Finite Group Databasebeta

← #30638532

Subgroups — #30638532

Order 16256.

284 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 113 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×#16861216C211yesmaximal, characteristic
2#22549860C211yesmaximal, characteristic
3#22549832C212yesmaximal, complemented
4#22549646C212yesmaximal, complemented
5#410×C127C211yesmaximal, characteristic
6#16861216—22no—
7#16861216—22no—
8C2×#14523232C412yescomplemented
9C2×#14523232C2212yes—
10C2×#14523232C2211yescharacteristic
11C22×C127⋊C8C412yescomplemented
12C22×C127⋊C8C2211yescharacteristic
13#16861128C412yescomplemented
14#16861126C412yescomplemented
15#16861120—22no—
16C4×C127⋊C8—22no—
17C2×#41×C127C2211yescharacteristic
18#117×C127C2212yes—
19#99×C127C414yes—
20#14523232—24no—
21#14523232—22no—
22#14523232—22no—
23#14523232C2×C414yes—
24C2×C127⋊C8—22no—
25C2×C127⋊C8—22no—
26C2×C127⋊C8—22no—
27C2×C127⋊C8C2×C412yes—
28C2×C127⋊C8C2×C411yescharacteristic
29C2×C127⋊C8C2×C411yescharacteristic
30C2×C127⋊C8D412yes—
31C2×C127⋊C8D412yes—
32#41×C127—22no—
33#41×C127—22no—
34C2×Q8×C127C2×C411yescharacteristic
35C2×D4×C127C2×C411yescharacteristic
36C2×D4×C127C2×C412yes—
37C22×C4×C127C2×C412yes—
38C22×C4×C127C2311yescharacteristic
39C4⋊C4×C127—22no—
40C22⋊C4×C127C2×C414yes—
41C42×C127—22no—
42C127⋊C8—22no—
43C127⋊C8—24no—
44C127⋊C8C22⋊C411yescharacteristic
45C127⋊C8C22⋊C411yescharacteristic
46C127⋊C8C22⋊C412yes—
47C23×C127C4212yes—
48C23×C127C22×C411yescharacteristic
49Q8×C127—22no—
50D4×C127—24no—
51D4×C127—22no—
52C2×C4×C127—22no—
53C2×C4×C127—22no—
54C2×C4×C127—24no—
55C2×C4×C127C4212yes—
56C2×C4×C127C22×C411yescharacteristic
57C2×C4×C127C22×C411yescharacteristic
58C2×C4×C127C2×D412yes—
59C2×C4×C127#4112yes—
60C22×C127—22no—
61C22×C127—41no—
62C22×C127—42no—
63C22×C127C2×C4211yescharacteristic
64C22×C127C4×D412yes—
65C4×C127—22no—
66C4×C127—44no—
67C4×C127C2×C22⋊C411yescharacteristic
68C4×C127#11711yescharacteristic
69C4×C127C4×D412yes—
70C2×C127—22no—
71C2×C127—21no—
72C2×C127—42no—
73C2×C127C4×C22⋊C411yescharacteristic
74#1757—1271nomaximal, Sylow
75C127#175711yescharacteristic, complemented, Sylow
76C2×#131—1271no—
77#413—1271no—
78#410D12711yescharacteristic
79#405—1272no—
80#344—1272no—
81C2×#41C2×D12711yescharacteristic
82#131—2542no—
83#131—2542no—
84C2×M16—1272no—
85C2×M16—1271no—
86C2×M16—1272no—
87C22×C8—1271no—
88C22×C8—1272no—
89#117C2×D12712yes—
90#101—1272no—
91#100—1272no—
92#99Dic12714yes—
93#97—2542no—
94C4×C8—2542no—
95#41—22no—
96#41—22no—
97C2×Q8C4×D12711yescharacteristic
98C2×D4C4×D12711yescharacteristic
99C2×D4C2×Dic12712yes—
100C22×C4C4×D12712yes—
101C22×C4C22×D12711yescharacteristic
102M16—1274no—
103M16—2544no—
104M16—2542no—
105M16—2542no—
106C2×C8—1271no—
107C2×C8—1271no—
108C2×C8—1272no—
109C2×C8—1272no—
110C2×C8—1272no—
111C2×C8—2542no—
112C2×C8—2542no—
113C2×C8—2542no—
114C4⋊C4—22no—
115C22⋊C4C2×Dic12714yes—
116C42—22no—
117C23C4×Dic12712yes—
118C23C22×Dic12711yescharacteristic
119Q8—22no—
120D4—24no—
121D4—22no—
122C2×C4—22no—
123C2×C4—22no—
124C2×C4—24no—
125C2×C4C4×Dic12712yes—
126C2×C4C2×C4×D12711yescharacteristic
127C2×C4C2×C4×D12711yescharacteristic
128C2×C4D4×D12712yes—
129C2×C4#1452324712yes—
130C8—1271no—
131C8—1271no—
132C8—1272no—
133C8—2542no—
134C8—2544no—
135C22—22no—
136C22—41no—
137C22—42no—
138C22C2×C4×Dic12711yescharacteristic
139C22D4×Dic12712yes—
140C4—22no—
141C4—44no—
142C4C22⋊C4×D12711yescharacteristic
143C4#1686116511yescharacteristic
144C4(C22⋊D127)⋊C412yes—
145C2—22no—
146C2—21no—
147C2—42no—
148C2C22⋊C4×Dic12711yescharacteristic
subgroupG/H
centerC4C22⋊C4×D127
commutatorC22×C127C2×C42
FrattiniC22×C4C22×D127
Fitting#410×C127C2
socleC2×C127C4×C22⋊C4
prime radicalC2×#41×C127C22
cosocleC22×C4×C127C23
hypercenter#410D127
hypocenterC127#1757
LCSG ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
3. #22549832C2semidirect
4. #22549646C2semidirect
8. C2×#14523232C4semidirect
11. C22×C127⋊C8C4semidirect
13. #16861128C4semidirect
14. #16861126C4semidirect
75. C127#1757semidirect

Maximal subgroups

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

subgroupindex
C2×#168612162
#225498602
#225498322
#225496462
#410×C1272
#1757127