Finite Group Databasebeta

← #30640942

Subgroups — #30640942

Order 16256.

134 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 45 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#22549925C211yesmaximal, characteristic, complemented
2#22549698C212yesmaximal, complemented
3#22549689C212yesmaximal, complemented
4#22549658C211yesmaximal, characteristic, complemented
5#469×C127C211yesmaximal, characteristic
6C2×#14523232—21no—
7#16861149C2212yes—
8#16861145—21no—
9#16861145C2212yes—
10C2×D8×C127C2211yescharacteristic
11C2×M16×C127C2211yescharacteristic
12#121×C127—21no—
13#106×C127C2211yescharacteristic
14#102×C127—21no—
15#14523232—22no—
16#14523232—41no—
17C2×C127⋊C8—41no—
18C127⋊C16—22no—
19C2×D4×C127—21no—
20C2×D4×C127—41no—
21C22×C4×C127D411yescharacteristic
22D8×C127—22no—
23M16×C127—22no—
24C2×C8×C127D411yescharacteristic
25C2×C8×C127C2311yescharacteristic
26C4⋊C4×C127—21no—
27C22⋊C4×C127—41no—
28C127⋊C8—42no—
29C23×C127—41no—
30C23×C127SD1611yescharacteristic
31D4×C127—41no—
32D4×C127—41no—
33D4×C127—82no—
34C2×C4×C127—21no—
35C2×C4×C127—21no—
36C2×C4×C127—41no—
37C2×C4×C127SD1611yescharacteristic
38C2×C4×C127C2×D411yescharacteristic
39C8×C127—21no—
40C8×C127#4112yes—
41C22×C127—21no—
42C22×C127—41no—
43C22×C127—81no—
44C22×C127—81no—
45C22×C127—81no—
46C22×C127C2×SD1611yescharacteristic
47C4×C127—21no—
48C4×C127—81no—
49C4×C127#12311yescharacteristic
50C4×C127AGL(1,ℤ/8)11yescharacteristic
51C2×C127—21no—
52C2×C127—41no—
53C2×C127—161no—
54C2×C127#48111yescharacteristic
55#2238—1271nomaximal, Sylow
56C127#223811yescharacteristic, complemented, Sylow
57#469D12711yescharacteristic
58#430—1271no—
59#365—1272no—
60#361—1272no—
61#349—1271no—
62C2×D8C2×D12711yescharacteristic
63C2×M16—2541no—
64C2×M16C2×D12711yescharacteristic
65#121—21no—
66M32—1272no—
67#108—1272no—
68#108—2541no—
69#106C2×D12711yescharacteristic
70#102—21no—
71C2×D4—21no—
72C2×D4—41no—
73C22×C4C22⋊D12711yescharacteristic
74D8—22no—
75M16—22no—
76M16—5081no—
77M16—2542no—
78C2×C8—5081no—
79C2×C8C22⋊D12711yescharacteristic
80C2×C8C22×D12711yescharacteristic
81C4⋊C4—21no—
82C22⋊C4—41no—
83C16—2542no—
84C23—41no—
85C23#1452323611yescharacteristic
86D4—41no—
87D4—41no—
88D4—82no—
89C2×C4—21no—
90C2×C4—21no—
91C2×C4—41no—
92C2×C4#1452323611yescharacteristic
93C2×C4C2×C22⋊D12711yescharacteristic
94C8—21no—
95C8—5082no—
96C8#1452324712yes—
97C22—21no—
98C22—41no—
99C22—81no—
100C22—81no—
101C22—81no—
102C22C2×#1452323611yescharacteristic
103C4—21no—
104C4—81no—
105C4#1686119011yescharacteristic
106C4#1686123411yescharacteristic
107C2—21no—
108C2—41no—
109C2—161no—
110C2#2255019211yescharacteristic
subgroupG/H
centerC2#22550192
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting, prime radical#469×C127C2
socleC2×C127#481
hypercenter#469D127
hypocenterC127#2238
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #22549925C2semidirect
2. #22549698C2semidirect
3. #22549689C2semidirect
4. #22549658C2semidirect
56. C127#2238semidirect

Maximal subgroups

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

subgroupindex
#225499252
#225496982
#225496892
#225496582
#469×C1272
#2238127