Finite Group Databasebeta

← #13041313

Subgroups — #13041313

Order 1280.

840 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65788C211yesmaximal, characteristic, complemented
2C2×#65758C211yesmaximal, characteristic
3Dic5×C22⋊Q8C211yesmaximal, characteristic
4#10604985C211yesmaximal, characteristic, complemented
5#10604897C211yesmaximal, characteristic
6#10604327C211yesmaximal, characteristic, complemented
7#10604327C211yesmaximal, characteristic, complemented
8#10604326C211yesmaximal, characteristic, complemented
9#10604323C211yesmaximal, characteristic, complemented
10#10604321C211yesmaximal, characteristic, complemented
11#10604316C211yesmaximal, characteristic, complemented
12#10604232C211yesmaximal, characteristic
13#10604232C211yesmaximal, characteristic
14#10604227C211yesmaximal, characteristic
15#10604227C211yesmaximal, characteristic
16C22×#704C2211yescharacteristic
17C22×#704C2211yescharacteristic
18C2×C5×C22⋊Q8C411yescharacteristic
19C2×Q8×Dic5C2211yescharacteristic
20C22×C4×Dic5C2211yescharacteristic
21C2×C22⋊#170C2211yescharacteristic
22C2×C22⋊#170C2211yescharacteristic
23#65788—24no—
24#65758—24no—
25C2×#4004C2211yescharacteristic
26C2×#4004C2211yescharacteristic
27C2×#4004C2211yescharacteristic
28C2×#4004C2211yescharacteristic
29C2×#4002C2211yescharacteristic
30C2×#4002C2211yescharacteristic
31C2×#4002C2211yescharacteristic
32C2×#4002C2211yescharacteristic
33C2×C4⋊#170C2211yescharacteristic
34C2×C4⋊#170C2211yescharacteristic
35C2×C4⋊#170C2211yescharacteristic
36C2×C4⋊#170C2211yescharacteristic
37C2×C4×#170C2211yescharacteristic
38C2×C4×#170C2211yescharacteristic
39#65519C411yescharacteristic
40#65501C411yescharacteristic
41#65274C411yescharacteristic
42#65270C411yescharacteristic
43#65270C411yescharacteristic
44C4⋊C4×Dic5C2211yescharacteristic
45C4⋊C4×Dic5C2211yescharacteristic
46C4⋊C4×Dic5C2211yescharacteristic
47#65239C411yescharacteristic
48#65239C411yescharacteristic
49C22⋊C4×Dic5C2211yescharacteristic
50C22⋊C4×Dic5C2211yescharacteristic
51#64918C2211yescharacteristic
52#64918C2211yescharacteristic
53#64918C2211yescharacteristic
54#64918C2211yescharacteristic
55#64918C2211yescharacteristic
56#64918C2211yescharacteristic
57#64918C2211yescharacteristic
58#64918C2211yescharacteristic
59#64918C2211yescharacteristic
60#64918C2211yescharacteristic
61#22955—51nomaximal, Sylow
62C22×C5×Q8C2×C411yescharacteristic
63C23×C4×C5C2×C411yescharacteristic
64C23×Dic5C2×C411yescharacteristic
65C23×Dic5C2311yescharacteristic
66C2×#704—22no—
67C2×#704—22no—
68C2×#704—22no—
69C2×#704—22no—
70C2×#704—41no—
71C2×#704—41no—
72C22×#170C2311yescharacteristic
73C22×#170C2311yescharacteristic
74C22×#170C2311yescharacteristic
75C22×#170C2311yescharacteristic
76C22×#170C2311yescharacteristic
77C22×#170C2311yescharacteristic
78C22×#170C2311yescharacteristic
79C22×#170C2311yescharacteristic
80C5×C22⋊Q8—24no—
81C2×C5×C4⋊C4C2×C411yescharacteristic
82C2×C5×C4⋊C4C2×C411yescharacteristic
83C2×C5×C4⋊C4C2×C411yescharacteristic
84C2×C5×C22⋊C4C2×C411yescharacteristic
85C2×C5×C22⋊C4C2×C411yescharacteristic
86C2×C22⋊Dic5C2×C411yescharacteristic
87C2×C22⋊Dic5C2×C411yescharacteristic
88Q8×Dic5—22no—
89Q8×Dic5—22no—
90C2×C4⋊Dic5C2×C411yescharacteristic
91C2×C4⋊Dic5C2×C411yescharacteristic
92C2×C4⋊Dic5C2×C411yescharacteristic
93C2×C4⋊Dic5C2×C411yescharacteristic
94C2×C4⋊Dic5C2×C411yescharacteristic
95C2×C4⋊Dic5C2×C411yescharacteristic
96C2×C4×Dic5—22no—
97C2×C4×Dic5—22no—
98C2×C4×Dic5C2×C411yescharacteristic
99C2×C4×Dic5C2×C411yescharacteristic
100C2×C4×Dic5C2311yescharacteristic
101C2×C4×Dic5C2311yescharacteristic
102C2×C4×Dic5C2311yescharacteristic
103C2×C4×Dic5C2311yescharacteristic
104C2×C4×Dic5C2311yescharacteristic
105C2×C4×Dic5C2311yescharacteristic
106C22⋊#170—22no—
107C22⋊#170Q814yescomplemented
108#4004—22no—
109#4004—22no—
110#4004—41no—
111#4004D414yes—
112#4002—22no—
113#4002—22no—
114#4002—22no—
115#4002—41no—
116C4⋊#170—22no—
117C4⋊#170—22no—
118C4⋊#170—22no—
119C4⋊#170—41no—
120C4×#170—22no—
121C4×#170—41no—
122#3963C2×C411yescharacteristic
123#3963C2×C411yescharacteristic
124#3963C2×C411yescharacteristic
125#3963C2×C411yescharacteristic
126#3963C2×C411yescharacteristic
127#3963C2×C411yescharacteristic
128#3963C2×C411yescharacteristic
129#3963C2×C411yescharacteristic
130#3963C2×C411yescharacteristic
131#3963C2×C411yescharacteristic
132C2×#446—51no—
133C2×#423—51no—
134C4×C22⋊Q8—51no—
135#1985—51no—
136#1940—51no—
137#1940—51no—
138#1939—51no—
139#1917—51no—
140#1916—51no—
141#1851—51no—
142#1851—51no—
143#1771—51no—
144#1753—51no—
145#1753—51no—
146#1749—51no—
147C24×C5C22×C411yescharacteristic
148C2×C5×Q8—22no—
149C2×C5×Q8—22no—
150C2×C5×Q8—41no—
151C22×C4×C5—22no—
152C22×C4×C5—22no—
153C22×C4×C5—41no—
154C22×C4×C5C22×C411yescharacteristic
155C22×C4×C5C22×C411yescharacteristic
156C22×C4×C5C22×C411yescharacteristic
157C22×C4×C5C22×C411yescharacteristic
158C22×C4×C5C22×C411yescharacteristic
159C22×C4×C5C22×C411yescharacteristic
160C22×Dic5—21no—
161C22×Dic5—21no—
162C22×Dic5—21no—
163C22×Dic5—21no—
164C22×Dic5—22no—
165C22×Dic5—41no—
166C22×Dic5—41no—
167C22×Dic5C22×C411yescharacteristic
168C22×Dic5C22×C411yescharacteristic
169C22×Dic5C22×C411yescharacteristic
170C22×Dic5C22×C411yescharacteristic
171C22×Dic5C22×C411yescharacteristic
172C22×Dic5C22×C411yescharacteristic
173C22×Dic5C22×C411yescharacteristic
174C22×Dic5C2×Q812yes—
175C22×Dic5#4112yes—
176C22×Dic5C2411yescharacteristic
177#704—44no—
178#704—44no—
179C2×#170—21no—
180C2×#170—21no—
181C2×#170—21no—
182C2×#170—21no—
183C2×#170—21no—
184C2×#170—21no—
185C2×#170—21no—
186C2×#170—21no—
187C2×#170—21no—
188C2×#170—21no—
189C2×#170—21no—
190C2×#170—21no—
191C2×#170—21no—
192C2×#170—21no—
193C2×#170—21no—
194C2×#170—21no—
195C2×#170—21no—
196C2×#170—21no—
197C2×#170—21no—
198C2×#170—21no—
199C2×#170C2×D412yes—
200C2×#170C2×D412yes—
201C2×#170C2×Q812yes—
202C2×#170C2×Q812yes—
203C5×C4⋊C4—22no—
204C5×C4⋊C4—22no—
205C5×C4⋊C4M1614yescomplemented
206C5×C22⋊C4—22no—
207C5×C22⋊C4—22no—
208C22⋊Dic5—41no—
209C22⋊Dic5—41no—
210C4⋊Dic5—22no—
211C4⋊Dic5—22no—
212C4⋊Dic5—22no—
213C4⋊Dic5—41no—
214C4⋊Dic5—41no—
215C4⋊Dic5—41no—
216C4×Dic5—22no—
217C4×Dic5—22no—
218C4×Dic5—21no—
219C4×Dic5—21no—
220C4×Dic5—21no—
221C4×Dic5—21no—
222C4×Dic5—21no—
223C4×Dic5—21no—
224C4×Dic5—21no—
225C4×Dic5—21no—
226C4×Dic5—21no—
227C4×Dic5—21no—
228C4×Dic5C2×D412yes—
229C4×Dic5#4112yes—
230C22×M16—51no—
231C22×M16—51no—
232C2×C22⋊Q8AGL(1,𝔽5)11yescharacteristic
233C2×C4×Q8—51no—
234C22×C42—51no—
235#446—104no—
236#423—104no—
237C2×C4⋊C8—51no—
238C2×C4⋊C8—51no—
239C2×C4⋊C8—51no—
240C2×C4⋊C8—51no—
241C2×C4⋊C8—51no—
242C2×C4⋊C8—51no—
243C2×C4⋊C8—51no—
244C2×C4⋊C8—51no—
245C2×C22⋊C8—51no—
246C2×C22⋊C8—51no—
247C2×#97—51no—
248C2×#97—51no—
249C2×#97—51no—
250C2×#97—51no—
251C2×C4×C8—51no—
252C2×C4×C8—51no—
253#391—51no—
254#389—51no—
255#387—51no—
256#387—51no—
257#385—51no—
258#385—51no—
259#380—51no—
260C4×C4⋊C4—51no—
261C4×C4⋊C4—51no—
262C4×C4⋊C4—51no—
263C4×C22⋊C4—51no—
264C4×C22⋊C4—51no—
265#336—51no—
266#336—51no—
267#336—51no—
268#336—51no—
269#336—51no—
270#336—51no—
271#336—51no—
272#336—51no—
273#336—51no—
274#336—51no—
275C23×C5—21no—
276C23×C5—21no—
277C23×C5—21no—
278C23×C5—21no—
279C23×C5—21no—
280C23×C5—21no—
281C23×C5C4×Q812yes—
282C23×C5C23×C411yescharacteristic
283C5×Q8—44no—
284C2×C4×C5—21no—
285C2×C4×C5—21no—
286C2×C4×C5—21no—
287C2×C4×C5—21no—
288C2×C4×C5—21no—
289C2×C4×C5—21no—
290C2×C4×C5—21no—
291C2×C4×C5—21no—
292C2×C4×C5—21no—
293C2×C4×C5—21no—
294C2×C4×C5—21no—
295C2×C4×C5—21no—
296C2×C4×C5—21no—
297C2×C4×C5—21no—
298C2×C4×C5—44no—
299C2×C4×C5C4×D412yes—
300C2×C4×C5C2×M1612yes—
301C2×C4×C5C2×M1612yes—
302C2×C4×C5C2×M1612yes—
303C2×Dic5—21no—
304C2×Dic5—21no—
305C2×Dic5—21no—
306C2×Dic5—21no—
307C2×Dic5—22no—
308C2×Dic5—22no—
309C2×Dic5—22no—
310C2×Dic5—22no—
311C2×Dic5—21no—
312C2×Dic5—21no—
313C2×Dic5—21no—
314C2×Dic5—21no—
315C2×Dic5—21no—
316C2×Dic5—21no—
317C2×Dic5—21no—
318C2×Dic5—21no—
319C2×Dic5—21no—
320C2×Dic5—21no—
321C2×Dic5—21no—
322C2×Dic5—21no—
323C2×Dic5—21no—
324C2×Dic5—21no—
325C2×Dic5—21no—
326C2×Dic5—44no—
327C2×Dic5C4×D412yes—
328C2×Dic5C4×Q812yes—
329C2×Dic5C22×D411yescharacteristic
330C2×Dic5C22×Q811yescharacteristic
331C2×Dic5C2×#4111yescharacteristic
332C2×Dic5C2×#4111yescharacteristic
333C2×Dic5H(5,𝔽2)11yescharacteristic
334C2×Dic52−1+411yescharacteristic
335#170—22no—
336#170—22no—
337#170—22no—
338#170—22no—
339#170—41no—
340#170—41no—
341#170—41no—
342#170—41no—
343C22×Q8C2×AGL(1,𝔽5)11yescharacteristic
344C23×C4—51no—
345C23×C4—51no—
346C23×C4C2×AGL(1,𝔽5)11yescharacteristic
347C2×M16—102no—
348C2×M16—102no—
349C2×M16—102no—
350C2×M16—102no—
351C2×M16—201no—
352C2×M16—201no—
353C22×C8—51no—
354C22×C8—51no—
355C22×C8—51no—
356C22×C8—51no—
357C22×C8—51no—
358C22×C8—51no—
359C22×C8—51no—
360C22×C8—51no—
361C22⋊Q8—24no—
362C4×Q8—102no—
363C4×Q8—102no—
364C2×C4⋊C4—51no—
365C2×C4⋊C4—51no—
366C2×C4⋊C4—51no—
367C2×C4⋊C4—51no—
368C2×C4⋊C4—51no—
369C2×C4⋊C4—51no—
370C2×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
371C2×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
372C2×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
373C2×C22⋊C4—51no—
374C2×C22⋊C4—51no—
375C2×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
376C2×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
377C2×C42—51no—
378C2×C42—51no—
379C2×C42—51no—
380C2×C42—51no—
381C2×C42—51no—
382C2×C42—51no—
383C2×C42—51no—
384C2×C42—51no—
385C2×C42—102no—
386C2×C42—102no—
387C4⋊C8—54no—
388C4⋊C8—102no—
389C4⋊C8—102no—
390C4⋊C8—102no—
391C4⋊C8—102no—
392C4⋊C8—102no—
393C4⋊C8—201no—
394C4⋊C8—201no—
395C22⋊C8—54no—
396C22⋊C8—102no—
397#97—102no—
398#97—102no—
399#97—102no—
400#97—201no—
401C4×C8—102no—
402C4×C8—201no—
403#95—51no—
404#95—51no—
405#95—51no—
406#95—51no—
407#95—51no—
408#95—51no—
409#95—51no—
410#95—51no—
411#95—51no—
412#95—51no—
413C22×C5—22no—
414C22×C5—22no—
415C22×C5—22no—
416C22×C5—41no—
417C22×C5—41no—
418C22×C5—41no—
419C22×C5—41no—
420C22×C5C2×C4×D411yescharacteristic
421C22×C5C2×C4×Q811yescharacteristic
422C22×C5#52011yescharacteristic
423C22×C5C22×M1611yescharacteristic
424C22×C5#56811yescharacteristic
425C22×C5#56811yescharacteristic
426C22×C5#56811yescharacteristic
427C4×C5—22no—
428C4×C5—22no—
429C4×C5—22no—
430C4×C5—22no—
431C4×C5—41no—
432C4×C5—41no—
433Dic5—22no—
434Dic5—22no—
435Dic5—41no—
436Dic5—41no—
437Dic5—41no—
438Dic5—41no—
439Dic5—41no—
440Dic5#54611yescharacteristic
441Dic5D4×Q811yescharacteristic
442Dic5#55211yescharacteristic
443Dic5#55411yescharacteristic
444C24C22×AGL(1,𝔽5)11yescharacteristic
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—41no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—41no—
451C22×C4—51no—
452C22×C4—52no—
453C22×C4—52no—
454C22×C4—51no—
455C22×C4—51no—
456C22×C4—51no—
457C22×C4—51no—
458C22×C4—51no—
459C22×C4—51no—
460C22×C4—51no—
461C22×C4—101no—
462C22×C4—101no—
463C22×C4—101no—
464C22×C4—101no—
465C22×C4—102no—
466C22×C4—201no—
467C22×C4—201no—
468C22×C4C22×AGL(1,𝔽5)11yescharacteristic
469C22×C4C22×AGL(1,𝔽5)11yescharacteristic
470C22×C4C22×AGL(1,𝔽5)11yescharacteristic
471C22×C4C22×AGL(1,𝔽5)11yescharacteristic
472C22×C4C22×AGL(1,𝔽5)11yescharacteristic
473C22×C4C22×AGL(1,𝔽5)11yescharacteristic
474M16—204no—
475M16—204no—
476C2×C8—52no—
477C2×C8—52no—
478C2×C8—52no—
479C2×C8—52no—
480C2×C8—101no—
481C2×C8—101no—
482C2×C8—101no—
483C2×C8—101no—
484C2×C8—101no—
485C2×C8—101no—
486C2×C8—101no—
487C2×C8—101no—
488C2×C8—101no—
489C2×C8—101no—
490C2×C8—101no—
491C2×C8—101no—
492C2×C8—101no—
493C2×C8—101no—
494C2×C8—101no—
495C2×C8—101no—
496C2×C8—101no—
497C2×C8—101no—
498C2×C8—101no—
499C2×C8—101no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—102no—
503C4⋊C4—102no—
504C4⋊C4—102no—
505C4⋊C4—201no—
506C4⋊C4—201no—
507C4⋊C4—201no—
508C4⋊C4#70414yescomplemented
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—201no—
512C22⋊C4—201no—
513C42—52no—
514C42—52no—
515C42—101no—
516C42—101no—
517C42—101no—
518C42—101no—
519C42—101no—
520C42—101no—
521C42—102no—
522C42—101no—
523C42—101no—
524C42—101no—
525C42—101no—
526C42—102no—
527C2×C5—42no—
528C2×C5#236011yescharacteristic
529C2×C5#293011yescharacteristic
530C2×C5D4×M1611yescharacteristic
531C2×C5#296111yescharacteristic
532C2×C5Q8×M1611yescharacteristic
533C2×C5#298911yescharacteristic
534C2×C5#299511yescharacteristic
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23Q8×AGL(1,𝔽5)12yes—
542C23C23×AGL(1,𝔽5)11yescharacteristic
543Q8—44no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—44no—
559C2×C4—51no—
560C2×C4—51no—
561C2×C4—51no—
562C2×C4—51no—
563C2×C4—51no—
564C2×C4—51no—
565C2×C4—52no—
566C2×C4—52no—
567C2×C4—102no—
568C2×C4—102no—
569C2×C4—102no—
570C2×C4—102no—
571C2×C4—101no—
572C2×C4—101no—
573C2×C4—101no—
574C2×C4—101no—
575C2×C4—101no—
576C2×C4—101no—
577C2×C4—101no—
578C2×C4—101no—
579C2×C4—101no—
580C2×C4—101no—
581C2×C4—101no—
582C2×C4—101no—
583C2×C4—101no—
584C2×C4—101no—
585C2×C4—101no—
586C2×C4—101no—
587C2×C4—101no—
588C2×C4—101no—
589C2×C4—101no—
590C2×C4—204no—
591C2×C4D4×AGL(1,𝔽5)12yes—
592C2×C4C2×#70412yes—
593C2×C4C2×#70412yes—
594C2×C4C2×#70412yes—
595C8—102no—
596C8—102no—
597C8—102no—
598C8—102no—
599C8—201no—
600C8—201no—
601C8—201no—
602C8—201no—
603C5#2295511yescharacteristic, complemented, Sylow
604C22—22no—
605C22—22no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22#6625611yescharacteristic
612C22C2×D4×AGL(1,𝔽5)11yescharacteristic
613C22#6626011yescharacteristic
614C22C2×Q8×AGL(1,𝔽5)11yescharacteristic
615C22#6626411yescharacteristic
616C22#6626611yescharacteristic
617C22C22×#70411yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—41no—
623C4—41no—
624C4—51no—
625C4—51no—
626C4—51no—
627C4—51no—
628C4—102no—
629C4—102no—
630C4—201no—
631C4—201no—
632C4—201no—
633C4—201no—
634C4—201no—
635C2—42no—
636C2#1061725811yescharacteristic
637C2#1061726211yescharacteristic
638C2#1061728011yescharacteristic
639C2#1061728511yescharacteristic
640C2D4×#70411yescharacteristic
641C2Q8×#70411yescharacteristic
642C2#1061754911yescharacteristic
subgroupG/H
center, FrattiniC23C23×AGL(1,𝔽5)
commutator, socleC23×C5C23×C4
FittingC2×C5×C22⋊Q8C4
prime radicalC24×C5C22×C4
cosocleC22×Dic5C24
hypercenterC2×C22⋊Q8AGL(1,𝔽5)
hypocenterC5#22955
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#65788C2semidirect
4. #10604985C2semidirect
6. #10604327C2semidirect
7. #10604327C2semidirect
8. #10604326C2semidirect
9. #10604323C2semidirect
10. #10604321C2semidirect
11. #10604316C2semidirect
107. C22⋊#170Q8semidirect
205. C5×C4⋊C4M16semidirect
508. C4⋊C4#704semidirect
603. C5#22955semidirect

Maximal subgroups

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

subgroupindex
C2×#657882
C2×#657582
Dic5×C22⋊Q82
#106049852
#106048972
#106043272
#106043262
#106043232
#106043212
#106043162
#106042322
#106042272
#229555