Finite Group Databasebeta

← C23×#8552 #19156178

Subgroups — C23×#8552 #19156178

Order 5888.

1154 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1154 conjugacy classes of subgroups — too many to draw. Only its 158 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
1C22×C23×#124C211yesmaximal, characteristic
2C23×#1784C212yesmaximal, complemented
3C23×#1462C214yesmaximal, complemented
4C23×Q8×C23C2211yescharacteristic
5C23×D4×C23C411yescharacteristic
6C2×C23×#124—24no—
7C2×C23×#124—28no—
8C22×C22⋊C4×C23C412yes—
9C22×C22⋊C4×C23C2212yes—
10C22×C42×C23C411yescharacteristic
11C2×C23×#101—24no—
12C23×#407C2214yes—
13C23×#323—28no—
14C25×C23C2×C412yes—
15C22×Q8×C23—24no—
16C22×Q8×C23—44no—
17C22×Q8×C23D414yes—
18C22×D4×C23—24no—
19C22×D4×C23—22no—
20C22×D4×C23—44no—
21C23×C4×C23C2×C411yescharacteristic
22C23×C4×C23C2×C411yescharacteristic
23C23×C4×C23C2×C412yes—
24C23×C4×C23C2311yescharacteristic
25C2×M16×C23—24no—
26C23×#124—232no—
27C2×C22⋊C4×C23—24no—
28C2×C22⋊C4×C23—24no—
29C2×C22⋊C4×C23—24no—
30C2×C22⋊C4×C23—24no—
31C2×C22⋊C4×C23—24no—
32C2×C22⋊C4×C23D418yes—
33C2×C42×C23—24no—
34C2×C42×C23—22no—
35C23×#101—48no—
36C23×C22⋊C8—28no—
37C24×C23—22no—
38C24×C23—22no—
39C24×C23—44no—
40C24×C23—44no—
41C24×C23—44no—
42C24×C23C22⋊C414yes—
43C24×C23C22×C411yescharacteristic
44C2×Q8×C23—28no—
45C2×Q8×C23—28no—
46C2×Q8×C23—44no—
47C2×Q8×C23—416no—
48C2×D4×C23—28no—
49C2×D4×C23—416no—
50C2×D4×C23—44no—
51C2×D4×C23—44no—
52C22×C4×C23—22no—
53C22×C4×C23—24no—
54C22×C4×C23—22no—
55C22×C4×C23—22no—
56C22×C4×C23—22no—
57C22×C4×C23—42no—
58C22×C4×C23—42no—
59C22×C4×C23—44no—
60C22×C4×C23C22⋊C412yes—
61C22×C4×C23C22⋊C412yes—
62C22×C4×C23C22⋊C414yes—
63C22×C4×C23C2×D412yes—
64C22×C4×C23C2×D414yes—
65M16×C23—48no—
66C2×C8×C23—44no—
67C22⋊C4×C23—216no—
68C22⋊C4×C23—216no—
69C42×C23—28no—
70#8552C2311yesmaximal, characteristic, direct product factor, Sylow
71C23×C23—22no—
72C23×C23—24no—
73C23×C23—24no—
74C23×C23—28no—
75C23×C23—44no—
76C23×C23—44no—
77C23×C23—44no—
78C23×C23—44no—
79C23×C23—44no—
80C23×C23—44no—
81C23×C23—44no—
82C23×C23—44no—
83C23×C23—816no—
84C23×C23C2×C22⋊C411yescharacteristic
85C23×C23C2×C22⋊C412yes—
86Q8×C23—416no—
87D4×C23—416no—
88C2×C4×C23—28no—
89C2×C4×C23—22no—
90C2×C4×C23—24no—
91C2×C4×C23—22no—
92C2×C4×C23—24no—
93C2×C4×C23—22no—
94C2×C4×C23—24no—
95C2×C4×C23—44no—
96C2×C4×C23—44no—
97C2×C4×C23—44no—
98C2×C4×C23—44no—
99C2×C4×C23—44no—
100C2×C4×C23#9914yes—
101C2×C4×C23#12014yes—
102C8×C23—84no—
103C22×#124C2×C2311yescharacteristic
104#1784C2×C2312yescomplemented
105#1462C2×C2314yescomplemented
106C22×C23—28no—
107C22×C23—22no—
108C22×C23—22no—
109C22×C23—22no—
110C22×C23—44no—
111C22×C23—44no—
112C22×C23—44no—
113C22×C23—88no—
114C22×C23—88no—
115C22×C23—88no—
116C22×C23#37911yescharacteristic
117C22×C23C2×#9912yes—
118C22×C23#41314yes—
119C4×C23—24no—
120C4×C23—24no—
121C4×C23—42no—
122C4×C23—44no—
123C23×Q8C22×C2311yescharacteristic
124C23×D4C4×C2311yescharacteristic
125C2×#124—24no—
126C2×#124—28no—
127C22×C22⋊C4C4×C2312yes—
128C22×C22⋊C4C22×C2312yes—
129C22×C42C4×C2311yescharacteristic
130C2×#101—24no—
131#407C22×C2314yes—
132#323—28no—
133C2×C23—22no—
134C2×C23—24no—
135C2×C23—84no—
136C2×C23#178111yescharacteristic
137C2×C23#178512yes—
138C25C2×C4×C2312yes—
139C22×Q8—24no—
140C22×Q8—44no—
141C22×Q8D4×C2314yes—
142C22×D4—24no—
143C22×D4—22no—
144C22×D4—44no—
145C23×C4C2×C4×C2311yescharacteristic
146C23×C4C2×C4×C2311yescharacteristic
147C23×C4C2×C4×C2312yes—
148C23×C4C23×C2311yescharacteristic
149C2×M16—24no—
150#124—232no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4D4×C2318yes—
157C2×C42—24no—
158C2×C42—22no—
159#101—48no—
160C22⋊C8—28no—
161C23#855211yescharacteristic, direct product factor, Sylow
162C24—22no—
163C24—22no—
164C24—44no—
165C24—44no—
166C24—44no—
167C24C22⋊C4×C2314yes—
168C24C22×C4×C2311yescharacteristic
169C2×Q8—28no—
170C2×Q8—28no—
171C2×Q8—44no—
172C2×Q8—416no—
173C2×D4—28no—
174C2×D4—416no—
175C2×D4—44no—
176C2×D4—44no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—44no—
185C22×C4C22⋊C4×C2312yes—
186C22×C4C22⋊C4×C2312yes—
187C22×C4C22⋊C4×C2314yes—
188C22×C4C2×D4×C2312yes—
189C22×C4C2×D4×C2314yes—
190M16—48no—
191C2×C8—44no—
192C22⋊C4—216no—
193C22⋊C4—216no—
194C42—28no—
195C23—22no—
196C23—24no—
197C23—24no—
198C23—28no—
199C23—44no—
200C23—44no—
201C23—44no—
202C23—44no—
203C23—44no—
204C23—44no—
205C23—44no—
206C23—44no—
207C23—816no—
208C23C2×C22⋊C4×C2311yescharacteristic
209C23C2×C22⋊C4×C2312yes—
210Q8—416no—
211D4—416no—
212C2×C4—28no—
213C2×C4—22no—
214C2×C4—24no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—44no—
223C2×C4—44no—
224C2×C4C23×#9914yes—
225C2×C4C23×#12014yes—
226C8—84no—
227C22—28no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—44no—
232C22—44no—
233C22—44no—
234C22—88no—
235C22—88no—
236C22—88no—
237C22C23×#37911yescharacteristic
238C22C2×C23×#9912yes—
239C22C23×#41314yes—
240C4—24no—
241C4—24no—
242C4—42no—
243C4—44no—
244C2—22no—
245C2—24no—
246C2—84no—
247C2C23×#178111yescharacteristic
248C2C23×#178512yes—
subgroupG/H
center, socleC22×C23#379
commutatorC24C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC23×D4×C23C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1784C2semidirect
3. C23×#1462C2semidirect
70. #8552C23direct (N × H)
104. #1784C2×C23semidirect
105. #1462C2×C23semidirect
161. C23#8552direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C23×#1242
C23×#17842
C23×#14622
#855223