Finite Group Databasebeta

← #12262762

Subgroups — #12262762

Order 1280.

2286 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66159C211yesmaximal, characteristic
2C5×#2311C211yesmaximal, characteristic
3#10615817C211yesmaximal, characteristic
4#10613974C212yesmaximal, complemented
5#10613893C212yesmaximal, complemented
6#10608217C214yesmaximal, complemented
7#10608040C212yesmaximal, complemented
8#10608039C212yesmaximal, complemented
9#10606415C212yesmaximal, complemented
10#10606411C212yesmaximal, complemented
11#10606401C214yesmaximal, complemented
12#10605612C212yesmaximal, complemented
13#10605607C212yesmaximal, complemented
14#10605602C212yesmaximal, complemented
15#10605597C212yesmaximal, complemented
16C22×C5×#41C2211yescharacteristic
17C23×Dic10C2211yescharacteristic
18C2×C5×#117C2211yescharacteristic
19C22×C5×C4⋊C4C2211yescharacteristic
20#66159C4116yescomplemented
21C2×#4090C2212yes—
22C2×#4082C2212yes—
23C2×#4081C2212yes—
24C2×#4081C2212yes—
25C2×#4072C2211yescharacteristic
26C2×#4072C2211yescharacteristic
27C2×#4070C2212yes—
28C2×#4070C2212yes—
29C22×C4×Dic5C2211yescharacteristic
30C2×#4033C2212yes—
31C2×#4033C2212yes—
32C5×#391C2212yes—
33C5×#390C2212yes—
34C5×#386C2212yes—
35C5×#386C2212yes—
36C5×#380C2212yes—
37C5×#380C2212yes—
38#65519C2212yes—
39#65510C2212yes—
40#65509C2211yescharacteristic
41#65509C2211yescharacteristic
42#65509C2211yescharacteristic
43#65509C2211yescharacteristic
44#65499C2214yes—
45C4×C22⋊Dic5C2212yes—
46C4×C22⋊Dic5C2212yes—
47C4×C22⋊Dic5C2214yes—
48#65267C2214yes—
49C4⋊C4×Dic5C2212yes—
50C4⋊C4×Dic5C2212yes—
51#65263C2212yes—
52#65263C2212yes—
53#65262C2212yes—
54#65262C2212yes—
55#65261C2214yes—
56#65235C2214yes—
57#65234C2214yes—
58#65234C2214yes—
59#65234C2214yes—
60#65234C2214yes—
61#65233C2214yes—
62#65233C2214yes—
63#65231C2214yes—
64C22⋊C4×Dic5C2214yes—
65C22⋊C4×Dic5C2214yes—
66#65222C2214yes—
67C4×#683C2214yes—
68#64944C2214yes—
69#64941C2214yes—
70#64939C2214yes—
71#64938C2214yes—
72#64937C2214yes—
73#64936C2214yes—
74#64935C2214yes—
75#64935C2214yes—
76#34412—51nomaximal, Sylow
77C2×C5×#41—42no—
78C2×C5×#41—42no—
79C2×C5×#41C2×C418yes—
80C22×C5×Q8C2311yescharacteristic
81C22×C5×D4C2311yescharacteristic
82C22×C5×D4C2311yescharacteristic
83C22×C5×D4C2311yescharacteristic
84C23×C4×C5C2311yescharacteristic
85C23×C4×C5C2311yescharacteristic
86C23×C4×C5C2311yescharacteristic
87C23×C4×C5C2312yes—
88C23×Dic5C2311yescharacteristic
89C23×Dic5C2311yescharacteristic
90C23×Dic5C2312yes—
91C22×Dic10—42no—
92C22×Dic10—42no—
93C22×Dic10C2×C418yescomplemented
94C22×Dic10C2312yes—
95C22×Dic10C2312yes—
96C5×#117—42no—
97C2×C5×C4⋊C4—42no—
98C2×C5×C4⋊C4C2312yes—
99C2×C5×C4⋊C4C2312yes—
100C2×C5×C4⋊C4C2312yes—
101C2×C5×C22⋊C4C2314yes—
102C2×C5×C22⋊C4C2314yes—
103C2×C5×C22⋊C4C2312yes—
104C2×C42×C5C2312yes—
105C2×C22⋊Dic5C2312yes—
106C2×C22⋊Dic5C2312yes—
107C2×C22⋊Dic5C2311yescharacteristic
108C2×C22⋊Dic5C2311yescharacteristic
109C2×C22⋊Dic5C2312yes—
110C2×C22⋊Dic5C2312yes—
111C2×C22⋊Dic5C2314yes—
112C2×C22⋊Dic5C2314yes—
113C2×C22⋊Dic5C2312yes—
114#4090C2×C4116yescomplemented
115#4082C2×C4116yes—
116#4081C2×C4116yescomplemented
117#4081C2×C4116yescomplemented
118#4072—42no—
119#4072C2×C418yescomplemented
120C2×C4⋊Dic5C2312yes—
121C2×C4⋊Dic5C2312yes—
122#4070C2×C4116yescomplemented
123#4070C2×C4116yescomplemented
124C2×#683C2314yes—
125C2×#683C2314yes—
126C2×#683C2314yes—
127C2×#683C2314yes—
128C2×#683C2314yes—
129C2×#683C2314yes—
130C2×C4×Dic5—42no—
131C2×C4×Dic5C2312yes—
132C2×C4×Dic5C2312yes—
133C2×C4×Dic5C2314yes—
134C2×C4×Dic5C2314yes—
135C2×C4×Dic5C2314yes—
136C2×C4×Dic5C2312yes—
137C2×C4×Dic5C2314yes—
138C2×C4×Dic5C2312yes—
139#4033—44no—
140#4033—44no—
141C5×#95C2314yes—
142C5×#95C2314yes—
143#3963C2312yes—
144#3963C2312yes—
145#3963C2314yes—
146#3963C2314yes—
147#3963C2314yes—
148#3963C2314yes—
149#3963C2314yes—
150#3963C2314yes—
151#3963C2314yes—
152#3963C2314yes—
153#3963C2312yes—
154#3963C2312yes—
155C2×#536—51no—
156#2379—52no—
157#2376—52no—
158#2374—52no—
159#2372—52no—
160#2362—52no—
161#2360—52no—
162#2346—52no—
163Q8×C22⋊C4—52no—
164#2335—54no—
165#2331—54no—
166#2323—52no—
167#2311D511yescharacteristic
168C4×C22⋊Q8—52no—
169#2297—51no—
170C24×C5C2411yescharacteristic
171C24×C5C2411yescharacteristic
172C24×C5C2411yescharacteristic
173C5×#41—416no—
174C2×C5×Q8—41no—
175C2×C5×Q8—41no—
176C2×C5×Q8C22×C414yes—
177C2×C5×D4—41no—
178C2×C5×D4—41no—
179C2×C5×D4—41no—
180C2×C5×D4—41no—
181C2×C5×D4—41no—
182C2×C5×D4—41no—
183C2×C5×D4C22×C414yes—
184C2×C5×D4C22×C414yes—
185C2×C5×D4C22×C414yes—
186C22×C4×C5—22no—
187C22×C4×C5—22no—
188C22×C4×C5—22no—
189C22×C4×C5—22no—
190C22×C4×C5—22no—
191C22×C4×C5—22no—
192C22×C4×C5—42no—
193C22×C4×C5—44no—
194C22×C4×C5C22×C414yes—
195C22×C4×C5C22×C414yes—
196C22×C4×C5C22×C414yes—
197C22×C4×C5C2414yes—
198C22×C4×C5C2414yes—
199C22×C4×C5C2411yescharacteristic
200C22×C4×C5C2411yescharacteristic
201C22×C4×C5C2411yescharacteristic
202C22×C4×C5C2411yescharacteristic
203C22×Dic5—24no—
204C22×Dic5—42no—
205C22×Dic5—42no—
206C22×Dic5—42no—
207C22×Dic5—41no—
208C22×Dic5—41no—
209C22×Dic5C22×C418yes—
210C22×Dic5C2412yes—
211C22×Dic5C2412yes—
212C22×Dic5C2412yes—
213C22×Dic5C2414yes—
214C22×Dic5C2414yes—
215C22×Dic5C2412yes—
216C2×Dic10—24no—
217C2×Dic10—24no—
218C2×Dic10—416no—
219C2×Dic10—42no—
220C2×Dic10—42no—
221C2×Dic10C22×C418yes—
222C2×Dic10C22×C418yes—
223C5×C4⋊C4—24no—
224C5×C4⋊C4—24no—
225C5×C4⋊C4—42no—
226C5×C22⋊C4—44no—
227C5×C22⋊C4—44no—
228C5×C22⋊C4—42no—
229C42×C5—42no—
230C22⋊Dic5—24no—
231C22⋊Dic5—24no—
232C22⋊Dic5—42no—
233C22⋊Dic5—42no—
234C22⋊Dic5—41no—
235C22⋊Dic5—41no—
236C22⋊Dic5C22×C4116yes—
237C22⋊Dic5C22×C4116yes—
238C22⋊Dic5C22×C418yes—
239C4⋊Dic5—42no—
240C4⋊Dic5C22×C418yes—
241#683—44no—
242#683—44no—
243#683—44no—
244#683—44no—
245#683C22×C4116yes—
246#683C22×C4116yes—
247C4×Dic5—24no—
248C4×Dic5—24no—
249C4×Dic5—28no—
250C4×Dic5—44no—
251C4×Dic5—44no—
252C4×Dic5—42no—
253C4×Dic5—44no—
254C4×Dic5C22×C418yes—
255C22×#41C2×D511yescharacteristic
256C23×Q8—51no—
257#536—516no—
258C2×C4⋊Q8—52no—
259C2×#124—52no—
260C2×#123—52no—
261C2×#123—52no—
262C2×C22⋊Q8—52no—
263C2×C22⋊Q8—52no—
264C2×C4×Q8—52no—
265C2×C4×Q8—52no—
266C2×#117—51no—
267C2×#117—51no—
268C2×#117C2×D511yescharacteristic
269C22×C4⋊C4C2×D511yescharacteristic
270C22×C42—51no—
271#391—52no—
272#391—52no—
273#391—52no—
274#391—54no—
275#391—54no—
276#391C2×D512yes—
277#390—52no—
278#390C2×D512yes—
279#389—52no—
280#389—54no—
281#389—52no—
282#389—54no—
283#388—54no—
284#388—54no—
285#388—54no—
286#388—54no—
287#388—54no—
288#388—54no—
289#387—54no—
290#387—54no—
291#387—54no—
292#387—54no—
293#386—51no—
294#386—51no—
295#386—51no—
296#386—51no—
297#386C2×D512yes—
298#386C2×D512yes—
299#385—54no—
300#385—54no—
301#382—54no—
302#380—54no—
303#380C2×D512yes—
304#380C2×D512yes—
305C4×C4⋊C4—52no—
306C4×C4⋊C4—52no—
307C4×C4⋊C4—54no—
308C4×C4⋊C4—54no—
309C4×C22⋊C4—54no—
310C4×C22⋊C4—52no—
311C4×C22⋊C4—52no—
312C4×C22⋊C4—54no—
313C4×C22⋊C4—54no—
314#376—54no—
315C23×C5—21no—
316C23×C5—21no—
317C23×C5—21no—
318C23×C5—21no—
319C23×C5—21no—
320C23×C5—21no—
321C23×C5—22no—
322C23×C5—22no—
323C23×C5—22no—
324C23×C5—22no—
325C23×C5—22no—
326C23×C5—22no—
327C23×C5C23×C412yes—
328C23×C5C23×C412yes—
329C23×C5C23×C412yes—
330C23×C5C2511yescharacteristic
331C5×Q8—44no—
332C5×D4—44no—
333C5×D4—44no—
334C5×D4—44no—
335C2×C4×C5—21no—
336C2×C4×C5—21no—
337C2×C4×C5—21no—
338C2×C4×C5—21no—
339C2×C4×C5—21no—
340C2×C4×C5—21no—
341C2×C4×C5—28no—
342C2×C4×C5—28no—
343C2×C4×C5—28no—
344C2×C4×C5—24no—
345C2×C4×C5—24no—
346C2×C4×C5—24no—
347C2×C4×C5—24no—
348C2×C4×C5—24no—
349C2×C4×C5—24no—
350C2×C4×C5—84no—
351C2×C4×C5C23×C412yes—
352C2×C4×C5C23×C412yes—
353C2×C4×C5C23×C412yes—
354C2×C4×C5C23×C412yes—
355C2×C4×C5H(5,𝔽2)12yes—
356C2×C4×C5H(5,𝔽2)12yes—
357C2×Dic5—22no—
358C2×Dic5—24no—
359C2×Dic5—24no—
360C2×Dic5—24no—
361C2×Dic5—22no—
362C2×Dic5—24no—
363C2×Dic5—22no—
364C2×Dic5—22no—
365C2×Dic5—22no—
366C2×Dic5—22no—
367C2×Dic5—22no—
368C2×Dic5—22no—
369C2×Dic5—22no—
370C2×Dic5—22no—
371C2×Dic5—22no—
372C2×Dic5—48no—
373C2×Dic5—44no—
374C2×Dic5—44no—
375C2×Dic5C23×C418yes—
376C2×Dic5C23×C418yes—
377C2×Dic52−1+414yes—
378Dic10—216no—
379Dic10—48no—
380C2×#41—42no—
381C2×#41—42no—
382C2×#41C4×D518yes—
383C22×Q8—58no—
384C22×Q8—52no—
385C22×Q8—52no—
386C22×Q8—202no—
387C22×Q8—202no—
388C22×Q8C22×D511yescharacteristic
389C22×D4C22×D511yescharacteristic
390C22×D4C22×D511yescharacteristic
391C22×D4C22×D511yescharacteristic
392C23×C4—52no—
393C23×C4—51no—
394C23×C4—51no—
395C23×C4C22×D511yescharacteristic
396C23×C4C22×D511yescharacteristic
397C23×C4C22×D511yescharacteristic
398C23×C4C22×D512yes—
399C4⋊Q8—516no—
400#124—516no—
401#123—516no—
402#123—516no—
403C22⋊Q8—516no—
404C22⋊Q8—516no—
405C4×Q8—204no—
406C4×Q8—204no—
407#117—42no—
408#117—58no—
409#117—202no—
410C2×C4⋊C4—42no—
411C2×C4⋊C4—52no—
412C2×C4⋊C4—54no—
413C2×C4⋊C4—54no—
414C2×C4⋊C4—52no—
415C2×C4⋊C4—54no—
416C2×C4⋊C4—54no—
417C2×C4⋊C4—54no—
418C2×C4⋊C4—54no—
419C2×C4⋊C4C22×D512yes—
420C2×C4⋊C4C22×D512yes—
421C2×C4⋊C4C22×D512yes—
422C2×C22⋊C4—52no—
423C2×C22⋊C4—54no—
424C2×C22⋊C4—54no—
425C2×C22⋊C4—52no—
426C2×C22⋊C4—52no—
427C2×C22⋊C4—51no—
428C2×C22⋊C4—51no—
429C2×C22⋊C4—52no—
430C2×C22⋊C4—52no—
431C2×C22⋊C4C22×D512yes—
432C2×C22⋊C4C22×D514yes—
433C2×C22⋊C4C22×D514yes—
434C2×C42—52no—
435C2×C42—52no—
436C2×C42—52no—
437C2×C42—54no—
438C2×C42—54no—
439C2×C42—54no—
440C2×C42—54no—
441C2×C42—52no—
442C2×C42—202no—
443C2×C42C22×D512yes—
444#95—52no—
445#95—54no—
446#95—54no—
447#95—54no—
448#95—52no—
449#95—54no—
450#95—52no—
451#95—54no—
452#95—52no—
453#95—54no—
454#95—54no—
455#95—54no—
456#95C22×D514yes—
457#95C22×D514yes—
458C22×C5—22no—
459C22×C5—22no—
460C22×C5—22no—
461C22×C5—24no—
462C22×C5—24no—
463C22×C5—24no—
464C22×C5—41no—
465C22×C5—41no—
466C22×C5—41no—
467C22×C5—41no—
468C22×C5—41no—
469C22×C5—41no—
470C22×C5—42no—
471C22×C5—42no—
472C22×C5—42no—
473C22×C5C24×C411yescharacteristic
474C22×C5C2×H(5,𝔽2)11yescharacteristic
475C22×C5C2×H(5,𝔽2)11yescharacteristic
476C22×C5C2×2−1+412yes—
477C22×C5#58512yes—
478C4×C5—22no—
479C4×C5—22no—
480C4×C5—22no—
481C4×C5—44no—
482C4×C5—44no—
483C4×C5#51814yes—
484Dic5—28no—
485Dic5—28no—
486Dic5—24no—
487Dic5—42no—
488Dic5—42no—
489Dic5—42no—
490C24C23×D511yescharacteristic
491C24C23×D511yescharacteristic
492C24C23×D511yescharacteristic
493#41—416no—
494C2×Q8—41no—
495C2×Q8—41no—
496C2×Q8—58no—
497C2×Q8—58no—
498C2×Q8—104no—
499C2×Q8—104no—
500C2×Q8—202no—
501C2×Q8—202no—
502C2×Q8—2016no—
503C2×Q8C2×C4×D514yes—
504C2×D4—41no—
505C2×D4—41no—
506C2×D4—41no—
507C2×D4—41no—
508C2×D4—41no—
509C2×D4—41no—
510C2×D4C2×C4×D514yes—
511C2×D4C2×C4×D514yes—
512C2×D4C2×C4×D514yes—
513C22×C4—22no—
514C22×C4—22no—
515C22×C4—22no—
516C22×C4—22no—
517C22×C4—22no—
518C22×C4—22no—
519C22×C4—42no—
520C22×C4—44no—
521C22×C4—52no—
522C22×C4—54no—
523C22×C4—54no—
524C22×C4—58no—
525C22×C4—52no—
526C22×C4—52no—
527C22×C4—52no—
528C22×C4—104no—
529C22×C4—201no—
530C22×C4—201no—
531C22×C4—202no—
532C22×C4—202no—
533C22×C4—202no—
534C22×C4C2×C4×D514yes—
535C22×C4C2×C4×D514yes—
536C22×C4C2×C4×D514yes—
537C22×C4C23×D514yes—
538C22×C4C23×D514yes—
539C22×C4C23×D511yescharacteristic
540C22×C4C23×D511yescharacteristic
541C22×C4C23×D511yescharacteristic
542C22×C4C23×D511yescharacteristic
543C4⋊C4—24no—
544C4⋊C4—24no—
545C4⋊C4—42no—
546C4⋊C4—58no—
547C4⋊C4—516no—
548C4⋊C4—516no—
549C4⋊C4—202no—
550C4⋊C4—204no—
551C4⋊C4—204no—
552C4⋊C4—204no—
553C4⋊C4—204no—
554C22⋊C4—42no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—58no—
558C22⋊C4—516no—
559C22⋊C4—516no—
560C22⋊C4—104no—
561C22⋊C4—104no—
562C22⋊C4—201no—
563C22⋊C4—201no—
564C22⋊C4—202no—
565C22⋊C4—202no—
566C42—42no—
567C42—58no—
568C42—108no—
569C42—104no—
570C42—104no—
571C42—202no—
572C42—204no—
573C42—204no—
574C42—204no—
575C2×C5—42no—
576C2×C5—42no—
577C2×C5—42no—
578C2×C5C2×#51811yescharacteristic
579C2×C5C4×2−1+412yes—
580C2×C5#353811yescharacteristic
581C2×C5#354011yescharacteristic
582C2×C5#354712yes—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—22no—
590C23—22no—
591C23—22no—
592C23—22no—
593C23—22no—
594C23—22no—
595C23C22×C4×D512yes—
596C23C22×C4×D512yes—
597C23C22×C4×D512yes—
598C23C24×D511yescharacteristic
599Q8—44no—
600Q8—1016no—
601Q8—208no—
602D4—44no—
603D4—44no—
604D4—44no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—28no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—58no—
621C2×C4—58no—
622C2×C4—54no—
623C2×C4—84no—
624C2×C4—102no—
625C2×C4—104no—
626C2×C4—104no—
627C2×C4—104no—
628C2×C4—104no—
629C2×C4—102no—
630C2×C4—102no—
631C2×C4—102no—
632C2×C4—102no—
633C2×C4—102no—
634C2×C4—102no—
635C2×C4—102no—
636C2×C4—102no—
637C2×C4—102no—
638C2×C4—102no—
639C2×C4—208no—
640C2×C4—204no—
641C2×C4—204no—
642C2×C4C22×C4×D512yes—
643C2×C4C22×C4×D512yes—
644C2×C4C22×C4×D512yes—
645C2×C4C22×C4×D512yes—
646C2×C4#414412yes—
647C2×C4#414412yes—
648C5#3441211yescharacteristic, complemented, Sylow
649C22—24no—
650C22—24no—
651C22—24no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22C23×C4×D511yescharacteristic
665C22C2×#414411yescharacteristic
666C22C2×#414411yescharacteristic
667C22#6628512yes—
668C22D5×2−1+412yes—
669C4—22no—
670C4—22no—
671C4—22no—
672C4—44no—
673C4—44no—
674C4—104no—
675C4—108no—
676C4—108no—
677C4—202no—
678C4—202no—
679C4—202no—
680C4#6582014yes—
681C2—42no—
682C2—42no—
683C2—42no—
684C2C2×#6582011yescharacteristic
685C2#1061778812yes—
686C2#1061822311yescharacteristic
687C2#1061826011yescharacteristic
688C2#1061827412yes—
subgroupG/H
center, FrattiniC23C24×D5
commutatorC22×C5C24×C4
FittingC5×#2311C2
socle, cosocleC23×C5C25
prime radicalC22×C5×#41C22
hypercenter#2311D5
hypocenterC5#34412
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ #2311

Complements

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

NH ≅ G/Ntype
4. #10613974C2semidirect
5. #10613893C2semidirect
6. #10608217C2semidirect
7. #10608040C2semidirect
8. #10608039C2semidirect
9. #10606415C2semidirect
10. #10606411C2semidirect
11. #10606401C2semidirect
12. #10605612C2semidirect
13. #10605607C2semidirect
14. #10605602C2semidirect
15. #10605597C2semidirect
20. #66159C4semidirect
93. C22×Dic10C2×C4semidirect
114. #4090C2×C4semidirect
116. #4081C2×C4semidirect
117. #4081C2×C4semidirect
119. #4072C2×C4semidirect
122. #4070C2×C4semidirect
123. #4070C2×C4semidirect
648. C5#34412semidirect

Maximal subgroups

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

subgroupindex
C2×#661592
C5×#23112
#106158172
#106139742
#106138932
#106082172
#106080402
#106080392
#106064152
#106064112
#106064012
#106056122
#106056072
#106056022
#106055972
#344125