Finite Group Databasebeta

← C3×C5×#10587717⋊C2 #30655314

Subgroups — C3×C5×#10587717⋊C2 #30655314

Order 16380.

240 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 80 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×C5×#10587717C211yesmaximal, characteristic, complemented
2C3×C5×#10587721C211yesmaximal, characteristic, complemented
3C3×C5×#10587723C211yesmaximal, characteristic, complemented
4C3×C5×D7×D13C311yesmaximal, characteristic, complemented
5C5×#10587717⋊C2C313yesmaximal, direct product factor
6C3×C5×#63271C2211yescharacteristic, complemented, Hall
7C3×#10587717⋊C2C511yesmaximal, characteristic, direct product factor, Hall
8C5×#10587723C2×C313yescomplemented
9C5×#10587721C2×C313yescomplemented
10C5×#10587717C2×C313yescomplemented
11C3×C5×D91C2×C311yescharacteristic, complemented
12C3×C5×C13×D7C2×C311yescharacteristic, complemented
13C3×C5×C7×D13C2×C311yescharacteristic, complemented
14C2×C3×C5×D13⋊C3—71nomaximal, Hall
15C5×D7×D13C3211yescharacteristic, complemented, Hall
16C3×#10587723C2×C511yescharacteristic, complemented
17C3×#10587721C2×C511yescharacteristic, complemented
18C3×#10587717C2×C511yescharacteristic, complemented
19C3×C5×C7×C13C22×C311yescharacteristic, complemented
20C5×#63271C22×C313yescomplemented
21C2×C3×C5×AGL(1,𝔽7)—131nomaximal, Hall
22C2×C3×C5×C13⋊C3—71no—
23C3×C5×D13⋊C3—71no—
24C3×C5×D13⋊C3—71no—
25C3×D7×D13C3×C511yescharacteristic, complemented
26#10587717⋊C2C3×C513yesdirect product factor
27C5×D91C2×C3211yescharacteristic, complemented
28C5×C13×D7C2×C3211yescharacteristic, complemented
29C5×C7×D13C2×C3211yescharacteristic, complemented
30C3×#63271C22×C511yescharacteristic, complemented, Hall
31C2×C3×C5×D13—211no—
32C2×C5×D13⋊C3—73no—
33C2×C3×C5×ΓL(1,𝔽8)—131no—
34C3×C5×AGL(1,𝔽7)—131no—
35C3×C5×AGL(1,𝔽7)—131no—
36C3×C5×C13⋊C3—71noHall
37C3×D91C2×C3×C511yescharacteristic, complemented
38C3×C13×D7C2×C3×C511yescharacteristic, complemented
39C3×C7×D13C2×C3×C511yescharacteristic, complemented
40#10587723C2×C3×C513yescomplemented
41#10587721C2×C3×C513yescomplemented
42#10587717C2×C3×C513yescomplemented
43C2×C3×D13⋊C3—71noHall
44C5×C7×C13C22×C3211yescharacteristic, complemented, Hall
45C2×C3×C5×D7—391no—
46C2×C5×AGL(1,𝔽7)—133no—
47C2×C3×C5×C13—211no—
48C3×C5×D13—211no—
49C3×C5×D13AGL(1,𝔽7)11yescharacteristic, complemented
50C2×C5×C13⋊C3—73no—
51C5×D13⋊C3—73no—
52C5×D13⋊C3—73no—
53D7×D13C32×C511yescharacteristic, complemented, Hall
54C3×C5×ΓL(1,𝔽8)—131noHall
55C3×C7×C13C22×C3×C511yescharacteristic, complemented
56#63271C22×C3×C513yescomplemented
57C2×C5×D13—211noHall
58C2×C3×AGL(1,𝔽7)—131noHall
59C2×C3×C13⋊C3—71no—
60C3×D13⋊C3—71no—
61C3×D13⋊C3—71no—
62C2×C3×C5×C7—391no—
63C3×C5×D7—391no—
64C3×C5×D7D13⋊C311yescharacteristic, complemented
65C2×C5×ΓL(1,𝔽8)—133no—
66C5×AGL(1,𝔽7)—133no—
67C5×AGL(1,𝔽7)—133no—
68C3×C5×C13C2×AGL(1,𝔽7)11yescharacteristic, complemented
69C5×C13⋊C3—73no—
70D91C2×C32×C511yescharacteristic, complemented
71C7×D13C2×C32×C511yescharacteristic, complemented
72C13×D7C2×C32×C511yescharacteristic, complemented
73C22×C32×C5—911noHall
74C2×C3×D13—211no—
75C2×D13⋊C3—73no—
76C2×C5×D7—391noHall
77C2×C5×C13—211no—
78C5×D13—211no—
79C5×D13C3×AGL(1,𝔽7)11yescharacteristic, complemented
80C2×C3×ΓL(1,𝔽8)—131no—
81C3×AGL(1,𝔽7)—131no—
82C3×AGL(1,𝔽7)—131no—
83C3×C13⋊C3—71noHall
84C3×C5×C7C2×D13⋊C311yescharacteristic, complemented
85C5×ΓL(1,𝔽8)—133no—
86C7×C13C22×C32×C511yescharacteristic, complemented, Hall
87C2×C32×C5—911no—
88C2×C32×C5—911no—
89C2×C32×C5—911no—
90C2×C3×D7—391no—
91C2×AGL(1,𝔽7)—133no—
92C2×C3×C13—71no—
93C3×D13—71no—
94C3×D13C5×AGL(1,𝔽7)11yescharacteristic, complemented
95C2×C13⋊C3—73no—
96D13⋊C3—73no—
97D13⋊C3—73no—
98C2×C5×C7—391no—
99C5×D7—391no—
100C5×D7C3×D13⋊C311yescharacteristic, complemented
101C5×C13C2×C3×AGL(1,𝔽7)11yescharacteristic, complemented, Hall
102C3×ΓL(1,𝔽8)—131noHall
103C22×C3×C5—2731no—
104C22×C3×C5—913no—
105C2×D13—211noHall
106C32×C5—911noHall
107C2×C3×C7—391no—
108C3×D7—391no—
109C3×D7C5×D13⋊C311yescharacteristic, complemented
110C2×ΓL(1,𝔽8)—133no—
111AGL(1,𝔽7)—133no—
112AGL(1,𝔽7)—133no—
113C3×C13C2×C5×AGL(1,𝔽7)11yescharacteristic, complemented
114C13⋊C3—73no—
115C22×C32—911noHall
116C5×C7C2×C3×D13⋊C311yescharacteristic, complemented, Hall
117C2×C3×C5—2731no—
118C2×C3×C5—211no—
119C2×C3×C5—391no—
120C2×C3×C5—913no—
121C2×C3×C5—913no—
122C2×C3×C5—913no—
123C2×D7—391noHall
124C2×C13—71no—
125D13—71no—
126D13C3×C5×AGL(1,𝔽7)11yescharacteristic, complemented
127C3×C7C2×C5×D13⋊C311yescharacteristic, complemented
128ΓL(1,𝔽8)—133no—
129C22×C5—2731noHall
130C2×C32—911no—
131C2×C32—911no—
132C2×C32—911no—
133C3×C5—913no—
134C3×C5#10587717⋊C211yescharacteristic, direct product factor
135C2×C7—391no—
136D7—391no—
137D7C3×C5×D13⋊C311yescharacteristic, complemented
138C13C2×C3×C5×AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
139C22×C3—2731no—
140C22×C3—913no—
141C2×C5—2731no—
142C2×C5—211no—
143C2×C5—391no—
144C32—911noSylow
145C7C2×C3×C5×D13⋊C311yescharacteristic, complemented, Sylow
146C2×C3—2731no—
147C2×C3—211no—
148C2×C3—391no—
149C2×C3—913no—
150C2×C3—913no—
151C2×C3—913no—
152C5C3×#10587717⋊C211yescharacteristic, direct product factor, Sylow
153C22—2731noSylow
154C3—913no—
155C3C5×#10587717⋊C211yescharacteristic, direct product factor
156C2—2731no—
157C2—211no—
158C2—391no—
subgroupG/H
center, hypercenterC3×C5#10587717⋊C2
commutator, cosocle, hypocenterC7×C13C22×C32×C5
FrattiniC1G
Fitting, socleC3×C5×C7×C13C22×C3
LCSG ▷ C7×C13
UCSC1 ◁ C3×C5

Complements

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

NH ≅ G/Ntype
1. C3×C5×#10587717C2semidirect
2. C3×C5×#10587721C2semidirect
3. C3×C5×#10587723C2semidirect
4. C3×C5×D7×D13C3semidirect
5. C5×#10587717⋊C2C3direct (N × H)
6. C3×C5×#63271C22semidirect
7. C3×#10587717⋊C2C5direct (N × H)
8. C5×#10587723C2×C3semidirect
9. C5×#10587721C2×C3semidirect
10. C5×#10587717C2×C3semidirect
11. C3×C5×D91C2×C3semidirect
12. C3×C5×C13×D7C2×C3semidirect
13. C3×C5×C7×D13C2×C3semidirect
15. C5×D7×D13C32semidirect
16. C3×#10587723C2×C5semidirect
17. C3×#10587721C2×C5semidirect
18. C3×#10587717C2×C5semidirect
19. C3×C5×C7×C13C22×C3semidirect
20. C5×#63271C22×C3semidirect
25. C3×D7×D13C3×C5semidirect
26. #10587717⋊C2C3×C5direct (N × H)
27. C5×D91C2×C32semidirect
28. C5×C13×D7C2×C32semidirect
29. C5×C7×D13C2×C32semidirect
30. C3×#63271C22×C5semidirect
37. C3×D91C2×C3×C5semidirect
38. C3×C13×D7C2×C3×C5semidirect
39. C3×C7×D13C2×C3×C5semidirect
40. #10587723C2×C3×C5semidirect
41. #10587721C2×C3×C5semidirect
42. #10587717C2×C3×C5semidirect
44. C5×C7×C13C22×C32semidirect
49. C3×C5×D13AGL(1,𝔽7)semidirect
53. D7×D13C32×C5semidirect
55. C3×C7×C13C22×C3×C5semidirect
56. #63271C22×C3×C5semidirect
64. C3×C5×D7D13⋊C3semidirect
68. C3×C5×C13C2×AGL(1,𝔽7)semidirect
70. D91C2×C32×C5semidirect
71. C7×D13C2×C32×C5semidirect
72. C13×D7C2×C32×C5semidirect
79. C5×D13C3×AGL(1,𝔽7)semidirect
84. C3×C5×C7C2×D13⋊C3semidirect
86. C7×C13C22×C32×C5semidirect
94. C3×D13C5×AGL(1,𝔽7)semidirect
100. C5×D7C3×D13⋊C3semidirect
101. C5×C13C2×C3×AGL(1,𝔽7)semidirect
109. C3×D7C5×D13⋊C3semidirect
113. C3×C13C2×C5×AGL(1,𝔽7)semidirect
116. C5×C7C2×C3×D13⋊C3semidirect
126. D13C3×C5×AGL(1,𝔽7)semidirect
127. C3×C7C2×C5×D13⋊C3semidirect
134. C3×C5#10587717⋊C2direct (N × H)
137. D7C3×C5×D13⋊C3semidirect
138. C13C2×C3×C5×AGL(1,𝔽7)semidirect
145. C7C2×C3×C5×D13⋊C3semidirect
152. C5C3×#10587717⋊C2direct (N × H)
155. C3C5×#10587717⋊C2direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#105877172
C3×C5×#105877212
C3×C5×#105877232
C3×C5×D7×D133
C5×#10587717⋊C23
C3×#10587717⋊C25
C2×C3×C5×D13⋊C37
C2×C3×C5×AGL(1,𝔽7)13