Finite Group Databasebeta

← C127×#12872 #44576545

Subgroups — C127×#12872 #44576545

Order 32512.

398 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 142 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×#427×C127C211yesmaximal, characteristic
2C127×#2112C211yesmaximal, characteristic
3C127×#1810C211yesmaximal, characteristic
4C127×#1387C214yesmaximal, complemented
5C23×C8×C127C2211yescharacteristic
6C2×#117×C127—21no—
7C2×M32×C127C2212yes—
8C2×#108×C127C2212yes—
9#427×C127—22no—
10#427×C127C414yes—
11C2×#107×C127C2212yes—
12C2×#106×C127—21no—
13#407×C127—21no—
14C22⋊C16×C127—22no—
15#341×C127—21no—
16C23×C4×C127D411yescharacteristic
17C2×M16×C127—22no—
18C22×C8×C127—24no—
19C22×C8×C127C2×C412yes—
20C22×C8×C127D411yescharacteristic
21C22×C8×C127D412yes—
22C22×C8×C127C2311yescharacteristic
23#117×C127—24no—
24#117×C127—42no—
25C2×C4⋊C4×C127—22no—
26C2×C22⋊C4×C127—41no—
27C2×C42×C127—41no—
28M32×C127—24no—
29C2×C16×C127—22no—
30#108×C127—24no—
31#107×C127—22no—
32#107×C127C2×C414yes—
33#106×C127—24no—
34C22⋊C8×C127—42no—
35C24×C127SD1611yescharacteristic
36C22×C4×C127—22no—
37C22×C4×C127—22no—
38C22×C4×C127—42no—
39C22×C4×C127C22⋊C412yes—
40C22×C4×C127D812yes—
41C22×C4×C127SD1611yescharacteristic
42C22×C4×C127C2×D411yescharacteristic
43M16×C127—42no—
44M16×C127—42no—
45C2×C8×C127—22no—
46C2×C8×C127—24no—
47C2×C8×C127—24no—
48C2×C8×C127—42no—
49C2×C8×C127C22⋊C412yes—
50C2×C8×C127C22×C411yescharacteristic
51C2×C8×C127C2×D411yescharacteristic
52C2×C8×C127#4114yes—
53C4⋊C4×C127—24no—
54C4⋊C4×C127—42no—
55C22⋊C4×C127—44no—
56C42×C127—44no—
57C16×C127—42no—
58C23×C127—22no—
59C23×C127—42no—
60C23×C127#10212yes—
61C23×C127C2×SD1611yescharacteristic
62C2×C4×C127—22no—
63C2×C4×C127—22no—
64C2×C4×C127—22no—
65C2×C4×C127—24no—
66C2×C4×C127—44no—
67C2×C4×C127—84no—
68C2×C4×C127#10212yes—
69C2×C4×C127C2×C22⋊C411yescharacteristic
70C2×C4×C127#12312yes—
71C2×C4×C127C2×D811yescharacteristic
72C2×C4×C127#13712yes—
73C8×C127—22no—
74C8×C127—82no—
75C8×C127#11712yes—
76C8×C127#12312yes—
77C22×C127—22no—
78C22×C127—22no—
79C22×C127—24no—
80C22×C127—44no—
81C22×C127C2×#10211yescharacteristic
82C22×C127#48312yes—
83C4×C127—22no—
84C4×C127—84no—
85C4×C127#38111yescharacteristic
86C4×C127#41911yescharacteristic
87C4×C127#48012yes—
88#12872C12711yesmaximal, characteristic, direct product factor, Sylow
89C2×C127—22no—
90C2×C127—42no—
91C2×C127#182311yescharacteristic
92C2×C127#214812yes—
93C2×#427C2×C12711yescharacteristic
94#2112C2×C12711yescharacteristic
95#1810C2×C12711yescharacteristic
96#1387C2×C12714yescomplemented
97C127#1287211yescharacteristic, direct product factor, Sylow
98C23×C8C22×C12711yescharacteristic
99C2×#117—21no—
100C2×M32C22×C12712yes—
101C2×#108C22×C12712yes—
102#427—22no—
103#427C4×C12714yes—
104C2×#107C22×C12712yes—
105C2×#106—21no—
106#407—21no—
107C22⋊C16—22no—
108#341—21no—
109C23×C4D4×C12711yescharacteristic
110C2×M16—22no—
111C22×C8—24no—
112C22×C8C2×C4×C12712yes—
113C22×C8D4×C12711yescharacteristic
114C22×C8D4×C12712yes—
115C22×C8C23×C12711yescharacteristic
116#117—24no—
117#117—42no—
118C2×C4⋊C4—22no—
119C2×C22⋊C4—41no—
120C2×C42—41no—
121M32—24no—
122C2×C16—22no—
123#108—24no—
124#107—22no—
125#107C2×C4×C12714yes—
126#106—24no—
127C22⋊C8—42no—
128C24SD16×C12711yescharacteristic
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—42no—
132C22×C4C22⋊C4×C12712yes—
133C22×C4D8×C12712yes—
134C22×C4SD16×C12711yescharacteristic
135C22×C4C2×D4×C12711yescharacteristic
136M16—42no—
137M16—42no—
138C2×C8—22no—
139C2×C8—24no—
140C2×C8—24no—
141C2×C8—42no—
142C2×C8C22⋊C4×C12712yes—
143C2×C8C22×C4×C12711yescharacteristic
144C2×C8C2×D4×C12711yescharacteristic
145C2×C8#41×C12714yes—
146C4⋊C4—24no—
147C4⋊C4—42no—
148C22⋊C4—44no—
149C42—44no—
150C16—42no—
151C23—22no—
152C23—42no—
153C23#102×C12712yes—
154C23C2×SD16×C12711yescharacteristic
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—24no—
159C2×C4—44no—
160C2×C4—84no—
161C2×C4#102×C12712yes—
162C2×C4C2×C22⋊C4×C12711yescharacteristic
163C2×C4#123×C12712yes—
164C2×C4C2×D8×C12711yescharacteristic
165C2×C4#137×C12712yes—
166C8—22no—
167C8—82no—
168C8#117×C12712yes—
169C8#123×C12712yes—
170C22—22no—
171C22—22no—
172C22—24no—
173C22—44no—
174C22C2×#102×C12711yescharacteristic
175C22#483×C12712yes—
176C4—22no—
177C4—84no—
178C4#381×C12711yescharacteristic
179C4#419×C12711yescharacteristic
180C4#480×C12712yes—
181C2—22no—
182C2—42no—
183C2C127×#182311yescharacteristic
184C2C127×#214812yes—
subgroupG/H
center, socleC22×C127C2×#102
commutatorC2×C8C22×C4×C127
Frattini, cosocleC22×C8C23×C127
prime radicalC24×C127SD16
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C4×C127 ◁ C23×C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C127×#1387C2semidirect
88. #12872C127direct (N × H)
96. #1387C2×C127semidirect
97. C127#12872direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#427×C1272
C127×#21122
C127×#18102
C127×#13872
#12872127