Finite Group Databasebeta

← #30640707

Subgroups — #30640707

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#22550689C211yesmaximal, characteristic
2#22550317C211yesmaximal, characteristic, complemented
3#22550148C211yesmaximal, characteristic, complemented
4#22549692C211yesmaximal, characteristic, complemented
5#22549684C211yesmaximal, characteristic, complemented
6#22549659C211yesmaximal, characteristic, complemented
7C2×Q32×C127C211yesmaximal, characteristic, complemented
8#16861257—21no—
9#16861241—22no—
10#16861240—21no—
11#16861240—21no—
12#16861232—21no—
13#16861231C2211yescharacteristic
14Q16×D127—21no—
15#16861215C2211yescharacteristic
16#16861155—21no—
17#16861152—21no—
18#16861149C2211yescharacteristic
19#16861145C2211yescharacteristic
20#16861128—21no—
21C2×Q16×C127C2211yescharacteristic
22C2×Q16×C127C2211yescharacteristic
23Q32×C127—22no—
24C2×C16×C127C2211yescharacteristic
25#14523249—22no—
26#14523249—41no—
27#14523248—21no—
28#14523248—42no—
29#14523248D411yescharacteristic
30Q8×D127—22no—
31Q8×D127—41no—
32Dic508D411yescharacteristic
33#14523238—22no—
34#14523238—41no—
35#14523237—21no—
36#14523235—22no—
37#14523235—41no—
38D1016D411yescharacteristic
39#14523232—21no—
40#14523232D411yescharacteristic
41#14523231—21no—
42C8×D127—21no—
43C127⋊C16—21no—
44C2×Q8×C127—21no—
45C2×Q8×C127—21no—
46Q16×C127—21no—
47Q16×C127—21no—
48Q16×C127—21no—
49Q16×C127D412yes—
50C2×C8×C127C2311yescharacteristic
51C16×C127—21no—
52Dic254—21no—
53Dic254—42no—
54Dic254D811yescharacteristic
55D508—21no—
56D508—42no—
57D508D811yescharacteristic
58C22⋊D127—21no—
59C22⋊D127—41no—
60C4×D127—21no—
61C4×D127—42no—
62C4×D127—42no—
63C4×D127—41no—
64C127⋊C8—21no—
65C127⋊C8—41no—
66Q8×C127—22no—
67Q8×C127—41no—
68Q8×C127—41no—
69Q8×C127—41no—
70C2×C4×C127—41no—
71C2×C4×C127—41no—
72C2×C4×C127C2×D411yescharacteristic
73C8×C127C2×D411yescharacteristic
74C8×C127C2×D411yescharacteristic
75C22×C127AGL(1,ℤ/8)11yescharacteristic
76C2×D127—21no—
77C2×D127—41no—
78C4×C127—42no—
79C4×C127—81no—
80C4×C127#12011yescharacteristic
81C4×C127C2×D811yescharacteristic
82Dic127—21no—
83Dic127—41no—
84C2×C127—21no—
85C2×C127#44711yescharacteristic
86D127—41no—
87D127—81no—
88#2195—1271nomaximal, Sylow
89C127#219511yescharacteristic, complemented, Sylow
90#578—1271no—
91#510—1271no—
92C2×Q32D12711yescharacteristic, complemented
93#473—1271no—
94#362—1271no—
95#359—1271no—
96#350—1271no—
972−1+4—2541no—
98#137—2541no—
99#137—2541no—
100#137—2542no—
101#135—1271no—
102#135—2541no—
103C2×Q16—2541no—
104C2×Q16C2×D12711yescharacteristic
105C2×Q16C2×D12711yescharacteristic
106#131—1271no—
107Q32—22no—
108Q32—2541no—
109SD32—2541no—
110M32—1271no—
111C2×C16C2×D12711yescharacteristic
112#108—1271no—
113#101—2541no—
114#41—1271no—
115#41—5082no—
116#41—5081no—
117#41—2542no—
118#41—2541no—
119C2×Q8—21no—
120C2×Q8—21no—
121C2×Q8—5081no—
122C2×Q8—2542no—
123Q16—21no—
124Q16—21no—
125Q16—21no—
126Q16—1271no—
127Q16—5081no—
128Q16—2542no—
129Q16C22⋊D12712yes—
130SD16—5081no—
131SD16—2542no—
132SD16—2541no—
133D8—1271no—
134M16—1271no—
135M16—2541no—
136M16—2541no—
137C2×C8—2541no—
138C2×C8C22×D12711yescharacteristic
139C16—21no—
140C16—2541no—
141Q8—22no—
142Q8—41no—
143Q8—41no—
144Q8—41no—
145Q8—1271no—
146Q8—5082no—
147Q8—2541no—
148D4—1271no—
149D4—5082no—
150D4—5081no—
151D4—2541no—
152D4—2541no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—5082no—
156C2×C4—5082no—
157C2×C4—5081no—
158C2×C4—2541no—
159C2×C4D4×D12711yescharacteristic
160C8—5081no—
161C8—2541no—
162C8D4×D12711yescharacteristic
163C8C2×C22⋊D12711yescharacteristic
164C22—5081no—
165C22—2541no—
166C22#1686123311yescharacteristic
167C4—42no—
168C4—81no—
169C4—5081no—
170C4—2541no—
171C4#1686117711yescharacteristic
172C4D8×D12711yescharacteristic
173C2—21no—
174C2—10161no—
175C2—5081no—
176C2#2255000811yescharacteristic
subgroupG/H
centerC2#22550008
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
FittingC2×Q32×C127C2
socleC2×C127#447
prime radical#22550689C2
hypercenterC2×Q32D127
hypocenterC127#2195
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×Q32

Complements

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

NH ≅ G/Ntype
2. #22550317C2semidirect
3. #22550148C2semidirect
4. #22549692C2semidirect
5. #22549684C2semidirect
6. #22549659C2semidirect
7. C2×Q32×C127C2semidirect
89. C127#2195semidirect
92. C2×Q32D127semidirect

Maximal subgroups

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

subgroupindex
#225506892
#225503172
#225501482
#225496922
#225496842
#225496592
C2×Q32×C1272
#2195127