Finite Group Databasebeta

← #30640704

Subgroups — #30640704

Order 16256.

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

normal subgroups only

This group has 216 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#22550677C211yesmaximal, characteristic
2#22550319C211yesmaximal, characteristic, complemented
3C2×#16861153C211yesmaximal, characteristic, complemented
4#22550152C211yesmaximal, characteristic, complemented
5#22549682C211yesmaximal, characteristic, complemented
6#22549661C211yesmaximal, characteristic, complemented
7#361×C127C211yesmaximal, characteristic, complemented
8#16861254—21no—
9#16861242—21no—
10#16861235—21no—
11#16861233—22no—
12#16861231C2211yescharacteristic
13#16861228—21no—
14C2×Dic508C2211yescharacteristic, complemented
15D8×D127—21no—
16#16861215C2211yescharacteristic
17Dic1016—21no—
18#16861154—21no—
19#16861153—22no—
20C2×C127⋊C16C2211yescharacteristic, complemented
21#16861127—21no—
22C2×D8×C127C2211yescharacteristic
23M32×C127C2211yescharacteristic, complemented
24#108×C127C2211yescharacteristic, complemented
25#14523248—21no—
26#14523248D411yescharacteristic
27#14523247—22no—
28#14523247—41no—
29C2×Dic254—21no—
30C2×C22⋊D127—42no—
31D4×D127—22no—
32D4×D127—41no—
33Dic508—21no—
34Dic508—21no—
35Dic508—41no—
36Dic508D411yescharacteristic
37#14523237—21no—
38#14523237—41no—
39#14523236—22no—
40D1016D411yescharacteristic
41#14523233—22no—
42#14523232D411yescharacteristic
43#14523231—21no—
44C8×D127—21no—
45C127⋊C16—21no—
46C2×D4×C127—21no—
47D8×C127—21no—
48D8×C127D412yes—
49M16×C127—21no—
50C2×C8×C127C2311yescharacteristic
51C16×C127—21no—
52C22×D127—42no—
53Dic254—21no—
54Dic254—41no—
55Dic254—41no—
56Dic254D811yescharacteristic
57D508—21no—
58D508D811yescharacteristic
59C22⋊D127—21no—
60C22⋊D127—42no—
61C22⋊D127—42no—
62C22⋊D127—41no—
63C2×Dic127—42no—
64C2×Dic127—41no—
65C4×D127—21no—
66C4×D127—41no—
67C127⋊C8—21no—
68C23×C127—41no—
69D4×C127—22no—
70D4×C127—41no—
71C2×C4×C127C2×D411yescharacteristic
72C8×C127—41no—
73C8×C127C2×D411yescharacteristic
74C8×C127C2×D411yescharacteristic
75C22×C127—42no—
76C22×C127—161no—
77C22×C127AGL(1,ℤ/8)11yescharacteristic
78C2×D127—21no—
79C2×D127—41no—
80C2×D127—162no—
81C4×C127#12011yescharacteristic
82C4×C127C2×D811yescharacteristic
83Dic127—21no—
84Dic127—41no—
85Dic127—81no—
86C2×C127—21no—
87C2×C127—82no—
88C2×C127#44711yescharacteristic
89D127—41no—
90D127—81no—
91#2194—1271nomaximal, Sylow
92C127#219411yescharacteristic, complemented, Sylow
93#576—1271no—
94#510—1271no—
95C2×SD32—1271no—
96#473—1271no—
97#361D12711yescharacteristic, complemented
98#359—1271no—
99#350—1271no—
100H(5,𝔽2)—2541no—
101#137—2541no—
102AGL(1,ℤ/8)—2541no—
103AGL(1,ℤ/8)—2542no—
104#135—1271no—
105#135—2541no—
106C2×Q16—1271no—
107C2×D8—2541no—
108C2×D8C2×D12711yescharacteristic
109#131—1271no—
110Q32—2541no—
111SD32—2541no—
112SD32—2542no—
113M32C2×D12711yescharacteristic, complemented
114C2×C16—1271no—
115#108C2×D12711yescharacteristic, complemented
116#101—2541no—
117#41—1271no—
118#41—5081no—
119#41—2542no—
120#41—2541no—
121C2×Q8—2541no—
122C2×D4—21no—
123C2×D4—5081no—
124C2×D4—5082no—
125C2×D4—2542no—
126Q16—1271no—
127Q16—5081no—
128Q16—2541no—
129Q16—2541no—
130SD16—5081no—
131SD16—2542no—
132SD16—2541no—
133D8—21no—
134D8—1271no—
135D8—2542no—
136D8D50812yes—
137M16—21no—
138M16—1271no—
139M16—2541no—
140C2×C8—2541no—
141C2×C8C22×D12711yescharacteristic
142C16—21no—
143C16—2541no—
144C23—41no—
145C23—5082no—
146Q8—1271no—
147Q8—5081no—
148Q8—5081no—
149Q8—2541no—
150D4—22no—
151D4—41no—
152D4—1271no—
153D4—5082no—
154D4—5081no—
155D4—5082no—
156D4—2541no—
157D4—2541no—
158C2×C4—5081no—
159C2×C4—5082no—
160C2×C4—5081no—
161C2×C4—2541no—
162C2×C4D4×D12711yescharacteristic
163C8—41no—
164C8—2541no—
165C8D4×D12711yescharacteristic
166C8C2×D50811yescharacteristic
167C22—42no—
168C22—161no—
169C22—20322no—
170C22—5081no—
171C22—2541no—
172C22#1686123611yescharacteristic
173C4—10161no—
174C4—5081no—
175C4—2541no—
176C4#1686117911yescharacteristic
177C4D8×D12711yescharacteristic
178C2—21no—
179C2—82no—
180C2—10161no—
181C2—5081no—
182C2#2255001111yescharacteristic
subgroupG/H
centerC2#22550011
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#361×C127C2
socleC2×C127#447
prime radical#22550677C2
hypercenter#361D127
hypocenterC127#2194
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #361

Complements

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

NH ≅ G/Ntype
2. #22550319C2semidirect
3. C2×#16861153C2semidirect
4. #22550152C2semidirect
5. #22549682C2semidirect
6. #22549661C2semidirect
7. #361×C127C2semidirect
14. C2×Dic508C22semidirect
20. C2×C127⋊C16C22semidirect
23. M32×C127C22semidirect
24. #108×C127C22semidirect
92. C127#2194semidirect
97. #361D127semidirect
113. M32C2×D127semidirect
115. #108C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506772
#225503192
C2×#168611532
#225501522
#225496822
#225496612
#361×C1272
#2194127