Finite Group Databasebeta

← #30640798

Subgroups — #30640798

Order 16256.

154 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 43 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#22550319C211yesmaximal, characteristic, complemented
2#22550312C211yesmaximal, characteristic
3#22550259C211yesmaximal, characteristic, complemented
4#22549992C211yesmaximal, characteristic, complemented
5#22549696C211yesmaximal, characteristic, complemented
6#22549691C211yesmaximal, characteristic, complemented
7#361×C127C211yesmaximal, characteristic, complemented
8#16861231C2211yescharacteristic
9C2×Dic508C2211yescharacteristic
10C2×#14523236—21no—
11#16861217—21no—
12#16861196—21no—
13Dic1016—21no—
14#16861154—21no—
15#16861153—21no—
16#16861150—21no—
17#16861149C2211yescharacteristic
18#16861135—21no—
19C8×Dic127C2211yescharacteristic
20C2×D8×C127C2211yescharacteristic
21M32×C127C2211yescharacteristic
22#108×C127C2211yescharacteristic
23#14523248—21no—
24C2×Dic254—21no—
25Dic508—21no—
26Dic508—21no—
27Dic508D411yescharacteristic
28#14523237—21no—
29#14523236—41no—
30#14523236—41no—
31D1016D411yescharacteristic
32#14523231—41no—
33C2×C127⋊C8D411yescharacteristic
34C8×D127—41no—
35C22⋊Dic127—42no—
36C4×Dic127D411yescharacteristic
37C127⋊C16—21no—
38C2×D4×C127—21no—
39D8×C127—21no—
40D8×C127—21no—
41M16×C127—21no—
42C2×C8×C127C2311yescharacteristic
43C16×C127—21no—
44Dic254—21no—
45Dic254—41no—
46Dic254—41no—
47D508—21no—
48C22⋊D127—41no—
49C2×Dic127—21no—
50C2×Dic127—41no—
51C4×D127—41no—
52C127⋊C8D811yescharacteristic
53C127⋊C8D811yescharacteristic
54C23×C127—41no—
55D4×C127—41no—
56D4×C127—41no—
57C2×C4×C127C2×D411yescharacteristic
58C8×C127—41no—
59C8×C127C2×D411yescharacteristic
60C8×C127#4111yescharacteristic
61C22×C127—82no—
62C22×C127—81no—
63C22×C127AGL(1,ℤ/8)11yescharacteristic
64C2×D127—41no—
65C4×C127#12111yescharacteristic
66C4×C127C2×D811yescharacteristic
67Dic127—41no—
68Dic127—41no—
69Dic127—81no—
70C2×C127—21no—
71C2×C127—161no—
72C2×C127#45911yescharacteristic
73D127—81no—
74#2212—1271nomaximal, Sylow
75C127#221211yescharacteristic, complemented, Sylow
76#510—1271no—
77#509—1271no—
78#495—1271no—
79#443—1271no—
80#364—1271no—
81#362—1271no—
82#361D12711yescharacteristic, complemented
83#135—1271no—
84C2×Q16—1271no—
85C2×SD16—2541no—
86C2×D8C2×D12711yescharacteristic
87#131—2541no—
88#124—2541no—
89Q32—2541no—
90SD32—2541no—
91SD32—2541no—
92D16—2541no—
93M32—1271no—
94M32C2×D12711yescharacteristic
95#108C2×D12711yescharacteristic
96#104—2541no—
97C4×C8—1271no—
98#41—2541no—
99C2×Q8—2541no—
100C2×D4—21no—
101Q16—1271no—
102Q16—2541no—
103Q16—2541no—
104SD16—5081no—
105SD16—5081no—
106SD16—2541no—
107D8—21no—
108D8—21no—
109D8—1271no—
110M16—21no—
111M16—5081no—
112C2×C8—1271no—
113C2×C8—5081no—
114C2×C8C22×D12711yescharacteristic
115C22⋊C4—5082no—
116C42—1271no—
117C16—21no—
118C16—2541no—
119C23—41no—
120Q8—5081no—
121Q8—5081no—
122Q8—2541no—
123D4—41no—
124D4—41no—
125D4—5081no—
126D4—2541no—
127C2×C4—5081no—
128C2×C4—5081no—
129C2×C4—2541no—
130C2×C4D4×D12711yescharacteristic
131C8—41no—
132C8—1271no—
133C8—1271no—
134C8D4×D12711yescharacteristic
135C8#1452324811yescharacteristic
136C22—82no—
137C22—81no—
138C22—5081no—
139C22#1686123611yescharacteristic
140C4—10161no—
141C4—5081no—
142C4—5081no—
143C4#1686118211yescharacteristic
144C4D8×D12711yescharacteristic
145C2—21no—
146C2—161no—
147C2—10161no—
148C2#2255007011yescharacteristic
subgroupG/H
centerC2#22550070
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#361×C127C2
socleC2×C127#459
prime radical#22550312C2
hypercenter#361D127
hypocenterC127#2212
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #361

Complements

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

NH ≅ G/Ntype
1. #22550319C2semidirect
3. #22550259C2semidirect
4. #22549992C2semidirect
5. #22549696C2semidirect
6. #22549691C2semidirect
7. #361×C127C2semidirect
75. C127#2212semidirect
82. #361D127semidirect

Maximal subgroups

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

subgroupindex
#225503192
#225503122
#225502592
#225499922
#225496962
#225496912
#361×C1272
#2212127