Finite Group Databasebeta

← C53×#12209 #27219721

Subgroups — C53×#12209 #27219721

Order 13568.

404 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2724C211yesmaximal, characteristic
2C53×#2027C211yesmaximal, characteristic
3C53×#1611C212yesmaximal, complemented
4C53×#1346C212yesmaximal, complemented
5C53×#1318C211yesmaximal, characteristic
6C53×#544C2211yescharacteristic
7C2×#123×C53—21no—
8C53×#483—22no—
9C53×#418—21no—
10C53×#398C2212yes—
11C53×#397—22no—
12C53×#390C2211yescharacteristic
13C53×#389—21no—
14C53×#385—22no—
15C53×#343C2211yescharacteristic
16C53×#339—21no—
17C53×#328—22no—
18C53×#324C2212yes—
19C22×D4×C53D411yescharacteristic
20C23×C4×C53—21no—
21C2×M16×C53—21no—
22C4⋊Q8×C53—41no—
23#125×C53—41no—
24#123×C53—42no—
25#123×C53—42no—
26C22⋊Q8×C53—22no—
27#117×C53—21no—
28C2×C4⋊C4×C53—21no—
29C2×C4⋊C4×C53—21no—
30C2×C4⋊C4×C53—22no—
31C2×C4⋊C4×C53D412yes—
32C2×C4⋊C4×C53C2311yescharacteristic
33C2×C22⋊C4×C53—22no—
34C2×C22⋊C4×C53—21no—
35C2×C22⋊C4×C53—22no—
36C2×C42×C53D411yescharacteristic
37#106×C53—42no—
38#103×C53—42no—
39#102×C53—41no—
40#102×C53—41no—
41C22⋊C8×C53—22no—
42#95×C53—22no—
43#95×C53—22no—
44#95×C53D412yes—
45C24×C53—21no—
46C2×Q8×C53—22no—
47C2×D4×C53—21no—
48C2×D4×C53—41no—
49C2×D4×C53—41no—
50C2×D4×C53SD1612yes—
51C22×C4×C53—21no—
52C22×C4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53—42no—
55C22×C4×C53—81no—
56C22×C4×C53—81no—
57C22×C4×C53C2×D411yescharacteristic
58C22×C4×C53C2×D412yes—
59M16×C53—41no—
60M16×C53—41no—
61C2×C8×C53—42no—
62C2×C8×C53—41no—
63C4⋊C4×C53—21no—
64C4⋊C4×C53—21no—
65C4⋊C4×C53—22no—
66C4⋊C4×C53—41no—
67C4⋊C4×C53—42no—
68C4⋊C4×C53—42no—
69C4⋊C4×C53—42no—
70C4⋊C4×C53—42no—
71C4⋊C4×C53—42no—
72C4⋊C4×C53—81no—
73C22⋊C4×C53—44no—
74C22⋊C4×C53—44no—
75C22⋊C4×C53—42no—
76C22⋊C4×C53—41no—
77C42×C53—41no—
78C42×C53—41no—
79C23×C53—22no—
80C23×C53—42no—
81C23×C53—42no—
82C23×C53—41no—
83C23×C53—41no—
84C23×C53#12011yescharacteristic
85Q8×C53—82no—
86D4×C53—42no—
87D4×C53—81no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—42no—
97C2×C4×C53—42no—
98C2×C4×C53—42no—
99C2×C4×C53—42no—
100C2×C4×C53—42no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—84no—
104C2×C4×C53C2×SD1611yescharacteristic
105C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
106C8×C53—81no—
107C8×C53—82no—
108#12209C5311yesmaximal, characteristic, direct product factor, Sylow
109C22×C53—21no—
110C22×C53—21no—
111C22×C53—44no—
112C22×C53—42no—
113C22×C53—82no—
114C22×C53—81no—
115C22×C53—81no—
116C22×C53#45011yescharacteristic
117C22×C53#45511yescharacteristic
118C22×C53B(4,𝔽2)11yescharacteristic
119C4×C53—21no—
120C4×C53—21no—
121C4×C53—42no—
122C4×C53—42no—
123C4×C53—82no—
124C4×C53—82no—
125C4×C53—81no—
126C4×C53—81no—
127#2724C2×C5311yescharacteristic
128#2027C2×C5311yescharacteristic
129#1611C2×C5312yescomplemented
130#1346C2×C5312yescomplemented
131#1318C2×C5311yescharacteristic
132C2×C53—21no—
133C2×C53—21no—
134C2×C53—82no—
135C2×C53#160111yescharacteristic
136C2×C53#219811yescharacteristic
137C2×C53#220311yescharacteristic
138#544C22×C5311yescharacteristic
139C2×#123—21no—
140#483—22no—
141#418—21no—
142#398C22×C5312yes—
143#397—22no—
144#390C22×C5311yescharacteristic
145#389—21no—
146#385—22no—
147#343C22×C5311yescharacteristic
148#339—21no—
149#328—22no—
150#324C22×C5312yes—
151C53#1220911yescharacteristic, direct product factor, Sylow
152C22×D4D4×C5311yescharacteristic
153C23×C4—21no—
154C2×M16—21no—
155C4⋊Q8—41no—
156#125—41no—
157#123—42no—
158#123—42no—
159C22⋊Q8—22no—
160#117—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4D4×C5312yes—
165C2×C4⋊C4C23×C5311yescharacteristic
166C2×C22⋊C4—22no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—22no—
169C2×C42D4×C5311yescharacteristic
170#106—42no—
171#103—42no—
172#102—41no—
173#102—41no—
174C22⋊C8—22no—
175#95—22no—
176#95—22no—
177#95D4×C5312yes—
178C24—21no—
179C2×Q8—22no—
180C2×D4—21no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4SD16×C5312yes—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—42no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4C2×D4×C5311yescharacteristic
191C22×C4C2×D4×C5312yes—
192M16—41no—
193M16—41no—
194C2×C8—42no—
195C2×C8—41no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—22no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—81no—
206C22⋊C4—44no—
207C22⋊C4—42no—
208C22⋊C4—44no—
209C22⋊C4—41no—
210C42—41no—
211C42—41no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23—41no—
216C23—41no—
217C23#120×C5311yescharacteristic
218Q8—82no—
219D4—42no—
220D4—81no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—84no—
237C2×C4C2×SD16×C5311yescharacteristic
238C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
239C8—81no—
240C8—82no—
241C22—21no—
242C22—21no—
243C22—44no—
244C22—42no—
245C22—82no—
246C22—81no—
247C22—81no—
248C22C53×#45011yescharacteristic
249C22C53×#45511yescharacteristic
250C22C53×B(4,𝔽2)11yescharacteristic
251C4—21no—
252C4—21no—
253C4—42no—
254C4—42no—
255C4—82no—
256C4—81no—
257C4—81no—
258C4—82no—
259C2—21no—
260C2—21no—
261C2—82no—
262C2C53×#160111yescharacteristic
263C2C53×#219811yescharacteristic
264C2C53×#220311yescharacteristic
subgroupG/H
center, socleC22×C53B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C53
prime radicalC22×D4×C53D4
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C53 ◁ C2×C4⋊C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
3. C53×#1611C2semidirect
4. C53×#1346C2semidirect
108. #12209C53direct (N × H)
129. #1611C2×C53semidirect
130. #1346C2×C53semidirect
151. C53#12209direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#27242
C53×#20272
C53×#16112
C53×#13462
C53×#13182
#1220953