Finite Group Databasebeta

← #35213860

Subgroups — #35213860

Order 20864.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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#23782413C211yesmaximal, characteristic
2#23782035C211yesmaximal, characteristic, complemented
3#23782025C212yesmaximal, complemented
4#23781404C212yesmaximal, complemented
5#330×C163C211yesmaximal, characteristic, complemented
6#19104735C2211yescharacteristic
7#19104723C2212yes—
8#19104720—22no—
9#19104713—22no—
10#19104710—22no—
11#19104660—22no—
12#19104658—22no—
13#19104657—22no—
14#19104646C2211yescharacteristic
15#125×C163C2211yescharacteristic
16#97×C163C2212yes—
17C2×Dic326—21no—
18C2×Dic326D412yes—
19C2×D652—22no—
20C2×D652D412yes—
21C2×C4×D163—22no—
22C2×C163⋊C8—21no—
23C4⋊Dic163—22no—
24Dic163⋊C4—22no—
25(C2×D163)⋊C4—22no—
26(C2×D163)⋊C4—22no—
27(C2×D163)⋊C4—22no—
28C2×C8×C163—22no—
29C4⋊C4×C163—22no—
30C4⋊C4×C163D412yes—
31C42×C163C2311yescharacteristic
32C22×D163—22no—
33Dic326—42no—
34Dic326—42no—
35D652—42no—
36D652—82no—
37C2×Dic163—22no—
38C2×Dic163—41no—
39C4×D163—82no—
40C163⋊C8—41no—
41C2×C4×C163—22no—
42C2×C4×C163C2×D412yes—
43C2×C4×C163C2×D411yescharacteristic
44C8×C163—42no—
45C22×C163#12011yescharacteristic
46C2×D163—42no—
47C2×D163—42no—
48C2×D163—42no—
49C4×C163—21no—
50C4×C163—22no—
51C4×C163—42no—
52Dic163—42no—
53Dic163—81no—
54C2×C163#45411yescharacteristic
55C2×C163#45512yes—
56D163—82no—
57C163#166211yescharacteristic, complemented, Sylow
58#1662—1631nomaximal, Sylow
59#562—1631no—
60#491—1631no—
61#489—1632no—
62#330D16311yescharacteristic, complemented
63#329—1632no—
64C4⋊Q8—1631no—
65#125C2×D16311yescharacteristic
66#124—1632no—
67#123—3262no—
68C22⋊Q8—3262no—
69#121—3262no—
70#103—3262no—
71#103—3262no—
72#102—3262no—
73#97—1631no—
74#97C2×D16312yes—
75C2×Q8—3261no—
76C2×Q8—1632no—
77C2×D4—3262no—
78C2×D4—1632no—
79C22×C4—3262no—
80C2×C8—22no—
81C2×C8—3261no—
82C4⋊C4—22no—
83C4⋊C4—3262no—
84C4⋊C4—3262no—
85C4⋊C4D65212yes—
86C22⋊C4—3262no—
87C22⋊C4—3262no—
88C22⋊C4—3262no—
89C42C22×D16311yescharacteristic
90C23—3262no—
91Q8—6522no—
92Q8—6522no—
93D4—13042no—
94D4—6522no—
95C2×C4—22no—
96C2×C4—13042no—
97C2×C4—3262no—
98C2×C4—6521no—
99C2×C4D4×D16312yes—
100C2×C4C2×D65211yescharacteristic
101C8—42no—
102C8—6521no—
103C22—6522no—
104C22—6522no—
105C22—6522no—
106C22#1910470511yescharacteristic
107C4—21no—
108C4—22no—
109C4—42no—
110C4—13041no—
111C4—6522no—
112C2—13042no—
113C2#2378184011yescharacteristic
114C2#2378184412yes—
subgroupG/H
centerC22#19104705
commutator, cosocleC42×C163C23
FrattiniC42C22×D163
Fitting#330×C163C2
socleC22×C163#120
prime radical#23782413C2
hypercenter#330D163
hypocenterC163#1662
LCSG ▷ C42×C163 ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
2. #23782035C2semidirect
3. #23782025C2semidirect
4. #23781404C2semidirect
5. #330×C163C2semidirect
57. C163#1662semidirect
62. #330D163semidirect

Maximal subgroups

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

subgroupindex
#237824132
#237820352
#237820252
#237814042
#330×C1632
#1662163