Finite Group Databasebeta

← C23×#9478 #19157104

Subgroups — C23×#9478 #19157104

Order 5888.

610 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 130 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×#2484C211yesmaximal, characteristic
2C23×#2274C211yesmaximal, characteristic
3C23×#1826C212yesmaximal
4C23×#1463C211yesmaximal, characteristic
5C23×#1284C212yesmaximal
6C2×C23×#117—21no—
7C2×C23×#117C2211yescharacteristic
8C22×C4⋊C4×C23C2211yescharacteristic
9C23×#407C2212yes—
10C23×#400C412yescomplemented
11C23×#398C412yescomplemented
12C23×#387—22no—
13C23×#385—21no—
14C23×#381—21no—
15C23×#381C2211yescharacteristic
16C4×C4⋊C4×C23—24no—
17C4×C22⋊C4×C23—21no—
18C4×C22⋊C4×C23C2212yes—
19C23×#339—24no—
20C23×#324—22no—
21C23×C4×C23D411yescharacteristic
22C23×C4×C23D411yescharacteristic
23C23×C4×C23C2311yescharacteristic
24C2×M16×C23—22no—
25C23×#117—28no—
26C23×#117—22no—
27C23×#117—22no—
28C2×C4⋊C4×C23—22no—
29C2×C4⋊C4×C23—22no—
30C2×C4⋊C4×C23—22no—
31C2×C4⋊C4×C23—22no—
32C2×C4⋊C4×C23C2×C411yescharacteristic
33C2×C4⋊C4×C23C2×C411yescharacteristic
34C2×C4⋊C4×C23C2×C412yes—
35C2×C4⋊C4×C23D412yes—
36C2×C22⋊C4×C23—22no—
37C2×C22⋊C4×C23—21no—
38C2×C22⋊C4×C23—21no—
39C2×C42×C23—22no—
40C2×C42×C23—21no—
41C2×C42×C23—22no—
42C2×C42×C23D414yes—
43C23×C22⋊C8—22no—
44C23×C22⋊C8—22no—
45C23×#95—22no—
46C23×#95—22no—
47C23×#95—22no—
48C23×#95—22no—
49C23×#95—22no—
50C23×#95—21no—
51C23×#95C2×C412yes—
52C24×C23C2×D411yescharacteristic
53C22×C4×C23—21no—
54C22×C4×C23—21no—
55C22×C4×C23—24no—
56C22×C4×C23—24no—
57C22×C4×C23—21no—
58C22×C4×C23—21no—
59C22×C4×C23—21no—
60C22×C4×C23—21no—
61C22×C4×C23—22no—
62C22×C4×C23—42no—
63C22×C4×C23—42no—
64C22×C4×C23C22⋊C412yes—
65C22×C4×C23C22⋊C412yes—
66C22×C4×C23C22×C411yescharacteristic
67C22×C4×C23C2×D412yes—
68C22×C4×C23C2×D411yescharacteristic
69C22×C4×C23#4112yes—
70M16×C23—44no—
71C2×C8×C23—42no—
72C4⋊C4×C23—28no—
73C4⋊C4×C23—24no—
74C4⋊C4×C23—24no—
75C4⋊C4×C23—22no—
76C4⋊C4×C23—22no—
77C4⋊C4×C23—44no—
78C22⋊C4×C23—28no—
79C22⋊C4×C23—42no—
80C22⋊C4×C23—41no—
81C22⋊C4×C23—41no—
82C42×C23—28no—
83C42×C23—44no—
84C42×C23—42no—
85C42×C23—82no—
86#9478C2311yesmaximal, characteristic, direct product factor, Sylow
87C23×C23—22no—
88C23×C23—22no—
89C23×C23—41no—
90C23×C23C2×C22⋊C411yescharacteristic
91C23×C23#12011yescharacteristic
92C23×C23#12311yescharacteristic
93C2×C4×C23—22no—
94C2×C4×C23—22no—
95C2×C4×C23—24no—
96C2×C4×C23—22no—
97C2×C4×C23—22no—
98C2×C4×C23—24no—
99C2×C4×C23—28no—
100C2×C4×C23—22no—
101C2×C4×C23—41no—
102C2×C4×C23—41no—
103C2×C4×C23—42no—
104C2×C4×C23—44no—
105C2×C4×C23—44no—
106C2×C4×C23—42no—
107C2×C4×C23—42no—
108C2×C4×C23—42no—
109C2×C4×C23—42no—
110C2×C4×C23—42no—
111C2×C4×C23—41no—
112C2×C4×C23—41no—
113C2×C4×C23—42no—
114C2×C4×C23—42no—
115C2×C4×C23—81no—
116C2×C4×C23#10414yes—
117C2×C4×C23C4×D412yes—
118C2×C4×C23#12112yes—
119C8×C23—82no—
120#2484C2×C2311yescharacteristic
121#2274C2×C2311yescharacteristic
122#1826C2×C2312yes—
123#1463C2×C2311yescharacteristic
124#1284C2×C2312yes—
125C22×C23—21no—
126C22×C23—21no—
127C22×C23—21no—
128C22×C23—21no—
129C22×C23—21no—
130C22×C23—21no—
131C22×C23—21no—
132C22×C23—44no—
133C22×C23#38611yescharacteristic
134C22×C23C2×#10412yes—
135C22×C23#45812yes—
136C4×C23—22no—
137C4×C23—22no—
138C4×C23—24no—
139C4×C23—48no—
140C4×C23—42no—
141C4×C23—82no—
142C4×C23—81no—
143C4×C23—81no—
144C2×#117—21no—
145C2×#117C22×C2311yescharacteristic
146C22×C4⋊C4C22×C2311yescharacteristic
147#407C22×C2312yes—
148#400C4×C2312yescomplemented
149#398C4×C2312yescomplemented
150#387—22no—
151#385—21no—
152#381—21no—
153#381C22×C2311yescharacteristic
154C4×C4⋊C4—24no—
155C4×C22⋊C4—21no—
156C4×C22⋊C4C22×C2312yes—
157#339—24no—
158#324—22no—
159C2×C23—22no—
160C2×C23—22no—
161C2×C23—41no—
162C2×C23#189011yescharacteristic
163C2×C23#189611yescharacteristic
164C2×C23#191211yescharacteristic
165C23×C4D4×C2311yescharacteristic
166C23×C4D4×C2311yescharacteristic
167C23×C4C23×C2311yescharacteristic
168C2×M16—22no—
169#117—28no—
170#117—22no—
171#117—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4C2×C4×C2311yescharacteristic
177C2×C4⋊C4C2×C4×C2311yescharacteristic
178C2×C4⋊C4C2×C4×C2312yes—
179C2×C4⋊C4D4×C2312yes—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C42—22no—
184C2×C42—21no—
185C2×C42—22no—
186C2×C42D4×C2314yes—
187C22⋊C8—22no—
188C22⋊C8—22no—
189#95—22no—
190#95—22no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—21no—
195#95C2×C4×C2312yes—
196C23#947811yescharacteristic, direct product factor, Sylow
197C24C2×D4×C2311yescharacteristic
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4C22⋊C4×C2312yes—
210C22×C4C22⋊C4×C2312yes—
211C22×C4C22×C4×C2311yescharacteristic
212C22×C4C2×D4×C2312yes—
213C22×C4C2×D4×C2311yescharacteristic
214C22×C4#41×C2312yes—
215M16—44no—
216C2×C8—42no—
217C4⋊C4—28no—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—44no—
223C22⋊C4—28no—
224C22⋊C4—42no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C42—28no—
228C42—44no—
229C42—42no—
230C42—82no—
231C23—22no—
232C23—22no—
233C23—41no—
234C23C2×C22⋊C4×C2311yescharacteristic
235C23C23×#12011yescharacteristic
236C23C23×#12311yescharacteristic
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—24no—
243C2×C4—28no—
244C2×C4—22no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—81no—
260C2×C4C23×#10414yes—
261C2×C4C4×D4×C2312yes—
262C2×C4C23×#12112yes—
263C8—82no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22C23×#38611yescharacteristic
273C22C2×C23×#10412yes—
274C22C23×#45812yes—
275C4—22no—
276C4—22no—
277C4—24no—
278C4—48no—
279C4—42no—
280C4—82no—
281C4—81no—
282C4—81no—
283C2—22no—
284C2—22no—
285C2—41no—
286C2C23×#189011yescharacteristic
287C2C23×#189611yescharacteristic
288C2C23×#191211yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC24×C23C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
10. C23×#400C4semidirect
11. C23×#398C4semidirect
86. #9478C23direct (N × H)
148. #400C4×C23semidirect
149. #398C4×C23semidirect
196. C23#9478direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#24842
C23×#22742
C23×#18262
C23×#14632
C23×#12842
#947823