Finite Group Databasebeta

← #27186251

Subgroups — #27186251

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#20265345C211yesmaximal, characteristic
4#15692322—21no—
5#15692314—21no—
6#15692280—21no—
7#15692251C2211yescharacteristic
8#15692182—21no—
9#15692172C411yescharacteristic, complemented
10H(5,𝔽2)×C101C411yescharacteristic, complemented
11C2×#11717581—41no—
12C22⋊#87938—41no—
13C2×D404D411yescharacteristic
14C2×C22⋊D101—41no—
15#13108061—41no—
16#13108059—41no—
17#13108058—21no—
18C4×Dic101D411yescharacteristic
19#41×C101—21no—
20#41×C101—41no—
21C2×D4×C101—41no—
22C2×D4×C101—41no—
23C2×D4×C101C2×C411yescharacteristic
24C23×C101—21no—
25C23×C101—41no—
26C22×D101—21no—
27C2×#87938—81no—
28Q8×C101—21no—
29D4×C101—21no—
30D4×C101—41no—
31D4×C101—42no—
32D4×C101—41no—
33C2×C4×C101—41no—
34C2×C4×C101—41no—
35C2×C4×C101C22⋊C411yescharacteristic
36C22⋊D101—81no—
37C22⋊D101—81no—
38C2×Dic101—21no—
39D404—41no—
40#11717581—81no—
41#11717579—41no—
42C22×C101—41no—
43C22×C101—41no—
44C22×C101—81no—
45C22×C101—161no—
46C22×C101#9911yescharacteristic
47C2×D101—41no—
48C2×D101—81no—
49#87938—161no—
50C4×C101—21no—
51C4×C101—41no—
52Dic101—21no—
53Dic101—21no—
54C2×C101—21no—
55C2×C101—81no—
56C2×C101—81no—
57C2×C101AΣL(1,𝔽16)11yescharacteristic
58D101—81no—
59#1402—1011nomaximal, Sylow
60C101#140211yescharacteristic, complemented, Sylow
61#453—1011no—
62#353—1011no—
63#331—1011no—
64H(5,𝔽2)#8793811yescharacteristic, complemented
65AGL(1,ℤ/8)—2021no—
66#127—1011no—
67C4⋊C8—2021no—
68#104—2021no—
69#100—1011no—
70#99—2021no—
71#41—21no—
72#41—41no—
73C2×D4—41no—
74C2×D4—41no—
75C2×D4—1011no—
76C2×D4—4041no—
77C2×D4C2×#8793811yescharacteristic
78SD16—4041no—
79D8—4041no—
80M16—2021no—
81C2×C8—4041no—
82C22⋊C4—4041no—
83C42—1011no—
84C23—21no—
85C23—41no—
86C23—2021no—
87Q8—21no—
88D4—21no—
89D4—41no—
90D4—42no—
91D4—41no—
92D4—8081no—
93D4—8081no—
94D4—4041no—
95C2×C4—41no—
96C2×C4—41no—
97C2×C4—8081no—
98C2×C4—2021no—
99C2×C4C22⋊#8793811yescharacteristic
100C8—8081no—
101C8—4041no—
102C22—41no—
103C22—41no—
104C22—81no—
105C22—161no—
106C22—8081no—
107C22—4041no—
108C22#1569231311yescharacteristic
109C4—21no—
110C4—41no—
111C4—16161no—
112C4—2021no—
113C4—2021no—
114C2—21no—
115C2—81no—
116C2—81no—
117C2—8081no—
118C2#2026607711yescharacteristic
subgroupG/H
centerC2#20266077
commutatorC2×D4×C101C2×C4
FrattiniC2×D4C2×#87938
FittingH(5,𝔽2)×C101C4
socleC2×C101AΣL(1,𝔽16)
prime radical#20265345C2
cosocle#15692251C22
hypercenterH(5,𝔽2)#87938
hypocenterC101#1402
LCSG ▷ C2×D4×C101 ▷ C2×C4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C2×D4 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #20266081C2semidirect
2. #20266035C2semidirect
9. #15692172C4semidirect
10. H(5,𝔽2)×C101C4semidirect
60. C101#1402semidirect
64. H(5,𝔽2)#87938semidirect

Maximal subgroups

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

subgroupindex
#202660812
#202660352
#202653452
#1402101