Finite Group Databasebeta

← #30636882

Subgroups — #30636882

Order 16256.

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#22550048C211yesmaximal, characteristic
2#22549671C211yesmaximal, characteristic, complemented
3#329×C127C211yesmaximal, characteristic, complemented
4#16861257C411yescharacteristic, complemented
5#16861235—21no—
6#16861137—21no—
7#16861128C411yescharacteristic, complemented
8#16861124—21no—
9#124×C127C2211yescharacteristic
10#97×C127—21no—
11#14523249—41no—
12#14523248—21no—
13#14523248—41no—
14Q8×D127—41no—
15#14523236—41no—
16#14523233—41no—
17#14523232—21no—
18C22⋊Dic127—41no—
19C2×Q8×C127C2×C411yescharacteristic
20C2×D4×C127D411yescharacteristic
21C2×C8×C127—41no—
22C22⋊C4×C127—41no—
23C42×C127D411yescharacteristic
24Dic254—21no—
25Dic254—41no—
26D508—21no—
27D508—41no—
28C22⋊D127—41no—
29C2×Dic127—81no—
30C4×D127—41no—
31C4×D127—42no—
32C127⋊C8—41no—
33C23×C127—21no—
34Q8×C127—41no—
35D4×C127—41no—
36C2×C4×C127—21no—
37C2×C4×C127—21no—
38C2×C4×C127C22⋊C411yescharacteristic
39C8×C127—42no—
40C22×C127—41no—
41C22×C127—81no—
42C22×C127#9911yescharacteristic
43C2×D127—41no—
44C4×C127—21no—
45C4×C127—41no—
46C4×C127—41no—
47Dic127—41no—
48Dic127—161no—
49C2×C127—21no—
50C2×C127—81no—
51C2×C127AΣL(1,𝔽16)11yescharacteristic
52D127—81no—
53#1404—1271nomaximal, Sylow
54C127#140411yescharacteristic, complemented, Sylow
55#454—1271no—
56#354—1271no—
57#329D12711yescharacteristic, complemented
582−1+4—1271no—
59AGL(1,ℤ/8)—2541no—
60#124C2×D12711yescharacteristic
61#104—2541no—
62#101—1271no—
63#99—2541no—
64#97—21no—
65#41—5081no—
66#41—5081no—
67#41—2541no—
68C2×Q8—5081no—
69C2×Q8C4×D12711yescharacteristic
70C2×D4D50811yescharacteristic
71SD16—5081no—
72D8—5081no—
73M16—2541no—
74C2×C8—41no—
75C22⋊C4—41no—
76C22⋊C4—5081no—
77C42C22⋊D12711yescharacteristic
78C23—21no—
79Q8—41no—
80Q8—5081no—
81Q8—2541no—
82D4—41no—
83D4—5081no—
84D4—5081no—
85D4—2541no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—10161no—
89C2×C4—5081no—
90C2×C4—5082no—
91C2×C4(C2×D127)⋊C411yescharacteristic
92C8—42no—
93C8—5081no—
94C22—41no—
95C22—81no—
96C22—5081no—
97C22#1686112311yescharacteristic
98C4—21no—
99C4—41no—
100C4—41no—
101C4—20321no—
102C4—5081no—
103C2—21no—
104C2—81no—
105C2—10161no—
106C2#2254966411yescharacteristic
subgroupG/H
centerC2#22549664
commutatorC2×Q8×C127C2×C4
Frattini#124C2×D127
Fitting#329×C127C2
socleC2×C127AΣL(1,𝔽16)
prime radical#22550048C2
cosocle#124×C127C22
hypercenter#329D127
hypocenterC127#1404
LCSG ▷ C2×Q8×C127 ▷ C2×C4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #22549671C2semidirect
3. #329×C127C2semidirect
4. #16861257C4semidirect
7. #16861128C4semidirect
54. C127#1404semidirect
57. #329D127semidirect

Maximal subgroups

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

subgroupindex
#225500482
#225496712
#329×C1272
#1404127