Finite Group Databasebeta

← C53×#11511 #27219023

Subgroups — C53×#11511 #27219023

Order 13568.

490 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 118 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
1C2×C53×#382C211yesmaximal, characteristic
2C53×#1463C212yesmaximal
3C53×#1324C214yesmaximal, complemented
4C22×C4⋊C4×C53C411yescharacteristic
5C22×C4⋊C4×C53C2211yescharacteristic
6C22×C42×C53C411yescharacteristic
7C53×#407—22no—
8C53×#382—22no—
9C53×#382C412yes—
10C53×#382C2212yes—
11C2×#95×C53—22no—
12C53×#324—22no—
13C53×#324C2214yes—
14C23×C4×C53—21no—
15C23×C4×C53—21no—
16C23×C4×C53C2×C411yescharacteristic
17C23×C4×C53D412yes—
18C2×M16×C53—42no—
19C2×C4⋊C4×C53—21no—
20C2×C4⋊C4×C53—41no—
21C2×C4⋊C4×C53—41no—
22C2×C4⋊C4×C53—41no—
23C2×C4⋊C4×C53—41no—
24C2×C4⋊C4×C53C2×C412yes—
25C2×C4⋊C4×C53C2×C411yescharacteristic
26C2×C4⋊C4×C53D412yes—
27C2×C4⋊C4×C53C2311yescharacteristic
28C2×C42×C53—24no—
29C2×C42×C53—21no—
30C2×C42×C53C2×C412yes—
31C22⋊C8×C53—24no—
32C22⋊C8×C53—42no—
33#95×C53—24no—
34#95×C53—24no—
35C24×C53C22⋊C411yescharacteristic
36C22×C4×C53—21no—
37C22×C4×C53—21no—
38C22×C4×C53—22no—
39C22×C4×C53—22no—
40C22×C4×C53—24no—
41C22×C4×C53—22no—
42C22×C4×C53—22no—
43C22×C4×C53—22no—
44C22×C4×C53—22no—
45C22×C4×C53—42no—
46C22×C4×C53—42no—
47C22×C4×C53—44no—
48C22×C4×C53—44no—
49C22×C4×C53—41no—
50C22×C4×C53—41no—
51C22×C4×C53C22⋊C411yescharacteristic
52C22×C4×C53C22⋊C412yes—
53C22×C4×C53C22×C411yescharacteristic
54C22×C4×C53C2×D412yes—
55M16×C53—84no—
56C2×C8×C53—44no—
57C4⋊C4×C53—41no—
58C4⋊C4×C53—42no—
59C4⋊C4×C53—41no—
60C4⋊C4×C53—42no—
61C4⋊C4×C53—42no—
62C42×C53—24no—
63C42×C53—22no—
64C42×C53M1614yescomplemented
65C23×C53—22no—
66C23×C53—22no—
67C23×C53—21no—
68C23×C53—21no—
69C23×C53#9911yescharacteristic
70C23×C53#9911yescharacteristic
71C23×C53C2×C22⋊C411yescharacteristic
72C2×C4×C53—24no—
73C2×C4×C53—24no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—24no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—48no—
82C2×C4×C53—42no—
83C2×C4×C53—42no—
84C2×C4×C53—44no—
85C2×C4×C53—42no—
86C2×C4×C53—42no—
87C2×C4×C53—44no—
88C2×C4×C53—42no—
89C2×C4×C53—82no—
90C2×C4×C53—82no—
91C2×C4×C53—82no—
92C2×C4×C53#10112yes—
93C2×C4×C53C2×M1612yes—
94C8×C53—84no—
95#11511C5311yesmaximal, characteristic, direct product factor, Sylow
96C22×C53—24no—
97C22×C53—24no—
98C22×C53—21no—
99C22×C53—21no—
100C22×C53—21no—
101C22×C53—21no—
102C22×C53—21no—
103C22×C53—21no—
104C22×C53#35412yes—
105C22×C53#35512yes—
106C22×C53#40711yescharacteristic
107C22×C53C2×#9911yescharacteristic
108C22×C53C2×#10111yescharacteristic
109C4×C53—22no—
110C4×C53—22no—
111C4×C53—44no—
112C4×C53—42no—
113C4×C53—42no—
114C4×C53—82no—
115C2×#382C2×C5311yescharacteristic
116#1463C2×C5312yes—
117#1324C2×C5314yescomplemented
118C2×C53—21no—
119C2×C53—22no—
120C2×C53—22no—
121C2×C53—21no—
122C2×C53#146311yescharacteristic
123C2×C53C2×#35411yescharacteristic
124C2×C53C2×#35511yescharacteristic
125C22×C4⋊C4C4×C5311yescharacteristic
126C22×C4⋊C4C22×C5311yescharacteristic
127C22×C42C4×C5311yescharacteristic
128#407—22no—
129#382—22no—
130#382C4×C5312yes—
131#382C22×C5312yes—
132C2×#95—22no—
133#324—22no—
134#324C22×C5314yes—
135C53#1151111yescharacteristic, direct product factor, Sylow
136C23×C4—21no—
137C23×C4—21no—
138C23×C4C2×C4×C5311yescharacteristic
139C23×C4D4×C5312yes—
140C2×M16—42no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4C2×C4×C5312yes—
147C2×C4⋊C4C2×C4×C5311yescharacteristic
148C2×C4⋊C4D4×C5312yes—
149C2×C4⋊C4C23×C5311yescharacteristic
150C2×C42—21no—
151C2×C42—24no—
152C2×C42C2×C4×C5312yes—
153C22⋊C8—24no—
154C22⋊C8—42no—
155#95—24no—
156#95—24no—
157C24C22⋊C4×C5311yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—24no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4C22⋊C4×C5311yescharacteristic
174C22×C4C22⋊C4×C5312yes—
175C22×C4C22×C4×C5311yescharacteristic
176C22×C4C2×D4×C5312yes—
177M16—84no—
178C2×C8—44no—
179C4⋊C4—41no—
180C4⋊C4—42no—
181C4⋊C4—41no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C42—24no—
185C42—22no—
186C42M16×C5314yescomplemented
187C23—22no—
188C23—22no—
189C23—21no—
190C23—21no—
191C23#99×C5311yescharacteristic
192C23#99×C5311yescharacteristic
193C23C2×C22⋊C4×C5311yescharacteristic
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—24no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—48no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—44no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—44no—
210C2×C4—42no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4#101×C5312yes—
215C2×C4C2×M16×C5312yes—
216C8—84no—
217C22—24no—
218C22—24no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22C53×#35412yes—
226C22C53×#35512yes—
227C22C53×#40711yescharacteristic
228C22C2×#99×C5311yescharacteristic
229C22C2×#101×C5311yescharacteristic
230C4—22no—
231C4—22no—
232C4—44no—
233C4—42no—
234C4—42no—
235C4—82no—
236C2—21no—
237C2—22no—
238C2—22no—
239C2—21no—
240C2C53×#146311yescharacteristic
241C2C2×C53×#35411yescharacteristic
242C2C2×C53×#35511yescharacteristic
subgroupG/H
center, socleC22×C53C2×#99
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×C4⋊C4C23×C53
prime radicalC24×C53C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C24×C53 ◁ C22×C4⋊C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
3. C53×#1324C2semidirect
64. C42×C53M16semidirect
95. #11511C53direct (N × H)
117. #1324C2×C53semidirect
135. C53#11511direct (N × H)
186. C42M16×C53semidirect

Maximal subgroups

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

subgroupindex
C2×C53×#3822
C53×#14632
C53×#13242
#1151153