Finite Group Databasebeta

← C2×#22550066 #30648118

Subgroups — C2×#22550066 #30648118

Order 16256.

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

normal subgroups only

This group has 794 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
1C2×#16861190C213yesmaximal, complemented
2#22550066C218yesmaximal, direct product factor
3C2×#16861124C213yesmaximal, complemented
4C2×H(5,𝔽2)×C127C211yesmaximal, characteristic
5C23×Dic127C2211yescharacteristic, complemented
6#16861190—26no—
7#16861190C22112yescomplemented
8C2×Dic127⋊C4—23no—
9C2×C22⋊Dic127—23no—
10C2×C22⋊Dic127C2213yescomplemented
11#16861124C22112yescomplemented
12H(5,𝔽2)×C127—26no—
13H(5,𝔽2)×C127C2214yes—
14C2×#41×C127—23no—
15C22×D4×C127—23no—
16C22×D4×C127C2213yes—
17C22×Dic127—23no—
18C22×Dic127—23no—
19C22×Dic127—23no—
20C22×Dic127C2312yescomplemented
21Dic127⋊C4—212no—
22C22⋊Dic127—23no—
23C22⋊Dic127—212no—
24C22⋊Dic127C2316yescomplemented
25C24×C127—21no—
26C24×C127D413yes—
27C24×C127C2311yescharacteristic
28#41×C127—212no—
29#41×C127—212no—
30C2×Q8×C127—21no—
31C2×D4×C127—23no—
32C2×D4×C127—212no—
33C2×D4×C127—26no—
34C2×D4×C127—212no—
35C2×D4×C127—26no—
36C2×D4×C127D416yes—
37C2×D4×C127C2316yes—
38C22×C4×C127—23no—
39C22×C4×C127D413yes—
40C2×Dic127—26no—
41C2×Dic127—26no—
42C2×Dic127—46no—
43C2×Dic127—43no—
44C2×Dic127—41no—
45C23×C127—24no—
46C23×C127—23no—
47C23×C127—23no—
48C23×C127—42no—
49C23×C127—46no—
50C23×C127—82no—
51C23×C127C2×D413yes—
52C23×C127C2×D416yes—
53C23×C127C2×D413yes—
54C23×C127C2411yescharacteristic
55Q8×C127—24no—
56D4×C127—212no—
57D4×C127—212no—
58D4×C127—212no—
59C2×C4×C127—212no—
60C2×C4×C127—23no—
61C2×C4×C127—23no—
62C2×C4×C127C2×D416yes—
63C22×C127—26no—
64C22×C127—26no—
65C22×C127—43no—
66C22×C127—41no—
67C22×C127—46no—
68C22×C127—46no—
69C22×C127—83no—
70C22×C127—84no—
71C22×C127#12013yes—
72C22×C127#12011yescharacteristic
73C22×C127C22×D413yes—
74C4×C127—26no—
75Dic127—46no—
76Dic127—42no—
77C2×C127—23no—
78C2×C127—23no—
79C2×C127—46no—
80C2×C127#45812yescomplemented
81C2×C127C2×#12011yescharacteristic
82C2×#458—1271nomaximal, Sylow
83C127C2×#45811yescharacteristic, complemented, Sylow
84C2×H(5,𝔽2)D12711yescharacteristic
85C2×#123—1273no—
86#458—1278no—
87C2×#99—1273no—
88H(5,𝔽2)—26no—
89H(5,𝔽2)C2×D12714yes—
90C2×#41—23no—
91C22×D4—23no—
92C22×D4C2×D12713yes—
93C23×C4—1271no—
94#123—12712no—
95#123—2546no—
96C2×C4⋊C4—2543no—
97C2×C22⋊C4—1273no—
98C2×C22⋊C4—2543no—
99#99—12712no—
100C24—21no—
101C24C22⋊D12713yes—
102C24C22×D12711yescharacteristic
103#41—212no—
104#41—212no—
105C2×Q8—21no—
106C2×D4—23no—
107C2×D4—212no—
108C2×D4—26no—
109C2×D4—212no—
110C2×D4—26no—
111C2×D4C22⋊D12716yes—
112C2×D4C22×D12716yes—
113C22×C4—23no—
114C22×C4—1272no—
115C22×C4—2543no—
116C22×C4—2543no—
117C22×C4—2543no—
118C22×C4C22⋊D12713yes—
119C4⋊C4—25412no—
120C22⋊C4—1276no—
121C22⋊C4—2543no—
122C22⋊C4—25412no—
123C23—24no—
124C23—23no—
125C23—23no—
126C23—42no—
127C23—46no—
128C23—82no—
129C23C2×C22⋊D12713yes—
130C23C2×C22⋊D12716yes—
131C23C2×C22⋊D12713yes—
132C23C23×D12711yescharacteristic
133Q8—24no—
134D4—212no—
135D4—212no—
136D4—212no—
137C2×C4—212no—
138C2×C4—23no—
139C2×C4—23no—
140C2×C4—5086no—
141C2×C4—5083no—
142C2×C4—5081no—
143C2×C4—2546no—
144C2×C4—2546no—
145C2×C4C2×C22⋊D12716yes—
146C22—26no—
147C22—26no—
148C22—43no—
149C22—41no—
150C22—46no—
151C22—46no—
152C22—83no—
153C22—84no—
154C22#1686117813yes—
155C22#1686117811yescharacteristic
156C22C22×C22⋊D12713yes—
157C4—26no—
158C4—5086no—
159C4—5082no—
160C2—23no—
161C2—23no—
162C2—46no—
163C2#2255006612yesdirect product factor
164C2C2×#1686117811yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC23×C127C24
FrattiniC23C23×D127
Fitting, prime radicalC2×H(5,𝔽2)×C127C2
socleC22×C127#120
hypercenterC2×H(5,𝔽2)D127
hypocenterC127C2×#458
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C24 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#16861190C2semidirect
2. #22550066C2direct (N × H)
3. C2×#16861124C2semidirect
5. C23×Dic127C22semidirect
7. #16861190C22semidirect
10. C2×C22⋊Dic127C22semidirect
11. #16861124C22semidirect
20. C22×Dic127C23semidirect
24. C22⋊Dic127C23semidirect
80. C2×C127#458semidirect
83. C127C2×#458semidirect
163. C2#22550066direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168611902
#225500662
C2×#168611242
C2×H(5,𝔽2)×C1272
C2×#458127