563C2×C4—202no—
564C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
565C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
566C2×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
567C2×C4C23×AGL(1,𝔽5)11yescharacteristic
568C8—204no—
569C8—204no—
570C5#1770811yescharacteristic, complemented, Sylow
571C22—21no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—44no—
578C22C2×#400912yes—
579C22C2×#400912yes—
580C22C2×#400912yes—
581C22C22×C22⋊AGL(1,𝔽5)11yescharacteristic
582C4—21no—
583C4—21no—
584C4—21no—
585C4—42no—
586C4—42no—
587C4—51no—
588C4—51no—
589C4—52no—
590C4—52no—
591C4—52no—
592C4—52no—
593C4—101no—
594C4—101no—
595C4—101no—
596C4—202no—
597C4—202no—
598C4#6577712yes—
599C2—22no—
600C2—22no—
601C2—22no—
602C2C2×#6577711yescharacteristic
603C2C22×#400911yescharacteristic
604C2#1061753311yescharacteristic
subgroupG/H
centerC22C22×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C23×C4
FrattiniC2×C4C23×AGL(1,𝔽5)
FittingC2×C5×C22⋊Q8C4
socleC22×C5C22×C22⋊C4
prime radicalC24×C5C22⋊C4
cosocleC4×Dic5C24
hypercenterC2×C22⋊Q8AGL(1,𝔽5)
hypocenterC5#17708
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
4. #10604971C2semidirect
5. #10604518C2semidirect
6. #10604517C2semidirect
7. #10604278C2semidirect
41. #64931C22semidirect
42. #64873C22semidirect
570. C5#17708semidirect

Maximal subgroups

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

subgroupindex
C2×#659562
C2×#657542
C2×#657582
#106049712
#106045182
#106045172
#106042782
#177085