Finite Group Databasebeta

← C2×#356×C127 #30635083

Subgroups — C2×#356×C127 #30635083

Order 16256.

240 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 88 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×C4⋊Q8×C127C211yesmaximal, characteristic
2C2×#101×C127C212yesmaximal
3#356×C127C214yesmaximal, direct product factor
4C22×Q8×C127C411yescharacteristic
5C22×Q8×C127C2211yescharacteristic
6C2×M16×C127—22no—
7C4⋊Q8×C127—22no—
8C4⋊Q8×C127C412yes—
9C4⋊Q8×C127C2212yes—
10C2×C4⋊C4×C127—22no—
11C2×C42×C127C411yescharacteristic
12#101×C127—22no—
13#101×C127C2214yes—
14C2×Q8×C127—21no—
15C2×Q8×C127—41no—
16C2×Q8×C127—41no—
17C2×Q8×C127—41no—
18C2×Q8×C127—41no—
19C2×Q8×C127C2×C412yes—
20C2×Q8×C127C2×C411yescharacteristic
21C2×Q8×C127D412yes—
22C2×Q8×C127C2311yescharacteristic
23C22×C4×C127—21no—
24C22×C4×C127—21no—
25C22×C4×C127C2×C411yescharacteristic
26C22×C4×C127D412yes—
27M16×C127—24no—
28M16×C127—42no—
29C2×C8×C127—42no—
30C4⋊C4×C127—24no—
31C4⋊C4×C127—24no—
32C42×C127—21no—
33C42×C127C2×C412yes—
34C23×C127C22⋊C411yescharacteristic
35Q8×C127—41no—
36Q8×C127—42no—
37Q8×C127—41no—
38Q8×C127—42no—
39Q8×C127—42no—
40C2×C4×C127—24no—
41C2×C4×C127—22no—
42C2×C4×C127—21no—
43C2×C4×C127—22no—
44C2×C4×C127—21no—
45C2×C4×C127—22no—
46C2×C4×C127—22no—
47C2×C4×C127—41no—
48C2×C4×C127—41no—
49C2×C4×C127C22⋊C411yescharacteristic
50C2×C4×C127C22⋊C412yes—
51C2×C4×C127C22×C411yescharacteristic
52C2×C4×C127C2×D412yes—
53C8×C127—44no—
54C22×C127—22no—
55C22×C127#9911yescharacteristic
56C22×C127#9911yescharacteristic
57C22×C127C2×C22⋊C411yescharacteristic
58C4×C127—24no—
59C4×C127—22no—
60C4×C127—22no—
61C4×C127—21no—
62C4×C127—21no—
63C4×C127—42no—
64C2×C127—21no—
65C2×C127—21no—
66C2×C127#35612yesdirect product factor
67C2×C127C2×#9911yescharacteristic
68C2×#356C12711yesmaximal, characteristic, direct product factor, Sylow
69C127C2×#35611yescharacteristic, direct product factor, Sylow
70C2×C4⋊Q8C2×C12711yescharacteristic
71C2×#101C2×C12712yes—
72#356C2×C12714yesdirect product factor
73C22×Q8C4×C12711yescharacteristic
74C22×Q8C22×C12711yescharacteristic
75C2×M16—22no—
76C4⋊Q8—22no—
77C4⋊Q8C4×C12712yes—
78C4⋊Q8C22×C12712yes—
79C2×C4⋊C4—22no—
80C2×C42C4×C12711yescharacteristic
81#101—22no—
82#101C22×C12714yes—
83C2×Q8—21no—
84C2×Q8—41no—
85C2×Q8—41no—
86C2×Q8—41no—
87C2×Q8—41no—
88C2×Q8C2×C4×C12712yes—
89C2×Q8C2×C4×C12711yescharacteristic
90C2×Q8D4×C12712yes—
91C2×Q8C23×C12711yescharacteristic
92C22×C4—21no—
93C22×C4—21no—
94C22×C4C2×C4×C12711yescharacteristic
95C22×C4D4×C12712yes—
96M16—24no—
97M16—42no—
98C2×C8—42no—
99C4⋊C4—24no—
100C4⋊C4—24no—
101C42—21no—
102C42C2×C4×C12712yes—
103C23C22⋊C4×C12711yescharacteristic
104Q8—41no—
105Q8—42no—
106Q8—41no—
107Q8—42no—
108Q8—42no—
109C2×C4—24no—
110C2×C4—22no—
111C2×C4—22no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—22no—
115C2×C4—22no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4C22⋊C4×C12711yescharacteristic
119C2×C4C22⋊C4×C12712yes—
120C2×C4C22×C4×C12711yescharacteristic
121C2×C4C2×D4×C12712yes—
122C8—44no—
123C22—22no—
124C22#99×C12711yescharacteristic
125C22#99×C12711yescharacteristic
126C22C2×C22⋊C4×C12711yescharacteristic
127C4—24no—
128C4—22no—
129C4—21no—
130C4—22no—
131C4—21no—
132C4—42no—
133C2—21no—
134C2—21no—
135C2#356×C12712yesdirect product factor
136C2C2×#99×C12711yescharacteristic
subgroupG/H
center, socleC22×C127#99
commutatorC2×C4C22×C4×C127
Frattini, cosocleC2×Q8C23×C127
prime radicalC23×C127C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C127 ◁ C22×Q8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. #356×C127C2direct (N × H)
66. C2×C127#356direct (N × H)
68. C2×#356C127direct (N × H)
69. C127C2×#356direct (N × H)
72. #356C2×C127direct (N × H)
135. C2#356×C127direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4⋊Q8×C1272
C2×#101×C1272
#356×C1272
C2×#356127