Finite Group Databasebeta

← #30638839

Subgroups — #30638839

Order 16256.

272 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 81 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#22550351C211yesmaximal, characteristic
2C2×#16861126C211yesmaximal, characteristic
3#22549648C212yesmaximal, complemented
4#22549647C212yesmaximal, complemented
5C2×#100×C127C211yesmaximal, characteristic
6C2×#14523232C2212yes—
7D4×Dic127—22no—
8D4×Dic127—22no—
9#16861168C2211yescharacteristic
10#16861168C2211yescharacteristic
11C2×C22⋊Dic127C414yes—
12#16861126—22no—
13C22×D4×C127C2211yescharacteristic
14C2×M16×C127C2212yes—
15#100×C127—22no—
16C22×Dic127—24no—
17#14523232—22no—
18#14523232—22no—
19C2×C127⋊C8—22no—
20C4⋊Dic127—21no—
21C4⋊Dic127—21no—
22C22⋊Dic127—24no—
23C22⋊Dic127C2×C412yes—
24C22⋊Dic127C2×C412yes—
25C4×Dic127—21no—
26C4×Dic127—21no—
27C24×C127C2×C412yes—
28C2×D4×C127—22no—
29C2×D4×C127—22no—
30C2×D4×C127D412yes—
31C2×D4×C127D412yes—
32C22×C4×C127C2311yescharacteristic
33M16×C127—22no—
34M16×C127—22no—
35C2×C8×C127—22no—
36C2×Dic127—22no—
37C2×Dic127—22no—
38C2×Dic127—84no—
39C127⋊C8—42no—
40C23×C127—22no—
41C23×C127—22no—
42C23×C127—82no—
43C23×C127C22⋊C414yes—
44C23×C127C22×C411yescharacteristic
45D4×C127—24no—
46D4×C127—24no—
47C2×C4×C127C2×D411yescharacteristic
48C2×C4×C127C2×D411yescharacteristic
49C2×C4×C127#4112yes—
50C2×C4×C127#4112yes—
51C8×C127—42no—
52C22×C127—24no—
53C22×C127—41no—
54C22×C127—44no—
55C22×C127—82no—
56C22×C127—82no—
57C22×C127C2×C22⋊C411yescharacteristic
58C22×C127#11711yescharacteristic
59C22×C127#11711yescharacteristic
60C4×C127#12312yes—
61C4×C127#12312yes—
62Dic127—42no—
63Dic127—42no—
64C2×C127—21no—
65C2×C127—21no—
66C2×C127—21no—
67C2×C127—44no—
68C2×C127#38111yescharacteristic
69#1829—1271nomaximal, Sylow
70C127#182911yescharacteristic, complemented, Sylow
71#518—1271no—
72C2×#100—1271no—
73C2×#100D12711yescharacteristic
74#344—1272no—
75#344—1272no—
76C22×D4C2×D12711yescharacteristic
77C2×M16—1272no—
78C2×M16C2×D12712yes—
79C4×D4—2542no—
80C4×D4—2542no—
81#117—1271no—
82#117—1271no—
83C2×C22⋊C4—1274no—
84#100—22no—
85#100—2542no—
86C24C4×D12712yes—
87C2×D4—22no—
88C2×D4—22no—
89C2×D4C22⋊D12712yes—
90C2×D4D50812yes—
91C22×C4—2544no—
92C22×C4C22×D12711yescharacteristic
93M16—22no—
94M16—22no—
95M16—2542no—
96M16—2542no—
97C2×C8—22no—
98C2×C8—2542no—
99C4⋊C4—2541no—
100C4⋊C4—2541no—
101C22⋊C4—1272no—
102C22⋊C4—1272no—
103C22⋊C4—2544no—
104C42—2541no—
105C42—2541no—
106C23—22no—
107C23—22no—
108C23—82no—
109C23(C2×D127)⋊C414yes—
110C23C2×C4×D12711yescharacteristic
111D4—24no—
112D4—24no—
113C2×C4—10164no—
114C2×C4—2542no—
115C2×C4—2542no—
116C2×C4C2×C22⋊D12711yescharacteristic
117C2×C4C2×D50811yescharacteristic
118C2×C4#1452324712yes—
119C2×C4#1452324712yes—
120C8—42no—
121C8—5082no—
122C22—24no—
123C22—41no—
124C22—44no—
125C22—82no—
126C22—82no—
127C22C2×(C2×D127)⋊C411yescharacteristic
128C22#1686116511yescharacteristic
129C22#1686116511yescharacteristic
130C4—5082no—
131C4—5082no—
132C4#1686119012yes—
133C4#1686119312yes—
134C2—21no—
135C2—21no—
136C2—21no—
137C2—44no—
138C2#2254973711yescharacteristic
subgroupG/H
centerC2#22549737
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#100×C127C2
socleC2×C127#381
prime radicalC22×D4×C127C22
cosocleC22×C4×C127C23
hypercenterC2×#100D127
hypocenterC127#1829
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
3. #22549648C2semidirect
4. #22549647C2semidirect
70. C127#1829semidirect

Maximal subgroups

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

subgroupindex
#225503512
C2×#168611262
#225496482
#225496472
C2×#100×C1272
#1829127