Finite Group Databasebeta

← C53×#34012 #27241524

Subgroups — C53×#34012 #27241524

Order 13568.

3158 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C42×C53C211yesmaximal, characteristic
2C53×#2285C216yesmaximal, complemented
3C53×#2283C2112yesmaximal, complemented
4C53×#2280C216yesmaximal
5C4×#117×C53C216yesmaximal, complemented
6C24×C4×C53C2213yes—
7C2×#117×C53C22112yes—
8C22×C42×C53C4116yescomplemented
9C22×C42×C53C2216yes—
10C22×C42×C53C2213yes—
11C22×C42×C53C2213yes—
12C53×#384C2214yescomplemented
13C53×#383C2218yescomplemented
14C53×#382C22112yescomplemented
15C53×#381C22112yes—
16C53×#380C22124yes—
17C53×#379C2214yes—
18C4×C4⋊C4×C53C22124yescomplemented
19C4×C22⋊C4×C53C22124yes—
20C53×#376C22112yescomplemented
21C43×C53C2214yescomplemented
22C25×C53C2311yescharacteristic
23C23×C4×C53—224no—
24C23×C4×C53C2×C4124yes—
25C23×C4×C53C2316yes—
26C23×C4×C53C2316yes—
27C23×C4×C53C2313yes—
28C23×C4×C53C2313yes—
29#117×C53—248no—
30C2×C4⋊C4×C53C23124yes—
31C2×C22⋊C4×C53C23124yes—
32C2×C42×C53C2×C4148yes—
33C2×C42×C53C2×C4124yes—
34C2×C42×C53C2×C4124yes—
35C2×C42×C53C23124yes—
36C2×C42×C53C2313yes—
37C2×C42×C53C2316yes—
38C2×C42×C53C2313yes—
39C2×C42×C53C2312yes—
40C2×C42×C53C2311yescharacteristic
41C2×C42×C53C2311yescharacteristic
42#95×C53C23148yes—
43C24×C53—212no—
44C24×C53C22×C414yes—
45C24×C53C2413yes—
46C22×C4×C53—212no—
47C22×C4×C53—296no—
48C22×C4×C53—224no—
49C22×C4×C53—224no—
50C22×C4×C53C22×C4112yes—
51C22×C4×C53C22×C4124yes—
52C22×C4×C53C22×C4124yes—
53C22×C4×C53C22×C4112yes—
54C22×C4×C53#41124yes—
55C22×C4×C53C24116yes—
56C22×C4×C53C2416yes—
57C22×C4×C53C2413yes—
58C22×C4×C53C2413yes—
59C4⋊C4×C53—248no—
60C22⋊C4×C53—248no—
61C42×C53—248no—
62C42×C53C22×C418yes—
63C42×C53C22×C4112yes—
64C42×C53C22×C4112yes—
65C42×C53C22×C4124yes—
66C42×C53C22×C414yes—
67C42×C53C22×C414yes—
68C23×C53—232no—
69C23×C53—224no—
70C23×C53—212no—
71C23×C53C23×C416yes—
72C23×C53C2×#41112yes—
73C23×C53C2511yescharacteristic
74C2×C4×C53—224no—
75C2×C4×C53—212no—
76C2×C4×C53—248no—
77C2×C4×C53—248no—
78C2×C4×C53—248no—
79C2×C4×C53#117148yescomplemented
80C2×C4×C53C23×C4112yes—
81C2×C4×C53C23×C416yes—
82C2×C4×C53C23×C416yes—
83C2×C4×C53C2×#41112yes—
84C2×C4×C53C2×#41112yes—
85#34012C5311yesmaximal, characteristic, direct product factor, Sylow
86C22×C53—232no—
87C22×C53—212no—
88C22×C53—224no—
89C22×C53C2×#117112yes—
90C22×C53C24×C411yescharacteristic
91C22×C53C22×#4116yes—
92C4×C53—212no—
93C4×C53—416no—
94C4×C53C2×#117112yes—
95C4×C53C2×#117112yes—
96C23×C42C2×C5311yescharacteristic
97#2285C2×C5316yescomplemented
98#2283C2×C53112yescomplemented
99#2280C2×C5316yes—
100C4×#117C2×C5316yescomplemented
101C2×C53—212no—
102C2×C53C22×#11713yes—
103C2×C53#344414yes—
104C24×C4C22×C5313yes—
105C2×#117C22×C53112yes—
106C22×C42C4×C53116yescomplemented
107C22×C42C22×C5313yes—
108C22×C42C22×C5316yes—
109C22×C42C22×C5313yes—
110#384C22×C5314yescomplemented
111#383C22×C5318yescomplemented
112#382C22×C53112yescomplemented
113#381C22×C53112yes—
114#380C22×C53124yes—
115#379C22×C5314yes—
116C4×C4⋊C4C22×C53124yescomplemented
117C4×C22⋊C4C22×C53124yes—
118#376C22×C53112yescomplemented
119C43C22×C5314yescomplemented
120C53#3401211yescharacteristic, direct product factor, Sylow
121C25C23×C5311yescharacteristic
122C23×C4—224no—
123C23×C4C2×C4×C53124yes—
124C23×C4C23×C5316yes—
125C23×C4C23×C5316yes—
126C23×C4C23×C5313yes—
127C23×C4C23×C5313yes—
128#117—248no—
129C2×C4⋊C4C23×C53124yes—
130C2×C22⋊C4C23×C53124yes—
131C2×C42C2×C4×C53148yes—
132C2×C42C2×C4×C53124yes—
133C2×C42C2×C4×C53124yes—
134C2×C42C23×C53124yes—
135C2×C42C23×C5313yes—
136C2×C42C23×C5316yes—
137C2×C42C23×C5313yes—
138C2×C42C23×C5312yes—
139C2×C42C23×C5311yescharacteristic
140C2×C42C23×C5311yescharacteristic
141#95C23×C53148yes—
142C24—212no—
143C24C22×C4×C5314yes—
144C24C24×C5313yes—
145C22×C4—212no—
146C22×C4—296no—
147C22×C4—224no—
148C22×C4—224no—
149C22×C4C22×C4×C53112yes—
150C22×C4C22×C4×C53124yes—
151C22×C4C22×C4×C53124yes—
152C22×C4C22×C4×C53112yes—
153C22×C4#41×C53124yes—
154C22×C4C24×C53116yes—
155C22×C4C24×C5316yes—
156C22×C4C24×C5313yes—
157C22×C4C24×C5313yes—
158C4⋊C4—248no—
159C22⋊C4—248no—
160C42—248no—
161C42C22×C4×C5318yes—
162C42C22×C4×C53112yes—
163C42C22×C4×C53112yes—
164C42C22×C4×C53124yes—
165C42C22×C4×C5314yes—
166C42C22×C4×C5314yes—
167C23—232no—
168C23—224no—
169C23—212no—
170C23C23×C4×C5316yes—
171C23C2×#41×C53112yes—
172C23C25×C5311yescharacteristic
173C2×C4—224no—
174C2×C4—212no—
175C2×C4—248no—
176C2×C4—248no—
177C2×C4—248no—
178C2×C4#117×C53148yescomplemented
179C2×C4C23×C4×C53112yes—
180C2×C4C23×C4×C5316yes—
181C2×C4C23×C4×C5316yes—
182C2×C4C2×#41×C53112yes—
183C2×C4C2×#41×C53112yes—
184C22—232no—
185C22—212no—
186C22—224no—
187C22C2×#117×C53112yes—
188C22C24×C4×C5311yescharacteristic
189C22C22×#41×C5316yes—
190C4—212no—
191C4—416no—
192C4C2×#117×C53112yes—
193C4C2×#117×C53112yes—
194C2—212no—
195C2C22×#117×C5313yes—
196C2C53×#344414yes—
subgroupG/H
centerC2×C42×C53C23
commutatorC22C24×C4×C53
Frattini, cosocleC23C25×C53
socleC23×C53C25
prime radicalC25×C53C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2285C2semidirect
3. C53×#2283C2semidirect
5. C4×#117×C53C2semidirect
8. C22×C42×C53C4semidirect
12. C53×#384C22semidirect
13. C53×#383C22semidirect
14. C53×#382C22semidirect
18. C4×C4⋊C4×C53C22semidirect
20. C53×#376C22semidirect
21. C43×C53C22semidirect
79. C2×C4×C53#117semidirect
85. #34012C53direct (N × H)
97. #2285C2×C53semidirect
98. #2283C2×C53semidirect
100. C4×#117C2×C53semidirect
106. C22×C42C4×C53semidirect
110. #384C22×C53semidirect
111. #383C22×C53semidirect
112. #382C22×C53semidirect
116. C4×C4⋊C4C22×C53semidirect
118. #376C22×C53semidirect
119. C43C22×C53semidirect
120. C53#34012direct (N × H)
178. C2×C4#117×C53semidirect

Maximal subgroups

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

subgroupindex
C23×C42×C532
C53×#22852
C53×#22832
C53×#22802
C4×#117×C532
#3401253