Finite Group Databasebeta

← #30640680

Subgroups — #30640680

Order 16256.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#22550676C211yesmaximal, characteristic
2#22550317C212yesmaximal, complemented
3#22549685C212yesmaximal, complemented
4#22549658C211yesmaximal, characteristic, complemented
5#470×C127C211yesmaximal, characteristic, complemented
6C2×#14523248—21no—
7#16861242—22no—
8#16861237—22no—
9#16861231—21no—
10#16861231C2212yes—
11#16861155—22no—
12#16861152—22no—
13#16861149C2212yescomplemented
14C2×Q16×C127C2211yescharacteristic, complemented
15C2×M16×C127C2211yescharacteristic
16C22⋊Q8×C127—21no—
17#106×C127C2211yescharacteristic, complemented
18#103×C127—21no—
19#14523248—22no—
20#14523248—22no—
21#14523248—41no—
22#14523248—41no—
23C2×Dic254—21no—
24C2×D508—21no—
25C2×C22⋊D127—41no—
26C2×C4×D127—41no—
27Dic508—21no—
28Dic508D412yes—
29#14523237—22no—
30#14523237—22no—
31D1016—21no—
32D1016D412yes—
33C127⋊C16—22no—
34C2×Q8×C127—21no—
35C22×C4×C127D411yescharacteristic
36Q16×C127—22no—
37M16×C127—22no—
38C2×C8×C127D411yescharacteristic
39C2×C8×C127C2311yescharacteristic
40C4⋊C4×C127—21no—
41C4⋊C4×C127—41no—
42C22⋊C4×C127—41no—
43C22×D127—41no—
44Dic254—22no—
45Dic254—41no—
46D508—22no—
47D508—41no—
48C22⋊D127—42no—
49C22⋊D127—42no—
50C2×Dic127—41no—
51C4×D127—42no—
52C4×D127—42no—
53C23×C127D811yescharacteristic
54Q8×C127—41no—
55Q8×C127—41no—
56C2×C4×C127—21no—
57C2×C4×C127—21no—
58C2×C4×C127—41no—
59C2×C4×C127—41no—
60C2×C4×C127D811yescharacteristic
61C2×C4×C127C2×D411yescharacteristic
62C8×C127—21no—
63C8×C127C2×D412yes—
64C22×C127—21no—
65C22×C127—41no—
66C22×C127C2×D811yescharacteristic
67C2×D127—42no—
68C2×D127—82no—
69C4×C127—21no—
70C4×C127—81no—
71C4×C127—81no—
72C4×C127#12011yescharacteristic
73C4×C127AGL(1,ℤ/8)11yescharacteristic
74Dic127—42no—
75C2×C127—21no—
76C2×C127—41no—
77C2×C127#44711yescharacteristic
78D127—82no—
79#2191—1271nomaximal, Sylow
80C127#219111yescharacteristic, complemented, Sylow
81#575—1271no—
82#510—1272no—
83#470D12711yescharacteristic, complemented
84#360—1272no—
85#349—1271no—
86C2×#41—2541no—
87#137—2542no—
88AGL(1,ℤ/8)—2542no—
89#135—1272no—
90#135—2541no—
91C2×Q16C2×D12711yescharacteristic, complemented
92C2×M16C2×D12711yescharacteristic
93C22⋊Q8—21no—
94Q32—2542no—
95SD32—2542no—
96M32—1272no—
97#106C2×D12711yescharacteristic, complemented
98#103—21no—
99#41—5081no—
100#41—5081no—
101#41—2542no—
102#41—2542no—
103C2×Q8—21no—
104C2×Q8—2541no—
105C2×D4—5081no—
106C2×D4—2541no—
107C22×C4—5081no—
108C22×C4C22⋊D12711yescharacteristic
109Q16—22no—
110Q16—1272no—
111Q16—2541no—
112SD16—2542no—
113SD16—2542no—
114D8—1272no—
115D8—2541no—
116M16—22no—
117C2×C8C22⋊D12711yescharacteristic
118C2×C8C22×D12711yescharacteristic
119C4⋊C4—21no—
120C4⋊C4—41no—
121C22⋊C4—41no—
122C16—2542no—
123C23—5081no—
124C23#1452323311yescharacteristic
125Q8—41no—
126Q8—41no—
127Q8—5081no—
128Q8—2542no—
129D4—5081no—
130D4—5082no—
131D4—5082no—
132D4—2542no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—5082no—
138C2×C4—5081no—
139C2×C4—5082no—
140C2×C4#1452323311yescharacteristic
141C2×C4C2×C22⋊D12711yescharacteristic
142C8—21no—
143C8D4×D12712yes—
144C22—21no—
145C22—41no—
146C22—10162no—
147C22—5082no—
148C22C2×#1452323311yescharacteristic
149C4—21no—
150C4—81no—
151C4—81no—
152C4—5082no—
153C4#1686117711yescharacteristic
154C4#1686123411yescharacteristic
155C2—21no—
156C2—41no—
157C2—10162no—
158C2#2255000911yescharacteristic
subgroupG/H
centerC2#22550009
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#470×C127C2
socleC2×C127#447
prime radical#22550676C2
hypercenter#470D127
hypocenterC127#2191
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #470

Complements

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

NH ≅ G/Ntype
2. #22550317C2semidirect
3. #22549685C2semidirect
4. #22549658C2semidirect
5. #470×C127C2semidirect
13. #16861149C22semidirect
14. C2×Q16×C127C22semidirect
17. #106×C127C22semidirect
80. C127#2191semidirect
83. #470D127semidirect
91. C2×Q16C2×D127semidirect
97. #106C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506762
#225503172
#225496852
#225496582
#470×C1272
#2191127