Finite Group Databasebeta

← C23×#3022 #15670641

Subgroups — C23×#3022 #15670641

Order 2944.

704 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 212 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×#585C211yesmaximal, characteristic
2C2×C23×#137C212yesmaximal, complemented
3C23×#536C211yesmaximal, characteristic, complemented
4C23×#456C214yesmaximal, complemented
5C23×#455C214yesmaximal, complemented
6C23×#421C212yesmaximal, complemented
7C23×#413C211yesmaximal, characteristic, complemented
8C23×2−1+4—22no—
9C23×2−1+4C2212yes—
10C23×H(5,𝔽2)—22no—
11C23×H(5,𝔽2)—22no—
12C23×H(5,𝔽2)C2212yes—
13C2×#41×C23—24no—
14C2×#41×C23—22no—
15C2×#41×C23C2212yes—
16C2×#41×C23C2211yescharacteristic
17C22×Q8×C23C2211yescharacteristic, complemented
18C23×#137—24no—
19C23×#137C2218yescomplemented
20C2×Q16×C23—22no—
21C2×SD16×C23—22no—
22C2×M16×C23C2212yescomplemented
23C23×C4⋊Q8C2212yescomplemented
24C23×#124C2212yescomplemented
25C23×#123—22no—
26C23×C22⋊Q8—22no—
27C23×#117C2211yescharacteristic, complemented
28C23×#104C2218yescomplemented
29C23×#101C2212yescomplemented
30C23×#100C2212yescomplemented
31#41×C23—24no—
32#41×C23—28no—
33#41×C23—24no—
34#41×C23—24no—
35#41×C23—24no—
36#41×C23—22no—
37#41×C23—22no—
38#41×C23—24no—
39#41×C23—22no—
40#41×C23—22no—
41#41×C23D414yes—
42#41×C23C2314yes—
43C2×Q8×C23—22no—
44C2×Q8×C23—24no—
45C2×Q8×C23—21no—
46C2×Q8×C23—22no—
47C2×Q8×C23—22no—
48C2×Q8×C23D412yes—
49C2×Q8×C23C2312yes—
50C2×Q8×C23C2311yescharacteristic
51C2×D4×C23—28no—
52C2×D4×C23—22no—
53C2×D4×C23—24no—
54C2×D4×C23—22no—
55C2×D4×C23—24no—
56C2×D4×C23D412yes—
57C2×D4×C23D412yes—
58C2×D4×C23C2311yescharacteristic
59C22×C4×C23—24no—
60C22×C4×C23—22no—
61C22×C4×C23—21no—
62C22×C4×C23D412yes—
63C22×C4×C23C2311yescharacteristic
64Q16×C23—28no—
65SD16×C23—28no—
66M16×C23—22no—
67M16×C23C2314yes—
68C2×C8×C23—22no—
69C4⋊C4×C23—21no—
70C4⋊C4×C23—24no—
71C22⋊C4×C23—21no—
72C22⋊C4×C23—24no—
73C42×C23C2312yes—
74C23×C23—24no—
75C23×C23—22no—
76C23×C23C2×D412yes—
77C23×C23C2×D411yescharacteristic
78Q8×C23—24no—
79Q8×C23—22no—
80Q8×C23—22no—
81Q8×C23—24no—
82Q8×C23—41no—
83Q8×C23—41no—
84Q8×C23C2×D414yes—
85D4×C23—28no—
86D4×C23—24no—
87D4×C23—24no—
88D4×C23—24no—
89D4×C23—24no—
90D4×C23—22no—
91D4×C23—22no—
92D4×C23C2×D414yes—
93C2×C4×C23—24no—
94C2×C4×C23—28no—
95C2×C4×C23—22no—
96C2×C4×C23—24no—
97C2×C4×C23—22no—
98C2×C4×C23—22no—
99C2×C4×C23—24no—
100C2×C4×C23—22no—
101C2×C4×C23—22no—
102C2×C4×C23—42no—
103C2×C4×C23C2×D412yes—
104C2×C4×C23C2×D411yescharacteristic
105C2×C4×C23C2×D412yes—
106C2×C4×C23C2×D412yes—
107C2×C4×C23C2411yescharacteristic
108C8×C23—24no—
109#3022C2311yesmaximal, characteristic, direct product factor, Sylow
110C22×C23—22no—
111C22×C23—24no—
112C22×C23—41no—
113C22×C23—42no—
114C22×C23—82no—
115C22×C23—84no—
116C22×C23#12012yes—
117C22×C23C22×D411yescharacteristic
118C4×C23—24no—
119C4×C23—22no—
120C4×C23—42no—
121C4×C23—42no—
122C4×C23#12011yescharacteristic
123C4×C23#12011yescharacteristic
124C4×C23C22×D412yes—
125#585C2×C2311yescharacteristic
126C2×#137C2×C2312yescomplemented
127#536C2×C2311yescharacteristic, complemented
128#456C2×C2314yescomplemented
129#455C2×C2314yescomplemented
130#421C2×C2312yescomplemented
131#413C2×C2311yescharacteristic, complemented
132C2×C23—21no—
133C2×C23—22no—
134C2×C23—42no—
135C2×C23—44no—
136C2×C23C2×#12011yescharacteristic
1372−1+4—22no—
1382−1+4C22×C2312yes—
139H(5,𝔽2)—22no—
140H(5,𝔽2)—22no—
141H(5,𝔽2)C22×C2312yes—
142C2×#41—24no—
143C2×#41—22no—
144C2×#41C22×C2312yes—
145C2×#41C22×C2311yescharacteristic
146C22×Q8C22×C2311yescharacteristic, complemented
147#137—24no—
148#137C22×C2318yescomplemented
149C2×Q16—22no—
150C2×SD16—22no—
151C2×M16C22×C2312yescomplemented
152C4⋊Q8C22×C2312yescomplemented
153#124C22×C2312yescomplemented
154#123—22no—
155C22⋊Q8—22no—
156#117C22×C2311yescharacteristic, complemented
157#104C22×C2318yescomplemented
158#101C22×C2312yescomplemented
159#100C22×C2312yescomplemented
160C23#302211yescharacteristic, direct product factor, Sylow
161#41—24no—
162#41—28no—
163#41—24no—
164#41—24no—
165#41—24no—
166#41—22no—
167#41—22no—
168#41—24no—
169#41—22no—
170#41—22no—
171#41D4×C2314yes—
172#41C23×C2314yes—
173C2×Q8—22no—
174C2×Q8—24no—
175C2×Q8—21no—
176C2×Q8—22no—
177C2×Q8—22no—
178C2×Q8D4×C2312yes—
179C2×Q8C23×C2312yes—
180C2×Q8C23×C2311yescharacteristic
181C2×D4—28no—
182C2×D4—22no—
183C2×D4—24no—
184C2×D4—22no—
185C2×D4—24no—
186C2×D4D4×C2312yes—
187C2×D4D4×C2312yes—
188C2×D4C23×C2311yescharacteristic
189C22×C4—24no—
190C22×C4—22no—
191C22×C4—21no—
192C22×C4D4×C2312yes—
193C22×C4C23×C2311yescharacteristic
194Q16—28no—
195SD16—28no—
196M16—22no—
197M16C23×C2314yes—
198C2×C8—22no—
199C4⋊C4—21no—
200C4⋊C4—24no—
201C22⋊C4—21no—
202C22⋊C4—24no—
203C42C23×C2312yes—
204C23—24no—
205C23—22no—
206C23C2×D4×C2312yes—
207C23C2×D4×C2311yescharacteristic
208Q8—24no—
209Q8—22no—
210Q8—22no—
211Q8—24no—
212Q8—41no—
213Q8—41no—
214Q8C2×D4×C2314yes—
215D4—28no—
216D4—24no—
217D4—24no—
218D4—24no—
219D4—24no—
220D4—22no—
221D4—22no—
222D4C2×D4×C2314yes—
223C2×C4—24no—
224C2×C4—28no—
225C2×C4—22no—
226C2×C4—24no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—42no—
233C2×C4C2×D4×C2312yes—
234C2×C4C2×D4×C2311yescharacteristic
235C2×C4C2×D4×C2312yes—
236C2×C4C2×D4×C2312yes—
237C2×C4C24×C2311yescharacteristic
238C8—24no—
239C22—22no—
240C22—24no—
241C22—41no—
242C22—42no—
243C22—82no—
244C22—84no—
245C22C23×#12012yes—
246C22C22×D4×C2311yescharacteristic
247C4—24no—
248C4—22no—
249C4—42no—
250C4—42no—
251C4C23×#12011yescharacteristic
252C4C23×#12011yescharacteristic
253C4C22×D4×C2312yes—
254C2—21no—
255C2—22no—
256C2—42no—
257C2—44no—
258C2C2×C23×#12011yescharacteristic
subgroupG/H
center, socleC2×C23C2×#120
commutator, Frattini, cosocleC2×C4C24×C23
prime radicalC23×#585C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C22×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C23×#137C2semidirect
3. C23×#536C2semidirect
4. C23×#456C2semidirect
5. C23×#455C2semidirect
6. C23×#421C2semidirect
7. C23×#413C2semidirect
17. C22×Q8×C23C22semidirect
19. C23×#137C22semidirect
22. C2×M16×C23C22semidirect
23. C23×C4⋊Q8C22semidirect
24. C23×#124C22semidirect
27. C23×#117C22semidirect
28. C23×#104C22semidirect
29. C23×#101C22semidirect
30. C23×#100C22semidirect
109. #3022C23direct (N × H)
126. C2×#137C2×C23semidirect
127. #536C2×C23semidirect
128. #456C2×C23semidirect
129. #455C2×C23semidirect
130. #421C2×C23semidirect
131. #413C2×C23semidirect
146. C22×Q8C22×C23semidirect
148. #137C22×C23semidirect
151. C2×M16C22×C23semidirect
152. C4⋊Q8C22×C23semidirect
153. #124C22×C23semidirect
156. #117C22×C23semidirect
157. #104C22×C23semidirect
158. #101C22×C23semidirect
159. #100C22×C23semidirect
160. C23#3022direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#5852
C2×C23×#1372
C23×#5362
C23×#4562
C23×#4552
C23×#4212
C23×#4132
#302223