Finite Group Databasebeta

← C23×#12862 #19160488

Subgroups — C23×#12862 #19160488

Order 5888.

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

normal subgroups only

This group has 566 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×#2310C211yesmaximal, characteristic
2C23×#1784C212yesmaximal
3C23×#1394C214yesmaximal, complemented
4C23×Q8×C23C2211yescharacteristic
5C2×C23×#117C411yescharacteristic
6C2×C23×#117C411yescharacteristic
7C2×C23×#101—22no—
8C2×C23×#101C2214yes—
9C23×#407—22no—
10C23×#391—22no—
11C23×#391C412yes—
12C23×#391C2212yes—
13C23×#381—22no—
14C22×Q8×C23—24no—
15C22×Q8×C23—42no—
16C22×Q8×C23—42no—
17C22×Q8×C23C2×C411yescharacteristic
18C22×Q8×C23D412yes—
19C22×Q8×C23C2311yescharacteristic
20C23×C4×C23C2×C411yescharacteristic
21C23×C4×C23D412yes—
22C2×M16×C23—24no—
23C23×#117—42no—
24C23×#117—42no—
25C2×C4⋊C4×C23—21no—
26C2×C4⋊C4×C23C2×C412yes—
27C2×C22⋊C4×C23—21no—
28C2×C22⋊C4×C23—21no—
29C2×C42×C23—21no—
30C2×C42×C23C2×C412yes—
31C23×#101—28no—
32C23×#101—44no—
33C23×C22⋊C8—44no—
34C23×#95—24no—
35C23×#95—24no—
36C24×C23C22⋊C411yescharacteristic
37C2×Q8×C23—24no—
38C2×Q8×C23—22no—
39C2×Q8×C23—24no—
40C2×Q8×C23—41no—
41C2×Q8×C23—41no—
42C2×Q8×C23—42no—
43C2×Q8×C23—41no—
44C2×Q8×C23—41no—
45C2×Q8×C23—416no—
46C2×Q8×C23#4114yes—
47C22×C4×C23—22no—
48C22×C4×C23—22no—
49C22×C4×C23—22no—
50C22×C4×C23—24no—
51C22×C4×C23—24no—
52C22×C4×C23—22no—
53C22×C4×C23—22no—
54C22×C4×C23—42no—
55C22×C4×C23C22⋊C412yes—
56C22×C4×C23C22⋊C411yescharacteristic
57C22×C4×C23C22×C411yescharacteristic
58C22×C4×C23C2×D412yes—
59M16×C23—44no—
60M16×C23—44no—
61C2×C8×C23—44no—
62C4⋊C4×C23—42no—
63C4⋊C4×C23—42no—
64C22⋊C4×C23—42no—
65C22⋊C4×C23—42no—
66C42×C23—42no—
67C42×C23—42no—
68#12862C2311yesmaximal, characteristic, direct product factor, Sylow
69C23×C23—21no—
70C23×C23—21no—
71C23×C23—42no—
72C23×C23#9912yes—
73C23×C23C2×C22⋊C411yescharacteristic
74Q8×C23—44no—
75Q8×C23—48no—
76Q8×C23—44no—
77C2×C4×C23—28no—
78C2×C4×C23—21no—
79C2×C4×C23—21no—
80C2×C4×C23—22no—
81C2×C4×C23—21no—
82C2×C4×C23—21no—
83C2×C4×C23—22no—
84C2×C4×C23—22no—
85C2×C4×C23—22no—
86C2×C4×C23—22no—
87C2×C4×C23—48no—
88C2×C4×C23—44no—
89C2×C4×C23—44no—
90C2×C4×C23—42no—
91C2×C4×C23—42no—
92C2×C4×C23#9912yes—
93C2×C4×C23#11712yes—
94C2×C4×C23#12314yes—
95C8×C23—84no—
96#2310C2×C2311yescharacteristic
97#1784C2×C2312yes—
98#1394C2×C2314yescomplemented
99C22×C23—22no—
100C22×C23—22no—
101C22×C23—24no—
102C22×C23—42no—
103C22×C23—42no—
104C22×C23#38111yescharacteristic
105C22×C23C2×#9911yescharacteristic
106C22×C23C2×#9911yescharacteristic
107C4×C23—24no—
108C4×C23—22no—
109C4×C23—22no—
110C4×C23—42no—
111C4×C23—82no—
112C4×C23—82no—
113C23×Q8C22×C2311yescharacteristic
114C2×#117C4×C2311yescharacteristic
115C2×#117C4×C2311yescharacteristic
116C2×#101—22no—
117C2×#101C22×C2314yes—
118#407—22no—
119#391—22no—
120#391C4×C2312yes—
121#391C22×C2312yes—
122#381—22no—
123C2×C23—21no—
124C2×C23—21no—
125C2×C23—42no—
126C2×C23#181911yescharacteristic
127C2×C23#213312yes—
128C22×Q8—24no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8C2×C4×C2311yescharacteristic
132C22×Q8D4×C2312yes—
133C22×Q8C23×C2311yescharacteristic
134C23×C4C2×C4×C2311yescharacteristic
135C23×C4D4×C2312yes—
136C2×M16—24no—
137#117—42no—
138#117—42no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4C2×C4×C2312yes—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C42—21no—
144C2×C42C2×C4×C2312yes—
145#101—28no—
146#101—44no—
147C22⋊C8—44no—
148#95—24no—
149#95—24no—
150C23#1286211yescharacteristic, direct product factor, Sylow
151C24C22⋊C4×C2311yescharacteristic
152C2×Q8—24no—
153C2×Q8—22no—
154C2×Q8—24no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8—42no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8—416no—
161C2×Q8#41×C2314yes—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—42no—
170C22×C4C22⋊C4×C2312yes—
171C22×C4C22⋊C4×C2311yescharacteristic
172C22×C4C22×C4×C2311yescharacteristic
173C22×C4C2×D4×C2312yes—
174M16—44no—
175M16—44no—
176C2×C8—44no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C22⋊C4—42no—
180C22⋊C4—42no—
181C42—42no—
182C42—42no—
183C23—21no—
184C23—21no—
185C23—42no—
186C23C23×#9912yes—
187C23C2×C22⋊C4×C2311yescharacteristic
188Q8—44no—
189Q8—48no—
190Q8—44no—
191C2×C4—28no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—48no—
202C2×C4—44no—
203C2×C4—44no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4C23×#9912yes—
207C2×C4C23×#11712yes—
208C2×C4C23×#12314yes—
209C8—84no—
210C22—22no—
211C22—22no—
212C22—24no—
213C22—42no—
214C22—42no—
215C22C23×#38111yescharacteristic
216C22C2×C23×#9911yescharacteristic
217C22C2×C23×#9911yescharacteristic
218C4—24no—
219C4—22no—
220C4—22no—
221C4—42no—
222C4—82no—
223C4—82no—
224C2—21no—
225C2—21no—
226C2—42no—
227C2C23×#181911yescharacteristic
228C2C23×#213312yes—
subgroupG/H
center, socleC22×C23C2×#99
commutatorC22×C4C22×C4×C23
Frattini, cosocleC22×Q8C23×C23
prime radicalC24×C23C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C24×C23 ◁ C23×Q8×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C23×#1394C2semidirect
68. #12862C23direct (N × H)
98. #1394C2×C23semidirect
150. C23#12862direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#23102
C23×#17842
C23×#13942
#1286223