Finite Group Databasebeta

← #27369272

Subgroups — #27369272

Order 13568.

792 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285630C211yesmaximal, characteristic, complemented
2C2×#16810818C211yesmaximal, characteristic, complemented
3#20283739C211yesmaximal, characteristic, complemented
4#20274154C214yesmaximal, complemented
5#20274068C214yesmaximal, complemented
6#20273491C211yesmaximal, characteristic, complemented
7#20273384C211yesmaximal, characteristic, complemented
8#20273364C211yesmaximal, characteristic, complemented
9C53×#3036C211yesmaximal, characteristic, complemented
10C22×#11719922—21no—
11C2×#13133214C2211yescharacteristic
12#16810999C2212yescomplemented
13#16810971C2212yescomplemented
14#16810818—22no—
15#16810818—22no—
16#16810818C2214yes—
17#16810756—22no—
18D4×#88322—22no—
19D4×#88322C2214yescomplemented
20#16810676C2211yescharacteristic
21#16810676C2211yescharacteristic
22C2×C4⋊#88322C2211yescharacteristic
23C2×#13133137—21no—
24C2×C22⋊#88322—22no—
25#16810375C2214yescomplemented
26#16810369C2212yescomplemented
27#16810367C2212yescomplemented
28#16810366C2214yescomplemented
29C2×AGL(1,ℤ/8)×C53—21no—
30C53×#532C2211yescharacteristic
31C2×C4×D4×C53C2211yescharacteristic
32C53×#463—22no—
33C53×#461C2212yescomplemented
34C53×#459C2212yescomplemented
35C53×#423C2211yescharacteristic
36C53×#417—21no—
37C22×D212—21no—
38C22×D212—21no—
39C22×D212—21no—
40C2×#11719922—24no—
41C2×#11719922—24no—
42C2×#11719922—44no—
43C22×#88322—24no—
44C22×#88322—21no—
45#13133214—21no—
46#13133214—21no—
47#13133214—21no—
48#13133214C2312yes—
49C4⋊#88322—21no—
50C4⋊#88322D412yes—
51C4⋊#88322C2312yes—
52C4⋊#88322C2312yes—
53#13133137—24no—
54#13133137—24no—
55C22⋊#88322—24no—
56C22⋊#88322—21no—
57C22⋊#88322D412yes—
58C4×#88322—21no—
59C4×#88322C2312yes—
60C2×#41×C53—21no—
61C22×D4×C53D411yescharacteristic
62C23×C4×C53—21no—
63AGL(1,ℤ/8)×C53—44no—
64C2×SD16×C53—22no—
65C2×D8×C53—22no—
66C2×M16×C53—21no—
67C4⋊Q8×C53C2311yescharacteristic
68#127×C53C2311yescharacteristic
69#124×C53—21no—
70#121×C53—21no—
71#121×C53—21no—
72C4×D4×C53—21no—
73C4×D4×C53—21no—
74C4×D4×C53—21no—
75C4×D4×C53—21no—
76C4×D4×C53D412yes—
77C4×D4×C53C2312yes—
78C2×C4⋊C4×C53D411yescharacteristic
79C2×C22⋊C4×C53—21no—
80C2×C42×C53C2311yescharacteristic
81C4⋊C8×C53—21no—
82C4⋊C8×C53C2312yes—
83#103×C53—22no—
84#102×C53—22no—
85C24×C53—21no—
86C23×D53—41no—
87#41×C53—42no—
88#41×C53—42no—
89C2×Q8×C53—21no—
90C2×D4×C53—21no—
91C2×D4×C53—22no—
92C2×D4×C53—21no—
93C2×D4×C53—21no—
94C2×D4×C53—21no—
95C2×D4×C53—41no—
96C2×D4×C53D814yescomplemented
97C2×D4×C53C2×D412yes—
98C22×C4×C53—21no—
99C22×C4×C53—22no—
100C22×C4×C53—22no—
101C22×C4×C53—41no—
102C22×C4×C53—82no—
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53#4111yescharacteristic
106C2×D212—22no—
107C2×D212—22no—
108C2×D212—22no—
109C2×D212—42no—
110C2×D212—42no—
111C2×D212—41no—
112C2×D212—41no—
113C2×D212—41no—
114C2×D212—82no—
115C2×D212—82no—
116SD16×C53—44no—
117D8×C53—44no—
118M16×C53—42no—
119C2×C8×C53—22no—
120C4⋊C4×C53—21no—
121C4⋊C4×C53—21no—
122C4⋊C4×C53—21no—
123C4⋊C4×C53C2×D412yes—
124C22⋊C4×C53—22no—
125C22⋊C4×C53—22no—
126C22⋊C4×C53—41no—
127C22⋊C4×C53—41no—
128C42×C53C2×D411yescharacteristic
129C42×C53C2×D411yescharacteristic
130C42×C53#4111yescharacteristic
131C42×C53C2411yescharacteristic
132#11719922—48no—
133#11719922—84no—
134C2×#88322—22no—
135C2×#88322—22no—
136C2×#88322—42no—
137C2×#88322—84no—
138C2×#88322C2×D414yes—
139C23×C53—22no—
140C23×C53—24no—
141C23×C53—41no—
142C23×C53—81no—
143C23×C53—81no—
144C23×C53#12111yescharacteristic
145C22×D53—42no—
146C22×D53—82no—
147C22×D53—82no—
148C22×D53—82no—
149Q8×C53—41no—
150Q8×C53—41no—
151D4×C53—22no—
152D4×C53—24no—
153D4×C53—41no—
154D4×C53—41no—
155D4×C53—42no—
156D4×C53—81no—
157D4×C53—82no—
158D4×C53C2×D814yes—
159C2×C4×C53—22no—
160C2×C4×C53—22no—
161C2×C4×C53—41no—
162C2×C4×C53—44no—
163C2×C4×C53—41no—
164C2×C4×C53—82no—
165C2×C4×C53—81no—
166C2×C4×C53—82no—
167C2×C4×C53#12111yescharacteristic
168C2×C4×C53#12111yescharacteristic
169C2×C4×C53#12111yescharacteristic
170C2×C4×C53C2×D812yes—
171C2×C4×C53#13512yes—
172C2×C4×C53C22×D411yescharacteristic
173C2×C4×C53C22×D411yescharacteristic
174C2×C4×C53C2×#4111yescharacteristic
175D212—42no—
176D212—42no—
177D212—44no—
178D212—82no—
179D212—82no—
180D212—82no—
181D212—82no—
182C8×C53—42no—
183#88322—24no—
184#88322—42no—
185#88322—42no—
186#19386—531nomaximal, Sylow
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—24no—
190C22×C53—42no—
191C22×C53—42no—
192C22×C53—84no—
193C22×C53—81no—
194C22×C53—81no—
195C22×C53—81no—
196C22×C53#46612yes—
197C22×C53C2×#12111yescharacteristic
198C2×D53—82no—
199C2×D53—82no—
200C2×D53—82no—
201C2×D53—164no—
202C4×C53—21no—
203C4×C53—21no—
204C4×C53—21no—
205C4×C53—42no—
206C4×C53—81no—
207C4×C53#46612yes—
208C4×C53C22×D811yescharacteristic
209C4×C53C2×#13511yescharacteristic
210C4×C53#57612yes—
211#3205—531no—
212#3036D5311yescharacteristic, complemented
213C2×#459—531no—
214#2972—531no—
215#1681—534no—
216#1667—534no—
217#1571—531no—
218#1545—531no—
219#1540—531no—
220C2×C53—22no—
221C2×C53—44no—
222C2×C53—161no—
223C2×C53#304111yescharacteristic
224C2×C53C2×#46611yescharacteristic
225C2×C53#306811yescharacteristic
226D53—162no—
227C2×AGL(1,ℤ/8)—21no—
228C22×D8—1061no—
229#532C2×D5311yescharacteristic
230C2×#127—531no—
231C2×C4×D4C2×D5311yescharacteristic
232#493—532no—
233#486—532no—
234#463—22no—
235#461C2×D5312yescomplemented
236#459—534no—
237#459—1062no—
238#459—1062no—
239#459C2×D5312yescomplemented
240#447—1062no—
241C8×D4—534no—
242C8×D4—1062no—
243#431—531no—
244#431—531no—
245#423C2×D5311yescharacteristic
246C2×C4⋊C8—531no—
247#417—21no—
248C2×#102—1061no—
249C2×C22⋊C8—1062no—
250#335—534no—
251#332—532no—
252#331—532no—
253#331—534no—
254C53#1938611yescharacteristic, complemented, Sylow
255C2×#41—21no—
256C22×D4—1061no—
257C22×D4—1061no—
258C22×D4—1061no—
259C22×D4C22⋊D5311yescharacteristic
260C23×C4—21no—
261AGL(1,ℤ/8)—44no—
262C2×SD16—22no—
263C2×D8—22no—
264C2×D8—1064no—
265C2×D8—1064no—
266C2×D8—2124no—
267C2×M16—21no—
268C22×C8—1064no—
269C22×C8—1061no—
270C4⋊Q8C22×D5311yescharacteristic
271#127—532no—
272#127—1061no—
273#127—1061no—
274#127—1061no—
275#127C22×D5311yescharacteristic
276#124—21no—
277#121—21no—
278#121—21no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4—21no—
282C4×D4—21no—
283C4×D4C22⋊D5312yes—
284C4×D4C22×D5312yes—
285C2×C4⋊C4C22⋊D5311yescharacteristic
286C2×C22⋊C4—21no—
287C2×C42C22×D5311yescharacteristic
288C4⋊C8—21no—
289C4⋊C8—532no—
290C4⋊C8—532no—
291C4⋊C8—532no—
292C4⋊C8—1061no—
293C4⋊C8C22×D5312yes—
294#103—22no—
295#102—22no—
296#102—1064no—
297#102—1064no—
298C22⋊C8—532no—
299C22⋊C8—1064no—
300C22⋊C8—1061no—
301C4×C8—532no—
302C4×C8—1061no—
303C24—21no—
304C24—2121no—
305#41—42no—
306#41—42no—
307C2×Q8—21no—
308C2×D4—21no—
309C2×D4—22no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—41no—
314C2×D4—1062no—
315C2×D4—1062no—
316C2×D4—1062no—
317C2×D4—4242no—
318C2×D4—4242no—
319C2×D4—2121no—
320C2×D4—2121no—
321C2×D4—2121no—
322C2×D4—2122no—
323C2×D4—2122no—
324C2×D4#1171992214yescomplemented
325C2×D4C2×C22⋊D5312yes—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—41no—
330C22×C4—82no—
331C22×C4D4×D5311yescharacteristic
332C22×C4#1171994711yescharacteristic
333C22×C4C2×C22⋊D5311yescharacteristic
334SD16—44no—
335D8—44no—
336D8—4244no—
337D8—2128no—
338M16—42no—
339C2×C8—22no—
340C2×C8—534no—
341C2×C8—1062no—
342C2×C8—1062no—
343C2×C8—4244no—
344C2×C8—2122no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4C2×C22⋊D5312yes—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C42D4×D5311yescharacteristic
354C42#1171994711yescharacteristic
355C42C2×C22⋊D5311yescharacteristic
356C42C23×D5311yescharacteristic
357C23—22no—
358C23—24no—
359C23—41no—
360C23—81no—
361C23—81no—
362C23—4242no—
363C23—4242no—
364C23—4242no—
365C23—2122no—
366C23#1313318911yescharacteristic
367Q8—41no—
368Q8—41no—
369D4—24no—
370D4—22no—
371D4—41no—
372D4—41no—
373D4—42no—
374D4—81no—
375D4—82no—
376D4—4242no—
377D4—4242no—
378D4—4242no—
379D4—4242no—
380D4—2122no—
381D4—2122no—
382D4—2124no—
383D4C2×#1171992214yes—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—41no—
387C2×C4—44no—
388C2×C4—41no—
389C2×C4—82no—
390C2×C4—81no—
391C2×C4—82no—
392C2×C4#1313318911yescharacteristic
393C2×C4#1313318911yescharacteristic
394C2×C4#1313318911yescharacteristic
395C2×C4C2×#1171992212yes—
396C2×C4#1313323712yes—
397C2×C4C2×D4×D5311yescharacteristic
398C2×C4C22×C22⋊D5311yescharacteristic
399C2×C4C2×#1171994711yescharacteristic
400C8—42no—
401C8—1064no—
402C8—2122no—
403C8—2122no—
404C22—21no—
405C22—21no—
406C22—24no—
407C22—42no—
408C22—42no—
409C22—84no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22—8484no—
414C22—4242no—
415C22—4242no—
416C22—4242no—
417C22#1681085712yes—
418C22C2×#1313318911yescharacteristic
419C4—21no—
420C4—21no—
421C4—21no—
422C4—42no—
423C4—81no—
424C4#1681085712yes—
425C4C22×#1171992211yescharacteristic
426C4C2×#1313323711yescharacteristic
427C4#1681142712yes—
428C2—22no—
429C2—44no—
430C2—161no—
431C2—8482no—
432C2#2028422011yescharacteristic
433C2C2×#1681085711yescharacteristic
434C2#2028444511yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3036C2
socleC22×C53C2×#121
hypercenter#3036D53
hypocenterC53#19386
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
1. #20285630C2semidirect
2. C2×#16810818C2semidirect
3. #20283739C2semidirect
4. #20274154C2semidirect
5. #20274068C2semidirect
6. #20273491C2semidirect
7. #20273384C2semidirect
8. #20273364C2semidirect
9. C53×#3036C2semidirect
12. #16810999C22semidirect
13. #16810971C22semidirect
19. D4×#88322C22semidirect
25. #16810375C22semidirect
26. #16810369C22semidirect
27. #16810367C22semidirect
28. #16810366C22semidirect
33. C53×#461C22semidirect
34. C53×#459C22semidirect
96. C2×D4×C53D8semidirect
212. #3036D53semidirect
235. #461C2×D53semidirect
239. #459C2×D53semidirect
254. C53#19386semidirect
324. C2×D4#11719922semidirect

Maximal subgroups

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

subgroupindex
#202856302
C2×#168108182
#202837392
#202741542
#202740682
#202734912
#202733842
#202733642
C53×#30362
#1938653