Finite Group Databasebeta

← #27465777

Subgroups — #27465777

Order 13568.

894 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810947C212yesmaximal
2C4×#13133191C211yesmaximal, characteristic
3#20276192C211yesmaximal, characteristic, complemented
4#20276187C211yesmaximal, characteristic, complemented
5#20276119C211yesmaximal, characteristic, complemented
6#20276119C211yesmaximal, characteristic, complemented
7#20275481C212yesmaximal, complemented
8#20274810C212yesmaximal, complemented
9#20274722C212yesmaximal, complemented
10#20274695C211yesmaximal, characteristic, complemented
11C53×#2285C211yesmaximal, characteristic, complemented
12C2×#13133191C2211yescharacteristic
13C2×C4×D212C2211yescharacteristic
14C2×C4×C22⋊D53—21no—
15#16810947—28no—
16C2×#13133150C2212yes—
17C2×#13133148C2212yes—
18C2×C4⋊#88322C2212yes—
19C2×#13133137C2212yes—
20C2×#13133137C2212yes—
21C2×C22⋊#88322C2212yes—
22#16810525C2211yescharacteristic
23#16810521C2211yescharacteristic
24#16810514—21no—
25#16810502—21no—
26#16810483C2211yescharacteristic
27C4×C4⋊Dic53C2211yescharacteristic
28C4×#11719921—21no—
29#16810385C2212yes—
30#16810385C2212yes—
31#16810385C2212yes—
32#16810385C2212yes—
33#16810378C2211yescharacteristic
34#16810378C2211yescharacteristic
35C2×#117×C53C2212yes—
36C22×C42×C53C2211yescharacteristic
37C53×#384C2211yescharacteristic
38C53×#384C2211yescharacteristic
39C53×#382C2212yes—
40C53×#380—21no—
41C53×#380—21no—
42C4×C4⋊C4×C53—21no—
43C4×C4⋊C4×C53—21no—
44C22×D212C2311yescharacteristic
45C22×C22⋊D53—21no—
46C22×C4×D53—21no—
47C22×#88322C2312yes—
48C22×#88322C2312yes—
49#13133191—22no—
50#13133191D414yes—
51C4×D212—24no—
52C4×C22⋊D53—44no—
53C2×C4⋊Dic53C2311yescharacteristic
54C2×C4⋊Dic53C2311yescharacteristic
55C2×C4⋊Dic53C2311yescharacteristic
56C2×#11719919—21no—
57C2×C22⋊Dic53—21no—
58C2×#11719921—21no—
59C2×#11719921—21no—
60C2×#11719921—21no—
61C2×#11719921—21no—
62C2×C4×Dic53—21no—
63#13133150—24no—
64#13133148—24no—
65C4⋊#88322—24no—
66#13133137—24no—
67#13133137—24no—
68C22⋊#88322—24no—
69#13133124—21no—
70#13133124—21no—
71#13133124—21no—
72C23×C4×C53D411yescharacteristic
73C23×C4×C53D411yescharacteristic
74C23×C4×C53C2311yescharacteristic
75#117×C53—24no—
76#117×C53—24no—
77C2×C4⋊C4×C53—21no—
78C2×C4⋊C4×C53—21no—
79C2×C4⋊C4×C53—21no—
80C2×C4⋊C4×C53—21no—
81C2×C4⋊C4×C53C2312yes—
82C2×C4⋊C4×C53C2312yes—
83C2×C22⋊C4×C53—22no—
84C2×C42×C53—24no—
85C2×C42×C53—22no—
86C2×C42×C53D411yescharacteristic
87C2×C42×C53D411yescharacteristic
88C2×C42×C53C2311yescharacteristic
89C2×C42×C53C2311yescharacteristic
90#95×C53—22no—
91#95×C53—22no—
92C24×C53C2×D411yescharacteristic
93C23×D53—21no—
94C22×C4×C53—22no—
95C22×C4×C53—22no—
96C22×C4×C53—22no—
97C22×C4×C53—22no—
98C22×C4×C53—22no—
99C22×C4×C53—22no—
100C22×C4×C53—22no—
101C22×C4×C53—22no—
102C22×C4×C53—41no—
103C22×C4×C53—41no—
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D412yes—
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53#4112yes—
111C22×C4×C53C2411yescharacteristic
112C2×C22⋊D53—24no—
113C2×C22⋊D53—42no—
114C2×C22⋊D53—42no—
115C22×Dic53—21no—
116C22×Dic53—21no—
117C22×Dic53—21no—
118C2×D212—22no—
119C2×D212—22no—
120C2×D212—21no—
121C2×D212C2×D412yes—
122C2×C4×D53—42no—
123C2×C4×D53—42no—
124C2×C4×D53—42no—
125C4⋊C4×C53—22no—
126C4⋊C4×C53—22no—
127C4⋊C4×C53—22no—
128C4⋊C4×C53—22no—
129C4⋊C4×C53—42no—
130C4⋊C4×C53—42no—
131C4⋊C4×C53—42no—
132C4⋊C4×C53—42no—
133C22⋊C4×C53—44no—
134C42×C53—22no—
135C42×C53—22no—
136C42×C53—22no—
137C42×C53—44no—
138C42×C53#4114yes—
139C22⋊Dic53—42no—
140#11719921—24no—
141#11719921—42no—
142#11719921—42no—
143#11719921—42no—
144C4⋊Dic53—21no—
145C4⋊Dic53—22no—
146C4⋊Dic53—22no—
147C4⋊Dic53C2×D412yes—
148#11719919—42no—
149C4×Dic53—42no—
150C2×#88322—22no—
151C2×#88322—22no—
152C2×#88322—22no—
153C2×#88322—22no—
154C2×#88322—22no—
155C2×#88322—22no—
156C23×C53—21no—
157C23×C53—21no—
158C23×C53—21no—
159C23×C53—21no—
160C23×C53—21no—
161C23×C53—21no—
162C23×C53#12112yes—
163C23×C53C22×D411yescharacteristic
164C22×D53—22no—
165C22×D53—41no—
166C22×D53—41no—
167C22×D53—41no—
168C22×D53—41no—
169C22×D53—41no—
170C22×D53—41no—
171C2×C4×C53—21no—
172C2×C4×C53—21no—
173C2×C4×C53—21no—
174C2×C4×C53—21no—
175C2×C4×C53—21no—
176C2×C4×C53—21no—
177C2×C4×C53—21no—
178C2×C4×C53—21no—
179C2×C4×C53—21no—
180C2×C4×C53—21no—
181C2×C4×C53—24no—
182C2×C4×C53—24no—
183C2×C4×C53—44no—
184C2×C4×C53—44no—
185C2×C4×C53—42no—
186C2×C4×C53—42no—
187C2×C4×C53—42no—
188C2×C4×C53—42no—
189C2×C4×C53—42no—
190C2×C4×C53—42no—
191C2×C4×C53#12112yes—
192C2×C4×C53#12312yes—
193C2×C4×C53#12312yes—
194C2×C4×C53#13512yes—
195C2×C4×C53AGL(1,ℤ/8)12yes—
196C2×C4×C53C22×D411yescharacteristic
197C2×C4×C53C2×#4111yescharacteristic
198C2×C4×C53C2×#4112yes—
199C2×C4×C532−1+412yes—
200C22⋊D53—48no—
201C2×Dic53—22no—
202C2×Dic53—41no—
203C2×Dic53—41no—
204C2×Dic53—41no—
205C2×Dic53—41no—
206C2×Dic53—41no—
207C2×Dic53—41no—
208C2×Dic53—41no—
209C2×Dic53—41no—
210D212—24no—
211D212—42no—
212C4×D53—84no—
213#88322—42no—
214#88322—42no—
215#27650—531nomaximal, Sylow
216C22×C53—22no—
217C22×C53—22no—
218C22×C53—22no—
219C22×C53—41no—
220C22×C53—41no—
221C22×C53—41no—
222C22×C53—41no—
223C22×C53C2×#12111yescharacteristic
224C22×C53C2×#12311yescharacteristic
225C22×C53#53611yescharacteristic
226C22×C53C2×#13511yescharacteristic
227C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
228C22×C53#57511yescharacteristic
229C22×C53#57511yescharacteristic
230C2×D53—42no—
231C2×D53—42no—
232C2×D53—42no—
233C2×D53—81no—
234C2×D53—81no—
235C2×D53—81no—
236C2×D53—81no—
237C4×C53—22no—
238C4×C53—22no—
239C4×C53—22no—
240C4×C53—22no—
241C4×C53—22no—
242C4×C53—82no—
243C4×C53—82no—
244C4×C53#54712yes—
245C4×C53#55212yes—
246Dic53—42no—
247Dic53—81no—
248Dic53—81no—
249C2×#482—532no—
250C4×#121—531no—
251#2285D5311yescharacteristic, complemented
252#2084—531no—
253#2083—531no—
254#2071—531no—
255#2071—531no—
256#1949—532no—
257#1825—532no—
258#1804—532no—
259#1797—531no—
260C2×C53—42no—
261C2×C53#248611yescharacteristic
262C2×C53#303511yescharacteristic
263C2×C53#303611yescharacteristic
264C2×C53#309111yescharacteristic
265C2×C53#309411yescharacteristic
266C2×C53#322811yescharacteristic
267C2×C53#324011yescharacteristic
268D53—82no—
269C2×#121—531no—
270C2×C4×D4—531no—
271C2×C4×D4—1061no—
272C2×#117C2×D5312yes—
273C22×C42C2×D5311yescharacteristic
274#482—1068no—
275C2×#107—532no—
276C2×#106—532no—
277C2×C4⋊C8—532no—
278C2×#102—532no—
279C2×#102—532no—
280C2×C22⋊C8—532no—
281#390—531no—
282#389—531no—
283#388—1061no—
284#386—1061no—
285#384C2×D5311yescharacteristic
286#384C2×D5311yescharacteristic
287#382C2×D5312yes—
288#380—21no—
289#380—21no—
290#380—531no—
291C4×C4⋊C4—21no—
292C4×C4⋊C4—21no—
293C4×C4⋊C4—531no—
294C4×C22⋊C4—1061no—
295#340—532no—
296#340—532no—
297#340—532no—
298#340—532no—
299#336—531no—
300#336—531no—
301C53#2765011yescharacteristic, complemented, Sylow
302C22×D4—531no—
303C22×D4—1061no—
304C23×C4—1061no—
305C23×C4C22⋊D5311yescharacteristic
306C23×C4C22⋊D5311yescharacteristic
307C23×C4C22×D5311yescharacteristic
308C22×C8—532no—
309C22×C8—532no—
310#121—534no—
311#121—1062no—
312C4×D4—1064no—
313C4×D4—2124no—
314#117—24no—
315#117—24no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—531no—
321C2×C4⋊C4—531no—
322C2×C4⋊C4—531no—
323C2×C4⋊C4—1061no—
324C2×C4⋊C4C22×D5312yes—
325C2×C4⋊C4C22×D5312yes—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—1061no—
328C2×C22⋊C4—1061no—
329C2×C22⋊C4—1061no—
330C2×C22⋊C4—1061no—
331C2×C22⋊C4—1061no—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42—1061no—
335C2×C42C22⋊D5311yescharacteristic
336C2×C42C22⋊D5311yescharacteristic
337C2×C42C22×D5311yescharacteristic
338C2×C42C22×D5311yescharacteristic
339#107—1064no—
340#106—1064no—
341C4⋊C8—1064no—
342#102—1064no—
343#102—1064no—
344C22⋊C8—1064no—
345#95—22no—
346#95—22no—
347#95—1061no—
348#95—1061no—
349#95—1061no—
350C24—1061no—
351C24C2×C22⋊D5311yescharacteristic
352C2×D4—532no—
353C2×D4—1064no—
354C2×D4—1061no—
355C2×D4—1062no—
356C2×D4—1062no—
357C2×D4—2122no—
358C2×D4—2122no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—1061no—
370C22×C4—1061no—
371C22×C4—1061no—
372C22×C4—2122no—
373C22×C4—2122no—
374C22×C4—2122no—
375C22×C4D4×D5312yes—
376C22×C4#1171994712yes—
377C22×C4C2×C22⋊D5311yescharacteristic
378C22×C4C2×C22⋊D5311yescharacteristic
379C22×C4C2×C22⋊D5311yescharacteristic
380C22×C4C2×C22⋊D5311yescharacteristic
381C22×C4C2×C22⋊D5311yescharacteristic
382C22×C4C23×D5311yescharacteristic
383C2×C8—1062no—
384C2×C8—1062no—
385C2×C8—1062no—
386C2×C8—1062no—
387C2×C8—1062no—
388C2×C8—1062no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—532no—
398C4⋊C4—1061no—
399C4⋊C4—1062no—
400C4⋊C4—1062no—
401C4⋊C4—2122no—
402C22⋊C4—44no—
403C22⋊C4—1064no—
404C22⋊C4—2122no—
405C22⋊C4—2122no—
406C22⋊C4—2122no—
407C22⋊C4—2122no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—44no—
412C42—2122no—
413C42#1171994514yes—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—1062no—
421C23—2121no—
422C23—2121no—
423C23—2121no—
424C23—2121no—
425C23—2121no—
426C23—2121no—
427C23#1313318812yes—
428C23C22×C22⋊D5311yescharacteristic
429D4—1064no—
430D4—2128no—
431D4—2122no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—1062no—
453C2×C4—4244no—
454C2×C4—2121no—
455C2×C4—2121no—
456C2×C4—2121no—
457C2×C4—2121no—
458C2×C4—2121no—
459C2×C4—2121no—
460C2×C4—2121no—
461C2×C4—2121no—
462C2×C4#1313318812yes—
463C2×C4#1313320012yes—
464C2×C4#1313320012yes—
465C2×C4#1313323712yes—
466C2×C4#1313324212yes—
467C2×C4C2×D4×D5311yescharacteristic
468C2×C4C2×#1171994711yescharacteristic
469C2×C4C2×#1171994512yes—
470C2×C4#1313326512yes—
471C8—2122no—
472C8—2122no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22—4241no—
481C22—4241no—
482C22—4241no—
483C22—4241no—
484C22—2122no—
485C22—2122no—
486C22—2122no—
487C22C2×#1313318811yescharacteristic
488C22C2×#1313320011yescharacteristic
489C22#1681119711yescharacteristic
490C22C2×#1313323711yescharacteristic
491C22C2×#1313324211yescharacteristic
492C22#1681142111yescharacteristic
493C22#1681142111yescharacteristic
494C4—22no—
495C4—22no—
496C4—22no—
497C4—22no—
498C4—22no—
499C4—82no—
500C4—82no—
501C4—4241no—
502C4—4241no—
503C4—2122no—
504C4#1681127412yes—
505C4#1681130812yes—
506C2—42no—
507C2—4242no—
508C2#2027901711yescharacteristic
509C2#2028416411yescharacteristic
510C2#2028417511yescharacteristic
511C2#2028462811yescharacteristic
512C2#2028465511yescharacteristic
513C2#2028585111yescharacteristic
514C2#2028595511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2285C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#2285D53
hypocenterC53#27650
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2285

Complements

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

NH ≅ G/Ntype
3. #20276192C2semidirect
4. #20276187C2semidirect
5. #20276119C2semidirect
6. #20276119C2semidirect
7. #20275481C2semidirect
8. #20274810C2semidirect
9. #20274722C2semidirect
10. #20274695C2semidirect
11. C53×#2285C2semidirect
251. #2285D53semidirect
301. C53#27650semidirect

Maximal subgroups

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

subgroupindex
C2×#168109472
C4×#131331912
#202761922
#202761872
#202761192
#202754812
#202748102
#202747222
#202746952
C53×#22852
#2765053