Finite Group Databasebeta

← #27186256

Subgroups — #27186256

Order 12928.

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

normal subgroups only

This group has 112 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#20266083C211yesmaximal, characteristic, complemented
2#20266033C211yesmaximal, characteristic, complemented
3#20265352C211yesmaximal, characteristic
4#15692314—21no—
5#15692310—21no—
6#15692303C411yescharacteristic, complemented
7#15692283—21no—
8#15692244C2211yescharacteristic
9#15692181—21no—
10#101×C101C411yescharacteristic, complemented
11C2×#11717581—41no—
12C22⋊#87938—41no—
13#13108075—41no—
14#13108074—21no—
15#13108074—41no—
16Q8×D101—41no—
17C2×D404D411yescharacteristic
18#13108063—41no—
19D808—41no—
20#13108051—41no—
21C4×Dic101D411yescharacteristic
22C2×Q8×C101C2×C411yescharacteristic
23M16×C101—21no—
24C22×D101—21no—
25C2×#87938—81no—
26Q8×C101—41no—
27C2×C4×C101—21no—
28C2×C4×C101C22⋊C411yescharacteristic
29C22⋊D101—41no—
30C2×Dic101—21no—
31D404—21no—
32D404—41no—
33D404—41no—
34C4×D101—41no—
35C4×D101—42no—
36Dic202—21no—
37Dic202—41no—
38#11717581—42no—
39C8×C101—41no—
40C22×C101#9911yescharacteristic
41C2×D101—41no—
42C2×D101—41no—
43C2×D101—81no—
44#87938—161no—
45C4×C101—21no—
46C4×C101—41no—
47Dic101—41no—
48Dic101—41no—
49C2×C101—21no—
50C2×C101AΣL(1,𝔽16)11yescharacteristic
51D101—81no—
52D101—81no—
53#1404—1011nomaximal, Sylow
54C101#140411yescharacteristic, complemented, Sylow
55#454—1011no—
56#354—1011no—
57#329—1011no—
582−1+4—1011no—
59AGL(1,ℤ/8)—2021no—
60#124—1011no—
61#104—2021no—
62#101#8793811yescharacteristic, complemented
63#99—2021no—
64#97—2021no—
65#41—4041no—
66#41—4041no—
67#41—2021no—
68C2×Q8—4041no—
69C2×Q8C2×#8793811yescharacteristic
70C2×D4—1011no—
71SD16—4041no—
72D8—4041no—
73M16—21no—
74C2×C8—4041no—
75C22⋊C4—4041no—
76C22⋊C4—4041no—
77C42—1011no—
78C23—2021no—
79Q8—41no—
80Q8—4041no—
81Q8—2021no—
82D4—4041no—
83D4—4041no—
84D4—4041no—
85D4—2021no—
86C2×C4—21no—
87C2×C4—8081no—
88C2×C4—4041no—
89C2×C4—4042no—
90C2×C4—2021no—
91C2×C4C22⋊#8793811yescharacteristic
92C8—41no—
93C8—4042no—
94C22—8081no—
95C22—4041no—
96C22—4041no—
97C22#1569231411yescharacteristic
98C4—21no—
99C4—41no—
100C4—16161no—
101C4—4041no—
102C4—4041no—
103C2—21no—
104C2—8081no—
105C2—8081no—
106C2#2026607811yescharacteristic
subgroupG/H
centerC2#20266078
commutatorC2×Q8×C101C2×C4
FrattiniC2×Q8C2×#87938
Fitting#101×C101C4
socleC2×C101AΣL(1,𝔽16)
prime radical#20265352C2
cosocle#15692244C22
hypercenter#101#87938
hypocenterC101#1404
LCSG ▷ C2×Q8×C101 ▷ C2×C4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C2×Q8 ◁ #101

Complements

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

NH ≅ G/Ntype
1. #20266083C2semidirect
2. #20266033C2semidirect
6. #15692303C4semidirect
10. #101×C101C4semidirect
54. C101#1404semidirect
62. #101#87938semidirect

Maximal subgroups

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

subgroupindex
#202660832
#202660332
#202653522
#1404101