Finite Group Databasebeta

← C4×C53×#519 #27241514

Subgroups — C4×C53×#519 #27241514

Order 13568.

3402 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C53×#519C212yesmaximal
2C22×C4×Q8×C53C211yesmaximal, characteristic
3C53×#2276C218yesmaximal, complemented
4C53×#2274C216yesmaximal
5C42×Q8×C53C218yesmaximal, complemented
6C4×#117×C53C216yesmaximal
7C23×Q8×C53C2211yescharacteristic
8C53×#519C4132yesdirect product factor
9C2×C4×Q8×C53C4116yescomplemented
10C2×C4×Q8×C53C2214yes—
11C2×C4×Q8×C53C2214yes—
12C2×C4×Q8×C53C22116yes—
13C2×#117×C53C22112yes—
14C22×C4⋊C4×C53C2213yes—
15C22×C42×C53C2213yes—
16C4×C4⋊C4×C53C22112yes—
17C4×C4⋊C4×C53C22112yes—
18C4×C4⋊C4×C53C22148yes—
19C4×C22⋊C4×C53C2214yes—
20C4×C22⋊C4×C53C22112yes—
21C53×#376C22112yes—
22C43×C53C22112yes—
23C22×Q8×C53C2×C418yes—
24C22×Q8×C53C2×C4116yes—
25C22×Q8×C53C2314yes—
26C23×C4×C53C2311yescharacteristic
27C23×C4×C53C2313yes—
28C23×C4×C53C2313yes—
29C4×Q8×C53C2×C4132yes—
30C4×Q8×C53C2×C4132yes—
31C4×Q8×C53C2×C41128yescomplemented
32#117×C53C2×C4196yes—
33C2×C4⋊C4×C53C2×C4124yes—
34C2×C4⋊C4×C53C23112yes—
35C2×C4⋊C4×C53C23124yes—
36C2×C22⋊C4×C53C2318yes—
37C2×C42×C53C2×C4124yes—
38C2×C42×C53C23124yes—
39C2×C42×C53C2316yes—
40C2×C42×C53C23124yes—
41C2×C42×C53C2318yes—
42C2×C42×C53C2316yes—
43#95×C53C2318yes—
44#95×C53C23124yes—
45C24×C53C2411yescharacteristic
46C2×Q8×C53C42164yescomplemented
47C2×Q8×C53C22×C4116yes—
48C2×Q8×C53C22×C4132yes—
49C22×C4×C53—24no—
50C22×C4×C53—46no—
51C22×C4×C53C22×C414yes—
52C22×C4×C53C22×C4112yes—
53C22×C4×C53C22×C4112yes—
54C22×C4×C53C22×C4124yes—
55C22×C4×C53C2411yescharacteristic
56C22×C4×C53C24116yes—
57C22×C4×C53C2416yes—
58C22×C4×C53C2416yes—
59C22×C4×C53C2411yescharacteristic
60C4⋊C4×C53C22×C4148yes—
61C4⋊C4×C53C22×C4196yes—
62C22⋊C4×C53C22×C4132yes—
63C42×C53—216no—
64C42×C53—424no—
65C42×C53C22×C4124yes—
66C42×C53C22×C4124yes—
67C42×C53C22×C4196yes—
68C23×C53—22no—
69C23×C53—22no—
70C23×C53C23×C412yes—
71C23×C53C23×C414yes—
72C23×C53C2511yescharacteristic
73Q8×C53C2×C42164yes—
74C2×C4×C53—22no—
75C2×C4×C53—216no—
76C2×C4×C53—212no—
77C2×C4×C53—216no—
78C2×C4×C53—28no—
79C2×C4×C53—412no—
80C2×C4×C53C2×C42148yes—
81C2×C4×C53C23×C412yes—
82C2×C4×C53C23×C4112yes—
83C2×C4×C53C23×C4132yes—
84C2×C4×C53C23×C412yes—
85C2×C4×C53C23×C4112yes—
86C2×C4×C53C23×C4124yes—
87C2×C4×C532−1+414yes—
88C4×#519C5311yesmaximal, characteristic, direct product factor, Sylow
89C22×C53—22no—
90C22×C53—22no—
91C22×C53—28no—
92C22×C53C22×C4214yes—
93C22×C53C24×C411yescharacteristic
94C22×C53C24×C412yes—
95C22×C53C2×2−1+412yes—
96C22×C53#58512yes—
97C4×C53—22no—
98C4×C53—212no—
99C4×C53—232no—
100C4×C53C22×C42124yes—
101C4×C53#51914yesdirect product factor
102C2×#519C2×C5312yes—
103C22×C4×Q8C2×C5311yescharacteristic
104#2276C2×C5318yescomplemented
105#2274C2×C5316yes—
106C42×Q8C2×C5318yescomplemented
107C4×#117C2×C5316yes—
108C2×C53—24no—
109C2×C53C23×C4211yescharacteristic
110C2×C53C2×#51911yescharacteristic
111C2×C53#342811yescharacteristic
112C2×C53C4×2−1+414yes—
113C23×Q8C22×C5311yescharacteristic
114#519C4×C53132yesdirect product factor
115C2×C4×Q8C4×C53116yescomplemented
116C2×C4×Q8C22×C5314yes—
117C2×C4×Q8C22×C5314yes—
118C2×C4×Q8C22×C53116yes—
119C2×#117C22×C53112yes—
120C22×C4⋊C4C22×C5313yes—
121C22×C42C22×C5313yes—
122C4×C4⋊C4C22×C53112yes—
123C4×C4⋊C4C22×C53112yes—
124C4×C4⋊C4C22×C53148yes—
125C4×C22⋊C4C22×C5314yes—
126C4×C22⋊C4C22×C53112yes—
127#376C22×C53112yes—
128C43C22×C53112yes—
129C53C4×#51911yescharacteristic, direct product factor, Sylow
130C22×Q8C2×C4×C5318yes—
131C22×Q8C2×C4×C53116yes—
132C22×Q8C23×C5314yes—
133C23×C4C23×C5311yescharacteristic
134C23×C4C23×C5313yes—
135C23×C4C23×C5313yes—
136C4×Q8C2×C4×C53132yes—
137C4×Q8C2×C4×C53132yes—
138C4×Q8C2×C4×C531128yescomplemented
139#117C2×C4×C53196yes—
140C2×C4⋊C4C2×C4×C53124yes—
141C2×C4⋊C4C23×C53112yes—
142C2×C4⋊C4C23×C53124yes—
143C2×C22⋊C4C23×C5318yes—
144C2×C42C2×C4×C53124yes—
145C2×C42C23×C5318yes—
146C2×C42C23×C53124yes—
147C2×C42C23×C53124yes—
148C2×C42C23×C5316yes—
149C2×C42C23×C5316yes—
150#95C23×C5318yes—
151#95C23×C53124yes—
152C24C24×C5311yescharacteristic
153C2×Q8C42×C53164yescomplemented
154C2×Q8C22×C4×C53116yes—
155C2×Q8C22×C4×C53132yes—
156C22×C4—24no—
157C22×C4—46no—
158C22×C4C22×C4×C5314yes—
159C22×C4C22×C4×C53112yes—
160C22×C4C22×C4×C53112yes—
161C22×C4C22×C4×C53124yes—
162C22×C4C24×C5311yescharacteristic
163C22×C4C24×C53116yes—
164C22×C4C24×C5316yes—
165C22×C4C24×C5316yes—
166C22×C4C24×C5311yescharacteristic
167C4⋊C4C22×C4×C53148yes—
168C4⋊C4C22×C4×C53196yes—
169C22⋊C4C22×C4×C53132yes—
170C42—216no—
171C42—424no—
172C42C22×C4×C53124yes—
173C42C22×C4×C53124yes—
174C42C22×C4×C53196yes—
175C23—22no—
176C23—22no—
177C23C23×C4×C5314yes—
178C23C23×C4×C5312yes—
179C23C25×C5311yescharacteristic
180Q8C2×C42×C53164yes—
181C2×C4—216no—
182C2×C4—22no—
183C2×C4—216no—
184C2×C4—212no—
185C2×C4—28no—
186C2×C4—412no—
187C2×C4C2×C42×C53148yes—
188C2×C4C23×C4×C5312yes—
189C2×C4C23×C4×C53132yes—
190C2×C4C23×C4×C53112yes—
191C2×C4C23×C4×C53112yes—
192C2×C4C23×C4×C53124yes—
193C2×C4C23×C4×C5312yes—
194C2×C42−1+4×C5314yes—
195C22—22no—
196C22—22no—
197C22—28no—
198C22C22×C42×C5314yes—
199C22C24×C4×C5311yescharacteristic
200C22C24×C4×C5312yes—
201C22C2×2−1+4×C5312yes—
202C22C53×#58512yes—
203C4—22no—
204C4—212no—
205C4—232no—
206C4C22×C42×C53124yes—
207C4C53×#51914yesdirect product factor
208C2—24no—
209C2C23×C42×C5311yescharacteristic
210C2C2×C53×#51911yescharacteristic
211C2C53×#342811yescharacteristic
212C2C4×2−1+4×C5314yes—
subgroupG/H
centerC22×C4×C53C24
commutatorC2C23×C42×C53
Frattini, cosocleC23C25×C53
socleC23×C53C25
prime radicalC24×C53C24
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
3. C53×#2276C2semidirect
5. C42×Q8×C53C2semidirect
8. C53×#519C4direct (N × H)
9. C2×C4×Q8×C53C4semidirect
31. C4×Q8×C53C2×C4semidirect
46. C2×Q8×C53C42semidirect
88. C4×#519C53direct (N × H)
101. C4×C53#519direct (N × H)
104. #2276C2×C53semidirect
106. C42×Q8C2×C53semidirect
114. #519C4×C53direct (N × H)
115. C2×C4×Q8C4×C53semidirect
129. C53C4×#519direct (N × H)
138. C4×Q8C2×C4×C53semidirect
153. C2×Q8C42×C53semidirect
207. C4C53×#519direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C53×#5192
C22×C4×Q8×C532
C53×#22762
C53×#22742
C42×Q8×C532
C4×#117×C532
C4×#51953