Finite Group Databasebeta

← #30638041

Subgroups — #30638041

Order 16256.

196 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 49 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#22550286C212yesmaximal, complemented
2#22550271C211yesmaximal, characteristic
3#22549618C212yesmaximal, complemented
4#22549616C211yesmaximal, characteristic
5#563×C127C211yesmaximal, characteristic
6C2×#14523238—21no—
7C2×#14523236—21no—
8#16861209C2211yescharacteristic
9#16861142—22no—
10#16861140—22no—
11#16861120C2212yes—
12#16861120C2211yescharacteristic
13C4⋊Q8×C127—21no—
14#126×C127—22no—
15#125×C127C2212yes—
16#124×C127C2211yescharacteristic
17#123×C127—21no—
18C22⋊Q8×C127—22no—
19C2×Dic254—21no—
20#14523238—42no—
21#14523236—42no—
22C2×C127⋊C8—21no—
23C2×C127⋊C8—22no—
24C4⋊Dic127—22no—
25C2×Q8×C127—21no—
26C2×Q8×C127D411yescharacteristic
27C2×D4×C127D411yescharacteristic
28C22×C4×C127—21no—
29C4⋊C4×C127—22no—
30C4⋊C4×C127—22no—
31C4⋊C4×C127—22no—
32C4⋊C4×C127D412yes—
33C4⋊C4×C127D412yes—
34C22⋊C4×C127—22no—
35C22⋊C4×C127—21no—
36C22⋊C4×C127—22no—
37C42×C127—21no—
38C42×C127C2311yescharacteristic
39Dic254—41no—
40Dic254—41no—
41C2×Dic127—41no—
42C127⋊C8—24no—
43C127⋊C8—22no—
44C23×C127—21no—
45Q8×C127—41no—
46Q8×C127—81no—
47D4×C127—41no—
48C2×C4×C127—22no—
49C2×C4×C127—21no—
50C2×C4×C127—22no—
51C2×C4×C127—81no—
52C2×C4×C127C2×D412yes—
53C2×C4×C127C2×D411yescharacteristic
54C22×C127—41no—
55C22×C127—42no—
56C22×C127#12011yescharacteristic
57C4×C127—21no—
58C4×C127—22no—
59C4×C127—42no—
60C4×C127—41no—
61C4×C127—42no—
62Dic127—81no—
63C2×C127—81no—
64C2×C127#45311yescharacteristic
65C2×C127#45612yes—
66#1660—1271nomaximal, Sylow
67C127#166011yescharacteristic, complemented, Sylow
68#563D12711yescharacteristic
69#501—1272no—
70#497—1271no—
71#330—1272no—
72#329—1271no—
73C2×Q16—2541no—
74C2×SD16—2541no—
75C4⋊Q8—21no—
76C4⋊Q8—1271no—
77#126—22no—
78#125C2×D12712yes—
79#124C2×D12711yescharacteristic
80#123—21no—
81C22⋊Q8—22no—
82#107—2542no—
83#106—2542no—
84#97—1272no—
85#97—1271no—
86C2×Q8—21no—
87C2×Q8—2541no—
88C2×Q8C22⋊D12711yescharacteristic
89C2×D4C22⋊D12711yescharacteristic
90C22×C4—21no—
91Q16—5082no—
92SD16—5082no—
93C2×C8—2542no—
94C2×C8—2541no—
95C4⋊C4—22no—
96C4⋊C4—22no—
97C4⋊C4—22no—
98C4⋊C4—2542no—
99C4⋊C4C22⋊D12712yes—
100C4⋊C4C22⋊D12712yes—
101C22⋊C4—22no—
102C22⋊C4—21no—
103C22⋊C4—22no—
104C42—21no—
105C42C22×D12711yescharacteristic
106C23—21no—
107Q8—41no—
108Q8—81no—
109Q8—5081no—
110Q8—5081no—
111D4—41no—
112C2×C4—22no—
113C2×C4—21no—
114C2×C4—22no—
115C2×C4—81no—
116C2×C4—5081no—
117C2×C4C2×C22⋊D12712yes—
118C2×C4C2×C22⋊D12711yescharacteristic
119C8—2542no—
120C8—2544no—
121C22—41no—
122C22—42no—
123C22#1686117811yescharacteristic
124C4—21no—
125C4—22no—
126C4—42no—
127C4—41no—
128C4—42no—
129C4—10161no—
130C2—81no—
131C2#2255004411yescharacteristic
132C2#2255005912yes—
subgroupG/H
centerC22#16861178
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#563×C127C2
socleC22×C127#120
prime radicalC2×D4×C127D4
hypercenter#563D127
hypocenterC127#1660
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #22550286C2semidirect
3. #22549618C2semidirect
67. C127#1660semidirect

Maximal subgroups

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

subgroupindex
#225502862
#225502712
#225496182
#225496162
#563×C1272
#1660127