Finite Group Databasebeta

← C5×D15×#3862 #36348040

Subgroups — C5×D15×#3862 #36348040

Order 22050.

232 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 50 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
1C3×C52×#3862C211yesmaximal, characteristic, complemented, Hall
2C5×C72×D15C311yesmaximal, characteristic, complemented
3C5×D5×#3862—31nomaximal
4D15×#3862C511yesmaximal, characteristic, direct product factor
5C5×D3×#3862—51nomaximal
6C52×#3862—21no—
7C52×#3862D311yescharacteristic, complemented
8C3×C52×C72C2×C311yescharacteristic, complemented
9C5×ΓL(1,𝔽8)×D15—72nomaximal
10C5×C72×D5—31noHall
11C3×C5×#3862—22no—
12C3×C5×#3862D511yescharacteristic, complemented
13C3×C5×#3862C2×C511yescharacteristic, complemented
14C3×C52×ΓL(1,𝔽8)—72no—
15C72×D15C3×C511yescharacteristic, complemented
16C5×D3×C72—51no—
17C2×C5×#3862—151no—
18D5×#3862—31no—
19C52×C72C3×D311yescharacteristic, complemented, Hall
20C5×C7×D15—32no—
21C5×C7×D15ΓL(1,𝔽8)12yescomplemented
22C5×D5×ΓL(1,𝔽8)—212no—
23D3×#3862—51noHall
24C3×C5×C72—22no—
25C3×C5×C72C3×D511yescharacteristic, complemented
26C3×C5×C72C2×C3×C511yescharacteristic, complemented
27C5×#3862—24no—
28C5×#3862—22no—
29C5×#3862—21no—
30C5×#3862—21no—
31C5×#3862C5×D311yescharacteristic, complemented
32C5×#3862D1511yescharacteristic, direct product factor
33ΓL(1,𝔽8)×D15—72no—
34C5×D3×ΓL(1,𝔽8)—352no—
35C3×C52×C7—32no—
36C3×C52×C7C2×ΓL(1,𝔽8)12yescomplemented
37C52×ΓL(1,𝔽8)—72no—
38C52×ΓL(1,𝔽8)—142no—
39C2×C5×C72—151no—
40C72×D5—31no—
41C3×C5×D15—491noHall
42C3×#3862C5×D511yescharacteristic, complemented, Hall
43C5×C7×D5—32no—
44C5×C7×D5—92no—
45C3×C5×ΓL(1,𝔽8)—72no—
46C3×C5×ΓL(1,𝔽8)—72no—
47C3×C5×ΓL(1,𝔽8)—144no—
48D3×C72—51no—
49C2×#3862—151no—
50C5×C72—22no—
51C5×C72C3×C5×D311yescharacteristic, complemented
52C5×C72C3×D1511yescharacteristic, complemented
53C32×C52—491noHall
54C7×D15—32no—
55C7×D15C5×ΓL(1,𝔽8)12yescomplemented
56C5×D3×C7—52no—
57C5×D3×C7—152no—
58C2×C5×ΓL(1,𝔽8)—1052no—
59D5×ΓL(1,𝔽8)—212no—
60C52×C7—32no—
61C52×C7D3×ΓL(1,𝔽8)12yescomplemented
62C5×D15#386211yescharacteristic, direct product factor
63C3×C5×D5—1471no—
64C3×C72C3×C5×D511yescharacteristic, complemented
65#3862—21no—
66#3862C5×D1511yescharacteristic, direct product factor
67D3×ΓL(1,𝔽8)—352no—
68C3×C5×C7—24no—
69C3×C5×C7—32no—
70C3×C5×C7—32no—
71C3×C5×C7—64no—
72C3×C5×C7D5×ΓL(1,𝔽8)12yescomplemented
73C3×C5×C7C2×C5×ΓL(1,𝔽8)12yescomplemented
74C5×ΓL(1,𝔽8)—72no—
75C5×ΓL(1,𝔽8)—72no—
76C5×ΓL(1,𝔽8)—144no—
77C5×ΓL(1,𝔽8)—148no—
78C5×ΓL(1,𝔽8)—142no—
79C5×ΓL(1,𝔽8)—142no—
80C2×C72—151noHall
81C3×D15—491no—
82C3×C5×D3—2451no—
83C3×C52—491no—
84C3×C52—981no—
85C3×C52C2×#386211yescharacteristic, complemented
86C2×C5×C7—152no—
87C2×C5×C7—452no—
88C7×D5—32no—
89C7×D5—92no—
90C3×ΓL(1,𝔽8)—72no—
91C5×D5—31noHall
92C72C3×C5×D1511yescharacteristic, complemented, Sylow
93C32×C5—491no—
94C32×C5—491no—
95C32×C5—982no—
96D3×C7—52no—
97D3×C7—152no—
98C2×ΓL(1,𝔽8)—1052no—
99C5×C7—24no—
100C5×C7—32no—
101C5×C7—32no—
102C5×C7—64no—
103C5×C7C5×D3×ΓL(1,𝔽8)12yescomplemented
104C5×C7ΓL(1,𝔽8)×D1512yescomplemented
105C2×C3×C5—7351no—
106D15C5×#386211yescharacteristic, direct product factor
107C3×D5—1471no—
108C5×D3—51no—
109C52D3×#386211yescharacteristic, complemented, Sylow
110C3×C7—32no—
111C3×C7C5×D5×ΓL(1,𝔽8)12yescomplemented
112ΓL(1,𝔽8)—72no—
113ΓL(1,𝔽8)—142no—
114C3×D3—2451noHall
115C3×C5—22no—
116C3×C5—491no—
117C3×C5—491no—
118C3×C5—982no—
119C3×C5—984no—
120C3×C5—981no—
121C3×C5—981no—
122C3×C5D5×#386211yescharacteristic, complemented
123C3×C5C2×C5×#386211yescharacteristic, complemented
124C2×C7—152no—
125C2×C7—452no—
126C2×C5—151no—
127D5—31no—
128C32—491noSylow
129C7—32no—
130C7C5×ΓL(1,𝔽8)×D1512yescomplemented
131C2×C3—7351no—
132D3—51no—
133C5—22no—
134C5C5×D3×#386211yescharacteristic, complemented
135C5D15×#386211yescharacteristic, direct product factor
136C3—491no—
137C3—981no—
138C3C5×D5×#386211yescharacteristic, complemented
139C2—151noSylow
subgroupG/H
center, hypercenterC5D15×#3862
commutator, cosocle, hypocenterC3×C5×C72C2×C3×C5
FrattiniC1G
Fitting, socleC3×C52×C72C2×C3
LCSG ▷ C3×C5×C72
UCSC1 ◁ C5

Complements

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

NH ≅ G/Ntype
1. C3×C52×#3862C2semidirect
2. C5×C72×D15C3semidirect
4. D15×#3862C5direct (N × H)
7. C52×#3862D3semidirect
8. C3×C52×C72C2×C3semidirect
12. C3×C5×#3862D5semidirect
13. C3×C5×#3862C2×C5semidirect
15. C72×D15C3×C5semidirect
19. C52×C72C3×D3semidirect
21. C5×C7×D15ΓL(1,𝔽8)semidirect
25. C3×C5×C72C3×D5semidirect
26. C3×C5×C72C2×C3×C5semidirect
31. C5×#3862C5×D3semidirect
32. C5×#3862D15direct (N × H)
36. C3×C52×C7C2×ΓL(1,𝔽8)semidirect
42. C3×#3862C5×D5semidirect
51. C5×C72C3×C5×D3semidirect
52. C5×C72C3×D15semidirect
55. C7×D15C5×ΓL(1,𝔽8)semidirect
61. C52×C7D3×ΓL(1,𝔽8)semidirect
62. C5×D15#3862direct (N × H)
64. C3×C72C3×C5×D5semidirect
66. #3862C5×D15direct (N × H)
72. C3×C5×C7D5×ΓL(1,𝔽8)semidirect
73. C3×C5×C7C2×C5×ΓL(1,𝔽8)semidirect
85. C3×C52C2×#3862semidirect
92. C72C3×C5×D15semidirect
103. C5×C7C5×D3×ΓL(1,𝔽8)semidirect
104. C5×C7ΓL(1,𝔽8)×D15semidirect
106. D15C5×#3862direct (N × H)
109. C52D3×#3862semidirect
111. C3×C7C5×D5×ΓL(1,𝔽8)semidirect
122. C3×C5D5×#3862semidirect
123. C3×C5C2×C5×#3862semidirect
130. C7C5×ΓL(1,𝔽8)×D15semidirect
134. C5C5×D3×#3862semidirect
135. C5D15×#3862direct (N × H)
138. C3C5×D5×#3862semidirect

Maximal subgroups

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

subgroupindex
C3×C52×#38622
C5×C72×D153
C5×D5×#38623
D15×#38625
C5×D3×#38625
C5×ΓL(1,𝔽8)×D157