Finite Group Databasebeta

← #30640805

Subgroups — #30640805

Order 16256.

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

normal subgroups only

This group has 170 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#22550314C211yesmaximal, characteristic, complemented
2#22550312C211yesmaximal, characteristic, complemented
3#22550251C211yesmaximal, characteristic, complemented
4#22549992C211yesmaximal, characteristic, complemented
5#22549696C211yesmaximal, characteristic, complemented
6#22549688C211yesmaximal, characteristic, complemented
7#361×C127C211yesmaximal, characteristic, complemented
8#16861231C2211yescharacteristic
9C2×D1016C2211yescharacteristic
10C2×#14523233—21no—
11#16861217—21no—
12#16861205—21no—
13#16861154—21no—
14#16861153—21no—
15D2032—21no—
16#16861150—21no—
17#16861149C2211yescharacteristic
18#16861135—21no—
19C8×Dic127C2211yescharacteristic
20C2×D8×C127C2211yescharacteristic
21M32×C127C2211yescharacteristic
22#108×C127C2211yescharacteristic
23#14523248—21no—
24C2×D508—21no—
25C2×C22⋊D127—42no—
26Dic508D411yescharacteristic, complemented
27#14523237—21no—
28D1016—21no—
29D1016—21no—
30D1016D411yescharacteristic, complemented
31#14523233—41no—
32#14523233—41no—
33#14523231—41no—
34C2×C127⋊C8D411yescharacteristic
35C8×D127—41no—
36C4×Dic127D411yescharacteristic
37C127⋊C16—21no—
38C2×D4×C127—21no—
39D8×C127—21no—
40D8×C127—21no—
41M16×C127—21no—
42C2×C8×C127C2311yescharacteristic
43C16×C127—21no—
44C22×D127—41no—
45Dic254—21no—
46D508—21no—
47D508—41no—
48D508—41no—
49C22⋊D127—41no—
50C22⋊D127—84no—
51C2×Dic127—21no—
52C4×D127—41no—
53C127⋊C8D811yescharacteristic
54C127⋊C8D811yescharacteristic
55C23×C127—41no—
56D4×C127—41no—
57D4×C127—41no—
58C2×C4×C127C2×D411yescharacteristic
59C8×C127—41no—
60C8×C127C2×D411yescharacteristic
61C8×C127#4111yescharacteristic
62C22×C127—82no—
63C22×C127—81no—
64C22×C127AGL(1,ℤ/8)11yescharacteristic
65C2×D127—41no—
66C2×D127—82no—
67C2×D127—81no—
68C4×C127#12111yescharacteristic
69C4×C127C2×D811yescharacteristic
70Dic127—22no—
71Dic127—41no—
72C2×C127—21no—
73C2×C127—161no—
74C2×C127#45911yescharacteristic
75D127—81no—
76D127—161no—
77#2213—1271nomaximal, Sylow
78C127#221311yescharacteristic, complemented, Sylow
79#509—1271no—
80#509—1271no—
81#493—1271no—
82#443—1271no—
83#364—1271no—
84#361—1271no—
85#361D12711yescharacteristic, complemented
86#135—1271no—
87C2×D8—1271no—
88C2×D8—2541no—
89C2×D8C2×D12711yescharacteristic
90#131—2541no—
91#127—2541no—
92SD32—2541no—
93SD32—2541no—
94D16—2541no—
95D16—2541no—
96M32—1271no—
97M32C2×D12711yescharacteristic
98#108C2×D12711yescharacteristic
99#104—2541no—
100C4×C8—1271no—
101#41—2541no—
102C2×D4—21no—
103C2×D4—5082no—
104C2×D4—2541no—
105Q16—1271no—
106SD16—2541no—
107D8—21no—
108D8—21no—
109D8—1271no—
110D8—5081no—
111D8—5081no—
112D8—2541no—
113D8—2541no—
114M16—21no—
115M16—5081no—
116C2×C8—1271no—
117C2×C8—5081no—
118C2×C8C22×D12711yescharacteristic
119C42—1271no—
120C16—21no—
121C16—2541no—
122C23—41no—
123C23—5081no—
124Q8—2541no—
125D4—41no—
126D4—41no—
127D4—10164no—
128D4—5081no—
129D4—5081no—
130D4—5081no—
131D4—2541no—
132C2×C4—5081no—
133C2×C4—2541no—
134C2×C4D4×D12711yescharacteristic
135C8—41no—
136C8—1271no—
137C8—1271no—
138C8D4×D12711yescharacteristic
139C8#1452324811yescharacteristic
140C22—82no—
141C22—81no—
142C22—10162no—
143C22—10161no—
144C22—5081no—
145C22#1686123611yescharacteristic
146C4—5081no—
147C4—2542no—
148C4#1686118211yescharacteristic
149C4D8×D12711yescharacteristic
150C2—21no—
151C2—161no—
152C2—20321no—
153C2—10161no—
154C2#2255007011yescharacteristic
subgroupG/H
centerC2#22550070
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#361×C127C2
socleC2×C127#459
hypercenter#361D127
hypocenterC127#2213
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. #22550314C2semidirect
2. #22550312C2semidirect
3. #22550251C2semidirect
4. #22549992C2semidirect
5. #22549696C2semidirect
6. #22549688C2semidirect
7. #361×C127C2semidirect
26. Dic508D4semidirect
30. D1016D4semidirect
78. C127#2213semidirect
85. #361D127semidirect

Maximal subgroups

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

subgroupindex
#225503142
#225503122
#225502512
#225499922
#225496962
#225496882
#361×C1272
#2213127