Finite Group Databasebeta

← #27277592

Subgroups — #27277592

Order 13568.

814 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 99 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
1#20277293C211yesmaximal, characteristic
2#20274630C211yesmaximal, characteristic
3#20272619C214yesmaximal, complemented
4C53×#1781C211yesmaximal, characteristic
5C22×C22⋊Dic53C412yescomplemented
6C2×#13133132—22no—
7#16810503—22no—
8#16810503—22no—
9#16810486C412yescomplemented
10#16810479C2212yes—
11#16810479C2212yes—
12C23×D4×C53C2211yescharacteristic
13C2×#99×C53—22no—
14C53×#379C2212yes—
15C23×Dic53—22no—
16C2×C22⋊Dic53—22no—
17C2×C22⋊Dic53—22no—
18C2×C22⋊Dic53—22no—
19C2×C22⋊Dic53—22no—
20C2×C22⋊Dic53—24no—
21C2×C22⋊Dic53—44no—
22C2×C22⋊Dic53C2×C414yes—
23#13133132—44no—
24#13133124—24no—
25C25×C53C2×C411yescharacteristic
26C25×C53C2311yescharacteristic
27C22×D4×C53—24no—
28C22×D4×C53—44no—
29C22×D4×C53D412yes—
30C22×D4×C53D412yes—
31C23×C4×C53C2×C411yescharacteristic
32C2×C22⋊C4×C53—22no—
33C2×C22⋊C4×C53—24no—
34#99×C53—44no—
35C24×C53—24no—
36C24×C53—22no—
37C24×C53—42no—
38C24×C53—82no—
39C24×C53—82no—
40C24×C53C22⋊C412yes—
41C24×C53C22×C411yescharacteristic
42C24×C53C2×D411yescharacteristic
43C24×C53C2×D411yescharacteristic
44C24×C53#4114yes—
45C2×D4×C53—24no—
46C2×D4×C53—24no—
47C2×D4×C53—24no—
48C2×D4×C53—24no—
49C2×D4×C53—42no—
50C2×D4×C53—42no—
51C2×D4×C53—416no—
52C22×C4×C53—22no—
53C22×C4×C53—42no—
54C22×C4×C53—42no—
55C22×C4×C53C22⋊C412yes—
56C22×Dic53—24no—
57C22×Dic53—44no—
58C22×Dic53—42no—
59C22×Dic53—84no—
60C22⋊C4×C53—44no—
61C22⋊C4×C53—84no—
62C22⋊Dic53—44no—
63C22⋊Dic53—44no—
64C22⋊Dic53—44no—
65C22⋊Dic53—82no—
66C22⋊Dic53—82no—
67C22⋊Dic53—84no—
68C23×C53—28no—
69C23×C53—22no—
70C23×C53—22no—
71C23×C53—22no—
72C23×C53—22no—
73C23×C53—22no—
74C23×C53—42no—
75C23×C53—44no—
76C23×C53—42no—
77C23×C53—42no—
78C23×C53—44no—
79C23×C53—44no—
80C23×C53—44no—
81C23×C53—82no—
82C23×C53—82no—
83C23×C53—88no—
84C23×C53—88no—
85C23×C53#9914yes—
86C23×C53C2×C22⋊C411yescharacteristic
87C23×C53#11712yes—
88C23×C53#12312yes—
89C23×C53#12312yes—
90D4×C53—48no—
91D4×C53—48no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—44no—
96C2×C4×C53—82no—
97C2×C4×C53—84no—
98C2×Dic53—44no—
99C2×Dic53—44no—
100C2×Dic53—44no—
101C2×Dic53—88no—
102C2×Dic53—82no—
103C2×Dic53—84no—
104#8747—531nomaximal, Sylow
105C22×C53—22no—
106C22×C53—22no—
107C22×C53—22no—
108C22×C53—22no—
109C22×C53—24no—
110C22×C53—24no—
111C22×C53—24no—
112C22×C53—22no—
113C22×C53—22no—
114C22×C53—44no—
115C22×C53—44no—
116C22×C53—44no—
117C22×C53—44no—
118C22×C53—82no—
119C22×C53—82no—
120C22×C53—88no—
121C22×C53—88no—
122C22×C53#38111yescharacteristic
123C22×C53C2×#9912yes—
124C4×C53—42no—
125C4×C53—162no—
126Dic53—84no—
127Dic53—162no—
128#2308—531no—
129#1781—531no—
130#1781D5311yescharacteristic
131#1304—534no—
132C2×C53—22no—
133C2×C53—42no—
134C2×C53—42no—
135C2×C53—42no—
136C2×C53—44no—
137C2×C53#181911yescharacteristic
138C2×C53#182812yes—
139C23×D4C2×D5311yescharacteristic
140C22×C22⋊C4—532no—
141C2×#99—22no—
142C2×#99—1062no—
143#386—1062no—
144#386—1062no—
145#381—532no—
146#379—532no—
147#379—532no—
148#379C2×D5312yes—
149C53#874711yescharacteristic, complemented, Sylow
150C25C4×D5311yescharacteristic
151C25C22×D5311yescharacteristic
152C22×D4—24no—
153C22×D4—44no—
154C22×D4D21212yes—
155C22×D4C22⋊D5312yes—
156C23×C4—1062no—
157C23×C4C4×D5311yescharacteristic
158C2×C22⋊C4—22no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—534no—
161C2×C22⋊C4—1062no—
162C2×C22⋊C4—1062no—
163C2×C22⋊C4—1064no—
164C2×C22⋊C4—1062no—
165C2×C22⋊C4—1062no—
166C2×C22⋊C4—2124no—
167#99—44no—
168#99—2124no—
169#95—1064no—
170C24—24no—
171C24—22no—
172C24—42no—
173C24—82no—
174C24—82no—
175C24#1171992112yes—
176C24C2×C4×D5311yescharacteristic
177C24C2×D21211yescharacteristic
178C24#1171994714yes—
179C24C2×C22⋊D5311yescharacteristic
180C2×D4—24no—
181C2×D4—24no—
182C2×D4—24no—
183C2×D4—24no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—416no—
187C22×C4—22no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—1064no—
191C22×C4—4244no—
192C22×C4—2122no—
193C22×C4—2124no—
194C22×C4#1171992112yes—
195C22⋊C4—44no—
196C22⋊C4—84no—
197C22⋊C4—4244no—
198C22⋊C4—4242no—
199C22⋊C4—4242no—
200C22⋊C4—2124no—
201C22⋊C4—2124no—
202C22⋊C4—2124no—
203C23—28no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—42no—
210C23—44no—
211C23—42no—
212C23—44no—
213C23—42no—
214C23—44no—
215C23—44no—
216C23—82no—
217C23—82no—
218C23—88no—
219C23—88no—
220C23#1313313114yes—
221C23C2×#1171992111yescharacteristic
222C23#1313317312yes—
223C23#1313319812yes—
224C23#1313320112yes—
225D4—48no—
226D4—48no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—44no—
231C2×C4—82no—
232C2×C4—84no—
233C2×C4—4244no—
234C2×C4—4242no—
235C2×C4—4248no—
236C2×C4—2124no—
237C2×C4—2124no—
238C2×C4—2124no—
239C22—22no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—24no—
244C22—24no—
245C22—24no—
246C22—22no—
247C22—22no—
248C22—44no—
249C22—44no—
250C22—44no—
251C22—44no—
252C22—82no—
253C22—82no—
254C22—88no—
255C22—88no—
256C22#1681048411yescharacteristic
257C22C2×#1313313112yes—
258C4—42no—
259C4—162no—
260C4—8482no—
261C4—4244no—
262C2—22no—
263C2—42no—
264C2—42no—
265C2—42no—
266C2—44no—
267C2#2027478711yescharacteristic
268C2#2027482112yes—
subgroupG/H
centerC22#16810484
commutatorC24×C53C22×C4
FrattiniC25C22×D53
FittingC53×#1781C2
socleC22×C53#381
prime radicalC23×D4×C53C22
cosocleC25×C53C23
hypercenter#1781D53
hypocenterC53#8747
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C25 ◁ #1781

Complements

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

NH ≅ G/Ntype
3. #20272619C2semidirect
5. C22×C22⋊Dic53C4semidirect
9. #16810486C4semidirect
149. C53#8747semidirect

Maximal subgroups

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

subgroupindex
#202772932
#202746302
#202726192
C53×#17812
#874753