Finite Group Databasebeta

← C23×#8704 #19156330

Subgroups — C23×#8704 #19156330

Order 5888.

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

normal subgroups only

This group has 830 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
1C2×C23×#380C211yesmaximal, characteristic
2C23×#1779C212yesmaximal
3C23×#1280C214yesmaximal, complemented
4C24×C4×C23C2211yescharacteristic
5C22×C4⋊C4×C23C412yes—
6C22×C22⋊C4×C23C412yes—
7C2×C23×C22⋊C8—22no—
8C2×C23×C22⋊C8—22no—
9C2×C23×C22⋊C8C2214yes—
10C23×#380—22no—
11C23×#380—22no—
12C2×C23×#95C2212yes—
13C25×C23C2×C411yescharacteristic
14C23×C4×C23—24no—
15C23×C4×C23—44no—
16C23×C4×C23C2×C411yescharacteristic
17C23×C4×C23C2×C414yes—
18C23×C4×C23D412yes—
19C23×C4×C23D412yes—
20C23×C4×C23C2311yescharacteristic
21C22×C8×C23—24no—
22C2×C4⋊C4×C23—22no—
23C2×C4⋊C4×C23—22no—
24C2×C4⋊C4×C23—44no—
25C2×C22⋊C4×C23—22no—
26C2×C22⋊C4×C23—22no—
27C2×C22⋊C4×C23—44no—
28C23×C22⋊C8—28no—
29C23×C22⋊C8—44no—
30C23×C22⋊C8—44no—
31C23×#95—24no—
32C24×C23—22no—
33C24×C23—22no—
34C24×C23—22no—
35C24×C23—24no—
36C24×C23—42no—
37C24×C23C22⋊C411yescharacteristic
38C24×C23C22⋊C411yescharacteristic
39C24×C23C22×C411yescharacteristic
40C22×C4×C23—22no—
41C22×C4×C23—24no—
42C22×C4×C23—24no—
43C22×C4×C23—24no—
44C22×C4×C23—24no—
45C22×C4×C23—24no—
46C22×C4×C23—24no—
47C22×C4×C23—28no—
48C22×C4×C23—42no—
49C22×C4×C23—42no—
50C22×C4×C23—42no—
51C22×C4×C23—416no—
52C22×C4×C23—44no—
53C22×C4×C23—44no—
54C22×C4×C23C22⋊C412yes—
55C22×C4×C23C2×D412yes—
56C22×C4×C23#4114yes—
57C2×C8×C23—44no—
58C2×C8×C23—44no—
59C2×C8×C23—44no—
60C4⋊C4×C23—44no—
61C4⋊C4×C23—44no—
62C22⋊C4×C23—44no—
63C22⋊C4×C23—44no—
64#8704C2311yesmaximal, characteristic, direct product factor, Sylow
65C23×C23—22no—
66C23×C23—21no—
67C23×C23—21no—
68C23×C23—28no—
69C23×C23—21no—
70C23×C23—21no—
71C23×C23—22no—
72C23×C23—22no—
73C23×C23—22no—
74C23×C23—22no—
75C23×C23—21no—
76C23×C23—21no—
77C23×C23—48no—
78C23×C23—48no—
79C23×C23—44no—
80C23×C23—44no—
81C23×C23#10012yes—
82C23×C23#10112yes—
83C23×C23C2×C22⋊C411yescharacteristic
84C23×C23#11712yes—
85C23×C23#13114yes—
86C2×C4×C23—28no—
87C2×C4×C23—22no—
88C2×C4×C23—22no—
89C2×C4×C23—22no—
90C2×C4×C23—22no—
91C2×C4×C23—44no—
92C2×C4×C23—48no—
93C2×C4×C23—48no—
94C2×C4×C23—44no—
95C2×C4×C23—44no—
96C2×C4×C23—44no—
97C2×C4×C23—88no—
98C2×C4×C23#9914yes—
99C2×C4×C23#12314yes—
100C8×C23—84no—
101C2×#380C2×C2311yescharacteristic
102#1779C2×C2312yes—
103#1280C2×C2314yescomplemented
104C22×C23—22no—
105C22×C23—22no—
106C22×C23—24no—
107C22×C23—24no—
108C22×C23—22no—
109C22×C23—22no—
110C22×C23—24no—
111C22×C23—24no—
112C22×C23—416no—
113C22×C23—41no—
114C22×C23—41no—
115C22×C23—41no—
116C22×C23—41no—
117C22×C23—42no—
118C22×C23—42no—
119C22×C23#38111yescharacteristic
120C22×C23#40812yes—
121C22×C23C2×#9912yes—
122C22×C23C2×#10011yescharacteristic
123C22×C23C2×#10111yescharacteristic
124C22×C23#43314yes—
125C4×C23—24no—
126C4×C23—24no—
127C4×C23—84no—
128C24×C4C22×C2311yescharacteristic
129C22×C4⋊C4C4×C2312yes—
130C22×C22⋊C4C4×C2312yes—
131C2×C22⋊C8—22no—
132C2×C22⋊C8—22no—
133C2×C22⋊C8C22×C2314yes—
134#380—22no—
135#380—22no—
136C2×#95C22×C2312yes—
137C2×C23—24no—
138C2×C23—42no—
139C2×C23—42no—
140C2×C23#147112yes—
141C2×C23#147212yes—
142C2×C23#181811yescharacteristic
143C2×C23#181911yescharacteristic
144C2×C23#182111yescharacteristic
145C25C2×C4×C2311yescharacteristic
146C23×C4—24no—
147C23×C4—44no—
148C23×C4C2×C4×C2311yescharacteristic
149C23×C4C2×C4×C2314yes—
150C23×C4D4×C2312yes—
151C23×C4D4×C2312yes—
152C23×C4C23×C2311yescharacteristic
153C22×C8—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—44no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—44no—
160C22⋊C8—28no—
161C22⋊C8—44no—
162C22⋊C8—44no—
163#95—24no—
164C23#870411yescharacteristic, direct product factor, Sylow
165C24—22no—
166C24—22no—
167C24—22no—
168C24—24no—
169C24—42no—
170C24C22⋊C4×C2311yescharacteristic
171C24C22⋊C4×C2311yescharacteristic
172C24C22×C4×C2311yescharacteristic
173C22×C4—22no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—28no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—416no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4C22⋊C4×C2312yes—
188C22×C4C2×D4×C2312yes—
189C22×C4#41×C2314yes—
190C2×C8—44no—
191C2×C8—44no—
192C2×C8—44no—
193C4⋊C4—44no—
194C4⋊C4—44no—
195C22⋊C4—44no—
196C22⋊C4—44no—
197C23—22no—
198C23—21no—
199C23—21no—
200C23—28no—
201C23—21no—
202C23—21no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—21no—
208C23—21no—
209C23—48no—
210C23—48no—
211C23—44no—
212C23—44no—
213C23C23×#10012yes—
214C23C23×#10112yes—
215C23C2×C22⋊C4×C2311yescharacteristic
216C23C23×#11712yes—
217C23C23×#13114yes—
218C2×C4—28no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—44no—
224C2×C4—48no—
225C2×C4—48no—
226C2×C4—44no—
227C2×C4—44no—
228C2×C4—44no—
229C2×C4—88no—
230C2×C4C23×#9914yes—
231C2×C4C23×#12314yes—
232C8—84no—
233C22—24no—
234C22—24no—
235C22—22no—
236C22—22no—
237C22—24no—
238C22—24no—
239C22—22no—
240C22—22no—
241C22—416no—
242C22—42no—
243C22—42no—
244C22—41no—
245C22—41no—
246C22—41no—
247C22—41no—
248C22C23×#38111yescharacteristic
249C22C23×#40812yes—
250C22C2×C23×#9912yes—
251C22C2×C23×#10011yescharacteristic
252C22C2×C23×#10111yescharacteristic
253C22C23×#43314yes—
254C4—24no—
255C4—24no—
256C4—84no—
257C2—24no—
258C2—42no—
259C2—42no—
260C2C23×#147112yes—
261C2C23×#147212yes—
262C2C23×#181811yescharacteristic
263C2C23×#181911yescharacteristic
264C2C23×#182111yescharacteristic
subgroupG/H
center, socleC23×C23C2×C22⋊C4
commutatorC24C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC25×C23C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23×C23 ◁ C24×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C23×#1280C2semidirect
64. #8704C23direct (N × H)
103. #1280C2×C23semidirect
164. C23#8704direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#3802
C23×#17792
C23×#12802
#870423