Finite Group Databasebeta

← C2×#22550126 #30648323

Subgroups — C2×#22550126 #30648323

Order 16256.

596 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 157 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×#14523231C211yesmaximal, characteristic, complemented
2C2×#16861180C212yesmaximal, complemented
3#22550126C218yesmaximal, direct product factor
4C2×#16861141C211yesmaximal, characteristic, complemented
5C2×#16861130C212yesmaximal, complemented
6C22×D8×C127C211yesmaximal, characteristic, complemented
7C22×C22⋊D127—22no—
8C22×C4×D127C2211yescharacteristic
9C2×#14523231—24no—
10C2×#14523231C2214yescomplemented
11C22×C127⋊C8C2211yescharacteristic
12#16861180—24no—
13#16861180C2218yescomplemented
14C2×C4⋊Dic127C2212yes—
15C2×C22⋊Dic127—22no—
16#16861141C2214yescomplemented
17#16861130C2218yescomplemented
18C22×D4×C127C2212yes—
19C2×D8×C127—24no—
20C2×D8×C127C2214yescomplemented
21C22×C8×C127C2211yescharacteristic
22C23×D127D411yescharacteristic
23C2×C22⋊D127—28no—
24C2×C22⋊D127—48no—
25C22×Dic127—22no—
26C22×Dic127D411yescharacteristic
27C2×C4×D127—22no—
28C2×C4×D127—22no—
29C2×C4×D127D412yes—
30C2×C4×D127C2312yes—
31#14523231—28no—
32C2×C127⋊C8—22no—
33C2×C127⋊C8C2312yes—
34C4⋊Dic127—22no—
35C4⋊Dic127C2314yes—
36C22⋊Dic127—28no—
37C24×C127—22no—
38C2×D4×C127—24no—
39C2×D4×C127—22no—
40C2×D4×C127—24no—
41C2×D4×C127C2314yes—
42C22×C4×C127C2311yescharacteristic
43D8×C127—28no—
44C2×C8×C127D414yescomplemented
45C2×C8×C127C2312yes—
46C22×D127—22no—
47C22×D127—24no—
48C22×D127C2×D412yes—
49C22⋊D127—416no—
50C2×Dic127—22no—
51C2×Dic127—24no—
52C2×Dic127—44no—
53C2×Dic127C2×D412yes—
54C4×D127—24no—
55C4×D127—22no—
56C4×D127—22no—
57C127⋊C8—22no—
58C23×C127—24no—
59C23×C127—44no—
60C23×C127—44no—
61C23×C127—44no—
62C23×C127C2×D411yescharacteristic
63D4×C127—24no—
64D4×C127—24no—
65D4×C127—44no—
66C2×C4×C127C2×D411yescharacteristic
67C2×C4×C127C2×D412yes—
68C2×C4×C127#4112yes—
69C2×C4×C127C2411yescharacteristic
70C8×C127C2×D414yescomplemented
71C22×C127—44no—
72C22×C127—44no—
73C22×C127—44no—
74C22×C127—88no—
75C22×C127#12112yes—
76C22×C127AGL(1,ℤ/8)14yes—
77C22×C127C22×D411yescharacteristic
78C2×D127—22no—
79C2×D127—24no—
80C2×D127—44no—
81C4×C127#12112yes—
82C4×C127C22×D411yescharacteristic
83C4×C127C2×#4111yescharacteristic
84Dic127—22no—
85Dic127—44no—
86C2×C127—84no—
87C2×C127#46914yescomplemented
88C2×C127C2×#12111yescharacteristic
89C2×C127C2×AGL(1,ℤ/8)12yes—
90D127—42no—
91C2×#469—1271nomaximal, Sylow
92C127C2×#46911yescharacteristic, complemented, Sylow
93C22×D8D12711yescharacteristic, complemented
94C22×M16—1271no—
95C2×#121—1272no—
96#469—1278no—
97C2×#106—1271no—
98C2×#102—1272no—
99C22×D4—2542no—
100C22×D4C2×D12712yes—
101C23×C4—1271no—
102C2×D8—24no—
103C2×D8C2×D12714yescomplemented
104C2×M16—1274no—
105C2×M16—2544no—
106C22×C8—1271no—
107C22×C8C2×D12711yescharacteristic
108#121—1278no—
109#121—2544no—
110C2×C4⋊C4—1272no—
111C2×C22⋊C4—2542no—
112#106—1274no—
113#102—1278no—
114C24—22no—
115C24—1271no—
116C2×D4—22no—
117C2×D4—24no—
118C2×D4—24no—
119C2×D4—5088no—
120C2×D4—2548no—
121C2×D4C22×D12714yes—
122C22×C4—1271no—
123C22×C4—1272no—
124C22×C4—1272no—
125C22×C4—2542no—
126C22×C4—2542no—
127C22×C4—2542no—
128C22×C4C22×D12711yescharacteristic
129D8—28no—
130M16—2548no—
131C2×C8—1272no—
132C2×C8—2542no—
133C2×C8C22⋊D12714yescomplemented
134C2×C8C22×D12712yes—
135C4⋊C4—1274no—
136C4⋊C4—2542no—
137C22⋊C4—2548no—
138C23—24no—
139C23—44no—
140C23—44no—
141C23—44no—
142C23—1272no—
143C23—2544no—
144C23—2542no—
145C23D4×D12711yescharacteristic
146D4—24no—
147D4—24no—
148D4—44no—
149D4—50816no—
150C2×C4—1272no—
151C2×C4—5084no—
152C2×C4—2544no—
153C2×C4—2542no—
154C2×C4—2542no—
155C2×C4—2542no—
156C2×C4—2544no—
157C2×C4D4×D12711yescharacteristic
158C2×C4C2×C22⋊D12712yes—
159C2×C4#1452324712yes—
160C2×C4C23×D12711yescharacteristic
161C8—2542no—
162C8C2×C22⋊D12714yescomplemented
163C22—44no—
164C22—44no—
165C22—44no—
166C22—88no—
167C22—5084no—
168C22—2542no—
169C22—2544no—
170C22#1686118012yes—
171C22#1686123314yes—
172C22C2×D4×D12711yescharacteristic
173C4—5084no—
174C4—2542no—
175C4#1686118012yes—
176C4C22×C22⋊D12711yescharacteristic
177C4C2×#1452324711yescharacteristic
178C2—84no—
179C2—5082no—
180C2#2255012614yesdirect product factor
181C2C2×#1686118011yescharacteristic
182C2C2×#1686123312yes—
subgroupG/H
centerC23D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC22×D8×C127C2
socleC23×C127C2×D4
hypercenterC22×D8D127
hypocenterC127C2×#469
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C22×#14523231C2semidirect
2. C2×#16861180C2semidirect
3. #22550126C2direct (N × H)
4. C2×#16861141C2semidirect
5. C2×#16861130C2semidirect
6. C22×D8×C127C2semidirect
10. C2×#14523231C22semidirect
13. #16861180C22semidirect
16. #16861141C22semidirect
17. #16861130C22semidirect
20. C2×D8×C127C22semidirect
44. C2×C8×C127D4semidirect
70. C8×C127C2×D4semidirect
87. C2×C127#469semidirect
92. C127C2×#469semidirect
93. C22×D8D127semidirect
103. C2×D8C2×D127semidirect
133. C2×C8C22⋊D127semidirect
162. C8C2×C22⋊D127semidirect
180. C2#22550126direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145232312
C2×#168611802
#225501262
C2×#168611412
C2×#168611302
C22×D8×C1272
C2×#469127