Finite Group Databasebeta

← #27279810

Subgroups — #27279810

Order 13568.

738 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283054C211yesmaximal, characteristic
2C2×C4×C4⋊Dic53C211yesmaximal, characteristic
3C2×#16810361C212yesmaximal
4#20272595C212yesmaximal
5C53×#1817C211yesmaximal, characteristic
6C2×#13133196C2211yescharacteristic
7C22×C4⋊Dic53—21no—
8C22×C4×Dic53—21no—
9#16810529C412yescomplemented
10#16810521C412yescomplemented
11#16810507—22no—
12#16810496—21no—
13#16810486—21no—
14#16810483C2211yescharacteristic
15C4×C4⋊Dic53—24no—
16C2×#13133124C2212yes—
17#16810361—24no—
18C22×C42×C53C2211yescharacteristic
19C2×C22⋊C8×C53C2212yes—
20C53×#336—22no—
21C22×Dic106C2×C411yescharacteristic
22C23×Dic53—22no—
23#13133196—22no—
24#13133196—22no—
25C2×C4⋊Dic53—28no—
26C2×C4⋊Dic53—24no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×C4⋊Dic53C2×C412yes—
29C2×#11719919—22no—
30C2×#11719919—21no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53—21no—
33C2×C4×Dic53—24no—
34C2×C4×Dic53—42no—
35C2×C4×Dic53—42no—
36#13133124—22no—
37#13133124—22no—
38#13133124—22no—
39#13133124—21no—
40#13133124—22no—
41#13133124D414yes—
42C23×C4×C53D411yescharacteristic
43C23×C4×C53D411yescharacteristic
44C23×C4×C53C2311yescharacteristic
45C22×C8×C53—22no—
46C2×C42×C53—22no—
47C2×C42×C53—22no—
48C2×C42×C53C2×C412yes—
49C2×C42×C53D412yes—
50C22⋊C8×C53—22no—
51C22⋊C8×C53—22no—
52C24×C53C2×D411yescharacteristic
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—24no—
60C22×C4×C53—42no—
61C22×C4×C53C22⋊C412yes—
62C22×C4×C53C22⋊C412yes—
63C22×C4×C53SD1612yes—
64C22×C4×C53Q1612yes—
65C22×C4×C53C22×C411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53C2×D412yes—
68C22×C4×C53#4112yes—
69C22×Dic53—28no—
70C22×Dic53—24no—
71C22×Dic53—22no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—44no—
75C22×Dic53—42no—
76C22×Dic53—42no—
77C2×Dic106—22no—
78C2×Dic106—42no—
79C2×C8×C53—44no—
80C2×C8×C53—42no—
81C42×C53—24no—
82C42×C53—24no—
83C22⋊Dic53—41no—
84C22⋊Dic53—41no—
85C22⋊Dic53—41no—
86C22⋊Dic53—41no—
87C4⋊Dic53—216no—
88C4⋊Dic53—22no—
89C4⋊Dic53—42no—
90#11719919—44no—
91#11719919—42no—
92C4×Dic53—48no—
93C23×C53—22no—
94C23×C53—22no—
95C23×C53C2×C22⋊C411yescharacteristic
96C23×C53#12011yescharacteristic
97C23×C53#12311yescharacteristic
98C23×C53C2×SD1611yescharacteristic
99C23×C53C2×Q1611yescharacteristic
100C23×C53AGL(1,ℤ/8)11yescharacteristic
101C23×C53#13711yescharacteristic
102C2×C4×C53—24no—
103C2×C4×C53—24no—
104C2×C4×C53—22no—
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—22no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—22no—
113C2×C4×C53—22no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53#10314yes—
117C2×C4×C53#10414yes—
118C2×C4×C53C4×D412yes—
119C2×C4×C53#12112yes—
120C2×Dic53—28no—
121C2×Dic53—44no—
122C2×Dic53—48no—
123C2×Dic53—48no—
124C2×Dic53—44no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—42no—
128C2×Dic53—42no—
129C2×Dic53—42no—
130C2×Dic53—42no—
131C2×Dic53—42no—
132Dic106—82no—
133C8×C53—82no—
134#9144—531nomaximal, Sylow
135C22×C53—21no—
136C22×C53—21no—
137C22×C53—21no—
138C22×C53—21no—
139C22×C53—22no—
140C22×C53—22no—
141C22×C53—22no—
142C22×C53—22no—
143C22×C53#38611yescharacteristic
144C22×C53#41012yes—
145C22×C53C2×#10311yescharacteristic
146C22×C53#41711yescharacteristic
147C22×C53C2×#10412yes—
148C22×C53#45011yescharacteristic
149C22×C53#45111yescharacteristic
150C22×C53#48311yescharacteristic
151C22×C53#48411yescharacteristic
152C4×C53—24no—
153C4×C53—24no—
154C4×C53—22no—
155C4×C53—22no—
156Dic53—48no—
157Dic53—81no—
158Dic53—81no—
159Dic53—82no—
160#2855—531no—
161C2×C4×C4⋊C4—531no—
162#1817D5311yescharacteristic
163C2×#328—532no—
164#1294—532no—
165C2×C53—21no—
166C2×C53—21no—
167C2×C53—21no—
168C2×C53—21no—
169C2×C53#150312yes—
170C2×C53#150912yes—
171C2×C53#187211yescharacteristic
172C2×C53#189611yescharacteristic
173C2×C53#189911yescharacteristic
174C2×C22⋊Q8—531no—
175C22×C4⋊C4—1061no—
176C22×C42—1061no—
177C22×C42C2×D5311yescharacteristic
178C2×C22⋊C8C2×D5312yes—
179#391—532no—
180#389—532no—
181#387—1062no—
182#385—1061no—
183#381—1061no—
184#380—531no—
185C4×C4⋊C4—1064no—
186C2×#95—532no—
187#336—22no—
188#328—1064no—
189C53#914411yescharacteristic, complemented, Sylow
190C22×Q8—531no—
191C23×C4—1062no—
192C23×C4D21211yescharacteristic
193C23×C4D21211yescharacteristic
194C23×C4C22×D5311yescharacteristic
195C22×C8—22no—
196C22⋊Q8—1062no—
197C22⋊Q8—1062no—
198C2×C4⋊C4—531no—
199C2×C4⋊C4—532no—
200C2×C4⋊C4—1068no—
201C2×C4⋊C4—1064no—
202C2×C4⋊C4—1061no—
203C2×C4⋊C4—1062no—
204C2×C22⋊C4—1061no—
205C2×C22⋊C4—1061no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—1064no—
209C2×C42—2122no—
210C2×C42—2122no—
211C2×C42C4×D5312yes—
212C2×C42C22⋊D5312yes—
213C22⋊C8—22no—
214C22⋊C8—22no—
215#95—534no—
216#95—1062no—
217#95—1061no—
218#95—1062no—
219#95—1062no—
220#95—1062no—
221C24C2×D21211yescharacteristic
222C2×Q8—1062no—
223C2×Q8—2122no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—24no—
231C22×C4—42no—
232C22×C4—1068no—
233C22×C4—1064no—
234C22×C4—1062no—
235C22×C4—1061no—
236C22×C4—1061no—
237C22×C4—2124no—
238C22×C4—2122no—
239C22×C4—2122no—
240C22×C4#1171991312yes—
241C22×C4Dic21212yes—
242C22×C4#1171992112yes—
243C22×C4#1171992112yes—
244C22×C4C2×C4×D5311yescharacteristic
245C22×C4D4×D5312yes—
246C22×C4#1171994712yes—
247C22×C4C2×C22⋊D5311yescharacteristic
248C2×C8—44no—
249C2×C8—42no—
250C4⋊C4—10616no—
251C4⋊C4—1062no—
252C4⋊C4—2122no—
253C4⋊C4—2124no—
254C4⋊C4—2122no—
255C22⋊C4—2121no—
256C22⋊C4—2121no—
257C22⋊C4—2121no—
258C22⋊C4—2121no—
259C42—24no—
260C42—24no—
261C42—2128no—
262C23—22no—
263C23—22no—
264C23C2×#1171992111yescharacteristic
265C23#1313318711yescharacteristic
266C23#1313320111yescharacteristic
267C23C2×#1171991311yescharacteristic
268C23C2×Dic21211yescharacteristic
269C23#1313324511yescharacteristic
270C23#1313325011yescharacteristic
271Q8—4242no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—1068no—
287C2×C4—2121no—
288C2×C4—2121no—
289C2×C4—2122no—
290C2×C4—2122no—
291C2×C4—2122no—
292C2×C4—2122no—
293C2×C4—2122no—
294C2×C4—2128no—
295C2×C4—2128no—
296C2×C4—2124no—
297C2×C4—2124no—
298C2×C4#1313314214yes—
299C2×C4#1313314314yes—
300C2×C4#1313317712yes—
301C2×C4#1313318912yes—
302C8—82no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22#1681050011yescharacteristic
312C22#1681059612yes—
313C22C2×#1313314211yescharacteristic
314C22#1681062511yescharacteristic
315C22C2×#1313314312yes—
316C22#1681077611yescharacteristic
317C22#1681078111yescharacteristic
318C22#1681095411yescharacteristic
319C22#1681095911yescharacteristic
320C4—24no—
321C4—24no—
322C4—22no—
323C4—22no—
324C4—4242no—
325C4—4241no—
326C4—4241no—
327C4—2128no—
328C2—21no—
329C2—21no—
330C2—21no—
331C2—21no—
332C2#2027321512yes—
333C2#2027324112yes—
334C2#2027502011yescharacteristic
335C2#2027515611yescharacteristic
336C2#2027517111yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1817C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#1817D53
hypocenterC53#9144
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
9. #16810529C4semidirect
10. #16810521C4semidirect
189. C53#9144semidirect

Maximal subgroups

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

subgroupindex
#202830542
C2×C4×C4⋊Dic532
C2×#168103612
#202725952
C53×#18172
#914453