Finite Group Databasebeta

← #27277609

Subgroups — #27277609

Order 13568.

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

normal subgroups only

This group has 754 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#20274636C211yesmaximal, characteristic
3#20272624C214yesmaximal, complemented
4C53×#1783C211yesmaximal, characteristic
5C22×C22⋊Dic53C412yes—
6C2×#13133134—22no—
7#16810587C2212yes—
8#16810503—22no—
9#16810503—22no—
10#16810486C2212yes—
11#16810479C412yes—
12C23×D4×C53C2211yescharacteristic
13C2×#100×C53—22no—
14C53×#407C2212yes—
15C23×Dic53—22no—
16C2×#11719917—22no—
17C2×C22⋊Dic53—22no—
18C2×C22⋊Dic53—24no—
19C2×C22⋊Dic53—44no—
20C2×C22⋊Dic53C2×C414yes—
21#13133134—44no—
22C22⋊#88322—24no—
23#13133124—22no—
24#13133124—22no—
25C25×C53C2×C411yescharacteristic
26C25×C53C2×C411yescharacteristic
27C22×D4×C53—24no—
28C22×D4×C53—44no—
29C22×D4×C53D412yes—
30C22×D4×C53D412yes—
31C23×C4×C53C2311yescharacteristic
32C2×M16×C53—22no—
33#100×C53—44no—
34C22⋊C8×C53—24no—
35C24×C53—24no—
36C24×C53—22no—
37C24×C53—22no—
38C24×C53—42no—
39C24×C53—82no—
40C24×C53—82no—
41C24×C53C22⋊C412yes—
42C24×C53C22⋊C412yes—
43C24×C53C22×C411yescharacteristic
44C2×D4×C53—24no—
45C2×D4×C53—24no—
46C2×D4×C53—24no—
47C2×D4×C53—24no—
48C2×D4×C53—42no—
49C2×D4×C53—42no—
50C2×D4×C53—416no—
51C22×C4×C53—42no—
52C22×C4×C53C2×D411yescharacteristic
53C22×C4×C53C2×D411yescharacteristic
54C22×C4×C53#4114yes—
55C22×Dic53—24no—
56C22×Dic53—44no—
57C22×Dic53—84no—
58M16×C53—44no—
59C2×C8×C53—42no—
60C22⋊Dic53—44no—
61C22⋊Dic53—44no—
62C22⋊Dic53—84no—
63#11719917—44no—
64C2×#88322—42no—
65C23×C53—28no—
66C23×C53—22no—
67C23×C53—22no—
68C23×C53—22no—
69C23×C53—22no—
70C23×C53—22no—
71C23×C53—22no—
72C23×C53—42no—
73C23×C53—44no—
74C23×C53—44no—
75C23×C53—44no—
76C23×C53—84no—
77C23×C53—84no—
78C23×C53—88no—
79C23×C53—88no—
80C23×C53#9914yes—
81C23×C53C2×C22⋊C411yescharacteristic
82C23×C53#11712yes—
83D4×C53—48no—
84D4×C53—48no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—44no—
88C2×C4×C53#12312yes—
89C2×C4×C53#12312yes—
90C2×Dic53—48no—
91C2×Dic53—44no—
92C2×Dic53—88no—
93C8×C53—82no—
94#88322—82no—
95#8752—531nomaximal, Sylow
96C22×C53—22no—
97C22×C53—22no—
98C22×C53—24no—
99C22×C53—24no—
100C22×C53—44no—
101C22×C53—44no—
102C22×C53—44no—
103C22×C53—44no—
104C22×C53—42no—
105C22×C53—44no—
106C22×C53—82no—
107C22×C53—82no—
108C22×C53—88no—
109C22×C53—88no—
110C22×C53#38111yescharacteristic
111C22×C53C2×#9912yes—
112C4×C53—22no—
113C4×C53—22no—
114Dic53—84no—
115#2308—531no—
116#1783—531no—
117#1783D5311yescharacteristic
118#1306—534no—
119C2×C53—22no—
120C2×C53—42no—
121C2×C53—44no—
122C2×C53—82no—
123C2×C53#181911yescharacteristic
124C2×C53#182912yes—
125C23×D4C2×D5311yescharacteristic
126C22×C22⋊C4—532no—
127C2×#100—22no—
128C2×#100—1062no—
129#407—532no—
130#407C2×D5312yes—
131#386—1062no—
132#386—1062no—
133#381—532no—
134#379—532no—
135C53#875211yescharacteristic, complemented, Sylow
136C25C4×D5311yescharacteristic
137C25C4×D5311yescharacteristic
138C22×D4—24no—
139C22×D4—44no—
140C22×D4D21212yes—
141C22×D4C22⋊D5312yes—
142C23×C4—1062no—
143C23×C4C22×D5311yescharacteristic
144C2×M16—22no—
145C2×M16—1062no—
146C2×C22⋊C4—534no—
147C2×C22⋊C4—1064no—
148C2×C22⋊C4—1062no—
149C2×C22⋊C4—2124no—
150#100—44no—
151#100—2124no—
152C22⋊C8—24no—
153C22⋊C8—1064no—
154#95—1062no—
155#95—1062no—
156C24—24no—
157C24—22no—
158C24—22no—
159C24—42no—
160C24—82no—
161C24—82no—
162C24#1171992112yes—
163C24#1171992112yes—
164C24C2×C4×D5311yescharacteristic
165C2×D4—24no—
166C2×D4—24no—
167C2×D4—24no—
168C2×D4—24no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—416no—
172C22×C4—42no—
173C22×C4—1064no—
174C22×C4—4244no—
175C22×C4—2124no—
176C22×C4C2×D21211yescharacteristic
177C22×C4#1171994714yes—
178C22×C4C2×C22⋊D5311yescharacteristic
179M16—44no—
180M16—2124no—
181C2×C8—42no—
182C2×C8—2122no—
183C22⋊C4—4244no—
184C22⋊C4—2124no—
185C22⋊C4—2124no—
186C23—28no—
187C23—22no—
188C23—22no—
189C23—22no—
190C23—22no—
191C23—22no—
192C23—22no—
193C23—42no—
194C23—44no—
195C23—44no—
196C23—44no—
197C23—84no—
198C23—84no—
199C23—88no—
200C23—88no—
201C23#1313313114yes—
202C23C2×#1171992111yescharacteristic
203C23#1313317312yes—
204D4—48no—
205D4—48no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—44no—
209C2×C4—4248no—
210C2×C4—2128no—
211C2×C4—2124no—
212C2×C4#1313319812yes—
213C2×C4#1313320112yes—
214C8—82no—
215C8—4242no—
216C22—22no—
217C22—22no—
218C22—24no—
219C22—24no—
220C22—44no—
221C22—44no—
222C22—44no—
223C22—44no—
224C22—42no—
225C22—44no—
226C22—82no—
227C22—82no—
228C22—88no—
229C22—88no—
230C22#1681048411yescharacteristic
231C22C2×#1313313112yes—
232C4—22no—
233C4—22no—
234C4—4244no—
235C2—22no—
236C2—42no—
237C2—44no—
238C2—82no—
239C2#2027478711yescharacteristic
240C2#2027482412yes—
subgroupG/H
centerC22#16810484
commutatorC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1783C2
socleC22×C53#381
prime radicalC23×D4×C53C22
cosocleC23×C4×C53C23
hypercenter#1783D53
hypocenterC53#8752
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1783

Complements

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

NH ≅ G/Ntype
3. #20272624C2semidirect
135. C53#8752semidirect

Maximal subgroups

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

subgroupindex
#202772932
#202746362
#202726242
C53×#17832
#875253