Finite Group Databasebeta

← C127×#13686 #44577359

Subgroups — C127×#13686 #44577359

Order 32512.

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

normal subgroups only

This group has 220 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, complemented
2C127×#2246C211yesmaximal, characteristic
3C4×Q32×C127C211yesmaximal, characteristic, complemented
4C127×C4⋊C32C211yesmaximal, characteristic, complemented
5C127×#1415C212yesmaximal, complemented
6C2×Q32×C127C2211yescharacteristic
7C2×D16×C127—21no—
8C2×D16×C127C2212yes—
9#493×C127—21no—
10C4×Q16×C127—21no—
11SD64×C127—24no—
12C2×C32×C127C2212yes—
13#366×C127C2211yescharacteristic
14#358×C127—21no—
15C4×C16×C127C2211yescharacteristic
16C2×Q16×C127—21no—
17C2×D8×C127—22no—
18C2×D8×C127—41no—
19#127×C127—41no—
20C4×Q8×C127—41no—
21Q32×C127—21no—
22Q32×C127D412yescomplemented
23D16×C127—22no—
24D16×C127—22no—
25D16×C127—42no—
26C2×C16×C127D411yescharacteristic
27C2×C16×C127C2311yescharacteristic
28#107×C127—21no—
29#103×C127—41no—
30C4×C8×C127D411yescharacteristic
31C32×C127—22no—
32C2×Q8×C127—41no—
33C2×D4×C127—42no—
34C2×D4×C127—81no—
35C2×D4×C127—81no—
36Q16×C127—22no—
37Q16×C127—41no—
38D8×C127—42no—
39D8×C127—42no—
40D8×C127—82no—
41C2×C8×C127D811yescharacteristic
42C2×C8×C127C2×D411yescharacteristic
43C4⋊C4×C127—41no—
44C4⋊C4×C127—81no—
45C42×C127—81no—
46C42×C127D811yescharacteristic
47C16×C127—21no—
48C16×C127C2×D411yescharacteristic
49C16×C127#4111yescharacteristic
50C23×C127—82no—
51Q8×C127—42no—
52Q8×C127—81no—
53D4×C127—82no—
54D4×C127—82no—
55D4×C127—162no—
56D4×C127—162no—
57C2×C4×C127—81no—
58C2×C4×C127—81no—
59C2×C4×C127D1611yescharacteristic
60C2×C4×C127D1611yescharacteristic
61C2×C4×C127C2×D811yescharacteristic
62C8×C127—21no—
63C8×C127#12111yescharacteristic
64C8×C127AGL(1,ℤ/8)11yescharacteristic
65C22×C127—162no—
66C22×C127—162no—
67C22×C127—162no—
68C22×C127C2×D1611yescharacteristic
69C4×C127—21no—
70C4×C127—82no—
71C4×C127—161no—
72C4×C127SD6412yescomplemented
73C4×C127#45911yescharacteristic
74C4×C127#50911yescharacteristic
75#13686C12711yesmaximal, characteristic, direct product factor, Sylow
76C2×C127—322no—
77C2×C127#220611yescharacteristic
78C2×C127C2×SD6411yescharacteristic
79C2×C127#226311yescharacteristic
80C2×SD64C2×C12712yescomplemented
81#2246C2×C12711yescharacteristic
82C4×Q32C2×C12711yescharacteristic, complemented
83C4⋊C32C2×C12711yescharacteristic, complemented
84#1415C2×C12712yescomplemented
85C127#1368611yescharacteristic, direct product factor, Sylow
86C2×Q32C22×C12711yescharacteristic
87C2×D16—21no—
88C2×D16C22×C12712yes—
89#493—21no—
90C4×Q16—21no—
91SD64—24no—
92C2×C32C22×C12712yes—
93#366C22×C12711yescharacteristic
94#358—21no—
95C4×C16C22×C12711yescharacteristic
96C2×Q16—21no—
97C2×D8—22no—
98C2×D8—41no—
99#127—41no—
100C4×Q8—41no—
101Q32—21no—
102Q32D4×C12712yescomplemented
103D16—22no—
104D16—22no—
105D16—42no—
106C2×C16D4×C12711yescharacteristic
107C2×C16C23×C12711yescharacteristic
108#107—21no—
109#103—41no—
110C4×C8D4×C12711yescharacteristic
111C32—22no—
112C2×Q8—41no—
113C2×D4—42no—
114C2×D4—81no—
115C2×D4—81no—
116Q16—22no—
117Q16—41no—
118D8—42no—
119D8—42no—
120D8—82no—
121C2×C8D8×C12711yescharacteristic
122C2×C8C2×D4×C12711yescharacteristic
123C4⋊C4—41no—
124C4⋊C4—81no—
125C42—81no—
126C42D8×C12711yescharacteristic
127C16—21no—
128C16C2×D4×C12711yescharacteristic
129C16#41×C12711yescharacteristic
130C23—82no—
131Q8—42no—
132Q8—81no—
133D4—82no—
134D4—82no—
135D4—162no—
136D4—162no—
137C2×C4—81no—
138C2×C4—81no—
139C2×C4D16×C12711yescharacteristic
140C2×C4D16×C12711yescharacteristic
141C2×C4C2×D8×C12711yescharacteristic
142C8—21no—
143C8#121×C12711yescharacteristic
144C8AGL(1,ℤ/8)×C12711yescharacteristic
145C22—162no—
146C22—162no—
147C22—162no—
148C22C2×D16×C12711yescharacteristic
149C4—21no—
150C4—82no—
151C4—161no—
152C4SD64×C12712yescomplemented
153C4#459×C12711yescharacteristic
154C4#509×C12711yescharacteristic
155C2—322no—
156C2C127×#220611yescharacteristic
157C2C2×SD64×C12711yescharacteristic
158C2C127×#226311yescharacteristic
subgroupG/H
center, socleC22×C127C2×D16
commutator, Frattini, cosocleC2×C16C23×C127
prime radicalC127×#2246C2
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
1. C2×SD64×C127C2semidirect
3. C4×Q32×C127C2semidirect
4. C127×C4⋊C32C2semidirect
5. C127×#1415C2semidirect
22. Q32×C127D4semidirect
72. C4×C127SD64semidirect
75. #13686C127direct (N × H)
80. C2×SD64C2×C127semidirect
82. C4×Q32C2×C127semidirect
83. C4⋊C32C2×C127semidirect
84. #1415C2×C127semidirect
85. C127#13686direct (N × H)
102. Q32D4×C127semidirect
152. 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×#22462
C4×Q32×C1272
C127×C4⋊C322
C127×#14152
#13686127