Finite Group Databasebeta

← C23×#9026 #19156652

Subgroups — C23×#9026 #19156652

Order 5888.

616 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C2×C23×#398C211yesmaximal, characteristic
2C23×#2313C211yesmaximal, characteristic
3C23×#1823C212yesmaximal
4C2×C23×#324C211yesmaximal, characteristic
5C23×#1294C212yesmaximal
6C22×C4⋊C4×C23—21no—
7C22×C4⋊C4×C23C2211yescharacteristic
8C22×C4⋊C4×C23C2211yescharacteristic
9C2×C23×C22⋊C8C2212yes—
10C23×#398—22no—
11C23×#398C414yescomplemented
12C23×#387—24no—
13C23×#381—21no—
14C23×#380C2212yes—
15C4×C22⋊C4×C23—21no—
16C2×C23×#95—21no—
17C2×C23×#95C2211yescharacteristic
18C23×#340—24no—
19C23×#324—22no—
20C23×C4×C23—21no—
21C23×C4×C23—21no—
22C23×C4×C23D411yescharacteristic
23C23×C4×C23D411yescharacteristic
24C23×C4×C23C2311yescharacteristic
25C22×C8×C23—22no—
26C2×C4⋊C4×C23—22no—
27C2×C4⋊C4×C23—22no—
28C2×C4⋊C4×C23—22no—
29C2×C4⋊C4×C23—22no—
30C2×C4⋊C4×C23—24no—
31C2×C4⋊C4×C23—21no—
32C2×C4⋊C4×C23—44no—
33C2×C4⋊C4×C23C2×C412yes—
34C2×C4⋊C4×C23C2×C412yes—
35C2×C4⋊C4×C23D412yes—
36C2×C4⋊C4×C23D414yes—
37C2×C22⋊C4×C23—22no—
38C2×C42×C23—22no—
39C23×C22⋊C8—22no—
40C23×C22⋊C8—22no—
41C23×#95—22no—
42C23×#95—24no—
43C23×#95—24no—
44C23×#95—21no—
45C23×#95—22no—
46C23×#95C2×C412yes—
47C24×C23C2×D411yescharacteristic
48C22×C4×C23—24no—
49C22×C4×C23—22no—
50C22×C4×C23—21no—
51C22×C4×C23—21no—
52C22×C4×C23—21no—
53C22×C4×C23—22no—
54C22×C4×C23—22no—
55C22×C4×C23—21no—
56C22×C4×C23—22no—
57C22×C4×C23—24no—
58C22×C4×C23—42no—
59C22×C4×C23—42no—
60C22×C4×C23—42no—
61C22×C4×C23—42no—
62C22×C4×C23—41no—
63C22×C4×C23—41no—
64C22×C4×C23—42no—
65C22×C4×C23—42no—
66C22×C4×C23C22⋊C412yes—
67C22×C4×C23C22⋊C412yes—
68C22×C4×C23C22×C411yescharacteristic
69C22×C4×C23C2×D412yes—
70C22×C4×C23C2×D411yescharacteristic
71C22×C4×C23#4112yes—
72C2×C8×C23—44no—
73C2×C8×C23—42no—
74C4⋊C4×C23—24no—
75C4⋊C4×C23—24no—
76C4⋊C4×C23—24no—
77C4⋊C4×C23—44no—
78C4⋊C4×C23—48no—
79C4⋊C4×C23—42no—
80C22⋊C4×C23—42no—
81C22⋊C4×C23—42no—
82C42×C23—82no—
83#9026C2311yesmaximal, characteristic, direct product factor, Sylow
84C23×C23—22no—
85C23×C23—22no—
86C23×C23C2×C22⋊C411yescharacteristic
87C23×C23#12011yescharacteristic
88C23×C23#12311yescharacteristic
89C23×C23#13512yes—
90C23×C23AGL(1,ℤ/8)12yes—
91C2×C4×C23—24no—
92C2×C4×C23—24no—
93C2×C4×C23—22no—
94C2×C4×C23—22no—
95C2×C4×C23—22no—
96C2×C4×C23—22no—
97C2×C4×C23—41no—
98C2×C4×C23—41no—
99C2×C4×C23—42no—
100C2×C4×C23—48no—
101C2×C4×C23—42no—
102C2×C4×C23—42no—
103C2×C4×C23—42no—
104C2×C4×C23—42no—
105C2×C4×C23—42no—
106C2×C4×C23—42no—
107C2×C4×C23—42no—
108C2×C4×C23—42no—
109C2×C4×C23—42no—
110C2×C4×C23—44no—
111C2×C4×C23—44no—
112C2×C4×C23—44no—
113C2×C4×C23—81no—
114C2×C4×C23—82no—
115C2×C4×C23—82no—
116C2×C4×C23C4×D412yes—
117C2×C4×C23#12112yes—
118C8×C23—82no—
119C2×#398C2×C2311yescharacteristic
120#2313C2×C2311yescharacteristic
121#1823C2×C2312yes—
122C2×#324C2×C2311yescharacteristic
123#1294C2×C2312yes—
124C22×C23—24no—
125C22×C23—24no—
126C22×C23—21no—
127C22×C23—21no—
128C22×C23—21no—
129C22×C23—21no—
130C22×C23#38611yescharacteristic
131C22×C23#41611yescharacteristic
132C22×C23#41811yescharacteristic
133C22×C23#44912yes—
134C22×C23#45612yes—
135C22×C23#45812yes—
136C22×C23#48212yes—
137C4×C23—22no—
138C4×C23—22no—
139C4×C23—44no—
140C4×C23—42no—
141C4×C23—42no—
142C4×C23—84no—
143C4×C23—82no—
144C22×C4⋊C4—21no—
145C22×C4⋊C4C22×C2311yescharacteristic
146C22×C4⋊C4C22×C2311yescharacteristic
147C2×C22⋊C8C22×C2312yes—
148#398—22no—
149#398C4×C2314yescomplemented
150#387—24no—
151#381—21no—
152#380C22×C2312yes—
153C4×C22⋊C4—21no—
154C2×#95—21no—
155C2×#95C22×C2311yescharacteristic
156#340—24no—
157#324—22no—
158C2×C23—21no—
159C2×C23—21no—
160C2×C23—22no—
161C2×C23#160914yes—
162C2×C23#187511yescharacteristic
163C2×C23#188611yescharacteristic
164C2×C23#189011yescharacteristic
165C23×C4—21no—
166C23×C4—21no—
167C23×C4D4×C2311yescharacteristic
168C23×C4D4×C2311yescharacteristic
169C23×C4C23×C2311yescharacteristic
170C22×C8—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4C2×C4×C2312yes—
179C2×C4⋊C4C2×C4×C2312yes—
180C2×C4⋊C4D4×C2312yes—
181C2×C4⋊C4D4×C2314yes—
182C2×C22⋊C4—22no—
183C2×C42—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186#95—22no—
187#95—24no—
188#95—24no—
189#95—21no—
190#95—22no—
191#95C2×C4×C2312yes—
192C23#902611yescharacteristic, direct product factor, Sylow
193C24C2×D4×C2311yescharacteristic
194C22×C4—24no—
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—24no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4C22⋊C4×C2312yes—
213C22×C4C22⋊C4×C2312yes—
214C22×C4C22×C4×C2311yescharacteristic
215C22×C4C2×D4×C2312yes—
216C22×C4C2×D4×C2311yescharacteristic
217C22×C4#41×C2312yes—
218C2×C8—44no—
219C2×C8—42no—
220C4⋊C4—24no—
221C4⋊C4—24no—
222C4⋊C4—24no—
223C4⋊C4—44no—
224C4⋊C4—48no—
225C4⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C42—82no—
229C23—22no—
230C23—22no—
231C23C2×C22⋊C4×C2311yescharacteristic
232C23C23×#12011yescharacteristic
233C23C23×#12311yescharacteristic
234C23C23×#13512yes—
235C23C23×AGL(1,ℤ/8)12yes—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—48no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—44no—
256C2×C4—44no—
257C2×C4—44no—
258C2×C4—81no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4C4×D4×C2312yes—
262C2×C4C23×#12112yes—
263C8—82no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—24no—
269C22—24no—
270C22C23×#38611yescharacteristic
271C22C23×#41611yescharacteristic
272C22C23×#41811yescharacteristic
273C22C23×#44912yes—
274C22C23×#45612yes—
275C22C23×#45812yes—
276C22C23×#48212yes—
277C4—22no—
278C4—22no—
279C4—44no—
280C4—42no—
281C4—42no—
282C4—84no—
283C4—82no—
284C2—21no—
285C2—21no—
286C2—22no—
287C2C23×#160914yes—
288C2C23×#187511yescharacteristic
289C2C23×#188611yescharacteristic
290C2C23×#189011yescharacteristic
subgroupG/H
center, socleC23×C23#120
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC24×C23C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
11. C23×#398C4semidirect
83. #9026C23direct (N × H)
149. #398C4×C23semidirect
192. C23#9026direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#3982
C23×#23132
C23×#18232
C2×C23×#3242
C23×#12942
#902623