Finite Group Databasebeta

← #27186255

Subgroups — #27186255

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#20265350C211yesmaximal, characteristic
4#15692314—21no—
5#15692310—21no—
6#15692280—21no—
7#15692244C2211yescharacteristic
8#15692182—21no—
9#15692174C411yescharacteristic, complemented
102−1+4×C101C411yescharacteristic, complemented
11C2×#11717581—41no—
12C22⋊#87938—41no—
13C2×D404D411yescharacteristic
14#13108061—41no—
15#13108059—41no—
16#13108058—21no—
17#13108051—41no—
18C4×Dic101D411yescharacteristic
19#41×C101—21no—
20#41×C101—41no—
21#41×C101—41no—
22C2×Q8×C101—41no—
23C2×Q8×C101C2×C411yescharacteristic
24C22×D101—21no—
25C2×#87938—81no—
26Q8×C101—21no—
27Q8×C101—41no—
28Q8×C101—41no—
29D4×C101—21no—
30D4×C101—41no—
31D4×C101—41no—
32C2×C4×C101—21no—
33C2×C4×C101—41no—
34C2×C4×C101—42no—
35C2×C4×C101C22⋊C411yescharacteristic
36C2×Dic101—21no—
37D404—41no—
38#11717581—42no—
39#11717579—41no—
40C22×C101—41no—
41C22×C101#9911yescharacteristic
42C2×D101—41no—
43C2×D101—81no—
44#87938—161no—
45C4×C101—21no—
46C4×C101—41no—
47C4×C101—41no—
48Dic101—41no—
49C2×C101—21no—
50C2×C101—81no—
51C2×C101AΣL(1,𝔽16)11yescharacteristic
52D101—81no—
53#1404—1011nomaximal, Sylow
54C101#140411yescharacteristic, complemented, Sylow
55#454—1011no—
56#354—1011no—
57#329—1011no—
582−1+4#8793811yescharacteristic, complemented
59AGL(1,ℤ/8)—2021no—
60#124—1011no—
61#104—2021no—
62#101—1011no—
63#99—2021no—
64#97—2021no—
65#41—21no—
66#41—41no—
67#41—41no—
68C2×Q8—41no—
69C2×Q8C2×#8793811yescharacteristic
70C2×D4—1011no—
71SD16—4041no—
72D8—4041no—
73M16—2021no—
74C2×C8—4041no—
75C22⋊C4—4041no—
76C22⋊C4—4041no—
77C42—1011no—
78C23—2021no—
79Q8—21no—
80Q8—41no—
81Q8—41no—
82D4—21no—
83D4—41no—
84D4—41no—
85D4—4041no—
86C2×C4—21no—
87C2×C4—41no—
88C2×C4—42no—
89C2×C4—8081no—
90C2×C4—2021no—
91C2×C4C22⋊#8793811yescharacteristic
92C8—4041no—
93C8—4042no—
94C22—41no—
95C22—8081no—
96C22—4041no—
97C22#1569231311yescharacteristic
98C4—21no—
99C4—41no—
100C4—41no—
101C4—16161no—
102C4—4041no—
103C2—21no—
104C2—81no—
105C2—8081no—
106C2#2026607711yescharacteristic
subgroupG/H
centerC2#20266077
commutatorC2×Q8×C101C2×C4
FrattiniC2×Q8C2×#87938
Fitting2−1+4×C101C4
socleC2×C101AΣL(1,𝔽16)
prime radical#20265350C2
cosocle#15692244C22
hypercenter2−1+4#87938
hypocenterC101#1404
LCSG ▷ C2×Q8×C101 ▷ C2×C4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C2×Q8 ◁ 2−1+4

Complements

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

NH ≅ G/Ntype
1. #20266083C2semidirect
2. #20266033C2semidirect
9. #15692174C4semidirect
10. 2−1+4×C101C4semidirect
54. C101#1404semidirect
58. 2−1+4#87938semidirect

Maximal subgroups

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

subgroupindex
#202660832
#202660332
#202653502
#1404101