Finite Group Databasebeta

← C53×#34348 #27241860

Subgroups — C53×#34348 #27241860

Order 13568.

1898 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C53×#533C211yesmaximal, characteristic
2C53×#2382C214yesmaximal, complemented
3C53×#2381C214yesmaximal, complemented
4C53×#2378C218yesmaximal
5C53×#2298C214yesmaximal
6C53×#2295C218yesmaximal, complemented
7C53×#2283C212yesmaximal
8C53×#533C4116yescomplemented
9C2×C4⋊Q8×C53C2212yes—
10C2×#125×C53C2212yes—
11C2×C22⋊Q8×C53C2214yes—
12C2×C4×Q8×C53C2214yes—
13C2×#117×C53C2212yes—
14C22×C42×C53C2211yescharacteristic
15C53×#391C22116yes—
16C53×#389C22116yes—
17C53×#388C22116yes—
18C53×#387C22116yes—
19C53×#387C22116yes—
20C53×#387C22116yes—
21C53×#383C2214yes—
22C53×#382C2214yes—
23C53×#381C2214yes—
24C53×#380C2214yes—
25C4×C4⋊C4×C53C2214yes—
26C4×C4⋊C4×C53C2218yes—
27C4×C22⋊C4×C53C2214yes—
28C53×#376C2214yes—
29C53×#376C2214yes—
30C43×C53C2214yes—
31C22×Q8×C53C2314yes—
32C23×C4×C53C2311yescharacteristic
33C23×C4×C53C2312yes—
34C4⋊Q8×C53C2×C4116yes—
35#125×C53C2×C4116yes—
36C22⋊Q8×C53C2×C4132yes—
37C4×Q8×C53C2×C4132yes—
38#117×C53C2×C4116yes—
39C2×C4⋊C4×C53C2318yes—
40C2×C4⋊C4×C53C23116yes—
41C2×C4⋊C4×C53C2314yes—
42C2×C4⋊C4×C53C2318yes—
43C2×C4⋊C4×C53C2314yes—
44C2×C22⋊C4×C53C2318yes—
45C2×C22⋊C4×C53C2314yes—
46C2×C42×C53C2×C418yes—
47C2×C42×C53C23116yes—
48C2×C42×C53C2311yescharacteristic
49C2×C42×C53C2311yescharacteristic
50C2×C42×C53C2318yes—
51C2×C42×C53C2312yes—
52C2×C42×C53C2314yes—
53#95×C53C2318yes—
54#95×C53C2318yes—
55#95×C53C23116yes—
56#95×C53C2318yes—
57#95×C53C2318yes—
58#95×C53C23116yes—
59C24×C53C2411yescharacteristic
60C2×Q8×C53—48no—
61C2×Q8×C53C22×C4116yes—
62C22×C4×C53—24no—
63C22×C4×C53—44no—
64C22×C4×C53C22×C414yes—
65C22×C4×C53C22×C418yes—
66C22×C4×C53C24116yes—
67C22×C4×C53C2411yescharacteristic
68C22×C4×C53C2412yes—
69C22×C4×C53C2418yes—
70C22×C4×C53C2412yes—
71C22×C4×C53C2411yescharacteristic
72C4⋊C4×C53—48no—
73C4⋊C4×C53—416no—
74C4⋊C4×C53C22×C4116yes—
75C4⋊C4×C53C22×C4132yes—
76C4⋊C4×C53C22×C4116yes—
77C22⋊C4×C53—48no—
78C22⋊C4×C53C22×C4116yes—
79C42×C53—216no—
80C42×C53—416no—
81C42×C53C22×C418yes—
82C42×C53C22×C414yes—
83C42×C53C22×C414yes—
84C42×C53C22×C4116yes—
85C23×C53—22no—
86C23×C53—24no—
87C23×C53C23×C412yes—
88C23×C53C2511yescharacteristic
89Q8×C53—416no—
90C2×C4×C53—22no—
91C2×C4×C53—24no—
92C2×C4×C53—24no—
93C2×C4×C53—216no—
94C2×C4×C53—28no—
95C2×C4×C53—216no—
96C2×C4×C53—216no—
97C2×C4×C53—216no—
98C2×C4×C53—48no—
99C2×C4×C53C23×C414yes—
100C2×C4×C53C23×C412yes—
101C2×C4×C53C23×C414yes—
102C2×C4×C53C23×C4116yes—
103C2×C4×C53C23×C412yes—
104C2×C4×C532−1+414yes—
105#34348C5311yesmaximal, characteristic, direct product factor, Sylow
106C22×C53—22no—
107C22×C53—24no—
108C22×C53—44no—
109C22×C53C24×C411yescharacteristic
110C22×C53C2×2−1+412yes—
111C22×C53#58514yes—
112C4×C53—22no—
113C4×C53—24no—
114C4×C53—24no—
115C4×C53—216no—
116C4×C53—416no—
117C4×C53#51914yes—
118C2×#533C2×C5311yescharacteristic
119#2382C2×C5314yescomplemented
120#2381C2×C5314yescomplemented
121#2378C2×C5318yes—
122#2298C2×C5314yes—
123#2295C2×C5318yescomplemented
124#2283C2×C5312yes—
125C2×C53—42no—
126C2×C53C2×#51911yescharacteristic
127C2×C53#342812yes—
128C2×C53#353114yes—
129#533C4×C53116yescomplemented
130C2×C4⋊Q8C22×C5312yes—
131C2×#125C22×C5312yes—
132C2×C22⋊Q8C22×C5314yes—
133C2×C4×Q8C22×C5314yes—
134C2×#117C22×C5312yes—
135C22×C42C22×C5311yescharacteristic
136#391C22×C53116yes—
137#389C22×C53116yes—
138#388C22×C53116yes—
139#387C22×C53116yes—
140#387C22×C53116yes—
141#387C22×C53116yes—
142#383C22×C5314yes—
143#382C22×C5314yes—
144#381C22×C5314yes—
145#380C22×C5314yes—
146C4×C4⋊C4C22×C5314yes—
147C4×C4⋊C4C22×C5318yes—
148C4×C22⋊C4C22×C5314yes—
149#376C22×C5314yes—
150#376C22×C5314yes—
151C43C22×C5314yes—
152C53#3434811yescharacteristic, direct product factor, Sylow
153C22×Q8C23×C5314yes—
154C23×C4C23×C5311yescharacteristic
155C23×C4C23×C5312yes—
156C4⋊Q8C2×C4×C53116yes—
157#125C2×C4×C53116yes—
158C22⋊Q8C2×C4×C53132yes—
159C4×Q8C2×C4×C53132yes—
160#117C2×C4×C53116yes—
161C2×C4⋊C4C23×C5318yes—
162C2×C4⋊C4C23×C53116yes—
163C2×C4⋊C4C23×C5314yes—
164C2×C4⋊C4C23×C5318yes—
165C2×C4⋊C4C23×C5314yes—
166C2×C22⋊C4C23×C5318yes—
167C2×C22⋊C4C23×C5314yes—
168C2×C42C2×C4×C5318yes—
169C2×C42C23×C5318yes—
170C2×C42C23×C53116yes—
171C2×C42C23×C5314yes—
172C2×C42C23×C5312yes—
173C2×C42C23×C5311yescharacteristic
174C2×C42C23×C5311yescharacteristic
175#95C23×C5318yes—
176#95C23×C5318yes—
177#95C23×C53116yes—
178#95C23×C5318yes—
179#95C23×C5318yes—
180#95C23×C53116yes—
181C24C24×C5311yescharacteristic
182C2×Q8—48no—
183C2×Q8C22×C4×C53116yes—
184C22×C4—24no—
185C22×C4—44no—
186C22×C4C22×C4×C5314yes—
187C22×C4C22×C4×C5318yes—
188C22×C4C24×C53116yes—
189C22×C4C24×C5311yescharacteristic
190C22×C4C24×C5312yes—
191C22×C4C24×C5318yes—
192C22×C4C24×C5312yes—
193C22×C4C24×C5311yescharacteristic
194C4⋊C4—416no—
195C4⋊C4—48no—
196C4⋊C4C22×C4×C53116yes—
197C4⋊C4C22×C4×C53132yes—
198C4⋊C4C22×C4×C53116yes—
199C22⋊C4—48no—
200C22⋊C4C22×C4×C53116yes—
201C42—216no—
202C42—416no—
203C42C22×C4×C5318yes—
204C42C22×C4×C5314yes—
205C42C22×C4×C5314yes—
206C42C22×C4×C53116yes—
207C23—22no—
208C23—24no—
209C23C23×C4×C5312yes—
210C23C25×C5311yescharacteristic
211Q8—416no—
212C2×C4—22no—
213C2×C4—216no—
214C2×C4—24no—
215C2×C4—24no—
216C2×C4—28no—
217C2×C4—216no—
218C2×C4—216no—
219C2×C4—216no—
220C2×C4—48no—
221C2×C4C23×C4×C5312yes—
222C2×C4C23×C4×C5314yes—
223C2×C4C23×C4×C5314yes—
224C2×C4C23×C4×C53116yes—
225C2×C4C23×C4×C5312yes—
226C2×C42−1+4×C5314yes—
227C22—22no—
228C22—24no—
229C22—44no—
230C22C24×C4×C5311yescharacteristic
231C22C2×2−1+4×C5312yes—
232C22C53×#58514yes—
233C4—22no—
234C4—24no—
235C4—24no—
236C4—216no—
237C4—416no—
238C4C53×#51914yes—
239C2—42no—
240C2C2×C53×#51911yescharacteristic
241C2C53×#342812yes—
242C2C53×#353114yes—
subgroupG/H
centerC22×C4×C53C24
commutatorC22C24×C4×C53
Frattini, cosocleC23C25×C53
socleC23×C53C25
prime radicalC24×C53C24
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2382C2semidirect
3. C53×#2381C2semidirect
6. C53×#2295C2semidirect
8. C53×#533C4semidirect
105. #34348C53direct (N × H)
119. #2382C2×C53semidirect
120. #2381C2×C53semidirect
123. #2295C2×C53semidirect
129. #533C4×C53semidirect
152. C53#34348direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C53×#5332
C53×#23822
C53×#23812
C53×#23782
C53×#22982
C53×#22952
C53×#22832
#3434853