Finite Group Databasebeta

← #27169282

Subgroups — #27169282

Order 12928.

112 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 23 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#20265349C211yesmaximal, characteristic
2#20264917C211yesmaximal, characteristic, complemented
3#354×C101C211yesmaximal, characteristic, complemented
4#15692303C411yescharacteristic, complemented
5#15692281—21no—
6#15692183—21no—
7#15692174C411yescharacteristic, complemented
8#15692166—21no—
9#124×C101C2211yescharacteristic
10#99×C101—21no—
11#13108075—41no—
12#13108074—21no—
13#13108074—41no—
14Q8×D101—41no—
15#13108062—41no—
16#13108059—41no—
17#13108058—21no—
18C2×#11717579—41no—
19C2×Q8×C101C2×C411yescharacteristic
20C2×D4×C101D411yescharacteristic
21C22⋊C4×C101—41no—
22C22⋊C4×C101—41no—
23C42×C101D411yescharacteristic
24C23×C101—21no—
25Q8×C101—41no—
26D4×C101—41no—
27C2×C4×C101—21no—
28C2×C4×C101—21no—
29C2×C4×C101—81no—
30C2×C4×C101C22⋊C411yescharacteristic
31C22⋊D101—41no—
32D404—21no—
33D404—41no—
34C4×D101—41no—
35C4×D101—42no—
36Dic202—21no—
37Dic202—41no—
38#11717579—41no—
39#11717579—42no—
40C22×C101—41no—
41C22×C101—81no—
42C22×C101#9911yescharacteristic
43C2×D101—41no—
44C4×C101—21no—
45C4×C101—41no—
46C4×C101—41no—
47C4×C101—161no—
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#354D10111yescharacteristic, complemented
57#329—1011no—
582−1+4—1011no—
59AGL(1,ℤ/8)—2021no—
60#124C2×D10111yescharacteristic
61#104—2021no—
62#101—1011no—
63#99—21no—
64#97—2021no—
65#41—4041no—
66#41—4041no—
67#41—2021no—
68C2×Q8—4041no—
69C2×Q8C4×D10111yescharacteristic
70C2×D4C22⋊D10111yescharacteristic
71SD16—4041no—
72D8—4041no—
73M16—2021no—
74C2×C8—4041no—
75C22⋊C4—41no—
76C22⋊C4—41no—
77C42D40411yescharacteristic
78C23—21no—
79Q8—41no—
80Q8—4041no—
81Q8—2021no—
82D4—41no—
83D4—4041no—
84D4—4041no—
85D4—2021no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—81no—
89C2×C4—4041no—
90C2×C4—4042no—
91C2×C4#1310805111yescharacteristic
92C8—4041no—
93C8—4042no—
94C22—41no—
95C22—81no—
96C22—4041no—
97C22#1569216911yescharacteristic
98C4—21no—
99C4—41no—
100C4—41no—
101C4—161no—
102C4—4041no—
103C2—21no—
104C2—81no—
105C2—8081no—
106C2#2026496311yescharacteristic
subgroupG/H
centerC2#20264963
commutatorC2×Q8×C101C2×C4
Frattini#124C2×D101
Fitting#354×C101C2
socleC2×C101AΣL(1,𝔽16)
prime radical#20265349C2
cosocle#124×C101C22
hypercenter#354D101
hypocenterC101#1404
LCSG ▷ C2×Q8×C101 ▷ C2×C4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ #354

Complements

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

NH ≅ G/Ntype
2. #20264917C2semidirect
3. #354×C101C2semidirect
4. #15692303C4semidirect
7. #15692174C4semidirect
54. C101#1404semidirect
56. #354D101semidirect

Maximal subgroups

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

subgroupindex
#202653492
#202649172
#354×C1012
#1404101