Finite Group Databasebeta

← #27369267

Subgroups — #27369267

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#20285633C211yesmaximal, characteristic, complemented
2#20284179C211yesmaximal, characteristic, complemented
3#20283739C211yesmaximal, characteristic, complemented
4#20274150C214yesmaximal, complemented
5#20274068C214yesmaximal, complemented
6#20273489C211yesmaximal, characteristic, complemented
7#20273383C211yesmaximal, characteristic, complemented
8#20273365C211yesmaximal, characteristic, complemented
9C2×C53×#459C211yesmaximal, characteristic, complemented
10C2×#13133243—21no—
11#16811173C2211yescharacteristic
12#16810999C2212yescomplemented
13#16810969C2212yescomplemented
14#16810838—22no—
15#16810828C2212yescomplemented
16#16810818C2212yescomplemented
17#16810815—22no—
18#16810757—22no—
19D4×#88322—22no—
20D4×#88322C2214yescomplemented
21#16810676C2211yescharacteristic
22#16810676C2211yescharacteristic
23#16810647C2211yescharacteristic
24#16810622—21no—
25C2×C22⋊#88322—22no—
26#16810375C2214yescomplemented
27#16810370C2212yescomplemented
28#16810367C2214yescomplemented
29#16810366C2212yescomplemented
30C22×D8×C53—21no—
31C2×#127×C53C2211yescharacteristic
32C2×C4×D4×C53C2211yescharacteristic
33C53×#459—22no—
34C53×#459C2214yes—
35C2×C4⋊C8×C53C2211yescharacteristic
36C2×#102×C53—21no—
37C2×#11719945—21no—
38#13133243—44no—
39C2×#11719923—22no—
40C2×#11719922—24no—
41C2×#11719922—22no—
42C2×#11719917—21no—
43C22×#88322—24no—
44#13133217C2311yescharacteristic
45#13133214C2311yescharacteristic
46#13133205—21no—
47#13133191—21no—
48#13133191—21no—
49C4×D212—21no—
50C4×D212—21no—
51C4⋊#88322—21no—
52C4⋊#88322D412yes—
53C4⋊#88322C2312yes—
54C4⋊#88322C2312yes—
55#13133140—22no—
56#13133138—24no—
57#13133137—22no—
58C22⋊#88322—24no—
59C22⋊#88322—21no—
60C22⋊#88322D412yes—
61C4×#88322—21no—
62C4×#88322C2312yes—
63C22×D4×C53—21no—
64C22×D4×C53—21no—
65C22×D4×C53—21no—
66C22×D4×C53D411yescharacteristic
67C23×C4×C53—21no—
68C2×D8×C53—24no—
69C2×D8×C53—44no—
70C22×C8×C53—21no—
71#127×C53—21no—
72#127×C53—21no—
73#127×C53—21no—
74#127×C53C2312yes—
75C4×D4×C53—21no—
76C4×D4×C53—21no—
77C4×D4×C53D412yes—
78C4×D4×C53C2312yes—
79C2×C4⋊C4×C53D411yescharacteristic
80C2×C22⋊C4×C53—21no—
81C2×C42×C53C2311yescharacteristic
82C4⋊C8×C53—21no—
83C4⋊C8×C53C2312yes—
84#102×C53—24no—
85C24×C53—21no—
86C24×C53—41no—
87C2×D4×C53—21no—
88C2×D4×C53—22no—
89C2×D4×C53—22no—
90C2×D4×C53—22no—
91C2×D4×C53—22no—
92C2×D4×C53—41no—
93C2×D4×C53—41no—
94C2×D4×C53—41no—
95C2×D4×C53—42no—
96C2×D4×C53—42no—
97C2×D4×C53—82no—
98C2×D4×C53—82no—
99C2×D4×C53D814yescomplemented
100C2×D4×C53C2×D412yes—
101C22×C4×C53—21no—
102C22×C4×C53—22no—
103C22×C4×C53—22no—
104C22×C4×C53—82no—
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53#4111yescharacteristic
108C2×C22⋊D53—41no—
109#11719945—42no—
110#11719945—42no—
111C2×D212—21no—
112C2×D212—21no—
113C2×D212—21no—
114C2×C4×D53—41no—
115C2×Dic106—21no—
116D8×C53—48no—
117C2×C8×C53—22no—
118C2×C8×C53—42no—
119C4⋊C4×C53—21no—
120C4⋊C4×C53C2×D412yes—
121C22⋊C4×C53—22no—
122C22⋊C4×C53—22no—
123C42×C53C2×D411yescharacteristic
124C42×C53C2×D411yescharacteristic
125C42×C53#4111yescharacteristic
126C42×C53C2411yescharacteristic
127#11719923—44no—
128#11719922—44no—
129#11719922—84no—
130#11719921—41no—
131#11719921—41no—
132C4⋊Dic53—21no—
133C4⋊Dic53—21no—
134#11719917—42no—
135C2×#88322—22no—
136C2×#88322—22no—
137C2×#88322—84no—
138C2×#88322C2×D414yes—
139C23×C53—22no—
140C23×C53—24no—
141C23×C53—42no—
142C23×C53—81no—
143C23×C53—81no—
144C23×C53—82no—
145C23×C53—82no—
146C23×C53—82no—
147C23×C53#12111yescharacteristic
148C22×D53—41no—
149D4×C53—22no—
150D4×C53—24no—
151D4×C53—42no—
152D4×C53—42no—
153D4×C53—44no—
154D4×C53—82no—
155D4×C53—82no—
156D4×C53—82no—
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—82no—
164C2×C4×C53—81no—
165C2×C4×C53#12111yescharacteristic
166C2×C4×C53#12111yescharacteristic
167C2×C4×C53#12111yescharacteristic
168C2×C4×C53C2×D812yes—
169C2×C4×C53#13512yes—
170C2×C4×C53C22×D411yescharacteristic
171C2×C4×C53C22×D411yescharacteristic
172C2×C4×C53C2×#4111yescharacteristic
173C22⋊D53—82no—
174C2×Dic53—41no—
175D212—41no—
176D212—41no—
177D212—42no—
178D212—81no—
179C4×D53—82no—
180Dic106—41no—
181Dic106—41no—
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—82no—
194C22×C53—82no—
195C22×C53—82no—
196C22×C53—164no—
197C22×C53#46612yes—
198C22×C53C2×#12111yescharacteristic
199C2×D53—81no—
200C2×D53—81no—
201C2×D53—81no—
202C4×C53—21no—
203C4×C53—21no—
204C4×C53—21no—
205C4×C53—42no—
206C4×C53#46612yes—
207C4×C53C22×D811yescharacteristic
208C4×C53C2×#13511yescharacteristic
209C4×C53#57612yes—
210Dic53—81no—
211#3205—531no—
212#3036—531no—
213C2×#459D5311yescharacteristic, complemented
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—162no—
223C2×C53#304111yescharacteristic
224C2×C53C2×#46611yescharacteristic
225C2×C53#306811yescharacteristic
226D53—161no—
227C2×AGL(1,ℤ/8)—1061no—
228C22×D8—21no—
229#532—531no—
230C2×#127C2×D5311yescharacteristic
231C2×C4×D4C2×D5311yescharacteristic
232#493—532no—
233#486—532no—
234#463—1062no—
235#461—532no—
236#459—22no—
237#459—532no—
238#459—1062no—
239#459C2×D5314yes—
240#447—1062no—
241C8×D4—534no—
242C8×D4—1062no—
243#431—531no—
244#431—531no—
245#423—531no—
246C2×C4⋊C8C2×D5311yescharacteristic
247#417—1061no—
248C2×#102—21no—
249C2×C22⋊C8—1062no—
250#335—534no—
251#332—532no—
252#331—534no—
253#331—532no—
254C53#1938611yescharacteristic, complemented, Sylow
255C2×#41—1061no—
256C22×D4—21no—
257C22×D4—21no—
258C22×D4—21no—
259C22×D4C22⋊D5311yescharacteristic
260C23×C4—21no—
261AGL(1,ℤ/8)—2124no—
262C2×SD16—1062no—
263C2×D8—24no—
264C2×D8—44no—
265C2×D8—1064no—
266C2×D8—1062no—
267C2×M16—1061no—
268C22×C8—21no—
269C22×C8—1064no—
270C4⋊Q8—531no—
271#127—21no—
272#127—21no—
273#127—21no—
274#127—531no—
275#127C22×D5312yes—
276#124—1061no—
277#121—1061no—
278#121—1061no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4—1061no—
282C4×D4—1061no—
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—1062no—
295#102—24no—
296#102—1064no—
297#102—1062no—
298C22⋊C8—532no—
299C22⋊C8—1064no—
300C22⋊C8—1061no—
301C4×C8—532no—
302C4×C8—1061no—
303C24—21no—
304C24—41no—
305#41—2122no—
306#41—2122no—
307C2×Q8—1061no—
308C2×D4—21no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—82no—
319C2×D4—82no—
320C2×D4—1061no—
321C2×D4—1061no—
322C2×D4—1061no—
323C2×D4—2121no—
324C2×D4#1171992214yescomplemented
325C2×D4C2×C22⋊D5312yes—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—82no—
330C22×C4—2121no—
331C22×C4D4×D5311yescharacteristic
332C22×C4#1171994711yescharacteristic
333C22×C4C2×C22⋊D5311yescharacteristic
334SD16—2124no—
335D8—48no—
336D8—4244no—
337D8—2124no—
338M16—2122no—
339C2×C8—22no—
340C2×C8—42no—
341C2×C8—534no—
342C2×C8—1062no—
343C2×C8—1062no—
344C2×C8—4244no—
345C4⋊C4—21no—
346C4⋊C4—1061no—
347C4⋊C4—1061no—
348C4⋊C4C2×C22⋊D5312yes—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—2121no—
352C22⋊C4—2121no—
353C42D4×D5311yescharacteristic
354C42#1171994711yescharacteristic
355C42C2×C22⋊D5311yescharacteristic
356C42C23×D5311yescharacteristic
357C23—22no—
358C23—24no—
359C23—42no—
360C23—81no—
361C23—81no—
362C23—82no—
363C23—82no—
364C23—82no—
365C23—2121no—
366C23#1313318911yescharacteristic
367Q8—2121no—
368Q8—2121no—
369D4—24no—
370D4—22no—
371D4—42no—
372D4—42no—
373D4—44no—
374D4—82no—
375D4—82no—
376D4—82no—
377D4—82no—
378D4—4241no—
379D4—4242no—
380D4—2121no—
381D4—2121no—
382D4—2122no—
383D4C2×#1171992214yes—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—41no—
387C2×C4—44no—
388C2×C4—82no—
389C2×C4—81no—
390C2×C4—4242no—
391C2×C4—2121no—
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—82no—
411C22—82no—
412C22—82no—
413C22—164no—
414C22—4241no—
415C22—4241no—
416C22—4241no—
417C22#1681085712yes—
418C22C2×#1313318911yescharacteristic
419C4—21no—
420C4—21no—
421C4—21no—
422C4—42no—
423C4—4241no—
424C4#1681085712yes—
425C4C22×#1171992211yescharacteristic
426C4C2×#1313323711yescharacteristic
427C4#1681142412yes—
428C2—22no—
429C2—44no—
430C2—162no—
431C2—8481no—
432C2#2028421511yescharacteristic
433C2C2×#1681085711yescharacteristic
434C2#2028444011yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#459C2
socleC22×C53C2×#121
hypercenterC2×#459D53
hypocenterC53#19386
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
1. #20285633C2semidirect
2. #20284179C2semidirect
3. #20283739C2semidirect
4. #20274150C2semidirect
5. #20274068C2semidirect
6. #20273489C2semidirect
7. #20273383C2semidirect
8. #20273365C2semidirect
9. C2×C53×#459C2semidirect
12. #16810999C22semidirect
13. #16810969C22semidirect
15. #16810828C22semidirect
16. #16810818C22semidirect
20. D4×#88322C22semidirect
26. #16810375C22semidirect
27. #16810370C22semidirect
28. #16810367C22semidirect
29. #16810366C22semidirect
99. C2×D4×C53D8semidirect
213. C2×#459D53semidirect
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
#202856332
#202841792
#202837392
#202741502
#202740682
#202734892
#202733832
#202733652
C2×C53×#4592
#1938653