Finite Group Databasebeta

← C2×C23×#2986 #19207872

Subgroups — C2×C23×#2986 #19207872

Order 5888.

1794 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#533C211yesmaximal, characteristic
2C23×#2986C2116yesmaximal, direct product factor
3C2×C23×#446C214yesmaximal, complemented
4C2×C8×Q8×C23C214yesmaximal, complemented
5C2×C23×#433C214yesmaximal
6C2×C23×#431C212yesmaximal
7C23×#533C418yes—
8C23×#533C2218yes—
9C2×C23×C4⋊Q8C412yes—
10C2×C23×#125C412yes—
11C2×C23×C22⋊Q8C414yes—
12C2×C4×Q8×C23C2214yes—
13C2×C23×#117C2212yes—
14C22×C42×C23C2211yescharacteristic
15C23×#446C22132yescomplemented
16C8×Q8×C23C22132yescomplemented
17C23×#433C22132yes—
18C23×#431C22116yes—
19C2×C23×C4⋊C8C2212yes—
20C2×C23×C4⋊C8C2212yes—
21C2×C23×C4⋊C8C2218yes—
22C2×C23×C22⋊C8C2214yes—
23C2×C23×#97C2214yes—
24C2×C4×C8×C23C2214yes—
25C2×C4×C8×C23C2214yes—
26C22×Q8×C23C2×C414yes—
27C23×C4×C23C2×C412yes—
28C23×C4×C23C2311yescharacteristic
29C22×C8×C23C2314yes—
30C22×C8×C23C2314yes—
31C23×C4⋊Q8C2×C4116yes—
32C23×#125C2×C4116yes—
33C23×C22⋊Q8C2×C4132yes—
34C4×Q8×C23C2×C4116yes—
35C4×Q8×C23C23116yes—
36C23×#117C2×C418yes—
37C23×#117C2318yes—
38C2×C4⋊C4×C23C2×C414yes—
39C2×C4⋊C4×C23C2×C418yes—
40C2×C4⋊C4×C23C2×C414yes—
41C2×C22⋊C4×C23C2×C414yes—
42C2×C42×C23C2×C411yescharacteristic
43C2×C42×C23C2×C411yescharacteristic
44C2×C42×C23C2×C414yes—
45C2×C42×C23C2314yes—
46C2×C42×C23C2312yes—
47C2×C42×C23C2314yes—
48C23×C4⋊C8C2318yes—
49C23×C4⋊C8C2318yes—
50C23×C4⋊C8C23132yes—
51C23×C22⋊C8C23116yes—
52C23×#97C23116yes—
53C4×C8×C23C23116yes—
54C4×C8×C23C23116yes—
55C24×C23C22×C411yescharacteristic
56C2×Q8×C23—216no—
57C2×Q8×C23C22×C4116yes—
58C22×C4×C23—28no—
59C22×C4×C23—24no—
60C22×C4×C23C22×C411yescharacteristic
61C22×C4×C23C22×C412yes—
62C22×C4×C23C22×C412yes—
63C22×C4×C23C22×C418yes—
64C22×C4×C23C22×C418yes—
65C22×C4×C23C22×C412yes—
66C22×C4×C23C2412yes—
67C22×C4×C23C2411yescharacteristic
68C2×C8×C23—28no—
69C2×C8×C23—28no—
70C2×C8×C23C2418yes—
71C2×C8×C23C2418yes—
72C4⋊C4×C23C22×C4116yes—
73C4⋊C4×C23C22×C4132yes—
74C4⋊C4×C23C22×C4116yes—
75C22⋊C4×C23C22×C4116yes—
76C42×C23C22×C414yes—
77C42×C23C22×C414yes—
78C42×C23C22×C414yes—
79C42×C23C22×C418yes—
80C42×C23C2414yes—
81C42×C23C2418yes—
82C2×#2986C2311yesmaximal, characteristic, direct product factor, Sylow
83C23×C23—22no—
84C23×C23—24no—
85C23×C23C23×C412yes—
86C23×C23C23×C411yescharacteristic
87Q8×C23—232no—
88C2×C4×C23—22no—
89C2×C4×C23—24no—
90C2×C4×C23—216no—
91C2×C4×C23—24no—
92C2×C4×C23—216no—
93C2×C4×C23—24no—
94C2×C4×C23#13118yes—
95C2×C4×C23C23×C412yes—
96C2×C4×C23C23×C414yes—
97C2×C4×C23C23×C4116yes—
98C2×C4×C23C23×C411yescharacteristic
99C2×C4×C23C23×C414yes—
100C2×C4×C232−1+414yes—
101C2×C4×C23C2511yescharacteristic
102C8×C23—28no—
103C8×C23—28no—
104C2×#533C2×C2311yescharacteristic
105#2986C2×C23116yesdirect product factor
106C2×#446C2×C2314yescomplemented
107C2×C8×Q8C2×C2314yescomplemented
108C2×#433C2×C2314yes—
109C2×#431C2×C2312yes—
110C22×C23—22no—
111C22×C23—24no—
112C22×C23—44no—
113C22×C23#51912yes—
114C22×C23C2×#13114yes—
115C22×C23C24×C411yescharacteristic
116C4×C23—22no—
117C4×C23—24no—
118C4×C23—216no—
119C4×C23#51912yes—
120C4×C23C2×#13118yes—
121C4×C23C2×2−1+412yes—
122#533C4×C2318yes—
123#533C22×C2318yes—
124C2×C4⋊Q8C4×C2312yes—
125C2×#125C4×C2312yes—
126C2×C22⋊Q8C4×C2314yes—
127C2×C4×Q8C22×C2314yes—
128C2×#117C22×C2312yes—
129C22×C42C22×C2311yescharacteristic
130#446C22×C23132yescomplemented
131C8×Q8C22×C23132yescomplemented
132#433C22×C23132yes—
133#431C22×C23116yes—
134C2×C4⋊C8C22×C2312yes—
135C2×C4⋊C8C22×C2312yes—
136C2×C4⋊C8C22×C2318yes—
137C2×C22⋊C8C22×C2314yes—
138C2×#97C22×C2314yes—
139C2×C4×C8C22×C2314yes—
140C2×C4×C8C22×C2314yes—
141C2×C23—42no—
142C2×C23#298614yesdirect product factor
143C2×C23C2×#51911yescharacteristic
144C2×C23C22×#13112yes—
145C22×Q8C2×C4×C2314yes—
146C23×C4C2×C4×C2312yes—
147C23×C4C23×C2311yescharacteristic
148C22×C8C23×C2314yes—
149C22×C8C23×C2314yes—
150C4⋊Q8C2×C4×C23116yes—
151#125C2×C4×C23116yes—
152C22⋊Q8C2×C4×C23132yes—
153C4×Q8C2×C4×C23116yes—
154C4×Q8C23×C23116yes—
155#117C2×C4×C2318yes—
156#117C23×C2318yes—
157C2×C4⋊C4C2×C4×C2314yes—
158C2×C4⋊C4C2×C4×C2318yes—
159C2×C4⋊C4C2×C4×C2314yes—
160C2×C22⋊C4C2×C4×C2314yes—
161C2×C42C2×C4×C2311yescharacteristic
162C2×C42C2×C4×C2311yescharacteristic
163C2×C42C2×C4×C2314yes—
164C2×C42C23×C2314yes—
165C2×C42C23×C2312yes—
166C2×C42C23×C2314yes—
167C4⋊C8C23×C2318yes—
168C4⋊C8C23×C2318yes—
169C4⋊C8C23×C23132yes—
170C22⋊C8C23×C23116yes—
171#97C23×C23116yes—
172C4×C8C23×C23116yes—
173C4×C8C23×C23116yes—
174C23C2×#298611yescharacteristic, direct product factor, Sylow
175C24C22×C4×C2311yescharacteristic
176C2×Q8—216no—
177C2×Q8C22×C4×C23116yes—
178C22×C4—28no—
179C22×C4—24no—
180C22×C4C22×C4×C2311yescharacteristic
181C22×C4C22×C4×C2312yes—
182C22×C4C22×C4×C2312yes—
183C22×C4C22×C4×C2318yes—
184C22×C4C22×C4×C2318yes—
185C22×C4C22×C4×C2312yes—
186C22×C4C24×C2312yes—
187C22×C4C24×C2311yescharacteristic
188C2×C8—28no—
189C2×C8—28no—
190C2×C8C24×C2318yes—
191C2×C8C24×C2318yes—
192C4⋊C4C22×C4×C23116yes—
193C4⋊C4C22×C4×C23132yes—
194C4⋊C4C22×C4×C23116yes—
195C22⋊C4C22×C4×C23116yes—
196C42C22×C4×C2314yes—
197C42C22×C4×C2314yes—
198C42C22×C4×C2314yes—
199C42C22×C4×C2318yes—
200C42C24×C2314yes—
201C42C24×C2318yes—
202C23—24no—
203C23—22no—
204C23C23×C4×C2312yes—
205C23C23×C4×C2311yescharacteristic
206Q8—232no—
207C2×C4—24no—
208C2×C4—216no—
209C2×C4—22no—
210C2×C4—216no—
211C2×C4—24no—
212C2×C4—24no—
213C2×C4C23×#13118yes—
214C2×C4C23×C4×C2312yes—
215C2×C4C23×C4×C2314yes—
216C2×C4C23×C4×C23116yes—
217C2×C4C23×C4×C2314yes—
218C2×C4C23×C4×C2311yescharacteristic
219C2×C4C23×2−1+414yes—
220C2×C4C25×C2311yescharacteristic
221C8—28no—
222C8—28no—
223C22—22no—
224C22—24no—
225C22—44no—
226C22C23×#51912yes—
227C22C2×C23×#13114yes—
228C22C24×C4×C2311yescharacteristic
229C4—22no—
230C4—24no—
231C4—216no—
232C4C23×#51912yes—
233C4C2×C23×#13118yes—
234C4C2×C23×2−1+412yes—
235C2—42no—
236C2C23×#298614yesdirect product factor
237C2C2×C23×#51911yescharacteristic
238C2C22×C23×#13112yes—
subgroupG/H
centerC22×C4×C23C24
commutatorC22C24×C4×C23
Frattini, cosocleC2×C4C25×C23
socleC23×C23C23×C4
prime radicalC24×C23C22×C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2986C2direct (N × H)
3. C2×C23×#446C2semidirect
4. C2×C8×Q8×C23C2semidirect
15. C23×#446C22semidirect
16. C8×Q8×C23C22semidirect
82. C2×#2986C23direct (N × H)
105. #2986C2×C23direct (N × H)
106. C2×#446C2×C23semidirect
107. C2×C8×Q8C2×C23semidirect
130. #446C22×C23semidirect
131. C8×Q8C22×C23semidirect
142. C2×C23#2986direct (N × H)
174. C23C2×#2986direct (N × H)
236. C2C23×#2986direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#5332
C23×#29862
C2×C23×#4462
C2×C8×Q8×C232
C2×C23×#4332
C2×C23×#4312
C2×#298623