Finite Group Databasebeta

← #30640033

Subgroups — #30640033

Order 16256.

264 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 67 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#22549904C211yesmaximal, characteristic, complemented
2#22549872C212yesmaximal, complemented
3#22549850C212yesmaximal, complemented
4#22549633C211yesmaximal, characteristic, complemented
5#575×C127C211yesmaximal, characteristic
6C22×C127⋊C8C2211yescharacteristic
7#16861168C2212yes—
8#16861168C2211yescharacteristic
9Dic127⋊C8—22no—
10#16861133—22no—
11#16861130—22no—
12#16861124—24no—
13C2×#41×C127C2212yes—
14#137×C127—22no—
15AGL(1,ℤ/8)×C127—22no—
16#135×C127—22no—
17C2×M16×C127C2211yescharacteristic
18C2×C127⋊C8—22no—
19C2×C127⋊C8—21no—
20C4⋊Dic127—22no—
21C4⋊Dic127—21no—
22C22⋊Dic127—22no—
23C22⋊Dic127D412yes—
24C4×Dic127—21no—
25C4×Dic127—22no—
26#41×C127—22no—
27#41×C127—22no—
28#41×C127—22no—
29#41×C127—22no—
30C2×Q8×C127D412yes—
31C2×D4×C127—22no—
32C2×D4×C127D412yes—
33C22×C4×C127—22no—
34C22×C4×C127C2311yescharacteristic
35Q16×C127—22no—
36SD16×C127—24no—
37D8×C127—22no—
38M16×C127—22no—
39C2×C8×C127D412yes—
40C2×Dic127—22no—
41C2×Dic127—42no—
42C127⋊C8—41no—
43C23×C127—22no—
44C23×C127C2×D411yescharacteristic
45Q8×C127—22no—
46Q8×C127—22no—
47D4×C127—22no—
48D4×C127—22no—
49D4×C127—42no—
50D4×C127—42no—
51C2×C4×C127—22no—
52C2×C4×C127—42no—
53C2×C4×C127—42no—
54C2×C4×C127C2×D412yes—
55C2×C4×C127C2×D411yescharacteristic
56C2×C4×C127#4111yescharacteristic
57C2×C4×C127#4112yes—
58C8×C127—22no—
59C22×C127—41no—
60C22×C127—42no—
61C22×C127—82no—
62C22×C127#12111yescharacteristic
63C22×C127#12112yes—
64C4×C127—42no—
65C4×C127#12011yescharacteristic
66C4×C127#12312yes—
67C4×C127#12411yescharacteristic
68Dic127—42no—
69Dic127—82no—
70C2×C127—21no—
71C2×C127—22no—
72C2×C127—82no—
73C2×C127#39411yescharacteristic
74#2053—1271nomaximal, Sylow
75C127#205311yescharacteristic, complemented, Sylow
76#575D12711yescharacteristic
77#424—1271no—
78#416—1272no—
79#410—1272no—
80H(3,ℤ/4)—1271no—
81C2×#41C2×D12712yes—
82#137—22no—
83AGL(1,ℤ/8)—22no—
84#135—22no—
85C2×M16C2×D12711yescharacteristic
86C22×C8—1271no—
87#117—1271no—
88#117—1272no—
89C4⋊C8—2542no—
90#103—2542no—
91#102—2542no—
92#99—2544no—
93#41—22no—
94#41—22no—
95#41—22no—
96#41—22no—
97C2×Q8C22⋊D12712yes—
98C2×D4—22no—
99C2×D4C22⋊D12712yes—
100C22×C4—22no—
101C22×C4C22×D12711yescharacteristic
102Q16—22no—
103SD16—24no—
104D8—22no—
105M16—22no—
106C2×C8—2542no—
107C2×C8—2541no—
108C2×C8C22⋊D12712yes—
109C4⋊C4—2542no—
110C4⋊C4—2541no—
111C22⋊C4—1272no—
112C22⋊C4—2542no—
113C42—2542no—
114C42—2541no—
115C23—22no—
116C23D4×D12711yescharacteristic
117Q8—22no—
118Q8—22no—
119D4—22no—
120D4—22no—
121D4—42no—
122D4—42no—
123C2×C4—22no—
124C2×C4—42no—
125C2×C4—42no—
126C2×C4—5082no—
127C2×C4—2542no—
128C2×C4C2×C22⋊D12712yes—
129C2×C4C2×C22⋊D12711yescharacteristic
130C2×C4#1452324711yescharacteristic
131C2×C4#1452324712yes—
132C8—22no—
133C8—5081no—
134C22—41no—
135C22—42no—
136C22—82no—
137C22#1686118011yescharacteristic
138C22#1686118112yes—
139C4—42no—
140C4—10162no—
141C4—5082no—
142C4#1686117811yescharacteristic
143C4#1686119012yes—
144C4#1686119611yescharacteristic
145C2—22no—
146C2—21no—
147C2—82no—
148C2#2254979011yescharacteristic
subgroupG/H
centerC4#16861196
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting, prime radical#575×C127C2
socleC2×C127#394
hypercenter#575D127
hypocenterC127#2053
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #22549904C2semidirect
2. #22549872C2semidirect
3. #22549850C2semidirect
4. #22549633C2semidirect
75. C127#2053semidirect

Maximal subgroups

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

subgroupindex
#225499042
#225498722
#225498502
#225496332
#575×C1272
#2053127