Finite Group Databasebeta

← #30640681

Subgroups — #30640681

Order 16256.

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

normal subgroups only

This group has 216 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#22550316C212yesmaximal, complemented
2#22550134C211yesmaximal, characteristic, complemented
3#22549686C212yesmaximal, complemented
4#22549658C211yesmaximal, characteristic, complemented
5#575×C127C211yesmaximal, characteristic
6C2×Dic508C2211yescharacteristic, complemented
7#16861188—21no—
8#16861155—22no—
9#16861153—22no—
10#16861149C2212yescomplemented
11#16861141C2211yescharacteristic, complemented
12#16861134—21no—
13C2×#41×C127—21no—
14#137×C127—22no—
15AGL(1,ℤ/8)×C127—22no—
16#135×C127—21no—
17#135×C127C2212yes—
18C2×M16×C127C2211yescharacteristic
19C2×Dic254—21no—
20Dic508—22no—
21C4⋊Dic127—21no—
22Dic127⋊C4—41no—
23C22⋊Dic127—41no—
24C127⋊C16—22no—
25#41×C127—22no—
26#41×C127—22no—
27#41×C127—41no—
28#41×C127—41no—
29C2×Q8×C127—21no—
30C2×D4×C127—21no—
31C2×D4×C127—41no—
32C22×C4×C127—41no—
33C22×C4×C127D411yescharacteristic
34Q16×C127—21no—
35Q16×C127D412yes—
36SD16×C127—22no—
37SD16×C127—22no—
38D8×C127—21no—
39D8×C127D412yes—
40M16×C127—22no—
41C2×C8×C127D411yescharacteristic
42C2×C8×C127C2311yescharacteristic
43Dic254—41no—
44Dic254—41no—
45C2×Dic127—41no—
46C2×Dic127—41no—
47C23×C127—41no—
48C23×C127D811yescharacteristic
49Q8×C127—22no—
50Q8×C127—41no—
51D4×C127—22no—
52D4×C127—41no—
53D4×C127—42no—
54D4×C127—42no—
55C2×C4×C127—21no—
56C2×C4×C127—21no—
57C2×C4×C127—42no—
58C2×C4×C127—41no—
59C2×C4×C127—42no—
60C2×C4×C127D811yescharacteristic
61C2×C4×C127C2×D411yescharacteristic
62C8×C127—21no—
63C8×C127C2×D412yes—
64C22×C127—21no—
65C22×C127—41no—
66C22×C127—42no—
67C22×C127—82no—
68C22×C127C2×D811yescharacteristic
69C4×C127—21no—
70C4×C127—42no—
71C4×C127#12011yescharacteristic
72C4×C127AGL(1,ℤ/8)11yescharacteristic
73Dic127—81no—
74Dic127—81no—
75C2×C127—21no—
76C2×C127—41no—
77C2×C127—82no—
78C2×C127#44711yescharacteristic
79#2191—1271nomaximal, Sylow
80C127#219111yescharacteristic, complemented, Sylow
81#575D12711yescharacteristic
82#510—1272no—
83#470—1271no—
84#360—1272no—
85#349—1271no—
86C2×#41—21no—
87#137—22no—
88AGL(1,ℤ/8)—22no—
89#135—21no—
90#135C2×D12712yes—
91C2×Q16—1271no—
92C2×M16C2×D12711yescharacteristic
93C22⋊Q8—2541no—
94Q32—2542no—
95SD32—2542no—
96M32—1272no—
97#106—1271no—
98#103—2541no—
99#41—22no—
100#41—22no—
101#41—41no—
102#41—41no—
103C2×Q8—21no—
104C2×Q8—2541no—
105C2×D4—21no—
106C2×D4—41no—
107C22×C4—41no—
108C22×C4C22⋊D12711yescharacteristic
109Q16—21no—
110Q16—2542no—
111Q16C22⋊D12712yes—
112SD16—22no—
113SD16—22no—
114D8—21no—
115D8C22⋊D12712yes—
116M16—22no—
117C2×C8C22⋊D12711yescharacteristic
118C2×C8C22×D12711yescharacteristic
119C4⋊C4—5081no—
120C4⋊C4—2541no—
121C22⋊C4—5081no—
122C16—2542no—
123C23—41no—
124C23#1452323311yescharacteristic
125Q8—22no—
126Q8—41no—
127Q8—5081no—
128Q8—5081no—
129D4—22no—
130D4—41no—
131D4—42no—
132D4—42no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—42no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—5081no—
139C2×C4—5081no—
140C2×C4#1452323311yescharacteristic
141C2×C4C2×C22⋊D12711yescharacteristic
142C8—21no—
143C8C2×C22⋊D12712yes—
144C22—21no—
145C22—41no—
146C22—42no—
147C22—82no—
148C22C2×#1452323311yescharacteristic
149C4—21no—
150C4—42no—
151C4—10161no—
152C4—10161no—
153C4#1686117811yescharacteristic
154C4#1686123511yescharacteristic
155C2—21no—
156C2—41no—
157C2—82no—
158C2#2255001011yescharacteristic
subgroupG/H
centerC2#22550010
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting, prime radical#575×C127C2
socleC2×C127#447
hypercenter#575D127
hypocenterC127#2191
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #22550316C2semidirect
2. #22550134C2semidirect
3. #22549686C2semidirect
4. #22549658C2semidirect
6. C2×Dic508C22semidirect
10. #16861149C22semidirect
11. #16861141C22semidirect
80. C127#2191semidirect

Maximal subgroups

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

subgroupindex
#225503162
#225501342
#225496862
#225496582
#575×C1272
#2191127