Finite Group Databasebeta

← C23×#8757 #19156383

Subgroups — C23×#8757 #19156383

Order 5888.

438 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2316C211yesmaximal, characteristic
2C23×#1463C212yesmaximal
3C23×#1305C214yesmaximal
4C2×C23×#117C411yescharacteristic
5C2×C23×#117C411yescharacteristic
6C22×C4⋊C4×C23C2211yescharacteristic
7C23×#407C2214yes—
8C23×#389—22no—
9C23×#389—22no—
10C23×#380C412yes—
11C23×#380C2212yes—
12C23×#324—22no—
13C23×#324—22no—
14C23×C4×C23C2×C411yescharacteristic
15C23×C4×C23C2×C411yescharacteristic
16C23×C4×C23C2311yescharacteristic
17C2×M16×C23—24no—
18C23×#117—42no—
19C23×#117—42no—
20C2×C4⋊C4×C23—24no—
21C2×C4⋊C4×C23—21no—
22C2×C4⋊C4×C23—21no—
23C2×C4⋊C4×C23—22no—
24C2×C4⋊C4×C23—22no—
25C2×C4⋊C4×C23D412yes—
26C2×C4⋊C4×C23D412yes—
27C2×C22⋊C4×C23C2×C412yes—
28C2×C22⋊C4×C23C2×C412yes—
29C2×C42×C23—21no—
30C2×C42×C23—21no—
31C23×C22⋊C8—24no—
32C23×C22⋊C8—24no—
33C23×#95—22no—
34C23×#95—22no—
35C24×C23C22×C411yescharacteristic
36C22×C4×C23—22no—
37C22×C4×C23—22no—
38C22×C4×C23—22no—
39C22×C4×C23—22no—
40C22×C4×C23—22no—
41C22×C4×C23—22no—
42C22×C4×C23—81no—
43C22×C4×C23—81no—
44C22×C4×C23C22⋊C411yescharacteristic
45C22×C4×C23C22⋊C411yescharacteristic
46C22×C4×C23C22⋊C412yes—
47C22×C4×C23C2×D412yes—
48C22×C4×C23#4114yes—
49M16×C23—84no—
50C2×C8×C23—44no—
51C4⋊C4×C23—24no—
52C4⋊C4×C23—24no—
53C4⋊C4×C23—42no—
54C4⋊C4×C23—42no—
55C4⋊C4×C23—82no—
56C4⋊C4×C23—82no—
57C22⋊C4×C23—42no—
58C22⋊C4×C23—42no—
59C42×C23—42no—
60C42×C23—42no—
61#8757C2311yesmaximal, characteristic, direct product factor, Sylow
62C23×C23—21no—
63C23×C23—21no—
64C23×C23—41no—
65C23×C23—41no—
66C23×C23C2×C22⋊C411yescharacteristic
67C23×C23#11712yes—
68C2×C4×C23—28no—
69C2×C4×C23—21no—
70C2×C4×C23—21no—
71C2×C4×C23—21no—
72C2×C4×C23—21no—
73C2×C4×C23—22no—
74C2×C4×C23—42no—
75C2×C4×C23—44no—
76C2×C4×C23—42no—
77C2×C4×C23—42no—
78C2×C4×C23—42no—
79C2×C4×C23—42no—
80C2×C4×C23—42no—
81C2×C4×C23—42no—
82C2×C4×C23—42no—
83C2×C4×C23—82no—
84C2×C4×C23#10012yes—
85C2×C4×C23#10112yes—
86C2×C4×C23#12314yes—
87C8×C23—84no—
88#2316C2×C2311yescharacteristic
89#1463C2×C2312yes—
90#1305C2×C2314yes—
91C22×C23—21no—
92C22×C23—21no—
93C22×C23—22no—
94C22×C23—22no—
95C22×C23—22no—
96C22×C23—42no—
97C22×C23—42no—
98C22×C23#38111yescharacteristic
99C22×C23C2×#10011yescharacteristic
100C22×C23C2×#10111yescharacteristic
101C4×C23—22no—
102C4×C23—22no—
103C4×C23—24no—
104C4×C23—42no—
105C4×C23—82no—
106C4×C23—82no—
107C2×#117C4×C2311yescharacteristic
108C2×#117C4×C2311yescharacteristic
109C22×C4⋊C4C22×C2311yescharacteristic
110#407C22×C2314yes—
111#389—22no—
112#389—22no—
113#380C4×C2312yes—
114#380C22×C2312yes—
115#324—22no—
116#324—22no—
117C2×C23—21no—
118C2×C23—21no—
119C2×C23—42no—
120C2×C23#182111yescharacteristic
121C2×C23#182811yescharacteristic
122C2×C23#183011yescharacteristic
123C23×C4C2×C4×C2311yescharacteristic
124C23×C4C2×C4×C2311yescharacteristic
125C23×C4C23×C2311yescharacteristic
126C2×M16—24no—
127#117—42no—
128#117—42no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4D4×C2312yes—
135C2×C4⋊C4D4×C2312yes—
136C2×C22⋊C4C2×C4×C2312yes—
137C2×C22⋊C4C2×C4×C2312yes—
138C2×C42—21no—
139C2×C42—21no—
140C22⋊C8—24no—
141C22⋊C8—24no—
142#95—22no—
143#95—22no—
144C23#875711yescharacteristic, direct product factor, Sylow
145C24C22×C4×C2311yescharacteristic
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—81no—
153C22×C4—81no—
154C22×C4C22⋊C4×C2311yescharacteristic
155C22×C4C22⋊C4×C2311yescharacteristic
156C22×C4C22⋊C4×C2312yes—
157C22×C4C2×D4×C2312yes—
158C22×C4#41×C2314yes—
159M16—84no—
160C2×C8—44no—
161C4⋊C4—24no—
162C4⋊C4—24no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—82no—
166C4⋊C4—82no—
167C22⋊C4—42no—
168C22⋊C4—42no—
169C42—42no—
170C42—42no—
171C23—21no—
172C23—21no—
173C23—41no—
174C23—41no—
175C23C2×C22⋊C4×C2311yescharacteristic
176C23C23×#11712yes—
177C2×C4—28no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—42no—
184C2×C4—44no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—82no—
193C2×C4C23×#10012yes—
194C2×C4C23×#10112yes—
195C2×C4C23×#12314yes—
196C8—84no—
197C22—21no—
198C22—21no—
199C22—22no—
200C22—22no—
201C22—22no—
202C22—42no—
203C22—42no—
204C22C23×#38111yescharacteristic
205C22C2×C23×#10011yescharacteristic
206C22C2×C23×#10111yescharacteristic
207C4—22no—
208C4—22no—
209C4—24no—
210C4—42no—
211C4—82no—
212C4—82no—
213C2—21no—
214C2—21no—
215C2—42no—
216C2C23×#182111yescharacteristic
217C2C23×#182811yescharacteristic
218C2C23×#183011yescharacteristic
subgroupG/H
center, socleC22×C23#381
commutatorC24C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC24×C23C22×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
61. #8757C23direct (N × H)
144. C23#8757direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#23162
C23×#14632
C23×#13052
#875723