Finite Group Databasebeta

← C23×#11453 #19159079

Subgroups — C23×#11453 #19159079

Order 5888.

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

normal subgroups only

This group has 562 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×#2315C211yesmaximal, characteristic
2C2×C23×#323C212yesmaximal
3C23×#1326C214yesmaximal, complemented
4C22×C22⋊C4×C23C411yescharacteristic
5C22×C22⋊C4×C23C411yescharacteristic
6C22×C22⋊C4×C23C2211yescharacteristic
7C2×C23×C22⋊C8—22no—
8C23×#385—22no—
9C23×#385C412yes—
10C23×#385C2212yes—
11C23×#380—22no—
12C23×#323—22no—
13C23×#323C2214yes—
14C25×C23C2×C411yescharacteristic
15C23×C4×C23—21no—
16C23×C4×C23C2×C412yes—
17C23×C4×C23D412yes—
18C22×C8×C23—42no—
19C2×C4⋊C4×C23—22no—
20C2×C4⋊C4×C23—22no—
21C2×C22⋊C4×C23—21no—
22C2×C22⋊C4×C23—22no—
23C2×C22⋊C4×C23—21no—
24C2×C22⋊C4×C23—21no—
25C2×C22⋊C4×C23—42no—
26C2×C22⋊C4×C23—42no—
27C2×C22⋊C4×C23C814yescomplemented
28C2×C22⋊C4×C23C2×C412yes—
29C2×C22⋊C4×C23C2×C411yescharacteristic
30C2×C22⋊C4×C23D412yes—
31C2×C22⋊C4×C23C2311yescharacteristic
32C23×C22⋊C8—24no—
33C23×C22⋊C8—42no—
34C23×#95—22no—
35C23×#95—22no—
36C24×C23—22no—
37C24×C23—22no—
38C24×C23—82no—
39C24×C23C22⋊C411yescharacteristic
40C24×C23C22⋊C411yescharacteristic
41C24×C23C2×C812yes—
42C24×C23M1612yes—
43C24×C23C22×C411yescharacteristic
44C22×C4×C23—22no—
45C22×C4×C23—22no—
46C22×C4×C23—24no—
47C22×C4×C23—22no—
48C22×C4×C23—22no—
49C22×C4×C23—42no—
50C22×C4×C23—82no—
51C22×C4×C23—82no—
52C22×C4×C23—82no—
53C22×C4×C23C22⋊C412yes—
54C22×C4×C23C2×D412yes—
55C2×C8×C23—44no—
56C2×C8×C23—84no—
57C4⋊C4×C23—82no—
58C4⋊C4×C23—82no—
59C22⋊C4×C23—24no—
60C22⋊C4×C23—22no—
61C22⋊C4×C23—41no—
62C22⋊C4×C23—42no—
63C22⋊C4×C23—41no—
64C22⋊C4×C23—42no—
65C22⋊C4×C23—42no—
66C22⋊C4×C23C2×C814yescomplemented
67#11453C2311yesmaximal, characteristic, direct product factor, Sylow
68C23×C23—21no—
69C23×C23—21no—
70C23×C23—21no—
71C23×C23—21no—
72C23×C23—24no—
73C23×C23—24no—
74C23×C23—42no—
75C23×C23—42no—
76C23×C23—42no—
77C23×C23—82no—
78C23×C23—82no—
79C23×C23—82no—
80C23×C23—82no—
81C23×C23—84no—
82C23×C23C22⋊C812yes—
83C23×C23C22⋊C812yes—
84C23×C23#9912yes—
85C23×C23#10112yes—
86C23×C23C2×C22⋊C411yescharacteristic
87C23×C23C22×C811yescharacteristic
88C23×C23C2×M1611yescharacteristic
89C2×C4×C23—24no—
90C2×C4×C23—22no—
91C2×C4×C23—22no—
92C2×C4×C23—42no—
93C2×C4×C23—42no—
94C2×C4×C23—44no—
95C2×C4×C23—42no—
96C2×C4×C23—42no—
97C2×C4×C23—42no—
98C2×C4×C23—42no—
99C2×C4×C23—42no—
100C2×C4×C23—82no—
101C2×C4×C23—84no—
102C2×C4×C23—84no—
103C8×C23—84no—
104#2315C2×C2311yescharacteristic
105C2×#323C2×C2312yes—
106#1326C2×C2314yescomplemented
107C22×C23—22no—
108C22×C23—22no—
109C22×C23—22no—
110C22×C23—22no—
111C22×C23—21no—
112C22×C23—21no—
113C22×C23—42no—
114C22×C23—42no—
115C22×C23—42no—
116C22×C23—41no—
117C22×C23—41no—
118C22×C23—42no—
119C22×C23—42no—
120C22×C23—42no—
121C22×C23—42no—
122C22×C23—84no—
123C22×C23—84no—
124C22×C23#32414yes—
125C22×C23C2×C22⋊C811yescharacteristic
126C22×C23C2×#9911yescharacteristic
127C22×C23C2×#10111yescharacteristic
128C4×C23—44no—
129C4×C23—42no—
130C4×C23—42no—
131C4×C23—82no—
132C22×C22⋊C4C4×C2311yescharacteristic
133C22×C22⋊C4C4×C2311yescharacteristic
134C22×C22⋊C4C22×C2311yescharacteristic
135C2×C22⋊C8—22no—
136#385—22no—
137#385C4×C2312yes—
138#385C22×C2312yes—
139#380—22no—
140#323—22no—
141#323C22×C2314yes—
142C2×C23—21no—
143C2×C23—21no—
144C2×C23—24no—
145C2×C23—42no—
146C2×C23—41no—
147C2×C23—41no—
148C2×C23C2×#32411yescharacteristic
149C2×C23#212112yes—
150C25C2×C4×C2311yescharacteristic
151C23×C4—21no—
152C23×C4C2×C4×C2312yes—
153C23×C4D4×C2312yes—
154C22×C8—42no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—42no—
163C2×C22⋊C4C8×C2314yescomplemented
164C2×C22⋊C4C2×C4×C2312yes—
165C2×C22⋊C4C2×C4×C2311yescharacteristic
166C2×C22⋊C4D4×C2312yes—
167C2×C22⋊C4C23×C2311yescharacteristic
168C22⋊C8—24no—
169C22⋊C8—42no—
170#95—22no—
171#95—22no—
172C23#1145311yescharacteristic, direct product factor, Sylow
173C24—22no—
174C24—22no—
175C24—82no—
176C24C22⋊C4×C2311yescharacteristic
177C24C22⋊C4×C2311yescharacteristic
178C24C2×C8×C2312yes—
179C24M16×C2312yes—
180C24C22×C4×C2311yescharacteristic
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—24no—
185C22×C4—22no—
186C22×C4—42no—
187C22×C4—82no—
188C22×C4—82no—
189C22×C4—82no—
190C22×C4C22⋊C4×C2312yes—
191C22×C4C2×D4×C2312yes—
192C2×C8—44no—
193C2×C8—84no—
194C4⋊C4—82no—
195C4⋊C4—82no—
196C22⋊C4—22no—
197C22⋊C4—24no—
198C22⋊C4—41no—
199C22⋊C4—42no—
200C22⋊C4—41no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4C2×C8×C2314yescomplemented
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—24no—
209C23—24no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23—82no—
214C23—82no—
215C23—82no—
216C23—82no—
217C23—84no—
218C23C23×C22⋊C812yes—
219C23C23×C22⋊C812yes—
220C23C23×#9912yes—
221C23C23×#10112yes—
222C23C2×C22⋊C4×C2311yescharacteristic
223C23C22×C8×C2311yescharacteristic
224C23C2×M16×C2311yescharacteristic
225C2×C4—24no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—44no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—82no—
237C2×C4—84no—
238C2×C4—84no—
239C8—84no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—21no—
244C22—21no—
245C22—22no—
246C22—42no—
247C22—42no—
248C22—42no—
249C22—41no—
250C22—41no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—84no—
256C22—84no—
257C22C23×#32414yes—
258C22C2×C23×C22⋊C811yescharacteristic
259C22C2×C23×#9911yescharacteristic
260C22C2×C23×#10111yescharacteristic
261C4—44no—
262C4—42no—
263C4—42no—
264C4—82no—
265C2—21no—
266C2—21no—
267C2—24no—
268C2—41no—
269C2—41no—
270C2—42no—
271C2C2×C23×#32411yescharacteristic
272C2C23×#212112yes—
subgroupG/H
center, socleC22×C23C2×#99
commutatorC23C22×C8×C23
Frattini, cosocleC2×C22⋊C4C23×C23
prime radicalC25×C23C2×C4
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C24×C23 ◁ C22×C22⋊C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C23×#1326C2semidirect
27. C2×C22⋊C4×C23C8semidirect
66. C22⋊C4×C23C2×C8semidirect
67. #11453C23direct (N × H)
106. #1326C2×C23semidirect
163. C2×C22⋊C4C8×C23semidirect
172. C23#11453direct (N × H)
203. C22⋊C4C2×C8×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#23152
C2×C23×#3232
C23×#13262
#1145323