Finite Group Databasebeta

← C23×#9460 #19157086

Subgroups — C23×#9460 #19157086

Order 5888.

512 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 122 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×#2503C211yesmaximal, characteristic
2C23×#2313C211yesmaximal, characteristic
3C23×#1821C211yesmaximal, characteristic
4C23×#1511C212yesmaximal
5C23×#1284C212yesmaximal
6C2×C23×C22⋊Q8C2211yescharacteristic
7C22×C4⋊C4×C23C2211yescharacteristic
8C23×#407C2212yes—
9C23×#400—22no—
10C23×#391C412yescomplemented
11C23×#387—24no—
12C23×#382C412yescomplemented
13C23×#381—21no—
14C23×#381—21no—
15C23×#381C2212yes—
16C23×#380—21no—
17C4×C22⋊C4×C23—21no—
18C4×C22⋊C4×C23C2211yescharacteristic
19C23×#343—22no—
20C23×#328—24no—
21C22×Q8×C23C2×C411yescharacteristic
22C23×C4×C23D411yescharacteristic
23C23×C4×C23D411yescharacteristic
24C23×C4×C23C2311yescharacteristic
25C2×M16×C23—22no—
26C23×C22⋊Q8—22no—
27C23×C22⋊Q8—22no—
28C2×C4⋊C4×C23—22no—
29C2×C4⋊C4×C23—22no—
30C2×C4⋊C4×C23—21no—
31C2×C4⋊C4×C23—22no—
32C2×C4⋊C4×C23C2×C411yescharacteristic
33C2×C4⋊C4×C23C2×C412yes—
34C2×C4⋊C4×C23D412yes—
35C2×C22⋊C4×C23—22no—
36C2×C22⋊C4×C23—21no—
37C2×C22⋊C4×C23—21no—
38C2×C42×C23—22no—
39C2×C42×C23C2×C412yes—
40C23×C22⋊C8—22no—
41C23×C22⋊C8—22no—
42C23×#95—22no—
43C23×#95—22no—
44C23×#95—22no—
45C23×#95—22no—
46C23×#95—22no—
47C23×#95—22no—
48C23×#95—21no—
49C23×#95—22no—
50C23×#95D414yes—
51C24×C23C2×D411yescharacteristic
52C2×Q8×C23—22no—
53C2×Q8×C23—42no—
54C22×C4×C23—21no—
55C22×C4×C23—21no—
56C22×C4×C23—21no—
57C22×C4×C23—21no—
58C22×C4×C23—22no—
59C22×C4×C23—21no—
60C22×C4×C23—21no—
61C22×C4×C23—24no—
62C22×C4×C23—42no—
63C22×C4×C23—42no—
64C22×C4×C23—42no—
65C22×C4×C23C22⋊C412yes—
66C22×C4×C23C22⋊C412yes—
67C22×C4×C23C22×C411yescharacteristic
68C22×C4×C23C2×D411yescharacteristic
69C22×C4×C23C2×D412yes—
70C22×C4×C23#4112yes—
71M16×C23—44no—
72C2×C8×C23—42no—
73C4⋊C4×C23—24no—
74C4⋊C4×C23—24no—
75C4⋊C4×C23—22no—
76C4⋊C4×C23—42no—
77C4⋊C4×C23—82no—
78C22⋊C4×C23—44no—
79C22⋊C4×C23—42no—
80C22⋊C4×C23—41no—
81C22⋊C4×C23—41no—
82C42×C23—42no—
83C42×C23—82no—
84#9460C2311yesmaximal, characteristic, direct product factor, Sylow
85C23×C23—22no—
86C23×C23—22no—
87C23×C23—41no—
88C23×C23C2×C22⋊C411yescharacteristic
89C23×C23#12011yescharacteristic
90C23×C23#12311yescharacteristic
91Q8×C23—82no—
92C2×C4×C23—22no—
93C2×C4×C23—22no—
94C2×C4×C23—22no—
95C2×C4×C23—22no—
96C2×C4×C23—42no—
97C2×C4×C23—42no—
98C2×C4×C23—42no—
99C2×C4×C23—42no—
100C2×C4×C23—42no—
101C2×C4×C23—42no—
102C2×C4×C23—42no—
103C2×C4×C23—44no—
104C2×C4×C23—42no—
105C2×C4×C23—44no—
106C2×C4×C23—42no—
107C2×C4×C23—41no—
108C2×C4×C23—41no—
109C2×C4×C23—42no—
110C2×C4×C23—42no—
111C2×C4×C23—44no—
112C2×C4×C23—81no—
113C2×C4×C23—81no—
114C2×C4×C23C4×D412yes—
115C2×C4×C23#12112yes—
116C8×C23—82no—
117#2503C2×C2311yescharacteristic
118#2313C2×C2311yescharacteristic
119#1821C2×C2311yescharacteristic
120#1511C2×C2312yes—
121#1284C2×C2312yes—
122C22×C23—21no—
123C22×C23—21no—
124C22×C23—21no—
125C22×C23—21no—
126C22×C23—21no—
127C22×C23—21no—
128C22×C23—21no—
129C22×C23—44no—
130C22×C23#38611yescharacteristic
131C22×C23#41012yes—
132C22×C23#45512yes—
133C4×C23—22no—
134C4×C23—22no—
135C4×C23—42no—
136C4×C23—42no—
137C4×C23—84no—
138C4×C23—82no—
139C4×C23—81no—
140C4×C23—81no—
141C2×C22⋊Q8C22×C2311yescharacteristic
142C22×C4⋊C4C22×C2311yescharacteristic
143#407C22×C2312yes—
144#400—22no—
145#391C4×C2312yescomplemented
146#387—24no—
147#382C4×C2312yescomplemented
148#381—21no—
149#381—21no—
150#381C22×C2312yes—
151#380—21no—
152C4×C22⋊C4—21no—
153C4×C22⋊C4C22×C2311yescharacteristic
154#343—22no—
155#328—24no—
156C2×C23—22no—
157C2×C23—22no—
158C2×C23—41no—
159C2×C23#188511yescharacteristic
160C2×C23#189911yescharacteristic
161C2×C23#191211yescharacteristic
162C22×Q8C2×C4×C2311yescharacteristic
163C23×C4D4×C2311yescharacteristic
164C23×C4D4×C2311yescharacteristic
165C23×C4C23×C2311yescharacteristic
166C2×M16—22no—
167C22⋊Q8—22no—
168C22⋊Q8—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4C2×C4×C2311yescharacteristic
174C2×C4⋊C4C2×C4×C2312yes—
175C2×C4⋊C4D4×C2312yes—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—21no—
179C2×C42—22no—
180C2×C42C2×C4×C2312yes—
181C22⋊C8—22no—
182C22⋊C8—22no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—22no—
189#95—21no—
190#95—22no—
191#95D4×C2314yes—
192C23#946011yescharacteristic, direct product factor, Sylow
193C24C2×D4×C2311yescharacteristic
194C2×Q8—22no—
195C2×Q8—42no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—24no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4C22⋊C4×C2312yes—
208C22×C4C22⋊C4×C2312yes—
209C22×C4C22×C4×C2311yescharacteristic
210C22×C4C2×D4×C2311yescharacteristic
211C22×C4C2×D4×C2312yes—
212C22×C4#41×C2312yes—
213M16—44no—
214C2×C8—42no—
215C4⋊C4—24no—
216C4⋊C4—24no—
217C4⋊C4—22no—
218C4⋊C4—42no—
219C4⋊C4—82no—
220C22⋊C4—44no—
221C22⋊C4—42no—
222C22⋊C4—41no—
223C22⋊C4—41no—
224C42—42no—
225C42—82no—
226C23—22no—
227C23—22no—
228C23—41no—
229C23C2×C22⋊C4×C2311yescharacteristic
230C23C23×#12011yescharacteristic
231C23C23×#12311yescharacteristic
232Q8—82no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—44no—
245C2×C4—42no—
246C2×C4—44no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—44no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4C4×D4×C2312yes—
256C2×C4C23×#12112yes—
257C8—82no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—44no—
266C22C23×#38611yescharacteristic
267C22C23×#41012yes—
268C22C23×#45512yes—
269C4—22no—
270C4—22no—
271C4—42no—
272C4—42no—
273C4—84no—
274C4—82no—
275C4—81no—
276C4—81no—
277C2—22no—
278C2—22no—
279C2—41no—
280C2C23×#188511yescharacteristic
281C2C23×#189911yescharacteristic
282C2C23×#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×#391C4semidirect
12. C23×#382C4semidirect
84. #9460C23direct (N × H)
145. #391C4×C23semidirect
147. #382C4×C23semidirect
192. C23#9460direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#25032
C23×#23132
C23×#18212
C23×#15112
C23×#12842
#946023