Finite Group Databasebeta

← #27299816

Subgroups — #27299816

Order 13568.

934 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810436C211yesmaximal, characteristic, complemented
2C2×#16810430C211yesmaximal, characteristic, complemented
3#20276427C211yesmaximal, characteristic, complemented
4C2×#16810424C211yesmaximal, characteristic, complemented
5C2×#16810407C211yesmaximal, characteristic, complemented
6#20274779C211yesmaximal, characteristic, complemented
7C2×C53×#576C211yesmaximal, characteristic
8C2×#13133157C2211yescharacteristic, complemented
9C22×#11719908C2211yescharacteristic, complemented
10C2×#13133153C2211yescharacteristic, complemented
11C2×#13133151C2211yescharacteristic, complemented
12#16810624—21no—
13#16810436—22no—
14#16810430—22no—
15#16810427—21no—
16#16810424—22no—
17#16810424—21no—
18#16810407—22no—
19#16810391—21no—
20C2×H(5,𝔽2)×C53—21no—
21C53×#576—22no—
22C53×#576—22no—
23C53×#576—22no—
24C53×#576C414yescomplemented
25C2×AGL(1,ℤ/8)×C53—22no—
26C2×AGL(1,ℤ/8)×C53—21no—
27C2×#135×C53—21no—
28C2×#135×C53C2211yescharacteristic
29C22×D8×C53—21no—
30C22×D8×C53C2211yescharacteristic
31C2×#131×C53C2211yescharacteristic
32C2×#11719917—21no—
33C2×C4⋊Dic53—21no—
34#13133157—21no—
35#13133157—21no—
36C2×#11719908—21no—
37C2×#11719908—21no—
38C2×#11719908—21no—
39C2×#11719908—21no—
40#13133153—21no—
41#13133153—21no—
42#13133151—21no—
43#13133151—21no—
44#13133141—41no—
45#13133138—41no—
46H(5,𝔽2)×C53—24no—
47H(5,𝔽2)×C53—24no—
48H(5,𝔽2)×C53—42no—
49H(5,𝔽2)×C53—42no—
50C2×#41×C53—22no—
51C2×#41×C53—21no—
52C2×#41×C53—41no—
53C2×#41×C53D411yescharacteristic
54C22×D4×C53—22no—
55C22×D4×C53—21no—
56C22×D4×C53—42no—
57C22×D4×C53—41no—
58AGL(1,ℤ/8)×C53—28no—
59AGL(1,ℤ/8)×C53—24no—
60AGL(1,ℤ/8)×C53—28no—
61AGL(1,ℤ/8)×C53—24no—
62#135×C53—24no—
63#135×C53—24no—
64#135×C53—21no—
65#135×C53—21no—
66#135×C53C2×C412yes—
67#135×C53D412yes—
68C2×Q16×C53D411yescharacteristic
69C2×SD16×C53—22no—
70C2×SD16×C53—21no—
71C2×D8×C53—22no—
72C2×D8×C53—24no—
73C2×D8×C53—24no—
74C2×D8×C53—21no—
75C2×D8×C53—21no—
76C2×D8×C53—21no—
77C2×D8×C53C2×C412yes—
78C2×D8×C53D411yescharacteristic
79C2×D8×C53D412yes—
80C2×D8×C53D412yes—
81#131×C53—21no—
82#131×C53—21no—
83#131×C53C2×C412yes—
84#131×C53D412yes—
85C2×M16×C53—21no—
86C2×M16×C53D411yescharacteristic
87C22×C8×C53—21no—
88C22×C8×C53C2311yescharacteristic
89C24×C53—42no—
90C24×C53—41no—
91#41×C53—21no—
92#41×C53—21no—
93#41×C53—28no—
94#41×C53—22no—
95#41×C53—22no—
96#41×C53—44no—
97#41×C53—44no—
98#41×C53—44no—
99#41×C53—41no—
100#41×C53—41no—
101#41×C53C22⋊C412yes—
102#41×C53SD1612yes—
103C2×Q8×C53—21no—
104C2×Q8×C53D811yescharacteristic
105C2×D4×C53—21no—
106C2×D4×C53—28no—
107C2×D4×C53—22no—
108C2×D4×C53—21no—
109C2×D4×C53—22no—
110C2×D4×C53—22no—
111C2×D4×C53—42no—
112C2×D4×C53—44no—
113C2×D4×C53—48no—
114C2×D4×C53—48no—
115C2×D4×C53—44no—
116C2×D4×C53—42no—
117C2×D4×C53—41no—
118C2×D4×C53—44no—
119C2×D4×C53—41no—
120C2×D4×C53—41no—
121C2×D4×C53—41no—
122C2×D4×C53D811yescharacteristic
123C22×C4×C53—21no—
124C22×C4×C53—42no—
125C22×C4×C53—41no—
126C22×C4×C53C2×D411yescharacteristic
127C22×Dic53—41no—
128Q16×C53—21no—
129Q16×C53C22⋊C412yes—
130SD16×C53—28no—
131SD16×C53—22no—
132SD16×C53—22no—
133D8×C53—28no—
134D8×C53—22no—
135D8×C53—21no—
136D8×C53—22no—
137D8×C53—22no—
138D8×C53C22⋊C412yes—
139D8×C53C22⋊C414yes—
140M16×C53—21no—
141M16×C53—22no—
142M16×C53—22no—
143M16×C53C22⋊C412yes—
144C2×C8×C53—21no—
145C2×C8×C53—22no—
146C2×C8×C53—22no—
147C2×C8×C53C22×C411yescharacteristic
148C2×C8×C53C2×D411yescharacteristic
149C2×C8×C53C2×D411yescharacteristic
150C2×C8×C53C2×D411yescharacteristic
151C2×C8×C53C2×D411yescharacteristic
152C2×C8×C53C2×D411yescharacteristic
153C4⋊Dic53—41no—
154C4⋊Dic53—41no—
155#11719917—41no—
156#11719917—41no—
157C2×#88322—41no—
158#11719908—41no—
159#11719908—41no—
160C23×C53—21no—
161C23×C53—48no—
162C23×C53—42no—
163C23×C53—41no—
164C23×C53—42no—
165C23×C53—42no—
166C23×C53—161no—
167C23×C53—161no—
168C23×C53—162no—
169C23×C53—162no—
170C23×C53AGL(1,ℤ/8)11yescharacteristic
171Q8×C53—21no—
172Q8×C53—22no—
173Q8×C53—41no—
174Q8×C53#10212yes—
175D4×C53—22no—
176D4×C53—22no—
177D4×C53—22no—
178D4×C53—24no—
179D4×C53—21no—
180D4×C53—48no—
181D4×C53—48no—
182D4×C53—42no—
183D4×C53—42no—
184D4×C53—42no—
185D4×C53—42no—
186D4×C53—41no—
187D4×C53—42no—
188D4×C53#10212yes—
189C2×C4×C53—21no—
190C2×C4×C53—22no—
191C2×C4×C53—22no—
192C2×C4×C53—48no—
193C2×C4×C53—41no—
194C2×C4×C53—42no—
195C2×C4×C53—42no—
196C2×C4×C53C2×C22⋊C411yescharacteristic
197C2×C4×C53#12011yescharacteristic
198C2×C4×C53#12011yescharacteristic
199C2×C4×C53C2×D811yescharacteristic
200C2×C4×C53C2×SD1611yescharacteristic
201C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
202C2×Dic53—81no—
203C2×Dic53—161no—
204C8×C53—22no—
205C8×C53C2×C22⋊C411yescharacteristic
206C8×C53C2×C22⋊C411yescharacteristic
207C8×C53#12011yescharacteristic
208C8×C53#12011yescharacteristic
209#88322—81no—
210#12748—531nomaximal, Sylow
211C22×C53—21no—
212C22×C53—21no—
213C22×C53—22no—
214C22×C53—41no—
215C22×C53—41no—
216C22×C53—44no—
217C22×C53—42no—
218C22×C53—82no—
219C22×C53—81no—
220C22×C53—82no—
221C22×C53—81no—
222C22×C53—162no—
223C22×C53—162no—
224C22×C53—168no—
225C22×C53#41811yescharacteristic
226C22×C53#44711yescharacteristic
227C22×C53#45011yescharacteristic
228C4×C53—22no—
229C4×C53—42no—
230C4×C53#37911yescharacteristic
231C4×C53C2×#10211yescharacteristic
232C4×C53#44711yescharacteristic
233C4×C53#45011yescharacteristic
234Dic53—161no—
235C2×#576D5311yescharacteristic
236C2×#361—531no—
237C2×#359—531no—
238#2139—531no—
239C2×#357—531no—
240C2×#350—531no—
241#1816—531no—
242C2×C53—21no—
243C2×C53—21no—
244C2×C53—42no—
245C2×C53—84no—
246C2×C53—82no—
247C2×C53#178611yescharacteristic
248C2×C53#219211yescharacteristic
249C2×C53#219411yescharacteristic
250C2×H(5,𝔽2)—21no—
251#576—22no—
252#576—22no—
253#576—22no—
254#576Dic5314yescomplemented
255C2×AGL(1,ℤ/8)—22no—
256C2×AGL(1,ℤ/8)—21no—
257C2×#135—21no—
258C2×#135C2×D5311yescharacteristic
259C22×D8—21no—
260C22×D8C2×D5311yescharacteristic
261C2×#131C2×D5311yescharacteristic
262C2×M32—531no—
263C22×C16—531no—
264C2×#108—531no—
265C2×#107—531no—
266#417—1061no—
267#361—1062no—
268#359—1062no—
269#358—1061no—
270#357—1062no—
271#357—1061no—
272#350—1062no—
273#343—1061no—
274C53#1274811yescharacteristic, complemented, Sylow
275H(5,𝔽2)—24no—
276H(5,𝔽2)—24no—
277H(5,𝔽2)—42no—
278H(5,𝔽2)—42no—
279C2×#41—22no—
280C2×#41—21no—
281C2×#41—41no—
282C2×#41C22⋊D5311yescharacteristic
283C22×D4—22no—
284C22×D4—21no—
285C22×D4—42no—
286C22×D4—41no—
287AGL(1,ℤ/8)—24no—
288AGL(1,ℤ/8)—28no—
289AGL(1,ℤ/8)—28no—
290AGL(1,ℤ/8)—24no—
291#135—24no—
292#135—24no—
293#135—21no—
294#135—21no—
295#135C2×Dic5312yes—
296#135C22⋊D5312yes—
297C2×Q16C22⋊D5311yescharacteristic
298C2×SD16—22no—
299C2×SD16—21no—
300C2×D8—22no—
301C2×D8—24no—
302C2×D8—24no—
303C2×D8—21no—
304C2×D8—21no—
305C2×D8—21no—
306C2×D8C2×Dic5312yes—
307C2×D8C22⋊D5311yescharacteristic
308C2×D8C22⋊D5312yes—
309C2×D8C22⋊D5312yes—
310#131—21no—
311#131—21no—
312#131C2×Dic5312yes—
313#131C22⋊D5312yes—
314C2×M16—21no—
315C2×M16—1061no—
316C2×M16C22⋊D5311yescharacteristic
317C22×C8—21no—
318C22×C8C22×D5311yescharacteristic
319C2×C4⋊C4—1061no—
320M32—1061no—
321M32—1061no—
322C2×C16—1061no—
323C2×C16—1061no—
324C2×C16—1061no—
325C2×C16—1061no—
326#108—1061no—
327#108—1061no—
328#107—1061no—
329#107—1061no—
330#103—2121no—
331#102—2121no—
332C24—42no—
333C24—41no—
334#41—21no—
335#41—21no—
336#41—28no—
337#41—22no—
338#41—22no—
339#41—44no—
340#41—44no—
341#41—44no—
342#41—41no—
343#41—41no—
344#41#1171992312yes—
345#41C22⋊Dic5312yes—
346C2×Q8—21no—
347C2×Q8#1171992211yescharacteristic
348C2×D4—21no—
349C2×D4—28no—
350C2×D4—22no—
351C2×D4—21no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—42no—
355C2×D4—44no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—44no—
359C2×D4—42no—
360C2×D4—41no—
361C2×D4—44no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4#1171992211yescharacteristic
366C22×C4—21no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—2121no—
370C22×C4C2×C22⋊D5311yescharacteristic
371Q16—21no—
372Q16C22⋊Dic5312yes—
373SD16—28no—
374SD16—22no—
375SD16—22no—
376D8—28no—
377D8—22no—
378D8—21no—
379D8—22no—
380D8—22no—
381D8C22⋊Dic5312yes—
382D8C22⋊Dic5314yes—
383M16—21no—
384M16—22no—
385M16—22no—
386M16—2121no—
387M16—2121no—
388M16C22⋊Dic5312yes—
389C2×C8—21no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—2121no—
393C2×C8C22×Dic5311yescharacteristic
394C2×C8C2×C22⋊D5311yescharacteristic
395C2×C8C2×C22⋊D5311yescharacteristic
396C2×C8C2×C22⋊D5311yescharacteristic
397C2×C8C2×C22⋊D5311yescharacteristic
398C2×C8C2×C22⋊D5311yescharacteristic
399C4⋊C4—2121no—
400C4⋊C4—2121no—
401C16—2121no—
402C16—2121no—
403C23—21no—
404C23—48no—
405C23—42no—
406C23—41no—
407C23—42no—
408C23—42no—
409C23—161no—
410C23—161no—
411C23—162no—
412C23—162no—
413C23#1313324311yescharacteristic
414Q8—21no—
415Q8—22no—
416Q8—41no—
417Q8#1313313812yes—
418D4—22no—
419D4—22no—
420D4—21no—
421D4—22no—
422D4—24no—
423D4—48no—
424D4—48no—
425D4—42no—
426D4—42no—
427D4—42no—
428D4—41no—
429D4—42no—
430D4—42no—
431D4#1313313812yes—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—48no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—8481no—
440C2×C4—4241no—
441C2×C4C2×C22⋊Dic5311yescharacteristic
442C2×C4#1313318611yescharacteristic
443C2×C4#1313318611yescharacteristic
444C2×C4C2×#1171992211yescharacteristic
445C2×C4C2×#1171992311yescharacteristic
446C2×C4#1313324311yescharacteristic
447C8—22no—
448C8—4241no—
449C8C2×C22⋊Dic5311yescharacteristic
450C8C2×C22⋊Dic5311yescharacteristic
451C8#1313318611yescharacteristic
452C8#1313318611yescharacteristic
453C22—21no—
454C22—21no—
455C22—22no—
456C22—41no—
457C22—41no—
458C22—44no—
459C22—42no—
460C22—82no—
461C22—81no—
462C22—82no—
463C22—81no—
464C22—162no—
465C22—162no—
466C22—168no—
467C22#1681062811yescharacteristic
468C22#1681075711yescharacteristic
469C22#1681077411yescharacteristic
470C4—22no—
471C4—42no—
472C4—8481no—
473C4#1681047911yescharacteristic
474C4C2×#1313313811yescharacteristic
475C4#1681075711yescharacteristic
476C4#1681077411yescharacteristic
477C2—21no—
478C2—21no—
479C2—42no—
480C2—84no—
481C2—82no—
482C2#2027464611yescharacteristic
483C2#2027667211yescharacteristic
484C2#2027668611yescharacteristic
subgroupG/H
centerC22#16810757
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
Fitting, prime radicalC2×C53×#576C2
socleC22×C53#447
cosocleC22×C8×C53C23
hypercenterC2×#576D53
hypocenterC53#12748
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ C2×#576

Complements

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

NH ≅ G/Ntype
1. C2×#16810436C2semidirect
2. C2×#16810430C2semidirect
3. #20276427C2semidirect
4. C2×#16810424C2semidirect
5. C2×#16810407C2semidirect
6. #20274779C2semidirect
8. C2×#13133157C22semidirect
9. C22×#11719908C22semidirect
10. C2×#13133153C22semidirect
11. C2×#13133151C22semidirect
24. C53×#576C4semidirect
254. #576Dic53semidirect
274. C53#12748semidirect

Maximal subgroups

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

subgroupindex
C2×#168104362
C2×#168104302
#202764272
C2×#168104242
C2×#168104072
#202747792
C2×C53×#5762
#1274853