Finite Group Databasebeta

← C53×#7649 #27215161

Subgroups — C53×#7649 #27215161

Order 13568.

468 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 130 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
1C4×#124×C53C211yesmaximal, characteristic
2C53×#1839C212yesmaximal, complemented
3C53×#1272C212yesmaximal, complemented
4C53×#1270C212yesmaximal
5C2×#124×C53C411yescharacteristic
6C2×C4×Q8×C53C411yescharacteristic
7C2×C4×D4×C53C411yescharacteristic
8C53×#391C2211yescharacteristic
9C53×#390C2211yescharacteristic
10C53×#388—22no—
11C53×#382C411yescharacteristic
12C4×C22⋊C4×C53—22no—
13C43×C53C2211yescharacteristic
14C53×#336—22no—
15C53×#336—22no—
16C53×#336C2214yes—
17C22×Q8×C53C2×C411yescharacteristic
18C22×D4×C53C2×C411yescharacteristic
19C23×C4×C53—21no—
20C22×C8×C53—24no—
21#124×C53—22no—
22#124×C53—22no—
23C4×Q8×C53—42no—
24C4×D4×C53—42no—
25C2×C4⋊C4×C53—21no—
26C2×C4⋊C4×C53C2×C411yescharacteristic
27C2×C4⋊C4×C53C2×C411yescharacteristic
28C2×C22⋊C4×C53—22no—
29C2×C22⋊C4×C53—21no—
30C2×C22⋊C4×C53—22no—
31C2×C42×C53—21no—
32C2×C42×C53C2×C411yescharacteristic
33C2×C42×C53C2×C411yescharacteristic
34C2×C42×C53D412yes—
35C2×C42×C53D412yes—
36C2×C42×C53C2311yescharacteristic
37#95×C53—22no—
38#95×C53—22no—
39C24×C53—21no—
40C2×Q8×C53—21no—
41C2×Q8×C53—41no—
42C2×Q8×C53—41no—
43C2×Q8×C53M1612yes—
44C2×D4×C53—21no—
45C2×D4×C53—41no—
46C2×D4×C53—41no—
47C2×D4×C53M1612yes—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53—21no—
51C22×C4×C53—44no—
52C22×C4×C53—42no—
53C22×C4×C53C22⋊C412yes—
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22×C411yescharacteristic
57C22×C4×C53C2×D412yes—
58C2×C8×C53—44no—
59C2×C8×C53—44no—
60C2×C8×C53—44no—
61C4⋊C4×C53—42no—
62C4⋊C4×C53—41no—
63C4⋊C4×C53—41no—
64C22⋊C4×C53—28no—
65C22⋊C4×C53—42no—
66C22⋊C4×C53—44no—
67C42×C53—22no—
68C42×C53—21no—
69C42×C53—21no—
70C42×C53—24no—
71C42×C53—42no—
72C42×C53—41no—
73C42×C53—42no—
74C23×C53—22no—
75C23×C53—41no—
76C23×C53—41no—
77C23×C53—42no—
78C23×C53—42no—
79C23×C53C2×C22⋊C411yescharacteristic
80Q8×C53—42no—
81Q8×C53—42no—
82D4×C53—42no—
83D4×C53—42no—
84C2×C4×C53—21no—
85C2×C4×C53—22no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—21no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53—42no—
100C2×C4×C53—42no—
101C2×C4×C53—84no—
102C2×C4×C53#10012yes—
103C2×C4×C53C2×M1611yescharacteristic
104C2×C4×C53C2×M1611yescharacteristic
105C2×C4×C53#13514yes—
106C8×C53—84no—
107#7649C5311yesmaximal, characteristic, direct product factor, Sylow
108C22×C53—42no—
109C22×C53—44no—
110C22×C53—81no—
111C22×C53—82no—
112C22×C53—81no—
113C22×C53#40711yescharacteristic
114C22×C53#41011yescharacteristic
115C22×C53C2×#10011yescharacteristic
116C22×C53#41612yes—
117C22×C53#42112yes—
118C4×C53—21no—
119C4×C53—21no—
120C4×C53—24no—
121C4×C53—22no—
122C4×C53—41no—
123C4×C53—42no—
124C4×C53—42no—
125C4×C53—81no—
126C4×C53—81no—
127C4×#124C2×C5311yescharacteristic
128#1839C2×C5312yescomplemented
129#1272C2×C5312yescomplemented
130#1270C2×C5312yes—
131C2×C53—82no—
132C2×C53#146611yescharacteristic
133C2×C53#148812yes—
134C2×C53#152112yes—
135C2×C53#153311yescharacteristic
136C2×C53#154311yescharacteristic
137C2×#124C4×C5311yescharacteristic
138C2×C4×Q8C4×C5311yescharacteristic
139C2×C4×D4C4×C5311yescharacteristic
140#391C22×C5311yescharacteristic
141#390C22×C5311yescharacteristic
142#388—22no—
143#382C4×C5311yescharacteristic
144C4×C22⋊C4—22no—
145C43C22×C5311yescharacteristic
146#336—22no—
147#336—22no—
148#336C22×C5314yes—
149C53#764911yescharacteristic, direct product factor, Sylow
150C22×Q8C2×C4×C5311yescharacteristic
151C22×D4C2×C4×C5311yescharacteristic
152C23×C4—21no—
153C22×C8—24no—
154#124—22no—
155#124—22no—
156C4×Q8—42no—
157C4×D4—42no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4C2×C4×C5311yescharacteristic
160C2×C4⋊C4C2×C4×C5311yescharacteristic
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—21no—
164C2×C42—21no—
165C2×C42C2×C4×C5311yescharacteristic
166C2×C42C2×C4×C5311yescharacteristic
167C2×C42D4×C5312yes—
168C2×C42D4×C5312yes—
169C2×C42C23×C5311yescharacteristic
170#95—22no—
171#95—22no—
172C24—21no—
173C2×Q8—21no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8M16×C5312yes—
177C2×D4—21no—
178C2×D4—41no—
179C2×D4—41no—
180C2×D4M16×C5312yes—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—44no—
185C22×C4—42no—
186C22×C4C22⋊C4×C5312yes—
187C22×C4C22⋊C4×C5311yescharacteristic
188C22×C4C22⋊C4×C5311yescharacteristic
189C22×C4C22×C4×C5311yescharacteristic
190C22×C4C2×D4×C5312yes—
191C2×C8—44no—
192C2×C8—44no—
193C2×C8—44no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—42no—
197C22⋊C4—28no—
198C22⋊C4—42no—
199C22⋊C4—44no—
200C42—22no—
201C42—21no—
202C42—21no—
203C42—24no—
204C42—42no—
205C42—41no—
206C42—42no—
207C23—22no—
208C23—41no—
209C23—41no—
210C23—42no—
211C23—42no—
212C23C2×C22⋊C4×C5311yescharacteristic
213Q8—42no—
214Q8—42no—
215D4—42no—
216D4—42no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—84no—
235C2×C4#100×C5312yes—
236C2×C4C2×M16×C5311yescharacteristic
237C2×C4C2×M16×C5311yescharacteristic
238C2×C4#135×C5314yes—
239C8—84no—
240C22—42no—
241C22—44no—
242C22—82no—
243C22—81no—
244C22—81no—
245C22C53×#40711yescharacteristic
246C22C53×#41011yescharacteristic
247C22C2×#100×C5311yescharacteristic
248C22C53×#41612yes—
249C22C53×#42112yes—
250C4—21no—
251C4—21no—
252C4—24no—
253C4—22no—
254C4—42no—
255C4—42no—
256C4—41no—
257C4—81no—
258C4—81no—
259C2—82no—
260C2C53×#146611yescharacteristic
261C2C53×#148812yes—
262C2C53×#152112yes—
263C2C53×#153311yescharacteristic
264C2C53×#154311yescharacteristic
subgroupG/H
center, socleC23×C53C2×C22⋊C4
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC22×D4×C53C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C53 ◁ C43×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#1839C2semidirect
3. C53×#1272C2semidirect
107. #7649C53direct (N × H)
128. #1839C2×C53semidirect
129. #1272C2×C53semidirect
149. C53#7649direct (N × H)

Maximal subgroups

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

subgroupindex
C4×#124×C532
C53×#18392
C53×#12722
C53×#12702
#764953