Finite Group Databasebeta

← #27391270

Subgroups — #27391270

Order 13568.

1048 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810937C211yesmaximal, characteristic
2#20277221C211yesmaximal, characteristic
3C2×C4⋊C4×Dic53C211yesmaximal, characteristic, complemented
4#20275341C212yesmaximal, complemented
5#20275306C212yesmaximal, complemented
6#20275032C211yesmaximal, characteristic
7#20274799C211yesmaximal, characteristic, complemented
8#20274615C212yesmaximal, complemented
9#20274539C212yesmaximal, complemented
10#20274479C211yesmaximal, characteristic, complemented
11C53×#1853C211yesmaximal, characteristic, complemented
12C2×#13133191C2211yescharacteristic
13C2×#13133179C2211yescharacteristic
14C2×#13133177—21no—
15C22×C4⋊Dic53C2211yescharacteristic
16C22×C4×Dic53—21no—
17C22×C4×Dic53C2211yescharacteristic
18#16810937C418yescomplemented
19C2×#13133151C2212yes—
20C2×#13133139C2212yes—
21C2×#13133137C2212yes—
22C2×C22⋊#88322C2211yescharacteristic
23C2×#13133127C2212yes—
24#16810524C2211yescharacteristic
25#16810519C2211yescharacteristic
26#16810512—21no—
27#16810500—21no—
28C4⋊C4×Dic53—21no—
29C4⋊C4×Dic53—21no—
30C4⋊C4×Dic53C2212yes—
31C4⋊C4×Dic53C2212yes—
32C4⋊C4×Dic53C2211yescharacteristic
33#16810470—21no—
34C2×#13133124—21no—
35#16810385C2212yes—
36#16810385C2212yes—
37#16810385C2212yes—
38#16810377C2212yes—
39C22×C4⋊C4×C53C2211yescharacteristic
40C2×#106×C53C2212yes—
41C2×C22⋊C8×C53C2211yescharacteristic
42C53×#380C2211yescharacteristic
43C53×#340C2212yes—
44C22×D212C2311yescharacteristic
45C22×C22⋊D53—21no—
46C23×Dic53—21no—
47C23×Dic53D411yescharacteristic
48C23×Dic53D411yescharacteristic
49C22×C4×D53—21no—
50C22×#88322C2312yes—
51#13133191—22no—
52#13133191C2×C414yes—
53#13133179—24no—
54#13133177—44no—
55C2×C4⋊Dic53—21no—
56C2×C4⋊Dic53—22no—
57C2×C4⋊Dic53C2×C414yescomplemented
58C2×C4⋊Dic53C2311yescharacteristic
59C2×C4⋊Dic53C2312yes—
60C2×#11719919—21no—
61C2×#11719921—21no—
62C2×#11719921—21no—
63C2×#11719921—21no—
64C2×#11719921—21no—
65C2×C4×Dic53—24no—
66C2×C4×Dic53—22no—
67C2×C4×Dic53—22no—
68C2×C4×Dic53—21no—
69C2×C4×Dic53—44no—
70C2×C4×Dic53D411yescharacteristic
71C2×C4×Dic53D411yescharacteristic
72C2×C4×Dic53C2311yescharacteristic
73C2×C4×Dic53C2311yescharacteristic
74#13133151C2×C418yescomplemented
75#13133139C2×C418yescomplemented
76#13133137—24no—
77C22⋊#88322—22no—
78#13133127—24no—
79#13133124—21no—
80#13133124—21no—
81#13133124—22no—
82#13133124—22no—
83C23×C4×C53—21no—
84C23×C4×C53C2311yescharacteristic
85C22×C8×C53C2312yes—
86C2×C4⋊C4×C53—24no—
87C2×C4⋊C4×C53—21no—
88C2×C4⋊C4×C53C2311yescharacteristic
89C2×C4⋊C4×C53C2311yescharacteristic
90C2×C4⋊C4×C53C2312yes—
91C2×C22⋊C4×C53—21no—
92#106×C53—24no—
93C22⋊C8×C53C2×C414yescomplemented
94#95×C53—21no—
95C24×C53C2×D411yescharacteristic
96C23×D53—21no—
97C22×C4×C53—22no—
98C22×C4×C53—22no—
99C22×C4×C53—24no—
100C22×C4×C53—22no—
101C22×C4×C53—21no—
102C22×C4×C53—21no—
103C22×C4×C53—42no—
104C22×C4×C53—42no—
105C22×C4×C53C22×C412yes—
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53#4112yes—
108C22×C4×C53C2411yescharacteristic
109C2×C22⋊D53—24no—
110C2×C22⋊D53—42no—
111C2×C22⋊D53—42no—
112C22×Dic53—22no—
113C22×Dic53—22no—
114C22×Dic53—22no—
115C22×Dic53—24no—
116C22×Dic53—22no—
117C22×Dic53—22no—
118C22×Dic53—21no—
119C22×Dic53—22no—
120C22×Dic53—41no—
121C22×Dic53—42no—
122C22×Dic53—42no—
123C22×Dic53C2×D411yescharacteristic
124C22×Dic53C2×D411yescharacteristic
125C22×Dic53C2×D411yescharacteristic
126C22×Dic53C2×D411yescharacteristic
127C2×D212—22no—
128C2×D212—22no—
129C2×D212—21no—
130C2×D212C22×C412yes—
131C2×C4×D53—42no—
132C2×C4×D53—42no—
133C2×C4×D53—42no—
134C2×C8×C53—22no—
135C2×C8×C53—22no—
136C2×C8×C53C22×C414yes—
137C4⋊C4×C53—24no—
138C4⋊C4×C53—22no—
139C4⋊C4×C53—22no—
140C4⋊C4×C53—22no—
141C4⋊C4×C53#4114yes—
142C22⋊C4×C53—42no—
143#11719921—24no—
144#11719921—42no—
145#11719921—42no—
146#11719921—42no—
147C4⋊Dic53—24no—
148C4⋊Dic53—22no—
149C4⋊Dic53—21no—
150C4⋊Dic53C22×C414yes—
151C4⋊Dic53C22×C412yes—
152#11719919—42no—
153C4×Dic53—22no—
154C4×Dic53—22no—
155C4×Dic53—22no—
156C4×Dic53—44no—
157C4×Dic53—44no—
158C4×Dic53—42no—
159C4×Dic53#4114yes—
160C2×#88322—22no—
161C2×#88322—22no—
162C2×#88322—22no—
163C23×C53—21no—
164C23×C53—21no—
165C23×C53—21no—
166C23×C53—21no—
167C23×C53C4×D412yes—
168C23×C53AGL(1,ℤ/8)12yes—
169C23×C53AGL(1,ℤ/8)12yes—
170C23×C53C22×D411yescharacteristic
171C22×D53—22no—
172C22×D53—41no—
173C22×D53—41no—
174C22×D53—41no—
175C22×D53—41no—
176C22×D53—41no—
177C22×D53—41no—
178C2×C4×C53—21no—
179C2×C4×C53—21no—
180C2×C4×C53—24no—
181C2×C4×C53—24no—
182C2×C4×C53—24no—
183C2×C4×C53—48no—
184C2×C4×C53—42no—
185C2×C4×C53—42no—
186C2×C4×C53—41no—
187C2×C4×C53—41no—
188C2×C4×C53—41no—
189C2×C4×C53—41no—
190C2×C4×C53—41no—
191C2×C4×C53—41no—
192C2×C4×C53C4×D412yes—
193C2×C4×C53C23×C411yescharacteristic
194C2×C4×C53C2×#4112yes—
195C2×C4×C53C2×#4112yes—
196C2×C4×C53C2×#4111yescharacteristic
197C22⋊D53—48no—
198C2×Dic53—28no—
199C2×Dic53—21no—
200C2×Dic53—21no—
201C2×Dic53—21no—
202C2×Dic53—21no—
203C2×Dic53—21no—
204C2×Dic53—21no—
205C2×Dic53—24no—
206C2×Dic53—22no—
207C2×Dic53—21no—
208C2×Dic53—21no—
209C2×Dic53—21no—
210C2×Dic53—44no—
211C2×Dic53—42no—
212C2×Dic53—42no—
213C2×Dic53—41no—
214C2×Dic53—41no—
215C2×Dic53—48no—
216C2×Dic53#12312yes—
217C2×Dic53#12312yes—
218C2×Dic53#13712yes—
219D212—24no—
220D212—42no—
221C4×D53—84no—
222C8×C53—24no—
223#88322—42no—
224#21578—531nomaximal, Sylow
225C22×C53—22no—
226C22×C53—22no—
227C22×C53—22no—
228C22×C53—22no—
229C22×C53—22no—
230C22×C53—22no—
231C22×C53#44214yes—
232C22×C53C2×C4×D411yescharacteristic
233C22×C53C2×#12311yescharacteristic
234C22×C53#52511yescharacteristic
235C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
236C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
237C22×C53C2×#13711yescharacteristic
238C22×C53#57511yescharacteristic
239C2×D53—42no—
240C2×D53—42no—
241C2×D53—42no—
242C2×D53—81no—
243C2×D53—81no—
244C2×D53—81no—
245C2×D53—81no—
246C4×C53—22no—
247C4×C53—44no—
248C4×C53—81no—
249C4×C53—81no—
250C4×C53C4×#4112yes—
251C4×C53#52912yes—
252Dic53—22no—
253Dic53—22no—
254Dic53—22no—
255Dic53—41no—
256Dic53—44no—
257Dic53—42no—
258C2×#480—531no—
259C4×#121—531no—
260C2×C4×C4⋊C4—531no—
261#1926—532no—
262#1921—532no—
263#1874—531no—
264#1853D5311yescharacteristic, complemented
265#1823—531no—
266#1776—532no—
267#1762—532no—
268#1749—531no—
269C2×C53—24no—
270C2×C53C4×#12311yescharacteristic
271C2×C53C2×#44211yescharacteristic
272C2×C53C4×#13711yescharacteristic
273C2×C53#309211yescharacteristic
274C2×C53#309411yescharacteristic
275C2×C53#311211yescharacteristic
276C2×C53#311411yescharacteristic
277D53—82no—
278C2×#121—531no—
279C2×C4×D4—531no—
280C2×C4×D4—1061no—
281C22×C4⋊C4—531no—
282C22×C4⋊C4C2×D5311yescharacteristic
283C22×C42—531no—
284C22×C42—1061no—
285#480—538no—
286C2×#107—532no—
287C2×#106C2×D5312yes—
288C2×#102—532no—
289C2×#102—532no—
290C2×C22⋊C8—531no—
291C2×C22⋊C8C2×D5311yescharacteristic
292C2×#97—532no—
293#390—531no—
294#389—531no—
295#388—1061no—
296#386—1061no—
297#380C2×D5311yescharacteristic
298C4×C4⋊C4—532no—
299C4×C4⋊C4—532no—
300C4×C4⋊C4—531no—
301C4×C4⋊C4—1061no—
302C4×C4⋊C4—1061no—
303C4×C22⋊C4—1061no—
304C2×#95—1061no—
305#340—532no—
306#340—532no—
307#340—532no—
308#340C2×D5312yes—
309#336—532no—
310C53#2157811yescharacteristic, complemented, Sylow
311C22×D4—531no—
312C22×D4—1061no—
313C23×C4—21no—
314C23×C4—531no—
315C23×C4—531no—
316C23×C4—1061no—
317C23×C4—1061no—
318C23×C4C22×D5311yescharacteristic
319C22×C8—532no—
320C22×C8C22×D5312yes—
321#121—534no—
322#121—1062no—
323C4×D4—1064no—
324C4×D4—2124no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—534no—
328C2×C4⋊C4—532no—
329C2×C4⋊C4—531no—
330C2×C4⋊C4—1061no—
331C2×C4⋊C4—1062no—
332C2×C4⋊C4—1061no—
333C2×C4⋊C4C22×D5311yescharacteristic
334C2×C4⋊C4C22×D5311yescharacteristic
335C2×C4⋊C4C22×D5312yes—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—1061no—
338C2×C22⋊C4—1061no—
339C2×C22⋊C4—1061no—
340C2×C22⋊C4—1061no—
341C2×C42—531no—
342C2×C42—531no—
343C2×C42—531no—
344C2×C42—531no—
345C2×C42—1064no—
346C2×C42—1061no—
347C2×C42—1062no—
348C2×C42—1062no—
349C2×C42—2124no—
350#107—538no—
351#106—24no—
352#102—538no—
353#102—1064no—
354C22⋊C8—1062no—
355C22⋊C8C4×D5314yescomplemented
356#97—1064no—
357#95—21no—
358#95—1062no—
359#95—1062no—
360#95—1061no—
361#95—1061no—
362C24—1061no—
363C24D4×D5311yescharacteristic
364C2×D4—532no—
365C2×D4—1064no—
366C2×D4—1062no—
367C2×D4—1062no—
368C2×D4—1061no—
369C2×D4—2122no—
370C2×D4—2122no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—531no—
380C22×C4—531no—
381C22×C4—531no—
382C22×C4—531no—
383C22×C4—1062no—
384C22×C4—1062no—
385C22×C4—1062no—
386C22×C4—1062no—
387C22×C4—1064no—
388C22×C4—1062no—
389C22×C4—1062no—
390C22×C4—1061no—
391C22×C4—2122no—
392C22×C4—2122no—
393C22×C4—2122no—
394C22×C4—2121no—
395C22×C4—2122no—
396C22×C4—2122no—
397C22×C4C2×C4×D5312yes—
398C22×C4D4×D5311yescharacteristic
399C22×C4#1171994912yes—
400C22×C4C23×D5311yescharacteristic
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—1062no—
404C2×C8—1062no—
405C2×C8—1062no—
406C2×C8C2×C4×D5314yes—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—534no—
412C4⋊C4—532no—
413C4⋊C4—1064no—
414C4⋊C4—1062no—
415C4⋊C4—1061no—
416C4⋊C4—2122no—
417C4⋊C4#1171994714yes—
418C22⋊C4—42no—
419C22⋊C4—1064no—
420C22⋊C4—2122no—
421C22⋊C4—2122no—
422C22⋊C4—2122no—
423C42—534no—
424C42—1062no—
425C42—1062no—
426C42—1062no—
427C42—2124no—
428C42—2124no—
429C42—2122no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—1062no—
435C23—2121no—
436C23—2121no—
437C23—2121no—
438C23—2121no—
439C23—2121no—
440C23—2121no—
441C23#1313317912yes—
442C23#1313324412yes—
443C23#1313324412yes—
444C23C2×D4×D5311yescharacteristic
445D4—1064no—
446D4—2128no—
447D4—2122no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—48no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—532no—
463C2×C4—532no—
464C2×C4—532no—
465C2×C4—1068no—
466C2×C4—1061no—
467C2×C4—1061no—
468C2×C4—1061no—
469C2×C4—1061no—
470C2×C4—1061no—
471C2×C4—1061no—
472C2×C4—1062no—
473C2×C4—1064no—
474C2×C4—1061no—
475C2×C4—1061no—
476C2×C4—1061no—
477C2×C4—4244no—
478C2×C4—2124no—
479C2×C4—2122no—
480C2×C4—2121no—
481C2×C4—2122no—
482C2×C4—2121no—
483C2×C4—2128no—
484C2×C4#1313317912yes—
485C2×C4C22×C4×D5311yescharacteristic
486C2×C4#41×D5312yes—
487C2×C4C2×#1171994712yes—
488C2×C4C2×#1171994911yescharacteristic
489C8—24no—
490C8—2122no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—4241no—
498C22—4241no—
499C22—4241no—
500C22—4241no—
501C22—2122no—
502C22—2122no—
503C22—2122no—
504C22#1681073414yes—
505C22C2×#1313317911yescharacteristic
506C22#123×D5311yescharacteristic
507C22#1681112911yescharacteristic
508C22C2×#1313324411yescharacteristic
509C22C2×#1313324411yescharacteristic
510C22#137×D5311yescharacteristic
511C22#1681141911yescharacteristic
512C4—22no—
513C4—44no—
514C4—81no—
515C4—81no—
516C4—1062no—
517C4—1062no—
518C4—1062no—
519C4—2124no—
520C4—2122no—
521C4—2121no—
522C4C4×#1171994712yes—
523C4#1681115912yes—
524C2—24no—
525C2—4242no—
526C2#2027723411yescharacteristic
527C2C2×#1681073411yescharacteristic
528C2#2028353311yescharacteristic
529C2#2028463911yescharacteristic
530C2#2028465211yescharacteristic
531C2#2028483211yescharacteristic
532C2#2028485411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1853C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
cosocleC22×C4×C53C24
hypercenter#1853D53
hypocenterC53#21578
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
3. C2×C4⋊C4×Dic53C2semidirect
4. #20275341C2semidirect
5. #20275306C2semidirect
7. #20274799C2semidirect
8. #20274615C2semidirect
9. #20274539C2semidirect
10. #20274479C2semidirect
11. C53×#1853C2semidirect
18. #16810937C4semidirect
57. C2×C4⋊Dic53C2×C4semidirect
74. #13133151C2×C4semidirect
75. #13133139C2×C4semidirect
93. C22⋊C8×C53C2×C4semidirect
264. #1853D53semidirect
310. C53#21578semidirect
355. C22⋊C8C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109372
#202772212
C2×C4⋊C4×Dic532
#202753412
#202753062
#202750322
#202747992
#202746152
#202745392
#202744792
C53×#18532
#2157853