Finite Group Databasebeta

← #27284192

Subgroups — #27284192

Order 13568.

432 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283102C211yesmaximal, characteristic
2#20274568C211yesmaximal, characteristic
3#20272597C212yesmaximal
4#20272597C212yesmaximal
5C4×#97×C53C211yesmaximal, characteristic
6C2×C4⋊#88322—21no—
7C2×#13133128C2212yes—
8#16810521C411yescharacteristic
9#16810521C411yescharacteristic
10#16810521C411yescharacteristic
11#16810521C411yescharacteristic
12#16810507—21no—
13#16810507—21no—
14#16810507—21no—
15#16810507—21no—
16#16810489C2211yescharacteristic
17#16810489C2211yescharacteristic
18#16810378—21no—
19C2×#97×C53—21no—
20C2×#97×C53C2212yes—
21C2×C4×C8×C53—21no—
22C43×C53C2211yescharacteristic
23C22×#88322—22no—
24C2×C4⋊Dic53C2×C411yescharacteristic
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×#11719919—21no—
32C4⋊#88322—24no—
33#13133128—24no—
34#13133124—21no—
35#13133124—21no—
36#13133124—21no—
37#13133124—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42C22×C8×C53—22no—
43C2×C42×C53C2×C411yescharacteristic
44C2×C42×C53C2×C411yescharacteristic
45C2×C42×C53D411yescharacteristic
46C2×C42×C53D411yescharacteristic
47C2×C42×C53D411yescharacteristic
48C2×C42×C53D411yescharacteristic
49C2×C42×C53C2311yescharacteristic
50#97×C53—24no—
51#97×C53—22no—
52#97×C53Q814yes—
53C4×C8×C53—24no—
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22×C411yescharacteristic
59C22×C4×C53C2×D411yescharacteristic
60C22×C4×C53C2×D411yescharacteristic
61C22×Dic53—21no—
62C22×Dic53—21no—
63C22×Dic53—21no—
64C22×Dic53—21no—
65C2×C8×C53—24no—
66C2×C8×C53—24no—
67C2×C8×C53—24no—
68C42×C53—22no—
69C42×C53—22no—
70C42×C53—22no—
71C42×C53—41no—
72C42×C53—41no—
73C42×C53—41no—
74C42×C53C2×Q812yes—
75C42×C53#4112yes—
76C4⋊Dic53—41no—
77C4⋊Dic53—41no—
78C4⋊Dic53—41no—
79C4⋊Dic53—41no—
80#11719919—42no—
81#11719919—42no—
82#11719919—42no—
83#11719919—42no—
84C2×#88322—24no—
85C2×#88322—42no—
86C2×#88322—42no—
87C23×C53C2×C22⋊C411yescharacteristic
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53#10412yes—
104C2×C4×C53#10412yes—
105C2×C4×C53#10412yes—
106C2×C4×C53C4×Q812yes—
107C2×C4×C53C22⋊Q812yes—
108C2×C4×C53#12411yescharacteristic
109C2×C4×C53C4⋊Q811yescharacteristic
110C2×Dic53—41no—
111C2×Dic53—41no—
112C2×Dic53—41no—
113C2×Dic53—41no—
114C2×Dic53—41no—
115C2×Dic53—41no—
116C2×Dic53—41no—
117C2×Dic53—41no—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C8×C53—28no—
123#88322—44no—
124#9892—531nomaximal, Sylow
125C22×C53#39111yescharacteristic
126C22×C53#41311yescharacteristic
127C22×C53#41311yescharacteristic
128C22×C53C2×#10411yescharacteristic
129C22×C53C2×#10411yescharacteristic
130C22×C53C2×#10411yescharacteristic
131C22×C53#42111yescharacteristic
132C4×C53—21no—
133C4×C53—21no—
134C4×C53—22no—
135C4×C53—22no—
136C4×C53—22no—
137C4×C53—22no—
138C4×C53—22no—
139C4×C53—41no—
140Dic53—81no—
141Dic53—81no—
142Dic53—81no—
143Dic53—81no—
144#2864—531no—
145#1767—531no—
146C4×#97D5311yescharacteristic
147#1295—532no—
148#1295—532no—
149C2×C53#152411yescharacteristic
150C2×C53#152411yescharacteristic
151C2×C53#152911yescharacteristic
152C2×C53#153211yescharacteristic
153C2×C53#198611yescharacteristic
154C2×C53#199411yescharacteristic
155C2×C53#199811yescharacteristic
156C2×C4⋊C8—1061no—
157C2×#97—21no—
158C2×#97—532no—
159C2×#97C2×D5312yes—
160C2×C4×C8—21no—
161#389—531no—
162#389—531no—
163#389—531no—
164#389—531no—
165#387—1061no—
166#387—1061no—
167#387—1061no—
168#387—1061no—
169#382—531no—
170#382—531no—
171C43C2×D5311yescharacteristic
172#336—1061no—
173C53#989211yescharacteristic, complemented, Sylow
174C22×C8—22no—
175C22×C8—1062no—
176C2×C4⋊C4—531no—
177C2×C4⋊C4—531no—
178C2×C4⋊C4—531no—
179C2×C4⋊C4—531no—
180C2×C4⋊C4—1061no—
181C2×C4⋊C4—1061no—
182C2×C4⋊C4—1061no—
183C2×C4⋊C4—1061no—
184C2×C42C4×D5311yescharacteristic
185C2×C42C4×D5311yescharacteristic
186C2×C42D21211yescharacteristic
187C2×C42D21211yescharacteristic
188C2×C42C22⋊D5311yescharacteristic
189C2×C42C22⋊D5311yescharacteristic
190C2×C42C22×D5311yescharacteristic
191C4⋊C8—1064no—
192#97—24no—
193#97—22no—
194#97—1064no—
195#97Dic10614yes—
196C4×C8—24no—
197#95—1061no—
198#95—1061no—
199#95—1061no—
200#95—1061no—
201#95—1061no—
202#95—1061no—
203#95—1061no—
204#95—1061no—
205C22×C4—1061no—
206C22×C4—1061no—
207C22×C4—1061no—
208C22×C4—1061no—
209C22×C4#1171992111yescharacteristic
210C22×C4#1171992111yescharacteristic
211C22×C4#1171992111yescharacteristic
212C22×C4#1171992111yescharacteristic
213C22×C4C2×C4×D5311yescharacteristic
214C22×C4C2×D21211yescharacteristic
215C22×C4C2×C22⋊D5311yescharacteristic
216C2×C8—24no—
217C2×C8—24no—
218C2×C8—24no—
219C2×C8—1064no—
220C2×C8—2122no—
221C2×C8—2122no—
222C4⋊C4—2121no—
223C4⋊C4—2121no—
224C4⋊C4—2121no—
225C4⋊C4—2121no—
226C4⋊C4—2122no—
227C4⋊C4—2122no—
228C4⋊C4—2122no—
229C4⋊C4—2122no—
230C42—22no—
231C42—22no—
232C42—22no—
233C42—41no—
234C42—41no—
235C42—41no—
236C42C2×Dic10612yes—
237C42#1171994512yes—
238C23C2×#1171992111yescharacteristic
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—2121no—
255C2×C4—2121no—
256C2×C4—2121no—
257C2×C4—2121no—
258C2×C4—2121no—
259C2×C4—2121no—
260C2×C4—2121no—
261C2×C4—2121no—
262C2×C4—2121no—
263C2×C4—2121no—
264C2×C4—2121no—
265C2×C4—2121no—
266C2×C4#1313314312yes—
267C2×C4#1313314512yes—
268C2×C4#1313314512yes—
269C2×C4C4×Dic10612yes—
270C2×C4#1313319612yes—
271C2×C4#1313320511yescharacteristic
272C2×C4#1313321711yescharacteristic
273C8—28no—
274C8—2124no—
275C22#1681052911yescharacteristic
276C22#1681060511yescharacteristic
277C22#1681060511yescharacteristic
278C22C2×#1313314311yescharacteristic
279C22C2×#1313314511yescharacteristic
280C22C2×#1313314511yescharacteristic
281C22#1681063911yescharacteristic
282C4—22no—
283C4—22no—
284C4—22no—
285C4—21no—
286C4—21no—
287C4—22no—
288C4—22no—
289C4—41no—
290C4—4241no—
291C4—4241no—
292C4—4241no—
293C4—4241no—
294C2#2027330111yescharacteristic
295C2#2027330411yescharacteristic
296C2#2027332011yescharacteristic
297C2#2027333311yescharacteristic
298C2#2027565911yescharacteristic
299C2#2027569211yescharacteristic
300C2#2027570811yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC4×#97×C53C2
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenterC4×#97D53
hypocenterC53#9892
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C43 ◁ C4×#97

Complements

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

NH ≅ G/Ntype
173. C53#9892semidirect

Maximal subgroups

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

subgroupindex
#202831022
#202745682
#202725972
C4×#97×C532
#989253