Finite Group Databasebeta

← C23×#32881 #19180507

Subgroups — C23×#32881 #19180507

Order 5888.

5942 conjugacy classes of subgroups; only the 438 normal subgroups are shown.

normal subgroups only

This group has 5942 conjugacy classes of subgroups — too many to draw. Only its 438 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
1C24×D4×C23C211yesmaximal, characteristic
2C2×C23×#450C216yesmaximal, complemented
3C23×#2848C211yesmaximal, characteristic, complemented
4C23×#1786C216yesmaximal, complemented
5C23×#1779C211yesmaximal, characteristic, complemented
6C26×C23—21no—
7C23×D4×C23—224no—
8C23×D4×C23—23no—
9C23×D4×C23C2216yes—
10C24×C4×C23C2211yescharacteristic
11C22×SD16×C23C2214yescomplemented
12C2×C23×C22⋊Q8C2213yescomplemented
13C23×#450—224no—
14C2×C23×#102C22112yescomplemented
15C2×C23×C22⋊C8C2216yescomplemented
16C23×#385—23no—
17C23×#380C2213yescomplemented
18C23×#379—21no—
19C25×C23—224no—
20C25×C23—26no—
21C25×C23—412no—
22C25×C23—44no—
23C25×C23D411yescharacteristic
24C22×Q8×C23C2311yescharacteristic, complemented
25C22×D4×C23—2128no—
26C22×D4×C23—224no—
27C22×D4×C23—248no—
28C22×D4×C23—248no—
29C22×D4×C23—26no—
30C22×D4×C23—212no—
31C22×D4×C23D4124yes—
32C22×D4×C23C2314yes—
33C23×C4×C23—212no—
34C23×C4×C23D413yes—
35C23×C4×C23C2313yes—
36C2×SD16×C23—224no—
37C22×C8×C23C2314yescomplemented
38C23×C22⋊Q8—212no—
39C2×C4⋊C4×C23—23no—
40C2×C4⋊C4×C23C2313yescomplemented
41C2×C22⋊C4×C23—23no—
42C2×C22⋊C4×C23—23no—
43C23×#102—224no—
44C23×C22⋊C8—212no—
45C23×#95—23no—
46C24×C23—26no—
47C24×C23—26no—
48C24×C23—264no—
49C24×C23—248no—
50C24×C23—224no—
51C24×C23—24no—
52C24×C23—496no—
53C24×C23—424no—
54C24×C23—496no—
55C24×C23—412no—
56C24×C23—412no—
57C24×C23SD1614yescomplemented
58C24×C23C2×D413yes—
59C2×Q8×C23—26no—
60C2×D4×C23—2192no—
61C2×D4×C23—248no—
62C2×D4×C23—296no—
63C2×D4×C23—296no—
64C2×D4×C23—224no—
65C2×D4×C23—212no—
66C2×D4×C23C2×D4124yes—
67C22×C4×C23—232no—
68C22×C4×C23—26no—
69C22×C4×C23—212no—
70C22×C4×C23—212no—
71C22×C4×C23—21no—
72C22×C4×C23—23no—
73C22×C4×C23C2×D413yes—
74C22×C4×C23C2×D4112yes—
75C22×C4×C23C2411yescharacteristic
76SD16×C23—416no—
77C2×C8×C23—212no—
78C4⋊C4×C23—26no—
79C4⋊C4×C23—46no—
80C22⋊C4×C23—46no—
81C22⋊C4×C23—46no—
82#32881C2311yesmaximal, characteristic, direct product factor, Sylow
83C23×C23—28no—
84C23×C23—224no—
85C23×C23—26no—
86C23×C23—26no—
87C23×C23—224no—
88C23×C23—296no—
89C23×C23—224no—
90C23×C23—4256no—
91C23×C23—496no—
92C23×C23—448no—
93C23×C23—496no—
94C23×C23—44no—
95C23×C23—412no—
96C23×C23—448no—
97C23×C23#12016yes—
98C23×C23C2×SD1616yes—
99C23×C23AGL(1,ℤ/8)16yes—
100C23×C23C22×D411yescharacteristic
101Q8×C23—44no—
102D4×C23—296no—
103D4×C23—224no—
104D4×C23#120116yes—
105C2×C4×C23—224no—
106C2×C4×C23—26no—
107C2×C4×C23—224no—
108C2×C4×C23—43no—
109C2×C4×C23—43no—
110C2×C4×C23—46no—
111C2×C4×C23#12016yes—
112C2×C4×C23C22×D416yes—
113C8×C23—44no—
114C24×D4C2×C2311yescharacteristic
115C2×#450C2×C2316yescomplemented
116#2848C2×C2311yescharacteristic, complemented
117#1786C2×C2316yescomplemented
118#1779C2×C2311yescharacteristic, complemented
119C22×C23—232no—
120C22×C23—26no—
121C22×C23—212no—
122C22×C23—26no—
123C22×C23—212no—
124C22×C23—216no—
125C22×C23—4192no—
126C22×C23—424no—
127C22×C23—424no—
128C22×C23#450112yescomplemented
129C22×C23C2×#12013yes—
130C22×C23C22×SD1611yescharacteristic
131C22×C23C2×AGL(1,ℤ/8)13yes—
132C4×C23—26no—
133C4×C23—81no—
134C4×C23—83no—
135C4×C23C2×#12014yes—
136C26—21no—
137C23×D4—224no—
138C23×D4—23no—
139C23×D4C22×C2316yes—
140C24×C4C22×C2311yescharacteristic
141C22×SD16C22×C2314yescomplemented
142C2×C22⋊Q8C22×C2313yescomplemented
143#450—224no—
144C2×#102C22×C23112yescomplemented
145C2×C22⋊C8C22×C2316yescomplemented
146#385—23no—
147#380C22×C2313yescomplemented
148#379—21no—
149C2×C23—212no—
150C2×C23—416no—
151C2×C23#284611yescharacteristic
152C2×C23C2×#45013yes—
153C2×C23#300313yes—
154C25—224no—
155C25—26no—
156C25—412no—
157C25—44no—
158C25D4×C2311yescharacteristic
159C22×Q8C23×C2311yescharacteristic, complemented
160C22×D4—2128no—
161C22×D4—224no—
162C22×D4—248no—
163C22×D4—248no—
164C22×D4—26no—
165C22×D4—212no—
166C22×D4D4×C23124yes—
167C22×D4C23×C2314yes—
168C23×C4—212no—
169C23×C4D4×C2313yes—
170C23×C4C23×C2313yes—
171C2×SD16—224no—
172C22×C8C23×C2314yescomplemented
173C22⋊Q8—212no—
174C2×C4⋊C4—23no—
175C2×C4⋊C4C23×C2313yescomplemented
176C2×C22⋊C4—23no—
177C2×C22⋊C4—23no—
178#102—224no—
179C22⋊C8—212no—
180#95—23no—
181C23#3288111yescharacteristic, direct product factor, Sylow
182C24—26no—
183C24—26no—
184C24—264no—
185C24—248no—
186C24—224no—
187C24—24no—
188C24—496no—
189C24—424no—
190C24—496no—
191C24—412no—
192C24—412no—
193C24SD16×C2314yescomplemented
194C24C2×D4×C2313yes—
195C2×Q8—26no—
196C2×D4—2192no—
197C2×D4—248no—
198C2×D4—296no—
199C2×D4—296no—
200C2×D4—224no—
201C2×D4—212no—
202C2×D4C2×D4×C23124yes—
203C22×C4—232no—
204C22×C4—26no—
205C22×C4—212no—
206C22×C4—212no—
207C22×C4—23no—
208C22×C4—21no—
209C22×C4C2×D4×C2313yes—
210C22×C4C2×D4×C23112yes—
211C22×C4C24×C2311yescharacteristic
212SD16—416no—
213C2×C8—212no—
214C4⋊C4—26no—
215C4⋊C4—46no—
216C22⋊C4—46no—
217C22⋊C4—46no—
218C23—28no—
219C23—224no—
220C23—26no—
221C23—224no—
222C23—26no—
223C23—296no—
224C23—224no—
225C23—4256no—
226C23—496no—
227C23—448no—
228C23—496no—
229C23—44no—
230C23—412no—
231C23—448no—
232C23C23×#12016yes—
233C23C2×SD16×C2316yes—
234C23C23×AGL(1,ℤ/8)16yes—
235C23C22×D4×C2311yescharacteristic
236Q8—44no—
237D4—296no—
238D4—224no—
239D4C23×#120116yes—
240C2×C4—224no—
241C2×C4—26no—
242C2×C4—224no—
243C2×C4—46no—
244C2×C4—43no—
245C2×C4—43no—
246C2×C4C23×#12016yes—
247C2×C4C22×D4×C2316yes—
248C8—44no—
249C22—232no—
250C22—212no—
251C22—26no—
252C22—212no—
253C22—26no—
254C22—216no—
255C22—4192no—
256C22—424no—
257C22—424no—
258C22C23×#450112yescomplemented
259C22C2×C23×#12013yes—
260C22C22×SD16×C2311yescharacteristic
261C22C2×C23×AGL(1,ℤ/8)13yes—
262C4—26no—
263C4—83no—
264C4—81no—
265C4C2×C23×#12014yes—
266C2—212no—
267C2—416no—
268C2C23×#284611yescharacteristic
269C2C2×C23×#45013yes—
270C2C23×#300313yes—
subgroupG/H
center, socleC23×C23C22×D4
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC24×D4×C23C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C23 ◁ C24×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C23×#450C2semidirect
3. C23×#2848C2semidirect
4. C23×#1786C2semidirect
5. C23×#1779C2semidirect
11. C22×SD16×C23C22semidirect
12. C2×C23×C22⋊Q8C22semidirect
14. C2×C23×#102C22semidirect
15. C2×C23×C22⋊C8C22semidirect
17. C23×#380C22semidirect
24. C22×Q8×C23C23semidirect
37. C22×C8×C23C23semidirect
40. C2×C4⋊C4×C23C23semidirect
57. C24×C23SD16semidirect
82. #32881C23direct (N × H)
115. C2×#450C2×C23semidirect
116. #2848C2×C23semidirect
117. #1786C2×C23semidirect
118. #1779C2×C23semidirect
128. C22×C23#450semidirect
141. C22×SD16C22×C23semidirect
142. C2×C22⋊Q8C22×C23semidirect
144. C2×#102C22×C23semidirect
145. C2×C22⋊C8C22×C23semidirect
147. #380C22×C23semidirect
159. C22×Q8C23×C23semidirect
172. C22×C8C23×C23semidirect
175. C2×C4⋊C4C23×C23semidirect
181. C23#32881direct (N × H)
193. C24SD16×C23semidirect
258. C22C23×#450semidirect

Maximal subgroups

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

subgroupindex
C24×D4×C232
C2×C23×#4502
C23×#28482
C23×#17862
C23×#17792
#3288123