Finite Group Databasebeta

← C23×#34012 #19181638

Subgroups — C23×#34012 #19181638

Order 5888.

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×C23C211yesmaximal, characteristic
2C23×#2285C216yesmaximal, complemented
3C23×#2283C2112yesmaximal, complemented
4C23×#2280C216yesmaximal
5C4×C23×#117C216yesmaximal, complemented
6C24×C4×C23C2213yes—
7C2×C23×#117C22112yes—
8C22×C42×C23C4116yescomplemented
9C22×C42×C23C2216yes—
10C22×C42×C23C2213yes—
11C22×C42×C23C2213yes—
12C23×#384C2214yescomplemented
13C23×#383C2218yescomplemented
14C23×#382C22112yescomplemented
15C23×#381C22112yes—
16C23×#380C22124yes—
17C23×#379C2214yes—
18C4×C4⋊C4×C23C22124yescomplemented
19C4×C22⋊C4×C23C22124yes—
20C23×#376C22112yescomplemented
21C43×C23C2214yescomplemented
22C25×C23C2311yescharacteristic
23C23×C4×C23—224no—
24C23×C4×C23C2×C4124yes—
25C23×C4×C23C2316yes—
26C23×C4×C23C2316yes—
27C23×C4×C23C2313yes—
28C23×C4×C23C2313yes—
29C23×#117—248no—
30C2×C4⋊C4×C23C23124yes—
31C2×C22⋊C4×C23C23124yes—
32C2×C42×C23C2×C4148yes—
33C2×C42×C23C2×C4124yes—
34C2×C42×C23C2×C4124yes—
35C2×C42×C23C23124yes—
36C2×C42×C23C2313yes—
37C2×C42×C23C2316yes—
38C2×C42×C23C2313yes—
39C2×C42×C23C2312yes—
40C2×C42×C23C2311yescharacteristic
41C2×C42×C23C2311yescharacteristic
42C23×#95C23148yes—
43C24×C23—212no—
44C24×C23C22×C414yes—
45C24×C23C2413yes—
46C22×C4×C23—212no—
47C22×C4×C23—296no—
48C22×C4×C23—224no—
49C22×C4×C23—224no—
50C22×C4×C23C22×C4112yes—
51C22×C4×C23C22×C4124yes—
52C22×C4×C23C22×C4124yes—
53C22×C4×C23C22×C4112yes—
54C22×C4×C23#41124yes—
55C22×C4×C23C24116yes—
56C22×C4×C23C2416yes—
57C22×C4×C23C2413yes—
58C22×C4×C23C2413yes—
59C4⋊C4×C23—248no—
60C22⋊C4×C23—248no—
61C42×C23—248no—
62C42×C23C22×C418yes—
63C42×C23C22×C4112yes—
64C42×C23C22×C4112yes—
65C42×C23C22×C4124yes—
66C42×C23C22×C414yes—
67C42×C23C22×C414yes—
68#34012C2311yesmaximal, characteristic, direct product factor, Sylow
69C23×C23—232no—
70C23×C23—224no—
71C23×C23—212no—
72C23×C23C23×C416yes—
73C23×C23C2×#41112yes—
74C23×C23C2511yescharacteristic
75C2×C4×C23—224no—
76C2×C4×C23—212no—
77C2×C4×C23—248no—
78C2×C4×C23—248no—
79C2×C4×C23—248no—
80C2×C4×C23#117148yescomplemented
81C2×C4×C23C23×C4112yes—
82C2×C4×C23C23×C416yes—
83C2×C4×C23C23×C416yes—
84C2×C4×C23C2×#41112yes—
85C2×C4×C23C2×#41112yes—
86C23×C42C2×C2311yescharacteristic
87#2285C2×C2316yescomplemented
88#2283C2×C23112yescomplemented
89#2280C2×C2316yes—
90C4×#117C2×C2316yescomplemented
91C22×C23—232no—
92C22×C23—212no—
93C22×C23—224no—
94C22×C23C2×#117112yes—
95C22×C23C24×C411yescharacteristic
96C22×C23C22×#4116yes—
97C4×C23—212no—
98C4×C23—416no—
99C4×C23C2×#117112yes—
100C4×C23C2×#117112yes—
101C24×C4C22×C2313yes—
102C2×#117C22×C23112yes—
103C22×C42C4×C23116yescomplemented
104C22×C42C22×C2313yes—
105C22×C42C22×C2316yes—
106C22×C42C22×C2313yes—
107#384C22×C2314yescomplemented
108#383C22×C2318yescomplemented
109#382C22×C23112yescomplemented
110#381C22×C23112yes—
111#380C22×C23124yes—
112#379C22×C2314yes—
113C4×C4⋊C4C22×C23124yescomplemented
114C4×C22⋊C4C22×C23124yes—
115#376C22×C23112yescomplemented
116C43C22×C2314yescomplemented
117C2×C23—212no—
118C2×C23C22×#11713yes—
119C2×C23#344414yes—
120C25C23×C2311yescharacteristic
121C23×C4—224no—
122C23×C4C2×C4×C23124yes—
123C23×C4C23×C2316yes—
124C23×C4C23×C2316yes—
125C23×C4C23×C2313yes—
126C23×C4C23×C2313yes—
127#117—248no—
128C2×C4⋊C4C23×C23124yes—
129C2×C22⋊C4C23×C23124yes—
130C2×C42C2×C4×C23148yes—
131C2×C42C2×C4×C23124yes—
132C2×C42C2×C4×C23124yes—
133C2×C42C23×C23124yes—
134C2×C42C23×C2313yes—
135C2×C42C23×C2316yes—
136C2×C42C23×C2313yes—
137C2×C42C23×C2312yes—
138C2×C42C23×C2311yescharacteristic
139C2×C42C23×C2311yescharacteristic
140#95C23×C23148yes—
141C23#3401211yescharacteristic, direct product factor, Sylow
142C24—212no—
143C24C22×C4×C2314yes—
144C24C24×C2313yes—
145C22×C4—212no—
146C22×C4—296no—
147C22×C4—224no—
148C22×C4—224no—
149C22×C4C22×C4×C23112yes—
150C22×C4C22×C4×C23124yes—
151C22×C4C22×C4×C23124yes—
152C22×C4C22×C4×C23112yes—
153C22×C4#41×C23124yes—
154C22×C4C24×C23116yes—
155C22×C4C24×C2316yes—
156C22×C4C24×C2313yes—
157C22×C4C24×C2313yes—
158C4⋊C4—248no—
159C22⋊C4—248no—
160C42—248no—
161C42C22×C4×C2318yes—
162C42C22×C4×C23112yes—
163C42C22×C4×C23112yes—
164C42C22×C4×C23124yes—
165C42C22×C4×C2314yes—
166C42C22×C4×C2314yes—
167C23—232no—
168C23—212no—
169C23—224no—
170C23C23×C4×C2316yes—
171C23C2×#41×C23112yes—
172C23C25×C2311yescharacteristic
173C2×C4—212no—
174C2×C4—224no—
175C2×C4—248no—
176C2×C4—248no—
177C2×C4—248no—
178C2×C4C23×#117148yescomplemented
179C2×C4C23×C4×C23112yes—
180C2×C4C23×C4×C2316yes—
181C2×C4C23×C4×C2316yes—
182C2×C4C2×#41×C23112yes—
183C2×C4C2×#41×C23112yes—
184C22—232no—
185C22—212no—
186C22—224no—
187C22C2×C23×#117112yes—
188C22C24×C4×C2311yescharacteristic
189C22C22×#41×C2316yes—
190C4—212no—
191C4—416no—
192C4C2×C23×#117112yes—
193C4C2×C23×#117112yes—
194C2—212no—
195C2C22×C23×#11713yes—
196C2C23×#344414yes—
subgroupG/H
centerC2×C42×C23C23
commutatorC22C24×C4×C23
Frattini, cosocleC23C25×C23
socleC23×C23C25
prime radicalC25×C23C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2285C2semidirect
3. C23×#2283C2semidirect
5. C4×C23×#117C2semidirect
8. C22×C42×C23C4semidirect
12. C23×#384C22semidirect
13. C23×#383C22semidirect
14. C23×#382C22semidirect
18. C4×C4⋊C4×C23C22semidirect
20. C23×#376C22semidirect
21. C43×C23C22semidirect
68. #34012C23direct (N × H)
80. C2×C4×C23#117semidirect
87. #2285C2×C23semidirect
88. #2283C2×C23semidirect
90. C4×#117C2×C23semidirect
103. C22×C42C4×C23semidirect
107. #384C22×C23semidirect
108. #383C22×C23semidirect
109. #382C22×C23semidirect
113. C4×C4⋊C4C22×C23semidirect
115. #376C22×C23semidirect
116. C43C22×C23semidirect
141. C23#34012direct (N × H)
178. C2×C4C23×#117semidirect

Maximal subgroups

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

subgroupindex
C23×C42×C232
C23×#22852
C23×#22832
C23×#22802
C4×C23×#1172
#3401223