Finite Group Databasebeta

← #30640675

Subgroups — #30640675

Order 16256.

226 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 49 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#22550676C211yesmaximal, characteristic, complemented
2#22550312C212yesmaximal, complemented
3#22549685C212yesmaximal, complemented
4#22549658C211yesmaximal, characteristic, complemented
5#469×C127C211yesmaximal, characteristic, complemented
6C2×#14523248—21no—
7#16861242—22no—
8#16861237—22no—
9#16861231—21no—
10#16861231C2212yes—
11#16861153—22no—
12#16861150—22no—
13#16861149C2212yescomplemented
14C2×D8×C127C2211yescharacteristic, complemented
15C2×M16×C127C2211yescharacteristic
16#121×C127—21no—
17#106×C127C2211yescharacteristic, complemented
18#102×C127—21no—
19#14523248—22no—
20#14523248—22no—
21#14523248—41no—
22#14523248—41no—
23C2×Dic254—21no—
24C2×D508—21no—
25C2×C22⋊D127—41no—
26C2×C4×D127—41no—
27Dic508—21no—
28Dic508D412yescomplemented
29#14523237—22no—
30#14523237—22no—
31D1016—21no—
32D1016D412yescomplemented
33C127⋊C16—22no—
34C2×D4×C127—21no—
35C2×D4×C127—41no—
36C22×C4×C127D411yescharacteristic
37D8×C127—22no—
38M16×C127—22no—
39C2×C8×C127D411yescharacteristic
40C2×C8×C127C2311yescharacteristic
41C4⋊C4×C127—21no—
42C22⋊C4×C127—41no—
43C22×D127—41no—
44Dic254—22no—
45Dic254—41no—
46D508—22no—
47D508—41no—
48C22⋊D127—42no—
49C22⋊D127—42no—
50C2×Dic127—41no—
51C4×D127—42no—
52C4×D127—42no—
53C23×C127—41no—
54C23×C127D811yescharacteristic
55D4×C127—41no—
56D4×C127—41no—
57D4×C127—82no—
58C2×C4×C127—21no—
59C2×C4×C127—21no—
60C2×C4×C127—41no—
61C2×C4×C127D811yescharacteristic
62C2×C4×C127C2×D411yescharacteristic
63C8×C127—21no—
64C8×C127C2×D412yes—
65C22×C127—21no—
66C22×C127—41no—
67C22×C127—81no—
68C22×C127—81no—
69C22×C127—81no—
70C22×C127C2×D811yescharacteristic
71C2×D127—42no—
72C2×D127—82no—
73C4×C127—21no—
74C4×C127—81no—
75C4×C127#12011yescharacteristic
76C4×C127AGL(1,ℤ/8)11yescharacteristic
77Dic127—42no—
78C2×C127—21no—
79C2×C127—41no—
80C2×C127—161no—
81C2×C127#44711yescharacteristic
82D127—82no—
83#2190—1271nomaximal, Sylow
84C127#219011yescharacteristic, complemented, Sylow
85#575—1271no—
86#509—1272no—
87#469D12711yescharacteristic, complemented
88#360—1272no—
89#349—1271no—
90C2×#41—2541no—
91#137—2542no—
92AGL(1,ℤ/8)—2542no—
93#135—1272no—
94#135—2541no—
95C2×D8C2×D12711yescharacteristic, complemented
96C2×M16C2×D12711yescharacteristic
97#121—21no—
98SD32—2542no—
99D16—2542no—
100M32—1272no—
101#106C2×D12711yescharacteristic, complemented
102#102—21no—
103#41—5081no—
104#41—5081no—
105#41—2542no—
106#41—2542no—
107C2×Q8—2541no—
108C2×D4—21no—
109C2×D4—41no—
110C2×D4—5081no—
111C2×D4—2541no—
112C22×C4—5081no—
113C22×C4C22⋊D12711yescharacteristic
114Q16—1272no—
115Q16—2541no—
116SD16—2542no—
117SD16—2542no—
118D8—22no—
119D8—1272no—
120D8—2541no—
121M16—22no—
122C2×C8C22⋊D12711yescharacteristic
123C2×C8C22×D12711yescharacteristic
124C4⋊C4—21no—
125C22⋊C4—41no—
126C16—2542no—
127C23—41no—
128C23—5081no—
129C23#1452323311yescharacteristic
130Q8—5081no—
131Q8—2542no—
132D4—41no—
133D4—41no—
134D4—82no—
135D4—5081no—
136D4—5082no—
137D4—5082no—
138D4—2542no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—41no—
142C2×C4—5082no—
143C2×C4—5081no—
144C2×C4—5082no—
145C2×C4#1452323311yescharacteristic
146C2×C4C2×C22⋊D12711yescharacteristic
147C8—21no—
148C8D4×D12712yes—
149C22—21no—
150C22—41no—
151C22—81no—
152C22—81no—
153C22—81no—
154C22—10162no—
155C22—5082no—
156C22C2×#1452323311yescharacteristic
157C4—21no—
158C4—81no—
159C4—5082no—
160C4#1686117711yescharacteristic
161C4#1686123411yescharacteristic
162C2—21no—
163C2—41no—
164C2—161no—
165C2—10162no—
166C2#2255000911yescharacteristic
subgroupG/H
centerC2#22550009
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#469×C127C2
socleC2×C127#447
hypercenter#469D127
hypocenterC127#2190
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #22550676C2semidirect
2. #22550312C2semidirect
3. #22549685C2semidirect
4. #22549658C2semidirect
5. #469×C127C2semidirect
13. #16861149C22semidirect
14. C2×D8×C127C22semidirect
17. #106×C127C22semidirect
28. Dic508D4semidirect
32. D1016D4semidirect
84. C127#2190semidirect
87. #469D127semidirect
95. C2×D8C2×D127semidirect
101. #106C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506762
#225503122
#225496852
#225496582
#469×C1272
#2190127