Finite Group Databasebeta

← C23×#9022 #19156648

Subgroups — C23×#9022 #19156648

Order 5888.

572 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 150 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×#400C211yesmaximal, characteristic
2C23×#2313C211yesmaximal, characteristic
3C23×#1824C212yesmaximal
4C2×C23×#324C211yesmaximal, characteristic
5C23×#1294C212yesmaximal
6C22×C4⋊C4×C23C2211yescharacteristic
7C22×C4⋊C4×C23C2211yescharacteristic
8C2×C23×C22⋊C8C2212yes—
9C23×#400—22no—
10C23×#400C414yescomplemented
11C23×#387—24no—
12C23×#381—21no—
13C23×#380C2212yes—
14C4×C22⋊C4×C23—21no—
15C2×C23×#95—21no—
16C2×C23×#95—21no—
17C2×C23×#95C2211yescharacteristic
18C23×#340—24no—
19C23×#324—22no—
20C23×C4×C23—21no—
21C23×C4×C23—21no—
22C23×C4×C23D411yescharacteristic
23C23×C4×C23D411yescharacteristic
24C23×C4×C23C2311yescharacteristic
25C22×C8×C23—22no—
26C2×C4⋊C4×C23—22no—
27C2×C4⋊C4×C23—22no—
28C2×C4⋊C4×C23—22no—
29C2×C4⋊C4×C23—22no—
30C2×C4⋊C4×C23—21no—
31C2×C4⋊C4×C23C2×C412yes—
32C2×C4⋊C4×C23C2×C412yes—
33C2×C4⋊C4×C23D412yes—
34C2×C4⋊C4×C23D414yes—
35C2×C22⋊C4×C23—22no—
36C2×C42×C23—22no—
37C23×C22⋊C8—22no—
38C23×C22⋊C8—22no—
39C23×#95—22no—
40C23×#95—24no—
41C23×#95—24no—
42C23×#95—24no—
43C23×#95—21no—
44C23×#95—22no—
45C23×#95C2×C412yes—
46C24×C23C2×D411yescharacteristic
47C22×C4×C23—21no—
48C22×C4×C23—21no—
49C22×C4×C23—21no—
50C22×C4×C23—22no—
51C22×C4×C23—22no—
52C22×C4×C23—21no—
53C22×C4×C23—22no—
54C22×C4×C23—22no—
55C22×C4×C23—24no—
56C22×C4×C23—42no—
57C22×C4×C23—42no—
58C22×C4×C23—42no—
59C22×C4×C23—42no—
60C22×C4×C23—44no—
61C22×C4×C23—41no—
62C22×C4×C23—41no—
63C22×C4×C23—42no—
64C22×C4×C23C22⋊C412yes—
65C22×C4×C23C22⋊C412yes—
66C22×C4×C23C22×C411yescharacteristic
67C22×C4×C23C2×D412yes—
68C22×C4×C23C2×D411yescharacteristic
69C22×C4×C23#4112yes—
70C2×C8×C23—44no—
71C2×C8×C23—42no—
72C4⋊C4×C23—24no—
73C4⋊C4×C23—24no—
74C4⋊C4×C23—24no—
75C4⋊C4×C23—44no—
76C4⋊C4×C23—42no—
77C22⋊C4×C23—42no—
78C22⋊C4×C23—42no—
79C42×C23—82no—
80#9022C2311yesmaximal, characteristic, direct product factor, Sylow
81C23×C23—22no—
82C23×C23—22no—
83C23×C23C2×C22⋊C411yescharacteristic
84C23×C23#12011yescharacteristic
85C23×C23#12311yescharacteristic
86C23×C23#13512yes—
87C23×C23#13712yes—
88C2×C4×C23—24no—
89C2×C4×C23—22no—
90C2×C4×C23—22no—
91C2×C4×C23—22no—
92C2×C4×C23—22no—
93C2×C4×C23—41no—
94C2×C4×C23—41no—
95C2×C4×C23—42no—
96C2×C4×C23—42no—
97C2×C4×C23—42no—
98C2×C4×C23—42no—
99C2×C4×C23—42no—
100C2×C4×C23—42no—
101C2×C4×C23—42no—
102C2×C4×C23—44no—
103C2×C4×C23—42no—
104C2×C4×C23—42no—
105C2×C4×C23—44no—
106C2×C4×C23—44no—
107C2×C4×C23—44no—
108C2×C4×C23—81no—
109C2×C4×C23—84no—
110C2×C4×C23—82no—
111C2×C4×C23—82no—
112C2×C4×C23C4×D412yes—
113C2×C4×C23#12112yes—
114C8×C23—82no—
115C2×#400C2×C2311yescharacteristic
116#2313C2×C2311yescharacteristic
117#1824C2×C2312yes—
118C2×#324C2×C2311yescharacteristic
119#1294C2×C2312yes—
120C22×C23—24no—
121C22×C23—24no—
122C22×C23—21no—
123C22×C23—21no—
124C22×C23—21no—
125C22×C23—21no—
126C22×C23#38611yescharacteristic
127C22×C23#41611yescharacteristic
128C22×C23#41911yescharacteristic
129C22×C23#45212yes—
130C22×C23#45412yes—
131C22×C23#45812yes—
132C22×C23#48512yes—
133C4×C23—22no—
134C4×C23—22no—
135C4×C23—42no—
136C4×C23—42no—
137C4×C23—82no—
138C4×C23—84no—
139C4×C23—82no—
140C22×C4⋊C4C22×C2311yescharacteristic
141C22×C4⋊C4C22×C2311yescharacteristic
142C2×C22⋊C8C22×C2312yes—
143#400—22no—
144#400C4×C2314yescomplemented
145#387—24no—
146#381—21no—
147#380C22×C2312yes—
148C4×C22⋊C4—21no—
149C2×#95—21no—
150C2×#95—21no—
151C2×#95C22×C2311yescharacteristic
152#340—24no—
153#324—22no—
154C2×C23—21no—
155C2×C23—21no—
156C2×C23—22no—
157C2×C23#161514yes—
158C2×C23#187311yescharacteristic
159C2×C23#188811yescharacteristic
160C2×C23#189011yescharacteristic
161C23×C4—21no—
162C23×C4—21no—
163C23×C4D4×C2311yescharacteristic
164C23×C4D4×C2311yescharacteristic
165C23×C4C23×C2311yescharacteristic
166C22×C8—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4C2×C4×C2312yes—
173C2×C4⋊C4C2×C4×C2312yes—
174C2×C4⋊C4D4×C2312yes—
175C2×C4⋊C4D4×C2314yes—
176C2×C22⋊C4—22no—
177C2×C42—22no—
178C22⋊C8—22no—
179C22⋊C8—22no—
180#95—22no—
181#95—24no—
182#95—24no—
183#95—24no—
184#95—21no—
185#95—22no—
186#95C2×C4×C2312yes—
187C23#902211yescharacteristic, direct product factor, Sylow
188C24C2×D4×C2311yescharacteristic
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—24no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—44no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—42no—
206C22×C4C22⋊C4×C2312yes—
207C22×C4C22⋊C4×C2312yes—
208C22×C4C22×C4×C2311yescharacteristic
209C22×C4C2×D4×C2312yes—
210C22×C4C2×D4×C2311yescharacteristic
211C22×C4#41×C2312yes—
212C2×C8—44no—
213C2×C8—42no—
214C4⋊C4—24no—
215C4⋊C4—24no—
216C4⋊C4—24no—
217C4⋊C4—44no—
218C4⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C42—82no—
222C23—22no—
223C23—22no—
224C23C2×C22⋊C4×C2311yescharacteristic
225C23C23×#12011yescharacteristic
226C23C23×#12311yescharacteristic
227C23C23×#13512yes—
228C23C23×#13712yes—
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—44no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—81no—
250C2×C4—84no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4C4×D4×C2312yes—
254C2×C4C23×#12112yes—
255C8—82no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—24no—
261C22—24no—
262C22C23×#38611yescharacteristic
263C22C23×#41611yescharacteristic
264C22C23×#41911yescharacteristic
265C22C23×#45212yes—
266C22C23×#45412yes—
267C22C23×#45812yes—
268C22C23×#48512yes—
269C4—22no—
270C4—22no—
271C4—42no—
272C4—42no—
273C4—82no—
274C4—82no—
275C4—84no—
276C2—21no—
277C2—21no—
278C2—22no—
279C2C23×#161514yes—
280C2C23×#187311yescharacteristic
281C2C23×#188811yescharacteristic
282C2C23×#189011yescharacteristic
subgroupG/H
center, socleC23×C23#120
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC24×C23C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
10. C23×#400C4semidirect
80. #9022C23direct (N × H)
144. #400C4×C23semidirect
187. C23#9022direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#4002
C23×#23132
C23×#18242
C2×C23×#3242
C23×#12942
#902223