Finite Group Databasebeta

← #35213620

Subgroups — #35213620

Order 20864.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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#23782381C211yesmaximal, characteristic, complemented
2#23782039C211yesmaximal, characteristic, complemented
3#23781861C212yesmaximal, complemented
4#23781392C212yesmaximal, complemented
5#333×C163C211yesmaximal, characteristic, complemented
6C2×#14579686—22no—
7C2×D1304—22no—
8#19104732C2211yescharacteristic
9#19104724—22no—
10#19104710—22no—
11C4×D652C2212yes—
12#19104693—22no—
13#19104657—22no—
14C4×C163⋊C8C2211yescharacteristic
15C4⋊Q8×C163C2211yescharacteristic
16C4⋊C8×C163C2212yes—
17C2×D652—22no—
18C2×D652—22no—
19C2×D652—21no—
20C2×D652D412yes—
21C2×C4×D163—22no—
22#14579686—44no—
23D1304—44no—
24C2×C163⋊C8—21no—
25C4⋊Dic163D412yes—
26(C2×D163)⋊C4—24no—
27(C2×D163)⋊C4—22no—
28C4×Dic163—22no—
29C2×Q8×C163D412yes—
30C2×C8×C163—22no—
31C4⋊C4×C163—22no—
32C42×C163C2311yescharacteristic
33C22×D163—22no—
34C22×D163—41no—
35D652—22no—
36D652—22no—
37D652—42no—
38D652—42no—
39D652—42no—
40D652—82no—
41C2×Dic163—22no—
42C4×D163—42no—
43C4×D163—42no—
44C163⋊C8—22no—
45Q8×C163—42no—
46C2×C4×C163—22no—
47C2×C4×C163C2×D412yes—
48C2×C4×C163C2×D411yescharacteristic
49C8×C163—42no—
50C22×C163#12011yescharacteristic
51C2×D163—42no—
52C2×D163—42no—
53C2×D163—42no—
54C2×D163—82no—
55C2×D163—81no—
56C4×C163—21no—
57C4×C163—42no—
58C4×C163#13512yes—
59C4×C163AGL(1,ℤ/8)12yes—
60Dic163—42no—
61C2×C163#44912yes—
62C2×C163#45311yescharacteristic
63D163—82no—
64D163—161no—
65C163#162411yescharacteristic, complemented, Sylow
66#1624—1631nomaximal, Sylow
67#555—1631no—
68#492—1631no—
69#459—1632no—
70#333D16311yescharacteristic, complemented
71D4⋊C8—1632no—
72C2×SD16—3262no—
73C2×D8—3262no—
74C4⋊Q8C2×D16311yescharacteristic
75#127—1631no—
76#124—3262no—
77#121—3262no—
78C4×D4—1632no—
79#117—3262no—
80C4⋊C8C2×D16312yes—
81#102—3262no—
82C4×C8—1631no—
83C2×Q8D65212yes—
84C2×D4—3262no—
85C2×D4—3262no—
86C2×D4—3261no—
87C2×D4—1632no—
88C22×C4—3262no—
89SD16—6524no—
90D8—6524no—
91C2×C8—22no—
92C2×C8—3261no—
93C4⋊C4—22no—
94C4⋊C4—1632no—
95C22⋊C4—3264no—
96C22⋊C4—3262no—
97C42—3262no—
98C42C22×D16311yescharacteristic
99C23—3262no—
100C23—6521no—
101Q8—42no—
102D4—13042no—
103D4—3262no—
104D4—3262no—
105D4—6522no—
106D4—6522no—
107D4—6522no—
108C2×C4—22no—
109C2×C4—3262no—
110C2×C4—6522no—
111C2×C4—6522no—
112C2×C4D4×D16312yes—
113C2×C4C2×D65211yescharacteristic
114C8—42no—
115C8—3262no—
116C22—13042no—
117C22—13041no—
118C22—6522no—
119C22—6522no—
120C22—6522no—
121C22#1910470511yescharacteristic
122C4—21no—
123C4—42no—
124C4—6522no—
125C4#1910475812yes—
126C4#1910476212yes—
127C2—13042no—
128C2—26081no—
129C2#2378181312yes—
130C2#2378183511yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC42×C163C23
FrattiniC42C22×D163
Fitting#333×C163C2
socleC22×C163#120
hypercenter#333D163
hypocenterC163#1624
LCSG ▷ C42×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
1. #23782381C2semidirect
2. #23782039C2semidirect
3. #23781861C2semidirect
4. #23781392C2semidirect
5. #333×C163C2semidirect
65. C163#1624semidirect
70. #333D163semidirect

Maximal subgroups

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

subgroupindex
#237823812
#237820392
#237818612
#237813922
#333×C1632
#1624163