Finite Group Databasebeta

← #27416020

Subgroups — #27416020

Order 13568.

1446 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 1446 conjugacy classes of subgroups — too many to draw. Only its 207 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810756C212yesmaximal, complemented
2#20277295C211yesmaximal, characteristic, complemented
3#20275858C212yesmaximal, complemented
4#20275786C212yesmaximal, complemented
5#20275063C214yesmaximal, complemented
6#20275038C212yesmaximal, complemented
7#20274786C211yesmaximal, characteristic, complemented
8C2×C53×#540C211yesmaximal, characteristic, complemented
9C23×D212C2211yescharacteristic
10C22×#11719922C2214yes—
11C2×#13133176C2211yescharacteristic
12#16810756—28no—
13C2×#13133138C2214yescomplemented
14C2×#13133137C2214yes—
15C2×C22⋊#88322C2212yes—
16#16810525C2212yes—
17#16810525C2212yes—
18#16810524—22no—
19#16810478—22no—
20#16810385C2214yescomplemented
21#16810378C2212yescomplemented
22#16810378C2212yescomplemented
23C53×#540—28no—
24C2×#127×C53C2211yescharacteristic, complemented
25C2×#125×C53C2211yescharacteristic, complemented
26C2×#123×C53—22no—
27C2×#121×C53—22no—
28C2×#121×C53C2212yes—
29C2×C4×D4×C53C2212yescomplemented
30C2×#117×C53C2211yescharacteristic
31C24×D53—21no—
32C22×D212—21no—
33C22×D212—24no—
34C22×D212—24no—
35C22×D212—42no—
36C22×D212C2312yes—
37C2×#11719922—28no—
38C2×#11719922—28no—
39C2×#11719922—44no—
40C22×#88322C2314yes—
41#13133176—42no—
42C2×C4⋊Dic53C2312yes—
43C2×C22⋊Dic53—21no—
44C2×#11719921—24no—
45C2×#11719921—24no—
46C2×C4×Dic53—21no—
47#13133138—44no—
48#13133137—28no—
49C22⋊#88322—24no—
50C22×D4×C53—22no—
51C22×D4×C53—22no—
52C22×D4×C53C2312yes—
53C23×C4×C53—22no—
54C23×C4×C53C2311yescharacteristic
55#127×C53—24no—
56#125×C53—24no—
57#123×C53—216no—
58#121×C53—216no—
59#121×C53—24no—
60#121×C53D418yes—
61C4×D4×C53—24no—
62C4×D4×C53—24no—
63#117×C53—24no—
64C2×C4⋊C4×C53—22no—
65C2×C4⋊C4×C53D412yes—
66C2×C4⋊C4×C53C2312yes—
67C2×C22⋊C4×C53—22no—
68C2×C22⋊C4×C53—22no—
69C2×C22⋊C4×C53D412yes—
70C2×C42×C53C2311yescharacteristic
71C2×C42×C53C2311yescharacteristic
72C24×C53—22no—
73C24×C53C2×D411yescharacteristic
74C23×D53—22no—
75C23×D53—24no—
76C23×D53—42no—
77C23×D53—42no—
78C23×D53—84no—
79C2×D4×C53—28no—
80C2×D4×C53—28no—
81C2×D4×C53—22no—
82C2×D4×C53—24no—
83C2×D4×C53—24no—
84C2×D4×C53—84no—
85C2×D4×C53—84no—
86C2×D4×C53C2×D414yes—
87C22×C4×C53—22no—
88C22×C4×C53—22no—
89C22×C4×C53—22no—
90C22×C4×C53—28no—
91C22×C4×C53—84no—
92C22×C4×C53C2×D412yes—
93C22×C4×C53C2×D412yes—
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D414yes—
96C22×C4×C53C2411yescharacteristic
97C22×Dic53—22no—
98C2×D212—24no—
99C2×D212—24no—
100C2×D212—24no—
101C2×D212—22no—
102C2×D212—416no—
103C2×D212—42no—
104C2×D212—42no—
105C2×D212#4114yes—
106C4⋊C4×C53—28no—
107C4⋊C4×C53—24no—
108C4⋊C4×C53—22no—
109C4⋊C4×C53C2×D414yes—
110C22⋊C4×C53—28no—
111C22⋊C4×C53—24no—
112C22⋊C4×C53—28no—
113C42×C53—22no—
114C42×C53—22no—
115C22⋊Dic53—42no—
116#11719922—416no—
117#11719921—84no—
118#11719921—84no—
119C4⋊Dic53—24no—
120C4×Dic53—42no—
121C2×#88322—24no—
122C2×#88322—24no—
123C2×#88322—24no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53—24no—
127C23×C53—24no—
128C23×C53—42no—
129C23×C53—42no—
130C23×C53—44no—
131C23×C53—44no—
132C23×C53#12012yes—
133C23×C53C22×D411yescharacteristic
134C22×D53—24no—
135C22×D53—48no—
136C22×D53—42no—
137C22×D53—42no—
138C22×D53—44no—
139C22×D53—44no—
140C22×D53—88no—
141C22×D53—88no—
142C22×D53—88no—
143D4×C53—28no—
144D4×C53—44no—
145D4×C53—88no—
146D4×C53—88no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—28no—
150C2×C4×C53—22no—
151C2×C4×C53—24no—
152C2×C4×C53—22no—
153C2×C4×C53—24no—
154C2×C4×C53—24no—
155C2×C4×C53—44no—
156C2×C4×C53—88no—
157C2×C4×C53#12012yes—
158C2×C4×C53C22×D412yes—
159C2×C4×C53C2×#4112yes—
160C2×C4×C53H(5,𝔽2)12yes—
161C2×Dic53—42no—
162C2×Dic53—44no—
163D212—28no—
164D212—48no—
165D212—44no—
166#88322—44no—
167#23939—531nomaximal, Sylow
168C22×C53—22no—
169C22×C53—24no—
170C22×C53—42no—
171C22×C53—42no—
172C22×C53—44no—
173C22×C53—44no—
174C22×C53—44no—
175C22×C53—84no—
176C22×C53—84no—
177C22×C53C2×#12011yescharacteristic
178C22×C53#52511yescharacteristic
179C22×C53#53511yescharacteristic
180C22×C53#57614yes—
181C2×D53—48no—
182C2×D53—44no—
183C2×D53—816no—
184C2×D53—82no—
185C2×D53—82no—
186C2×D53—84no—
187C4×C53—22no—
188C4×C53—42no—
189C4×C53—42no—
190C4×C53—44no—
191C4×C53#54614yes—
192Dic53—82no—
193C2×#540D5311yescharacteristic, complemented
194C2×#447—532no—
195#2309—531no—
196#2025—532no—
197#2011—532no—
198#1879—534no—
199#1875—532no—
200#1818—531no—
201C2×C53—42no—
202C2×C53—84no—
203C2×C53#240411yescharacteristic
204C2×C53#300812yes—
205C2×C53#312212yes—
206C2×C53#322012yes—
207D53—84no—
208C23×D4—531no—
209C22×D8—534no—
210#540—28no—
211C2×#127C2×D5311yescharacteristic, complemented
212C2×#125C2×D5311yescharacteristic, complemented
213C2×#123—22no—
214C2×#121—22no—
215C2×#121C2×D5312yes—
216C2×C4×D4C2×D5312yescomplemented
217C2×#117—531no—
218C2×#117C2×D5311yescharacteristic
219#447—1068no—
220C2×#102—534no—
221C2×#102—534no—
222C2×C22⋊C8—532no—
223#390—532no—
224#390—532no—
225#390—1062no—
226#379—1062no—
227#340—534no—
228#336—532no—
229#336—532no—
230C53#2393911yescharacteristic, complemented, Sylow
231C25—1061no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—532no—
235C22×D4—1064no—
236C22×D4—1061no—
237C22×D4—1064no—
238C22×D4—2122no—
239C22×D4C22×D5312yes—
240C23×C4—22no—
241C23×C4C22×D5311yescharacteristic
242C2×D8—1068no—
243C2×D8—1068no—
244C2×D8—2124no—
245C22×C8—534no—
246#127—24no—
247#125—24no—
248#123—216no—
249#121—216no—
250#121—24no—
251#121C22⋊D5318yes—
252C4×D4—24no—
253C4×D4—24no—
254#117—24no—
255#117—2122no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—532no—
258C2×C4⋊C4C22⋊D5312yes—
259C2×C4⋊C4C22×D5312yes—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—1064no—
263C2×C22⋊C4—1064no—
264C2×C22⋊C4—1061no—
265C2×C22⋊C4C22⋊D5312yes—
266C2×C42—1061no—
267C2×C42C22×D5311yescharacteristic
268C2×C42C22×D5311yescharacteristic
269#102—1068no—
270#102—2124no—
271C22⋊C8—1064no—
272C24—22no—
273C24—1064no—
274C24—1062no—
275C24—4244no—
276C24—2122no—
277C24—2122no—
278C24C2×C22⋊D5311yescharacteristic
279C2×D4—28no—
280C2×D4—28no—
281C2×D4—22no—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—84no—
285C2×D4—84no—
286C2×D4—534no—
287C2×D4—1064no—
288C2×D4—1064no—
289C2×D4—1062no—
290C2×D4—1064no—
291C2×D4—21216no—
292C2×D4—2122no—
293C2×D4—2122no—
294C2×D4C2×C22⋊D5314yes—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—28no—
299C22×C4—84no—
300C22×C4—1062no—
301C22×C4C2×C22⋊D5312yes—
302C22×C4C2×C22⋊D5312yes—
303C22×C4C2×C22⋊D5311yescharacteristic
304C22×C4C2×C22⋊D5314yes—
305C22×C4C23×D5311yescharacteristic
306D8—21216no—
307C2×C8—1064no—
308C2×C8—1064no—
309C2×C8—1064no—
310C4⋊C4—28no—
311C4⋊C4—24no—
312C4⋊C4—22no—
313C4⋊C4—1064no—
314C4⋊C4C2×C22⋊D5314yes—
315C22⋊C4—28no—
316C22⋊C4—24no—
317C22⋊C4—28no—
318C22⋊C4—4244no—
319C22⋊C4—4244no—
320C22⋊C4—2122no—
321C42—22no—
322C42—22no—
323C42—2122no—
324C23—21no—
325C23—24no—
326C23—21no—
327C23—24no—
328C23—42no—
329C23—44no—
330C23—44no—
331C23—42no—
332C23—1064no—
333C23—4248no—
334C23—4248no—
335C23—4248no—
336C23—2128no—
337C23—2122no—
338C23—2124no—
339C23—2124no—
340C23—2122no—
341C23#1313318612yes—
342C23C22×C22⋊D5311yescharacteristic
343D4—28no—
344D4—44no—
345D4—88no—
346D4—88no—
347D4—1068no—
348D4—2128no—
349D4—2124no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—28no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—44no—
359C2×C4—88no—
360C2×C4—2124no—
361C2×C4—2122no—
362C2×C4#1313318612yes—
363C2×C4C22×C22⋊D5312yes—
364C2×C4#41×D5312yes—
365C2×C4#1313326312yes—
366C8—2124no—
367C22—22no—
368C22—24no—
369C22—42no—
370C22—42no—
371C22—44no—
372C22—44no—
373C22—44no—
374C22—84no—
375C22—84no—
376C22—42416no—
377C22—4242no—
378C22—4242no—
379C22—4244no—
380C22—2124no—
381C22—2128no—
382C22C2×#1313318611yescharacteristic
383C22#1681113011yescharacteristic
384C22#1681118711yescharacteristic
385C22#1681142614yes—
386C4—22no—
387C4—42no—
388C4—42no—
389C4—44no—
390C4—4242no—
391C4#1681126114yes—
392C2—42no—
393C2—84no—
394C2—4244no—
395C2#2027803011yescharacteristic
396C2#2028398312yes—
397C2#2028495312yes—
398C2#2028575912yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#540C2
socleC23×C53C22×D4
hypercenterC2×#540D53
hypocenterC53#23939
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#540

Complements

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

NH ≅ G/Ntype
1. C2×#16810756C2semidirect
2. #20277295C2semidirect
3. #20275858C2semidirect
4. #20275786C2semidirect
5. #20275063C2semidirect
6. #20275038C2semidirect
7. #20274786C2semidirect
8. C2×C53×#540C2semidirect
13. C2×#13133138C22semidirect
20. #16810385C22semidirect
21. #16810378C22semidirect
22. #16810378C22semidirect
24. C2×#127×C53C22semidirect
25. C2×#125×C53C22semidirect
29. C2×C4×D4×C53C22semidirect
193. C2×#540D53semidirect
211. C2×#127C2×D53semidirect
212. C2×#125C2×D53semidirect
216. C2×C4×D4C2×D53semidirect
230. C53#23939semidirect

Maximal subgroups

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

subgroupindex
C2×#168107562
#202772952
#202758582
#202757862
#202750632
#202750382
#202747862
C2×C53×#5402
#2393953