Finite Group Databasebeta

← C23×#1510 #15669129

Subgroups — C23×#1510 #15669129

Order 2944.

326 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 92 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
1C2×C23×#121C211yesmaximal, characteristic
2C23×#407C211yesmaximal, characteristic, complemented
3C23×#381C211yesmaximal, characteristic, complemented
4C23×#327C214yesmaximal, complemented
5C22×D4×C23—21no—
6C22×D4×C23C411yescharacteristic, complemented
7C23×C4×C23C2211yescharacteristic
8C2×M16×C23—21no—
9C23×#121—21no—
10C23×#121—21no—
11C23×#121C412yescomplemented
12C23×#121C2212yes—
13C2×C4⋊C4×C23C411yescharacteristic, complemented
14C2×C22⋊C4×C23—21no—
15C2×C22⋊C4×C23—21no—
16C23×C22⋊C8—21no—
17C23×C22⋊C8C2212yescomplemented
18C23×#95—21no—
19C23×#95C2212yescomplemented
20C24×C23—21no—
21C24×C23D411yescharacteristic
22C2×D4×C23—22no—
23C2×D4×C23—22no—
24C2×D4×C23—21no—
25C2×D4×C23—42no—
26C2×D4×C23—44no—
27C2×D4×C23C2×C412yes—
28C22×C4×C23—21no—
29C22×C4×C23—22no—
30C22×C4×C23—42no—
31C22×C4×C23C2×C411yescharacteristic
32C22×C4×C23C2×C411yescharacteristic
33C22×C4×C23D411yescharacteristic
34C22×C4×C23D411yescharacteristic
35C22×C4×C23D411yescharacteristic
36C22×C4×C23C2311yescharacteristic
37M16×C23—42no—
38C2×C8×C23—22no—
39C4⋊C4×C23—21no—
40C4⋊C4×C23C2×C412yes—
41C22⋊C4×C23—22no—
42C22⋊C4×C23—22no—
43C22⋊C4×C23—42no—
44C23×C23—22no—
45C23×C23—22no—
46C23×C23—22no—
47C23×C23—41no—
48C23×C23—42no—
49C23×C23—42no—
50C23×C23—42no—
51C23×C23C22⋊C411yescharacteristic
52C23×C23C22⋊C411yescharacteristic
53C23×C23C2×D411yescharacteristic
54D4×C23—42no—
55D4×C23—42no—
56D4×C23—44no—
57D4×C23—44no—
58C2×C4×C23—22no—
59C2×C4×C23—22no—
60C2×C4×C23—22no—
61C2×C4×C23—41no—
62C2×C4×C23—41no—
63C2×C4×C23—41no—
64C2×C4×C23—41no—
65C2×C4×C23—42no—
66C2×C4×C23—42no—
67C2×C4×C23—42no—
68C2×C4×C23—42no—
69C2×C4×C23C22⋊C411yescharacteristic
70C2×C4×C23C22⋊C411yescharacteristic
71C2×C4×C23C22×C411yescharacteristic
72C2×C4×C23C2×D411yescharacteristic
73C8×C23—42no—
74#1510C2311yesmaximal, characteristic, direct product factor, Sylow
75C22×C23—21no—
76C22×C23—21no—
77C22×C23—21no—
78C22×C23—21no—
79C22×C23—21no—
80C22×C23—21no—
81C22×C23—21no—
82C22×C23—44no—
83C22×C23—42no—
84C22×C23—42no—
85C22×C23—42no—
86C22×C23—84no—
87C22×C23#9912yes—
88C22×C23C2×C22⋊C411yescharacteristic
89C22×C23AGL(1,ℤ/8)12yes—
90C4×C23—21no—
91C4×C23—21no—
92C4×C23—21no—
93C4×C23—21no—
94C4×C23—42no—
95C4×C23—82no—
96C2×#121C2×C2311yescharacteristic
97#407C2×C2311yescharacteristic, complemented
98#381C2×C2311yescharacteristic, complemented
99#327C2×C2314yescomplemented
100C2×C23—22no—
101C2×C23—22no—
102C2×C23—41no—
103C2×C23—82no—
104C2×C23C2×#9911yescharacteristic
105C2×C23#41811yescharacteristic
106C2×C23#42111yescharacteristic
107C22×D4—21no—
108C22×D4C4×C2311yescharacteristic, complemented
109C23×C4C22×C2311yescharacteristic
110C2×M16—21no—
111#121—21no—
112#121—21no—
113#121C4×C2312yescomplemented
114#121C22×C2312yes—
115C2×C4⋊C4C4×C2311yescharacteristic, complemented
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C22⋊C8—21no—
119C22⋊C8C22×C2312yescomplemented
120#95—21no—
121#95C22×C2312yescomplemented
122C23#151011yescharacteristic, direct product factor, Sylow
123C24—21no—
124C24D4×C2311yescharacteristic
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—21no—
128C2×D4—42no—
129C2×D4—44no—
130C2×D4C2×C4×C2312yes—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—42no—
134C22×C4C2×C4×C2311yescharacteristic
135C22×C4C2×C4×C2311yescharacteristic
136C22×C4D4×C2311yescharacteristic
137C22×C4D4×C2311yescharacteristic
138C22×C4D4×C2311yescharacteristic
139C22×C4C23×C2311yescharacteristic
140M16—42no—
141C2×C8—22no—
142C4⋊C4—21no—
143C4⋊C4C2×C4×C2312yes—
144C22⋊C4—22no—
145C22⋊C4—22no—
146C22⋊C4—42no—
147C23—22no—
148C23—22no—
149C23—22no—
150C23—41no—
151C23—42no—
152C23—42no—
153C23—42no—
154C23C22⋊C4×C2311yescharacteristic
155C23C22⋊C4×C2311yescharacteristic
156C23C2×D4×C2311yescharacteristic
157D4—42no—
158D4—42no—
159D4—44no—
160D4—44no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4C22⋊C4×C2311yescharacteristic
173C2×C4C22⋊C4×C2311yescharacteristic
174C2×C4C22×C4×C2311yescharacteristic
175C2×C4C2×D4×C2311yescharacteristic
176C8—42no—
177C22—21no—
178C22—21no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—44no—
185C22—42no—
186C22—42no—
187C22—42no—
188C22—84no—
189C22C23×#9912yes—
190C22C2×C22⋊C4×C2311yescharacteristic
191C22C23×AGL(1,ℤ/8)12yes—
192C4—21no—
193C4—21no—
194C4—21no—
195C4—21no—
196C4—42no—
197C4—82no—
198C2—22no—
199C2—22no—
200C2—41no—
201C2—82no—
202C2C2×C23×#9911yescharacteristic
203C2C23×#41811yescharacteristic
204C2C23×#42111yescharacteristic
subgroupG/H
center, socleC22×C23C2×C22⋊C4
commutatorC2×C4C22×C4×C23
Frattini, cosocleC22×C4C23×C23
prime radicalC2×C23×#121C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#407C2semidirect
3. C23×#381C2semidirect
4. C23×#327C2semidirect
6. C22×D4×C23C4semidirect
11. C23×#121C4semidirect
13. C2×C4⋊C4×C23C4semidirect
17. C23×C22⋊C8C22semidirect
19. C23×#95C22semidirect
74. #1510C23direct (N × H)
97. #407C2×C23semidirect
98. #381C2×C23semidirect
99. #327C2×C23semidirect
108. C22×D4C4×C23semidirect
113. #121C4×C23semidirect
115. C2×C4⋊C4C4×C23semidirect
119. C22⋊C8C22×C23semidirect
121. #95C22×C23semidirect
122. C23#1510direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#1212
C23×#4072
C23×#3812
C23×#3272
#151023