Finite Group Databasebeta

← C23×#7819 #19155445

Subgroups — C23×#7819 #19155445

Order 5888.

434 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2724C211yesmaximal, characteristic
2C23×#2093C211yesmaximal, characteristic
3C23×#2084C212yesmaximal, complemented
4C23×#1271C212yesmaximal, complemented
5C23×#1270C211yesmaximal, characteristic
6C2×C23×#123—21no—
7C2×C23×#107—21no—
8C2×C23×#106—22no—
9C2×C23×#102—21no—
10C23×#398—21no—
11C23×#397—22no—
12C23×#390C2211yescharacteristic
13C23×#389C2212yes—
14C23×#385—22no—
15C23×#384C2211yescharacteristic
16C23×#340—22no—
17C23×#336C2212yes—
18C23×#336C2211yescharacteristic
19C22×D4×C23D411yescharacteristic
20C23×C4×C23—21no—
21C22×C8×C23—22no—
22C22×C8×C23—21no—
23C23×#123—42no—
24C23×#123—42no—
25C2×C4⋊C4×C23—22no—
26C2×C4⋊C4×C23—22no—
27C2×C4⋊C4×C23—21no—
28C2×C4⋊C4×C23—21no—
29C2×C4⋊C4×C23D412yes—
30C2×C4⋊C4×C23D412yes—
31C2×C4⋊C4×C23D411yescharacteristic
32C2×C22⋊C4×C23—22no—
33C2×C22⋊C4×C23—21no—
34C2×C22⋊C4×C23—22no—
35C2×C42×C23C2311yescharacteristic
36C23×#107—42no—
37C23×#106—44no—
38C23×#102—42no—
39C23×#95—22no—
40C23×#95—21no—
41C23×#95—22no—
42C24×C23—21no—
43C2×D4×C23—21no—
44C2×D4×C23—41no—
45C2×D4×C23—41no—
46C2×D4×C23SD1612yes—
47C22×C4×C23—21no—
48C22×C4×C23—22no—
49C22×C4×C23—22no—
50C22×C4×C23—42no—
51C22×C4×C23—41no—
52C22×C4×C23—81no—
53C22×C4×C23—81no—
54C22×C4×C23C2×D411yescharacteristic
55C22×C4×C23C2×D412yes—
56C2×C8×C23—22no—
57C2×C8×C23—24no—
58C2×C8×C23—42no—
59C2×C8×C23—41no—
60C2×C8×C23—42no—
61C2×C8×C23—41no—
62C4⋊C4×C23—22no—
63C4⋊C4×C23—22no—
64C4⋊C4×C23—41no—
65C4⋊C4×C23—42no—
66C4⋊C4×C23—41no—
67C4⋊C4×C23—41no—
68C4⋊C4×C23—42no—
69C4⋊C4×C23—42no—
70C4⋊C4×C23—42no—
71C4⋊C4×C23—42no—
72C4⋊C4×C23—81no—
73C22⋊C4×C23—42no—
74C22⋊C4×C23—44no—
75C22⋊C4×C23—42no—
76C22⋊C4×C23—42no—
77C42×C23—41no—
78#7819C2311yesmaximal, characteristic, direct product factor, Sylow
79C23×C23—22no—
80C23×C23—42no—
81C23×C23—41no—
82C23×C23—42no—
83C23×C23—41no—
84C23×C23#12011yescharacteristic
85D4×C23—42no—
86D4×C23—81no—
87C2×C4×C23—21no—
88C2×C4×C23—22no—
89C2×C4×C23—22no—
90C2×C4×C23—21no—
91C2×C4×C23—42no—
92C2×C4×C23—42no—
93C2×C4×C23—42no—
94C2×C4×C23—42no—
95C2×C4×C23—41no—
96C2×C4×C23—42no—
97C2×C4×C23—42no—
98C2×C4×C23—42no—
99C2×C4×C23—84no—
100C2×C4×C23—82no—
101C2×C4×C23—81no—
102C2×C4×C23C2×SD1611yescharacteristic
103C2×C4×C23#13512yes—
104C2×C4×C23AGL(1,ℤ/8)12yes—
105C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
106C8×C23—44no—
107C8×C23—42no—
108#2724C2×C2311yescharacteristic
109#2093C2×C2311yescharacteristic
110#2084C2×C2312yescomplemented
111#1271C2×C2312yescomplemented
112#1270C2×C2311yescharacteristic
113C22×C23—42no—
114C22×C23—44no—
115C22×C23—82no—
116C22×C23—81no—
117C22×C23—81no—
118C22×C23#44912yes—
119C22×C23#45011yescharacteristic
120C22×C23#45311yescharacteristic
121C22×C23#45612yes—
122C22×C23#45811yescharacteristic
123C4×C23—22no—
124C4×C23—22no—
125C4×C23—21no—
126C4×C23—21no—
127C4×C23—82no—
128C4×C23—81no—
129C4×C23—82no—
130C4×C23—161no—
131C2×#123—21no—
132C2×#107—21no—
133C2×#106—22no—
134C2×#102—21no—
135#398—21no—
136#397—22no—
137#390C22×C2311yescharacteristic
138#389C22×C2312yes—
139#385—22no—
140#384C22×C2311yescharacteristic
141#340—22no—
142#336C22×C2312yes—
143#336C22×C2311yescharacteristic
144C2×C23—82no—
145C2×C23#160912yes—
146C2×C23#161311yescharacteristic
147C2×C23#162011yescharacteristic
148C2×C23#165412yes—
149C2×C23#166011yescharacteristic
150C22×D4D4×C2311yescharacteristic
151C23×C4—21no—
152C22×C8—22no—
153C22×C8—21no—
154#123—42no—
155#123—42no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4D4×C2312yes—
161C2×C4⋊C4D4×C2312yes—
162C2×C4⋊C4D4×C2311yescharacteristic
163C2×C22⋊C4—22no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—22no—
166C2×C42C23×C2311yescharacteristic
167#107—42no—
168#106—44no—
169#102—42no—
170#95—22no—
171#95—21no—
172#95—22no—
173C23#781911yescharacteristic, direct product factor, Sylow
174C24—21no—
175C2×D4—21no—
176C2×D4—41no—
177C2×D4—41no—
178C2×D4SD16×C2312yes—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—42no—
183C22×C4—41no—
184C22×C4—81no—
185C22×C4—81no—
186C22×C4C2×D4×C2311yescharacteristic
187C22×C4C2×D4×C2312yes—
188C2×C8—22no—
189C2×C8—24no—
190C2×C8—42no—
191C2×C8—41no—
192C2×C8—42no—
193C2×C8—41no—
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—81no—
205C22⋊C4—42no—
206C22⋊C4—44no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C42—41no—
210C23—22no—
211C23—42no—
212C23—41no—
213C23—42no—
214C23—41no—
215C23C23×#12011yescharacteristic
216D4—42no—
217D4—81no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—84no—
231C2×C4—82no—
232C2×C4—81no—
233C2×C4C2×SD16×C2311yescharacteristic
234C2×C4C23×#13512yes—
235C2×C4C23×AGL(1,ℤ/8)12yes—
236C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
237C8—44no—
238C8—42no—
239C22—44no—
240C22—42no—
241C22—81no—
242C22—82no—
243C22—81no—
244C22C23×#44912yes—
245C22C23×#45011yescharacteristic
246C22C23×#45311yescharacteristic
247C22C23×#45612yes—
248C22C23×#45811yescharacteristic
249C4—22no—
250C4—22no—
251C4—21no—
252C4—21no—
253C4—82no—
254C4—82no—
255C4—81no—
256C4—161no—
257C2—82no—
258C2C23×#160912yes—
259C2C23×#161311yescharacteristic
260C2C23×#162011yescharacteristic
261C2C23×#165412yes—
262C2C23×#166011yescharacteristic
subgroupG/H
center, socleC23×C23#120
commutator, Frattini, cosocleC2×C42C23×C23
prime radicalC22×D4×C23D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C23×#2084C2semidirect
4. C23×#1271C2semidirect
78. #7819C23direct (N × H)
110. #2084C2×C23semidirect
111. #1271C2×C23semidirect
173. C23#7819direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#27242
C23×#20932
C23×#20842
C23×#12712
C23×#12702
#781923