Finite Group Databasebeta

← C23×#8759 #19156385

Subgroups — C23×#8759 #19156385

Order 5888.

474 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 110 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×#2312C211yesmaximal, characteristic
2C23×#1462C212yesmaximal
3C23×#1305C214yesmaximal, complemented
4C22×C22⋊C4×C23C2211yescharacteristic
5C23×#407C2214yes—
6C23×#388—24no—
7C23×#380C2212yes—
8C23×#379C412yes—
9C4×C22⋊C4×C23C412yes—
10C23×#323—24no—
11C25×C23C2×C411yescharacteristic
12C23×C4×C23C2×C411yescharacteristic
13C23×C4×C23C2311yescharacteristic
14C2×M16×C23—24no—
15C2×C4⋊C4×C23—22no—
16C2×C22⋊C4×C23—24no—
17C2×C22⋊C4×C23—22no—
18C2×C22⋊C4×C23—22no—
19C2×C22⋊C4×C23C2×C414yes—
20C2×C22⋊C4×C23D414yes—
21C2×C42×C23—22no—
22C23×C22⋊C8—24no—
23C23×C22⋊C8—24no—
24C23×#95—22no—
25C23×#95—22no—
26C24×C23—22no—
27C24×C23—42no—
28C24×C23—81no—
29C24×C23—81no—
30C24×C23C22⋊C412yes—
31C24×C23C22×C411yescharacteristic
32C22×C4×C23—22no—
33C22×C4×C23—24no—
34C22×C4×C23—81no—
35C22×C4×C23—81no—
36C22×C4×C23C22⋊C412yes—
37C22×C4×C23C2×D412yes—
38C22×C4×C23#4114yes—
39M16×C23—84no—
40C2×C8×C23—44no—
41C4⋊C4×C23—82no—
42C22⋊C4×C23—28no—
43C22⋊C4×C23—44no—
44C22⋊C4×C23—82no—
45C22⋊C4×C23—82no—
46C42×C23—82no—
47#8759C2311yesmaximal, characteristic, direct product factor, Sylow
48C23×C23—22no—
49C23×C23—22no—
50C23×C23—24no—
51C23×C23—41no—
52C23×C23—41no—
53C23×C23—44no—
54C23×C23—42no—
55C23×C23—42no—
56C23×C23—82no—
57C23×C23—82no—
58C23×C23—88no—
59C23×C23C2×C22⋊C411yescharacteristic
60C23×C23#11712yes—
61C2×C4×C23—22no—
62C2×C4×C23—42no—
63C2×C4×C23—44no—
64C2×C4×C23—42no—
65C2×C4×C23—44no—
66C2×C4×C23—44no—
67C2×C4×C23—44no—
68C2×C4×C23—82no—
69C2×C4×C23#12314yes—
70C8×C23—84no—
71#2312C2×C2311yescharacteristic
72#1462C2×C2312yes—
73#1305C2×C2314yescomplemented
74C22×C23—21no—
75C22×C23—21no—
76C22×C23—22no—
77C22×C23—22no—
78C22×C23—22no—
79C22×C23—44no—
80C22×C23—42no—
81C22×C23—42no—
82C22×C23—42no—
83C22×C23—84no—
84C22×C23—88no—
85C22×C23#38111yescharacteristic
86C22×C23#41312yes—
87C4×C23—24no—
88C4×C23—42no—
89C4×C23—84no—
90C22×C22⋊C4C22×C2311yescharacteristic
91#407C22×C2314yes—
92#388—24no—
93#380C22×C2312yes—
94#379C4×C2312yes—
95C4×C22⋊C4C4×C2312yes—
96#323—24no—
97C2×C23—22no—
98C2×C23—42no—
99C2×C23—82no—
100C2×C23#182211yescharacteristic
101C2×C23#182811yescharacteristic
102C2×C23#182911yescharacteristic
103C25C2×C4×C2311yescharacteristic
104C23×C4C2×C4×C2311yescharacteristic
105C23×C4C23×C2311yescharacteristic
106C2×M16—24no—
107C2×C4⋊C4—22no—
108C2×C22⋊C4—24no—
109C2×C22⋊C4—22no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4C2×C4×C2314yes—
112C2×C22⋊C4D4×C2314yes—
113C2×C42—22no—
114C22⋊C8—24no—
115C22⋊C8—24no—
116#95—22no—
117#95—22no—
118C23#875911yescharacteristic, direct product factor, Sylow
119C24—22no—
120C24—42no—
121C24—81no—
122C24—81no—
123C24C22⋊C4×C2312yes—
124C24C22×C4×C2311yescharacteristic
125C22×C4—22no—
126C22×C4—24no—
127C22×C4—81no—
128C22×C4—81no—
129C22×C4C22⋊C4×C2312yes—
130C22×C4C2×D4×C2312yes—
131C22×C4#41×C2314yes—
132M16—84no—
133C2×C8—44no—
134C4⋊C4—82no—
135C22⋊C4—28no—
136C22⋊C4—44no—
137C22⋊C4—82no—
138C22⋊C4—82no—
139C42—82no—
140C23—22no—
141C23—22no—
142C23—24no—
143C23—41no—
144C23—41no—
145C23—44no—
146C23—42no—
147C23—42no—
148C23—82no—
149C23—82no—
150C23—88no—
151C23C2×C22⋊C4×C2311yescharacteristic
152C23C23×#11712yes—
153C2×C4—22no—
154C2×C4—42no—
155C2×C4—44no—
156C2×C4—42no—
157C2×C4—44no—
158C2×C4—44no—
159C2×C4—44no—
160C2×C4—82no—
161C2×C4C23×#12314yes—
162C8—84no—
163C22—21no—
164C22—21no—
165C22—22no—
166C22—22no—
167C22—22no—
168C22—44no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22—84no—
173C22—88no—
174C22C23×#38111yescharacteristic
175C22C23×#41312yes—
176C4—24no—
177C4—42no—
178C4—84no—
179C2—22no—
180C2—42no—
181C2—82no—
182C2C23×#182211yescharacteristic
183C2C23×#182811yescharacteristic
184C2C23×#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
47. #8759C23direct (N × H)
73. #1305C2×C23semidirect
118. C23#8759direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#23122
C23×#14622
C23×#13052
#875923