Finite Group Databasebeta

← C23×#11452 #19159078

Subgroups — C23×#11452 #19159078

Order 5888.

462 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 142 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×#2296C211yesmaximal, characteristic
2C23×#1458C212yesmaximal
3C23×#1326C214yesmaximal
4C2×C23×#117C411yescharacteristic
5C2×C23×#117C2211yescharacteristic
6C22×C42×C23C411yescharacteristic
7C23×#431—22no—
8C23×#389—22no—
9C23×#385C412yes—
10C23×#385C2212yes—
11C4×C4⋊C4×C23—22no—
12C23×#324—22no—
13C23×#323C2214yes—
14C23×C4×C23C2×C411yescharacteristic
15C23×C4×C23C2×C412yes—
16C23×#117—22no—
17C23×#117—41no—
18C23×#117—41no—
19C23×#117C814yescomplemented
20C2×C4⋊C4×C23—21no—
21C2×C4⋊C4×C23—22no—
22C2×C4⋊C4×C23—22no—
23C2×C4⋊C4×C23D412yes—
24C2×C22⋊C4×C23C2×C412yes—
25C2×C22⋊C4×C23C2×C411yescharacteristic
26C2×C22⋊C4×C23C2311yescharacteristic
27C2×C42×C23—21no—
28C2×C42×C23—21no—
29C2×C42×C23—21no—
30C2×C42×C23—41no—
31C2×C42×C23—41no—
32C2×C42×C23D412yes—
33C23×C4⋊C8—42no—
34C23×C22⋊C8—24no—
35C4×C8×C23—42no—
36C23×#95—22no—
37C23×#95—22no—
38C24×C23C22×C411yescharacteristic
39C22×C4×C23—21no—
40C22×C4×C23—21no—
41C22×C4×C23—22no—
42C22×C4×C23—22no—
43C22×C4×C23—22no—
44C22×C4×C23—44no—
45C22×C4×C23—42no—
46C22×C4×C23C22⋊C411yescharacteristic
47C22×C4×C23C22⋊C412yes—
48C22×C4×C23C22⋊C411yescharacteristic
49C22×C4×C23C2×C812yes—
50C22×C4×C23M1612yes—
51C22×C4×C23C2×D412yes—
52C2×C8×C23—44no—
53C4⋊C4×C23—24no—
54C4⋊C4×C23—42no—
55C4⋊C4×C23—44no—
56C4⋊C4×C23—44no—
57C22⋊C4×C23—22no—
58C22⋊C4×C23—41no—
59C22⋊C4×C23—41no—
60C22⋊C4×C23C2×C814yes—
61C42×C23—24no—
62C42×C23—22no—
63C42×C23—22no—
64C42×C23—42no—
65C42×C23—41no—
66C42×C23—41no—
67#11452C2311yesmaximal, characteristic, direct product factor, Sylow
68C23×C23—21no—
69C23×C23—21no—
70C23×C23—42no—
71C23×C23C2×C22⋊C411yescharacteristic
72C23×C23C22×C811yescharacteristic
73C23×C23C2×M1611yescharacteristic
74C2×C4×C23—21no—
75C2×C4×C23—21no—
76C2×C4×C23—22no—
77C2×C4×C23—22no—
78C2×C4×C23—22no—
79C2×C4×C23—22no—
80C2×C4×C23—24no—
81C2×C4×C23—24no—
82C2×C4×C23—22no—
83C2×C4×C23—22no—
84C2×C4×C23—42no—
85C2×C4×C23—42no—
86C2×C4×C23—42no—
87C2×C4×C23—44no—
88C2×C4×C23—42no—
89C2×C4×C23—42no—
90C2×C4×C23—42no—
91C2×C4×C23—44no—
92C2×C4×C23C22⋊C812yes—
93C2×C4×C23C22⋊C812yes—
94C2×C4×C23#9911yescharacteristic
95C2×C4×C23#9911yescharacteristic
96C2×C4×C23#10112yes—
97C8×C23—84no—
98#2296C2×C2311yescharacteristic
99#1458C2×C2312yes—
100#1326C2×C2314yes—
101C22×C23—22no—
102C22×C23—21no—
103C22×C23—21no—
104C22×C23—41no—
105C22×C23—41no—
106C22×C23—42no—
107C22×C23—42no—
108C22×C23C2×C22⋊C811yescharacteristic
109C22×C23C2×#9911yescharacteristic
110C22×C23C2×#10111yescharacteristic
111C4×C23—22no—
112C4×C23—44no—
113C4×C23—42no—
114C4×C23—42no—
115C4×C23—44no—
116C4×C23#32412yes—
117C4×C23#32412yes—
118C2×#117C4×C2311yescharacteristic
119C2×#117C22×C2311yescharacteristic
120C22×C42C4×C2311yescharacteristic
121#431—22no—
122#389—22no—
123#385C4×C2312yes—
124#385C22×C2312yes—
125C4×C4⋊C4—22no—
126#324—22no—
127#323C22×C2314yes—
128C2×C23—21no—
129C2×C23—21no—
130C2×C23—41no—
131C2×C23—41no—
132C2×C23C2×#32411yescharacteristic
133C2×C23#212012yes—
134C23×C4C2×C4×C2311yescharacteristic
135C23×C4C2×C4×C2312yes—
136#117—22no—
137#117—41no—
138#117—41no—
139#117C8×C2314yescomplemented
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4D4×C2312yes—
144C2×C22⋊C4C2×C4×C2312yes—
145C2×C22⋊C4C2×C4×C2311yescharacteristic
146C2×C22⋊C4C23×C2311yescharacteristic
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—41no—
151C2×C42—41no—
152C2×C42D4×C2312yes—
153C4⋊C8—42no—
154C22⋊C8—24no—
155C4×C8—42no—
156#95—22no—
157#95—22no—
158C23#1145211yescharacteristic, direct product factor, Sylow
159C24C22×C4×C2311yescharacteristic
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—44no—
166C22×C4—42no—
167C22×C4C22⋊C4×C2311yescharacteristic
168C22×C4C22⋊C4×C2312yes—
169C22×C4C22⋊C4×C2311yescharacteristic
170C22×C4C2×C8×C2312yes—
171C22×C4M16×C2312yes—
172C22×C4C2×D4×C2312yes—
173C2×C8—44no—
174C4⋊C4—24no—
175C4⋊C4—42no—
176C4⋊C4—44no—
177C4⋊C4—44no—
178C22⋊C4—22no—
179C22⋊C4—41no—
180C22⋊C4—41no—
181C22⋊C4C2×C8×C2314yes—
182C42—22no—
183C42—22no—
184C42—24no—
185C42—42no—
186C42—41no—
187C42—41no—
188C23—21no—
189C23—21no—
190C23—42no—
191C23C2×C22⋊C4×C2311yescharacteristic
192C23C22×C8×C2311yescharacteristic
193C23C2×M16×C2311yescharacteristic
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—44no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—44no—
212C2×C4C23×C22⋊C812yes—
213C2×C4C23×C22⋊C812yes—
214C2×C4C23×#9911yescharacteristic
215C2×C4C23×#9911yescharacteristic
216C2×C4C23×#10112yes—
217C8—84no—
218C22—22no—
219C22—21no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—42no—
224C22—42no—
225C22C2×C23×C22⋊C811yescharacteristic
226C22C2×C23×#9911yescharacteristic
227C22C2×C23×#10111yescharacteristic
228C4—22no—
229C4—44no—
230C4—42no—
231C4—44no—
232C4—42no—
233C4C23×#32412yes—
234C4C23×#32412yes—
235C2—21no—
236C2—21no—
237C2—41no—
238C2—41no—
239C2C2×C23×#32411yescharacteristic
240C2C23×#212012yes—
subgroupG/H
centerC2×C4×C23#99
commutatorC23C22×C8×C23
Frattini, cosocleC2×C22⋊C4C23×C23
socleC22×C23C2×#99
prime radicalC24×C23C22×C4
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C23 ◁ C22×C4×C23 ◁ C2×C23×#117 ◁ G

Complements

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

NH ≅ G/Ntype
19. C23×#117C8semidirect
67. #11452C23direct (N × H)
139. #117C8×C23semidirect
158. C23#11452direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#22962
C23×#14582
C23×#13262
#1145223