Finite Group Databasebeta

← C2×#451×C127 #30635951

Subgroups — C2×#451×C127 #30635951

Order 16256.

740 conjugacy classes of subgroups; only the 248 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 248 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
1C23×Q8×C127C211yesmaximal, characteristic
2C22×Q16×C127C212yesmaximal, complemented
3C2×C22⋊Q8×C127C211yesmaximal, characteristic
4#451×C127C218yesmaximal, direct product factor
5C2×#103×C127C212yesmaximal, complemented
6C2×C22⋊C8×C127C211yesmaximal, characteristic
7C22×Q8×C127—28no—
8C22×Q8×C127—22no—
9C22×Q8×C127—21no—
10C22×Q8×C127C2211yescharacteristic
11C22×Q8×C127C2214yes—
12C22×Q8×C127C2212yes—
13C23×C4×C127—21no—
14C23×C4×C127C2211yescharacteristic
15C2×Q16×C127—28no—
16C2×Q16×C127C2218yescomplemented
17C22×C8×C127C2212yes—
18C22⋊Q8×C127—22no—
19C22⋊Q8×C127C2214yes—
20C2×C4⋊C4×C127—21no—
21C2×C4⋊C4×C127C2211yescharacteristic
22C2×C22⋊C4×C127—21no—
23#103×C127C2218yescomplemented
24C22⋊C8×C127C2214yes—
25C24×C127D411yescharacteristic
26C2×Q8×C127—216no—
27C2×Q8×C127—28no—
28C2×Q8×C127—216no—
29C2×Q8×C127—24no—
30C2×Q8×C127—24no—
31C2×Q8×C127—22no—
32C2×Q8×C127—21no—
33C2×Q8×C127—24no—
34C2×Q8×C127D418yes—
35C2×Q8×C127C2312yes—
36C2×Q8×C127C2314yes—
37C22×C4×C127—28no—
38C22×C4×C127—24no—
39C22×C4×C127—22no—
40C22×C4×C127—24no—
41C22×C4×C127—21no—
42C22×C4×C127—21no—
43C22×C4×C127D412yes—
44C22×C4×C127D411yescharacteristic
45C22×C4×C127C2312yes—
46C22×C4×C127C2311yescharacteristic
47Q16×C127—216no—
48C2×C8×C127—24no—
49C2×C8×C127C2314yes—
50C4⋊C4×C127—24no—
51C4⋊C4×C127—21no—
52C4⋊C4×C127C2312yes—
53C22⋊C4×C127—24no—
54C23×C127—22no—
55C23×C127—22no—
56C23×C127Q1614yescomplemented
57C23×C127C2×D412yes—
58C23×C127C2×D411yescharacteristic
59Q8×C127—216no—
60Q8×C127—28no—
61Q8×C127—24no—
62Q8×C127—24no—
63Q8×C127—42no—
64Q8×C127C2×D418yes—
65C2×C4×C127—216no—
66C2×C4×C127—28no—
67C2×C4×C127—22no—
68C2×C4×C127—24no—
69C2×C4×C127—24no—
70C2×C4×C127—24no—
71C2×C4×C127—22no—
72C2×C4×C127—22no—
73C2×C4×C127—42no—
74C2×C4×C127—42no—
75C2×C4×C127C2×D412yes—
76C2×C4×C127C2×D411yescharacteristic
77C2×C4×C127C2×D414yes—
78C2×C4×C127C2411yescharacteristic
79C8×C127—24no—
80C22×C127—22no—
81C22×C127—22no—
82C22×C127—28no—
83C22×C127#12012yes—
84C22×C127C2×Q1612yes—
85C22×C127C2×Q1614yescomplemented
86C22×C127#13712yes—
87C22×C127C22×D411yescharacteristic
88C4×C127—28no—
89C4×C127—22no—
90C4×C127—42no—
91C4×C127—42no—
92C4×C127#12012yes—
93C4×C127C22×D412yes—
94C2×C127—24no—
95C2×C127#45114yesdirect product factor
96C2×C127C2×#12011yescharacteristic
97C2×C127C22×Q1611yescharacteristic
98C2×C127C2×#13711yescharacteristic
99C2×#451C12711yesmaximal, characteristic, direct product factor, Sylow
100C127C2×#45111yescharacteristic, direct product factor, Sylow
101C23×Q8C2×C12711yescharacteristic
102C22×Q16C2×C12712yescomplemented
103C2×C22⋊Q8C2×C12711yescharacteristic
104#451C2×C12718yesdirect product factor
105C2×#103C2×C12712yescomplemented
106C2×C22⋊C8C2×C12711yescharacteristic
107C22×Q8—28no—
108C22×Q8—21no—
109C22×Q8—22no—
110C22×Q8C22×C12711yescharacteristic
111C22×Q8C22×C12714yes—
112C22×Q8C22×C12712yes—
113C23×C4—21no—
114C23×C4C22×C12711yescharacteristic
115C2×Q16—28no—
116C2×Q16C22×C12718yescomplemented
117C22×C8C22×C12712yes—
118C22⋊Q8—22no—
119C22⋊Q8C22×C12714yes—
120C2×C4⋊C4—21no—
121C2×C4⋊C4C22×C12711yescharacteristic
122C2×C22⋊C4—21no—
123#103C22×C12718yescomplemented
124C22⋊C8C22×C12714yes—
125C24D4×C12711yescharacteristic
126C2×Q8—216no—
127C2×Q8—28no—
128C2×Q8—216no—
129C2×Q8—24no—
130C2×Q8—24no—
131C2×Q8—22no—
132C2×Q8—21no—
133C2×Q8—24no—
134C2×Q8D4×C12718yes—
135C2×Q8C23×C12712yes—
136C2×Q8C23×C12714yes—
137C22×C4—28no—
138C22×C4—24no—
139C22×C4—22no—
140C22×C4—24no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4D4×C12711yescharacteristic
144C22×C4D4×C12712yes—
145C22×C4C23×C12712yes—
146C22×C4C23×C12711yescharacteristic
147Q16—216no—
148C2×C8—24no—
149C2×C8C23×C12714yes—
150C4⋊C4—24no—
151C4⋊C4—21no—
152C4⋊C4C23×C12712yes—
153C22⋊C4—24no—
154C23—22no—
155C23—22no—
156C23Q16×C12714yescomplemented
157C23C2×D4×C12712yes—
158C23C2×D4×C12711yescharacteristic
159Q8—216no—
160Q8—28no—
161Q8—24no—
162Q8—24no—
163Q8—42no—
164Q8C2×D4×C12718yes—
165C2×C4—216no—
166C2×C4—28no—
167C2×C4—22no—
168C2×C4—24no—
169C2×C4—24no—
170C2×C4—24no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4C2×D4×C12712yes—
176C2×C4C2×D4×C12711yescharacteristic
177C2×C4C2×D4×C12714yes—
178C2×C4C24×C12711yescharacteristic
179C8—24no—
180C22—22no—
181C22—22no—
182C22—28no—
183C22#120×C12712yes—
184C22C2×Q16×C12712yes—
185C22C2×Q16×C12714yescomplemented
186C22#137×C12712yes—
187C22C22×D4×C12711yescharacteristic
188C4—28no—
189C4—22no—
190C4—42no—
191C4—42no—
192C4#120×C12712yes—
193C4C22×D4×C12712yes—
194C2—24no—
195C2#451×C12714yesdirect product factor
196C2C2×#120×C12711yescharacteristic
197C2C22×Q16×C12711yescharacteristic
198C2C2×#137×C12711yescharacteristic
subgroupG/H
center, socleC23×C127C2×D4
commutator, Frattini, cosocleC2×C4C24×C127
prime radicalC24×C127D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×Q16×C127C2semidirect
4. #451×C127C2direct (N × H)
5. C2×#103×C127C2semidirect
16. C2×Q16×C127C22semidirect
23. #103×C127C22semidirect
56. C23×C127Q16semidirect
85. C22×C127C2×Q16semidirect
95. C2×C127#451direct (N × H)
99. C2×#451C127direct (N × H)
100. C127C2×#451direct (N × H)
102. C22×Q16C2×C127semidirect
104. #451C2×C127direct (N × H)
105. C2×#103C2×C127semidirect
116. C2×Q16C22×C127semidirect
123. #103C22×C127semidirect
156. C23Q16×C127semidirect
185. C22C2×Q16×C127semidirect
195. C2#451×C127direct (N × H)

Maximal subgroups

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

subgroupindex
C23×Q8×C1272
C22×Q16×C1272
C2×C22⋊Q8×C1272
#451×C1272
C2×#103×C1272
C2×C22⋊C8×C1272
C2×#451127