Finite Group Databasebeta

← C127×#7523 #44571196

Subgroups — C127×#7523 #44571196

Order 32512.

158 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 30 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×#1656C211yesmaximal, characteristic
2C127×#1412C211yesmaximal, characteristic, complemented
3C127×#1355C211yesmaximal, characteristic, complemented
4#562×C127C411yescharacteristic, complemented
5#496×C127—21no—
6#490×C127C2211yescharacteristic
7#353×C127—21no—
8#347×C127—21no—
9#330×C127C411yescharacteristic, complemented
10#329×C127—21no—
11C2×SD16×C127—41no—
12C2×D8×C127—41no—
13#127×C127D411yescharacteristic
14#125×C127C2×C411yescharacteristic
15#124×C127—21no—
16#123×C127—41no—
17C22⋊Q8×C127—41no—
18#121×C127—41no—
19M32×C127—41no—
20#102×C127—41no—
21#99×C127—41no—
22#97×C127—21no—
23#97×C127D411yescharacteristic
24C2×Q8×C127—21no—
25C2×D4×C127—21no—
26C2×D4×C127—21no—
27C2×D4×C127—41no—
28C2×D4×C127—41no—
29C22×C4×C127—41no—
30SD16×C127—82no—
31D8×C127—82no—
32C2×C8×C127—21no—
33C2×C8×C127—41no—
34C4⋊C4×C127—21no—
35C4⋊C4×C127—41no—
36C4⋊C4×C127—41no—
37C22⋊C4×C127—41no—
38C22⋊C4×C127—41no—
39C22⋊C4×C127—41no—
40C22⋊C4×C127—81no—
41C42×C127C22⋊C411yescharacteristic
42C16×C127—42no—
43C23×C127—41no—
44C23×C127—41no—
45Q8×C127—81no—
46D4×C127—81no—
47D4×C127—81no—
48D4×C127—81no—
49D4×C127—81no—
50D4×C127—161no—
51C2×C4×C127—21no—
52C2×C4×C127—41no—
53C2×C4×C127—41no—
54C2×C4×C127—161no—
55C2×C4×C127—161no—
56C2×C4×C127#9911yescharacteristic
57C8×C127—41no—
58C8×C127—42no—
59C22×C127—81no—
60C22×C127—81no—
61C22×C127—81no—
62C22×C127—81no—
63C22×C127—161no—
64C22×C127AΣL(1,𝔽16)11yescharacteristic
65C4×C127—21no—
66C4×C127—41no—
67C4×C127—81no—
68C4×C127—81no—
69C4×C127—321no—
70#7523C12711yesmaximal, characteristic, direct product factor, Sylow
71C2×C127—21no—
72C2×C127—161no—
73C2×C127—161no—
74C2×C127#140411yescharacteristic
75#1656C2×C12711yescharacteristic
76#1412C2×C12711yescharacteristic, complemented
77#1355C2×C12711yescharacteristic, complemented
78C127#752311yescharacteristic, direct product factor, Sylow
79#562C4×C12711yescharacteristic, complemented
80#496—21no—
81#490C22×C12711yescharacteristic
82#353—21no—
83#347—21no—
84#330C4×C12711yescharacteristic, complemented
85#329—21no—
86C2×SD16—41no—
87C2×D8—41no—
88#127D4×C12711yescharacteristic
89#125C2×C4×C12711yescharacteristic
90#124—21no—
91#123—41no—
92C22⋊Q8—41no—
93#121—41no—
94M32—41no—
95#102—41no—
96#99—41no—
97#97—21no—
98#97D4×C12711yescharacteristic
99C2×Q8—21no—
100C2×D4—21no—
101C2×D4—21no—
102C2×D4—41no—
103C2×D4—41no—
104C22×C4—41no—
105SD16—82no—
106D8—82no—
107C2×C8—21no—
108C2×C8—41no—
109C4⋊C4—21no—
110C4⋊C4—41no—
111C4⋊C4—41no—
112C22⋊C4—41no—
113C22⋊C4—41no—
114C22⋊C4—41no—
115C22⋊C4—81no—
116C42C22⋊C4×C12711yescharacteristic
117C16—42no—
118C23—41no—
119C23—41no—
120Q8—81no—
121D4—81no—
122D4—81no—
123D4—81no—
124D4—81no—
125D4—161no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—161no—
130C2×C4—161no—
131C2×C4#99×C12711yescharacteristic
132C8—41no—
133C8—42no—
134C22—81no—
135C22—81no—
136C22—81no—
137C22—81no—
138C22—161no—
139C22AΣL(1,𝔽16)×C12711yescharacteristic
140C4—21no—
141C4—41no—
142C4—81no—
143C4—81no—
144C4—321no—
145C2—21no—
146C2—161no—
147C2—161no—
148C2C127×#140411yescharacteristic
subgroupG/H
center, socleC2×C127#1404
commutator#125C2×C4×C127
Frattini, cosocle#490C22×C127
prime radicalC127×#1656C2
LCSG ▷ #125 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C22×C127 ◁ C2×C4×C127 ◁ C42×C127 ◁ #490×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#1412C2semidirect
3. C127×#1355C2semidirect
4. #562×C127C4semidirect
9. #330×C127C4semidirect
70. #7523C127direct (N × H)
76. #1412C2×C127semidirect
77. #1355C2×C127semidirect
78. C127#7523direct (N × H)
79. #562C4×C127semidirect
84. #330C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#16562
C127×#14122
C127×#13552
#7523127