Finite Group Databasebeta

← #35213546

Subgroups — #35213546

Order 20864.

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

normal subgroups only

This group has 210 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#23782296C211yesmaximal, characteristic
2#23781996C212yesmaximal
3#23781595C211yesmaximal, characteristic
4#23781403C212yesmaximal
5#324×C163C211yesmaximal, characteristic
6#19104734—22no—
7#19104726—22no—
8#19104714—21no—
9#19104714C2212yes—
10#19104694—21no—
11C2×C4⋊Dic163C2211yescharacteristic
12C2×Dic163⋊C4—21no—
13#19104667—22no—
14#19104660—22no—
15C163⋊#95—21no—
16C163⋊#95C2211yescharacteristic
17C2×C4⋊C4×C163C2211yescharacteristic
18C22⋊C8×C163C2212yes—
19C2×Dic326—21no—
20C2×Dic326D412yes—
21C22×Dic163—21no—
22C22×Dic163—21no—
23C4⋊Dic163—21no—
24C4⋊Dic163—21no—
25C4⋊Dic163—21no—
26C4⋊Dic163—22no—
27C4⋊Dic163D412yes—
28Dic163⋊C4—22no—
29Dic163⋊C4—22no—
30Dic163⋊C4—42no—
31C22⋊Dic163—22no—
32C4×Dic163—22no—
33C22×C4×C163D411yescharacteristic
34C22×C4×C163D411yescharacteristic
35C22×C4×C163C2311yescharacteristic
36C2×C8×C163—22no—
37C4⋊C4×C163—22no—
38Dic326—42no—
39Dic326—81no—
40C2×Dic163—22no—
41C2×Dic163—22no—
42C2×Dic163—22no—
43C2×Dic163—41no—
44C2×Dic163—41no—
45C2×Dic163—41no—
46C2×Dic163—41no—
47C2×Dic163—41no—
48C23×C163C2×D411yescharacteristic
49C2×C4×C163—21no—
50C2×C4×C163—21no—
51C2×C4×C163—21no—
52C2×C4×C163—41no—
53C2×C4×C163—41no—
54C2×C4×C163SD1612yes—
55C2×C4×C163C2×D412yes—
56C8×C163—42no—
57C22×C163—21no—
58C22×C163—21no—
59C22×C163#12011yescharacteristic
60C22×C163C2×SD1611yescharacteristic
61C22×C163#13711yescharacteristic
62C4×C163—22no—
63C4×C163—22no—
64C4×C163—41no—
65Dic163—42no—
66Dic163—42no—
67Dic163—42no—
68Dic163—81no—
69C2×C163—21no—
70C2×C163—21no—
71C2×C163#44811yescharacteristic
72C2×C163#45611yescharacteristic
73C2×C163#45811yescharacteristic
74C163#161111yescharacteristic, complemented, Sylow
75#1611—1631nomaximal, Sylow
76#544—1631no—
77#483—1632no—
78#398—1631no—
79#328—1632no—
80#324D16311yescharacteristic
81C4⋊Q8—3262no—
82#125—3262no—
83C22⋊Q8—3261no—
84C22⋊Q8—1632no—
85#117—3261no—
86C2×C4⋊C4—3261no—
87C2×C4⋊C4—1631no—
88C2×C4⋊C4C2×D16311yescharacteristic
89#106—3262no—
90#103—3262no—
91C22⋊C8C2×D16312yes—
92#95—3261no—
93#95—1631no—
94C2×Q8—3261no—
95C2×Q8—1632no—
96C22×C4—3261no—
97C22×C4—3261no—
98C22×C4D65211yescharacteristic
99C22×C4D65211yescharacteristic
100C22×C4C22×D16311yescharacteristic
101C2×C8—22no—
102C4⋊C4—22no—
103C4⋊C4—3261no—
104C4⋊C4—3261no—
105C4⋊C4—3261no—
106C4⋊C4—3262no—
107C4⋊C4—3262no—
108C4⋊C4—3262no—
109C4⋊C4—6522no—
110C4⋊C4—1632no—
111C22⋊C4—3262no—
112C42—3262no—
113C23C2×D65211yescharacteristic
114Q8—13041no—
115Q8—6522no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—3262no—
122C2×C4—3262no—
123C2×C4—3262no—
124C2×C4—6521no—
125C2×C4—6521no—
126C2×C4—6521no—
127C2×C4—6521no—
128C2×C4—6521no—
129C2×C4#1457968812yes—
130C2×C4D4×D16312yes—
131C8—42no—
132C22—21no—
133C22—21no—
134C22#1910470511yescharacteristic
135C22C2×#1457968811yescharacteristic
136C22#1910476811yescharacteristic
137C4—22no—
138C4—22no—
139C4—41no—
140C4—13041no—
141C4—6522no—
142C4—6522no—
143C4—6522no—
144C2—21no—
145C2—21no—
146C2#2378180611yescharacteristic
147C2#2378185011yescharacteristic
148C2#2378185611yescharacteristic
subgroupG/H
centerC22#19104705
commutator, cosocleC22×C4×C163C23
FrattiniC22×C4C22×D163
Fitting#324×C163C2
socleC22×C163#120
prime radicalC23×C163C2×D4
hypercenter#324D163
hypocenterC163#1611
LCSG ▷ C22×C4×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
74. C163#1611semidirect

Maximal subgroups

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

subgroupindex
#237822962
#237819962
#237815952
#237814032
#324×C1632
#1611163