Finite Group Databasebeta

← #27391295

Subgroups — #27391295

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×#16810941C211yesmaximal, characteristic
2C4×#13133191C211yesmaximal, characteristic
3#20275335C212yesmaximal, complemented
4#20275300C212yesmaximal, complemented
5#20275032C211yesmaximal, characteristic
6#20274903C211yesmaximal, characteristic, complemented
7#20274805C211yesmaximal, characteristic, complemented
8#20274613C212yesmaximal, complemented
9#20274543C212yesmaximal, complemented
10#20274477C211yesmaximal, characteristic, complemented
11C2×C4×C4⋊C4×C53C211yesmaximal, characteristic, complemented
12C2×#13133191C2211yescharacteristic
13C2×C4×D212C2211yescharacteristic
14C2×C4×C22⋊D53—21no—
15#16810941C418yescomplemented
16C2×#13133150C2212yes—
17C2×#13133148C2212yes—
18C2×#13133137C2212yes—
19C2×#13133137C2212yes—
20C2×C22⋊#88322C2211yescharacteristic
21C2×C22⋊#88322C2211yescharacteristic
22C2×#13133128C2212yes—
23#16810525C2211yescharacteristic
24#16810521C2211yescharacteristic
25#16810514—21no—
26#16810502—21no—
27#16810483C2211yescharacteristic
28C4×C4⋊Dic53C2211yescharacteristic
29C4×#11719921—21no—
30#16810385C2212yes—
31#16810385C2212yes—
32#16810385C2212yes—
33#16810385C2212yes—
34#16810378C2212yes—
35C22×C4⋊C4×C53C2211yescharacteristic
36C22×C4⋊C4×C53C2211yescharacteristic
37C22×C42×C53—21no—
38C22×C42×C53C2211yescharacteristic
39C4×C4⋊C4×C53—21no—
40C4×C4⋊C4×C53—21no—
41C4×C4⋊C4×C53C2212yes—
42C4×C4⋊C4×C53C2212yes—
43C2×#95×C53—21no—
44C22×D212C2311yescharacteristic
45C22×C22⋊D53—21no—
46C22×C4×D53—21no—
47C22×#88322C2312yes—
48C22×#88322C2312yes—
49#13133191—22no—
50#13133191C2×C414yes—
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#13133148C2×C418yescomplemented
65#13133137—24no—
66#13133137C2×C418yescomplemented
67C22⋊#88322—22no—
68C22⋊#88322C2×C414yescomplemented
69#13133128—24no—
70#13133124—21no—
71#13133124—21no—
72#13133124—21no—
73C23×C4×C53—21no—
74C23×C4×C53—21no—
75C23×C4×C53D411yescharacteristic
76C23×C4×C53D411yescharacteristic
77C23×C4×C53C2311yescharacteristic
78C2×C4⋊C4×C53—24no—
79C2×C4⋊C4×C53—21no—
80C2×C4⋊C4×C53—21no—
81C2×C4⋊C4×C53—22no—
82C2×C4⋊C4×C53C2×C414yescomplemented
83C2×C4⋊C4×C53C2312yes—
84C2×C4⋊C4×C53C2312yes—
85C2×C42×C53—24no—
86C2×C42×C53—22no—
87C2×C42×C53—22no—
88C2×C42×C53—44no—
89C2×C42×C53D411yescharacteristic
90C2×C42×C53D411yescharacteristic
91C2×C42×C53C2311yescharacteristic
92C2×C42×C53C2311yescharacteristic
93#95×C53—22no—
94#95×C53—22no—
95C24×C53C2×D411yescharacteristic
96C23×D53—21no—
97C22×C4×C53—22no—
98C22×C4×C53—22no—
99C22×C4×C53—22no—
100C22×C4×C53—22no—
101C22×C4×C53—22no—
102C22×C4×C53—24no—
103C22×C4×C53—24no—
104C22×C4×C53—22no—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—41no—
109C22×C4×C53—42no—
110C22×C4×C53—42no—
111C22×C4×C53—42no—
112C22×C4×C53—42no—
113C22×C4×C53C22×C412yes—
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53C2×D411yescharacteristic
118C22×C4×C53C2×D411yescharacteristic
119C22×C4×C53#4112yes—
120C22×C4×C53C2411yescharacteristic
121C2×C22⋊D53—24no—
122C2×C22⋊D53—42no—
123C2×C22⋊D53—42no—
124C22×Dic53—21no—
125C22×Dic53—21no—
126C22×Dic53—21no—
127C2×D212—22no—
128C2×D212—22no—
129C2×D212—21no—
130C2×D212C22×C412yes—
131C2×C4×D53—42no—
132C2×C4×D53—42no—
133C2×C4×D53—42no—
134C4⋊C4×C53—24no—
135C4⋊C4×C53—24no—
136C4⋊C4×C53—22no—
137C4⋊C4×C53—22no—
138C4⋊C4×C53C22×C414yes—
139C4⋊C4×C53#4114yes—
140C42×C53—22no—
141C42×C53—22no—
142C42×C53—22no—
143C42×C53—44no—
144C42×C53—44no—
145C42×C53#4114yes—
146C22⋊Dic53—42no—
147#11719921—24no—
148#11719921—42no—
149#11719921—42no—
150#11719921—42no—
151C4⋊Dic53—21no—
152C4⋊Dic53—22no—
153C4⋊Dic53—22no—
154C4⋊Dic53C22×C412yes—
155#11719919—42no—
156C4×Dic53—42no—
157C2×#88322—22no—
158C2×#88322—22no—
159C2×#88322—22no—
160C2×#88322—22no—
161C2×#88322—22no—
162C2×#88322C22×C414yes—
163C23×C53—21no—
164C23×C53—21no—
165C23×C53—21no—
166C23×C53—21no—
167C23×C53C4×D412yes—
168C23×C53AGL(1,ℤ/8)12yes—
169C23×C53#13712yes—
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—28no—
179C2×C4×C53—21no—
180C2×C4×C53—21no—
181C2×C4×C53—21no—
182C2×C4×C53—21no—
183C2×C4×C53—21no—
184C2×C4×C53—21no—
185C2×C4×C53—21no—
186C2×C4×C53—21no—
187C2×C4×C53—24no—
188C2×C4×C53—24no—
189C2×C4×C53—24no—
190C2×C4×C53—21no—
191C2×C4×C53—24no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—44no—
195C2×C4×C53—48no—
196C2×C4×C53—48no—
197C2×C4×C53—42no—
198C2×C4×C53—42no—
199C2×C4×C53—42no—
200C2×C4×C53—42no—
201C2×C4×C53C4×D412yes—
202C2×C4×C53#12312yes—
203C2×C4×C53#12312yes—
204C2×C4×C53AGL(1,ℤ/8)12yes—
205C2×C4×C53C23×C411yescharacteristic
206C2×C4×C53C2×#4112yes—
207C2×C4×C53C2×#4112yes—
208C2×C4×C53C2×#4111yescharacteristic
209C22⋊D53—48no—
210C2×Dic53—22no—
211C2×Dic53—41no—
212C2×Dic53—41no—
213C2×Dic53—41no—
214C2×Dic53—41no—
215C2×Dic53—41no—
216C2×Dic53—41no—
217C2×Dic53—41no—
218C2×Dic53—41no—
219D212—24no—
220D212—42no—
221C4×D53—84no—
222#88322—24no—
223#88322—42no—
224#21580—531nomaximal, Sylow
225C22×C53—22no—
226C22×C53—22no—
227C22×C53—22no—
228C22×C53—22no—
229C22×C53—22no—
230C22×C53—22no—
231C22×C53#44014yes—
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—22no—
248C4×C53—22no—
249C4×C53—22no—
250C4×C53—41no—
251C4×C53—44no—
252C4×C53—44no—
253C4×C53C4×#4112yes—
254C4×C53#52912yes—
255Dic53—42no—
256Dic53—81no—
257Dic53—81no—
258C2×#481—531no—
259C4×#121—531no—
260C2×C4×C4⋊C4D5311yescharacteristic, complemented
261#1925—532no—
262#1920—532no—
263#1874—531no—
264#1852—531no—
265#1824—531no—
266#1776—532no—
267#1762—532no—
268#1749—531no—
269C2×C53—24no—
270C2×C53C4×#12311yescharacteristic
271C2×C53C2×#44011yescharacteristic
272C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
273C2×C53#309211yescharacteristic
274C2×C53#309411yescharacteristic
275C2×C53#310911yescharacteristic
276C2×C53#311711yescharacteristic
277D53—82no—
278C2×#121—531no—
279C2×C4×D4—531no—
280C2×C4×D4—1061no—
281C22×C4⋊C4C2×D5311yescharacteristic
282C22×C4⋊C4C2×D5311yescharacteristic
283C22×C42—21no—
284C22×C42C2×D5311yescharacteristic
285#481—538no—
286C2×#107—532no—
287C2×#106—532no—
288C2×#102—532no—
289C2×#102—532no—
290C2×C22⋊C8—531no—
291C2×C22⋊C8—531no—
292C2×#97—532no—
293#390—531no—
294#389—531no—
295#388—1061no—
296#386—1061no—
297#380—531no—
298C4×C4⋊C4—21no—
299C4×C4⋊C4—21no—
300C4×C4⋊C4—531no—
301C4×C4⋊C4C2×D5312yes—
302C4×C4⋊C4C2×D5312yes—
303C4×C22⋊C4—1061no—
304C2×#95—21no—
305#340—532no—
306#340—532no—
307#340—532no—
308#340—532no—
309#336—532no—
310C53#2158011yescharacteristic, complemented, Sylow
311C22×D4—531no—
312C22×D4—1061no—
313C23×C4—21no—
314C23×C4—21no—
315C23×C4—1061no—
316C23×C4C22⋊D5311yescharacteristic
317C23×C4C22⋊D5311yescharacteristic
318C23×C4C22×D5311yescharacteristic
319C22×C8—532no—
320C22×C8—532no—
321#121—534no—
322#121—1062no—
323C4×D4—1064no—
324C4×D4—2124no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—531no—
330C2×C4⋊C4—531no—
331C2×C4⋊C4—531no—
332C2×C4⋊C4—1061no—
333C2×C4⋊C4C4×D5314yescomplemented
334C2×C4⋊C4C22×D5312yes—
335C2×C4⋊C4C22×D5312yes—
336C2×C22⋊C4—1061no—
337C2×C22⋊C4—1061no—
338C2×C22⋊C4—1061no—
339C2×C22⋊C4—1061no—
340C2×C22⋊C4—1061no—
341C2×C42—24no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—44no—
345C2×C42—1061no—
346C2×C42C22⋊D5311yescharacteristic
347C2×C42C22⋊D5311yescharacteristic
348C2×C42C22×D5311yescharacteristic
349C2×C42C22×D5311yescharacteristic
350#107—1064no—
351#106—538no—
352#102—538no—
353#102—1064no—
354C22⋊C8—534no—
355C22⋊C8—1062no—
356#97—1064no—
357#95—22no—
358#95—22no—
359#95—1061no—
360#95—1061no—
361#95—1061no—
362C24—1061no—
363C24C2×C22⋊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—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—1061no—
388C22×C4—1061no—
389C22×C4—1061no—
390C22×C4—2122no—
391C22×C4—2122no—
392C22×C4—2122no—
393C22×C4C2×C4×D5312yes—
394C22×C4#1171994512yes—
395C22×C4C2×C22⋊D5311yescharacteristic
396C22×C4C2×C22⋊D5311yescharacteristic
397C22×C4C2×C22⋊D5311yescharacteristic
398C22×C4C2×C22⋊D5311yescharacteristic
399C22×C4C2×C22⋊D5311yescharacteristic
400C22×C4C23×D5311yescharacteristic
401C2×C8—534no—
402C2×C8—1062no—
403C2×C8—1062no—
404C2×C8—1062no—
405C2×C8—1062no—
406C2×C8—1062no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—532no—
412C4⋊C4—1061no—
413C4⋊C4—1062no—
414C4⋊C4—1062no—
415C4⋊C4—2122no—
416C4⋊C4C2×C4×D5314yes—
417C4⋊C4#1171994514yes—
418C22⋊C4—1064no—
419C22⋊C4—2122no—
420C22⋊C4—2122no—
421C22⋊C4—2122no—
422C22⋊C4—2122no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—44no—
427C42—44no—
428C42—2122no—
429C42#1171994514yes—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—1062no—
435C23—2121no—
436C23—2121no—
437C23—2121no—
438C23—2121no—
439C23—2121no—
440C23—2121no—
441C23C4×C22⋊D5312yes—
442C23#1313324212yes—
443C23#1313324812yes—
444C23C22×C22⋊D5311yescharacteristic
445D4—1064no—
446D4—2128no—
447D4—2122no—
448C2×C4—28no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—21no—
461C2×C4—24no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—44no—
465C2×C4—48no—
466C2×C4—48no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—1062no—
472C2×C4—4244no—
473C2×C4—2121no—
474C2×C4—2121no—
475C2×C4—2121no—
476C2×C4—2121no—
477C2×C4—2121no—
478C2×C4—2121no—
479C2×C4—2121no—
480C2×C4—2121no—
481C2×C4C4×C22⋊D5312yes—
482C2×C4#1313320012yes—
483C2×C4#1313320012yes—
484C2×C4#1313324312yes—
485C2×C4C22×C4×D5311yescharacteristic
486C2×C4C2×#1171994512yes—
487C2×C4C2×#1171994512yes—
488C2×C4C2×#1171994511yescharacteristic
489C8—1064no—
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#1681072114yes—
505C22C2×C4×C22⋊D5311yescharacteristic
506C22C2×#1313320011yescharacteristic
507C22#1681113411yescharacteristic
508C22C2×#1313324211yescharacteristic
509C22C2×#1313324311yescharacteristic
510C22C2×#1313324811yescharacteristic
511C22#1681142111yescharacteristic
512C4—22no—
513C4—22no—
514C4—22no—
515C4—22no—
516C4—44no—
517C4—44no—
518C4—41no—
519C4—4241no—
520C4—4241no—
521C4—2122no—
522C4C4×#1171994512yes—
523C4#1681115712yes—
524C2—24no—
525C2—4242no—
526C2C4×#1313320011yescharacteristic
527C2C2×#1681072111yescharacteristic
528C2C4×#1313324311yescharacteristic
529C2#2028463811yescharacteristic
530C2#2028465511yescharacteristic
531C2#2028480611yescharacteristic
532C2#2028488611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C4×C4⋊C4×C53C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
cosocleC22×C4×C53C24
hypercenterC2×C4×C4⋊C4D53
hypocenterC53#21580
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #20275335C2semidirect
4. #20275300C2semidirect
6. #20274903C2semidirect
7. #20274805C2semidirect
8. #20274613C2semidirect
9. #20274543C2semidirect
10. #20274477C2semidirect
11. C2×C4×C4⋊C4×C53C2semidirect
15. #16810941C4semidirect
64. #13133148C2×C4semidirect
66. #13133137C2×C4semidirect
68. C22⋊#88322C2×C4semidirect
82. C2×C4⋊C4×C53C2×C4semidirect
260. C2×C4×C4⋊C4D53semidirect
310. C53#21580semidirect
333. C2×C4⋊C4C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109412
C4×#131331912
#202753352
#202753002
#202750322
#202749032
#202748052
#202746132
#202745432
#202744772
C2×C4×C4⋊C4×C532
#2158053