Finite Group Databasebeta

← #30640713

Subgroups — #30640713

Order 16256.

354 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 41 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#22550511C211yesmaximal, characteristic, complemented
2#22550063C212yesmaximal, complemented
3#22549669C211yesmaximal, characteristic, complemented
4#22549664C212yesmaximal, complemented
5#453×C127C211yesmaximal, characteristic, complemented
6C2×D4×D127—22no—
7#16861180—22no—
8#16861177—22no—
9#16861177—21no—
10#16861177C2212yes—
11C4×D508—21no—
12#16861124—22no—
13#16861124C2212yescomplemented
14H(5,𝔽2)×C127C2211yescharacteristic, complemented
15AGL(1,ℤ/8)×C127—21no—
16#127×C127C2211yescharacteristic
17#104×C127—21no—
18#100×C127C2211yescharacteristic, complemented
19C23×D127—21no—
20C23×D127D412yescomplemented
21C2×D508—21no—
22C2×C22⋊D127—22no—
23C2×C22⋊D127—22no—
24C2×C22⋊D127—42no—
25D4×D127—48no—
26C2×C4×D127—22no—
27C4⋊Dic127—21no—
28C22⋊Dic127—21no—
29C22⋊Dic127—22no—
30C22⋊Dic127D412yescomplemented
31(C2×D127)⋊C4—22no—
32(C2×D127)⋊C4—42no—
33#41×C127—21no—
34#41×C127—41no—
35C2×D4×C127—22no—
36C2×D4×C127—22no—
37C2×D4×C127—41no—
38C2×D4×C127D411yescharacteristic, complemented
39C2×D4×C127C2311yescharacteristic
40SD16×C127—41no—
41D8×C127—41no—
42M16×C127—21no—
43C42×C127D411yescharacteristic, complemented
44C22×D127—22no—
45C22×D127—44no—
46C22×D127—44no—
47C22×D127—42no—
48C22×D127—81no—
49D508—24no—
50C22⋊D127—44no—
51C22⋊D127—44no—
52C22⋊D127—84no—
53C2×Dic127—22no—
54C2×Dic127—42no—
55C4×D127—44no—
56C23×C127—22no—
57C23×C127—21no—
58C23×C127C2×D412yes—
59Q8×C127—21no—
60D4×C127—21no—
61D4×C127—41no—
62D4×C127—41no—
63D4×C127—44no—
64D4×C127—42no—
65D4×C127—41no—
66C2×C4×C127—21no—
67C2×C4×C127—22no—
68C2×C4×C127—41no—
69C2×C4×C127C2×D411yescharacteristic
70C8×C127—41no—
71C22×C127—22no—
72C22×C127—42no—
73C22×C127—42no—
74C22×C127—41no—
75C22×C127—41no—
76C22×C127—82no—
77C22×C127#12011yescharacteristic
78C2×D127—24no—
79C2×D127—42no—
80C2×D127—42no—
81C2×D127—44no—
82C2×D127—82no—
83C2×D127—84no—
84C4×C127—22no—
85C4×C127—21no—
86C4×C127—41no—
87Dic127—42no—
88Dic127—82no—
89C2×C127—21no—
90C2×C127—42no—
91C2×C127—81no—
92C2×C127—81no—
93C2×C127B(4,𝔽2)11yescharacteristic
94D127—44no—
95#2196—1271nomaximal, Sylow
96C127#219611yescharacteristic, complemented, Sylow
97D42—1271no—
98B(4,𝔽2)—1272no—
99#453D12711yescharacteristic, complemented
100#353—1271no—
101AΣL(1,𝔽16)—1272no—
102H(5,𝔽2)C2×D12711yescharacteristic, complemented
103C22×D4—2542no—
104AGL(1,ℤ/8)—21no—
105#127C2×D12711yescharacteristic
106#121—2542no—
107#120—1272no—
108#120—2542no—
109#120—2541no—
110C4×D4—2541no—
111#104—21no—
112#100C2×D12711yescharacteristic, complemented
113#99—1272no—
114#99—2542no—
115C24—1272no—
116C24—2541no—
117#41—21no—
118#41—41no—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4—41no—
122C2×D4—5088no—
123C2×D4—5082no—
124C2×D4—2541no—
125C2×D4—2542no—
126C2×D4—2542no—
127C2×D4C22⋊D12711yescharacteristic, complemented
128C2×D4C22×D12711yescharacteristic
129C22×C4—2542no—
130SD16—41no—
131D8—41no—
132M16—21no—
133C4⋊C4—2541no—
134C22⋊C4—1272no—
135C22⋊C4—5082no—
136C22⋊C4—2542no—
137C22⋊C4—2541no—
138C22⋊C4—2542no—
139C42C22⋊D12711yescharacteristic, complemented
140C23—22no—
141C23—21no—
142C23—10161no—
143C23—5082no—
144C23—5084no—
145C23—5084no—
146C23—2542no—
147C23D4×D12712yes—
148Q8—21no—
149D4—21no—
150D4—41no—
151D4—41no—
152D4—44no—
153D4—42no—
154D4—41no—
155D4—10164no—
156D4—5084no—
157D4—5084no—
158D4—2544no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—41no—
162C2×C4—5084no—
163C2×C4—5082no—
164C2×C4—2542no—
165C2×C4C2×C22⋊D12711yescharacteristic
166C8—41no—
167C22—22no—
168C22—42no—
169C22—42no—
170C22—41no—
171C22—41no—
172C22—82no—
173C22—10162no—
174C22—10164no—
175C22—5082no—
176C22—5084no—
177C22—5082no—
178C22—2544no—
179C22#1686117711yescharacteristic
180C4—22no—
181C4—21no—
182C4—41no—
183C4—10162no—
184C4—5082no—
185C2—21no—
186C2—42no—
187C2—81no—
188C2—81no—
189C2—5084no—
190C2#2255006211yescharacteristic
subgroupG/H
centerC2#22550062
commutator, cosocleC2×D4×C127C23
FrattiniC2×D4C22×D127
Fitting#453×C127C2
socleC2×C127B(4,𝔽2)
hypercenter#453D127
hypocenterC127#2196
LCSG ▷ C2×D4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #22550511C2semidirect
2. #22550063C2semidirect
3. #22549669C2semidirect
4. #22549664C2semidirect
5. #453×C127C2semidirect
13. #16861124C22semidirect
14. H(5,𝔽2)×C127C22semidirect
18. #100×C127C22semidirect
20. C23×D127D4semidirect
30. C22⋊Dic127D4semidirect
38. C2×D4×C127D4semidirect
43. C42×C127D4semidirect
96. C127#2196semidirect
99. #453D127semidirect
102. H(5,𝔽2)C2×D127semidirect
112. #100C2×D127semidirect
127. C2×D4C22⋊D127semidirect
139. C42C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225505112
#225500632
#225496692
#225496642
#453×C1272
#2196127