Finite Group Databasebeta

← C23×#12863 #19160489

Subgroups — C23×#12863 #19160489

Order 5888.

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

normal subgroups only

This group has 614 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×#2311C211yesmaximal, characteristic
2C23×#1785C212yesmaximal
3C23×#1394C214yesmaximal, complemented
4C22×#41×C23C2211yescharacteristic
5C2×C23×#117C411yescharacteristic
6C22×C4⋊C4×C23C411yescharacteristic
7C2×C23×#101C2214yes—
8C2×C23×#100—22no—
9C23×#407—22no—
10C23×#391C2212yes—
11C23×#390C412yes—
12C23×#386—22no—
13C23×#380—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×C23—21no—
26C2×C4⋊C4×C23—21no—
27C2×C4⋊C4×C23—42no—
28C2×C4⋊C4×C23C2×C412yes—
29C2×C22⋊C4×C23—21no—
30C2×C22⋊C4×C23—24no—
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—24no—
52C22×C4×C23—24no—
53C22×C4×C23—22no—
54C22×C4×C23—22no—
55C22×C4×C23—42no—
56C22×C4×C23—42no—
57C22×C4×C23—82no—
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—
65C4⋊C4×C23—42no—
66C4⋊C4×C23—42no—
67C22⋊C4×C23—42no—
68C22⋊C4×C23—84no—
69C42×C23—42no—
70#12863C2311yesmaximal, characteristic, direct product factor, Sylow
71C23×C23—21no—
72C23×C23—21no—
73C23×C23—22no—
74C23×C23—22no—
75C23×C23—22no—
76C23×C23—41no—
77C23×C23—41no—
78C23×C23—42no—
79C23×C23—42no—
80C23×C23#9912yes—
81C23×C23C2×C22⋊C411yescharacteristic
82Q8×C23—44no—
83D4×C23—48no—
84D4×C23—44no—
85C2×C4×C23—28no—
86C2×C4×C23—21no—
87C2×C4×C23—21no—
88C2×C4×C23—21no—
89C2×C4×C23—22no—
90C2×C4×C23—21no—
91C2×C4×C23—22no—
92C2×C4×C23—48no—
93C2×C4×C23—42no—
94C2×C4×C23—44no—
95C2×C4×C23—42no—
96C2×C4×C23—42no—
97C2×C4×C23—42no—
98C2×C4×C23—84no—
99C2×C4×C23#9912yes—
100C2×C4×C23#11712yes—
101C2×C4×C23#12314yes—
102C8×C23—84no—
103#2311C2×C2311yescharacteristic
104#1785C2×C2312yes—
105#1394C2×C2314yescomplemented
106C22×C23—21no—
107C22×C23—21no—
108C22×C23—22no—
109C22×C23—42no—
110C22×C23—42no—
111C22×C23—44no—
112C22×C23—41no—
113C22×C23—41no—
114C22×C23—42no—
115C22×C23—42no—
116C22×C23—42no—
117C22×C23—82no—
118C22×C23#38111yescharacteristic
119C22×C23C2×#9911yescharacteristic
120C22×C23C2×#9911yescharacteristic
121C4×C23—24no—
122C4×C23—22no—
123C4×C23—22no—
124C4×C23—82no—
125C4×C23—82no—
126C22×#41C22×C2311yescharacteristic
127C2×#117C4×C2311yescharacteristic
128C22×C4⋊C4C4×C2311yescharacteristic
129C2×#101C22×C2314yes—
130C2×#100—22no—
131#407—22no—
132#391C22×C2312yes—
133#390C4×C2312yes—
134#386—22no—
135#380—22no—
136C2×C23—21no—
137C2×C23—21no—
138C2×C23—42no—
139C2×C23—82no—
140C2×C23#181911yescharacteristic
141C2×C23#213411yescharacteristic
142C2×C23#213511yescharacteristic
143C2×#41—24no—
144C2×#41—42no—
145C2×#41—42no—
146C22×Q8C23×C2311yescharacteristic
147C22×D4C2×C4×C2311yescharacteristic
148C22×D4D4×C2312yes—
149C23×C4—21no—
150C23×C4C2×C4×C2311yescharacteristic
151C23×C4D4×C2312yes—
152C2×M16—24no—
153#117—42no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4C2×C4×C2312yes—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—24no—
160C2×C42C2×C4×C2312yes—
161#101—28no—
162#100—44no—
163C22⋊C8—44no—
164#95—24no—
165C23#1286311yescharacteristic, direct product factor, Sylow
166C24C22⋊C4×C2312yes—
167C24C22⋊C4×C2311yescharacteristic
168#41—416no—
169C2×Q8—41no—
170C2×Q8—41no—
171C2×Q8#41×C2314yes—
172C2×D4—24no—
173C2×D4—22no—
174C2×D4—24no—
175C2×D4—41no—
176C2×D4—41no—
177C2×D4—42no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—82no—
188C22×C4C22⋊C4×C2311yescharacteristic
189C22×C4C22×C4×C2311yescharacteristic
190C22×C4C2×D4×C2312yes—
191M16—44no—
192M16—44no—
193C2×C8—44no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C22⋊C4—42no—
198C22⋊C4—84no—
199C42—42no—
200C23—21no—
201C23—21no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—41no—
206C23—41no—
207C23—42no—
208C23—42no—
209C23C23×#9912yes—
210C23C2×C22⋊C4×C2311yescharacteristic
211Q8—44no—
212D4—48no—
213D4—44no—
214C2×C4—28no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—48no—
222C2×C4—42no—
223C2×C4—44no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—84no—
228C2×C4C23×#9912yes—
229C2×C4C23×#11712yes—
230C2×C4C23×#12314yes—
231C8—84no—
232C22—21no—
233C22—21no—
234C22—22no—
235C22—42no—
236C22—42no—
237C22—44no—
238C22—41no—
239C22—41no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—82no—
244C22C23×#38111yescharacteristic
245C22C2×C23×#9911yescharacteristic
246C22C2×C23×#9911yescharacteristic
247C4—24no—
248C4—22no—
249C4—22no—
250C4—82no—
251C4—82no—
252C2—21no—
253C2—21no—
254C2—42no—
255C2—82no—
256C2C23×#181911yescharacteristic
257C2C23×#213411yescharacteristic
258C2C23×#213511yescharacteristic
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. #12863C23direct (N × H)
105. #1394C2×C23semidirect
165. C23#12863direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#23112
C23×#17852
C23×#13942
#1286323