Finite Group Databasebeta

← C127×#12437 #44576110

Subgroups — C127×#12437 #44576110

Order 32512.

214 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C127×#2247C211yesmaximal, characteristic, complemented
2C127×#2214C212yesmaximal
3C127×#2170C211yesmaximal, characteristic
4C127×#1373C211yesmaximal, characteristic, complemented
5C127×#1358C212yesmaximal, complemented
6#510×C127—22no—
7C2×Q32×C127—21no—
8C2×Q32×C127—21no—
9#504×C127—21no—
10#494×C127C2212yes—
11#443×C127C2211yescharacteristic
12#364×C127C2211yescharacteristic
13#364×C127C2212yes—
14#362×C127—22no—
15#350×C127—21no—
16C4×C16×C127C2211yescharacteristic
17#135×C127D411yescharacteristic
18C2×Q16×C127—22no—
19C2×Q16×C127—22no—
20#131×C127—21no—
21C4⋊Q8×C127—22no—
22Q32×C127—42no—
23Q32×C127—42no—
24Q32×C127—41no—
25Q32×C127—41no—
26SD32×C127—42no—
27M32×C127—21no—
28M32×C127—22no—
29M32×C127—41no—
30C2×C16×C127—41no—
31C2×C16×C127D411yescharacteristic
32C2×C16×C127D411yescharacteristic
33#108×C127D411yescharacteristic
34#104×C127—21no—
35C4×C8×C127C2311yescharacteristic
36#41×C127—21no—
37C2×Q8×C127—42no—
38Q16×C127—42no—
39Q16×C127—42no—
40Q16×C127—42no—
41Q16×C127—42no—
42Q16×C127D811yescharacteristic
43SD16×C127—21no—
44D8×C127D811yescharacteristic
45M16×C127—21no—
46M16×C127—41no—
47C2×C8×C127—41no—
48C2×C8×C127C2×D411yescharacteristic
49C2×C8×C127C2×D411yescharacteristic
50C4⋊C4×C127—82no—
51C42×C127#4111yescharacteristic
52C16×C127—21no—
53C16×C127—41no—
54C16×C127—42no—
55C16×C127D811yescharacteristic
56C16×C127D811yescharacteristic
57Q8×C127—21no—
58Q8×C127—82no—
59Q8×C127—82no—
60D4×C127—21no—
61D4×C127—41no—
62C2×C4×C127—21no—
63C2×C4×C127—41no—
64C2×C4×C127—82no—
65C2×C4×C127#12111yescharacteristic
66C8×C127—41no—
67C8×C127C2×D811yescharacteristic
68C8×C127C2×D811yescharacteristic
69C8×C127#13511yescharacteristic
70C8×C127AGL(1,ℤ/8)11yescharacteristic
71C22×C127—41no—
72C22×C127#47211yescharacteristic
73C4×C127—22no—
74C4×C127—41no—
75C4×C127—162no—
76C4×C127#45911yescharacteristic
77C4×C127#46611yescharacteristic
78#12437C12711yesmaximal, characteristic, direct product factor, Sylow
79C2×C127—21no—
80C2×C127—81no—
81C2×C127#166711yescharacteristic
82#2247C2×C12711yescharacteristic, complemented
83#2214C2×C12712yes—
84#2170C2×C12711yescharacteristic
85#1373C2×C12711yescharacteristic, complemented
86#1358C2×C12712yescomplemented
87C127#1243711yescharacteristic, direct product factor, Sylow
88#510—22no—
89C2×Q32—21no—
90C2×Q32—21no—
91#504—21no—
92#494C22×C12712yes—
93#443C22×C12711yescharacteristic
94#364C22×C12711yescharacteristic
95#364C22×C12712yes—
96#362—22no—
97#350—21no—
98C4×C16C22×C12711yescharacteristic
99#135D4×C12711yescharacteristic
100C2×Q16—22no—
101C2×Q16—22no—
102#131—21no—
103C4⋊Q8—22no—
104Q32—42no—
105Q32—42no—
106Q32—41no—
107Q32—41no—
108SD32—42no—
109M32—21no—
110M32—22no—
111M32—41no—
112C2×C16—41no—
113C2×C16D4×C12711yescharacteristic
114C2×C16D4×C12711yescharacteristic
115#108D4×C12711yescharacteristic
116#104—21no—
117C4×C8C23×C12711yescharacteristic
118#41—21no—
119C2×Q8—42no—
120Q16—42no—
121Q16—42no—
122Q16—42no—
123Q16—42no—
124Q16D8×C12711yescharacteristic
125SD16—21no—
126D8D8×C12711yescharacteristic
127M16—21no—
128M16—41no—
129C2×C8—41no—
130C2×C8C2×D4×C12711yescharacteristic
131C2×C8C2×D4×C12711yescharacteristic
132C4⋊C4—82no—
133C42#41×C12711yescharacteristic
134C16—21no—
135C16—41no—
136C16—42no—
137C16D8×C12711yescharacteristic
138C16D8×C12711yescharacteristic
139Q8—21no—
140Q8—82no—
141Q8—82no—
142D4—21no—
143D4—41no—
144C2×C4—21no—
145C2×C4—41no—
146C2×C4—82no—
147C2×C4#121×C12711yescharacteristic
148C8—41no—
149C8C2×D8×C12711yescharacteristic
150C8C2×D8×C12711yescharacteristic
151C8#135×C12711yescharacteristic
152C8AGL(1,ℤ/8)×C12711yescharacteristic
153C22—41no—
154C22#472×C12711yescharacteristic
155C4—22no—
156C4—41no—
157C4—162no—
158C4#459×C12711yescharacteristic
159C4#466×C12711yescharacteristic
160C2—21no—
161C2—81no—
162C2C127×#166711yescharacteristic
subgroupG/H
center, socleC2×C127#1667
commutator, Frattini, cosocleC4×C8C23×C127
prime radical#135×C127D4
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C2×C4×C127 ◁ C4×C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#2247C2semidirect
4. C127×#1373C2semidirect
5. C127×#1358C2semidirect
78. #12437C127direct (N × H)
82. #2247C2×C127semidirect
85. #1373C2×C127semidirect
86. #1358C2×C127semidirect
87. C127#12437direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#22472
C127×#22142
C127×#21702
C127×#13732
C127×#13582
#12437127