Finite Group Databasebeta

← C23×#12861 #19160487

Subgroups — C23×#12861 #19160487

Order 5888.

642 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2297C211yesmaximal, characteristic
2C23×#1785C212yesmaximal
3C23×#1394C214yesmaximal, complemented
4C22×#41×C23C2211yescharacteristic
5C2×C23×#117C411yescharacteristic
6C22×C42×C23C411yescharacteristic
7C2×C23×#101C2214yes—
8C2×C23×#100—22no—
9C23×#407—22no—
10C23×#391C2212yes—
11C23×#390C412yes—
12C23×#386—22no—
13C4×C22⋊C4×C23—22no—
14C2×#41×C23—24no—
15C2×#41×C23—42no—
16C2×#41×C23—42no—
17C22×Q8×C23C2311yescharacteristic
18C22×D4×C23C2×C411yescharacteristic
19C22×D4×C23D412yes—
20C23×C4×C23—21no—
21C23×C4×C23C2×C411yescharacteristic
22C23×C4×C23D412yes—
23C2×M16×C23—24no—
24C23×#117—42no—
25C2×C4⋊C4×C23C2×C412yes—
26C2×C22⋊C4×C23—24no—
27C2×C22⋊C4×C23—21no—
28C2×C42×C23—21no—
29C2×C42×C23—21no—
30C2×C42×C23—42no—
31C2×C42×C23C2×C412yes—
32C23×#101—28no—
33C23×#100—44no—
34C23×C22⋊C8—44no—
35C23×#95—24no—
36C24×C23C22⋊C412yes—
37C24×C23C22⋊C411yescharacteristic
38#41×C23—416no—
39C2×Q8×C23—41no—
40C2×Q8×C23—41no—
41C2×Q8×C23#4114yes—
42C2×D4×C23—24no—
43C2×D4×C23—22no—
44C2×D4×C23—24no—
45C2×D4×C23—41no—
46C2×D4×C23—41no—
47C2×D4×C23—42no—
48C22×C4×C23—22no—
49C22×C4×C23—22no—
50C22×C4×C23—22no—
51C22×C4×C23—22no—
52C22×C4×C23—24no—
53C22×C4×C23—24no—
54C22×C4×C23—22no—
55C22×C4×C23—42no—
56C22×C4×C23—44no—
57C22×C4×C23—42no—
58C22×C4×C23C22⋊C411yescharacteristic
59C22×C4×C23C22×C411yescharacteristic
60C22×C4×C23C2×D412yes—
61M16×C23—44no—
62M16×C23—44no—
63C2×C8×C23—44no—
64C4⋊C4×C23—42no—
65C22⋊C4×C23—48no—
66C22⋊C4×C23—42no—
67C42×C23—42no—
68C42×C23—42no—
69C42×C23—42no—
70#12861C2311yesmaximal, characteristic, direct product factor, Sylow
71C23×C23—21no—
72C23×C23—21no—
73C23×C23—22no—
74C23×C23—22no—
75C23×C23—22no—
76C23×C23—42no—
77C23×C23—44no—
78C23×C23#9912yes—
79C23×C23C2×C22⋊C411yescharacteristic
80Q8×C23—44no—
81D4×C23—48no—
82D4×C23—44no—
83C2×C4×C23—28no—
84C2×C4×C23—21no—
85C2×C4×C23—21no—
86C2×C4×C23—21no—
87C2×C4×C23—22no—
88C2×C4×C23—21no—
89C2×C4×C23—22no—
90C2×C4×C23—24no—
91C2×C4×C23—48no—
92C2×C4×C23—42no—
93C2×C4×C23—48no—
94C2×C4×C23—42no—
95C2×C4×C23—44no—
96C2×C4×C23—42no—
97C2×C4×C23#9912yes—
98C2×C4×C23#11712yes—
99C2×C4×C23#12314yes—
100C8×C23—84no—
101#2297C2×C2311yescharacteristic
102#1785C2×C2312yes—
103#1394C2×C2314yescomplemented
104C22×C23—22no—
105C22×C23—22no—
106C22×C23—42no—
107C22×C23—42no—
108C22×C23—44no—
109C22×C23—42no—
110C22×C23—42no—
111C22×C23—44no—
112C22×C23—82no—
113C22×C23#38111yescharacteristic
114C22×C23C2×#9911yescharacteristic
115C22×C23C2×#9911yescharacteristic
116C4×C23—24no—
117C4×C23—22no—
118C4×C23—22no—
119C4×C23—44no—
120C4×C23—82no—
121C22×#41C22×C2311yescharacteristic
122C2×#117C4×C2311yescharacteristic
123C22×C42C4×C2311yescharacteristic
124C2×#101C22×C2314yes—
125C2×#100—22no—
126#407—22no—
127#391C22×C2312yes—
128#390C4×C2312yes—
129#386—22no—
130C4×C22⋊C4—22no—
131C2×C23—21no—
132C2×C23—21no—
133C2×C23—42no—
134C2×C23—82no—
135C2×C23#181911yescharacteristic
136C2×C23#213212yes—
137C2×#41—24no—
138C2×#41—42no—
139C2×#41—42no—
140C22×Q8C23×C2311yescharacteristic
141C22×D4C2×C4×C2311yescharacteristic
142C22×D4D4×C2312yes—
143C23×C4—21no—
144C23×C4C2×C4×C2311yescharacteristic
145C23×C4D4×C2312yes—
146C2×M16—24no—
147#117—42no—
148C2×C4⋊C4C2×C4×C2312yes—
149C2×C22⋊C4—21no—
150C2×C22⋊C4—24no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42—42no—
154C2×C42C2×C4×C2312yes—
155#101—28no—
156#100—44no—
157C22⋊C8—44no—
158#95—24no—
159C23#1286111yescharacteristic, direct product factor, Sylow
160C24C22⋊C4×C2312yes—
161C24C22⋊C4×C2311yescharacteristic
162#41—416no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8#41×C2314yes—
166C2×D4—24no—
167C2×D4—22no—
168C2×D4—24no—
169C2×D4—41no—
170C2×D4—41no—
171C2×D4—42no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—22no—
179C22×C4—42no—
180C22×C4—44no—
181C22×C4—42no—
182C22×C4C22⋊C4×C2311yescharacteristic
183C22×C4C22×C4×C2311yescharacteristic
184C22×C4C2×D4×C2312yes—
185M16—44no—
186M16—44no—
187C2×C8—44no—
188C4⋊C4—42no—
189C22⋊C4—48no—
190C22⋊C4—42no—
191C42—42no—
192C42—42no—
193C42—42no—
194C23—21no—
195C23—21no—
196C23—22no—
197C23—22no—
198C23—22no—
199C23—42no—
200C23—44no—
201C23C23×#9912yes—
202C23C2×C22⋊C4×C2311yescharacteristic
203Q8—44no—
204D4—48no—
205D4—44no—
206C2×C4—28no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—24no—
214C2×C4—48no—
215C2×C4—42no—
216C2×C4—48no—
217C2×C4—42no—
218C2×C4—44no—
219C2×C4—42no—
220C2×C4C23×#9912yes—
221C2×C4C23×#11712yes—
222C2×C4C23×#12314yes—
223C8—84no—
224C22—22no—
225C22—22no—
226C22—42no—
227C22—42no—
228C22—44no—
229C22—42no—
230C22—42no—
231C22—44no—
232C22—82no—
233C22C23×#38111yescharacteristic
234C22C2×C23×#9911yescharacteristic
235C22C2×C23×#9911yescharacteristic
236C4—24no—
237C4—22no—
238C4—22no—
239C4—44no—
240C4—82no—
241C2—21no—
242C2—21no—
243C2—42no—
244C2—82no—
245C2C23×#181911yescharacteristic
246C2C23×#213212yes—
subgroupG/H
center, socleC22×C23C2×#99
commutatorC22×C4C22×C4×C23
Frattini, cosocleC22×Q8C23×C23
prime radicalC22×#41×C23C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C22×#41×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C23×#1394C2semidirect
70. #12861C23direct (N × H)
103. #1394C2×C23semidirect
159. C23#12861direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#22972
C23×#17852
C23×#13942
#1286123