Finite Group Databasebeta

← C127×#13509 #44577182

Subgroups — C127×#13509 #44577182

Order 32512.

268 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 114 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×#2159C212yesmaximal
2C127×#1945C211yesmaximal, characteristic
3C127×#1846C211yesmaximal, characteristic
4C127×#1387C212yesmaximal
5C127×#1376C211yesmaximal, characteristic
6C2×M32×C127C2212yes—
7C2×#108×C127C2211yescharacteristic
8#427×C127C2211yescharacteristic
9C2×#107×C127C2212yes—
10C2×#106×C127—21no—
11#424×C127—21no—
12C2×C4×C8×C127C2211yescharacteristic
13#382×C127—21no—
14#367×C127—22no—
15#366×C127—22no—
16#339×C127—21no—
17C2×M16×C127—21no—
18C22×C8×C127D411yescharacteristic
19C22×C8×C127C2311yescharacteristic
20#117×C127—21no—
21C2×C4⋊C4×C127—22no—
22C2×C42×C127D411yescharacteristic
23M32×C127—22no—
24M32×C127Q814yes—
25C2×C16×C127—22no—
26#108×C127—22no—
27#107×C127—22no—
28#107×C127—22no—
29#107×C127D412yes—
30#106×C127—21no—
31#106×C127—42no—
32C4⋊C8×C127—42no—
33C4×C8×C127—22no—
34#95×C127—42no—
35C22×C4×C127—21no—
36C22×C4×C127—42no—
37C22×C4×C127C2×D411yescharacteristic
38M16×C127—41no—
39M16×C127—41no—
40C2×C8×C127—21no—
41C2×C8×C127—21no—
42C2×C8×C127—41no—
43C2×C8×C127D812yes—
44C2×C8×C127C2×D411yescharacteristic
45C2×C8×C127C2×Q812yes—
46C2×C8×C127#4112yes—
47C2×C8×C127#4111yescharacteristic
48C4⋊C4×C127—22no—
49C4⋊C4×C127—42no—
50C4⋊C4×C127—42no—
51C22⋊C4×C127—41no—
52C42×C127—21no—
53C42×C127—41no—
54C42×C127Q1612yes—
55C16×C127—24no—
56C23×C127AGL(1,ℤ/8)11yescharacteristic
57C2×C4×C127—22no—
58C2×C4×C127—42no—
59C2×C4×C127—42no—
60C2×C4×C127—84no—
61C2×C4×C127—82no—
62C2×C4×C127#12111yescharacteristic
63C2×C4×C127#12311yescharacteristic
64C2×C4×C127C4⋊Q811yescharacteristic
65C2×C4×C127C2×D811yescharacteristic
66C2×C4×C127C2×Q1611yescharacteristic
67C2×C4×C127#13711yescharacteristic
68C8×C127—22no—
69C8×C127—81no—
70C8×C127C22⋊Q812yes—
71C8×C127#12512yes—
72C22×C127—22no—
73C22×C127#45911yescharacteristic
74C22×C127#48411yescharacteristic
75C22×C127#50111yescharacteristic
76C4×C127—42no—
77C4×C127—82no—
78C4×C127—162no—
79C4×C127#39811yescharacteristic
80C4×C127#46211yescharacteristic
81C4×C127#48011yescharacteristic
82C4×C127#50011yescharacteristic
83#13509C12711yesmaximal, characteristic, direct product factor, Sylow
84C2×C127—21no—
85C2×C127—21no—
86C2×C127#208311yescharacteristic
87C2×C127#221212yes—
88#2159C2×C12712yes—
89#1945C2×C12711yescharacteristic
90#1846C2×C12711yescharacteristic
91#1387C2×C12712yes—
92#1376C2×C12711yescharacteristic
93C127#1350911yescharacteristic, direct product factor, Sylow
94C2×M32C22×C12712yes—
95C2×#108C22×C12711yescharacteristic
96#427C22×C12711yescharacteristic
97C2×#107C22×C12712yes—
98C2×#106—21no—
99#424—21no—
100C2×C4×C8C22×C12711yescharacteristic
101#382—21no—
102#367—22no—
103#366—22no—
104#339—21no—
105C2×M16—21no—
106C22×C8D4×C12711yescharacteristic
107C22×C8C23×C12711yescharacteristic
108#117—21no—
109C2×C4⋊C4—22no—
110C2×C42D4×C12711yescharacteristic
111M32—22no—
112M32Q8×C12714yes—
113C2×C16—22no—
114#108—22no—
115#107—22no—
116#107—22no—
117#107D4×C12712yes—
118#106—21no—
119#106—42no—
120C4⋊C8—42no—
121C4×C8—22no—
122#95—42no—
123C22×C4—21no—
124C22×C4—42no—
125C22×C4C2×D4×C12711yescharacteristic
126M16—41no—
127M16—41no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—41no—
131C2×C8D8×C12712yes—
132C2×C8C2×D4×C12711yescharacteristic
133C2×C8C2×Q8×C12712yes—
134C2×C8#41×C12712yes—
135C2×C8#41×C12711yescharacteristic
136C4⋊C4—22no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C22⋊C4—41no—
140C42—21no—
141C42—41no—
142C42Q16×C12712yes—
143C16—24no—
144C23AGL(1,ℤ/8)×C12711yescharacteristic
145C2×C4—22no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—84no—
149C2×C4—82no—
150C2×C4#121×C12711yescharacteristic
151C2×C4#123×C12711yescharacteristic
152C2×C4C4⋊Q8×C12711yescharacteristic
153C2×C4C2×D8×C12711yescharacteristic
154C2×C4C2×Q16×C12711yescharacteristic
155C2×C4#137×C12711yescharacteristic
156C8—22no—
157C8—81no—
158C8C22⋊Q8×C12712yes—
159C8#125×C12712yes—
160C22—22no—
161C22#459×C12711yescharacteristic
162C22#484×C12711yescharacteristic
163C22#501×C12711yescharacteristic
164C4—42no—
165C4—82no—
166C4—162no—
167C4#398×C12711yescharacteristic
168C4#462×C12711yescharacteristic
169C4#480×C12711yescharacteristic
170C4#500×C12711yescharacteristic
171C2—21no—
172C2—21no—
173C2C127×#208311yescharacteristic
174C2C127×#221212yes—
subgroupG/H
center, socleC22×C127#459
commutator, Frattini, cosocleC22×C8C23×C127
prime radicalC23×C127AGL(1,ℤ/8)
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×C4×C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
83. #13509C127direct (N × H)
93. C127#13509direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#21592
C127×#19452
C127×#18462
C127×#13872
C127×#13762
#13509127