Finite Group Databasebeta

← #30637699

Subgroups — #30637699

Order 16256.

282 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 57 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#22550014C212yesmaximal, complemented
2#22549796C211yesmaximal, characteristic, complemented
3#22549610C212yesmaximal, complemented
4#22549602C211yesmaximal, characteristic, complemented
5#537×C127C211yesmaximal, characteristic
6C22×Dic254C2211yescharacteristic
7C2×#14523236—22no—
8C2×Dic127⋊C4—21no—
9#16861132—22no—
10C22⋊(C127⋊C8)C2212yes—
11C127⋊#95—21no—
12C127⋊#95C2211yescharacteristic
13C2×#41×C127—21no—
14C22⋊Q8×C127—22no—
15#121×C127—22no—
16#121×C127—21no—
17#121×C127C2212yes—
18C2×C4⋊C4×C127C2211yescharacteristic
19C2×Dic254—22no—
20C2×Dic254—42no—
21C22×Dic127—21no—
22C22×Dic127—21no—
23#14523236—44no—
24C2×C127⋊C8—22no—
25Dic127⋊C4—42no—
26#41×C127—42no—
27#41×C127—42no—
28C2×Q8×C127—21no—
29C2×D4×C127—21no—
30C2×D4×C127—22no—
31C2×D4×C127—21no—
32C2×D4×C127D412yes—
33C22×C4×C127—22no—
34C22×C4×C127D411yescharacteristic
35C22×C4×C127D411yescharacteristic
36C22×C4×C127C2311yescharacteristic
37C4⋊C4×C127—21no—
38C4⋊C4×C127—22no—
39C4⋊C4×C127D412yes—
40C22⋊C4×C127—22no—
41C22⋊C4×C127—22no—
42Dic254—24no—
43Dic254—42no—
44C2×Dic127—22no—
45C2×Dic127—22no—
46C2×Dic127—41no—
47C2×Dic127—41no—
48C2×Dic127—41no—
49C2×Dic127—41no—
50C127⋊C8—42no—
51C23×C127—22no—
52C23×C127C2×D411yescharacteristic
53Q8×C127—42no—
54D4×C127—42no—
55D4×C127—42no—
56D4×C127—42no—
57D4×C127—42no—
58D4×C127—81no—
59C2×C4×C127—21no—
60C2×C4×C127—21no—
61C2×C4×C127—21no—
62C2×C4×C127—22no—
63C2×C4×C127—41no—
64C2×C4×C127—44no—
65C2×C4×C127—41no—
66C2×C4×C127SD1612yescomplemented
67C2×C4×C127C2×D412yes—
68C22×C127—21no—
69C22×C127—21no—
70C22×C127—42no—
71C22×C127—42no—
72C22×C127—42no—
73C22×C127#12011yescharacteristic
74C22×C127C2×SD1611yescharacteristic
75C22×C127AGL(1,ℤ/8)11yescharacteristic
76C4×C127—22no—
77C4×C127—22no—
78C4×C127—42no—
79C4×C127—41no—
80Dic127—42no—
81Dic127—42no—
82C2×C127—21no—
83C2×C127—21no—
84C2×C127—82no—
85C2×C127#45011yescharacteristic
86C2×C127#45611yescharacteristic
87C2×C127B(4,𝔽2)11yescharacteristic
88#1604—1271nomaximal, Sylow
89C127#160411yescharacteristic, complemented, Sylow
90#537D12711yescharacteristic
91#448—1272no—
92#395—1271no—
93#327—1272no—
94#324—1271no—
95C2×#41—21no—
96C22×Q8—1271no—
97C2×SD16—2542no—
98C22⋊Q8—22no—
99#121—22no—
100#121—21no—
101#121C2×D12712yes—
102C2×C4⋊C4—2541no—
103C2×C4⋊C4C2×D12711yescharacteristic
104#103—2542no—
105C22⋊C8—1272no—
106#95—1271no—
107#95—2541no—
108#41—42no—
109#41—42no—
110C2×Q8—21no—
111C2×Q8—5082no—
112C2×Q8—2542no—
113C2×D4—21no—
114C2×D4—22no—
115C2×D4—21no—
116C2×D4C22⋊D12712yes—
117C22×C4—22no—
118C22×C4—2541no—
119C22×C4—2541no—
120C22×C4C22⋊D12711yescharacteristic
121C22×C4C22⋊D12711yescharacteristic
122C22×C4C22×D12711yescharacteristic
123SD16—5084no—
124C2×C8—2542no—
125C4⋊C4—21no—
126C4⋊C4—22no—
127C4⋊C4—5082no—
128C4⋊C4C22⋊D12712yes—
129C22⋊C4—22no—
130C22⋊C4—22no—
131C23—22no—
132C23C2×C22⋊D12711yescharacteristic
133Q8—42no—
134Q8—5082no—
135Q8—2544no—
136D4—42no—
137D4—42no—
138D4—42no—
139D4—42no—
140D4—81no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—22no—
145C2×C4—41no—
146C2×C4—44no—
147C2×C4—41no—
148C2×C4—5081no—
149C2×C4—5081no—
150C2×C4—5081no—
151C2×C4—5081no—
152C2×C4—2542no—
153C2×C4—2542no—
154C2×C4#1452323612yescomplemented
155C2×C4C2×C22⋊D12712yes—
156C8—5082no—
157C22—21no—
158C22—21no—
159C22—42no—
160C22—42no—
161C22—42no—
162C22#1686117811yescharacteristic
163C22C2×#1452323611yescharacteristic
164C22#1686123511yescharacteristic
165C4—22no—
166C4—22no—
167C4—42no—
168C4—41no—
169C4—5082no—
170C4—5082no—
171C2—21no—
172C2—21no—
173C2—82no—
174C2#2255002711yescharacteristic
175C2#2255005911yescharacteristic
176C2#2255006311yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting, prime radical#537×C127C2
socleC22×C127#120
hypercenter#537D127
hypocenterC127#1604
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #22550014C2semidirect
2. #22549796C2semidirect
3. #22549610C2semidirect
4. #22549602C2semidirect
66. C2×C4×C127SD16semidirect
89. C127#1604semidirect
154. C2×C4#14523236semidirect

Maximal subgroups

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

subgroupindex
#225500142
#225497962
#225496102
#225496022
#537×C1272
#1604127