Finite Group Databasebeta

← #30640705

Subgroups — #30640705

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
2C2×#16861155C211yesmaximal, characteristic, complemented
3#22550148C211yesmaximal, characteristic, complemented
4#22549693C211yesmaximal, characteristic, complemented
5#22549683C211yesmaximal, characteristic, complemented
6#22549661C211yesmaximal, characteristic, complemented
7#510×C127C211yesmaximal, characteristic, complemented
8#16861256—21no—
9#16861240—21no—
10#16861239—21no—
11#16861238—22no—
12#16861228—21no—
13C2×Dic508C2211yescharacteristic
14Q16×D127—21no—
15#16861215C2211yescharacteristic
16#16861155—22no—
17C2×C127⋊C16C2211yescharacteristic
18#16861145C2211yescharacteristic
19#16861128—21no—
20#135×C127C2211yescharacteristic
21C2×Q16×C127C2211yescharacteristic
22Q32×C127—21no—
23SD32×C127—21no—
24M32×C127C2211yescharacteristic
25#14523248—41no—
26#14523248D411yescharacteristic
27#14523247—22no—
28#14523247—42no—
29C2×Dic254—21no—
30C2×Dic254—41no—
31Q8×D127—22no—
32Dic508—21no—
33Dic508D412yes—
34#14523238—22no—
35#14523238—41no—
36#14523236—22no—
37#14523235—41no—
38#14523232—21no—
39#14523232D411yescharacteristic
40#14523231—21no—
41C8×D127—21no—
42C127⋊C16—21no—
43#41×C127—21no—
44C2×Q8×C127—21no—
45Q16×C127—21no—
46Q16×C127—21no—
47Q16×C127D411yescharacteristic
48SD16×C127—21no—
49D8×C127D411yescharacteristic
50C2×C8×C127C2311yescharacteristic
51C16×C127—21no—
52Dic254—22no—
53Dic254—42no—
54Dic254—41no—
55Dic254D811yescharacteristic
56D508D811yescharacteristic
57C22⋊D127—21no—
58C22⋊D127—42no—
59C2×Dic127—42no—
60C2×Dic127—41no—
61C4×D127—21no—
62C4×D127—42no—
63C127⋊C8—21no—
64C127⋊C8—41no—
65Q8×C127—21no—
66Q8×C127—41no—
67Q8×C127—41no—
68D4×C127—21no—
69D4×C127—41no—
70C2×C4×C127—41no—
71C2×C4×C127—41no—
72C2×C4×C127C2×D411yescharacteristic
73C8×C127C2×D411yescharacteristic
74C8×C127C2×D411yescharacteristic
75C22×C127—41no—
76C22×C127AGL(1,ℤ/8)11yescharacteristic
77C2×D127—21no—
78C4×C127—41no—
79C4×C127—81no—
80C4×C127#12011yescharacteristic
81C4×C127C2×D811yescharacteristic
82Dic127—21no—
83Dic127—42no—
84C2×C127—21no—
85C2×C127—81no—
86C2×C127#44711yescharacteristic
87D127—41no—
88#2195—1271nomaximal, Sylow
89C127#219511yescharacteristic, complemented, Sylow
90#578—1271no—
91#510D12711yescharacteristic, complemented
92C2×Q32—1271no—
93#473—1271no—
94#362—1271no—
95#359—1271no—
96#350—1271no—
972−1+4—2541no—
98#137—2541no—
99#137—2542no—
100#137—2541no—
101#135—2541no—
102#135C2×D12711yescharacteristic
103C2×Q16—1271no—
104C2×Q16—2541no—
105C2×Q16C2×D12711yescharacteristic
106#131—1271no—
107Q32—21no—
108Q32—2542no—
109SD32—21no—
110M32C2×D12711yescharacteristic
111C2×C16—1271no—
112#108—1271no—
113#101—2541no—
114#41—21no—
115#41—1271no—
116#41—5081no—
117#41—5082no—
118#41—2542no—
119C2×Q8—21no—
120C2×Q8—5081no—
121C2×Q8—2542no—
122C2×Q8—2541no—
123Q16—21no—
124Q16—21no—
125Q16—1272no—
126Q16—5081no—
127Q16—2542no—
128Q16—2541no—
129Q16C22⋊D12711yescharacteristic
130SD16—21no—
131SD16—5081no—
132SD16—2542no—
133D8C22⋊D12711yescharacteristic
134M16—1271no—
135M16—2541no—
136M16—2541no—
137C2×C8—2541no—
138C2×C8C22×D12711yescharacteristic
139C16—21no—
140C16—2541no—
141Q8—21no—
142Q8—41no—
143Q8—41no—
144Q8—1271no—
145Q8—5082no—
146Q8—5081no—
147Q8—2542no—
148D4—21no—
149D4—41no—
150D4—1271no—
151D4—5082no—
152D4—2541no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—5082no—
156C2×C4—5081no—
157C2×C4—5082no—
158C2×C4—2541no—
159C2×C4D4×D12711yescharacteristic
160C8—5081no—
161C8—2541no—
162C8D4×D12711yescharacteristic
163C8C2×C22⋊D12711yescharacteristic
164C22—41no—
165C22—2541no—
166C22#1686123311yescharacteristic
167C4—41no—
168C4—81no—
169C4—5082no—
170C4—2541no—
171C4#1686117711yescharacteristic
172C4D8×D12711yescharacteristic
173C2—21no—
174C2—81no—
175C2—5081no—
176C2#2255000811yescharacteristic
subgroupG/H
centerC2#22550008
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#510×C127C2
socleC2×C127#447
prime radical#22550690C2
hypercenter#510D127
hypocenterC127#2195
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
2. C2×#16861155C2semidirect
3. #22550148C2semidirect
4. #22549693C2semidirect
5. #22549683C2semidirect
6. #22549661C2semidirect
7. #510×C127C2semidirect
89. C127#2195semidirect
91. #510D127semidirect

Maximal subgroups

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

subgroupindex
#225506902
C2×#168611552
#225501482
#225496932
#225496832
#225496612
#510×C1272
#2195127