Finite Group Databasebeta

← C23×#11534 #19159160

Subgroups — C23×#11534 #19159160

Order 5888.

424 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2326C211yesmaximal, characteristic
2C23×#1467C212yesmaximal, complemented
3C23×#1325C212yesmaximal, complemented
4C23×#1318C212yesmaximal
5C2×C23×#124C411yescharacteristic
6C2×C4×Q8×C23C2211yescharacteristic
7C2×C4×D4×C23C411yescharacteristic
8C23×#432—22no—
9C23×#391C411yescharacteristic
10C23×#390C2211yescharacteristic
11C23×#388—22no—
12C23×#385—22no—
13C23×#384C2211yescharacteristic
14C23×#376C411yescharacteristic
15C23×#324—22no—
16C23×#324C2214yes—
17C22×Q8×C23C2×C411yescharacteristic
18C22×D4×C23C2×C411yescharacteristic
19C23×C4×C23—21no—
20C23×#124—22no—
21C23×#124—22no—
22C4×Q8×C23—42no—
23C4×D4×C23—42no—
24C2×C4⋊C4×C23—22no—
25C2×C4⋊C4×C23C2×C411yescharacteristic
26C2×C4⋊C4×C23D412yes—
27C2×C4⋊C4×C23C2311yescharacteristic
28C2×C22⋊C4×C23—22no—
29C2×C22⋊C4×C23—21no—
30C2×C22⋊C4×C23—22no—
31C2×C42×C23C2×C411yescharacteristic
32C2×C42×C23C2×C411yescharacteristic
33C2×C42×C23C2×C411yescharacteristic
34C2×C42×C23D412yes—
35C23×C4⋊C8—42no—
36C23×C22⋊C8—24no—
37C23×#97—42no—
38C23×#95—21no—
39C23×#95—22no—
40C23×#95—21no—
41C24×C23—21no—
42C2×Q8×C23—21no—
43C2×Q8×C23—21no—
44C2×Q8×C23—41no—
45C2×Q8×C23—41no—
46C2×D4×C23—21no—
47C2×D4×C23—41no—
48C2×D4×C23—41no—
49C2×D4×C23M1612yes—
50C22×C4×C23—21no—
51C22×C4×C23—21no—
52C22×C4×C23—21no—
53C22×C4×C23—42no—
54C22×C4×C23—82no—
55C22×C4×C23C22⋊C412yes—
56C22×C4×C23C22⋊C411yescharacteristic
57C22×C4×C23C22⋊C411yescharacteristic
58C22×C4×C23C22×C411yescharacteristic
59C22×C4×C23C2×D412yes—
60C2×C8×C23—44no—
61C4⋊C4×C23—42no—
62C4⋊C4×C23—41no—
63C4⋊C4×C23—41no—
64C4⋊C4×C23—82no—
65C22⋊C4×C23—28no—
66C22⋊C4×C23—42no—
67C22⋊C4×C23—82no—
68C42×C23—22no—
69C42×C23—22no—
70C42×C23—22no—
71C42×C23—21no—
72C42×C23—41no—
73C42×C23M1612yes—
74#11534C2311yesmaximal, characteristic, direct product factor, Sylow
75C23×C23—22no—
76C23×C23—42no—
77C23×C23—42no—
78C23×C23—41no—
79C23×C23—41no—
80C23×C23C2×C22⋊C411yescharacteristic
81Q8×C23—42no—
82Q8×C23—42no—
83D4×C23—42no—
84D4×C23—42no—
85C2×C4×C23—21no—
86C2×C4×C23—21no—
87C2×C4×C23—22no—
88C2×C4×C23—22no—
89C2×C4×C23—22no—
90C2×C4×C23—21no—
91C2×C4×C23—22no—
92C2×C4×C23—22no—
93C2×C4×C23—41no—
94C2×C4×C23—41no—
95C2×C4×C23—41no—
96C2×C4×C23—42no—
97C2×C4×C23—41no—
98C2×C4×C23—41no—
99C2×C4×C23—42no—
100C2×C4×C23—41no—
101C2×C4×C23—41no—
102C2×C4×C23—41no—
103C2×C4×C23—84no—
104C2×C4×C23#9912yes—
105C2×C4×C23C2×M1611yescharacteristic
106C2×C4×C23C2×M1611yescharacteristic
107C8×C23—84no—
108#2326C2×C2311yescharacteristic
109#1467C2×C2312yescomplemented
110#1325C2×C2312yescomplemented
111#1318C2×C2312yes—
112C22×C23—21no—
113C22×C23—21no—
114C22×C23—44no—
115C22×C23—42no—
116C22×C23—82no—
117C22×C23—81no—
118C22×C23—81no—
119C22×C23#40711yescharacteristic
120C22×C23C2×#9911yescharacteristic
121C22×C23#41311yescharacteristic
122C4×C23—22no—
123C4×C23—21no—
124C4×C23—21no—
125C4×C23—42no—
126C4×C23—42no—
127C4×C23—41no—
128C4×C23—42no—
129C4×C23—81no—
130C4×C23—81no—
131C2×#124C4×C2311yescharacteristic
132C2×C4×Q8C22×C2311yescharacteristic
133C2×C4×D4C4×C2311yescharacteristic
134#432—22no—
135#391C4×C2311yescharacteristic
136#390C22×C2311yescharacteristic
137#388—22no—
138#385—22no—
139#384C22×C2311yescharacteristic
140#376C4×C2311yescharacteristic
141#324—22no—
142#324C22×C2314yes—
143C2×C23—21no—
144C2×C23—21no—
145C2×C23—82no—
146C2×C23#146511yescharacteristic
147C2×C23#212811yescharacteristic
148C2×C23#213311yescharacteristic
149C22×Q8C2×C4×C2311yescharacteristic
150C22×D4C2×C4×C2311yescharacteristic
151C23×C4—21no—
152#124—22no—
153#124—22no—
154C4×Q8—42no—
155C4×D4—42no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4C2×C4×C2311yescharacteristic
158C2×C4⋊C4D4×C2312yes—
159C2×C4⋊C4C23×C2311yescharacteristic
160C2×C22⋊C4—22no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—22no—
163C2×C42C2×C4×C2311yescharacteristic
164C2×C42C2×C4×C2311yescharacteristic
165C2×C42C2×C4×C2311yescharacteristic
166C2×C42D4×C2312yes—
167C4⋊C8—42no—
168C22⋊C8—24no—
169#97—42no—
170#95—21no—
171#95—21no—
172#95—22no—
173C23#1153411yescharacteristic, direct product factor, Sylow
174C24—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×D4—21no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4M16×C2312yes—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—42no—
187C22×C4—82no—
188C22×C4C22⋊C4×C2312yes—
189C22×C4C22⋊C4×C2311yescharacteristic
190C22×C4C22⋊C4×C2311yescharacteristic
191C22×C4C22×C4×C2311yescharacteristic
192C22×C4C2×D4×C2312yes—
193C2×C8—44no—
194C4⋊C4—42no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—82no—
198C22⋊C4—28no—
199C22⋊C4—42no—
200C22⋊C4—82no—
201C42—22no—
202C42—22no—
203C42—22no—
204C42—21no—
205C42—41no—
206C42M16×C2312yes—
207C23—22no—
208C23—42no—
209C23—42no—
210C23—41no—
211C23—41no—
212C23C2×C22⋊C4×C2311yescharacteristic
213Q8—42no—
214Q8—42no—
215D4—42no—
216D4—42no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—84no—
236C2×C4C23×#9912yes—
237C2×C4C2×M16×C2311yescharacteristic
238C2×C4C2×M16×C2311yescharacteristic
239C8—84no—
240C22—21no—
241C22—21no—
242C22—44no—
243C22—42no—
244C22—82no—
245C22—81no—
246C22—81no—
247C22C23×#40711yescharacteristic
248C22C2×C23×#9911yescharacteristic
249C22C23×#41311yescharacteristic
250C4—21no—
251C4—21no—
252C4—22no—
253C4—42no—
254C4—42no—
255C4—42no—
256C4—41no—
257C4—81no—
258C4—81no—
259C2—21no—
260C2—21no—
261C2—82no—
262C2C23×#146511yescharacteristic
263C2C23×#212811yescharacteristic
264C2C23×#213311yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C4⋊C4C23×C23
prime radicalC22×D4×C23C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C2×C4×Q8×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1467C2semidirect
3. C23×#1325C2semidirect
74. #11534C23direct (N × H)
109. #1467C2×C23semidirect
110. #1325C2×C23semidirect
173. C23#11534direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#23262
C23×#14672
C23×#13252
C23×#13182
#1153423