Finite Group Databasebeta

← #30640803

Subgroups — #30640803

Order 16256.

170 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 47 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#22550312C212yesmaximal, complemented
2#22549994C211yesmaximal, characteristic, complemented
3#22549697C211yesmaximal, characteristic, complemented
4#22549689C212yesmaximal, complemented
5#493×C127C211yesmaximal, characteristic, complemented
6#16861231C2211yescharacteristic
7#16861215—21no—
8#16861153—22no—
9#16861150—22no—
10#16861149C2212yes—
11#16861145C2211yescharacteristic
12#16861137—21no—
13C2×D8×C127—21no—
14C2×D8×C127C2212yes—
15#127×C127—21no—
16C4×C8×C127C2211yescharacteristic
17#14523248—21no—
18Dic508D411yescharacteristic, complemented
19#14523237—21no—
20D1016D411yescharacteristic, complemented
21#14523232—21no—
22#14523231—41no—
23C8×D127—41no—
24C127⋊C16—22no—
25C2×D4×C127—22no—
26C2×D4×C127—42no—
27D8×C127—22no—
28D8×C127—22no—
29D8×C127—41no—
30D8×C127—41no—
31C2×C8×C127D411yescharacteristic
32C2×C8×C127C2311yescharacteristic
33C42×C127D411yescharacteristic
34Dic254—21no—
35D508—21no—
36C22⋊D127—41no—
37C4×D127—41no—
38C127⋊C8—41no—
39C23×C127—42no—
40D4×C127—42no—
41D4×C127—42no—
42D4×C127—84no—
43C2×C4×C127—21no—
44C2×C4×C127C2×D411yescharacteristic
45C8×C127D811yescharacteristic
46C8×C127D811yescharacteristic
47C8×C127C2×D411yescharacteristic
48C8×C127#4111yescharacteristic
49C22×C127—84no—
50C22×C127—82no—
51C22×C127AGL(1,ℤ/8)11yescharacteristic
52C2×D127—41no—
53C4×C127—22no—
54C4×C127#12111yescharacteristic
55C4×C127C2×D811yescharacteristic
56Dic127—41no—
57C2×C127—21no—
58C2×C127—162no—
59C2×C127#45911yescharacteristic
60D127—81no—
61#2213—1271nomaximal, Sylow
62C127#221311yescharacteristic, complemented, Sylow
63#509—1272no—
64#493D12711yescharacteristic, complemented
65#443—1271no—
66#364—1271no—
67#361—1272no—
68#135—1271no—
69C2×D8—21no—
70C2×D8C2×D12712yes—
71#131—2541no—
72#127—21no—
73SD32—2542no—
74D16—2542no—
75M32—1272no—
76#108—1271no—
77#104—2541no—
78C4×C8C2×D12711yescharacteristic
79#41—2541no—
80C2×D4—22no—
81C2×D4—42no—
82Q16—1271no—
83SD16—2541no—
84D8—22no—
85D8—22no—
86D8—41no—
87D8—41no—
88D8—1271no—
89M16—5081no—
90M16—2541no—
91C2×C8—5081no—
92C2×C8C22⋊D12711yescharacteristic
93C2×C8C22×D12711yescharacteristic
94C42C22⋊D12711yescharacteristic
95C16—2542no—
96C23—42no—
97Q8—2541no—
98D4—42no—
99D4—42no—
100D4—84no—
101D4—5081no—
102D4—2541no—
103C2×C4—21no—
104C2×C4—5081no—
105C2×C4C2×C22⋊D12711yescharacteristic
106C8—5081no—
107C8#1452323311yescharacteristic
108C8#1452323311yescharacteristic
109C8D4×D12711yescharacteristic
110C8#1452324711yescharacteristic
111C22—84no—
112C22—82no—
113C22—5081no—
114C22#1686123511yescharacteristic
115C4—22no—
116C4—5081no—
117C4#1686118011yescharacteristic
118C4C2×#1452323311yescharacteristic
119C2—21no—
120C2—162no—
121C2—10161no—
122C2#2255006811yescharacteristic
subgroupG/H
centerC2#22550068
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#493×C127C2
socleC2×C127#459
hypercenter#493D127
hypocenterC127#2213
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #493

Complements

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

NH ≅ G/Ntype
1. #22550312C2semidirect
2. #22549994C2semidirect
3. #22549697C2semidirect
4. #22549689C2semidirect
5. #493×C127C2semidirect
18. Dic508D4semidirect
20. D1016D4semidirect
62. C127#2213semidirect
64. #493D127semidirect

Maximal subgroups

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

subgroupindex
#225503122
#225499942
#225496972
#225496892
#493×C1272
#2213127