Finite Group Databasebeta

← C127×#7306 #44570979

Subgroups — C127×#7306 #44570979

Order 32512.

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

normal subgroups only

This group has 456 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
1C127×#1785C212yesmaximal
2C127×#1780C211yesmaximal, characteristic
3C22×#41×C127C2211yescharacteristic
4C2×#101×C127—22no—
5C2×#100×C127C414yes—
6#407×C127—22no—
7C2×C22⋊C8×C127—22no—
8C2×C22⋊C8×C127C412yes—
9C2×#41×C127—42no—
10C2×#41×C127—44no—
11C22×Q8×C127D411yescharacteristic
12C22×D4×C127C2×C412yes—
13C22×D4×C127Q811yescharacteristic
14C23×C4×C127C2×C411yescharacteristic
15C23×C4×C127D412yes—
16C2×M16×C127—24no—
17C22×C8×C127—22no—
18#101×C127—44no—
19#100×C127—44no—
20C22⋊C8×C127—44no—
21C22⋊C8×C127—44no—
22C22⋊C8×C127—42no—
23C24×C127C4211yescharacteristic
24C24×C127C4⋊C412yes—
25#41×C127—416no—
26C2×Q8×C127—24no—
27C2×Q8×C127—41no—
28C2×D4×C127—24no—
29C2×D4×C127—42no—
30C2×D4×C127—42no—
31C2×D4×C127—42no—
32C2×D4×C127—41no—
33C22×C4×C127—22no—
34C22×C4×C127—24no—
35C22×C4×C127—24no—
36C22×C4×C127—42no—
37C22×C4×C127—42no—
38C22×C4×C127C22⋊C412yes—
39C22×C4×C127C22⋊C411yescharacteristic
40C22×C4×C127C4⋊C411yescharacteristic
41M16×C127—44no—
42M16×C127—44no—
43C2×C8×C127—44no—
44C2×C8×C127—44no—
45C2×C8×C127—42no—
46C23×C127—22no—
47C23×C127—21no—
48C23×C127—22no—
49C23×C127—22no—
50C23×C127—22no—
51C23×C127—44no—
52C23×C127—42no—
53C23×C127#9511yescharacteristic
54Q8×C127—44no—
55D4×C127—48no—
56D4×C127—44no—
57C2×C4×C127—22no—
58C2×C4×C127—22no—
59C2×C4×C127—22no—
60C2×C4×C127—22no—
61C2×C4×C127—21no—
62C2×C4×C127—48no—
63C2×C4×C127—44no—
64C2×C4×C127#9914yes—
65C2×C4×C127#9912yes—
66C8×C127—82no—
67C8×C127—84no—
68C22×C127—22no—
69C22×C127—44no—
70C22×C127—42no—
71C22×C127—41no—
72C22×C127—44no—
73C22×C127—42no—
74C22×C127—82no—
75C22×C127—82no—
76C22×C127#34212yes—
77C22×C127#34211yescharacteristic
78C4×C127—22no—
79C4×C127—24no—
80C4×C127—41no—
81#7306C12711yesmaximal, characteristic, direct product factor, Sylow
82C2×C127—22no—
83C2×C127—82no—
84C2×C127—81no—
85C2×C127#130411yescharacteristic
86C2×C127#139212yes—
87#1785C2×C12712yes—
88#1780C2×C12711yescharacteristic
89C127#730611yescharacteristic, direct product factor, Sylow
90C22×#41C22×C12711yescharacteristic
91C2×#101—22no—
92C2×#100C4×C12714yes—
93#407—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8C4×C12712yes—
96C2×#41—42no—
97C2×#41—44no—
98C22×Q8D4×C12711yescharacteristic
99C22×D4C2×C4×C12712yes—
100C22×D4Q8×C12711yescharacteristic
101C23×C4C2×C4×C12711yescharacteristic
102C23×C4D4×C12712yes—
103C2×M16—24no—
104C22×C8—22no—
105#101—44no—
106#100—44no—
107C22⋊C8—44no—
108C22⋊C8—44no—
109C22⋊C8—42no—
110C24C42×C12711yescharacteristic
111C24C4⋊C4×C12712yes—
112#41—416no—
113C2×Q8—24no—
114C2×Q8—41no—
115C2×D4—24no—
116C2×D4—42no—
117C2×D4—42no—
118C2×D4—41no—
119C2×D4—42no—
120C22×C4—22no—
121C22×C4—24no—
122C22×C4—24no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4C22⋊C4×C12712yes—
126C22×C4C22⋊C4×C12711yescharacteristic
127C22×C4C4⋊C4×C12711yescharacteristic
128M16—44no—
129M16—44no—
130C2×C8—44no—
131C2×C8—44no—
132C2×C8—42no—
133C23—21no—
134C23—22no—
135C23—22no—
136C23—22no—
137C23—22no—
138C23—44no—
139C23—42no—
140C23#95×C12711yescharacteristic
141Q8—44no—
142D4—48no—
143D4—44no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—21no—
148C2×C4—22no—
149C2×C4—48no—
150C2×C4—44no—
151C2×C4#99×C12712yes—
152C2×C4#99×C12714yes—
153C8—82no—
154C8—84no—
155C22—22no—
156C22—44no—
157C22—42no—
158C22—41no—
159C22—44no—
160C22—42no—
161C22—82no—
162C22—82no—
163C22#342×C12712yes—
164C22#342×C12711yescharacteristic
165C4—22no—
166C4—24no—
167C4—41no—
168C2—22no—
169C2—82no—
170C2—81no—
171C2C127×#130411yescharacteristic
172C2C127×#139212yes—
subgroupG/H
center, socleC22×C127#342
commutatorC24C42×C127
Frattini, cosocleC22×#41C22×C127
prime radicalC22×#41×C127C22
LCSG ▷ C24 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C22×#41×C127 ◁ G

Complements

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

NH ≅ G/Ntype
81. #7306C127direct (N × H)
89. C127#7306direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#17852
C127×#17802
#7306127