Finite Group Databasebeta

← C53×#11535 #27219047

Subgroups — C53×#11535 #27219047

Order 13568.

440 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2326C211yesmaximal, characteristic
2C53×#1465C212yesmaximal
3C53×#1325C212yesmaximal, complemented
4C53×#1319C212yesmaximal, complemented
5C2×#124×C53C411yescharacteristic
6C2×C4×Q8×C53C411yescharacteristic
7C2×C4×D4×C53C2211yescharacteristic
8C53×#407—22no—
9C53×#391C2211yescharacteristic
10C53×#390C411yescharacteristic
11C53×#388—22no—
12C53×#385—22no—
13C53×#384C2211yescharacteristic
14C53×#376C411yescharacteristic
15C53×#324C2214yes—
16C53×#323—22no—
17C22×Q8×C53C2×C411yescharacteristic
18C22×D4×C53C2×C411yescharacteristic
19C23×C4×C53D412yes—
20C2×M16×C53—42no—
21#124×C53—22no—
22#124×C53—22no—
23C4×Q8×C53—42no—
24C4×D4×C53—42no—
25C2×C4⋊C4×C53—21no—
26C2×C4⋊C4×C53—22no—
27C2×C4⋊C4×C53C2×C411yescharacteristic
28C2×C4⋊C4×C53C2311yescharacteristic
29C2×C22⋊C4×C53—22no—
30C2×C22⋊C4×C53—22no—
31C2×C22⋊C4×C53D412yes—
32C2×C42×C53—21no—
33C2×C42×C53C2×C411yescharacteristic
34C2×C42×C53C2×C411yescharacteristic
35C2×C42×C53C2×C411yescharacteristic
36C22⋊C8×C53—24no—
37C22⋊C8×C53—42no—
38#95×C53—21no—
39#95×C53—21no—
40#95×C53—22no—
41C24×C53C22⋊C412yes—
42C2×Q8×C53—21no—
43C2×Q8×C53—41no—
44C2×Q8×C53—41no—
45C2×Q8×C53M1612yescomplemented
46C2×D4×C53—21no—
47C2×D4×C53—21no—
48C2×D4×C53—41no—
49C2×D4×C53—41no—
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—82no—
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22×C411yescharacteristic
60C22×C4×C53C2×D412yes—
61M16×C53—84no—
62C2×C8×C53—44no—
63C4⋊C4×C53—41no—
64C4⋊C4×C53—41no—
65C4⋊C4×C53—42no—
66C4⋊C4×C53—82no—
67C22⋊C4×C53—28no—
68C22⋊C4×C53—42no—
69C22⋊C4×C53—82no—
70C42×C53—22no—
71C42×C53—21no—
72C42×C53—42no—
73C42×C53—41no—
74C42×C53M1612yescomplemented
75C23×C53—22no—
76C23×C53—22no—
77C23×C53—22no—
78C23×C53—42no—
79C23×C53—42no—
80C23×C53C2×C22⋊C411yescharacteristic
81Q8×C53—42no—
82Q8×C53—42no—
83D4×C53—42no—
84D4×C53—42no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—21no—
91C2×C4×C53—41no—
92C2×C4×C53—44no—
93C2×C4×C53—41no—
94C2×C4×C53—41no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53—42no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53—42no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—82no—
105C2×C4×C53#9912yes—
106C2×C4×C53C2×M1611yescharacteristic
107C2×C4×C53C2×M1611yescharacteristic
108C8×C53—84no—
109#11535C5311yesmaximal, characteristic, direct product factor, Sylow
110C22×C53—21no—
111C22×C53—21no—
112C22×C53—44no—
113C22×C53—42no—
114C22×C53—42no—
115C22×C53—42no—
116C22×C53—82no—
117C22×C53#40711yescharacteristic
118C22×C53C2×#9911yescharacteristic
119C22×C53#41311yescharacteristic
120C4×C53—22no—
121C4×C53—21no—
122C4×C53—21no—
123C4×C53—42no—
124C4×C53—42no—
125C4×C53—41no—
126C4×C53—42no—
127C4×C53—81no—
128C4×C53—81no—
129#2326C2×C5311yescharacteristic
130#1465C2×C5312yes—
131#1325C2×C5312yescomplemented
132#1319C2×C5312yescomplemented
133C2×C53—21no—
134C2×C53—21no—
135C2×C53—82no—
136C2×C53#146511yescharacteristic
137C2×C53#212711yescharacteristic
138C2×C53#213511yescharacteristic
139C2×#124C4×C5311yescharacteristic
140C2×C4×Q8C4×C5311yescharacteristic
141C2×C4×D4C22×C5311yescharacteristic
142#407—22no—
143#391C22×C5311yescharacteristic
144#390C4×C5311yescharacteristic
145#388—22no—
146#385—22no—
147#384C22×C5311yescharacteristic
148#376C4×C5311yescharacteristic
149#324C22×C5314yes—
150#323—22no—
151C53#1153511yescharacteristic, direct product factor, Sylow
152C22×Q8C2×C4×C5311yescharacteristic
153C22×D4C2×C4×C5311yescharacteristic
154C23×C4D4×C5312yes—
155C2×M16—42no—
156#124—22no—
157#124—22no—
158C4×Q8—42no—
159C4×D4—42no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4C2×C4×C5311yescharacteristic
163C2×C4⋊C4C23×C5311yescharacteristic
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4D4×C5312yes—
167C2×C42—21no—
168C2×C42C2×C4×C5311yescharacteristic
169C2×C42C2×C4×C5311yescharacteristic
170C2×C42C2×C4×C5311yescharacteristic
171C22⋊C8—24no—
172C22⋊C8—42no—
173#95—22no—
174#95—21no—
175#95—21no—
176C24C22⋊C4×C5312yes—
177C2×Q8—21no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8M16×C5312yescomplemented
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—41no—
184C2×D4—41no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—82no—
192C22×C4C22⋊C4×C5311yescharacteristic
193C22×C4C22⋊C4×C5311yescharacteristic
194C22×C4C22×C4×C5311yescharacteristic
195C22×C4C2×D4×C5312yes—
196M16—84no—
197C2×C8—44no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—82no—
202C22⋊C4—28no—
203C22⋊C4—42no—
204C22⋊C4—82no—
205C42—22no—
206C42—21no—
207C42—41no—
208C42—42no—
209C42M16×C5312yescomplemented
210C23—22no—
211C23—22no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23C2×C22⋊C4×C5311yescharacteristic
216Q8—42no—
217Q8—42no—
218D4—42no—
219D4—42no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—44no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—82no—
240C2×C4#99×C5312yes—
241C2×C4C2×M16×C5311yescharacteristic
242C2×C4C2×M16×C5311yescharacteristic
243C8—84no—
244C22—21no—
245C22—21no—
246C22—44no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—82no—
251C22C53×#40711yescharacteristic
252C22C2×#99×C5311yescharacteristic
253C22C53×#41311yescharacteristic
254C4—21no—
255C4—21no—
256C4—22no—
257C4—42no—
258C4—42no—
259C4—42no—
260C4—41no—
261C4—81no—
262C4—81no—
263C2—21no—
264C2—21no—
265C2—82no—
266C2C53×#146511yescharacteristic
267C2C53×#212711yescharacteristic
268C2C53×#213511yescharacteristic
subgroupG/H
center, socleC22×C53C2×#99
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×C4⋊C4C23×C53
prime radicalC22×D4×C53C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C2×C4×D4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
3. C53×#1325C2semidirect
4. C53×#1319C2semidirect
45. C2×Q8×C53M16semidirect
74. C42×C53M16semidirect
109. #11535C53direct (N × H)
131. #1325C2×C53semidirect
132. #1319C2×C53semidirect
151. C53#11535direct (N × H)
180. C2×Q8M16×C53semidirect
209. C42M16×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#23262
C53×#14652
C53×#13252
C53×#13192
#1153553