Finite Group Databasebeta

← C22×C23×#430 #19207737

Subgroups — C22×C23×#430 #19207737

Order 5888.

2718 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×M16×C23C212yesmaximal
2C23×M16×C23C211yesmaximal, characteristic
3C2×C23×#430C2124yesmaximal, direct product factor
4C22×C23×#108C214yesmaximal, complemented
5C24×C4×C23C2211yescharacteristic
6C22×M16×C23—212no—
7C22×M16×C23—22no—
8C22×M16×C23C414yes—
9C22×M16×C23C4112yes—
10C22×M16×C23C2214yes—
11C22×M16×C23C22124yes—
12C22×M16×C23C22112yes—
13C23×C8×C23—22no—
14C23×C8×C23C2212yes—
15C23×#430C22164yesdirect product factor
16C2×C23×#108C22148yescomplemented
17C25×C23Q811yescharacteristic
18C23×C4×C23C2×C412yes—
19C23×C4×C23C2×C416yes—
20C23×C4×C23D416yes—
21C23×C4×C23D412yes—
22C23×C4×C23Q811yescharacteristic
23C23×C4×C23Q816yes—
24C23×C4×C23C2316yes—
25C23×C4×C23C2311yescharacteristic
26C2×M16×C23—224no—
27C2×M16×C23—212no—
28C2×M16×C23—248no—
29C2×M16×C23C2×C4148yes—
30C2×M16×C23C2×C4148yes—
31C2×M16×C23C23124yes—
32C2×M16×C23C23132yes—
33C2×M16×C23C23116yes—
34C22×C8×C23—224no—
35C22×C8×C23—24no—
36C22×C8×C23C2×C414yes—
37C22×C8×C23C2×C4112yes—
38C22×C8×C23C23112yes—
39C23×#108C23164yescomplemented
40C24×C23—44no—
41C24×C23C4⋊C412yes—
42C24×C23C4⋊C416yes—
43C24×C23C2×Q811yescharacteristic
44C24×C23C2×Q816yes—
45C22×C4×C23C4⋊C4112yes—
46C22×C4×C23C4⋊C416yes—
47C22×C4×C23C4⋊C4124yes—
48C22×C4×C23C4⋊C412yes—
49C22×C4×C23C4⋊C4112yes—
50C22×C4×C23C22×C411yescharacteristic
51C22×C4×C23C22×C413yes—
52C22×C4×C23C22×C4112yes—
53C22×C4×C23C22×C4112yes—
54C22×C4×C23C2×D4112yes—
55C22×C4×C23C2×D413yes—
56C22×C4×C23C2×D4112yes—
57C22×C4×C23C2×D411yescharacteristic
58C22×C4×C23C2×Q816yes—
59C22×C4×C23C2×Q8112yes—
60C22×C4×C23C2×Q813yes—
61C22×C4×C23C2413yes—
62C22×C4×C23C2414yes—
63M16×C23—232no—
64M16×C23—224no—
65M16×C23C22×C4164yes—
66M16×C23C24116yes—
67C2×C8×C23—224no—
68C2×C8×C23—232no—
69C2×C8×C23C22×C4124yes—
70C2×C8×C23C22×C4124yes—
71C2×C8×C23C2418yes—
72C22×#430C2311yesmaximal, characteristic, direct product factor, Sylow
73C23×C23—28no—
74C23×C23—24no—
75C23×C23—412no—
76C23×C23—412no—
77C23×C23C2×C4⋊C413yes—
78C23×C23C2×C4⋊C4112yes—
79C23×C23C2×C4⋊C4112yes—
80C23×C23C2×C4⋊C411yescharacteristic
81C23×C23C22×Q813yes—
82C23×C23C22×Q814yes—
83C2×C4×C23C2×C4⋊C416yes—
84C2×C4×C23C2×C4⋊C4124yes—
85C2×C4×C23C2×C4⋊C4124yes—
86C2×C4×C23C2×C4⋊C4112yes—
87C2×C4×C23C2×C4⋊C413yes—
88C2×C4×C23C2×C4⋊C4112yes—
89C2×C4×C23C2×C4⋊C413yes—
90C2×C4×C23C23×C413yes—
91C2×C4×C23C23×C413yes—
92C2×C4×C23C23×C418yes—
93C2×C4×C23C22×D413yes—
94C2×C4×C23C22×D418yes—
95C2×C4×C23C22×D413yes—
96C2×C4×C23C22×Q813yes—
97C2×C4×C23C22×Q814yes—
98C2×C4×C23C2511yescharacteristic
99C8×C23—216no—
100C8×C23C23×C4116yes—
101C23×M16C2×C2312yes—
102C23×M16C2×C2311yescharacteristic
103C2×#430C2×C23124yesdirect product factor
104C22×#108C2×C2314yescomplemented
105C22×C23—224no—
106C22×C23—26no—
107C22×C23—26no—
108C22×C23—412no—
109C22×C23—44no—
110C22×C23#43014yesdirect product factor
111C22×C23C22×C4⋊C413yes—
112C22×C23C22×C4⋊C418yes—
113C22×C23C22×C4⋊C413yes—
114C22×C23C23×Q811yescharacteristic
115C4×C23C22×C4⋊C413yes—
116C4×C23C22×C4⋊C418yes—
117C4×C23C22×C4⋊C413yes—
118C4×C23C24×C411yescharacteristic
119C4×C23C23×D411yescharacteristic
120C24×C4C22×C2311yescharacteristic
121C22×M16—22no—
122C22×M16—212no—
123C22×M16C4×C2314yes—
124C22×M16C4×C23112yes—
125C22×M16C22×C2314yes—
126C22×M16C22×C23124yes—
127C22×M16C22×C23112yes—
128C23×C8—22no—
129C23×C8C22×C2312yes—
130#430C22×C23164yesdirect product factor
131C2×#108C22×C23148yescomplemented
132C2×C23—28no—
133C2×C23—23no—
134C2×C23—21no—
135C2×C23C2×#43016yesdirect product factor
136C2×C23C23×C4⋊C411yescharacteristic
137C25Q8×C2311yescharacteristic
138C23×C4C2×C4×C2312yes—
139C23×C4C2×C4×C2316yes—
140C23×C4D4×C2312yes—
141C23×C4D4×C2316yes—
142C23×C4Q8×C2311yescharacteristic
143C23×C4Q8×C2316yes—
144C23×C4C23×C2311yescharacteristic
145C23×C4C23×C2316yes—
146C2×M16—224no—
147C2×M16—212no—
148C2×M16—248no—
149C2×M16C2×C4×C23148yes—
150C2×M16C2×C4×C23148yes—
151C2×M16C23×C23124yes—
152C2×M16C23×C23132yes—
153C2×M16C23×C23116yes—
154C22×C8—24no—
155C22×C8—224no—
156C22×C8C2×C4×C23112yes—
157C22×C8C2×C4×C2314yes—
158C22×C8C23×C23112yes—
159#108C23×C23164yescomplemented
160C23C22×#43011yescharacteristic, direct product factor, Sylow
161C24—44no—
162C24C4⋊C4×C2312yes—
163C24C4⋊C4×C2316yes—
164C24C2×Q8×C2311yescharacteristic
165C24C2×Q8×C2316yes—
166C22×C4C4⋊C4×C23112yes—
167C22×C4C4⋊C4×C23112yes—
168C22×C4C4⋊C4×C2316yes—
169C22×C4C4⋊C4×C23124yes—
170C22×C4C4⋊C4×C2312yes—
171C22×C4C22×C4×C2313yes—
172C22×C4C22×C4×C2311yescharacteristic
173C22×C4C22×C4×C23112yes—
174C22×C4C22×C4×C23112yes—
175C22×C4C2×D4×C2311yescharacteristic
176C22×C4C2×D4×C23112yes—
177C22×C4C2×D4×C2313yes—
178C22×C4C2×D4×C23112yes—
179C22×C4C2×Q8×C2313yes—
180C22×C4C2×Q8×C2316yes—
181C22×C4C2×Q8×C23112yes—
182C22×C4C24×C2313yes—
183C22×C4C24×C2314yes—
184M16—232no—
185M16—224no—
186M16C22×C4×C23164yes—
187M16C24×C23116yes—
188C2×C8—224no—
189C2×C8—232no—
190C2×C8C22×C4×C23124yes—
191C2×C8C22×C4×C23124yes—
192C2×C8C24×C2318yes—
193C23—28no—
194C23—24no—
195C23—412no—
196C23—412no—
197C23C2×C4⋊C4×C2313yes—
198C23C2×C4⋊C4×C23112yes—
199C23C2×C4⋊C4×C23112yes—
200C23C2×C4⋊C4×C2311yescharacteristic
201C23C22×Q8×C2313yes—
202C23C22×Q8×C2314yes—
203C2×C4C2×C4⋊C4×C2316yes—
204C2×C4C2×C4⋊C4×C23124yes—
205C2×C4C2×C4⋊C4×C23124yes—
206C2×C4C2×C4⋊C4×C23112yes—
207C2×C4C2×C4⋊C4×C2313yes—
208C2×C4C2×C4⋊C4×C2313yes—
209C2×C4C2×C4⋊C4×C23112yes—
210C2×C4C23×C4×C2313yes—
211C2×C4C23×C4×C2313yes—
212C2×C4C23×C4×C2318yes—
213C2×C4C22×D4×C2313yes—
214C2×C4C22×D4×C2318yes—
215C2×C4C22×D4×C2313yes—
216C2×C4C22×Q8×C2313yes—
217C2×C4C22×Q8×C2314yes—
218C2×C4C25×C2311yescharacteristic
219C8—216no—
220C8C23×C4×C23116yes—
221C22—224no—
222C22—26no—
223C22—26no—
224C22—412no—
225C22—44no—
226C22C23×#43014yesdirect product factor
227C22C22×C4⋊C4×C2313yes—
228C22C22×C4⋊C4×C2318yes—
229C22C22×C4⋊C4×C2313yes—
230C22C23×Q8×C2311yescharacteristic
231C4C22×C4⋊C4×C2313yes—
232C4C22×C4⋊C4×C2313yes—
233C4C22×C4⋊C4×C2318yes—
234C4C24×C4×C2311yescharacteristic
235C4C23×D4×C2311yescharacteristic
236C2—21no—
237C2—28no—
238C2—23no—
239C2C2×C23×#43016yesdirect product factor
240C2C23×C4⋊C4×C2311yescharacteristic
subgroupG/H
centerC22×C4×C23C2×D4
commutatorC4C24×C4×C23
Frattini, cosocleC2×C4C25×C23
socleC23×C23C2×C4⋊C4
prime radicalC25×C23Q8
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4×C23 ◁ C24×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C23×#430C2direct (N × H)
4. C22×C23×#108C2semidirect
15. C23×#430C22direct (N × H)
16. C2×C23×#108C22semidirect
39. C23×#108C23semidirect
72. C22×#430C23direct (N × H)
103. C2×#430C2×C23direct (N × H)
104. C22×#108C2×C23semidirect
110. C22×C23#430direct (N × H)
130. #430C22×C23direct (N × H)
131. C2×#108C22×C23semidirect
135. C2×C23C2×#430direct (N × H)
159. #108C23×C23semidirect
160. C23C22×#430direct (N × H)
226. C22C23×#430direct (N × H)
239. C2C2×C23×#430direct (N × H)

Maximal subgroups

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

subgroupindex
C23×M16×C232
C2×C23×#4302
C22×C23×#1082
C22×#43023