Finite Group Databasebeta

← C53×#7324 #27214836

Subgroups — C53×#7324 #27214836

Order 13568.

482 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 78 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
1C53×#1462C212yesmaximal
2C53×#1440C211yesmaximal, characteristic
3C22×C22⋊C4×C53C412yes—
4C22×C22⋊C4×C53C2211yescharacteristic
5C53×#407—22no—
6C2×#95×C53—22no—
7C53×#323—22no—
8C53×#323C414yescomplemented
9C25×C53C2×C411yescharacteristic
10C23×C4×C53—22no—
11C23×C4×C53D412yes—
12C2×M16×C53—42no—
13C2×C22⋊C4×C53—42no—
14C2×C22⋊C4×C53—44no—
15C2×C22⋊C4×C53—42no—
16C2×C22⋊C4×C53C2×C412yes—
17C2×C22⋊C4×C53D411yescharacteristic
18C2×C22⋊C4×C53Q811yescharacteristic
19C22⋊C8×C53—24no—
20C22⋊C8×C53—42no—
21#95×C53—44no—
22C24×C53—22no—
23C24×C53—42no—
24C24×C53—42no—
25C24×C53—42no—
26C24×C53C4211yescharacteristic
27C24×C53C22⋊C411yescharacteristic
28C24×C53C4⋊C411yescharacteristic
29C22×C4×C53—22no—
30C22×C4×C53—22no—
31C22×C4×C53—24no—
32C22×C4×C53—42no—
33C22×C4×C53—42no—
34C22×C4×C53—42no—
35C22×C4×C53—42no—
36C22×C4×C53—42no—
37C22×C4×C53—42no—
38C22×C4×C53—42no—
39C22×C4×C53—42no—
40C22×C4×C53C22⋊C412yes—
41C22×C4×C53C4⋊C412yes—
42M16×C53—82no—
43M16×C53—82no—
44C2×C8×C53—44no—
45C22⋊C4×C53—42no—
46C22⋊C4×C53—41no—
47C22⋊C4×C53—41no—
48C22⋊C4×C53—44no—
49C22⋊C4×C53—44no—
50C23×C53—22no—
51C23×C53—22no—
52C23×C53—22no—
53C23×C53—21no—
54C23×C53—21no—
55C23×C53—22no—
56C23×C53—22no—
57C23×C53—44no—
58C23×C53—44no—
59C23×C53—42no—
60C23×C53—42no—
61C23×C53—42no—
62C23×C53—42no—
63C23×C53—82no—
64C23×C53—82no—
65C23×C53—82no—
66C23×C53—82no—
67C23×C53#9511yescharacteristic
68C23×C53#9912yes—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—42no—
72C2×C4×C53—42no—
73C2×C4×C53—42no—
74C2×C4×C53—42no—
75C2×C4×C53—44no—
76C2×C4×C53—44no—
77C2×C4×C53—44no—
78C2×C4×C53—82no—
79C2×C4×C53—82no—
80C2×C4×C53—82no—
81C2×C4×C53—82no—
82C2×C4×C53—82no—
83C8×C53—84no—
84#7324C5311yesmaximal, characteristic, direct product factor, Sylow
85C22×C53—22no—
86C22×C53—22no—
87C22×C53—22no—
88C22×C53—22no—
89C22×C53—22no—
90C22×C53—22no—
91C22×C53—22no—
92C22×C53—41no—
93C22×C53—41no—
94C22×C53—42no—
95C22×C53—42no—
96C22×C53—82no—
97C22×C53—82no—
98C22×C53—84no—
99C22×C53—84no—
100C22×C53#34211yescharacteristic
101C22×C53#34412yes—
102C22×C53AΣL(1,𝔽16)12yes—
103C22×C53#35212yes—
104C4×C53—42no—
105C4×C53—42no—
106C4×C53—84no—
107#1462C2×C5312yes—
108#1440C2×C5311yescharacteristic
109C2×C53—22no—
110C2×C53—22no—
111C2×C53—22no—
112C2×C53—81no—
113C2×C53—81no—
114C2×C53#130611yescharacteristic
115C2×C53#139012yes—
116C22×C22⋊C4C4×C5312yes—
117C22×C22⋊C4C22×C5311yescharacteristic
118#407—22no—
119C2×#95—22no—
120#323—22no—
121#323C4×C5314yescomplemented
122C53#732411yescharacteristic, direct product factor, Sylow
123C25C2×C4×C5311yescharacteristic
124C23×C4—22no—
125C23×C4D4×C5312yes—
126C2×M16—42no—
127C2×C22⋊C4—42no—
128C2×C22⋊C4—42no—
129C2×C22⋊C4—44no—
130C2×C22⋊C4C2×C4×C5312yes—
131C2×C22⋊C4D4×C5311yescharacteristic
132C2×C22⋊C4Q8×C5311yescharacteristic
133C22⋊C8—24no—
134C22⋊C8—42no—
135#95—44no—
136C24—22no—
137C24—42no—
138C24—42no—
139C24—42no—
140C24C42×C5311yescharacteristic
141C24C22⋊C4×C5311yescharacteristic
142C24C4⋊C4×C5311yescharacteristic
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—24no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4C22⋊C4×C5312yes—
155C22×C4C4⋊C4×C5312yes—
156M16—82no—
157M16—82no—
158C2×C8—44no—
159C22⋊C4—42no—
160C22⋊C4—41no—
161C22⋊C4—41no—
162C22⋊C4—44no—
163C22⋊C4—44no—
164C23—22no—
165C23—22no—
166C23—22no—
167C23—21no—
168C23—21no—
169C23—22no—
170C23—22no—
171C23—44no—
172C23—44no—
173C23—42no—
174C23—42no—
175C23—42no—
176C23—42no—
177C23—82no—
178C23—82no—
179C23—82no—
180C23—82no—
181C23#95×C5311yescharacteristic
182C23#99×C5312yes—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—44no—
190C2×C4—44no—
191C2×C4—44no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4—82no—
195C2×C4—82no—
196C2×C4—82no—
197C8—84no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—22no—
202C22—22no—
203C22—22no—
204C22—22no—
205C22—41no—
206C22—41no—
207C22—42no—
208C22—42no—
209C22—82no—
210C22—82no—
211C22—84no—
212C22—84no—
213C22C53×#34211yescharacteristic
214C22C53×#34412yes—
215C22C53×AΣL(1,𝔽16)12yes—
216C22C53×#35212yes—
217C4—42no—
218C4—42no—
219C4—84no—
220C2—22no—
221C2—22no—
222C2—22no—
223C2—81no—
224C2—81no—
225C2C53×#130611yescharacteristic
226C2C53×#139012yes—
subgroupG/H
center, socleC22×C53#342
commutatorC24C42×C53
Frattini, cosocleC22×C22⋊C4C22×C53
prime radicalC25×C53C2×C4
LCSG ▷ C24 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C24×C53 ◁ C22×C22⋊C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
8. C53×#323C4semidirect
84. #7324C53direct (N × H)
121. #323C4×C53semidirect
122. C53#7324direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#14622
C53×#14402
#732453