Finite Group Databasebeta

← #12121318

Subgroups — #12121318

Order 1280.

850 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2369C211yesmaximal, characteristic, complemented
2C2×#65376C211yesmaximal, characteristic
3#10605501C211yesmaximal, characteristic
4#10602581C211yesmaximal, characteristic, complemented
5#10600419C211yesmaximal, characteristic
6#10600418C211yesmaximal, characteristic
7#10600413C211yesmaximal, characteristic, complemented
8#10600412C211yesmaximal, characteristic, complemented
9#10600271C211yesmaximal, characteristic, complemented
10#10600267C211yesmaximal, characteristic
11#10600267C211yesmaximal, characteristic
12C4×#3941C211yesmaximal, characteristic
13#10600261C211yesmaximal, characteristic, complemented
14#10600254C211yesmaximal, characteristic, complemented
15#10600251C211yesmaximal, characteristic, complemented
16C2×C5×C4⋊Q8C2211yescharacteristic
17C2×C4×D20C2211yescharacteristic
18C2×C4×D20C2211yescharacteristic
19C2×C4×D20C2211yescharacteristic
20C5×#391—21no—
21C5×#391—21no—
22C5×#389—21no—
23C5×#389C2211yescharacteristic
24C5×#389C2211yescharacteristic
25C4×C5×C4⋊C4—21no—
26C4×C5×C4⋊C4C2211yescharacteristic
27C4×C5×C4⋊C4C2211yescharacteristic
28C4×C5×C4⋊C4C2211yescharacteristic
29C4×C5×C4⋊C4C2211yescharacteristic
30#65376C418yescomplemented
31#65278—21no—
32#65278—21no—
33C2×#3941C2211yescharacteristic
34C2×#3941C2211yescharacteristic
35C2×#3941C2211yescharacteristic
36C2×#3941C2211yescharacteristic
37C2×#3940C2211yescharacteristic
38C2×#3939C2211yescharacteristic
39C2×#3939C2211yescharacteristic
40C2×#3939C2211yescharacteristic
41#65225C2211yescharacteristic
42#65217C2211yescharacteristic
43#65217C2211yescharacteristic
44C2×C4⋊#168C2211yescharacteristic
45C2×#3935C2211yescharacteristic
46C2×#3935C2211yescharacteristic
47C2×C4×#168C2211yescharacteristic
48#64956—21no—
49D5×#95—21no—
50#64749C2211yescharacteristic
51#64749C2211yescharacteristic
52#64749C2211yescharacteristic
53#64749C2211yescharacteristic
54#64749C2211yescharacteristic
55#64749C2211yescharacteristic
56#64749C2211yescharacteristic
57#64749C2211yescharacteristic
58#64744C2211yescharacteristic
59#64744C2211yescharacteristic
60#22979—51nomaximal, Sylow
61C22×C5×Q8—21no—
62C22×D20C2311yescharacteristic
63C22×C4×D5—21no—
64C22×C4×D5—21no—
65C22×C4×D5—21no—
66C5×C4⋊Q8—22no—
67C5×C4⋊Q8C2×C414yescomplemented
68C2×C5×C4⋊C4—21no—
69C2×C5×C4⋊C4—21no—
70C2×C5×C4⋊C4—21no—
71C2×C5×C4⋊C4D411yescharacteristic
72C2×C5×C4⋊C4D411yescharacteristic
73C2×C5×C4⋊C4C2311yescharacteristic
74C2×C5×C4⋊C4C2311yescharacteristic
75C2×C5×C4⋊C4C2311yescharacteristic
76C2×C5×C4⋊C4C2311yescharacteristic
77C2×C42×C5—21no—
78C2×C42×C5—21no—
79C2×C42×C5—21no—
80C2×C42×C5C2311yescharacteristic
81C2×C42×C5C2311yescharacteristic
82C2×C42×C5C2311yescharacteristic
83C2×#685—21no—
84C2×#685—21no—
85C2×#685—21no—
86C2×C4⋊Dic5C2311yescharacteristic
87C2×C4⋊Dic5C2311yescharacteristic
88C2×C4⋊Dic5C2311yescharacteristic
89C2×#683—21no—
90C2×#683—21no—
91C22×#168C2311yescharacteristic
92C22×#168C2311yescharacteristic
93C22×#168C2311yescharacteristic
94C22×#168C2311yescharacteristic
95C4×D20—22no—
96C4×D20—22no—
97C4×D20—22no—
98C4×D20—41no—
99C4×D20—41no—
100C4×D20C2×C414yes—
101C5×#95—21no—
102C5×#95—21no—
103C5×#95—21no—
104C5×#95—21no—
105C5×#95—21no—
106C5×#95D411yescharacteristic
107C5×#95D411yescharacteristic
108#3963—21no—
109#3963—21no—
110#3963—21no—
111#3963—21no—
112#3941—22no—
113#3941C2×C414yes—
114#3941C2×C414yes—
115#3941Q814yescomplemented
116#3940—41no—
117#3939—41no—
118#3939C2×C414yescomplemented
119#3939C2×C414yescomplemented
120C4⋊#168C2×C414yescomplemented
121#3935—22no—
122#3935—41no—
123C4×#168—41no—
124C2×#474—51no—
125#2369D511yescharacteristic, complemented
126#2349—51no—
127#1993—51no—
128#1949—51no—
129#1943—51no—
130#1942—51no—
131#1921—51no—
132#1918—51no—
133#1864—51no—
134#1862—51no—
135#1771—51no—
136#1762—51no—
137#1762—51no—
138C4×#102—51no—
139C23×D5—21no—
140C2×C5×Q8—24no—
141C2×C5×Q8—81no—
142C2×C5×Q8—81no—
143C22×C4×C5—21no—
144C22×C4×C5—21no—
145C22×C4×C5—21no—
146C22×C4×C5—21no—
147C22×C4×C5C2×D411yescharacteristic
148C22×C4×C5C2×D411yescharacteristic
149C22×C4×C5C2×D411yescharacteristic
150C22×C4×C5C2×D411yescharacteristic
151C22×C4×C5C2×D411yescharacteristic
152C22×C4×C5C2×D411yescharacteristic
153C22×C4×C5C2411yescharacteristic
154C22×Dic5—21no—
155C22×Dic5—21no—
156C22×Dic5—21no—
157C2×D20—21no—
158C2×D20—21no—
159C2×D20—21no—
160C2×D20C22×C412yes—
161C2×D20C2×Q812yes—
162C2×D20#4112yes—
163C2×C4×D5—24no—
164C2×C4×D5—42no—
165C2×C4×D5—42no—
166C2×C4×D5—42no—
167C2×C4×D5—42no—
168C2×C4×D5—81no—
169C2×C4×D5—81no—
170C2×C4×D5—81no—
171C2×C4×D5—81no—
172C5×C4⋊C4—24no—
173C5×C4⋊C4—22no—
174C5×C4⋊C4—21no—
175C5×C4⋊C4—21no—
176C5×C4⋊C4—21no—
177C5×C4⋊C4—21no—
178C5×C4⋊C4—21no—
179C5×C4⋊C4—41no—
180C5×C4⋊C4—81no—
181C5×C4⋊C4—81no—
182C5×C4⋊C4C22×C412yes—
183C5×C4⋊C4C22×C412yes—
184C5×C4⋊C4C2×Q812yes—
185C42×C5—21no—
186C42×C5—21no—
187C42×C5—21no—
188C42×C5—21no—
189C42×C5—42no—
190C42×C5—81no—
191C42×C5—81no—
192C42×C5C22×C412yes—
193C42×C5#4112yes—
194#685—24no—
195#685—42no—
196#685—42no—
197C4⋊Dic5—21no—
198C4⋊Dic5—21no—
199C4⋊Dic5—21no—
200C4⋊Dic5—21no—
201C4⋊Dic5—21no—
202C4⋊Dic5C22×C412yes—
203#683—81no—
204#683—81no—
205C2×#168—21no—
206C2×#168—21no—
207C2×#168—21no—
208C2×#168—21no—
209C2×#168—21no—
210C2×#168—21no—
211C2×#168—21no—
212C2×#168—21no—
213C2×#168—21no—
214C2×#168C22×C412yes—
215C2×#168C22×C412yes—
216C2×#168C2×Q812yes—
217C2×C4⋊Q8C2×D511yescharacteristic
218C2×C4×D4—51no—
219C2×C4×D4—51no—
220C2×C4×D4—51no—
221#474—58no—
222C2×#107—51no—
223C2×#107—51no—
224C2×#107—51no—
225C2×#106—51no—
226C2×C4⋊C8—51no—
227C2×#102—51no—
228C2×#102—51no—
229C2×#102—51no—
230C2×#102—51no—
231C2×#97—51no—
232C2×#97—51no—
233C2×C4×C8—51no—
234#391—21no—
235#391—21no—
236#389—21no—
237#389—51no—
238#389—51no—
239#389C2×D511yescharacteristic
240#389C2×D511yescharacteristic
241#385—101no—
242#385—101no—
243#382—51no—
244#380—101no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4C2×D511yescharacteristic
247C4×C4⋊C4C2×D511yescharacteristic
248C4×C4⋊C4C2×D511yescharacteristic
249C4×C4⋊C4C2×D511yescharacteristic
250C2×#95—101no—
251#340—51no—
252#340—51no—
253#340—51no—
254#340—51no—
255#340—51no—
256#340—51no—
257#340—51no—
258#340—51no—
259#336—51no—
260#336—51no—
261C23×C5C22×D411yescharacteristic
262C22×D5—22no—
263C22×D5—22no—
264C22×D5—22no—
265C22×D5—41no—
266C22×D5—41no—
267C22×D5—41no—
268C22×D5—41no—
269C5×Q8—84no—
270C2×C4×C5—21no—
271C2×C4×C5—21no—
272C2×C4×C5—21no—
273C2×C4×C5—21no—
274C2×C4×C5—21no—
275C2×C4×C5—21no—
276C2×C4×C5—22no—
277C2×C4×C5—22no—
278C2×C4×C5—21no—
279C2×C4×C5—21no—
280C2×C4×C5—21no—
281C2×C4×C5—22no—
282C2×C4×C5—21no—
283C2×C4×C5—21no—
284C2×C4×C5—21no—
285C2×C4×C5—21no—
286C2×C4×C5—21no—
287C2×C4×C5—21no—
288C2×C4×C5—41no—
289C2×C4×C5—41no—
290C2×C4×C5—41no—
291C2×C4×C5—41no—
292C2×C4×C5—41no—
293C2×C4×C5—41no—
294C2×C4×C5—41no—
295C2×C4×C5—41no—
296C2×C4×C5—41no—
297C2×C4×C5C4×D412yes—
298C2×C4×C5C4×D412yes—
299C2×C4×C5#13712yes—
300C2×C4×C5C23×C411yescharacteristic
301C2×C4×C5C22×Q811yescharacteristic
302C2×C4×C5C2×#4111yescharacteristic
303C2×C4×C5C2×#4111yescharacteristic
304C2×C4×C5H(5,𝔽2)11yescharacteristic
305C2×C4×C52−1+411yescharacteristic
306C2×Dic5—22no—
307C2×Dic5—41no—
308C2×Dic5—41no—
309C2×Dic5—41no—
310C2×Dic5—41no—
311C2×Dic5—41no—
312C2×Dic5—41no—
313C2×Dic5—41no—
314C2×Dic5—41no—
315D20—22no—
316D20—22no—
317D20—22no—
318D20C4×Q814yes—
319C4×D5—84no—
320C4×D5—84no—
321C4×D5—84no—
322#168—22no—
323#168—22no—
324#168—22no—
325#168—41no—
326C22×Q8—21no—
327C22×D4—51no—
328C23×C4—101no—
329C23×C4—101no—
330C23×C4—101no—
331C22×C8—51no—
332C22×C8—51no—
333C22×C8—51no—
334C22×C8—51no—
335C4⋊Q8—22no—
336C4⋊Q8C4×D514yescomplemented
337C4×D4—54no—
338C4×D4—102no—
339C4×D4—102no—
340C4×D4—102no—
341C4×D4—201no—
342C4×D4—201no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—51no—
347C2×C4⋊C4—51no—
348C2×C4⋊C4—51no—
349C2×C4⋊C4—101no—
350C2×C4⋊C4—101no—
351C2×C4⋊C4C22⋊D511yescharacteristic
352C2×C4⋊C4C22⋊D511yescharacteristic
353C2×C4⋊C4C22×D511yescharacteristic
354C2×C4⋊C4C22×D511yescharacteristic
355C2×C4⋊C4C22×D511yescharacteristic
356C2×C4⋊C4C22×D511yescharacteristic
357C2×C22⋊C4—101no—
358C2×C22⋊C4—101no—
359C2×C22⋊C4—101no—
360C2×C42—21no—
361C2×C42—21no—
362C2×C42—21no—
363C2×C42C22×D511yescharacteristic
364C2×C42C22×D511yescharacteristic
365C2×C42C22×D511yescharacteristic
366#107—54no—
367#107—54no—
368#107—201no—
369#106—201no—
370C4⋊C8—54no—
371#102—54no—
372#102—54no—
373#102—54no—
374#102—102no—
375#97—102no—
376#97—201no—
377C4×C8—201no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—21no—
383#95—101no—
384#95—101no—
385#95—101no—
386#95—101no—
387#95C22⋊D511yescharacteristic
388#95C22⋊D511yescharacteristic
389C22×C5C2×C4×D411yescharacteristic
390C22×C5#54611yescharacteristic
391C22×C5D4×Q811yescharacteristic
392C22×C5C2×#13711yescharacteristic
393C22×C5#57511yescharacteristic
394C22×C5#57611yescharacteristic
395C22×C5#57611yescharacteristic
396C2×D5—24no—
397C2×D5—42no—
398C2×D5—42no—
399C2×D5—42no—
400C2×D5—42no—
401C2×D5—42no—
402C2×D5—42no—
403C4×C5—22no—
404C4×C5—22no—
405C4×C5—22no—
406C4×C5—42no—
407C4×C5—42no—
408C4×C5—42no—
409C4×C5—41no—
410C4×C5—41no—
411C4×C5—41no—
412C4×C5—81no—
413C4×C5C2×C4×Q811yescharacteristic
414C4×C5#52011yescharacteristic
415C4×C5#55211yescharacteristic
416C4×C5#55411yescharacteristic
417Dic5—42no—
418Dic5—81no—
419Dic5—81no—
420C24—101no—
421C2×Q8—24no—
422C2×Q8—81no—
423C2×Q8—81no—
424C2×D4—52no—
425C2×D4—52no—
426C2×D4—52no—
427C2×D4—101no—
428C2×D4—101no—
429C2×D4—101no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—101no—
435C22×C4—104no—
436C22×C4—101no—
437C22×C4—101no—
438C22×C4—202no—
439C22×C4—202no—
440C22×C4—202no—
441C22×C4—202no—
442C22×C4—401no—
443C22×C4—401no—
444C22×C4—401no—
445C22×C4—401no—
446C22×C4C2×C22⋊D511yescharacteristic
447C22×C4C2×C22⋊D511yescharacteristic
448C22×C4C2×C22⋊D511yescharacteristic
449C22×C4C2×C22⋊D511yescharacteristic
450C22×C4C2×C22⋊D511yescharacteristic
451C22×C4C2×C22⋊D511yescharacteristic
452C22×C4C23×D511yescharacteristic
453C2×C8—52no—
454C2×C8—52no—
455C2×C8—52no—
456C2×C8—101no—
457C2×C8—101no—
458C2×C8—101no—
459C2×C8—101no—
460C2×C8—101no—
461C2×C8—101no—
462C2×C8—101no—
463C2×C8—101no—
464C2×C8—101no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—41no—
473C4⋊C4—52no—
474C4⋊C4—81no—
475C4⋊C4—81no—
476C4⋊C4—101no—
477C4⋊C4—101no—
478C4⋊C4—101no—
479C4⋊C4—101no—
480C4⋊C4—101no—
481C4⋊C4—401no—
482C4⋊C4—401no—
483C4⋊C4C2×C4×D512yes—
484C4⋊C4C2×C4×D512yes—
485C4⋊C4Q8×D512yes—
486C22⋊C4—104no—
487C22⋊C4—202no—
488C22⋊C4—202no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—42no—
494C42—81no—
495C42—81no—
496C42C2×C4×D512yes—
497C42#70912yes—
498C2×C5#236011yescharacteristic
499C2×C5C4×#13711yescharacteristic
500C2×C5#294811yescharacteristic
501C2×C5#331411yescharacteristic
502C2×C5#331911yescharacteristic
503C2×C5#338511yescharacteristic
504C2×C5#338911yescharacteristic
505D5—44no—
506C23—102no—
507C23—102no—
508C23—102no—
509C23—201no—
510C23—201no—
511C23—201no—
512C23—201no—
513C23C22×C22⋊D511yescharacteristic
514Q8—84no—
515D4—54no—
516D4—102no—
517D4—102no—
518D4—102no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—102no—
547C2×C4—201no—
548C2×C4—201no—
549C2×C4—201no—
550C2×C4—201no—
551C2×C4—201no—
552C2×C4—201no—
553C2×C4—201no—
554C2×C4—201no—
555C2×C4—404no—
556C2×C4—404no—
557C2×C4—404no—
558C2×C4C4×C22⋊D512yes—
559C2×C4C4×C22⋊D512yes—
560C2×C4#408812yes—
561C2×C4C22×C4×D511yescharacteristic
562C2×C4C2×#70911yescharacteristic
563C2×C4C2×Q8×D511yescharacteristic
564C2×C4#414711yescharacteristic
565C2×C4D5×#4111yescharacteristic
566C2×C4#414911yescharacteristic
567C8—102no—
568C8—102no—
569C8—102no—
570C8—201no—
571C5#2297911yescharacteristic, complemented, Sylow
572C22—104no—
573C22—202no—
574C22—202no—
575C22—202no—
576C22—202no—
577C22—202no—
578C22—202no—
579C22C2×C4×C22⋊D511yescharacteristic
580C22C2×#408811yescharacteristic
581C22Q8×C22⋊D511yescharacteristic
582C22#6615611yescharacteristic
583C22#6616311yescharacteristic
584C22#6616911yescharacteristic
585C22#6616911yescharacteristic
586C4—22no—
587C4—22no—
588C4—22no—
589C4—42no—
590C4—41no—
591C4—41no—
592C4—41no—
593C4—42no—
594C4—42no—
595C4—81no—
596C4—202no—
597C4—401no—
598C4—401no—
599C4C4×Q8×D511yescharacteristic
600C4#6590811yescharacteristic
601C4#6591311yescharacteristic
602C4#6591511yescharacteristic
603C2—204no—
604C2C4×#408811yescharacteristic
605C2#1061422111yescharacteristic
606C2#1061426911yescharacteristic
607C2#1061536411yescharacteristic
608C2#1061536611yescharacteristic
609C2#1061538411yescharacteristic
610C2#1061543711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#2369C2
socleC23×C5C22×D4
prime radicalC22×D20C23
cosocleC22×C4×C5C24
hypercenter#2369D5
hypocenterC5#22979
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2369

Complements

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

NH ≅ G/Ntype
1. C5×#2369C2semidirect
4. #10602581C2semidirect
7. #10600413C2semidirect
8. #10600412C2semidirect
9. #10600271C2semidirect
13. #10600261C2semidirect
14. #10600254C2semidirect
15. #10600251C2semidirect
30. #65376C4semidirect
67. C5×C4⋊Q8C2×C4semidirect
115. #3941Q8semidirect
118. #3939C2×C4semidirect
119. #3939C2×C4semidirect
120. C4⋊#168C2×C4semidirect
125. #2369D5semidirect
336. C4⋊Q8C4×D5semidirect
571. C5#22979semidirect

Maximal subgroups

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

subgroupindex
C5×#23692
C2×#653762
#106055012
#106025812
#106004192
#106004182
#106004132
#106004122
#106002712
#106002672
C4×#39412
#106002612
#106002542
#106002512
#229795