Finite Group Databasebeta

← #27416024

Subgroups — #27416024

Order 13568.

1446 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 1446 conjugacy classes of subgroups — too many to draw. Only its 195 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×#16811228C211yesmaximal, characteristic, complemented
2C2×#16810757C212yesmaximal, complemented
3#20275861C212yesmaximal, complemented
4#20275785C212yesmaximal, complemented
5#20275066C214yesmaximal, complemented
6#20275037C212yesmaximal, complemented
7#20274786C211yesmaximal, characteristic, complemented
8C53×#2309C211yesmaximal, characteristic, complemented
9C22×#11719922C2214yes—
10#16811228—28no—
11C2×#13133214C2211yescharacteristic, complemented
12C2×#13133209C2211yescharacteristic, complemented
13C2×#13133202—22no—
14C2×#13133191C2212yes—
15C2×#13133190—22no—
16C2×C4×D212C2212yescomplemented
17#16810757—28no—
18C2×#13133138C2214yes—
19C2×#13133137C2214yescomplemented
20C2×C22⋊#88322C2212yes—
21#16810385C2214yescomplemented
22#16810378C2212yescomplemented
23#16810378C2212yescomplemented
24C23×D4×C53C2211yescharacteristic
25C2×#117×C53C2211yescharacteristic
26C2×#117×C53C2211yescharacteristic
27C53×#390—22no—
28C53×#390C2212yes—
29C53×#390C2212yes—
30C53×#379—22no—
31C22×D212—22no—
32C22×D212C2312yes—
33C22×C22⋊D53—22no—
34C22×C4×D53—22no—
35C2×#11719922—28no—
36C2×#11719922—28no—
37C2×#11719922—44no—
38C22×#88322C2314yes—
39#13133214—24no—
40#13133209—24no—
41#13133202—216no—
42#13133191—24no—
43#13133191D418yes—
44#13133190—216no—
45C4×D212—24no—
46C4×D212—24no—
47C2×C4⋊Dic53C2312yes—
48C2×#11719919—22no—
49C2×#11719921—22no—
50C2×#11719921—22no—
51#13133138—28no—
52#13133137—44no—
53C22⋊#88322—24no—
54C25×C53—21no—
55C22×D4×C53—24no—
56C22×D4×C53—21no—
57C22×D4×C53—24no—
58C22×D4×C53—42no—
59C22×D4×C53C2312yes—
60C23×C4×C53C2311yescharacteristic
61#117×C53—24no—
62#117×C53—42no—
63C2×C4⋊C4×C53D412yes—
64C2×C4⋊C4×C53C2312yes—
65C2×C22⋊C4×C53—21no—
66C2×C22⋊C4×C53—24no—
67C2×C22⋊C4×C53—24no—
68C2×C22⋊C4×C53D412yes—
69C2×C42×C53—21no—
70C2×C42×C53C2311yescharacteristic
71C2×C42×C53C2311yescharacteristic
72C24×C53—24no—
73C24×C53—22no—
74C24×C53—42no—
75C24×C53—42no—
76C24×C53—84no—
77C24×C53C2×D411yescharacteristic
78C23×D53—22no—
79C2×D4×C53—24no—
80C2×D4×C53—24no—
81C2×D4×C53—22no—
82C2×D4×C53—24no—
83C2×D4×C53—416no—
84C2×D4×C53—42no—
85C2×D4×C53—42no—
86C2×D4×C53#4114yes—
87C22×C4×C53—22no—
88C22×C4×C53—22no—
89C22×C4×C53—22no—
90C22×C4×C53C2×D412yes—
91C22×C4×C53C2×D412yes—
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D414yes—
94C22×C4×C53C2411yescharacteristic
95C2×C22⋊D53—28no—
96C2×C22⋊D53—84no—
97C22×Dic53—22no—
98C2×D212—24no—
99C2×D212—24no—
100C2×D212—22no—
101C2×D212—28no—
102C2×D212—84no—
103C2×D212C2×D414yes—
104C2×C4×D53—28no—
105C2×C4×D53—84no—
106C4⋊C4×C53—24no—
107C4⋊C4×C53—24no—
108C22⋊C4×C53—24no—
109C22⋊C4×C53—42no—
110C22⋊C4×C53—84no—
111C22⋊C4×C53—84no—
112C42×C53—22no—
113C42×C53—22no—
114C42×C53—42no—
115#11719922—416no—
116#11719921—28no—
117#11719921—28no—
118C4⋊Dic53—22no—
119C4⋊Dic53C2×D414yes—
120#11719919—28no—
121C2×#88322—24no—
122C2×#88322—24no—
123C2×#88322—24no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53—24no—
127C23×C53—24no—
128C23×C53—48no—
129C23×C53—42no—
130C23×C53—42no—
131C23×C53—44no—
132C23×C53—44no—
133C23×C53—88no—
134C23×C53—88no—
135C23×C53—88no—
136C23×C53#12012yes—
137C23×C53C22×D411yescharacteristic
138C22×D53—24no—
139C22×D53—42no—
140C22×D53—42no—
141C22×D53—44no—
142C22×D53—44no—
143D4×C53—28no—
144D4×C53—48no—
145D4×C53—44no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—28no—
149C2×C4×C53—22no—
150C2×C4×C53—22no—
151C2×C4×C53—24no—
152C2×C4×C53—24no—
153C2×C4×C53—42no—
154C2×C4×C53—44no—
155C2×C4×C53#12012yes—
156C2×C4×C53C22×D412yes—
157C2×C4×C53C2×#4112yes—
158C2×C4×C53H(5,𝔽2)12yes—
159C22⋊D53—88no—
160C2×Dic53—24no—
161C2×Dic53—44no—
162D212—28no—
163D212—44no—
164D212—88no—
165C4×D53—88no—
166#88322—44no—
167#23939—531nomaximal, Sylow
168C22×C53—22no—
169C22×C53—24no—
170C22×C53—42no—
171C22×C53—42no—
172C22×C53—48no—
173C22×C53—44no—
174C22×C53—816no—
175C22×C53—82no—
176C22×C53—82no—
177C22×C53—84no—
178C22×C53C2×#12011yescharacteristic
179C22×C53#52511yescharacteristic
180C22×C53#53511yescharacteristic
181C22×C53#57614yes—
182C2×D53—44no—
183C2×D53—44no—
184C2×D53—44no—
185C2×D53—84no—
186C2×D53—84no—
187C4×C53—22no—
188C4×C53—42no—
189C4×C53—42no—
190C4×C53—82no—
191C4×C53#54614yes—
192Dic53—44no—
193C2×#540—531no—
194C2×#447—532no—
195#2309D5311yescharacteristic, complemented
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×D4C2×D5311yescharacteristic
209C22×D8—534no—
210#540—1068no—
211C2×#127—531no—
212C2×#125—531no—
213C2×#123—1062no—
214C2×#121—532no—
215C2×#121—1062no—
216C2×C4×D4—532no—
217C2×#117C2×D5311yescharacteristic
218C2×#117C2×D5311yescharacteristic
219#447—1068no—
220C2×#102—534no—
221C2×#102—534no—
222C2×C22⋊C8—532no—
223#390—22no—
224#390C2×D5312yes—
225#390C2×D5312yes—
226#379—22no—
227#340—534no—
228#336—532no—
229#336—532no—
230C53#2393911yescharacteristic, complemented, Sylow
231C25—21no—
232C22×D4—24no—
233C22×D4—24no—
234C22×D4—21no—
235C22×D4—42no—
236C22×D4—532no—
237C22×D4—1062no—
238C22×D4—1062no—
239C22×D4C22×D5312yes—
240C23×C4—1062no—
241C23×C4C22×D5311yescharacteristic
242C2×D8—1068no—
243C2×D8—1068no—
244C2×D8—2124no—
245C22×C8—534no—
246#127—1064no—
247#125—1064no—
248#123—10616no—
249#121—538no—
250#121—10616no—
251#121—1064no—
252C4×D4—1064no—
253C4×D4—1064no—
254#117—24no—
255#117—42no—
256C2×C4⋊C4—532no—
257C2×C4⋊C4—1062no—
258C2×C4⋊C4C22⋊D5312yes—
259C2×C4⋊C4C22×D5312yes—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—1062no—
264C2×C22⋊C4—1062no—
265C2×C22⋊C4C22⋊D5312yes—
266C2×C42—21no—
267C2×C42C22×D5311yescharacteristic
268C2×C42C22×D5311yescharacteristic
269#102—1068no—
270#102—2124no—
271C22⋊C8—1064no—
272C24—24no—
273C24—22no—
274C24—42no—
275C24—42no—
276C24—84no—
277C24—1062no—
278C24C2×C22⋊D5311yescharacteristic
279C2×D4—24no—
280C2×D4—24no—
281C2×D4—22no—
282C2×D4—24no—
283C2×D4—416no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—534no—
287C2×D4—1068no—
288C2×D4—1068no—
289C2×D4—1062no—
290C2×D4—1064no—
291C2×D4—1064no—
292C2×D4—4244no—
293C2×D4—4244no—
294C2×D4#1171994514yes—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—1068no—
299C22×C4—1062no—
300C22×C4—4244no—
301C22×C4D4×D5314yes—
302C22×C4C2×C22⋊D5312yes—
303C22×C4C2×C22⋊D5312yes—
304C22×C4C2×C22⋊D5311yescharacteristic
305C22×C4C23×D5311yescharacteristic
306D8—21216no—
307C2×C8—1064no—
308C2×C8—1064no—
309C2×C8—1064no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—534no—
313C4⋊C4—1068no—
314C4⋊C4—1062no—
315C22⋊C4—24no—
316C22⋊C4—42no—
317C22⋊C4—84no—
318C22⋊C4—84no—
319C22⋊C4—1068no—
320C22⋊C4—1068no—
321C42—22no—
322C42—22no—
323C42—42no—
324C23—21no—
325C23—24no—
326C23—21no—
327C23—24no—
328C23—48no—
329C23—42no—
330C23—44no—
331C23—44no—
332C23—42no—
333C23—88no—
334C23—88no—
335C23—88no—
336C23—1064no—
337C23—2122no—
338C23—2124no—
339C23—2124no—
340C23—2122no—
341C23#1313318512yes—
342C23C22×C22⋊D5311yescharacteristic
343D4—28no—
344D4—48no—
345D4—44no—
346D4—1068no—
347D4—4248no—
348D4—4248no—
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—44no—
358C2×C4—42no—
359C2×C4—1064no—
360C2×C4—4248no—
361C2×C4—2124no—
362C2×C4#1313318512yes—
363C2×C4C2×D4×D5312yes—
364C2×C4C2×#1171994512yes—
365C2×C4#1313326212yes—
366C8—2124no—
367C22—22no—
368C22—24no—
369C22—42no—
370C22—42no—
371C22—44no—
372C22—48no—
373C22—816no—
374C22—82no—
375C22—82no—
376C22—84no—
377C22—4244no—
378C22—4244no—
379C22—2124no—
380C22—2124no—
381C22—2124no—
382C22C2×#1313318511yescharacteristic
383C22#1681113411yescharacteristic
384C22#1681119011yescharacteristic
385C22#1681142614yes—
386C4—22no—
387C4—42no—
388C4—42no—
389C4—82no—
390C4—2124no—
391C4#1681126314yes—
392C2—42no—
393C2—84no—
394C2—4244no—
395C2#2027803411yescharacteristic
396C2#2028398112yes—
397C2#2028495812yes—
398C2#2028576512yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2309C2
socleC23×C53C22×D4
hypercenter#2309D53
hypocenterC53#23939
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2309

Complements

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

NH ≅ G/Ntype
1. C2×#16811228C2semidirect
2. C2×#16810757C2semidirect
3. #20275861C2semidirect
4. #20275785C2semidirect
5. #20275066C2semidirect
6. #20275037C2semidirect
7. #20274786C2semidirect
8. C53×#2309C2semidirect
11. C2×#13133214C22semidirect
12. C2×#13133209C22semidirect
16. C2×C4×D212C22semidirect
19. C2×#13133137C22semidirect
21. #16810385C22semidirect
22. #16810378C22semidirect
23. #16810378C22semidirect
195. #2309D53semidirect
230. C53#23939semidirect

Maximal subgroups

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

subgroupindex
C2×#168112282
C2×#168107572
#202758612
#202757852
#202750662
#202750372
#202747862
C53×#23092
#2393953