Finite Group Databasebeta

← C2×#22550129 #30648326

Subgroups — C2×#22550129 #30648326

Order 16256.

596 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 165 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×D1016C211yesmaximal, characteristic, complemented
2C2×#16861183C212yesmaximal, complemented
3#22550129C218yesmaximal, direct product factor
4C2×#16861141C211yesmaximal, characteristic, complemented
5C2×#16861131C212yesmaximal, complemented
6C22×M16×C127C211yesmaximal, characteristic, complemented
7C22×D508C2212yes—
8C22×C22⋊D127—22no—
9C2×D1016—24no—
10C2×D1016C2214yescomplemented
11#16861183—24no—
12#16861183C2218yescomplemented
13C2×C4⋊Dic127C2212yes—
14C2×(C2×D127)⋊C4—22no—
15#16861141C2214yescomplemented
16#16861131C2218yescomplemented
17C23×C4×C127C2211yescharacteristic
18C2×M16×C127—24no—
19C2×M16×C127C2214yescomplemented
20C22×C8×C127C2211yescharacteristic
21C22×C8×C127C2211yescharacteristic
22C23×D127—22no—
23C2×D508—24no—
24C2×D508—22no—
25C2×D508—24no—
26C2×D508C2314yes—
27C2×C22⋊D127—28no—
28C2×C22⋊D127—48no—
29C22×Dic127—22no—
30D1016—28no—
31C4⋊Dic127—22no—
32C4⋊Dic127C2314yes—
33(C2×D127)⋊C4—28no—
34C24×C127D411yescharacteristic
35C22×C4×C127—22no—
36C22×C4×C127—22no—
37C22×C4×C127D411yescharacteristic
38C22×C4×C127D412yes—
39C22×C4×C127C2312yes—
40C22×C4×C127C2311yescharacteristic
41M16×C127—28no—
42C2×C8×C127—22no—
43C2×C8×C127D414yescomplemented
44C2×C8×C127C2312yes—
45C2×C8×C127C2312yes—
46C22×D127—24no—
47C22×D127—44no—
48C22×D127—44no—
49C22×D127—44no—
50D508—24no—
51D508—24no—
52D508—44no—
53C22⋊D127—416no—
54C2×Dic127—24no—
55C2×Dic127—44no—
56C23×C127—22no—
57C23×C127—24no—
58C23×C127C2×D412yes—
59C23×C127C2×D411yescharacteristic
60C2×C4×C127—22no—
61C2×C4×C127—24no—
62C2×C4×C127—22no—
63C2×C4×C127—22no—
64C2×C4×C127C2×D412yes—
65C2×C4×C127C2×D412yes—
66C2×C4×C127C2×D411yescharacteristic
67C2×C4×C127#4112yes—
68C2×C4×C127C2411yescharacteristic
69C8×C127—22no—
70C8×C127C2×D414yescomplemented
71C22×C127—22no—
72C22×C127—24no—
73C22×C127—44no—
74C22×C127#12112yes—
75C22×C127AGL(1,ℤ/8)14yes—
76C22×C127C22×D411yescharacteristic
77C2×D127—44no—
78C2×D127—44no—
79C2×D127—44no—
80C2×D127—88no—
81C4×C127—22no—
82C4×C127#12112yes—
83C4×C127C22×D411yescharacteristic
84C4×C127C2×#4111yescharacteristic
85Dic127—44no—
86C2×C127—42no—
87C2×C127#46914yescomplemented
88C2×C127C2×#12111yescharacteristic
89C2×C127C2×AGL(1,ℤ/8)12yes—
90D127—84no—
91C2×#469—1271nomaximal, Sylow
92C127C2×#46911yescharacteristic, complemented, Sylow
93C22×D8—1271no—
94C22×M16D12711yescharacteristic, complemented
95C2×#121—1272no—
96#469—1278no—
97C2×#106—1271no—
98C2×#102—1272no—
99C22×D4—1272no—
100C22×D4—2542no—
101C23×C4C2×D12711yescharacteristic
102C2×D8—1274no—
103C2×D8—2544no—
104C2×M16—24no—
105C2×M16C2×D12714yescomplemented
106C22×C8C2×D12711yescharacteristic
107C22×C8C2×D12711yescharacteristic
108#121—1278no—
109#121—2544no—
110C2×C4⋊C4—1272no—
111C2×C22⋊C4—2542no—
112#106—1274no—
113#102—1278no—
114C24—2542no—
115C24D50811yescharacteristic
116C2×D4—1274no—
117C2×D4—5088no—
118C2×D4—2548no—
119C2×D4—2544no—
120C2×D4—2542no—
121C2×D4—2544no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—2542no—
125C22×C4D50812yes—
126C22×C4D50811yescharacteristic
127C22×C4C22×D12712yes—
128C22×C4C22×D12711yescharacteristic
129D8—2548no—
130M16—28no—
131C2×C8—22no—
132C2×C8C22⋊D12714yescomplemented
133C2×C8C22×D12712yes—
134C2×C8C22×D12712yes—
135C4⋊C4—1274no—
136C4⋊C4—2542no—
137C22⋊C4—2548no—
138C23—24no—
139C23—22no—
140C23—5084no—
141C23—5084no—
142C23—5084no—
143C23—2544no—
144C23C2×D50812yes—
145C23C2×D50811yescharacteristic
146D4—50816no—
147D4—5084no—
148D4—2544no—
149D4—2544no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—24no—
154C2×C4—5084no—
155C2×C4—2544no—
156C2×C4C2×C22⋊D12712yes—
157C2×C4C2×D50812yes—
158C2×C4C2×D50811yescharacteristic
159C2×C4#1452324812yes—
160C2×C4C23×D12711yescharacteristic
161C8—22no—
162C8C2×C22⋊D12714yescomplemented
163C22—22no—
164C22—24no—
165C22—44no—
166C22—10168no—
167C22—5084no—
168C22—5084no—
169C22—5084no—
170C22#1686118312yes—
171C22#1686123714yes—
172C22C22×D50811yescharacteristic
173C4—22no—
174C4—5084no—
175C4#1686118312yes—
176C4C22×C22⋊D12711yescharacteristic
177C4C2×#1452324811yescharacteristic
178C2—42no—
179C2—10164no—
180C2#2255012914yesdirect product factor
181C2C2×#1686118311yescharacteristic
182C2C2×#1686123712yes—
subgroupG/H
centerC23C2×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC22×M16×C127C2
socleC23×C127C2×D4
hypercenterC22×M16D127
hypocenterC127C2×#469
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
1. C22×D1016C2semidirect
2. C2×#16861183C2semidirect
3. #22550129C2direct (N × H)
4. C2×#16861141C2semidirect
5. C2×#16861131C2semidirect
6. C22×M16×C127C2semidirect
10. C2×D1016C22semidirect
12. #16861183C22semidirect
15. #16861141C22semidirect
16. #16861131C22semidirect
19. C2×M16×C127C22semidirect
43. C2×C8×C127D4semidirect
70. C8×C127C2×D4semidirect
87. C2×C127#469semidirect
92. C127C2×#469semidirect
94. C22×M16D127semidirect
105. C2×M16C2×D127semidirect
132. C2×C8C22⋊D127semidirect
162. C8C2×C22⋊D127semidirect
180. C2#22550129direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D10162
C2×#168611832
#225501292
C2×#168611412
C2×#168611312
C22×M16×C1272
C2×#469127