Finite Group Databasebeta

← C23×#1508 #15669127

Subgroups — C23×#1508 #15669127

Order 2944.

294 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 96 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×C22⋊Q8C211yesmaximal, characteristic
2C23×#407C211yesmaximal, characteristic
3C4×C22⋊C4×C23C211yesmaximal, characteristic
4C23×#328C214yesmaximal, complemented
5C22×Q8×C23C411yescharacteristic, complemented
6C23×C4×C23C2211yescharacteristic
7C2×M16×C23—21no—
8C23×C22⋊Q8—21no—
9C23×C22⋊Q8—21no—
10C23×C22⋊Q8C412yescomplemented
11C23×C22⋊Q8C2212yes—
12C2×C4⋊C4×C23—21no—
13C2×C4⋊C4×C23C411yescharacteristic, complemented
14C2×C22⋊C4×C23—21no—
15C2×C22⋊C4×C23—21no—
16C2×C42×C23—21no—
17C23×C22⋊C8—21no—
18C23×C22⋊C8C2212yes—
19C23×#95C2212yes—
20C24×C23D411yescharacteristic
21C2×Q8×C23—21no—
22C2×Q8×C23—42no—
23C2×Q8×C23C2×C412yes—
24C22×C4×C23—21no—
25C22×C4×C23—21no—
26C22×C4×C23—22no—
27C22×C4×C23—42no—
28C22×C4×C23C2×C411yescharacteristic
29C22×C4×C23C2×C411yescharacteristic
30C22×C4×C23D411yescharacteristic
31C22×C4×C23D411yescharacteristic
32C22×C4×C23D411yescharacteristic
33C22×C4×C23C2311yescharacteristic
34M16×C23—42no—
35C2×C8×C23—22no—
36C4⋊C4×C23—22no—
37C4⋊C4×C23—21no—
38C4⋊C4×C23—22no—
39C4⋊C4×C23C2×C412yes—
40C22⋊C4×C23—24no—
41C22⋊C4×C23—22no—
42C22⋊C4×C23—22no—
43C42×C23—42no—
44C23×C23—22no—
45C23×C23—22no—
46C23×C23—41no—
47C23×C23C22⋊C411yescharacteristic
48C23×C23C22⋊C411yescharacteristic
49C23×C23C2×D411yescharacteristic
50Q8×C23—42no—
51Q8×C23—42no—
52C2×C4×C23—22no—
53C2×C4×C23—22no—
54C2×C4×C23—22no—
55C2×C4×C23—24no—
56C2×C4×C23—22no—
57C2×C4×C23—22no—
58C2×C4×C23—41no—
59C2×C4×C23—41no—
60C2×C4×C23—41no—
61C2×C4×C23—42no—
62C2×C4×C23—42no—
63C2×C4×C23—42no—
64C2×C4×C23—42no—
65C2×C4×C23C22⋊C411yescharacteristic
66C2×C4×C23C22⋊C411yescharacteristic
67C2×C4×C23C22×C411yescharacteristic
68C2×C4×C23C2×D411yescharacteristic
69C8×C23—42no—
70#1508C2311yesmaximal, characteristic, direct product factor, Sylow
71C22×C23—21no—
72C22×C23—21no—
73C22×C23—21no—
74C22×C23—21no—
75C22×C23—21no—
76C22×C23—21no—
77C22×C23—21no—
78C22×C23—44no—
79C22×C23#10412yes—
80C22×C23C2×C22⋊C411yescharacteristic
81C22×C23#13712yes—
82C4×C23—21no—
83C4×C23—21no—
84C4×C23—21no—
85C4×C23—44no—
86C4×C23—42no—
87C4×C23—42no—
88C4×C23#9912yes—
89C2×C22⋊Q8C2×C2311yescharacteristic
90#407C2×C2311yescharacteristic
91C4×C22⋊C4C2×C2311yescharacteristic
92#328C2×C2314yescomplemented
93C2×C23—22no—
94C2×C23—22no—
95C2×C23—41no—
96C2×C23C2×#9911yescharacteristic
97C2×C23#41911yescharacteristic
98C2×C23C2×#10411yescharacteristic
99C22×Q8C4×C2311yescharacteristic, complemented
100C23×C4C22×C2311yescharacteristic
101C2×M16—21no—
102C22⋊Q8—21no—
103C22⋊Q8—21no—
104C22⋊Q8C4×C2312yescomplemented
105C22⋊Q8C22×C2312yes—
106C2×C4⋊C4—21no—
107C2×C4⋊C4C4×C2311yescharacteristic, complemented
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C42—21no—
111C22⋊C8—21no—
112C22⋊C8C22×C2312yes—
113#95C22×C2312yes—
114C23#150811yescharacteristic, direct product factor, Sylow
115C24D4×C2311yescharacteristic
116C2×Q8—21no—
117C2×Q8—42no—
118C2×Q8C2×C4×C2312yes—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—42no—
123C22×C4C2×C4×C2311yescharacteristic
124C22×C4C2×C4×C2311yescharacteristic
125C22×C4D4×C2311yescharacteristic
126C22×C4D4×C2311yescharacteristic
127C22×C4D4×C2311yescharacteristic
128C22×C4C23×C2311yescharacteristic
129M16—42no—
130C2×C8—22no—
131C4⋊C4—22no—
132C4⋊C4—21no—
133C4⋊C4—22no—
134C4⋊C4C2×C4×C2312yes—
135C22⋊C4—24no—
136C22⋊C4—22no—
137C22⋊C4—22no—
138C42—42no—
139C23—22no—
140C23—22no—
141C23—41no—
142C23C22⋊C4×C2311yescharacteristic
143C23C22⋊C4×C2311yescharacteristic
144C23C2×D4×C2311yescharacteristic
145Q8—42no—
146Q8—42no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—24no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4C22⋊C4×C2311yescharacteristic
161C2×C4C22⋊C4×C2311yescharacteristic
162C2×C4C22×C4×C2311yescharacteristic
163C2×C4C2×D4×C2311yescharacteristic
164C8—42no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—21no—
169C22—21no—
170C22—21no—
171C22—21no—
172C22—44no—
173C22C23×#10412yes—
174C22C2×C22⋊C4×C2311yescharacteristic
175C22C23×#13712yes—
176C4—21no—
177C4—21no—
178C4—21no—
179C4—44no—
180C4—42no—
181C4—42no—
182C4C23×#9912yes—
183C2—22no—
184C2—22no—
185C2—41no—
186C2C2×C23×#9911yescharacteristic
187C2C23×#41911yescharacteristic
188C2C2×C23×#10411yescharacteristic
subgroupG/H
center, socleC22×C23C2×C22⋊C4
commutatorC2×C4C22×C4×C23
Frattini, cosocleC22×C4C23×C23
prime radicalC24×C23D4
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
4. C23×#328C2semidirect
5. C22×Q8×C23C4semidirect
10. C23×C22⋊Q8C4semidirect
13. C2×C4⋊C4×C23C4semidirect
70. #1508C23direct (N × H)
92. #328C2×C23semidirect
99. C22×Q8C4×C23semidirect
104. C22⋊Q8C4×C23semidirect
107. C2×C4⋊C4C4×C23semidirect
114. C23#1508direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×C22⋊Q82
C23×#4072
C4×C22⋊C4×C232
C23×#3282
#150823