Finite Group Databasebeta

← #30638945

Subgroups — #30638945

Order 16256.

272 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 93 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#22549919C212yesmaximal
2C2×#16861128C211yesmaximal, characteristic
3#22549783C211yesmaximal, characteristic
4#22549635C212yesmaximal
5#519×C127C211yesmaximal, characteristic
6C2×#14523232C2212yes—
7C2×C4⋊Dic127C2211yescharacteristic
8C2×C4×Dic127C2211yescharacteristic
9#16861142—22no—
10#16861140—22no—
11#16861128C414yescomplemented
12C127⋊#95—22no—
13C22×Q8×C127C2211yescharacteristic
14C4×Q8×C127—22no—
15C4×Q8×C127—22no—
16#117×C127—22no—
17#117×C127C2212yes—
18C22×Dic127—21no—
19C22×Dic127—21no—
20#14523232—22no—
21#14523232C2×C414yes—
22C2×C127⋊C8—22no—
23C4⋊Dic127—22no—
24C4×Dic127—22no—
25C2×Q8×C127—22no—
26C2×Q8×C127—22no—
27C2×Q8×C127C2×C412yes—
28C2×Q8×C127Q812yes—
29C22×C4×C127D412yes—
30C22×C4×C127C2311yescharacteristic
31C4⋊C4×C127—22no—
32C4⋊C4×C127—22no—
33C4⋊C4×C127D412yes—
34C4⋊C4×C127D412yes—
35C22⋊C4×C127—22no—
36C42×C127—24no—
37C42×C127—22no—
38C2×Dic127—22no—
39C2×Dic127—41no—
40C2×Dic127—41no—
41C2×Dic127—42no—
42C2×Dic127—41no—
43C127⋊C8—24no—
44C23×C127C2×D411yescharacteristic
45Q8×C127—24no—
46Q8×C127—24no—
47C2×C4×C127—22no—
48C2×C4×C127—22no—
49C2×C4×C127—22no—
50C2×C4×C127—22no—
51C2×C4×C127C4⋊C414yes—
52C2×C4×C127C22×C411yescharacteristic
53C2×C4×C127C2×D412yes—
54C2×C4×C127C2×Q811yescharacteristic
55C2×C4×C127#4112yes—
56C22×C127—21no—
57C22×C127—21no—
58C22×C127C2×C4⋊C411yescharacteristic
59C22×C127#12011yescharacteristic
60C22×C127#12311yescharacteristic
61C4×C127—24no—
62C4×C127—42no—
63C4×C127—42no—
64C4×C127C4×D412yes—
65C4×C127C22⋊Q812yes—
66Dic127—42no—
67Dic127—81no—
68C2×C127—21no—
69C2×C127—21no—
70C2×C127#38511yescharacteristic
71C2×C127#45411yescharacteristic
72C2×C127#45611yescharacteristic
73#1857—1271nomaximal, Sylow
74C127#185711yescharacteristic, complemented, Sylow
75#519D12711yescharacteristic
76#428—1272no—
77C2×#101—1271no—
78#391—1271no—
79#339—1272no—
80C22×Q8C2×D12711yescharacteristic
81C2×M16—1272no—
82C4×Q8—22no—
83C4×Q8—22no—
84#117—22no—
85#117C2×D12712yes—
86C2×C4⋊C4—1271no—
87C2×C42—1271no—
88#107—2542no—
89#106—2542no—
90#101—1274no—
91#95—2542no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8C4×D12712yes—
95C2×Q8Dic25412yes—
96C22×C4—2541no—
97C22×C4—2541no—
98C22×C4C22⋊D12712yes—
99C22×C4C22×D12711yescharacteristic
100M16—1274no—
101M16—2542no—
102C2×C8—2542no—
103C4⋊C4—22no—
104C4⋊C4—22no—
105C4⋊C4—2542no—
106C4⋊C4C22⋊D12712yes—
107C4⋊C4C22⋊D12712yes—
108C22⋊C4—22no—
109C42—24no—
110C42—22no—
111C42—2542no—
112C23C2×C22⋊D12711yescharacteristic
113Q8—24no—
114Q8—24no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—22no—
119C2×C4—5082no—
120C2×C4—5081no—
121C2×C4—5081no—
122C2×C4—5081no—
123C2×C4—2542no—
124C2×C4Dic127⋊C414yes—
125C2×C4C2×C4×D12711yescharacteristic
126C2×C4C2×C22⋊D12712yes—
127C2×C4C2×Dic25411yescharacteristic
128C2×C4#1452324812yes—
129C8—2544no—
130C22—21no—
131C22—21no—
132C22C2×Dic127⋊C411yescharacteristic
133C22#1686117811yescharacteristic
134C22#1686119211yescharacteristic
135C4—24no—
136C4—42no—
137C4—42no—
138C4—10161no—
139C4—5082no—
140C4C4×C22⋊D12712yes—
141C4#1686118812yes—
142C2—21no—
143C2—21no—
144C2#2254974911yescharacteristic
145C2#2255004911yescharacteristic
146C2#2255005911yescharacteristic
subgroupG/H
centerC22#16861178
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#519×C127C2
socleC22×C127#120
prime radicalC23×C127C2×D4
cosocleC22×C4×C127C23
hypercenter#519D127
hypocenterC127#1857
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #519

Complements

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

NH ≅ G/Ntype
11. #16861128C4semidirect
74. C127#1857semidirect

Maximal subgroups

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

subgroupindex
#225499192
C2×#168611282
#225497832
#225496352
#519×C1272
#1857127