Finite Group Databasebeta

← #19212025

Subgroups — #19212025

Order 5888.

162 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 27 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
1#15671560C211yesmaximal, characteristic
2#15671434C211yesmaximal, characteristic
3C23×#1620C211yesmaximal, characteristic
4#13092238—21no—
5#13092210—21no—
6C23×#559C411yescharacteristic, complemented
7C23×#493C2211yescharacteristic
8C23×#460—21no—
9C23×#331C411yescharacteristic, complemented
10C23×Q8⋊C8—21no—
11C2×SD16×C23—41no—
12C2×D8×C23—41no—
13C23×#127C2×C411yescharacteristic
14C23×#127D411yescharacteristic
15C23×#124—41no—
16C23×#123—41no—
17C4×Q8×C23—21no—
18C4×D4×C23—41no—
19C23×C4⋊C8—21no—
20C23×#102—41no—
21C4×C8×C23D411yescharacteristic
22#10625016—41no—
23#10624994—41no—
24C2×Q8×C23—21no—
25C2×D4×C23—21no—
26C2×D4×C23—21no—
27C2×D4×C23—41no—
28C2×D4×C23—41no—
29C22×C4×C23—41no—
30SD16×C23—82no—
31D8×C23—81no—
32D8×C23—81no—
33C2×C8×C23—21no—
34C2×C8×C23—41no—
35C4⋊C4×C23—21no—
36C4⋊C4×C23—41no—
37C22⋊C4×C23—42no—
38C22⋊C4×C23—41no—
39C42×C23—41no—
40C42×C23C22⋊C411yescharacteristic
41C22⋊Dic23—81no—
42C23⋊C16—81no—
43#7516—231nomaximal, Sylow
44C23×C23—41no—
45C23×C23—41no—
46Q8×C23—41no—
47Q8×C23—41no—
48D4×C23—81no—
49D4×C23—81no—
50D4×C23—81no—
51D4×C23—81no—
52D4×C23—81no—
53D4×C23—81no—
54C2×C4×C23—21no—
55C2×C4×C23—41no—
56C2×C4×C23—41no—
57C2×C4×C23—161no—
58C2×C4×C23#9911yescharacteristic
59C2×Dic23—161no—
60C8×C23—21no—
61C8×C23—21no—
62C8×C23—81no—
63#1620D2311yescharacteristic
64#1408—231no—
65#1357—231no—
66C22×C23—82no—
67C22×C23—81no—
68C22×C23—81no—
69C22×C23—161no—
70C22×C23AΣL(1,𝔽16)11yescharacteristic
71C4×C23—21no—
72C4×C23—21no—
73C4×C23—21no—
74C4×C23—81no—
75C4×C23—81no—
76Dic23—321no—
77#559Dic2311yescharacteristic, complemented
78#493C2×D2311yescharacteristic
79#460—21no—
80#364—461no—
81#353—461no—
82#331Dic2311yescharacteristic, complemented
83Q8⋊C8—21no—
84C2×C23—21no—
85C2×C23—161no—
86C2×C23—161no—
87C2×C23#140211yescharacteristic
88C2×SD16—41no—
89C2×D8—41no—
90#127C2×Dic2311yescharacteristic
91#127C22⋊D2311yescharacteristic
92#124—41no—
93#123—41no—
94C4×Q8—21no—
95C4×D4—41no—
96M32—921no—
97C4⋊C8—21no—
98#102—41no—
99#99—921no—
100C4×C8C22⋊D2311yescharacteristic
101C23#751611yescharacteristic, complemented, Sylow
102C2×Q8—21no—
103C2×D4—21no—
104C2×D4—21no—
105C2×D4—41no—
106C2×D4—41no—
107C22×C4—41no—
108SD16—82no—
109D8—81no—
110D8—81no—
111C2×C8—21no—
112C2×C8—41no—
113C4⋊C4—21no—
114C4⋊C4—41no—
115C22⋊C4—42no—
116C22⋊C4—41no—
117C22⋊C4—1841no—
118C42—41no—
119C42C22⋊Dic2311yescharacteristic
120C16—1841no—
121C23—41no—
122C23—41no—
123Q8—41no—
124Q8—41no—
125D4—81no—
126D4—81no—
127D4—81no—
128D4—81no—
129D4—81no—
130D4—81no—
131C2×C4—21no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—161no—
135C2×C4—3681no—
136C2×C4#1062501611yescharacteristic
137C8—21no—
138C8—21no—
139C8—81no—
140C22—82no—
141C22—81no—
142C22—81no—
143C22—161no—
144C22#1309220411yescharacteristic
145C4—21no—
146C4—21no—
147C4—21no—
148C4—81no—
149C4—81no—
150C4—7361no—
151C2—21no—
152C2—161no—
153C2—161no—
154C2#1567154211yescharacteristic
subgroupG/H
centerC2#15671542
commutatorC23×#127C2×C4
Frattini#493C2×D23
FittingC23×#1620C2
socleC2×C23#1402
hypercenter#1620D23
hypocenterC23#7516
LCSG ▷ C23×#127 ▷ C42×C23 ▷ C2×C4×C23 ▷ C22×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #493 ◁ #1620

Complements

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

NH ≅ G/Ntype
6. C23×#559C4semidirect
9. C23×#331C4semidirect
77. #559Dic23semidirect
82. #331Dic23semidirect
101. C23#7516semidirect