Finite Group Databasebeta

← C23×#12627 #19160253

Subgroups — C23×#12627 #19160253

Order 5888.

564 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 162 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×#2326C211yesmaximal, characteristic
2C2×C23×#354C212yesmaximal
3C4×C23×#99C212yesmaximal
4C23×#1393C212yesmaximal
5C2×C23×#124C2211yescharacteristic
6C2×C4×Q8×C23C411yescharacteristic, complemented
7C2×C4×D4×C23C2211yescharacteristic
8C2×C23×#99C2214yes—
9C23×#391C411yescharacteristic, complemented
10C23×#390C2211yescharacteristic
11C23×#388—22no—
12C23×#385—22no—
13C23×#384C411yescharacteristic, complemented
14C4×C22⋊C4×C23—22no—
15C23×#376C411yescharacteristic, complemented
16C23×#354C418yescomplemented
17C23×#342—22no—
18C22×Q8×C23C2×C411yescharacteristic
19C22×D4×C23C2311yescharacteristic
20C23×C4×C23D412yes—
21C23×#124—22no—
22C23×#124C2×C414yes—
23C4×Q8×C23—42no—
24C4×D4×C23—42no—
25C2×C4⋊C4×C23—21no—
26C2×C4⋊C4×C23—22no—
27C2×C4⋊C4×C23C2×C411yescharacteristic
28C2×C4⋊C4×C23C2×C411yescharacteristic
29C2×C22⋊C4×C23—22no—
30C2×C22⋊C4×C23—24no—
31C2×C22⋊C4×C23—22no—
32C2×C22⋊C4×C23D412yes—
33C2×C42×C23—21no—
34C2×C42×C23—42no—
35C2×C42×C23C2×C411yescharacteristic
36C2×C42×C23C2×C411yescharacteristic
37C2×C42×C23C2×C411yescharacteristic
38C23×#99—24no—
39C23×#99C2×C418yes—
40C23×#95—21no—
41C23×#95—22no—
42C23×#95—21no—
43C23×#95—42no—
44C24×C23C2×D412yes—
45C2×Q8×C23—21no—
46C2×Q8×C23—41no—
47C2×Q8×C23—41no—
48C2×Q8×C23C4212yescomplemented
49C2×D4×C23—41no—
50C2×D4×C23—41no—
51C2×D4×C23C22×C412yes—
52C2×D4×C23#4112yes—
53C22×C4×C23—21no—
54C22×C4×C23—21no—
55C22×C4×C23—22no—
56C22×C4×C23—22no—
57C22×C4×C23—21no—
58C22×C4×C23—21no—
59C22×C4×C23—44no—
60C22×C4×C23—82no—
61C22×C4×C23C22⋊C411yescharacteristic
62C22×C4×C23C22⋊C412yes—
63C22×C4×C23C22⋊C411yescharacteristic
64C22×C4×C23C22×C411yescharacteristic
65C4⋊C4×C23—41no—
66C4⋊C4×C23—41no—
67C4⋊C4×C23—42no—
68C4⋊C4×C23—82no—
69C22⋊C4×C23—28no—
70C22⋊C4×C23—28no—
71C22⋊C4×C23—42no—
72C22⋊C4×C23—44no—
73C22⋊C4×C23—82no—
74C42×C23—22no—
75C42×C23—21no—
76C42×C23—41no—
77C42×C23—42no—
78C42×C23—84no—
79C42×C23C4212yescomplemented
80#12627C2311yesmaximal, characteristic, direct product factor, Sylow
81C23×C23—22no—
82C23×C23—22no—
83C23×C23—42no—
84C23×C23—42no—
85C23×C23C2×C22⋊C411yescharacteristic
86C23×C23C4×D414yes—
87Q8×C23—42no—
88Q8×C23—42no—
89D4×C23—42no—
90D4×C23—42no—
91C2×C4×C23—21no—
92C2×C4×C23—21no—
93C2×C4×C23—22no—
94C2×C4×C23—22no—
95C2×C4×C23—21no—
96C2×C4×C23—22no—
97C2×C4×C23—41no—
98C2×C4×C23—44no—
99C2×C4×C23—41no—
100C2×C4×C23—41no—
101C2×C4×C23—41no—
102C2×C4×C23—41no—
103C2×C4×C23—42no—
104C2×C4×C23—41no—
105C2×C4×C23—48no—
106C2×C4×C23—41no—
107C2×C4×C23—42no—
108C2×C4×C23—41no—
109C2×C4×C23—41no—
110C2×C4×C23—41no—
111C2×C4×C23—82no—
112C2×C4×C23—84no—
113C2×C4×C23—84no—
114C2×C4×C23#9912yes—
115C2×C4×C23C2×C4211yescharacteristic
116C2×C4×C23#11711yescharacteristic
117#2326C2×C2311yescharacteristic
118C2×#354C2×C2312yes—
119C4×#99C2×C2312yes—
120#1393C2×C2312yes—
121C22×C23—21no—
122C22×C23—21no—
123C22×C23—24no—
124C22×C23—44no—
125C22×C23—42no—
126C22×C23—42no—
127C22×C23—82no—
128C22×C23C4×C22⋊C411yescharacteristic
129C22×C23C2×#9911yescharacteristic
130C22×C23#41011yescharacteristic
131C4×C23—22no—
132C4×C23—21no—
133C4×C23—21no—
134C4×C23—42no—
135C4×C23—42no—
136C4×C23—41no—
137C4×C23—42no—
138C4×C23—81no—
139C4×C23—88no—
140C4×C23—81no—
141C2×#124C22×C2311yescharacteristic
142C2×C4×Q8C4×C2311yescharacteristic, complemented
143C2×C4×D4C22×C2311yescharacteristic
144C2×#99C22×C2314yes—
145#391C4×C2311yescharacteristic, complemented
146#390C22×C2311yescharacteristic
147#388—22no—
148#385—22no—
149#384C4×C2311yescharacteristic, complemented
150C4×C22⋊C4—22no—
151#376C4×C2311yescharacteristic, complemented
152#354C4×C2318yescomplemented
153#342—22no—
154C2×C23—21no—
155C2×C23—21no—
156C2×C23—44no—
157C2×C23C4×#9911yescharacteristic
158C2×C23#212711yescharacteristic
159C2×C23#212911yescharacteristic
160C22×Q8C2×C4×C2311yescharacteristic
161C22×D4C23×C2311yescharacteristic
162C23×C4D4×C2312yes—
163#124—22no—
164#124C2×C4×C2314yes—
165C4×Q8—42no—
166C4×D4—42no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4C2×C4×C2311yescharacteristic
170C2×C4⋊C4C2×C4×C2311yescharacteristic
171C2×C22⋊C4—22no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4D4×C2312yes—
175C2×C42—21no—
176C2×C42—42no—
177C2×C42C2×C4×C2311yescharacteristic
178C2×C42C2×C4×C2311yescharacteristic
179C2×C42C2×C4×C2311yescharacteristic
180#99—24no—
181#99C2×C4×C2318yes—
182#95—21no—
183#95—21no—
184#95—22no—
185#95—42no—
186C23#1262711yescharacteristic, direct product factor, Sylow
187C24C2×D4×C2312yes—
188C2×Q8—21no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8C42×C2312yescomplemented
192C2×D4—41no—
193C2×D4—41no—
194C2×D4C22×C4×C2312yes—
195C2×D4#41×C2312yes—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—44no—
203C22×C4—82no—
204C22×C4C22⋊C4×C2311yescharacteristic
205C22×C4C22⋊C4×C2312yes—
206C22×C4C22⋊C4×C2311yescharacteristic
207C22×C4C22×C4×C2311yescharacteristic
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—42no—
211C4⋊C4—82no—
212C22⋊C4—28no—
213C22⋊C4—28no—
214C22⋊C4—42no—
215C22⋊C4—44no—
216C22⋊C4—82no—
217C42—22no—
218C42—21no—
219C42—41no—
220C42—42no—
221C42—84no—
222C42C42×C2312yescomplemented
223C23—22no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23C2×C22⋊C4×C2311yescharacteristic
228C23C4×D4×C2314yes—
229Q8—42no—
230Q8—42no—
231D4—42no—
232D4—42no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—41no—
240C2×C4—44no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—48no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—82no—
254C2×C4—84no—
255C2×C4—84no—
256C2×C4C23×#9912yes—
257C2×C4C2×C42×C2311yescharacteristic
258C2×C4C23×#11711yescharacteristic
259C22—21no—
260C22—21no—
261C22—24no—
262C22—44no—
263C22—42no—
264C22—42no—
265C22—82no—
266C22C4×C22⋊C4×C2311yescharacteristic
267C22C2×C23×#9911yescharacteristic
268C22C23×#41011yescharacteristic
269C4—21no—
270C4—21no—
271C4—22no—
272C4—42no—
273C4—42no—
274C4—42no—
275C4—41no—
276C4—81no—
277C4—88no—
278C4—81no—
279C2—21no—
280C2—21no—
281C2—44no—
282C2C4×C23×#9911yescharacteristic
283C2C23×#212711yescharacteristic
284C2C23×#212911yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC2×C4C2×C42×C23
Frattini, cosocleC22×D4C23×C23
prime radicalC22×D4×C23C23
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C2×C4×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
6. C2×C4×Q8×C23C4semidirect
9. C23×#391C4semidirect
13. C23×#384C4semidirect
15. C23×#376C4semidirect
16. C23×#354C4semidirect
48. C2×Q8×C23C42semidirect
79. C42×C23C42semidirect
80. #12627C23direct (N × H)
142. C2×C4×Q8C4×C23semidirect
145. #391C4×C23semidirect
149. #384C4×C23semidirect
151. #376C4×C23semidirect
152. #354C4×C23semidirect
186. C23#12627direct (N × H)
191. C2×Q8C42×C23semidirect
222. C42C42×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#23262
C2×C23×#3542
C4×C23×#992
C23×#13932
#1262723