Finite Group Databasebeta

← #27169277

Subgroups — #27169277

Order 12928.

122 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 122 conjugacy classes of subgroups — too many to draw. Only its 25 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#20264970C211yesmaximal, characteristic, complemented
2#20264921C211yesmaximal, characteristic, complemented
3#453×C101C211yesmaximal, characteristic
4C4⋊#11717579—21no—
5#15692170—21no—
6H(5,𝔽2)×C101C411yescharacteristic, complemented
7AGL(1,ℤ/8)×C101—21no—
8#127×C101C2211yescharacteristic
9#104×C101—21no—
10#100×C101C411yescharacteristic, complemented
11C2×#11717579—41no—
12C22⋊Dic101—41no—
13#41×C101—21no—
14#41×C101—41no—
15C2×D4×C101—41no—
16C2×D4×C101—41no—
17C2×D4×C101—41no—
18C2×D4×C101C2×C411yescharacteristic
19C2×D4×C101D411yescharacteristic
20SD16×C101—41no—
21D8×C101—41no—
22M16×C101—21no—
23C42×C101D411yescharacteristic
24C23×C101—21no—
25C23×C101—21no—
26C23×C101—41no—
27Q8×C101—21no—
28D4×C101—21no—
29D4×C101—41no—
30D4×C101—41no—
31D4×C101—42no—
32D4×C101—41no—
33D4×C101—81no—
34D4×C101—81no—
35C2×C4×C101—21no—
36C2×C4×C101—41no—
37C2×C4×C101—41no—
38C2×C4×C101C22⋊C411yescharacteristic
39C2×Dic101—81no—
40C8×C101—41no—
41#11717579—81no—
42C22×C101—41no—
43C22×C101—41no—
44C22×C101—41no—
45C22×C101—81no—
46C22×C101—81no—
47C22×C101—161no—
48C22×C101#9911yescharacteristic
49C4×C101—21no—
50C4×C101—21no—
51C4×C101—21no—
52C4×C101—41no—
53Dic101—161no—
54C2×C101—21no—
55C2×C101—81no—
56C2×C101—81no—
57C2×C101—81no—
58C2×C101AΣL(1,𝔽16)11yescharacteristic
59#1402—1011nomaximal, Sylow
60C101#140211yescharacteristic, complemented, Sylow
61#453D10111yescharacteristic
62#353—1011no—
63#331—1011no—
64H(5,𝔽2)Dic10111yescharacteristic, complemented
65AGL(1,ℤ/8)—21no—
66#127C2×D10111yescharacteristic
67C4⋊C8—2021no—
68#104—21no—
69#100Dic10111yescharacteristic, complemented
70#99—2021no—
71#41—21no—
72#41—41no—
73C2×D4—41no—
74C2×D4—41no—
75C2×D4—41no—
76C2×D4C2×Dic10111yescharacteristic
77C2×D4C22⋊D10111yescharacteristic
78SD16—41no—
79D8—41no—
80M16—21no—
81C2×C8—4041no—
82C22⋊C4—4041no—
83C42C22⋊D10111yescharacteristic
84C23—21no—
85C23—21no—
86C23—41no—
87Q8—21no—
88D4—21no—
89D4—41no—
90D4—41no—
91D4—42no—
92D4—41no—
93D4—81no—
94D4—81no—
95C2×C4—21no—
96C2×C4—41no—
97C2×C4—41no—
98C2×C4—8081no—
99C2×C4C22⋊Dic10111yescharacteristic
100C8—41no—
101C8—8081no—
102C22—41no—
103C22—41no—
104C22—41no—
105C22—81no—
106C22—81no—
107C22—161no—
108C22#1569217011yescharacteristic
109C4—21no—
110C4—21no—
111C4—21no—
112C4—41no—
113C4—16161no—
114C2—21no—
115C2—81no—
116C2—81no—
117C2—81no—
118C2#2026496411yescharacteristic
subgroupG/H
centerC2#20264964
commutatorC2×D4×C101C2×C4
Frattini#127C2×D101
Fitting, prime radical#453×C101C2
socleC2×C101AΣL(1,𝔽16)
cosocle#127×C101C22
hypercenter#453D101
hypocenterC101#1402
LCSG ▷ C2×D4×C101 ▷ C2×C4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #127 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #20264970C2semidirect
2. #20264921C2semidirect
6. H(5,𝔽2)×C101C4semidirect
10. #100×C101C4semidirect
60. C101#1402semidirect
64. H(5,𝔽2)Dic101semidirect
69. #100Dic101semidirect

Maximal subgroups

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

subgroupindex
#202649702
#202649212
#453×C1012
#1402101