Finite Group Databasebeta

← C23×#8756 #19156382

Subgroups — C23×#8756 #19156382

Order 5888.

546 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2315C211yesmaximal, characteristic
2C23×#1462C212yesmaximal
3C23×#1305C214yesmaximal, complemented
4C22×C22⋊C4×C23C411yescharacteristic
5C22×C22⋊C4×C23C411yescharacteristic
6C22×C22⋊C4×C23C2211yescharacteristic
7C23×#407C2214yes—
8C23×#385—22no—
9C23×#385—22no—
10C23×#380C412yes—
11C23×#380C2212yes—
12C23×#323—22no—
13C23×#323—22no—
14C25×C23C2×C411yescharacteristic
15C23×C4×C23—21no—
16C23×C4×C23—21no—
17C23×C4×C23C2×C411yescharacteristic
18C23×C4×C23C2311yescharacteristic
19C2×M16×C23—24no—
20C2×C4⋊C4×C23—22no—
21C2×C4⋊C4×C23—22no—
22C2×C22⋊C4×C23—24no—
23C2×C22⋊C4×C23—21no—
24C2×C22⋊C4×C23—21no—
25C2×C22⋊C4×C23—42no—
26C2×C22⋊C4×C23—42no—
27C2×C22⋊C4×C23C2×C412yes—
28C2×C22⋊C4×C23C2×C412yes—
29C2×C22⋊C4×C23D412yes—
30C2×C22⋊C4×C23D412yes—
31C23×C22⋊C8—24no—
32C23×C22⋊C8—24no—
33C23×#95—22no—
34C23×#95—22no—
35C24×C23—22no—
36C24×C23—22no—
37C24×C23—22no—
38C24×C23—81no—
39C24×C23—81no—
40C24×C23C22⋊C411yescharacteristic
41C24×C23C22⋊C411yescharacteristic
42C24×C23C22×C411yescharacteristic
43C22×C4×C23—22no—
44C22×C4×C23—22no—
45C22×C4×C23—22no—
46C22×C4×C23—42no—
47C22×C4×C23—42no—
48C22×C4×C23—81no—
49C22×C4×C23—81no—
50C22×C4×C23—81no—
51C22×C4×C23—81no—
52C22×C4×C23—81no—
53C22×C4×C23—81no—
54C22×C4×C23C22⋊C412yes—
55C22×C4×C23C2×D412yes—
56C22×C4×C23#4114yes—
57M16×C23—84no—
58C2×C8×C23—44no—
59C4⋊C4×C23—82no—
60C4⋊C4×C23—82no—
61C22⋊C4×C23—24no—
62C22⋊C4×C23—24no—
63C22⋊C4×C23—42no—
64C22⋊C4×C23—42no—
65C22⋊C4×C23—42no—
66C22⋊C4×C23—42no—
67#8756C2311yesmaximal, characteristic, direct product factor, Sylow
68C23×C23—28no—
69C23×C23—21no—
70C23×C23—21no—
71C23×C23—21no—
72C23×C23—21no—
73C23×C23—21no—
74C23×C23—21no—
75C23×C23—41no—
76C23×C23—41no—
77C23×C23—41no—
78C23×C23—41no—
79C23×C23—41no—
80C23×C23—41no—
81C23×C23—82no—
82C23×C23—82no—
83C23×C23—84no—
84C23×C23—84no—
85C23×C23#10012yes—
86C23×C23#10112yes—
87C23×C23C2×C22⋊C411yescharacteristic
88C23×C23#11712yes—
89C2×C4×C23—22no—
90C2×C4×C23—42no—
91C2×C4×C23—44no—
92C2×C4×C23—42no—
93C2×C4×C23—42no—
94C2×C4×C23—42no—
95C2×C4×C23—42no—
96C2×C4×C23—42no—
97C2×C4×C23—42no—
98C2×C4×C23—42no—
99C2×C4×C23—82no—
100C2×C4×C23—84no—
101C2×C4×C23—84no—
102C2×C4×C23#12314yes—
103C8×C23—84no—
104#2315C2×C2311yescharacteristic
105#1462C2×C2312yes—
106#1305C2×C2314yescomplemented
107C22×C23—21no—
108C22×C23—21no—
109C22×C23—22no—
110C22×C23—22no—
111C22×C23—22no—
112C22×C23—22no—
113C22×C23—22no—
114C22×C23—42no—
115C22×C23—42no—
116C22×C23—42no—
117C22×C23—42no—
118C22×C23—42no—
119C22×C23—42no—
120C22×C23—41no—
121C22×C23—41no—
122C22×C23—41no—
123C22×C23—41no—
124C22×C23—88no—
125C22×C23#38111yescharacteristic
126C22×C23C2×#10011yescharacteristic
127C22×C23C2×#10111yescharacteristic
128C4×C23—24no—
129C4×C23—42no—
130C4×C23—82no—
131C4×C23—82no—
132C22×C22⋊C4C4×C2311yescharacteristic
133C22×C22⋊C4C4×C2311yescharacteristic
134C22×C22⋊C4C22×C2311yescharacteristic
135#407C22×C2314yes—
136#385—22no—
137#385—22no—
138#380C4×C2312yes—
139#380C22×C2312yes—
140#323—22no—
141#323—22no—
142C2×C23—21no—
143C2×C23—21no—
144C2×C23—42no—
145C2×C23—42no—
146C2×C23—42no—
147C2×C23#182111yescharacteristic
148C2×C23#182811yescharacteristic
149C2×C23#182911yescharacteristic
150C25C2×C4×C2311yescharacteristic
151C23×C4—21no—
152C23×C4—21no—
153C23×C4C2×C4×C2311yescharacteristic
154C23×C4C23×C2311yescharacteristic
155C2×M16—24no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—42no—
163C2×C22⋊C4C2×C4×C2312yes—
164C2×C22⋊C4C2×C4×C2312yes—
165C2×C22⋊C4D4×C2312yes—
166C2×C22⋊C4D4×C2312yes—
167C22⋊C8—24no—
168C22⋊C8—24no—
169#95—22no—
170#95—22no—
171C23#875611yescharacteristic, direct product factor, Sylow
172C24—22no—
173C24—22no—
174C24—22no—
175C24—81no—
176C24—81no—
177C24C22⋊C4×C2311yescharacteristic
178C24C22⋊C4×C2311yescharacteristic
179C24C22×C4×C2311yescharacteristic
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—81no—
186C22×C4—81no—
187C22×C4—81no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4—81no—
191C22×C4C22⋊C4×C2312yes—
192C22×C4C2×D4×C2312yes—
193C22×C4#41×C2314yes—
194M16—84no—
195C2×C8—44no—
196C4⋊C4—82no—
197C4⋊C4—82no—
198C22⋊C4—24no—
199C22⋊C4—24no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C23—28no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—82no—
218C23—82no—
219C23—84no—
220C23—84no—
221C23C23×#10012yes—
222C23C23×#10112yes—
223C23C2×C22⋊C4×C2311yescharacteristic
224C23C23×#11712yes—
225C2×C4—22no—
226C2×C4—42no—
227C2×C4—44no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—82no—
236C2×C4—84no—
237C2×C4—84no—
238C2×C4C23×#12314yes—
239C8—84no—
240C22—21no—
241C22—21no—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—41no—
254C22—41no—
255C22—41no—
256C22—41no—
257C22—88no—
258C22C23×#38111yescharacteristic
259C22C2×C23×#10011yescharacteristic
260C22C2×C23×#10111yescharacteristic
261C4—24no—
262C4—42no—
263C4—82no—
264C4—82no—
265C2—21no—
266C2—21no—
267C2—42no—
268C2—42no—
269C2—42no—
270C2C23×#182111yescharacteristic
271C2C23×#182811yescharacteristic
272C2C23×#182911yescharacteristic
subgroupG/H
center, socleC22×C23#381
commutatorC24C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC25×C23C2×C4
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
3. C23×#1305C2semidirect
67. #8756C23direct (N × H)
106. #1305C2×C23semidirect
171. C23#8756direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#23152
C23×#14622
C23×#13052
#875623