Finite Group Databasebeta

← #30640726

Subgroups — #30640726

Order 16256.

246 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 45 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#22550065C212yesmaximal, complemented
2#22550043C211yesmaximal, characteristic
3#22549669C211yesmaximal, characteristic, complemented
4#22549666C212yesmaximal, complemented
5#559×C127C211yesmaximal, characteristic, complemented
6#16861254C2211yescharacteristic
7#16861235—21no—
8#16861192—22no—
9#16861137—21no—
10#16861126C2211yescharacteristic
11#16861124C2212yes—
12#16861123—22no—
13#127×C127C2211yescharacteristic
14#124×C127—22no—
15#123×C127—21no—
16#123×C127C2212yes—
17C4×Q8×C127—21no—
18C4×D4×C127—22no—
19#14523248—21no—
20#14523247—41no—
21C2×C22⋊D127—22no—
22D4×D127—41no—
23#14523236—41no—
24#14523233—41no—
25#14523232—21no—
26Dic127⋊C4—42no—
27C22⋊Dic127—22no—
28(C2×D127)⋊C4—42no—
29C2×Q8×C127—21no—
30C2×D4×C127—22no—
31C2×D4×C127D411yescharacteristic
32C2×D4×C127C2311yescharacteristic
33C22×C4×C127—21no—
34C22×C4×C127D412yes—
35C4⋊C4×C127—21no—
36C4⋊C4×C127—22no—
37C22⋊C4×C127—22no—
38C22⋊C4×C127—21no—
39C22⋊C4×C127—22no—
40C22⋊C4×C127D412yes—
41C42×C127—22no—
42C42×C127D411yescharacteristic
43C22×D127—22no—
44Dic254—21no—
45D508—21no—
46C22⋊D127—42no—
47C22⋊D127—41no—
48C2×Dic127—22no—
49C2×Dic127—42no—
50C4×D127—41no—
51C127⋊C8—41no—
52C23×C127—21no—
53C23×C127C2×D412yes—
54Q8×C127—42no—
55D4×C127—41no—
56D4×C127—41no—
57D4×C127—44no—
58C2×C4×C127—21no—
59C2×C4×C127—22no—
60C2×C4×C127—22no—
61C2×C4×C127—42no—
62C2×C4×C127—81no—
63C2×C4×C127C2×D411yescharacteristic
64C22×C127—22no—
65C22×C127—42no—
66C22×C127—42no—
67C22×C127—41no—
68C22×C127#12011yescharacteristic
69C2×D127—41no—
70C2×D127—82no—
71C4×C127—22no—
72C4×C127—21no—
73C4×C127—42no—
74C4×C127—42no—
75Dic127—41no—
76Dic127—82no—
77C2×C127—21no—
78C2×C127—42no—
79C2×C127—81no—
80C2×C127B(4,𝔽2)11yescharacteristic
81D127—81no—
82#2198—1271nomaximal, Sylow
83C127#219811yescharacteristic, complemented, Sylow
84#559D12711yescharacteristic, complemented
85#458—1272no—
86#453—1271no—
87#353—1271no—
88#352—1272no—
89H(5,𝔽2)—1271no—
90AGL(1,ℤ/8)—2541no—
91#127C2×D12711yescharacteristic
92#124—22no—
93#123—21no—
94#123—2542no—
95#123C2×D12712yes—
96C4×Q8—21no—
97C4×D4—22no—
98#104—2541no—
99#100—1271no—
100#99—1272no—
101#99—2542no—
102#41—5081no—
103#41—2541no—
104C2×Q8—21no—
105C2×D4—22no—
106C2×D4—5081no—
107C2×D4—2542no—
108C2×D4C22⋊D12711yescharacteristic
109C2×D4C22×D12711yescharacteristic
110C22×C4—21no—
111C22×C4C22⋊D12712yes—
112SD16—5081no—
113D8—5081no—
114M16—2541no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C4⋊C4—5082no—
118C22⋊C4—22no—
119C22⋊C4—21no—
120C22⋊C4—22no—
121C22⋊C4—5082no—
122C22⋊C4—2542no—
123C22⋊C4C22⋊D12712yes—
124C42—22no—
125C42C22⋊D12711yescharacteristic
126C23—21no—
127C23—2542no—
128C23C2×C22⋊D12712yes—
129Q8—42no—
130Q8—2541no—
131D4—41no—
132D4—41no—
133D4—44no—
134D4—5082no—
135D4—5081no—
136D4—2541no—
137C2×C4—21no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—42no—
141C2×C4—81no—
142C2×C4—5081no—
143C2×C4—5082no—
144C2×C4—2542no—
145C2×C4C2×C22⋊D12711yescharacteristic
146C8—5081no—
147C22—22no—
148C22—42no—
149C22—42no—
150C22—41no—
151C22—10162no—
152C22—5081no—
153C22#1686117811yescharacteristic
154C4—22no—
155C4—21no—
156C4—42no—
157C4—42no—
158C4—10162no—
159C4—5081no—
160C2—21no—
161C2—42no—
162C2—81no—
163C2—10161no—
164C2#2255006311yescharacteristic
subgroupG/H
centerC2#22550063
commutator, cosocleC2×D4×C127C23
FrattiniC2×D4C22×D127
Fitting#559×C127C2
socleC2×C127B(4,𝔽2)
prime radical#22550043C2
hypercenter#559D127
hypocenterC127#2198
LCSG ▷ C2×D4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #559

Complements

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

NH ≅ G/Ntype
1. #22550065C2semidirect
3. #22549669C2semidirect
4. #22549666C2semidirect
5. #559×C127C2semidirect
83. C127#2198semidirect
84. #559D127semidirect

Maximal subgroups

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

subgroupindex
#225500652
#225500432
#225496692
#225496662
#559×C1272
#2198127