Finite Group Databasebeta

← #27186252

Subgroups — #27186252

Order 12928.

122 conjugacy classes of subgroups; only the 20 normal subgroups are shown.

normal subgroups only

This group has 122 conjugacy classes of subgroups — too many to draw. Only its 20 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#20266081C211yesmaximal, characteristic, complemented
2#20266035C211yesmaximal, characteristic, complemented
3#20265347C211yesmaximal, characteristic
4#15692322—21no—
5#15692314—21no—
6#15692300C411yescharacteristic, complemented
7#15692283—21no—
8#15692251C2211yescharacteristic
9#15692181—21no—
10#100×C101C411yescharacteristic, complemented
11C2×#11717581—41no—
12C22⋊#87938—41no—
13#13108074—21no—
14#13108073—41no—
15C2×D404D411yescharacteristic
16C2×C22⋊D101—41no—
17C2×C22⋊D101—41no—
18D4×D101—41no—
19#13108063—41no—
20D808—41no—
21C4×Dic101D411yescharacteristic
22C2×D4×C101C2×C411yescharacteristic
23M16×C101—21no—
24C23×C101—21no—
25C22×D101—21no—
26C22×D101—41no—
27C2×#87938—81no—
28D4×C101—41no—
29C2×C4×C101C22⋊C411yescharacteristic
30C22⋊D101—42no—
31C22⋊D101—41no—
32C22⋊D101—81no—
33C22⋊D101—81no—
34C2×Dic101—21no—
35C2×Dic101—41no—
36D404—21no—
37D404—41no—
38C4×D101—41no—
39Dic202—21no—
40#11717581—81no—
41C8×C101—41no—
42C22×C101—41no—
43C22×C101—81no—
44C22×C101#9911yescharacteristic
45C2×D101—41no—
46C2×D101—41no—
47C2×D101—81no—
48C2×D101—161no—
49#87938—161no—
50C4×C101—21no—
51Dic101—21no—
52Dic101—21no—
53Dic101—41no—
54C2×C101—21no—
55C2×C101—81no—
56C2×C101AΣL(1,𝔽16)11yescharacteristic
57D101—81no—
58D101—81no—
59#1402—1011nomaximal, Sylow
60C101#140211yescharacteristic, complemented, Sylow
61#453—1011no—
62#353—1011no—
63#331—1011no—
64H(5,𝔽2)—1011no—
65AGL(1,ℤ/8)—2021no—
66#127—1011no—
67C4⋊C8—2021no—
68#104—2021no—
69#100#8793811yescharacteristic, complemented
70#99—2021no—
71#41—4041no—
72#41—2021no—
73C2×D4—1011no—
74C2×D4—4041no—
75C2×D4—4041no—
76C2×D4—4041no—
77C2×D4C2×#8793811yescharacteristic
78SD16—4041no—
79D8—4041no—
80M16—21no—
81C2×C8—4041no—
82C22⋊C4—4041no—
83C42—1011no—
84C23—21no—
85C23—4041no—
86C23—2021no—
87Q8—2021no—
88D4—41no—
89D4—8081no—
90D4—8081no—
91D4—4041no—
92D4—4042no—
93D4—4041no—
94D4—2021no—
95C2×C4—8081no—
96C2×C4—4041no—
97C2×C4—4041no—
98C2×C4—2021no—
99C2×C4C22⋊#8793811yescharacteristic
100C8—41no—
101C8—8081no—
102C22—41no—
103C22—81no—
104C22—8081no—
105C22—16161no—
106C22—4041no—
107C22—4041no—
108C22#1569231411yescharacteristic
109C4—21no—
110C4—16161no—
111C4—4041no—
112C4—2021no—
113C4—2021no—
114C2—21no—
115C2—81no—
116C2—8081no—
117C2—8081no—
118C2#2026607811yescharacteristic
subgroupG/H
centerC2#20266078
commutatorC2×D4×C101C2×C4
FrattiniC2×D4C2×#87938
Fitting#100×C101C4
socleC2×C101AΣL(1,𝔽16)
prime radical#20265347C2
cosocle#15692251C22
hypercenter#100#87938
hypocenterC101#1402
LCSG ▷ C2×D4×C101 ▷ C2×C4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C2×D4 ◁ #100

Complements

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

NH ≅ G/Ntype
1. #20266081C2semidirect
2. #20266035C2semidirect
6. #15692300C4semidirect
10. #100×C101C4semidirect
60. C101#1402semidirect
69. #100#87938semidirect

Maximal subgroups

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

subgroupindex
#202660812
#202660352
#202653472
#1402101