Finite Group Databasebeta

← #27169278

Subgroups — #27169278

Order 12928.

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

normal subgroups only

This group has 122 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#20265344C211yesmaximal, characteristic
2#20264970C211yesmaximal, characteristic, complemented
3#331×C101C211yesmaximal, characteristic, complemented
4#15692300C411yescharacteristic, complemented
5#15692281—21no—
6#15692183—21no—
7#15692172C411yescharacteristic, complemented
8#15692170—21no—
9#127×C101C2211yescharacteristic
10C4⋊C8×C101—21no—
11#13108074—21no—
12#13108073—41no—
13C2×C22⋊D101—41no—
14D4×D101—41no—
15#13108062—41no—
16#13108059—41no—
17#13108058—21no—
18C22⋊Dic101—41no—
19C2×D4×C101—41no—
20C2×D4×C101C2×C411yescharacteristic
21C2×D4×C101D411yescharacteristic
22C2×C8×C101—41no—
23C42×C101D411yescharacteristic
24C23×C101—21no—
25C23×C101—21no—
26C22×D101—41no—
27D4×C101—41no—
28D4×C101—41no—
29D4×C101—81no—
30D4×C101—81no—
31C2×C4×C101—21no—
32C2×C4×C101C22⋊C411yescharacteristic
33C22⋊D101—42no—
34C22⋊D101—41no—
35C2×Dic101—41no—
36C2×Dic101—81no—
37D404—21no—
38C4×D101—41no—
39Dic202—21no—
40C8×C101—81no—
41#11717579—41no—
42C22×C101—41no—
43C22×C101—41no—
44C22×C101—81no—
45C22×C101—81no—
46C22×C101#9911yescharacteristic
47C2×D101—41no—
48C2×D101—161no—
49C4×C101—21no—
50C4×C101—21no—
51C4×C101—21no—
52Dic101—41no—
53Dic101—161no—
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#331D10111yescharacteristic, complemented
64H(5,𝔽2)—1011no—
65AGL(1,ℤ/8)—2021no—
66#127C2×D10111yescharacteristic
67C4⋊C8—21no—
68#104—2021no—
69#100—1011no—
70#99—2021no—
71#41—4041no—
72#41—2021no—
73C2×D4—41no—
74C2×D4—4041no—
75C2×D4—4041no—
76C2×D4C4×D10111yescharacteristic
77C2×D4D40411yescharacteristic
78SD16—4041no—
79D8—4041no—
80M16—2021no—
81C2×C8—41no—
82C22⋊C4—4041no—
83C42C22⋊D10111yescharacteristic
84C23—21no—
85C23—21no—
86C23—4041no—
87Q8—2021no—
88D4—41no—
89D4—41no—
90D4—81no—
91D4—81no—
92D4—4042no—
93D4—4041no—
94D4—2021no—
95C2×C4—21no—
96C2×C4—8081no—
97C2×C4—4041no—
98C2×C4—4041no—
99C2×C4#1310805111yescharacteristic
100C8—81no—
101C8—4041no—
102C22—41no—
103C22—41no—
104C22—81no—
105C22—81no—
106C22—16161no—
107C22—4041no—
108C22#1569216911yescharacteristic
109C4—21no—
110C4—21no—
111C4—21no—
112C4—16161no—
113C4—4041no—
114C2—21no—
115C2—81no—
116C2—81no—
117C2—8081no—
118C2#2026496511yescharacteristic
subgroupG/H
centerC2#20264965
commutatorC2×D4×C101C2×C4
Frattini#127C2×D101
Fitting#331×C101C2
socleC2×C101AΣL(1,𝔽16)
prime radical#20265344C2
cosocle#127×C101C22
hypercenter#331D101
hypocenterC101#1402
LCSG ▷ C2×D4×C101 ▷ C2×C4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #127 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #20264970C2semidirect
3. #331×C101C2semidirect
4. #15692300C4semidirect
7. #15692172C4semidirect
60. C101#1402semidirect
63. #331D101semidirect

Maximal subgroups

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

subgroupindex
#202653442
#202649702
#331×C1012
#1402101