Finite Group Databasebeta

← C53×#11530 #27219042

Subgroups — C53×#11530 #27219042

Order 13568.

444 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 110 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×#1467C212yesmaximal, complemented
3C53×#1324C212yesmaximal, complemented
4C53×#1318C212yesmaximal
5C2×#124×C53C411yescharacteristic
6C2×C4×Q8×C53C2211yescharacteristic
7C2×C4×D4×C53C411yescharacteristic
8C53×#432—22no—
9C53×#391C411yescharacteristic
10C53×#390C2211yescharacteristic
11C53×#388—22no—
12C53×#382C2211yescharacteristic
13C4×C22⋊C4×C53—22no—
14C43×C53C411yescharacteristic
15C53×#324—22no—
16C53×#324C2214yes—
17C22×Q8×C53C2×C411yescharacteristic
18C22×D4×C53C2×C411yescharacteristic
19C23×C4×C53—21no—
20#124×C53—22no—
21#124×C53—22no—
22C4×Q8×C53—41no—
23C4×Q8×C53—41no—
24C4×D4×C53—41no—
25C4×D4×C53—41no—
26C2×C4⋊C4×C53C2×C411yescharacteristic
27C2×C4⋊C4×C53D412yes—
28C2×C4⋊C4×C53C2311yescharacteristic
29C2×C22⋊C4×C53—22no—
30C2×C22⋊C4×C53—21no—
31C2×C22⋊C4×C53—22no—
32C2×C42×C53—21no—
33C2×C42×C53—21no—
34C2×C42×C53C2×C411yescharacteristic
35C2×C42×C53C2×C411yescharacteristic
36C2×C42×C53C2×C411yescharacteristic
37C2×C42×C53D412yes—
38C4⋊C8×C53—42no—
39C22⋊C8×C53—24no—
40#97×C53—42no—
41#95×C53—22no—
42#95×C53—22no—
43C24×C53—21no—
44C2×Q8×C53—21no—
45C2×Q8×C53—21no—
46C2×Q8×C53—41no—
47C2×Q8×C53—41no—
48C2×D4×C53—21no—
49C2×D4×C53—41no—
50C2×D4×C53—41no—
51C2×D4×C53M1612yes—
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—44no—
56C22×C4×C53—41no—
57C22×C4×C53—41no—
58C22×C4×C53C22⋊C412yes—
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22×C411yescharacteristic
62C22×C4×C53C2×D412yes—
63C2×C8×C53—44no—
64C4⋊C4×C53—42no—
65C4⋊C4×C53—41no—
66C4⋊C4×C53—41no—
67C22⋊C4×C53—28no—
68C22⋊C4×C53—44no—
69C22⋊C4×C53—42no—
70C42×C53—22no—
71C42×C53—22no—
72C42×C53—21no—
73C42×C53—21no—
74C42×C53—21no—
75C42×C53—42no—
76C42×C53—42no—
77C42×C53—41no—
78C42×C53M1612yes—
79C23×C53—22no—
80C23×C53—42no—
81C23×C53—42no—
82C23×C53—41no—
83C23×C53—41no—
84C23×C53C2×C22⋊C411yescharacteristic
85Q8×C53—42no—
86Q8×C53—42no—
87D4×C53—42no—
88D4×C53—42no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—41no—
107C2×C4×C53—82no—
108C2×C4×C53—82no—
109C2×C4×C53#9911yescharacteristic
110C2×C4×C53#9911yescharacteristic
111C2×C4×C53C2×M1611yescharacteristic
112C2×C4×C53C2×M1611yescharacteristic
113C8×C53—84no—
114#11530C5311yesmaximal, characteristic, direct product factor, Sylow
115C22×C53—21no—
116C22×C53—21no—
117C22×C53—44no—
118C22×C53—42no—
119C22×C53—82no—
120C22×C53—81no—
121C22×C53—81no—
122C22×C53#40711yescharacteristic
123C22×C53C2×#9911yescharacteristic
124C22×C53#41311yescharacteristic
125C4×C53—21no—
126C4×C53—21no—
127C4×C53—21no—
128C4×C53—42no—
129C4×C53—42no—
130C4×C53—42no—
131C4×C53—41no—
132C4×C53—42no—
133C4×C53—81no—
134C4×C53#35512yes—
135C4×#124C2×C5311yescharacteristic
136#1467C2×C5312yescomplemented
137#1324C2×C5312yescomplemented
138#1318C2×C5312yes—
139C2×C53—21no—
140C2×C53—21no—
141C2×C53—82no—
142C2×C53#146511yescharacteristic
143C2×C53#212611yescharacteristic
144C2×C53C2×#35511yescharacteristic
145C2×#124C4×C5311yescharacteristic
146C2×C4×Q8C22×C5311yescharacteristic
147C2×C4×D4C4×C5311yescharacteristic
148#432—22no—
149#391C4×C5311yescharacteristic
150#390C22×C5311yescharacteristic
151#388—22no—
152#382C22×C5311yescharacteristic
153C4×C22⋊C4—22no—
154C43C4×C5311yescharacteristic
155#324—22no—
156#324C22×C5314yes—
157C53#1153011yescharacteristic, direct product factor, Sylow
158C22×Q8C2×C4×C5311yescharacteristic
159C22×D4C2×C4×C5311yescharacteristic
160C23×C4—21no—
161#124—22no—
162#124—22no—
163C4×Q8—41no—
164C4×Q8—41no—
165C4×D4—41no—
166C4×D4—41no—
167C2×C4⋊C4C2×C4×C5311yescharacteristic
168C2×C4⋊C4D4×C5312yes—
169C2×C4⋊C4C23×C5311yescharacteristic
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—21no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42C2×C4×C5311yescharacteristic
176C2×C42C2×C4×C5311yescharacteristic
177C2×C42C2×C4×C5311yescharacteristic
178C2×C42D4×C5312yes—
179C4⋊C8—42no—
180C22⋊C8—24no—
181#97—42no—
182#95—22no—
183#95—22no—
184C24—21no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×D4—21no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4M16×C5312yes—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—44no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4C22⋊C4×C5312yes—
200C22×C4C22⋊C4×C5311yescharacteristic
201C22×C4C22⋊C4×C5311yescharacteristic
202C22×C4C22×C4×C5311yescharacteristic
203C22×C4C2×D4×C5312yes—
204C2×C8—44no—
205C4⋊C4—42no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C22⋊C4—28no—
209C22⋊C4—42no—
210C22⋊C4—44no—
211C42—22no—
212C42—22no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—41no—
217C42—42no—
218C42—42no—
219C42M16×C5312yes—
220C23—22no—
221C23—42no—
222C23—42no—
223C23—41no—
224C23—41no—
225C23C2×C22⋊C4×C5311yescharacteristic
226Q8—42no—
227Q8—42no—
228D4—42no—
229D4—42no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4#99×C5311yescharacteristic
251C2×C4#99×C5311yescharacteristic
252C2×C4C2×M16×C5311yescharacteristic
253C2×C4C2×M16×C5311yescharacteristic
254C8—84no—
255C22—21no—
256C22—21no—
257C22—44no—
258C22—42no—
259C22—82no—
260C22—81no—
261C22—81no—
262C22C53×#40711yescharacteristic
263C22C2×#99×C5311yescharacteristic
264C22C53×#41311yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—42no—
269C4—42no—
270C4—42no—
271C4—42no—
272C4—41no—
273C4—81no—
274C4C53×#35512yes—
275C2—21no—
276C2—21no—
277C2—82no—
278C2C53×#146511yescharacteristic
279C2C53×#212611yescharacteristic
280C2C2×C53×#35511yescharacteristic
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×Q8×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#1467C2semidirect
3. C53×#1324C2semidirect
114. #11530C53direct (N × H)
136. #1467C2×C53semidirect
137. #1324C2×C53semidirect
157. C53#11530direct (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×#14672
C53×#13242
C53×#13182
#1153053