Finite Group Databasebeta

← #27395305

Subgroups — #27395305

Order 13568.

1080 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810762C211yesmaximal, characteristic
2#20277221C211yesmaximal, characteristic
3C2×#16810530C211yesmaximal, characteristic, complemented
4#20275593C212yesmaximal, complemented
5#20275402C212yesmaximal, complemented
6#20275359C212yesmaximal, complemented
7#20275046C212yesmaximal, complemented
8#20275030C211yesmaximal, characteristic
9#20274902C211yesmaximal, characteristic, complemented
10#20274479C211yesmaximal, characteristic, complemented
11C53×#1787C211yesmaximal, characteristic, complemented
12C22×#11719924C2212yes—
13C2×#13133191C2211yescharacteristic
14C2×#13133179C2211yescharacteristic
15C2×#13133177—21no—
16C22×C4⋊Dic53C2211yescharacteristic
17C22×C4×Dic53C2211yescharacteristic
18#16810762C418yescomplemented
19C2×#13133150C2212yes—
20C2×#13133141C2212yes—
21C2×#13133139C2212yes—
22C2×C22⋊#88322C2211yescharacteristic
23C2×#13133127C2212yes—
24#16810530—21no—
25#16810530—21no—
26#16810530C2212yes—
27#16810530C2212yes—
28#16810524C2211yescharacteristic
29#16810519C2211yescharacteristic
30#16810512—21no—
31#16810500—21no—
32C4⋊C4×Dic53C2211yescharacteristic
33#16810470—21no—
34C2×#13133124—21no—
35C2×#13133124—21no—
36#16810386C2212yes—
37#16810385C2212yes—
38#16810377C2212yes—
39C23×Q8×C53C2211yescharacteristic
40C2×#103×C53C2212yes—
41C2×#103×C53C2212yes—
42C2×C22⋊C8×C53C2211yescharacteristic
43C53×#380C2211yescharacteristic
44C22×D212C2311yescharacteristic
45C22×C22⋊D53—21no—
46C23×Dic53—21no—
47C23×Dic53D411yescharacteristic
48C23×Dic53D411yescharacteristic
49C22×C4×D53—21no—
50C2×#11719924—42no—
51C2×#11719924—42no—
52C2×#11719924C2×C418yescomplemented
53C22×#88322C2312yes—
54#13133191—22no—
55#13133191C2×C414yes—
56#13133179—42no—
57#13133177—44no—
58C2×C4⋊Dic53—21no—
59C2×C4⋊Dic53—42no—
60C2×C4⋊Dic53C2312yes—
61C2×C4⋊Dic53C2311yescharacteristic
62C2×#11719919—21no—
63C2×#11719921—21no—
64C2×#11719921—21no—
65C2×#11719921—21no—
66C2×#11719921—21no—
67C2×C4×Dic53—21no—
68C2×C4×Dic53—42no—
69C2×C4×Dic53D411yescharacteristic
70C2×C4×Dic53D411yescharacteristic
71C2×C4×Dic53C2311yescharacteristic
72C2×C4×Dic53C2311yescharacteristic
73#13133150—42no—
74#13133141C2×C418yescomplemented
75#13133139—42no—
76C22⋊#88322C2×C414yescomplemented
77#13133127—42no—
78#13133124—21no—
79#13133124—21no—
80#13133124—22no—
81#13133124—22no—
82#13133124—22no—
83#13133124—22no—
84C22×Q8×C53—21no—
85C22×Q8×C53—22no—
86C22×Q8×C53—42no—
87C22×Q8×C53—42no—
88C22×Q8×C53C2×C414yescomplemented
89C22×Q8×C53C2312yes—
90C23×C4×C53—21no—
91C23×C4×C53C2311yescharacteristic
92C22×C8×C53C2312yes—
93C2×C4⋊C4×C53C2311yescharacteristic
94C2×C4⋊C4×C53C2311yescharacteristic
95C2×C22⋊C4×C53—21no—
96#103×C53—42no—
97#103×C53—42no—
98C22⋊C8×C53—22no—
99#95×C53—21no—
100C24×C53C2×D411yescharacteristic
101C23×D53—21no—
102C2×Q8×C53—24no—
103C2×Q8×C53—22no—
104C2×Q8×C53—22no—
105C2×Q8×C53—22no—
106C2×Q8×C53—22no—
107C2×Q8×C53—22no—
108C2×Q8×C53—48no—
109C2×Q8×C53—42no—
110C2×Q8×C53—42no—
111C2×Q8×C53—48no—
112C2×Q8×C53C22×C414yes—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—24no—
116C22×C4×C53—22no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—42no—
120C22×C4×C53—42no—
121C22×C4×C53C22×C412yes—
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53#4112yes—
124C22×C4×C53C2411yescharacteristic
125C2×C22⋊D53—24no—
126C2×C22⋊D53—42no—
127C2×C22⋊D53—42no—
128C22×Dic53—22no—
129C22×Dic53—22no—
130C22×Dic53—21no—
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—41no—
134C22×Dic53—41no—
135C22×Dic53—41no—
136C22×Dic53—42no—
137C22×Dic53—42no—
138C22×Dic53—42no—
139C22×Dic53C2×D411yescharacteristic
140C22×Dic53C2×D411yescharacteristic
141C22×Dic53C2×D411yescharacteristic
142C22×Dic53C2×D411yescharacteristic
143C2×D212—22no—
144C2×D212—22no—
145C2×D212—21no—
146C2×D212C22×C412yes—
147C2×C4×D53—42no—
148C2×C4×D53—42no—
149C2×C4×D53—42no—
150C2×C8×C53—22no—
151C2×C8×C53—22no—
152C2×C8×C53—22no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—22no—
155C22⋊C4×C53—42no—
156#11719924—48no—
157#11719921—24no—
158#11719921—42no—
159#11719921—42no—
160#11719921—42no—
161C4⋊Dic53—21no—
162C4⋊Dic53—22no—
163C4⋊Dic53—22no—
164C4⋊Dic53—42no—
165C4⋊Dic53C22×C412yes—
166#11719919—42no—
167C4×Dic53—22no—
168C4×Dic53—22no—
169C4×Dic53—41no—
170C4×Dic53—41no—
171C4×Dic53—42no—
172C2×#88322—22no—
173C2×#88322—22no—
174C2×#88322C22×C414yes—
175C23×C53—21no—
176C23×C53—21no—
177C23×C53—21no—
178C23×C53—21no—
179C23×C53C4×D412yes—
180C23×C53AGL(1,ℤ/8)12yes—
181C23×C53#13712yes—
182C23×C53C22×D411yescharacteristic
183C22×D53—22no—
184C22×D53—41no—
185C22×D53—41no—
186C22×D53—41no—
187C22×D53—41no—
188C22×D53—41no—
189C22×D53—41no—
190Q8×C53—24no—
191Q8×C53—24no—
192Q8×C53—48no—
193Q8×C53—42no—
194Q8×C53—42no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53—24no—
198C2×C4×C53—24no—
199C2×C4×C53—24no—
200C2×C4×C53—48no—
201C2×C4×C53—42no—
202C2×C4×C53—42no—
203C2×C4×C53—41no—
204C2×C4×C53—41no—
205C2×C4×C53—41no—
206C2×C4×C53—41no—
207C2×C4×C53—41no—
208C2×C4×C53—41no—
209C2×C4×C53C4×D412yes—
210C2×C4×C53C23×C411yescharacteristic
211C2×C4×C53C2×#4111yescharacteristic
212C2×C4×C53H(5,𝔽2)12yes—
213C2×C4×C53H(5,𝔽2)12yes—
214C22⋊D53—48no—
215C2×Dic53—21no—
216C2×Dic53—21no—
217C2×Dic53—22no—
218C2×Dic53—21no—
219C2×Dic53—21no—
220C2×Dic53—21no—
221C2×Dic53—21no—
222C2×Dic53—21no—
223C2×Dic53—21no—
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—21no—
227C2×Dic53—44no—
228C2×Dic53—44no—
229C2×Dic53—41no—
230C2×Dic53—41no—
231C2×Dic53—42no—
232C2×Dic53—42no—
233C2×Dic53—42no—
234C2×Dic53—84no—
235C2×Dic53#13712yes—
236D212—24no—
237D212—42no—
238C4×D53—84no—
239C8×C53—42no—
240#88322—24no—
241#22047—531nomaximal, Sylow
242C22×C53—22no—
243C22×C53—22no—
244C22×C53—22no—
245C22×C53—22no—
246C22×C53—22no—
247C22×C53—22no—
248C22×C53#44014yes—
249C22×C53C2×C4×D411yescharacteristic
250C22×C53#53411yescharacteristic
251C22×C53#53511yescharacteristic
252C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
253C22×C53C2×#13711yescharacteristic
254C22×C53C2×#13711yescharacteristic
255C22×C53#57511yescharacteristic
256C2×D53—42no—
257C2×D53—42no—
258C2×D53—42no—
259C2×D53—81no—
260C2×D53—81no—
261C2×D53—81no—
262C2×D53—81no—
263C4×C53—22no—
264C4×C53—44no—
265C4×C53—81no—
266C4×C53—81no—
267C4×C53#51812yes—
268C4×C53#53812yes—
269Dic53—22no—
270Dic53—41no—
271Dic53—42no—
272Dic53—41no—
273Dic53—41no—
274Dic53—82no—
275C2×#448—531no—
276C4×#121—531no—
277C2×#391—531no—
278#1973—532no—
279#1935—532no—
280#1928—532no—
281#1877—532no—
282#1874—531no—
283#1852—531no—
284#1787D5311yescharacteristic, complemented
285#1749—531no—
286C2×C53—24no—
287C2×C53#232111yescharacteristic
288C2×C53C2×#44011yescharacteristic
289C2×C53C4×#13711yescharacteristic
290C2×C53#319311yescharacteristic
291C2×C53#319611yescharacteristic
292C2×C53#321311yescharacteristic
293C2×C53#321811yescharacteristic
294D53—82no—
295C23×Q8C2×D5311yescharacteristic
296C22×SD16—532no—
297C2×#121—531no—
298C2×C4×D4—531no—
299C2×C4×D4—1061no—
300C22×C4⋊C4—531no—
301C22×C42—531no—
302#448—538no—
303C2×#107—532no—
304C2×#103—532no—
305C2×#103C2×D5312yes—
306C2×#103C2×D5312yes—
307C2×#102—532no—
308C2×C22⋊C8—531no—
309C2×C22⋊C8C2×D5311yescharacteristic
310C2×#97—532no—
311#391—532no—
312#391—532no—
313#391—1061no—
314#391—1061no—
315#390—531no—
316#389—531no—
317#388—1061no—
318#386—1061no—
319#380C2×D5311yescharacteristic
320C4×C4⋊C4—531no—
321C4×C22⋊C4—1061no—
322C2×#95—1061no—
323C2×#95—1061no—
324#340—532no—
325#340—532no—
326#336—532no—
327C53#2204711yescharacteristic, complemented, Sylow
328C22×Q8—21no—
329C22×Q8—22no—
330C22×Q8—42no—
331C22×Q8—42no—
332C22×Q8C4×D5314yescomplemented
333C22×Q8C22×D5312yes—
334C22×D4—531no—
335C22×D4—1061no—
336C23×C4—21no—
337C23×C4—531no—
338C23×C4—531no—
339C23×C4—1061no—
340C23×C4—1061no—
341C23×C4C22×D5311yescharacteristic
342C2×SD16—538no—
343C2×SD16—2122no—
344C2×SD16—2122no—
345C22×C8—532no—
346C22×C8C22×D5312yes—
347#121—534no—
348#121—1062no—
349C4×D4—2122no—
350C4×D4—2124no—
351C2×C4⋊C4—531no—
352C2×C4⋊C4—532no—
353C2×C4⋊C4—1061no—
354C2×C4⋊C4—1061no—
355C2×C4⋊C4—2122no—
356C2×C4⋊C4C22×D5311yescharacteristic
357C2×C4⋊C4C22×D5311yescharacteristic
358C2×C22⋊C4—21no—
359C2×C22⋊C4—1061no—
360C2×C22⋊C4—1061no—
361C2×C22⋊C4—1061no—
362C2×C22⋊C4—1061no—
363C2×C42—531no—
364C2×C42—531no—
365C2×C42—531no—
366C2×C42—531no—
367C2×C42—1061no—
368C2×C42—2122no—
369#107—2122no—
370#103—42no—
371#103—42no—
372#103—538no—
373#102—2122no—
374C22⋊C8—22no—
375C22⋊C8—534no—
376#97—2122no—
377#95—21no—
378#95—1062no—
379#95—1062no—
380#95—1062no—
381#95—1062no—
382#95—1061no—
383#95—1061no—
384C24—1061no—
385C24D4×D5311yescharacteristic
386C2×Q8—24no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—48no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—48no—
396C2×Q8C2×C4×D5314yes—
397C2×D4—532no—
398C2×D4—1064no—
399C2×D4—1061no—
400C2×D4—1062no—
401C2×D4—1062no—
402C2×D4—2122no—
403C2×D4—2122no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—531no—
413C22×C4—531no—
414C22×C4—531no—
415C22×C4—531no—
416C22×C4—1062no—
417C22×C4—1062no—
418C22×C4—1062no—
419C22×C4—1061no—
420C22×C4—1062no—
421C22×C4—2122no—
422C22×C4—2122no—
423C22×C4—2122no—
424C22×C4—2121no—
425C22×C4—2121no—
426C22×C4—2121no—
427C22×C4—2122no—
428C22×C4—2122no—
429C22×C4—2122no—
430C22×C4C2×C4×D5312yes—
431C22×C4D4×D5311yescharacteristic
432C22×C4#1171994712yes—
433C22×C4C23×D5311yescharacteristic
434SD16—2128no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—534no—
439C2×C8—1062no—
440C2×C8—1062no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—532no—
444C4⋊C4—1061no—
445C4⋊C4—1062no—
446C4⋊C4—1062no—
447C4⋊C4—2122no—
448C4⋊C4—2122no—
449C22⋊C4—42no—
450C22⋊C4—1064no—
451C22⋊C4—2122no—
452C22⋊C4—2122no—
453C22⋊C4—2122no—
454C42—1062no—
455C42—1062no—
456C42—2121no—
457C42—2121no—
458C42—2122no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—1062no—
464C23—2121no—
465C23—2121no—
466C23—2121no—
467C23—2121no—
468C23—2121no—
469C23—2121no—
470C23#1313317712yes—
471C23#1313324412yes—
472C23#1313324912yes—
473C23C2×D4×D5311yescharacteristic
474Q8—24no—
475Q8—24no—
476Q8—48no—
477Q8—42no—
478Q8—42no—
479D4—1064no—
480D4—2128no—
481D4—2122no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—48no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—532no—
497C2×C4—1061no—
498C2×C4—1061no—
499C2×C4—1062no—
500C2×C4—1061no—
501C2×C4—1061no—
502C2×C4—1061no—
503C2×C4—1061no—
504C2×C4—1061no—
505C2×C4—1061no—
506C2×C4—1061no—
507C2×C4—1061no—
508C2×C4—1061no—
509C2×C4—4244no—
510C2×C4—4244no—
511C2×C4—2124no—
512C2×C4—2122no—
513C2×C4—2122no—
514C2×C4—2122no—
515C2×C4—2124no—
516C2×C4—2121no—
517C2×C4—2121no—
518C2×C4#1313317712yes—
519C2×C4C22×C4×D5311yescharacteristic
520C2×C4C2×#1171994711yescharacteristic
521C2×C4#1313326212yes—
522C2×C4#1313326212yes—
523C8—42no—
524C8—1064no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—4241no—
532C22—4241no—
533C22—4241no—
534C22—4241no—
535C22—2122no—
536C22—2122no—
537C22—2122no—
538C22#1681072614yes—
539C22C2×#1313317711yescharacteristic
540C22#1681118311yescharacteristic
541C22#1681118811yescharacteristic
542C22C2×#1313324411yescharacteristic
543C22#137×D5311yescharacteristic
544C22C2×#1313324911yescharacteristic
545C22#1681142211yescharacteristic
546C4—22no—
547C4—44no—
548C4—81no—
549C4—81no—
550C4—1062no—
551C4—4242no—
552C4—2121no—
553C4—2122no—
554C4—2121no—
555C4—2121no—
556C4#1681109612yes—
557C4#1681120812yes—
558C2—24no—
559C2—4242no—
560C2#2027737011yescharacteristic
561C2C2×#1681072611yescharacteristic
562C2#2028353011yescharacteristic
563C2#2028551911yescharacteristic
564C2#2028554811yescharacteristic
565C2#2028570111yescharacteristic
566C2#2028574511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1787C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
cosocleC22×C4×C53C24
hypercenter#1787D53
hypocenterC53#22047
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
3. C2×#16810530C2semidirect
4. #20275593C2semidirect
5. #20275402C2semidirect
6. #20275359C2semidirect
7. #20275046C2semidirect
9. #20274902C2semidirect
10. #20274479C2semidirect
11. C53×#1787C2semidirect
18. #16810762C4semidirect
52. C2×#11719924C2×C4semidirect
74. #13133141C2×C4semidirect
76. C22⋊#88322C2×C4semidirect
88. C22×Q8×C53C2×C4semidirect
284. #1787D53semidirect
327. C53#22047semidirect
332. C22×Q8C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168107622
#202772212
C2×#168105302
#202755932
#202754022
#202753592
#202750462
#202750302
#202749022
#202744792
C53×#17872
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