Finite Group Databasebeta

← #30640689

Subgroups — #30640689

Order 16256.

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

normal subgroups only

This group has 220 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#22550681C211yesmaximal, characteristic, complemented
2#22550314C211yesmaximal, characteristic, complemented
3C2×D2032C211yesmaximal, characteristic, complemented
4#22550146C211yesmaximal, characteristic, complemented
5#22549690C211yesmaximal, characteristic, complemented
6#22549684C211yesmaximal, characteristic, complemented
7#350×C127C211yesmaximal, characteristic, complemented
8#16861255—21no—
9#16861237—22no—
10#16861237—21no—
11#16861234—21no—
12#16861231—21no—
13#16861231C2211yescharacteristic
14C2×D1016—21no—
15C2×D1016C2211yescharacteristic
16C2×D1016C2211yescharacteristic, complemented
17#16861154—21no—
18D2032—22no—
19D2032—21no—
20#16861145C2211yescharacteristic, complemented
21#16861125—21no—
22#131×C127C2211yescharacteristic
23M32×C127C2211yescharacteristic, complemented
24C2×C16×C127C2211yescharacteristic, complemented
25#14523249—41no—
26#14523248—22no—
27#14523248—21no—
28C2×D508—22no—
29C2×D508—21no—
30C2×D508—21no—
31D4×D127—42no—
32D4×D127—41no—
33Dic508D411yescharacteristic
34#14523237—22no—
35#14523237—21no—
36#14523235—41no—
37D1016—22no—
38D1016—21no—
39D1016—21no—
40D1016—21no—
41D1016D411yescharacteristic
42D1016D412yes—
43#14523233—41no—
44#14523232—21no—
45#41×C127D411yescharacteristic
46M16×C127—21no—
47M16×C127D411yescharacteristic
48C2×C8×C127—21no—
49C2×C8×C127C2311yescharacteristic
50C16×C127—21no—
51C16×C127—21no—
52C22×D127—42no—
53C22×D127—41no—
54C22×D127—41no—
55Dic254—21no—
56D508—21no—
57D508—22no—
58D508—42no—
59D508—41no—
60D508—41no—
61D508—41no—
62C22⋊D127—42no—
63C22⋊D127—41no—
64C4×D127—42no—
65C4×D127—41no—
66C127⋊C8—41no—
67Q8×C127D811yescharacteristic
68D4×C127—21no—
69D4×C127D811yescharacteristic
70C2×C4×C127—21no—
71C2×C4×C127C2×D411yescharacteristic
72C8×C127—21no—
73C8×C127C2×D411yescharacteristic
74C8×C127C2×D411yescharacteristic
75C22×C127—21no—
76C22×C127AGL(1,ℤ/8)11yescharacteristic
77C2×D127—42no—
78C2×D127—41no—
79C2×D127—81no—
80C2×D127—161no—
81C2×D127—162no—
82C2×D127—161no—
83C4×C127—21no—
84C4×C127#12011yescharacteristic
85C4×C127C2×D811yescharacteristic
86Dic127—41no—
87C2×C127—21no—
88C2×C127—41no—
89C2×C127#44711yescharacteristic
90D127—82no—
91D127—81no—
92D127—161no—
93#2192—1271nomaximal, Sylow
94C127#219211yescharacteristic, complemented, Sylow
95#576—1271no—
96#509—1271no—
97C2×D16—1271no—
98#472—1271no—
99#361—1271no—
100#359—1271no—
101#350D12711yescharacteristic, complemented
102H(5,𝔽2)—2541no—
103AGL(1,ℤ/8)—2542no—
104AGL(1,ℤ/8)—2541no—
105AGL(1,ℤ/8)—2541no—
106#135—1271no—
107#135—2541no—
108C2×D8—1271no—
109C2×D8—1271no—
110C2×D8—2541no—
111#131C2×D12711yescharacteristic
112SD32—2541no—
113D16—2542no—
114D16—2541no—
115M32C2×D12711yescharacteristic, complemented
116C2×C16C2×D12711yescharacteristic, complemented
117#108—1271no—
118#100—2541no—
119#41—5081no—
120#41—2542no—
121#41—2541no—
122#41D50811yescharacteristic
123C2×D4—5082no—
124C2×D4—5081no—
125C2×D4—2542no—
126C2×D4—2541no—
127C2×D4—2541no—
128Q16—1271no—
129SD16—5081no—
130SD16—2542no—
131SD16—2541no—
132D8—1272no—
133D8—1271no—
134D8—5081no—
135D8—2542no—
136D8—2541no—
137D8—2541no—
138D8—2541no—
139M16—21no—
140M16—2541no—
141M16D50811yescharacteristic
142C2×C8—21no—
143C2×C8C22×D12711yescharacteristic
144C16—21no—
145C16—21no—
146C23—5082no—
147C23—5081no—
148C23—5081no—
149Q8—2541no—
150Q8D101611yescharacteristic
151D4—21no—
152D4—5082no—
153D4—5082no—
154D4—5081no—
155D4—5081no—
156D4—5081no—
157D4—5081no—
158D4—2542no—
159D4—2541no—
160D4D101611yescharacteristic
161C2×C4—21no—
162C2×C4—5082no—
163C2×C4—5081no—
164C2×C4C2×D50811yescharacteristic
165C8—21no—
166C8—5081no—
167C8D4×D12711yescharacteristic
168C8D4×D12711yescharacteristic
169C22—21no—
170C22—20321no—
171C22—20321no—
172C22—20322no—
173C22—10161no—
174C22—5082no—
175C22—5081no—
176C22#1686123711yescharacteristic
177C4—21no—
178C4—5081no—
179C4#1686117911yescharacteristic
180C4C2×D101611yescharacteristic
181C2—21no—
182C2—41no—
183C2—20321no—
184C2—10162no—
185C2—10161no—
186C2#2255001211yescharacteristic
subgroupG/H
centerC2#22550012
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#350×C127C2
socleC2×C127#447
hypercenter#350D127
hypocenterC127#2192
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #350

Complements

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

NH ≅ G/Ntype
1. #22550681C2semidirect
2. #22550314C2semidirect
3. C2×D2032C2semidirect
4. #22550146C2semidirect
5. #22549690C2semidirect
6. #22549684C2semidirect
7. #350×C127C2semidirect
16. C2×D1016C22semidirect
20. #16861145C22semidirect
23. M32×C127C22semidirect
24. C2×C16×C127C22semidirect
94. C127#2192semidirect
101. #350D127semidirect
115. M32C2×D127semidirect
116. C2×C16C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506812
#225503142
C2×D20322
#225501462
#225496902
#225496842
#350×C1272
#2192127