Finite Group Databasebeta

← C23×#12860 #19160486

Subgroups — C23×#12860 #19160486

Order 5888.

638 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 126 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×#391C211yesmaximal, characteristic
2C23×#1784C212yesmaximal
3C23×#1394C214yesmaximal, complemented
4C23×Q8×C23C2211yescharacteristic
5C22×C4⋊C4×C23C411yescharacteristic
6C22×C42×C23C411yescharacteristic
7C2×C23×#101—22no—
8C2×C23×#101C2214yes—
9C23×#407—22no—
10C23×#391—22no—
11C23×#391C412yes—
12C23×#391C2212yes—
13C2×C23×#95—22no—
14C22×Q8×C23—24no—
15C22×Q8×C23—41no—
16C22×Q8×C23—41no—
17C22×Q8×C23—42no—
18C22×Q8×C23C2×C411yescharacteristic
19C22×Q8×C23D412yes—
20C22×Q8×C23C2311yescharacteristic
21C23×C4×C23—21no—
22C23×C4×C23—21no—
23C23×C4×C23C2×C411yescharacteristic
24C23×C4×C23D412yes—
25C2×M16×C23—24no—
26C2×C4⋊C4×C23—21no—
27C2×C4⋊C4×C23—41no—
28C2×C4⋊C4×C23—41no—
29C2×C4⋊C4×C23C2×C412yes—
30C2×C42×C23—21no—
31C2×C42×C23—42no—
32C2×C42×C23C2×C412yes—
33C23×#101—28no—
34C23×#101—44no—
35C23×C22⋊C8—42no—
36C23×C22⋊C8—42no—
37C23×#95—24no—
38C23×#95—24no—
39C24×C23C22⋊C411yescharacteristic
40C2×Q8×C23—24no—
41C2×Q8×C23—22no—
42C2×Q8×C23—24no—
43C2×Q8×C23—41no—
44C2×Q8×C23—41no—
45C2×Q8×C23—42no—
46C2×Q8×C23—41no—
47C2×Q8×C23—41no—
48C2×Q8×C23—48no—
49C2×Q8×C23—48no—
50C2×Q8×C23#4114yes—
51C22×C4×C23—21no—
52C22×C4×C23—21no—
53C22×C4×C23—22no—
54C22×C4×C23—22no—
55C22×C4×C23—24no—
56C22×C4×C23—22no—
57C22×C4×C23—22no—
58C22×C4×C23—22no—
59C22×C4×C23—24no—
60C22×C4×C23—22no—
61C22×C4×C23—42no—
62C22×C4×C23—42no—
63C22×C4×C23—42no—
64C22×C4×C23—44no—
65C22×C4×C23—41no—
66C22×C4×C23—41no—
67C22×C4×C23C22⋊C412yes—
68C22×C4×C23C22⋊C411yescharacteristic
69C22×C4×C23C22×C411yescharacteristic
70C22×C4×C23C2×D412yes—
71M16×C23—44no—
72M16×C23—44no—
73C2×C8×C23—44no—
74C4⋊C4×C23—42no—
75C4⋊C4×C23—42no—
76C42×C23—42no—
77C42×C23—42no—
78#12860C2311yesmaximal, characteristic, direct product factor, Sylow
79C23×C23—22no—
80C23×C23—22no—
81C23×C23—21no—
82C23×C23—21no—
83C23×C23#9911yescharacteristic
84C23×C23#9911yescharacteristic
85C23×C23C2×C22⋊C411yescharacteristic
86Q8×C23—44no—
87Q8×C23—48no—
88Q8×C23—44no—
89C2×C4×C23—24no—
90C2×C4×C23—24no—
91C2×C4×C23—21no—
92C2×C4×C23—21no—
93C2×C4×C23—22no—
94C2×C4×C23—21no—
95C2×C4×C23—21no—
96C2×C4×C23—22no—
97C2×C4×C23—24no—
98C2×C4×C23—22no—
99C2×C4×C23—22no—
100C2×C4×C23—22no—
101C2×C4×C23—44no—
102C2×C4×C23—44no—
103C2×C4×C23—48no—
104C2×C4×C23—42no—
105C2×C4×C23—42no—
106C2×C4×C23—44no—
107C2×C4×C23—42no—
108C2×C4×C23—82no—
109C2×C4×C23—82no—
110C2×C4×C23#9912yes—
111C2×C4×C23#11712yes—
112C2×C4×C23#12314yes—
113C8×C23—84no—
114C2×#391C2×C2311yescharacteristic
115#1784C2×C2312yes—
116#1394C2×C2314yescomplemented
117C22×C23—24no—
118C22×C23—24no—
119C22×C23—21no—
120C22×C23—21no—
121C22×C23—21no—
122C22×C23—21no—
123C22×C23—21no—
124C22×C23—21no—
125C22×C23#35512yes—
126C22×C23#35612yes—
127C22×C23#38111yescharacteristic
128C22×C23C2×#9911yescharacteristic
129C22×C23C2×#9911yescharacteristic
130C4×C23—24no—
131C4×C23—22no—
132C4×C23—22no—
133C4×C23—44no—
134C4×C23—42no—
135C4×C23—82no—
136C23×Q8C22×C2311yescharacteristic
137C22×C4⋊C4C4×C2311yescharacteristic
138C22×C42C4×C2311yescharacteristic
139C2×#101—22no—
140C2×#101C22×C2314yes—
141#407—22no—
142#391—22no—
143#391C4×C2312yes—
144#391C22×C2312yes—
145C2×#95—22no—
146C2×C23—21no—
147C2×C23—22no—
148C2×C23—22no—
149C2×C23—21no—
150C2×C23#181911yescharacteristic
151C2×C23C2×#35511yescharacteristic
152C2×C23C2×#35611yescharacteristic
153C22×Q8—24no—
154C22×Q8—41no—
155C22×Q8—41no—
156C22×Q8—42no—
157C22×Q8C2×C4×C2311yescharacteristic
158C22×Q8D4×C2312yes—
159C22×Q8C23×C2311yescharacteristic
160C23×C4—21no—
161C23×C4—21no—
162C23×C4C2×C4×C2311yescharacteristic
163C23×C4D4×C2312yes—
164C2×M16—24no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4C2×C4×C2312yes—
169C2×C42—21no—
170C2×C42—42no—
171C2×C42C2×C4×C2312yes—
172#101—28no—
173#101—44no—
174C22⋊C8—42no—
175C22⋊C8—42no—
176#95—24no—
177#95—24no—
178C23#1286011yescharacteristic, direct product factor, Sylow
179C24C22⋊C4×C2311yescharacteristic
180C2×Q8—24no—
181C2×Q8—22no—
182C2×Q8—24no—
183C2×Q8—41no—
184C2×Q8—41no—
185C2×Q8—42no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8—48no—
189C2×Q8—48no—
190C2×Q8#41×C2314yes—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—24no—
200C22×C4—22no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—44no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4C22⋊C4×C2312yes—
208C22×C4C22⋊C4×C2311yescharacteristic
209C22×C4C22×C4×C2311yescharacteristic
210C22×C4C2×D4×C2312yes—
211M16—44no—
212M16—44no—
213C2×C8—44no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C42—42no—
217C42—42no—
218C23—22no—
219C23—22no—
220C23—21no—
221C23—21no—
222C23C23×#9911yescharacteristic
223C23C23×#9911yescharacteristic
224C23C2×C22⋊C4×C2311yescharacteristic
225Q8—44no—
226Q8—48no—
227Q8—44no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—24no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—48no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4C23×#9912yes—
250C2×C4C23×#11712yes—
251C2×C4C23×#12314yes—
252C8—84no—
253C22—24no—
254C22—24no—
255C22—21no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22C23×#35512yes—
262C22C23×#35612yes—
263C22C23×#38111yescharacteristic
264C22C2×C23×#9911yescharacteristic
265C22C2×C23×#9911yescharacteristic
266C4—24no—
267C4—22no—
268C4—22no—
269C4—44no—
270C4—42no—
271C4—82no—
272C2—21no—
273C2—22no—
274C2—22no—
275C2—21no—
276C2C23×#181911yescharacteristic
277C2C2×C23×#35511yescharacteristic
278C2C2×C23×#35611yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC22×C4C22×C4×C23
Frattini, cosocleC22×Q8C23×C23
prime radicalC24×C23C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C24×C23 ◁ C23×Q8×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C23×#1394C2semidirect
78. #12860C23direct (N × H)
116. #1394C2×C23semidirect
178. C23#12860direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#3912
C23×#17842
C23×#13942
#1286023