Finite Group Databasebeta

← #30640708

Subgroups — #30640708

Order 16256.

206 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 53 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#22550316C211yesmaximal, characteristic, complemented
2C2×#16861155C211yesmaximal, characteristic, complemented
3#22550151C211yesmaximal, characteristic, complemented
4#22549692C211yesmaximal, characteristic, complemented
5#22549683C211yesmaximal, characteristic, complemented
6#22549660C211yesmaximal, characteristic, complemented
7#578×C127C211yesmaximal, characteristic
8#16861239—21no—
9C2×Dic508C2211yescharacteristic
10#16861155—22no—
11#16861155—21no—
12#16861153—21no—
13#16861149C2211yescharacteristic
14C2×C127⋊C16C2211yescharacteristic
15#16861145C2211yescharacteristic
16#16861127—21no—
172−1+4×C127—21no—
18#137×C127—22no—
19#137×C127—21no—
20#135×C127—21no—
21#135×C127C2211yescharacteristic
22C2×Q16×C127—21no—
23C2×Q16×C127C2211yescharacteristic
24#131×C127C2211yescharacteristic
25C2×Dic254—21no—
26Dic508—21no—
27Dic508—21no—
28#14523238—41no—
29#14523236—41no—
30#14523232—21no—
31C127⋊C16—21no—
32C127⋊C16—21no—
33#41×C127—22no—
34#41×C127—21no—
35#41×C127—42no—
36#41×C127—41no—
37#41×C127D411yescharacteristic
38C2×Q8×C127—22no—
39C2×Q8×C127—21no—
40C2×Q8×C127—41no—
41Q16×C127—22no—
42Q16×C127—21no—
43Q16×C127D411yescharacteristic
44Q16×C127D412yes—
45SD16×C127—22no—
46SD16×C127—21no—
47D8×C127D411yescharacteristic
48M16×C127—21no—
49M16×C127D411yescharacteristic
50C2×C8×C127—21no—
51C2×C8×C127C2311yescharacteristic
52Dic254—41no—
53Dic254—41no—
54C2×Dic127—41no—
55C127⋊C8—41no—
56Q8×C127—22no—
57Q8×C127—21no—
58Q8×C127—42no—
59Q8×C127—41no—
60Q8×C127D811yescharacteristic
61D4×C127—21no—
62D4×C127—21no—
63D4×C127—42no—
64D4×C127—41no—
65D4×C127D811yescharacteristic
66C2×C4×C127—21no—
67C2×C4×C127—42no—
68C2×C4×C127—42no—
69C2×C4×C127—41no—
70C2×C4×C127—41no—
71C2×C4×C127C2×D411yescharacteristic
72C8×C127—21no—
73C8×C127C2×D411yescharacteristic
74C8×C127C2×D411yescharacteristic
75C22×C127—21no—
76C22×C127—41no—
77C22×C127AGL(1,ℤ/8)11yescharacteristic
78C4×C127—21no—
79C4×C127—42no—
80C4×C127—41no—
81C4×C127#12011yescharacteristic
82C4×C127C2×D811yescharacteristic
83Dic127—81no—
84C2×C127—21no—
85C2×C127—41no—
86C2×C127—81no—
87C2×C127#44711yescharacteristic
88#2195—1271nomaximal, Sylow
89C127#219511yescharacteristic, complemented, Sylow
90#578D12711yescharacteristic
91#510—1271no—
92C2×Q32—1271no—
93#473—1271no—
94#362—1271no—
95#359—1271no—
96#350—1271no—
972−1+4—21no—
98#137—22no—
99#137—21no—
100#137—2541no—
101#135—21no—
102#135C2×D12711yescharacteristic
103C2×Q16—21no—
104C2×Q16—1271no—
105C2×Q16C2×D12711yescharacteristic
106#131C2×D12711yescharacteristic
107Q32—2542no—
108Q32—2541no—
109SD32—2541no—
110M32—1271no—
111C2×C16—1271no—
112#108—1271no—
113#101—2541no—
114#41—22no—
115#41—21no—
116#41—42no—
117#41—41no—
118#41C22⋊D12711yescharacteristic
119C2×Q8—22no—
120C2×Q8—21no—
121C2×Q8—41no—
122C2×Q8—2541no—
123Q16—22no—
124Q16—21no—
125Q16—5081no—
126Q16—2541no—
127Q16—2541no—
128Q16C22⋊D12711yescharacteristic
129Q16C22⋊D12712yes—
130SD16—22no—
131SD16—21no—
132SD16—5081no—
133D8C22⋊D12711yescharacteristic
134M16—21no—
135M16—2541no—
136M16C22⋊D12711yescharacteristic
137C2×C8—21no—
138C2×C8C22×D12711yescharacteristic
139C16—2541no—
140C16—2541no—
141Q8—22no—
142Q8—21no—
143Q8—42no—
144Q8—41no—
145Q8—5081no—
146Q8—5081no—
147Q8#1452323311yescharacteristic
148D4—21no—
149D4—21no—
150D4—42no—
151D4—41no—
152D4#1452323311yescharacteristic
153C2×C4—21no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—5081no—
159C2×C4C2×C22⋊D12711yescharacteristic
160C8—21no—
161C8—5081no—
162C8C2×C22⋊D12711yescharacteristic
163C8C2×C22⋊D12711yescharacteristic
164C22—21no—
165C22—41no—
166C22#1686123511yescharacteristic
167C4—21no—
168C4—42no—
169C4—41no—
170C4—10161no—
171C4#1686117811yescharacteristic
172C4C2×#1452323311yescharacteristic
173C2—21no—
174C2—41no—
175C2—81no—
176C2#2255001011yescharacteristic
subgroupG/H
centerC2#22550010
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting, prime radical#578×C127C2
socleC2×C127#447
hypercenter#578D127
hypocenterC127#2195
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #578

Complements

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

NH ≅ G/Ntype
1. #22550316C2semidirect
2. C2×#16861155C2semidirect
3. #22550151C2semidirect
4. #22549692C2semidirect
5. #22549683C2semidirect
6. #22549660C2semidirect
89. C127#2195semidirect

Maximal subgroups

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

subgroupindex
#225503162
C2×#168611552
#225501512
#225496922
#225496832
#225496602
#578×C1272
#2195127