Finite Group Databasebeta

← C23×#2010 #15669629

Subgroups — C23×#2010 #15669629

Order 2944.

394 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 88 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×#575C211yesmaximal, characteristic, complemented
2C23×#525C212yesmaximal, complemented
3C23×#421C212yesmaximal, complemented
4C23×#407C211yesmaximal, characteristic, complemented
5C23×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
6C2×#41×C23C2212yes—
7C23×C4×C23C2211yescharacteristic
8C23×#137—22no—
9C23×AGL(1,ℤ/8)—22no—
10C23×#135—22no—
11C2×M16×C23C2211yescharacteristic
12C2×M16×C23C2211yescharacteristic
13C23×#123—22no—
14C23×C22⋊Q8—22no—
15C23×#121—22no—
16C23×#120—22no—
17C4×D4×C23—24no—
18C23×#117C2212yes—
19C23×#104—24no—
20C23×C22⋊C8—22no—
21C24×C23D411yescharacteristic
22#41×C23—22no—
23#41×C23—22no—
24#41×C23—22no—
25#41×C23—22no—
26C2×Q8×C23D412yescomplemented
27C2×D4×C23—24no—
28C2×D4×C23—22no—
29C2×D4×C23D412yescomplemented
30C22×C4×C23—22no—
31C22×C4×C23—22no—
32C22×C4×C23—22no—
33C22×C4×C23—21no—
34C22×C4×C23—21no—
35C22×C4×C23D411yescharacteristic
36C22×C4×C23C2311yescharacteristic
37Q16×C23—22no—
38SD16×C23—24no—
39D8×C23—22no—
40M16×C23—22no—
41M16×C23—22no—
42C2×C8×C23—21no—
43C2×C8×C23D412yescomplemented
44C4⋊C4×C23—24no—
45C4⋊C4×C23—22no—
46C22⋊C4×C23—24no—
47C22⋊C4×C23—24no—
48C22⋊C4×C23—22no—
49C42×C23D414yescomplemented
50C23×C23—22no—
51C23×C23—21no—
52C23×C23—22no—
53C23×C23—42no—
54C23×C23C2×D411yescharacteristic
55Q8×C23—22no—
56Q8×C23—22no—
57D4×C23—22no—
58D4×C23—22no—
59D4×C23—48no—
60D4×C23—42no—
61D4×C23—42no—
62C2×C4×C23—24no—
63C2×C4×C23—22no—
64C2×C4×C23—22no—
65C2×C4×C23—21no—
66C2×C4×C23—41no—
67C2×C4×C23—42no—
68C2×C4×C23—42no—
69C2×C4×C23—41no—
70C2×C4×C23—42no—
71C2×C4×C23C2×D412yes—
72C2×C4×C23C2×D411yescharacteristic
73C2×C4×C23C2×D412yes—
74C2×C4×C23#4111yescharacteristic
75C8×C23—22no—
76C8×C23—41no—
77#2010C2311yesmaximal, characteristic, direct product factor, Sylow
78C22×C23—24no—
79C22×C23—22no—
80C22×C23—41no—
81C22×C23—42no—
82C22×C23—41no—
83C22×C23—44no—
84C22×C23—41no—
85C22×C23—42no—
86C22×C23#12012yes—
87C22×C23#12111yescharacteristic
88C4×C23—44no—
89C4×C23—41no—
90C4×C23—42no—
91C4×C23#12011yescharacteristic
92C4×C23#12112yes—
93C4×C23#12711yescharacteristic
94#575C2×C2311yescharacteristic, complemented
95#525C2×C2312yescomplemented
96#421C2×C2312yescomplemented
97#407C2×C2311yescharacteristic, complemented
98H(3,ℤ/4)C2×C2311yescharacteristic, complemented
99C2×C23—22no—
100C2×C23—21no—
101C2×C23—42no—
102C2×C23—82no—
103C2×C23#39211yescharacteristic
104C2×#41C22×C2312yes—
105C23×C4C22×C2311yescharacteristic
106#137—22no—
107AGL(1,ℤ/8)—22no—
108#135—22no—
109C2×M16C22×C2311yescharacteristic
110C2×M16C22×C2311yescharacteristic
111#123—22no—
112C22⋊Q8—22no—
113#121—22no—
114#120—22no—
115C4×D4—24no—
116#117C22×C2312yes—
117#104—24no—
118C22⋊C8—22no—
119C23#201011yescharacteristic, direct product factor, Sylow
120C24D4×C2311yescharacteristic
121#41—22no—
122#41—22no—
123#41—22no—
124#41—22no—
125C2×Q8D4×C2312yescomplemented
126C2×D4—24no—
127C2×D4—22no—
128C2×D4D4×C2312yescomplemented
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4D4×C2311yescharacteristic
135C22×C4C23×C2311yescharacteristic
136Q16—22no—
137SD16—24no—
138D8—22no—
139M16—22no—
140M16—22no—
141C2×C8—21no—
142C2×C8D4×C2312yescomplemented
143C4⋊C4—24no—
144C4⋊C4—22no—
145C22⋊C4—24no—
146C22⋊C4—24no—
147C22⋊C4—22no—
148C42D4×C2314yescomplemented
149C23—22no—
150C23—21no—
151C23—22no—
152C23—42no—
153C23C2×D4×C2311yescharacteristic
154Q8—22no—
155Q8—22no—
156D4—22no—
157D4—22no—
158D4—48no—
159D4—42no—
160D4—42no—
161C2×C4—24no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4C2×D4×C2312yes—
171C2×C4C2×D4×C2311yescharacteristic
172C2×C4C2×D4×C2312yes—
173C2×C4#41×C2311yescharacteristic
174C8—22no—
175C8—41no—
176C22—24no—
177C22—22no—
178C22—41no—
179C22—42no—
180C22—41no—
181C22—44no—
182C22—41no—
183C22—42no—
184C22C23×#12012yes—
185C22C23×#12111yescharacteristic
186C4—44no—
187C4—41no—
188C4—42no—
189C4C23×#12011yescharacteristic
190C4C23×#12112yes—
191C4C23×#12711yescharacteristic
192C2—22no—
193C2—21no—
194C2—42no—
195C2—82no—
196C2C23×#39211yescharacteristic
subgroupG/H
centerC4×C23#127
commutator, Frattini, cosocleC22×C4C23×C23
socleC2×C23#392
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C23 ◁ C22×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#575C2semidirect
2. C23×#525C2semidirect
3. C23×#421C2semidirect
4. C23×#407C2semidirect
5. C23×H(3,ℤ/4)C2semidirect
26. C2×Q8×C23D4semidirect
29. C2×D4×C23D4semidirect
43. C2×C8×C23D4semidirect
49. C42×C23D4semidirect
77. #2010C23direct (N × H)
94. #575C2×C23semidirect
95. #525C2×C23semidirect
96. #421C2×C23semidirect
97. #407C2×C23semidirect
98. H(3,ℤ/4)C2×C23semidirect
119. C23#2010direct (N × H)
125. C2×Q8D4×C23semidirect
128. C2×D4D4×C23semidirect
142. C2×C8D4×C23semidirect
148. C42D4×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#5752
C23×#5252
C23×#4212
C23×#4072
C23×H(3,ℤ/4)2
#201023