Finite Group Databasebeta

← C23×#7499 #19155125

Subgroups — C23×#7499 #19155125

Order 5888.

272 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 70 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×#356C212yesmaximal
2C23×#1956C211yesmaximal, characteristic
3C2×C23×C4⋊Q8C2211yescharacteristic
4C2×C23×#103—22no—
5C2×C23×#101—22no—
6C2×C4×C8×C23C411yescharacteristic
7C23×#384C411yescharacteristic
8C23×#356—22no—
9C23×#356C414yescomplemented
10C22×Q8×C23D412yes—
11C2×M16×C23—42no—
12C22×C8×C23—21no—
13C23×C4⋊Q8—41no—
14C23×C4⋊Q8C2×C412yes—
15C23×C4⋊Q8D411yescharacteristic
16C23×C4⋊Q8Q811yescharacteristic
17C2×C4⋊C4×C23—21no—
18C2×C4⋊C4×C23—41no—
19C2×C4⋊C4×C23—41no—
20C2×C42×C23C2×C411yescharacteristic
21C23×#103—44no—
22C23×#101—24no—
23C23×#101—42no—
24C4×C8×C23—41no—
25C2×Q8×C23—22no—
26C2×Q8×C23—42no—
27C2×Q8×C23—42no—
28C2×Q8×C23C22⋊C412yes—
29C2×Q8×C23C4⋊C412yes—
30C22×C4×C23—21no—
31C22×C4×C23—22no—
32C22×C4×C23—41no—
33C22×C4×C23C22⋊C411yescharacteristic
34M16×C23—44no—
35M16×C23—82no—
36C2×C8×C23—22no—
37C2×C8×C23—41no—
38C2×C8×C23—41no—
39C2×C8×C23—82no—
40C4⋊C4×C23—42no—
41C4⋊C4×C23—41no—
42C4⋊C4×C23—41no—
43C4⋊C4×C23—42no—
44C4⋊C4×C23—82no—
45C42×C23—21no—
46C42×C23C4211yescharacteristic
47C42×C23C4⋊C411yescharacteristic
48#7499C2311yesmaximal, characteristic, direct product factor, Sylow
49C23×C23#9911yescharacteristic
50Q8×C23—42no—
51Q8×C23—42no—
52Q8×C23—82no—
53C2×C4×C23—22no—
54C2×C4×C23—22no—
55C2×C4×C23—21no—
56C2×C4×C23—21no—
57C2×C4×C23—21no—
58C2×C4×C23—21no—
59C2×C4×C23—22no—
60C2×C4×C23—22no—
61C2×C4×C23—42no—
62C2×C4×C23—42no—
63C2×C4×C23—82no—
64C2×C4×C23—81no—
65C2×C4×C23#9511yescharacteristic
66C2×C4×C23#9911yescharacteristic
67C8×C23—42no—
68C8×C23—84no—
69C2×#356C2×C2312yes—
70#1956C2×C2311yescharacteristic
71C22×C23—21no—
72C22×C23—21no—
73C22×C23#34211yescharacteristic
74C22×C23#35311yescharacteristic
75C22×C23#35411yescharacteristic
76C4×C23—22no—
77C4×C23—22no—
78C4×C23—21no—
79C4×C23—21no—
80C4×C23—42no—
81C4×C23—42no—
82C4×C23—161no—
83C2×C4⋊Q8C22×C2311yescharacteristic
84C2×#103—22no—
85C2×#101—22no—
86C2×C4×C8C4×C2311yescharacteristic
87#384C4×C2311yescharacteristic
88#356—22no—
89#356C4×C2314yescomplemented
90C2×C23—21no—
91C2×C23—21no—
92C2×C23#139311yescharacteristic
93C2×C23#141011yescharacteristic
94C2×C23#141111yescharacteristic
95C22×Q8D4×C2312yes—
96C2×M16—42no—
97C22×C8—21no—
98C4⋊Q8—41no—
99C4⋊Q8C2×C4×C2312yes—
100C4⋊Q8D4×C2311yescharacteristic
101C4⋊Q8Q8×C2311yescharacteristic
102C2×C4⋊C4—21no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C42C2×C4×C2311yescharacteristic
106#103—44no—
107#101—24no—
108#101—42no—
109C4×C8—41no—
110C23#749911yescharacteristic, direct product factor, Sylow
111C2×Q8—22no—
112C2×Q8—42no—
113C2×Q8—42no—
114C2×Q8C22⋊C4×C2312yes—
115C2×Q8C4⋊C4×C2312yes—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—41no—
119C22×C4C22⋊C4×C2311yescharacteristic
120M16—44no—
121M16—82no—
122C2×C8—22no—
123C2×C8—41no—
124C2×C8—41no—
125C2×C8—82no—
126C4⋊C4—42no—
127C4⋊C4—41no—
128C4⋊C4—41no—
129C4⋊C4—42no—
130C4⋊C4—82no—
131C42—21no—
132C42C42×C2311yescharacteristic
133C42C4⋊C4×C2311yescharacteristic
134C23C23×#9911yescharacteristic
135Q8—42no—
136Q8—42no—
137Q8—82no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—82no—
149C2×C4—81no—
150C2×C4C23×#9511yescharacteristic
151C2×C4C23×#9911yescharacteristic
152C8—42no—
153C8—84no—
154C22—21no—
155C22—21no—
156C22C23×#34211yescharacteristic
157C22C23×#35311yescharacteristic
158C22C23×#35411yescharacteristic
159C4—22no—
160C4—22no—
161C4—21no—
162C4—21no—
163C4—42no—
164C4—42no—
165C4—161no—
166C2—21no—
167C2—21no—
168C2C23×#139311yescharacteristic
169C2C23×#141011yescharacteristic
170C2C23×#141111yescharacteristic
subgroupG/H
center, socleC22×C23#353
commutatorC42C42×C23
Frattini, cosocleC2×C4⋊Q8C22×C23
prime radicalC23×C23#99
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C23 ◁ C22×C4×C23 ◁ C2×C23×C4⋊Q8 ◁ G

Complements

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

NH ≅ G/Ntype
9. C23×#356C4semidirect
48. #7499C23direct (N × H)
89. #356C4×C23semidirect
110. C23#7499direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#3562
C23×#19562
#749923