Finite Group Databasebeta

← #27369266

Subgroups — #27369266

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#20284175C211yesmaximal, characteristic, complemented
2C2×#16810815C211yesmaximal, characteristic, complemented
3#20283738C211yesmaximal, characteristic, complemented
4#20274149C214yesmaximal, complemented
5#20274067C214yesmaximal, complemented
6#20273490C211yesmaximal, characteristic, complemented
7#20273383C211yesmaximal, characteristic, complemented
8#20273364C211yesmaximal, characteristic, complemented
9C53×#3205C211yesmaximal, characteristic, complemented
10C2×#13133242—21no—
11C22×#11719922—21no—
12C2×C4×D212C2211yescharacteristic
13#16810835—22no—
14#16810825C2212yescomplemented
15#16810815—22no—
16#16810815C2214yes—
17#16810815C2212yescomplemented
18C8×D212C2214yescomplemented
19#16810688—22no—
20#16810676C2211yescharacteristic
21#16810647C2211yescharacteristic
22C2×C4⋊#88322C2211yescharacteristic
23#16810624—21no—
24C2×#13133138—21no—
25C22⋊C8×D53—22no—
26#16810376C2214yescomplemented
27#16810369C2212yescomplemented
28#16810367C2212yescomplemented
29#16810367C2214yescomplemented
30C53×#532C2211yescharacteristic
31C2×#127×C53C2211yescharacteristic
32C53×#493C2212yescomplemented
33C53×#486C2212yescomplemented
34C53×#459—22no—
35C53×#447—22no—
36C53×#431C2211yescharacteristic
37C22×D212D411yescharacteristic
38C22×C4×D53—21no—
39#13133242—44no—
40C2×#11719924—22no—
41C2×#11719922—24no—
42C2×#11719922—22no—
43C2×#11719922—44no—
44C2×#11719917—21no—
45C22×#88322—21no—
46C2×C8×D53—24no—
47C4×D212—21no—
48C4×D212—21no—
49C4×D212D412yes—
50C4×D212C2312yes—
51C2×C4⋊Dic53D411yescharacteristic
52C2×#11719921—21no—
53C4⋊#88322—21no—
54C4⋊#88322—21no—
55C4⋊#88322C2312yes—
56C4⋊#88322C2312yes—
57C4⋊#88322C2312yes—
58#13133141—22no—
59#13133138—24no—
60#13133138—22no—
61C22⋊#88322—21no—
62#13133129—24no—
63C4×#88322—21no—
64C2×#41×C53—21no—
65C22×D4×C53—21no—
66C22×D4×C53—21no—
67C22×D4×C53—21no—
68C2×D8×C53—24no—
69C4⋊Q8×C53C2311yescharacteristic
70#127×C53—21no—
71#127×C53—21no—
72#127×C53—21no—
73#127×C53C2312yes—
74#127×C53C2311yescharacteristic
75#124×C53—21no—
76#121×C53—21no—
77#121×C53—21no—
78C4×D4×C53—21no—
79C4×D4×C53—21no—
80C2×C42×C53C2311yescharacteristic
81C4⋊C8×C53D412yes—
82#102×C53—24no—
83C22⋊C8×C53D412yes—
84C4×C8×C53C2312yes—
85C24×C53—41no—
86C23×D53—21no—
87#41×C53—42no—
88#41×C53—42no—
89C2×Q8×C53—21no—
90C2×D4×C53—22no—
91C2×D4×C53—21no—
92C2×D4×C53—21no—
93C2×D4×C53—22no—
94C2×D4×C53—21no—
95C2×D4×C53—22no—
96C2×D4×C53—41no—
97C2×D4×C53—41no—
98C2×D4×C53—41no—
99C2×D4×C53—41no—
100C2×D4×C53—42no—
101C2×D4×C53—42no—
102C2×D4×C53—82no—
103C2×D4×C53—82no—
104C22×C4×C53—41no—
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53#4111yescharacteristic
108C22×Dic53—21no—
109C2×D212—21no—
110C2×D212—22no—
111C2×D212D814yescomplemented
112C2×D212C2×D412yes—
113C2×C4×D53—22no—
114C2×C4×D53—22no—
115C2×C4×D53—82no—
116D8×C53—84no—
117C2×C8×C53C2×D414yes—
118C4⋊C4×C53—21no—
119C4⋊C4×C53—21no—
120C22⋊C4×C53—41no—
121C22⋊C4×C53—41no—
122C42×C53C2×D411yescharacteristic
123C42×C53C2×D411yescharacteristic
124C42×C53#4111yescharacteristic
125C42×C53C2411yescharacteristic
126#11719924—44no—
127#11719922—48no—
128#11719922—44no—
129#11719921—22no—
130#11719921—22no—
131C4⋊Dic53—21no—
132C4⋊Dic53C2×D412yes—
133#11719917—42no—
134C2×#88322—22no—
135C2×#88322—22no—
136C2×#88322—22no—
137C2×#88322—42no—
138C8×D53—84no—
139C23×C53—41no—
140C23×C53—42no—
141C23×C53—82no—
142C23×C53—82no—
143C23×C53—82no—
144C23×C53#12111yescharacteristic
145C22×D53—22no—
146C22×D53—24no—
147C22×D53—81no—
148C22×D53—81no—
149Q8×C53—41no—
150Q8×C53—41no—
151D4×C53—42no—
152D4×C53—41no—
153D4×C53—42no—
154D4×C53—41no—
155D4×C53—44no—
156D4×C53—42no—
157D4×C53—81no—
158D4×C53—82no—
159D4×C53—82no—
160D4×C53—82no—
161D4×C53—82no—
162D4×C53—82no—
163C2×C4×C53—22no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
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
175C2×Dic53—22no—
176C2×Dic53—81no—
177D212—22no—
178D212—24no—
179D212C2×D814yes—
180C4×D53—44no—
181C4×D53—82no—
182C8×C53—24no—
183#88322—42no—
184#88322—42no—
185#88322—42no—
186#19386—531nomaximal, Sylow
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—82no—
190C22×C53—81no—
191C22×C53—81no—
192C22×C53—82no—
193C22×C53—82no—
194C22×C53—81no—
195C22×C53—164no—
196C22×C53#46612yes—
197C22×C53C2×#12111yescharacteristic
198C2×D53—24no—
199C2×D53—42no—
200C2×D53—42no—
201C2×D53—84no—
202C4×C53—21no—
203C4×C53—21no—
204C4×C53—21no—
205C4×C53—81no—
206C4×C53#46612yes—
207C4×C53C22×D811yescharacteristic
208C4×C53C2×#13511yescharacteristic
209C4×C53#57612yes—
210Dic53—42no—
211#3205D5311yescharacteristic, complemented
212#3036—531no—
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—161no—
222C2×C53—162no—
223C2×C53#304111yescharacteristic
224C2×C53C2×#46611yescharacteristic
225C2×C53#306811yescharacteristic
226D53—44no—
227C2×AGL(1,ℤ/8)—1061no—
228C22×D8—1061no—
229#532C2×D5311yescharacteristic
230C2×#127C2×D5311yescharacteristic
231C2×C4×D4—531no—
232#493C2×D5312yescomplemented
233#486C2×D5312yescomplemented
234#463—1062no—
235#461—532no—
236#459—22no—
237#459—534no—
238#459—532no—
239#459—1062no—
240#447—22no—
241C8×D4—534no—
242C8×D4—1062no—
243#431—531no—
244#431C2×D5311yescharacteristic
245#423—531no—
246C2×C4⋊C8—531no—
247#417—1061no—
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—21no—
257C22×D4—21no—
258C22×D4—21no—
259C22×D4—531no—
260C23×C4—1061no—
261AGL(1,ℤ/8)—2124no—
262C2×SD16—1062no—
263C2×D8—24no—
264C2×D8—1064no—
265C2×D8—1062no—
266C2×D8—2124no—
267C2×M16—1061no—
268C22×C8—1061no—
269C22×C8—1064no—
270C4⋊Q8C22×D5311yescharacteristic
271#127—21no—
272#127—21no—
273#127—21no—
274#127C22×D5312yes—
275#127C22×D5311yescharacteristic
276#124—21no—
277#121—21no—
278#121—21no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4—532no—
282C4×D4—532no—
283C4×D4—1061no—
284C4×D4—1061no—
285C2×C4⋊C4—531no—
286C2×C22⋊C4—1061no—
287C2×C42C22×D5311yescharacteristic
288C4⋊C8—532no—
289C4⋊C8—532no—
290C4⋊C8—532no—
291C4⋊C8—1061no—
292C4⋊C8—1061no—
293C4⋊C8C22⋊D5312yes—
294#103—1062no—
295#102—24no—
296#102—1064no—
297#102—1062no—
298C22⋊C8—1061no—
299C22⋊C8—1064no—
300C22⋊C8C22⋊D5312yes—
301C4×C8—1061no—
302C4×C8C22×D5312yes—
303C24—41no—
304C24—1061no—
305#41—42no—
306#41—42no—
307C2×Q8—21no—
308C2×D4—21no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—22no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—82no—
321C2×D4—82no—
322C2×D4—534no—
323C2×D4—532no—
324C2×D4—1061no—
325C2×D4—1062no—
326C22×C4—41no—
327C22×C4—1061no—
328C22×C4—1062no—
329C22×C4—1062no—
330C22×C4—4242no—
331C22×C4D4×D5311yescharacteristic
332C22×C4#1171994711yescharacteristic
333C22×C4C2×C22⋊D5311yescharacteristic
334SD16—2124no—
335D8—84no—
336D8—2128no—
337D8—2124no—
338M16—2122no—
339C2×C8—1062no—
340C2×C8—1062no—
341C2×C8—1062no—
342C2×C8—4244no—
343C2×C8—2122no—
344C2×C8C2×C22⋊D5314yes—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—532no—
348C4⋊C4—1061no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—1062no—
352C22⋊C4—1062no—
353C42D4×D5311yescharacteristic
354C42#1171994711yescharacteristic
355C42C2×C22⋊D5311yescharacteristic
356C42C23×D5311yescharacteristic
357C23—41no—
358C23—42no—
359C23—82no—
360C23—82no—
361C23—82no—
362C23—1064no—
363C23—1062no—
364C23—4241no—
365C23—4241no—
366C23#1313318811yescharacteristic
367Q8—41no—
368Q8—41no—
369D4—42no—
370D4—41no—
371D4—42no—
372D4—41no—
373D4—42no—
374D4—44no—
375D4—81no—
376D4—82no—
377D4—82no—
378D4—82no—
379D4—82no—
380D4—82no—
381D4—534no—
382D4—1064no—
383D4—1062no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—82no—
388C2×C4—1062no—
389C2×C4—4242no—
390C2×C4—4241no—
391C2×C4—2124no—
392C2×C4#1313318811yescharacteristic
393C2×C4#1313318811yescharacteristic
394C2×C4#1313318811yescharacteristic
395C2×C4D8×D5312yes—
396C2×C4#1313323612yes—
397C2×C4C2×D4×D5311yescharacteristic
398C2×C4C22×C22⋊D5311yescharacteristic
399C2×C4C2×#1171994711yescharacteristic
400C8—24no—
401C8—2122no—
402C8—2122no—
403C8—2122no—
404C22—21no—
405C22—21no—
406C22—82no—
407C22—81no—
408C22—82no—
409C22—81no—
410C22—81no—
411C22—82no—
412C22—164no—
413C22—1064no—
414C22—4244no—
415C22—2122no—
416C22—2122no—
417C22#1681085612yes—
418C22C2×#1313318811yescharacteristic
419C4—21no—
420C4—21no—
421C4—21no—
422C4—81no—
423C4—2122no—
424C4#1681085612yes—
425C4C2×D8×D5311yescharacteristic
426C4C2×#1313323611yescharacteristic
427C4#1681142612yes—
428C2—22no—
429C2—162no—
430C2—161no—
431C2—2124no—
432C2#2028421411yescharacteristic
433C2C2×#1681085611yescharacteristic
434C2#2028443911yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3205C2
socleC22×C53C2×#121
hypercenter#3205D53
hypocenterC53#19386
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3205

Complements

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

NH ≅ G/Ntype
1. #20284175C2semidirect
2. C2×#16810815C2semidirect
3. #20283738C2semidirect
4. #20274149C2semidirect
5. #20274067C2semidirect
6. #20273490C2semidirect
7. #20273383C2semidirect
8. #20273364C2semidirect
9. C53×#3205C2semidirect
14. #16810825C22semidirect
17. #16810815C22semidirect
18. C8×D212C22semidirect
26. #16810376C22semidirect
27. #16810369C22semidirect
28. #16810367C22semidirect
29. #16810367C22semidirect
32. C53×#493C22semidirect
33. C53×#486C22semidirect
111. C2×D212D8semidirect
211. #3205D53semidirect
232. #493C2×D53semidirect
233. #486C2×D53semidirect
254. C53#19386semidirect

Maximal subgroups

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

subgroupindex
#202841752
C2×#168108152
#202837382
#202741492
#202740672
#202734902
#202733832
#202733642
C53×#32052
#1938653