Finite Group Databasebeta

← #12162215

Subgroups — #12162215

Order 1280.

896 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2361C211yesmaximal, characteristic
2C2×#65298C211yesmaximal, characteristic, complemented
3C2×#65253C211yesmaximal, characteristic, complemented
4#10605225C211yesmaximal, characteristic, complemented
5#10602475C211yesmaximal, characteristic
6#10602471C211yesmaximal, characteristic
7#10602400C211yesmaximal, characteristic, complemented
8#10602349C211yesmaximal, characteristic, complemented
9#10600449C211yesmaximal, characteristic, complemented
10#10600446C211yesmaximal, characteristic, complemented
11#10600403C211yesmaximal, characteristic, complemented
12#10600399C211yesmaximal, characteristic, complemented
13#10600260C211yesmaximal, characteristic, complemented
14#10600251C211yesmaximal, characteristic, complemented
15#10600243C211yesmaximal, characteristic, complemented
16C2×C5×#121C2211yescharacteristic
17C2×C4×C5×D4—21no—
18C2×C4×C5×D4C2211yescharacteristic
19C22×C5×C4⋊C4C2211yescharacteristic
20C2×#4072C2211yescharacteristic
21C22×#681C2211yescharacteristic
22C2×#4016C2211yescharacteristic
23C5×#390C2211yescharacteristic
24C5×#389C2211yescharacteristic
25C5×#388—21no—
26C5×#386—21no—
27C5×#385—21no—
28C4×C5×C4⋊C4C2211yescharacteristic
29C4×C5×C22⋊C4C2211yescharacteristic
30C2×#3964C2211yescharacteristic
31C2×#3964C2211yescharacteristic
32C2×#3964C2211yescharacteristic
33C2×#3964C2211yescharacteristic
34#65501C2211yescharacteristic
35C2×C22⋊#168C2211yescharacteristic
36#65298—24no—
37#65253—24no—
38C2×#3940C2211yescharacteristic
39C2×#3940C2211yescharacteristic
40C2×#3940C2211yescharacteristic
41C2×#3940C2211yescharacteristic
42C2×#3939C2211yescharacteristic
43C2×#3939C2211yescharacteristic
44#65238—21no—
45#65237—21no—
46#65224C2211yescharacteristic
47#65217C2211yescharacteristic
48#65217C2211yescharacteristic
49C2×C4⋊#168C2211yescharacteristic
50C2×C4⋊#168C2211yescharacteristic
51#64943—21no—
52#64936—21no—
53#64749C2211yescharacteristic
54#64749C2211yescharacteristic
55#64749C2211yescharacteristic
56#64749C2211yescharacteristic
57#64749C2211yescharacteristic
58#64749C2211yescharacteristic
59#64744C2211yescharacteristic
60#64744C2211yescharacteristic
61#26366—51nomaximal, Sylow
62C22×C5×D4—21no—
63C22×C5×D4C2311yescharacteristic
64C23×C4×C5—21no—
65C23×C4×C5—21no—
66C23×C4×C5C2311yescharacteristic
67C5×#121—22no—
68C5×#121Q814yes—
69C4×C5×D4—22no—
70C4×C5×D4—41no—
71C4×C5×D4—44no—
72C2×C5×C4⋊C4—22no—
73C2×C5×C4⋊C4—21no—
74C2×C5×C4⋊C4—21no—
75C2×C5×C4⋊C4—41no—
76C2×C5×C4⋊C4C2311yescharacteristic
77C2×C5×C4⋊C4C2311yescharacteristic
78C2×C5×C4⋊C4C2311yescharacteristic
79C2×C5×C22⋊C4—21no—
80C2×C5×C22⋊C4—21no—
81C2×C5×C22⋊C4—21no—
82C2×C5×C22⋊C4—21no—
83C2×C5×C22⋊C4D411yescharacteristic
84C2×C5×C22⋊C4D411yescharacteristic
85C2×C42×C5—21no—
86C2×C42×C5C2311yescharacteristic
87C2×C42×C5C2311yescharacteristic
88C2×C22⋊Dic5—21no—
89C2×C22⋊Dic5—21no—
90#4072—22no—
91#4072—41no—
92C2×C4⋊Dic5—21no—
93C2×C4⋊Dic5—21no—
94C2×C4⋊Dic5C2311yescharacteristic
95C2×C4⋊Dic5C2311yescharacteristic
96C2×C4⋊Dic5C2311yescharacteristic
97C2×C4⋊Dic5C2311yescharacteristic
98C2×#683—21no—
99C2×#683—21no—
100C2×#683—21no—
101C2×#683—21no—
102C2×C4×Dic5—21no—
103C2×#681—22no—
104C2×#681—22no—
105C2×#681—22no—
106C22×#168C2311yescharacteristic
107C22×#168C2311yescharacteristic
108C22×#168C2311yescharacteristic
109C22×#168C2311yescharacteristic
110#4016—22no—
111#4016—41no—
112C5×#95—21no—
113C5×#95—21no—
114C5×#95D411yescharacteristic
115C5×#95D411yescharacteristic
116#3964—22no—
117#3964—22no—
118#3964—22no—
119#3964—41no—
120#3963—21no—
121#3963—21no—
122#3963—21no—
123C22⋊#168—22no—
124#3940—22no—
125#3940—22no—
126#3940—22no—
127#3940—41no—
128#3939—22no—
129#3939D414yescomplemented
130C4⋊#168—22no—
131C4⋊#168—22no—
132C2×#478—51no—
133C2×#428—51no—
134#2441—51no—
135#2361D511yescharacteristic
136#2094—51no—
137#2088—51no—
138#2067—51no—
139#2059—51no—
140#1948—51no—
141#1918—51no—
142#1895—51no—
143#1875—51no—
144#1864—51no—
145#1853—51no—
146#1851—51no—
147C24×C5—21no—
148C24×C5C2×D411yescharacteristic
149C2×C5×D4—24no—
150C2×C5×D4—21no—
151C2×C5×D4—21no—
152C2×C5×D4—21no—
153C2×C5×D4—21no—
154C2×C5×D4—21no—
155C2×C5×D4—42no—
156C2×C5×D4—42no—
157C2×C5×D4C2×Q812yes—
158C22×C4×C5—21no—
159C22×C4×C5—21no—
160C22×C4×C5—21no—
161C22×C4×C5—21no—
162C22×C4×C5—21no—
163C22×C4×C5—21no—
164C22×C4×C5—21no—
165C22×C4×C5—42no—
166C22×C4×C5—42no—
167C22×C4×C5—42no—
168C22×C4×C5—42no—
169C22×C4×C5—81no—
170C22×C4×C5—81no—
171C22×C4×C5C2×D411yescharacteristic
172C22×C4×C5C2×D411yescharacteristic
173C22×C4×C5C2×D411yescharacteristic
174C22×C4×C5C2×D411yescharacteristic
175C22×C4×C5C2×D411yescharacteristic
176C22×C4×C5C2×Q812yes—
177C22×C4×C5#4112yes—
178C22×C4×C5C2411yescharacteristic
179C22×Dic5—21no—
180C22×Dic5—21no—
181C22×Dic5—21no—
182C22×Dic5—21no—
183C5×C4⋊C4—21no—
184C5×C4⋊C4—21no—
185C5×C4⋊C4—21no—
186C5×C4⋊C4—21no—
187C5×C4⋊C4—42no—
188C5×C4⋊C4—42no—
189C5×C4⋊C4C2×D412yes—
190C5×C4⋊C4C2×Q812yes—
191C5×C22⋊C4—24no—
192C5×C22⋊C4—22no—
193C5×C22⋊C4—41no—
194C5×C22⋊C4—42no—
195C5×C22⋊C4—42no—
196C5×C22⋊C4—81no—
197C42×C5—21no—
198C42×C5—21no—
199C42×C5—21no—
200C42×C5—21no—
201C42×C5—42no—
202C22⋊Dic5—42no—
203C22⋊Dic5—81no—
204C4⋊Dic5—21no—
205C4⋊Dic5—21no—
206C4⋊Dic5—21no—
207C4⋊Dic5—21no—
208C4⋊Dic5—21no—
209C4⋊Dic5—21no—
210C4⋊Dic5—21no—
211C4⋊Dic5—42no—
212C4⋊Dic5—81no—
213C4⋊Dic5C2×D412yes—
214#683—42no—
215#683—42no—
216#683—42no—
217#683—81no—
218C4×Dic5—42no—
219#681—28no—
220C2×#168—21no—
221C2×#168—21no—
222C2×#168—21no—
223C2×#168—21no—
224C2×#168—21no—
225C2×#168—21no—
226C2×#168—21no—
227C2×#168—21no—
228C2×#168—21no—
229C2×#168—21no—
230C2×#168C2×D412yes—
231C2×#168#4112yes—
232C22×M16—51no—
233C2×#125—51no—
234C2×#121C2×D511yescharacteristic
235C2×C4×D4—21no—
236C2×C4×D4C2×D511yescharacteristic
237C2×#117—51no—
238C22×C4⋊C4C2×D511yescharacteristic
239#478—104no—
240#428—104no—
241C2×#107—51no—
242C2×#107—51no—
243C2×#106—51no—
244C2×#106—51no—
245C2×#106—51no—
246C2×#106—51no—
247C2×C4⋊C8—51no—
248C2×C4⋊C8—51no—
249C2×#102—51no—
250C2×#102—51no—
251C2×#102—51no—
252C2×#102—51no—
253C2×C22⋊C8—51no—
254#399—101no—
255#396—101no—
256#390C2×D511yescharacteristic
257#389—51no—
258#389—51no—
259#389—101no—
260#389C2×D511yescharacteristic
261#388—21no—
262#387—101no—
263#386—21no—
264#385—21no—
265#384—51no—
266#380—51no—
267C4×C4⋊C4C2×D511yescharacteristic
268C4×C22⋊C4C2×D511yescharacteristic
269#340—51no—
270#340—51no—
271#340—51no—
272#340—51no—
273#340—51no—
274#340—51no—
275#336—51no—
276#336—51no—
277C23×C5—21no—
278C23×C5—21no—
279C23×C5—21no—
280C23×C5—21no—
281C23×C5—21no—
282C23×C5—21no—
283C23×C5—22no—
284C23×C5—41no—
285C23×C5—41no—
286C23×C5—41no—
287C23×C5—41no—
288C23×C5—41no—
289C23×C5—41no—
290C23×C5C22⋊Q812yes—
291C23×C5C22×D411yescharacteristic
292C5×D4—22no—
293C5×D4—22no—
294C5×D4—48no—
295C5×D4—41no—
296C5×D4—41no—
297C2×C4×C5—21no—
298C2×C4×C5—21no—
299C2×C4×C5—21no—
300C2×C4×C5—21no—
301C2×C4×C5—22no—
302C2×C4×C5—22no—
303C2×C4×C5—22no—
304C2×C4×C5—22no—
305C2×C4×C5—22no—
306C2×C4×C5—21no—
307C2×C4×C5—21no—
308C2×C4×C5—22no—
309C2×C4×C5—21no—
310C2×C4×C5—21no—
311C2×C4×C5—21no—
312C2×C4×C5—21no—
313C2×C4×C5—21no—
314C2×C4×C5—21no—
315C2×C4×C5—21no—
316C2×C4×C5—41no—
317C2×C4×C5—41no—
318C2×C4×C5—41no—
319C2×C4×C5—41no—
320C2×C4×C5—41no—
321C2×C4×C5—41no—
322C2×C4×C5—41no—
323C2×C4×C5—84no—
324C2×C4×C5—84no—
325C2×C4×C5C22⋊Q812yes—
326C2×C4×C5AGL(1,ℤ/8)12yes—
327C2×C4×C5C22×D411yescharacteristic
328C2×C4×C5C22×Q811yescharacteristic
329C2×C4×C5C2×#4111yescharacteristic
330C2×C4×C5C2×#4111yescharacteristic
331C2×C4×C5H(5,𝔽2)11yescharacteristic
332C2×C4×C52−1+411yescharacteristic
333C2×Dic5—22no—
334C2×Dic5—41no—
335C2×Dic5—41no—
336C2×Dic5—41no—
337C2×Dic5—41no—
338C2×Dic5—41no—
339C2×Dic5—41no—
340C2×Dic5—41no—
341C2×Dic5—41no—
342C2×Dic5—41no—
343C2×Dic5—41no—
344C2×Dic5—41no—
345#168—22no—
346#168—22no—
347#168—41no—
348#168—41no—
349C22×D4—21no—
350C22×D4C22×D511yescharacteristic
351C23×C4—21no—
352C23×C4—21no—
353C23×C4C22×D511yescharacteristic
354C2×M16—102no—
355C2×M16—102no—
356C2×M16—102no—
357C22×C8—51no—
358C22×C8—51no—
359C22×C8—51no—
360C22×C8—51no—
361#125—102no—
362#125—201no—
363#121—22no—
364#121Dic1014yes—
365C4×D4—22no—
366C4×D4—41no—
367C4×D4—44no—
368#117—102no—
369#117—201no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—41no—
374C2×C4⋊C4—51no—
375C2×C4⋊C4—51no—
376C2×C4⋊C4—51no—
377C2×C4⋊C4—51no—
378C2×C4⋊C4—101no—
379C2×C4⋊C4—101no—
380C2×C4⋊C4—101no—
381C2×C4⋊C4—101no—
382C2×C4⋊C4—101no—
383C2×C4⋊C4—101no—
384C2×C4⋊C4C22×D511yescharacteristic
385C2×C4⋊C4C22×D511yescharacteristic
386C2×C4⋊C4C22×D511yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—101no—
392C2×C22⋊C4—101no—
393C2×C22⋊C4C22⋊D511yescharacteristic
394C2×C22⋊C4C22⋊D511yescharacteristic
395C2×C42—21no—
396C2×C42—101no—
397C2×C42C22×D511yescharacteristic
398C2×C42C22×D511yescharacteristic
399#107—54no—
400#107—102no—
401#106—102no—
402#106—102no—
403#106—102no—
404#106—201no—
405C4⋊C8—102no—
406C4⋊C8—102no—
407#102—102no—
408#102—102no—
409#102—102no—
410#102—201no—
411C22⋊C8—102no—
412#95—21no—
413#95—21no—
414#95—101no—
415#95—101no—
416#95—101no—
417#95C22⋊D511yescharacteristic
418#95C22⋊D511yescharacteristic
419C22×C5—22no—
420C22×C5—22no—
421C22×C5—22no—
422C22×C5—41no—
423C22×C5—41no—
424C22×C5—41no—
425C22×C5—41no—
426C22×C5—42no—
427C22×C5—42no—
428C22×C5—42no—
429C22×C5—81no—
430C22×C5—81no—
431C22×C5—81no—
432C22×C5—81no—
433C22×C5C2×C22⋊Q811yescharacteristic
434C22×C5D4211yescharacteristic
435C22×C5#54711yescharacteristic
436C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
437C22×C5#57511yescharacteristic
438C22×C5#57711yescharacteristic
439C22×C5#57711yescharacteristic
440C4×C5—22no—
441C4×C5—22no—
442C4×C5—42no—
443C4×C5—41no—
444C4×C5—41no—
445C4×C5—41no—
446C4×C5—42no—
447C4×C5—81no—
448C4×C5#52511yescharacteristic
449C4×C5#53911yescharacteristic
450C4×C5D4×Q811yescharacteristic
451C4×C5#55411yescharacteristic
452Dic5—42no—
453Dic5—81no—
454Dic5—81no—
455Dic5—81no—
456C24—21no—
457C24C2×C22⋊D511yescharacteristic
458C2×D4—24no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4C2×Dic1012yes—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—81no—
479C22×C4—81no—
480C22×C4—101no—
481C22×C4—101no—
482C22×C4—101no—
483C22×C4—101no—
484C22×C4C2×Dic1012yes—
485C22×C4#70912yes—
486C22×C4C2×C22⋊D511yescharacteristic
487C22×C4C2×C22⋊D511yescharacteristic
488C22×C4C2×C22⋊D511yescharacteristic
489C22×C4C2×C22⋊D511yescharacteristic
490C22×C4C2×C22⋊D511yescharacteristic
491C22×C4C23×D511yescharacteristic
492M16—108no—
493C2×C8—52no—
494C2×C8—52no—
495C2×C8—101no—
496C2×C8—101no—
497C2×C8—101no—
498C2×C8—101no—
499C2×C8—101no—
500C2×C8—101no—
501C2×C8—101no—
502C2×C8—101no—
503C2×C8—101no—
504C2×C8—101no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—52no—
512C4⋊C4—101no—
513C4⋊C4—101no—
514C4⋊C4—101no—
515C4⋊C4—101no—
516C4⋊C4—101no—
517C4⋊C4—101no—
518C4⋊C4—101no—
519C4⋊C4—202no—
520C4⋊C4—202no—
521C4⋊C4—202no—
522C4⋊C4—202no—
523C4⋊C4—401no—
524C4⋊C4—401no—
525C4⋊C4C2×Dic1012yes—
526C4⋊C4D4×D512yes—
527C22⋊C4—24no—
528C22⋊C4—22no—
529C22⋊C4—41no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—81no—
533C22⋊C4—202no—
534C22⋊C4—401no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—42no—
540C42—202no—
541C2×C5—42no—
542C2×C5—82no—
543C2×C5#244911yescharacteristic
544C2×C5#307711yescharacteristic
545C2×C5#308411yescharacteristic
546C2×C5#327111yescharacteristic
547C2×C5#327211yescharacteristic
548C2×C5#334511yescharacteristic
549C2×C5#335111yescharacteristic
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—22no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23#407012yes—
564C23C22×C22⋊D511yescharacteristic
565D4—22no—
566D4—22no—
567D4—48no—
568D4—41no—
569D4—41no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—102no—
599C2×C4—201no—
600C2×C4—201no—
601C2×C4—201no—
602C2×C4—201no—
603C2×C4—201no—
604C2×C4—201no—
605C2×C4—201no—
606C2×C4—201no—
607C2×C4—201no—
608C2×C4—201no—
609C2×C4—201no—
610C2×C4#407012yes—
611C2×C4#407812yes—
612C2×C4C22×Dic1011yescharacteristic
613C2×C4C2×#70911yescharacteristic
614C2×C4C2×D4×D511yescharacteristic
615C2×C4#414411yescharacteristic
616C2×C4D5×#4111yescharacteristic
617C2×C4#415011yescharacteristic
618C8—102no—
619C8—102no—
620C8—201no—
621C8—201no—
622C5#2636611yescharacteristic, complemented, Sylow
623C22—22no—
624C22—22no—
625C22—22no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22C2×#407011yescharacteristic
638C22C2×#407811yescharacteristic
639C22D4×C22⋊D511yescharacteristic
640C22#6615611yescharacteristic
641C22#6615811yescharacteristic
642C22#6617011yescharacteristic
643C22#6617111yescharacteristic
644C4—22no—
645C4—22no—
646C4—42no—
647C4—41no—
648C4—41no—
649C4—42no—
650C4—41no—
651C4—81no—
652C4—202no—
653C4—401no—
654C4—401no—
655C4—401no—
656C4D4×Dic1011yescharacteristic
657C4#6587311yescharacteristic
658C4#6588111yescharacteristic
659C4#6589811yescharacteristic
660C2—42no—
661C2—82no—
662C2#1061402711yescharacteristic
663C2#1061409111yescharacteristic
664C2#1061429111yescharacteristic
665C2#1061522011yescharacteristic
666C2#1061525011yescharacteristic
667C2#1061527811yescharacteristic
668C2#1061534311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC5×#2361C2
socleC23×C5C22×D4
prime radicalC2×C5×#121C22
hypercenter#2361D5
hypocenterC5#26366
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2361

Complements

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

NH ≅ G/Ntype
2. C2×#65298C2semidirect
3. C2×#65253C2semidirect
4. #10605225C2semidirect
7. #10602400C2semidirect
8. #10602349C2semidirect
9. #10600449C2semidirect
10. #10600446C2semidirect
11. #10600403C2semidirect
12. #10600399C2semidirect
13. #10600260C2semidirect
14. #10600251C2semidirect
15. #10600243C2semidirect
129. #3939D4semidirect
622. C5#26366semidirect

Maximal subgroups

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

subgroupindex
C5×#23612
C2×#652982
C2×#652532
#106052252
#106024752
#106024712
#106024002
#106023492
#106004492
#106004462
#106004032
#106003992
#106002602
#106002512
#106002432
#263665