Finite Group Databasebeta

← #30640684

Subgroups — #30640684

Order 16256.

220 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 47 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#22550678C211yesmaximal, characteristic, complemented
2C2×#16861150C211yesmaximal, characteristic, complemented
3#22550143C211yesmaximal, characteristic, complemented
4#22549690C211yesmaximal, characteristic, complemented
5#22549683C211yesmaximal, characteristic, complemented
6#22549661C211yesmaximal, characteristic, complemented
7#509×C127C211yesmaximal, characteristic, complemented
8#16861255—21no—
9#16861236—22no—
10#16861235—21no—
11#16861234—21no—
12#16861232—21no—
13C2×D1016C2211yescharacteristic
14D8×D127—21no—
15#16861215C2211yescharacteristic
16#16861150—22no—
17C2×C127⋊C16C2211yescharacteristic, complemented
18#16861145C2211yescharacteristic, complemented
19#16861126—21no—
20#135×C127C2211yescharacteristic
21C2×D8×C127C2211yescharacteristic, complemented
22SD32×C127—21no—
23D16×C127—21no—
24M32×C127C2211yescharacteristic, complemented
25#14523249—22no—
26#14523248—41no—
27#14523248D411yescharacteristic
28C2×D508—21no—
29C2×D508—41no—
30D4×D127—22no—
31D4×D127—42no—
32#14523236—41no—
33#14523235—22no—
34D1016—21no—
35D1016D412yes—
36#14523233—22no—
37#14523233—41no—
38#14523232—21no—
39#14523232D411yescharacteristic
40#14523231—21no—
41C8×D127—21no—
42C127⋊C16—21no—
43#41×C127—21no—
44C2×D4×C127—21no—
45Q16×C127D411yescharacteristic
46SD16×C127—21no—
47D8×C127—21no—
48D8×C127—21no—
49D8×C127D411yescharacteristic
50C2×C8×C127C2311yescharacteristic
51C16×C127—21no—
52C22×D127—42no—
53C22×D127—41no—
54Dic254D811yescharacteristic
55D508—22no—
56D508—42no—
57D508—41no—
58D508D811yescharacteristic
59C22⋊D127—21no—
60C22⋊D127—42no—
61C4×D127—21no—
62C4×D127—42no—
63C127⋊C8—21no—
64C127⋊C8—41no—
65C23×C127—41no—
66Q8×C127—21no—
67D4×C127—21no—
68D4×C127—41no—
69D4×C127—41no—
70D4×C127—41no—
71C2×C4×C127—41no—
72C2×C4×C127C2×D411yescharacteristic
73C8×C127C2×D411yescharacteristic
74C8×C127C2×D411yescharacteristic
75C22×C127—41no—
76C22×C127—81no—
77C22×C127—161no—
78C22×C127AGL(1,ℤ/8)11yescharacteristic
79C2×D127—21no—
80C2×D127—42no—
81C2×D127—162no—
82C2×D127—161no—
83C4×C127—41no—
84C4×C127#12011yescharacteristic
85C4×C127C2×D811yescharacteristic
86Dic127—21no—
87C2×C127—21no—
88C2×C127—81no—
89C2×C127—161no—
90C2×C127#44711yescharacteristic
91D127—41no—
92D127—82no—
93#2192—1271nomaximal, Sylow
94C127#219211yescharacteristic, complemented, Sylow
95#576—1271no—
96#509D12711yescharacteristic, complemented
97C2×D16—1271no—
98#472—1271no—
99#361—1271no—
100#359—1271no—
101#350—1271no—
102H(5,𝔽2)—2541no—
103AGL(1,ℤ/8)—2541no—
104AGL(1,ℤ/8)—2542no—
105AGL(1,ℤ/8)—2541no—
106#135—2541no—
107#135C2×D12711yescharacteristic
108C2×D8—1271no—
109C2×D8—2541no—
110C2×D8C2×D12711yescharacteristic, complemented
111#131—1271no—
112SD32—21no—
113D16—21no—
114D16—2542no—
115M32C2×D12711yescharacteristic, complemented
116C2×C16—1271no—
117#108—1271no—
118#100—2541no—
119#41—21no—
120#41—1271no—
121#41—5081no—
122#41—2542no—
123C2×D4—21no—
124C2×D4—5081no—
125C2×D4—5082no—
126C2×D4—2542no—
127C2×D4—2541no—
128Q16C22⋊D12711yescharacteristic
129SD16—21no—
130SD16—5081no—
131SD16—2542no—
132D8—21no—
133D8—21no—
134D8—1272no—
135D8—5081no—
136D8—2542no—
137D8—2541no—
138D8C22⋊D12711yescharacteristic
139M16—1271no—
140M16—2541no—
141M16—2541no—
142C2×C8—2541no—
143C2×C8C22×D12711yescharacteristic
144C16—21no—
145C16—2541no—
146C23—41no—
147C23—5082no—
148C23—5081no—
149Q8—21no—
150Q8—1271no—
151D4—21no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—1271no—
156D4—5082no—
157D4—5082no—
158D4—5081no—
159D4—2541no—
160D4—2542no—
161C2×C4—41no—
162C2×C4—5082no—
163C2×C4—2541no—
164C2×C4D4×D12711yescharacteristic
165C8—5081no—
166C8—2541no—
167C8D4×D12711yescharacteristic
168C8C2×C22⋊D12711yescharacteristic
169C22—41no—
170C22—81no—
171C22—161no—
172C22—20321no—
173C22—20322no—
174C22—5082no—
175C22—2541no—
176C22#1686123311yescharacteristic
177C4—41no—
178C4—2541no—
179C4#1686117711yescharacteristic
180C4D8×D12711yescharacteristic
181C2—21no—
182C2—81no—
183C2—161no—
184C2—10162no—
185C2—5081no—
186C2#2255000811yescharacteristic
subgroupG/H
centerC2#22550008
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#509×C127C2
socleC2×C127#447
hypercenter#509D127
hypocenterC127#2192
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #22550678C2semidirect
2. C2×#16861150C2semidirect
3. #22550143C2semidirect
4. #22549690C2semidirect
5. #22549683C2semidirect
6. #22549661C2semidirect
7. #509×C127C2semidirect
17. C2×C127⋊C16C22semidirect
18. #16861145C22semidirect
21. C2×D8×C127C22semidirect
24. M32×C127C22semidirect
94. C127#2192semidirect
96. #509D127semidirect
110. C2×D8C2×D127semidirect
115. M32C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506782
C2×#168611502
#225501432
#225496902
#225496832
#225496612
#509×C1272
#2192127