Finite Group Databasebeta

← C22×C23×#427 #19207729

Subgroups — C22×C23×#427 #19207729

Order 5888.

3006 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×C8×C23C211yesmaximal, characteristic
2C22×C23×#117C212yesmaximal
3C2×C23×#427C2124yesmaximal, direct product factor
4C22×C23×#107C212yesmaximal, complemented
5C22×C23×#106C212yesmaximal, complemented
6C24×C4×C23C2211yescharacteristic
7C23×C8×C23C4116yescomplemented
8C23×C8×C23C2211yescharacteristic
9C23×C8×C23C2211yescharacteristic
10C23×C8×C23C22112yes—
11C2×C23×#117—212no—
12C2×C23×#117C22124yes—
13C22×C4⋊C4×C23C2214yes—
14C22×C22⋊C4×C23—22no—
15C22×C42×C23—22no—
16C23×#427C22164yesdirect product factor
17C2×C23×#107C22124yescomplemented
18C2×C23×#106C22124yescomplemented
19C25×C23D411yescharacteristic
20C23×C4×C23—24no—
21C23×C4×C23C2×C418yes—
22C23×C4×C23D416yes—
23C23×C4×C23D411yescharacteristic
24C23×C4×C23Q818yes—
25C23×C4×C23C2311yescharacteristic
26C23×C4×C23C2316yes—
27C22×C8×C23C2×C418yes—
28C22×C8×C23C2×C418yes—
29C22×C8×C23C2×C4196yescomplemented
30C22×C8×C23C2316yes—
31C22×C8×C23C2316yes—
32C22×C8×C23C23116yes—
33C23×#117—248no—
34C23×#117C23132yes—
35C2×C4⋊C4×C23—212no—
36C2×C4⋊C4×C23C23124yes—
37C2×C22⋊C4×C23—224no—
38C2×C42×C23—224no—
39C23×#107C23132yescomplemented
40C23×#106C23132yescomplemented
41C24×C23—28no—
42C24×C23C4⋊C418yes—
43C24×C23C2×D416yes—
44C24×C23C2×D411yescharacteristic
45C22×C4×C23—224no—
46C22×C4×C23—416no—
47C22×C4×C23C4⋊C4148yes—
48C22×C4×C23C4⋊C418yes—
49C22×C4×C23C22×C414yes—
50C22×C4×C23C22×C4124yes—
51C22×C4×C23C2×D416yes—
52C22×C4×C23C2×D413yes—
53C22×C4×C23C2×D4112yes—
54C22×C4×C23C2×Q8124yes—
55C22×C4×C23C2×Q814yes—
56C22×C4×C23C2413yes—
57C22×C4×C23C2414yes—
58C2×C8×C23C22×C4124yes—
59C2×C8×C23C22×C4124yes—
60C2×C8×C23C22×C4164yescomplemented
61C2×C8×C23C2414yes—
62C2×C8×C23C2414yes—
63C4⋊C4×C23—224no—
64C4⋊C4×C23C24116yes—
65C22⋊C4×C23—232no—
66C42×C23—232no—
67C22×#427C2311yesmaximal, characteristic, direct product factor, Sylow
68C23×C23—224no—
69C23×C23—232no—
70C23×C23C2×C4⋊C4124yes—
71C23×C23C2×C4⋊C414yes—
72C23×C23#13518yes—
73C23×C23C22×D414yes—
74C23×C23C22×D413yes—
75C2×C4×C23—216no—
76C2×C4×C23—448no—
77C2×C4×C23C2×C4⋊C4148yes—
78C2×C4×C23C2×C4⋊C4112yes—
79C2×C4×C23C2×C4⋊C4124yes—
80C2×C4×C23C23×C416yes—
81C2×C4×C23C23×C418yes—
82C2×C4×C23C22×D414yes—
83C2×C4×C23C22×D413yes—
84C2×C4×C23C22×Q818yes—
85C2×C4×C23C22×Q816yes—
86C2×C4×C23C2511yescharacteristic
87C8×C23C23×C418yes—
88C8×C23C23×C418yes—
89C24×C8C2×C2311yescharacteristic
90C22×#117C2×C2312yes—
91C2×#427C2×C23124yesdirect product factor
92C22×#107C2×C2312yescomplemented
93C22×#106C2×C2312yescomplemented
94C22×C23—248no—
95C22×C23—28no—
96C22×C23#427116yesdirect product factor
97C22×C23C22×C4⋊C418yes—
98C22×C23C22×C4⋊C416yes—
99C22×C23C2×#135112yes—
100C22×C23C23×D411yescharacteristic
101C4×C23—416no—
102C4×C23C22×C4⋊C416yes—
103C4×C23C22×C4⋊C418yes—
104C4×C23C24×C411yescharacteristic
105C4×C23C23×Q811yescharacteristic
106C24×C4C22×C2311yescharacteristic
107C23×C8C4×C23116yescomplemented
108C23×C8C22×C2311yescharacteristic
109C23×C8C22×C2311yescharacteristic
110C23×C8C22×C23112yes—
111C2×#117—212no—
112C2×#117C22×C23124yes—
113C22×C4⋊C4C22×C2314yes—
114C22×C22⋊C4—22no—
115C22×C42—22no—
116#427C22×C23164yesdirect product factor
117C2×#107C22×C23124yescomplemented
118C2×#106C22×C23124yescomplemented
119C2×C23—28no—
120C2×C23C2×#427112yesdirect product factor
121C2×C23C23×C4⋊C411yescharacteristic
122C2×C23C22×#13512yes—
123C25D4×C2311yescharacteristic
124C23×C4—24no—
125C23×C4C2×C4×C2318yes—
126C23×C4D4×C2316yes—
127C23×C4D4×C2311yescharacteristic
128C23×C4Q8×C2318yes—
129C23×C4C23×C2311yescharacteristic
130C23×C4C23×C2316yes—
131C22×C8C2×C4×C2318yes—
132C22×C8C2×C4×C2318yes—
133C22×C8C2×C4×C23196yescomplemented
134C22×C8C23×C2316yes—
135C22×C8C23×C2316yes—
136C22×C8C23×C23116yes—
137#117—248no—
138#117C23×C23132yes—
139C2×C4⋊C4—212no—
140C2×C4⋊C4C23×C23124yes—
141C2×C22⋊C4—224no—
142C2×C42—224no—
143#107C23×C23132yescomplemented
144#106C23×C23132yescomplemented
145C23C22×#42711yescharacteristic, direct product factor, Sylow
146C24—28no—
147C24C4⋊C4×C2318yes—
148C24C2×D4×C2316yes—
149C24C2×D4×C2311yescharacteristic
150C22×C4—224no—
151C22×C4—416no—
152C22×C4C4⋊C4×C23148yes—
153C22×C4C4⋊C4×C2318yes—
154C22×C4C22×C4×C2314yes—
155C22×C4C22×C4×C23124yes—
156C22×C4C2×D4×C2316yes—
157C22×C4C2×D4×C2313yes—
158C22×C4C2×D4×C23112yes—
159C22×C4C2×Q8×C23124yes—
160C22×C4C2×Q8×C2314yes—
161C22×C4C24×C2313yes—
162C22×C4C24×C2314yes—
163C2×C8C22×C4×C23124yes—
164C2×C8C22×C4×C23124yes—
165C2×C8C22×C4×C23164yescomplemented
166C2×C8C24×C2314yes—
167C2×C8C24×C2314yes—
168C4⋊C4—224no—
169C4⋊C4C24×C23116yes—
170C22⋊C4—232no—
171C42—232no—
172C23—224no—
173C23—232no—
174C23C2×C4⋊C4×C23124yes—
175C23C2×C4⋊C4×C2314yes—
176C23C23×#13518yes—
177C23C22×D4×C2314yes—
178C23C22×D4×C2313yes—
179C2×C4—216no—
180C2×C4—448no—
181C2×C4C2×C4⋊C4×C23148yes—
182C2×C4C2×C4⋊C4×C23112yes—
183C2×C4C2×C4⋊C4×C23124yes—
184C2×C4C23×C4×C2316yes—
185C2×C4C23×C4×C2318yes—
186C2×C4C22×D4×C2314yes—
187C2×C4C22×D4×C2313yes—
188C2×C4C22×Q8×C2318yes—
189C2×C4C22×Q8×C2316yes—
190C2×C4C25×C2311yescharacteristic
191C8C23×C4×C2318yes—
192C8C23×C4×C2318yes—
193C22—248no—
194C22—28no—
195C22C23×#427116yesdirect product factor
196C22C22×C4⋊C4×C2318yes—
197C22C22×C4⋊C4×C2316yes—
198C22C2×C23×#135112yes—
199C22C23×D4×C2311yescharacteristic
200C4—416no—
201C4C22×C4⋊C4×C2316yes—
202C4C22×C4⋊C4×C2318yes—
203C4C24×C4×C2311yescharacteristic
204C4C23×Q8×C2311yescharacteristic
205C2—28no—
206C2C2×C23×#427112yesdirect product factor
207C2C23×C4⋊C4×C2311yescharacteristic
208C2C22×C23×#13512yes—
subgroupG/H
centerC23×C4×C23D4
commutatorC4C24×C4×C23
Frattini, cosocleC2×C4C25×C23
socleC24×C23C2×D4
prime radicalC25×C23D4
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4×C23 ◁ C24×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C23×#427C2direct (N × H)
4. C22×C23×#107C2semidirect
5. C22×C23×#106C2semidirect
7. C23×C8×C23C4semidirect
16. C23×#427C22direct (N × H)
17. C2×C23×#107C22semidirect
18. C2×C23×#106C22semidirect
29. C22×C8×C23C2×C4semidirect
39. C23×#107C23semidirect
40. C23×#106C23semidirect
60. C2×C8×C23C22×C4semidirect
67. C22×#427C23direct (N × H)
91. C2×#427C2×C23direct (N × H)
92. C22×#107C2×C23semidirect
93. C22×#106C2×C23semidirect
96. C22×C23#427direct (N × H)
107. C23×C8C4×C23semidirect
116. #427C22×C23direct (N × H)
117. C2×#107C22×C23semidirect
118. C2×#106C22×C23semidirect
120. C2×C23C2×#427direct (N × H)
133. C22×C8C2×C4×C23semidirect
143. #107C23×C23semidirect
144. #106C23×C23semidirect
145. C23C22×#427direct (N × H)
165. C2×C8C22×C4×C23semidirect
195. C22C23×#427direct (N × H)
206. C2C2×C23×#427direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C8×C232
C22×C23×#1172
C2×C23×#4272
C22×C23×#1072
C22×C23×#1062
C22×#42723