Finite Group Databasebeta

← C3×C5×D5×#3862 #36347977

Subgroups — C3×C5×D5×#3862 #36347977

Order 22050.

280 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 72 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
2C3×C5×C72×D5C311yesmaximal, characteristic, complemented
3C5×D5×#3862C313yesmaximal, direct product factor
4C2×C3×C5×#3862—51nomaximal
5C3×D5×#3862C511yesmaximal, characteristic, direct product factor
6C52×#3862C2×C313yescomplemented
7C3×C52×C72C2×C311yescharacteristic, complemented
8C3×C5×D5×ΓL(1,𝔽8)—72nomaximal
9C5×C72×D5C3211yescharacteristic, complemented, Hall
10C3×C5×#3862—22no—
11C3×C5×#3862D511yescharacteristic, direct product factor
12C3×C5×#3862C2×C511yescharacteristic, complemented
13C3×C52×ΓL(1,𝔽8)—72no—
14C2×C3×C5×C72—51no—
15C3×C72×D5C3×C511yescharacteristic, complemented
16C2×C5×#3862—53no—
17D5×#3862C3×C513yesdirect product factor
18C52×C72C2×C3211yescharacteristic, complemented, Hall
19C3×C5×C7×D5—32no—
20C3×C5×C7×D5ΓL(1,𝔽8)12yescomplemented
21C5×D5×ΓL(1,𝔽8)—76no—
22C2×C3×#3862—51noHall
23C3×C5×C72—22no—
24C3×C5×C72C3×D511yescharacteristic, complemented
25C3×C5×C72C2×C3×C511yescharacteristic, complemented
26C5×#3862—26no—
27C5×#3862C3×D513yesdirect product factor
28C5×#3862C2×C3×C513yescomplemented
29C2×C3×C5×ΓL(1,𝔽8)—352no—
30C3×D5×ΓL(1,𝔽8)—72no—
31C3×C52×C7—32no—
32C3×C52×C7C2×ΓL(1,𝔽8)12yescomplemented
33C52×ΓL(1,𝔽8)—76no—
34C2×C5×C72—51no—
35C72×D5C32×C511yescharacteristic, complemented
36C32×C5×D5—491noHall
37C3×#3862C5×D511yescharacteristic, direct product factor, Hall
38C5×C7×D5—32no—
39C5×C7×D5C3×ΓL(1,𝔽8)12yescomplemented
40C3×C5×ΓL(1,𝔽8)—72no—
41C3×C5×ΓL(1,𝔽8)—72no—
42C3×C5×ΓL(1,𝔽8)—144no—
43C2×C3×C72—51no—
44C2×#3862—53no—
45C5×C72—22no—
46C5×C72C32×D511yescharacteristic, complemented
47C5×C72C2×C32×C511yescharacteristic, complemented
48C32×C52—491noHall
49C2×C3×C5×C7—52no—
50C2×C3×C5×C7—152no—
51C3×C7×D5—32no—
52C3×C7×D5C5×ΓL(1,𝔽8)12yescomplemented
53C2×C5×ΓL(1,𝔽8)—356no—
54D5×ΓL(1,𝔽8)—76no—
55C52×C7—32no—
56C52×C7C2×C3×ΓL(1,𝔽8)12yescomplemented
57C3×C5×D5—493no—
58C3×C5×D5#386211yescharacteristic, direct product factor
59C3×C72C3×C5×D511yescharacteristic, complemented
60#3862C3×C5×D513yesdirect product factor
61C2×C3×ΓL(1,𝔽8)—352no—
62C3×C5×C7—24no—
63C3×C5×C7—32no—
64C3×C5×C7—32no—
65C3×C5×C7—64no—
66C3×C5×C7D5×ΓL(1,𝔽8)12yescomplemented
67C3×C5×C7C2×C5×ΓL(1,𝔽8)12yescomplemented
68C5×ΓL(1,𝔽8)—76no—
69C5×ΓL(1,𝔽8)—76no—
70C5×ΓL(1,𝔽8)—1412no—
71C2×C72—51noHall
72C2×C32×C5—2451no—
73C32×D5—491no—
74C3×C52—493no—
75C3×C52C2×#386211yescharacteristic, complemented
76C2×C5×C7—52no—
77C2×C5×C7—152no—
78C7×D5—32no—
79C7×D5C3×C5×ΓL(1,𝔽8)12yescomplemented
80C3×ΓL(1,𝔽8)—72no—
81C5×D5C3×#386211yescharacteristic, direct product factor, Hall
82C72C32×C5×D511yescharacteristic, complemented, Sylow
83C32×C5—491no—
84C32×C5—491no—
85C32×C5—982no—
86C2×C3×C7—52no—
87C2×C3×C7—152no—
88C2×ΓL(1,𝔽8)—356no—
89C5×C7—24no—
90C5×C7—32no—
91C5×C7—32no—
92C5×C7—64no—
93C5×C7C3×D5×ΓL(1,𝔽8)12yescomplemented
94C5×C7C2×C3×C5×ΓL(1,𝔽8)12yescomplemented
95C2×C3×C5—51no—
96C2×C3×C5—2453no—
97C3×D5—493no—
98C3×D5C5×#386211yescharacteristic, direct product factor
99C52C2×C3×#386211yescharacteristic, complemented, Sylow
100C3×C7—32no—
101C3×C7C5×D5×ΓL(1,𝔽8)12yescomplemented
102ΓL(1,𝔽8)—76no—
103C2×C32—2451noHall
104C3×C5—22no—
105C3×C5—493no—
106C3×C5—493no—
107C3×C5—986no—
108C3×C5D5×#386211yescharacteristic, direct product factor
109C3×C5C2×C5×#386211yescharacteristic, complemented
110C2×C7—52no—
111C2×C7—152no—
112C2×C5—51no—
113D5C3×C5×#386211yescharacteristic, direct product factor
114C32—491noSylow
115C7—32no—
116C7C3×C5×D5×ΓL(1,𝔽8)12yescomplemented
117C2×C3—51no—
118C2×C3—2453no—
119C5—22no—
120C5C3×D5×#386211yescharacteristic, direct product factor
121C5C2×C3×C5×#386211yescharacteristic, complemented
122C3—493no—
123C3C5×D5×#386211yescharacteristic, direct product factor
124C2—51noSylow
subgroupG/H
center, hypercenterC3×C5D5×#3862
commutator, cosocle, hypocenterC5×C72C2×C32×C5
FrattiniC1G
Fitting, socleC3×C52×C72C2×C3
LCSG ▷ C5×C72
UCSC1 ◁ C3×C5

Complements

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

NH ≅ G/Ntype
1. C3×C52×#3862C2semidirect
2. C3×C5×C72×D5C3semidirect
3. C5×D5×#3862C3direct (N × H)
5. C3×D5×#3862C5direct (N × H)
6. C52×#3862C2×C3semidirect
7. C3×C52×C72C2×C3semidirect
9. C5×C72×D5C32semidirect
11. C3×C5×#3862D5direct (N × H)
12. C3×C5×#3862C2×C5semidirect
15. C3×C72×D5C3×C5semidirect
17. D5×#3862C3×C5direct (N × H)
18. C52×C72C2×C32semidirect
20. C3×C5×C7×D5ΓL(1,𝔽8)semidirect
24. C3×C5×C72C3×D5semidirect
25. C3×C5×C72C2×C3×C5semidirect
27. C5×#3862C3×D5direct (N × H)
28. C5×#3862C2×C3×C5semidirect
32. C3×C52×C7C2×ΓL(1,𝔽8)semidirect
35. C72×D5C32×C5semidirect
37. C3×#3862C5×D5direct (N × H)
39. C5×C7×D5C3×ΓL(1,𝔽8)semidirect
46. C5×C72C32×D5semidirect
47. C5×C72C2×C32×C5semidirect
52. C3×C7×D5C5×ΓL(1,𝔽8)semidirect
56. C52×C7C2×C3×ΓL(1,𝔽8)semidirect
58. C3×C5×D5#3862direct (N × H)
59. C3×C72C3×C5×D5semidirect
60. #3862C3×C5×D5direct (N × H)
66. C3×C5×C7D5×ΓL(1,𝔽8)semidirect
67. C3×C5×C7C2×C5×ΓL(1,𝔽8)semidirect
75. C3×C52C2×#3862semidirect
79. C7×D5C3×C5×ΓL(1,𝔽8)semidirect
81. C5×D5C3×#3862direct (N × H)
82. C72C32×C5×D5semidirect
93. C5×C7C3×D5×ΓL(1,𝔽8)semidirect
94. C5×C7C2×C3×C5×ΓL(1,𝔽8)semidirect
98. C3×D5C5×#3862direct (N × H)
99. C52C2×C3×#3862semidirect
101. C3×C7C5×D5×ΓL(1,𝔽8)semidirect
108. C3×C5D5×#3862direct (N × H)
109. C3×C5C2×C5×#3862semidirect
113. D5C3×C5×#3862direct (N × H)
116. C7C3×C5×D5×ΓL(1,𝔽8)semidirect
120. C5C3×D5×#3862direct (N × H)
121. C5C2×C3×C5×#3862semidirect
123. C3C5×D5×#3862direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C52×#38622
C3×C5×C72×D53
C5×D5×#38623
C2×C3×C5×#38625
C3×D5×#38625
C3×C5×D5×ΓL(1,𝔽8)7