Finite Group Databasebeta

← C23×#8744 #19156370

Subgroups — C23×#8744 #19156370

Order 5888.

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

normal subgroups only

This group has 384 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×#2298C211yesmaximal, characteristic
2C23×#1464C212yesmaximal
3C23×#1285C214yesmaximal, complemented
4C2×C23×#117C2211yescharacteristic
5C23×#432C2214yes—
6C23×#388—22no—
7C23×#387—22no—
8C4×C4⋊C4×C23C2212yes—
9C4×C22⋊C4×C23C412yes—
10C23×#376C411yescharacteristic
11C43×C23C411yescharacteristic
12C23×#324—22no—
13C23×#323—22no—
14C23×C4×C23C2×C411yescharacteristic
15C23×#117—24no—
16C2×C4⋊C4×C23—22no—
17C2×C4⋊C4×C23D412yes—
18C2×C22⋊C4×C23—22no—
19C2×C22⋊C4×C23D412yes—
20C2×C42×C23—22no—
21C2×C42×C23C2×C411yescharacteristic
22C2×C42×C23C2×C412yes—
23C2×C42×C23C2×C412yes—
24C2×C42×C23C2311yescharacteristic
25C23×C4⋊C8—24no—
26C23×C22⋊C8—24no—
27C23×#97—24no—
28C23×#95—22no—
29C23×#95—22no—
30C23×#95—22no—
31C24×C23C22⋊C411yescharacteristic
32C22×C4×C23—22no—
33C22×C4×C23—22no—
34C22×C4×C23—22no—
35C22×C4×C23—42no—
36C22×C4×C23C22⋊C411yescharacteristic
37C22×C4×C23C22⋊C412yes—
38C22×C4×C23C22×C411yescharacteristic
39C22×C4×C23C2×D412yes—
40C2×C8×C23—44no—
41C4⋊C4×C23—24no—
42C4⋊C4×C23—44no—
43C22⋊C4×C23—24no—
44C22⋊C4×C23—44no—
45C42×C23—22no—
46C42×C23—42no—
47C42×C23—42no—
48C42×C23—44no—
49C42×C23#4114yes—
50#8744C2311yesmaximal, characteristic, direct product factor, Sylow
51C23×C23—21no—
52C23×C23—22no—
53C23×C23—42no—
54C23×C23C2×C22⋊C411yescharacteristic
55C2×C4×C23—22no—
56C2×C4×C23—24no—
57C2×C4×C23—22no—
58C2×C4×C23—21no—
59C2×C4×C23—22no—
60C2×C4×C23—42no—
61C2×C4×C23—44no—
62C2×C4×C23—42no—
63C2×C4×C23—42no—
64C2×C4×C23—42no—
65C2×C4×C23—42no—
66C2×C4×C23—42no—
67C2×C4×C23—42no—
68C2×C4×C23#11712yes—
69C2×C4×C23#12314yes—
70C8×C23—84no—
71#2298C2×C2311yescharacteristic
72#1464C2×C2312yes—
73#1285C2×C2314yescomplemented
74C22×C23—22no—
75C22×C23—41no—
76C22×C23—42no—
77C22×C23—82no—
78C22×C23#38111yescharacteristic
79C22×C23#41312yes—
80C4×C23—22no—
81C4×C23—24no—
82C4×C23—44no—
83C4×C23—41no—
84C4×C23—42no—
85C4×C23—82no—
86C2×#117C22×C2311yescharacteristic
87#432C22×C2314yes—
88#388—22no—
89#387—22no—
90C4×C4⋊C4C22×C2312yes—
91C4×C22⋊C4C4×C2312yes—
92#376C4×C2311yescharacteristic
93C43C4×C2311yescharacteristic
94#324—22no—
95#323—22no—
96C2×C23—22no—
97C2×C23—81no—
98C2×C23#182211yescharacteristic
99C2×C23#182712yes—
100C23×C4C2×C4×C2311yescharacteristic
101#117—24no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4D4×C2312yes—
104C2×C22⋊C4—22no—
105C2×C22⋊C4D4×C2312yes—
106C2×C42—22no—
107C2×C42C2×C4×C2311yescharacteristic
108C2×C42C2×C4×C2312yes—
109C2×C42C2×C4×C2312yes—
110C2×C42C23×C2311yescharacteristic
111C4⋊C8—24no—
112C22⋊C8—24no—
113#97—24no—
114#95—22no—
115#95—22no—
116#95—22no—
117C23#874411yescharacteristic, direct product factor, Sylow
118C24C22⋊C4×C2311yescharacteristic
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—42no—
123C22×C4C22⋊C4×C2311yescharacteristic
124C22×C4C22⋊C4×C2312yes—
125C22×C4C22×C4×C2311yescharacteristic
126C22×C4C2×D4×C2312yes—
127C2×C8—44no—
128C4⋊C4—24no—
129C4⋊C4—44no—
130C22⋊C4—24no—
131C22⋊C4—44no—
132C42—22no—
133C42—42no—
134C42—42no—
135C42—44no—
136C42#41×C2314yes—
137C23—21no—
138C23—22no—
139C23—42no—
140C23C2×C22⋊C4×C2311yescharacteristic
141C2×C4—22no—
142C2×C4—24no—
143C2×C4—22no—
144C2×C4—21no—
145C2×C4—22no—
146C2×C4—42no—
147C2×C4—44no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4C23×#11712yes—
155C2×C4C23×#12314yes—
156C8—84no—
157C22—22no—
158C22—41no—
159C22—42no—
160C22—82no—
161C22C23×#38111yescharacteristic
162C22C23×#41312yes—
163C4—22no—
164C4—24no—
165C4—44no—
166C4—41no—
167C4—42no—
168C4—82no—
169C2—22no—
170C2—81no—
171C2C23×#182211yescharacteristic
172C2C23×#182712yes—
subgroupG/H
center, socleC22×C23#381
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC24×C23C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C23×#1285C2semidirect
50. #8744C23direct (N × H)
73. #1285C2×C23semidirect
117. C23#8744direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#22982
C23×#14642
C23×#12852
#874423