Finite Group Databasebeta

← #30640948

Subgroups — #30640948

Order 16256.

124 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 41 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#22550131C211yesmaximal, characteristic
2#22549923C211yesmaximal, characteristic
3#22549700C211yesmaximal, characteristic, complemented
4#22549699C211yesmaximal, characteristic, complemented
5#22549692C211yesmaximal, characteristic, complemented
6#22549657C211yesmaximal, characteristic
7#362×C127C211yesmaximal, characteristic, complemented
8C2×#14523231C2211yescharacteristic
9M16×D127—21no—
10#16861185—21no—
11#16861149C2211yescharacteristic
12#16861145C2211yescharacteristic
13#16861144—21no—
14#16861141C2211yescharacteristic
15#16861133—21no—
16C2×Q16×C127C2211yescharacteristic
17M32×C127C2211yescharacteristic
18#108×C127C2211yescharacteristic
19C2×C4×D127D411yescharacteristic
20#14523232—21no—
21#14523231—21no—
22#14523231—21no—
23#14523231—41no—
24C2×C127⋊C8D411yescharacteristic
25C8×D127—41no—
26C4⋊Dic127—21no—
27Dic127⋊C4—41no—
28(C2×D127)⋊C4—41no—
29C127⋊C16—21no—
30C2×Q8×C127—21no—
31Q16×C127—21no—
32Q16×C127—21no—
33M16×C127—21no—
34C2×C8×C127C2311yescharacteristic
35C16×C127—21no—
36C22×D127SD1611yescharacteristic
37C2×Dic127—41no—
38C2×Dic127SD1611yescharacteristic
39C4×D127—21no—
40C4×D127—21no—
41C127⋊C8—21no—
42C127⋊C8—41no—
43Q8×C127—41no—
44Q8×C127—41no—
45C2×C4×C127—41no—
46C2×C4×C127C2×D411yescharacteristic
47C8×C127—41no—
48C8×C127#4111yescharacteristic
49C8×C127#4111yescharacteristic
50C22×C127C2×SD1611yescharacteristic
51C2×D127—21no—
52C2×D127—41no—
53C4×C127—81no—
54C4×C127#12311yescharacteristic
55C4×C127AGL(1,ℤ/8)11yescharacteristic
56Dic127—21no—
57Dic127—81no—
58C2×C127—21no—
59C2×C127#48111yescharacteristic
60D127—41no—
61#2239—1271nomaximal, Sylow
62C127#223911yescharacteristic, complemented, Sylow
63#470—1271no—
64#430—1271no—
65#365—1271no—
66#365—1271no—
67#362—1271no—
68#362D12711yescharacteristic, complemented
69#349—1271no—
70C2×Q16C2×D12711yescharacteristic
71C2×M16—1271no—
72C2×M16—2541no—
73C22⋊Q8—2541no—
74M32—1271no—
75M32C2×D12711yescharacteristic
76#108—1271no—
77#108—2541no—
78#108C2×D12711yescharacteristic
79#106—1271no—
80#103—2541no—
81C2×Q8—21no—
82C22×C4—1271no—
83Q16—21no—
84Q16—21no—
85M16—21no—
86M16—5081no—
87M16—2541no—
88M16—2541no—
89M16—2541no—
90C2×C8—1271no—
91C2×C8—5081no—
92C2×C8C22×D12711yescharacteristic
93C4⋊C4—5081no—
94C4⋊C4—2541no—
95C22⋊C4—5081no—
96C16—21no—
97C16—2541no—
98C23—1271no—
99Q8—41no—
100Q8—41no—
101C2×C4—41no—
102C2×C4—1271no—
103C2×C4—5081no—
104C2×C4—2541no—
105C2×C4—2541no—
106C2×C4D4×D12711yescharacteristic
107C8—41no—
108C8—5081no—
109C8—2541no—
110C8#1452324711yescharacteristic
111C8#1452324811yescharacteristic
112C22—5081no—
113C22—2541no—
114C22SD16×D12711yescharacteristic
115C4—81no—
116C4—10161no—
117C4—2541no—
118C4#1686119111yescharacteristic
119C4#1686123311yescharacteristic
120C2—21no—
121C2—5081no—
122C2#2255019311yescharacteristic
subgroupG/H
centerC2#22550193
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#362×C127C2
socleC2×C127#481
prime radicalC22×D127SD16
hypercenter#362D127
hypocenterC127#2239
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #362

Complements

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

NH ≅ G/Ntype
3. #22549700C2semidirect
4. #22549699C2semidirect
5. #22549692C2semidirect
7. #362×C127C2semidirect
62. C127#2239semidirect
68. #362D127semidirect

Maximal subgroups

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

subgroupindex
#225501312
#225499232
#225497002
#225496992
#225496922
#225496572
#362×C1272
#2239127