Finite Group Databasebeta

← C127×#13687 #44577360

Subgroups — C127×#13687 #44577360

Order 32512.

212 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 78 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×SD64×C127C212yesmaximal
2C127×#2247C211yesmaximal, characteristic, complemented
3C4×D16×C127C211yesmaximal, characteristic
4C127×C4⋊C32C211yesmaximal, characteristic, complemented
5C127×#1416C212yesmaximal, complemented
6C2×Q32×C127—21no—
7C2×Q32×C127C2212yes—
8C2×D16×C127C2211yescharacteristic
9#494×C127—21no—
10C4×D8×C127—21no—
11SD64×C127—24no—
12C2×C32×C127C2212yes—
13#366×C127C2211yescharacteristic
14#357×C127—21no—
15C4×C16×C127C2211yescharacteristic
16C2×Q16×C127—22no—
17C2×Q16×C127—41no—
18C2×D8×C127—21no—
19C4⋊Q8×C127—41no—
20C4×D4×C127—41no—
21Q32×C127—22no—
22Q32×C127—22no—
23Q32×C127—42no—
24D16×C127—21no—
25D16×C127D412yes—
26C2×C16×C127D411yescharacteristic
27C2×C16×C127C2311yescharacteristic
28#107×C127—21no—
29#102×C127—41no—
30C4×C8×C127D411yescharacteristic
31C32×C127—22no—
32C2×Q8×C127—42no—
33C2×D4×C127—41no—
34C22×C4×C127—81no—
35Q16×C127—42no—
36Q16×C127—42no—
37Q16×C127—82no—
38D8×C127—22no—
39D8×C127—41no—
40C2×C8×C127D811yescharacteristic
41C2×C8×C127C2×D411yescharacteristic
42C4⋊C4×C127—41no—
43C4⋊C4×C127—81no—
44C4⋊C4×C127—81no—
45C22⋊C4×C127—81no—
46C42×C127D811yescharacteristic
47C16×C127—21no—
48C16×C127C2×D411yescharacteristic
49C16×C127#4111yescharacteristic
50C23×C127—81no—
51Q8×C127—82no—
52Q8×C127—82no—
53D4×C127—42no—
54D4×C127—81no—
55C2×C4×C127—81no—
56C2×C4×C127—82no—
57C2×C4×C127—162no—
58C2×C4×C127D1611yescharacteristic
59C2×C4×C127D1611yescharacteristic
60C2×C4×C127C2×D811yescharacteristic
61C8×C127—21no—
62C8×C127#12111yescharacteristic
63C8×C127AGL(1,ℤ/8)11yescharacteristic
64C22×C127—82no—
65C22×C127—161no—
66C22×C127—161no—
67C22×C127C2×D1611yescharacteristic
68C4×C127—21no—
69C4×C127—162no—
70C4×C127—161no—
71C4×C127SD6412yescomplemented
72C4×C127#45911yescharacteristic
73C4×C127#50911yescharacteristic
74#13687C12711yesmaximal, characteristic, direct product factor, Sylow
75C2×C127—162no—
76C2×C127#220611yescharacteristic
77C2×C127C2×SD6411yescharacteristic
78C2×C127#226411yescharacteristic
79C2×SD64C2×C12712yes—
80#2247C2×C12711yescharacteristic, complemented
81C4×D16C2×C12711yescharacteristic
82C4⋊C32C2×C12711yescharacteristic, complemented
83#1416C2×C12712yescomplemented
84C127#1368711yescharacteristic, direct product factor, Sylow
85C2×Q32—21no—
86C2×Q32C22×C12712yes—
87C2×D16C22×C12711yescharacteristic
88#494—21no—
89C4×D8—21no—
90SD64—24no—
91C2×C32C22×C12712yes—
92#366C22×C12711yescharacteristic
93#357—21no—
94C4×C16C22×C12711yescharacteristic
95C2×Q16—22no—
96C2×Q16—41no—
97C2×D8—21no—
98C4⋊Q8—41no—
99C4×D4—41no—
100Q32—22no—
101Q32—22no—
102Q32—42no—
103D16—21no—
104D16D4×C12712yes—
105C2×C16D4×C12711yescharacteristic
106C2×C16C23×C12711yescharacteristic
107#107—21no—
108#102—41no—
109C4×C8D4×C12711yescharacteristic
110C32—22no—
111C2×Q8—42no—
112C2×D4—41no—
113C22×C4—81no—
114Q16—42no—
115Q16—42no—
116Q16—82no—
117D8—22no—
118D8—41no—
119C2×C8D8×C12711yescharacteristic
120C2×C8C2×D4×C12711yescharacteristic
121C4⋊C4—41no—
122C4⋊C4—81no—
123C4⋊C4—81no—
124C22⋊C4—81no—
125C42D8×C12711yescharacteristic
126C16—21no—
127C16C2×D4×C12711yescharacteristic
128C16#41×C12711yescharacteristic
129C23—81no—
130Q8—82no—
131Q8—82no—
132D4—42no—
133D4—81no—
134C2×C4—81no—
135C2×C4—82no—
136C2×C4—162no—
137C2×C4D16×C12711yescharacteristic
138C2×C4D16×C12711yescharacteristic
139C2×C4C2×D8×C12711yescharacteristic
140C8—21no—
141C8#121×C12711yescharacteristic
142C8AGL(1,ℤ/8)×C12711yescharacteristic
143C22—82no—
144C22—161no—
145C22—161no—
146C22C2×D16×C12711yescharacteristic
147C4—21no—
148C4—162no—
149C4—161no—
150C4SD64×C12712yescomplemented
151C4#459×C12711yescharacteristic
152C4#509×C12711yescharacteristic
153C2—162no—
154C2C127×#220611yescharacteristic
155C2C2×SD64×C12711yescharacteristic
156C2C127×#226411yescharacteristic
subgroupG/H
center, socleC22×C127C2×D16
commutator, Frattini, cosocleC2×C16C23×C127
prime radicalC2×D16×C127C22
LCSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C42×C127 ◁ C4×C8×C127 ◁ C4×C16×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2247C2semidirect
4. C127×C4⋊C32C2semidirect
5. C127×#1416C2semidirect
71. C4×C127SD64semidirect
74. #13687C127direct (N × H)
80. #2247C2×C127semidirect
82. C4⋊C32C2×C127semidirect
83. #1416C2×C127semidirect
84. C127#13687direct (N × H)
150. C4SD64×C127semidirect

Maximal subgroups

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

subgroupindex
C2×SD64×C1272
C127×#22472
C4×D16×C1272
C127×C4⋊C322
C127×#14162
#13687127