Finite Group Databasebeta

← #27369284

Subgroups — #27369284

Order 13568.

662 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284810C211yesmaximal, characteristic, complemented
2#20284711C211yesmaximal, characteristic
3#20283738C211yesmaximal, characteristic, complemented
4#20274149C212yesmaximal, complemented
5#20274135C212yesmaximal, complemented
6#20274077C212yesmaximal, complemented
7#20274053C212yesmaximal, complemented
8#20273495C211yesmaximal, characteristic, complemented
9#20273420C211yesmaximal, characteristic, complemented
10#20273380C211yesmaximal, characteristic, complemented
11C53×#3210C211yesmaximal, characteristic, complemented
12C2×C4×D212C2211yescharacteristic
13#16810956—21no—
14#16810934—21no—
15#16810900C2212yes—
16#16810900C2212yes—
17#16810825C2212yes—
18#16810815C2212yes—
19#16810773—21no—
20#16810756—21no—
21#16810721—22no—
22C4×#11719922—22no—
23C8×D212C2212yes—
24C8×D212C2212yes—
25#16810688—22no—
26#16810680C2211yescharacteristic
27#16810676C2211yescharacteristic
28#16810676C2211yescharacteristic
29C22⋊C8×D53—22no—
30#16810376C2212yescomplemented
31#16810374C2212yescomplemented
32#16810369C2212yescomplemented
33#16810369C2212yescomplemented
34#16810369C2212yescomplemented
35#16810367C2212yescomplemented
36C53×#533C2211yescharacteristic
37C53×#532C2211yescharacteristic
38C53×#494C2211yescharacteristic, complemented
39C53×#492C2211yescharacteristic, complemented
40C53×#487C2211yescharacteristic, complemented
41C53×#486C2211yescharacteristic, complemented
42C53×#464—22no—
43C53×#452—22no—
44C53×#431C2211yescharacteristic
45C22×D212D411yescharacteristic
46C22×C4×D53—21no—
47C2×#11719924—22no—
48C2×#11719922—22no—
49C2×C8×D53—22no—
50C2×C8×D53—22no—
51C4×D212—21no—
52C4×D212—21no—
53C4×D212D412yes—
54C4×D212C2312yes—
55C2×C4⋊Dic53D411yescharacteristic
56C2×#11719921—21no—
57#13133150—22no—
58#13133150—22no—
59C4⋊#88322C2312yes—
60C4⋊#88322C2312yes—
61C4⋊#88322C2312yes—
62#13133141—22no—
63#13133138—22no—
64#13133137—22no—
65#13133137—22no—
66C22⋊#88322—21no—
67C22⋊#88322—21no—
68C22⋊#88322—21no—
69#13133129—22no—
70#13133129—22no—
71#13133128—21no—
72C4×#88322—21no—
73C4×#88322—21no—
74C2×#41×C53—21no—
75C2×Q16×C53—22no—
76C2×SD16×C53—22no—
77C4⋊Q8×C53C2311yescharacteristic
78C4⋊Q8×C53C2311yescharacteristic
79C4⋊Q8×C53C2311yescharacteristic
80#127×C53C2311yescharacteristic
81#125×C53—21no—
82#124×C53—21no—
83C22⋊Q8×C53—21no—
84C22⋊Q8×C53—21no—
85#121×C53—21no—
86#121×C53—21no—
87C4×Q8×C53—21no—
88C4×Q8×C53—21no—
89C4×D4×C53—21no—
90C4×D4×C53—21no—
91#117×C53—21no—
92C2×C42×C53C2311yescharacteristic
93C4⋊C8×C53D412yes—
94#103×C53—22no—
95#102×C53—22no—
96C22⋊C8×C53D412yes—
97C4×C8×C53C2311yescharacteristic
98C4×C8×C53C2311yescharacteristic
99C23×D53—21no—
100#41×C53—82no—
101C2×Q8×C53—21no—
102C2×Q8×C53—21no—
103C2×Q8×C53—21no—
104C2×D4×C53—21no—
105C2×D4×C53—21no—
106C2×D4×C53—21no—
107C2×D4×C53—41no—
108C22×C4×C53—41no—
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53#4111yescharacteristic
112C22×Dic53—21no—
113C2×D212—21no—
114C2×D212—24no—
115C2×D212C2×D412yes—
116C2×C4×D53—22no—
117C2×C4×D53—22no—
118C2×C4×D53—82no—
119Q16×C53—82no—
120SD16×C53—82no—
121C2×C8×C53C2×D412yes—
122C2×C8×C53C2×D412yes—
123C4⋊C4×C53—21no—
124C4⋊C4×C53—21no—
125C4⋊C4×C53—21no—
126C4⋊C4×C53—21no—
127C4⋊C4×C53—21no—
128C4⋊C4×C53—21no—
129C4⋊C4×C53—41no—
130C4⋊C4×C53—41no—
131C22⋊C4×C53—41no—
132C22⋊C4×C53—41no—
133C22⋊C4×C53—41no—
134C42×C53—41no—
135C42×C53C2×D411yescharacteristic
136C42×C53C2×D411yescharacteristic
137C42×C53#4111yescharacteristic
138C42×C53C2411yescharacteristic
139#11719924—44no—
140#11719922—44no—
141#11719921—22no—
142#11719921—22no—
143C4⋊Dic53—21no—
144C4⋊Dic53C2×D412yes—
145C2×#88322—22no—
146C2×#88322—22no—
147C2×#88322—22no—
148C8×D53—82no—
149C8×D53—82no—
150C23×C53—41no—
151C23×C53#12111yescharacteristic
152C22×D53—22no—
153C22×D53—24no—
154C22×D53—81no—
155C22×D53—81no—
156Q8×C53—42no—
157Q8×C53—42no—
158Q8×C53—42no—
159D4×C53—42no—
160D4×C53—42no—
161D4×C53—81no—
162D4×C53—82no—
163C2×C4×C53—22no—
164C2×C4×C53—22no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—82no—
170C2×C4×C53#12111yescharacteristic
171C2×C4×C53#12111yescharacteristic
172C2×C4×C53#12111yescharacteristic
173C2×C4×C53C22×D411yescharacteristic
174C2×C4×C53C22×D411yescharacteristic
175C2×C4×C53C2×#4111yescharacteristic
176C2×Dic53—22no—
177C2×Dic53—81no—
178D212—24no—
179D212—22no—
180D212#13514yes—
181C4×D53—42no—
182C4×D53—42no—
183C4×D53—82no—
184C8×C53—22no—
185C8×C53—22no—
186#88322—42no—
187#88322—42no—
188#88322—42no—
189#19388—531nomaximal, Sylow
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—21no—
193C22×C53—81no—
194C22×C53—81no—
195C22×C53—81no—
196C22×C53C2×#12111yescharacteristic
197C2×D53—24no—
198C2×D53—42no—
199C2×D53—42no—
200C2×D53—84no—
201C4×C53—21no—
202C4×C53—21no—
203C4×C53—21no—
204C4×C53—21no—
205C4×C53—81no—
206C4×C53—81no—
207C4×C53—81no—
208C4×C53C2×#13512yes—
209C4×C53#57612yes—
210Dic53—42no—
211#3210D5311yescharacteristic, complemented
212#3110—531no—
213#3100—531no—
214#2972—531no—
215#1681—532no—
216#1679—532no—
217#1669—532no—
218#1665—532no—
219#1572—531no—
220#1553—531no—
221#1544—531no—
222C2×C53—41no—
223C2×C53—161no—
224C2×C53#304111yescharacteristic
225C2×C53#305011yescharacteristic
226C2×C53#306911yescharacteristic
227D53—44no—
228#533C2×D5311yescharacteristic
229#532C2×D5311yescharacteristic
230C2×C4×D4—531no—
231#494C2×D5311yescharacteristic, complemented
232#492C2×D5311yescharacteristic, complemented
233#487C2×D5311yescharacteristic, complemented
234#486C2×D5311yescharacteristic, complemented
235#484—1061no—
236#480—1061no—
237#474—532no—
238#474—532no—
239#464—22no—
240#461—532no—
241#459—532no—
242#452—22no—
243#450—1061no—
244#447—1061no—
245#440—1062no—
246C4×D8—1062no—
247C8×D4—532no—
248C8×D4—532no—
249C8×D4—1062no—
250#432—531no—
251#431—531no—
252#431—531no—
253#431C2×D5311yescharacteristic
254C2×C22⋊C8—1062no—
255#335—532no—
256#334—532no—
257#332—532no—
258#332—532no—
259#332—532no—
260#331—532no—
261C53#1938811yescharacteristic, complemented, Sylow
262C2×#41—21no—
263C22×D4—531no—
264C23×C4—1061no—
265C2×Q16—22no—
266C2×SD16—22no—
267C2×SD16—1062no—
268C2×D8—1062no—
269C22×C8—1062no—
270C22×C8—1062no—
271C4⋊Q8C22×D5311yescharacteristic
272C4⋊Q8C22×D5311yescharacteristic
273C4⋊Q8C22×D5311yescharacteristic
274#127C22×D5311yescharacteristic
275#125—21no—
276#124—21no—
277C22⋊Q8—21no—
278C22⋊Q8—21no—
279#121—21no—
280#121—21no—
281C4×Q8—21no—
282C4×Q8—21no—
283C4×D4—21no—
284C4×D4—21no—
285C4×D4—532no—
286C4×D4—532no—
287C4×D4—1061no—
288C4×D4—1061no—
289#117—21no—
290C2×C4⋊C4—531no—
291C2×C22⋊C4—1061no—
292C2×C42C22×D5311yescharacteristic
293#107—1062no—
294#107—1062no—
295C4⋊C8—532no—
296C4⋊C8—532no—
297C4⋊C8—532no—
298C4⋊C8C22⋊D5312yes—
299#103—22no—
300#103—1062no—
301#102—22no—
302#102—1062no—
303#102—1062no—
304#102—1062no—
305C22⋊C8—1061no—
306C22⋊C8—1061no—
307C22⋊C8—1061no—
308C22⋊C8—1062no—
309C22⋊C8—1062no—
310C22⋊C8C22⋊D5312yes—
311#97—1061no—
312C4×C8—1061no—
313C4×C8—1061no—
314C4×C8C22×D5311yescharacteristic
315C4×C8C22×D5311yescharacteristic
316C24—1061no—
317#41—82no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—41no—
325C2×D4—532no—
326C2×D4—1064no—
327C2×D4—1061no—
328C22×C4—41no—
329C22×C4—1061no—
330C22×C4—1062no—
331C22×C4—1062no—
332C22×C4—4242no—
333C22×C4D4×D5311yescharacteristic
334C22×C4#1171994711yescharacteristic
335C22×C4C2×C22⋊D5311yescharacteristic
336Q16—82no—
337SD16—82no—
338SD16—2124no—
339D8—2124no—
340C2×C8—1062no—
341C2×C8—1062no—
342C2×C8—1062no—
343C2×C8—4242no—
344C2×C8—4242no—
345C2×C8C2×C22⋊D5312yes—
346C2×C8C2×C22⋊D5312yes—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—532no—
356C4⋊C4—1061no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C22⋊C4—1062no—
361C22⋊C4—1062no—
362C42—41no—
363C42D4×D5311yescharacteristic
364C42#1171994711yescharacteristic
365C42C2×C22⋊D5311yescharacteristic
366C42C23×D5311yescharacteristic
367C23—41no—
368C23—1064no—
369C23—1062no—
370C23—4241no—
371C23—4241no—
372C23#1313318811yescharacteristic
373Q8—42no—
374Q8—42no—
375Q8—42no—
376D4—42no—
377D4—42no—
378D4—81no—
379D4—82no—
380D4—534no—
381D4—1064no—
382D4—1062no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—82no—
390C2×C4—1062no—
391C2×C4—4242no—
392C2×C4—4241no—
393C2×C4—2122no—
394C2×C4—2122no—
395C2×C4#1313318811yescharacteristic
396C2×C4#1313318811yescharacteristic
397C2×C4#1313318811yescharacteristic
398C2×C4C2×D4×D5311yescharacteristic
399C2×C4C22×C22⋊D5311yescharacteristic
400C2×C4C2×#1171994711yescharacteristic
401C8—22no—
402C8—22no—
403C8—2122no—
404C8—2122no—
405C8—2122no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—1064no—
413C22—4244no—
414C22—2122no—
415C22—2122no—
416C22C2×#1313318811yescharacteristic
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—81no—
422C4—81no—
423C4—81no—
424C4—2122no—
425C4#135×D5312yes—
426C4#1681142612yes—
427C2—41no—
428C2—161no—
429C2—2124no—
430C2#2028421411yescharacteristic
431C2#2028429211yescharacteristic
432C2#2028444811yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3210C2
socleC22×C53C2×#121
prime radical#20284711C2
hypercenter#3210D53
hypocenterC53#19388
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3210

Complements

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

NH ≅ G/Ntype
1. #20284810C2semidirect
3. #20283738C2semidirect
4. #20274149C2semidirect
5. #20274135C2semidirect
6. #20274077C2semidirect
7. #20274053C2semidirect
8. #20273495C2semidirect
9. #20273420C2semidirect
10. #20273380C2semidirect
11. C53×#3210C2semidirect
30. #16810376C22semidirect
31. #16810374C22semidirect
32. #16810369C22semidirect
33. #16810369C22semidirect
34. #16810369C22semidirect
35. #16810367C22semidirect
38. C53×#494C22semidirect
39. C53×#492C22semidirect
40. C53×#487C22semidirect
41. C53×#486C22semidirect
211. #3210D53semidirect
231. #494C2×D53semidirect
232. #492C2×D53semidirect
233. #487C2×D53semidirect
234. #486C2×D53semidirect
261. C53#19388semidirect

Maximal subgroups

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

subgroupindex
#202848102
#202847112
#202837382
#202741492
#202741352
#202740772
#202740532
#202734952
#202734202
#202733802
C53×#32102
#1938853