Finite Group Databasebeta

← C23×#8750 #19156376

Subgroups — C23×#8750 #19156376

Order 5888.

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

normal subgroups only

This group has 538 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×#1819C212yesmaximal
3C23×#1304C214yesmaximal
4C22×C22⋊C4×C23C2211yescharacteristic
5C23×#388—24no—
6C23×#380C412yescomplemented
7C23×#379C2214yes—
8C23×#379C2212yes—
9C4×C22⋊C4×C23C412yescomplemented
10C23×#342—24no—
11C25×C23C2311yescharacteristic
12C23×C4×C23C2×C411yescharacteristic
13C23×C4×C23C2×C411yescharacteristic
14C2×C4⋊C4×C23—22no—
15C2×C22⋊C4×C23—24no—
16C2×C22⋊C4×C23—24no—
17C2×C22⋊C4×C23—22no—
18C2×C22⋊C4×C23—22no—
19C2×C22⋊C4×C23—24no—
20C2×C22⋊C4×C23—24no—
21C2×C22⋊C4×C23C2×C414yes—
22C2×C22⋊C4×C23D414yes—
23C2×C42×C23—22no—
24C23×#95—22no—
25C23×#95—22no—
26C24×C23—42no—
27C24×C23—81no—
28C24×C23—81no—
29C24×C23C22×C411yescharacteristic
30C24×C23C2×D412yes—
31C24×C23#4114yes—
32C22×C4×C23—22no—
33C22×C4×C23—22no—
34C22×C4×C23—24no—
35C22×C4×C23—44no—
36C22×C4×C23—81no—
37C22×C4×C23—81no—
38C22×C4×C23C22⋊C412yes—
39C22×C4×C23C22⋊C412yes—
40C4⋊C4×C23—82no—
41C22⋊C4×C23—28no—
42C22⋊C4×C23—44no—
43C22⋊C4×C23—48no—
44C22⋊C4×C23—82no—
45C22⋊C4×C23—82no—
46C22⋊C4×C23—84no—
47C22⋊C4×C23—84no—
48C42×C23—82no—
49#8750C2311yesmaximal, characteristic, direct product factor, Sylow
50C23×C23—22no—
51C23×C23—24no—
52C23×C23—41no—
53C23×C23—41no—
54C23×C23—42no—
55C23×C23—42no—
56C23×C23—44no—
57C23×C23—44no—
58C23×C23—44no—
59C23×C23—88no—
60C23×C23C2×C22⋊C411yescharacteristic
61C23×C23#11712yes—
62C23×C23#12314yes—
63C2×C4×C23—22no—
64C2×C4×C23—22no—
65C2×C4×C23—42no—
66C2×C4×C23—42no—
67C2×C4×C23—44no—
68C2×C4×C23—44no—
69C2×C4×C23—44no—
70C2×C4×C23—82no—
71C2×C4×C23—82no—
72C2×C4×C23—84no—
73C2×C4×C23—84no—
74C2×C4×C23—84no—
75#2312C2×C2311yescharacteristic
76#1819C2×C2312yes—
77#1304C2×C2314yes—
78C22×C23—21no—
79C22×C23—21no—
80C22×C23—24no—
81C22×C23—22no—
82C22×C23—22no—
83C22×C23—22no—
84C22×C23—24no—
85C22×C23—24no—
86C22×C23—44no—
87C22×C23—84no—
88C22×C23—88no—
89C22×C23#38111yescharacteristic
90C22×C23#41012yes—
91C4×C23—42no—
92C4×C23—42no—
93C4×C23—84no—
94C4×C23—164no—
95C22×C22⋊C4C22×C2311yescharacteristic
96#388—24no—
97#380C4×C2312yescomplemented
98#379C22×C2314yes—
99#379C22×C2312yes—
100C4×C22⋊C4C4×C2312yescomplemented
101#342—24no—
102C2×C23—22no—
103C2×C23—44no—
104C2×C23—42no—
105C2×C23#182011yescharacteristic
106C2×C23#182811yescharacteristic
107C2×C23#182811yescharacteristic
108C25C23×C2311yescharacteristic
109C23×C4C2×C4×C2311yescharacteristic
110C23×C4C2×C4×C2311yescharacteristic
111C2×C4⋊C4—22no—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—24no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—24no—
118C2×C22⋊C4C2×C4×C2314yes—
119C2×C22⋊C4D4×C2314yes—
120C2×C42—22no—
121#95—22no—
122#95—22no—
123C23#875011yescharacteristic, direct product factor, Sylow
124C24—42no—
125C24—81no—
126C24—81no—
127C24C22×C4×C2311yescharacteristic
128C24C2×D4×C2312yes—
129C24#41×C2314yes—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—24no—
133C22×C4—44no—
134C22×C4—81no—
135C22×C4—81no—
136C22×C4C22⋊C4×C2312yes—
137C22×C4C22⋊C4×C2312yes—
138C4⋊C4—82no—
139C22⋊C4—28no—
140C22⋊C4—44no—
141C22⋊C4—48no—
142C22⋊C4—82no—
143C22⋊C4—82no—
144C22⋊C4—84no—
145C22⋊C4—84no—
146C42—82no—
147C23—22no—
148C23—24no—
149C23—41no—
150C23—41no—
151C23—42no—
152C23—42no—
153C23—44no—
154C23—44no—
155C23—44no—
156C23—88no—
157C23C2×C22⋊C4×C2311yescharacteristic
158C23C23×#11712yes—
159C23C23×#12314yes—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—44no—
165C2×C4—44no—
166C2×C4—44no—
167C2×C4—82no—
168C2×C4—82no—
169C2×C4—84no—
170C2×C4—84no—
171C2×C4—84no—
172C22—21no—
173C22—21no—
174C22—24no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—24no—
179C22—24no—
180C22—44no—
181C22—84no—
182C22—88no—
183C22C23×#38111yescharacteristic
184C22C23×#41012yes—
185C4—42no—
186C4—42no—
187C4—84no—
188C4—164no—
189C2—22no—
190C2—44no—
191C2—42no—
192C2C23×#182011yescharacteristic
193C2C23×#182811yescharacteristic
194C2C23×#182811yescharacteristic
subgroupG/H
center, socleC22×C23#381
commutatorC24C22×C4×C23
Frattini, cosocleC25C23×C23
prime radicalC25×C23C23
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C25×C23 ◁ G

Complements

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

NH ≅ G/Ntype
6. C23×#380C4semidirect
9. C4×C22⋊C4×C23C4semidirect
49. #8750C23direct (N × H)
97. #380C4×C23semidirect
100. C4×C22⋊C4C4×C23semidirect
123. C23#8750direct (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×#18192
C23×#13042
#875023