Finite Group Databasebeta

← #27278877

Subgroups — #27278877

Order 13568.

692 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810815C211yesmaximal, characteristic
2#20277662C211yesmaximal, characteristic, complemented
3#20275522C211yesmaximal, characteristic, complemented
4C2×#16810350C211yesmaximal, characteristic, complemented
5#20272599C211yesmaximal, characteristic, complemented
6#20272556C211yesmaximal, characteristic, complemented
7C53×#1982C211yesmaximal, characteristic, complemented
8C22×#11719922—21no—
9C2×C4×D212C2211yescharacteristic
10C2×#13133175—21no—
11#16810815—22no—
12#16810815C414yescomplemented
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133138—21no—
15C2×#13133138—22no—
16C2×C4×#88322C2211yescharacteristic
17#16810525—21no—
18#16810515—22no—
19#16810493C2211yescharacteristic
20#16810480—21no—
21C4×C4⋊Dic53—21no—
22#16810350—22no—
23C2×#127×C53C2211yescharacteristic
24C2×C4⋊C8×C53C2211yescharacteristic
25C2×#102×C53—21no—
26C2×#102×C53—21no—
27C4×C4⋊C4×C53C2211yescharacteristic
28C22×D212D411yescharacteristic
29C22×C4×D53—21no—
30C2×#11719922—24no—
31C2×#11719922—42no—
32C2×#11719922—42no—
33C22×#88322—21no—
34C22×#88322—21no—
35C4×D212—21no—
36C4×D212—21no—
37C4×D212C2×C412yes—
38C4×D212D412yes—
39#13133175—42no—
40#13133175—42no—
41C2×C4⋊Dic53—21no—
42C2×C4⋊Dic53—21no—
43C2×C4⋊Dic53D411yescharacteristic
44C2×C4⋊Dic53D412yes—
45C2×#11719921—22no—
46C2×#11719921—22no—
47C2×#11719921—21no—
48C2×C4×Dic53—22no—
49C4⋊#88322—21no—
50C4⋊#88322C2×C412yes—
51#13133138—24no—
52#13133138—44no—
53C4×#88322—21no—
54C4×#88322—21no—
55#13133124—22no—
56C22×D4×C53—21no—
57C22×D4×C53—21no—
58C22×D4×C53—21no—
59C22×C8×C53—21no—
60#127×C53—21no—
61#127×C53—21no—
62#127×C53—21no—
63#127×C53C2×C412yes—
64C2×C4⋊C4×C53D411yescharacteristic
65C2×C4⋊C4×C53D411yescharacteristic
66C2×C42×C53—21no—
67C2×C42×C53C2311yescharacteristic
68C4⋊C8×C53—21no—
69C4⋊C8×C53—21no—
70#102×C53—42no—
71#102×C53—42no—
72#95×C53—21no—
73C24×C53—41no—
74C23×D53—21no—
75C2×D4×C53—22no—
76C2×D4×C53—22no—
77C2×D4×C53—22no—
78C2×D4×C53—42no—
79C2×D4×C53—42no—
80C2×D4×C53—41no—
81C2×D4×C53—41no—
82C2×D4×C53—41no—
83C2×D4×C53—41no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C2×D4×C53—41no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53—41no—
90C2×D4×C53—41no—
91C22×C4×C53—21no—
92C22×C4×C53—21no—
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×Dic53—21no—
97C22×Dic53—22no—
98C2×D212—21no—
99C2×D212—21no—
100C2×D212—21no—
101C2×D212—21no—
102C2×D212—21no—
103C2×D212C22⋊C412yes—
104C2×C4×D53—22no—
105C2×C4×D53—22no—
106C2×C4×D53—41no—
107C2×C4×D53—41no—
108C2×C4×D53—41no—
109C2×C4×D53—41no—
110C2×C8×C53—41no—
111C2×C8×C53—41no—
112C2×C8×C53—41no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53—21no—
116C4⋊C4×C53D812yescomplemented
117C42×C53—41no—
118C42×C53—41no—
119C42×C53C22×C411yescharacteristic
120C42×C53C2×D411yescharacteristic
121C42×C53#4111yescharacteristic
122C42×C53#4111yescharacteristic
123#11719922—48no—
124#11719921—22no—
125#11719921—22no—
126#11719921—44no—
127#11719921—82no—
128C4⋊Dic53—21no—
129C4⋊Dic53—41no—
130C4⋊Dic53—41no—
131C4⋊Dic53—41no—
132C4⋊Dic53—41no—
133C4⋊Dic53—42no—
134C4⋊Dic53C22⋊C412yes—
135C4×Dic53—44no—
136C2×#88322—22no—
137C2×#88322—22no—
138C2×#88322—41no—
139C2×#88322—41no—
140C2×#88322—41no—
141C2×#88322—41no—
142C23×C53—42no—
143C23×C53—81no—
144C23×C53—81no—
145C23×C53—81no—
146C23×C53—81no—
147C23×C53—81no—
148C23×C53—81no—
149C23×C53#12011yescharacteristic
150C22×D53—22no—
151C22×D53—41no—
152C22×D53—41no—
153C22×D53—41no—
154C22×D53—41no—
155C22×D53—41no—
156C22×D53—41no—
157D4×C53—44no—
158D4×C53—42no—
159D4×C53—42no—
160D4×C53—42no—
161D4×C53—42no—
162D4×C53—82no—
163D4×C53—81no—
164D4×C53—81no—
165D4×C53—81no—
166D4×C53—81no—
167C2×C4×C53—21no—
168C2×C4×C53—21no—
169C2×C4×C53—22no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53C2×C22⋊C411yescharacteristic
176C2×C4×C53C4×D411yescharacteristic
177C2×C4×C53C4×D411yescharacteristic
178C2×C4×C53#12111yescharacteristic
179C2×C4×C53#12111yescharacteristic
180C2×C4×C53#12311yescharacteristic
181C2×C4×C53C2×D811yescharacteristic
182C2×C4×C53#13511yescharacteristic
183C2×C4×C53#13511yescharacteristic
184C2×C4×C53#13511yescharacteristic
185C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
186C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
187C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
188C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
189C2×Dic53—22no—
190C2×Dic53—41no—
191C2×Dic53—41no—
192C2×Dic53—42no—
193C2×Dic53—42no—
194C2×Dic53—42no—
195D212—22no—
196D212—22no—
197D212—41no—
198D212—41no—
199C4×D53—42no—
200C4×D53—42no—
201C4×D53—42no—
202C4×D53—42no—
203C8×C53—81no—
204#88322—42no—
205#88322—42no—
206#8980—531nomaximal, Sylow
207C22×C53—82no—
208C22×C53—82no—
209C22×C53—82no—
210C22×C53—161no—
211C22×C53—161no—
212C22×C53—161no—
213C22×C53—161no—
214C22×C53#38611yescharacteristic
215C22×C53#44711yescharacteristic
216C22×C53#44911yescharacteristic
217C22×C53#44911yescharacteristic
218C22×C53#44911yescharacteristic
219C22×C53#45311yescharacteristic
220C22×C53#45411yescharacteristic
221C2×D53—42no—
222C2×D53—42no—
223C2×D53—42no—
224C2×D53—81no—
225C2×D53—81no—
226C2×D53—81no—
227C2×D53—81no—
228C4×C53—21no—
229C4×C53—21no—
230C4×C53—42no—
231C4×C53—81no—
232C4×C53#41611yescharacteristic
233C4×C53#41811yescharacteristic
234C4×C53C4×D811yescharacteristic
235C4×C53#44211yescharacteristic
236C4×C53#45911yescharacteristic
237C4×C53#46311yescharacteristic
238C4×C53#48211yescharacteristic
239C4×C53#48211yescharacteristic
240Dic53—42no—
241Dic53—82no—
242C2×#459—531no—
243#2354—531no—
244#1982D5311yescharacteristic, complemented
245#1957—531no—
246C2×D4⋊C8—531no—
247#1296—531no—
248#1279—531no—
249C2×C53—162no—
250C2×C53#162111yescharacteristic
251C2×C53#162411yescharacteristic
252C2×C53#162511yescharacteristic
253C2×C53#162811yescharacteristic
254C2×C53#187511yescharacteristic
255C2×C53#187911yescharacteristic
256C2×C53#188311yescharacteristic
257D53—82no—
258C22×D8—1061no—
259C2×#127C2×D5311yescharacteristic
260C2×C4×D4—531no—
261C2×#117—1061no—
262#459—534no—
263#459—1062no—
264C2×C4⋊C8—531no—
265C2×C4⋊C8C2×D5311yescharacteristic
266C2×#102—21no—
267C2×#102—21no—
268C2×#102—1061no—
269C2×#102—1062no—
270C2×C4×C8—531no—
271#390—1061no—
272#388—1062no—
273#384—531no—
274#380—1061no—
275C4×C4⋊C4—1061no—
276C4×C4⋊C4C2×D5311yescharacteristic
277D4⋊C8—1062no—
278C53#898011yescharacteristic, complemented, Sylow
279C22×D4—21no—
280C22×D4—21no—
281C22×D4—21no—
282C22×D4—531no—
283C23×C4—1061no—
284C2×D8—1064no—
285C2×D8—2122no—
286C2×D8—2122no—
287C22×C8—21no—
288C22×C8—1061no—
289C22×C8—1061no—
290#127—21no—
291#127—21no—
292#127—21no—
293#127C4×D5312yes—
294C4×D4—532no—
295C4×D4—532no—
296C4×D4—1061no—
297C4×D4—1061no—
298#117—2122no—
299#117—2122no—
300C2×C4⋊C4—531no—
301C2×C4⋊C4—532no—
302C2×C4⋊C4—1061no—
303C2×C4⋊C4—1061no—
304C2×C4⋊C4C22⋊D5311yescharacteristic
305C2×C4⋊C4C22⋊D5311yescharacteristic
306C2×C22⋊C4—1061no—
307C2×C22⋊C4—1062no—
308C2×C22⋊C4—1062no—
309C2×C42—21no—
310C2×C42—1062no—
311C2×C42C22×D5311yescharacteristic
312C4⋊C8—21no—
313C4⋊C8—21no—
314C4⋊C8—532no—
315C4⋊C8—1061no—
316#102—42no—
317#102—42no—
318#102—1064no—
319#102—2124no—
320C4×C8—1061no—
321C4×C8—1061no—
322#95—21no—
323#95—1062no—
324C24—41no—
325C24—1061no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—532no—
343C2×D4—1061no—
344C2×D4—1061no—
345C2×D4—1061no—
346C2×D4—1061no—
347C2×D4—1061no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—1061no—
351C22×C4—1062no—
352C22×C4—1062no—
353C22×C4—1062no—
354C22×C4—2121no—
355C22×C4—2121no—
356C22×C4—2121no—
357C22×C4—2121no—
358C22×C4D4×D5311yescharacteristic
359C22×C4D4×D5311yescharacteristic
360C22×C4C2×C22⋊D5311yescharacteristic
361D8—2128no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—1062no—
366C2×C8—1062no—
367C2×C8—2121no—
368C2×C8—2121no—
369C2×C8—2121no—
370C2×C8—2121no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—532no—
375C4⋊C4—1061no—
376C4⋊C4—2122no—
377C4⋊C4—2121no—
378C4⋊C4—2121no—
379C4⋊C4—2121no—
380C4⋊C4—2121no—
381C4⋊C4#1171992212yescomplemented
382C22⋊C4—1062no—
383C22⋊C4—1062no—
384C22⋊C4—4242no—
385C22⋊C4—2124no—
386C42—41no—
387C42—41no—
388C42—2124no—
389C42C2×C4×D5311yescharacteristic
390C42#1171994511yescharacteristic
391C42D4×D5311yescharacteristic
392C42#1171994711yescharacteristic
393C23—42no—
394C23—81no—
395C23—81no—
396C23—81no—
397C23—81no—
398C23—81no—
399C23—81no—
400C23—1062no—
401C23—2121no—
402C23—2121no—
403C23—2121no—
404C23—2121no—
405C23—2121no—
406C23—2121no—
407C23#1313318511yescharacteristic
408D4—44no—
409D4—42no—
410D4—42no—
411D4—42no—
412D4—42no—
413D4—82no—
414D4—81no—
415D4—81no—
416D4—81no—
417D4—81no—
418D4—1062no—
419D4—1062no—
420D4—2121no—
421D4—2121no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—1062no—
431C2×C4—2122no—
432C2×C4—2122no—
433C2×C4—2122no—
434C2×C4—2121no—
435C2×C4—2121no—
436C2×C4—2122no—
437C2×C4—2122no—
438C2×C4—2122no—
439C2×C4—2122no—
440C2×C4C22⋊C4×D5311yescharacteristic
441C2×C4#1313317711yescharacteristic
442C2×C4C4×C22⋊D5311yescharacteristic
443C2×C4#1313318811yescharacteristic
444C2×C4#1313319011yescharacteristic
445C2×C4#1313319911yescharacteristic
446C2×C4C2×#1171992211yescharacteristic
447C2×C4#1313323611yescharacteristic
448C2×C4#1313323711yescharacteristic
449C2×C4#1313323811yescharacteristic
450C2×C4#1313324111yescharacteristic
451C2×C4#1313324211yescharacteristic
452C2×C4#1313324311yescharacteristic
453C2×C4#1313324411yescharacteristic
454C8—81no—
455C8—2122no—
456C8—2122no—
457C22—82no—
458C22—82no—
459C22—82no—
460C22—161no—
461C22—161no—
462C22—161no—
463C22—161no—
464C22—4241no—
465C22—4241no—
466C22—4241no—
467C22—4241no—
468C22—2122no—
469C22—2122no—
470C22—2122no—
471C22#1681049911yescharacteristic
472C22#1681075611yescharacteristic
473C22#1681076511yescharacteristic
474C22#1681076611yescharacteristic
475C22#1681076711yescharacteristic
476C22#1681078911yescharacteristic
477C22#1681079411yescharacteristic
478C4—21no—
479C4—21no—
480C4—42no—
481C4—81no—
482C4—4242no—
483C4—2122no—
484C4#1681061611yescharacteristic
485C4#1681062611yescharacteristic
486C4C4×#1171992211yescharacteristic
487C4#1681073311yescharacteristic
488C4#1681081511yescharacteristic
489C4#1681083511yescharacteristic
490C4#1681094511yescharacteristic
491C4#1681094611yescharacteristic
492C2—162no—
493C2—4242no—
494C2#2027377811yescharacteristic
495C2#2027379711yescharacteristic
496C2#2027380211yescharacteristic
497C2#2027382411yescharacteristic
498C2#2027503411yescharacteristic
499C2#2027506611yescharacteristic
500C2#2027508811yescharacteristic
subgroupG/H
centerC23#13133185
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1982C2
socleC23×C53#120
prime radicalC2×#16810815C2
cosocleC2×C42×C53C23
hypercenter#1982D53
hypocenterC53#8980
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1982

Complements

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

NH ≅ G/Ntype
2. #20277662C2semidirect
3. #20275522C2semidirect
4. C2×#16810350C2semidirect
5. #20272599C2semidirect
6. #20272556C2semidirect
7. C53×#1982C2semidirect
12. #16810815C4semidirect
116. C4⋊C4×C53D8semidirect
244. #1982D53semidirect
278. C53#8980semidirect
381. C4⋊C4#11719922semidirect

Maximal subgroups

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

subgroupindex
C2×#168108152
#202776622
#202755222
C2×#168103502
#202725992
#202725562
C53×#19822
#898053