Finite Group Databasebeta

← C127×#2818 #30635770

Subgroups — C127×#2818 #30635770

Order 16256.

630 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 184 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
1C2×#127×C127C211yesmaximal, characteristic, complemented
2C2×#126×C127C211yesmaximal, characteristic, complemented
3#394×C127C216yesmaximal, complemented
4#392×C127C216yesmaximal, complemented
5#383×C127C211yesmaximal, characteristic, complemented
6C22×D4×C127C2216yes—
7C22×D4×C127C2214yes—
8#127×C127—24no—
9#126×C127—24no—
10C2×C4⋊C4×C127C2213yes—
11C2×C22⋊C4×C127C2213yes—
12C2×C22⋊C4×C127C22112yes—
13C2×C42×C127C2211yescharacteristic
14#95×C127C2216yes—
15C24×C127C2311yescharacteristic
16C24×C127C2314yes—
17C2×D4×C127—212no—
18C2×D4×C127—412no—
19C2×D4×C127—412no—
20C22×C4×C127C2314yes—
21C22×C4×C127C2313yes—
22C22×C4×C127C2313yes—
23C4⋊C4×C127—26no—
24C22⋊C4×C127—26no—
25C22⋊C4×C127—412no—
26C42×C127D414yescomplemented
27C23×C127—21no—
28C23×C127—24no—
29C23×C127—212no—
30C23×C127—212no—
31C23×C127—26no—
32C23×C127C2411yescharacteristic
33D4×C127—424no—
34D4×C127—416no—
35C2×C4×C127—26no—
36C2×C4×C127—23no—
37C2×C4×C127—26no—
38C2×C4×C127—212no—
39C2×C4×C127C2×D416yes—
40C22×C127—412no—
41C22×C127—43no—
42C22×C127—44no—
43C22×C127—44no—
44C22×C127—412no—
45C22×C127C22×D411yescharacteristic
46C22×C127H(5,𝔽2)16yes—
47C4×C127—26no—
48C4×C127—43no—
49C4×C127—44no—
50C2×C127—84no—
51C2×C127—81no—
52C2×C127#53513yes—
53C2×C127#56014yes—
54#2818C12711yesmaximal, characteristic, direct product factor, Sylow
55C127#281811yescharacteristic, direct product factor, Sylow
56C2×#127C2×C12711yescharacteristic, complemented
57C2×#126C2×C12711yescharacteristic, complemented
58#394C2×C12716yescomplemented
59#392C2×C12716yescomplemented
60#383C2×C12711yescharacteristic, complemented
61C22×D4C22×C12716yes—
62C22×D4C22×C12714yes—
63#127—24no—
64#126—24no—
65C2×C4⋊C4C22×C12713yes—
66C2×C22⋊C4C22×C12713yes—
67C2×C22⋊C4C22×C127112yes—
68C2×C42C22×C12711yescharacteristic
69#95C22×C12716yes—
70C24C23×C12711yescharacteristic
71C24C23×C12714yes—
72C2×D4—212no—
73C2×D4—412no—
74C2×D4—412no—
75C22×C4C23×C12714yes—
76C22×C4C23×C12713yes—
77C22×C4C23×C12713yes—
78C4⋊C4—26no—
79C22⋊C4—26no—
80C22⋊C4—412no—
81C42D4×C12714yescomplemented
82C23—21no—
83C23—24no—
84C23—212no—
85C23—212no—
86C23—26no—
87C23C24×C12711yescharacteristic
88D4—424no—
89D4—416no—
90C2×C4—26no—
91C2×C4—23no—
92C2×C4—26no—
93C2×C4—212no—
94C2×C4C2×D4×C12716yes—
95C22—412no—
96C22—43no—
97C22—44no—
98C22—44no—
99C22—412no—
100C22C22×D4×C12711yescharacteristic
101C22H(5,𝔽2)×C12716yes—
102C4—26no—
103C4—43no—
104C4—44no—
105C2—84no—
106C2—81no—
107C2#535×C12713yes—
108C2#560×C12714yes—
subgroupG/H
center, socleC23×C127C24
commutator, Frattini, cosocleC23C24×C127
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#127×C127C2semidirect
2. C2×#126×C127C2semidirect
3. #394×C127C2semidirect
4. #392×C127C2semidirect
5. #383×C127C2semidirect
26. C42×C127D4semidirect
54. #2818C127direct (N × H)
55. C127#2818direct (N × H)
56. C2×#127C2×C127semidirect
57. C2×#126C2×C127semidirect
58. #394C2×C127semidirect
59. #392C2×C127semidirect
60. #383C2×C127semidirect
81. C42D4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#127×C1272
C2×#126×C1272
#394×C1272
#392×C1272
#383×C1272
#2818127