Finite Group Databasebeta

← #30640733

Subgroups — #30640733

Order 16256.

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

normal subgroups only

This group has 270 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#22550618C211yesmaximal, characteristic, complemented
2#22550064C212yesmaximal, complemented
3#22550053C211yesmaximal, characteristic, complemented
4#22549662C212yesmaximal, complemented
5#354×C127C211yesmaximal, characteristic, complemented
6#16861254C2211yescharacteristic
7#16861240—21no—
8#16861198—21no—
9#16861197—21no—
10#16861195—22no—
11#16861179—22no—
12#16861179—22no—
13#16861177C2212yes—
14#16861137—21no—
15#16861126C2211yescharacteristic
16#16861123—22no—
17#124×C127C2211yescharacteristic
18#99×C127C2212yes—
19C23×D127D412yescomplemented
20#14523248—21no—
21#14523247—41no—
22C2×Dic254—21no—
23C2×D508—21no—
24C2×D508—21no—
25C2×D508—42no—
26C2×C22⋊D127—22no—
27C2×C22⋊D127—22no—
28D4×D127—41no—
29#14523238—41no—
30#14523235—41no—
31#14523232—21no—
32C22⋊Dic127D412yescomplemented
33(C2×D127)⋊C4—22no—
34(C2×D127)⋊C4—22no—
35(C2×D127)⋊C4—22no—
36(C2×D127)⋊C4—42no—
37C4×Dic127—21no—
38C2×Q8×C127D411yescharacteristic, complemented
39C2×D4×C127C2311yescharacteristic
40C22⋊C4×C127—22no—
41C22⋊C4×C127—22no—
42C42×C127D411yescharacteristic, complemented
43C22×D127—22no—
44C22×D127—22no—
45C22×D127—42no—
46C22×D127—82no—
47Dic254—21no—
48Dic254—81no—
49D508—21no—
50D508—81no—
51D508—81no—
52D508—84no—
53C22⋊D127—42no—
54C22⋊D127—41no—
55C22⋊D127—82no—
56C2×Dic127—22no—
57C2×Dic127—22no—
58C4×D127—41no—
59C127⋊C8—41no—
60C23×C127C2×D412yes—
61Q8×C127—41no—
62D4×C127—41no—
63C2×C4×C127—21no—
64C2×C4×C127—21no—
65C2×C4×C127—42no—
66C2×C4×C127C2×D411yescharacteristic
67C22×C127—22no—
68C22×C127—42no—
69C22×C127#12011yescharacteristic
70C2×D127—42no—
71C2×D127—42no—
72C2×D127—44no—
73C2×D127—41no—
74C2×D127—42no—
75C2×D127—82no—
76C2×D127—82no—
77C4×C127—21no—
78C4×C127—41no—
79C4×C127—41no—
80C4×C127—82no—
81Dic127—42no—
82Dic127—41no—
83C2×C127—21no—
84C2×C127—42no—
85C2×C127B(4,𝔽2)11yescharacteristic
86D127—82no—
87D127—81no—
88#2199—1271nomaximal, Sylow
89C127#219911yescharacteristic, complemented, Sylow
90H(3,𝔽4)—1271no—
91B(4,𝔽2)—1272no—
92#455—1271no—
93#354D12711yescharacteristic, complemented
94AΣL(1,𝔽16)—1272no—
95H(5,𝔽2)—1271no—
96#137—2541no—
97#124—2542no—
98#124—2541no—
99#124—2541no—
100#124C2×D12711yescharacteristic
101#120—1272no—
102#120—2542no—
103#120—2542no—
104#104—2541no—
105#100—1271no—
106#99—2542no—
107#99C2×D12712yes—
108C24—1272no—
109#41—5081no—
110#41—2541no—
111C2×Q8—2541no—
112C2×Q8D50811yescharacteristic, complemented
113C2×D4—5082no—
114C2×D4—5081no—
115C2×D4—2541no—
116C2×D4—2541no—
117C2×D4—2542no—
118C2×D4—2542no—
119C2×D4C22×D12711yescharacteristic
120Q16—5081no—
121SD16—5081no—
122M16—2541no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—1272no—
126C22⋊C4—5082no—
127C22⋊C4—2542no—
128C22⋊C4—2542no—
129C22⋊C4—2542no—
130C42—2541no—
131C42D50811yescharacteristic, complemented
132C23—10162no—
133C23—5082no—
134C23—2542no—
135C23—2542no—
136C23D4×D12712yes—
137Q8—41no—
138Q8—10161no—
139Q8—2541no—
140D4—41no—
141D4—10161no—
142D4—10161no—
143D4—10164no—
144D4—10162no—
145D4—5082no—
146D4—5081no—
147D4—2541no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—42no—
151C2×C4—5081no—
152C2×C4—2542no—
153C2×C4—2542no—
154C2×C4C2×D50811yescharacteristic
155C8—5081no—
156C22—22no—
157C22—42no—
158C22—10162no—
159C22—10162no—
160C22—5082no—
161C22—5082no—
162C22—5084no—
163C22—5081no—
164C22—5082no—
165C22#1686117911yescharacteristic
166C4—21no—
167C4—41no—
168C4—41no—
169C4—82no—
170C4—5082no—
171C4—5081no—
172C2—21no—
173C2—42no—
174C2—10161no—
175C2—10162no—
176C2#2255006411yescharacteristic
subgroupG/H
centerC2#22550064
commutator, cosocleC2×D4×C127C23
FrattiniC2×D4C22×D127
Fitting#354×C127C2
socleC2×C127B(4,𝔽2)
hypercenter#354D127
hypocenterC127#2199
LCSG ▷ C2×D4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
1. #22550618C2semidirect
2. #22550064C2semidirect
3. #22550053C2semidirect
4. #22549662C2semidirect
5. #354×C127C2semidirect
19. C23×D127D4semidirect
32. C22⋊Dic127D4semidirect
38. C2×Q8×C127D4semidirect
42. C42×C127D4semidirect
89. C127#2199semidirect
93. #354D127semidirect
112. C2×Q8D508semidirect
131. C42D508semidirect

Maximal subgroups

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

subgroupindex
#225506182
#225500642
#225500532
#225496622
#354×C1272
#2199127