Finite Group Databasebeta

← #12101706

Subgroups — #12101706

Order 1280.

1218 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C5×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#65255C211yesmaximal, characteristic
3C2×#65255C211yesmaximal, characteristic
4C2×#65253C211yesmaximal, characteristic, complemented
5#10613801C211yesmaximal, characteristic
6#10602351C211yesmaximal, characteristic, complemented
7#10602320C211yesmaximal, characteristic, complemented
8#10600258C212yesmaximal, complemented
9#10600257C212yesmaximal, complemented
10#10600256C212yesmaximal, complemented
11#10600255C212yesmaximal, complemented
12C22×#709C2211yescharacteristic
13C22×C5×C4⋊C4C2211yescharacteristic
14C22×C5×C4⋊C4C2211yescharacteristic
15C22×C42×C5—21no—
16C22×C42×C5C2211yescharacteristic
17C2×#4072C2211yescharacteristic
18C22×#681C2211yescharacteristic
19C22×#681C2211yescharacteristic
20C4×C5×C4⋊C4—21no—
21C4×C5×C4⋊C4—21no—
22C4×C5×C4⋊C4C2212yes—
23C4×C5×C4⋊C4C2212yes—
24C2×C5×#95—21no—
25#65502—21no—
26#65502—21no—
27C2×#3942C2212yes—
28C2×#3942C2212yes—
29#65255—24no—
30#65255C418yescomplemented
31C2×#3941C2212yes—
32C2×#3941C2212yes—
33#65253—24no—
34C2×#3940C2212yes—
35C2×#3939C2212yes—
36#65228C2211yescharacteristic
37#65228C2211yescharacteristic
38C4×#685—21no—
39C4×#685—21no—
40#65218C2211yescharacteristic
41#65218C2211yescharacteristic
42#64749C2212yes—
43#64749C2212yes—
44#64744C2212yes—
45#64744C2212yes—
46#21059—51nomaximal, Sylow
47C23×C4×C5—21no—
48C23×C4×C5—21no—
49C23×C4×C5D411yescharacteristic
50C23×C4×C5D411yescharacteristic
51C23×C4×C5C2311yescharacteristic
52C22×C22⋊D5—21no—
53C2×#709—22no—
54C2×#709—22no—
55C2×#709—22no—
56C2×#709—22no—
57C2×#709—22no—
58C2×#709C2×C414yes—
59C22×D20C2311yescharacteristic
60C22×C4×D5—21no—
61C22×Dic10C2311yescharacteristic
62C2×C5×C4⋊C4—21no—
63C2×C5×C4⋊C4—21no—
64C2×C5×C4⋊C4—22no—
65C2×C5×C4⋊C4—22no—
66C2×C5×C4⋊C4C2×C414yes—
67C2×C5×C4⋊C4D414yes—
68C2×C5×C4⋊C4C2312yes—
69C2×C5×C4⋊C4C2312yes—
70C2×C42×C5—22no—
71C2×C42×C5—22no—
72C2×C42×C5—22no—
73C2×C42×C5—22no—
74C2×C42×C5—44no—
75C2×C42×C5D411yescharacteristic
76C2×C42×C5D411yescharacteristic
77C2×C42×C5C2311yescharacteristic
78C2×C42×C5C2311yescharacteristic
79C2×C22⋊Dic5—21no—
80C2×#685—21no—
81C2×#685—21no—
82C2×#685—21no—
83C2×#685—21no—
84#4072—22no—
85#4072—22no—
86C2×C4⋊Dic5C2311yescharacteristic
87C2×C4⋊Dic5C2311yescharacteristic
88C2×C4×Dic5—21no—
89C2×#681—22no—
90C2×#681—22no—
91C2×#681—22no—
92C2×#681—22no—
93C2×#681—22no—
94C2×#681C2×C414yes—
95C22×#168C2312yes—
96C22×#168C2312yes—
97C5×#95—22no—
98C5×#95—22no—
99#3963—21no—
100#3963—21no—
101#3963—21no—
102#3963—21no—
103#3942—24no—
104#3942C2×C418yescomplemented
105#3941—24no—
106#3941C2×C418yescomplemented
107#3940—24no—
108#3939—24no—
109C2×#428—51no—
110C2×#417—51no—
111C2×#417—51no—
112#2297—51no—
113C2×C4×C4⋊C4D511yescharacteristic, complemented
114#1928—52no—
115#1927—52no—
116#1926—52no—
117#1925—52no—
118#1736—51no—
119#1728—51no—
120C24×C5C2×D411yescharacteristic
121C23×D5—21no—
122C22×C4×C5—22no—
123C22×C4×C5—22no—
124C22×C4×C5—22no—
125C22×C4×C5—24no—
126C22×C4×C5—24no—
127C22×C4×C5—22no—
128C22×C4×C5—22no—
129C22×C4×C5—22no—
130C22×C4×C5—41no—
131C22×C4×C5—41no—
132C22×C4×C5—42no—
133C22×C4×C5—42no—
134C22×C4×C5—42no—
135C22×C4×C5—42no—
136C22×C4×C5C22×C412yes—
137C22×C4×C5C2×D412yes—
138C22×C4×C5C2×D411yescharacteristic
139C22×C4×C5C2×D411yescharacteristic
140C22×C4×C5C2×D411yescharacteristic
141C22×C4×C5C2×D411yescharacteristic
142C22×C4×C5C2×D411yescharacteristic
143C22×C4×C5#4112yes—
144C22×C4×C5#4112yes—
145C22×C4×C5#4112yes—
146C22×C4×C5#4112yes—
147C22×C4×C5C2411yescharacteristic
148C2×C22⋊D5—24no—
149C2×C22⋊D5—42no—
150C2×C22⋊D5—42no—
151C22×Dic5—21no—
152C22×Dic5—21no—
153C22×Dic5—21no—
154#709—28no—
155#709—28no—
156#709—44no—
157#709—44no—
158C2×D20—21no—
159C2×D20—21no—
160C2×D20—21no—
161C2×D20—21no—
162C2×D20—21no—
163C2×D20C22×C412yes—
164C2×C4×D5—24no—
165C2×C4×D5—42no—
166C2×C4×D5—42no—
167C2×Dic10—21no—
168C2×Dic10—21no—
169C2×Dic10—21no—
170C2×Dic10—21no—
171C2×Dic10—21no—
172C2×Dic10C22×C412yes—
173C5×C4⋊C4—24no—
174C5×C4⋊C4—24no—
175C5×C4⋊C4—22no—
176C5×C4⋊C4—22no—
177C5×C4⋊C4C22×C414yes—
178C5×C4⋊C4C2×D414yes—
179C42×C5—22no—
180C42×C5—22no—
181C42×C5—21no—
182C42×C5—21no—
183C42×C5—21no—
184C42×C5—21no—
185C42×C5—44no—
186C42×C5—44no—
187C22⋊Dic5—42no—
188#685—42no—
189#685—42no—
190#685—42no—
191#685—42no—
192C4⋊Dic5—21no—
193C4⋊Dic5—21no—
194C4⋊Dic5—21no—
195C4⋊Dic5—21no—
196C4×Dic5—42no—
197#681—28no—
198#681—44no—
199C2×#168—22no—
200C2×#168—22no—
201C2×#168—22no—
202C2×#168—22no—
203C2×#168—22no—
204C2×#168C22×C414yes—
205C22×#41—51no—
206C22×M16—51no—
207C22×M16—51no—
208C2×#117—51no—
209C22×C4⋊C4C2×D511yescharacteristic
210C22×C4⋊C4C2×D511yescharacteristic
211C22×C42—21no—
212C22×C42C2×D511yescharacteristic
213#428—104no—
214C2×#107—52no—
215C2×#106—52no—
216#417—58no—
217#417—104no—
218C2×#103—52no—
219C2×#103—52no—
220C2×#102—52no—
221C2×#102—52no—
222#391—51no—
223#391—51no—
224#390—51no—
225#390—51no—
226#386—101no—
227#386—101no—
228C4×C4⋊C4—21no—
229C4×C4⋊C4—21no—
230C4×C4⋊C4C2×D512yes—
231C4×C4⋊C4C2×D512yes—
232C4×C22⋊C4—101no—
233C4×C22⋊C4—101no—
234C2×#95—21no—
235#340—52no—
236#340—52no—
237#336—52no—
238#336—52no—
239C23×C5—21no—
240C23×C5—21no—
241C23×C5—21no—
242C23×C5—21no—
243C23×C5C4×D412yes—
244C23×C5#12112yes—
245C23×C5#12312yes—
246C23×C5C22×D411yescharacteristic
247C22×D5—22no—
248C22×D5—41no—
249C22×D5—41no—
250C22×D5—41no—
251C22×D5—41no—
252C22×D5—41no—
253C22×D5—41no—
254C2×C4×C5—28no—
255C2×C4×C5—21no—
256C2×C4×C5—21no—
257C2×C4×C5—21no—
258C2×C4×C5—21no—
259C2×C4×C5—24no—
260C2×C4×C5—24no—
261C2×C4×C5—21no—
262C2×C4×C5—21no—
263C2×C4×C5—21no—
264C2×C4×C5—21no—
265C2×C4×C5—21no—
266C2×C4×C5—21no—
267C2×C4×C5—44no—
268C2×C4×C5—48no—
269C2×C4×C5—48no—
270C2×C4×C5—42no—
271C2×C4×C5—42no—
272C2×C4×C5—42no—
273C2×C4×C5—42no—
274C2×C4×C5#11714yes—
275C2×C4×C5C4×D414yes—
276C2×C4×C5C4×D412yes—
277C2×C4×C5#12112yes—
278C2×C4×C5#12312yes—
279C2×C4×C5#12414yes—
280C2×C4×C5#12614yes—
281C2×C4×C5AGL(1,ℤ/8)12yes—
282C2×C4×C5#13712yes—
283C2×C4×C5C23×C411yescharacteristic
284C2×C4×C5C22×D411yescharacteristic
285C2×C4×C5C2×#4111yescharacteristic
286C2×C4×C5C2×#4111yescharacteristic
287C2×C4×C5C2×#4111yescharacteristic
288C2×C4×C5C2×#4111yescharacteristic
289C22⋊D5—48no—
290C2×Dic5—22no—
291C2×Dic5—41no—
292C2×Dic5—41no—
293C2×Dic5—41no—
294C2×Dic5—41no—
295C2×Dic5—41no—
296C2×Dic5—41no—
297C2×Dic5—41no—
298C2×Dic5—41no—
299D20—22no—
300D20—22no—
301D20—41no—
302D20—41no—
303C4×D5—48no—
304Dic10—22no—
305Dic10—22no—
306Dic10—41no—
307Dic10—41no—
308#168—24no—
309#168—42no—
310C2×#41—54no—
311C2×#41—102no—
312C2×#41—102no—
313C2×#41—102no—
314C2×#41—102no—
315C2×#41—102no—
316C22×Q8—51no—
317C22×D4—51no—
318C22×D4—101no—
319C23×C4—21no—
320C23×C4—21no—
321C23×C4—101no—
322C23×C4C22⋊D511yescharacteristic
323C23×C4C22⋊D511yescharacteristic
324C23×C4C22×D511yescharacteristic
325C2×M16—54no—
326C2×M16—102no—
327C2×M16—102no—
328C2×M16—102no—
329C2×M16—102no—
330C2×M16—102no—
331C22×C8—52no—
332C22×C8—52no—
333#117—102no—
334#117—102no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—51no—
340C2×C4⋊C4—51no—
341C2×C4⋊C4C4×D514yes—
342C2×C4⋊C4D2014yes—
343C2×C4⋊C4C22×D512yes—
344C2×C4⋊C4C22×D512yes—
345C2×C22⋊C4—101no—
346C2×C22⋊C4—101no—
347C2×C22⋊C4—101no—
348C2×C22⋊C4—101no—
349C2×C22⋊C4—101no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—44no—
355C2×C42—101no—
356C2×C42C22⋊D511yescharacteristic
357C2×C42C22⋊D511yescharacteristic
358C2×C42C22×D511yescharacteristic
359C2×C42C22×D511yescharacteristic
360#107—104no—
361#106—104no—
362#103—58no—
363#103—104no—
364#102—58no—
365#102—104no—
366#95—22no—
367#95—22no—
368#95—101no—
369#95—101no—
370#95—101no—
371#95—101no—
372C22×C5—22no—
373C22×C5—22no—
374C22×C5—22no—
375C22×C5—22no—
376C22×C5—22no—
377C22×C5—22no—
378C22×C5#38814yes—
379C22×C5C2×C4×D411yescharacteristic
380C22×C5C2×#12111yescharacteristic
381C22×C5C2×#12311yescharacteristic
382C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
383C22×C5C2×#13711yescharacteristic
384C22×C5#57511yescharacteristic
385C22×C5#57511yescharacteristic
386C2×D5—42no—
387C2×D5—42no—
388C2×D5—42no—
389C2×D5—81no—
390C2×D5—81no—
391C2×D5—81no—
392C2×D5—81no—
393C4×C5—22no—
394C4×C5—22no—
395C4×C5—41no—
396C4×C5—41no—
397C4×C5—44no—
398C4×C5—44no—
399C4×C5#38814yes—
400C4×C5C2×#11711yescharacteristic
401C4×C5C2×C4×D411yescharacteristic
402C4×C5C2×#12411yescharacteristic
403C4×C5C2×#12611yescharacteristic
404Dic5—42no—
405Dic5—81no—
406Dic5—81no—
407C24—101no—
408C24C2×C22⋊D511yescharacteristic
409#41—108no—
410#41—108no—
411#41—204no—
412#41—204no—
413C2×Q8—52no—
414C2×Q8—101no—
415C2×Q8—101no—
416C2×Q8—101no—
417C2×Q8—101no—
418C2×Q8—101no—
419C2×D4—52no—
420C2×D4—104no—
421C2×D4—101no—
422C2×D4—101no—
423C2×D4—101no—
424C2×D4—101no—
425C2×D4—101no—
426C2×D4—202no—
427C2×D4—202no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—101no—
443C22×C4—104no—
444C22×C4—101no—
445C22×C4—101no—
446C22×C4—202no—
447C22×C4—202no—
448C22×C4C2×C4×D512yes—
449C22×C4C2×D2012yes—
450C22×C4#70912yes—
451C22×C4#70912yes—
452C22×C4#70912yes—
453C22×C4#70912yes—
454C22×C4C2×C22⋊D511yescharacteristic
455C22×C4C2×C22⋊D511yescharacteristic
456C22×C4C2×C22⋊D511yescharacteristic
457C22×C4C2×C22⋊D511yescharacteristic
458C22×C4C2×C22⋊D511yescharacteristic
459C22×C4C23×D511yescharacteristic
460M16—108no—
461M16—204no—
462C2×C8—54no—
463C2×C8—102no—
464C2×C8—102no—
465C2×C8—102no—
466C2×C8—102no—
467C2×C8—102no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—101no—
473C4⋊C4—101no—
474C4⋊C4—101no—
475C4⋊C4—101no—
476C4⋊C4C2×C4×D514yes—
477C4⋊C4C2×D2014yes—
478C22⋊C4—202no—
479C22⋊C4—202no—
480C22⋊C4—202no—
481C22⋊C4—202no—
482C22⋊C4—202no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—22no—
488C42—22no—
489C42—44no—
490C42—44no—
491C42—202no—
492C2×C5—24no—
493C2×C5C2×#38811yescharacteristic
494C2×C5C4×AGL(1,ℤ/8)11yescharacteristic
495C2×C5C4×#13711yescharacteristic
496C2×C5#303611yescharacteristic
497C2×C5#303711yescharacteristic
498C2×C5#309211yescharacteristic
499C2×C5#309311yescharacteristic
500D5—82no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—102no—
506C23—201no—
507C23—201no—
508C23—201no—
509C23—201no—
510C23—201no—
511C23—201no—
512C23C4×C22⋊D512yes—
513C23#407512yes—
514C23#407612yes—
515C23C22×C22⋊D511yescharacteristic
516Q8—102no—
517Q8—102no—
518Q8—201no—
519Q8—201no—
520D4—102no—
521D4—102no—
522D4—208no—
523D4—201no—
524D4—201no—
525C2×C4—28no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—24no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—24no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—48no—
539C2×C4—48no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—102no—
546C2×C4—201no—
547C2×C4—201no—
548C2×C4—208no—
549C2×C4—201no—
550C2×C4—201no—
551C2×C4—201no—
552C2×C4—201no—
553C2×C4—201no—
554C2×C4—201no—
555C2×C4#401814yes—
556C2×C4C4×D2014yes—
557C2×C4#402114yes—
558C2×C4#402214yes—
559C2×C4C4×C22⋊D512yes—
560C2×C4#407512yes—
561C2×C4#407612yes—
562C2×C4#407812yes—
563C2×C4#408812yes—
564C2×C4C22×C4×D511yescharacteristic
565C2×C4C22×D2011yescharacteristic
566C2×C4C2×#70911yescharacteristic
567C2×C4C2×#70911yescharacteristic
568C2×C4C2×#70911yescharacteristic
569C2×C4C2×#70911yescharacteristic
570C8—104no—
571C8—202no—
572C5#2105911yescharacteristic, complemented, Sylow
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—202no—
580C22—202no—
581C22—202no—
582C22—401no—
583C22—401no—
584C22—401no—
585C22—401no—
586C22#6522914yes—
587C22C2×C4×C22⋊D511yescharacteristic
588C22C2×#407511yescharacteristic
589C22C2×#407611yescharacteristic
590C22C2×#407811yescharacteristic
591C22C2×#408811yescharacteristic
592C22#6615611yescharacteristic
593C22#6615611yescharacteristic
594C4—22no—
595C4—22no—
596C4—44no—
597C4—44no—
598C4—41no—
599C4—41no—
600C4—202no—
601C4—401no—
602C4—401no—
603C4#6522914yes—
604C4C2×#401811yescharacteristic
605C4C2×C4×D2011yescharacteristic
606C4C2×#402111yescharacteristic
607C4C2×#402211yescharacteristic
608C2—24no—
609C2—402no—
610C2C2×#6522911yescharacteristic
611C2#1061387011yescharacteristic
612C2#1061394711yescharacteristic
613C2C4×#407811yescharacteristic
614C2#1061403811yescharacteristic
615C2C4×#408811yescharacteristic
616C2#1061417211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC2×C4×C5×C4⋊C4C2
socleC23×C5C22×D4
prime radicalC22×#709C22
cosocleC22×C4×C5C24
hypercenterC2×C4×C4⋊C4D5
hypocenterC5#21059
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4×C5×C4⋊C4C2semidirect
4. C2×#65253C2semidirect
6. #10602351C2semidirect
7. #10602320C2semidirect
8. #10600258C2semidirect
9. #10600257C2semidirect
10. #10600256C2semidirect
11. #10600255C2semidirect
30. #65255C4semidirect
104. #3942C2×C4semidirect
106. #3941C2×C4semidirect
113. C2×C4×C4⋊C4D5semidirect
572. C5#21059semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C5×C4⋊C42
C2×#652552
C2×#652532
#106138012
#106023512
#106023202
#106002582
#106002572
#106002562
#106002552
#210595