Finite Group Databasebeta

← #12023013

Subgroups — #12023013

Order 1280.

2006 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65499C211yesmaximal, characteristic
2C22×#3963C211yesmaximal, characteristic
3C2×#65239C211yesmaximal, characteristic
4C2×#65235C211yesmaximal, characteristic
5C2×C22⋊C4×Dic5C211yesmaximal, characteristic
6C5×C22⋊#95C211yesmaximal, characteristic
7#10602337C211yesmaximal, characteristic
8#10600207C212yesmaximal, complemented
9#10600203C212yesmaximal, complemented
10#10600200C212yesmaximal, complemented
11#10600197C212yesmaximal, complemented
12C24×C4×C5C2211yescharacteristic
13C24×Dic5C2211yescharacteristic
14C24×Dic5C2211yescharacteristic
15C22×C5×C22⋊C4C2211yescharacteristic
16C22×C5×C22⋊C4C2211yescharacteristic
17C22×C22⋊Dic5C2211yescharacteristic
18C22×C22⋊Dic5C2211yescharacteristic
19C22×C4⋊Dic5C2211yescharacteristic
20C22×C4⋊Dic5C2211yescharacteristic
21C22×#683C2212yes—
22C22×#683C2212yes—
23C22×C4×Dic5C2212yes—
24C2×C5×#95C2212yes—
25C2×C5×#95C2212yes—
26#65499—24no—
27C2×#3963—24no—
28C2×#3963C2211yescharacteristic
29C2×#3963C2212yes—
30C2×#3963C2212yes—
31C2×#3963C2212yes—
32C2×#3963C2212yes—
33C2×#3963C2211yescharacteristic
34C2×#3963C2212yes—
35C2×#3963C2212yes—
36C2×#3963C2212yes—
37#65239—24no—
38#65235C418yescomplemented
39C22⋊C4×Dic5—24no—
40#14044—51nomaximal, Sylow
41C25×C5C2311yescharacteristic
42C23×C4×C5—22no—
43C23×C4×C5—24no—
44C23×C4×C5—22no—
45C23×C4×C5—22no—
46C23×C4×C5—42no—
47C23×C4×C5C2311yescharacteristic
48C23×C4×C5C2311yescharacteristic
49C23×C4×C5C2312yes—
50C23×C4×C5C2312yes—
51C23×Dic5—22no—
52C23×Dic5—22no—
53C23×Dic5—22no—
54C23×Dic5—22no—
55C23×Dic5—24no—
56C23×Dic5—42no—
57C23×Dic5—42no—
58C23×Dic5—42no—
59C23×Dic5C2×C414yes—
60C23×Dic5D414yes—
61C23×Dic5C2311yescharacteristic
62C23×Dic5C2311yescharacteristic
63C23×Dic5C2312yes—
64C23×Dic5C2311yescharacteristic
65C23×Dic5C2312yes—
66C23×Dic5C2311yescharacteristic
67C2×C5×C22⋊C4—22no—
68C2×C5×C22⋊C4—22no—
69C2×C5×C22⋊C4—22no—
70C2×C5×C22⋊C4—22no—
71C2×C5×C22⋊C4C2×C414yes—
72C2×C5×C22⋊C4Q814yes—
73C2×C22⋊Dic5—22no—
74C2×C22⋊Dic5—22no—
75C2×C22⋊Dic5—22no—
76C2×C22⋊Dic5C2×C414yes—
77C2×C22⋊Dic5D414yes—
78C2×C22⋊Dic5Q814yes—
79C2×C4⋊Dic5—22no—
80C2×C4⋊Dic5—22no—
81C2×C4⋊Dic5—24no—
82C2×C4⋊Dic5—42no—
83C2×#683—24no—
84C2×#683—24no—
85C2×#683—42no—
86C2×#683—42no—
87C2×#683—42no—
88C2×#683C2×C418yescomplemented
89C2×C4×Dic5—24no—
90C2×C4×Dic5—24no—
91C2×C4×Dic5—42no—
92C5×#95—24no—
93C5×#95—42no—
94#3963—24no—
95#3963—22no—
96#3963—22no—
97#3963—24no—
98#3963—24no—
99#3963—24no—
100#3963—42no—
101#3963—42no—
102#3963C2×C418yescomplemented
103C2×#385—51no—
104C2×#385—51no—
105C2×#385—51no—
106C2×C4×C22⋊C4—51no—
107C22×#95—51no—
108#1442—52no—
109#1441—52no—
110#1436—52no—
111C4⋊#95—52no—
112C22⋊#95—51no—
113C22⋊#95D511yescharacteristic
114C24×C5—22no—
115C24×C5—22no—
116C24×C5—22no—
117C24×C5—22no—
118C24×C5C22×C412yes—
119C24×C5C2×D412yes—
120C24×C5C2×D412yes—
121C24×C5C2×Q812yes—
122C24×C5C2×Q812yes—
123C24×C5#4112yes—
124C24×C5#4112yes—
125C24×C5C2411yescharacteristic
126C22×C4×C5—28no—
127C22×C4×C5—22no—
128C22×C4×C5—21no—
129C22×C4×C5—21no—
130C22×C4×C5—22no—
131C22×C4×C5—22no—
132C22×C4×C5—21no—
133C22×C4×C5—21no—
134C22×C4×C5—22no—
135C22×C4×C5—22no—
136C22×C4×C5—22no—
137C22×C4×C5—22no—
138C22×C4×C5—22no—
139C22×C4×C5—22no—
140C22×C4×C5—22no—
141C22×C4×C5—22no—
142C22×C4×C5—22no—
143C22×C4×C5—22no—
144C22×C4×C5—22no—
145C22×C4×C5—22no—
146C22×C4×C5—22no—
147C22×C4×C5—48no—
148C22×C4×C5—48no—
149C22×C4×C5—44no—
150C22×C4×C5—44no—
151C22×C4×C5C22×C412yes—
152C22×C4×C5C22×C412yes—
153C22×C4×C5C2×Q814yes—
154C22×C4×C5#4114yes—
155C22×Dic5—21no—
156C22×Dic5—21no—
157C22×Dic5—22no—
158C22×Dic5—22no—
159C22×Dic5—22no—
160C22×Dic5—22no—
161C22×Dic5—22no—
162C22×Dic5—28no—
163C22×Dic5—28no—
164C22×Dic5—22no—
165C22×Dic5—21no—
166C22×Dic5—21no—
167C22×Dic5—22no—
168C22×Dic5—22no—
169C22×Dic5—22no—
170C22×Dic5—22no—
171C22×Dic5—22no—
172C22×Dic5—22no—
173C22×Dic5—22no—
174C22×Dic5—28no—
175C22×Dic5—21no—
176C22×Dic5—21no—
177C22×Dic5—21no—
178C22×Dic5—21no—
179C22×Dic5—21no—
180C22×Dic5—21no—
181C22×Dic5—22no—
182C22×Dic5—22no—
183C22×Dic5—22no—
184C22×Dic5—22no—
185C22×Dic5—48no—
186C22×Dic5—44no—
187C22×Dic5—48no—
188C22×Dic5—48no—
189C22×Dic5—44no—
190C22×Dic5—44no—
191C22×Dic5—48no—
192C22×Dic5C22×C414yes—
193C22×Dic5C22×C414yes—
194C22×Dic5C2×D412yes—
195C22×Dic5C2×D414yes—
196C22×Dic5C2×D412yes—
197C22×Dic5C2×Q814yes—
198C22×Dic5#4114yes—
199C5×C22⋊C4—28no—
200C5×C22⋊C4—28no—
201C22⋊Dic5—44no—
202C22⋊Dic5C4⋊C4116yescomplemented
203C4⋊Dic5—44no—
204C4⋊Dic5—44no—
205#683—48no—
206#683—48no—
207C4×Dic5—48no—
208C24×C4—51no—
209C24×C4—51no—
210C24×C4C2×D511yescharacteristic
211C22×C4⋊C4—52no—
212C22×C4⋊C4—51no—
213C22×C4⋊C4—52no—
214C22×C4⋊C4—51no—
215C22×C22⋊C4—51no—
216C22×C22⋊C4—51no—
217C22×C22⋊C4C2×D511yescharacteristic
218C22×C22⋊C4C2×D511yescharacteristic
219C22×C42—52no—
220#385—58no—
221#385—104no—
222#385—104no—
223C4×C22⋊C4—104no—
224C2×#95—52no—
225C2×#95—51no—
226C2×#95—52no—
227C2×#95—52no—
228C2×#95—52no—
229C2×#95—52no—
230C2×#95—51no—
231C2×#95—52no—
232C2×#95—52no—
233C2×#95—104no—
234C2×#95C2×D512yes—
235C2×#95C2×D512yes—
236C23×C5—24no—
237C23×C5—24no—
238C23×C5—22no—
239C23×C5—22no—
240C23×C5—24no—
241C23×C5—22no—
242C23×C5—22no—
243C23×C5—24no—
244C23×C5—24no—
245C23×C5—24no—
246C23×C5—24no—
247C23×C5—24no—
248C23×C5—24no—
249C23×C5—24no—
250C23×C5—24no—
251C23×C5—24no—
252C23×C5C2×C4⋊C414yes—
253C23×C5C4×D414yes—
254C23×C5C4×Q814yes—
255C23×C5#12114yes—
256C23×C5C22⋊Q814yes—
257C23×C5#12514yes—
258C23×C5C4⋊Q814yes—
259C23×C5C23×C411yescharacteristic
260C23×C5C22×D411yescharacteristic
261C23×C5C22×D411yescharacteristic
262C23×C5C22×Q811yescharacteristic
263C23×C5C22×Q811yescharacteristic
264C23×C5C2×#4111yescharacteristic
265C23×C5C2×#4111yescharacteristic
266C23×C5C2×#4112yes—
267C23×C5C2×#4112yes—
268C23×C5H(5,𝔽2)12yes—
269C23×C5H(5,𝔽2)12yes—
270C2×C4×C5—22no—
271C2×C4×C5—22no—
272C2×C4×C5—24no—
273C2×C4×C5—24no—
274C2×C4×C5—24no—
275C2×C4×C5—24no—
276C2×C4×C5—24no—
277C2×C4×C5—24no—
278C2×C4×C5—24no—
279C2×C4×C5—24no—
280C2×C4×C5—416no—
281C2×C4×C5—42no—
282C2×C4×C5—41no—
283C2×C4×C5—41no—
284C2×C4×C5—42no—
285C2×C4×C5—42no—
286C2×C4×C5—41no—
287C2×C4×C5—41no—
288C2×C4×C5—42no—
289C2×C4×C5—42no—
290C2×C4×C5—42no—
291C2×C4×C5—42no—
292C2×C4×C5—42no—
293C2×C4×C5—42no—
294C2×C4×C5—42no—
295C2×Dic5—24no—
296C2×Dic5—24no—
297C2×Dic5—22no—
298C2×Dic5—22no—
299C2×Dic5—24no—
300C2×Dic5—24no—
301C2×Dic5—24no—
302C2×Dic5—24no—
303C2×Dic5—24no—
304C2×Dic5—24no—
305C2×Dic5—24no—
306C2×Dic5—24no—
307C2×Dic5—24no—
308C2×Dic5—22no—
309C2×Dic5—22no—
310C2×Dic5—22no—
311C2×Dic5—416no—
312C2×Dic5—416no—
313C2×Dic5—42no—
314C2×Dic5—42no—
315C2×Dic5—42no—
316C2×Dic5—42no—
317C2×Dic5—42no—
318C2×Dic5—42no—
319C2×Dic5—41no—
320C2×Dic5—41no—
321C2×Dic5—41no—
322C2×Dic5—41no—
323C2×Dic5—41no—
324C2×Dic5—41no—
325C2×Dic5—42no—
326C2×Dic5—42no—
327C2×Dic5—42no—
328C2×Dic5—42no—
329C2×Dic5C2×C4⋊C418yes—
330C2×Dic5#12314yes—
331C25C22×D511yescharacteristic
332C23×C4—22no—
333C23×C4—24no—
334C23×C4—22no—
335C23×C4—22no—
336C23×C4—42no—
337C23×C4—52no—
338C23×C4—51no—
339C23×C4—52no—
340C23×C4—51no—
341C23×C4—54no—
342C23×C4—54no—
343C23×C4—51no—
344C23×C4—51no—
345C23×C4—104no—
346C23×C4—102no—
347C23×C4—102no—
348C23×C4—102no—
349C23×C4—102no—
350C23×C4—202no—
351C23×C4—202no—
352C23×C4—202no—
353C23×C4C22×D512yes—
354C23×C4C22×D511yescharacteristic
355C23×C4C22×D511yescharacteristic
356C23×C4C22×D512yes—
357C2×C4⋊C4—58no—
358C2×C4⋊C4—104no—
359C2×C4⋊C4—102no—
360C2×C4⋊C4—102no—
361C2×C4⋊C4—104no—
362C2×C4⋊C4—104no—
363C2×C4⋊C4—202no—
364C2×C4⋊C4—202no—
365C2×C4⋊C4—202no—
366C2×C4⋊C4—202no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—54no—
372C2×C22⋊C4—54no—
373C2×C22⋊C4—54no—
374C2×C22⋊C4—102no—
375C2×C22⋊C4—102no—
376C2×C22⋊C4—102no—
377C2×C22⋊C4Dic1014yes—
378C2×C22⋊C4C4×D514yes—
379C2×C42—104no—
380C2×C42—104no—
381C2×C42—202no—
382#95—24no—
383#95—42no—
384#95—58no—
385#95—104no—
386#95—104no—
387#95—104no—
388#95—102no—
389#95—102no—
390#95—104no—
391#95—202no—
392#95—202no—
393C22×C5—24no—
394C22×C5—24no—
395C22×C5—28no—
396C22×C5—28no—
397C22×C5—24no—
398C22×C5—24no—
399C22×C5—28no—
400C22×C5—24no—
401C22×C5—28no—
402C22×C5—24no—
403C22×C5#38918yes—
404C22×C5C22×C4⋊C411yescharacteristic
405C22×C5C2×C4×D411yescharacteristic
406C22×C5C2×C4×Q811yescharacteristic
407C22×C5C4×#4112yes—
408C22×C5#51812yes—
409C22×C5C2×#12111yescharacteristic
410C22×C5C2×C22⋊Q811yescharacteristic
411C22×C5C2×#12311yescharacteristic
412C22×C5#52511yescharacteristic
413C22×C5C2×#12511yescharacteristic
414C22×C5#52912yes—
415C22×C5C2×C4⋊Q811yescharacteristic
416C22×C5#53411yescharacteristic
417C22×C5#53511yescharacteristic
418C22×C5#53812yes—
419C22×C5#54612yes—
420C22×C5#54612yes—
421C22×C5#55112yes—
422C22×C5#55112yes—
423C22×C5#55412yes—
424C22×C5#55412yes—
425C22×C5#55412yes—
426C22×C5#55412yes—
427C4×C5—42no—
428C4×C5—44no—
429C4×C5—44no—
430C4×C5—42no—
431Dic5—28no—
432Dic5—24no—
433Dic5—44no—
434Dic5—42no—
435Dic5—81no—
436Dic5—81no—
437C24—22no—
438C24—22no—
439C24—22no—
440C24—22no—
441C24C2×Dic1012yes—
442C24C2×C4×D512yes—
443C24#70912yes—
444C24D4×D512yes—
445C24D4×D512yes—
446C24#71112yes—
447C24Q8×D512yes—
448C24C23×D511yescharacteristic
449C22×C4—28no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—54no—
475C22×C4—52no—
476C22×C4—54no—
477C22×C4—54no—
478C22×C4—54no—
479C22×C4—54no—
480C22×C4—52no—
481C22×C4—108no—
482C22×C4—101no—
483C22×C4—101no—
484C22×C4—102no—
485C22×C4—102no—
486C22×C4—102no—
487C22×C4—102no—
488C22×C4—102no—
489C22×C4—101no—
490C22×C4—102no—
491C22×C4—101no—
492C22×C4—102no—
493C22×C4—102no—
494C22×C4—102no—
495C22×C4—102no—
496C22×C4—102no—
497C22×C4—102no—
498C22×C4—102no—
499C22×C4—108no—
500C22×C4—108no—
501C22×C4—101no—
502C22×C4—101no—
503C22×C4—101no—
504C22×C4—101no—
505C22×C4—101no—
506C22×C4—101no—
507C22×C4—102no—
508C22×C4—102no—
509C22×C4—102no—
510C22×C4—102no—
511C22×C4—204no—
512C22×C4—208no—
513C22×C4—208no—
514C22×C4—208no—
515C22×C4—204no—
516C22×C4—208no—
517C22×C4—204no—
518C22×C4C2×Dic1014yes—
519C22×C4C2×C4×D512yes—
520C22×C4C2×C4×D512yes—
521C22×C4#71114yes—
522C4⋊C4—208no—
523C4⋊C4—208no—
524C4⋊C4—204no—
525C4⋊C4—204no—
526C22⋊C4—28no—
527C22⋊C4—28no—
528C22⋊C4—516no—
529C22⋊C4—204no—
530C42—208no—
531C2×C5—28no—
532C2×C5C2×#38911yescharacteristic
533C2×C5C4×#12311yescharacteristic
534C2×C5#232111yescharacteristic
535C2×C5#234912yes—
536C2×C5#236812yes—
537C2×C5#241111yescharacteristic
538C2×C5#241211yescharacteristic
539C2×C5#245012yes—
540C2×C5#250211yescharacteristic
541C2×C5#250512yes—
542C2×C5#258311yescharacteristic
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—22no—
550C23—24no—
551C23—22no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—22no—
557C23—22no—
558C23—24no—
559C23C4×Dic1014yes—
560C23#402414yes—
561C23#402714yes—
562C23#403014yes—
563C23#403414yes—
564C23#403514yes—
565C23D5×C4⋊C414yes—
566C23C22×Dic1011yescharacteristic
567C23C22×C4×D511yescharacteristic
568C23C2×#70911yescharacteristic
569C23C2×D4×D511yescharacteristic
570C23C2×D4×D511yescharacteristic
571C23C2×#71111yescharacteristic
572C23C2×#71112yes—
573C23#414412yes—
574C23C2×Q8×D511yescharacteristic
575C23D5×#4112yes—
576C23#414912yes—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—416no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—58no—
603C2×C4—54no—
604C2×C4—102no—
605C2×C4—104no—
606C2×C4—104no—
607C2×C4—104no—
608C2×C4—104no—
609C2×C4—104no—
610C2×C4—104no—
611C2×C4—102no—
612C2×C4—104no—
613C2×C4—104no—
614C2×C4—104no—
615C2×C4—102no—
616C2×C4—104no—
617C2×C4—104no—
618C2×C4—102no—
619C2×C4—102no—
620C2×C4—2016no—
621C2×C4—2016no—
622C2×C4—202no—
623C2×C4—202no—
624C2×C4—202no—
625C2×C4—202no—
626C2×C4—202no—
627C2×C4—202no—
628C2×C4—201no—
629C2×C4—201no—
630C2×C4—202no—
631C2×C4—202no—
632C2×C4—201no—
633C2×C4—202no—
634C2×C4—201no—
635C2×C4—201no—
636C2×C4—201no—
637C2×C4—202no—
638C5#1404411yescharacteristic, complemented, Sylow
639C22—24no—
640C22—24no—
641C22—28no—
642C22—24no—
643C22—28no—
644C22—28no—
645C22—24no—
646C22—24no—
647C22—28no—
648C22—24no—
649C22#6493618yes—
650C22C2×C4×Dic1011yescharacteristic
651C22C2×#402411yescharacteristic
652C22C2×#402711yescharacteristic
653C22C2×#403011yescharacteristic
654C22#6582912yes—
655C22C2×#403411yescharacteristic
656C22C2×#403511yescharacteristic
657C22C2×D5×C4⋊C411yescharacteristic
658C22#6586112yes—
659C22C4×#71112yes—
660C22#6587312yes—
661C22#6587712yes—
662C22#6587912yes—
663C22#6592011yescharacteristic
664C22#6592512yes—
665C22#6592611yescharacteristic
666C22#6593412yes—
667C22#6595112yes—
668C22#6596912yes—
669C22#6597012yes—
670C22D5×#12311yescharacteristic
671C22#6598711yescharacteristic
672C22#6598812yes—
673C4—42no—
674C4—42no—
675C4—44no—
676C4—44no—
677C4—104no—
678C4—108no—
679C4—204no—
680C4—202no—
681C4—401no—
682C4—401no—
683C2—28no—
684C2C2×#6493611yescharacteristic
685C2#1060520512yes—
686C2#1060555611yescharacteristic
687C2#1060565812yes—
688C2#1060585711yescharacteristic
689C2#1060612512yes—
690C2#1060619111yescharacteristic
691C2#1060621911yescharacteristic
692C2#1060639211yescharacteristic
693C2#1060644212yes—
694C2#1060660611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC5×C22⋊#95C2
socle, cosocleC24×C5C24
prime radicalC25×C5C23
hypercenterC22⋊#95D5
hypocenterC5#14044
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
8. #10600207C2semidirect
9. #10600203C2semidirect
10. #10600200C2semidirect
11. #10600197C2semidirect
38. #65235C4semidirect
88. C2×#683C2×C4semidirect
102. #3963C2×C4semidirect
202. C22⋊Dic5C4⋊C4semidirect
638. C5#14044semidirect

Maximal subgroups

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

subgroupindex
C2×#654992
C22×#39632
C2×#652392
C2×#652352
C2×C22⋊C4×Dic52
C5×C22⋊#952
#106023372
#106002072
#106002032
#106002002
#106001972
#140445