Finite Group Databasebeta

← #30640245

Subgroups — #30640245

Order 16256.

222 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 63 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#22549924C212yesmaximal, complemented
2#22549839C211yesmaximal, characteristic
3#22549839C212yesmaximal
4C127⋊#338C211yesmaximal, characteristic, complemented
5#430×C127C211yesmaximal, characteristic, complemented
6C22×C4×D127C2211yescharacteristic
7C2×#14523232C2211yescharacteristic
8C2×#14523232C2212yes—
9#16861144—24no—
10(C2×D127)⋊C8—24no—
11(C2×D127)⋊C8—22no—
12C2×M16×C127C2212yes—
13C2×M16×C127C2211yescharacteristic
14#108×C127—22no—
15C23×D127Q811yescharacteristic
16C22×Dic127Q811yescharacteristic
17C2×C4×D127—22no—
18C2×C4×D127—21no—
19C2×C4×D127—22no—
20C2×C4×D127—21no—
21#14523232—22no—
22#14523232—22no—
23#14523232—22no—
24C2×C127⋊C8—21no—
25C2×C127⋊C8D414yes—
26C22×C4×C127C2311yescharacteristic
27M16×C127—22no—
28M16×C127—22no—
29M16×C127—22no—
30C2×C8×C127—22no—
31C2×C8×C127D412yes—
32C22×D127—21no—
33C22×D127—22no—
34C22×D127—42no—
35C2×Dic127—21no—
36C2×Dic127—22no—
37C4×D127—41no—
38C4×D127—42no—
39C4×D127—41no—
40C127⋊C8—24no—
41C127⋊C8—41no—
42C23×C127C2×Q811yescharacteristic
43C2×C4×C127C2×D412yes—
44C2×C4×C127C2×D411yescharacteristic
45C2×C4×C127#4112yes—
46C2×C4×C127#4111yescharacteristic
47C8×C127—22no—
48C8×C127—42no—
49C22×C127—41no—
50C22×C127C22⋊Q811yescharacteristic
51C22×C127C22⋊Q812yes—
52C2×D127—22no—
53C2×D127—24no—
54C2×D127—41no—
55C2×D127—42no—
56C2×D127—41no—
57C4×C127#12311yescharacteristic
58C4×C127#12312yes—
59C4×C127#12711yescharacteristic
60Dic127—41no—
61C2×C127—21no—
62C2×C127—22no—
63C2×C127#39911yescharacteristic
64D127—42no—
65#2091—1271nomaximal, Sylow
66C127#209111yescharacteristic, complemented, Sylow
67#430—1272no—
68#430D12711yescharacteristic, complemented
69#407—1272no—
70#407—1271no—
71#338—1271no—
72C23×C4—1271no—
73C2×M16—1272no—
74C2×M16—1271no—
75C2×M16C2×D12712yes—
76C2×M16C2×D12711yescharacteristic
77#108—22no—
78#108—2544no—
79C22⋊C8—2544no—
80C22⋊C8—2542no—
81C24—1271no—
82C22×C4—1271no—
83C22×C4—2542no—
84C22×C4—2541no—
85C22×C4—2542no—
86C22×C4—2541no—
87C22×C4C22×D12711yescharacteristic
88M16—22no—
89M16—22no—
90M16—22no—
91M16—2542no—
92M16—2542no—
93M16—2542no—
94C2×C8—22no—
95C2×C8—1274no—
96C2×C8—2541no—
97C2×C8C22⋊D12712yes—
98C23—5082no—
99C23—2541no—
100C23—2542no—
101C23Q8×D12711yescharacteristic
102C2×C4—5081no—
103C2×C4—5082no—
104C2×C4—5081no—
105C2×C4—2541no—
106C2×C4—2542no—
107C2×C4D4×D12712yes—
108C2×C4C2×C22⋊D12711yescharacteristic
109C2×C4#1452324812yes—
110C2×C4#1452324911yescharacteristic
111C8—22no—
112C8—42no—
113C8—5081no—
114C8—2544no—
115C22—41no—
116C22—5081no—
117C22—5082no—
118C22—5081no—
119C22—2542no—
120C22—2544no—
121C22#1686118511yescharacteristic
122C22#1686118712yes—
123C4—5081no—
124C4#1686119211yescharacteristic
125C4#1686119412yes—
126C4#1686120511yescharacteristic
127C2—21no—
128C2—22no—
129C2—5082no—
130C2#2254981111yescharacteristic
subgroupG/H
centerC4#16861205
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#430×C127C2
socleC2×C127#399
prime radicalC23×D127Q8
hypercenter#430D127
hypocenterC127#2091
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #430

Complements

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

NH ≅ G/Ntype
1. #22549924C2semidirect
4. C127⋊#338C2semidirect
5. #430×C127C2semidirect
66. C127#2091semidirect
68. #430D127semidirect

Maximal subgroups

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

subgroupindex
#225499242
#225498392
C127⋊#3382
#430×C1272
#2091127