Finite Group Databasebeta

← #27271448

Subgroups — #27271448

Order 13568.

324 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 39 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#20274638C211yesmaximal, characteristic
2#20272651C211yesmaximal, characteristic
3C53×#1320C211yesmaximal, characteristic
4C23×Dic106C411yescharacteristic
5C2×#13133135—21no—
6C2×#13133135—21no—
7#16810587C411yescharacteristic
8#16810405—21no—
9C53×#407C2211yescharacteristic
10C53×#349—21no—
11C22×Dic106—21no—
12C22×Dic106—42no—
13C22×Dic106—42no—
14C22×Dic106—41no—
15C22×Dic106—41no—
16C22×Dic106C812yes—
17C23×Dic53—21no—
18C2×#11719917—21no—
19#13133157—41no—
20#13133135—42no—
21#13133135—42no—
22C22⋊#88322—41no—
23C23×C4×C53C2×C411yescharacteristic
24C2×M16×C53D411yescharacteristic
25C2×M16×C53D411yescharacteristic
26M32×C53—41no—
27C22⋊C8×C53—41no—
28C24×C53M1611yescharacteristic
29C22×C4×C53—21no—
30C22×C4×C53—21no—
31C22×C4×C53—41no—
32C22×C4×C53—41no—
33C22×C4×C53C22⋊C411yescharacteristic
34C22×C4×C53C2×C811yescharacteristic
35C22×Dic53—22no—
36C22×Dic53—44no—
37C22×Dic53—42no—
38C2×Dic106—22no—
39C2×Dic106—22no—
40C2×Dic106—41no—
41C2×Dic106—41no—
42C2×Dic106—48no—
43C2×Dic106—42no—
44C2×Dic106—42no—
45C2×Dic106—41no—
46C2×Dic106—41no—
47C2×Dic106—42no—
48C2×Dic106—44no—
49C2×Dic106—42no—
50C2×Dic106—42no—
51M16×C53—41no—
52M16×C53—41no—
53C2×C8×C53—21no—
54C2×C8×C53—21no—
55#11719917—41no—
56#11719917—41no—
57C2×#88322—41no—
58C16×C53—81no—
59#11719908—81no—
60C23×C53—21no—
61C23×C53—41no—
62C23×C53—42no—
63C23×C53C22⋊C811yescharacteristic
64C2×C4×C53—21no—
65C2×C4×C53—21no—
66C2×C4×C53—21no—
67C2×C4×C53—42no—
68C2×C4×C53—41no—
69C2×C4×C53—41no—
70C2×C4×C53—41no—
71C2×C4×C53#9911yescharacteristic
72C2×C4×C53#10011yescharacteristic
73C2×Dic53—24no—
74C2×Dic53—42no—
75C2×Dic53—44no—
76C2×Dic53—44no—
77C2×Dic53—42no—
78Dic106—42no—
79Dic106—42no—
80Dic106—44no—
81Dic106—42no—
82Dic106—42no—
83Dic106—44no—
84C8×C53—41no—
85C8×C53—41no—
86#88322—81no—
87#7343—531nomaximal, Sylow
88C22×C53—22no—
89C22×C53—21no—
90C22×C53—41no—
91C22×C53—41no—
92C22×C53—42no—
93C22×C53—42no—
94C22×C53—41no—
95C22×C53#32311yescharacteristic
96C4×C53—21no—
97C4×C53—41no—
98C4×C53AΣL(1,𝔽16)11yescharacteristic
99C4×C53AΣL(1,𝔽16)11yescharacteristic
100Dic53—44no—
101#1784—531no—
102#1320—531no—
103#1320D5311yescharacteristic
104C2×C53—21no—
105C2×C53—42no—
106C2×C53—41no—
107C2×C53#131611yescharacteristic
108C23×Q8—531no—
109C2×#101—1061no—
110C2×#101—1061no—
111#407—531no—
112#407C2×D5311yescharacteristic
113#349—21no—
114#349—1061no—
115C53#734311yescharacteristic, complemented, Sylow
116C22×Q8—532no—
117C22×Q8—1061no—
118C22×Q8—2121no—
119C22×Q8—2121no—
120C22×Q8—2122no—
121C22×Q8—2122no—
122C23×C4—1061no—
123C23×C4C4×D5311yescharacteristic
124C2×M16—1061no—
125C2×M16D21211yescharacteristic
126C2×M16C22⋊D5311yescharacteristic
127M32—41no—
128M32—2121no—
129#101—2122no—
130#101—2122no—
131C22⋊C8—41no—
132C22⋊C8—2121no—
133C24#1171991211yescharacteristic
134C2×Q8—1062no—
135C2×Q8—1062no—
136C2×Q8—2121no—
137C2×Q8—2121no—
138C2×Q8—2122no—
139C2×Q8—2124no—
140C2×Q8—2122no—
141C2×Q8—2122no—
142C2×Q8—2128no—
143C2×Q8—2122no—
144C2×Q8—2122no—
145C2×Q8—2121no—
146C2×Q8—2121no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—1062no—
152C22×C4—2124no—
153C22×C4—2122no—
154C22×C4C8×D5311yescharacteristic
155C22×C4#1171992111yescharacteristic
156M16—41no—
157M16—41no—
158M16—2121no—
159M16—2121no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—2121no—
163C16—81no—
164C16—4241no—
165C23—21no—
166C23—41no—
167C23—42no—
168C23#1313312911yescharacteristic
169Q8—2122no—
170Q8—2122no—
171Q8—2124no—
172Q8—2124no—
173Q8—2122no—
174Q8—2122no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—42no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—1064no—
183C2×C4—2122no—
184C2×C4—2124no—
185C2×C4—2124no—
186C2×C4—2122no—
187C2×C4#1313313111yescharacteristic
188C2×C4#1313313311yescharacteristic
189C8—41no—
190C8—41no—
191C8—4241no—
192C22—22no—
193C22—21no—
194C22—41no—
195C22—41no—
196C22—42no—
197C22—42no—
198C22—41no—
199C22#1681034611yescharacteristic
200C4—21no—
201C4—41no—
202C4—2124no—
203C4#1681040911yescharacteristic
204C4#1681041111yescharacteristic
205C2—21no—
206C2—42no—
207C2—41no—
208C2#2027264211yescharacteristic
subgroupG/H
centerC2#20272642
commutatorC22×C4×C53C2×C8
Frattini#407C2×D53
FittingC53×#1320C2
socleC2×C53#1316
prime radicalC24×C53M16
cosocleC53×#407C22
hypercenter#1320D53
hypocenterC53#7343
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #407 ◁ #1320

Complements

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

NH ≅ G/Ntype
115. C53#7343semidirect

Maximal subgroups

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

subgroupindex
#202746382
#202726512
C53×#13202
#734353