Finite Group Databasebeta

← C22×C5×#126⋊C3 #14511327

Subgroups — C22×C5×#126⋊C3 #14511327

Order 1920.

600 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 106 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×C5×#126⋊C3C216yesmaximal, direct product factor
2C22×C5×C42⋊C3C211yesmaximal, characteristic, complemented
3C22×C5×#126C311yesmaximal, characteristic, complemented, Hall
4C23×C5×A4—41nomaximal
5C5×#126⋊C3C2214yesdirect product factor
6C2×C5×C42⋊C3C2213yescomplemented
7C22×#126⋊C3C511yesmaximal, characteristic, direct product factor, Hall
8C2×C5×#126—36no—
9C2×C5×#126C2×C316yescomplemented
10C22×C5×C4⋊C4—31no—
11C22×C5×C22⋊C4—31no—
12C22×C42×C5C2×C311yescharacteristic, complemented
13C22×C5×A4—41no—
14C22×C5×A4—46no—
15C5×C42⋊C3C2311yescharacteristic, complemented
16C2×#126⋊C3C2×C516yesdirect product factor
17C22×C42⋊C3C2×C511yescharacteristic, complemented
18C25×C5A411yescharacteristic
19C23×C4×C5—31no—
20C23×C4×C5—31no—
21C5×#126—38no—
22C5×#126—312no—
23C5×#126C22×C314yescomplemented
24C2×C5×C4⋊C4—36no—
25C2×C5×C4⋊C4—36no—
26C2×C5×C22⋊C4—36no—
27C2×C5×C22⋊C4—36no—
28C2×C42×C5—33no—
29C2×C42×C5C22×C313yescomplemented
30C22×#126C3×C511yescharacteristic, complemented, Sylow
31C23×C3×C5—161no—
32C2×C5×A4—43no—
33C2×C5×A4—44no—
34C23×A4—41no—
35#126⋊C3C22×C514yesdirect product factor
36C2×C42⋊C3C22×C513yescomplemented
37C24×C5—61no—
38C24×C5—63no—
39C24×C5C2×A411yescharacteristic
40C24×C5C2×A416yes—
41C22×C4×C5—33no—
42C22×C4×C5—33no—
43C22×C4×C5—36no—
44C22×C4×C5—63no—
45C22×C4×C5—63no—
46C22×C4×C5—61no—
47C22×C4×C5—61no—
48C5×C4⋊C4—34no—
49C5×C4⋊C4—312no—
50C5×C22⋊C4—312no—
51C5×C22⋊C4—34no—
52C42×C5—33no—
53C42×C5—32no—
54C42×C5C23×C311yescharacteristic, complemented
55C2×#126—36no—
56C2×#126C2×C3×C516yescomplemented
57C22×C4⋊C4—31no—
58C22×C22⋊C4—31no—
59C22×C42C2×C3×C511yescharacteristic, complemented
60C22×C3×C5—166no—
61C22×C3×C5—161no—
62C5×A4—41no—
63C22×A4—41no—
64C22×A4—46no—
65C42⋊C3C23×C511yescharacteristic, complemented
66C23×C5—33no—
67C23×C5—31no—
68C23×C5—41no—
69C23×C5—66no—
70C23×C5—66no—
71C23×C5—123no—
72C23×C5—122no—
73C23×C5C22×A413yes—
74C23×C5C22×A414yes—
75C2×C4×C5—31no—
76C2×C4×C5—33no—
77C2×C4×C5—34no—
78C2×C4×C5—63no—
79C2×C4×C5—63no—
80C2×C4×C5—66no—
81C2×C4×C5—63no—
82C2×C4×C5—63no—
83C2×C4×C5—66no—
84C25C5×A411yescharacteristic
85C23×C4—31no—
86C23×C4—31no—
87#126—38no—
88#126—312no—
89#126C22×C3×C514yescomplemented
90C2×C4⋊C4—36no—
91C2×C4⋊C4—36no—
92C2×C22⋊C4—36no—
93C2×C22⋊C4—36no—
94C2×C42—33no—
95C2×C42C22×C3×C513yescomplemented
96C2×C3×C5—164no—
97C2×C3×C5—163no—
98C23×C3—161no—
99C2×A4—43no—
100C2×A4—44no—
101C22×C5—33no—
102C22×C5—32no—
103C22×C5—33no—
104C22×C5—33no—
105C22×C5—46no—
106C22×C5—64no—
107C22×C5—126no—
108C22×C5#126⋊C311yescharacteristic, direct product factor
109C22×C5C23×A411yescharacteristic
110C4×C5—61no—
111C4×C5—63no—
112C4×C5—64no—
113C24—61no—
114C24—63no—
115C24C2×C5×A411yescharacteristic
116C24C2×C5×A416yes—
117C22×C4—33no—
118C22×C4—33no—
119C22×C4—36no—
120C22×C4—63no—
121C22×C4—63no—
122C22×C4—61no—
123C22×C4—61no—
124C4⋊C4—34no—
125C4⋊C4—312no—
126C22⋊C4—34no—
127C22⋊C4—312no—
128C42—33no—
129C42—32no—
130C42C23×C3×C511yescharacteristic, complemented
131C3×C5—161noHall
132C22×C3—166no—
133C22×C3—161no—
134A4—41no—
135C2×C5—31no—
136C2×C5—33no—
137C2×C5—44no—
138C2×C5C2×#126⋊C313yesdirect product factor
139C23—33no—
140C23—31no—
141C23—41no—
142C23—66no—
143C23—66no—
144C23—123no—
145C23—122no—
146C23C22×C5×A413yes—
147C23C22×C5×A414yes—
148C2×C4—31no—
149C2×C4—33no—
150C2×C4—34no—
151C2×C4—63no—
152C2×C4—63no—
153C2×C4—66no—
154C2×C4—63no—
155C2×C4—63no—
156C2×C4—66no—
157C2×C3—164no—
158C2×C3—163no—
159C5C22×#126⋊C311yescharacteristic, direct product factor, Sylow
160C22—33no—
161C22—32no—
162C22—33no—
163C22—33no—
164C22—46no—
165C22—64no—
166C22—126no—
167C22C5×#126⋊C311yescharacteristic, direct product factor
168C22C23×C5×A411yescharacteristic
169C4—61no—
170C4—63no—
171C4—64no—
172C3—161noSylow
173C2—31no—
174C2—33no—
175C2—44no—
176C2C2×C5×#126⋊C313yesdirect product factor
subgroupG/H
center, hypercenterC22×C5#126⋊C3
commutator, cosocle, hypocenterC42C23×C3×C5
FrattiniC22C23×C5×A4
FittingC22×C5×#126C3
socleC24×C5C2×A4
LCSG ▷ C42
UCSC1 ◁ C22×C5

Complements

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

NH ≅ G/Ntype
1. C2×C5×#126⋊C3C2direct (N × H)
2. C22×C5×C42⋊C3C2semidirect
3. C22×C5×#126C3semidirect
5. C5×#126⋊C3C22direct (N × H)
6. C2×C5×C42⋊C3C22semidirect
7. C22×#126⋊C3C5direct (N × H)
9. C2×C5×#126C2×C3semidirect
12. C22×C42×C5C2×C3semidirect
15. C5×C42⋊C3C23semidirect
16. C2×#126⋊C3C2×C5direct (N × H)
17. C22×C42⋊C3C2×C5semidirect
23. C5×#126C22×C3semidirect
29. C2×C42×C5C22×C3semidirect
30. C22×#126C3×C5semidirect
35. #126⋊C3C22×C5direct (N × H)
36. C2×C42⋊C3C22×C5semidirect
54. C42×C5C23×C3semidirect
56. C2×#126C2×C3×C5semidirect
59. C22×C42C2×C3×C5semidirect
65. C42⋊C3C23×C5semidirect
89. #126C22×C3×C5semidirect
95. C2×C42C22×C3×C5semidirect
108. C22×C5#126⋊C3direct (N × H)
130. C42C23×C3×C5semidirect
138. C2×C5C2×#126⋊C3direct (N × H)
159. C5C22×#126⋊C3direct (N × H)
167. C22C5×#126⋊C3direct (N × H)
176. C2C2×C5×#126⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#126⋊C32
C22×C5×C42⋊C32
C22×C5×#1263
C23×C5×A44
C22×#126⋊C35