Finite Group Databasebeta

← #30640854

Subgroups — #30640854

Order 16256.

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

normal subgroups only

This group has 140 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
1C2×#16861155C211yesmaximal, characteristic, complemented
2#22550294C211yesmaximal, characteristic
3#22550152C211yesmaximal, characteristic
4#22550148C211yesmaximal, characteristic
5#22549705C211yesmaximal, characteristic, complemented
6#22549693C211yesmaximal, characteristic, complemented
7#362×C127C211yesmaximal, characteristic, complemented
8#16861242—21no—
9#16861240—21no—
10C2×Dic508C2211yescharacteristic
11#16861215C2211yescharacteristic
12#16861155—21no—
13#16861155—21no—
14#16861148—21no—
15C2×C127⋊C16C2211yescharacteristic
16C16×D127—21no—
17#16861145C2211yescharacteristic
18#16861128—21no—
19#16861127—21no—
20C2×Q16×C127C2211yescharacteristic
21M32×C127C2211yescharacteristic
22#108×C127C2211yescharacteristic
23#14523248D411yescharacteristic
24C2×Dic254—21no—
25Dic508—21no—
26Dic508—21no—
27Dic508—41no—
28#14523238—41no—
29#14523237—41no—
30#14523235—41no—
31#14523232—21no—
32#14523232D411yescharacteristic
33#14523231—21no—
34C8×D127—21no—
35C127⋊C16D411yescharacteristic
36C127⋊C16D411yescharacteristic
37C2×Q8×C127—21no—
38Q16×C127—21no—
39Q16×C127—21no—
40M16×C127—21no—
41C2×C8×C127C2311yescharacteristic
42C16×C127—21no—
43Dic254—41no—
44Dic254—41no—
45Dic254D811yescharacteristic
46D508D811yescharacteristic
47C22⋊D127—21no—
48C2×Dic127—41no—
49C4×D127—21no—
50C127⋊C8—21no—
51C127⋊C8—41no—
52Q8×C127—41no—
53Q8×C127—41no—
54C2×C4×C127—41no—
55C2×C4×C127C2×D411yescharacteristic
56C8×C127—41no—
57C8×C127C2×D411yescharacteristic
58C8×C127#4111yescharacteristic
59C22×C127#13511yescharacteristic
60C2×D127—21no—
61C4×C127—81no—
62C4×C127#12111yescharacteristic
63C4×C127C2×D811yescharacteristic
64Dic127—21no—
65Dic127—81no—
66C2×C127—21no—
67C2×C127#46611yescharacteristic
68D127—41no—
69#2222—1271nomaximal, Sylow
70C127#222211yescharacteristic, complemented, Sylow
71C2×Q32—1271no—
72#504—1271no—
73#473—1271no—
74#473—1271no—
75#368—1271no—
76#362—1271no—
77#362D12711yescharacteristic, complemented
78#137—2541no—
79#137—2541no—
80C2×Q16—1271no—
81C2×Q16C2×D12711yescharacteristic
82#131—1271no—
83Q32—2541no—
84Q32—2541no—
85M32—2541no—
86M32C2×D12711yescharacteristic
87C2×C16—1271no—
88C2×C16—2541no—
89#108—1271no—
90#108C2×D12711yescharacteristic
91#101—2541no—
92#101—2541no—
93#41—1271no—
94C2×Q8—21no—
95C2×Q8—2541no—
96Q16—21no—
97Q16—21no—
98Q16—5081no—
99Q16—5081no—
100Q16—2541no—
101Q16—2541no—
102SD16—5081no—
103SD16—5081no—
104M16—21no—
105M16—1271no—
106M16—2541no—
107M16—2541no—
108C2×C8—2541no—
109C2×C8C22×D12711yescharacteristic
110C16—21no—
111C16—1271no—
112C16—1271no—
113Q8—41no—
114Q8—41no—
115Q8—1271no—
116Q8—5081no—
117Q8—5081no—
118D4—1271no—
119D4—2541no—
120C2×C4—41no—
121C2×C4—5081no—
122C2×C4—2541no—
123C2×C4D4×D12711yescharacteristic
124C8—41no—
125C8—5081no—
126C8—2541no—
127C8D4×D12711yescharacteristic
128C8#1452324811yescharacteristic
129C22—2541no—
130C22#1686123011yescharacteristic
131C4—81no—
132C4—10161no—
133C4—2541no—
134C4#1686118211yescharacteristic
135C4D8×D12711yescharacteristic
136C2—21no—
137C2—5081no—
138C2#2255011111yescharacteristic
subgroupG/H
centerC2#22550111
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#362×C127C2
socleC2×C127#466
prime radical#14523248D4
hypercenter#362D127
hypocenterC127#2222
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
1. C2×#16861155C2semidirect
5. #22549705C2semidirect
6. #22549693C2semidirect
7. #362×C127C2semidirect
70. C127#2222semidirect
77. #362D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611552
#225502942
#225501522
#225501482
#225497052
#225496932
#362×C1272
#2222127