Finite Group Databasebeta

← C127×#1721 #30634673

Subgroups — C127×#1721 #30634673

Order 16256.

242 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 80 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#493×C127C212yesmaximal, complemented
2#493×C127C211yesmaximal, characteristic, complemented
3#492×C127C211yesmaximal, characteristic, complemented
4#486×C127C212yesmaximal, complemented
5#322×C127C211yesmaximal, characteristic, complemented
6C2×SD16×C127—22no—
7C2×D8×C127—24no—
8C2×D8×C127—22no—
9C4⋊Q8×C127C2211yescharacteristic
10#127×C127C2212yes—
11#127×C127C2211yescharacteristic
12#102×C127—24no—
13C4×C8×C127C2212yes—
14C4×C8×C127C2211yescharacteristic
15C2×Q8×C127—21no—
16C2×D4×C127—22no—
17C2×D4×C127—22no—
18C2×D4×C127—22no—
19C2×D4×C127—22no—
20C2×D4×C127—21no—
21SD16×C127—44no—
22D8×C127—48no—
23D8×C127—44no—
24C2×C8×C127D414yes—
25C2×C8×C127D412yes—
26C4⋊C4×C127—22no—
27C42×C127C2311yescharacteristic
28C23×C127—42no—
29C23×C127—41no—
30Q8×C127—42no—
31D4×C127—44no—
32D4×C127—42no—
33D4×C127—44no—
34D4×C127—44no—
35D4×C127—44no—
36C2×C4×C127—41no—
37C2×C4×C127C2×D411yescharacteristic
38C2×C4×C127C2×D412yes—
39C8×C127—24no—
40C8×C127D814yescomplemented
41C22×C127—82no—
42C22×C127—81no—
43C22×C127—82no—
44C22×C127—82no—
45C22×C127—82no—
46C22×C127#12711yescharacteristic
47C4×C127—81no—
48C4×C127C2×D812yes—
49C4×C127AGL(1,ℤ/8)14yes—
50C2×C127—162no—
51C2×C127—161no—
52C2×C127#49311yescharacteristic
53C2×C127#49612yes—
54#1721C12711yesmaximal, characteristic, direct product factor, Sylow
55C127#172111yescharacteristic, direct product factor, Sylow
56#493C2×C12712yescomplemented
57#493C2×C12711yescharacteristic, complemented
58#492C2×C12711yescharacteristic, complemented
59#486C2×C12712yescomplemented
60#322C2×C12711yescharacteristic, complemented
61C2×SD16—22no—
62C2×D8—24no—
63C2×D8—22no—
64C4⋊Q8C22×C12711yescharacteristic
65#127C22×C12712yes—
66#127C22×C12711yescharacteristic
67#102—24no—
68C4×C8C22×C12712yes—
69C4×C8C22×C12711yescharacteristic
70C2×Q8—21no—
71C2×D4—22no—
72C2×D4—22no—
73C2×D4—22no—
74C2×D4—22no—
75C2×D4—21no—
76SD16—44no—
77D8—48no—
78D8—44no—
79C2×C8D4×C12714yes—
80C2×C8D4×C12712yes—
81C4⋊C4—22no—
82C42C23×C12711yescharacteristic
83C23—42no—
84C23—41no—
85Q8—42no—
86D4—44no—
87D4—42no—
88D4—44no—
89D4—44no—
90D4—44no—
91C2×C4—41no—
92C2×C4C2×D4×C12711yescharacteristic
93C2×C4C2×D4×C12712yes—
94C8—24no—
95C8D8×C12714yescomplemented
96C22—82no—
97C22—82no—
98C22—82no—
99C22—82no—
100C22—81no—
101C22#127×C12711yescharacteristic
102C4—81no—
103C4C2×D8×C12712yes—
104C4AGL(1,ℤ/8)×C12714yes—
105C2—161no—
106C2—162no—
107C2#493×C12711yescharacteristic
108C2#496×C12712yes—
subgroupG/H
center, socleC22×C127#127
commutator, Frattini, cosocleC42C23×C127
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. #493×C127C2semidirect
2. #493×C127C2semidirect
3. #492×C127C2semidirect
4. #486×C127C2semidirect
5. #322×C127C2semidirect
40. C8×C127D8semidirect
54. #1721C127direct (N × H)
55. C127#1721direct (N × H)
56. #493C2×C127semidirect
57. #493C2×C127semidirect
58. #492C2×C127semidirect
59. #486C2×C127semidirect
60. #322C2×C127semidirect
95. C8D8×C127semidirect

Maximal subgroups

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

subgroupindex
#493×C1272
#492×C1272
#486×C1272
#322×C1272
#1721127