Finite Group Databasebeta

← C127×#1800 #30634752

Subgroups — C127×#1800 #30634752

Order 16256.

336 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 124 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×D4×C127C211yesmaximal, characteristic
2#423×C127C212yesmaximal, complemented
3C2×#100×C127C212yesmaximal
4#343×C127C212yesmaximal, complemented
5C22×D4×C127C2211yescharacteristic
6C23×C4×C127C412yes—
7C2×M16×C127C2214yes—
8C4×D4×C127—22no—
9C4×D4×C127—22no—
10C2×C4⋊C4×C127C2211yescharacteristic
11C2×C22⋊C4×C127C412yes—
12C2×C42×C127C2211yescharacteristic
13C4⋊C8×C127—22no—
14C4⋊C8×C127—22no—
15#100×C127—24no—
16C24×C127C2×C412yes—
17C2×D4×C127—22no—
18C2×D4×C127—22no—
19C2×D4×C127D412yes—
20C2×D4×C127Q812yes—
21C22×C4×C127—22no—
22C22×C4×C127—22no—
23C22×C4×C127—44no—
24C22×C4×C127C2×C411yescharacteristic
25C22×C4×C127C2×C412yes—
26C22×C4×C127C2×C411yescharacteristic
27C22×C4×C127C2311yescharacteristic
28M16×C127—24no—
29M16×C127—24no—
30C2×C8×C127—24no—
31C4⋊C4×C127—22no—
32C22⋊C4×C127—24no—
33C42×C127D412yes—
34C42×C127D412yes—
35C23×C127—22no—
36C23×C127—22no—
37C23×C127—42no—
38C23×C127—42no—
39C23×C127C4⋊C414yes—
40C23×C127C22×C411yescharacteristic
41D4×C127—24no—
42D4×C127—24no—
43C2×C4×C127—22no—
44C2×C4×C127—22no—
45C2×C4×C127—41no—
46C2×C4×C127—44no—
47C2×C4×C127—44no—
48C2×C4×C127—42no—
49C2×C4×C127C22⋊C412yes—
50C2×C4×C127C22⋊C411yescharacteristic
51C2×C4×C127C22⋊C411yescharacteristic
52C2×C4×C127C2×D411yescharacteristic
53C2×C4×C127C2×D412yes—
54C2×C4×C127C2×Q811yescharacteristic
55C2×C4×C127#4112yes—
56C8×C127—44no—
57C22×C127—21no—
58C22×C127—21no—
59C22×C127—24no—
60C22×C127—44no—
61C22×C127—44no—
62C22×C127—42no—
63C22×C127—42no—
64C22×C127C2×C22⋊C411yescharacteristic
65C22×C127C2×C4⋊C411yescharacteristic
66C22×C127#11711yescharacteristic
67C4×C127—21no—
68C4×C127—22no—
69C4×C127—42no—
70C4×C127#10012yescomplemented
71C4×C127#12112yes—
72C4×C127C22⋊Q812yes—
73C2×C127—21no—
74C2×C127—21no—
75C2×C127—44no—
76C2×C127#38011yescharacteristic
77C2×C127C2×#10011yescharacteristic
78C2×C127#41311yescharacteristic
79#1800C12711yesmaximal, characteristic, direct product factor, Sylow
80C127#180011yescharacteristic, direct product factor, Sylow
81C2×C4×D4C2×C12711yescharacteristic
82#423C2×C12712yescomplemented
83C2×#100C2×C12712yes—
84#343C2×C12712yescomplemented
85C22×D4C22×C12711yescharacteristic
86C23×C4C4×C12712yes—
87C2×M16C22×C12714yes—
88C4×D4—22no—
89C4×D4—22no—
90C2×C4⋊C4C22×C12711yescharacteristic
91C2×C22⋊C4C4×C12712yes—
92C2×C42C22×C12711yescharacteristic
93C4⋊C8—22no—
94C4⋊C8—22no—
95#100—24no—
96C24C2×C4×C12712yes—
97C2×D4—22no—
98C2×D4—22no—
99C2×D4D4×C12712yes—
100C2×D4Q8×C12712yes—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—44no—
104C22×C4C2×C4×C12711yescharacteristic
105C22×C4C2×C4×C12712yes—
106C22×C4C2×C4×C12711yescharacteristic
107C22×C4C23×C12711yescharacteristic
108M16—24no—
109M16—24no—
110C2×C8—24no—
111C4⋊C4—22no—
112C22⋊C4—24no—
113C42D4×C12712yes—
114C42D4×C12712yes—
115C23—22no—
116C23—22no—
117C23—42no—
118C23—42no—
119C23C4⋊C4×C12714yes—
120C23C22×C4×C12711yescharacteristic
121D4—24no—
122D4—24no—
123C2×C4—22no—
124C2×C4—22no—
125C2×C4—41no—
126C2×C4—44no—
127C2×C4—44no—
128C2×C4—42no—
129C2×C4C22⋊C4×C12712yes—
130C2×C4C22⋊C4×C12711yescharacteristic
131C2×C4C22⋊C4×C12711yescharacteristic
132C2×C4C2×D4×C12711yescharacteristic
133C2×C4C2×D4×C12712yes—
134C2×C4C2×Q8×C12711yescharacteristic
135C2×C4#41×C12712yes—
136C8—44no—
137C22—21no—
138C22—21no—
139C22—24no—
140C22—44no—
141C22—44no—
142C22—42no—
143C22—42no—
144C22C2×C22⋊C4×C12711yescharacteristic
145C22C2×C4⋊C4×C12711yescharacteristic
146C22#117×C12711yescharacteristic
147C4—21no—
148C4—22no—
149C4—42no—
150C4#100×C12712yescomplemented
151C4#121×C12712yes—
152C4C22⋊Q8×C12712yes—
153C2—21no—
154C2—21no—
155C2—44no—
156C2#380×C12711yescharacteristic
157C2C2×#100×C12711yescharacteristic
158C2#413×C12711yescharacteristic
subgroupG/H
center, socleC22×C127C2×C22⋊C4
commutatorC23C22×C4×C127
Frattini, cosocleC22×C4C23×C127
prime radicalC22×D4×C127C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. #423×C127C2semidirect
4. #343×C127C2semidirect
70. C4×C127#100semidirect
79. #1800C127direct (N × H)
80. C127#1800direct (N × H)
82. #423C2×C127semidirect
84. #343C2×C127semidirect
150. C4#100×C127semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D4×C1272
#423×C1272
C2×#100×C1272
#343×C1272
#1800127