Finite Group Databasebeta

← #30640676

Subgroups — #30640676

Order 16256.

226 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 53 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#22550311C212yesmaximal, complemented
2#22550129C211yesmaximal, characteristic, complemented
3#22549686C212yesmaximal, complemented
4#22549658C211yesmaximal, characteristic, complemented
5#575×C127C211yesmaximal, characteristic, complemented
6C2×D1016C2211yescharacteristic, complemented
7#16861183—21no—
8#16861152—22no—
9#16861150—22no—
10#16861149C2212yescomplemented
11#16861141C2211yescharacteristic, complemented
12#16861131—21no—
13C2×#41×C127—21no—
14#137×C127—22no—
15AGL(1,ℤ/8)×C127—22no—
16#135×C127—21no—
17#135×C127C2212yes—
18C2×M16×C127C2211yescharacteristic
19C2×D508—21no—
20C2×C22⋊D127—41no—
21D1016—22no—
22C4⋊Dic127—21no—
23(C2×D127)⋊C4—41no—
24C127⋊C16—22no—
25#41×C127—22no—
26#41×C127—22no—
27#41×C127—41no—
28#41×C127—41no—
29C2×Q8×C127—21no—
30C2×D4×C127—21no—
31C2×D4×C127—41no—
32C22×C4×C127—41no—
33C22×C4×C127D411yescharacteristic
34Q16×C127—21no—
35Q16×C127D412yescomplemented
36SD16×C127—22no—
37SD16×C127—22no—
38D8×C127—21no—
39D8×C127D412yescomplemented
40M16×C127—22no—
41C2×C8×C127D411yescharacteristic
42C2×C8×C127C2311yescharacteristic
43C22×D127—41no—
44D508—41no—
45D508—41no—
46C22⋊D127—82no—
47C2×Dic127—41no—
48C23×C127—41no—
49C23×C127D811yescharacteristic
50Q8×C127—22no—
51Q8×C127—41no—
52D4×C127—22no—
53D4×C127—41no—
54D4×C127—42no—
55D4×C127—42no—
56C2×C4×C127—21no—
57C2×C4×C127—21no—
58C2×C4×C127—42no—
59C2×C4×C127—41no—
60C2×C4×C127—42no—
61C2×C4×C127D811yescharacteristic
62C2×C4×C127C2×D411yescharacteristic
63C8×C127—21no—
64C8×C127C2×D412yes—
65C22×C127—21no—
66C22×C127—41no—
67C22×C127—42no—
68C22×C127—82no—
69C22×C127C2×D811yescharacteristic
70C2×D127—81no—
71C2×D127—81no—
72C2×D127—81no—
73C4×C127—21no—
74C4×C127—42no—
75C4×C127#12011yescharacteristic
76C4×C127AGL(1,ℤ/8)11yescharacteristic
77Dic127—81no—
78C2×C127—21no—
79C2×C127—41no—
80C2×C127—82no—
81C2×C127#44711yescharacteristic
82D127—161no—
83#2190—1271nomaximal, Sylow
84C127#219011yescharacteristic, complemented, Sylow
85#575D12711yescharacteristic, complemented
86#509—1272no—
87#469—1271no—
88#360—1272no—
89#349—1271no—
90C2×#41—21no—
91#137—22no—
92AGL(1,ℤ/8)—22no—
93#135—21no—
94#135C2×D12712yes—
95C2×D8—1271no—
96C2×M16C2×D12711yescharacteristic
97#121—2541no—
98SD32—2542no—
99D16—2542no—
100M32—1272no—
101#106—1271no—
102#102—2541no—
103#41—22no—
104#41—22no—
105#41—41no—
106#41—41no—
107C2×Q8—21no—
108C2×D4—21no—
109C2×D4—41no—
110C2×D4—5081no—
111C2×D4—2541no—
112C22×C4—41no—
113C22×C4C22⋊D12711yescharacteristic
114Q16—21no—
115Q16C22⋊D12712yescomplemented
116SD16—22no—
117SD16—22no—
118D8—21no—
119D8—2542no—
120D8C22⋊D12712yescomplemented
121M16—22no—
122C2×C8C22⋊D12711yescharacteristic
123C2×C8C22×D12711yescharacteristic
124C4⋊C4—2541no—
125C22⋊C4—5081no—
126C16—2542no—
127C23—41no—
128C23—5081no—
129C23#1452323311yescharacteristic
130Q8—22no—
131Q8—41no—
132D4—22no—
133D4—41no—
134D4—42no—
135D4—42no—
136D4—10162no—
137D4—5081no—
138D4—5081no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—42no—
142C2×C4—41no—
143C2×C4—42no—
144C2×C4—5081no—
145C2×C4#1452323311yescharacteristic
146C2×C4C2×C22⋊D12711yescharacteristic
147C8—21no—
148C8C2×C22⋊D12712yes—
149C22—21no—
150C22—41no—
151C22—42no—
152C22—82no—
153C22—10161no—
154C22—10161no—
155C22—10161no—
156C22C2×#1452323311yescharacteristic
157C4—21no—
158C4—42no—
159C4—10161no—
160C4#1686117811yescharacteristic
161C4#1686123511yescharacteristic
162C2—21no—
163C2—41no—
164C2—82no—
165C2—20321no—
166C2#2255001011yescharacteristic
subgroupG/H
centerC2#22550010
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#575×C127C2
socleC2×C127#447
hypercenter#575D127
hypocenterC127#2190
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #22550311C2semidirect
2. #22550129C2semidirect
3. #22549686C2semidirect
4. #22549658C2semidirect
5. #575×C127C2semidirect
6. C2×D1016C22semidirect
10. #16861149C22semidirect
11. #16861141C22semidirect
35. Q16×C127D4semidirect
39. D8×C127D4semidirect
84. C127#2190semidirect
85. #575D127semidirect
115. Q16C22⋊D127semidirect
120. D8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225503112
#225501292
#225496862
#225496582
#575×C1272
#2190127