Finite Group Databasebeta

← C127×#3027 #30635979

Subgroups — C127×#3027 #30635979

Order 16256.

772 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 212 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#585×C127C211yesmaximal, characteristic, complemented
2#535×C127C213yesmaximal, complemented
3B(4,𝔽2)×C127C218yesmaximal, complemented
4#410×C127C213yesmaximal, complemented
52−1+4×C127—23no—
6H(5,𝔽2)×C127—23no—
7H(5,𝔽2)×C127C2214yes—
8C2×#41×C127—26no—
9C2×#41×C127C2213yes—
10C22×D4×C127C2211yescharacteristic, complemented
11#127×C127—23no—
12#124×C127—23no—
13#121×C127—26no—
14#120×C127C22112yescomplemented
15#117×C127C2213yescomplemented
16#99×C127C22112yescomplemented
17C24×C127C2312yes—
18#41×C127—24no—
19#41×C127—212no—
20#41×C127—26no—
21#41×C127—212no—
22#41×C127—26no—
23C2×Q8×C127—26no—
24C2×Q8×C127D413yescomplemented
25C2×D4×C127—26no—
26C2×D4×C127—212no—
27C2×D4×C127—212no—
28C2×D4×C127—23no—
29C2×D4×C127—23no—
30C2×D4×C127D413yescomplemented
31C2×D4×C127C2316yes—
32C22×C4×C127—24no—
33C22×C4×C127D416yescomplemented
34C22×C4×C127C2311yescharacteristic
35C4⋊C4×C127—23no—
36C22⋊C4×C127—212no—
37C22⋊C4×C127C2316yes—
38C42×C127—23no—
39C23×C127—24no—
40C23×C127—26no—
41C23×C127—82no—
42C23×C127C2×D416yes—
43C23×C127C2411yescharacteristic
44Q8×C127—24no—
45Q8×C127—26no—
46D4×C127—212no—
47D4×C127—212no—
48D4×C127—26no—
49D4×C127—41no—
50D4×C127—43no—
51D4×C127—812no—
52C2×C4×C127—212no—
53C2×C4×C127—26no—
54C2×C4×C127—26no—
55C2×C4×C127—26no—
56C2×C4×C127C2×D416yes—
57C2×C4×C127C2×D413yes—
58C2×C4×C127C2×D413yes—
59C22×C127—26no—
60C22×C127—41no—
61C22×C127—46no—
62C22×C127—412no—
63C22×C127—42no—
64C22×C127—84no—
65C22×C127C22×D413yes—
66C4×C127—26no—
67C4×C127—46no—
68C4×C127#12011yescharacteristic
69C4×C127#12013yes—
70C2×C127—23no—
71C2×C127—46no—
72C2×C127—82no—
73C2×C127C2×#12011yescharacteristic
74#3027C12711yesmaximal, characteristic, direct product factor, Sylow
75C127#302711yescharacteristic, direct product factor, Sylow
76#585C2×C12711yescharacteristic, complemented
77#535C2×C12713yescomplemented
78B(4,𝔽2)C2×C12718yescomplemented
79#410C2×C12713yescomplemented
802−1+4—23no—
81H(5,𝔽2)—23no—
82H(5,𝔽2)C22×C12714yes—
83C2×#41—26no—
84C2×#41C22×C12713yes—
85C22×D4C22×C12711yescharacteristic, complemented
86#127—23no—
87#124—23no—
88#121—26no—
89#120C22×C127112yescomplemented
90#117C22×C12713yescomplemented
91#99C22×C127112yescomplemented
92C24C23×C12712yes—
93#41—24no—
94#41—212no—
95#41—26no—
96#41—212no—
97#41—26no—
98C2×Q8—26no—
99C2×Q8D4×C12713yescomplemented
100C2×D4—26no—
101C2×D4—212no—
102C2×D4—212no—
103C2×D4—23no—
104C2×D4—23no—
105C2×D4D4×C12713yescomplemented
106C2×D4C23×C12716yes—
107C22×C4—24no—
108C22×C4D4×C12716yescomplemented
109C22×C4C23×C12711yescharacteristic
110C4⋊C4—23no—
111C22⋊C4—212no—
112C22⋊C4C23×C12716yes—
113C42—23no—
114C23—24no—
115C23—26no—
116C23—82no—
117C23C2×D4×C12716yes—
118C23C24×C12711yescharacteristic
119Q8—24no—
120Q8—26no—
121D4—212no—
122D4—212no—
123D4—26no—
124D4—41no—
125D4—43no—
126D4—812no—
127C2×C4—212no—
128C2×C4—26no—
129C2×C4—26no—
130C2×C4—26no—
131C2×C4C2×D4×C12716yes—
132C2×C4C2×D4×C12713yes—
133C2×C4C2×D4×C12713yes—
134C22—26no—
135C22—41no—
136C22—46no—
137C22—412no—
138C22—42no—
139C22—84no—
140C22C22×D4×C12713yes—
141C4—26no—
142C4—46no—
143C4#120×C12711yescharacteristic
144C4#120×C12713yes—
145C2—23no—
146C2—46no—
147C2—82no—
148C2C2×#120×C12711yescharacteristic
subgroupG/H
center, socleC2×C127C2×#120
commutator, Frattini, cosocleC23C24×C127
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C22×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. #585×C127C2semidirect
2. #535×C127C2semidirect
3. B(4,𝔽2)×C127C2semidirect
4. #410×C127C2semidirect
10. C22×D4×C127C22semidirect
14. #120×C127C22semidirect
15. #117×C127C22semidirect
16. #99×C127C22semidirect
24. C2×Q8×C127D4semidirect
30. C2×D4×C127D4semidirect
33. C22×C4×C127D4semidirect
74. #3027C127direct (N × H)
75. C127#3027direct (N × H)
76. #585C2×C127semidirect
77. #535C2×C127semidirect
78. B(4,𝔽2)C2×C127semidirect
79. #410C2×C127semidirect
85. C22×D4C22×C127semidirect
89. #120C22×C127semidirect
90. #117C22×C127semidirect
91. #99C22×C127semidirect
99. C2×Q8D4×C127semidirect
105. C2×D4D4×C127semidirect
108. C22×C4D4×C127semidirect

Maximal subgroups

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

subgroupindex
#585×C1272
#535×C1272
B(4,𝔽2)×C1272
#410×C1272
#3027127