Finite Group Databasebeta

← #30640709

Subgroups — #30640709

Order 16256.

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

normal subgroups only

This group has 206 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#22550690C211yesmaximal, characteristic
2#22550316C211yesmaximal, characteristic, complemented
3C2×Dic1016C211yesmaximal, characteristic, complemented
4#22550152C211yesmaximal, characteristic, complemented
5#22549691C211yesmaximal, characteristic, complemented
6#22549659C211yesmaximal, characteristic, complemented
7#359×C127C211yesmaximal, characteristic, complemented
8#16861256—21no—
9#16861242—21no—
10#16861239—21no—
11#16861238—22no—
12#16861228—21no—
13C2×Dic508C2211yescharacteristic
14C2×Dic508C2211yescharacteristic
15Q16×D127—21no—
16#16861215C2211yescharacteristic
17Dic1016—22no—
18#16861155—21no—
19#16861153—21no—
20#16861149C2211yescharacteristic
21#16861127—21no—
22#135×C127C2211yescharacteristic
23C2×C16×C127C2211yescharacteristic
24#108×C127C2211yescharacteristic
25#14523248—41no—
26#14523248D411yescharacteristic
27#14523247—22no—
28#14523247—42no—
29C2×Dic254—21no—
30C2×Dic254—21no—
31C2×Dic254—41no—
32Q8×D127—22no—
33Dic508—21no—
34Dic508—21no—
35Dic508—21no—
36Dic508—41no—
37Dic508D412yes—
38#14523238—22no—
39#14523237—41no—
40#14523236—22no—
41#14523232D411yescharacteristic
42#14523231—21no—
43C8×D127—21no—
44C127⋊C16—21no—
45#41×C127—21no—
46Q16×C127D411yescharacteristic
47SD16×C127—21no—
48D8×C127D411yescharacteristic
49M16×C127—21no—
50C2×C8×C127C2311yescharacteristic
51C16×C127—21no—
52Dic254—22no—
53Dic254—42no—
54Dic254—41no—
55Dic254—41no—
56Dic254—41no—
57Dic254D811yescharacteristic
58D508D811yescharacteristic
59C22⋊D127—21no—
60C22⋊D127—42no—
61C2×Dic127—42no—
62C2×Dic127—41no—
63C2×Dic127—41no—
64C4×D127—21no—
65C4×D127—42no—
66C127⋊C8—21no—
67Q8×C127—21no—
68D4×C127—21no—
69D4×C127—41no—
70C2×C4×C127—41no—
71C2×C4×C127C2×D411yescharacteristic
72C8×C127—41no—
73C8×C127C2×D411yescharacteristic
74C8×C127C2×D411yescharacteristic
75C22×C127—41no—
76C22×C127AGL(1,ℤ/8)11yescharacteristic
77C2×D127—21no—
78C4×C127—41no—
79C4×C127#12011yescharacteristic
80C4×C127C2×D811yescharacteristic
81Dic127—21no—
82Dic127—42no—
83Dic127—81no—
84C2×C127—21no—
85C2×C127—81no—
86C2×C127#44711yescharacteristic
87D127—41no—
88#2195—1271nomaximal, Sylow
89C127#219511yescharacteristic, complemented, Sylow
90#578—1271no—
91#510—1271no—
92C2×Q32—1271no—
93#473—1271no—
94#362—1271no—
95#359D12711yescharacteristic, complemented
96#350—1271no—
972−1+4—2541no—
98#137—2541no—
99#137—2541no—
100#137—2542no—
101#135—2541no—
102#135C2×D12711yescharacteristic
103C2×Q16—1271no—
104C2×Q16—1271no—
105C2×Q16—2541no—
106#131—1271no—
107Q32—2542no—
108Q32—2541no—
109SD32—2541no—
110M32—1271no—
111C2×C16C2×D12711yescharacteristic
112#108C2×D12711yescharacteristic
113#101—2541no—
114#41—21no—
115#41—1271no—
116#41—5081no—
117#41—5082no—
118#41—2542no—
119C2×Q8—5081no—
120C2×Q8—2542no—
121C2×Q8—2541no—
122C2×Q8—2541no—
123Q16—1272no—
124Q16—5081no—
125Q16—2542no—
126Q16—2541no—
127Q16—2541no—
128Q16—2541no—
129Q16D50811yescharacteristic
130SD16—21no—
131SD16—5081no—
132SD16—2542no—
133D8D50811yescharacteristic
134M16—21no—
135M16—1271no—
136M16—2541no—
137C2×C8—2541no—
138C2×C8C22×D12711yescharacteristic
139C16—21no—
140C16—2541no—
141Q8—21no—
142Q8—1271no—
143Q8—5082no—
144Q8—5081no—
145Q8—5081no—
146Q8—5081no—
147Q8—2542no—
148D4—21no—
149D4—41no—
150D4—1271no—
151D4—5082no—
152D4—2541no—
153C2×C4—41no—
154C2×C4—5082no—
155C2×C4—5081no—
156C2×C4—5081no—
157C2×C4—5082no—
158C2×C4—2541no—
159C2×C4D4×D12711yescharacteristic
160C8—41no—
161C8—2541no—
162C8D4×D12711yescharacteristic
163C8C2×D50811yescharacteristic
164C22—41no—
165C22—2541no—
166C22#1686123611yescharacteristic
167C4—41no—
168C4—10161no—
169C4—5082no—
170C4—2541no—
171C4#1686117911yescharacteristic
172C4D8×D12711yescharacteristic
173C2—21no—
174C2—81no—
175C2—5081no—
176C2#2255001111yescharacteristic
subgroupG/H
centerC2#22550011
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#359×C127C2
socleC2×C127#447
prime radical#22550690C2
hypercenter#359D127
hypocenterC127#2195
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #359

Complements

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

NH ≅ G/Ntype
2. #22550316C2semidirect
3. C2×Dic1016C2semidirect
4. #22550152C2semidirect
5. #22549691C2semidirect
6. #22549659C2semidirect
7. #359×C127C2semidirect
89. C127#2195semidirect
95. #359D127semidirect

Maximal subgroups

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

subgroupindex
#225506902
#225503162
C2×Dic10162
#225501522
#225496912
#225496592
#359×C1272
#2195127