Finite Group Databasebeta

← C2×#22550032 #30647924

Subgroups — C2×#22550032 #30647924

Order 16256.

740 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 197 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
1C22×#14523238C212yesmaximal, complemented
2C2×#16861188C211yesmaximal, characteristic
3#22550032C218yesmaximal, direct product factor
4C2×#16861133C212yesmaximal, complemented
5C2×C22⋊(C127⋊C8)C211yesmaximal, characteristic
6C23×Q8×C127C211yesmaximal, characteristic
7C22×Dic254C2211yescharacteristic
8C2×#14523238—28no—
9C2×#14523238C2218yescomplemented
10C22×C127⋊C8C2212yes—
11#16861188—22no—
12#16861188C2214yes—
13C2×C4⋊Dic127C2211yescharacteristic
14C2×Dic127⋊C4—21no—
15C2×C22⋊Dic127—21no—
16#16861133C2218yescomplemented
17C22⋊(C127⋊C8)C2214yes—
18C22×Q8×C127—28no—
19C22×Q8×C127—22no—
20C22×Q8×C127—21no—
21C22×Q8×C127C2214yes—
22C22×Q8×C127C2212yes—
23C23×C4×C127—21no—
24C23×C4×C127C2211yescharacteristic
25C2×Dic254—21no—
26C2×Dic254—24no—
27C2×Dic254C2312yes—
28C22×Dic127—21no—
29C22×Dic127—21no—
30#14523238—216no—
31C2×C127⋊C8—24no—
32C2×C127⋊C8C2314yes—
33C4⋊Dic127—21no—
34C4⋊Dic127C2312yes—
35Dic127⋊C4—24no—
36C22⋊Dic127—24no—
37C24×C127D411yescharacteristic
38C2×Q8×C127—216no—
39C2×Q8×C127—28no—
40C2×Q8×C127—216no—
41C2×Q8×C127—24no—
42C2×Q8×C127—24no—
43C2×Q8×C127—22no—
44C2×Q8×C127D418yes—
45C2×Q8×C127C2314yes—
46C22×C4×C127—28no—
47C22×C4×C127—24no—
48C22×C4×C127—22no—
49C22×C4×C127—24no—
50C22×C4×C127D412yes—
51C22×C4×C127D411yescharacteristic
52C22×C4×C127C2312yes—
53C22×C4×C127C2311yescharacteristic
54Dic254—24no—
55Dic254—42no—
56C2×Dic127—22no—
57C2×Dic127—22no—
58C2×Dic127—42no—
59C2×Dic127—42no—
60C127⋊C8—24no—
61C23×C127—22no—
62C23×C127—22no—
63C23×C127Q1614yescomplemented
64C23×C127C2×D412yes—
65C23×C127C2×D411yescharacteristic
66Q8×C127—216no—
67Q8×C127—28no—
68Q8×C127—24no—
69Q8×C127C2×D418yes—
70C2×C4×C127—216no—
71C2×C4×C127—28no—
72C2×C4×C127—22no—
73C2×C4×C127—24no—
74C2×C4×C127—24no—
75C2×C4×C127—24no—
76C2×C4×C127C2×D412yes—
77C2×C4×C127C2×D411yescharacteristic
78C2×C4×C127C2×D414yes—
79C2×C4×C127C2411yescharacteristic
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#12012yes—
91C4×C127C22×D412yes—
92Dic127—42no—
93Dic127—42no—
94C2×C127—24no—
95C2×C127#45114yescomplemented
96C2×C127C2×#12011yescharacteristic
97C2×C127C22×Q1611yescharacteristic
98C2×C127C2×#13711yescharacteristic
99C2×#451—1271nomaximal, Sylow
100C127C2×#45111yescharacteristic, complemented, Sylow
101C23×Q8D12711yescharacteristic
102C22×Q16—1272no—
103C2×C22⋊Q8—1271no—
104#451—1278no—
105C2×#103—1272no—
106C2×C22⋊C8—1271no—
107C22×Q8—28no—
108C22×Q8—22no—
109C22×Q8—21no—
110C22×Q8—1271no—
111C22×Q8C2×D12714yes—
112C22×Q8C2×D12712yes—
113C23×C4—21no—
114C23×C4C2×D12711yescharacteristic
115C2×Q16—1278no—
116C2×Q16—2548no—
117C22×C8—1272no—
118C22⋊Q8—1274no—
119C22⋊Q8—2542no—
120C2×C4⋊C4—1271no—
121C2×C4⋊C4—2541no—
122C2×C22⋊C4—2541no—
123#103—1278no—
124C22⋊C8—1274no—
125C24C22⋊D12711yescharacteristic
126C2×Q8—216no—
127C2×Q8—28no—
128C2×Q8—216no—
129C2×Q8—24no—
130C2×Q8—22no—
131C2×Q8—24no—
132C2×Q8—1272no—
133C2×Q8—2544no—
134C2×Q8—2541no—
135C2×Q8C22⋊D12718yes—
136C2×Q8C22×D12714yes—
137C22×C4—28no—
138C22×C4—24no—
139C22×C4—22no—
140C22×C4—24no—
141C22×C4—2541no—
142C22×C4—2541no—
143C22×C4C22⋊D12712yes—
144C22×C4C22⋊D12711yescharacteristic
145C22×C4C22×D12712yes—
146C22×C4C22×D12711yescharacteristic
147Q16—25416no—
148C2×C8—1274no—
149C2×C8—2544no—
150C4⋊C4—1272no—
151C4⋊C4—2544no—
152C4⋊C4—2541no—
153C22⋊C4—2544no—
154C23—22no—
155C23—22no—
156C23#1452323814yescomplemented
157C23C2×C22⋊D12712yes—
158C23C2×C22⋊D12711yescharacteristic
159Q8—216no—
160Q8—24no—
161Q8—28no—
162Q8—5082no—
163Q8—2544no—
164Q8C2×C22⋊D12718yes—
165C2×C4—216no—
166C2×C4—24no—
167C2×C4—28no—
168C2×C4—22no—
169C2×C4—24no—
170C2×C4—24no—
171C2×C4—5082no—
172C2×C4—5082no—
173C2×C4—2542no—
174C2×C4—2542no—
175C2×C4C2×C22⋊D12712yes—
176C2×C4C2×C22⋊D12714yes—
177C2×C4C2×C22⋊D12711yescharacteristic
178C2×C4C23×D12711yescharacteristic
179C8—2544no—
180C22—28no—
181C22—22no—
182C22—22no—
183C22#1686117812yes—
184C22C2×#1452323812yes—
185C22C2×#1452323814yescomplemented
186C22#1686124012yes—
187C22C22×C22⋊D12711yescharacteristic
188C4—28no—
189C4—22no—
190C4—5082no—
191C4—5082no—
192C4#1686117812yes—
193C4C22×C22⋊D12712yes—
194C2—24no—
195C2#2255003214yesdirect product factor
196C2C2×#1686117811yescharacteristic
197C2C22×#1452323811yescharacteristic
198C2C2×#1686124011yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC23×Q8×C127C2
socleC23×C127C2×D4
prime radicalC24×C127D4
hypercenterC23×Q8D127
hypocenterC127C2×#451
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
1. C22×#14523238C2semidirect
3. #22550032C2direct (N × H)
4. C2×#16861133C2semidirect
9. C2×#14523238C22semidirect
16. #16861133C22semidirect
63. C23×C127Q16semidirect
85. C22×C127C2×Q16semidirect
95. C2×C127#451semidirect
100. C127C2×#451semidirect
156. C23#14523238semidirect
185. C22C2×#14523238semidirect
195. C2#22550032direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145232382
C2×#168611882
#225500322
C2×#168611332
C2×C22⋊(C127⋊C8)2
C23×Q8×C1272
C2×#451127